The Finite Element Method for Boundary Value Problems: Mathematics and Computations [1 ed.] 1498780504, 978-1-4987-8050-6

Table of contents :
Content: Introduction Basic Elements from Applied Mathematics Classical Methods of Approximation The Finite Element Method Finite Element Method for Self Adjoint Operators The Finite Element Method for Non-Self Adjoint Operators Finite Element Method for Non-linear operators Basic Elements of Mapping and Interpolation Theory Finite element processes in linear solid and structural mechanics Finite element formulations using principle of virtual work Appendix

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The Finite Element Method for Boundary Value Problems Mathematics and Computations

The Finite Element Method for Boundary Value Problems Mathematics and Computations

Karan S. Surana • J. N. Reddy

Boca Raton London New York

CRC Press is an imprint of the Taylor & Francis Group, an informa business

CRC Press Taylor & Francis Group 6000 Broken Sound Parkway NW, Suite 300 Boca Raton, FL 33487-2742 © 2017 by Taylor & Francis Group, LLC CRC Press is an imprint of Taylor & Francis Group, an Informa business No claim to original U.S. Government works Printed on acid-free paper Version Date: 20160930 International Standard Book Number-13: 978-1-4987-8050-6 (Hardback) This book contains information obtained from authentic and highly regarded sources. Reasonable efforts have been made to publish reliable data and information, but the author and publisher cannot assume responsibility for the validity of all materials or the consequences of their use. The authors and publishers have attempted to trace the copyright holders of all material reproduced in this publication and apologize to copyright holders if permission to publish in this form has not been obtained. If any copyright material has not been acknowledged please write and let us know so we may rectify in any future reprint. Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying, microfilming, and recording, or in any information storage or retrieval system, without written permission from the publishers. For permission to photocopy or use material electronically from this work, please access www.copyright.com (http:// www.copyright.com/) or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and registration for a variety of users. For organizations that have been granted a photocopy license by the CCC, a separate system of payment has been arranged. Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used only for identification and explanation without intent to infringe. Library of Congress Cataloging‑in‑Publication Data Names: Surana, Karan S. | Reddy, J. N. (Junuthula Narasimha), 1945Title: The finite element method for boundary value problems : mathematics and computations / Karan S. Surana and J.N. Reddy. Description: Boca Raton : CRC Press, 2017. Identifiers: LCCN 2016035534| ISBN 9781498780506 (hardback : alk. paper) | ISBN 9781315365718 (ebook) Subjects: LCSH: Boundary value problems--Numerical solutions. | Finite element method. Classification: LCC QA379 .S87 2017 | DDC 515/.62--dc23 Visit the Taylor & Francis Web site at http://www.taylorandfrancis.com and the CRC Press Web site at http://www.crcpress.com

To My (KSS) beloved family Abha, Deepak, Rishi, and Yogini

My (JNR) loving grandchildren Rohan, Asha, and Mira

Contents Preface

xix

About the Authors

xxv

1 Introduction 1.1 General Comments and Basic Philosophy . . . . . 1.2 Basic Concepts of the Finite Element Method . . 1.2.1 Discretization . . . . . . . . . . . . . . . . 1.2.2 Local approximation . . . . . . . . . . . . 1.2.3 Integral forms and algebraic equations over 1.2.4 Assembly of element equations . . . . . . . 1.2.5 Computation of the solution . . . . . . . . 1.2.6 Post-processing . . . . . . . . . . . . . . . 1.2.7 Remarks . . . . . . . . . . . . . . . . . . . 1.2.8 k-version of the finite element method and hpk framework . . . . . . . . . . . . . 1.3 Summary . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . an . . . . . . . .

. . . . . . . . . . . . . . . . element . . . . . . . . . . . . . . . .

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2 Concepts from Functional Analysis 2.1 General Comments . . . . . . . . . . . . . . . . . . . . . 2.2 Sets, Spaces, Functions, Functions Spaces, and Operators ¯ . . . . . . . . . . . . . . . . 2.2.1 Hilbert spaces H k (Ω) ¯ space . . . 2.2.2 Definition of scalar product in H k (Ω)

2.3

2.2.3 Properties of scalar product . . . . . . ¯ . . . 2.2.4 Norm of u in Hilbert space H k (Ω) ¯ 2.2.5 Seminorm of u in Hilbert space H k (Ω) 2.2.6 Function spaces . . . . . . . . . . . . . 2.2.7 Operators . . . . . . . . . . . . . . . . 2.2.8 Types of operators . . . . . . . . . . . 2.2.9 Energy product . . . . . . . . . . . . . 2.2.10 Integration by parts (IBP) . . . . . . . Elements of Calculus of Variations . . . . . . . vii

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1 1 3 3 5 7 8 9 9 9 11 13 15 15 15 18 19 19 20 20 22 23 24 27 27 31

CONTENTS

viii

2.4

2.5

2.3.1 Concept of the variation of a functional . . . . . 2.3.2 Euler’s equation: Simplest variational problem . 2.3.3 Variation of a functional: some practical aspects 2.3.4 Riemann and Lebesgue integrals . . . . . . . . . Examples of Differential Operators and their Properties . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Self-adjoint differential operators . . . . . . . . 2.4.2 Non-self-adjoint differential operators . . . . . . 2.4.3 Non-linear differential operators . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

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33 34 41 42

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44 44 58 63 64

3 Classical Methods of Approximation 69 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.2 Basic Steps in Classical Methods of Approximation based on Integral Forms . . . . . . . . . . . . 70 3.3 Integral forms using the Fundamental Lemma of the Calculus of Variations . . . . . . . . . . . . . . . . . . 72 3.3.1 The Galerkin method . . . . . . . . . . . . . . . . . . 73 3.3.1.1 Self-adjoint and non-self-adjoint linear differential operators . . . . . . . . . . . . . . 74 3.3.1.2 Non-linear differential operators . . . . . . . 77 3.3.2 The Petrov–Galerkin and weighted-residual methods 78 3.3.2.1 Self-adjoint and non-self-adjoint linear differential operators . . . . . . . . . . . . . . 78 3.3.2.2 Non-linear differential operators . . . . . . . 80 3.3.3 The Galerkin method with weak form . . . . . . . . . 81 3.3.3.1 Linear differential operators . . . . . . . . . . 85 3.3.3.2 Non-linear differential operators . . . . . . . 86 3.3.4 The least-squares method . . . . . . . . . . . . . . . . 87 3.3.4.1 Self-adjoint and non-self-adjoint linear differential operators . . . . . . . . . . . . . . 92 3.3.4.2 Non-linear differential operators . . . . . . . 93 3.3.5 Collocation method . . . . . . . . . . . . . . . . . . . 94 3.4 Approximation Spaces for Various Methods of Approximation 95 3.5 Integral Formulations of BVPs using the Classical Methods of Approximations . . . . . . . . . . . 97 3.5.1 Self-adjoint differential operators . . . . . . . . . . . 98 3.5.2 Non-self-adjoint Differential Operators . . . . . . . . 127 3.5.3 Non-linear Differential Operators . . . . . . . . . . . 141 3.6 Numerical Examples . . . . . . . . . . . . . . . . . . . . . . . 153 3.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 4 The Finite Element Method

193

CONTENTS

4.1 4.2

4.3

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . Basic steps in the finite element method . . . . . . . . . . . . 4.2.1 Discretization . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Construction of integral forms over an element . . . . 4.2.2.1 Integral forms for GM, PGM, and WRM . . 4.2.2.2 Integral form for GM/WF . . . . . . . . . . 4.2.2.3 Integral form based on residual functional . . 4.2.3 The local approximation φeh of φ over an element . . 4.2.4 Element matrices and vectors resulting from the integral form and the local approximation . . . . . . 4.2.4.1 Galerkin method, Petrov–Galerkin method, and weighted residual method . . . . . . . . 4.2.4.2 Galerkin method with weak form . . . . . . . 4.2.4.3 Least-squares process based on residual functional . . . . . . . . . . . . . . . . . . . . 4.2.5 Assembly of element equations: GM, PGM, WRM, GM/WF and LSP when A is linear . . . . . . . . . . 4.2.6 Consideration of boundary conditions in the assembled equations . . . . . . . . . . . . . . . . . . . 4.2.6.1 GM, PGM, WRM, and LSP based on the residual functional . . . . . . . . . . . . . . . 4.2.6.2 GM/WF . . . . . . . . . . . . . . . . . . . . 4.2.7 Computation of the solution: finite element processes based on all methods of approximation except LSP for non-linear operators . . . . . . . . . . . . . . . . . 4.2.8 Assembly of element equations and their solution in finite element processes based on residual functional (LSP) when the differential operator A is non-linear . 4.2.9 Post processing of the solution . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

5 Self-Adjoint Differential Operators 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 GM/WF . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 LSP based on residual functional . . . . . . . . . . . 5.2 One-dimensional BVPs in a single dependent variable . . . . 5.2.1 1D steady-state diffusion equation: finite element processes based on GM/WF . . . . . . . . . . . . . . 5.2.1.1 Discretization . . . . . . . . . . . . . . . . . 5.2.1.2 Integral form using GM/WF (weak form) of ¯e . the BVP for an element e with domain Ω

ix

193 194 194 198 199 200 202 203 204 205 206 210 212 214 214 215

216

217 220 221 223 223 224 225 226 226 227 227

CONTENTS

x

Approximation space Vh , test function space V and local approximation φeh . . . . . . . . ¯ e to 5.2.1.4 Local approximation φeh and mapping Ω ξ ¯ . . . . . . . . . . . . . . . . . . . . . . . . Ω 5.2.1.5 Element equations . . . . . . . . . . . . . . . 5.2.1.6 Assembly of element equations and computation of the solution . . . . . . . . . . 5.2.1.7 Inter-element continuity conditions on PVs or dependent variables . . . . . . . . . . . . . 5.2.1.8 Rules for assembling element matrices and vectors . . . . . . . . . . . . . . . . . . . . . 5.2.1.9 Inter-element continuity conditions on the sum of secondary variables . . . . . . . . . . . . . 5.2.1.10 Imposition of EBCs . . . . . . . . . . . . . . 5.2.1.11 Solving for unknown degrees of freedom . . . 5.2.1.12 Special case: numerical study . . . . . . . . . 5.2.1.13 Post-processing of solution . . . . . . . . . . 5.2.1.14 Analytical solution and comparison with finite element solutions . . . . . . . . . . . . 5.2.2 1D steady-state diffusion equation . . . . . . . . . . . 5.2.2.1 Case (a): a = 1, L = 1, q(x) = xn , n is a positive integer . . . . . . . . . . . . . . . . . 5.2.2.2 Case (b): a = 1, L = 1, q(x) = sin nπx, n = 4 5.2.3 Least-squares finite element formulation . . . . . . . 5.2.3.1 Approximation space Vh . . . . . . . . . . . . 5.2.3.2 Numerical studies . . . . . . . . . . . . . . . 5.2.4 LSFEP using auxiliary variables and auxiliary equations . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.4.1 Approximation spaces for φeh and τhe . . . . . 5.2.4.2 Numerical studies . . . . . . . . . . . . . . . 5.2.5 One-dimensional heat conduction with convective boundary . . . . . . . . . . . . . . . . . . 5.2.5.1 Approximation space Vh . . . . . . . . . . . . 5.2.5.2 Numerical study . . . . . . . . . . . . . . . . 5.2.6 1D axisymmetric heat conduction . . . . . . . . . . . 5.2.6.1 Galerkin method with weak form . . . . . . . 5.2.6.2 LSM based on residual functional . . . . . . 5.2.7 A 1D BVP governed by a fourth-order differential operator . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.7.1 Approximation space Vh . . . . . . . . . . . . Two-dimensional boundary value problems . . . . . . . . . . 5.3.1 A general 2D BVP in a single dependent variable . . 5.2.1.3

5.3

231 233 234 235 237 237 242 243 243 244 246 246 251 252 260 265 275 275 278 285 285 289 296 297 300 301 303 305 309 310 310

xi

CONTENTS

¯ e : element geometry . . . . . 314 Definition of Ω Approximation space Vh . . . . . . . . . . . . 316 Computation of the element matrix [K e ] and vector {F e } . . . . . . . . . . . . . . . . . . . 317 5.3.1.4 Details of secondary variable vector {P e } . . 317 5.3.2 2D Poisson’s equation: numerical studies . . . . . . . 320 5.3.2.1 Case (a): f = 1 with BCs φ(±1, y) = φ(x, ±1) = 0; GM/WF . . . . . . . . . . . . . . . . . . . 320 5.3.2.2 Case (b): BCs φ(±1, y) = φ(x, ±1) = 1.0; GM/WF . . . . . . . . . . . . . . . . . . . . 327 5.3.3 Two-dimensional boundary value problems in multi-variables: 2D plane elasticity . . . . . . . . . 328 5.3.3.1 Galerkin method with weak form . . . . . . . 332 5.3.3.2 Least-squares method using residual functional . . . . . . . . . . . . . . . . . . . . 337 Three-dimensional boundary value problems . . . . . . . . . 344 5.4.1 Three-dimensional boundary value problems in a single dependent variable . . . . . . . . . . . . . . . . 344 5.4.1.1 Galerkin method with weak form . . . . . . . 345 5.4.1.2 Approximation space . . . . . . . . . . . . . 347 5.4.1.3 Local approximation The . . . . . . . . . . . . 347 ¯ e : Element geometry . . . . . 348 5.4.1.4 Definition of Ω 5.4.1.5 Computations of element matrix [K e ] and vector {F e } . . . . . . . . . . . . . . . . . . . 350 5.4.1.6 Details of secondary variable vector {P e } . . 350 5.4.2 Three-dimensional boundary value problems in multivariables . . . . . . . . . . . . . . . . . . . . . 353 5.4.2.1 Galerkin method with weak form . . . . . . . 355 5.4.2.2 Approximation spaces . . . . . . . . . . . . . 357 5.4.2.3 Local approximation . . . . . . . . . . . . . . 357 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 5.3.1.1 5.3.1.2 5.3.1.3

5.4

5.5

6 Non-Self-Adjoint Differential Operators 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 1D convection-diffusion equation . . . . . . . . . . . . . . . . 6.2.1 Analytical solution . . . . . . . . . . . . . . . . . . . 6.2.2 The Galerkin method with weak form (GM/WF) . . 6.2.3 Least squares finite element formulation . . . . . . . 6.2.4 Least squares formulation: first order system . . . . . 6.3 2D convection-diffusion equation . . . . . . . . . . . . . . . . 6.3.1 Least squares finite element formulation based on the residual functional . . . . . . . . . . . . . . . . . . . .

363 363 365 365 367 378 380 390 395

CONTENTS

xii

6.3.2

6.4

Least squares finite element formulation of (6.113) by recasting it as a system of first order PDEs . . . . . . 6.3.3 Convection dominated thermal flow (advection skewed to a square domain) . . . . . . . . . . . . . . . . . . . 6.3.4 Advection of a cosine hill in a rotating flow field . . . 6.3.5 Thermal boundary layer . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

397 401 408 411 413

7 Non-Linear Differential Operators 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 One dimensional Burgers equation . . . . . . . . . . . . . . . 7.2.1 The Galerkin method with weak form . . . . . . . . . 7.2.2 LSP based on residual functional . . . . . . . . . . . 7.2.3 LSP based on residual functional: first order system of equations . . . . . . . . . . . . . . . . . . . . . . . 7.3 Fully developed flow of Giesekus fluid between parallel plates (polymer flow) . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 2D steady-state Navier–Stokes equations . . . . . . . . . . . 7.4.1 LSP based on residual functional: first order system of PDEs . . . . . . . . . . . . . . . . . . . . . . . . . 7.4.2 LSP based on residual functional: higher order systems of PDEs . . . . . . . . . . . . . . . . . . . . . 7.4.3 Slider bearing; flow of a viscous lubricant . . . . . . . 7.4.4 A square lid-driven cavity . . . . . . . . . . . . . . . 7.4.5 Asymmetric backward facing step . . . . . . . . . . . 7.4.6 Flow past a circular cylinder . . . . . . . . . . . . . . 7.5 2D compressible Newtonian fluid flow . . . . . . . . . . . . . 7.5.1 Carter’s plate . . . . . . . . . . . . . . . . . . . . . . 7.5.1.1 Mach 1 flow . . . . . . . . . . . . . . . . . . 7.5.1.2 General consideration for higher Mach number flows . . . . . . . . . . . . . . . . . . 7.5.1.3 Mach 2 flow . . . . . . . . . . . . . . . . . . 7.5.1.4 Mach 3 flow . . . . . . . . . . . . . . . . . . 7.5.1.5 Mach 5 flow . . . . . . . . . . . . . . . . . . 7.5.2 Mach 1 flow past a circular cylinder . . . . . . . . . . 7.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

419 419 422 423 428

8 Basic Elements of Mapping and Interpolation Theory 8.1 Mapping in one dimension . . . . . . . . . . . . . . . . 8.1.1 Mapping of points . . . . . . . . . . . . . . . . . 8.1.2 Mapping of lengths . . . . . . . . . . . . . . . . ¯e . . . . 8.1.3 Behavior of dependent variable φ over Ω ¯ ξ = [−1, 1] . . . 8.2 Elements of interpolation theory over Ω

493 493 494 494 495 495

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429 442 450 452 454 455 457 461 467 471 474 476 478 479 480 481 483 486

CONTENTS

8.2.1 8.2.2 8.2.3 8.2.4

8.3 8.4

8.5

8.6

8.7

8.8

A polynomial approximation in one dimension . . . . Lagrange interpolating polynomials in one dimension p-version hierarchical functions in one dimension . . . Higher order global differentiability approximations in one dimension: p-version . . . . . . . . . . . . . . . ¯ e) . . . . . 8.2.4.1 Local approximation of class C 1 (Ω 8.2.4.2 Interpolations or local approximations of class ¯ e ): . . . . . . . . . . . . . . . . . . . . . C 2 (Ω ¯ e) . . . . 8.2.4.3 Local approximations of class C i (Ω Mapping in two dimensions: quadrilateral elements . . . . . ¯ m : quadrilateral elements . . . . Local approximation over Ω 00 ¯ ξη : polynomial 8.4.1 C local approximations over Ω approach . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.2 C 00 Lagrange type local approximation using tensor product . . . . . . . . . . . . . . . . . . . . . . 8.4.3 C 00 p-version hierarchical local approximations based on Lagrange polynomials . . . . . . . . . . . . . . . . ij ¯ e ) p-version local approximations . . . . . . . . . . 2D C (Ω ¯ e ) with p-levels 8.5.1 2D interpolations of type C 11 (Ω of pξ and pη . . . . . . . . . . . . . . . . . . . . . . . ¯ e ) with p-levels 8.5.2 2D interpolations of type C 22 (Ω of pξ and pη . . . . . . . . . . . . . . . . . . . . . . . ¯ e ) interpolations of p-levels pξ and pη . . . . 8.5.3 2D C ij (Ω ¯ e ) approximations for quadrilateral elements . . . . 2D C ij (Ω 8.6.1 C 11 HGDA for 2D distorted quadrilateral elements in xy space . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.2 C 22 HGDA for 2D distorted quadrilateral elements in xy space . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.3 C 33 HGDA for 2D distorted quadrilateral elements in xy space . . . . . . . . . . . . . . . . . . . . . . . . . 8.6.4 Derivation of C ij approximations for distorted quadrilateral elements . . . . . . . . . . . . . . . . . . 8.6.5 Limitations of 2D C 11 global differentiability local approximations for distorted quadrilateral elements . Interpolation theory for 2D triangular elements . . . . . . . . 8.7.1 Langrange family C 00 basis functions based on Pascal triangle . . . . . . . . . . . . . . . . . . . . . . . . . . 8.7.2 Lagrange family C 00 basis functions based on area coordinates . . . . . . . . . . . . . . . . . . . . . 8.7.3 Higher degree C 00 basis functions using area coordinates . . . . . . . . . . . . . . . . . . . . . . . . 1D and 2D approximations based on Legendre polynomials .

xiii

495 497 501 507 508 511 515 516 520 522 525 529 532 538 540 542 542 549 550 551 553 554 556 556 558 559 566

CONTENTS

xiv

8.8.1 8.8.2

Legendre polynomials . . . . . . . . . . . . . . . . . . 1D p-version C 0 hierarchical approximation functions (Legendre polynomials) . . . . . . . . . . . . . . . . . 8.8.3 2D p-version C 00 hierarchical interpolation functions for quadrilateral elements (Legendre polynomials) . . 8.8.4 2D C ij p-version interpolations functions for quadrilateral elements (Legendre polynomials) . . . . . . . 8.8.5 2D C 00 p-version interpolation functions for triangular elements (Legendre polynomials) . . . . . . . . . . 8.8.6 2D C ij interpolation functions for triangular elements (Legendre polynomials) . . . . . . . . . . . . . . . . . 8.9 1D and 2D interpolations based on Chebyshev polynomials . 8.9.1 Chebyshev polynomials . . . . . . . . . . . . . . . . . 8.9.2 1D C 0 p-version hierarchical interpolations based on Chebyshev polynomials . . . . . . . . . . . . . . . . . 8.9.3 2D p-version C 00 hierarchical interpolation functions for quadrilateral elements (Chebyshev polynomials) . 8.9.4 2D C ij p-version interpolation functions for quadrilateral elements (Chebyshev polynomials) . . . . . . . 8.10 Serendipity family of C 00 interpolations . . . . . . . . . . . . 8.10.1 Method of deriving serendipity interpolation functions 8.11 Interpolation functions for 3D elements . . . . . . . . . . . . 8.11.1 Hexahedron elements . . . . . . . . . . . . . . . . . . 8.11.1.1 Mapping of points . . . . . . . . . . . . . . . 8.11.1.2 Mapping of lengths . . . . . . . . . . . . . . 8.11.1.3 Mapping of volumes . . . . . . . . . . . . . . 8.11.1.4 Obtaining derivatives of φeh (ξ, η, ζ) with respect to x, y, z . . . . . . . . . . . . . . . . 8.11.2 Local approximation for a dependent variable ¯m . . . . . . . . . . . . . . . . . . . . . . . . φ over Ω 8.11.2.1 Hexahedron elements . . . . . . . . . . . . . ¯m . 8.11.2.2 Higher degree approximations of φ over Ω 8.11.2.3 C 000 Lagrange type local approximations using tensor product . . . . . . . . . . . . . . 8.11.2.4 C 000 p-version 3D hierarchical local approximations: using tensor product . . . . ¯ e ) p-version local approximations: 8.11.2.5 3D C ijk (Ω Hexahedron elements . . . . . . . . . . . . . ¯ e ) p-version interpolations for 8.11.2.6 3D C ijk (Ω distorted hexahedron elements: 27 node element . . . . . . . . . . . . . . . .

566 567 567 568 568 571 577 577 577 578 578 578 579 584 584 584 586 586 587 588 588 590 592 597 599

600

xv

CONTENTS

8.11.2.7 Interpolation theory for 3D tetrahedron ¯ e) elements: basis functions of class C 000 (Ω based on Lagrange interpolations . . . . . . . 8.11.2.8 Lagrange family C 000 interpolations based on volume coordinates . . . . . . . . . . . . . 8.11.2.9 Higher degree C 000 basis functions using volume coordinates . . . . . . . . . . . . . . 8.11.2.10 Four-node linear tetrahedron element (p-level of one) . . . . . . . . . . . . . . . . . . . . . 8.11.2.11 A ten-node tetrahedron element (p-level of 2) 8.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

600 601 603 604 604 604

9 Linear Elasticity using the Principle of Minimum Total Potential Energy 609 9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 609 9.2 New notation . . . . . . . . . . . . . . . . . . . . . . . . . . . 610 9.3 Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 611 9.4 Element equations . . . . . . . . . . . . . . . . . . . . . . . . 611 9.4.1 Local approximation of the displacement field . . . . 611 9.4.2 Stresses and strains . . . . . . . . . . . . . . . . . . . 612 9.4.3 Strain energy Πe1 and potential energy of loads Πe2 . . 612 9.4.4 Total potential energy Πe for an element e . . . . . . 614 9.5 Finite element formulation for 2D linear elasticity . . . . . . 617 ¯ e or Ω ¯ ξη . . . 618 9.5.1 Local approximation of u and v over Ω 9.5.2 Stresses, strains and constitutive equations . . . . . . 618 9.5.3 [B] matrix relating strains to nodal degrees of freedom 619 9.5.4 Element stiffness matrix [K e ] . . . . . . . . . . . . . 619 9.5.5 Transformations from (ξ, η) to (x, y) space . . . . . . 620 9.5.6 Body forces . . . . . . . . . . . . . . . . . . . . . . . 620 9.5.7 Initial strains (thermal loads) . . . . . . . . . . . . . 621 9.5.8 Equivalent nodal loads {F e }p due to pressure acting normal to the element faces . . . . . . . . . . . . . . 622 9.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 624 10 Linear and Nonlinear Solid Mechanics using the Principle Virtual Displacements 10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 10.2 Principle of virtual displacements . . . . . . . . . . . . . . 10.3 Virtual work statements . . . . . . . . . . . . . . . . . . . . 10.3.1 Stiffness matrix . . . . . . . . . . . . . . . . . . . . 10.4 Solution method . . . . . . . . . . . . . . . . . . . . . . . . 10.4.1 Summary of solution procedure . . . . . . . . . . . 10.5 Finite element formulation for 2D solid continua . . . . . .

of 625 . 625 . 626 . 627 . 633 . 635 . 636 . 637

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CONTENTS 10.6 Finite element formulation for 3D solid continua . . . . . . . 641 10.7 Axisymmetric solid finite elements . . . . . . . . . . . . . . . 644 10.8 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 648

11 Additional Topics in Linear Structural Mechanics 11.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . 11.2 1D axial spar or rod element in R1 (1D space) . . . . . 11.2.1 Stresses and strains . . . . . . . . . . . . . . . . 11.2.2 Total potential energy: Πe . . . . . . . . . . . . 11.3 1D axial spar or rod element in R2 . . . . . . . . . . . . 11.3.1 Coordinate transformation . . . . . . . . . . . . 11.3.2 A two member truss . . . . . . . . . . . . . . . 11.3.2.1 Computations . . . . . . . . . . . . . . 11.3.2.2 Post-processing . . . . . . . . . . . . . . 11.4 1D axial spar or rod element in R3 (3D space) . . . . . 11.5 The Euler–Bernoulli beam element . . . . . . . . . . . . 11.5.1 Derivation of the element equations (GM/WF) . 11.5.2 Local approximation . . . . . . . . . . . . . . . 11.6 Euler-Bernoulli frame elements in R2 . . . . . . . . . . 11.7 The Timoshenko beam elements . . . . . . . . . . . . . 11.7.1 Element equations: GM/WF . . . . . . . . . . . 11.8 Finite element formulations in R2 and R3 . . . . . . . . 11.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

. . . . . . . . . . . . . . . . . .

651 651 651 653 653 656 656 659 660 665 666 668 670 671 675 677 678 681 681

12 Convergence, Error Estimation, and Adaptivity 683 12.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 683 12.2 h-, p-, k-versions of FEM and their convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684 12.2.1 h-version of FEM and h-convergence . . . . . . . . . 685 12.2.2 p-version of FEM and p-convergence . . . . . . . . . 686 12.2.3 hp-version of FEM and hp-convergence . . . . . . . . 686 12.2.4 k-version of FEM and k-convergence . . . . . . . . . 687 12.3 Convergence and convergence rate . . . . . . . . . . . . . . . 689 12.3.1 Convergence behavior of computations . . . . . . . . 690 12.3.2 Convergence rates . . . . . . . . . . . . . . . . . . . . 692 12.4 Error estimation and error computation . . . . . . . . . . . . 693 12.5 A priori error estimation . . . . . . . . . . . . . . . . . . . . 694 12.5.1 Galerkin method with weak form (GM/WF): self-adjoint operators . . . . . . . . . . . . . . . . . . 694 12.5.2 GM/WF for non-self adjoint and non-linear operators 697 12.5.3 Least-squares method based on residual functional: self-adjoint and non-self-adjoint operators . . . . . . 698

CONTENTS

12.5.4 Least-squares method based on residual functional for non-linear operators . . . . . . . . . . . . . . . . . . . 12.5.5 Integral forms based on other methods of approximation . . . . . . . . . . . . . . . . . . . . . . 12.5.6 General remarks . . . . . . . . . . . . . . . . . . . . . 12.5.7 A priori error estimates: GM/WF and LSP . . . . . 12.5.7.1 Model problem 1: GM/WF . . . . . . . . . . 12.5.7.2 Model problem 2: LSP . . . . . . . . . . . . 12.5.7.3 Proposition and proof . . . . . . . . . . . . . 12.5.7.4 Proposition and proof . . . . . . . . . . . . . 12.5.7.5 Convergence rates . . . . . . . . . . . . . . . 12.5.7.6 Proposition and proof . . . . . . . . . . . . . 12.5.7.7 General Remarks . . . . . . . . . . . . . . . . 12.6 Model problems . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1 Model problem 1: Self-adjoint operator, 1D diffusion equation . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.1.1 GM/WF . . . . . . . . . . . . . . . . . . . . 12.6.1.2 LSP, higher-order system (no auxiliary equation) . . . . . . . . . . . . . . . . . . . . 12.6.2 Model problem 2: Non-self-adjoint operator, 1D convection-diffusion equation . . . . . . . . . . . . 12.6.2.1 LSP: First order system . . . . . . . . . . . . 12.6.2.2 GM/WF . . . . . . . . . . . . . . . . . . . . 12.6.2.3 LSP: Higher order system (without auxiliary equation) . . . . . . . . . . . . . . . . . . . . 12.6.3 Model problem 3: Non-linear operator, 1D Burgers equation . . . . . . . . . . . . . . . . . . . . . . . . . 12.6.3.1 LSP: Higher-order system (without auxiliary equation) . . . . . . . . . . . . . . . . . . . . 12.6.3.2 GM/WF . . . . . . . . . . . . . . . . . . . . 12.7 A posteriori error estimation and computation . . . . . . . . 12.7.1 A posteriori error estimation . . . . . . . . . . . . . . 12.7.2 A posteriori error computation . . . . . . . . . . . . . 12.8 Adaptive processes in finite element computations . . . . . . 12.8.1 Adaptive processes for 1D convection-diffusion equation: non-self adjoint operator . . . . . . . . . . 12.8.1.1 Adaptivity in the pre-asymptotic range: uniform h-refinement . . . . . . . . . . . . . 12.8.1.2 Adaptivity in the pre-asymptotic range: adaptive h-refinement . . . . . . . . . . . . . . . . . .

xvii

699 702 702 703 703 704 706 712 714 716 719 719 720 721 727 730 731 737 738 741 743 747 747 747 749 751 752 752

753

xviii

CONTENTS

12.8.1.3 Adaptivity in the pre-asymptotic range: graded h-rediscretizations . . . . . . . . . . . . . . . 754 12.8.1.4 General Remarks . . . . . . . . . . . . . . . . 757 12.8.1.5 Adaptivity in the onset of asymptotic and asymptotic ranges: uniform p-refinement . . 758 12.8.1.6 Adaptivity in the onset of asymptotic and asymptotic ranges: adaptive p-refinement . . 758 12.8.1.7 Adaptivity in the onset of asymptotic and asymptotic ranges: adaptive h-refinement . . 760 12.8.1.8 Adaptivity using higher geometric ratios for h-rediscretization at P e = 1000 and P e = 106 760 12.8.2 Adaptive processes for 1D Burgers equation: non-linear operator . . . . . . . . . . . . . . . . . . . 761 12.8.3 Adaptive processes for 1D diffusion equation: self-adjoint operator . . . . . . . . . . . . . . . . . . . 765 12.8.4 General Remarks . . . . . . . . . . . . . . . . . . . . 767 12.9 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 767 Appendix A: Numerical Integration using Gauss Quadrature A.1 Gauss quadrature in R1 , R2 and R3 . . . . . . . . . . . . . . ¯m = Ω ¯ ξ = [−1, 1] . . . . . . . . . A.1.1 Line integrals over Ω ¯m = Ω ¯ ξη = [−1, 1] × [−1, 1] . . . A.1.2 Area integrals over Ω m ¯ ¯ ξηζ = [−1, 1]×[−1, 1]× A.1.3 Volume Integrals over Ω = Ω [−1, 1] . . . . . . . . . . . . . . . . . . . . . . . . . . A.2 Gauss quadrature over triangular domains . . . . . . . . . .

771 771 771 772

INDEX

779

773 775

Preface Since there are already many textbooks and monographs on the finite element method, it is perhaps natural to ask “why another book?” This question can be answered if one examines the published material on the subject. Broadly speaking, the books on the subject can be classified into two categories: those that present the finite element method as a study in applied mathematics and those that approach the subject using specific applications. Finite element books on linear elasticity, stress analysis, heat transfer, fluid mechanics, and so on are examples of application based approach. Both types of writings have their own strengths and weaknesses from the point of view of the students wanting to learn the subject. The applied mathematics approach requires more rigorous mathematical background and preparation and as a consequence graduate students in engineering and sciences shy away from learning the subject through this approach. Secondly, these writings often lack application aspects of the subject that are generally helpful for engineering and science students. The writings that are highly focused in presenting the subject through specific applications obviously result in loss of generality and as a consequence the students often learn the subject as a technique for a specific class of problems. For example, a finite element book on linear elasticity may generally focus on minimization of total potential energy and as a consequence the students may never realize the much broader impact of the subject on all BVPs in the other areas of mechanics and applied sciences. Distinct demarcation in writings and teaching of the subject for linear processes, non-linear processes, solids, liquids and gasses often leaves the students confused and unclear not only regarding the mathematical foundations of the subject, but also its much broader impact in applications in all areas of engineering and sciences. This book is intended to bridge the gap between the applied mathematics and strictly application-oriented books. The material in this book is presented in a mathematically rigorous fashion but with sufficient examples, applications, and illustrations in various areas of engineering, sciences, and mathematical physics so that students are able to grasp the mathematical foundation of the subject as well as its versatility of applications in all areas of engineering, sciences, and mathematical physics. The book is aimed for a first semester graduate study of the finite element method for boundary values problems (BVPs). The finite element method is introduced and presented as a method of approximation for obtaining numerical solutions of differential and partial differential equations describing time-independent xix

xx

PREFACE

processes (BVPs) regardless of their origin or field of application. In order to address the totality of all BVPs rigorously and in an application-independent fashion, the differential operators appearing in all BVPs are classified mathematically into three categories: self-adjoint, non-self adjoint, and non-linear operators, and their properties are established. These are then utilized with various methods of approximation such as the Galerkin method, PetrovGalerkin method, weighted residual method, Galerkin method with weak form, least squares processes, and other methods from which the details of the finite element processes are derived. A correspondence is established between the methods of approximation and hence finite element method and the elements of the calculus of variations. This is then utilized for various methods of approximation for the three classes of differential operators to determine which methods of approximation yield unconditionally stable computational processes. Chapter 1 provides a brief introduction of the subject of the mathematics of computations and the finite element method for boundary value problems. Concepts of discretization, local approximations, integral forms, element algebraic equations, assembly of element equations, computations of solutions, and post-processing of solutions are introduced. An introduction to the kversion of the finite element method and hpk framework for computations of the solutions of the boundary value problems is also presented. Basic elements from applied mathematics: spaces, scalar product spaces, scalar product and its significance, function spaces, differential operators and their mathematical classifications, and energy product are presented in chapter 2. Chapter 2 also contains elements of calculus of variations and functional analysis: concept of variation of a functional, Euler’s equations, correspondence between extrema of functionals and solutions of boundary value problems, fundamental and other lemmas in calculus of variations and their proofs, Riemann and Lebesgue integrals, properties of self adjoint, non-self adjoint, and non-linear differential operators including examples. Concepts and definitions of variationally consistent (VC) and variationally inconsistent (VIC) integral forms are introduced to establish when the integral forms yield unconditionally stable computational processes. Chapter 3 contains classical methods of approximation based on fundamental lemma such as the Galerkin method, Galerkin method with weak form, Petrov-Galerkin method, and method of weighted residuals (GM, GM/WF, PGM, WRM), and least squares method based on residual functional (LSM) for all three classes of differential operators. Many theorems and their proofs related to VC and VIC integral forms from these methods are presented for the three classes of differential operators. Applications and model problems are considered to illustrate various concepts. Serious shortcomings of classical methods of approximation for practical applica-

xxi

tions are discussed. Chapter 4 introduces the finite element model details for GM, GM/WF, PGM, WRM, and LSM for all three classes of differential operators. Specific details and formulations for model problems in R1 , R2 , and R3 , proofs of VC and VIC integral forms for the model problems, C 0 solutions and solutions of higher classes in hpk framework for finite element formulations and processes for self adjoint, non-self adjoint, and non-linear differential operators are presented in chapters 5, 6, and 7, respectively. Chapter 8 contains basic elements of mapping and interpolation theory. Details of mapping of points, lengths, areas, and volumes are discussed. Local approximations are presented for elements in R1 , R2 , and R3 using Lagrange, Legendre, and Chebyshev polynomials. Local approximations of class C 0 and higher classes, p-version hierarchical local approximations of class C 0 and higher classes are presented in R1 , R2 , and R3 . Area and volume coordinates are introduced and utilized for triangular and tetrahedral family of elements in R2 and R3 to derive local approximations of various classes. Chapter 9 presents finite element formulations in linear solid and structural mechanics, derived using principle of minimum total potential energy, formulated directly from the physics of deformation without utilizing the underlying differential equations. A general derivation applicable to finite element formulations in R1 , R2 , and R3 is presented first and then followed by specific examples in R2 such as plane stress. Chapter 10 contains derivations of finite element formulations using principle of virtual work. This approach is specially meritorious for finite deformation and finite strain reversible processes. A general derivation in R3 is presented first that is specialized for applications in R1 and R2 . Some additional finite element formulations related to axial deformation of rods (or spars) in R1 , R2 , and R3 , use of element local coordinate systems, and finite element formulations for Euler–Bernoulli and Timoshenko beams are presented in chapter 11. Appendix A contains details related to Gauss quadrature in R1 , R2 , and R3 . The computer program, Finesse (“Finite element system”), used to solve the problems in this book is available free of cost from the first author upon request. The material in this book is self-contained and requires no supplementary reading or any other reference material. The students learning the subject through this book are expected to have two semesters of calculus, an undergraduate course in differential and partial differential equations, a course in linear algebra and an undergraduate level course in numerical methods. An advanced course in partial differential equations and a course in calculus of variations are helpful but are not prerequisite to learning the material presented in this book. This book is a result of the evolution of the first author’s thirty years of teaching and research of the subject in the Department of Mechanical Engineering at the University of Kansas. The author’s own research work in mathematics of computations and the finite element subject

xxii

PREFACE

and continuum mechanics has contributed heavily to this unique approach of presenting the mathematical details of the foundation of the finite element method with simplicity while maintaining its versatility and transparency for applications. The first author has successfully utilized this material in educating graduate students on the subject as well as preparing them for post-graduate studies and research. Both authors over the last twenty years have been engaged in joint research grants, publications and collaborative research efforts that have resulted in many new concepts such as operator classifications, the k-version of finite element method, variationally consistent integral forms, and so on that form the foundation of much of the material in this book. Authors’ long friendship and research collaborations have been extremely enjoyable and fruitful in bringing focus, depth, clarity, and in developing this unique approach of presenting the mathematics and computations related to the finite element method for boundary value problems presented in this book. The DEPSCoR/AFOSR grant to the first author and the joint research grants to the authors from the U.S. Army (ARO) related to k-version of finite element method, operator classifications, VC and VIC integral forms, and unconditionally stable computational processes resulted in a significant number of joint fundamental publications that form the core of the material in many chapters of this book. The authors are truly grateful to many of their graduate students whose Ph.D. and M.S. theses in many areas of computational mathematics and finite element method have contributed immensely in bringing the subject matter to its present level of maturity. The first author is extremely thankful to his Ph.D. student Mr. Tyler Stone who prepared the first draft of the manuscript of this book single handedly, performed many numerical studies contained in the book, and also helped in many subsequent versions. Thanks are also due to Dr. Daniel Nunez, a former Ph.D. student of the first author, who encouraged and supported such writing endeavors and contributed heavily in many portions of chapters 6, 7, and 8 including numerical studies for model problems contained in these chapters. A very special thanks to Mr. Aaron D. Joy, the first author’s current Ph.D. student who has typeset the book and has typed and retyped many portions of the book, reorganized and in many cases redid the graphs and illustrations to bring the manuscript of the book to its present level. His interest and knowledge of the subject, hard work, and commitment to this book project have been instrumental in the completion of the book. This book would not have been possible without the research grant from DEPSCoR/AFOSR to the first author and the joint research grants: W911NF-09-1-0548 (FED0065623), W-911NF-11-10471 (FED0061541), and W911NF-12-1-0463 from the U.S. Army Research Office (ARO) to the authors that led to research in various areas of com-

xxiii

putational mathematics and finite element processes. Sincere thanks to Dr. Joseph Myers, Division Chief, Mathematical Sciences Division, Information Science Directorate, ARO, for his interest and support of some of the research results included in this book. This book contains so many equations, derivations, and mathematical details that it is hardly possible to avoid some typographical and other errors. Authors would be grateful to those readers who are willing to draw attention to the errors using the emails: [email protected] or [email protected]. Karan S. Surana, Lawrence, KS J. N. Reddy, College Station, TX

About the Authors Karan S. Surana, born in India, went to undergraduate school at Birla Institute of Technology and Science (BITS), Pilani, India, and received a B.E. degree in Mechanical Engineering in 1965. He then attended the University of Wisconsin, Madison, where he obtained M.S. and Ph.D. degrees in Mechanical Engineering in 1967 and 1970, respectively. He worked in industry, in research and development in various areas of computational mechanics and software development, for fifteen years: SDRC, Cincinnati (1970–1973), EMRC, Detroit (1973–1978); and McDonnell Douglas, St. Louis (1978– 1984). In 1984, he joined the Department of Mechanical Engineering faculty at University of Kansas, where he is currently the Deane E. Ackers University Distinguished Professor of Mechanical Engineering. His areas of interest and expertise are computational mathematics, computational mechanics, and continuum mechanics. He is author of over 350 research reports, conference papers, and journal articles. He has served as advisor and chairman of 50 M.S. students and 22 Ph.D. students in various areas of Computational Mathematics and Continuum Mechanics. He has delivered many plenary and keynote lectures in various national and international conferences and congresses on computational mathematics, computational mechanics, and continuum mechanics. He has served on international advisory committees of many conferences and has co-organized minisymposia on k-version of the finite element method, computational methods, and constitutive theories at US National Congresses of Computational Mechanics organized by the US Association of Computational Mechanics (USACM). He is a member of International Association of Computational Mechanics (IACM) and USACM, and a fellow and life member of ASME. Dr. Surana’s most notable contributions include: large deformation finite element formulations of shells, the k-version of the finite element method, operator classification and variationally consistent integral forms in methods of approximations, and ordered rate constitutive theories for solid and fluent continua. His most recent and present research work is in non-classical internal polar continuum theories and non-classical Cosserat continuum theories for solid and fluent continua and associated ordered rate constitutive theories. He is author of recently published continuum mechanics textbook, Advanced Mechanics of Continua, CRC/Taylor & Francis, Boca Raton, Florida, 2015.

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ABOUT THE AUTHORS

J. N. Reddy was born in Telangana, India, and obtained his B,E. (Mech) from Osmania University, Hyderabad, in 1968. He obtained M.S. from Oklahoma State University (1970) and Ph.D. from University of Alabama in Huntsville (1973). He worked at Lockheed Missiles and Space Company for a short period in 1974 and served on the faculty in the School of Aerospace, Mechanical, and Nuclear Engineering at University of Oklahoma (1975–1980), in the Department of Engineering Science and Mechanics at Virginia Polytechnic Institute and State University (1980–1992). Since 1992 he has been in the Department of Mechanical Engineering at Texas A&M University, currently holding the titles University Distinguished Professor, Regents Professor, and the Oscar S. Wyatt Endowed Chair Professor. Professor Reddy is internationally-recognized for his research on mechanics of composite materials and for computational methods. The shear deformation plate and shell theories that he developed bear his name (Reddy third-order shear deformation theory and Reddy layerwise theory) in the literature. The finite element formulations and models he developed have been implemented into commercial software like ABAQUS, NISA, and HyperXtrude. He is the author of numerous journal papers and 20 textbooks, several of them with multiple editions. Dr. Reddy is one of the original top 100 ISI Highly Cited Researchers in Engineering around world with over 19,200 citations with h-index of over 66 as per Web of Science; the number of citations is over 47,000 with h-index of 89 and i10-index of 405 (i.e., 405 papers are cited at least 10 times) as per Google Scholar. Most significant awards and honors he received to date are: 1992 Worcester Reed Warner Medal and 1995 Charles Russ Richards Memorial Award of the American Society of Mechanical Engineers (ASME); 1997 Archie Higdon Distinguished Educator Award from the Mechanics Division of the American Society of Engineering Education; 1998 Nathan M. Newmark Medal from the American Society of Civil Engineers (ASCE); 2000 Excellence in the Field of Composites and 2004 Distinguished Research Award from the American Society for Composites (ASC); 2003 Computational Solid Mechanics award from the US Association of Computational Mechanics; 2014 The IACM O.C. Zienkiewicz Award from the International Association of Computational Mechanics; 2014 Raymond D. Mindlin Medal from the Engineering Mechanics Institute of ASCE; and 2016 William Prager Medal of the Society of Engineering Science. He is an elected member of the US National Academy of Engineering for contributions to composite structures and to engineering education and practice and elected as a Foreign Fellow of the Indian National Academy of Engineering. He is a fellow of many professional societies (e.g., AIAA, ASC, ASCE, ASME, AAM, USACM, IACM), and serves on the editorial boards of two dozen journals. A more complete resume can be found at http://www.tamu.edu/acml/.

1

Introduction 1.1 General Comments and Basic Philosophy The physical processes encountered in all branches of sciences and engineering can be classified into two categories: time-dependent processes and stationary processes. Time-dependent processes describe evolutions in which quantities of interest change as time elapses. If the quantities of interest cease to change in an evolution, then the evolution is said to have a stationary state. Not all evolutions reach stationary states. The evolutions without a stationary states are often referred to as unsteady processes. Stationary processes are those in which the quantities of interest do not depend on time. For a stationary process to be valid or viable it must correspond to the stationary state of an evolution. Every process in nature is an evolution, but nonetheless it is sometimes convenient to consider their stationary state. In this book, we consider only time-independent or stationary processes. A mathematical description of most stationary processes in sciences and engineering often leads to a system of ordinary or partial differential equations. The mathematical descriptions of the stationary processes are referred to as boundary value problems (BVPs). Since stationary processes are independent of time, the partial differential equations describing their behavior only involve dependent variables and spatial coordinates as independent variables. On the other hand, mathematical descriptions of evolutions leads to partial differential equations in dependent variables and space coordinates as well as time as independent variables and are referred to as initial value problems (IVPs). The numerical solutions of the BVPs using the finite element method is the subject of study in this book. In case of simple physical systems, the BVPs may be simple enough to permit analytical solutions, however most physical processes of interest may be quite complicated and their mathematical descriptions (BVPs) may be complex enough not to permit analytical solutions. In such case, one could undertake simplification of the mathematical description to a point that analytical solutions are possible. In this approach, the simplified forms may not describe the actual behavior and sometimes this simplification may not be possible. In the second alternative, we abandon the possibility of analytical solutions all together as viable means of solving complex practical problems and instead resort to numerical methods for obtaining numerical 1

2

INTRODUCTION

solutions of BVPs. The finite element method is one such method of solving BVPs numerically and constitutes the subject matter for this book. Before proceeding with the details of the FEM for BVPs, it is perhaps fitting to discuss some of the commonly used numerical methods for obtaining numerical solutions of the BVPs. These are: 1. finite difference method 2. finite volume method 3. finite element method 4. meshless and element free methods 5. boundary element method 6. and other methods, including hybrid methods that are a combination of above methods A comprehensive and detailed description, merits and shortcomings of all of these methods are not considered in this book but can be found in many published works. In this book we consider only details of FEM as a technique for obtaining numerical solutions of the BVPs. Regardless of the method used for obtaining numerical solutions of the BVPs, all of the methods have one feature in common: in each case, the calculated numerical solution is an approximation of the true or theoretical solution of the BVP. This is perhaps the reason that all numerical methods for BVPs are referred to as methods of approximation. In the development of the numerical methods for BVPs one must exercise utmost care in ensuring that the methods are general and are not problem dependent, as the host of BVPs in various areas of engineering, sciences, and mathematical physics is so vast that the study, development, and implementation of problem dependent computational methods is a never-ending and exhausting undertaking with total lack of generality. It is for this reason that we must consider totality of all possible BVPs in all areas of engineering, sciences, and mathematical physics and perhaps classify them mathematically into groups so that specialized, accurate and efficient numerical methods only need to be developed for each of these groups. In engineering and physical sciences, it has been customary and perhaps more appealing to classify BVPs as: elliptic, parabolic, hyperbolic, mixed type etc. due to the fact that such classification may be more meaningful in revealing the physics of the processes described by the BVPs. However, if we intend to borrow various tools from different branches of applied mathematics to pursue the development of a comprehensive and general mathematical and computational framework to solve BVPs numerically, then the classification of BVPs depending upon the strict mathematical nature of the differential operators involved in their description is certainly the most prudent choice. All BVPs

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD

3

can be classified mathematically into three categories, those described by: (1) self-adjoint differential operators, (2) non-self-adjoint differential operators, and (3) non-linear differential operators. This classification is vital in understanding the mathematical properties of these operators and development of general and problem independent numerical methods to address their solutions. In this book, we consider FEM for the BVPs described by these three classes of differential operators. This approach addresses numerical solutions of the totality of all BVPs using finite element method. When addressing numerical methods for BVPs in addition to the fact that all such methods are methods of approximation there is one more feature that is common in all methods: the BVPs described by the differential operators must somehow be converted into a system of algebraic equations. The manner in which this is accomplished differs in different methods of approximation. The nature of the coefficient matrix in the resulting algebraic systems depends upon: a) the method of approximation and b) the nature of the differential operators. A solution of the algebraic equation provides the approximate solution. Thus, in principle this process is rather simple but the ingenuity of one method of approximation over the others lies in the manner in which this is accomplished.

1.2 Basic Concepts of the Finite Element Method Symbolically we can represent all BVPs by writing Aφ − f = 0 in Ω, where A is the differential operator, φ are the dependent variables and f is the nonhomogeneous part. Here Ω is the domain of definition of the BVP. It is a subset of R1 ≡ R, R2 or R3 , that is, it is a collection of x; x, y or x, y, z (in the continuum sense) for which the BVP is valid. In addition, the BVP description has boundary conditions on the part or the whole of ¯ as the closure Ω ¯ = Ω ∪ Γ. the boundary Γ of the domain Ω. We define Ω This symbolic representation of the BVPs is helpful in two ways. First, it makes the presentation of the development of methodologies for their solutions compact and concise and secondly with this representation we can concentrate on the mathematical properties of the operator A as opposed to getting entangled with the physics.

1.2.1 Discretization When addressing numerical solutions of BVPs using the finite element ¯ over which the BVP is defined is subdimethod, the domain of definition Ω vided into smaller domains called subdomains. Now, the original domain of ¯ can be visualized as the assembly of these subdomains intercondefinition Ω nected with each other through their common boundaries. The subdomains are obviously of finite sizes. Each subdomain is referred to as a finite element.

4

INTRODUCTION

The assembly of these subdomains, that is, the finite elements is referred to as a finite element mesh or discretization. The precise nature and the shapes of these finite elements depend upon the number of independent variables in the PDEs, that is, 1-D, 2-D or 3-D boundary value problems and also to some extent on our choice. For example, for 1-D BVPs the finite elements are simply line segments interconnected at the two ends with adjacent line segments to constitute the entire domain of definition of the 1-D BVP which is also a line segment. For 2-D BVPs the domain of definition is an area and hence in this case many choices exist for choosing a finite element shape: triangular with linear sides, rectangular with linear sides, triangular with distorted sides, quadrilateral with distorted sides and others. For 3-D BVPs the finite elements are three-dimensional as well. Tetrahedron and hexahedron shapes with undistorted and distorted edges and faces are commonly used. When all finite elements of a discretization are of the same shape and size, the discretization is called a uniform discretization. On the other hand, when the elements of the discretization vary in size, the discretization is called nonuniform or graded. Regardless of the type of discretization, the consequence of discretization is a finite element mesh. For each finite element of the discretization we generally identify a finite number of points on the boundary of the element (sometimes in the interior of the element as well). These points are called “nodes.” An element must be connected to the neighbors at the node points as well as common boundaries. Thus, in this process we have identified a finite number of points for the whole discretization of the ¯ The collection of all of these points for the whole discretization is domain Ω. sometimes referred to as grid points. The numerical values of the dependent variables (and/or their derivatives) at these grid points are the quantities of interest. ¯ of the BVPs In summary, discretization of the domain of definition of Ω in the finite element processes yields subdomains, that is, finite elements, a finite element mesh, which is a collection of all finite elements, node points on the boundaries and interiors of the elements and the grid points. SymM ¯T = S Ω ¯ e as the discretization of Ω, ¯ the closure of the bolically we denote Ω e

¯ e = Ωe ∪ Γe is the domain of domain of definition of the BVP in which Ω e definition of an element e, Γ is the closed boundary of the element e and M ¯ T . Figure 1.1(a) shows is the total number of elements in the discretization Ω an axial rod fixed at one end and subjected to an axial load P at the other end. The area of cross-section of the rod is A and the modulus of elasticity of the rod material is E. The mathematical idealization of the rod and the two three-element discretizations employing two-node and three-node line elements are shown in Figs. 1.1(b)-(d). Figure 1.2(a) shows a thin plate in

5

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD

L A, E P

(a) Physical system ¯ Ω P (b) Mathematical idealization

y

¯T Ω

1 1

¯e Ω

2 2

3 3

x

4

y

xe

xe+1 x

P a typical element e

¯ T using two-node elements (c) Discretization Ω y

¯e Ω

¯T Ω

1 1

2 2

3 3

x

4

y

xe

xe 1

xe+2 x

P a typical element e

¯ T using three-node elements (d) Discretization Ω

Figure 1.1: Axial rod with end load P

tension (plane stress problem) with thickness t, modulus of elasticity E and Poisson’s ratio ν. The finite element discretizations of the plate using threenode and six-node triangular elements and nine-node quadrilateral elements are shown in Figs. 1.2(b)-(d). Similarly, for the three-dimensional domains of definition of the BVPs one could construct discretizations using tetrahedron or hexahedron with desired numbers of nodes on the edges, faces and the interiors of elements.

1.2.2 Local approximation For each finite element of the discretization one must define the behaviors of the dependent variables. This is accomplished by using basis functions

6

INTRODUCTION

y t, E, ν

¯ Ω

σx

x (a) Physical system y ¯T Ω

¯e Ω σx a typical element e x (b) Discretization using 3-node triangular elements y ¯T Ω

¯e Ω

σx

a typical element e x (c) Discretization using 6-node triangular elements y ¯T Ω

¯e Ω σx a typical element e x (d) Discretization using 9-node quadrilateral elements

Figure 1.2: Thin plate in tension

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD

7

or local approximation functions and the dependent variable and possibly their derivative values at the nodes of the elements, often referred to as the degrees of freedom. One constructs a linear combination using the local approximation functions and the nodal degrees of freedom in which the local approximation functions are known but the nodal degrees of freedom are unknown. Thus, each dependent variable has a local approximation over each element of the discretization with its own local approximation functions and the nodal degrees of freedom. We can write φeh = [N ]{φe } in which φeh ¯ e , [N ] is the is the local approximation of the dependent variable φ over Ω collection of all local approximation functions for the dependent variables φ and {φe } is the collection of all nodal degrees of freedom for the dependent variable φ. Such local approximations are constructed for each dependent variable φ in the description of the BVP. In the following, for simplicity, we consider φ to be the only dependent variable. In the process of constructing φeh , there are two important points to note: ¯ e of an element e, that is first, φeh is constructed locally over the domain Ω without regard to the connecting neighboring elements to it, secondly, φeh ¯ e locally. Hence, is an approximation of the theoretical solution φ over Ω e the reason for referring to φh as a local approximation of φ. The choices of [N ] and {φe } are not arbitrary and bear great consequences on the resulting finite element processes. The subjects of interpolation theory and mapping in applied mathematics are vital in this respect. The basic concepts and required details from both of these areas are covered in chapter 8. Here we ¯ T is given by φh = S φe only remark that the approximation φh of φ over Ω e h ¯ e . φh is referred to as the global and is naturally dependent on φeh over Ω ¯T . approximation of φ over the discretization Ω If one chooses monomials to construct local approximation φeh , then the functions in [N ], that is, local approximation functions are algebraic polynomials. By choosing appropriate degree of these polynomials, one can control ¯ e . However, the global the differentiability of local approximations φeh over Ω T ¯ differentiability of φh over Ω is dependent on the differentiability of φeh over the elements as well as their inter-element behaviors. The choice of the ¯ e plays a crucial role in controlling the global nodal degrees of freedom for Ω T ¯ differentiability of φh over Ω .

1.2.3 Integral forms and algebraic equations over an element Now we have the mathematical model of the BVP, Aφ − f = 0 in Ω, a ¯ T of Ω ¯ = Ω ∪ Γ, and a local approximation φe of φ over Ω ¯ e in discretization Ω h e e which the degrees of freedom {φ } for φh are unknown for each element of the discretization. Determining numerical values of the degrees of freedom {φe } ¯ T is our objective. First, we note that and thereby φeh for each element of Ω ¯ e one cannot solve for {φe } due to the fact using Aφ − f = 0 and φeh over Ω

8

INTRODUCTION

¯ T so that boundary conditions can that solutions of BVPs must consider Ω be applied without which the solution is non-unique. However, we can take ¯T = S Ω ¯ e , and since Ω ¯ T is the discretization of advantage of the fact that Ω e ¯ it suggests that perhaps we must examine φ˜n , the approximation of φ over Ω, ¯ Ω and establish how we must proceed to determine the unknown constants in φ˜n . This is the subject of classical methods of approximations for BVPs. In these methods of approximation, we consider approximation φ˜n of φ over ¯ that is, global approximation of φ over the non-discretized domain Ω. ¯ Ω, The finite element method is simply an application of these methods over ¯ T with local approximations φe over Ω ¯ e . Let us first consider discretization Ω h classical methods of approximation. In all such methods we eventually have ¯ This is valid based an integral from corresponding to Aφ − f = 0 over Ω. on the fundamental Lemma of the calculus of variations. These integral forms contain a function called a test function. When the approximation φ˜n is substituted in the integral form and when we choose as many test functions as the number of basis functions in the approximation φ˜n , the integral form results into a system of as many algebraic equations as the number of basis functions or the number of unknowns in the approximation φ˜n . The solution of these algebraic equations yields the desired numerical values of the constants in the approximation φ˜n . The restrictions on the choices of basis functions and the test functions are subject of detailed study in the various methods of approximation. ¯ e , the In the finite element method, we construct the integral form over Ω domain of an element e. This is justified by the fact that integral forms over ¯ results in functionals that are scalars and hence the functional over Ω ¯T Ω ¯ (the discretization of Ω) can be written as the sum of the functionals over the individual elements. The construction of the integral form over an element e results in a system of algebraic equation for the element. Thus, we have a system of algebraic equations for each element of the discretization containing nodal dofs of the element and the right hand side. (details postponed for later). These are generally referred to as element equations, discretized equations of equilibrium, element stiffness equations etc. depending upon the specific discipline of engineering, sciences, and mathematical physics.

1.2.4 Assembly of element equations ¯T The fact that the functionals resulting from the integral form over Ω can be written as the sum of the functionals over the individual elements, and since the construction of the integral form over an element e results in a system of algebraic equations in the element dofs, suggests that the algebraic equations of the elements must be summed or assembled in order to obtain ¯ T . The a system of algebraic equations that is valid for the discretization Ω rules of assembling the element equations are derived using the facts that: (1)

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD

9

¯ T is the sum of the functionals the functional for the whole discretization Ω for the elements and (2) the dofs at a node on the common boundaries between the elements are unique. Precise details of how to accomplish the assembly will be considered in the subsequent chapters. The outcome of the assembly process is a set of algebraic equations that are valid for the whole discretization in which the dofs at the grid points are the unknowns.

1.2.5 Computation of the solution The assembled equations in Section 1.2.4 must be subjected to boundary conditions (transformed in terms of the dofs at the nodes on the boundaries) and then solved for the remaining unknown degrees of freedom. If the algebraic system consists of a system of linear simultaneous equations (when A in Aφ − f = 0 is linear), the elimination methods provide the most straightforward and efficient means of finding the solution. When the system of algebraic equations are non-linear (i.e., A in Aφ − f = 0 is non-linear), iterative methods such as fixed-point method, Newton’s method, etc. must be used to find the solution. In either case we have the numerical values of the dofs at all of the grid points of the discretization.

1.2.6 Post-processing Once we have the numerical values of the degrees of freedom at the grid points and thus at the node points of each element, the element local approximation provides an analytical expression for the behaviors of the dependent variables over each element. Using such descriptions one could easily calculate the derivatives of the dependent variables or any other desired quantities elementwise, hence for the whole discretization. This phase of the computations is referred to as post-processing of the solution as it is the step after the computation of the solution at the grid points of the whole discretization.

1.2.7 Remarks 1. The most crucial steps in the finite element process are: (a) Discretization. The finite element mesh. (b) Local approximation. The interpolation theory is crucial for a good understanding of this area (c) Algebraic equations for an element. The methods of approximation and calculus of variations are essential in this regard. 2. The finite element method is a piecewise application of the classical methods of approximation in which we consider the entire domain of definition of the BVP without discretization, a piece being a finite element of the

10

INTRODUCTION

discretization. Thus, in order to gain a good understanding of the finite element method one must have a thorough knowledge of the classical methods of approximation. 3. The accuracy of the finite element method depends upon: (a) The nature of the discretization, that is, the number of elements and their sizes and locations. If he is the measure of the size of element e, then we define h = max he (1.1) e

h is referred to as the characteristic length of the discretization. (b) The nature of the local approximation, that is, the behaviors of the dependent variables over each element of the discretization. If we use monomials to construct the local approximation functions, then the local approximation functions are algebraic polynomials of some degree. If p is the degree of local approximation and if this is the same for all elements of the discretization, then we can say that the accuracy of the finite element solution also depends upon p. We refer to degree p of the polynomials defining local approximation functions as p-level. 4. There are three basic sources of errors in the finite element processes. Errors can be due to: ¯ of the BVP. (a) Approximation of the shape of the domain of definition Ω This can be completely eliminated by choosing appropriate element shapes with curved sides, edges and distorted faces so that any desired geometry of ¯ can be discretized accurately. Ω ¯ e may not be an accurate description of (b) Local approximations over each Ω the true behaviors of the dependent variables. This error is generally always present in all finite element solutions but can be reduced by reducing h and increasing p. (c) Numerical computations. Such errors are generally a results of the lack of desired precision during computations due to inadequate word size of the computer and due to the use of faulty or inadequate algorithms. Obviously, such errors can also be eliminated completely from the numerically computed solutions. Thus, the subject of study of error in the finite element process is predominantly the study of errors in the computed solution due to local approximation, thus it is dependent on h and p and of course the characteristics of the theoretical solution of the BVP. 5. In all finite element processes, as more degrees of freedom are added, the solution accuracy improves, that is, the errors reduce. The degrees of

1.2. BASIC CONCEPTS OF THE FINITE ELEMENT METHOD

11

freedom can be added to a finite element computational process in one of three ways: (a) One could progressively reduce h, that is, refine the mesh thereby adding more elements, hence, more grid points and therefore more degrees of freedom. This process is know as the h-version of the finite element method and the process of improvement in the calculated solution and eventually approaching the theoretical solution is referred to as h-convergence. In this process one generally chooses a value of p and keeps it fixed while h is reduced so the mesh is refined. (b) In the other approach, one could keep a fixed discretization, that is, not change h but increase p-level, thereby adding more dofs at the nodes and hence, the grid points. This process is known as the p-version of the finite element method. The process of reducing error in the computed solution by only increasing p-levels for the elements of the discretization while keeping h constant and eventually converging to the theoretical solution is known as p-convergence. (c) The finite element processes in which h is reduced and p is increased simultaneously are known as the hp-version of the finite element method which leads to hp-convergence.

1.2.8 k-version of the finite element method and hpk framework Now since we understand the basic concept of the finite element method (even though in the most rudimentary way), it is perhaps fitting to ask what features the numerically computed solutions must have. The answer, of course, is that it should be as close to the theoretical solution in as many aspects as possible. This is rather vague but can be made more concrete if we examine the features of the theoretical solutions of the BVPs. When the theoretical solutions of BVPs are analytic, then we observe that the derivatives of the theoretical solution up to certain order (say j) are continuous over Ω, the domain of definition of the BVP, and we say that such solutions are of class C j (Ω) in which j can be infinity. However, j can never be less than the highest orders of the derivatives of the dependent variables apprearing in the GDEs of the BVP. We say that the theoretical solution has global differentiability of order j. In order to incorporate this feature of the theoretical solution in the finite element computational processes, the ¯ T , the discretization of Ω ¯ must be of global approximation S φh of φ over Ω j e e class C . Since φh = e φh , the local approximations φh must yield global ¯ T . The global differentiability of φh over differentiability of order j for Ω T ¯ ¯ e which Ω depends upon: (1) The global differentiability of φh over each Ω ¯ e and can be controlled by is the same as the differentiability of φeh over Ω

12

INTRODUCTION

choosing appropriate p-levels of the local approximation functions [N ] and can be greater than j. That is, φh over each Ωe can be of class C p in which p can be greater than j. (2) At the inter-element boundaries the global differentiability of φh is controlled by the basis functions and the dofs of the mating elements. The basis functions for the elements and the dofs at the nodes of the elements must be specifically chosen to ensure that at the mating boundaries φh has the desired global differentiability. In the currently used finite element processes, h and p are considered as the only independent parameters in the computational processes. The local approximations are generally chosen to be of class C 0 , that is in such cases φh is of class C p over each Ωe but of class C 0 at the inter-element boundaries ¯ T remains of class C 0 regardless of p-level. and as a consequence, φh over Ω Secondly, the change in h (mesh refinement) obviously does not influence global differentiability of φh as each φeh remains of class C 0 regardless of its size. Surana and coworkers [1–3] have shown that global differentiability of approximations in finite element processes can not be increased (or decreased) by changing h or p, hence global differentiability is an independent parameter in all finite element processes in addition to h and p. The global differentiability of φh is intrinsic in local approximations φeh , hence the local approximations φeh must be constructed specifically for a desired global differentiability at the inter-element boundaries. In order to give the idea of global differentiability a more concrete mathematical form, we proceed as follows. Let the local approximation [N ] or Ni belong to an approximation space Vh which is a subspace of H k,p (a bigger space). At this point, it suffices to say that k is the order of the space and this space contains basis functions of degree p (p-level) and of global differentiability k − 1. That is, when the basis functions from Vh ⊂ H k,p are used in constructing local approximations φeh , we are ensured global differentiability of φh of order k − 1 at the inter-element boundaries and of order p > (k − 1) over each Ωe . Thus, with the basis functions from Vh , φh has global differentiability of order k − 1. Hence, the order of the approximation space is an independent parameter in addition to h and p used currently. Hence, the k-version of the finite element method in addition to the h-version and p-version and associated k, hk, pk and hpk processes in addition to h, p and hp processes used currently. Thus, when one looks at the mathematical framework for finite element processes, h, p and k become independent parameters as opposed to h and p used presently. To emphasize this, we say hpk mathematical framework as opposed to hp framework. This terminology emphasizes the fact that global differentiability of approximation has been considered as an independent parameter in the development and construction of the mathematical framework for the finite element processes.

1.3. SUMMARY

13

The material presented in this book uses hpk framework. During the developments of the details of the finite element processes, clear distinctions are made between hpk framework and hp framework with local approximations of class C 0 so that the reader clearly sees the benefits of k as an independent parameter in the design and development of the finite element processes. Obviously, hp framework with C 0 local approximations is a subset of hpk framework.

1.3 Summary In the material presented in the various sections of this chapter, it is perhaps clear that in the development of the mathematical foundation and computational infrastructure for finite element processes we utilize concepts and principles from various areas of applied mathematics such as theory of differential operators, calculus of variations, theory of functions or real analysis, functional analysis, interpolation theory and approximation theory. Basic understanding of these areas of applied mathematics helps in developing a clearer and deeper understanding of the mathematical concepts involved in the finite element method. This book covers only the essential elements from these areas but in sufficient details to gain a good theoretical and working knowledge of the subject of finite element method for BVPs. [1–23]

References for additional reading [1] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002. [2] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [3] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004. [4] G. Hellwig. Differential Operators of Mathematical Physics. Addison-Wesley Publishing Co., 1967. [5] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964. [6] C. Johnson. Numerical Solutions of Partial Differential Equations by Finite Element Method. Cambridge University Press, 1987. [7] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000. [8] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering. McGraw Hill Company, 1986. [9] M. Becker. The Principles and Applications of Variational Methods. MIT Press, 1964. [10] M. Forray. Variational Calculus in Science and Engineering. McGraw-Hill, 1968. [11] K. Rektorys. Variational Methods in Mathematics, Science and Engineering. Reidel, 1977.

14

REFERENCES FOR ADDITIONAL READING

[12] R. S. Schechter. The Variational Methods in Engineering. McGraw-Hill, 1967. [13] K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 3nd edition, 1982. [14] R. Weinstock. Calculus of Variations with Applications to Physics and Engineering. McGraw-Hill, 1952. [15] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966. [16] J. T. Oden and L. Demkowicz. Applied Funtional Analysis. CRC-Press, 1996. [17] W. R. Clough. The finite element method in plane stress analysis. In Proceedings of 2nd Conference on Electronic Computation, pages 345–378. Journal of Structures Division, ASCE, 1960. [18] A. Hrenikoff. Solution of problems in elasticity by the frame work method. J. Applied Math., Trans of the ASME, 8:169–175, 1941. [19] J. T. Oden. Finite Elements of Nonlinear Continua. McGraw-Hill, New York, 1972. [20] J. T. Oden and J. N. Reddy. An Introduction to the Mathematical Theory of Finite Elements. John Wiley, New York, 1976. [21] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., New York, 3rd edition, 2006. [22] G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, New Jersey, 1973. [23] M. J. Turner, R. W. Clough, H. C. Martin, and L. J. Topp. Stiffness and deflection analysis of complex structures. J. Aero. Sci., 23:805–823, 1956.

2

Concepts from Functional Analysis

2.1 General Comments In this chapter we present some basic elements from various areas of linear algebra and functional analysis such as sets, function spaces, theory of functions, differential operators and calculus of variations. These elements provide the necessary concepts and basic principles for developing a mathematical framework and associated computational infrastructure for finite element processes. The foundation of the finite element method for boundary value problems (BVPs) relies heavily on these ideas. We begin with some basic definitions. Many theorems and lemmas are given, many of them with proofs. Proofs that are not included here may be found in references cited at the end of the chapter. The material presented here is not meant to be a crash course in various areas of applied mathematics but its intention is that of review so that the readers are refreshed. References are cited for those who may be interested in further reading on these topics.

2.2 Sets, Spaces, Functions, Functions Spaces, and Operators The material presented here contains basic definitions, theorems (many of them without proofs) related to sets, spaces, functions, function spaces and operators. The objective here is to help readers refresh the material essential for the further development of the concepts and principles related to the mathematics of computations and finite element subject for BVPs. Definition 2.1 (Set). A set is a collection of objects that share a certain common feature or property. Sets could be open or closed. In an open set, the boundary points (limit points) are not included in the set. For a closed set, boundary points are part of the set. 15

16

CONCEPTS FROM FUNCTIONAL ANALYSIS

Notation for sets of elements from the real number field, R: a < x < b ⇒ x ∈ (a, b),

open set; open on left as well as right

a ≤ x < b ⇒ x ∈ [a, b),

closed on left but open on right

a < x ≤ b ⇒ x ∈ (a, b],

open on left but closed on right

a ≤ x ≤ b ⇒ x ∈ [a, b],

closed set; closed on left as well as right

Definition 2.2 (Space). A set could also be called a space, except that spaces are more restricted (i.e., obey certain rules) than sets. A space has a basis and hence dimension, but a set may not. Definition 2.3 (Linear space). A set S is called a linear space if the following rules of addition and multiplication by a scalar are satisfied by the elements of the set: (i) w = u + v ∈ S ∀u, v ∈ S (defines the sum of u and v). (ii) αu =∈ S ∀u ∈ S, α ∈ R (defines the product of α and u where R is the space of real numbers). (iii) In addition, sums and products obey the following laws: (a) u + v = v + u (commutative) (b) (u + v) + w = u + (v + w) (associative) (c) For any u ∈ S there exists a unique element z ∈ S independent of u such that u + z = u (existence of the zero element, z = 0) (d) For any u ∈ S there exists a unique element w ∈ S that depends on u such that u + w = z (existence of the negative element, w = −u) (e) 1 · u = u (f) α(βu) = (αβ)u ∀α, β ∈ R (g) (α + β)u = αu + βu ∀α, β ∈ R (h) α(u + v) = αu + αv ∀α ∈ R Definition 2.4 (Linear relation). For u1 , u2 , . . . , un ∈ S, a linear space, and α1 , α2 , . . . , αn ∈ R an expression of the form α1 u1 + α2 u2 + . . . + αn un =

n X

αi ui ∈ S

i=1

is Pncalled a linear combination of u1 , u2 , . . . , un . An expression of the form i=1 αi ui = 0 is called a linear relation. Definition 2.5 (Linear dependence and independence). A linear relation among ui ∈ S α1 u1 + α2 u2 + · · · + αn un = 0

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

17

with all αi = 0 (i = 1, 2, . . . , n) is called a trivial relation among ui (i = 1, 2, . . . , n). A set of elements ui ∈ S (i = 1, 2, . . . , n) is called linearly dependent if there exists a nontrivial relation among them (i.e., at least one αi is nonzero); otherwise, the elements are said to be linearly independent. In other Pn words a set of elements is linearly independent when a linear relation i=1 αi ui = 0 holds with all αi = 0 (i = 1, 2, . . . , n). Definition 2.6 (Finite-dimensional space). A linear space S is said to be finite-dimensional or more precisely n-dimensional if there are n linearly independent elements in S, and n + 1st element in S is linearly dependent. The n linearly independent elements of an n-dimensional space form a basis. Theorem 2.1. If S is an n-dimensional linear space and if ui ∈ S (i = 1, 2, . . . , n) are linearly independent, then every element u ∈ S can be written as a linear combination of the n elements u = α1 u1 + α2 u2 + . . . + αn un ,

αi ∈ R

where αi (i = 1, 2, . . . , n) are uniquely determined by u. Definition 2.7 (Function). A function defines a rule, law or mapping. Consider f (x), x ∈ Ω = [a, b]. The function f defining a rule, maps every element of the set Ω called its domain of definition to another set R called its range of definition; that is, f (x) maps set Ω to set R or we simply write f : Ω → R. Definition 2.8 (Metric space). A linear space M of elements u, v, w, . . . is said to be a metric space if to each pair of elements u, v ∈ M there corresponds a number ρ(u, v) called the distance between u and v with the following properties: (i) ρ(u, v) = ρ(v, u) (ii) ρ(u, w) ≤ ρ(u, v) + ρ(v, w) triangle inequality (iii) ρ(u, v) ≥ 0 and ρ(u, v) = 0 ⇔ u = v Example: n-dimensional Euclidean space. Definition 2.9 (Normed space). A linear space S is said to be a normed space if for every u ∈ S there corresponds a real number ||u||, called the norm of u, which satisfies the following properties: (i) ||αu|| = |α| ||u||

∀α ∈ R (homogeneous)

(ii) ||u + v|| ≤ ||u|| + ||v|| triangle inequality (iii) ||u|| ≥ 0 and ||u|| = 0 ⇔ u = 0 (positive-definite)

18

CONCEPTS FROM FUNCTIONAL ANALYSIS

Definition 2.10 (Banach space). A Banach space B is a complete normed space (the notion of completeness is defined later). Definition 2.11 (Inner product space). A linear space S with elements u, v, w, . . . is said to be an inner product space if for every pair of elements u, v ∈ S there corresponds a real number (u, v), called the scalar product of u and v, which satisfies the following properties: (i) (u, v) = (v, u) (symmetry) (ii) (αu + βv, w) = α(u, w) + β(v, w) ∀α, β ∈ R (iii) (u, u) ≥ 0 and (u, u) = 0 ⇔ u = 0 An inner product space becomes a normed space when the norm is defined with respect to the inner product. Definition 2.12 (Hilbert space). A Hilbert space is a complete inner product space (completeness is defined later). In the definition of Hilbert spaces we generally also specify the differentiability of the functions contained in them. This is doen in section 2.2.1. Remarks. 1. The scalar product of u and v given by (u, v) has not been defined yet and could have any convenient definition as long as it satisfies the properties defined above. 2. In the solutions of BVPs, we are interested in Hilbert spaces of functions. 3. We only consider Hilbert spaces that are separable (defined later).

¯ 2.2.1 Hilbert spaces H k (Ω) ¯ is denoted A Hilbert space of square-integrable functions defined over Ω 0 k ¯ ¯ ¯ by H (Ω) = L2 (Ω). A Hilbert space H (Ω) is a space of functions that ¯ (i.e., possesses continuous derivatives up to order k − 1 defined over a set Ω the kth derivatives exist and are square-integrable in Lebesgue sense). Ob¯ is the viously, H k ⊂ H k−1 ⊂ · · · ⊂ H 2 ⊂ H 1 ⊂ H 0 ≡ L2 ; that is, H 0 (Ω) i ¯ (i = 1, 2, . . . , k) and largest and least restrictive of all of the spaces H (Ω) 1 2 k H , H , . . . , H (k > 2) are progressively more and more restricted spaces ¯ is more restrictive than space H n (Ω) ¯ than H 0 = L2 . Thus, space H k (Ω) when k > n.

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

19

¯ space 2.2.2 Definition of scalar product in H k (Ω) In the theory of differential operators and hence finite element method, the following definition of scalar product (·, ·) is meaningful. If u and v are two elements of a Hilbert space H 0 = L2 , then we could define (called the L2 -inner product) Z (u, v)0 =

¯ ∀u, v ∈ L2 (Ω)

uv dΩ, ¯ Ω

¯ is the domain of definition of the functions u and v. We note that and Ω the Hilbert space H 1 contains functions that are continuous but the first derivatives may be discontinuous. However, if the first derivatives are square integrable, the definition of the scalar product Z  (u, v)1 =

uv +

du dv dx dx

 dΩ,

¯ ∀u, v ∈ H 1 (Ω)

¯ Ω

is meaningful (assuming u = u(x), v = v(x), . . .). Likewise, the scalar prod¯ can be defined as uct (u, v) in H k (Ω)

(u, v)k =

k Z X di u di v dΩ dxi dxi

¯ ∀u, v ∈ H k (Ω)

i=0 ¯ Ω

The extension of the above definition for functions of more than one variable is rather straight forward.

2.2.3 Properties of scalar product One could easily verify that the definition of (u, v) in section 2.2.2 satisfies all of the properties of the scalar product, i.e. (i) (u, v) = (v, u) (ii) (a1 u1 + a2 u2 , b1 v1 + b2 v2 ) = a1 b1 (u1 , v1 ) + a1 b2 (u1 , v2 ) + a2 b1 (u2 , v1 ) + a2 b2 (u2 , v2 ) ∀a, b ∈ R k R  i 2 P i du dΩ ≥ 0 and (u, u) = 0 ⇔ ddxui = 0; i = 0, . . . , k (iii) (u, u) = dxi ¯ i=0 Ω

20

CONCEPTS FROM FUNCTIONAL ANALYSIS

¯ 2.2.4 Norm of u in Hilbert space H k (Ω) ¯ Hilbert Based on the definition of the scalar product (u, v)k over H k (Ω) k ¯ space in section 2.2.2, we define the norm of u in H (Ω) by

||u||H k

1  2 k Z  i 2 X du   dΩ ≥ 0 = ||u||k = dxi i=0 ¯ Ω

||u||H k = ||u||k = 0 ⇔ Thus ||u||H 0 = ||u||0 = ||u||L2

di u ¯ = 0; i = 0, . . . , k ∀x ∈ Ω dxi

 1 2 Z 2 =  u dΩ ≥ 0 ;

¯ ||u||H 0 = 0 ⇔ u = 0 ∀x ∈ Ω

¯ Ω

 Z  ||u||H 1 = ||u||1 =

1 2  2 ! du du 2 ¯  ∀x ∈ Ω dΩ ≥ 0 ; ||u||H 1 = 0 ⇔ u = 0 = u + dx dx

¯ Ω

and so on.

¯ 2.2.5 Seminorm of u in Hilbert space H k (Ω) ¯ is defined by Seminorm of u in H k (Ω)

Thus

1  2 Z  k 2 d u   |u|H k = |u|k = dΩ ≥0 dxk ¯ Ω  1 2 Z 2 |u|H 0 = ||u||H 0 = ||u||L2 =  u dΩ ≥ 0 ¯ Ω

¯ where equality implies that u = 0 ∀x ∈ Ω. Theorem 2.2 (Cauchy-Schwarz inequality). Let u and v be arbitrary elements of an inner product space. Then 1

1

|(u, v)| ≤ (u, u) 2 (v, v) 2 Proof. For v = 0, the inequality is obviously satisfied. Suppose v 6= 0. Then for arbitrary α ∈ R, define f (α) = (u + αv, u + αv) = α2 (v, v) + 2α(u, v) + (u, u) > 0

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

Choose α = α1 = − (u,v) (v,v) , then (u, v)2 (u, v)2 −2 + (u, u) (v, v) (v, v) (u, v)2 =− + (u, u) > 0 (v, v)

f (α1 ) =

Hence, (u, v)2 ≤ (u, u)(v, v) and

1

1

|(u, v)| ≤ (u, u) 2 (v, v) 2 This completes the proof. Theorem 2.3 (Triangle inequality). Assume that u, v ∈ H 0 (Ω), then ||u + v||0 ≤ ||u||0 + ||v||0 Proof. ||u + v||20 = ||u||20 + ||v||20 + 2(u, v) ≤ ||u||20 + ||v||20 + 2|(u, v)| Using Cauchy-Schwarz inequality for |(u, v)| ||u + v||20 ≤ ||u||20 + ||v||20 + 2 ||u||0 ||v||0 = (||u||0 + ||v||0 )2 Taking square root of both sides ||u + v||0 ≤ ||u||0 + ||v||0 This completes the proof. Theorem 2.4 (Friedrichs inequality). If u ∈ H 1 (0, l) and u(0) = 0, then 

Zl

 0

 21



|u(x)|2 dx ≤ l 

Zl

 21 |u0 (x)|2 dx

0

Proof. Since

Zx u(x) = u(x) − u(0) = 0

u0 (t) dt

21

22

CONCEPTS FROM FUNCTIONAL ANALYSIS

using Cauchy-Schwarz inequality x 2  x  x  Z Z Z Zl |u(x)|2 = 1u0 (t) dt ≤  1 dt  |u0 (t)|2 dt ≤ l |u0 (x)|2 dx 0

0

0

0

Integrating both sides Zl

|u(x)|2 dx ≤ l

0

Zl

 l  Z  |u0 (x)|2 dx dx

0

≤ l2

0

Zl

|u0 (x)|2 dx

0

Taking square root of both sides 

Zl



 l  21 Z |u(x)|2  ≤ l  |u0 (x)|2 dx  21

0

0

This completes the proof. Theorem 2.5 (Continuity of scalar product). If lim un = u and lim vn = n→∞ n→∞ v, then lim (un , vn ) = (u, v) n→∞

2.2.6 Function spaces When the elements of a space are functions, we refer to such spaces as function spaces. In the theory of differential operators, function spaces contain desired functions. In the definition of the scalar product (section 2.2.2) u, v ∈ H where H can be a Hilbert space containing functions. Definition 2.13 (Fundamental sequence). A sequence u1 , u2 , . . . of elements, ui ∈ M , is said to be a fundamental sequence if for any  > 0 there exists a positive number N () such that ρ(un , um ) < ,

∀n, m > N ()

Theorem 2.6. A convergent sequence of elements u1 , u2 , . . . is a fundamental sequence. (Note: Not every fundamental sequence is a convergent sequence.)

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

23

Definition 2.14 (Complete Hilbert space). A Hilbert space H is said to be complete if every fundamental sequence in it converges. Definition 2.15 (Separable Hilbert space). A Hilbert space H is said to be separable if there exists a sequence of elements u1 , u2 , . . . ∈ H such that for every u ∈ H and every  > 0 we can find an element ul in the sequence for which ||u − ul || <  holds. Definition 2.16 (Orthogonality). Two elements u, v ∈ H are said to be orthogonal if (u, v) = 0 The element u ∈ H is said to be orthogonal to a subspace S ⊆ H if u is orthogonal to every v ∈ S. Theorem 2.7 (Dense subspace). The subspace S ⊆ H is dense in H if and only if there is no element in H except null element which is orthogonal to S.

2.2.7 Operators • An operator also defines a rule, law or mapping. • An operator acts on functions and hence must be defined for a set or a space of functions called the domain of definition of the operator. • An operator is denoted by a symbol together with the functions on which it acts. • For each element in its domain of definition, an operator produces another element. The collection of these elements is called the range of the operator. Thus, if V ⊂ H is the linear space containing functions u, v, w, . . . constituting the domain of definition of an operator A then the collection of elements Au, Av, Aw, . . . is the range of A, denoted A V , and we have A:V →

A

V, ∀u ∈ V

That is, operator A maps V into A V (∀u ∈ V , A(u) ∈ A V ). Even though V is a linear space containing functions u, v, w, . . ., the nature of space A V depends upon the specific nature of the operator A, which can be linear or nonlinear.

24

CONCEPTS FROM FUNCTIONAL ANALYSIS

2.2.8 Types of operators Many different types of operators are encountered in mathematical physics, engineering, and sciences. In this book we are interested in differential operators. Differential operators define operation of differentiation on functions constituting its domain of definition. Differential operators transform elements of higher-order spaces to elements in lower order spaces. Definition 2.17 (Linear operator). Let A be an operator, A : V ⊂ H → H. Then A is linear if the following relation holds: A(a1 u1 +· · ·+an un ) = a1 Au1 +· · ·+an Aun

∀u1 , . . . , un ∈ V, ∀a1 , . . . , an ∈ R

Example 2.1. Consider the differential equation ∂2φ ∂2φ + 2 − f (x, y) = 0, ∂x2 ∂y 2

(x, y) ∈ Ω

2

∂ ∂ Define operator A as A = ∂x 2 + ∂y 2 so that we have Aφ − f = 0 over Ω. This compact notation is very useful in the mathematical developments associated with the finite element processes. We note that since the differential equation is linear, the operator A is not a function of φ; hence A is a linear operator:  2  ∂ ∂2 A(αu + βv) = + (αu + βv) ∂x2 ∂y 2  2   2  ∂ u ∂2u ∂ v ∂2v =α + 2 +β + ∂x2 ∂y ∂x2 ∂y 2 = αA(u) + βA(v)

Example 2.2. Consider the differential equation φ

dφ d2 φ − k 2 = 0 ∀x ∈ Ω dx ∂x 2

d d Defining operator A as A = φ dx − k dx 2 , we see that A is a function of φ. Hence, the differential operator A is not linear:

d d2 (αφ1 + βφ2 ) − k 2 (αφ1 + βφ2 ) dx   dx 2 dφ1 dφ2 d φ1 d2 φ2 = (αφ1 + βφ2 ) α +β − kα 2 − kβ 2 dx dx dx dx 6= αA(φ1 ) + βA(φ2 )

A(αφ1 + βφ2 ) = (αφ1 + βφ2 )

Definition 2.18 (Adjoint of an operator). Let A be a linear (differential) operator and let V ⊂ H be its domain of definition. If (Au, v) = (u, A∗ v) + hAu, viΓ

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

25

holds, then A∗ is called the adjoint of A. Here hAu, viΓ is called the concomitant, and it is only a symbol that represents the boundary terms that are obtained in the process of moving operator A from u to operator A∗ on v. Thus, the concomitant is a collection of boundary terms obtained as a consequence of integration by parts. This is amply illustrated in examples presented shortly. Definition 2.19 (Symmetric operator). Let A be a linear operator with its domain of definition V ⊂ H, then A is symmetric if (Au, v) = (u, Av), ∀u, v ∈ V If we use the definition of the scalar product in section 2.2.2, the Symmetry of A implies Z Z (Au)v dΩ = u(Av) dΩ, ∀u, v, ∈ V ¯ Ω

¯ Ω

A∗

for Au, Av ∈ V and = A, that is, for symmetric operators the adjoint of the operator is the same as the operator. Definition 2.20 (Self-adjoint operator). If an operator A is linear and symmetric, then it is self-adjoint. Thus, for self-adjoint operators we have A(αu + βv) = αAu + βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R (Au, v) = (u, Av), ∀u, v ∈ V ⊂ H Definition 2.21 (Non-self-adjoint operator). If an operator A is linear but not symmetric, then it is non-self-adjoint. Thus, for non-self-adjoint operators we have A(αu + βv) = αAu + βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R = (Au, v) 6= (u, Av), ∀u, v ∈ V ⊂ H For non-self-adjoint operators the adjoint A∗ of the operator is not the same as the operator A. Definition 2.22 (Non-linear operator). If an operator A is neither linear nor symmetric, then it is non-linear. Thus, for non-linear operators we have A(αu + βv) 6= αAu + βAv, ∀u, v ∈ V ⊂ H, ∀α, β ∈ R (Au, v) 6= (u, Av), ∀u, v ∈ V ⊂ H Since linearity of the operator is essential for symmetry, it suffices to say that if the operator A is not linear, then it is also not symmetric. The definition of the adjoint A∗ of A in such cases is not meaningful.

26

CONCEPTS FROM FUNCTIONAL ANALYSIS

Definition 2.23 (Positive-definite operator). A linear symmetric operator (hence, self-adjoint) A is positive-definite if, (Au, u) > 0 ∀u ∈ V ⊂ H\{0} Definition 2.24 (Positive bounded below operator). A linear symmetric operator A is positive positive bounded below if, (Au, u) ≥ γ 2 ||u||2 ∀u ∈ V ⊂ H where γ is a positive constant. Definition 2.25 (Functional). For each function in its field or domain of definition V if an operator A generates a function identically equal to a constant then this operator is known as a functional. Functionals are a restricted class of operators. Functionals encountered in the theory of differential operators are integrals corresponding to the differential operators. Definition 2.26 (Linear functionals). A functional L(u) with a single argument u is linear if (i) its field of definition V ⊂ H is linear. (ii) for ui ; i = 1, . . . , n ∈ V ⊂ H and ai ; i = 1, 2, . . . , n ∈ R, L(a1 u1 + a2 u2 + · · · + an un ) = a1 L(u1 ) + a2 L(u2 ) + · · · + an L(un ) holds. Definition 2.27 (Bilinear functionals). A functional B(u, v) with two arguments u and v is bilinear if (i) its field of definition V ⊂ H is linear. (ii) it is linear in u as well as v, B(a1 u1 + a2 u2 + · · · + an un , v) = a1 B(u1 , v) + a2 B(u2 , v) + · · · + an B(un , v) B(u, b1 v1 + b2 v2 + · · · + bn vn ) = b1 B(u, v1 ) + b2 B(u, v2 ) + · · · + bn B(u, vn ) holds for all ui ∈ V ⊂ H, ai ∈ R, vi ∈ V ⊂ H, and bi ∈ R for i = 1, 2, . . . , n. Definition 2.28 (Symmetric functional). A functional B(u, v) is symmetric if B(u, v) = B(v, u)

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

27

Definition 2.29 (Quadratic functional). If A is a linear and symmetric operator with domain of definition V1 ⊂ H1 and if L is a linear functional with domain of definition V2 ⊂ H2 , then functionals of the form F (u) = (Au, u) + L(u) + c ∀u ∈ V1 ∩ V2 , c being a constant is called a quadratic functional.

2.2.9 Energy product If A is a positive-definite differential operator with its domain of definition V ⊂ H, then Z uA(u) dΩ

(Au, u) =

∀u ∈ V ⊂ H

¯ Ω

is proportional to the amount of energy stored in the system described by A. We generalize the definition the energy product. Consider (Au, v), which is meaningful if u ∈ V ⊂ H and v is an arbitrary function with finite norm, ||v|| = (v, v)1/2 < ∞. v is called a test function. Now we consider (Au, v) in which u, v ∈ V ⊂ H and v satisfies the same boundary conditions as u. Under these conditions, the scalar product (Au, v) is called energy product of functions u and v and we denote Z [u, v] = (Au, v) = vA(u) dΩ ∀u, v ∈ V ⊂ H ¯ Ω

Obviously, the energy product possesses the same properties as the scalar product of functions.

2.2.10 Integration by parts (IBP) Integration by parts is an important calculus tool that allows one to transfer differentiation from one function to another in an integral representation. In the following we consider line, surface and volume integrals that correspond to R ≡ R1 , R2 , and R3 . Example 2.3. (Line integrals in 1D) Consider the linear differential operator A : S ⊂ H 1 (a, b) → H 0 (a, b) defined by dφ , x ∈ (a, b) dx Then we have by integration by parts the relation (here Ω = (a, b) and Γ = a and b) Z b Z b dφ dψ ψ dx = φ dx + [φψ]ba (Aφ, ψ) = − dx a dx a = (φ, A∗ ψ) + hAφ, ψiΓ Aφ = −

28

CONCEPTS FROM FUNCTIONAL ANALYSIS

Clearly, A∗ = d/dx and hAφ, ψiΓ = [φψ]ba . Thus, d/dx is the adjoint of −d/dx, and A = −d/dx is not self-adjoint. hAφ, ψiΓ is called concomitant. Example 2.4. (Line integrals in 1D) Now consider the linear differential operator A : S ⊂ H 2 (a, b) → H 0 (a, b) defined by Aφ =

d2 φ , dx2

x ∈ (a, b)

Then using the integration by parts twice we obtain the relation (Γ = a and b)   Z b 2 Z b 2 d φ d ψ dφ dψ b φ 2 dx + (Aφ, ψ) = ψ dx = ψ−φ 2 dx dx dx a a dx a = (φ, A∗ ψ) + hAφ, ψiΓ Clearly, A∗ = d2 /dx2 and the concomitant is defined by   dφ dψ b hAφ, ψiΓ = ψ−φ dx dx a Thus, A = A∗ = d2 /dx2 is self-adjoint. If S is defined to be a linear space of elements φ and ψ satisfying the conditions φ(a) = 0 and φ(b) = 0 as well as ψ(a) = 0 and ψ(b) = 0, then we have (Aφ, ψ) = (φ, Aψ), φ, ψ ∈ S

Example 2.5. (Area integrals in 2D) Consider the linear differential oper∂ ∂ ator A = ∂x + ∂y , A : S ⊂ H 1 (Ω) → H 0 (Ω) (Ω ⊂ R2 ), and Z  ∂φ ∂φ  (Aφ, ψ) = + ψ dΩ ∂x ∂y ¯ Ω  Z  I ∂ψ ∂ψ =− φ + dx dy + (nx + ny ) φ ψ dΓ ∂x ∂y ¯ Ω

Γ ∗

= (φ, A ψ) + hAφ, ψiΓ where Γ is the closed contour constituting the boundary of domain Ω and (nx , ny ) are the direction cosines of the unit exterior normal to the boundary Γ (see Fig. 2.1). We have transferred all of the differentiation from φ to ψ using integration by parts. The concomitant is defined by I hAφ, ψiΓ = (nx + ny ) φ ψ dΓ Γ

2.2. SETS, SPACES, FUNCTIONS, FUNCTIONS SPACES, AND OPERATORS

29

ny nx Ω

n2x + n2y = 1

y Γ x Figure 2.1: Domain of definition of a 2D BVP

Example 2.6. (Area integrals in 2D) Consider the Laplace operator A = ∂2 ∂2 ∇2 = ∂x 2 + ∂y 2 and (Aφ, ψ) =

Z  2 ∂ φ ∂2φ  + 2 ψ dΩ ∂x2 ∂y ¯ Ω

 ∂2ψ ∂2ψ = φ + 2 dx dy ∂x2 ∂ y ¯ Ω   I  I  ∂φ ∂φ ∂ψ ∂ψ nx + ny ψ dΓ − φ nx + ny dΓ + ∂x ∂y ∂x ∂y 

Z

Γ

Γ ∗

= (φ, A ψ)Ω¯ + hAφ, ψiΓ where two orders of differentiations are transferred from φ to ψ (using the gradient theorem), and the concomitant < Aφ, ψ >Γ is given by  I  ∂φ ∂ψ hAφ, ψiΓ = ψ−φ dΓ ∂n Γ ∂n where the normal derivative

∂ ∂n

in 2-D is defined by

∂ ∂ ∂ = nx + ny ∂n ∂x ∂y The direction cosines (nx , ny ) and Γ have the same meaning as in Example 2.5. Thus, the adjoint of A is A∗ =

∂2 ∂2 + = ∇2 = A ∂x2 ∂y 2

Thus, the operator is self-adjoint.

30

CONCEPTS FROM FUNCTIONAL ANALYSIS

Example 2.7. (Volume integrals in 3D) Consider the operator A = ∂ ∂ ∂y + ∂z and  Z  ∂φ ∂φ ∂φ ψ dΩ (Aφ, ψ) = + + ∂x ∂y ∂z ¯ Ω  Z  ∂ψ ∂ψ ∂ψ =− φ dx dy dz + + ∂x ∂y ∂z ¯ Ω I + (nx + ny + nz ) φ ψ dΓ

∂ ∂x

+

Γ

= (φ, A∗ ψ)Ω¯ + hAφ, ψiΓ where one order of differentiation is transferred from φ to ψ with respect to x, y and z in each of the terms in the integrand using integration by parts. ∂ ∂ ∂ Thus, the adjoint of A is A∗ = − ∂x − ∂y − ∂z 6= A and the concomitant hAφ, ψiΓ is given by I hAφ, ψiΓ = (nx + ny + nz )φ ψ dΓ Γ

Example 2.8. (Volume integrals in 3D) Consider the operator A = ∂2 ∂2 + ∂z 2 , and ∂y 2  Z  2 ∂ φ ∂2φ ∂2φ (Aφ, ψ) = + 2 + 2 ψ dΩ ∂x2 ∂y ∂z

∂2 ∂x2

+

¯ Ω

∂2ψ ∂2ψ ∂2ψ + 2 + 2 = φ ∂x2 ∂ y ∂ z Ω  I  ∂φ ∂ψ + ψ−φ dΓ ∂n Γ ∂n = (φ, A∗ ψ)Ω¯ + hAφ, ψiΓ Z



 dx dy dz

where two orders of differentiation are transferred from φ to ψ with respect to x, y and z in each of the three terms in the integrand and the normal derivative is defined by ∂ ∂ ∂ ∂ = nx + ny + nz ∂n ∂x ∂y ∂z 2

2

2

∂ ∂ ∂ Thus, the adjoint of A is A∗ = ∂x 2 + ∂y 2 + ∂z 2 = A and the concomitant is given by  I  ∂φ ∂ψ hAφ, ψiΓ = ψ−φ dΓ ∂n Γ ∂n

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

31

2.3 Elements of Calculus of Variations In an earlier section we introduced the concept of ’functionals’. In the abstract sense, we mean a mapping or a correspondence that assigns a definite real number to each function belonging to some class or space. Functionals are variable quantities and play a very important role in mathematical physics, sciences, and engineering. Calculus of variations is a branch of applied mathematics that deals with extrema of functionals, that is, maximum, saddle points, and minimum. First, we need to establish a connection between the differential equations and their solutions, which is the subject of interest to us, and the functionals and their extrema, that is calculus of variations. There are four basic lemmas that are important in this regard. We state these and provide their proofs. Lemma 2.1 (Fundamental lemma). If α(x) ∈ H 1 (a, b) and Zb α(x)h(x)dx = 0 a

holds ∀h(x) ∈ H 1 (a, b), then α(x) = 0 ∀x ∈ (a, b). Proof. We construct the proof of this lemma by contradiction. Suppose α(x) is nonzero, say positive at some point [x1 , x2 ] in [a, b]. If we let h(x) = (x − x1 )(x2 − x) for some x ∈ [x1 , x2 ] and h(x) = 0 otherwise, then h(x) obviously satisfies the conditions of the lemma. However, Zx2

Zb α(x)h(x)dx = a

α(x)(x − x1 )(x2 − x)dx > 0 x1

since the integrand is positive (except at x1 and x2 , where it is zero), which contradicts the lemma. Conversely, since h(x) is arbitrary we can choose it to be equal to α(x). Then an integral of a positive integrand is zero only if the integrand itself is equal to zero, proving that α(x) = 0. Lemma 2.2. If α(x) ∈ H 2 (a, b) and Zb

α(x)h0 (x)dx = 0

a

holds ∀h(x) ∈ H 2 (a, b) such that h(a) = h(b) = 0, then α0 (x) = 0 or α(x) = c, ∀x ∈ (a, b)

32

CONCEPTS FROM FUNCTIONAL ANALYSIS

where c is a constant. Proof. Consider Zb

α(x)h0 (x)dx = 0

a

Transfer of differentiation from h(x) to α(x) using integration by parts gives Zb

Zb

0

α(x)h (x)dx = − a


b α0 (x)h(x)dx + α(x)h(x) = 0 a

a


b Since h(a) = h(b) = 0, α(x)h(x) = 0 and we have a

Zb

α0 (x)h(x)dx = 0

a

Using lemma 2.1, we have α0 (x) = 0, which implies that α(x) is a constant. This completes the proof of the lemma. Lemma 2.3. If α(x) ∈ H 3 (a, b) and Zb

α(x)h00 (x)dx = 0

a

holds ∀h(x) ∈ H 3 (a, b) such that h(a) = h(b) = h0 (a) = h0 (b) = 0, then α00 (x) = 0 or α(x) = c0 + c1 x, ∀x ∈ (a, b) where c0 and c1 are constants. Proof. Consider Zb

α(x)h00 (x)dx = 0

a

Transferring the differentiation from h(x) to α(x) using integration by parts twice, we obtain Zb

00

Zb

α(x)h (x)dx = − a

h ib h ib α00 (x)h(x)dx + α(x)h0 (x) − α0 (x)h(x) = 0 a

a

a

33

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

Since h(a) = h(b) = h0 (a) = h0 (b) = 0, the boundary terms vanish and we obtain Zb Zb 00 α(x)h (x)dx = α00 (x)h(x)dx = 0 a

a

α00 (x)

Using lemma 2.1, we have = 0 which yields α(x) = c0 + c1 x. This completes the proof of the lemma. Lemma 2.4. If α(x) ∈ H 1 (a, b) and β(x) ∈ H 2 (a, b) and Zb



 α(x)h(x) + β(x)h0 (x) dx = 0

a

holds ∀h(x) ∈ H 2 (a, b) with the property h(a) = h(b) = 0, then α(x) − β 0 (x) = 0 ∀x ∈ (a, b) Proof. Transferring differentiation from h(x) to β(x) using integration by parts once, we arrive at Zb



0



Zb

α(x)h(x) + β(x)h (x) dx =

h ib   α(x) − β 0 (x) h(x) dx + β(x)h(x) = 0 a

a

a

Since h(a) = h(b) = 0, we obtain Zb a

  α(x)h(x) + β(x)h0 (x) dx =

Zb



 α(x) − β 0 (x) h(x)dx = 0

a

Using lemma 2.1, we obtain α(x) − β 0 (x) = 0 ∀x ∈ (a, b) must hold. Hence α(x) − β 0 (x) = 0. This completes the proof of the lemma.

2.3.1 Concept of the variation of a functional Variation means change or in the sense of calculus, differential. Let I(y) with y = y(x) be a functional defined over some normed linear space and let ∆I(h) = I(y + h) − I(y) be the increment in I corresponding to increment h = h(x) of the dependent variable y = y(x). If y is fixed, then ∆I(h) is a functional of h, in general a non-linear functional. Suppose that ∆I(h) = Φ(h) +  ||h||

34

CONCEPTS FROM FUNCTIONAL ANALYSIS

where Φ(h) is a linear functional and  → 0 as ||h|| → 0. Then the functional I is said to be differentiable and the principal linear part of the increment ∆I(h), that is, the linear functional Φ(h) which differs from ∆I(h) by an infinitesimal of order higher than 1 relative to ||h|| is called the variation of I(y) denoted by δI or δI(h). Theorem 2.8. The variation of a differentiable functional is unique (see reference [1] for proof ). Theorem 2.9. A necessary condition for the differentiable functional I(y) to have an extremum for y = y∗ is that its variation vanishes for y = y∗; that is, δI(h) = 0 for y = y∗ for all admissible h (see reference [1] for proof ).

2.3.2 Euler’s equation: Simplest variational problem Let F (x, y, y 0 ) be a function with continuous first and second derivatives with respect to all of its arguments. Then among all functions y(x) which are differentiable for a ≤ x ≤ b and satisfy boundary conditions y(a) = A and y(b) = B, find the function y(x) for which the functional Zb I(y) =

F (x, y, y 0 ) dx

a

has an extremum. Details are provided in the following. Let us consider an increment h(x) in y(x) and h0 (x) in y 0 (x); that is, let y(x) change to y(x) + h(x) and y 0 (x) to y 0 (x) + h0 (x). In order for the new values of y(x) to satisfy boundary conditions we must have h(a) = h(b) = 0 (homogeneous part of the boundary conditions). Due to increments in y(x) and y 0 (x), there must be a change in I(y). Let ∆I be the incremental change in I. Then

∆I = I y(x) + h(x) − I y(x) or Zb ∆I = a

Zb =

0

0


F (x, y(x) + h(x), y (x) + h (x)) dx −

Zb

F (x, y, y 0 ) dx

a


F (x, y(x) + h(x), y 0 (x) + h0 (x)) − F (x, y, y 0 ) dx

a

Expanding F (x, y(x) + h(x), y 0 (x) + h0 (x)) in Taylor series about y, y 0 (x being fixed) Zb ∆I = a


F (x, y, y 0 ) +Fy (x, y, y 0 ) h+Fy0 (x, y, y 0 ) h0 +O(h2 , (h0 )2 )−F (x, y, y 0 ) dx

35

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

or Zb ∆I =


Fy (x, y, y 0 ) h + Fy0 (x, y, y 0 ) h0 dx +

a

Zb

O(h2 , (h0 )2 ) dx

a

O(h2 , (h0 )2 )

where is a measure of the remaining terms in the Taylor series that are quadratic and of higher degrees in h and h0 . Let R b expansion 2 0 2 a O(h , (h ) dx =  ||h|| in which as  → 0, ||h|| → 0. Hence Zb


Fy (x, y, y 0 ) h + Fy0 (x, y, y 0 ) h0 dx +  ||h||

∆I = a

we note that the integral in the above expression is a linear functional in h and h0 and satisfies the definition of the variation of I. Therefore, we have Zb δI =


Fy (x, y, y 0 ) h + Fy0 (x, y, y 0 ) h0 dx

a

Based on Theorem 2.9, a necessary condition for the functional I(y) to have an extremum at y = y ∗ is that δI must vanish at y = y ∗ . Thus Zb δI =


Fy (x, y, y 0 ) h + Fy0 (x, y, y 0 ) h0 dx = 0

a

for all admissible h(x). Using Lemma 2.4 and comparing with the above integral we find that α(x) = Fy

and

β(x) = Fy0

and β(x) must be differentiable and α(x) = β 0 (x) must hold which gives Fy = or Fy −

d (Fy0 ) dx

d (Fy0 ) = 0 dx

must hold for a y(x) obtained from δI = 0. This is a differential equation and is known as Euler’s equation. Thus, a y(x) obtained from δI = 0 gives a unique extremum of I(y) and also satisfies Euler’s equation which is a differential equation. We note that based on lemma 2.4 Fy0 is differentiable, a necessary condition for the second term in Euler’s equation.

36

CONCEPTS FROM FUNCTIONAL ANALYSIS

Remarks. The Euler’s equation can also be derived in an alternate manner using δI = 0. Consider Zb δI =


Fy (x, y, y 0 )h(x) + Fy0 (x, y, y 0 )h0 (x) dx = 0

a

Transfer one order of differentiation with respect to x from h to Fy0 in the second term in the integrand.  Zb  h ib d(Fy0 ) 0 δI = Fy (x, y, y )h(x) − h(x) dx + Fy0 h(x) = 0 dx a a

h ib Since h(a) = h(b) = 0, Fy0 h(x) = 0 and we have a

 Zb  d(Fy0 ) 0 h(x)dx δI = Fy (x, y, y ) − dx a

Using lemma 2.1 we obtain Fy (x, y, y 0 ) −

d(Fy0 ) =0 dx

which is Euler’s equation. Theorem 2.10. Let I(y) be a functional of the form Zb I(y) =

F (x, y, y 0 ) dx

(2.1)

a

defined on some set of functions y(x), which have continuous first derivatives in [a, b] and satisfy boundary conditions y(a) = A and y(b) = B. Then a necessary condition for I(y) to have an extremum is that δI = 0 must hold and a y(x) determined from δI = 0 must satisfy the Euler’s equation Fy −

d (Fy0 ) = 0 dx

(2.2)

or conversely if y(x) is a solutions to (2.2) then it yields an extremum of the functional (2.1) and δI = 0 holds for this y(x). Remarks.

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

37

(1) For the first time we observe a correspondence between solutions of BVPs and the associated functionals and their extrema. (2) Section 2.3.2 and Theorem 2.10 suggest that one could use the following approach for obtaining solutions of boundary value problems: If Aφ − f = 0 is defined over Ω, then the problem of finding a solution of Aφ − f = 0 is equivalent to finding the extremum of a functional I(φ) associated with Aφ − f = 0 such that Euler’s equation resulting from δI = 0 is the differential equation Aφ − f = 0. (3) We note that Theorem 2.10 only provides necessary conditions. Sufficient conditions that ensure the uniqueness of the solutions are yet to be established. (4) From the definition of δI, we observe that δI involves differentiation of the integrand of I with respect to dependent variables only. The following theorem generalizes the concepts presented here. Theorem 2.11 (Variationally Consistent (VC) integral form of a boundary value problem). Let Aφ−f = 0 in Ω be a system of differential or partial differential equations with some boundary conditions describing a boundary value problem. (i) Existence of a functional I(φ) Let there exist a functional I(φ) (an integral) corresponding to the BVP Aφ − f = 0. This is generally by construction. (ii) Necessary Condition If I(φ) is differentiable in φ, then the integral form given by δI = 0 gives the necessary condition from which we determine a function φ that yields an extremum of I(φ). Let the Euler’s equation resulting from δI = 0 be Aφ − f = 0, then a φ obtained from δI = 0 is also a solution of Aφ − f = 0, hence a solution of the BVP. (iii) Sufficient condition or extremum principle The second variation of the functional, δ 2 I, provides extremum principle or sufficient condition. A unique extremum principle ensures a unique φ from δI = 0, hence a unique extremum of I(φ) and a unique solution of the Euler’s equation resulting from δI(φ) = 0 which is the BVP Aφ − f = 0. δ 2 I > 0, = 0, < 0 correspond to minimum, saddle point, and maximum of I(φ). If for a BVP Aφ − f = 0 (i), (ii), and (iii) exist or hold then the integral form δI = 0 is called variationally consistent (VC) integral form associated with the BVP Aφ−f = 0. We also state this as a definition in the following.

38

CONCEPTS FROM FUNCTIONAL ANALYSIS

Definition 2.30 (Variationally consistent (VC) integral form of a BVP). A variationally consistent integral form corresponding to the BVP Aφ − f = 0 consists of (1) Existence of a functional I(φ) corresponding to the BVP Aφ − f = 0. This is generally by construction (or is assumed). (2) Necessary condition for the existence of an extremum of I(φ) is given by δI(φ) = 0. The integral form δI(φ) = 0 is used to determine φ. The Euler’s equation resulting from δI(φ) = 0 must be the BVP Aφ − f = 0. (3) δ 2 I > 0, = 0, < 0 (minimum, saddle point, maximum of I(φ)) is the sufficient condition or extremum principle. Extremum principle ensures that a φ obtained from δI(φ) = 0 is unique. Extremum principle also establishes whether φ from δI(φ) = 0 minimizes or maximizes I(φ) or yields a saddle point of I(φ). When all these three elements are present in an integral formulation of the BVP Aφ − f = 0, then the integral form (resulting from δI(φ) = 0 or otherwise) is called a variationally consistent integral form of the BVP Aφ − f = 0 (or simply VC integral process). VC integral form or process yields unique extremum of the functional I(φ) corresponding to Aφ − f = 0, hence a unique solution of the BVP Aφ − f = 0 (the Euler’s equation resulting from δI(φ) = 0). Definition 2.31 (Variationally inconsistent integral form (VIC) of a BVP). If an integral form of a BVP (resulting from δI(φ) = 0 or otherwise) is not variationally consistent, then it is variationally inconsistent. A variationally inconsistent integral form or process violates one or more of the three requirements needed for variational consistency of the integral form. Remarks. (1) Thus, we see that a variationally consistent integral form of a BVP Aφ − f = 0 emerges as a method of obtaining a unique solution of the BVP Aφ − f = 0. (2) The necessary condition (the integral form resulting from δI(φ) = 0 or otherwise) provides a system of algebraic equations from which the solution φ is determined. (3) The sufficient condition or unique extremum principle ensures that a φ obtained from the integral form (δI(φ) = 0 or otherwise) is unique, hence this φ yields a unique extremum of I(φ) as well as a unique solution of the Euler’s equation which is the BVP under consideration. (4) Variationally consistent integral forms yield symmetric coefficient matrices in the algebraic systems and the coefficient matrices are positivedefinite, hence have real, positive eigenvalues and real eigenvectors (ba-

39

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

sis). Such coefficient matrices are invertible , hence yield unique values of the unknowns in the corresponding algebraic systems. (5) When the integral form is variationally inconsistent, a unique extremum principle does not exist. In such cases the coefficient matrix in the algebraic system resulting from the integral form is not symmetric, hence is not ensured to be positive-definite. A unique solution of the unknowns in such algebraic systems is not ensured. A consequence of the nonpositive-definite coefficient matrix in the algebraic system is that such coefficient matrices may have zero or negative eigenvalues or the eigenvalues and eigenvectors may be complex. In summary, variationally inconsistent integral forms must be avoided at all cost due to the fact that when using such integral forms a unique solution of the BVP is not ensured. In other words when obtaining solution of BVPs, variationally consistent integral forms are essential to ensure unique solutions of the BVPs. (6) The theorem stated above can be applied to any BVP provided we can show existence of a functional I(φ) corresponding to the BVP Aφ−f = 0 such that δI = 0 and δ 2 I are necessary and sufficient conditions for the existence of extremum of I(φ). A φ yielding unique extremum of I(φ) is also a unique solution of Aφ − f = 0. (7) When the operator A in Aφ − f = 0 has some specific properties, then we can obtain a special form of Theorem 2.11 (shown later). (8) Consider Zb δI =

(Fy (x, y, y 0 ) h + Fy0 (x, y, y 0 ) h0 ) dx

a

where Zb I=

F (x, y, y 0 ) dx

a

we note that h(a) = h(b) = 0 must hold when y(a) = A and y(b) = B are the boundary conditions; that is, h satisfies the homogeneous part of the boundary conditions on y(x) and  → 0 as ||h|| → 0, other than these, h is arbitrary. Thus, h is a virtual change in y. We define h = δy (variation of y) for fixed x. Similarly, we define h0 = δy 0 (variation of y 0 ) also for fixed x. We note that for fixed x, δy and δy 0 are not functions of x. Substituting h = δy and h0 = δy 0 in δI we can write  b   b  Z Z δI =  Fy (x, y, y 0 ) dx δy +  Fy0 (x, y, y 0 ) dx δy 0 a

a

40

CONCEPTS FROM FUNCTIONAL ANALYSIS

Differentiation of F (·) with respect to y and y 0 can be taken outside the integral as the integral is with respect to x:  b  b   Z Z ∂  ∂ δI = F (x, y, y 0 ) dx δy + 0  F (x, y, y 0 ) dx δy 0 ∂y ∂y a

a

or δI =

∂I ∂I δy + 0 δy 0 ∂y ∂y

Thus, the variation of a functional I(x, y, y 0 ) requires differentiation of I with respect to dependent variables y and y 0 and variations in y and y 0 (i.e. δy, δy 0 ) for fixed position coordinates x. This relationship is istrumental in relating δI to dI, the differential of I. Treating I(·) as a function of x, y, and y 0 we can write the following for the differential of I   ∂I ∂I ∂I dI = dy + 0 dy 0 + dx ∂y ∂y ∂x If the positions x are fixed, then the term in the bracket in the expression for dI becomes zero and we obtain dI =

∂I ∂I dy + 0 dy 0 ∂y ∂y

Comparing with δI we conclude that the variational operator δ acts on I as a differential operator with respect to the dependent variables. In the following we state two important theorems related to positivedefinite differential operators and provide their proofs. These theorems are important in relation to the methods of approximation in chapters 3 and 4. Theorem 2.12. If the linear differential operator A in Aφ − f = 0 is positive-definite, then Aφ − f = 0 can not have more than one solution. Proof. We construct the proof of this theorem by contradiction. Let φ1 and φ2 be the two solutions of Aφ − f = 0. Then, Aφ1 − f = 0 and Aφ2 − f = 0. We can write Aφ1 − Aφ2 = A(φ1 − φ2 ) = 0 due to linearity ˜ we can write Aφ˜ = 0. Taking scalar product, we of A. Using φ1 − φ2 = φ, ˜ ˜ ˜ φ) ˜ > 0 and obtain (Aφ, φ) = 0. Since the operator is positive-definite (Aφ, ˜ φ) ˜ = 0 if and only if (iff) φ˜ = 0, which implies φ˜ = φ1 − φ2 = 0 or (Aφ, φ1 = φ2 . This proves that Aφ − f = 0 can only have one solution, that is, the solution of Aφ − f = 0 is unique. Theorem 2.13 (Minimal functional theorem.). Let Aφ − f = 0 have a solution where A is a positive-definite operator. We construct the quadratic functional I(φ) = 21 (Aφ, φ) − (f, φ). Then, out of all values given to the

2.3. ELEMENTS OF CALCULUS OF VARIATIONS

41

quadratic functional I(φ) by all possible functions φ from the domain of definition of A, the least is the value given to I(φ) by a φ that constitutes the solution to Aφ − f = 0. Conversely, if there exists a function φ in the domain of definition of A that gives the minimum value to I(φ), then this function is the solution of BVP Aφ − f = 0. Proof. Let φ0 be the solution of Aφ − f = 0 which is unique by virtue of theorem 2.12 so that Aφ0 − f = 0. Consider I(φ) = 12 (Aφ, φ) − (f, φ) and replace f by Aφ0 . 1 I(φ) = ((Aφ, φ) − 2(Aφ0 , φ)) 2 Add and subtract 12 (Aφ0 , φ0 ) to write I(φ) as
1 (Aφ, φ) − (Aφ0 , φ) − (Aφ0 , φ) + (Aφ0 , φ0 ) − (Aφ0 , φ0 ) 2 

1 = A(φ − φ0 ), φ − (Aφ0 , φ) − (Aφ0 , φ0 ) − (Aφ0 , φ0 ) 2 
1 = A(φ − φ0 ), φ − (Aφ0 , φ − φ0 ) − (Aφ0 , φ0 ) 2

I(φ) =

Since A is positive-definite, it is symmetric, hence

(Aφ0 , φ − φ0 ) = φ0 , A(φ − φ0 ) = A(φ − φ0 ), φ0 Substituting in I(φ) I(φ) =



1 A(φ − φ0 ), φ − A(φ − φ0 ), φ0 − (Aφ0 , φ0 ) 2

or


1 A(φ − φ0 ), φ − φ0 − (Aφ0 , φ0 ) 2
Since A is positive-definite A(φ − φ0 ), φ − φ0 > 0 and (Aφ0 , φ0 ) > 0, hence I(φ) assumes its least value when and only when φ = φ0 , that is, a solution φ0 of Aφ − f = 0 yields minimum value of I(φ) and I(φ) =

1 min I(φ) = − (Aφ0 , φ0 ) 2 This completes the proof.

2.3.3 Variation of a functional: some practical aspects Consider a functional I = I(x, u, u0 ) in which x is the independent coordinate and u is the dependent variable. For fixed values of coordinate x, I depends upon u and u0 . Let v be an arbitrary change in u; that is,

42

CONCEPTS FROM FUNCTIONAL ANALYSIS

let v = δu (variation of u). The symbol variational symbol δ is a differential operator with respect to dependent variables. The following properties regarding v = δu hold. (a) δu represents an admissible change in u for fixed position coordinates i.e fixed x. Boundary conditions, loads, and their points of application do not change due to admissible change δu. (b) If u is specified at some points in the domain (usually the boundary of the domain) then v = δu = 0 at such points because the specified values of u are fixed, hence cannot be changed or varied. Thus, if u = u0 on some boundary Γ, then v = δu0 = 0 on Γ, i.e v = δu satisfies the homogeneous form of the boundary conditions on u which is u = 0. In other words, v = δu vanishes on Γ where u is specified and is arbitrary everywhere else. So v = δu can be thought of as virtual change in u. v is called test function. Hence, the methods or the techniques based on this approach are also referred to as the methods based on the principle of virtual work. (c) As shown earlier the variational operator δ acts as a differential operator with respect to dependent variables. (d) Thus, the laws of variations of sums, products, ratios and powers of functionals are completely analogous to the corresponding laws of differentiations. If F and G are two functionals then (i) (ii) (iii) (iv) (v)

δ(F ± G) = δF ± δG δ(F G) )G + F (δG)
=G(δF F δG δ G = δFG−F 2 δ(F )n = nF n−1 δF Variational and differential operators can commute, that is, change positions, and the same is true for variational and integral operators. This is obviously due to the fact that variation is differentiation with respect to dependent variables whereas the integral or differential operators contain operations of integration or differentiation with respect to independent variables, that is, position coordinates. Thus   d du (δu) = δ dx dx b b Z Z δ u(x) dx = δu(x) dx a

a

2.3.4 Riemann and Lebesgue integrals In finite element processes we encounter definite integrals over the discretized domains of definition of the differential operators. These integrals

43

2.3. ELEMENTS OF CALCULUS OF VARIATIONS f (x)

f (x)

fu fl

a

c

x

b

a

Figure 2.2: f (x) versus x

c

b

Figure 2.3: f (x) versus x

must be expressed as the sum of the integrals over the subdomains (finite elements). In doing so, the continuity of the integrand (or lack of it) over the whole domain (discretization) is crucial in understanding what these integrals mean and or represent. Consider a simple definite integral in one dimension Zb I=

f (x) dx

(2.3)

a

In the strict sense of calculus of continuous and differentiable functions, the integral in (2.3) is valid if and only if f (x) is continuous for all x ∈ [a, b]. When this is the case, the above integral is called Riemann. Consider f (x) versus x shown in Fig. 2.2. The figure shows two different behaviors of f (x) versus x. In both cases, f (x) is continuous and, hence, (2.3) is a Riemann integral in both cases. In this case we can write 2.3 as, I=

f (x) dx = a

Zb

Zc

Zb

f (x) dx

f (x) dx + a

(2.4)

c

All integrals in (2.4) are Riemann and (2.4) holds precisely in the strict sense of calculus of continuous and differentiable functions. Consider f (x) versus x shown in Fig. 2.3; f (x) is continuous for any x ∈ [a, c) and x ∈ (c, b]. However, at x = c, f (x) is discontinuous, that is, f (x) changes from fl to fu at x = c; that is, there is a jump in f (x). In this case the integral in (2.3) is not valid in the Riemann sense and we cannot express (2.3) as a sum of integrals over the subintervals [a, c] and [c, b]. We note that change in f (x) from fl to fu is at a point which is a set of measure zero. Thus, if we decide to ignore the integral of f (x) over a set of measure zero then we can write (2.4) in this case also. Thus, the behavior of f (x) at c is ignored. Such integrals in which the discontinuous integrand behavior

x

44

CONCEPTS FROM FUNCTIONAL ANALYSIS

over sets of measure zero are neglected are called Lebesgue. In summary, for f (x) versus x in Fig. 2.2 the integrals in (2.3) and (2.4) are Riemann whereas for f (x) versus x in Fig. 2.3 the integrals are in the Lebesgue sense. Remarks. (1) Riemann integrals are based on calculus of continuous and differentiable functions. (2) Lebesgue integrals are based on theory of distributions. (3) Use of Lebesgue integrals over Riemann integrals must be done with care. If such an assumption disturbs the physics then the consequences may be serious. We will learn more about these and their use in the finite element processes in later chapters.

2.4 Examples of Differential Operators and their Properties In this section we consider specific examples of boundary value problems to examine the mathematical properties of the differential operators. This will permit us to classify them into three categories: self-adjoint, nonself-adjoint and non-linear. The linearity and symmetry of the differential operators (or lack there of) are two important properties that permit their mathematical classification into these three categories.

2.4.1 Self-adjoint differential operators The self-adjoint differential operators are linear and symmetric. In this section we consider some examples of BVPs in which we show that the differential operators are self-adjoint under certain restrictions. Example 2.9 (2D Poisson’s Equation). Consider the BVP     ∂ ∂φ ∂ ∂φ − − + Q(x, y) = 0 ∀(x, y) ∈ Ω ⊂ R2 ∂x ∂x ∂y ∂y

(2.5)

with boundary conditions φ = φ0 ∂φ ∂φ nx + ny = g(x, y) ∂x ∂y

on

Γ1

(2.6)

on

Γ2

(2.7)

where φ = φ(x, y) is the dependent variable, Q(x, y) is the source term, Ω is the domain of definition, φ0 is a specified value on Γ1 , g(x, y) on Γ2 is a known function, and Γ = Γ1 ∪ Γ2 is a closed contour defining the boundary

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

45

n ny nx

Γ2 Ω y

n2x + n2y = 1

Γ1 x

ˆ to boundary Figure 2.4: Domain Ω, boundaries Γ1 and Γ2 , and unit exterior normal n Γ2

¯ = Γ ∪ Ω; nx and ny are the direction cosines of the unit of Ω such that Ω exterior normal to the boundary Γ2 . Figure 2.4 shows the details. First, we write (2.5) symbolically as Aφ − f = 0 ∀(x, y) ∈ Ω ⊂ R2 in which

∂ A=− ∂x



∂ ∂x



∂ − ∂y



∂ ∂y

(2.8)

 (2.9)

and f = −Q(x, y)

(2.10)

In the following we determine if the differential operator A defined by (2.9) is linear and symmetric or not.

Linearity of A Let V ⊂ H be a subspace containing functions defined over Ω that are admissible in (2.5). Then, to establish linearity of the differential operator A we must show that A(αu + βv) = α(Au) + β(Av)

∀u, v ∈ V, ∀α, β ∈ R.

(2.11)

The proof is straight forward. Substitute A from (2.9) in the left side of (2.11), expand and regroup  2  ∂ ∂2 A(αu + βv) = − + (αu + βv) ∂x2 ∂y 2  2   2  ∂ u ∂2u ∂ v ∂2v = α − 2 − 2 +β − 2 − 2 ∂x ∂y ∂x ∂y = α(Au) + β(Av)

46

CONCEPTS FROM FUNCTIONAL ANALYSIS

Hence, the operator A defined by (2.9) is linear.

Symmetry of A Let φ, v ∈ V . Then to establish the symmetry of the differential operator A defined by (2.9), we must show that (Aφ, v) = (φ, Av)

(2.12)

¯ First, we note that based on The scalar products in (2.12) are over Ω. fundamental Lemma 2.1 with α = Aφ − f and h = v, Z (Aφ − f )v dΩ = (Aφ, v)Ω¯ − (f, v)Ω¯ = 0 is valid (2.13) ¯ Ω

in which dΩ = dx dy and the restriction that v = 0 where φ is specified. Thus, based on (2.6) v = 0 on Γ1 . v is commonly known as a test function. We note that v = δφ (variation of φ) is also an admissible choice, but v does not necessarily have to be δφ. This property of v is important in establishing symmetry of the differential operator A. Consider (2.12) in which v = 0 on Γ1 due to (2.6). In order to show whether (2.12) holds or not, we can proceed in two ways. The intermediate details are somewhat different, but the end result and conclusion is not effected.

Method I We can transfer all of the differentiation from φ to v in the scalar product on the left side of (2.12) using integration by parts, thereby obtaining (Aφ, v) = (φ, A∗ v) + hAφ, viΓ in which hAφ, viΓ is the concomitant containing terms resulting from integration by parts. Thus, for symmetry of A we must show that A = A∗ and hAφ, viΓ = 0. Establishing hAφ, viΓ = 0 requires the use of boundary conditions (2.6) and (2.7) and their variations. This approach is most general and is recommended.

Method II This approach is meaningful when the differential operator has derivatives of even order as in (2.9). In (Aφ, v), we transfer half the order of differentiation (one order in this case) from φ to v to obtain ˜ (Aφ, v) = B(φ, v) + hAφ, viΓ

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

47

Likewise, in (φ, Av) we transfer half the order of differentiation (one order in this case also) from v to φ to obtain (φ, Av) = B (φ, v) + hφ, AviΓ e

So symmetry of A in (2.12) reduces to showing that the following relation holds: ˜ B(φ, v) + hAφ, viΓ = B (φ, v) + hφ, AviΓ (2.14) e

Thus, A is symmetric if ˜ hAφ, viΓ − hφ, AviΓ = 0, B(φ, v) = B (φ, v)

(2.15)

e

If we can show that (2.15) holds using boundary conditions (2.6) and (2.7) and their variations, the symmetry of the operator A is established.

Details of Method I Consider (Aφ, v) in (2.12):  Z  ∂  ∂φ  ∂  ∂φ  − v dx dy (Aφ, v)Ω¯ = − ∂x ∂x ∂y ∂y

(2.16)

¯ Ω

Transfer one order of differentiation from φ to v with respect to x and y using integration by parts   Z  I  ∂φ ∂v ∂φ ∂v ∂φ ∂φ (Aφ, v)Ω¯ = + dx dy − nx + ny v dΓ (2.17) ∂x ∂x ∂y ∂y ∂x ∂y ¯ Ω

Γ

Transfer one more order of differentiation from φ to v in the first integral in the right side of (2.17)  Z  ∂  ∂v  ∂  ∂v  (Aφ, v)Ω¯ = − − φ dx dy ∂x ∂x ∂y ∂y ¯ Ω   I  I  ∂φ ∂φ ∂v ∂v − nx + ny v dΓ + nx + ny φ dΓ (2.18) ∂x ∂y ∂x ∂y Γ

Γ



∂ ∂ ∂ ∂ or (Aφ, v)Ω¯ = (φ, A∗ v)Ω¯ + hAφ, viΓ where A∗ = − ∂x ∂x − ∂y ∂y is the adjoint of A, which is obviously the same as A and the concomitant hAφ, viΓ is given by   I  I  ∂φ ∂φ ∂v ∂v hAφ, viΓ = − nx + ny v dΓ + nx + ny φ dΓ (2.19) ∂x ∂y ∂x ∂y Γ

Γ

48

CONCEPTS FROM FUNCTIONAL ANALYSIS

Thus, A is symmetric if hAφ, viΓ in (2.19) can be shown to be zero. We use boundary conditions (2.6) and (2.7) and their variations to simplify (2.19). First, we note that integrals over Γ can be written as the sum of the integrals over Γ1 and Γ2 . This is necessitated due to the fact that BCs are only given on Γ1 and Γ2 Z 

  Z  ∂φ ∂φ ∂φ ∂φ hAφ, viΓ = − nx + ny v dΓ − nx + ny v dΓ ∂x ∂y ∂x ∂y Γ1 Γ2   Z  Z  ∂v ∂v ∂v ∂v + φ nx + ny dΓ + φ nx + ny dΓ (2.20) ∂x ∂y ∂x ∂y Γ1

Γ2

The expressions in (2.20) can be simplified using φ = φ0 and v = 0 on Γ1 ∂v ∂v and nx + ny = 0 on Γ2 ∂x ∂y

∂φ ∂φ nx + ny = g ∂x ∂y

(2.21)

Substituting from (2.21) in (2.20) we get Z hAφ, viΓ = − Γ2



Z vg dΓ +

φ0

∂v ∂v nx + ny ∂x ∂y

 dΓ

(2.22)

Γ1

Remarks. (1) When g = 0 and φ0 = 0, hAφ, viΓ = 0 and the operator A is symmetric and, hence, self-adjoint. (2) When g 6= 0 or φ0 6= 0, the operator A is not symmetric in the strict mathematical sense. However, we note that hAφ, viΓ in (2.20) only contains v and known functions or quantities φ0 and g on boundaries Γ1 and Γ2 . In the classical methods of approximation presented in chapter 3 and the finite element processes in the subsequent chapters, we shall see that hAφ, viΓ only contributes to the right hand side vector and does not influence the coefficient matrix. That is, when (2.19) is not zero and when hAφ, viΓ is not a function of φ, the operator A is as good as being symmetric with regard to methods of approximations and finite element processes. Thus, the most important properties in the proof of symmetry is the fact that A = A∗ must hold and hAφ, viΓ must not be a function of φ. (3) In this case the adjoint of A is the same as A, A∗ = A. The determination of adjoint A∗ requires that we transfer all of the differentiation from φ to v which is only possible in this method labelled as method I.

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

49

Details of Method II Here, we consider both (Aφ, v) and (φ, Av) in (2.12). First, consider (Aφ, v)  Z  ∂  ∂φ  ∂  ∂φ  v dx dy (2.23) (Aφ, v)Ω¯ = − − ∂x ∂x ∂y ∂y ¯ Ω

Transfer one order of differentiation with respect to x and y from φ to v using integration by parts   Z  I  ∂v ∂φ ∂v ∂φ ∂φ ∂φ (Aφ, v)Ω¯ = dx dy − + nx + ny v dΓ (2.24) ∂x ∂x ∂y ∂y ∂x ∂y ¯ Ω

Γ

or ˜ (Aφ, v)Ω¯ = B(φ, v) + hAφ, viΓ

(2.25)

 Z  ∂  ∂v  ∂  ∂v  (φ, Av)Ω¯ = − − φ dx dy ∂x ∂x ∂y ∂y

(2.26)

Now consider (φ, Av)Ω¯

¯ Ω

Transfer one order of differentiation with respect to x and y from v to φ using integration by parts   Z  I  ∂φ ∂v ∂φ ∂v ∂v ∂v (φ, Av)Ω¯ = + dx dy − φ nx + ny dΓ (2.27) ∂x ∂x ∂y ∂y ∂x ∂y ¯ Ω

Γ

or (φ, Av)Ω¯ = B (φ, v) + hφ, AviΓ

(2.28)

e

The symmetry of A requires that (2.25) and (2.28) be equal: ˜ B(φ, v) + hAφ, viΓ = B (φ, v) + hφ, AviΓ

(2.29)

e

˜ From (2.24) and (2.27) we note that B(φ, v) = B (φ, v) holds, hence, syme

metry of A requires that hAφ, viΓ − hφ, AviΓ = 0

must hold

or I  − Γ

  I  ∂φ ∂v ∂v ∂φ nx + ny v dΓ + φ nx + ny dΓ = 0 ∂x ∂y ∂x ∂y Γ

(2.30)

50

CONCEPTS FROM FUNCTIONAL ANALYSIS

which is the same as the concomitant in Method I. Thus   Z Z ∂v ∂v hAφ, viΓ − hφ, AviΓ = − vg dΓ + φ0 nx + ny dΓ ∂x ∂y Γ2

(2.31)

Γ1

must hold for the symmetry of A. Since this is identical to Method I, the remarks presented there hold precisely here as well, except that in this approach it is not possible to determine the adjoint A∗ of A due to the fact that we have not transferred all of the differentiation from φ to v. Example 2.10 (1D Diffusion Equation). Consider the BVP d − dx

  dφ a(x) − c(x)φ + x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx

(2.32)

with boundary conditions φ(0) = φ0 dφ a(1) = q1 dx 1

(2.33) (2.34)

where φ is the dependent variable, a(x) and c(x) are known coefficients and φ0 and q1 are given. First, we write (2.32) symbolically as Aφ − f = 0 ∀x ∈ Ω

(2.35)

where d A=− dx

  d a(x) − c(x) dx

(2.36)

f = −x2

(2.37)

and In the following we determine if the differential operator A is linear and symmetric or not.

Linearity of A Let V ⊂ H be a subspace containing functions that are admissible in (2.32) over Ω. Then, ∀u, v ∈ V and ∀α, β ∈ R we must show that A(αu + βv) = α(Au) + β(Av)

(2.38)

holds to establish the linearity of the differential operator A defined by (2.36). To prove (2.38), we substitute for A from (2.36) in the left side of (2.38),

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

51

expand and regroup   d(αu + βv) d a(x) − c(x)(αu + βv) A(αu + βv) = − dx dx         d du d dv =α − a(x) − c(x)u + β − a(x) − c(x)v dx dx dx dx = α(Au) + β(Av) (2.39) Hence, the operator A defined by (2.36) is linear.

Symmetry of A Let φ, v ∈ V . Then to establish the symmetry of the differential operator A in (2.36) we must show that (Aφ, v)Ω¯ = (φ, Av)Ω¯

(2.40)

First, we note that based on fundamental Lemma 2.1 with α = Aφ − f and v=h Z (Aφ − f )v dΩ = (Aφ, v)Ω¯ − (f, v)Ω¯ = 0 is valid (2.41) ¯ Ω

in which dΩ = dx and the restriction that v = 0 at x = 0 where φ = φ0 . As in example 1, v is the test function and the choice v = δφ is admissible as well. In this case also, the differential operator has even order derivatives of φ. Hence, we can use Method I as well as Method II to determine symmetry of A.

Details of Method I Consider (Aφ, v) in (2.40)    Z  d dφ (Aφ, v)Ω¯ = − a(x) − c(x)φ v dx dx dx

(2.42)

¯ Ω

Transfer one order of differentiation with respect to x from φ to v in the first term of the integrand in (2.42) using integration by parts Z  (Aφ, v)Ω¯ = ¯ Ω

dv dφ a(x) − c(x)φv dx dx



   dφ  1 dx − v a(x) dx 0

(2.43)

52

CONCEPTS FROM FUNCTIONAL ANALYSIS

Transfer one more order of differentiation with respect to x from φ to v in the first term of the integrand in (2.43) using integration by parts  Z  d dv  − (Aφ, v)Ω¯ = a(x) φ − c(x)φv dx dx dx ¯ Ω

      dv  1 dφ  1 + φ a(x) − v a(x) dx 0 dx 0

(2.44)

or (Aφ, v)Ω¯ = (φ, A∗ v)Ω¯ + hAφ, viΓ (2.45)
d d where A∗ = − dx a(x) dx − c(x) is the adjoint of A which is clearly same as the differential operator A and the concomitant hAφ, viΓ is given by       dφ  1 dv  1 hAφ, viΓ = − v a(x) + φ a(x) dx 0 dx 0

(2.46)

Thus, A is symmetric if hAφ, viΓ = 0 in (2.46). We use boundary conditions (2.33) and (2.34) and their variations to simplify (2.46). Expanding terms on the right side of (2.46) dφ dφ dv dv hAφ, viΓ = −v(1)a(1) + v(0)a(0) + φ(1)(a(1) − φ(0)a(0) dx 1 dx 0 dx 1 dx 0 (2.47) The expression on the right hand side of (2.47) can be simplified using φ(0) = φ0 ⇒ v(0) = 0 dφ dv a(1) = q1 ⇒ a(1) = 0 dx 1 dx 1

(2.48)

Substituting (2.48) into (2.47) dv hAφ, viΓ = −v(1)q1 − φ0 a(0) dx 0

(2.49)

Remarks. 1. When q1 = 0 and φ0 =0, hAφ, viΓ = 0 and, hence, the operator A is symmetric and, thus, self-adjoint. 2. When q1 6= 0 or φ0 6= 0, the operator A is not symmetric in the strict mathematical sense. However, we note that hAφ, viΓ in (2.49) only constains v and known φ0 and q1 . In the methods of approximation presented in chapter 3 and the finite element processes subsequently, we shall see that hAφ, viΓ only contributes to the right hand side vector and does not influence the coefficient matrix. That is, when (2.45) holds with A∗ = A and when

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

53

hAφ, viΓ is not a function of φ, the operator A is as good as being symmetric from the point of view of the methods of approximation and finite element processes. 3. The adjoint A∗ of A is the same as A. Determination of the adjoint A∗ requires that we transfer all of the differentiation from φ to v which is only possible in this method.

Details of Method II Here we consider both (Aφ, v)Ω¯ and (φ, Av)Ω¯ in (2.40). First, consider (Aφ, v)Ω¯  Z  d dφ  − (Aφ, v)Ω¯ = a(x) − c(x)φ v dx (2.50) dx dx ¯ Ω

Transfer one order of differentiation with respect to x from φ to v in the first term of the integrand in (2.50) using integration by parts     Z  dφ dφ  1 dv a(x) − c(x)φv dx − v a(x) (2.51) (Aφ, v)Ω¯ = dx dx dx 0 ¯ Ω

or ˜ (Aφ, v)Ω¯ = B(φ, v) + hAφ, viΓ

(2.52)

Now consider (φ, Av) Z (φ, Av)Ω¯ =



 d dv  φ − a(x) − c(x)v dx dx dx

(2.53)

¯ Ω

Transfer one order of differentiation with respect to x from v φ in the first term of the integrand in (2.54) using integration by parts     Z  dφ dv dv  1 (2.54) (φ, Av)Ω¯ = a(x) − c(x)vφ dx − φ a(x) dx dx dx 0 ¯ Ω

or (φ, Av)Ω¯ = B (φ, v) + hφ, AviΓ

(2.55)

e

Symmetry of A requires that (2.52) and (2.55) be equal ˜ B(φ, v) + hAφ, viΓ = B (φ, v) + hφ, AviΓ

(2.56)

e

˜ From (2.78) and (2.54) we note that B(φ, v) = B (φ, v) holds. Hence, symmetry of A requires that hAφ, viΓ − hφ, AviΓ = 0

e

must hold

(2.57)

54

CONCEPTS FROM FUNCTIONAL ANALYSIS

or

      dv  1 dφ  1 + φ a(x) =0 − v a(x) dx 0 dx 0

(2.58)

which is the same as the concomitant in Method I and hence following the details presented in Method I we obtain dv hAφ, viΓ − hφ, AviΓ = −v(1)q1 − φ0 a(0) = 0 (2.59) dx 0 Thus, for the symmetry of A (2.59) must hold. Since (2.59) is the same as (2.49) in Method I, the remarks presented there hold here as well. In this method, determination of A∗ is not possible in this method as it requires transferring all of the differentiation from φ to v. Example 2.11 (1D Beam Equation). Consider the BVP   d2 d2 φ b(x) 2 + Q(x) = 0 ∀x ∈ Ω = (0, L) ⊂ R dx2 dx

(2.60)

with boundary conditions φ(0) = φ0 ;  d2 φ  b 2 = ML ; dx L

dφ = q0 dx 0 2  d φ d b 2 = QL dx dx L

(2.61) (2.62)

where φ is the dependent variable, Q(x) is the source terms and φ0 ,q0 ,ML and QL are known data. First, we write (2.60) symbolically as Aφ − f = 0 ∀x ∈ Ω

(2.63)

d2  d2  a(x) dx2 dx2

(2.64)

f = −Q(x)

(2.65)

where A= and

In the following we determine if the differential operator A defined by (2.60) is linear and symmetric or not.

Linearity of A Let V ⊂ H be a subspace containing functions that are admissible in (2.60) over Ω. Then, ∀u, v ∈ V and ∀α, β ∈ R we must show that A(αu + βv) = α(Au) + β(Av)

(2.66)

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

55

holds to establish the linearity of the differential operator A defined by (2.64). To prove (2.66), we substitute for A from (2.64) in the left side of (2.66), expand and regroup,   d2 d2 (αu + βv) A(αu + βv) = 2 b(x) dx dx2   2    2  d d d2 u  d2 v  +β =α b(x) 2 b(x) 2 dx2 dx dx2 dx = α(Au) + β(Av)

(2.67)

Hence, the operator A defined by (2.64) is linear.

Symmetry of A Let φ, v ∈ V . Then to establish the symmetry of the differential operator A in (2.64) we must show that (Aφ, v)Ω¯ = (φ, Av)Ω¯

(2.68)

First, we note that based on fundamental Lemma 2.1 with α = Aφ − f and v=h Z (Aφ − f )v dΩ = (Aφ, v)Ω¯ − (f, v)Ω¯ = 0 (2.69) ¯ Ω

in which dΩ = dx and the restriction that v = 0 at x = 0 where φ = φ0 . As in previous examples, v is the test function and the choice v = δφ is admissible as well. In this case also, the differential operator has even order derivatives of φ. Hence, we can use Method I as well as Method II to determine symmetry of A.

Details of Method I Consider (Aφ, v)Ω¯ in (2.68), Z (Aφ, v)Ω¯ =

d2  d2 φ  b(x) v dx dx2 dx2

(2.70)

¯ Ω

Transfer all orders of differentiation from the dependent variable φ to the test function v by using integration by parts. This results in four boundary

56

CONCEPTS FROM FUNCTIONAL ANALYSIS

terms: Z (Aφ, v)Ω¯ =

 d2  d2 v  dx φ b(x) 2 dx2 dx 

¯ Ω

L L  d d2 φ  d2 φ dv + b(x) 2 v − b(x) 2 dx dx dx dx 0 0 L   L  2 dφ d v d d2 v  b(x) + − b(x) 2 φ dx2 dx 0 dx dx 0 

(2.71)

or (Aφ, v)Ω¯ = (φ, A∗ v)Ω¯ + hAφ, viΓ

(2.72)

2

2

d d where A∗ = dx 2 (a(x) dx2 ) is the adjoint of A which is clearly the same as A and the concomitant hAφ, viΓ is given by

L  L dv d d2 φ  d2 φ hAφ, viΓ = b(x) 2 v − b(x) 2 dx dx dx dx 0 0  2 L   L 2 d v dφ d d v + b(x) − b(x) 2 φ dx2 dx 0 dx dx 0 

(2.73)

Expanding terms on the right side of (2.73)     d2 φ  d d2 φ  d b(x) 2 v − b(x) 2 v dx dx dx dx x=L x=0     2 2 dv d φ dv d φ − b(x) 2 + b(x) 2 dx dx x=L dx dx x=0  2   2  d v dφ d v dφ + b(x) − b(x) dx2 dx x=L dx2 dx x=0        2 d d v d d2 v  − b(x) 2 φ + b(x) 2 φ dx dx dx dx x=L x=0 

hAφ, viΓ =

(2.74)

The terms on the right hand side of (2.74) can be simplified using φ(0) = φ0 ⇒ v(0) = 0 dφ dv = q ⇒ =0 0 dx 0 dx 0  d2 v   d2 φ  b 2 = ML ⇒ b 2 = 0 dx L dx L   2 d d  d2 v  d φ b b = QL ⇒ =0 dx dx2 dx dx2 L

L

(2.75)

57

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

Substituting (2.75) into (2.74)     dv   2 v  2  d − q0 b(x) + φ0 d b(x) d v hAφ, viΓ = QL v(L) − ML dx L dx2 0 dx dx2 0 (2.76) Remarks. (1) When φ0 = 0, q0 = 0, ML = 0 and QL = 0 the differential operator is symmetric and, hence self-adjoint. (2) When φ0 6= 0 or q0 6= 0 or ML 6= 0 or QL 6= 0, the operator A is not symmetric in the strict mathematical sense. However, we note that hAφ, viΓ in (2.76) only constains v and known φ0 , q0 , ML and QL . In the methods of approximation presented in chapter 3 and the finite element processes subsequently, we shall see that hAφ, viΓ only contributes to the right hand side vector and does not influence the coefficient matrix. That is, when (2.45) holds with A∗ = A and when hAφ, viΓ is not a function of φ, the operator A is as good as being symmetric from the point of view of the methods of approximation and finite element processes. (3) The adjoint A∗ of A is the same as A. Determination of the adjoint A∗ requires that we transfer all of the differentiation from φ to v which is only possible in this method.

Details of Method II Here we consider both (Aφ, v)Ω¯ and (φ, Av)Ω¯ in (2.68). First, consider (Aφ, v)  Z  2  d d2 φ  (Aφ, v)Ω¯ = b(x) 2 v dx (2.77) dx2 dx ¯ Ω

Transfer two orders of differentiation with respect to x from φ to v in (2.77) using integration by parts Z  (Aφ, v)Ω¯ =

d2 φ d2 v b(x) dx2 dx2



d d2 φ  dx + b(x) 2 v dx dx 

¯ Ω

L 0

dv d2 φ − b(x) 2 dx dx 

L 0

(2.78) or ˜ (Aφ, v)Ω¯ = B(φ, v) + hAφ, viΓ in which d d2 φ  hAφ, viΓ = b(x) 2 v dx dx 

L 0

dv d2 φ − b(x) 2 dx dx 

(2.79) L (2.80) 0

58

CONCEPTS FROM FUNCTIONAL ANALYSIS

Now consider (φ, Av)Ω¯ Z  (φ, Av)Ω¯ =

 d2  d2 v  b(x) 2 φ dx dx2 dx

(2.81)

¯ Ω

Transfer two orders of differentiation with respect to x from v to φ in (2.81) using integration by parts    L L  Z  2 d φ d d2 v d2 v  d2 v dφ (φ, Av)Ω¯ = b(x) 2 dx + b(x) 2 φ − b(x) 2 dx2 dx dx dx dx dx 0 0 ¯ Ω

(2.82) or (φ, Av)Ω¯ = B (φ, v) + hφ, AviΓ

(2.83)

e

in which d d2 v  hφ, AviΓ = b(x) 2 φ dx dx 

L 0

dφ d2 v b(x) 2 − dx dx 

L (2.84) 0

Symmetry of A requires that (2.79) and (2.83) be equal ˜ B(φ, v) + hAφ, viΓ = B (φ, v) + hφ, AviΓ

(2.85)

e

˜ From (2.78) and (2.54) we note that B(φ, v) = B (φ, v) holds. Hence, symmetry of A requires that

e

hAφ, viΓ − hφ, AviΓ = 0

(2.86)

Substituting (2.80) and (2.80) into (2.86) and comparing the resulting expression with the concomitant in Method I, we note that they are exactly the same. Thus, the remaining details and remarks are the same as in Method I. In this method, the determination of A∗ is not possible for the same reasons as in previous examples.

2.4.2 Non-self-adjoint differential operators The non-self-adjoint differential operators are linear but not symmetric. For these operators, the adjoint A∗ of the operator is never the same as the operator A itself. This is a fundamental difference between self-adjoint and non-self-adjoint operators. In this section we consider two examples of non-self-adjoint operators. Example 2.12 (1D convection diffusion equation). Consider the BVP d2 φ dφ − k 2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx dx

(2.87)

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

59

with boundary conditions φ(0) = 1

and

φ(1) = 0

(2.88)

where φ(x) is the dependent variable and k is a known diffusion coefficient and is constant. First, we write (2.87) symbolically as Aφ − f = 0 ∀x ∈ Ω

(2.89)

where A=

d2 d −k 2 dx dx

(2.90)

f =0

(2.91)

and In the following we determine if the differential operator A defined by (2.87) is linear and symmetric or not.

Linearity of A Let V ⊂ H be a subspace containing functions that are admissible in (2.87) over Ω. Then, ∀u, v ∈ V and ∀α, β ∈ R we must show that A(αu + βv) = α(Au) + β(Av)

(2.92)

holds to establish the linearity of the differential operator A defined by (2.90). To prove (2.92), we substitute for A from (2.90) in the left side of (2.92), expand and regroup d d2 (αu + βv) (αu + βv) − k 2 dx  dx  2 du dv dv 2 du =α −k 2 +β −k 2 dx dx dx dx = α(Au) + β(Av)

A(αu + βv) =

(2.93)

Hence, the operator A defined by (2.90) is linear.

Symmetry of A Let φ, v ∈ V . Then to establish the symmetry of the differential operator A in (2.90) we must show that (Aφ, v)Ω¯ = (φ, Av)Ω¯

(2.94)

60

CONCEPTS FROM FUNCTIONAL ANALYSIS

As in the case of previous examples, here also we note that based on fundamental Lemma 2.1 with α = Aφ − f and v = h Z (Aφ − f )v dΩ = (Aφ, v)Ω¯ − (f, v)Ω¯ = 0 is valid (2.95) ¯ Ω

with the restriction that v = 0 where φ is specified. v is the test function and v = δφ is admissible. Since in this case the differential operator contains both odd and even order derivatives, we can only consider Method I in which all of the differentiation is transfered from φ to v in (Aφ, v)Ω¯ . Consider (Aφ, v)Ω¯ in (2.94) Z  (Aφ, v)Ω¯ =

dφ d2 φ v − k 2v dx dx

 dx

(2.96)

¯ Ω

Transfer one order of differentiation from φ to v in the first term and and two orders in the second term from φ to v in the integrand of (2.96) using integration by parts       Z  dv 1 dv d2 v dφ  1 1 + k φ (2.97) (Aφ, v)Ω¯ = − − k 2 φ dx + [vφ]0 − v k dx dx dx 0 dx 0 ¯ Ω

or (Aφ, v)Ω¯ = (φ, A∗ v)Ω¯ + hAφ, viΓ

(2.98)

2

d d ∗ is not the same where A∗ = − dx − k dx 2 is the adjoint of A. Clearly, A as the differential operator A, hence, the differential operator A cannot be symmetric regardless of whether the concomitant hAφ, viΓ is zero or not. We present the remaining details of simplifying hAφ, viΓ as these are helpful in the methods of approximation and the finite element processes presented in subsequent chapters. Comparing (2.98) with (2.97) we see that

hAφ, viΓ =

[vφ]10

     dφ  1 dv 1 − v k + k φ dx 0 dx 0

(2.99)

The terms on the right side of (2.99) can be simplified using the boundary conditions and v(0) = 0, v(1) = 0.     dv dv hAφ, viΓ = − k φ(0) = − k (2.100) dx 0 dx 0 Clearly, the differential operator A is not symmetric as A∗ 6= A and hAφ, viΓ 6= 0, hence the operator A is non-self-adjoint.

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

61

Example 2.13 (2D convection diffusion equation). Consider the BVP u

∂φ ∂2φ ∂φ ∂2φ +v −k 2 −k 2 =f ∂x ∂y ∂x ∂y

∀(x, y) ∈ Ω ⊂ R2

(2.101)

with boundary conditions φ = φ0 ∂φ ∂φ k nx + k ny = q ∂x ∂y

on

Γ1

(2.102)

on

Γ2

(2.103)

where Γ = Γ1 ∪ Γ2 is the closed contour constituting the boundary of the ¯ = Γ ∪ Ω. φ(x, y) is the dependent variable and, u and domain Ω such that Ω v are known and are constants and the diffusion coefficient k is also known and is constant. First, we write (2.101) symbolically as Aφ − f = 0 ∀x ∈ Ω ⊂ R2

(2.104)

where

∂ ∂ ∂2 ∂2 +v −k 2 −k 2 (2.105) ∂x ∂y ∂x ∂y In the following we determine if the differential operator A defined by (2.105) is linear and symmetric or not. A=u

Linearity of A Let V ⊂ H be a subspace containing functions that are admissible in (2.101) over Ω. Then, ∀w1 , w2 ∈ V and ∀α, β ∈ R we must show that A(αw1 + βw2 ) = α(Aw1 ) + β(Aw2 )

(2.106)

holds to establish the linearity of the differential operator A defined by (2.105). To prove (2.106), we substitute for A from (2.105) in the left hand side of (2.106), expand and regroup ∂ ∂ (αw1 + βw2 ) + v (αw1 + βw2 ) ∂x ∂y 2 ∂ ∂2 − k 2 (αw1 + βw2 ) − k 2 (αw1 + βw2 ) ∂x ∂y   ∂ ∂ ∂2 ∂2 = α u w1 + v w1 − k 2 w1 − k 2 w1 ∂x ∂y ∂x ∂y   2 ∂ ∂ ∂ ∂2 + β u w2 + v w2 − k 2 w2 − k 2 w2 ∂x ∂y ∂x ∂y = α(Aw1 ) + β(Aw2 )

A(αw1 + βw2 ) = u

Hence, the operator A defined by (2.105) is linear.

(2.107)

62

CONCEPTS FROM FUNCTIONAL ANALYSIS

Symmetry of A Let φ, w ∈ V . Then to establish the symmetry of the differential operator A in (2.105) we must show that (Aφ, w)Ω¯ = (φ, Aw)Ω¯ .

(2.108)

As in the case of previous examples, here also we note that based on fundamental Lemma 2.1 with α = Aφ − f and w = h Z (Aφ − f )w dΩ = (Aφ, w)Ω¯ − (f, w)Ω¯ = 0 (2.109) ¯ Ω

with the restriction that w = 0 where φ is specified. w is the test function and w = δφ is admissible. Since in this case the differential operator contains both odd and even order derivatives, we can only consider Method I in which all of the differentiation is transferred from φ to w in (Aφ, w)Ω¯ . Consider (Aφ, w)Ω¯ in (2.108)  Z  ∂φ ∂φ ∂2φ ∂2φ (Aφ, w)Ω¯ = u +v − k 2 − k 2 w dx dy ∂x ∂y ∂x ∂y

(2.110)

¯ Ω

Transfer all of the differentiation from φ to w in the integrand of (2.110) using integration by parts  Z  ∂w ∂2w ∂2w ∂w (Aφ, w)Ω¯ = −v − k 2 − k 2 φ dx dy −u ∂x ∂y ∂x ∂y ¯ Ω I   + (unx + vny )φw dΓ Γ

 I  ∂φ ∂φ − k nx + k ny w dΓ ∂x ∂y Γ  I  ∂w ∂w + k nx + k ny φ dΓ ∂x ∂y

(2.111)

Γ

or (Aφ, v)Ω¯ = (φ, A∗ v)Ω¯ + hAφ, viΓ 2

2

(2.112)

∂ ∂ ∂ ∂ ∗ where A∗ = −u ∂x − v ∂y − k ∂x 2 − k ∂y 2 is the adjoint of A. Clearly, A is not same as A, hence the differential operator is not symmetric. The concomitant hAφ, viΓ can be simplified using boundary conditions and the properties of w. We leave this as an exercise.

2.4. EXAMPLES OF DIFFERENTIAL OPERATORS AND THEIR PROPERTIES

63

2.4.3 Non-linear differential operators The non-linear differential operators are not linear and hence cannot be symmetric. In the following we consider two examples. Here, we only need to check for the linearity of the operator. Example 2.14 (1D Burgers equation). Consider the BVP φ

dφ d2 φ − k 2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx dx

(2.113)

with boundary conditions φ(0) = 1

and

φ(1) = 0

(2.114)

1 is a known coefficient (Re where φ(x) is the dependent variable and k = Re is Reynolds number). First, we write (2.113) symbolically as

Aφ − f = 0 ∀x ∈ Ω ⊂ R where

(2.115)

d d2 −k 2 dx dx

(2.116)

f =0

(2.117)

A=φ and

We only check for linearity of A in the following. Let V ⊂ H be a subspace containing functions that are admissible in (2.113) over Ω. Then, ∀u, v ∈ V and ∀α, β ∈ R we must show that A(αu + βv) = α A(u) + β A(v).

(2.118)

We substitute for A from (2.116) in the left hand side of (2.118) A(αu + βv) = (αu + βv)

d d2 (αu + βv) (αu + βv) − k dx dx2

For the terms on the right side of (2.118) we can write,   du 1 d2 u α A(u) = α u − dx Re dx2 and

dv 1 d2 v β A(v) = β v − dx Re dx2 

(2.119)

(2.120)

 (2.121)

Comparing (2.119) with the sum of (2.120) and (2.121), we clearly see that (2.118) does not hold. Hence, the operator A is not linear [because of the first term in (2.116)] and, therefore, cannot be symmetric either.

64

CONCEPTS FROM FUNCTIONAL ANALYSIS

Example 2.15 (2D Burgers equation). Consider the BVP   ∂φ 1 ∂2φ ∂2φ ∂φ +φ − + 2 = f ∀(x, y) ∈ Ω φ ∂x ∂y Re ∂x2 ∂y

(2.122)

with boundary conditions φ = φ0 ∂φ ∂φ k nx + k ny = q ∂x ∂y

on

Γ1

(2.123)

on

Γ2

(2.124)

where Re is the Reynolds number (a constant), φ0 and q are known data and Γ = Γ1 ∪ Γ2 is the closed contour constituting the boundary of the domain ¯ = Γ ∪ Ω. First, we write (2.122) symbolically as Ω such that Ω Aφ − f = 0 ∀x ∈ Ω

(2.125)

where A=φ

∂ ∂ 1 +φ − ∂x ∂y Re



∂2 ∂2 + 2 2 ∂x ∂y

 ∀(x, y) ∈ Ω

(2.126)

Following previous examples, it is rather obvious that A is not linear [because of the first two terms in (2.126)] and, hence, also not symmetric. Remarks. (1) We note that when the description of a BVP contains product terms in the dependent variables and/or their derivatives, the operator is bound to be non-linear. (2) When a differential operator is not linear it cannot be symmetric as linearity of an operator is an essential property for the symmetry of the operator.

2.5 Summary In this chapter basic elements of applied mathematics that are pertinent in understanding the mathematical details of the methods of approximations and the finite element method are presented. Sets, spaces, functions, function spaces, operators, differential operators, Hilbert spaces, scalar products in Hilbert spaces, properties of scalar product, norm of a function, and mathematical classifications of differential operators are defined with examples. Details of the integration by parts in R, R2 , and R3 are presented with examples. Basic elements of the calculus of variations, derivation of the variation of a functional, necessary condition for extrema of a functional, and associated Euler’s equation are presented and derived. Riemann and Lebesgue integrals are defined and illustrated. Examples of differential operators: self-adjoint, non-self-adjoint, and non-linear appearing in BVPs are presented and their properties are established.

65

2.5. SUMMARY

Problems In Problems 2.1 to 2.3 show whether the differential operator is linear or not. 2.1 Consider the dimensionless form of the one dimensional steady state convection diffusion equation dφ 1 d2 φ − = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx P e dx2 with φ(0) = 1, φ(1) = 0 where P e is called the Peclet number and is known. 2.2 Consider one dimensional axial deformation of a rod (with variable properties). d  du  a(x) = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx dx h du i with u(0) = 0, a(x) = P1 dx x=1 2.3 Consider the dimensional form of the one dimensional steady state Burgers equation. φ

dφ 1 d2 φ − =0 dx Re dx2

∀x ∈ Ω = (0, 1) ⊂ R

with φ(0) = 1,

φ(1) = 0

where Re is the Reynolds number. In Problems 2.4 to 2.7 show whether the functionals are bilinear and symmetric or not. 2.4 Consider the following functional: Z2 B(φ, v) =

a(x)

dφ dv dx + b(2) φ(2) v(2) dx dx

1

where a(·) and b(·) are given functions of x. 2.5 Consider the functional Z1 B(φ, v) =

d2 φ d2 v a(x) 2 2 dx + dx dx

Z1 c(x) φv dx

−1

−1

where a(·), c(·) are known functions of x. 2.6 The functional B(·, ·) is given by Z1 B(φ, v) =

φ

dφ dv dx dx dx

−1

2.7 Consider the following functional Z1 B(φ, v) = −1

where P e is the Peclet number.

In problems 2.8 to 2.11 determine

dφ 1 v dx + dx Pe

Z1 −1

dφ dv dx dx dx

REFERENCES FOR ADDITIONAL READING

66

(a) the differential operator (b) if the differential operator is linear or not. (c) if the differential operator is symmetric or not. If not, then under what conditions would it be symmetric. (d) the adjoint of the operator. 2.8 Consider the following BVP:   d du − a(x) + cu = q(x) ∀x ∈ Ω = (0, 1) ⊂ R dx dx   du + β(u − u∞ ) = Q1 with u(0) = u0 , a(x) dx x=1 Here a(x) and q(x) are known functions of x and β, c, u∞ , u0 , and Q1 are known constants. 2.9 Consider the BVP d  dφ  b(x) = q(x) ∀x ∈ Ω = (0, L) ⊂ R dx dx   dφ + kφ = PL with φ(0) = 0, b(x) dx x=L

−

where b(x) and q(x) are known functions of x and k and PL are known constants. 2.10 Consider the following BVP: d2  d2 ψ  d  dψ  b(x) 2 + c(x) + α(x) ψ = f (x) 2 dx dx dx dx

∀x ∈ Ω = (0, L) ⊂ R

ψ(0) = ψ0 , ψ(L) = ψL i h d ψ d2 ψ i b(x) 2 = M0 , b(x) 2 = ML dx x=0 dx x=L where b, c, α and f are known functions of x and ψ0 , ψL , M0 , and ML are known constants. with

h

2

2.11 Consider the BVP −

d  dφ  φ + f (x) = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx dx √ dφ with = q0 and φ(1) = 2 dx x=0

where f (·) is a known function of x and q0 is a constant. [1–5, 5, 5–9, 9, 9, 10, 10, 11, 11–20]

References for additional reading [1] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000. [2] W. Rudin. Real and Complex Analysis. McGraw-Hill, 1966. [3] F. B. Hildebrand. Methods of Applied Mathematics. Prentice-Hall, New York, 1965. [4] J. N. Reddy and M. L. Rasmussen. Advanced Engineering Analysis. John Wiley, New York, 1982. [5] G. Hellwig. Differential Operators of Mathematical Physics. Addison-Wesley Publishing Co., 1967. [6] E. C. Titchmarsh. The Theory of Functions. Oxford University Press, 2nd edition, 1939.

REFERENCES FOR ADDITIONAL READING

67

[7] I. S. Sokolnikoff and R. M. Redheffer. Mathematics of Physics and Modern Engineering. McGraw-Hill, 2nd edition, 1966. [8] J. T. Oden and L. Demkowicz. Applied Funtional Analysis. CRC-Press, 1996. [9] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002. [10] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [11] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004. [12] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964. [13] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering. McGraw Hill Company, 1986. [14] M. Becker. The Principles and Applications of Variational Methods. MIT Press, 1964. [15] M. Forray. Variational Calculus in Science and Engineering. McGraw-Hill, 1968. [16] K. Rektorys. Variational Methods in Mathematics, Science and Engineering. Reidel, 1977. [17] R. S. Schechter. The Variational Methods in Engineering. McGraw-Hill, 1967. [18] K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 3nd edition, 1982. [19] R. Weinstock. Calculus of Variations with Applications to Physics and Engineering. McGraw-Hill, 1952. [20] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., New York, 3rd edition, 2006.

3

Classical Methods of Approximation

3.1 Introduction We consider an abstract boundary value problem Aφ − f = 0 defined ¯ Let the differential operator contain derivatives up to over a domain Ω. order 2m of the dependent variables φ. We seek an approximation φn of ¯ In doing so, if the domain of definition Ω ¯ is not discretized, φ over Ω. then we have what are called “classical methods” of approximation for the ¯ is discretized into BVPs. On the other hand, if the domain of definition Ω T ¯ ¯ Ω consisting of subdomains, then the entire domain Ω can be thought of as an assembly of the subdomains and the approximation of the solution may ¯T be constructed over each subdomain. Then behavior of the solution over Ω in this process is obviously dependent on the behavior of the solution over each subdomain as well as inter-subdomain boundaries. The finite element method is a method in which one does precisely so. Thus, the finite element method can be viewed as application of classical methods of approximation ¯ T , i.e. piecewise application. Hence, herein lies the for each subdomain of Ω motivation for studying classical methods of approximation. ¯ of In the classical methods of approximation the domain of definition Ω a BVP Aφ − f = 0 is not discretized while seeking an approximation φn ¯ such as finite difference, of φ. In the methods utilizing discretization of Ω, ¯ of finite volume, and finite element methods the domain of definition Ω a BVP is obviously discretized in some way. There are some methods of approximation in which one constructs an integral form using the description ¯ This integral form is then utilized to find Aφ − f = 0 of the BVP over Ω. the approximation φn of φ. Such methods in the published literature are called “variational methods.” We simply refer to them as “methods based on integral forms.” Then, there are other methods of approximation in ¯ Such which one does not construct an integral form using Aφ−f = 0 over Ω. methods in the published work are referred to as “non-variational methods.” We simply refer to them as “methods not based on integral forms.” We keep in mind that a method of approximation based on an integral form may or may not be a variational method. This of course depends upon whether 69

70

CLASSICAL METHODS OF APPROXIMATION

the integral form at hand satisfies the criteria to be called variational. In this chapter we consider the following classical methods of approximation in which one either constructs an integral form directly using Aφ − f = 0 ¯ or the integral based on fundamental lemma of calculus of variations in Ω form eventually results from the first variation of a functional (set to zero) ¯ constructed using Aφ − f = 0 in Ω. 1. the Galerkin method (GM) 2. the Petrov–Galerkin method (PGM) 3. the method of weighted-residuals (WRM) 4. the Galerkin method with weak form (GM/WF) 5. Based on variation of residual functional: least-squares method/process (LSM/LSP) Methods 1 though 4 are very closely related in the sense that the fundamental lemma of calculus of variations forms the basis for them. This is also the case in method 4 but in this case integration by parts plays a crucial role. Method 5 is based on construction of a functional and its variation. In the following sections of this chapter we consider details of the various methods listed above for self-adjoint, non-self-adjoint and nonlinear differential operators. Our aim is to determine which methods of approximation yield variationally consistent integral forms for which differential operators.

3.2 Basic Steps in Classical Methods of Approximation based on Integral Forms Step 1: One considers Aφ − f = 0 in Ω and constructs an integral either directly using the fundamental Lemma of the calculus of variations or by taking the first variation of a functional associated with Aφ − f = 0 in Ω and setting it to zero. Step 2: In all methods of approximation considered in the following sections, the theoretical solution φ(x) is approximated by φn (x) in the form φn (x) = N0 (x) +

n X

Ci Ni (x)

(3.1)

i=1

where N0 (x) and Ni (x) are known functions and Ci (i = 1, 2, . . . , n) are unknown parameters to be determined. The functions Ni (x) are called the basis functions or approximation functions. Thus, φn is a linear combination of the basis functions Ni (x). The approximation φn of φ must satisfy either all or only some of boundary conditions of the BVP, depending upon the method of approximation.

71

3.2. BASIC STEPS IN CLASSICAL METHODS OF APPROXIMATION

Determination of N0 (x) and Ni (x) (i = 1, 2, . . . , n) We consider some guidelines that help us in determining N0 (x) and Ni (x) (i = 1, 2, . . . , n) in (3.1). We classify all specified BCs of the BVP in two types, homogeneous and nonhomogeneous, and consider the following: (a) We choose each Ni (x) such that each Ni (x) satisfies required homogeneous boundary conditions as well as homogeneous forms of the required nonhomogeneous boundary conditions. With n P this choice of Ni (x) their linear combination Ni (x) Ci will i=1

(b)

(c)

(d)

(e)

(f)

also satisfy these boundary conditions. With this choice of Ni (x) (i = 1, 2, . . . , n) in (a), we must choose N0 (x) such that it satisfies required nonhomogeneous boundary conditions as well as required homogeneous boundary conditions. If we choose Ni (x) (i = 1, 2, . . . , n) as in (a) and N0 (x) as in (b), then φn (x) in (3.1) is ensured to satisfy all required boundary conditions of the BVP. Ni (x) (i = 1, 2, . . . , n) must constitute a complete and linearly independent set so that the rows and columns in the resulting algebraic systems remain linearly independent. N0 (x) and Ni (x) (i = 1, 2, . . . , n) must also possess desired continuity and differentiability properties so that φn (x) in (3.1) remains admissible in the BVP. Out of all BCs (homogeneous and nonhomogeneous) of the BVP, the required set of BCs are those that remain to be satisfied by φn (x) after constructing the integral form. In other words if some boundary conditions are absorbed or used in constructing the integral form (as in weak form GM), then these must be removed from the total set of BCs to determine the required set of BCs to be satisfied by φn .

Step 3: Determine the unknown constants Ci by substituting φn in the integral form. When φn is substituted in the integral form and when we choose as many test functions as the number of unknowns Ci (i = 1, 2, . . . , n), we obtain a system of n algebraic equations from which Ci ’s can be calculated. Knowing the Ci ’s, the approximation ¯ Since the approximation φn φn in step 2 gives the solution over Ω. contains n basis functions Ni (i = 1, 2, . . . , n), it also is referred to as an n-parameter (Ci ’s being the parameters) approximation of φ, hence the reason for the subscript n on φ.

72

CLASSICAL METHODS OF APPROXIMATION

Thus, the classical methods of approximation are rather simple in principle and the basic steps are straightforward. The difficulty, however, lies in ¯ is geometthe determination of Ni (i = 0, 1, . . . , n). When the domain Ω rically complicated with involved boundary conditions, determination of Ni in most cases may not be possible. Remarks. In the approach described above, the integral form when compared with the elements of calculus of variations presented in chapter 2 (section 2.3) is only the necessary condition, i.e. variation of some functional I(φn ) (assumed to exist) set to zero. Extremum principle or sufficient condition is yet to be considered as without this we can not determine if the solution φn obtained from the integral form is unique or not. This aspect is considered for various methods of approximation for the three classes of differential operators in later sections by establishing variational consistency or variational inconsistency of the integral forms by using their definitions presented in chapter 2 (section 2.3).

3.3 Integral forms using the Fundamental Lemma of the Calculus of Variations Let Aφ − f = 0 in Ω be an abstract boundary value problem. Then, ¯ using the the integral form for Aφ − f = 0 can be constructed over Ω Fundamental Lemma (lemma 2.1). We recall the Fundamental Lemma first. Lemma. If α(x) is continuous on [a, b] and if Zb

α(x)h(x)dx = 0, ∀h(x) ∈ H 1 (a, b)

a

then α(x) = 0 ∀x ∈ (a, b). Since Aφ − f = 0, ∀x ∈ Ω, then Z (Aφ − f )v dΩ = 0

(3.2)

¯ Ω

holds for any v. In (3.2), when compared to Fundamental Lemma, Aφ−f = 0 takes the place of α(x) and v replaces h(x). If A is a differential operator containing highest order derivative of order 2m, then for the integrand in (3.2) to be continuous the following must hold ¯ is the Hilbert space of functions that have continuous [recall that H 2m+1 (Ω)

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

73

derivatives up to order 2m and square-integrable, and the 2m+1st derivative exists but may not be continuous]: ¯ ⊂ H 1 (Ω) ¯ φ ∈ Vn ⊂ H 2m+1 (Ω) ¯ v ∈ V ⊂ H 1 (Ω)

¯ ∀x ∈ Ω ¯ ∀x ∈ Ω

(3.3)

where v is called a test function. The choice of test function v must be such that (i) v = 0 on Γ if φ = φ0 (specified) on Γ and (ii) v = δφ is a valid choice since φ = φ0 on Γ implies that v = δφ0 = 0 on Γ. The space Vn of functions is called the approximation space and V is called the space of test functions. We also note that (3.2) can now be written as Z (Aφ − f )v dΩ = (Aφ − f, v)Ω¯ = 0 (3.4) ¯ Ω

Thus, the integral form is the scalar product of Aφ − f and the test function ¯ We can also write (3.4) as v over Ω. B(φ, v) = l(v)

(3.5)

where B(φ, v) = (Aφ, v) l(v) = (f, v)

(3.6)

Both B(·, ·) and l(·) are functionals. B(·, ·) contains both functions φ and v whereas l(·) only contains the test functions v. In the following we consider various classical methods of approximation based on the integral form (3.4) or (3.5) constructed using the fundamental lemma of the calculus of variations.

3.3.1 The Galerkin method We replace φ by φn in (3.4), an approximation of φ (Aφn − f, v) = 0

(3.7)

In the Galerkin method, we choose v = δφn = Nj (j = 1, 2, . . . , n)

(3.8)

with v = Nj (j = 1, 2, . . . , n) satisfying the homogeneous form of all specified boundary conditions on Γ, while φ0 satisfies all of the actual specified

74

CLASSICAL METHODS OF APPROXIMATION

boundary conditions of the BVP. Substituting φn from (3.1) and v from (3.8) into (3.7) we obtain 

A N0 (x) +

n X


Ci Ni (x) − f, Nj = 0, j = 1, 2, . . . , n

(3.9)

i=1

In this method φn (x) must satisfy all specified BCs of the BVP; hence, the required boundary conditions to be considered in determining N0 (x) and Ni (x) (i = 1, 2, . . . , n) are all specified BCs of the BVP. Whether (3.9) can be further simplified or not depends upon the nature of the differential operator (e.g., linear or nonlinear). We consider different cases in the following. 3.3.1.1 Self-adjoint and non-self-adjoint linear differential operators Using the linearity of the operator, (3.9) can be simplified as n X

 Ci ANi (x) + AN0 (x), Nj (x) = (f, Nj (x)), j = 1, 2, . . . , n

(3.10)

i=1

or

n X

ANi , Nj ) Ci = (f, Nj ) − (AN0 , Nj ), j = 1, 2, . . . , n

(3.11)

i=1

In matrix form, we can write (3.11) as [K]{C} = {F }

(3.12)

In (3.12), [K] is an n × n matrix and {C} and {F } are n × 1 vectors; Kij of [K] and Fi of {F } are given by Kij = (ANj , Ni ), i, j = 1, 2, . . . , n Fi = (f, Ni ) − (AN0 , Ni ), i = 1, 2, . . . , n

(3.13)

Remarks. (1) Obviously, Kij 6= Kji , that is, the coefficient matrix [K] is not symmetric. Hence, [K] is not ensured to be positive-definite for all admissible choices of Ni for i = 1, 2, . . . , n. (2) Whether the integral form in (3.7) is VC or VIC needs to be established. Theorem 3.1. If Aφ − f = 0 in Ω is a BVP with some boundary conditions in which A is self-adjoint, then there exists a functional I(φ) given by 1 I(φ) = (Aφ, φ) − (f, φ), φ ∈ Vn 2

(3.14)

75

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

corresponding to Aφ − f = 0 such that (Aφ − f, v) with v = δφ represents the first variation of I(φ) (i.e., δI(φ)) and if I(φ) is differentiable in its arguments then δI(φ) = 0 is a necessary condition for an extremum of I(φ). Proof. Taking the first variation of I(φ)

or

1 1 δI(φ) = (δ(Aφ), φ) + (Aφ, δφ) − (f, δφ) 2 2

(3.15)

1 1 δI(φ) = (A(δφ), φ) + (Aφ, δφ) − (f, δφ) 2 2

(3.16)

Since A is self-adjoint
A(δφ), φ = (δφ, Aφ) = (Aφ, δφ)

(3.17)

Substituting (3.17) into (3.16) we obtain 1 1 δI(φ) = (Aφ, δφ) + (Aφ, δφ) − (f, δφ) 2 2 = (Aφ, v) − (f, v) = (Aφ − f, v)

(3.18)

If I is differentiable in its arguments, then δI(φ) is continuous and we have δI = (Aφ − f, v) = 0, a necessary condition for an extremum of I(φ). Theorem 3.2. If Aφ − f = 0 in Ω is a BVP in which A is a self-adjoint differential operator, then the integral form resulting from the Galerkin method with v = δφ is VIC. Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ − f, v) = 0

with v = δφ, φ, v ∈ Vn

(3.19)

Then δ 2 I(φ) = δ(δI(φ)) = δ(Aφ − f, v) = (δ(Aφ − f ), v) + (Aφ − f, δv)

∀v ∈ V

(3.20)

Since A is linear and hence not a function of φ, we have δ(Aφ) = A(δφ) = Av

(3.21)

and δv = 0 (also note that δf = 0 and δA = 0). Using these in (3.20) we obtain δ 2 I(φ) = ((δA)φ + A(δφ) − δf, v) + (Aφ − f, δv) (3.22)

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CLASSICAL METHODS OF APPROXIMATION

Substituting from (3.21) into (3.22)   ≯ 0, minimum 2 δ I(φ) = (Av, v) 6= 0, saddle point v ∈ Vn   ≮ 0, maximum

(3.23)

Hence, δ 2 I(φ) is not a unique extremum principle. Thus, the integral form (Aφ − f, v) = 0 with v = δφ is VIC when the differential operator is selfadjoint. When the differential operator is non-self-adjoint it is linear, hence (3.10) - (3.13) must hold. We state two theorems in the following for non-selfadjoint operators and provide their proofs. Theorem 3.3. If Aφ − f in Ω is a BVP with some boundary conditions in which A is a non-self-adjoint operator, then the first variation of the functional I(φ) defined by 1 I(φ) = (Aφ, φ) − (f, φ), φ ∈ Vn 2 does not yield the integral form (Aφ − f, v) = 0 in which v = δφ. Proof. 1 δI = δ(Aφ, φ) − δ(f, φ) 2 1 1 = (A(δφ), φ) + (Aφ, δφ) − (f, δφ) 2 2 1 1 = (Av, φ) + (Aφ, v) − (f, v) 2 2

(3.24)

Since A is non-self-adjoint, no further simplification is possible in (3.24). Clearly (3.24), that is δI, is not the same as (Aφ − f, v). Theorem 3.4. If Aφ − f = 0 in Ω is a BVP in which A is a non-selfadjoint operator, then the integral form resulting from the Galerkin method (i.e. (Aφ − f, v) = 0 with v = δφ) is VIC. Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ − f, v) = 0 with v = δφ, φ, v ∈ Vn

(3.25)

δ 2 I(φ) = δ(δI(φ))

(3.26)

Then Following the proof of Theorem 3.2, δ 2 I(φ) = (Av, v), v = δφ

(3.27)

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3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

Hence

δ 2 I(φ) = (Av, v) =

Z ¯ Ω

  > 0, minimum (Av)v dΩ = 0, saddle point   < 0, maximum

∀v ∈ Vn

(3.28)

is not ensured. Hence, δ 2 I(φ) is not a unique extremum principle. Thus, the integral form (Aφ − f, v) = 0 with v = δφ is VIC when the differential operator is non-self-adjoint. 3.3.1.2 Non-linear differential operators In this case also we begin with (Aφ − f, v) = 0 and substitute the approximation φn to obtain 

A N0 (x) +

n X


Ci Ni (x) − f, Nj = 0, j = 1, 2, . . . , n

(3.29)

i=1

Since the differential operator A is non-linear, A is a function of φn . We can write (3.29) as 

A N0 (x) +

n X


Ci Ni (x) , Nj = (f, Nj ), j = 1, 2, . . . , n

(3.30)

i=1

(3.30) represents a system of n non-linear algebraic equations in n unknowns, Ci . Specific forms of these equations depends on the specific form of the operator A. Theorem 3.5. If Aφ − f = 0 in Ω is a BVP in which A is a non-linear differential operator, then the integral form resulting from the Galerkin method, that is (Aφ − f, v) = 0 with v = δφ, is VIC. Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ − f, v) = 0

with v = δφ, φ, v ∈ Vn

(3.31)

Then δ 2 I(φ) = δ(δI(φ)) = δ(Aφ − f, v)

(3.32)

= ((δA)φ + A(δφ) − δf, v) + (Aφ − f, δv) since δf = 0 δ 2 I(φ) = ((δA)φ + Av, v)

(3.33)

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CLASSICAL METHODS OF APPROXIMATION

Obviously   > 0, minimum 2 δ I(φ) = 0, saddle point ∀v ∈ Vn   < 0, maximum

(3.34)

does not hold for this case. Hence, δ 2 I(φ) is not a unique extremum principle. Thus, the integral form (Aφ − f, v) = 0 with v = δφ is VIC when the differential operator is non-linear.

3.3.2 The Petrov–Galerkin and weighted-residual methods In these methods we substitute φn from (3.1) into (3.4), or (3.6) with v = w, a weight function (in weighted-residual method) or test function (in the Petrov–Galerkin method), to obtain (Aφn − f, w) = 0

(3.35)

w = Ψj (x), j = 1, 2, . . . , n

(3.36)

and choose where w 6= δφn but w = Ψj (x) = 0 on Γ where φ = φ0 . Substituting φn from (3.1) and w from (3.36) into (3.35) we obtain (A(N0 (x) +

n X

Ci Ni (x)), Ψj ) = (f, Ψj ) = 0, j = 1, 2, . . . , n

(3.37)

i=1

In this method φn (x) must satisfy all BCs of the BVP, hence the required BCs to be considered in determining N0 (x) and Ni (x) (i = 1, 2, . . . , n) are all BCs of the BVP. 3.3.2.1 Self-adjoint and non-self-adjoint linear differential operators When the differential operator is linear, that is self-adjoint or non-selfadjoint, (3.37) can be written as 

A

n X


Ci Ni (x) , Ψj = (f, Ψj ) − (AN0 , Ψj ), j = 1, 2, . . . , n

(3.38)

i=1

or n X


ANi (x), Ψj Ci = (f, Ψj ) − (AN0 , Ψj ), j = 1, 2, . . . , n

(3.39)

i=1

In matrix form we can write (3.39) [K]{C} = {F }

(3.40)

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

79

Kij of [K] and Fi of {F } are given by Kij = (ANj (x), Ψi ), i, j = 1, 2, . . . , n Fi = (f, Ψi ) − (AN0 , Ψi ), i = 1, 2, . . . , n

(3.41)

Ψi (x) are referred to as weight functions and hence the name weighted residual due to the fact that (3.35) is a statement of the integral of weightedresiduals with weight functions Ψi . In the Petrov–Galerkin method, Ψi are called test functions or weight functions. Remarks. (1) Obviously, Kij 6= Kji , that is the coefficient matrix [K], is not symmetric and, hence, [K] cannot be ensured to be positive-definite for all admissible choices of Ni s and Ψj s. (2) Whether the integral form (3.39) is variationally consistent or not needs to be established. Theorem 3.6. If Aφ − f = 0 in Ω is a boundary value problem in which A is linear, then the first variation of the functional 1 I(φ) = (Aφ, φ) − (f, φ) 2

(3.42)

set to zero does not correspond to the integral form (Aφ − f, w) = 0 in PGM or WRM. Proof. Taking the first variation of I(φ) in (3.42) and setting it to zero we obtain
1
1 δI(φ) = A(δφ), φ + Aφ, δφ − (f, δφ) (3.43) 2 2 Let v = δφ 6= w, then 1 1 δI(φ) = (Av, φ) + (Aφ, v) − (f, δφ) 6= (Aφ − f, w) 2 2

(3.44)

Theorem 3.7. If Aφ−f = 0 in Ω is a BVP in which A is a linear differential operator, then the integral form resulting from the Petrov–Galerkin method and weighted-residual method, (Aφ − f, w) = 0 with w 6= δφ is VIC. Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ − f, w) = 0. Then δ 2 I(φ) = δ(δI(φ)) = δ(Aφ − f, w) = (A(δφ) − δf, w) + (Aφ − f, δw)

(3.45)

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CLASSICAL METHODS OF APPROXIMATION

or δ 2 I(φ) = (Av, w) because δf = 0 and δw = 0 with v = δφ and w is a weight function.   Z > 0, 2 δ I(φ) = (Av, w) = (Av)w dΩ = 0,   ¯ < 0, Ω

(3.46)

Hence,

minimum saddle point ∀v ∈ Vn , ∀w ∈ V maximum (3.47) is not possible. Hence, δ 2 I(φ) is not a unique extremum principle. Thus, the integral form (Aφ − f, w) = 0 with w 6= δφ is VIC when the differential operator A is linear. 3.3.2.2 Non-linear differential operators In this case we begin with (Aφ − f, w) = 0 and substitute Ψj for w and φn for φ (A(N0 (x) +

n X

Ci Ni (x)), Ψj ) = (f, Ψj ) = 0, j = 1, 2, . . . , n

(3.48)

i=1

Since the differential operator A is non-linear, A is a function of φn . (3.48) represents a system of non-linear algebraic equations in unknown Ci . The specific form of these equations depends upon the specific form of A. Theorem 3.8. If Aφ − f = 0 in Ω is a BVP in which A is a non-linear differential operator, then the integral form (Aφ − f, w) = 0 resulting from the Petrov–Galerkin method or weighted-residual method is VIC. Proof. Let there exist a functional I(φ) such that δI(φ) = (Aφ − f, w)

(3.49)

where w 6= δφ. Then δ 2 I(φ) = δ(δI(φ)) = δ(Aφ − f, w)

(3.50)


= (δA)φ + A(δφ) − δf, w + (Aφ − f, δw) or
δ 2 I(φ) = (δA)φ + Av − δf, w + (Aφ − f, δw)

(3.51)

since δw = 0 and δf = 0. We have  > 0, minimum
 δ 2 I(φ) = (δA)v + Av, w = 0, saddle point ∀v ∈ Vn , ∀w ∈ V (3.52)   < 0, maximum

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

81

is not possible. Hence, δ 2 I(φ) is not a unique extremum principle. Thus, the integral form (Aφ − f, w) = 0 with w 6= δφ is VIC when the differential operator is non-linear.

3.3.3 The Galerkin method with weak form Let Aφ − f = 0 in Ω be a boundary value problem, then in this method we begin with (as in GM) integral statement based on fundamental lemma (Aφ − f, v) = 0

(3.53)

in which v = δφ, and hence, v = 0 on Γ where φ = φ0 (specified). Let the differential operator contain highest order derivatives of order 2m. In this method, we transfer some differentiation form φ to v using integration by parts. The reasons for doing so are not clear at this stage, but will be explained shortly in the following sections. In doing so we observe the following. (1) We have lowered some differentiation on φ but increased by the same on the test function v. (2) When a differential operator contains even order derivatives, it is possible to transfer half of the differentiation from φ to v. In such cases, the integrand contains terms that have the same orders of derivatives of φ and v. This feature has special consequences in relating the resulting integral form to the elements of calculus of variations (i.e. variational consistency). (3) In the process of transferring differentiation from φ to v by using integration by parts, concomitant consisting of boundary terms, boundary integrals or surface integrals (concomitant, henceforth may also be referred to as boundary integrals or boundary terms) results. (4) Using the concomitant, the dependent variables and their derivatives are classified into two groups. (a) primary variables: PVs (b) secondary variables: SVs The definition of PVs and SVs result in the classification of the boundary conditions into two groups as well. (a) essential boundary conditions (EBCs) - those corresponding to the specified values of PVs. (b) natural boundary conditions (NBCs) - those corresponding to the specified values of SVs.

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CLASSICAL METHODS OF APPROXIMATION

In the following we consider some definitions that facilitate this process of determining PVs, SVs, EBCs, and NBCs. Definition 3.1 (Primary Variables (PVs)). Defining dependent variables in the same manner in which the test function (and its derivatives) appear in the concomitant defines primary variables (PVs). Definition 3.2 (Secondary Variables (SVs)). The coefficient of the test function (and its derivatives) in the concomitant defines secondary variables (SVs). Definition 3.3 (Essential boundary conditions (EBCs)). Specifications of primary variables on some boundaries constitutes essential boundary conditions (EBCs). Definition 3.4 (Natural boundary conditions (NBCs)). Specification of secondary variables on some boundaries constitutes natural boundary conditions. At this point we can examine the BCs of the BVP to determine which ones are EBCs and which ones are NBCs. We note that determination of PVs, SVs, EBCs, and NBCs is strictly using concomitant and not using the BCs of the BVP. Thus, in other methods of approximation that do not result in concomitant as there is no integration by parts, there is no concept of PVs, SVs, EBCs, and NBCs. Only the GM/WF has these due to integration by parts. (5) The concomitant is simplified using boundary conditions of the BVP and their variations. (6) In step (5) some boundary conditions of the BVP (NBCs) are absorbed (or used). This is important to note due to the fact that in the selection of Ni , these boundary conditions should not be considered. The resulting expressions are arranged in the form B(φ, v) = l(v)

(3.54)

In (3.54), the functional B(·, ·) contains those terms that contain both φ and v and the functional l(·) constains terms that have v only. The integral form (3.54) is referred to as the weak form of the integral form (3.53). Since the starting point in this method is (3.53), we refer to this method more specifically as “the Galerkin method with weak form.” In the following we consider three different classes of differential operators: self-adjoint, nonself-adjoint and nonlinear for this method of approximation. We first present some important theorems related to the weak form of the BVP for the three classes of differential operators.

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

83

Theorem 3.9. Let Aφ − f = 0 in Ω be a BVP in which A is a self-adjoint differential operator and out of all possible weak forms let B(φ, v) − l(v) = 0 be a weak form in which B(φ, v) is bilinear and symmetric and l(v) is linear. Then there exists a functional 1 I(φ) = B(φ, φ) − l(φ) 2 such that δI = 0 yields the weak form B(φ, v) − l(v) = 0 and δ 2 I yields a unique extremum principle and, hence, the weak form B(φ, v) − l(v) = 0 is variationally consistent. Proof. Let B(φ, v) − l(v) = 0 be a weak form of Aφ − f = 0 in Ω in which B(φ, v) is bilinear and symmetric and l(v) is linear. Consider the functional 1 I(φ) = B(φ, φ) − l(φ) 2

(existence of functional I(φ))

(3.55)

Then 1 δI(φ) = δ(B(φ, φ) − δ(l(φ)) = 0 2 1 1 = (B(δφ, φ) + (B(φ, δφ) − (l(δφ)) = 0 2 2 1 1 = (B(v, φ) + (B(φ, v) − (l(v)) = 0 2 2 Since B(·, ·) is symmetric, B(v, φ) = B(φ, v), therefore δI(φ) = B(φ, v) − l(v) = 0

(necessary condition)

(3.56)

and δ 2 I(φ) = δ(δI(φ)) = δ(B(φ, v) − l(v)) = B(δφ, v) = B(v, v), v = δφ Since B(·, ·) is symmetric δ 2 I(φ) = B(v, v) > 0

∀v ∈ Vn

(unique extremum principle)

(3.57)

Hence, the weak form B(φ, v) − l(v) = 0 is VC. Remarks. (1) A solution φ obtained from B(φ, v) − l(v) = 0 minimizes I(φ) due to the fact that δ 2 I(φ) > 0. (2) In linear solid mechanics applications, I(φ) represents the total potential energy, 21 B(φ, φ) is the strain energy and l(v) is the potential energy of

84

CLASSICAL METHODS OF APPROXIMATION

loads. Thus, setting the first variation of I(φ) to zero is the well-known principle of minimum total potential energy. (3) The Galerkin method with weak form for self-adjoint operators in which B(φ, v) is symmetric is known as the Ritz method (actually, one may call the weak form Galerkin method as the Ritz method). Theorem 3.10. Let Aφ − f = 0 in Ω be a BVP in which A is a nonself-adjoint differential operator. Let B(φ, v) − l(v) = 0 be all possible weak forms. Then all such integral forms are variationally inconsistent. Proof. Let there exist a functional I(φ) such that δI(φ) = 0 yield the weak form B(φ, v) − l(v) = 0. Since A is non-self-adjoint, B(φ, v) is bilinear but not symmetric (i.e. B(φ, v) 6= B(v, φ)), hence δ 2 I(φ) = δ(B(φ, v) − l(v)) = B(δφ, v)    >0 = B(v, v) =0   0 2 2 δ I(φn ) = 2(δE, δE) + 2(E, δ E) =0   0 ∀v ∈ Vn .

(3.74)

Hence, δ 2 I represents a unique extremum principle. Thus, the integral form resulting from δI(φn ) = 0, i.e. B(φn , v) = l(v) in (3.73) is VC. Remarks. (1) Since δ 2 I(φn ) > 0, a φn from δI(φn ) = 0 minimizes I(φn ), and the minimum of I(φn ) is zero. ¯ that is, Aφn − f = 0 is (2) When I(φn ) → 0, E = Aφn − f → 0 ∀x ∈ Ω, ¯ satisfied in the pointwise sense in Ω if the integrals are Riemann. (3) Since the differential operator is linear, B(φn , v) in the integral form (3.73) is bilinear and is also symmetric, B(φn , v) = B(v, φn )

(3.75)

(4) The algebraic system resulting from δI(φn ) = 0 yields a symmetric and positive-definite coefficient matrix. (5) The property of the symmetry of the operator is not required in the proof or the construction of any of the variations. Only the linearity of the operator A is needed. Theorem 3.13. Let Aφ − f = 0 in Ω be a boundary value problem in which ¯ and let A is a non-self-adjoint operator, let φn be an approximation of φ in Ω ¯ Then the integral form resulting Aφn − f = E be the residual function in Ω.

89

3.3. INTEGRAL FORMS USING THE FUNDAMENTAL LEMMA

from the first variation of the residual functional I(φn ) = (E, E) set to zero is VC. Proof. Since A is linear, E is a linear function of φn and δE is not a function of φn and hence δ 2 E = 0. I(φn ) = (E, E) = (Aφn − f, Aφn − f )

(existence of functional I(φn ))

If I(φn ) is differentiable in φn , then δI(φn ) = (δE, E) + (E, δE) = 2(E, δE) = 2g = 0 or g=0

with δE = Av

is necessary condition

or (Aφn , Av) = (f, Av)

(3.76)

B(φn , v) = l(v) and δ 2 I(φn ) = δ(δI(φn )) = 2(δE, δE) = 2(Av, Av) = 2B(v, v) > 0 ∀v ∈ Vn Hence, δ 2 I represents a unique extremum principle. Thus, the integral form resulting from δI(φn ) = 0, that is, B(φn , v) = l(v) in (3.76) is VC. The remarks following theorem 3.12 are precisely applicable here also. Theorem 3.14. Let Aφ − f = 0 in Ω be a boundary value problem in ¯ and which A is a non-linear operator, let φn be an approximation of φ in Ω ¯ let Aφn − f = E be the residual function in Ω. Then the integral form resulting from the first variation of the residual functional I(φn ) = (E, E) set to zero is VC provided δ 2 I ∼ = (δE, δE) and the system of non-linear algebraic equations resulting from δI(φn ) = 0 are solved iteratively using Newton-Raphson or Newton’s linear method. Proof. Since A is non-linear, E is a non-linear function of φn and so δE is also a function of φn . I(φn ) = (E, E) = (Aφn − f, Aφn − f )

(existence of functional I(φn )) (3.77)

If I(φn ) is differentiable in φn , then δI(φn ) = (δE, E) + (E, δE) = 2(E, δE) = 2g(φn ) = 0 or g(φn ) = 0

is necessary condition

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CLASSICAL METHODS OF APPROXIMATION

Since δE = δ(Aφn − f ) = δA(φn ) + A(δφn )

(3.78)

= δA(φn ) + Av we have g(φn ) = (Aφn − f, δA(φn ) + Av) = 0

(3.79)

(Aφn , δA(φn ) + Av) = (f, δA(φn ) + Av)

(3.80)

B(φn , v) = l(v)

(3.81)

or or Also 2

2

δ I(φn ) = 2(δE, δE) + 2(E, δ E)

    

>0 =0 0 B(v, v) = 0 ∀v ∈ Vn   0 B(v, w) = 0 ∀v ∈ Vn , w ∈ V   0 2 δ I(φ) = 0 ∀φ ∈ Vn , ∀v ∈ Vn   0 ∀v ∈ Vn , φn ∈ Vn (3.466) = ¯ Ω

Hence, δ 2 I(φn ) > 0 yields a unique extremum principle and so the integral form (3.463) is VC. We find a φn using Newton’s linear method with line search that satisfies g(φn ) = 0.

3.6 Numerical Examples In this section we present numerical solutions for a variety of 1D and 2D boundary value problems described by self-adjoint, non-self-adjoint and nonlinear differential operators using the Galerkin method (GM), the Petrov– Galerkin method (PGM), the weighted-residual method (WRM), the Galerkin method with weak form (GM/WF), and the least-squares processes (LSP). First we consider some general guidelines (also discussed in earlier sections) that are helpful in considering various methods of approximations.

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CLASSICAL METHODS OF APPROXIMATION

In all methods of approximation, we always approximate the theoretical solution φ by φn , an n-parameter approximation of φ given by φn = N0 (x) +

n X

Ci Ni (x)

(3.467)

i=1

where x could be x, (x, y), or (x, y, z) depending upon whether the BVP is 1D, 2D, or 3D, respectively; N0 (x) and Ni (x) are known functions. Guidelines for determining N0 (x) and Ni (x) for various methods of approximation have already been discussed. Convergence of the approximation φn to the theoretical solution φ Establishing the convergence of φn to φ and the rate at which φn converges is an important and essential aspect of the computations. However, the convergence of φn to φ implies that we must show that the error between φ and φn in some norm diminishes at some rate as n is increased. In the simple cases, it may be possible to determine the theoretical solution φ, but in general this may not be possible. We follow an alternate approach described in the following to assess error in the computed solution φn . Let φn be an n-parameter solution of a BVP. Then using the error function E = Aφ − f ∀x ∈ Ω we define the residual functional and I = (E, E), ¯ the square of the L2 -norm of the residual E over the domain of definition Ω. We note that the residual functional I is quadratic in E and, hence, is always positive and its minimum is zero. When φn → φ, the E → 0 and I → 0. Thus, the proximity of I to zero is an absolute measure of the accuracy of φn . This approach does not require the theoretical solution φ but clearly establishes the accuracy of φn . How quickly or slowly φn approaches φ as n is increased in different methods of√approximation can be easily established by studying the behavior of I or I versus n. We follow this approach in the numerical studies to determine the convergence rates and the superiority of one method of approximation over others. Example 3.9. Consider the 1D diffusion equation −

d2 φ − φ + x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 dx2

(3.468)

with φ(0) = 0 and φ(1) = 0

(3.469)

The Galerkin method Following the details presented earlier, for this case we have the integral form B(φ, v) = l(v) (3.470)

155

3.6. NUMERICAL EXAMPLES

where B(φ, v) =

 d2 φ

+ φ, v



dx2 l(v) = (x2 , v)

and v = δφ. Let φn = N0 (x) +

n X

(3.471) (3.472)

Ci Ni (x)

(3.473)

x=1

where φn is the approximation of φ and v = δφn = Nj (x) (j = 1, 2, . . . , n). In this case the differential operator is self-adjoint. Substituting φn in place of φ and Nj (x) in place of v in (3.471) and (3.472) and regrouping the terms we obtain the following from (3.470), n  2 X d Ni

  d2 N  0 2 + N , N C = (x , N ) − + N , N , j = 1, 2, . . . , n i j i j 0 j dx2 dx2 i=1 (3.474) Equations (3.474) can be written in matrix form as [K]{C} = {F }

(3.475)

in which Kij of [K] and Fi of {F } are given by  d2 N

 + Nj , Ni , i = 1, 2, . . . , n, j = 1, 2, . . . , n   d2 N 0 + N , N , i = 1, 2, . . . , n Fi = (x2 , Ni ) − 0 i dx2

Kij =

j 2 dx

(3.476) (3.477)

Obviously, Kij 6= Kji , i.e. [K] is not symmetric. This is a consequence of the VIC integral. We have N0 (x) = 0 due to the fact that both boundary conditions (3.469) are homogeneous. We choose Ni (x) such that each Ni (x) satisfies the homogeneous form of all boundary conditions of the problem. Therefore Ni (0) = Ni (1) = 0, i = 1, 2, . . . , n must hold

(3.478)

In choosing Ni (x), we may proceed as follows. Let N1 (x) = a0 +a1 x in which a0 and a1 are evaluated using N1 (0) = 0 and N1 (1) = 0 which gives a0 = 0 and a1 = 0. So this choice is of no consequence. Let N1 (x) = a0 +a1 x+a2 x2 . Then using N1 (0) = 0 and N1 (1) = 0 we have a0 = 0 and a1 + a2 = 0. Thus, we can choose either a1 = −a2 or a2 = −a1 , which yields N1 (x) = a2 x(x − 1)

or

N1 (x) = a1 x(1 − x)

(3.479)

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CLASSICAL METHODS OF APPROXIMATION

From (3.479) we note that the function in one choice i.e. N1 (x) is the negative of the function in other choice. Furthermore a1 , a2 , or −a1 , −a2 can all be absorbed in the constant C1 in the approximation φn . Thus, we can choose either of the two forms in (3.479). Consider N1 (x) = x(1 − x)

(3.480)

Next we consider N2 (x). First, we note that a general expression for N2 (x) must contain an x3 term (based on (3.480)) and can only have three arbitrary constants due to the fact that: (1) we have only two boundary conditions, i.e. N2 (0) = 0 and N2 (1) = 0, to evaluate the constants and (2) the determination of N2 (x) within an arbitrary constant is satisfactory due to the fact that the constant gets absorbed in C2 in the approximation φn . Thus, we have two choices: N2 (x) = a0 + a1 x + a3 x3 (3.481) or N2 (x) = a0 + a2 x2 + a3 x3

(3.482)

Using (3.481) with N2 (0) = 0 and N2 (1) = 0 we obtain a0 = 0 a1 = −a3 ⇒ N2 (x) = a3 x(x2 − 1)

(3.483)

2

a3 = −a1 ⇒ N2 (x) = a1 x(1 − x ) or using (3.482) with N2 (0) = 0 and N2 (1) = 0 we obtain a0 = 0 a2 = −a3 ⇒ N2 (x) = a3 x(x2 − 1)

(3.484)

2

a3 = −a2 ⇒ N2 (x) = a2 x(1 − x ) Since the polynomial terms in the two choices of N2 (x) in (3.483) are the negative of each other and the sign and the constants a3 or a1 can be absorbed in C2 in (3.473), either of the two choices are satisfactory. We choose the following for N2 (x): N2 (x) = x(1 − x2 ) (3.485) or N2 (x) = x2 (1 − x)

(3.486)

The constants a1 and a2 in (3.483) and (3.486) are absorbed in C2 in the approximation φn . Thus, we see that for a choice of N1 (x) in (3.480) there are two choices of N2 (x) ((3.485) and (3.486)). N1 (x) is a polynomial of up to degree two in x whereas both choices of N2 (x) are polynomials of up to degree three in x. A continuation of this procedure used for establishing

157

3.6. NUMERICAL EXAMPLES

N1 (x) and N2 (x) to determine Ni (x), i = 3, . . . , n is obviously difficult due to the fact that an expression for Ni (x) must contain terms a0 and ai xi but can only have one more term of degree lower than xi . This is due to the fact that we only have two conditions, namely Ni (0) = 0 and Ni (1) = 0, to evaluate Ni (x) within an arbitrary constant. Thus, there are many choices possible. This simple exercise points out the difficulty in establishing Ni (x) in the classical methods of approximation. At this stage, we abandon this approach but pursue a slightly different line of thinking. We note that N1 (x) and N2 (x) are algebraic polynomials of up to highest degrees of two and three. Thus, by induction, we expect Ni (x) to be an algebraic polynomial of highest degree i in x. Hence, using induction we can write Choice 1: Ni (x) = x(1 − xi ), i = 1, 2, . . . , n

(3.487)

Choice 2: Ni (x) = xi (1 − x), i = 1, 2, . . . , n

(3.488)

or with N0 (x) = 0. Thus, in this case there are at least two possible choices of Ni (x) (i = 1, 2, . . . , n). Other possible choices may also be admissible. For example, Ni (x) = x(1 − x)i is also admissible, though at this stage we do not have a way to derive these directly as we have done in case of (3.487) and (3.488). One parameter approximation: For this case choices 1 and 2 are the same, thus we can use N0 (x) = 0 N1 (x) = x(1 − x), v = N1 (x) The computed [K], {F }, and {C} are [K] = [−0.3],

{F } = {0.05},

{C} = {−0.166667}.

Hence φn = (−0.166667)x(1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.8796296 × 10−1 Two parameter approximation: Here, we choose N0 (x) = 0 N1 (x) = x(1 − x) N2 (x) = x(1 − x2 ) : choice 1 N2 (x) = x2 (1 − x) : choice 2

(3.489)

158

CLASSICAL METHODS OF APPROXIMATION

with v = Nj (x) (j = 1, 2). Choice 1:

  −0.30 −0.4500 [K] = , −0.45 −0.7238

and

 {C} =

 {F } =

 0.0500 0.0833

 0.08943 −0.17073

Hence φn = (0.08943)x(1 − x) + (−0.17073)x(1 − x2 ) I = (E, E) = (Aφn − f, Aφn − f ) = 0.4644502 × 10−2 Choice 2:

  −0.30 −0.1500 , [K] = −0.15 −0.1238

and

 {C} =

 {F } =

 0.0500 0.0333

 −0.08130 −0.17073

Hence φn = (−0.081301)x(1 − x) + (−0.17073)x2 (1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.4644501 × 10−2

1e+00 GM, GM/WF, PGM LSP

Residual functional I

1e-05

1e-10

1e-15

1e-20

1e-25 1

2

3

4 5 6 Number of terms n

7

8

9

Figure 3.1: Example 3.9: Residual functional I versus the number of terms n

159

3.6. NUMERICAL EXAMPLES

0.00 GM, PGM, GM/WF

n= 1 n= 2 n =3-6

Solution φn

-0.01

-0.02

-0.03

-0.04

-0.05 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.2: Example 3.9: Solution φn versus distance x

Remarks. (1) Residual functional I decreases by one order of magnitude when n is increased from one to two, clearly confirming the much improved approximation φn for n = 2. (2) For both choices of N2 (x), identical values of I confirm identical φn ∀x ∈ Ω = (0, 1). Hence, either choice 1 or choice 2 can be used for higher values of n. We consider choice 1 and compute solutions for values of n up to 9. (3) With progressively increasing n, we expect to obtain progressively reduced I values which would indicate progressively improved approximation φn . (4) Figure 3.1 shows a plot of residual functional I versus the number of terms n. I reduces from O(10−1 ) to O(10−16 ) for n = 1 to n = 9. Almost constant slope of the I versus n for n up to 8 indicates no change in the convergence rate of I with increasing n. (5) Figure 3.2 shows plots of φn versus x for n = 1, . . . , 6. For n > 3, the solution φn agrees quite well with the theoretical solution. (6) From (3.476) we note that the coefficient matrix [K] is not symmetric. However, for both of these choices of Ni (x), calculated [K] is symmetric as evident for n = 2 (choices 1 and 2).

160

CLASSICAL METHODS OF APPROXIMATION

The Petrov–Galerkin method In this method we have B(φ, w) = l(w)

(3.490)

in which B(φ, w) =

 d2 φ

+ φ, w



dx2 l(w) = (x2 , w)

(3.491)

where w 6= δφ. Here, w must be zero where φ is specified and, hence, we can ˜j (x) (j = 1, 2, . . . , n). We approximate φ by φn given by choose w = N φn = N0 (x) +

n X

Ci Ni (x)

(3.492)

i=1

Clearly, φn must satisfy all BCs of the BVP. Thus, the N0 (x) and Ni (x) established in the Galerkin method are applicable here as well. Secondly, the choice of w = Nj (x) (j = 1, 2, . . . , n) is admissible as well. With the choice of PGM and WRM are the same as the Galerkin method. For the choice ˜j , different than Nj (x), the PGM or WRM will obviously yield of w = N different φn compared to GM. However, the superiority of one choice over the other can be established by examining I = (E, E) = (Aφn − f, Aφn − f ) resulting from the different choices. Results shown in Figs. 3.1 and 3.2 are for w = Nj (x) (j = 1, 2, . . . , n), same as in case of the Galerkin method. The Galerkin method with weak form In this method we have B(φ, v) = l(v)

(3.493)

in which B(φ, v) =

Z   dφ dv − φv dΩ dx dx

(3.494)

¯ Ω

l(v) = −(x2 , v)

(3.495)

where v = δφ. We choose φn = N0 (x) +

N X i=1

Ci Ni (x),

v = Nj (x), j = 1, 2, . . . , n

(3.496)

161

3.6. NUMERICAL EXAMPLES

Substituting (3.496) into (3.493) n Z1   X dNi dNj − Ni Nj Ci dx dx dx i=1 0

Z1 =−

2

x Nj dx − 0

Z1 

 dN0 dNj − N0 Nj Ci dx, j = 1, 2, . . . , n (3.497) dx dx

0

Equation (3.497) can be written in the matrix form as [K]{C} = {F }

(3.498)

in which Kij of [K] and Fi of {F } are given by Kij =

Z1 

 dNj dNi − Nj Ni dx, i, j = 1, 2, . . . , n dx dx

(3.499)

0

Z1 Fi = −

2

x Ni dx − 0

Z1 

 dN0 dNi − N0 Ni dx, i = 1, 2, . . . , n dx dx

(3.500)

0

Clearly, Kij = Kji and, therefore, [K] is symmetric (a consequence of VC integral form). The functions N0 (x) and Ni (x) determined in the Galerkin method are applicable here also as both BCs are EBCs with v = Nj (x) (j = 1, 2, . . . , n). One parameter approximation: N0 (x) = 0 N1 (x) = x(1 − x),

v = N1 (x)

The computed [K], {F } and {C} are [K] = [0.3],

{F } = {−0.05},

{C} = {−0.166667}

Hence, the approximation φn becomes φn = (−0.166667)x(1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.8796296 × 10−1 Two parameter approximation: Here, we choose N0 (x) = 0 N1 (x) = x(1 − x) N2 (x) = x(1 − x2 ) : choice 1 N2 (x) = x2 (1 − x) : choice 2

162

CLASSICAL METHODS OF APPROXIMATION

with v = Nj (x) (j = 1, 2). Choice 1:



 0.30 0.4500 [K] = , 0.45 0.7328

and

 {C} =

 {F } =

 −0.0500 −0.0833

 0.08943 −0.17073

Hence φn = (0.089431)x(1 − x) + (−0.17073)x(1 − x2 ) I = (E, E) = (Aφn − f, Aφn − f ) = 0.4644502 × 10−2 Choice 2:

 [K] =

 0.30 0.1500 , 0.15 0.1238

and

 {C} =

{F } =

  −0.0500 −0.0333

 −0.08130 −0.17073

Hence φn = (−0.0813)x(1 − x) + (−0.17073)x2 (1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.4644501 × 10−2 Remarks. (1) Residual functional I decreases by one order of magnitude when n is increased from one to two, clearly confirming the much improved approximation φn for n = 2. (2) For both choices of N2 (x), identical values of I confirm identical φn ∀x ∈ Ω = (0, 1) for both choices (can be seen in the graphs in Figs. 3.2 and 3.1). Hence, for higher values of n we can use either choice. We consider choice 1 and compute solutions for up to n = 9. (3) With progressively increasing n, we obtain progressively reduced I values indicating progressively improved approximation φn . (4) A comparison of [K] and {f } in this method with the Galerkin method clearly confirms differences. The functional I for n = 1 and n = 2 matches exactly with the Galerkin method for n = 1 and n = 2, confirming that φn in both methods of approximation for n = 1 and n = 2 are identical. This is also confirmed by identical values of Ci in this method and the Galerkin method for n = 1 and n = 2. (5) Plots of the solution φn versus x and the residual functional I versus n are shown in Figs. 3.2 and 3.1.

163

3.6. NUMERICAL EXAMPLES

The least-squares method In this method we have B(φ, v) = l(v)

(3.501)

with B(φ, v) = (Aφ, Av) =

 d2 φ

dx2 2 l(v) = (f, v) = (x , v)

+ φ,

 d2 v + v dx2

(3.502) (3.503)

with v = δφ. We choose φn = N0 (x) +

N X

Ci Ni (x),

v = Nj (x), j = 1, 2, . . . , n

(3.504)

i=1

Substituting φn from (3.504) into (3.501), we obtain n  2 X d Ni i=1

dx2

+ Ni ,

 d2 Nj + N Ci dx j dx2

 d2 N  2 0 2 d Nj − N + x , + N = − 0 j , j = 1, 2, . . . , n (3.505) dx2 dx2 Equation (3.505) can be written in matrix form as [K] = {C} = {F }

(3.506)

in which Kij of [K] and Fi of {F } are given by  d2 N

 d2 Ni + N i , i, j = 1, 2, . . . , n dx2  d2 N  2 0 2 d Ni Fi = − − N + x , + N 0 i , i = 1, 2, . . . , n dx2 dx2

Kij =

j dx2

+ Nj ,

(3.507) (3.508)

Clearly, Kij = Kji , therefore, [K] is symmetric (a consequence of the VC integral form in the least-squares method). The functions N0 (x) and Ni (x) determined in the Galerkin method are applicable here also with v = Nj (x) (j = 1, 2, . . . , n). One parameter approximation: N0 (x) = 0 N1 (x) = x(1 − x),

v = N1 (x)

The computed [K], {F } and {C} are [K] = [3.3667],

{F } = {−0.61667},

{C} = {−0.18317}

164

CLASSICAL METHODS OF APPROXIMATION

Hence, the approximation φn becomes φn = (−0.18317)x(1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.8704621 × 10−1 Two parameter approximation: Here, we choose N0 (x) = 0 N1 (x) = x(1 − x) N2 (x) = x(1 − x2 ), choice 1 N2 (x) = x2 (1 − x), choice 2 with v = Nj (x) (j = 1, 2). Choice 1:

 3.3667 5.050 , [K] = 5.0500 10.476





and

 {C} =

{F } =

 −0.6167 −1.4167

 0.07104 −0.16947

Hence φn = (0.017038)x(1 − x) + (−0.16947)x(1 − x2 ) I = (E, E) = (Aφn − f, Aφn − f ) = 0.37231298 × 10−2 Choice 2:

 3.3667 1.6833 , [K] = 1.6833 3.7429





and

 {C} =

{F } =

 −0.6167 −0.8000

 −0.09843 −0.16947

Hence φn = (−0.098433)x(1 − x) + (−0.16947)x2 (1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.37231298 × 10−2 Remarks. (1) Residual functional I decreases by one order of magnitude when n is increased from one to two, clearly confirming the much improved approximation φn for n = 2. (2) For both choices of N2 (x), identical values of I confirm identical φn ∀x ∈ Ω = (0, 1) for both choices. For higher values of n it suffices to consider choice 1.

165

3.6. NUMERICAL EXAMPLES

(3) With progressively increasing n, we obtain progressively reduced I values indicating progressively improved approximation φn . (4) For both n = 1 and n = 2, we note that least-squares method yields lowest values of I compared to all other methods of approximation. This is of course no surprise due to the fact that least-squares processes are based on minimization of I, whereas other methods of approximation are not. The significance of this aspect of least-squares processes is that if our objective is to satisfy the GDEs as accurately as we desire (by ¯ then a φn from least-squares method gives us the increasing n) in Ω, best solution compared to all other methods of approximation. (5) A plot of the residual functional I versus n is shown in Fig. 3.1. We confirm that least-squares method yields the lowest value of I for each n compared to all other methods of approximation. (6) Graphs of solution φn versus x for different values of n are shown in Fig. 3.3. Beyond n = 3, the approximation φn agrees well with the theoretical solution φ. 0.00 n= 1 n= 2 n =3-6

LSM

Solution φn

-0.01

-0.02

-0.03

-0.04

-0.05 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.3: Example 3.9, LSM: Solution φn versus distance x

Example 3.10. Consider the 1D diffusion equation −

d2 φ − φ + x2 = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 dx2

with φ(0) = 0

and

dφ =1 dx x=1

(3.509)

(3.510)

166

CLASSICAL METHODS OF APPROXIMATION

This boundary value problem is exactly the same as in example 17 except that at x = 1 we have a Neumann boundary condition as opposed to Dirichlet boundary condition in example 17. The main purpose of this example is to illustrate that a simple change in boundary conditions has a significant influence on the choice of functions N0 (x) and Ni (x) (i = 1, 2, . . . , n). The Galerkin method The integral form remains the same as in the previous example. B(φ, v) = l(v)

(3.511)

where B(φ, v) =

 d2 φ

+ φ, v



dx2 l(v) = (x2 , v)

Let φn = N0 (x) +

n X

(3.512) (3.513)

Ci Ni (x)

(3.514)

v = δφn = Nj (x), 1, . . . , n

(3.515)

x=1

and Substituting (3.514) and (3.515) into (3.511) and rearranging terms, we obtain n  2 X d Ni

  d2 N  0 2 + N , N C = (x , N ) − + N , N , j = 1, 2, . . . , n i j i j 0 j dx2 dx2 i=1 (3.516) Equations (3.516) can be written in matrix form as [K]{C} = {F }

(3.517)

in which Kij of [K] and Fi of {F } are given by  d2 N

 + Nj , Ni ; i, j = 1, 2, . . . , n  d2 N  0 Fi = (x2 , Ni ) − + N , N , i = 1, 2, . . . , n 0 i dx2

Kij =

j 2 dx

(3.518) (3.519)

Obviously, Kij 6= Kji , i.e. [K] is not symmetric. This is a consequence of the VIC integral.

167

3.6. NUMERICAL EXAMPLES

The function N0 (x) must satisfy all of the boundary conditions of the dφn dN0 = 1 as φn (0) = 0 and dx = 1 must hold. BVP, N0 (0) = 0, and dx x=1 x=1 Thus, the choice of N0 (x) = x (3.520) is admissible. Next, each Ni (x) must satisfy the homogeneous form of all boundary conditions. Therefore Ni (0) = 0; i = 1, 2, . . . , n dNi = 0, i = 1, 2, . . . , n dx

(3.521)

x=1

Following the procedure described in the previous example, we find that if we choose N1 (x) = a0 + a1 x (3.522) then both a0 and a1 are zero due to the fact that the two boundary conditions to be used to evaluate a0 and a1 in (3.521) are homogeneous. If we choose N1 (x) = a0 + a1 x + a2 x2

(3.523)

then following the previous example and by using (3.521) we obtain N1 (x) = x(2 − x)

(3.524)

For N2 (x) we have two possible choices N2 (x) = a0 + a1 x + a3 x3

(3.525)

1 N2 (x) = x(1 − x2 ) 3

(3.526)

N2 (x) = a0 + a2 x2 + a3 x3

(3.527)

which gives

and which gives 2 N2 (x) = x2 (1 − x) 3 Using (3.524), (3.526), (3.528) and (3.520) we can write

(3.528)

N0 (x) = x

(3.529)

N1 (x) = x(2 − x)  i − 1 i Ni (x) = x 1 − x , i = 2, 3, . . . , n i+1   i Ni (x) = xi 1 − x , i = 2, 3, . . . , n i+1

(3.530) choice I

(3.531)

choice II

(3.532)

168

CLASSICAL METHODS OF APPROXIMATION

We can use either of the expressions for Ni (x). We choose the first choice of Ni (x) given by (3.531) in the numerical studies. We note that as n → ∞, (3.531) and (3.532) both yields algebraic polynomials of degree infinity. One parameter approximation: Using N0 (x) = x N1 (x) = x(2 − x), v = N1 (x) The computed [K], {F } and {C} are [K] = [−0.80],

{F } = {0.11667},

{C} = {−0.14583}

Hence φn = (−0.14583)x(2 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.9837963 × 10−2 Two parameter approximation: Here, we choose N0 (x) = x N1 (x) = x(2 − x) 1 N2 (x) = x(1 − x2 ), choice I (3.531) 3 with v = Nj (x) (j = 1, 2). The computed [K], {F } and {C} are  −0.8000 0.1889 , [K] = 0.1889 −0.9206 × 10−1





{F } =

0.11667 −0.27778 × 10−1



and  {C} =

−0.14468 0.48769 × 10−2



Hence 1 φn = x + (−0.14468)x(2 − x) + (0.48769 × 10−2 )x(1 − x2 ) 3 −2 I = (E, E) = (Aφn − f, Aφn − f ) = 0.9272824 × 10 Figure 3.4 shows plots of the solution φn versus x for different values of n. It appears that for all values of n the computed solution φn is in good agreement with the theoretical solution beyond n = 5. This can also be confirmed by the graph of I versus n shown in Fig. 3.5. I is of the order of 10−10 for n = 5 conforms extremely good accuracy of φn .

169

3.6. NUMERICAL EXAMPLES

1.20 n = 1-6 GM 1.00

Solution φn

0.80

0.60

0.40

0.20

0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.4: Example 3.10, GM: Solution φn versus distance x

1e+00 GM GM/WF PGM LSP

Residual functional I

1e-05

1e-10

1e-15

1e-20

1e-25 1

2

3

4 5 6 Number of terms n

7

8

9

Figure 3.5: Example 3.10: Residual functional I versus number of terms n

The Petrov–Galerkin method The choice of N0 (x) and Ni (x) remain the same as in the Galerkin ˜j (j = 1, 2, . . . , n) is the method. However, we may choose v = w = N weight function, independent of the basis functions. Based on fundamental Lemma, w must be zero where φ is specified, hence the only requirement in

170

CLASSICAL METHODS OF APPROXIMATION

˜j (x) (j = 1, . . . , n) is that the choice of w = N ˜j (0) = 0, j = 1, 2, . . . , n N

(3.533)

˜j (x) = xj , j = 1, 2, . . . , n N

(3.534)

Thus is an admissible choice. The details of the integral forms are as follows. B(φ, w) = l(w)

(3.535)

in which B(φ, w) =

 d2 φ

+ φ, w



dx2 l(w) = (x2 , w) ˜j (x) = xj , j = 1, 2, . . . , n w 6= δφ = N n X φn = N0 (x) + Ci Ni (x)

(3.536) (3.537) (3.538) (3.539)

i=1

Substituting (3.538) and (3.539) into (3.536) and (3.537) we obtain n  2 X d Ni

  2  ˜j Ci = (x2 , N ˜j ) − d N0 + N0 , N ˜j , j = 1, 2, . . . , n + N , N i dx2 dx2 i=1 (3.540) Equations (3.540) can be written in matrix form [K] = {C} = {F }

(3.541)

in which Kij of [K] and Fi of {F } are given by  d2 N

 ˜i ; i, j = 1, 2, . . . , n + Nj , N  2  ˜i ) − d N0 + N0 , N ˜i , i = 1, 2, . . . , n Fi = (x2 , N dx2

Kij =

j dx2

One parameter approximation: N0 (x) = x N1 (x) = x(2 − x),

˜1 (x) = x N

The computed [K], {F } and {C} are [K] = [0.58333],

{F } = {−0.083333},

{C} = {−0.14286}

(3.542) (3.543)

171

3.6. NUMERICAL EXAMPLES

Hence, the approximation φn becomes φn = x + (−0.14286)x(2 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.95238095 × 10−2 Two parameter approximation: Here, we choose N0 (x) = x N1 (x) = x(2 − x) 1 N2 (x) = x(1 − x2 ), choice I (3.531) 3 ˜j (x) (j = 1, 2), N ˜1 (x) = x, and N ˜2 (x) = x2 . The computed [K], with w = N {F }, and {C} are  0.58333 −0.21667 , [K] = 0.36667 −0.24444 

  −0.8333 × 10−1 {F } = −0.0500

and  {C} =

−0.15103 −0.21994 × 10−1



Hence 1 φn = (−0.15103)x(2 − x) + (−0.21994 × 10−1 )x(1 − x2 ) 3 I = (E, E) = (Aφn − f, Aφn − f ) = 0.1324869 × 10−1 Plots of φn versus x for different values of n are shown in Fig. 3.6. Here also, all values of n produce reasonable approximations. Beyond n = 5, values of I of the order of 10−10 or lower (Fig. 3.5) confirm extremely good accuracy of φn . The Galerkin method with weak form In this method we have B(φ, v) = l(v)

(3.544)

in which Z B(φ, v) =

(

dφ dv + φv) dΩ dx dx

(3.545)

¯ Ω

l(v) = −(x2 , v) + v|1

(3.546)

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CLASSICAL METHODS OF APPROXIMATION

1.20 n = 1-6 PGM 1.00

Solution φn

0.80

0.60

0.40

0.20

0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.6: Example 3.10, PGM: Solution φn versus distance x

where v = δφ. We choose φn = N0 (x) +

N X

Ci Ni (x),

v = Nj (x); j = 1, 2, . . . , n

(3.547)

i=1

Substituting (3.547) into (3.544) n Z1   X dNi dNj −Ni Nj Ci dx = −(x2 , Nj )+Nj (1), j = 1, 2, . . . , n (3.548) dx dx i=1 0

Equation (3.548) can be written in matrix form as [K]{C} = {F }

(3.549)

in which Kij of [K] and Fi of {F } are given by Kij =

Z1 

 dNj dNi − Nj Ni dx dx dx

(3.550)

0

Z1 Fi = −

x2 Ni dx + Ni (1)

(3.551)

0

Clearly, Kij = Kji and, therefore, [K] is symmetric (a consequence of VC integral form).

173

3.6. NUMERICAL EXAMPLES

In this method the boundary condition



dφ dx x=1

= 1 is absorbed in the

weak form , hence we only have φ(0) = 0 left to be satisfied by φn . Since φ(0) = 0 is homogeneous, we have N0 (x) = 0 and the choice Ni (x) = xi (i = 1, 2, . . . , n) is admissible. Thus, in this method we have N0 (x) = 0

(3.552) i

Ni (x) = x , i = 1, 2, . . . , n

(3.553)

One parameter approximation: N0 (x) = 0 Ni (x) = xi , v = Nj (x), i, j = 1, 2, . . . , n The computed [K], {F } and {C} are [K] = [0.66667],

{F } = {0.75000},

{C} = {1.1250}

Hence, the approximation φn becomes φn = (1.1250)x I = (E, E) = (Aφn − f, Aφn − f ) = 0.59375 × 10−1 Two parameter approximation: Here, we choose N0 (x) = 0 N1 (x) = x N2 (x) = x2 with v = Nj (x) (j = 1, 2). The computed [K], {F }, and {C} are     0.6667 0.7500 0.75000 , {F } = [K] = 0.7500 1.1333 0.80000 and

 {C} =

 1.2950 −0.15108

Hence φn = (1.2950)x + (−0.15108)x2 I = (E, E) = (Aφn − f, Aφn − f ) = 0.1055846 × 10−1 Figure 3.7 shows plots of φn versus x for different values of n. For n > 2, the computed solutions are quite accurate. This can be confirmed by I versus n graph shown in Fig. 3.5.

174

CLASSICAL METHODS OF APPROXIMATION

1.20 n= 1 n =2-6

GM/WF

1.00

Solution φn

0.80

0.60

0.40

0.20

0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.7: Example 3.10, GM/WF: Solution φn versus distance x

The least-squares method In this method we have B(φ, v) = l(v)

(3.554)

with B(φ, v) = (Aφ, Av) =

 d2 φ

dx2 2 l(v) = (f, v) = (x , v)

+ φ,

 d2 v + v dx2

(3.555) (3.556)

with v = δφ. We choose φn = N0 (x) +

N X

Ci Ni (x),

v = Nj (x), j = 1, 2, . . . , n

(3.557)

i=1

Substituting φn from (3.557) into (3.554), we obtain n  2 X d Ni i=1

dx2

+ Ni ,

 d2 Nj + N Ci dx j dx2

  d2 N 2 0 2 d Nj = − − N + x , + N , j = 1, 2, . . . , n (3.558) 0 j dx2 dx2 Equation (3.558) can be written in matrix form as [K] = {C} = {F }

(3.559)

175

3.6. NUMERICAL EXAMPLES

in which Kij of [K] and Fi of {f } are given by  d2 N

 d2 Ni + N i , i, j = 1, 2, . . . , n dx2  d2 N  2 0 2 d Ni Fi = − − N + x , + N 0 i , i = 1, 2, . . . , n dx2 dx2

Kij =

j dx2

+ Nj ,

(3.560) (3.561)

Clearly, Kij = Kji , therefore, [K] is symmetric (a consequence of the VC integral form in the least-squares method). The choice of N0 (x) and Ni (x) (i = 1, 2, . . . , n) in this method is obviously the same as in the Galerkin method due to same required BCs. Thus, we can choose N0 (x) = x N1 (x) = x(2 − x) i − 1 i
Ni (x) = x 1 − x , i = 2, . . . , n i+1

(3.562) choice I (see GM)

One parameter approximation: N0 (x) = x N1 (x) = x(2 − x),

v = N1 (x)

The computed [K], {F } and {C} are [K] = [1.8667],

{F } = {−0.21667},

{C} = {−0.11607}.

Hence, the approximation φn becomes φn = x + (−0.11607)x(2 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.818145238 × 10−2 Two parameter approximation: Here, we choose N0 (x) = x N1 (x) = x(2 − x) 1 N2 (x) = x(1 − x2 ) 3 with v = Nj (x) (j = 1, 2). The computed [K], {F } and {C} are 

 1.8667 0.5222 [K] = , 0.5222 1.1079

 {F } =

 −0.21667 −0.02778

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CLASSICAL METHODS OF APPROXIMATION

and

 {C} =

−0.12562 0.34140 × 10−1



Hence 1 φn = x + (−0.12565)x(2 − x) + (0.34140 × 10−1 )x(1 − x2 ) 3 I = (E, E) = (Aφn − f, Aφn − f ) = 0.70634539 × 10−2 Graphs of φn versus x for different values of n are shown in Fig. 3.8. I versus n is shown in Fig. 3.5. We note that I is lowest for least-squares method for all values of n. This is of course no surprise as least-squares processes are based on minimization of I, hence will produce the lowest I compared to all other methods of approximation. 1.20 n= 1 n= 2 n =3-6

1.00

LSM (LSP)

Solution φn

0.80

0.60

0.40

0.20

0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.8: Example 3.10, LSM: Solution φn versus distance x

Example 3.11. Consider the 1D beam equation d2  d2 φ  EI 2 − q = 0, ∀x ∈ Ω = (0, L) ⊂ R1 dx2 dx

(3.563)

with φ(0) = 0, 

dφ =0 dx x=0

d2 φ  EI 2 = ML , dx x=L

(3.564) d  d2 φ  EI 2 = FL dx dx x=L

(3.565)

177

3.6. NUMERICAL EXAMPLES

The Galerkin method with weak form The operator is self-adjoint provided ML = 0 and FL = 0. In the following, we consider an approximate solution φn using the Galerkin method with weak form. As shown earlier, the integral form (weak form) in this case is variationally consistent. The weak form can be written as B(φ, v) = l(v)

(3.566)

d2 v d2 φ dΩ dx2 dx2

(3.567)

in which ZL B(φ, v) =

EI 0 ZL

l(v) =

 dv  vq dx − v(L)FL + dx

0

ML

(3.568)

x=L

Let φn be an approximation of φ given by φn = N0 (x) +

N X

Ci Ni (x),

v = Nj (x), j = 1, 2, . . . , n

(3.569)

i=1

Substituting (3.569) into (3.566), (3.567) and (3.568) yields ZL n X i=1

! ZL d2 Nj d2 N0 d2 Ni d2 Nj EI dx C = − EI dx i dx2 dx2 dx2 dx2

0

0

ZL +

 dN  j Nj (x)q dx − Nj (L)FL + dx

0

ML , j = 1, 2, . . . , n (3.570) x=L

Equation (3.570) can be written in matrix form as [K]{C} = {F }

(3.571)

in which Kij of [K] and Fi of {F } are given by ZL Kij =

EI

d2 Nj d2 Ni dx, i, j = 1, 2, . . . , n dx2 dx2

(3.572)

0

ZL Fi = − 0

d2 Ni d2 N0 EI dx + dx2 dx2

ZL 0

 dN  j Ni q dx − Ni (L)FL + dx

ML x=L

(3.573)

178

CLASSICAL METHODS OF APPROXIMATION

Clearly, Kij = Kji and, therefore, [K] is symmetric (a consequence of VC integral form). First, we note that boundary conditions (3.564) are essential boundary conditions whereas (3.565) are natural boundary conditions. Hence, the boundary conditions in (3.565) are absorbed in the weak form (3.566) and do not need to be considered in determining N0 (x) and Ni (x). Secondly, the essential boundary conditions in (3.564) are homogeneous, hence N0 (x) = 0

(3.574)

and Ni (x) must be chosen such that Ni (0) = 0  dN  i =0 dx

(3.575) (3.576)

x=0

Therefore Ni (x) = xi+1 , i = 1, 2, . . . , n

(3.577)

are admissible. For numerical studies, we choose L = 1, EI = 1, q = 1.0, FL = −1.0, and ML = −1.0. One parameter approximation: N0 (x) = 0 N1 (x) = x2 , v = N1 (x) The computed [K], {F } and {C} are [K] = [4.0],

{F } = {−0.66667},

{C} = {−0.16667}

Hence, the approximation φn becomes φn = (−0.16667)x2 I = (E, E) = (Aφn − f, Aφn − f ), meaningless in this case Two parameter approximation: Here, we choose N0 (x) = 0,

N1 (x) = x2 ,

N2 (x) = x3

with v = Nj (x) (j = 1, 2). The computed [K], {F } and {C} are   4.0 6.0 [K] = , 6.0 12.0

 {F } =

−0.66667 −0.17500 × 101



179

3.6. NUMERICAL EXAMPLES

and

 {C} =

 0.20833 −0.25000

Hence φn = (0.20833)x2 + (−0.25000)x3 I = (E, E) = (Aφn − f, Aφn − f ), meaningless in this case Three parameter approximation: Here, we choose N0 (x) = 0, N1 (x) = x2 , N2 (x) = x3 , N3 (x) = x4 with v = Nj (x) (j = 1, 2, 3). The computed [K], {F }, and {C} are     4.0 6.0 8.0  −0.66667  [K] = 6.0 12.0 18.0 , {F } = −0.17500 × 101   8.0 18.0 28.8 −0.28000 × 101 and

 

 0.25000  −0.33333 {C} =   0.41667 × 10−1

Hence φn = (0.2500)x2 + (−0.33333)x3 + (0.416667 × 10−1 )x4 I = (E, E) = (Aφn − f, Aφn − f ) = 0.2839899 × 10−26 Remarks. (1) For n = 1, 2, φn is not admissible in the BVP and, hence, computation of I is meaningless. (2) When n = 3, I = O(10−26 ) indicating that n = 3 corresponds to the theoretical solution of this BVP (this can be verified by integrating (3.563) and using (3.564) and (3.565) to find constants of integration). (3) Graphs of φn versus x for n = 1, 2, 3 are shown in Fig. 3.9. Example 3.12. Consider the 1D convection-diffusion equation 1 d2 φ dφ − = 0 ∀x ∈ Ω = (0, 1) ⊂ R1 dx P e dx2

(3.578)

φ(0) = 1 and φ(1) = 0

(3.579)

with The operator is non-self-adjoint, hence all methods of approximation except least-squares processes yield VIC integral forms. In the following, we consider the Galerkin method with weak form and least-squares method.

180

CLASSICAL METHODS OF APPROXIMATION

n=1 n=2 n=3

0.04 0.02 0.00

Solution φn

-0.02 -0.04

GM/WF

-0.06 -0.08 -0.10 -0.12 -0.14 -0.16 -0.18 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.9: Example 3.11: Solution φn versus distance x

The Galerkin method with weak form The weak form is given by B(φ, v) = l(v)

(3.580)

in which Z1  B(φ, v) =

1 dφ dv dφ v+ dx P e dx dx

 dΩ

(3.581)

0

l(v) = 0

(3.582)

Let φn be an approximation of φ given by φn = N0 (x) +

N X

Ci Ni (x), v = Nj (x), j = 1, 2, . . . , n

(3.583)

i=1

Substituting (3.583) into (3.580) ! Z1  n X dNi 1 dNi dNj  Nj + dx Ci dx P e dx dx i=1

0

=−

Z1  0

dN0 1 dN0 dNj  Nj + dx, j = 1, 2, . . . , n (3.584) dx P e dx dx

181

3.6. NUMERICAL EXAMPLES

Equation (3.584) can be written in matrix form as [K]{C} = {F }

(3.585)

in which Kij of [K] and Fi of {F } are given by

Kij =

Z1 

dNj 1 dNj dNi  Ni + dx, i = 1, 2, . . . , n, j = 1, 2, . . . , n dx P e dx dx

0

(3.586) Fi = −

Z1 

1 dN0 dNi  dN0 Ni + dx dx P e dx dx

(3.587)

0

Clearly, Kij 6= Kji and, therefore, [K] is non-symmetric (a consequence of VIC integral form). In the numerical studies we choose P e = 10. Since both boundary conditions are essential, φn (x) must satisfy both of them. Let N0 (x) be such that N0 (0) = 1 and N0 (1) = 0, hence the choice N0 (x) = 1 − x

(3.588)

is admissible. We choose Ni (x) such that Ni (0) = 0 and Ni (1) = 0

(3.589)

Therefore, we can choose Ni (x) = xi (1 − x), i = 1, 2, . . . , n One parameter approximation: N0 (x) = 1 − x N1 (x) = x(1 − x),

v = N1 (x)

The computed [K], {F } and {C} are [K] = [0.33333 × 10−1 ], {F } = {0.16667}, {C} = {5.0} Hence, the approximation φn becomes φn = (1 − x) + (5.0)x(1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.833333 × 101

(3.590)

182

CLASSICAL METHODS OF APPROXIMATION

Two parameter approximation: Here, we choose N0 (x) = 1 − x N1 (x) = x(1 − x) N2 (x) = x2 (1 − x) with v = Nj (x); j = 1, 2. The computed [K], {F }, and {C} are   0.033333 0.066667 [K] = , 0.033333 0.080000

 {F } =

0.16667 0.25000



and  {C} =

−0.75000 × 101 0.62500 × 101



Hence φn = (1 − x) + (−0.75 × 101 )x(1 − x) + (0.625 × 101 )x2 (1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.234375 × 101 The three, five, ten, and higher term approximate solutions are calculated as well. Figure 3.10 shows a plot of the solution φn versus x for different values of n. For n > 10, the computed solution is quite accurate. This can be confirmed by the I versus n graph shown in Fig. 3.11. 1.80 n= 1 n= 2 n= 3 n =5,10

1.60 1.40

Solution φn

1.20 1.00 0.80 0.60

GM/WF

0.40 0.20 0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.10: Example 3.12, GM/WF: Solution φn versus distance x

183

3.6. NUMERICAL EXAMPLES

1e+01 1e+00

Residual functional I

1e-01 1e-02 1e-03 1e-04 GM/WF 1e-05 1e-06 1e-07 1

2

3

4

5 6 7 Number of terms n

8

9

10

Figure 3.11: Example 3.12, GM/WF: Residual functional I versus n

The least-squares method The integral form is given by B(φ, v) = l(v)

(3.591)

with B(φ, v) =

 dφ dx

−

1 d2 φ dv 1 d2 v  , − P e dx2 dx P e dx2

l(v) = 0

(3.592) (3.593)

with v = δφ. We choose φn = N0 (x) +

N X

Ci Ni (x),

v = Nj (x), j = 1, 2, . . . , n

(3.594)

i=1

Substituting φn from (3.594) into (3.591), we obtain n  X dNi i=1

1 d2 Ni dNj 1 d2 Nj  − , − Ci = dx P e dx2 dx P e dx2  dN 1 d2 N0 dNj 1 d2 Nj  0 − + , − , j = 1, 2, . . . , n (3.595) dx P e dx2 dx P e dx2

Equations (3.595) can be written in matrix form as [K]{C} = {F }

(3.596)

184

CLASSICAL METHODS OF APPROXIMATION

in which Kij of [K] and Fi of {F } are given by 1 d2 Nj dNi 1 d2 Ni  , − , i = 1, 2, . . . , n; j = 1, 2, . . . , n dx P e dx2 dx P e dx2 (3.597)   dN 2 2 1 d N0 dNi 1 d Ni 0 , , i = 1, 2, . . . , n (3.598) + − Fi = − 2 dx P e dx dx P e dx2

Kij =

 dN

j

−

Clearly, Kij = Kji , therefore, [K] is symmetric (a consequence of the VC integral form in the least-squares method). The choice of N0 (x) and Ni (x) (i = 1, 2, . . . , n) in the Galerkin method holds here as well due to same required BCs. Therefore N0 (x) = 1 − x Ni (x) = xi (1 − x), i = 1, 2, . . . , n One parameter approximation: N0 (x) = 1 − x N1 (x) = x(1 − x), v = N1 (x) The computed [K], {F } and {C} are [K] = [0.37333],

{F } = {0.20000},

{C} = {0.53571}.

Hence, the approximation φn becomes φn = (1 − x) + (0.53571)x(1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.89285714 Two parameter approximation: Here, we choose N0 (x) = 1 − x N1 (x) = x(1 − x) N2 (x) = x2 (1 − x) with v = Nj (x) (j = 1, 2). The computed [K], {F }, and {C} are 

 0.37333 0.46000 [K] = , 0.46000 0.62000

 {F } =

and  {C} =

 −0.70470 1.0067

 0.20000 0.30000

(3.599)

185

3.7. SUMMARY

Hence φn = (1 − x) + (−0.70470)x(1 − x) + (1.0067)x2 (1 − x) I = (E, E) = (Aφn − f, Aφn − f ) = 0.83892617 The three, five, ten and higher term approximations are computed for this case as well. Graphs of φn versus x for different values of n are shown in Fig. 3.12; I versus n is shown in Fig. 3.13. Once again, we note that I in leastsquares method is always lower compared to the Galerkin method with weak form. For P e = 10, the theoretical solution is sufficiently diffused and, hence, smooth. Even then, up to ten terms are required for reasonable accuracy. As P e increases, a sharp front develops near x = 1. Computation of solutions for such cases using these classical methods will become prohibitive. 1.20 n=1 n=2 n=3 n=5 n =10

1.00

Solution φn

0.80

0.60 LSM (LSP) 0.40

0.20

0.00 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 3.12: Example 3.12, LSM: Solution φn versus distance x

3.7 Summary In this chapter, we have considered classical integral methods of approximation: the Galerkin method, the Petrov–Galerkin method, the weightedresidual method, the Galerkin method with weak form, and the least-squares process for boundary value problems containing self-adjoint, non-self-adjoint, and non-linear differential operators. For each method of approximation, formulation details are presented for each of the three types of differential operators including the determination of variational consistency or inconsistency of the resulting integral forms. It has been shown and proven that

186

CLASSICAL METHODS OF APPROXIMATION

1e+01 GM/WF LSP

1e+00

Residual functional I

1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 1e-07 1

2

3

4

5 6 7 Number of terms n

8

9

10

Figure 3.13: Example 3.12, LSM: Error functional I versus n

(i) the integral forms resulting from the Galerkin method, and the Petrov– Galerkin method are always VIC, (ii) the integral form resulting from the Galerkin method with weak form is only variationally consistent when the differential operator is self-adjoint and when B(·, ·) in the weak form is symmetric, (iii) the least-squares processes always yields variationally consistent integral forms for self-adjoint and non-self-adjoint operators and (iv) leastsquares processes also yield VC integral forms for non-linear operators if in δ 2 I, δ 2 E is neglected and if the resulting non-linear algebraic equations from δI = 0 are solved using Newton’s linear method (or Newton–Raphson method). It has been shown that VC integral forms yield computational processes in which the coefficient matrices are symmetric and positive-definite, hence have real and positive eigenvalues and real bases i.e. real eigenvectors. Hence, the computations remain unconditionally stable and non-degenerate. The VIC integral forms, on the other hand, yield non-symmetric coefficient matrices which are not always ensured to be positive-definite. Such coefficient matrices may have partial or completely complex bases. Unconditional stability of computations in such cases cannot always be ensured. One must show on a problem-by-problem basis whether the computations remain stable. The LBB condition, Lax-Milgram theorem and Banach theorem are means of accomplishing this. A significant point to note here is that when the integral forms are VC, the LBB condition, Lax-Milgram theorem and Banach theorem are automatically satisfied.

187

3.7. SUMMARY

For each method of approximation, approximation spaces are discussed in conjunction with Riemann and Lebesgue measures. In section 3.5.1, a variety of boundary value problems are considered to present the details of various methods of approximation for each of the three types of differential operators. In section 3.6, specific numerical studies are presented to illustrate various details of approximation φn , computations of constants Ci in φn as well as the residual functional I as a measure of the accuracy of the computed solution. Form the material presented in section 3.6, it is rather clear that a major difficulty in the classical methods of approximation is the determination of N0 (x) and Ni (x). Since their determination requires satisfying some or all BCs of the BVP (depending upon the method of ap¯ is two or three dimensional proximation), when the domain of definition Ω and when the boundary conditions are not simple, the determination of N0 (x) and Ni (x) is extremely difficult if not impossible. This is perhaps the main reason that these methods can rarely be applied to boundary values problems of practical interest. We note that in the methods of approximation discussed in this chapter ¯ of the boundary value problem is not discretized the domain of definition Ω ¯ and hence must satisfy approand the approximation φn of φ is global over Ω priate boundary conditions of the BVP. We emphasize that the methods of approximation despite their lack of usefulness in practical applications, form the mathematical foundations of the finite element processes due to the fact that all fundamental principles of the finite element method are derived from these methods except two important aspects: (i) in finite element processes ¯ is discretized and (ii) for a subdomain of the discretization we have local Ω approximations that are based on interpolation theory and are independent of the boundary conditions of the boundary value problem, thus avoiding all of the difficulties associated with the determination of N0 (x) and Ni (x) in the classical methods of approximation.

Problems Consider boundary value problems 3.1 to 3.7. For each BVP construct and/or show the following [(a) and (b)] using non-discretized domain of definitions of the boundary value problems. (a) Construct an integral form using the Galerkin method. Show whether the integral form is VC or VIC. Discuss continuity requirements on the functions for the integrals in the integral form to be (i) in Riemann sense (ii) in Lebesgue sense. Define the associated spaces. (b) Construct an integral form using the Galerkin method with weak form. (i) Identify PVs, SVs, EBCs, and NBCs. (ii) Simplify the expressions resulting from integration by parts using BCs and express the final results in the following form: B(·, ·) = l(·)

188

CLASSICAL METHODS OF APPROXIMATION

(iii) Determine the nature of the functionals B(·, ·) and l(·), that is bilinearity, symmetry, linearity, etc. (iv) Discuss minimally conforming spaces for: (1) the mathematical model (2) the weak form (3) the BVP, i.e. the mathematical model, integral form, weak form, etc. for equivalency between them. 3.1 Consider one dimensional heat conduction equation. d  dT  − a + f = 0, 0 < x < 1 = Ω ⊂ R1 dx dx dT + h(T − T∞ ) = q at x = 1 T (0) = 0 ; a dx where a = a(x), f = f (x) are known functions and h, T∞ and q are constants. 3.2 Consider a beam on elastic foundation. d2  d2 w  b + kw + f = 0, 0 < x < L = Ω ⊂ R1 dx2 dx2 w = 0 at x = 0 and x = L b

d2 w = 0 at x = 0 and x = L dx2

b = b(x) and f = f (x) are known functions and k is a constant. 3.3 Consider longitudinal deformation of a bar with end spring, an eigenvalue problem. d  du  − a + λu = 0, 0 < x < L = Ω ⊂ R1 dx dx u(0) = 0  du  a + ku =0 dx x=L

a = a(x) is a known function and λ and k are constants. 3.4 Consider a second order BVP ∂u ∂u  ∂u ∂u  ∂  ∂  − a11 + a12 − a21 + a22 + f = 0 ∀x, y ∈ Ω ⊂ R2 ∂x ∂x ∂y ∂y ∂x ∂y u = u0 on Γ1   ∂u ∂u  ∂u ∂u  a11 + a12 nx + a21 + a22 ny = t0 on Γ2 ∂x ∂y ∂x ∂y where aij = aji , i = 1, 2; j = 1, 2 f = f (x, y)

∀x, y ∈ Ω

Γ = Γ 1 ∪ Γ2 u0 and t0 are known functions on Γ1 and Γ2 respectively and nx , ny are the direction cosines of a unit exterior normal to the boundary Γ2 . 3.5 Consider one dimensional steady state convection diffusion equation. dφ 1 d2 φ − = 0, 0 < x < 1 = Ω ⊂ R1 dx P e dx2 φ(0) = 1, φ(1) = 0

189

3.7. SUMMARY

P e > 0 is known data. 3.6 Consider one dimensional steady state convection Burgers equation, a nonlinear ordinary differential equation. dφ 1 d2 φ − = 0, 0 < x < 1 = Ω ⊂ R1 dx Re dx2 φ(0) = 1, φ(1) = 0 φ

Re > 0 and is known. 3.7 Consider the following non-linear ordinary differential equation. d  du  − u + f = 0, 0 < x < 1 = Ω ⊂ R1 dx dx √ du = 0, u(1) = 2 dx x=0 f = f (x) is a known function. 3.8 Consider the following BVP: d  du  − (1 + x) = 0, 0 < x < 1 = Ω ⊂ R1 dx dx u(0) = 0, u(1) = 1 Construct an integral form of the BVP using the Galerkin method with weak form. Approximate u by un , an n-parameter approximation using un = N0 (x) +

n X

Ci Ni (x)

i=1

(a) (b) (c) (d)

Provide details fo the integral form. Express it in the form B(·, ·) = l(·). Establish VC or VIC of the integral form. Establish general expressions for the coefficient matrix and the right hand side vector. Specialize your results for n = 2 and compute coefficients C1 and C2 using N0 (x) = x Ni (x) = xi (x − 1), i = 1, 2

3.9 Consider the following BVP, bending of simply supported beam subjected to uniform loading. d2  d2 w  EI 2 − f = 0 ; 0 < x < L = Ω ⊂ R1 dx2 dx d2 w w = EI 2 = 0 at x = 0 and x = L dx (a) Construct an integral form of the BVP using the Galerkin method with weak form. (b) Establish VC or VIC of the weak form. (c) Approximate w by wn n X wn = N0 (x) + Ci Ni (x) i=1

(d) Establish general expressions for the coefficient matrix and right hand side vector. (e) Specialize your solution for n = 2, a two parameter approximation using N0 (x) = 0, Ni (x) = xi (L − x), i = 1, 2

190

CLASSICAL METHODS OF APPROXIMATION

Compute coefficients C1 and C2 . 3.10 Consider the following BVP: d2 u  du 2 + = 4, 0 < x < 1 = Ω ⊂ R1 dx2 dx u(0) = 1, u(1) = 0 − 2u

(a) Approximate u by un un = N0 (x) +

n X

Ci Ni (x)

i=1

(b) Find coefficient C1 (i.e. for n = 1) using (i) the Galerkin method (ii) the least-squares method (iii) the Petrov–Galerkin method (or the weighted-residual method) with w1 = 1. Use N0 (x) = (1 − x), N1 (x) = x(1 − x) in all cases. 3.11 Consider the following eigenvalue problem. d  du  − (1 + x) = λu, 0 < x < 1 = Ω ⊂ R1 dx dx u(0) = u(1) = 0 (a) Construct an integral form of the BVP using the Galerkin method with weak form. (b) Establish VC or VIC of the weak form. (c) Approximate u by n X un = N0 (x) + Ci Ni (x) i=1

(d) Establish general expressions for the coefficient matrix and right hand side vector. (e) Specialize your solution for n = 2 and find eigenvalues λ1 and λ2 and the corresponding normalized eigenvectors. Use N0 (x) = 0, Nj (x) = xj (x − 1), j = 1, 2, . . . , n 3.12 Consider the following BVP −

∂2u ∂2u − =1 2 ∂x ∂y 2

∀x, y ∈ Ω ⊂ R2

The domain of definition Ω is a unit square (shown in the figure). u = 0 on Γ, closed boundary of the unit square. (a) Construct an integral form of the BVP using the Galerkin method with weak form. (b) Establish VC or VIC of the weak form. (c) Approximate u by un un = N0 (x, y) +

n X

Ci Ni (x, y)

i=1

(d) Establish general expressions for the coefficient matrix and right hand side vector. (e) Specialize your results for n = 1 using N0 (x, y) = 0

and

N1 (x, y) = sin πx sin πy

191

REFERENCES FOR ADDITIONAL READING

y

1

x 1

and calculate the coefficient C1 . 3.13 Consider the following BVP, an eigenvalue problem. d2 u − λu = 0, 0 < x < 1 = Ω ⊂ R1 dx2 u(0) = u(1) = 0 −

Use collocation method with collocation points at x=

1 , 4

x=

1 , 2

x=

3 4

and calculate first three eigenvalues of the problem using n = 3 and N0 (x) = 0, Ni (x) = xi (x − 1), i = 1, 2, 3 [5–20]

References for additional reading [1] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous petrovgalerkin methods. part I: The transport equation. Comp. Meth. in Applied Mech. and Eng., 199:1558–1572, 2010. [2] L. Demkowicz and J. Gopalakrishnan. A class of discontinuous petrovgalerkin methods. II. optimal test functions. Num. Meth. for Partial Diff. Equations, 27:70–105, 2011. [3] L. Demkowicz, J. Gopalakrishnan, and A. H. Niemi. A class of discontinuous petrovgalerkin methods. part III: Adaptivity. Applied Num. Math., 62:396–427, 2011. [4] J. Zitelli, I. Muga, L. Demkowicz, J. Gopalakrishnan, D. Pardo, and V.M. Calo. A class of discontinuous petrovgalerkin methods. part IV: The optimal test norm and time-harmonic wave propagation in 1d. Applied Num. Math., 230:2406–2432, 2011. [5] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering. McGraw Hill Company, 1986. [6] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000. [7] G. Hellwig. Differential Operators of Mathematical Physics. Addison-Wesley Publishing Co., 1967. [8] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964.

192

REFERENCES FOR ADDITIONAL READING

[9] G. Mikhlin. Numerical Performance of Variational Methods. Svoningen, 1971.

Woltes-Noordhoff,

[10] J. T. Oden and J. N. Reddy. An Introduction to the Mathematical Theory of Finite Elements. John Wiley, New York, 1976. [11] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., New York, 3rd edition, 2006. [12] K. Rektorys. Variational Methods in Mathematics, Science and Engineering. Reidel, 1977. [13] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002. [14] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [15] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004. [16] K. Washizu. Variational Methods in Elasticity and Plasticity. Pergamon Press, 3nd edition, 1982. [17] J. T. Oden and J. N. Reddy. Variational Methods in Theoretical Mechanics. John Wiley, New York, 2nd edition, 2002. [18] J. N. Reddy. Energy Principles and Variational Methods in Applied Mechanics. John Wiley, New York, (3rd in print) 2nd edition, 2002. [19] C. Lanczos. The variational principles of mechanics. University of Toronto Press, Toronto, 4th edition, 1970. [20] G. Strang and G. Fix. An Analysis of the Finite Element Method. Prentice Hall, New Jersey, 1973.

4

The Finite Element Method 4.1 Introduction The classical methods of approximation of the solutions of BVPs described in Chapter 3 are hampered primarily due to the fact that determination of the basis functions used in approximating φ of Aφ − f = 0 in Ω ¯ becomes increasingly difficult for practical for the non-discretized domain Ω problems of interest. The basis functions, apart from satisfying continuity, linear independence, completeness, and appropriate boundary conditions depending upon the method, there exists no systematic method for establishing them. Yet, the accuracy of the approximation is highly dependent upon them. Thus, classical methods of approximation are not regarded as practical and competitive for approximating the solutions of BVPs of practical interest. We remark on some important features that are essential for a computational method to be effective in practical applications. (1) The method must have a sound mathematical foundation which must result in a transparent computational infrastructure without further assumptions or approximations. (2) The method should be independent of the geometric description, that is, the domain of definition of the BVP, the nature of physical or transport properties, boundary conditions and the nature of external disturbances. (3) The method must address the following three classes of differential operators that account for the totality of all BVPs. (a) self-adjoint (b) non-self-adjoint (c) non-linear without additional assumptions or approximations than those used in obtaining the mathematical descriptions of the corresponding physical processes. The resulting computational infrastructure must be mathematically consistent and should require no additional ad hoc remedies. (4) The method should be convergent and unconditionally stable regardless of the choice of the degree or the order of the basis functions as long as such choices are admissible. 193

194

THE FINITE ELEMENT METHOD

(5) The method must permit higher degree and higher order of approximations so that coarser discretizations may remain practical with the desired degree of global smoothness that is dictated by the desired physics in the computational process. (6) Lastly, the method must lend itself as a systematic procedure that can be automated for use on digital computers with inherent and built-in adaptivity. The finite element method is indeed one such method that permits us to satisfy all of the above requirements. In applying the finite element method for approximating solutions of BVPs, one must adhere to the following basic steps regardless of the type of operator and the method of approximation used in constructing the integral form.

4.2 Basic steps in the finite element method We recall that in classical methods of approximation, one approximates ¯ in the the solutions of the BVP Aφ − f = 0 over the domain of definition Ω continuum sense, i.e. in such methods there is no concept of discretization and desired regularity is incorporated in the process. The finite element method derives all of its basic mathematical principles from classical meth¯ of the BVP ods of approximation except that the domain of definition of Ω is discretized into subdomains and the principles of classical methods of approximation are applied to the discretization and thereby to the subdomains. Thus, the finite element method can be thought of as piecewise (a piece being a subdomain or a finite element) application of classical methods of approximation. This is the fundamental difference between classical methods of approximation and the finite element method. This concept of discretization and dealing with subdomains allows the method to incorporate flexibility in all aspects which overcomes all of the shortcomings of the classical methods of approximation. In the following, we provide a systematic description and details of all of the basic steps involved in the finite element method for approximating solutions of the BVPs. These basic steps are identical regardless of the type of differential operator and the specific physical processes from which they arise.

4.2.1 Discretization ¯ = Ω ∪ Γ of the BVP Aφ − f = 0 is subdivided into smaller The domain Ω subdomains. Each subdomain is of finite size and is considered connected to the neighboring subdomains along its boundaries. A subdomain is referred to as a finite element. Thus, a finite element is a subdomain of finite dimensions

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

195

¯ Symbolically, of the original domain Ω. ¯= Ω

M [

¯e Ω

(4.1)

e=1

¯ are open and closed domains of the BVP and Ωe and Ω ¯e in which Ω and Ω are open and closed domains of a typical finite element e. This process of ¯ into subdomains Ω ¯ e is called discretization. Now, subdividing a domain Ω ¯ as the assembly of the subdomains Ω ¯ e. one could think of the domain Ω ¯ e is called a finite element mesh. If all the The collection of finite elements Ω elements of the mesh are of the same size and shape, then the mesh is called uniform. Otherwise, we refer to them as non-uniform (or graded) meshes. Graded meshes contain elements of various sizes (but often the same shape) in order to generate a bias in the computational process. For each element ¯ e we define some points on the boundaries of the element, which are called Ω nodes. The choice of the nodal points is not arbitrary. The node points help us in (a) defining the mathematical description of the geometrical shape of the element as well as (b) the behaviors of dependent variables over the element. With these nodes on the boundaries of elements, one could now think ¯ e as being connected to the neighboring elements only at the of an element Ω nodes provided the connection at the nodes automatically ensures a continuous connection along all of its boundaries with the neighboring elements. The connection of an element to its neighbors is in the mathematical sense of continuity and differentiability of the dependent variables and, hence is dependent upon the inter-element behaviors of the dependent variables. In the finite element mesh, the nodes and the elements are uniquely numbered. The choice of numbers for the nodes is irrelevant as long as they are unique. One also establishes an origin of a unique coordinate system (at any convenient location) in which the position coordinates of the nodes are uniquely identified. Two typical examples of discretizations are shown in ¯ = [0, 1], a line segment (see Fig. 4.1). FigFigs. 4.1 and 4.2. We consider Ω ¯ using two-node elements, ure 4.1 (b) shows a four element discretization of Ω the process of disconnecting and reconnecting the elements at the boundaries ¯ e with of the neighboring elements and a typical element e with its domain Ω local node numbers 1 and 2 isolated from the discretization. In Fig. 4.2, we ¯ of a BVP, discretized using four four-node quadriconsider a 2D domain Ω lateral elements, the process of disconnecting and reconnecting the elements ¯ e and the local node numbers and a typical element e with its domain Ω 1, 2, 3, 4 isolated from the discretization. Use of local node numbers for an ¯ e is helpful in deriving the mathematical details. Correspondence element Ω between the element numbers, their local node numbers and the nodes of the whole discretization can be easily established.

196

THE FINITE ELEMENT METHOD

y ¯ = [0, 1] Ω

x

¯ (a) One dimensional domain Ω y

1

2

1

3

2

4

3

1

4

2

1

2

x 5

3

2

3

4

3

4

4

5

a typical inter-element boundary y

e

1

2 x

xe

(local node numbers 1 and 2)

xe 1 ¯ e = [xe , xe+1 ] Ω

a 2-node typical element e ¯ the assembly of sub(b) A four-element uniform discretization of domain Ω, domains, and a typical two-node element e with its local node numbers ¯e and its domain Ω y

1 1

2

2

3

3

4

1 1

x

6

5

7

2

2

3

3

3

4

5

5

7

6

a typical inter-element boundary y

1 xe

e 2

3

xe+1

xe+2

x

(local node numbers 1, 2 and 3)

¯ e = [xe , xe+2 ] Ω a 3-node typical element e ¯ the assembly of (c) A three-element uniform discretization of domain Ω, subdomains, and a typical three-node element e with its local node ¯e numbers and its domain Ω

¯ of the BVP and its discretization Figure 4.1: 1D domain Ω

197

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

y

b

O

x

a

¯ of a BVP (a) A 2D domain Ω

y 9 9

8 3

4

1 1

8

3

7

5

4

8

7 4

4

5

5

6

4

5

5

6

6

1

2 3

2

1

x

Mesh

2 2

2

3

Individual elements

4

3 e

y

1

2

a typical four-node quadrilateral element e with local node numbers

x (b)

¯ an assembly of elements, and a typical element e with Discretization of 2D domain Ω, ¯ e. its local node numbers and domain Ω ¯ of the BVP and its discretization Figure 4.2: 2D domain Ω

198

THE FINITE ELEMENT METHOD

The end result of discretization is that we have a finite element mesh with M elements and N nodes clearly identified. The position coordinates of the ¯e nodes are known and a typical representative element e with its domain Ω of the discretization clearly identified. The values of dependent variables and/or their derivatives at the nodes of the entire discretization, referred to as the degrees of freedom (dofs) are the quantities of interest.

4.2.2 Construction of integral forms over an element In the study of the classical methods of approximation, for the boundary value problem S Aφ − f = 0 in Ω, we construct an integral form over ¯ the domain Ω = Ω Γ, Γ being closed boundary of Ω. This was done one of the two ways: (1) starting with the fundamental lemma, which lead to GM, PGM, WRM, and GM/WF methods of approximation, or (2) using residual functional in which the first variation of the residual functional is set to zero to obtain the integral form. This method is called least-squares method or process (LSP). A correspondence between these integral forms and the calculus of variations leads to the definitions of variationally consistent or variationally inconsistent integral forms. The variationally consistent integral forms yield algebraic systems in which the coefficient matrices are symmetric and positive definite and, hence, the unconditional stability of the computational processes is ensured. The variationally inconsistent integral forms, on the other hand, yield algebraic systems in which the coefficient matrices are non-symmetric and, hence, unconditional stability of the computational processes is not always ensured. All of the concepts, principles and stated findings and conclusions in connection with classical methods of approximation presented in chapter 3 precisely hold for the finite element processes as well. In the following sections we consider details of the finite element processes that are directly derived from classical methods of approximation: GM, PGM, WRM, GM/WF and LSP. In each case, details are presented for a ¯ e of the discretization Ω ¯ T of the domain of typical element e with domain Ω ¯ of the BVP Aφ − f = 0. In all cases we consider the following. definition Ω ¯ T = SM Ω ¯ e be the discretization of Ω ¯ in which Ω ¯ e = Ωe ∪ Γe Let Ω e=1 e is the domain of an element e with closed boundary Γ . Let φh be the ¯ T and φe be the approximation of φ over Ω ¯ e , then approximation of φ over Ω h SM e ¯ in the φh = e=1 φh . Let φn be the n-term approximation of φ over Ω classical methods of approximation.

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

199

4.2.2.1 Integral forms for GM, PGM, and WRM ¯ for the BVP Aφ − f = 0, we can Using the approximation φn over Ω write the integral statement Z ¯ =Ω∪Γ (Aφn − f )v dΩ = (Aφn − f, v)Ω¯ = 0, Ω (4.2) ¯ Ω

based on the fundamental lemma, in which v is the test function such that v = 0 on Γ∗ if φ = φ0 on Γ∗ of Γ. In GM method v = δφn , hence, v = 0 on Γ∗ is satisfied. In PGM v = ψ 6= δφn , but ψ = 0 on Γ∗ if φ = φ0 on Γ∗ must hold. In WRM v = w 6= δφn (weight function), but w = 0 on Γ∗ must ¯ T , we hold if φ = φ0 on Γ∗ . Using (4.2) and approximation φh of φ over Ω ¯ T as can write the integral form over discretization Ω Z (Aφh − f )v dΩ = (Aφh − f, v)Ω¯ T = 0 (4.3) ¯T Ω

¯T = S Ω ¯ e , and φh = Since the integral in (4.3) is a functional and since Ω e S e e φh using (4.3) we can write X XZ (4.4) (Aφeh − f, v)Ω¯ e = 0 (Aφh − f, v)Ω¯ T = (Aφeh − f )v dΩ = e

e ¯e Ω

or X

(Aφeh , v)Ω¯ e =

X

e

(f, v)Ω¯ e

(4.5)

e

or X e

B e (φeh , v)

le (v)

B e (φeh , v) =

X

le (v)

(4.6)

e

and are functionals for an element e with domain In (4.6), ¯ e. Ω ¯ T is such that the inWe remark that if the approximation φh over Ω T ¯ tegrand in (4.3) is continuous everywhere in Ω , then (4.4) holds in the Riemann sense. Otherwise, it holds in the Lebesgue sense in which case discontinuity in the behavior of the integrand over the inter-element boundaries (sets of measure zero) is neglected. This, of course, depends upon the global differentiability of φh which obviously depends upon S the global differentiability achievable from φeh due to the fact that φh = e φeh . Here, φeh is ¯e called the local approximation as it is local to the element e with domain Ω ¯ T and boundary conditions of the without regard to the other elements in Ω e BVP. The local approximation φh is established using interpolation theory. Once we know φeh , details of B e (φeh , v) and le (v) for an element e can be

200

THE FINITE ELEMENT METHOD

established. We note that in GM v = δφeh , in PGM v = ψ 6= δφeh and in WRM v = w 6= φeh . The choices of ψ and w in PGM and WRM is not arbitrary and must conform to the same restrictions locally over Γe as in the case of classical PGM and WRM over Γ∗ of Γ the closed boundary of Ω. The details of B e (φeh , v) and le (v) are presented in the following sections for GM, PGM and WRM. The desired integrals in these methods for an element e are (Aφeh , v)Ω¯ e = B e (φeh , v) and le (v) = (f, v)Ω¯ e . Remarks. (1) In these methods B e (φeh , v) is always non-symmetric, i.e. B e (φeh , v) 6= B e (v, φeh ). This is of course due to the variational inconsistency of the integral forms as shown in the classical methods of approximation. The consequence of this is that the element matrix resulting from B e (φeh , v) is not symmetric. (2) When the differential operator A is linear, B e (φeh , v) results in linear simultaneous algebraic relations for the element e in the nodal degrees of freedom for the element due to the fact that B e (φeh , v) is bilinear. (3) On the other hand if the differential operator A is non-linear, then B e (φeh , v) is a non-linear function of φeh (but linear in v) and as a consequence the resulting algebraic relations for an element e are also nonlinear in the nodal degrees of freedom for the element e. The element matrix is not symmetric as well. ¯ T cannot be (4) The variational inconsistency of the integral forms over Ω restored by any mathematically justifiable means. (5) le (v) is independent of φeh and is linear in v regardless of the nature of the differential operator. 4.2.2.2 Integral form for GM/WF In this method we also begin with (4.2) and arrive at (4.3) and then ¯ T . We consider (Aφe − f, v) ¯ e over an (4.5) or (4.6) for the discretization Ω Ω h e ¯ in which v = δφe , element e with domain Ω h (Aφeh − f, v)Ω¯ e = (Aφeh , v)Ω¯ e − (f, v)Ω¯ e

(4.7)

We transfer some differentiation from φeh to v in (Aφeh , v)Ω¯ e using integration by parts to obtain (Aφeh , v) = B e (φeh , v) − le (v) (4.8) e e

in which l (v) is concomitant e

hAφeh , viΓe ,

consisting of boundary terms or

boundary integrals resulting as a consequence of integration by parts. The motivation for integration by parts is to see if B e (φeh , v) = B e (v, φeh ), i.e. if

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

201

symmetric B e (·, ·) is possible as it would lead to symmetric element matrices (see section 4.2.4). Substituting from (4.8) into (4.7) we obtain (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v) − (f, v)Ω¯ e

(4.9)

e

or (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(4.10)

le (v) = le (v) + (f, v)Ω¯ e

(4.11)

in which e

Using φeh and v = δφeh details of B e (·, ·) and le (·) can be easily established (see section 4.2.4). (4.10) is the desired integral form based on GM/WF for ¯ e. the eth element with domain Ω Remarks. (1) When the differential operator A in Aφ − f = 0 is self-adjoint, i.e. contains only even order derivatives of the dependent variable(s), then it is possible to transfer half of the orders of differentiation from φeh to v and obtain B e (φeh , v) that is symmetric and, hence B e (φeh , v) = B e (v, φeh ). This is obviously due to variational consistency of the integral form in the classical GM/WF for self-adjoint operators. The consequence of this is that the element matrix resulting from B e (φeh , v) is symmetric. Due to linearity of the differential operator A, the algebraic relations for an element e are linear in the nodal degrees of freedom for the element. (2) When the differential operator A is non-self-adjoint, B e (φeh , v) always remains non-symmetric due to variational inconsistency of the integral form in the classical GM/WF for non-self-adjoint operators. The element matrix resulting from B e (φeh , v) is non-symmetric but the algebraic relations for the element e are linear in the nodal degrees of freedom for the element due to the fact that B e (φeh , v) is bilinear. (3) If the differential operator A is non-linear, then B e (φeh , v) is a non-linear function of φeh (but linear in v) and, hence the algebraic relations for the element e are non-linear in nodal degrees of freedom for the element and the element matrix is non-symmetric. The variational inconsistency of the integral form in this method is clearly due to variationally inconsistent integral form in classical GM/WF. (4) When the differential operator A is non-self-adjoint or non-linear variational inconsistency of the integral forms cannot be restored by any mathematically justifiable means. (5) le (v) is independent of φeh and is linear in v regardless of the nature of the differential operator.

202

THE FINITE ELEMENT METHOD

(6) Using the concomitant we can identify PVs, SVs, EBCs, and NBCs in the same manner as in classical methods of approximation. However, the BCs of the BVP cannot be used to simplify the concomitant as the ¯ e of an element e of the discretization integral form is over the domain Ω S ¯T = Ω ¯ e whereas the BCs can only be used for the whole domain (i.e. Ω e

¯ T ) as in classical methods of approximation. This is a fundamental Ω ¯ and Ω ¯ e. difference in GM/WF over Ω 4.2.2.3 Integral form based on residual functional ¯ for the BVP Aφ − f = 0 we can Using the approximation φn over Ω ¯ define the residual function E over Ω E = Aφn − f

(4.12)

The residual functional I(φn ) can be constructed using E I(φn ) = (E, E)Ω¯

(4.13)

¯ T , we can write the using (4.12) and (4.13) and approximation φh of φ over Ω following for the residual function E(φh ) and the residual functional I(φh ) ¯T over the discretization Ω ¯T E = Aφh − f holds everywhere in Ω

(4.14)

I(φh ) = (E, E)Ω¯ T

(4.15)

and or I(φh ) =

M X

I e (φeh ) =

X

(E e , E e )Ω¯

(4.16)

e

e=1

in which I e (φeh ) is the residual functional for an element e and E e is the ¯ e . E e (φe ) and I e (φe ) are given by E e = Aφe − f residual function over Ω h h h and I e = (E e , E e )Ω¯ e . In least-squares process we take the first variation of the residual functional I(φh ) and set it to zero provided I(φh ) is differentiable in φh (based on calculus of variations). δI(φh ) =

M X e=1

δI e (φeh ) =

X

(δE e , E e )Ω¯ e + (E e , δE e )Ω¯ e




But (δE e , E e )Ω¯ e = (E e , δE e )Ω¯ e , hence X X δI(φh ) = 2 (E e , δE e ) = 2 {g e } = 2{g} = 0 or {g} = 0 e

(4.17)

e

e

(4.18)

203

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

Using (E e , δE e )Ω¯ e , we can write (E e , δE e )Ω¯ e = B e (φeh , v) − le (v)

(4.19)

¯ e based on This is the desired integral form for an element e with domain Ω the residual functional or least-squares method. Remarks. (1) When the differential operator is linear, i.e. self-adjoint or non-selfadjoint, B e (φeh , v) is linear in φeh as well as v, that is, bilinear and is also symmetric (B e (φeh , v) = B e (v, φeh )). This is due to variational consistency of the integral form in classical LSP for self-adjoint and non-selfadjoint differential operators. The element coefficient matrix resulting from B e (φeh , v) is symmetric and the algebraic relations for the element are linear in the nodal degrees of freedom for the element. (2) When the differential operator A is non-linear, then B e (φeh , v) (or {g}) is a non-linear function of φeh but is linear in v. The element matrix is non-symmetric and its coefficients are functions of φeh . However, as shown in classical least-squares method of approximation if we find a φh satisfying {g} = 0 using Newton’s linear method and neglect (E e , δ 2 E e ) in δ 2 I e (φen ), then the finite element process based on LSP becomes VC and the symmetry of the element matrices and the assemble coefficient matrix in the algebraic system is restored as well. (3) le (v) is independent of φeh and is always linear in v regardless of the nature of the differential operator.

4.2.3 The local approximation φeh of φ over an element In order to obtain the element matrices (i.e. the algebraic relations) from B e (φeh , v) and vectors from le (v) we need local approximations φeh of φ over ¯ e . The local approximations φe of φ over Ω ¯ e can be constructed using Ω h ¯ e of each element interpolation theory presented in chapter 8. The domain Ω is mapped into the natural coordinate space (chapter 8) in a two-unit length, two-unit square or a two-unit cube with the origin of the natural coordinate system at the center by utilizing mappings such as x = x(ξ) =

X

Ni (ξ)xi

in R1

(4.20)

e

x = x(ξ, η) =

X

y = y(ξ, η) =

X

 Ni (ξ, η)xi   e

 Ni (ξ, η)yi  e

in R2

(4.21)

204

THE FINITE ELEMENT METHOD

 Ni (ξ, η, ζ)xi     e   X y = y(ξ, η, ζ) = Ni (ξ, η, ζ)yi in R3 e   X    z = y(ξ, η, ζ) = Ni (ξ, η, ζ)zi 

x = x(ξ, η, ζ) =

X

(4.22)

e

In (4.20)–(4.22), Ni (·) are the local approximation functions or shape funce

tions in the natural coordinate space and xi , yi , zi are nodal coordinates of the element nodes in the physical coordinate space. The sum in (4.20)–(4.22) is over suitably chosen nodes (see chapter 8). Using (4.20)–(4.22), relationships for mapping of lengths, areas and volumes in the (ξ, η, ζ) and (x, y, z) coordinate systems can be derived. Also, the derivatives of the dependent variable(s) φ in the two coordinate systems can be related. Based on the interpolation theory presented in chapter 8, we can also ¯ e. write the following for the local approximation φeh of φ over Ω φeh (ξ)

=

n X

Ni (ξ)δie = [N (ξ)]{δ e }

in R1

(4.23)

i=1

φeh (ξ, η) =

n X

Ni (ξ, η)δie = [N (ξ, η)]{δ e } in R2

(4.24)

i=1

φeh (ξ, η, ζ) =

n X

Ni (ξ, η, ζ)δie = [N (ξ, η, ζ)]{δ e } in R3

(4.25)

i=1

when [N (·)] is the basis or approximation function matrix in the natural coordinate space and {δ e } are the nodal degrees of freedom containing values of φ and/or its derivatives at the nodes of the element. The choice of Ni (·) and the resulting {δ e } depend upon the degree of local approximation p (p-level) and the order k of the approximation space defining global differentiability S of order (k − 1) of φh = φeh . Thus, k controls the global smoothness of φh e

¯ T through φe . Equations (4.23)–(4.25) completely define the behavior over Ω h ¯ e of the of each dependent variable over a typical element e with domain Ω T ¯ discretization Ω .

4.2.4 Element matrices and vectors resulting from the integral form and the local approximation In this section we utilize the integral forms derived in section 4.2.2 using various methods of approximation and the local approximation defined in section 4.2.3 to derive the details of the element matrices and vectors, i.e the ¯ e. algebraic relations for an element e with domain Ω

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

205

4.2.4.1 Galerkin method, Petrov–Galerkin method, and weighted residual method In the integral forms resulting from Galerkin method (GM), Petrov– Galerkin method (PGM), and weighted residual method (WRM) for an el¯ e , we have (see section 4.2.2) ement e with domain Ω B e (φeh , v)Ω¯ e = (Aφeh , v)Ω¯ e

(4.26)

le (v) = (f, v)Ω¯ e

(4.27)

and Let the local approximation be defined by φeh

=

n X

Ni δie

(4.28)

i=1

Then v = δφeh = Nj , j = 1, 2, . . . , n GM ˜j , j = 1, 2, . . . , n PGM or WRM v=w=N

(4.29)

In the following we consider two cases: when the differential operator A is (1) linear (i.e. self-adjoint or non-self-adjoint) and (2) non-linear. (1) Linear differential operators In this case B e (φeh , v) is bilinear, that is linear in φeh as well as v. We substitute from (4.28) and (4.29) in (4.26) and (4.27) and consider GM, i.e. v = Nj ; j = 1, 2, . . . , n. n  X   B e (φeh , v) = A Ni δie , Nj , j = 1, 2, . . . , n

(4.30)

i=1

le (v) = (f, Nj )Ω¯ e , j = 1, 2, . . . , n

(4.31)

Since the operator A is linear, (4.26) can be written as B

e

(φeh , v)

= =

n X

(ANi )δie , Nj



i=1 n X

¯e Ω

(ANi , Nj )Ω¯ e δie , j = 1, 2, . . . , n

(4.32)

i=1

Equations (4.32) and (4.31) can be written using matrix and vector notation as B e (φeh , v) = [K e ]{δ e } e

e

l (v) = {F }

(4.33) (4.34)

206

THE FINITE ELEMENT METHOD e of [K e ] and F e of {F e } are given by in which Kij i e Kij = B e (Nj , Ni ) = (ANj , Ni )Ω¯ e , i, j = 1, 2, . . . , n

Fie

e

= l (Ni ) = (f, ANi )Ω¯ e , i = 1, 2, . . . , n

(4.35) (4.36)

Equations (4.33)–(4.36) provide details of the element matrix and vector e 6= K e , i.e. the for an element e of the discretization. We note that Kij ji element matrix [K e ] is not symmetric, a consequence of the VIC integral form in the classical method of approximation based on GM. In the case ˜j and, hence, the details in (4.30)–(4.36) of PGM and WRM, v = N ˜j . remain the same except that Nj are replaced by N (2) Non-linear differential operators We begin with (4.26) and (4.27). Since the differential operator A is non-linear, B e (φeh , v) is a non-linear function of φeh but linear in v. We substitute (4.28) and (4.29) into (4.26) and (4.27) and consider GM. n  X 
Ni δie , Nj e , j = 1, 2, . . . , n B e (φeh , v) = A ¯ Ω

i=1

le (v) = (f, Nj )Ω¯ e , j = 1, 2, . . . , n

(4.37) (4.38)

Since A is a non-linear differential operator, (4.37) is a system of n nonlinear algebraic relations in δie . We can write (4.37) and (4.38) in the matrix and vector form as B e (φeh , v) = [K e ]{δ e } e

e

l (v) = {F }

(4.39) (4.40)

e of [K e ] are a function of {δ e } and K e 6= K e ; that The coefficients Kij ij ji is, [K e ] is not symmetric, a consequence of the VIC integral form in e can be obtained once we know the GM. The explicit expression for Kij ˜j specific form of the operator A. In the case of PGM and WRM, v = N but the remaining details remain the same as in GM except that Nj are ˜j . replaced by N

4.2.4.2 Galerkin method with weak form In the Galerkin method with weak form (GM/WF), we have v = δφeh and we obtain a weak form using (Aφeh , v)Ω¯e (Aφeh , v)Ω¯ e = B e (φeh , v) − le (v)

(4.41)

e

Therefore (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v) − (f, v)Ω¯ e = B e (φeh , v) − le (v) e

(4.42)

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

207

where le (v) = (f, v)Ω¯ e + le (v)

(4.43)

e

The right side of (4.41) is a consequence of transferring some differentiation ¯e from φeh to v using integration by parts. B e (·, ·) is the integral over Ω e whereas l (v) are the boundary terms or boundary integrals (concomitant). e

We consider local approximations φeh φeh

=

n X

Ni δie

(4.44)

i=1

and v = δφeh = Nj , j = 1, 2, . . . , n

(4.45)

We consider three cases: when the differential operator A is self-adjoint, non-self-adjoint and non-linear. (1) Self-adjoint differential operators In this case, A is linear and symmetric and, hence, contains only even derivative terms. Therefore, we can transfer half of the differentiation from φeh to v in obtaining (4.41). In doing so, we find that B e (φeh , v) = B e (v, φeh )

(4.46)

That is, B e (·, ·) is symmetric. Furthermore, since the differential operator A is linear, B e (·, ·) is bilinear, i.e. linear in φeh as well as v. Substituting (4.44) and (4.45) into (4.41) and (4.42), we obtain (Aφeh , v)Ω¯ e

=B

e

n X

 Ni δie , Nj − le (Nj ), j = 1, 2, . . . , n

i=1

(4.47)

e

(f, v)Ω¯ e = (f, Nj )Ω¯ e , j = 1, 2, . . . , n

(4.48)

Since B e (·, ·) is bilinear and le (v) is linear, the right sides of (4.46) and (4.47) can be written in the following form (Aφeh , v)Ω¯ e = [K e ]{δ}e − {P e } e

(f, v)Ω¯ e = {F }

(4.49) (4.50)

e of [K e ] and F e of {F e } are given by in which Kij i e Kij = B e (Nj , Ni ) = B e (Ni , Nj ), i, j = 1, 2, . . . , n

Fie

= (f, Ni )Ω¯ e , i = 1, 2, . . . , n

(4.51) (4.52)

208

THE FINITE ELEMENT METHOD

Therefore (Aφen − f, v)Ω¯ e = [K e ]{δ e } − {P e } − {F e } The vector {P e } is due to le (Nj ) (j = 1, 2, . . . , n) and is called a vector e

e = K e ; that is, [K e ] is symmetric, of secondary variables. Obviously Kij ji a direct consequence of the VC integral form possible only due to the e and {P e } depend upon the fact that A∗ = A. The explicit forms of Kij differential operator A.

(2) Non-self-adjoint differential operators In this case the differential operator is linear but not symmetric. We may transfer some differentiation from φeh to v in (Aφeh , v)Ω¯ e . It is important to consider the possible situations in which it is advantageous to do so. First, when the differential operator is non-self-adjoint, it definitely contains some terms with odd derivatives but may (or may not) also contain some terms with even derivatives. When A contains some terms with even derivative terms, then in these terms we can transfer half of the differentiation from φeh to v in (Aφeh , v)Ω¯ e using integration by parts. By doing so, we ensure that the contribution of these resulting terms to the element matrix [K e ] becomes symmetric. This helps in restoring some stability of the resulting computational process. In the remaining terms with odd derivatives of the dependent variable, we may also transfer some differentiation from φeh to v in (Aφeh , v)Ω¯ e to lower differentiability requirements in the resulting integral. However, for such terms their contribution to [K e ] remains non-symmetric. The end result is that we have (similar to (4.41) and (4.42)) (Aφeh , v)Ω¯ e = B e (φeh , v) − le (v)

(4.53)

e

Therefore (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v) − (f, v)Ω¯ e = B e (φeh , v) − le (v) (4.54) e

in which le (v) is concomitant which only occurs if some differentiation e

is transferred from the dependent variable(s) to v. The local approximation can be written as φeh

=

n X

Ni δie

(4.55)

i=1

and v = δφeh = Nj , j = 1, 2, . . . , n

(4.56)

209

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

Since the operator A is linear (but not symmetric), B e (·, ·) is bilinear but not symmetric due to A being non-symmetric. Substituting from (4.55) and (4.56) into (4.53) and (4.54), we obtain (Aφeh , v)Ω¯ e = B e

n X

 Ni δi , Nj − le (Nj ), j = 1, 2, . . . , n

i=1

(4.57)

e

(f, v)Ω¯ e = (f, Nj )Ω¯ e , j = 1, 2, . . . , n

(4.58)

Since B e (φeh , v) is bilinear and le (v) is linear. From (4.57) and (4.58) we can write (Aφeh , v)Ω¯ e = [K e ]{δ e } − {P e } e

(f, v)Ω¯ e = {F }

(4.59) (4.60)

e of [K e ] and F e of {F e } are given by in which Kij i e Kij = B e (Nj , Ni ) 6= B e (Ni , Nj ), i, j = 1, 2, . . . , n

Fie

= (f, Ni )Ω¯ e , i = 1, 2, . . . , n

(4.61) (4.62)

Therefore (Aφeh − f, v)Ω¯ e = [K e ]{δ e } − {P e } − {F e } The vector {P e } is due to le (Nj ) (j = 1, 2, . . . , n) and is called vector e

e 6= K e ; that is, [K e ] is not symof secondary variables. Obviously, Kij ji metric, a consequence of the VIC integral form due to the fact that the e and differential operator A is not symmetric. The explicit forms of Kij e {P } depend upon the differential operator A. (3) Non-linear differential operators

Here the differential operator is neither linear nor symmetric. In such cases it is possible that the differential operator A contains terms with even and odd derivatives in addition to non-linear terms. The rules for transferring differentiation from φeh to v in (Aφeh , v)Ω¯ e for the terms containing even and odd derivatives of the dependent variables remain the same as discussed in (2) when the differential operator A is nonself-adjoint. For the non-linear terms, there is generally no incentive to perform integration by parts. The final integral form in this case also results in (Aφeh , v)Ω¯ e = B e (φeh , v) − le (v) (4.63) e

Therefore (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v) − (f, v)Ω¯ e = B e (φeh , v) − le (v) (4.64) e

210

THE FINITE ELEMENT METHOD

The local approximation φeh

=

v=

n X i−1 δφeh

Ni δie

(4.65)

= Nj , j = 1, 2, . . . , n

(4.66)

In this case, B e (φeh , v) is non-linear in φeh but linear in v and le (v) is linear in v. Substituting from (4.65) and (4.66) into (4.63) and (4.64) (Aφeh , v)Ω¯ e

=B

e

n X

 Ni δi , Nj − le (Nj ), j = 1, 2, . . . , n

i=1

(4.67)

e

(f, v)Ω¯ e = (f, Nj )Ω¯ e , j = 1, 2, . . . , n

(4.68)

(4.67) and (4.68) can also be written in the matrix and vector form (Aφeh , v)Ω¯ e = [K e ]{δ e } − {P e } e

(f, v)Ω¯ e = {F }

(4.69) (4.70)

Therefore (Aφeh − f, v)Ω¯ e = [K e ]{δ e } − {P e } − {F e } The coefficients of [K e ] are functions of {δ e } due to the fact that the e of [K e ] can differential operator A is non-linear. Explicit forms of Kij e be obtained when A is defined. In this case also {P } is a vector of sece 6= K e , i.e. [K e ] is not symmetric, a ondary variables. Furthermore, Kij ji consequence of the VIC integral form due to the fact that the differential operator A is non-linear. 4.2.4.3 Least-squares process based on residual functional In the least-squares process based on residual functional (LSP), we consider E e = Aφeh − f , δE e and (E e , δE e )Ω¯ e for an element e with domain ¯ e . We consider two cases: when the differential operator A is linear (i.e. Ω self-adjoint and non-self-adjoint) and when the differential operator A is non-linear. (1) Linear differential operators ¯ e we have In this case, for an element e with domain Ω E e = Aφeh − f e

δE = Av

(4.71) (4.72)

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

211

Hence, for the necessary condition we have {g e } = (E e , δE e )Ω¯ e = (Aφeh − f, Av)Ω¯ e = (Aφeh , Av)Ω¯ e − (f, Av)Ω¯ e = B e (φeh , v) − le (v) (4.73) B e (·, ·) is bilinear and symmetric and le (v) is linear. Consider φeh =

n X

Ni δie

(4.74)

i=1

and δφh = v = Nj , j = 1, 2, . . . , n

(4.75)

Substituting from (4.74) and (4.75) into (4.73) e



{g } = A

n X


Ni δie , ANj

i=1

¯e Ω

− (f, ANj )Ω¯ e , j = 1, 2, . . . , n (4.76)

Since A is linear, using (4.76) we can write {g e } =

n X

ANi , ANj




e ¯ e δi Ω

− (f, ANj )Ω¯ e , j = 1, 2, . . . , n

(4.77)

i=1

or in matrix and vector notation {g e } = [K e ]{δ e } − {F e }

(4.78)

e of [K e ] and F e of {F e } are given by in which Kij i e Kij = (ANj , ANi )Ω¯ e , i, j = 1, 2, . . . , n

Fie

= (f, ANi ), i = 1, 2, . . . , n

(4.79) (4.80)

e = K e ; that is, [K e ] is symmetric, a consequence of the VC Clearly, Kij ji integral form in LSP for linear differential operator. (2) Non-linear differential operators

Consider BVPs in which the differential operator is non-linear. In this case also, {g e } is given by {g e } = (E e , δE e )Ω¯ e

(4.81)

where E e = Aφeh − f e

δE = Av +

δA(φeh )

(4.82) (4.83)

212

THE FINITE ELEMENT METHOD

¯ e . We note that since A in nonholds for an element e with domain Ω linear, δA 6= 0. Substituting from (4.82) and (4.83) in (4.81)
{g e } = Aφeh − f, Av + δA(φeh ) (4.84) Consider φeh =

n X

Ni δie

(4.85)

i=1

δφen = v = Nj , j = 1, 2, . . . , n.

(4.86)

Substituting from (4.85) and (4.86) into (4.84) and noting that A is a function of φeh , n n  X X

 {g e } = A Ni δi − f, Av + δA Ni δie i=1

(4.87)

i=1

We note that A is a function of φeh and, hence, δA 6= 0. Thus, {g e } is a non-linear function of {δie }. (4.87) can be written in matrix and vector notation, but we do not do so. Instead as shown in the classical methods, we find a [ {δ} = {δ e } (4.88) e

that satisfies {g} =

M X

{g e } = 0

(4.89)

i=1

using iterative methods such as Newton’s linear method with line search. Details follow the classical least-squares method discussed earlier but are repeated for finite element processes based on LSP in section 4.2.8.

4.2.5 Assembly of element equations: GM, PGM, WRM, GM/WF and LSP when A is linear ¯T = S Ω ¯ e , we have in all finite element Recall that for the discretization Ω e processes based on various methods of approximation except LSP for nonlinear differential operator the following. GM, PGM, WRM: M X

(Aφeh − f, v)Ω¯ e =

e=1

M X

B e (φeh , v) −

e=1

=

M X e=1

M X

le (v) = 0

e=1

[K e ]{δ e } −

M X e=1

{F e } = 0

(4.90)

213

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

GM/WF: M X

(Aφeh − f, v)Ω¯ e =

e=1

=

M X e=1 M X

B e (φeh , v) − [K e ]{δ e } −

e=1

M X

e=1 M X

le (v) − e

{P e } −

e=1

M X

le (v) = 0

e=1 M X

{F e } = 0

(4.91)

e=1

LSP: When A is linear 2

M X

(E e , δE e )Ω¯ e = 2

e=1

or

M X

M X  (Aφeh − f, Av)Ω¯ e = 2 B e (φeh , v) − le (v) = 0

e=1

e=1

M X

(Aφeh , Av)Ω¯ e −

(f, Av)Ω¯ e = 0

e=1

e=1

or

M X

M X

[K e ]{δ e } −

e=1

M X

{F e } = 0

(4.92)

e=1

By examining (4.90)–(4.92), we note that (4.91) can be used when the finite element processes are derived using GM, PGM, WRM, GM/WF and LSP (for linear operators) except that {P e } only exists in the case GM/WF. Thus, we consider the following to hold for all finite element processes except those derived using LSP for non-linear differential operators. Consider M X

e

e

[K ]{δ } =

e=1

M X

e

{P } +

M X

e=1

{F e }

(4.93)

e=1

in which [K e ] is the element matrix, {δ e } are the nodal degrees of freedom for an element e and {P e }, {F e } are vectors of secondary variables and the load vector due to the non-homogeneous part in the mathematical description of ¯T = S Ω ¯ e we can write the BVP. From (4.93), for Ω e [K]{δ} = {P } + {F }

(4.94)

in which {δ} =

[ e

{δ e }, {P } =

M X e=1

{P e }

;

{F } =

M X e=1

{F e }, [K] =

M X

[K e ]

e=1

(4.95) where [K] is the result of the sum, assembly or addition of the element matrices [K e ] and likewise {P } and {F } are the sum, assembly or additions

214

THE FINITE ELEMENT METHOD

¯ T . (4.94) of {P e } and {F e }. {δ} are the nodal degrees of freedom for Ω ¯ T . The rules for the assembly of the element holds for the discretization Ω equations are derived using the following. (1) The functionals such as B(·, ·) and l(·) are the sum of the functionals B e (·, ·) and le (·), le (·) for the individual elements. e

(2) At the mating boundaries of the elements, {δ e } at the common nodes must have unique values. This is also expressed by {δ} =

[

{δ e }

e

These conditions are known as inter-element continuity conditions on the nodal variables and must be considered for each element of the dis¯T . cretization Ω More details on how to assemble the element equations will be presented in chapters 5, 6 and 7 in connection with specific model problems and numerical studies.

4.2.6 Consideration of boundary conditions in the assembled equations 4.2.6.1 GM, PGM, WRM, and LSP based on the residual functional First we note that in GM, PGM, WRM, and LSP based on residual functional there is no integration by parts hence the concepts of PVs, SVs, EBCs, and NBCs do not exist. Hence in the element equations and therefore the assembled equations there is no concept of secondary variables {P e } and {P }. The assembled equations in this case are of the form [K]{δ} = {F }

(4.96)

in which {F } is a known vector. In GM, PGM, WRM, and LSP (4.96) must be subjected to all boundary conditions of the BVP i.e. the known BCs of the BVP must be described in terms of some dofs of the vector {δ} and imposed in (4.96). In this process {δ} vector can be split into {δ}T = {{δ1 }T , {δ2 }T }

(4.97)

in which {δ1 } are now known due to the known BCs of the BVP and {δ2 } contains those dofs that are yet to be determined.

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

215

4.2.6.2 GM/WF ¯ e integration by In Galerkin method with weak form over an element Ω parts results into the definitions of PVs, SVs, EBCs, and NBCs and definition of secondary variables into vector {P e }. This is due to the fact that over an element the concomitant can not be simplified using BCs (more specifically NBCs) of the BVP. Thus, in GM/WF the assembled equations for the ¯ T will be of the form discretization Ω [K]{δ} = {P } + {F }

(4.98)

The secondary variable vector {P } is due to assembly of the secondary vari¯ T and is not known. ables resulting for each element of the discretization Ω As in other methods, in GM/WF also {F } is a known vector. Thus, in this method consideration of the BCs (EBCs and NBCs) in (4.98) will result in specification of some elements of {δ} as well as some elements of {P }. We consider details in the following. Consideration of NBCs and EBCs (1) The NBCs only affect the sum of the secondary variable vector {P } in the assembled equations. (2) The sum of the secondary variables at a node from all connecting elements must be equal to the specified external disturbance at the node. (3) If the specified disturbance at a node is zero then the sum of the secondary variables at that node must be zero. (4) Consideration of the EBCs that are specifications of PVs on some boundaries result in specification of some components of the degrees of freedom vector {δ} say {δ1 }. (5) If the primary variables are specified at a node then the sum of the secondary variables is unknown at that node. Thus, the sum of the secondary variables is known at all those grid points where the primary variables are unknown and their sum is not known where the primary variables are known. This clearly helps us in identifying the components of the vector {P } that are known, say {P2 }, and those that are not known, say {P1 }. Thus in GM/WF we have 

 {P1 } {P } = {P2 }   {δ1 } {δ} = {δ2 }

(4.99)

(4.100)

216

THE FINITE ELEMENT METHOD

We note that in (4.99) and (4.100) {P2 } and {δ1 } are known and {P1 } and {δ2 } are unknown. We note that if a component of {P } is known then the corresponding component of {δ} is not known and if a component of {δ} is known then the corresponding component of {P } is not known. This rule must always hold and there is no exception to this. As an example, in case of solid mechanics {δ} can be displacements and {P } can be forces, hence this rule in this case means that if at a point displacement is known then the force is not known (reaction) and if the force is known then the displacement is not known. At any point both displacements and forces can not be known or unknown. If one is known, then the other is defined by the response of the structure.

4.2.7 Computation of the solution: finite element processes based on all methods of approximation except LSP for non-linear operators It is sufficient to consider the assembled equations for GM/WF as for other methods we only have to set {P } to null vector. The assembled equa¯ T are partitioned in terms of known tions (4.94) for the whole discretization Ω degrees of freedom {δ1 } and the unknown degrees of freedom {δ2 } to be calculated,        {F1 } {P1 } {δ1 } [K11 ] [K12 ] (4.101) + = {F2 } {P2 } {δ2 } [K21 ] [K22 ] When the differential operator A is linear, (4.101) is a system of linear simultaneous algebraic equations in the nodal degrees of freedom. When the differential operator A is non-linear, (4.101) is naturally a system of non-linear algebraic equations in the nodal dofs and must be solved using iterative methods. We consider the case when (4.101) is a system of linear simultaneous algebraic equations to illustrate the details. We note that {δ1 }, {P2 }, {F1 } and {F2 } are all known, whereas {δ2 } and {P1 } are unknown. From (4.101), we could solve for {δ2 } using [K21 ]{δ1 } + [K22 ]{δ2 } = {P2 } + {F2 }

(4.102)

[K22 ]{δ2 } = {P2 } + {F2 } − [K21 {δ1 }

(4.103)

or We can use elimination methods such as Gauss elimination to solve for {δ2 } from (4.103). Symbolically, we can write   {δ2 } = [K22 ]−1 {P2 } + {F2 } − [K21 ]{δ1 } (4.104) Thus, now the entire solution [{δ1 }T , {δ2 }T ] is known. The unknown secondary variables {P1 } are calculated using {P1 } = [K11 ]{δ1 } + [K12 ]{δ2 } − {F1 }

(4.105)

217

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

4.2.8 Assembly of element equations and their solution in finite element processes based on residual functional (LSP) when the differential operator A is non-linear S ¯ T , the discretization of Let φh = e φeh be the approximation of φ over Ω ¯ Then in LSP, we have Ω. (1) Existence of a functional I(φh ) I(φh ) = (E, E)Ω¯ T

(4.106)

E = Aφh − f

(4.107)

(2) Necessary condition δI(φh ) = 2(E, δE)Ω¯ T = 2{g(φh )} = 0

{g(φh )} = 0 (4.108)

or

(3) Sufficient condition or extremum principle δ 2 I(φh ) = 2(δE, δE)Ω¯ T + 2(E, δ 2 E)Ω¯ T

(4.109)

Since the differential operator A is non-linear, {g(φh )} in (4.108) is a nonlinear function of φh and, hence, we must find a φh iteratively that satisfies (4.108). Let [ (4.110) φh = φeh e

where φeh

=

n X

Ni δie

(4.111)

i=1

is the local approximation. Then [   g(φh ) = g({δ}) = 0, {δ} = {δ}e

(4.112)

e

must be satisfied iteratively. Let {δ0 } be an assumed solution (initial guess of {δ}). Then  g({δ0 }) 6= 0 (4.113) Let {∆δ} be a change in {δ0 } such that  g({δ0 } + {∆δ}) = 0

(4.114)

Expanding {g(·)} in (4.114) in Taylor series about {δ0 } and retaining only up to linear terms in {∆δ} (Newton’s linear method or Newton Raphson method)   ∂{g} {∆δ} + · · · = 0 (4.115) g({δ0 } + {∆δ}) = g({δ0 }) + ∂{δ} {δ0 }

218

THE FINITE ELEMENT METHOD

Hence 

∂{g} {∆δ} ≈ − ∂{δ}

−1

 g({δ0 })

(4.116)

{δ0 }

However, ∂{g} 1 = δ{g} = δ 2 I = (δE, δE)Ω¯ T + (E, δ 2 E)Ω¯ T ∂{δ} 2

(4.117)

From (4.116) and (4.117), we obtain  1 g({δ0 }) {∆δ} = − [δ 2 I]−1 {δ } 0 2

(4.118)

For a unique {∆δ} from (4.118), the coefficient matrix [δ 2 I]{δ0 } must be positive definite. This is possible if we approximate δ 2 I by δ 2 I ≈ 2(δE, δE)Ω¯ T > 0

(4.119)

which yields a unique extremum principle or sufficient condition, and we have  1  2 −1  {∆δ} ≈ − [(δE, δE)Ω¯ T ]−1 {δ0 } g({δ0 }) = 2 δ I {δ0 } g({δ0 })

(4.120)

An improved solution {δ} is obtained using {δ} = {δ0 } + α{∆δ}

(4.121)

in which generally 0 < α ≤ 2 and is determined using I({δ}) ≤ I({δ0 }). That is, for series of values of α, we determine I({δ}) and choose an α for which I({δ}) is minimum. This is referred to as line search. The line search is helpful in accelerating the convergence of the Newton’s first-order iterative method due to the fact that when a right direction (i.e. {∆δ}) has been found using (4.120), we proceed in this direction as far as possible as long as I({δ}) is less than or equal to I({δ0 }). The details presented above ¯ T such that the contributions can be easily expressed for the discretization Ω of the individual elements are summed or assembled. We provide these in the following. Assume a starting solution {δ0 }. Then (1) I(φh ) = (E, E)Ω¯ T =

M X

e

e

(E , E )Ω¯ e =

e=1

in which E = Aφh − f and E e = Aφeh − f

M X i=1

Ie

(4.122)

4.2. BASIC STEPS IN THE FINITE ELEMENT METHOD

219

(2) δI(φh ) = 2(E, δE)Ω¯ e = 2

M X

(E e , δE e )Ω¯ e

e=1

=2

M X

{g e } = 2{g} = 0 or {g} = 0 (4.123)

e=1

We must find a solution that satisfies {g} = 0 iteratively. (3) δ 2 I(φh ) ≈ 2(δE, δE)Ω¯ T = 2

M X

(δE e , δE e )Ω¯ e > 0

(4.124)

e=1

(4) M hX i−1  1 e e {g({δ })} = − (δE , δE ) g({δ }) {∆δ} = − [δ 2 I]−1 e ¯ 0 0 Ω {δ } 0 2 {δ0 } e=1 (4.125)

or M hX i−1   {∆δ} = − [K e ] g({δ0 }) = −[K]−1 {δ0 } g({δ0 }) e=1

{δ0 }

(4.126)

in which [K e ] = (δE e , δE e )Ω¯ e is the element matrix for an element e of ¯ T and Ω M X [K] = [K e ] (4.127) e=1

¯ T . We also note that {g} = PM {g e }, i.e. is the assembled matrix for Ω i=1 {g} is the result of the assembly of {g e } for the individual elements of ¯ T . The improved solution {δ{ is given by Ω {δ} = {δ0 } + α∗ {∆δ}, 0 < α∗ ≤ 2 such that I({δ}) ≤ I({δ0 }) (4.128) (5) Compute  {g} using (4.123). If {δ} in (4.127) is the converged solution, then g({δ}) = 0 holds. We check the absolute values of the components of {g} to ensure that these are less than or equal to a preset tolerance ∆, a threshold value for numerically computed zero (generally ∆ ≤ 10−6 suffices) for the convergence of the iterative solution method. (6) If the criterion in 5. is satisfied, then we have a new {δ} that satisfies (4.123) and, hence is the desired converged solution from the Newton’s linear method with line search. If not, then we set {δ0 } equal to {δ}, the current solution and repeat steps 2 through 6.

220

THE FINITE ELEMENT METHOD

Remarks. (1) The iterative solution method presented here is known as Newton’s linear method with line search. (2) With the approximation (4.124), we have a least-squares finite element formulation for a BVP in which the differential operator A is non-linear but the integral form is variationally consistent. (3) The motivation for neglecting (E, δ 2 E)Ω¯ T in δ 2 I(φh ) is obviously to achieve variational consistency of the integral form. This approximation is not as crude or heuristic as it might appear. (a) When we are in the close proximity to the correct solution E ≈ 0, then (E, δ 2 E)Ω¯ T can be expected to be making only a small (i.e. negligible) contribution to δ 2 I(φh ). (b) We note that in finding the solution in fact solving for
{δ} we are 1 2 a root of {g({δ})} = 0 = 2 δ I(δ) , hence, δ I({δ}) represents a tangent plane to the hyper surface defined by δI({δ}) = 0 at {δ0 }. Thus, approximating δ 2 I amounts to changing the orientation (or slope) of the tangent plane to the hypersurface δI = 0 at {δ0 }. It has no effect on the least-squares process which ends with (4.122) and (4.123). (4) In view of (a) and (b) in (3), the approximation in (4.124) is not heuristic but well justified especially when its major benefit is variational consistency of the integral form that leads to unconditional stability of the computations.

4.2.9 Post processing of the solution Once the nodal values of the solution in {δ} are all known and since [ {δ} = {δ e } (4.129) e

{δ e },

we in fact know the degrees of freedom for each element of the dis¯ T . Using local the approximation cretization Ω φeh

=

n X

Ni δie = [N ]{δ e }

(4.130)

i=1

¯ e , i.e. Ωe as well as Γe . the solution is defined everywhere over each element Ω Using (4.130), any desired further processing of the solution can be done on an element by element basis. This may involve calculating derivatives of φeh ¯ e . This part of the finite element computations or any desired norms over Ω is referred to as the post-processing phase (the word post implying after the calculation of the solution {δ}).

4.3. SUMMARY

221

4.3 Summary Ideal and important features of a good computational method are discussed. Mathematical classification of the differential operator is shown to be essential to address finite element processes for stability of all BVPs in a consistent and rigorous manner. Basic steps in the finite element process: discretization, integral forms, element matrices resulting from the integral forms, assembly of element equations, computation of solution and post processing are introduced. Various methods of approximation such as GM, GM/WF, WRM, PGM, and LSP are considered for the three classes of differential operators and the VC or VIC of the resulting form is evaluated to establish unconditional stability of the resulting computational process or lack of it. [1–10]

References for additional reading [1] B. Jiang. The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, 1998. [2] K. S. Surana, L. R. Anthoni, S. Allu, and J. N. Reddy. Strong and weak form of governing differential equations in least squares finite element processes in hpk framework. Int. J. Comp. Meth. in Eng. Sci. and Mech., 2008. [3] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002. [4] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [5] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004. [6] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., New York, 3rd edition, 2006. [7] J. T. Oden and J. N. Reddy. An Introduction to the Mathematical Theory of Finite Elements. John Wiley, New York, 1976. [8] O. C. Zienkiewicz. The Finite Element Method, volume 1. McGrawhill, England, 4rd edition, 1989. [9] K. J. Bathe. Finite Element Procedures. Prentice Hall, New Jersey, 1996. [10] J. C. Heinrich and D. W. Pepper. Intermediate Finite Element Method. Taylor and Francis, Philadelphia, 1999.

5

Self-Adjoint Differential Operators 5.1 Introduction In this chapter we consider one, two and three dimensional boundary value problems in single and multi variables that are described by selfadjoint differential operators. The self-adjoint differential operators arise in many branches of engineering, science, and mathematical physics. Linear elastic solid mechanics, structural mechanics, and linear heat condition are some examples in engineering. First, we review some basic properties of these differential operators and the integral forms resulting from the classical methods of approximation. Consider Aφ − f = 0 in Ω. (1) The self-adjoint differential operators are linear and symmetric. These operators contain even order derivatives of the dependent variables. For these problems the adjoint A∗ of the operator is the same as the operator A. (2) Based on the material presented in chapter 3, the integral forms using ¯ are always variationally inconsistent. GM, PGM, and WRM over Ω (3) In GM/WF, we begin with fundamental lemma and transfer half of the differentiation from the dependent variable to the test function to obtain B(φn , v)Ω¯ = l(v)Ω¯

(5.1)

in which B(·, ·) is bilinear and symmetric and l(v) is linear. The weak form (5.1) is variationally consistent. (4) The integral form based on residual functional (LSM or LSP) is also variationally consistent. Here, we use E = Aφn − f I = (E, E)Ω¯ δI = 0 to obtain the desired integral form. 223

¯ in Ω (5.2)

224

SELF-ADJOINT DIFFERENTIAL OPERATORS

(5) Variationally consistent integral forms yield unconditionally stable computational processes and ensure unique φn . Based on (2) - (5) we only consider finite element processes derived using GM/WF and LSP. The details of classical GM/WF and LSP have been presented in chapter 3. ¯ is not discretized In the classical methods, the domain of definition Ω ¯ and hence the approximation φn is global over Ω. The details of the finite element processes derived using the classical methods of approximation have also been presented in chapter 3. In the following, we present the important steps involved in the finite element processes based on GM/WF and LSP (described in chapter 3). In all finite element processes, regardless of the choice of integral form (i.e., the method of approximation), the following details are common: ¯ e be the discretization of Ω ¯ in which Ω ¯ e = Ωe ∪ Γe = ¯ T = SM Ω Let Ω e=1 closure of Ωe is the domain of an element e and Γe is the closed boundary of ¯ T and φe , the approximation Ωe . Let φh be the approximation of φ over Ω h S e ¯ e such that φh = of φ over Ω e φh .

5.1.1 GM/WF In this method we begin with the integral statement (based on funda¯T , mental lemma) over Ω Z (Aφh − f, v)Ω¯ T = (Aφh − f )v dΩ = 0, v = δφh (5.3) ¯T Ω

or X e

(Aφeh − f, v)Ω¯ e = 0 or

X

(Aφeh , v)Ω¯ e −

e

X

(f, v)Ω¯ e = 0

(5.4)

e

For an element e consider (Aφeh − f, v)Ω¯ e = (Aφeh , v) − (f, v)Ω¯ e . Since the differential operator is self-adjoint (implying even order derivatives of φeh ), we transfer half of the differentiation from φeh to v to obtain (Aφeh , v)Ω¯ e = B e (φeh , v) − le (v)

(5.5)

e

in which B e (φeh , v) is bilinear and symmetric and le (v) is the concomitant e

resulting as a consequence of integration by parts. We note that since the ¯ e , the domain of an element e, the concomitant le (v) integrals are over Ω e

cannot be simplified using the boundary conditions of the BVP. Instead, the unknown secondary variables appearing in le (v) need to be symbolically e

identified for each node of the element. Using (5.5), for an element e we can write (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v) − (f, v)Ω¯ e (5.6) e

225

5.1. INTRODUCTION

upon substituting local approximation φeh given by φeh =

n X

Ni δie = [N ]{δ e }

(5.7)

i=1

and v = δφeh = Nj , j = 1, 2, . . . , n

(5.8)

into (5.6) and using the notations, B e (φeh , v) = [K e ]{δ e } (f, v)Ω¯ e = {F e } e

(5.9)

e

l (v) = {P } e

we can write (5.6) in the form (Aφeh − f, v)Ω¯ e = [K e ]{δ e } − {P e } − {F e }

(5.10)

In the numerical example considered in this chapter, the details of the element matrices and the vectors on the right side of (5.10) will be considered. Using (5.10), assembly of element equations follows standard procedure described in this chapter and symbolically described in chapter 4.

5.1.2 LSP based on residual functional In this method the residual functions E and E e are defined as ¯T E = Aφh − f in Ω ¯e E e = Aφe − f in Ω

(5.11)

h

Let the residual functional I(φh ) be given by X X I(φh ) = (E, E)Ω¯ T = Ie = (E e , E e )Ω¯ e , I e = I e (φeh ) e

(5.12)

e

Then the necessary condition for an extremum of I is X X X δI = 2(E, δE)Ω¯ T = δI e = 2 (E e , δE e )Ω¯ e = 0 = 2 {g e } = 2{g} e

e

e

(5.13) or {g} =

X

{g e } =

e

We consider

{g e }

=

(E e , δE e )Ω¯ e

X

(E e , δE e )Ω¯ e = 0

(5.14)

e

for an element e

δE e = Av, v = δφeh

(5.15)

226

SELF-ADJOINT DIFFERENTIAL OPERATORS

Thus {g e } = (Aφeh − f, Av)Ω¯ e = (Aφeh , Av)Ω¯ e − (f, Av)Ω¯ e

(5.16)

{g e {= (Aφeh − f, Av)Ω¯ e = B e (φeh , v) − le (v)

(5.17)

or Using local approximations (5.7) and (5.8), (5.17) can be written as {g e } = [K e ]{δ e } − {F e }

(5.18)

Details of [K e ], {δ e } and {F e } will be considered for each of the numerical examples presented in the following sections. Assembly of the element equations (5.18) and solution follows standard procedure.

5.2 One-dimensional BVPs in a single dependent variable 5.2.1 1D steady-state diffusion equation: finite element processes based on GM/WF Consider the following BVP: −

d dφ  a(x) + c φ − q(x) = 0 ∀x ∈ Ω = (0, L) ⊂ R1 dx dx

(5.19)

with the boundary conditions  dφ  φ(0) = φ0 and a =P dx x=L

(5.20)

where a(x), c, φ0 and q(x) are known (data). This BVP describes a variety of physical processes: axial deformation of a rod, 1D heat conduction, flow through channels and pipes, transverse deflection of cables, etc. A schematic of the problem (for axial deformation of a rod) is shown in Fig. 5.1. In this case d d a(x) +c dx dx

(5.21)

f = q(x)

(5.22)

Aφ − f = 0 ∀x ∈ Ω

(5.23)

A=− and Hence, we can write (5.19) as

In this case, we can show that the differential operator A is

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

227

(i) linear (ii) symmetric when φ0 = 0 and P = 0. However, we note that in (Aφ, v) = (φ, A∗ v) + hAφ, viΓ

(5.24)

A∗ = A holds and the concomitant hAφ, viΓ only becomes zero when φ0 and P are both zero. The physics of the problem in Fig. 5.1 (a) can be idealized as shown in Fig. 5.1 (b) due to the fact that φ = φ(x). That is, a line representation of ¯ = [0, L] is justified without any additional assumptions. the domain Ω 5.2.1.1 Discretization ¯ = [0, L] Figure 5.1 (c) shows a typical discretization of the domain Ω using two-node line elements with element lengths of h1 , h2 , h3 , h4 . A typical ¯ e = [xe , xe+1 ] isolated from the discretization is element e with domain Ω shown in Fig. 5.1 (d). A map of the element e in the natural coordinate space ξ is shown in Fig. 5.1 (e) in which the element is mapped into a two unit length with the origin of the ξ coordinate system located at the center ¯e → Ω ¯ ξ = [−1, 1]. We discuss the details of the validity of the element, i.e. Ω of the choice of the two-node element later. Let h = max(he )

(5.25)

e

in which h is referred to as the characteristic length of the discretization 5.2.1.2 Integral form using GM/WF (weak form) of the BVP for ¯e an element e with domain Ω We consider (Aφeh − f, v)Ω¯ e in which v = δφeh . (Aφeh

xZe+1

(Aφeh − f )v dx

− f, v)Ω¯ e =

(5.26)

xe

or (Aφeh −f, v)Ω¯ e

xZe+1

= xe

d − dx

xZe+1 xZe+1   dφeh e a(x) v dx+ c φh v dx− qv dx (5.27) dx xe

xe

228

SELF-ADJOINT DIFFERENTIAL OPERATORS

y q(x)   P = a(x) dφ dx

x

x=L

L (a) Schematic of the problem

y q(x) φ = φ0 = 0

x

  P = a(x) dφ dx

L

x=L

(b) Mathematical idealization of the problem

y

element length (element characteristic length) h1

1

h2

1

h3

2

2

h4

3

3

4

4

5 grid point

x element number

(c) A four element discretization of the domain Ω

y

1

η

Node numbers 1 and 2 are local node numbers

e

2 x

xe

−1

xe+1

e

ξ

1

he (e) Map of an element e in the natural coordinate system

(d) Isolating an element e from the discretization in (c)

Figure 5.1: Schematic and discretization for problem of section 5.2.1



Transferring one order of differentiation from term in (5.27) (Aφeh

xZe+1

− f, v)Ω¯ e = xe

c φeh v dx

xe

dφeh dx



to v in the first

   dφeh dφeh xe+1 dv a(x) dx − v a(x) dx dx dx xe

xZe+1

+

a(x)

xZe+1

−

qv dx xe

(5.28)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

229

In (5.28) the concomitant hAφeh , viΓe is    dφeh xe+1 e hAφh , viΓe = v a(x) dx xe Using the concomitant in (5.28) we identify PV, SV,EBC and NBC. PV : φeh dφeh dx e ˜1 EBC : φh = φ˜ on some Γ dφe ˜2 NBC : a(x) h = q˜ on some Γ dx SV : a(x)

(5.29)

where φ˜ and q˜ are known on some boundaries Γ˜1 and Γ˜2 of the element e. We note that the boundary term in (5.28) cannot be simplified using the boundary conditions of the BVP (as done in classical methods) due to the fact that the element e is any arbitrary element of the discretization and the BCs can only be used for the whole discretization i.e. the assemblage of the element equations. Let  dφe  − a(x) h = P1e dx xe (5.30)  dφeh  e a(x) = P2 dx xe+1 where P1e and P2e are the secondary variables at nodes 1 and 2 (local node numbers) of element e. We note that the secondary variables P1e and P2e are unknown. Substituting from (5.30) into (5.28) (Aφeh

xZe+1

− f, v)Ω¯ e = xe

dφe dv a(x) h dx − v(xe ) P1e − v(xe+1 ) P2e dx dx xZe+1

c φeh v dx

+ xe

xZe+1

−

qv dx

(5.31)

xe

Collecting the terms containing both φeh and v and those containing only v (Aφeh − f, v)Ω¯ e =

xZe+1

 dv

xe

dx

a(x)

 dφeh + c φeh v dx dx

x  Ze+1  − qv dx + v(xe ) P1e + v(xe+1 ) P2e xe

(5.32)

230

SELF-ADJOINT DIFFERENTIAL OPERATORS

We define B

e

(φeh , v)

xZe+1

 dv

=

le (v) =

xe xZe+1

dx

a(x)

 dφeh + c φeh v dx dx

qv dx + v(xe ) P1e + v(xe+1 ) P2e

(5.33)

(5.34)

xe

or (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(5.35)

Equation (5.35) is the desired weak form of the BVP (5.19). We note that (a) B e (φeh , v) is bilinear (b) B e (φeh , v) is symmetric, B e (φeh , v) = B e (v, φeh ) (c) le (v) is linear in v Due to these properties of B e (·, ·) and le (·), the quadratic functional I e (φeh ) for an element e is possible and is given by 1 I e (φeh ) = B e (φeh , φeh ) − le (φh ) 2

(5.36)

or 1 I e (φeh ) = 2

xZe+1



xe

 dφe 2  h a(x) + c(φeh )2 dx dx

xZe+1

q φeh dx − φeh (xe )P1e − φeh (xe+1 )P2e

−

(5.37)

xe

¯ e and We note that δI e (φeh ) yields the weak form (5.35) over an element Ω δ 2 I e (φeh ) = δ(B e (φeh , v) − le (v)) = B e (v, v)

(5.38)

or δ 2 I e (φeh ) = B e (v, v) xZe+1   dv 2  = + c(v)2 dx > 0 if a > 0, c > 0 a(x) dx xe

(5.39)

231

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

5.2.1.3 Approximation space Vh , test function space V and local approximation φeh First, we note that in GM/WF we begin with (Aφh − f, v)Ω¯ T =

X

(Aφeh − f, v) = 0, φh =

[

e

φeh

(5.40)

e

¯ e , we obtain After integration by parts for an element e with domain Ω (Aφh − f, v)Ω¯ T =

X

(Aφeh − f, v)Ω¯ e =

e

X

B e (φeh ) −

e

X

le (v) = 0

(5.41)

e

We consider the approximation space Vh and test function space V in the following for the BVP as well as the weak form. The local approximation φeh is generally a polynomial of degree p. Thus, ¯ e ). However, the global differentiability of φh is controlled φeh is of class C p (Ω ¯ e as well as the differentiability at the inby differentiability of φeh over Ω S terelement boundaries. Obviously, since φh = e φeh , the lower of these two is the global differentiability of φh . We reflect this in defining the class of φeh . For example, if φeh is a polynomial of degree p, but if at the interelement boundaries only φh is continuous then φh is of class C 0 and we say that ¯ e ). Thus, the class of φh is reflected in defining the class of φe . φeh ∈ C 0 (Ω h Vh for BVP Since the highest order of the derivative of φ in the BVP is two (2m = 2), ¯ T to be admissible in the BVP in the pointwise sense, the for φh over Ω following must hold: ¯ T ), k ≥ 3 hence φe ∈ Vh ⊂ H k (Ω ¯ e ), k ≥ 3 φh ∈ V h ⊂ H k (Ω h In which k = 3 corresponds to the minimally conforming space. For the choice of k ≥ 3, (Aφh − f, v)Ω¯ T is in the Riemann sense and all three forms in (5.41) are precisely equivalent. That is (Aφh − f, v)Ω¯ T

⇔ R

X e

(Aφeh − f, v)Ω¯ e

⇔ R

X


B e (φeh , v) − le (v) (5.42)

e

where R indicates that the integrals are Riemann. Since v = δφh , or v = δφeh , v ∈ Vh also holds.

232

SELF-ADJOINT DIFFERENTIAL OPERATORS

Vh of one order lower than minimally conforming for the BVP If we choose Vh to be one order lower than that of the BVP, i.e. if we choose the order of the space Vh to be k = 2. then X X
e e e e 8 (Aφ − f, v) B (φ , v) − l (v) (Aφh − f, v)Ω¯ T 8 e ¯ h h Ω → → L

e

L

e

|

{z

}

R

(5.43) where L indicates that the integrals are Lebesgue. For this choice, the last ¯ T as it form (i.e. weak form) in (5.43) holds in the Riemann sense over Ω e e contains only the first order derivatives of φh and since φh , φh are of class ¯ T to Ω ¯ e in Lebesgue sense and C 1 . For this choice it is possible to go from Ω then to the weak form but not possible to return back to the integral form ¯T . over Ω ¯ e) Vh of order one, i.e. local approximations φeh of class C 0 (Ω If we just examine the weak form B e (φeh , v) − le (v), we note that it only contains first order derivatives of the dependent variable and the test function as the highest orderPderivatives. Thus, if we choose φeh of class C 0 , i.e. e (v)) is in the Lebesgue sense. In e e ¯ e ) space, then in H 1 (Ω e (B (φh , v) − l P this case, obviously (Aφh − f, v)Ω¯ T and e (Aφeh − f, v)Ω¯ e are not defined and symbolically we can write X X
B e (φeh , v) − le (v) (5.44) (Aφh − f, v)Ω¯ T < (Aφeh − f, v)Ω¯ e < e

e

|

{z L

}

¯ e ), φh ∈ Vh ⊂ H 1 (Ω ¯ T ) all equivalences in That is, when φeh ∈ Vh ⊂ H 1 (Ω (5.44) are lost. A remark on the notation The definite integrals are always over the closed domain. For example, a line integral of f (x) for a ≤ x ≤ b is independent of the path and only depends upon a and b. Thus, the notation used in the book is correct (i.e. ¯ e, Ω ¯ T and not over Ωe , ΩT ). However, we note that the integrals are over Ω integrals can be in the Riemann or Lebesgue sense depending upon the choice of the approximation space Vh . To emphasize this fact we could change the ¯ e to Ωe meaning that these could have resulted from Lebesgue integrals over Ω measures. This may be the case in most writings on the subject. However, ¯ e, Ω ¯T in this book we adhere to the correct notation, i.e. integrals over Ω etc. and not over Ωe , ΩT etc.

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

233

¯ e to Ω ¯ξ 5.2.1.4 Local approximation φeh and mapping Ω From the weak form (5.32)–(5.35), we note that an approximation of φeh of φ of class C 0 (i.e. k = 1) must at least be differentiable once, hence, a ¯ e is the lowest possible requirement. Though linear approximation of φ over Ω such choice is completely inadmissible based on minimally conforming space for this BVP (k = 3), however we continue with this choice to illustrate various procedural details as these become much simplified with this choice. Based on the interpolation theory in chapter 8, a two-node linear element may be chosen to satisfy these requirements. Let φe1 and φe2 be the values of φ at the two nodes (local node numbers 1 and 2) of a typical element e (see Fig. 5.1). Since the element is a line segment, its map in the natural coordinate space ξ will also be a line segment ¯e of length two units, that is, a stretch mapping suffices for mapping of Ω ξ ¯ into Ω . That is 1 − ξ  1 + ξ  x(ξ) = xe + xe+1 (5.45) 2 2 and 1 − ξ  1 + ξ  φeh (ξ) = φe1 + φe2 = N1 (ξ) φe1 + N2 (ξ) φe2 (5.46) 2 2 or 2 2 X X e e Ni (ξ) φi = Ni (ξ)δie = [N ]{δ e } (5.47) φh (ξ) = i=1

i=1

where [N ] = [N1 , N2 ] is a row matrix of approximation functions and {δ e }T = [φe1 , φe2 ]. Equation (5.45) describes the mapping of points from the ξ space ¯ e in which φe or δ e to the x space. φeh is the local approximation of φ over Ω i i e or {δ } are called degrees of freedom for an element e. Using (5.45) dx =

dx dξ = J dξ dξ

(5.48)

where J is called the Jacobian of transformation. For this particular mapping (5.45), we have dx xe+1 − xe he dx = = = (5.49) dξ 2 2 he being the length of the element e in the Cartesian coordinate space. We dφe i also note that in the weak form, we require dxh , i.e. dN dx (i = 1, 2). Since Ni = Ni (ξ), we can write

Hence

dNi dNi dx dNi = =J , i = 1, 2 dξ dx dξ dx

(5.50)

dNi 1 dNi = , i = 1, 2 dx J dξ

(5.51)

234

SELF-ADJOINT DIFFERENTIAL OPERATORS

5.2.1.5 Element equations ¯e Recall the weak form of the BVP over Ω (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(5.52)

or xZe+1

 dv

(Aφeh − f, v)Ω¯ e =

xe xZe+1

dx

a(x)

 dφeh + c φeh v dx dx 

(5.53)

φeh = N1 (ξ) φe1 + N2 (ξ) φe2 , δφeh = v = Nj (ξ), j = 1, 2

(5.54)

−



qv dx − v(xe ) P1e − v(xe+1 ) P2e

xe

We have dφeh

dNj dN1 e dN2 e dv φ1 + φ2 , = , j = 1, 2, dx = J dξ (5.55) dx dx dx dx dx Substituting from (5.54) and (5.55) into (5.53) and changing the limits of integration to (−1, 1) yield (Aφeh

=

Z1 

− f, v)Ω¯ e =

2 2 X X   dNj dNi e  a(ξ) φi + c Ni φei Nj J dξ dx dx i=1

−1

Z1 −

i=1

q(ξ)Nj (ξ)J dξ − Nj (−1) P1e − Nj (1) P2e

(5.56)

−1

for j = 1, 2. We can expand (5.56) for j = 1, 2. This gives the two relations (Aφeh

− f, v)Ω¯ e =

Z1 

2 2 X X   dN1 dNi e  a(ξ) φi + c Ni φei N1 J dξ dx dx i=1

−1

Z1 −

i=1

q(ξ)N1 (ξ)J dξ − N1 (−1) P1e − N1 (1) P2e

(5.57)

−1

and (Aφeh

− f, v)Ω¯ e =

Z1 

2 2 X X   dN2 dNi e  a(ξ) φi + c Ni φei N2 J dξ dx dx i=1

−1

Z1 − −1

i=1

q(ξ)N2 (ξ)J dξ − N2 (−1) P1e − N2 (1) P2e

(5.58)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

235

Equations (5.57) and (5.58) can be arranged in matrix and vector form to give the following element equations (Aφeh − f, v)Ω¯ e = [K e ]{δ e } − {F e } − {P e }

(5.59)

where Kij of [K e ], δie of {δ e }, Fie of {F e } and {P e } are given by e Kij

=

Z1 

 dNj dNi a(ξ) + c Ni Nj J dξ, i, j = 1, 2 dx dx

−1

Fie

Z1 =

q(ξ) Ni J dξ, i = 1, 2

−1 e δi = φei , e

{P } =

{P1e ,

(5.60)

i = 1, 2 P2e }T

because

( N1 (−1) = 1, N1 (1) = 0 N2 (−1) = 0, N2 (1) = 1

For this particular choice of local approximation (5.54), we have dN1 1 dN2 1 =− , = dξ 2 dξ 2 dN1 1 dN1 1  1 1 = = − =− dx J dξ he /2 2 he 1 dN2 1 1 1 dN2 = = = dx J dξ he /2 2 he

(5.61)

Substituting from (5.61) into (5.60) and assuming a(ξ) = ae , c = ce and q(ξ) = q e constant for an element e and evaluating the integrals, we obtain       e  q e he 1 ae 1 −1 ce he 2 1 F1 e e (5.62) , {F } = = [K ] = + 1 F2e he −1 1 6 1 2 2 We note that if a = a(x), c = c(x) and q = q(x) then using the mapping x = x(ξ), we can easily obtain a = a(ξ), c = c(ξ) and q = q(ξ) which can then be used in (5.60) to evaluate the coefficients of [K e ] and {F e }. The element matrix [K e ] and the vectors {F e } and {P e } are valid for each element of the discretization; that is, by using e = 1, . . . , M we have element matrices [K e ] and the vectors {F e } and {P e } for each element of the discretization. 5.2.1.6 Assembly of element equations and computation of the solution For illustrating the various steps clearly, we consider a uniform discretization of four elements (see Fig. 5.2) with ae = a, ce = 0, q e = q, e =

236

SELF-ADJOINT DIFFERENTIAL OPERATORS

1, 2, . . . , 4. For this case, he = L/4 = h (e = 1, 2, . . . , 4). For each of the four elements of the discretization (noting that ce = 0 and using the local node numbers) we have the following element equations:

(Aφ1h

− f, v)Ω¯ 1

(Aφ2h − f, v)Ω¯ 2 (Aφ3h

  1  1  1 1 K1 K11 φ1 F1 P1 12 = 1 K1 1 − F1 − P1 ; K21 φ 22 2 2 2  2   2  2  2 2 K11 K12 φ1 F1 P1 = − − ; 2 K2 K21 φ22 F22 P22 22  3   3  3  3 3 K11 K12 φ1 F1 P1 = − − ; 3 3 3 3 K21 K22 φ2 F2 P23  4   4  4  4 4 K11 K12 φ1 F1 P1 = 4 − F4 − P4 ; 4 K4 K21 φ 22 2 2 2 

− f, v)Ω¯ 3

(Aφ4h − f, v)Ω¯ 4

1 1 K21 = K12 2 2 K21 = K12

(5.63) 3 K21

=

3 K12

4 4 K21 = K12

In the matrix representation of the weak forms in (5.63), the element local node numbers are used to identify dofs at the element nodes that are only intrinsic to each element. y q = constant

1 1 x

2

2

3

3

4

4

P

5

(a) A four element discretization of the domain Ω

1

2

3

4

1

2

1

2

1

2

1

2

φ11

φ12

φ21

φ22

φ31

φ32

φ41

φ42

P11

P21

P12

P22

P13

P23

P14

P24

F11

F21

F12

F22

F13

F23

F14

F24

(b) Primary variables, secondary variables and {F e } vector at each element node in the element local node numbering system. (he = L/4 = h) Figure 5.2: A four-element uniform discretization of problem of section 5.2.1

237

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

5.2.1.7 Inter-element continuity conditions on PVs or dependent variables For the discretization shown in Fig. 5.2 (a), we have four elements and five grid points, one through five. At each grid point, the function values φi ; i = 1, . . . , 5 are the quantities of interest. Thus, if the assembly of the four elements of Fig. 5.2 (b) is to yield Fig. 5.2 (a) then the following interelement behaviors of φeh must hold. These are called inter-element continuity conditions on φeh or primary variables: at node 1 of element 1 ) at node 2 of element 1 at node 1 of element 2 ) at node 2 of element 2 at node 1 of element 3 ) at node 2 of element 3 at node 1 of element 4 at node 2 of element 4

:

φ11 = φ1

:

φ12 = φ21 = φ2

:

φ22 = φ31 = φ3

:

φ32 = φ41 = φ4

:

φ42 = φ5

(5.64)

When the inter-element continuity conditions are substituted into (5.63), we obtain the following for elements one through four in which the connections of the elements to their neighbors are clearly evident due to the unique function values at common nodes: (Aφ1h

− f, v)Ω¯ 1

(Aφ2h − f, v)Ω¯ 2 (Aφ3h

    1  1 1 K1 φ1 P1 K11 F1 12 = − 1 K1 1 − P1 ; K21 φ F 2 2 22 2     2  2  2 2 P1 F1 K11 K12 φ2 − = 2 − P2 ; 2 K2 F K21 φ 3 2 2 22  3     3  3 3 K11 K12 φ3 F1 P1 = − 3 K3 3 − P3 ; K21 φ F 4 22 2 2  4     4  4 4 K11 K12 φ4 F1 P1 = − − ; 4 K4 K21 φ5 F24 P24 22 

− f, v)Ω¯ 3

(Aφ4h − f, v)Ω¯ 4

1 1 K21 = K12 2 2 K21 = K12

(5.65) 3 K21

=

3 K12

4 4 K21 = K12

5.2.1.8 Rules for assembling element matrices and vectors We note that (Aφh − f, v)Ω¯ T =

X e

(Aφeh − f, v)Ω¯ e =

X e

B e (φeh , v) −

X e

le (v) = 0 (5.66)

238

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which le (v) = le (v) + (f, v)Ω¯ e . le (v) is the concomitant containing sece

e

ondary variables. However

B e (φeh , v) = [K e ]{δ e }

(5.67)

le (v) = {F e } + {P e } Therefore, after substituting (5.67) into (5.66), we obtain X X X [K e ]{δ e } = {F e } + {P e } e

e

hX

i X X {P e } {F e } + K e {δ} = e

e

in which {δ} =

S

e {δ

[K] =

e}

X

(5.68)

e

(5.69)

e

denoting

[K e ], {F } =

e

X

{F e }, {P } =

e

X

{P e }

(5.70)

e

we obtain the following from (5.69) [K]{δ} = {F } + {P }

(5.71)

¯ T and, hence are referred to as assembled equaEquations (5.71) hold for Ω tions. (5.70) show that [K e ], {F e } and {P e } need to be added or assembled to obtain (5.71). For the four element discretization considered here, {δ} = [φ1 φ2 φ3 φ4 φ5 ]T ; that is, (5.71) represents a system of five linear simultaneous algebraic equations in unknowns φi (i = 1, . . . , 5). We also note that {P }, the assembled vector of secondary variables, is also unknown at this point. Method I We note that element equations (5.65) must be added based on (5.66) or (5.69) to yield (5.71). However, in (5.65) the unknowns in the element equations, i.e. nodal values φi , change from element to element. Thus, if we are to entertain a brute force matrix addition then we must ensure that equations for each element are expanded to contain all of the five unknowns, φ1 , . . . , φ5 . This can be easily done by introducing identities for those dofs that are not present in the element equations keeping in mind to maintain the same order of φ1 , . . . , φ5 for each element of the discretization. The result is that we can write    1  1  1 1 0 0 0 φ F1  P1  K11 K12      1         1 1 1 K 1 0 0 0      K21   P φ F       2 2 22 2   1   (5.72) (Aφh − f, v)Ω¯ 1 =  0 0 0 0 0 φ3 − 0 − 0             0 0 0 0 0  φ 0 0                4   0 0 0 0 000 φ5

239

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

(Aφ2h − f, v)Ω¯ 2

(Aφ3h − f, v)Ω¯ 3

(Aφ4h − f, v)Ω¯ 4

       0 0 0 00  0 0  φ1            2 K 2 0 0  2    0 K11   φ F P  2   1   12   12   2 K 2 0 0 φ 2 2 0 K F P − = − 3 21 22 2 2           0 0 0 0 0  φ4  0 0                   0 0 0 00 φ5 0 0        00 0 0 0  0 0   φ1            0 0 0 0 0       φ 0 0       2   3 3 3 3  φ3 − F 0 0 K K 0 P = − 11 12  1 1     3 3       0 0 K 3 K 3 0  φ F P       4 21 22 2 2             00 0 0 0 φ5 0 0        000 0 0  0 0   φ1            0 0 0 0 0           φ2   0   0   = 0 0 0 0 0  φ3  −  0  −  0  0 0 0 K 4 K 4   φ    F14    P14   11 12    4    4    4  4 4 0 0 0 K21 K22 φ5 F2 P2

(5.73)

(5.74)

(5.75)

Equations (5.72)–(5.75) can now be added in a straight forward manner (just plain matrix and vector addition) keeping in mind that {F e } add to {F e } and {P e } add to {P e }. The result is (Aφh − f, v)Ω¯ T =

X

(Aφeh − f, v)Ω¯ e

e

 1  1 K11 K12 0 0 0       1 φ1  F11  P11   1 K22    1       0  K21 +K 2 K12 0    1 + F 2 1 + P 2       φ F P         11 2 2 1 2 1 2   K22 3 2 3 2 2 3 =  0 K21 +K 3 K12 0  φ3 − F2 + F1 − P2 + P1 = 0 11      φ4      3 P23 + P14  K22     F23 + F14  3 4        0 K21 +K        0 4 K12   4 4 11 φ F P 5 2 2 4 4 0 0 0 K21 K22 (5.76) Columns

φ1

φ2

Row φ1

1 K11

1 K12

Row φ2

1 K21

1 K22 2 +K11

Row φ3

0

2 K21

Row φ4

0

0

Row φ5

0

0

φ3

φ4

φ5

0

0

0

0

0

3 K12

0

2 K12 2 K22

3 +K11 3 K21

0

3 K22

4 +K11 4 K21

4 K12 4 K22

     φ1                   φ 2        

φ

3              φ   4             φ5  

=

F11

P11

Row φ1

F21 + F12

P21 + P12

Row φ2

P22 + P13

Row φ3

P23 + P14

Row φ4

F22 + F13 F23 + F14 F24

+

P24

Row φ5

(5.77) ¯T . These are the assembled algebraic equations for the whole discretization Ω

240

SELF-ADJOINT DIFFERENTIAL OPERATORS

Method II In this example the differential operator is self-adjoint, hence, the quadratic functional I(φh ) is possible I(φh ) =

X

I(φeh )

=

e

X 1 e

2

B

e

(φeh , φeh )

−l

e



(φeh )

(5.78)

After substituting inter-element continuity conditions in (5.78) we note that I = I(φ1 , . . . , φ5 ), hence, minimization of I implies ∂I = 0, i = 1, 2, . . . , 5 ∂φi

(5.79)

(5.79), when arranged in the matrix and vector form, yields exactly the same equations as in (5.77). Method III (Preferred) From method I (and likewise method II) we observe that the assembled equation (5.77) can be obtained in a more prudent and efficient manner. From the element equations (5.65), we note that unknown function values at the nodes vary from element to element. Thus, for each element we can identify its rows and columns by using the degrees of freedom that appear in it. In this approach, for element 1 we identify the rows and column by φ1 and φ2 and similarly for the remaining three element. Thus, symbolically we can write φ1

φ2

Row φ1

1 K11

1 K12

      φ1  

Row φ2

1 K21

1 K22

    φ2  

φ2

φ3

Row φ2

2 K11

2 K12

      φ2  

Row φ3

2 K21

2 K22

    φ3  

φ3

φ4

Row φ3

3 K11

3 K12

      φ3  

Row φ4

3 K21

3 K22

    φ4  

Columns Element 1:

Columns

Element 2:

Columns Element 3:

      F11  

−  

−

  1     Row φ1  P1 

(5.80)

  F1  2

    P1   Row φ2 2

      F12  

  2     P1   Row φ2

−   

−

(5.81)

  F22 

    P2   Row φ3 2

      F13  

  3     P1   Row φ3

−  

 F3   2

−

    P3   Row φ4 2

(5.82)

241

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

Columns

φ4

φ5

Row φ4

4 K11

4 K12

      φ4  

Row φ5

4 K21

4 K22

    φ5  

Element 4:

      F14  

−   

−

  F24 

  4     P1   Row φ4     P4   Row φ5 2

(5.83)

We realize that the assembled equations must correspond to the degrees of freedom φ1 , . . . , φ5 . Hence, the assembled [K] will be a (5 × 5) matrix and the assembled {F } and {P } would be (5 × 1) vectors. We set aside a (5 × 5) (initialized to zero) space for assembly to obtain [K] and (5 × 1) spaces for assembly to obtain {F } and {P } and identify rows and columns by φ1 , . . . , φ5 (in any preferred order) as labels (just like the element matrices and vectors) giving us the following setup ready for assembly of element equations. φ1

φ2

φ3

Row φ1

0

0

0

0

0

Row φ2

0

0

0

0

0

Row φ3

0

0

0

0

0

Row φ4

0

0

0

0

0

Row φ5

0

0

0

0

0

Columns

φ5

φ4

     φ1                   φ 2        

φ

3             φ   4             φ5  

=

0

0

Row φ1

0

0

Row φ2

0

Row φ3

0

0

Row φ4

0

0

Row φ5

0

+

space for [K]

space for {P } space for {F }

(5.84) Let us consider element one, i.e. [K 1 ], {F 1 } and {P 1 } in (5.80) based on the 1 corresponding to row and column labels for the coefficients. That is, K11 1 row φ1 and column φ1 in [K ] is added to the corresponding position (row 1 to row φ , column φ ; K 1 to row φ1 , column φ1 ) in [K] and likewise K12 1 2 21 1 φ2 , column φ1 ; and K22 to row φ2 , column φ2 locations in [K]. Similarly, F11 and F21 corresponding to rows φ1 and φ2 in {F 1 } were added to the corresponding row locations in {F }. {P 1 } is assembled likewise. At the end of the assembly of (5.80) in (5.84), the locations in (5.84) now contain Columns

φ1

φ2

φ3

φ4

Row φ1

1 K11

Row φ2

φ5

1 K12

0

0

0

1 K21

1 K22

0

0

0

Row φ3

0

0

0

0

0

Row φ4

0

0

0

0

0

Row φ5

0

0

0

0

0

     φ1                   φ 2        

φ

3              φ   4             φ5  

=

F11

P11

Row φ1

F21

P21

Row φ2

0

Row φ3

0

0

Row φ4

0

0

Row φ5

0

+

(5.85)

242

SELF-ADJOINT DIFFERENTIAL OPERATORS

Equations for element two (5.81) are now assembled in (5.85) keeping in mind that rows and columns of this element are identified (or labeled) by φ2 and φ3 . After assembly of the equations for element two in (5.85), we have Columns

φ1

φ2

Row φ1

1 K11

1 K12

Row φ2

1 K21

1 K22

2 +K11

Row φ3

0

2 K21

Row φ4

0

0

Row φ5

0

0

φ3

φ4

φ5

0

0

0

0

0

2 K12 2 K22

0

0

0

0

0

0

0

0

      φ   1                 φ 2        

φ

3              φ   4             φ5  

=

F11

P11

Row φ1

F21 + F12

P21 + P12

Row φ2

F22

P22

Row φ3

0

0

Row φ4

0

0

Row φ5

+

(5.86) Likewise, after assembling element three, i.e. (5.82), in (5.86) we have Columns

φ1

φ2

Row φ1

1 K11

1 K12

Row φ2 Row φ3

1 K21

0

1 K22

2 +K11

2 K21

φ3

φ4

φ5

0

0

0

2 K12

0

0

3 K12

0

2 K22 3 +K11

Row φ4

0

0

3 K21

Row φ5

0

0

0

3 K22

0

0

0

      φ   1                 φ 2        

φ

3             φ   4             φ5  

=

F11

P11

Row φ1

F21 + F12

P21 + P12

Row φ2

P22 + P13

Row φ3

F23

P23

Row φ4

0

0

Row φ5

F22 + F13

+

(5.87) Finally, after assembling element four, i.e. (5.83), in (5.87) we obtain the final assembled equations, Columns

φ1

φ2

Row φ1

1 K11

1 K12

Row φ2

1 K21

1 K22 2 +K11

Row φ3 Row φ4 Row φ5

0 0 0

2 K21

0 0

φ3

φ4

φ5

0

0

0

0

0

3 K12

0

2 K12 2 K22

3 +K11 3 K21

0

3 K22

4 +K11 4 K21

4 K12 4 K22

     φ1                    φ 2        

φ

3             φ4                φ5  

=

F11

P11

Row φ1

F21 + F12

P21 + P12

Row φ2

P22 + P13

Row φ3

P23 + P14

Row φ4

F22 + F13 F23 + F14 F24

+

P24

Row φ5

(5.88) In (5.88), we observe that the coefficients in [K] and {F } are known but the coefficients of assembled {P e } in {P } are still unknown. 5.2.1.9 Inter-element continuity conditions on the sum of secondary variables Following the general discussion presented in chapter 4 (repeated here for convenience), we have the following.

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

243

(a) The sum of secondary variables is zero at a node at which there is no externally applied disturbance. (b) If there is an externally applied disturbance at a node, then the sum of the secondary variables at that node must be equal to the externally applied disturbance. (c) The sum of all secondary variables at a node at which PVs are specified (EBCs) is unknown. These conditions are derived based on equilibrium considerations at each node of the discretization. Referring to Fig. 5.2 (a), we can arrive at the following conditions for the assembled secondary variables in {P }. P11 : P21 P22 P23

+ P12 = 0 + P13 = 0 + P14 = 0 P24 = P

not known because φ1 = 0

(EBC)

:

no externally applied disturbance at node 2

:

no externally applied disturbance at node 3

:

no externally applied disturbance at node 4

:

where P is an externally applied disturbance at node 5

(5.89)

5.2.1.10 Imposition of EBCs The essential boundary conditions of the BVP must be described or specified in terms of the dofs at the nodes or grid points of the discretization. Referring to Fig. 5.1 (b) we have φ1 = 0

(EBC)

(5.90)

5.2.1.11 Solving for unknown degrees of freedom ¯ T in nodal dofs φi (i = Based on (5.90), the assembled equation for Ω 1, . . . , 5) have φi (i = 2, . . . , 5) as unknowns since φ1 = 0 is known due to the EBC (5.90). We partition (5.88) in terms of known nodal dofs and unknown nodal dofs        [K11 ] [K12 ] {δ}1 {F }1 {P }1 = + (5.91) [K21 ] [K22 ] {δ}2 {F }2 {P }2 in which {δ}1 = φ1 = 0 known primary degrees of freedom {F }1 and {F }2 both are known {P }1 = P11 unknown due to the fact that {δ}1 = φ1 = 0 is known {δ}2 = {φ2 φ3 φ4 φ5 }T unknown

(5.92)

244

SELF-ADJOINT DIFFERENTIAL OPERATORS

 1    P2 + P12   0    2      P2 + P13 0 {P }2 = = known 3 4  P2 + P1    0      P24 P  1  2  F22 + F13     F2 + F1 {F }2 = known F 3 + F 4    2 4 1  F2

(5.93)

We note that where the φi s are known, the sum of secondary variables is not known. Likewise, where the φi s are not known the sum of secondary variables is known. We first calculate {δ}2 and then {P }1 . Expanding (5.91) [K11 ]{δ}1 + [K12 ]{δ}2 = {F }1 + {P }1 [K21 ]{δ}1 + [K22 ]{δ}2 = {F }2 + {P }2

(5.94)

From the second set of equations in (5.94) in which the right side is completely known, we can write [K22 ]{δ}2 = −[K21 ]{δ}1 + {F }2 + {P }2

(5.95)

(5.95) represents a system of linear simultaneous equations in {δ}2 , from which {δ}2 can be calculated using elimination methods (such as Gauss elimination). Symbolically, we write the following from which {δ}2 can be calculated:   {δ}2 = [K22 ]−1 −[K21 ]{δ}1 + {F }2 + {P }2 (5.96) knowing {δ}2 , {P }1 can be calculated using the first set of equations in (5.94), {P }1 = [K11 ]{δ}1 + [K12 {δ}2 − {F }1 (5.97) ¯ T are completely Hence, the values of φ at the nodes of the discretization Ω 1 known as well as the secondary variables {P }1 = P1 at node one where φ = φ1 = 0 (EBC). 5.2.1.12 Special case: numerical study Consider a, a constant, c = 0 and q(x) = q e = q, a constant, then for the uniform four element discretization we have, using he = h = L/4    1    φ1  1 −1 0 0 0  1 P1               −1 2 −1 0 0        φ 0 2       2  4a  qL  0 −1 2 −1 0 φ3 = 0 2 + (5.98)  L  8  φ4    2   0  0 0 −1 2 −1                    0 0 0 −1 1 φ5 1 P

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

245

The partitioned equations are    φ2   φ3  qL 4a 4  {1}φ1 + a −1 0 0 0 {1} + P11 {1} = φ4   L L 8    φ5

(5.99)

and          2 −1 0 0  φ2  2 −1 0                      4 qL 2 4 −1 2 −1 0 φ3 0 0 a = + φ1 + a  0 −1 2 −1  φ  2  0 0 L  L 8      4            0 0 −1 1 φ5 1 0 P

(5.100)

Since {φ1 } = {0}, we have for {δ}2 using (5.100)        −1  2 −1 0 0 2 0 φ2                    L −1 2 −1 0  qL 2 0  φ3  + =  2 0 φ4    4a  0 −1 2 −1  8              0 0 −1 1 1 P φ5

(5.101)

     1 2 0             0  qL 2 0   +  3  8  2 0          4 1 P

(5.102)

or    1 φ   2      L 1 φ3 = 1 φ   4a   4  1 φ5

1 2 2 2

1 2 3 3

or       7 φ   1   2     PL        qL2  12 2 φ3 + = φ  32a  15 3  4a            4  16 4 φ5

(5.103)

And using (5.99) we have P11 = −

qL 4a + (−φ2 ) 8 L

(5.104)

or P11 = −

qL 4a  7qL2 P L  qL 7qL + + =− − −P 8 L 32a 4A 8 8

(5.105)

P11 = −qL − P

(5.106)

or

Thus, (5.103) and (5.106) provide the complete solution.

246

SELF-ADJOINT DIFFERENTIAL OPERATORS

5.2.1.13 Post-processing of solution Knowing the nodal values of φ, i.e. φi ; i = 1, . . . , 5, we can now postprocess the solution for each element of the discretization, i.e. using element local approximation functions (in natural coordinate system) and the nodal values of the solution, we can write the following. 1 − ξ  1 + ξ   1 + ξ  7qL2 P L  Element 1: φ(ξ) = φ1 + φ2 = + , 2 2 2 32a 4a −1≤ξ≤1 0≤x≤L/4

1 + ξ   1 − ξ  7qL2 P L  φ2 + φ3 = + 2 2 2 32a 4a  1 + ξ  12qL2 2P L  −1≤ξ≤1 + , L/4≤x≤L/2 + 2 32a 4a 1 − ξ  1 + ξ   1 − ξ  12qL2 2P L  Element 3: φ(ξ) = φ3 + φ4 = + 2 2 2 32a 4a  1 + ξ  15qL2 3P L  −1≤ξ≤1 + + , L/2≤x≤3L/4 2 32a 4a 1 − ξ  1 + ξ   1 − ξ  15qL2 3P L  Element 4: φ(ξ) = φ4 + φ5 = + 2 2 2 32a 4a  1 + ξ  16qL2 4P L  −1≤ξ≤1 + , 3L/4≤x≤L (5.107) + 2 32a 4a Likewise, the derivatives of φ with respect to x can also be obtained for each element of the discretization (note that J = h/2 = L/8). dφ 1 dφ 4  7qL2 P L  −1≤ξ≤1 Element 1: = = + , 0≤x≤L/4 dx J dξ L 32a 4a dφ 4  5qL2 P L  −1≤ξ≤1 Element 2: , L/4≤x≤L/2 = + dx L 32a 4a (5.108) dφ 4  3qL2 P L  −1≤ξ≤1 Element 3: = + , L/2≤x≤3L/4 dx L 32a 4a dφ 4  qL2 P L  −1≤ξ≤1 = + Element 4: , 3L/4≤x≤L dx L 32a 4a Element 2: φ(ξ) =

1 − ξ 

5.2.1.14 Analytical solution and comparison with finite element solutions When a and q are constants and φ = 0 at x = L and a dφ dx = P at x = L, the analytical solution φt is given by  P q  2Lx − x2 + , 0 ≤ x ≤ L (5.109) φt = 2a a
P dφt q = L−x + , 0≤x≤L (5.110) dx a a

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

247

First, we compare φ and φt at the nodes of the discretization and at the mid points of the elements. (1) Table 5.1 shows a comparison of φt and the finite element solution φ. We note that the finite element solution matches exactly with the analytical solution at the nodes, but is in error in the interior of each of the four elements due to the fact that φt is parabolic in x whereas the finite element solution is linear over each of the four elements. Table 5.1: Comparison of φt and φ

node 1

node 2

node 3

node 4

node 5

x

Analytical solution φt

Finite element solution φ

0

0

0

L 8

3.75 qL2 32 a

L 4

7 qL2 32 a

3L 8

9.75 qL2 32 a

L 2

12 qL2 32 a

5L 8

13.725 qL2 32 a

3L 4

15 qL2 32 a

7L 8

15.75 qL2 32 a

+

7 PL 8 a

L

16 qL2 32 a

+

PL a

1 PL 8 a

+ +

1 PL 4 a 3 PL 8 a

+ +

1 PL 2 a

+

+

5 PL 8 a

3 PL 4 a

3.5 qL2 32 a

+

1 PL 8 a

7 qL2 32 a

+

1 PL 4 a

9.5 qL2 32 a

+

3 PL 8 a

12 qL2 32 a

+

1 PL 2 a

13.5 qL2 32 a 15 qL2 32 a

+ +

5 PL 8 a 3 PL 4 a

15.5 qL2 32 a

+

7 PL 8 a

16 qL2 32 a

+

PL a

(2) In the numerical studies we choose q = 1, a = 1, L = 1, c = 0, and P = 1. Thus a dφ dx = P = 1 at x = L = 1.0. Figure 5.3 shows φt and the finite element solution of class C 0 graphically for the four element discretization with p = 1. (3) Since φ is linear over each element, dφ dx is constant over each element as dφt seen from (5.108), whereas dx in (5.110) is clearly linear in x. Table 5.2 and dφ dφ t Fig. 5.4 show a comparison of dφ dx and dx . We observe that dx has the correct value only at the center of each element. A more disturbing fact is that at all the inter-element boundaries (nodes 2, 3 and 4) dφ dx is discontinuous, a ¯ e is of class C 0 . direct consequence of the fact that φeh for each element Ω (4) With mesh refinement and/or by increasing the degree of approximation of φeh over each element, we hope to achieve progressively diminishing discontinuity of dφ dx at the inter-element nodes. Such behavior is referred to as weak convergence of C 0 solutions to class C 1 .

248

SELF-ADJOINT DIFFERENTIAL OPERATORS

1.6 (GM/WF) 1.4

C0, p=1: 4 el. mesh theoretical

1.2

Solution φ

1 q=1, P=1, L=1, c=0 0.8 0.6 0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.3: Solution φ versus x

Table 5.2: Comparison of x node 1

node 2

node 3

node 4

node 5

Analytical solution φt

dφt dx

and

dφ dx

Finite element solution φ

0

8 qL 8 a

+

P a

7 qL 8 a

+

P a

L 8

7 qL 8 a

+

P a

7 qL 8 a

+

P a

L 4

6 qL 8 a

+

P a

7 8 5 8

qL a qL a

+ +

P a P a

3L 8

5 qL 8 a

+

P a

5 qL 8 a

+

P a

L 2

4 qL 8 a

+

P a

5 8 3 8

qL a qL a

+ +

P a P a

5L 8

3 qL 8 a

+

P a

3 qL 8 a

+

P a

3L 4

2 qL 8 a

+

P a

3 8 1 8

qL a qL a

+ +

P a P a

7L 8

1 qL 8 a

+

P a

1 qL 8 a

+

P a

1 qL 8 a

+

P a

L

P a

249

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

2.4

C0, p=1: 4 el. mesh theoretical q=1, P=1, L=1, c=0

(GM/WF) 2.2

dφ/dx

2 1.8 1.6 1.4 1.2 1 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.4: Plots of

dφ dx

versus x

(5) Even when C 0 solutions are converged weakly to class C 1 , they still are inadmissible in the BVP Aφ − f = 0. (6) It is possible to employ higher degree local approximation functions than linear, but of class C 0 in which case, also, the converged solutions are of weak C 1 class. h-convergence: C 0 : p = 1 and C 0 : p = 2 solutions Next we consider mesh refinement studies (h-convergence) for solutions of class C 0 with p = 1 and 2. First we consider uniform mesh refinement at p = 1 beginning with a two-element mesh. Figure 5.5 shows solution φ versus x and Fig. 5.6 gives plots of dφ dx versus x for various uniform meshes at p = 1. We note that the solution φ agrees reasonably well with the theoretical solution φt for discretization utilizing four or more elements. However, dφ 0 dx exhibits inter-element discontinuity (inherent in C local approximation). dφ Upon mesh refinement the jump in dx at the inter-element boundaries diminishes but is never eliminated. Figures 5.5 and 5.6 also show φ versus x and dφ dx versus x for one and two element (uniform) discretizations with p = 2. Since the theoretical solution is quadratic in x, a single p-version element with p = 2 produces the theoretical solution. With two-element mesh (or more refined mesh), the computed solution remains in exact agreement with the theoretical solution. This also holds for meshes with more than two p-version elements with p = 2.

250

SELF-ADJOINT DIFFERENTIAL OPERATORS

1.6 (GM/WF) 1.4 1.2

Solution φ

1 0.8 0.6 2 el. mesh ; p = 1 4 el. mesh ; p = 1 8 el. mesh ; p = 1 C0, p=1 16 el. mesh ; p = 1 theoretical ; 1 el. mesh ; p = 2 theoretical ; 2 el. mesh ; p = 2

0.4 0.2 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.5: Solution φ versus x

2.4 2 el. mesh ; p = 1 4 el. mesh ; p = 1 8 el. mesh ; p = 1 16 el. mesh ; p = 1 theoretical ; 1 el. mesh ; p = 2 theoretical ; 2 el. mesh ; p = 2

(GM/WF)

0

C , p=1 2.2

dφ/dx

2 1.8 1.6 1.4 1.2 1 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.6: Plots of

dφ dx

versus x

¯ e ), Remarks. (1) We note that with local approximations of class C 0 (Ω the nodal degrees of freedom at the two end nodes of each element are ¯ e ) local approximation, the boundary function values. Thus for C 0 (Ω dφ condition a dφ dx = dx = 1 (in this case) can not be enforced using nodal degrees of freedom. (2) However, in GM/WF a dφ dx =

dφ dx

(as a = 1) is a secondary variable, hence

251

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

in case of GM/WF the condition dφ dx = 1 can be enforced through the secondary variable value equal to one at x = L = 1.0 as we have done in the computations. When the computed solutions are not converged, P = 1 at x = L = 1.0, as secondary variable is satisfied due to the fact that this condition is enforced. This can be confirmed by computing the secondary variables {P e } for the last element containing x = L = 1.0 coordinate using the following. {P e } = [K e ]{δ e } − {F e } dφe

However, when the computed solutions are not converged, dxh computed at x = L = 1.0 using the local approximation φeh (ξ) for the last element containing x = L = 1.0 may not be equal to one. We can confirm this dφe from dxh versus x plots in Fig. 5.4 for the four element discretization. dφe

We note that dxh at x = 1 is not equal to one even though P = 1 is enforced, hence holds. From Fig. 5.6 we note that with C 0 local dφe approximation with p = 1 progressive mesh refinement yields dxh at x = 1 that progressively approaches 1. (3) Thus in case of C 0 local approximation the condition dφ dx = P = 1 at x = 1 is satisfied for all discretization as P = 1 (secondary variable) dφe but only weakly enforced as dxh = 1 at x = 1 as confirmed in Figs. 5.4 dφe

and 5.6. That is as the discretization is progressively refined dxh at x = 1 progressively approaches 1. This is a serious drawback of C 0 local approximations that result in inadequate description of physics in coarser discretizations.

5.2.2 1D steady-state diffusion equation Here we consider a slightly modified form of the BVP used in the previous section with more complex q = q(x) and different boundary conditions. Consider d  dφ  − a = q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (5.111) dx dx with the following BCs: φ(0) = 0,

 dφ  a =0 dx x=L

(5.112)

We consider two cases. Case (a): q(x) = xn , n is a positive integer. Case (b): q(x) = sin nπx, n is a positive integer. In all numerical studies we consider a = 1 and L = 1. We consider finite element processes based on

252

SELF-ADJOINT DIFFERENTIAL OPERATORS

¯ e , we Galerkin method with weak form. For an element e with domain Ω have (following Example 5.2.1): φeh

=

v=

n X

Ni (ξ) δie , δie being nodal degrees of freedom

i=1 δφeh =

(5.113)

Nj (ξ), j = 1, 2, . . . , n

we can write the following using weak form. (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(5.114)

(Aφeh − f, v)Ω¯ e = [K e ]{δ e } − {P e } − {F e }

(5.115)

where e Kij

Z1 a

=

dNi dNj J dξ, i, j = 1, 2, . . . , nnotag dx dx

(5.116)

−1

Fie

Z1 q(ξ) Ni (ξ) J dξ, i = 1, 2, . . . , n

=

(5.117)

−1 e T

{P } = and

{P1e ,

0, ... , 0,

P2e },

P1e

 dφ   dφ  e , P2 = a =− a dx xe dx xe+1

dNi 1 dNi he = , J = , i = 1, 2, . . . , n dx J dξ 2

(5.118)

We remark that specific forms of Ni (ξ) and δie depend upon the choice of local approximation, p-level, and k, the order of the approximation space. In {P e } the two nonzero terms are obviously the secondary variables at the two end nodes of the element. Details of element computations, assembly, solution and post-processing remain the same as in example 5.2.1. In the following we present numerical studies for the two cases. 5.2.2.1 Case (a): a = 1, L = 1, q(x) = xn , n is a positive integer For this choice of parameters, the theoretical solution φt (x) of (5.111) ¯ = [0, L]) with BCs (5.112) is given by (for Ω h i h i 1 1 φt (x) = − (L)n+1 xn+2 + x (5.119) a(n + 1)(n + 2) a(n + 1) If we choose n = 6, then for this choice φt (x) is of class C 8 . Minimally conforming space for the BVP is obviously k = 3 (solutions of class c2 ). We consider the following numerical studies.

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

253

(I) h-refinement with C 0 p-version (p = 2) local approximations (II) h-refinement with C 1 p-version (p = 3, minimum for C 1 ) local approximations (III) h-refinement with C 2 p-version (p = 5, minimum for C 2 ) local approximations Details of the studies, results and discussion are presented in the following. We note that {F e } for an element e can be easily calculated for this choice of q(x) = x6 using (5.115). I. h-refinement with C 0 p-version with p = 2 In this choice of local approximation, the element is a three-node element with nodes 1, 2, 3 located at ξ = −1, 0, 1 and φeh is given by (see chapter 8) φeh (ξ) =

1 − ξ  2

φe1 +

1 + ξ  2

φe3 +

 ξ 2 − 1  d2 φ  2

dξ 2

2

(5.120)

or φeh (ξ) = N1 (ξ) δ1e + N3 (ξ) δ3e + N2 (ξ) δ2e and φeh =

3 X

Ni (ξ) δie

(5.121)

(5.122)

i=1

In this case, [K e ] is a (3 × 3) matrix with {δ e }T = {δ1e δ2e δ3e }

(5.123)

We begin with a two-element uniform mesh and perform uniform mesh refinement, i.e. subdivide each element into two for the next discretization leading to 2, 4, 8, 16, . . . element uniform discretizations. For each case we keep p = 2 fixed and compute solutions. Plots of φ versus x and dφ dx versus x for various discretizations are shown in Figs. 5.7 and 5.8. From Fig. 5.7, we note that the solution φ is reasonably good for all discretizations. In Fig. 5.8, we observe that dφ dx exhibits inter-element discontinuity which diminishes rapidly upon mesh refinement. The values of dφ dx for sixteen-element uniform discretization are quite close to the theoretical values for practical purposes. dφ We note that due to C 0 local approximation a dφ dx = dx = 0 at x = L = 1.0 can only be enforced using secondary variable (NBC) as dφ dx is not a dof at the nodes for C 0 approximation, hence can not be specified as EBC at dφe computed using local approximation for the element x = 1.0. Thus, dxh x=1 dφe containing x = 1.0 will only result in dxh = 0 when the solution is x=1 converged. From Fig. 5.8 we clearly note that computed solutions for 2, 4,

254

SELF-ADJOINT DIFFERENTIAL OPERATORS dφeh dx x=1



= 0 but progressively and 8 element discretizations do not yield dφeh approach dx = 0 upon mesh refinement. For 16 element discretization x=1 its value is close to zero. In this case B(·, ·) is bilinear and symmetricP and l(·) is linear hence the quadratic functional I(φh ) is given by I(φh ) = e I(φeh ) and minimization of I(φh ) yields X X le (v) (5.124) B e (φeh , v) = e

e

which is exactly same as what we obtain using weak form for discretiza¯ T . Thus, since in this case quadratic functional I(φh ) is minimized, tion Ω hence behavior of I(φh ) versus dofs obtained from various discretizations is instructive to examine. Figure 5.9 shows a plot of I(φh ) versus dofs. We note that beyond the eight-element uniform discretization, I(φh ) shows no significant change, whereas we note that dφ dx in Fig. 5.8 still has measurable error for this mesh including jumps at inter-element nodes in dφ dx . This indicates that minimization of I(φh ) is a good overall indicator of equilibrium but may not be as sensitive to local errors in the solution. We note that in this case L2 -norm of E, the residual is not possible due to C 0 nature of the local approximations. 0.14 (GM/WF) 0.12

Solution φ

0.1 0.08 0.06 n

q(x)=x , n=6

0.04

2 el. mesh, p = 2 4 el. mesh, p = 2 8 el. mesh, p = 2 close to theoretical: 16 el. mesh, p = 2 C0, p=2

0.02 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.7: Case (a): solution φ versus x

II. h-refinement with C 1 p-version with p = 3 (minimum for C 1 ) With this choice of local approximation we have a three-node element e with nodes 1, 2, 3 located at ξ = −1, 0 and 1 and the local approximation is

255

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.18 (GM/WF) 0.16 0.14

dφ/dx

0.12 0.1 0.08 0.06

q(x)=xn, n=6

0.04

2 el. mesh, p = 2 4 el. mesh, p = 2 8 el. mesh, p = 2 close to theoretical: 16 el. mesh, p = 2 C0, p=2

0.02 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.8: Case (a):

dφ dx

versus x

-0.00827 2

0

C ,p=2

(GM/WF)

q(x)=xn, n=6

Quadratic functional I

-0.00828 -0.00829 -0.00830 -0.00831 -0.00832 4

-0.00833

8 -0.00834 0

5

10

15

20

25

30

35

dofs

Figure 5.9: Case (a): quadratic functional I versus dofs

given by (see chapter 8) φeh (ξ) = N10 (ξ) φe1 + N11 (ξ)

 dφ 

dx ξ=−1  dφ  + N30 (ξ) φe3 + N31 (ξ) dx ξ=+1

(5.125)

256

SELF-ADJOINT DIFFERENTIAL OPERATORS

or φeh =

4 X

Ni (ξ) δie

(5.126)

i=1

in which N1 (ξ) = N10 (ξ) = N2 (ξ) = N11 (ξ) =

1 − ξ 

+  23 ξ − ξ

 ξ3 − ξ  4  ξ 2 − 1 

J − 4 4  1 + ξ   ξ3 − ξ  N3 (ξ) = N30 (ξ) = − 2 4   3   ξ2 − 1  ξ −ξ + J N4 (ξ) = N31 (ξ) = 4 4

(5.127)

and δ1e = φe1

,

δ2e =

 dφ  dx

ξ=−1

, δ3e = φe3 , δ4e =

 dφ  dx

ξ=1

(5.128)

with J = h2e . In this case the element matrix [K e ] is (4 × 4). The center node of each element has no degrees of freedom, only the end nodes (1 and 3) have two degrees of freedom each. We begin with a two-element uniform discretization and continue uniform mesh refinement leading to 2, 4, 8, . . . element uniform discretizations. For each case we keep p = 3 fixed and compute solutions. ¯ e ) the end nodes of each eleFor local approximations of class C 1 (Ω ment contain φ and dφ dx as nodal degrees of freedom. In this finite element formulation (GM/WF) we have two choices to impose BC a dφ dx = 0 at x = L = 1.0. This BC can be imposed by setting the sum of secondary variables at x = L = 1.0 to zero i.e. P3e = 0 at node three (at x = L = 1.0) of the last element of the discretization. In this approach dφ dx imposed at x = L = 1.0 is a natural boundary condition (NBC). In the second approach dφ dφ dx = 0 at x = L = 1.0 can be imposed by setting the dof dx at grid point located at x = L = 1 to zero. In this approach, specification of dφ dx falls dφ into EBC. Both of these approaches have different consequences. dx = 0 x=1

as EBC (imposed through dof) will be satisfied by the local approximation, hence will hold regardless of the discretization whereas dφ dx x=1= 0 imposed through NBC will only be satisfied upon convergence as shown earlier in case of local approximation of class C 0 . We present results for both cases here. Solutions φ versus x in Fig. 5.10 computed using both choices of BC at x = 1.0 remain almost the same. Figure 5.11(a) shows plots of dφ dx versus x using local approximations when

257

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE



dφ dx x=1=

0 is imposed as NBC. We note that for two element discretization when the solution is not converged, dφ dx x=1= 0 is not satisfied even though secondary variable at x = 1 is zero. Upon mesh refinement dφ = 0 dx x=1

is approached. Figure 5.11(b) also shows plots of dφ dx vs x for various disdφ 1 cretizations at p = 3 for C solutions in which dx = 0 is imposed as x=1 dφ EBC. We clearly observe that dx = 0 holds in this case regardless of the x=1

2

discretization. Figures 5.12(a) and (b) show plots of ddxφ2 versus x for progressively refined discretizations at p = 3 for solutions of class C 1 . Influence of weakly satisfying dφ dx x=1= 0 using NBC is clearly observed. Except the two-element discretization all other discretizations yield good d2 φ values of φ and dφ dx . Even dx2 for the eight-element discretization (and beyond) is in good agreement with the theoretical solution for both cases using dφ NBC and EBC to impose dx = 0. In this case computations of the residx=1 P ual functional I = (E, E)Ω¯ T = e (Aφeh − f, Aφeh − f )Ω¯ e is possible but the integrals are in Lebesgue sense. Computation of I is not significantly in dφ fluenced by the manner in which dx = 0 is imposed i.e. NBC or EBC. x=1 √ Figure 5.17 shows a plot of I versus dofs for both cases in which dφ dx x=1= 0 √ is imposed as NBC and as EBC. Fixed slope of I versus dofs indicates constant convergence rate. 0.14 (GM/WF) 0.12

Solution φ

0.1 0.08 0.06

q(x)=xn, n=6

0.04

2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 32 el. mesh, p = 3 64 el. mesh, p = 3

C1, p=3

0.02 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.10: Case (a): solution φ versus x (for

dφ = dx x=1



0 imposed as NBC or EBC)

258

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.18 (GM/WF) 0.16 0.14

dφ/dx

0.12 0.1 0.08 n

q(x)=x , n=6 0.06 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 32 el. mesh, p = 3 64 el. mesh, p = 3

0.04 0.02

C1, p=3

0 0

0.2

0.4

0.6

0.8

1

0.8

1

Distance x

(a)

dφ = dx x=1



0 imposed as NBC

0.18 (GM/WF) 0.16 0.14

dφ/dx

0.12 0.1 0.08 n

q(x)=x , n=6 0.06 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 32 el. mesh, p = 3 64 el. mesh, p = 3

0.04 0.02

1

C , p=3

0 0

0.2

0.4

0.6 Distance x

(b)

dφ = dx x=1



0 imposed as EBC

Figure 5.11: Case (a):

dφ dx

versus x

III. h-refinement with C 2 p-version with p = 5 (minimum for C 2 ) With this choice of local approximation we have a three-node element with nodes 1,2,3 located at ξ = −1, 0 and 1. The local approximation φeh is

259

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.2 (GM/WF) 0

d2φ/dx2

-0.2

-0.4 n

q(x)=x , n=6 -0.6 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 32 el. mesh, p = 3 64 el. mesh, p = 3

-0.8

C1, p=3

-1 0

0.2

0.4

0.6

0.8

1

0.8

1

Distance x

(a)



d2 φ dx2

= 0 imposed as NBC x=1

0.2 (GM/WF) 0

d2φ/dx2

-0.2

-0.4 n

q(x)=x , n=6 -0.6 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 32 el. mesh, p = 3 64 el. mesh, p = 3

-0.8

1

C , p=3

-1 0

0.2

0.4

0.6 Distance x

(b)



d2 φ dx2

= 0 imposed as EBC x=1

Figure 5.12: Case (a):

d2 φ dx2

versus x

given by (see chapter 8)  dφ   d2 φ  φeh (ξ) =N10 (ξ) φe1 + N11 (ξ) + N12 (ξ) dx ξ=−1 dx2 ξ=−1  dφ   d2 φ  + N30 (ξ) φe3 + N31 (ξ) + N32 (ξ) dx ξ=1 dx2 ξ=1

(5.129)

260

SELF-ADJOINT DIFFERENTIAL OPERATORS

or φeh (ξ) =

6 X

Ni (ξ) δie

(5.130)

i=1

The basis functions Ni (ξ) for this case are described in chapter 8. In this case also the center node of each element has no degrees of freedom. We repeat studies similar to the previous cases for uniform mesh refinement beginning d2 φ d3 φ with the two-element uniform discretization. Plots of φ, dφ dx , dx2 and dx3 versus x are shown in Fig. 5.13–5.16. For solutions of class C 2 , the imposition of the boundary condition dφ dx x=1= 0 as NBC or EBC has virtually no effect on the results. Even the two-element mesh shows good agreement with 3 the theoretical solution except ddxφ3 which improves dramatically with the four-element uniform discretization. For this choice P of elocal approximation functions the residual functional I = (E, E)√= e (Aφh − f, Aφeh − f ) is in Riemann sense. Figure 5.17 shows a plot of I versus dofs. In this√case also, constant slope indicates fixed convergence rate. Higher slope of I versus dofs graph for solutions of class C 2 at p = 5 indicates faster convergence rate compared√to solutions of class C 1 with p = 3. For a given dofs much lower values of I for this case are obvious from Fig. 5.17 compared to solutions of class C 1 with p = 3. From the results √ presented in Fig. 5.17, it is difficult to conclude whether the reduction in I for solutions of class C 2 at p = 5 compared to solutions of class C 1 at p = 3 for a fixed degrees of freedom is due to increase in the order of √ the space or due to increase in p-level or both. Figure 5.18 shows plots of I versus dofs for solutions of class C 1 and C 2 at p = 5 for progressively uniform mesh refinements in which element lengths are halved each time. From the results in Fig. 5.18 √ we note that for a fixed dofs the solutions of 2 have lower values of class C I compared to solutions of class C 1 . Slopes √ of I versus dofs for both cases are almost the same, even though a slight increase in the slope is observed for solutions of class C 2 with progressively refined discretizations. These results decisively demonstrate higher accuracy of the solution of class C 2 compared to C 1 for a fixed dofs. For all practical purposes we can assume approximately same convergence rate in both cases. This behavior shown in Fig. 5.18 holds true for each p-level greater than or equal to 5 (five being minimum p-level for solutions of class C 2 ). 5.2.2.2 Case (b): a = 1, L = 1, q(x) = sin nπx, n = 4 For this choice of parameters the theoretical solution φt (x) ∀x ∈ [0, L] of (5.111) with BCs (5.112) is given by φ=

  1 1 sin nπx + x − cos nπL a(nπ)2 anπ

(5.131)

261

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.14 (GM/WF) 0.12

Solution φ

0.1 0.08 0.06 q(x)=xn, n=6 0.04 2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

C2, p=5

0.02 0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.13: Case (a): solution φ versus x

0.18 (GM/WF) 0.16 0.14

dφ/dx

0.12 0.1 0.08 q(x)=xn, n=6

0.06 0.04

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

0.02

2

C , p=5

0 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.14: Case (a):

dφ dx

versus x

For n = 4 there are two periods of the sine wave of q(x) over [0, L] and d2 φ = − sin nπx is negative of q(x). For this choice of q(x) we also calculate dx2 {F e } using the same procedure as case (a). We perform numerical studies similar to case (a) using progressively refined uniform descretizations. We keep in mind that the BC φ(0) = 0 (EBC in GM/WF) poses no problem and can be imposed using degrees of freedom

262

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.2 (GM/WF) 0

d2φ/dx2

-0.2

-0.4 q(x)=xn, n=6

-0.6

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

-0.8

C2, p=5

-1 0

0.2

0.4

0.6

0.8

1

0.8

1

Distance x

Figure 5.15: Case (a):

d2 φ dx2

versus x

1 (GM/WF) 0

d3φ/dx3

-1 -2 -3 q(x)=xn, n=6 -4 2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

-5

2

C , p=5

-6 0

0.2

0.4

0.6 Distance x

Figure 5.16: Case (a):

d3 φ dx3

versus x

φ at the element nodes regardless of whether the local approximations are of class C 0 or higher classes. However the BC dφ = 0 can only be imposed dx x=1

by setting the secondary variable (NBC) at x = 1 to be zero when using C 0 approximation in which case dφ dx x=1= 0 will only be satisfied upon conver¯ e ), C 2 (Ω ¯ e ), and higher class local gence. As shown earlier in case of C 1 (Ω

263

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

-1

C1, p=3 NBC C2, p=5 1 EBC C2, p=3 C , p=5

(GM/WF) -2 -3

log(√I)

-4 -5 -6 -7 -8 -9 -10 0.5

1

1.5

Figure 5.17: Case (a): ||E||L2 = EBC)

√

2 log(dofs)

2.5

I versus dofs (for

3

dφ = dx x=1



-2

3.5

0 imposed as NBC and

1

C2, p=5 C , p=5

(GM/WF) -3 -4

log(√I)

-5 -6 -7 -8 -9 -10 0.8

1

1.2

1.4

Figure 5.18: Case (a): ||E||L2 =

1.6

1.8 log(dofs)

2

√ I versus dofs (for

2.2

2.4

dφ = dx x=1



2.6

2.8

0 imposed as NBC)

approximations dφ dx x=1= 0 can be imposed either using NBC or EBC. When dφ dx x=1= 0 is imposed using EBC this condition at x = 1 is always satisfied by the local approximation regardless of the discretization. We present ¯ e ) and C 2 (Ω ¯ e ) local approximations using both numerical results for C 1 (Ω approaches of imposing dφ dx x=1= 0 boundary condition. A summary of the results is given in the following.

264

SELF-ADJOINT DIFFERENTIAL OPERATORS

0 ¯e Figures 5.19 and 5.20 show plots of φ and dφ dx versus x for C (Ω ), p = 2 local approximation for progressively refined uniform discretizations. In this ¯ e ) local approximations, the boundary condition dφ = 0 case due to C 0 (Ω dx x=1 can only be imposed using NBC. Figure 5.21 shows a plot of quadratic functional I versus dofs for these solutions. ¯ e ) at p = 3 for Figures 5.22–5.24 show results for solutions of class C 1 (Ω progressively refined uniform discretizations. Figure 5.22 shows graphs of φ ¯ e ), p = 3 for both cases in which dφ = 0 is imposed versus x for C 1 (Ω dx x=1 using NBC and EBC. We do not observe any significant difference in the results regardless of how dφ = 0 is specified. Figure 5.23(a) and (b) dx x=1

2

d φ and Fig. 5.24(a) and (b) show plots of dφ dx versus x and dx2 versus x when dφ dx x=1= 0 is imposed either using NBC or using EBC. We clearly observe that dφ dx x=1= 0 is satisfied for all discretizations when this BC is imposed using EBC (see Fig. 5.23(b)), whereas in Fig. 5.23(a) dφ = 0 is only dx x=1

satisfied when the computed solution is converged. In Figs. 5.24(a) and (b) 2 we observe some differences in ddxφ2 versus x, specially at x = 1.0, for the two cases for coarser discretizations. ¯ e ), Figures 5.25–5.28 show results for local approximations of class C 2 (Ω p = 5 for progressively refined discretizations. Behavior of φ versus x shown in Fig. 5.25 is not affected appreciably by the two choices of defining dφ = 0. Figures 5.26(a) and (b), 5.27(a) and (b), and 5.28(a) and (b) dx x=1

2

3

d φ d φ show graphs of dφ dx , dx2 , dx3 versus x at p = 5 for progressively refined uniform discretizations. We note that the two sets of solutions are very close without much appreciable, measurable difference between them. √ Figure 5.29 shows plots of L2 -norm of E; that is, I versus dofs for 2 solutions of class C 1 at p = 3 and the solutions of class C at p = 5 for both NBC and EBC of imposing dφ dx x=1= 0. We clearly observe that the two choices do not influence these results significantly. From Fig. 5.29 we note that solutions of √ the class C 2 at p = 5 exhibit much higher convergence rate (higher slope of I versus√dofs) compared to solutions of class C 1 at p = 3 as well as lower values of I for a given dofs confirming higher accuracy of the solutions of class C 2 at p = 5 compared to solutions of class C 1 at p = 3. From the results √ presented in Fig. 5.29, it is difficult to conclude whether the reduction in I for solutions of class C 2 at p = 5 compared to solutions of class C 1 at p = 3 for a fixed degrees of freedom is due to increase in the order of √ the space or due to increase in p-level or both. Figure 5.30 shows plots of I versus dofs for solutions of class C 1 and C 2 at p = 5 for progressively uniform mesh refinements in which element lengths are halved each time.

265

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

From the results in Fig. 5.30 √ we note that for a fixed dofs the solutions of 2 1 class √ C have lower values of I compared to solutions of class C . Slopes of I versus dofs for both cases are almost the same, even though a slight increase in the slope is observed for solutions of class C 2 with progressively refined discretizations. These results decisively demonstrate higher accuracy of the solution of class C 2 compared to C 1 for a fixed dofs. For all practical purposes we can assume approximately same convergence rate in both cases. This behavior shown in Fig. 5.30 holds true for each p-level greater than or equal to 5 (five being minimum p-level for solutions of class C 2 ). 0.01 (GM/WF) 0

C0, p=2

-0.01

2 el. mesh, p = 2 4 el. mesh, p = 2 8 el. mesh, p = 2 16 el. mesh, p = 2 64 el. mesh, p = 2

-0.02 Solution φ

q(x)=sin nπx, n=4 -0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.19: Case (b): solution φ versus x

5.2.3 Least-squares finite element formulation Consider the 1D steady state diffusion equation (same as in example 5.2.2). d  dφ  a + cφ = q(x) ∀x ∈ (0, L) = Ω ⊂ R1 (5.132) − dx dx with the following boundary conditions. φ(0) = 0  dφ  (5.133) a =0 dx x=L a, c and q(x) are given data. We consider the least-squares finite element formulation of this BVP based on residual functional. In this case d d A=− a +c dx dx (5.134) f = q(x)

266

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.05

C0, p=2

(GM/WF)

2 el. mesh, p = 2 4 el. mesh, p = 2 8 el. mesh, p = 2 16 el. mesh, p = 2 64 el. mesh, p = 2

0

q(x)=sin nπx, n=4 dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

Distance x

Figure 5.20: Case (b):

dφ dx

versus x

-0.002

0

C ,p=2

(GM/WF)

q(x)=sin nπx, n=4

Quadratic functional I

-0.0025

-0.003

-0.0035

-0.004

-0.0045

-0.005 0

20

40

60

80

100

120

140

dofs

Figure 5.21: Case (b): quadratic functional I versus dofs

and we can write Aφ − f = 0

∀x ∈ Ω

(5.135)

¯ e or Ω ¯ ξ , an element e Let φeh (x) or φeh (ξ) be local approximation of φ over Ω S e ¯T = ¯e ¯e of discretization Ω e Ω . Then the residual function E over Ω can be defined by ¯e E e = Aφeh − f ∀x ∈ Ω (5.136)

267

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.01 (GM/WF) 0

C1, p=3

-0.01

2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3

Solution φ

-0.02

q(x)=sin nπx, n=4

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

Distance x

= 0 as NBC or EBC) Figure 5.22: Case (b): solution φ versus x ( dφ dx x=1

¯ T is constructed using The residual or least-squares functional I for Ω X X I= Ie = (E e , E e )Ω¯ e , existence of I (5.137) e

e

¯ e . First variation of I set to in which I e is the least-squares functional for Ω zero gives necessary conditions. X X X δI = δI e = 2(E e , δE e )Ω¯ e = 2 {g e } = {g} = 0 (5.138) e

e

e

in which {g e } = (E e , δE e )Ω¯ e

(5.139)

The element relations can be derived using {g e }. We note that d  dφeh  a + cφeh − q(x) E e = Aφeh − f = − dx dx d  dv
δE e = − a + cv, v = δφeh dx dx Let φeh

=

n X

Ni (ξ) δie

(5.140) (5.141)

(5.142)

i=1

in which Ni (ξ) are approximation functions and δie are nodal degrees of freedom. We could choose a three-node element with nodes 1, 2, 3 located

268

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.05 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 C1, p=3

(GM/WF) 0

q(x)=sin nπx, n=4 dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

Distance x

(a)

dφ = dx x=1



0 as NBC

0.05 2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 1 C , p=3

(GM/WF) 0

q(x)=sin nπx, n=4 dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

Distance x

(b)

dφ = dx x=1



0 as EBC

Figure 5.23: Case (b):

dφ dx

versus x

at ξ− = 1, 0, and 1. Explicit forms of Ni (ξ) and δie depend upon the class ¯ e ). We discuss of φeh , i.e. the order of the approximation space Vh ⊂ H k (Ω e details a little later. First, based on local approximation φh , we have v = δφeh = Nj (ξ), j = 1, 2, . . . , n

(5.143)

269

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

2 (GM/WF)

C1, p=3

1.5 1

2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 q(x)=sin nπx, n=4

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

Distance x

(a)



d2 φ dx2

= 0 as NBC x=1

2 (GM/WF)

C1, p=3

1.5 1

2 el. mesh, p = 3 4 el. mesh, p = 3 8 el. mesh, p = 3 16 el. mesh, p = 3 q(x)=sin nπx, n=4

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

Distance x

(b)



d2 φ = dx2 x=1

0 as EBC

Figure 5.24: Case (b):

d2 φ dx2

versus x

Substituting φeh and v in {g e } we have gje of {g e }

gje

=

!  X  X n n d dNi e d  dNj  e a δ +c Ni δi −f, − (5.144) − a +c Nj dx dx i dx dx ¯e i=1

i=1

Ω

270

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.01 (GM/WF) 0

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

C2, p=5

-0.01

Solution φ

-0.02

q(x)=sin nπx, n=4

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

Distance x

= 0 as NBC or EBC) Figure 5.25: Case (b): solution φ versus x ( dφ dx x=1

for j = 1, 2, . . . , n. Since the differential operator A is linear, gje (j = 1, 2, . . . , n) can be written as !  n   dN   dN  X d d j i gje = − a + c Ni δie − f, − a + c Nj (5.145) dx dx dx dx ¯e i=1

Ω

which is same as gje

=

X n i=1

(ANi ) δie

 − f, ANj

; j = 1, 2, . . . , n

(5.146)

¯e Ω

or {g e } = [K e ]{δ e } − {F e }

(5.147)

e Kij = (ANj , ANi )Ω¯ e

(5.148)

in which

Fie = (f, ANj )Ω¯ e

(5.149)

Substituting for A   d  dNj  d  dNi  e Kij = − a + c Nj , − a + c Ni , i, j = 1, 2, . . . , n dx dx dx dx ¯e Ω (5.150)     d dNi Fie = f, − , i = 1, 2, . . . , n (5.151) a + c Ni dx dx ¯e Ω

271

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.05 2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

(GM/WF) C2, p=5 0

q(x)=sin nπx, n=4

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

Distance x

(a)

dφ = dx x=1



0 as NBC

0.05 2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

(GM/WF) C2, p=5 0

q(x)=sin nπx, n=4

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

Distance x

(b)

dφ = dx x=1



0 as EBC

Figure 5.26: Case (b):

dφ dx

versus x

Using element map in the natural coordinate space and noting that dx = J dξ, we obtain (i, j = 1, 2, . . . , n)   Z1  d  dNj  d  dNi  e Kij = − a + cNj − a + cNi Jdξ dx dx dx dx

(5.152)

−1

Fie

 Z1  d  dNi  = f − a + c Ni J dξ dx dx −1

(5.153)

272

SELF-ADJOINT DIFFERENTIAL OPERATORS

2 (GM/WF)

C2, p=5

1.5 1

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5 q(x)=sin nπx, n=4

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

Distance x

(a)



d2 φ dx2

= 0 as NBC x=1

2 (GM/WF)

C2, p=5

1.5 1

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5 q(x)=sin nπx, n=4

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

Distance x

(b)



d2 φ = dx2 x=1

0 as EBC

Figure 5.27: Case (b):

Since Ni = Ni (ξ),

d2 φ dx2

versus x

dm Ni 1 dm Ni = , m = 1, . . . dxm J m dξ m

(5.154)

e and F e are explicitly defined. We use gauss quadraand J = h2e . Hence, Kij i ture to calculate numerical values of the coefficients of [K e ] and {F e }. Know-

273

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

20 (GM/WF) 15 10

d3φ/dx3

5 0 -5 q(x)=sin nπx, n=4 -10 2

C , p=5 -15

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

-20 -25 0

0.2

0.4

0.6

0.8

1

Distance x

(a)



d3 φ dx3

= 0 as NBC x=1

20 (GM/WF) 15 10

d3φ/dx3

5 0 -5 q(x)=sin nπx, n=4 -10 2

C , p=5 -15

2 el. mesh, p = 5 4 el. mesh, p = 5 8 el. mesh, p = 5 16 el. mesh, p = 5

-20 -25 0

0.2

0.4

0.6

0.8

1

Distance x



d3 φ = dx3 x=1

(b)

0 as EBC

Figure 5.28: Case (b):

d3 φ dx3

versus x

ing {g e } = [K e ]{δ e } − {F e }

(5.155)

¯ e , we can write the following for the whole discretization for an element Ω T ¯ Ω e X X X {g e } = [K e ]{δ e } − {F e } = 0 (5.156) i=1

e

e

274

SELF-ADJOINT DIFFERENTIAL OPERATORS

0

C1, p=3 NBC C2, p=5 1 EBC C2, p=3 C , p=5

(GM/WF) -1 -2

log(√I)

-3 -4 -5 -6 -7 -8 0.5

1

1.5

2 log(dofs)

2.5

Figure 5.29: Case (b): ||E||L2 =

3

3.5

√ I versus dofs

-1

1

C2, p=5 C , p=5

(GM/WF) -2

log(√I)

-3 -4 -5 -6 -7 -8 0.8

1

1.2

1.4

Figure 5.30: Case (b): ||E||L2 =

1.6

1.8 log(dofs)

2

√ I versus dofs (for

2.2

2.4

dφ = dx x=1



2.6

2.8

0 imposed as NBC)

or X e

[K e ]{δ e } =

X

{F e }

(5.157)

e

or hX e

i [ [K e ] {δ} = {F }; {δ} = {δ e } e

(5.158)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

275

or [K]{δ} = {F }

¯T for Ω

(5.159)

in which [K] =

X

{F } =

X

[K e ]

e

{F e }

(5.160)

e

[K] and {F } are obtained from bly described in example 5.2.1.

[K e ]

and {F e } using usual process of assem-

5.2.3.1 Approximation space Vh Since the highest order of the derivative is two in the operator A and ¯ e ) is minimally conforming choice for which all hence φeh ∈ Vh ⊂ H 3,p (Ω integrals remain Riemann in the entire computational process. In this choice ¯ e ) then all integrals ¯ e ). If we choose φe ∈ Vh ⊂ H 2,p (Ω φeh is of class C 2 (Ω h in the entire least-squares process are in Lebesgue sense. φeh of class C 0 are not admissible in this finite element formulation. 5.2.3.2 Numerical studies In this section we consider numerical studies. We choose the following parameters a = 1, L = 1, c = 0 and q(x) = sin nπx, n = 4. This choice is same as used in example 5.2.2 case (b). This will permit us to compare the solutions from GM/WF with those computed here using least-squares finite element formulations. When a is constant, (5.152) and (5.153) reduce to e Kij

=

Z1 

−a

 d2 N  d2 Nj i + c N −a + c N J dξ j i dx2 dx2

(5.161)

−1

Fie

Z1   d2 Ni = f −a 2 + c Ni J dξ dx

(5.162)

−1

for i, j = 1, 2, . . . , n. Using (5.154) and J = e Kij

Z1  = −1

Fie

   2   2 2 d2 N 2 2 d Ni he j −a + c Nj −a + c Ni dξ he dξ 2 he dξ 2 2

 Z1    2 2 2 d Ni he = f −a + c Ni dξ 2 he dξ 2 −1

he 2

(5.163)

(5.164)

276

SELF-ADJOINT DIFFERENTIAL OPERATORS

e and F e are obtained using gauss quadrature. We Numerical values of Kij i consider the following numerical studies:

I. h-convergence study with φeh of class C 1 using p = 3 II. h-convergence study with φeh of class C 2 using p = 5 III. p-convergence study with φeh of class C 1 for a fixed discretization with p = 3, 4, 5, . . . IV. p-convergence study with φeh of class C 2 for a fixed discretization with p = 5, 6, 7, . . . Results are summarized in the following: ¯ e ) and higher We note that in this formulation for solutions of classes C 1 (Ω the boundary condition dφ = 0 is satisfied regardless of the discretization dx x=1

as it is imposed using nodal degree of freedom dφ dx . Figures 5.31 to 5.33 d2 φ dφ show plots of φ, dx and dx2 versus x for solutions of class C 1 at p = 3 for progressively refined discretizations. Figures 5.34 to 5.37 show plots d2 φ d3 φ 2 of φ, dφ dx , dx2 and dx3 versus x for solutions of class C at p = 5. With progressive mesh refinement, solutions of both classes show convergence to 2 the theoretical solutions. Interelement jumps of ddxφ2 for solutions of class 3

C 1 and those of ddxφ3 for solutions of class C 2 shown in Figs. 5.33 and 5.37 √ progressively diminish with mesh refinement. Figure 5.38 shows plots of I versus dofs for solutions of both classes. Higher convergence rate and better accuracy of the solutions of class C 2 at p = 5 is quite clear from Fig. 5.38. From the results √ presented in Fig. 5.38, it is difficult to conclude whether the reduction in I for solutions of class C 2 at p = 5 compared to solutions of class C 1 at p = 3 for a fixed degrees of freedom is due to increase in the order √ of the space or due to increase in p-level both. Figure 5.39 shows plots of I versus dofs for solutions of class C 1 and C 2 at p = 5 for progressively uniform mesh refinements in which element lengths are halved each time. From the results in Fig. 5.39√we note that for a fixed dofs the solutions of class C 2 have lower values of I compared to solutions of class C 1 . Slopes of √ I versus dofs for both cases are almost the same. These results decisively demonstrate higher accuracy of the solution of class C 2 compared to C 1 for a fixed dofs. For all practical purposes we can assume approximately same convergence rate in both cases. This behavior shown in Fig. 5.39 holds true for each p-level greater than or equal to 5 (five being minimum p-level for solutions of class C 2 ). In the last study we consider a four-element uniform discretization with solutions of classes C 1 and C 2 and present a p-convergence study by progressively increasing the p-levels by one in both cases beginning with minimally conforming p-level. Better accuracy and higher convergence rates of the

277

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

solution of class C 2 with progressively increasing p-level is quite clear (see Fig. 5.40). 0.01 (LSP: single equation) C1, p=3

0 -0.01 -0.02

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

φ

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

x

Figure 5.31: Solution φ versus x

0.05 (LSP: single equation) C1, p=3 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6 x

Figure 5.32: Plots of

dφ dx

versus x

0.8

1

278

SELF-ADJOINT DIFFERENTIAL OPERATORS

2 (LSP: single equation) 1.5 1

d2φ/dx2

0.5 0 -0.5 2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

-1 C1, p=3

-1.5 -2 0

0.2

0.4

0.6

0.8

1

x

Figure 5.33: Plots of

d2 φ dx2

versus x

0.01 (LSP: single equation) C2, p=5

0 -0.01 -0.02

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

φ

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

x

Figure 5.34: Solution φ versus x

5.2.4 LSFEP using auxiliary variables and auxiliary equations In the least-squares finite element formulation of the second order diffusion equation we note C 0 local approximations are not admissible. The purpose of the material presented in this section is to provide an alternative

279

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.05 (LSP: single equation) C2, p=5 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

x

Figure 5.35: Plots of

dφ dx

versus x

2 (LSP: single equation) 1.5 1

d2φ/dx2

0.5 0 -0.5 2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

-1 2

-1.5

C , p=5

-2 0

0.2

0.4

0.6

0.8

1

x

Figure 5.36: Plots of

d2 φ dx2

versus x

least-squares finite element formulation that permits C 0 local approximation. First, we note that admissibility of C 0 local approximation in the least-squares process is possible only if the GDEs are a system of first order ordinary or partial differential equations. Consider the same BVP as in

280

SELF-ADJOINT DIFFERENTIAL OPERATORS

20 (LSP: single equation) 15 10

d3φ/dx3

5 0 -5 2

C , p=5 -10 2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

-15 -20 -25 0

0.2

0.4

0.6

0.8

1

x

Figure 5.37: Plots of

d3 φ dx3

versus x

0

1

C , p=3 (LSP: single equation) C2, p=5 1 C (GM/WF) 2, p=3 C , p=5

-1 -2

log(√I)

-3 -4 -5 -6 -7 -8 0

0.5

1

1.5 2 log(dofs)

Figure 5.38: Plots of ||E||L2 =

2.5

3

3.5

√ I versus dofs

example 5.2.3. −

d  dφ  a + cφ = q(x) ∀x ∈ (0, L) = Ω ⊂ R1 dx dx φ(0) = 0  dφ  a =0 dx x=L

(5.165)

(5.166)

281

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0

C1, p=5 (LSP: single equation) C2, p=5 1 (GM/WF) C2, p=5 C , p=5

-1 -2

log(√I)

-3 -4 -5 -6 -7 -8 0.5

1

1.5

2

2.5

3

log(dofs)

Figure 5.39: Plots of ||E||L2 =

√ I versus dofs

0

1

C2 C p-convergence study 4-element uniform mesh

-2

log(√I)

-4 -6 -8 -10 -12 -14 0.9

1

1.1

1.2

1.3 1.4 log(dofs)

Figure 5.40: Plots of ||E||L2 =

1.5

1.6

1.7

1.8

√ I versus dofs

We convert the second order ODE into a system of two first order ODEs using auxiliary variable and auxiliary equation. Let

τ =a

dφ dx

(5.167)

282

SELF-ADJOINT DIFFERENTIAL OPERATORS

Then we have the following for the BVP: −

dτ + cφ = q(x) dx dφ τ =a dx

(5.168)

and φ(0) = 0  dφ  = τ (L) = 0 a dx x=L

(5.169)

τ is called auxiliary variable and τ = a dφ dx is called auxiliary equation. Thus, (5.165), a second order ODE in φ, has been converted into two first order ODEs in dependent variables φ and τ in (5.168). We also note the appearance of τ in the boundary conditions in (5.168). We approximate φ and τ ¯ e independent of each other (even though the two are by φeh and τhe over Ω related by τ = a dφ dx ). We can now consider least-squares finite element formulation of (5.168) and (5.169) instead of (5.165) and (5.166). Let φeh and τhe be local approxi¯ e . Then we have mations of φ and τ over Ω  dτ e  E1e = − h + cφeh − q(x) dx ¯e ∀x ∈ Ω (5.170) e dφ  e e h  E2 = τh − a(x) dx in which E1e and E2e are residuals corresponding to the two ODEs in (5.168). We define I1e = (E1e , E1e )Ω¯ e

(5.171)

I2e = (E2e , E2e )Ω¯ e

as least-squares functionals corresponding to the residual equations E1e and ¯ T , we can write E2e . Then for the discretization Ω I=

2 XX e

Iie

 =

X

Ie ; Ie =

e

i=1

2 X i=1

Iie =

2 X

(Eie , Eie )Ω¯ e

(5.172)

i=1

This establishes existence of the least-squares functional I. The first variation of I set to zero provides the necessary conditions: δI =

X e

δI e =

2 XX e

i=1

δIie = 2

X e

{g e } = 2{g} = 0

(5.173)

283

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

Since the differential operator A is linear e

δI =

2 X

δIie

=2

i=1

Hence

2 X

(Eie , δEie )Ω¯ = 2{g e }

X X  2 X e e {g e } = 0 δI = 2 (Ei , δEi )Ω¯ e = 2 e

(5.174)

i=1

(5.175)

e

i=1

where e

{g } =

2 X

(Eie , δEie )Ω¯ e

(5.176)

i=1

Let φeh =

n ˜ X

Niφ (ξ) φ δie

i=1 n

τhe =

e X

(5.177) Niτ (ξ) τ δie

i=1

in which Niφ (ξ) and Niτ (ξ) are local approximation functions and φ δie and τ δ e are corresponding nodal degrees of freedom for φe and τ e , we note n ˜ and i h h e e n are total degrees of freedom for φh and τh . Thus, local approximations e

for φ and τ can be chosen differently. For convenience, we write the local approximations φeh and τhe as follows: φeh = [N φ ]{φe } τhe = [N τ ]{τ e }

(5.178)

Substitution of φeh and τhe in the residual equations E1e and E2e yields

Let

h dN τ i {τ e } + c[N φ ]{φe } − q(x) E1e = − dx  dN φ i e {φ } E2e = [N τ ]{τ e } − a(x) dx

(5.179)

 e  {φ } {δ } = {τ e }

(5.180)

n e  ( )  ∂Ee1  c[N φ ]T ∂{φ } h τ iT δE1e = n ∂E e = dN 1   − e dx

(5.181)

e

Therefore

∂{τ }

284

SELF-ADJOINT DIFFERENTIAL OPERATORS

and

n e  ( h φ iT )  ∂Ee2  ∂{φ } −a(x) dN dx δE2e = n ∂E e = 2   [N τ ]T e

(5.182)

∂{τ }

Recall that e

{g } =

2 X

(Eie , δEie )Ω¯ e

(5.183)

i=1

or e

{g } =

X 2



[δEie , δEie ]

{δ e } −

i=1

2 X

(q(x), δEie )

(5.184)

i=1

Since the differential operator A is linear, we can write {g e } = [K e ]{δ e } − {F e } where e

[K ] =

2 X

(δEie , δEie )Ω¯ e

(5.185)

(5.186)

i=1

or

1

e

[K ] =

2 XZ X i=1

(δEie )(δEie )T J dξ

(5.187)

q(x)(δEie ) J dξ

(5.188)

−1

and

1

2 Z X

e

{F } =

i=1 −1

¯T For the discretization Ω X X X {g} = {g e } = [K e ]{δ e } − {F e } = 0 e

or

e

hX

(5.189)

e

i [K e ] {δ} = {F }

(5.190)

e

or [K]{δ} = {F }

(5.191)

where [K] =

X

[K e ],

{F } =

e

X

{F e }

(5.192)

e

and {δ} =

[

{δ e }

(5.193)

e

Coefficients of [K e ] and {F e } are computed using gauss quadrature. The process of assembly for [K] and [F ] in (5.192) follows the usual procedure (as in example 5.2.1).

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

285

5.2.4.1 Approximation spaces for φeh and τhe Generally one chooses equal order, equal degree local approximations for both φ and τ . That is Nφ = Nτ = N

and n ˜=n=n e

It can be shown [1] that with this choice the least-squares process remains convergent. Thus, we can choose C 0 p-version local approximations of same degree (and same order) for both φeh and τhe . Hence, we have φeh , τhe ∈ ¯ e ). For this approximation space the integrals in the LSP are Vh ⊂ H 1,p (Ω in Lebesgue sense. Based on the usual definition of minimally conforming ¯ e ). For this choice all integrals spaces we must choose φeh , τhe ∈ Vh ⊂ H 2,p (Ω in the LSP are Riemann. The problems associated with this approach using auxiliary variables and auxiliary equations in LSP has been pointed out by Surana et al. [2]. This approach of constructing least-squares finite element processes using first order systems of GDEs has been a common practice in computational mechanics prior to the introduction of k-version of the finite element method by Surana et al. [3–5] 5.2.4.2 Numerical studies We consider the same data as used in example 5.2.3 and present the following numerical studies: First, we discuss the manner in which the boundary conditions are imposed. φ(0) = 0 is rather straightforward to specify as φ is a degree of freedom at the end nodes of the element regardless of the order of local approximation. However, imposition of the boundary condition dφ 0 dx x=1= 0 requires some considerations. We note that in case of C local approximation for φ and τ this boundary condition can not be imposed using nodal degrees of freedom for φ as dφ dx is not a degree of freedom at the end nodes of the element. However, τ = dφ dx is an auxiliary equation, thus dφ τ |x=1= 0 imposes dx = 0 indirectly or in the weak sense due to the fact that τ =

dφ dx

x=1

only holds when the residuals approach zero i.e. when the so lutions are sufficiently converged. Thus, in case of C 0 solutions dφ dx x=1= 0 calculated using element local approximations for φ will not be zero even though τ |x=1= 0 is a specified boundary condition. This is a serious drawback of local approximations based on approximation spaces that are not minimally conforming as dictated by the BVP. ¯ e ), C 2 (Ω ¯ e ) or higher When the local approximations are of classes C 1 (Ω we have two choices. In the first choice we can elect to impose dφ =0 dx x=1

286

SELF-ADJOINT DIFFERENTIAL OPERATORS

¯ e ) local approximation. This of course will using τ |x=1= 0 as in case of C 0 (Ω 0 ¯e have similar consequences as in case of C (Ω ). In the second choice we can dφ impose dx = 0 using degree of freedom dφ dx in the local approximation x=1

dφ at dx calculated yield dφ dx x=1= 0 but

for φ. In this case

x = 1 using local approximations for φ

τ |x=1= 0 will only hold when solutions will naturally are converged. That is when the residual functional corresponding to the auxiliary equation is sufficiently close to zero. In the computations of the numerical solutions we consider both approaches of imposing the boundary dφ condition dx = 0. We consider the following numerical studies. x=1

I. h-convergence study with φeh and τhe of class C 0 using p = 2 II. h-convergence study with φeh and τhe of class C 1 using p = 3 III. h-convergence study with φeh and τhe of class C 2 using p = 5 IV. p-convergence study (fixed discretization) with φeh and τhe of class C 1 and C 2 0 Figures 5.41 and 5.42 show plots of φ and dφ dx versus x for solutions of class C at p-level of two using progressively refined discretizations in which element length is halved each time. In Fig. 5.41 we note that except two element discretization all others yield good values of solution φ. In Fig. 5.42 accuracy dφ of dφ improves with progressive mesh refinement. We note that dx dx x=1= 0 is not satisfied for coarser discretizations as expected. Figures 5.43, 5.44(a) and (b), and 5.45(a) and (b) show plots of φ, dφ dx and 2 d φ 1 e ¯ ) at p-level of 3 using progressive versus x for solutions of class C (Ω dx2 mesh refinement. The behavior of φ versus x (see Fig. 5.43) is not influenced significantly whether the derivative boundary condition at x = 1 is specified using τ or dφ dx as degrees of freedom. Figures 5.44(a) and (b) show plots dφ dφ of dx versus x for τ |x=1= 0 and dx = 0, respectively, used for defining x=1 derivative boundary condition at x = 1. We note that dφ = 0 is not dx x=1

satisfied in figure 5.44(a) for coarser discretizations whereas in Fig. 5.44(b) this boundary condition holds for all discretizations. Likewise τ |x=1 = 0 holds for all plots in Fig. 5.44(a) but not in Fig. 5.44(b). However, when the discretizations are sufficiently refined the results are same for both cases. In 2 Fig. 5.45(a) and (b) graphs of ddxφ2 show differences in the solution only for coarser discretizations. d2 φ d3 φ 2 Graphs of φ, dφ dx , dx2 and dx3 versus x for solutions of class C at p=5 dφ are shown in Figs. 5.46–5.49. In these studies choice of τ |x=1= 0 or dx = 0 x=1 has virtually insignificant influence on the computed solutions.

287

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.01 (LSP: first order system) 0

0

C , p=2

-0.01 -0.02

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

φ

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

0.8

1

x Figure 5.41: Solution φ versus x

0.05 (LSP: first order system) C0, p=2 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6 x

Figure 5.42: Plots of

dφ dx

versus x

√ Figure 5.50 shows plots of I versus dofs for solutions of classes C 1 and C 2 . Higher convergence rate and better accuracy of the solutions of class C 2

288

SELF-ADJOINT DIFFERENTIAL OPERATORS

0.01 (LSP: first order system) C1, p=3

0 -0.01 -0.02

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

φ

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

x

Figure 5.43: Solution φ versus x ( τ |x=1= 0 or

dφ = dx x=1



0)

at p = 5 is quite clear from the figure. From the results √ presented in Fig. 5.50, it is difficult to conclude whether the reduction in I for solutions of class C 2 at p = 5 compared to solutions of class C 1 at p = 3 for a fixed degrees of freedom is due to increase in the order √ of the space or due to increase in plevel both. Figure 5.51 shows plots of I versus dofs for solutions of class C 1 and C 2 at p = 5 for progressively uniform mesh refinements in which element lengths are halved each time. From the results in Fig. 5.51 √ we note that for 2 have lower values of a fixed dofs the solutions of class C I compared to √ 1 solutions of class C . Slopes of I versus dofs for both cases are almost the same, even though a slight increase in the slope is observed for solutions of class C 2 with progressively refined discretizations. These results decisively demonstrate higher accuracy of the solution of class C 2 compared to C 1 for a fixed dofs. For all practical purposes we can assume approximately same convergence rate in both cases. This behavior shown in Fig. 5.51 holds true for each p-level greater than or equal to 5 (five being minimum p-level for solutions of class C 2 ). In the last study we consider a four-element uniform discretization with solutions of classes C 1 and C 2 and present a p-convergence study by progressively increasing the p-levels by one in both cases beginning with minimally conforming p-level. Better accuracy and higher convergence rates of the solution of class C 2 with progressively increasing p-level is quite clear from Fig. 5.52.

289

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.05 (LSP: first order system) C1, p=3 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

x

(a) τ |x=1= 0 0.05 (LSP: first order system) C1, p=3 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

x

(b)

dφ = dx x=1



Figure 5.44: Plots of

0 dφ dx

versus x

5.2.5 One-dimensional heat conduction with convective boundary Here we consider 1D heat conduction problem (also a diffusion equation) with a convective boundary condition (Neumann boundary condition) containing unknown function (temperature). The purpose of this example

290

SELF-ADJOINT DIFFERENTIAL OPERATORS

2

2 el. mesh 4 el. mesh 8 el. mesh C1, p=3 16 el. mesh 32 el. mesh 64 el. mesh

(LSP: first order system) 1.5 1

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

0.8

1

x

(a) τ |x=1= 0 2

2 el. mesh 4 el. mesh 8 el. mesh C1, p=3 16 el. mesh 32 el. mesh 64 el. mesh

(LSP: first order system) 1.5 1

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6 x

(b)

dφ = dx x=1



Figure 5.45: Plots of

0 d2 φ dx2

versus x

is to illustrate how the finite element computations are effected due to the Neumann boundary conditions containing unknown temperature. Consider

−

d  dT  ka = Q ∀x ∈ (0, L) = Ω ⊂ R1 dx dx

(5.194)

291

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0.01 2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

(LSP: first order system) C2, p=5

0 -0.01 -0.02

φ

-0.03 -0.04 -0.05 -0.06 -0.07 -0.08 -0.09 0

0.2

0.4

0.6

0.8

1

x

Figure 5.46: Solution φ versus x ( τ |x=1= 0 or

dφ = dx x=1



0)

0.05 (LSP: first order system) C2, p=5 0

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

dφ/dx

-0.05

-0.1

-0.15

-0.2 0

0.2

0.4

0.6

0.8

1

x

Figure 5.47: Plots of

dφ dx

versus x ( τ |x=1= 0 or

dφ = dx x=1



0)

with T (0) = T0 dT ka + β(T − T∞ ) + qˆ = 0 at x = L dx

(5.195)

292

SELF-ADJOINT DIFFERENTIAL OPERATORS

2

2 el. mesh 4 el. mesh 8 el. mesh C2, p=5 16 el. mesh 32 el. mesh 64 el. mesh

(LSP: first order system) 1.5 1

d2φ/dx2

0.5 0 -0.5 -1 -1.5 -2 0

0.2

0.4

0.6

0.8

1

x

Figure 5.48: Plots of

d2 φ dx2

dφ = dx x=1



versus x ( τ |x=1= 0 or

0)

20 (LSP: first order system) 15 10

d3φ/dx3

5 0 -5

C2, p=5

-10

2 el. mesh 4 el. mesh 8 el. mesh 16 el. mesh 32 el. mesh 64 el. mesh

-15 -20 -25 0

0.2

0.4

0.6

0.8

1

x

Figure 5.49: Plots of

d3 φ dx3

versus x ( τ |x=1= 0 or

dφ = dx x=1



0)

where k is thermal conductivity, a is area of cross section, β is heat transfer coefficient, T∞ is ambient temperature, qˆ is heat flux and Q is internal heat generation. These are data and are given. T = T (x) is temperature.

293

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

0

C1, p=3 (LSP: first order system) C2, p=5 1 (LSP: single equation) C2, p=3 C , p=5

-2

log(√I)

-4 -6 -8 -10 -12 -14 0

1

2

3 log(dofs)

Figure 5.50: Plots of ||E||L2 =

0

4

5

√ I versus dofs ( τ |x=1= 0 or

6

dφ = dx x=1



0)

C1, p=5 (LSP: first order system) C2, p=5 1 (LSP: single equation) C2, p=5 C , p=5

-2

log(√I)

-4 -6 -8 -10 -12 -14 0

1

2

3 log(dofs)

Figure 5.51: Plots of ||E||L2 =

4

5

6

√ I versus dofs

We consider the Galerkin method with weak form. In this case A=− f =Q

d  dT  ka dx dx

(5.196)

294

SELF-ADJOINT DIFFERENTIAL OPERATORS

0

C1 C2 p-convergence study 4-element uniform mesh

-2

log(√I)

-4 -6 -8 -10 -12 -14 1.2

1.3

1.4

1.5

1.6 1.7 log(dofs)

Figure 5.52: Plots of ||E||L2 =

1.8

1.9

2

2.1

√ I versus dofs

Hence, we can write (5.194) as AT − Q = 0 ∀x ∈ Ω

(5.197)

¯ e , an element In this case A∗ = A. Let TheSbe approximation of T over Ω e ¯e ¯ ¯T = e of the discretization Ω e Ω of Ω. Then, if v is variation of Th , we consider the following:  Z  d  dThe  e (ATh − Q, v)Ω¯ e = − ka − Q v dx (5.198) dx dx ¯e Ω

¯ e = [xe , xe+1 ] mapped into Ω ¯ ξ = [−1, 1]. Using integration by in which Ω parts for the first term in the integrand of (5.198)    Z  Z dThe  xe+1 dv dThe  e ka dx − v ka − Qv dx (5.199) (ATh − Q, v)Ω¯ e = dx dx dx xe ¯e Ω

¯e Ω

In (5.199), the concomitant hAThe − Q, viΓe is (with φeh = The and f = Q)    dThe  xe+1 e hATh − Q, viΓe = − v ka dx xe From (5.199), we conclude that ˜ 1 is EBC T is PV ⇒ T = T˜ on some Γ e e dT dT ˜ 2 is NBC ka h is SV ⇒ ka h = q˜ on some Γ dx dx

(5.200)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

295

Let  dThe − ka = P1e dx xe   dThe = P2e ka dx xe+1 

(5.201)

where P1e and P2e are the secondary variables at the two end nodes of the ¯ ξ . Substituting from (5.201) into (5.199) element located at ξ = ∓1 in Ω (AThe − Q, v)Ω¯ e =

Z  dv dThe  ka dx − v(1) P2e − v(−1) P1e dx dx ¯e Ω Z − Qv dx (5.202) ¯e Ω

or (AThe − Q, v)Ω¯ e = B e (The , v) − le (v)

(5.203)

dv dThe ka dx dx dx

(5.204)

where B

e

(The , v)

Z = ¯e Ω

Z

le (v) = v(−1) P1e + v(1) P2e +

Qv dx

(5.205)

¯e Ω

Let The

=

n X

Ni δie = [N ]{δ e }

(5.206)

i=1

¯ e . Then be local approximation of T over Ω v = δThe = Nj , j = 1, 2, . . . , n

(5.207)

Substituting from (5.206) and (5.207) into (5.204) and (5.205) B

e

(The , v)

Z =

X  n dNj dNi e ka δ dx, j = 1, 2, . . . , n dx dx i

¯e Ω e

l (v) =

Nj (−1) P1e

(5.208)

i=1

+

Nj (1) P2e

Z Q Nj dx, j = 1, 2, . . . , n

+ ¯e Ω

(5.209)

296

SELF-ADJOINT DIFFERENTIAL OPERATORS

(5.203) is the desired weak form resulting from the Galerkin method with weak form. (5.208) and (5.209) can be arranged in the matrix and vector form B e (The , v) = [K e ]{δ e } e

e

(5.210) e

l (v) = {P } + {F }

(5.211)

in which    dNj dNi e e  Kij = ka J dξ = Kji dx dx , i, j = 1, 2, . . . , n  −1    {P e }T = [P1e , 0 , . . . , 0 , P2e ] Z1

(5.212)

and Fie

Z1 Q Ni J dξ, i = 1, 2, . . . , n

=

(5.213)

−1

We also note that dm Ni 1 dm Ni he = ; J = , m = 1, 2, . . . , i = 1, 2, . . . , n dxm J m dξ n 2

(5.214)

Thus, knowing explicit expressions for Ni (ξ), the local approximation funce and F e can be calculated using Gauss quadrature. tions, the coefficients Kij i 5.2.5.1 Approximation space Vh Since the BVP contains second order derivative of temperature T and the weak form that contains only the first order derivatives of the temperature, we have the following for minimally conforming spaces: S (i) Admissibility of Th = e The in AT − f = 0 in the pointwise sense in ¯ T requires T e ∈ Vh ⊂ H k,p (Ω ¯ e ); k = 3 is minimally conforming. For Ω h this choice, the integrals in the following are all Riemann: (ATh − Q, v)Ω¯ T =

X e

(AThe − Q, v)Ω¯ e =

X  B e (The , v) − le (v) = 0 e

(5.215) ¯ e ), then all integrals are Riemann Thus, if we choose The of class C 2 (Ω and all forms in (5.215) are precisely equivalent.

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

297

¯ e ), that is, if we (ii) Based on the weak form, if we choose The of class C 1 (Ω ¯ e ), then the following holds: choose The ∈ Vh ⊂ H 2,p (Ω (AThe − Q, v)Ω¯ T

⇔ L

X

⇔ L

X

(AThe − Q, v)Ω¯ e

e


B e (The , v) − le (v) = 0

(5.216)

e

{z

|

}

R

P That is, for this choice of The , e [B e (·, ·)−le (·)] holds in Riemann sense but all other integral forms in (5.216) are in Lebesgue sense. ¯ e ), then P (B e (·, ·)−le (·)) holds only in (iii) If we choose The ∈ Vh ⊂ H 1,p (Ω e Lebesgue sense and the other two integral forms in (5.216) are meaningless.

5.2.5.2 Numerical study For illustrating the details of the computations it suffices to choose The of class C 0 with p = 1, i.e. a two-node element with linear approximation of ¯ e or Ω ¯ ξ . Let the element local nodes 1 and 2 be located at ξ = −1 The over Ω ¯ ¯ e defined by and ξ = 1 in Ωξ map of Ω x = x(ξ) =

1 − ξ  2

xe +

1 + ξ  2

xe+1

(5.217)

T2e

(5.218)

and The (ξ) =

1 − ξ  2

T1e +

1 + ξ  2

Thus, in this case N1 (ξ) =

1−ξ , 2

N2 (ξ) =

1+ξ 2

(5.219)

using (5.212) we can obtain the following for an element e with domain ¯e → Ω ¯ ξ (with the assumption that ka = ke ae = constant over Ω ¯ e and Ω Q = f0 , also a constant). (AThe , v)Ω¯ e

     e   ke ae f0 he 1 1 −1 T1e P1 = − − P2e 1 he −1 1 T2e 2

(5.220)

298

SELF-ADJOINT DIFFERENTIAL OPERATORS

Equations (5.220) are valid for each element of the discretization. Let us choose a two-element non-uniform discretization, L = 10 h1 = 4, h2 = 6 k1 a1 = 76, k2 a2 = 96 β = 10, T∞ = 30, qˆ = 0 T (0) = T0 = 100 Then, for the two element discretization shown below 1

1

2

3 x

6

4 x=0

2

x=4

x = 10

we have Ele. 1:

(ATh1 , v)Ω¯ 1

Ele. 2:

(ATh2 , v)Ω¯ 2

    1   1 −1 T11 P1 1 − − 2f0 = 19 −1 1 T21 P21 1      2  2 1 −1 T1 P1 1 = 16 − − 2f0 −1 1 T22 P22 1 

(5.221) (5.222)

Inter-element continuity conditions on PVs

T11 = T1 = 100 T21 = T12 = T2 T22 = T3

) not known

Element equations after imposing inter-element continuity conditions on PVs become Ele. 1:

(ATh1 , v)Ω¯ 1

Ele. 2:

(ATh2 , v)Ω¯ 2

    1   P1 1 1 −1 T1 = 19 − − 2f0 1 P2 1 −1 1 T2      2   1 1 −1 T2 P1 − 2f0 = 16 − 1 P22 −1 1 T3 

(5.223) (5.224)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

299

Assembled element equations       19 −19 0 T1  1  P11  = 0 ⇒ −19 35 −16 T2 = 2f0 2 + P21 + P12       0 −16 16 1 P22 T3 (5.225) 

X

(AThe −f0 , v)Ω¯ T

e

Inter-element continuity conditions on the sum of secondary variables T1 = 100 ⇒ P11 is unknown P21 + P 12 = 0 (5.226)  dT e  2 h = −β(L)(T (L) − T∞ ) + qˆ(L) = −10(T3 ) + 300 P2 = k dx x=L Substituting from (5.226) into (5.225)        19 −19 0 T1  P11 1   −19 35 −16 T2 = 2f0 2 + 0       0 −16 16 T3 1 −10T3 + 300

(5.227)

We can transfer −10T3 on the left side of (5.227), substitute T1 = 100 and if we choose f0 = 0, then we have      19 −19 0 100  P11  −19 35 −16 T2 = 0 (5.228)     0 −16 26 T3 300 or



    35 −16 T2 0 = −16 26 T3 300

(5.229)

giving T2 = 82.88, T3 = 62.54 and   100     P11 = 19 −19 0 82.88 = 325.28   62.54 Post processing: computations of T (x) and

dT dx

for 0 ≤ x ≤ 10

Temperature T (ξ) =

1 − ξ 

T1 +

1 + ξ 

T2 , −1 ≤ ξ ≤ 1, 0 ≤ x ≤ 4 2 2 1 − ξ  1 + ξ  T (ξ) = T2 + T3 , −1 ≤ ξ ≤ 1, 4 ≤ x ≤ 10 2 2

(5.230)

300

SELF-ADJOINT DIFFERENTIAL OPERATORS

Derivatives of temperature dT dT dx dT 1 dT = ⇒ = dξ dx dξ dx J dξ Element 1: J =

h1 2

=

4 2

=2

1 dT = (T2 − T1 ), 0 ≤ x ≤ 4 dx 4 Element 2: J =

h2 2

=

6 2

=3

dT 1 = (T3 − T2 ), 4 ≤ x ≤ 10 dx 6

5.2.6 1D axisymmetric heat conduction Consider axisymmetric heat conduction independent of circumferential coordinate Θ and axial coordinate z in r, Θ, z cylindrical coordinate system described in Example 3.3. Following Example 3.3, this BVP is described by radial heat conduction equation   1 d dθ − kr − f = 0 ∀ri ≤ r ≤ ro (5.231) r dr dr with BCs:θ(ri ) = θ0 (5.232) dθ kr + β(θ − θ∞ ) = 0 at r = ro (5.233) dr where θ is temperature, k is conductivity, r is radius, β is film or convective heat transfer coefficient, and θ∞ is ambient temperature. The differential operator A is defined as   1 d d A=− kr (5.234) r dr dr Thus we can write (5.231) as Aθ − f = 0

(5.235)

We can show that A is linear and that the adjoint of A, that is, A∗ is same as A. ¯ e , an element of the discretization Let θhe be an approximation of θ over Ω ¯ T = ∪Ω ¯ e of Ω ¯ = [ri , ro ]. We consider Galerkin method with weak form and Ω e ¯ T with domain the LSP based on residual functional for an element e of Ω ¯ e. Ω

301

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

5.2.6.1 Galerkin method with weak form Let v = δθhe , then we consider the following using dΩ = 2πr dr.    Z  dθhe 1 d e kr − f vr dr (5.236) (Aθh − f, v) = 2π − r dr dr ¯e Ω

or (Aθhe

Z − f, v) = 2π

d − dr



dθe kr h dr



Z v dr − 2π

f vr dr

(5.237)

¯e Ω

¯e Ω

Transfer one order of differentiaion from θhe to v in the first term on the right ¯ e = [re , re+1 ]): side of (5.237) (using Ω    Z Z dθhe re+1 dv dθhe e (Aθh − f, v) = 2π k r dr − v 2πrk − 2π f vr dr dr dr dr re ¯e Ω

¯e Ω

(5.238) In (5.238), concomitant is given by    dθe re+1 hAθhe , viΓe = v 2πrk h dr re Let

(5.239)

 dθhe − 2πrk = P1e dr re   dθhe 2πrk = P2e dr re+1 

(5.240)

Using (5.240) in (5.239) we can write hAθhe , viΓe = v(re+1 )P2e + v(re )P1e

(5.241)

Using (5.241) we can write (5.238) as follows (Aθhe − f, v) = 2π

rZe+1 re

dv dθhe k r dr − v(re+1 )P2e − v(re )P1e − 2π dr dr

rZe+1

f vr dr re

(5.242) or (Aθ)he − f, v) = B e (θhe , v) − le (v) in which B

e

(θhe , v)

rZe+1

k

= 2π re

dv dθhe r dr dr dr

(5.243)

(5.244)

302

SELF-ADJOINT DIFFERENTIAL OPERATORS

and e

v(re+1 )P2e

l (v) =

+

rZe+1

v(re )P1e

f vr dr

+ 2π

(5.245)

re

¯ e , be given by (using the element Let θhe , the local approximation of θ over Ω ¯ e map in Ω ¯ ξ = [−1, 1]): Ω θhe =

n P i=1

Ni (ξ)δie = [N ]{δ e }

(5.246)

v = δθhe = Nj (ξ), j = 1, 2, . . . , n in which Ni (ξ) are approximation functions and {δ e } are nodal degrees of freedom and     1−ξ 1+ξ r(ξ) = re + re+1 2 2 dr re+1 − re he =J = = (5.247) dξ 2 2 dr = J dξ 1 dNi dNi = (5.248) dr J dξ Substituting from (5.246)–(5.248) into (5.244) and (5.245) we obtain (v = Nj ) B e (θhe , Nj ) = 2π

n dNj X dNi e k δ dr dr i

Z ¯ξ Ω Z1

= 2π −1 e

! r(ξ)J dξ

i=1

1 dNj k J dξ

n 1 X dNi e δ J dξ i

! r(ξ)J dξ

i=1

e

= [K ]{δ }

(5.249)

where e Kij

2π = J

Z1 k

dNj dNi r(ξ) dξ, i, j = 1, 2, . . . , n dξ dξ

(5.250)

−1

Also, we have e

l (Nj ) =

Nj (re )P1e

+

Nj (re+1 )P2e

Z1 + 2π −1

f Nj (ξ)r(ξ)J dξ

(5.251)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

for j = 1, 2, . . . , n. We note that )

N1 (re ) = N1 ξ|−1 = 1, Nn (re ) = Nn ξ|−1 = 0

N1 (re+1 ) = N1 ξ|+1 = 0, Nn (re+1 ) = Nn ξ|+1 = 1 Hence

 e P1           0   .. e le (Nj ) = .  + {F }     0      e  P2

303

(5.252)

(5.253)

in which Fie

Z1 =

he f Ni (ξ)2πr(ξ) dξ = πhe 2

Z1 f Ni (ξ)r(ξ) dξ

(5.254)

−1

−1

for i = 1, 2, . . . , n. Hence, we have

(Aθhe , v)Ω¯ e

 e P     1      0   .. e = B e (θhe , v) − le (v) = [K e ]{δ e } − .  − {F }     0      e  P2

(5.255)

Equations (5.255) are the element equations resulting from GM/WF over an ¯e = Ω ¯ ξ = [−1, 1]. element e with domain Ω 5.2.6.2 LSM based on residual functional ¯ e = [re , re+1 ] or Ω ¯ ξ = [−1, 1] then If θhe is approximation of θ over Ω e e ¯ ¯ e is (assuming k to be constant k over Ω ) the residual function E e over Ω given by ¯e E e = Aθhe − f ∀r ∈ Ω (5.256) ¯ T is constructed using The residual functional I for Ω I=

P e P e e I = (E , E )Ω¯ e , existence of I e

(5.257)

e

¯ e . First variation of I set to in which I e is the least-squares functional for Ω zero gives necessary condition. δI =

P P e P δI = 2(E e , δE e )Ω¯ e = 2 {g e } = {g} = 0 e

e

e

(5.258)

304

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which {g e } = (E e , δE e )Ω¯ e

(5.259)

The element relation can be derived using {g e }. We note that 

1 d E = r dr e

1 d δE = r dr e

dθe kr h dr

 + f (r)

(5.260)

  dv kr , v = δθhe dr

Let θhe =

n P

(5.261)

Ni (ξ)δie

(5.262)

i=1

in which Ni (ξ) are approximation functions and δie are nodal degrees of freedom. We choose a three node element with nodes 1, 2, 3 located at ξ = −1, 0, 1. Explicit forms of Ni (ξ) and δie depend upon the class of θhe i.e. ¯ e ). We discuss details in a the order of the approximation space Vh ⊂ H k (Ω later section. First, based on local approximation θhe , we have v = δθhe = Nj (ξ), j = 1, 2, . . . , n

(5.263)

Substituting θhe and v in {g e } we obtain gje (j = 1, 2, . . . , n) as gje =

1 d r dr

kr

n X dNi i=1

dr

! δie

1 d + f (r), r dr



dNj kr dr

! (5.264) ¯e Ω

Since the differential operator A is linear, gje can be written as gje

=

   ! n X dNj 1 d dNi 1 d e kr δi + f (r), kr r dr dr r dr dr ¯e i=1

(5.265)

Ω

which is same as gje

 =

n P

i=1

(ANi )δie

 + f, ANj

(5.266) ¯e Ω

or {g e } = [K e ]{δ e } + {F e }

(5.267)

e Kij = (ANj , ANi )Ω¯ e

(5.268)

in which

Fie

= (f, ANi )Ω¯ e

(5.269)

305

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

e and F e (i, j = Substituting for A we can obtain explicit forms of Kij i 1, 2, . . . , n):      dNj 1 d dNi 1 d e kr , kr (5.270) Kij = r dr dr r dr dr ¯e Ω    dNi 1 d e Fi = f, (5.271) kr r dr dr ¯e Ω

¯ e = [re , re+1 ] → Ω ¯ξ = Using element map in natural coordinate space ξ i.e. Ω [−1, 1] we can write     1+ξ 1−ξ re + re+1 (5.272) r(ξ) = 2 2 dr 1 he = J = (re+1 − re ) = (5.273) dξ 2 2 dr = J dξ (5.274)  m m m m d Ni 1 d Ni (ξ) 2 d Ni (ξ) = m = (5.275) m m dr J dξ he dξ m and dΩ = 2πr dr = 2πr(ξ) dr. Additionally (assuming k to be constant)   dNi 1 dNi d2 Ni 1 d kr = k +k 2 (5.276) r dr dr r dr dr e and F e in (5.270) and (5.271). Hence, we can write the following for Kij i

e Kij

Z1  = 2π

d2 Nj 1 dNj k +k r(ξ) dr dr2



1 dNi d2 Ni k +k 2 r(ξ) dr dr

 r(ξ)J dξ

−1

(5.277) Fie

Z1

 f (ξ)

= 2π

1 dNi d2 Ni k +k 2 r(ξ) dr dr

 r(ξ)J dξ

(5.278)

−1 dNi dr

2

and ddrN2i can be substituted in (5.277) and (5.278) using (5.275). The usual process of assembly of (5.267) follows the standard procedure.

5.2.7 A 1D BVP governed by a fourth-order differential operator Consider the following BVP describing bending of beams: d2  d2 w  EI 2 − Q = 0 ∀x ∈ (0, L) = Ω ⊂ R1 dx2 dx

(5.279)

306

SELF-ADJOINT DIFFERENTIAL OPERATORS

with dw =0 w(0) = dx x=0 d  d2 w  b = FL dx dx2 x=L d2 w = ML EI dx2 x=L

(5.280)

where EI is the product of the modulus of elasticity and bending moment of inertia, Q is distributed load along the length of the beam, FL and ML are shear force and bending moments at x = L. w = w(x) is the transverse deflection of the beam. Q(x) FL

w(0) = 0 dw dx x=0

ML

L

A schematic of the BVP is shown in the figure above. In this case the differential operator A is given by A=

d2  d2
EI dx2 dx2

and f = Q

(5.281)

The operator is linear and symmetric (if the boundary conditions are homogeneous) and A∗ = A. Thus, we can write (5.279) as Aw − Q = 0 ∀x ∈ Ω

(5.282)

¯ e , an element of the discretization Let wheSbe approximation of w over Ω T e ¯ ¯ ¯ Ω = e Ω of Ω. We consider Galerkin with weak form. If v = δwhe then ¯ e: we consider the following for Ω (Awhe − Q, v)Ω¯ e =



d2  d2  EI − Q, v dx2 dx2

 (5.283) ¯e Ω

¯ ξ = [−1, 1]. We can also write ¯ e = [xe , xe+1 ] is mapped into Ω in which Ω (Awhe

Z  − Q, v)Ω¯ e = ¯e Ω

 d2  d2  EI 2 v − Qv dx dx2 dx

(5.284)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

307

Transfer two orders of differentiation to v in the first term of the integrand in (5.284).     Z  2 d2 whe  d2 whe  xe+1 d v d e (Awh − Q, v)Ω¯ e = EI dx + v EI dx2 dx2 dx dx2 xe ¯e Ω

dv  d2 whe  − EI dx dx2 

xe+1

Z −

xe

Qv dx

(5.285)

¯e Ω

In (5.285), the concomitant hAwhe , viΓe is given by        d2 whe  xe+1 d2 whe  xe+1 dv d e (Awh , v)Γe = v − EI EI dx dx2 dx dx2 xe xe Using concomitant ˜ 1 is an EBC whe is a PV; hence, whe = w ˜ on some Γ dwhe dwhe ˜ 2 is an EBC is also a PV; hence, = θ˜ on some Γ dx dx d  d2 whe  d  d2 whe  ˜ 3 is a NBC EI is a SV; hence, EI = F˜ on some Γ dx dx2 dx dx2 d2 whe d2 whe ˜ on some Γ ˜ 4 is a NBC is a SV; hence, EI =M EI dx2 dx2 Introducing the following notations (assuming the end nodes of the elements ¯ ξ) to be 1 and 2 and located at ξ = −1 and +1 in its map Ω    d2 whe  d e Q1 = EI dx dx2 x=xe   2 w e  d d h EI Qe2 = dx dx2 x=xe +1 (5.286)  d2 we  h M1e = b dx2 x=xe  d2 we  h M2e = b dx2 x=xe +1 Substituting from (5.286) into (5.285) (Awhe − Q, v)Ω¯ e =

Z  2 d2 whe  d v EI dx − v(xe ) Qe1 − v(xe+1 ) Qe2 dx2 dx2 ¯e Ω ! ! Z dv dv e e M1 − M2 − Qv dx (5.287) − dx xe dx xe+1 ¯e Ω

308

SELF-ADJOINT DIFFERENTIAL OPERATORS

or (Awhe − Q, v)Ω¯ e = B e (whe , v) − le (v)

(5.288)

Z  2 d2 whe  d v EI dx dx2 dx2

(5.289)

where B e (whe , v) =

¯e Ω e

l (v) =

v(xe ) Qe1

+

v(xe+1 ) Qe2

dv dv e + M + Me dx xe 1 dx xe+1 2

Z Qv dx

+

(5.290)

¯e Ω

¯e Let whe be local approximation of w over Ω whe =

n X

Ni (ξ) δie = [N ]{δ e }

(5.291)

i=1

Therefore δwhe = v = Nj (ξ), j = 1, 2, . . . , n

(5.292)

Substituting from (5.291) and (5.292) into (5.289) and (5.290)  Z  2 n X d Nj d2 Ni e  e e B (wh , v) = EI δ dx (5.293) dx2 dx2 i i=1 ¯e Ω dNj dNj e e e e l (v) = Nj (xe ) Q1 + Nj (xe+1 ) Q2 + M + Me dx xe 1 dx xe+1 2 Z + Qv dx (5.294) ¯e Ω

where j = 1, 2, . . . , n. Equation (5.288) is the weak form resulting from the Galerkin method with weak form. Equations (5.293) and (5.294) can be written in the matrix and vector form as B e (whe , v) = [K e ]{δ e },

le (v) = {P e } + {F e }

(5.295)

in which (i, j = 1, 2, . . . , n) e Kij

=

Z1 

d2 Nj d2 Ni  e EI J dξ = Kji dx2 dx2

−1

 T {P } = Qe1 M1e 0 0 . . . Qe2 M2e Z e Fj = QNj J dξ e T

¯e Ω

(5.296)

5.2. ONE-DIMENSIONAL BVPS IN A SINGLE DEPENDENT VARIABLE

309

We also note that dm Nj 1 dm Nj he = , J= m m m dx J dξ 2

(5.297)

Thus, knowing explicit expressions for Ni (ξ), the local approximation functions, the coefficients of [K e ] and {F e } can be calculated using Gauss quadrature. 5.2.7.1 Approximation space Vh Since the BVP that contains fourth order derivative of the deflection w and the weak form that only contains second order derivative of the deflection, we have the following for the minimally conforming spaces: S (i) Admissibility of wh = e whe in Aw − Q = 0 in the pointwise sense in ¯ T requires we ∈ Vh ⊂ H k,p (Ω ¯ e ); k = 5 is minimally conforming. For Ω h this choice, the integrals in the following are all Riemann: (Awh − Q, v)Ω¯ T =

X

(Awhe − Q, v)Ω¯ e =

e

X


B e (whe , v) − le (v) = 0

e

(5.298) ¯ e ), then all integrals are Riemann Thus, if we choose whe of class C 4 (Ω and all forms in (5.298) are precisely equivalent. ¯ e ); that is, if (1) Based on the weak form, if we choose whe of class C 2 (Ω ¯ e ), then the following holds: whe ∈ Vh ⊂ H 3,p (Ω (Awh − Q, v)Ω¯ T

and

X

(Awhe − Q, v)Ω¯ e are not defined, but

e

X  B e (·, ·) − le (·) = 0 e

holds in Riemann sense. For this choice, obviously there is no equivalence between the three integral forms in (5.298), as the first two are not defined. ¯ e ) then (2) If we choose whe of class C 1 (Ω X  B e (·, ·) − le (·) = 0 e

holds in Lebesgue sense but the first two integral forms in (5.298) are not defined.

310

SELF-ADJOINT DIFFERENTIAL OPERATORS

5.3 Two-dimensional boundary value problems In this section we consider two dimensional boundary value problems in single and multiple dependent variables described by self-adjoint differential operators. The finite element processes for these boundary value problems remain the same as those for one dimensional boundary value problems in concepts and basic steps except that for two dimensional boundary value problems: (i) The domain of definition of the BVP is in R2 , i.e. two dimensional (say in x, y) and, hence, consists of an area. Therefore, the discretization of the domain consisting of subdomains that is finite elements contains area elements that may be triangular, quadrilateral or any other desired shape. (ii) The boundary of the domain of definition of the BVP is a closed contour or curve Γ, hence, in the process of integration by parts the concomitant consists of boundary integrals as opposed to the boundary terms as in the case of one dimensional boundary value problems. First, we consider a two-dimensional BVP in a single dependent variable. This is followed by the BVPs in multivariables.

5.3.1 A general 2D BVP in a single dependent variable We consider a typical 2D boundary value problem to illustrate various concepts, principles and procedures involved in deriving finite element processes for such BVPs. Consider the boundary value problem −

∂  ∂φ ∂  ∂φ ∂φ  ∂φ  a11 − a21 + a00 φ − f = 0 + a12 + a22 ∂x ∂x ∂y ∂y ∂x ∂y

(5.299)

for all (x, y) ∈ Ω ⊂ R2 , with boundary conditions φ = φ0 on Γ1 

a11

∂φ 

∂φ ∂φ + a12 nx + a21 + a22 ny = qˆ on Γ2 ∂x ∂y ∂x ∂y 

∂φ 

(5.300) (5.301)

¯ = Ω ∪ Γ and Γ = Γ1 ∪ Γ2 is the closed boundary of the domain in which Ω Ω. nx and ny are direction cosines of a unit exterior normal to the boundary Γ2 (see Fig. 5.53). From (5.299) we note that the differential operator A is defined by A=−

∂ ∂  ∂  ∂ ∂  ∂  a11 + a12 − a21 + a22 + a00 ∂x ∂x ∂y ∂y ∂x ∂y

(5.302)

311

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

n ny Γ2

nx

Ω y

n2x + n2y = 1

Γ1 x Figure 5.53: A two-dimensional domain

Hence, we can write (5.299) as Aφ − f = 0 ∀(x, y) ∈ Ω

(5.303)

where a00 , aij , aij = aji are known functions of position (x, y). We can show that the differential operator A is linear and A∗ = A. Therefore, the Galerkin method with weak form will yield a variationally consistent integral form if the functional B(·, ·) is bilinear and symmetric and the functional l(·) is linear. We consider the finite element process for this boundary value problem based on Galerkin method with weak form. For an element e with ¯ e = Ωe ∪ Γe of the discretization Ω ¯ T = ∪e Ω ¯ e of Ω, ¯ we consider domain Ω e e ¯ e , v = δφe (Aφh − f, v)Ω¯ e in which φh is the local approximation of φ over Ω h e e and Γ is a closed boundary of Ω . Let Qx = a11

∂φ ∂φ ∂φ ∂φ + a12 , Qy = a21 + a22 ∂x ∂y ∂x ∂y

(5.304)

Then we can write (5.299) and (5.301) as Aφ − f = −

∂Qx ∂Qy − + a00 φ − Q = 0 ∂x ∂y

in

Ω

(5.305)

and Qx nx + Qy ny = qˆ on Γ2

(5.306)

Thus, with new notation, we have (Aφeh

− f, v)Ω¯ e

Z   ∂Qey ∂Qe = − x− + a00 φ − f v dx dy ∂x ∂y ¯e Ω

(5.307)

312

SELF-ADJOINT DIFFERENTIAL OPERATORS

or (Aφeh − f, v)Ω¯ e =

Z  ∂Qey  ∂Qe v dx dy − x− ∂x ∂y ¯e Ω Z Z e + a00 φh v dx dy − f v dx dy ¯e Ω

(5.308)

¯e Ω

Using integration by parts once for the first term and using Qex = a11

∂φeh ∂φe ∂φe ∂φe + a12 h , Qey = a21 h + a22 h ∂x ∂x ∂x ∂x

we obtain (Aφeh

− f, v)Ω¯ e

Z  I ∂u e ∂v e  = Q + Q dx dy − v(Qex nx + Qey ny ) dΓ ∂x x ∂y y e Γ ¯e Ω Z Z (5.309) + a00 φeh v dx dy − f v dx dy ¯e Ω

¯e Ω

In which the concomitant hAφeh , viΓe is given by hAφeh , viΓe = −

I Γe

v(Qex nx + Qey ny ) dΓ

nx and ny are direction cosines of a unit exterior normal to the boundary ¯ e . Using the concomitant in (5.309) we can identify PVs, Γe of an element Ω SVs, EBCs and NBCs. ˜ 1 is EBC φeh is PV ⇒ φeh = φ˜ on Γ (Qex nx

+

Qey ny )

is SV ⇒

(Qex nx

+

Qey ny )

(5.310)

˜ 2 is NBC = q˜ on Γ

(5.311)

where φ˜ and q˜ are specified or known. Let qne = Qex nx + Qey ny

(5.312)

be the flux normal to Γe . Substituting from (5.312) into (5.309) I Z Z   ∂v e ∂v e e e e (Aφh − f, v)Ω¯ e = Q + Q + a00 φh v dx dy − vqn dΓ − f v dx dy ∂x x ∂y y Γe

¯e Ω

(5.313) or (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(5.314)

313

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

where B

e

Z   ∂v e ∂v e = Qx + Qy + a00 φeh v ∂x ∂y ¯e Ω I Z le (v) = vqne dΓ + f v dx dy

(φeh , v)

Γe

(5.315)

(5.316)

¯e Ω

¯ e resulting from The right side of (5.314) is the weak form of the BVP over Ω e the Galerkin method with weak form. We note that B (·, ·) is bilinear and symmetric and le (·) is linear. These properties are obviously due to the fact that A∗ = A and the integration by parts. Let φeh =

n X

Ni δie = [N ]{δ e }

(5.317)

i=1

¯ e , element e of the discretization Ω ¯T . be the local approximation of φ over Ω Then v = δφeh = Nj , j = 1, 2, . . . , n

(5.318)

Substituting from (5.317) and (5.318) into (5.315) and (5.316)

B

e

n

X i ∂Nj e ∂Ni e Qx + Qy + a00 Nj = Ni δie dx dy ∂x ∂y i=1 ¯e Ω I Z le (v) = Nj qne dΓ + f Nj dx dy

(φeh , v)

Zh

Γe

(5.319)

(5.320)

¯e Ω

for (i, j = 1, 2, . . . , n) and Qex

= a11

Qey = a21

n X ∂Ni i=1 n X i=1

∂x

δie

+ a12

∂Ni e δ + a22 ∂x i

n X ∂Ni i=1 n X i=1

δie

(5.321)

∂Ni e δ ∂y i

(5.322)

∂y

or B e (φeh , v) = [K e ]{δ e }, le (v) = {P e } + {F e }

(5.323)

314

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which Z Kij =

∂Nj  ∂Ni  ∂Nj a11 + a12 ∂x ∂x ∂y

¯e Ω

∂Nj  ∂Ni  ∂Nj + a21 + a22 + a00 Nj Ni ∂y ∂x ∂y I Pie = Ni qne dΓ

! dΩ

(5.324) (5.325)

Γe

Fie

Z f Ni dΩ

=

(5.326)

¯e Ω e = K e when a We note that Kij αβ = aβα (α, β = 1, 2). ji

Remarks. ¯ e , i.e. the 1. {P e } is yet to be determined which requires definition of Ω domain of definition (geometry) of an element e. ¯ e is defined, we need to establish the local approximations φe . 2. Once Ω h 3. The approximation space Vh containing local approximation functions Ni needs to be defined as well. ¯ e : element geometry 5.3.1.1 Definition of Ω The element geometry in the physical coordinate space (say x, y) could be a triangular domain or a quadrilateral domain with distorted sides. These ¯ with minimum geometshapes enable discretizations of irregular domains Ω ric approximations. Figure 5.54 show such elements in the physical space x, y. We consider a distorted quadrilateral element for presenting details for this model problem (see Fig. 5.55). Following the details of mapping for 2D quadrilateral domain given in chapter 8, for the mapping of points we have x = x(ξ, η) = y = y(ξ, η) =

9 X i=1 9 X

¯i (ξ, η)xi N

(5.327)

¯i (ξ, η)yi N

(5.328)

i=1

Mapping of length (dx, dy) and (dξ, dη) is given by     dx dξ = [J] dy dη

(5.329)

315

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

5

6 7

4

9 8 y

y

3

2 1 x

x

Triangular element in x, y space

Quadrilateral element in x, y space

Figure 5.54: Triangular and quadrilateral elements in x, y space ¯e Ω

7

5

6

¯ ξη Ω

η

5

6

7

4

9

2

8

9

y

2

ξ

4

8 3

1

1 x

2

3

2 ¯ e in the natural coMap of Ω ¯ ξη ) ordinate space ξ, η (i.e. Ω

Quadrilateral element in x, y space

¯ e and its map Ω ¯ ξη in natural coordinate space Figure 5.55: A quadrilateral element Ω ξ, η

where " [J] =

∂x ∂ξ ∂y ∂ξ

∂x ∂η ∂y ∂η

# (5.330)

is the Jacobian of transformation dx dy = |J| dξ dη

(5.331)

and (

∂Ni ∂x ∂Ni ∂y

)

( = [J T ]−1

∂Ni ∂ξ ∂Ni ∂η

) (5.332)

316

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which Ni (ξ, η) are the local approximation functions (used in (5.317)). We also note that (xi , yi ) are the coordinates of the element nodes in the ¯i (ξ, η) have the following properties (x, y) space, and the shape functions N (see chapter 8) ( ¯i (ξj , ηj ) = 1 ; j = i N 0 ; j 6= i (5.333) 9 X ¯i (ξ, η) = 1 N i=1

We have intentionally chosen an element geometry with nine nodes. Midside nodes permit quadratic geometry description. Nine-node configuration ¯i (ξ, η) using tensor product (see chapter 8). permits easy determination of N 5.3.1.2 Approximation space Vh The governing differential equation describing the boundary value problem contains second order derivatives of the dependent variable in x and y whereas the weak form only contains first order derivatives of the dependent variable and the test function v. (i) Admissibility of φh = requires

S

e e φh

¯ T in the pointwise sense in Aφ − f = 0 in Ω

¯ e ); k = 3 is minimally conforming φeh ∈ Vh ⊂ H k,p (Ω

(5.334)

For this choice, the integrals in the following are Riemann. X (Aφh − f, v)Ω¯ T = (Aφeh − f, v)Ω¯ e e

=

X

B e (φeh , v) − le (v)




¯e Ω

=0

(5.335)

e

¯ e ) then all integrals are Riemann Thus, if we choose φeh of class C 2 (Ω and all three forms in (5.335) are precisely equivalent. ¯ e ) then (ii) Based on the weak form, if we choose φeh of class C 1 (Ω (Aφh − f, v)Ω¯ T

8 → L 8 →

X

(Aφeh − f, v)Ω¯ e

e

X

L

 B e (φeh , v) − le (v) = 0

e

|

{z R

}

(5.336)

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

317

P That is, for this choice of φeh , e (B e (·, ·) − le (·)) holds in the Riemann sense, but all other integral forms in (5.336) are in the Lebesgue sense. ¯ e ), then P (B e (·, ·) − le (·)) only holds (iii) If we choose φeh ∈ Vn ⊂ H 1,p (Ω e in the Lebesgue sense and the other two integral forms in (5.336) are meaningless. 5.3.1.3 Computation of the element matrix [K e ] and vector {F e } ¯ ξη , we can transform integrals for [K e ] and {F e } Using the element map Ω ξη ¯ . Using (5.336), we can write over Ω e Kij =

Z1 Z1 " −1 −1

∂Nj  ∂Ni  ∂Nj ∂Nj  ∂Ni  ∂Nj a11 + a21 (5.337) + a12 + a22 ∂x ∂x ∂y ∂x ∂x ∂y #

+ a00 Nj Ni |J| dξ dη Fie

(5.338)

Z1 Z1 f Ni |J| dξ dη

=

(5.339)

−1 −1

The derivatives of Ni s with respect to x and y can be obtained using (5.332), ∂Ni ∂Ni ∂ξ , ∂η can be easily obtained as Ni = Ni (ξ, η); det[J] = |J| is calculated using (5.330). The derivatives of x(ξ, η) and y(ξ, η) with respect to ξ and e and F e are obtained η are obtained using (5.327). Numerical values of Kij i using Gauss quadrature. 5.3.1.4 Details of secondary variable vector {P e } H We note that {P e } is defined by Γe v qne dΓ where v = NSj ; j = 1, 2, . . . , n. ¯ e = Ωe Γe . Consider a Details of {P e } obviously requires definition of Ω ¯ ξη (or Ω ¯ m , the four node quadrilateral element in x, y space mapped into Ω master element) shown in Fig. 5.56. In this case φeh =

4 X

Ni (ξ, η) φei

(5.340)

i=1

is a C 0 bilinear local approximation (see chapter 8), in which Ni (ξ, η) are the approximation functions and φei are nodal values of φ for element e. The ¯ e consists of boundary Γe of Ω Γe = Γe1

[

Γe2

[

Γe3

[

Γe4

(5.341)

318

SELF-ADJOINT DIFFERENTIAL OPERATORS

η Γe3

4

Γe4

y

4

3

¯e Ω

3

2

Γe2

ξ

¯ ξη Ω

1 Γe1

2 1

x

2 2

(a) A four node quadrilateral element

(b) Element map in the natural coordinate space ξη

Figure 5.56: A four-node quadrilateral element in x, y and ξ, η space

¯ e and φe , we have For this particular choice of Ω h   N1   I     N2 {P e } = qne dΓ N3     Γe N  4

(5.342)

we can write the integral in (5.342) over the closed boundary Γe as the sum of the four integrals over Γei ; i = 1, 2, . . . , 4   N1   Z Z     N2 e e {P } = qn dΓ+  N3     e Γ1 Γe2 N4

  N1    Z    N2 e qn dΓ+  N3     Γe3 N4

  N1    Z    N2 e qn dΓ+   N3    Γe4 N4

  N1       N2 qne dΓ N   3     N4 (5.343)

We note that

N1 (ξ, η) = N2 (ξ, η) = N3 (ξ, η) = N4 (ξ, η) =

 1 − ξ  1 − η  2 2  1 + ξ  1 − η  2 2  1 + ξ  1 + η  2 2  1 − ξ  1 + η  2

2

(5.344)

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

319

Hence, we have on Γe1 : N3 = N4 = 0 on Γe2 : N1 = N4 = 0 on Γe3 : N1 = N2 = 0

(5.345)

on Γe4 : N2 = N3 = 0 Substituting these into (5.343)     N1  0     Z  Z Z      N N 2 2 e e e {P } = qn dΓ+ qn dΓ+ 0 N3            e e Γ1 Γ2 Γe3 0 0

  0   Z    0 e qn dΓ+  N3     Γe4 N4

  N1       0 qne dΓ 0       N4 (5.346)

Let us define  e      e P  0   0   P        11  e        14   e  P P 0 0 21 22 {P e } = + + + e Pe  0    0    P32             33  e  e  0 0 P43 P44

(5.347)

In (5.347), the superscript e is for element e, the first subscript is the node number and the second subscript represents the side (or boundary) number i = 1, 2, . . . , 4 for Γi ; i = 1, 2, . . . , 4. since qne is normal to Γe (positive when e and P e are pointing in the same direction as the unit exterior normal), P11 21 e e at nodes 1 and 2 of element e, normal to the boundary Γ1 . Similarly, P33 e are at nodes 3 and 4 of element e normal to the boundary Γe and and P43 3 so on. Symbolically, we can write  e  e  e P1  P11 + P14         e  e e  + P P P e 2 21 22 {P } = = (5.348) e P e  P e + P33      3e     32  e + Pe P4 P43 44 We must keep in mind that the sum in (5.348) is only symbolic as the directions of the quantities in the sum are normal to the corresponding element sides on which they are defined. Remarks. ¯ e ), then φe ∈ Vh ⊂ H 1,p (Ω ¯ e ). (1) When the φeh of class C 0 (Ω h ¯ e ) local approximation (of any degree p) permit inter-element con(2) C 0 (Ω tinuity of the function φ and its derivative tangent to the inter-element boundaries, but the derivative of φ normal to the inter-element boundaries is discontinuous. Referring to Fig. 5.57, we note that φeh and φe+1 h

320

SELF-ADJOINT DIFFERENTIAL OPERATORS

are local approximations for elements e and e+1 with a common boundary (Γe2 for element e and Γe+1 for element e + 1). Let n and t be the 4 normal and tangent directions at a point on the common boundary between elements e and e+1, then, for each point on this common boundary we have e+1 φeh |Γe = φe+1 h Γ4 1 e+1 e ∂φh ∂φh = ∂t Γe ∂t e+1 (5.349) 1 Γ 4 ∂φe+1 ∂φeh h = 6 ∂n Γe ∂n e+1 Γ4

1

¯ e ) local approximations in two(5.349) are intrinsic properties of C 0 (Ω dimensional space x, y for each inter-element boundary like the one shown in Fig. 5.57. ¯ e ), then in (5.349) (3) When the local approximations φeh are of class C 1 (Ω the first two conditions hold but additionally equality also holds in the case of the third conditions, i.e. derivatives of the function or dependent variable normal to the inter-element boundary is continuous as well.

5.3.2 2D Poisson’s equation: numerical studies We consider a special case of the model problem in section 5.3.1.1 for numerical studies. If we choose a11 = 1, a12 = 0, a21 = 0, a22 = 1 and ¯ a two unit square, then we have a00 = 0 and Ω ∂2φ ∂2φ + 2 + f (x, y) = 0 ∀(x, y) ∈ Ω = (−1, 1) × (−1, 1) ⊂ R2 ∂x2 ∂y

(5.350)

5.3.2.1 Case (a): f = 1 with BCs φ(±1, y) = φ(x, ±1) = 0; GM/WF For this case we present details of element computations, assembly, and solution procedure using GM/WF. Galerkin method with weak form: Following section 5.3.1.1, if we ¯ e ), i.e. choose a four-node quadrilateral element with φeh ∈ Vh ⊂ H 1,1 (Ω 0 e e ¯ ) approximation of φ over Ω ¯ . Then k = 1, p = 1, a bilinear C (Ω φeh (ξ, η) =

4 X i=1

Ni (ξ, η) φei

(5.351)

321

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

4

5

6 Γe+1 3

Γe3 Γe4

Γe2

e

Γe+1 4

e+1

Γe+1 2

Γe+1 1

Γe1

1

2

3

∂φeh ∂t

4

5

e



∂φeh ∂η

Γe2

1

2

Γe2

5



Γe2

∂φeh ∂η

6

Γe+1 4

Γe+1 4

e+1

2

3

∂φeh ∂t

Γe+1 4

Figure 5.57: A four-node quadrilateral element with inter-element boundary in x, y

Ni (ξ, η) are given by (5.344). The element matrix [K e ] and vector {F e } are e Kij

=

Z1 Z1 

∂Ni ∂Nj ∂Ni ∂Nj  + |J| dξ, dη, i, j = 1, 2, . . . , 4 ∂x ∂x ∂y ∂y

(5.352)

−1 −1

Fie

Z1 (1) Ni |J| dξ dη

=

(5.353)

−1

¯ e and the boundary conditions A schematic of the domain of definition Ω are shown in Fig. 5.58 (a), and a four element uniform discretization is shown in Fig. 5.58 (b). Figure 5.58 shows each of the four elements of the discretization with local node numbers and the nodal dofs (using local node numbers). For each of the four elements of the discretization we calculate [K e ] and {F e } using (5.352) and (5.353). We can write    e 4 −1 −2 −1  φ    1e     1 −1 4 −1 −2  φ2 , e = 1, 2, . . . , 4 (5.354) B e (φeh , v) = [K e ]{δ e } =  e φ  6 −2 −1 4 −1    3e   −1 −2 −1 4 φ4

322

SELF-ADJOINT DIFFERENTIAL OPERATORS y

y

φ=0

7

9

8

φ=0

3 2

x

φ=0

5

4

1

6

x

2

1

φ=0

4

2

3

2 ¯ (a) Schematic of Ω

(b) A four-element uniform discretization

¯ and a four-element uniform discretization Figure 5.58: Domain Ω

e P43 e P44

e P14

e P33

4

Γe3

Γe4

e

1

Γe1

e P11

e 3 P32

Γe2 e 2 P22 e P21

Figure 5.59: Secondary variables for an element

 e    e P11 + P14   1    e  1    e P21 + P22 1 e e e l (v) = {P } + {F } = + ; e = 1, 2, . . . , 4 e + Pe P32 1   4 33      e   e  P43 + P44 1

(5.355)

Details of {P e } for an element e are shown in Fig. 5.59. We note that the secondary variables are normal to the faces of the element as shown in Fig. 5.59.

Inter-element continuity conditions on the nodal variables (PVs) of the elements Comparing the four element mesh with node numbers 1-9 with the elements with local node numbers (Fig. 5.60), if φ1 to φ9 are the nodal values of φ for the nine nodes of the discretization, then we have the following correspondence:

323

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

φ11 = φ1 ; φ12 = φ21 = φ2 ; φ22 = φ3 φ14 = φ31 = φ4 ; φ13 = φ32 = φ5 ; φ23 = φ42 = φ6

(5.356)

φ34 = φ7 ; φ33 = φ44 = φ8 ; φ43 = φ9 φ34

φ33

φ44

3

4

φ43

3 1

3

4 4 1

2

2

φ31

φ32

φ41

φ42

φ14

φ13

φ24

φ23

3

4 1 1 φ11

3

4 2 1

2 φ12

φ21

2 φ22

Figure 5.60: Each element of the discretization with local node numbers and degrees of freedom using local node numbers

Substituting (5.356) in (5.354) we obtain the following for each of the four elements (noting that [K 1 ] = [K 2 ] = [K 3 ] = [K 4 ]).   φ1       φ2 Element 1 : B 1 (φ1h , v) = [K 1 ] = [K 1 ]{δ 1 } φ   5     φ4    φ2     φ3 2 2 2 Element 2 : B (φh , v) = [K ] = [K 2 ]{δ 2 } φ     6  φ5   φ4       φ5 Element 3 : B 3 (φ3h , v) = [K 3 ] = [K 3 ]{δ 3 } φ8       φ7

(5.357)

(5.358)

(5.359)

324

SELF-ADJOINT DIFFERENTIAL OPERATORS

  φ5       φ6 Element 4 : B 4 (φ4h , v) = [K 4 ] = [K 4 ]{δ 4 } φ   9     φ8

(5.360)

(5.355) remains valid for e = 1, 2, . . . , 4. Boundary conditions Using the boundary conditions of the BVP, we have φ1 = φ2 = φ3 = φ4 = φ6 = φ7 = φ8 = φ9 = 0

(5.361)

Thus, only φ5 is unknown. Assembly of element equations The assembly of element matrices and vectors follow the standard procedure, i.e. (Aφh − f, v)Ω¯ T =

X

(Aφeh − f, v)Ω¯ e =

X

=

e

e

[K ]{δ } −

e

X




e

e

X

B e (φeh , v) − le (v)

e

{P } −

e

X

e

{F } = [K]{δ} − {P } − {F } = 0 (5.362)

e

where [K] =

X e

[K e ], {P } =

X e

{P e }, {F } =

X

{F e }, {δ} =

[

{δ e }

(5.363)

e

In this case, [K] is a (9 × 9) matrix, {F } is a (9 × 1) vector with known coefficients (from assembly of [K e ] and {F e }). However, {P } is a vector of secondary variables containing unknown secondary variables from each of the four elements. We consider details in the following. Conditions on the sum of secondary variables Figure 5.61 shows secondary variables at the nodes of each of the four elements. The node numbers inside each element are local node numbers, whereas those outside are global node numbers (i.e. node numbers of the ¯ T ). grid points of Ω We recall that (i) The sum of secondary variables at a node must be equal to the externally applied disturbance at that node. (ii) If the externally applied disturbance at a node is zero, then the sum of secondary variables at that node must be zero.

325

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

3 7 P43 3 4 P44

3 P14

3

1

Γ31

3 P11

4

1 P43

1 P14

1

Γ33

Γ34

4

1 P44

3 P33

Γ14

1

1

Γ11

8 4 P44

Γ32 3 2 P22

1 P33

Γ13

1 P11

3 3 P32

3 P21

4

8

5

5

1 3 P32

4 P14

5

1 P21 2

4

Γ43

Γ44

4

1

Γ41

2 P44

2 P14

2

Γ42 4 2 P22

2 P33

4

Γ23

Γ24

2

1

Γ21

2 P11

4 P33 9 4 3 P32

4 P21 6

4 P11

2 5 P43

Γe2 1 2 P22

4 P43

6

2 3 P32

Γ22 2 2 P22 2 P21

3

Figure 5.61: Secondary variables at the element nodes for the four-element discretization

(iii) At nodes where the dependent variable (or PV in this case) is specified, the sum of secondary variables is not known. An important fact to keep in mind is that secondary variables shown in Fig. 5.61 are normal to the faces of the elements. Thus, care must be taken in their sum at a node. In view of the BCs (5.361) we have the following At node 1: 1 P11 : not known because φ = 0 on Γ11 , i.e. φ = 0 at node 1 1 P14 : not known because φ = 0 on Γ14

At node 2: 1 2 P22 + P14 = 0 : equilibrate 1 2 + P11 ) not known because φ = 0 at node 2 (P21

At node 3: 2 P21 : not known because φ = 0 on Γ21 , i.e. φ = 0 at node 3 2 P22 : not known because φ = 0 on Γ22 , i.e. φ = 0 at node 3

At node 4: 1 3 P43 + P11 = 0 : equilibrate 1 3 (P44 + P14 ) not known because φ = 0 at node 4

326

At node 5:

SELF-ADJOINT DIFFERENTIAL OPERATORS

 1 2 P32 + P44 = 0    1 3 P33 + P21 = 0 equilibrate 2 4  P43 + P11 = 0    4 4 P22 + P14 =0

At node 6: 2 4 P33 + P21 = 0 : equilibrate 2 4 (P32 + P22 ) not known because φ = 0 at node 6

At node 7: 3 P44 : not known because φ = 0 on Γ34 3 P43 : not known because φ = 0 on Γ33

At node 8: 3 4 P32 + P44 = 0 : equilibrate 3 4 (P33 + P43 ) not known because φ = 0 at node 8

At node 9: 4 P32 : not known because φ = 0 on Γ42 4 P33 : not known because φ = 0 on Γ43

Then we have   {δ}T = φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ8 φ9 or   {δ}T = 0 0 0 0 φ5 0 0 0 0

(5.364)

and   {P }T = P1 P2 P3 P4 P5 P6 P7 P8 P9 or   {P }T = 0 0 0 0 P5 0 0 0 0

(5.365)

Solving for φ5 using the assembled equations gives φ5 = 0.375. Secondary variables can be easily calculated using the standard procedure discussed earlier.

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

327

5.3.2.2 Case (b): BCs φ(±1, y) = φ(x, ±1) = 1.0; GM/WF Consider the same BVP as in (5.350) with boundary conditions φ(±1, y) = φ(x, ±1) = 1.0

(5.366)

We choose f (x, y) such that the theoretical solution φt (x, y) of (5.350) and (5.366) is φ(x, y) = e(1−x

2 )(1−y 2 )

(5.367)

Galerkin method with weak form: Details of the element equations remain the same as in case (a). For numerical studies we consider a 4 × 4 ¯ using nine-node p-version two-dimensional finite uniform discretization of Ω elements. We consider local approximation φeh of φ over an element e with ¯ e to be in H k,p (Ω ¯ e ); k = 1, 2, 3; p ≥ 2k − 1 i.e. solutions of classes domain Ω ¯ T ); j = 0, 1, 2. For each class of local approximation the discretization C j (Ω is kept fixed and the p-levels are uniformly increased (pξ = pη = p) for each element. As seen in earlier examples the quadratic functional cannot be used to quantitatively assess the approximation errors but the residual functional can be. Unfortunately for local approximations of class C 0 it cannot be computed. Figure 5.62 show graphs of square root of quadratic functional ¯ T ) at p = 3, 5, 7, 9 and of class C 2 (Ω ¯T ) versus dofs for solutions of class C 1 (Ω √ at p = 5, 7, 9. We clearly observe that for a given dofs, I for solutions ¯ T ) is considerably lower than for the solution of class C 1 (Ω ¯T ) of class C 2 (Ω 2 confirming better accuracy of the solutions of class √ C . The same slopes of the two curves show that the rate of convergence of I for the two classes of local approximations are same. Figure 5.63 shows plots of solution φ versus x at y = 0 (same for y at x = 0) for solutions of classes C 0 , C 1 and C 2 for various p-levels. At p = 3 and beyond local approximation of all three d2 φ d3 φ classes produce quite accurate results. Graphs of dφ dx , dx2 and dx3 versus x at y = 0 are shown in Figs. 5.64–5.66. Interelement discontinuity of dφ dx diminishes with increasing p-level. At p = 3 and beyond all three classes of local approximations produce converged solutions that are in excellent agreement ¯ T ) at p = 3 with the theoretical solution. In Fig. 5.65 solutions of class C 1 (Ω 3 d2 φ show interelement jumps in dx2 which diminish upon increasing p-level. ddxφ3 ¯ T ) show interelement jumps that also versus x for solutions of class C 2 (Ω diminish and eventually converge with increasing p-levels. Importance of the higher order global differentiability approximations in achieving convergence and accurate values of the higher order solution derivatives is clearly demonstrated in the numerical studies presented in this example.

328

SELF-ADJOINT DIFFERENTIAL OPERATORS

0

C1 C2

(GM/WF) -2

log(√I)

-4

p=5

p=5

-6 -8 p=7

p=7

-10 -12 p=9

p=9

-14 1.8

2

2.2

2.4 2.6 log(dofs)

2.8

3

3.2

Figure 5.62: Case (b): square root of residual functional versus dofs

2.4

C0, p=1 C0, p=2 0 1 2 C , C , C , p≥3 theoretical

(GM/WF) 2.2

Solution φ

2 1.8 1.6 1.4 1.2 1 -1

-0.5

0 x

0.5

1

Figure 5.63: Case (b): solution φ versus x at y = 0 (same for y at x = 0)

5.3.3 Two-dimensional boundary value problems in multi-variables: 2D plane elasticity In this section we consider a boundary value problem containing more than one dependent variable. Examples of such BVPs are in linear elasticity such as plane stress, plane strain, axisymmetric deformation, bending of

329

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

2 1.5

C0, p=1 C0, p=2

1

C0, C1, C2, p≥3 theoretical

(GM/WF)

dφ/dx

0.5 0 -0.5 -1 -1.5 -2 -1

-0.5

Figure 5.64: Case (b):

0 x dφ dx

0.5

1

versus x at y = 0 (same for y at x = 0)

2 (GM/WF) 1

C1, p=3 C1, p=4 1 2 C , C , p≥5 theoretical

d2φ/dx2

0

-1

-2

-3 -1

-0.5

Figure 5.65: Case (b):

0 x d2 φ dx2

0.5

1

versus x at y = 0 (same for y at x = 0)

plates and shells, etc. Here, we specifically consider plane stress and plane strain problems in linear elasticity. We have the following.

330

SELF-ADJOINT DIFFERENTIAL OPERATORS

12

C2, p=5 C2, p=6

(GM/WF) 10

C2, p≥7 theoretical

8

d3φ/dx3

6 4 2 0 -2 -4 -6 -1

-0.5

Figure 5.66: Case (b):

0 x d3 φ dx3

0.5

1

versus x at y = 0 (same for y at x = 0)

1. Governing differential equations in terms of stresses ∂σx ∂τxy + + fx = 0 ∂x ∂y ∀x, y ∈ Ω ⊂ R2 τxy ∂σy + + fy = 0 ∂x ∂y

(5.368)

2. Strain displacement relations εx =

∂u , ∂x

εy =

∂v , ∂y

γxy =

∂u ∂v + ∂y ∂x

(5.369)

3. Constitutive equations σx = D11 εx + D12 εy σy = D12 εx + D22 εy

(5.370)

τxy = D33 γxy in which u, v are displacements in 0 − x and O − y directions of a fixed Cartesian coordinate frame O − xy, εx , εy and γxy are strains, σx , σy and τxy are stresses and Dij = Dji are coefficients containing material constants such as modulus of elasticity E, Poisson’s ratio v and shear modulus G (for the isotropic case), we can also use matrix and vector representation for (1) and (5.370). Let {σ} = [D]{ε} (5.371)

331

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

It is perhaps easier to express strains in terms of stresses, i.e. εx = C11 σx + C12 σy εy = C12 σx + C22 σy

(5.372)

γxy = C33 τxy or {ε} = [C]{σ};

[D] = [C]−1

(5.373)

in which for plane stress case (assuming isotropic material) we have C11 =

1 , E

C12 =

−v = C21 , E

C22 =

−1 , E

C33 =

1 , G

G=

E 2(1 + v) (5.374)

Boundary conditions ) u=u ˆ v = vˆ

on Γ2

and σx nx + τxy ny = tˆx τxy ny + σny = tˆy

(5.375)

) on Γ1

(5.376)

in which nx and ny are the direction cosines of a unit exterior normal to the boundary Γ1 . tˆx , tˆy are specified tractions on boundary Γ1 . u ˆ and vˆS are ¯ specified values ofSdisplacements on boundary Γ2 . We note that Ω = Ω Γ in which Γ = Γ1 Γ2 is a closed contour constituting the boundary of the ¯ domain Ω. GDEs in u and v We can substitute stresses in terms of strains and then strains in terms of derivatives of displacements u and v in the equation of equilibrium as well as boundary conditions (5.376).      ∂u ∂v  ∂u ∂v ∂ ∂  D11 + D12 − D33 + = fx  −  ∂x ∂x ∂y ∂y ∂y ∂y     ∀x, y ∈ Ω ⊂ R2  ∂u ∂v   ∂u ∂v ∂ ∂  − D33 + − D21 + D22 = fy  ∂x ∂y ∂y ∂x ∂x ∂y (5.377) ) u=u ˆ on Γ2 (5.378) v = vˆ

332

SELF-ADJOINT DIFFERENTIAL OPERATORS

 ∂u ∂v  ∂u ∂v  + D12 nx + D33 + ny = tˆx ∂x ∂y ∂y ∂y   ∂u ∂v   ∂u ∂v ny = tˆy D33 + nx + D21 + D22 ∂y ∂y ∂x ∂y 

D11

(5.379)

(5.377) are the desired GDEs in terms of displacements for 2D plane elasticity. We can also write (5.377) as Aφ − f = 0 in which the different operator A and φ are given by  ∂ ∂ ∂ ∂ ∂ ∂ − ∂x (D11 ∂x ) − ∂y (D33 ∂y ) − ∂x (D12 ∂y )−  A = [A] = ∂ ∂ ∂ ∂ ∂ ∂ − ∂x (D33 ∂y ) − ∂y (D12 ∂x ) − ∂x (D33 ∂x )−

(5.380)



∂ ∂ ∂y (D33 ∂x ) ∂ ∂ ∂y (D22 ∂y )

(5.381)   u φ = {φ} = v   fx f = {f } = fy

(5.382) (5.383)

Equation (5.380) can also be written as A11 u + A12 v = fx A21 u + A22 v = fy

(5.384)

We can show that the operator A is linear. We can also show that A∗ = A. Therefore, Galerkin method with weak form yielding variationally consistent integral forms is a desirable approach to develop finite element process for this BVP. 5.3.3.1 Galerkin method with weak form ¯e Let w1 = δu and w2 = δv be the test functions. Then, for an element Ω S T e ¯ = ¯ of the discretization Ω e Ω , we can write (A11 ueh + A12 vhe − fx , w1 )Ω¯ e =  Z    ∂u ∂v  ∂ ∂u ∂v  ∂  D11 + D12 + D33 + − fx w1 dx dy (5.385) ∂x ∂x ∂y ∂y ∂y ∂x ¯e Ω

and (A21 ueh + A22 vhe − fy , w2 )Ω¯ e =  Z    ∂u ∂v  ∂ ∂  ∂u ∂v  D33 + + D21 + D22 − fy w2 dx dy (5.386) ∂x ∂y ∂x ∂y ∂x ∂y ¯e Ω

333

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

Transferring one order of differentiation with respect to x and y in each term in (5.385) and (5.386) to w1 and w2 (A11 ueh + A12 vhe − fx , w1 )Ω¯ e Z  ∂w1  ∂u ∂v  = D11 + D12 + ∂x ∂x ∂y ¯e Ω  I ∂u ∂v  − w1 D11 + D12 nx + D33 ∂x ∂x Γe

 ∂u ∂v  ∂w1  dx dy D33 + ∂y ∂y ∂x  Z ∂u ∂v
ny dΓ − fx w1 dx dy + ∂y ∂x ¯e Ω

(5.387) In (5.387), the concomitant is hA11 ueh + A12 vhe − fx , w1 iΓe   I ∂u ∂v  ∂u ∂v
= − w1 D11 + D12 nx + D33 + ny dΓ ∂x ∂x ∂y ∂x Γe

and (A21 ueh + A22 vhe − fy , w2 )Ω¯ e  Z   ∂u ∂v  ∂w  ∂w2  ∂u ∂v  2 = D33 + + D21 + D22 dx dy ∂x ∂y ∂x ∂y ∂x ∂y e ¯ Ω   I Z  ∂u ∂v
∂u ∂v  − w2 D33 + nx + D21 + D22 ny dΓ − fy w2 dx dy ∂y ∂x ∂x ∂x Γe

¯e Ω

(5.388) Using concomitants in (5.387) and (5.388) we identify PVs, SVs, EBCs, and NBCs. ˜ 2 are EBCs (5.389) u, v are PVs, hence u = u ˜, v = v˜ on some boundary Γ In which the concomitant is hA21 ueh + A22 vhe − fx , w2 iΓe   I  ∂u ∂v
∂u ∂v  = − w2 D33 + + D22 ny dΓ nx + D21 ∂y ∂x ∂x ∂x Γe



D11

D33

 ∂u ∂v  ∂u ∂v
 + D12 + ny  nx + D33  ∂x ∂x ∂y ∂x are SVs   ∂u ∂v
∂u ∂v   + nx + D21 + D22 ny  ∂y ∂x ∂x ∂x

(5.390)

334

SELF-ADJOINT DIFFERENTIAL OPERATORS

Let us introduce the following notation  ∂u ∂v  ∂u ∂v
D11 + D12 nx + D33 + ny = qne x ∂x ∂x ∂y ∂x  ∂u ∂v  ∂u ∂v
D33 + nx + D21 + D22 ny = qne y ∂y ∂x ∂x ∂x

(5.391)

˜ 1 are NBCs. Then qne x and qne y with some given values on some boundary Γ Using the notations (5.391), we can rewrite (5.387) and (5.388) in the following form. (A11 ueh + A12 vhe − fx , w1 )Ω¯ e = I Z   ∂u ∂v  ∂w1  ∂u ∂v  ∂w1  dx dy − w1 qne x dΓ D11 + D12 + D33 + ∂x ∂x ∂y ∂y ∂y ∂x ¯e Γe Ω Z Z − fx w1 dx dy = B1e (ueh , vhe ; w1 ) − le (w1 ) − fx w1 dx dy (5.392) e1

¯e Ω

¯e Ω

and (A21 ueh + A22 vhe − fy , w2 )Ω¯ e =  Z  I  ∂u ∂v  ∂w  ∂w2  ∂u ∂v  2 D33 + + D21 + D22 dx dy − w2 qne y dΓ ∂x ∂y ∂x ∂y ∂x ∂y ¯e Γe Ω Z Z − fy w2 dx dy = B2e (ueh , vhe ; w2 ) − le (w2 ) − fy w2 dx dy (5.393) e2

¯e Ω

¯e Ω

Integrals (5.392) and (5.393) together constitute the desired weak form resulting from the Galerkin method with weak form. We combine (5.392) and (5.393) to give (Aφeh − f, w) = B e (ueh , vhe ; w1 , w2 ) − le (w1 , w2 ) in which

 B1e (ueh , vhe ; w1 ) B = B2e (ueh , vhe ; w2 )    R le (w1 )   fx w1 dx dy   ¯e le (lw1 , w2 ) = e1e + ΩR l (w2 )   fy w2 dx dy   e e

(ueh , vhe ; w1 , w2 )



2

and

(5.394)

(5.395)

(5.396)

¯e Ω

H  e  w1 qnx dΓ e le (lw1 , w2 ) = ΓH  w2 qne y dΓ e Γe

(5.397)

335

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

Approximation space ¯ contains second order The boundary value problem Aφ − f = 0 in Ω derivatives of displacements u and v, but the weak form only contains first order derivatives u and v. Furthermore, equal order, equal degree local approximations of u and v is justified based on the same order derivatives of both in the governing differential equations as well as the weak form. Thus, we have the following. ¯ T in the pointwise sense (i) Admissibility of ueh and vhe in Aφ − f = 0 in Ω requires ¯ e ), k = 3 is minimally conforming ueh , vhe (i.e. φeh ) ∈ Vh ⊂ H k,p (Ω For this choice, the integrals in the following are Riemann. X X (B e (φeh , w) − le (w)) = 0 (Aφ − f, w)Ω¯ T = (Aφeh − f, w)Ω¯ e = e

e

¯ e ), C 2 (Ω

φeh

(5.398) then all integrals are Riemann

of class That is, if we choose and all forms in (5.398) are equivalent. ¯ e ), that is, if (ii) Based on the weak form, if we choose φeh of class C 1 (Ω ¯ e ), then φeh ∈ Vh ⊂ H 2,p (Ω (Aφ − f, w)Ω¯ T ⇔

X

L

(Aφeh − f, w)Ω¯ e ⇔ L

e

X e

(B e (φeh , w) − le (w)) = 0 | {z } R

(5.399) Thus, for this choice of − holds in the Riemann e sense, but all other integral forms in (5.399) are in the Lebesgue sense. ¯ e ), then P (B e (·, ·) − le (·)) only holds (iii) If we choose φeh ∈ Vh ⊂ H 1,k (Ω e in the Lebesgue sense and the other two integral forms in (5.399) are meaningless. φeh ,

P

(B e (·, ·)

le (·))

Local approximation ueh and vhe Consider equal order, equal degree local approximations of class C 0 with ¯ ξη in p = 1 for a four node quadrilateral element in xy space with its map Ω the natural coordinate space. Then, we can write ueh vhe

=

=

4 X i=1 4 X i=1

Ni (ξ, η) uei = [N ]{ue } (5.400) Ni (ξ, η) vie

e

= [N ]{v }

336

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which Ni (ξ, η) are standard local approximation functions for the four ¯ e mapped in Ω ¯ ξη in a two unit square with the origin of corner nodes of Ω ξη coordinate system at the center of the element. {ue } and {v e } are nodal degrees of freedom for ueh and vhe at the four corner nodes, i.e. the values of ueh and vhe . From (5.400) w1 = δueh = Nj , j = 1, 2, . . . , 4 w2 = δvhe = Nj , j = 1, 2, . . . , 4

(5.401)

Element matrix [K e ] and vector {F e } Substituting from (5.400) and (5.401) in the weak form, we obtain Z 

B1e (ueh , vhe , w1 )

=

  4 4 X X ∂Nj ∂Ni e ∂Ni e D11 u + D12 v ∂x ∂x i ∂x i i=1

¯e Ω 4

4

i=1

i=1



+

X ∂N X ∂Ni ∂Nj i e D33 ui + ve ∂y ∂y ∂x i

B2e (ueh , vhe , w2 )

Z  =

i=1



dx dy, j = 1, 2, . . . , 4 (5.402)

  4 4 X ∂Nj ∂Ni e X ∂Ni e  D33 u + v ∂x ∂y i ∂x i i=1

¯e Ω

i=1

  4 4 X X ∂Nj ∂Ni e ∂Ni e + D21 u + D22 v dx dy, j = 1, 2, . . . , 4 (5.403) ∂y ∂x i ∂y i i=1

i=1

R    Nj fx dΩ  ¯e {F e } = ΩR , j = 1, 2, . . . , 4   Nj fy dΩ 

(5.404)

¯e Ω

We can write these in the matrix and vector form  e e e  B1 (uh , vh , w1 ) (Aφ − f, w)Ω¯ e = − {P e } − {F e } B2e (ueh , vhe , w2 )

(5.405)

or (Aφeh − f, w)Ω¯ e = [K e ]{δ} − {P e } − {F e }

(5.406)

The specific form of [K e ] depends upon how the dofs for ueh and vhe are arranged in {δ e } for the four nodes of the element. If we assume that {δ 3 } = [ue1 , v1e ; ue2 , v2e ; ue3 , v3e ; ue4 , v4e ]t

(5.407)

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

then

 e u1        v1e          . .  N 0 . . . N 0 . . . . 1 i φeh = [N ]{δ e } = e 0 N1 . . . 0 Ni . . .  ui   e  e    v  i      ..   .

and [K e ]ij , i.e. for node i and node j is a 2 × 2 matrix  e   e  e e ) K11 K12 (K11 )ij (K12 ij e [K ]ij = = e K2 e ) (K e ) K21 (K21 ij 22 ij 22 ij

337

(5.408)

(5.409)

where e (K11 )ij

Z  ∂Nj ∂Nj  ∂Ni ∂Ni D11 + D33 dΩ, i, j = 1, 2, . . . , 4 (5.410) = ∂x ∂x ∂y ∂y ¯e Ω

e (K12 )ij

Z  ∂Nj ∂Nj  ∂Ni ∂Ni = D12 + D33 dΩ, i, j = 1, 2, . . . , 4 (5.411) ∂x ∂y ∂x ∂y ¯e Ω

e (K21 )ij =

Z  ∂Nj ∂Nj  ∂Ni ∂Ni D33 + D21 dΩ, i, j = 1, 2, . . . , 4 (5.412) ∂x ∂y ∂y ∂x ¯e Ω

e (K22 )ij

Z  ∂Nj ∂Nj  ∂Ni ∂Ni = D33 + D22 dΩ, i, j = 1, 2, . . . , 4 (5.413) ∂x ∂x ∂y ∂y ¯e Ω

R  e Ni fx dΩ ¯ Ω (F )i = R , i = 1, 2, . . . , 4 ¯ e Ni fy dΩ Ω e

(5.414)

We note that [K e ] is symmetric. For this 4-node element [K e ] is 8 × 8 and {F e } is 8 × 1. Details of {P e } for each boundary integral follows the same details as presented for the model problem in section 5.3.1. Numerical values of the coefficients of [K e ] and {F e } are obtained using gauss quadrature. The ¯ e are converted to Ω ¯ ξη domain (as in integrals in (5.410) to (5.414) over Ω section 5.3.1) and then integrated using Gauss quadrature. 5.3.3.2 Least-squares method using residual functional In this section we present a least-squares finite element formulation of the 2D plane elasticity model problem defined by the equations (5.368)–(5.376). We rewrite these equations in a slightly different form. The momentum equations in the absence of body forces and the constitutive relations (linear

338

SELF-ADJOINT DIFFERENTIAL OPERATORS

elasticity) are all we need. Using the mathematical model in (5.368)–(5.376), we can write the following: ∂σx ∂τxy + =0 ∂x ∂y ∂τxy ∂σy + =0 ∂x ∂y

(5.415)

∂u ∂v + D12 ∂x ∂y ∂u ∂v σy = D21 + D22 ∂x ∂y   ∂u ∂v τxy = D33 + ∂y ∂x

(5.416)

σx = D11

Details of [D], Dij = Dji are given in (5.373) and (5.374). Equations (5.415) and (5.416) are a system of first order differential equations in the dependent variables u, v, σx , σy , and τxy . We can also substitute stresses from (5.416) into (5.415) and thereby obtain only two partial differential equations in u and v but at the expense of the appearance of up to second order derivatives of u and v them. These would obviously require higher order global differentiability local approximations. In LSP is is beneficial to non-dimensionalize the mathematical model so that all dependent variables have numerical values in the same range during computations. We rewrite (5.415) and (5.416) with ∧ (hat) on all variables (dependent as well as independent) implying that they have their usual dimension. We use the dimensionless length L0 , force F0 , and time t0 (t0 unnecessary for stationary problems i.e. BVPs) in the mathematical model to nondimenstionalize it. For the mathematical model (5.415) and (5.416) we choose: ˆ L0 = L, σ ˆy σy = , τ0

x= τxy

x ˆ , L0 τˆxy = , τ0

y=

yˆ , L0

E0 = τ0 ,

ˆ E σ ˆx , σx = E0 τ0 u ˆ vˆ u= , v= L0 L0

E=

(5.417)

Using (5.417) in (5.415) and (5.416) we find that these are in fact also the dimensionless form. We consider nine node p-version 2D element mapped in ξη space (two unit square). Let ueh , vhe , (σx )eh , (σy )eh , and (τxy )eh be the local approximations ¯ e of the discretization Ω ¯ T = SΩ ¯ e of of u, v, σx , σy , and τxy for an element Ω e

¯ ⊂ R2 . For generality, we consider unequal order and unequal the domain Ω

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

339

degree local approximation. u

ueh

=

vhe =

n X i=1 nv X

Niu uei = [N u ]{ue } Niv vie = [N v ]{v e }

i=1 σ

(σx )eh

=

(σy )eh = (τxy )eh =

n x X i=1 σy n X i=1 τxy n X

Niσx (σx )ei = [N σx ]{(σx )e }

(5.418)

σ

Ni y (σy )ei = [N σy ]{(σy )e } τ

Ni xy (τxy )ei = [N τxy ]{(τxy )e }

i=1

By replacing u, v, σx , σy , and τxy in (5.415) and (5.416) with their local approximations ueh , vhe , (σx )eh , (σy )eh , and (τxy )eh we obtain the residual equa¯ e ⊂ R2 : tions ∀x, y ∈ Ω ∂(σx )eh ∂(τxy )eh + ∂x ∂y e ∂(τxy )h ∂(σy )eh = + ∂x ∂y e ∂v e ∂u ¯ e ⊂ R2 = (σx )eh − D11 h − D12 h ∀x, y ∈ Ω ∂x ∂y ∂ue ∂v e = (σy )eh − D21 h − D22 h ∂x ∂y  e  ∂uh ∂vhe = (τxy )eh − D33 + ∂y ∂x

E1e = E2e E3e E4e E5e

(5.419)

Upon substituting the local approximations 5.418 into 5.419 we obtain the explicit form of the residual equations. Let us define nodal degrees of freedom {δ e } for an element e.   e T {δ e }T = {ue }T, {v e }T, {(σx )e }T, {(σy )e }T, {(τxy }

(5.420)

Then T

{δEi } =

"

T    T ∂Ei T ∂Ei , , , ∂{v e } ∂{(σx )e } T  T # ∂Ei ∂Ei , , i = 1, 2, . . . , 5 (5.421) ∂{(σy )e } ∂{(τxy )e }

∂Ei ∂{ue } 

340

SELF-ADJOINT DIFFERENTIAL OPERATORS

and the element matrix [K e ] is given by   5 Z  X ∂Eie T ∂Eie dΩ [K ] = ∂{δ e } ∂{δ e } e

(5.422)

i=1 ¯ Ω

Remaining details of numerical integration for [K e ], assembly of element matrices, etc. follow the standard procedure. Minimally conforming order of the space k is two (local approximations ¯ e )) for all dependent variables for which the integrals over Ω ¯T of class C 1 (Ω are Riemann. However, if the theoretical solution is smooth then k = 1 ¯ e )) may suffice, keeping in mind that for (local approximations of class C 0 (Ω ¯ T are in the Lebesgue sense. this choice of k the integrals over Ω Simply supported and clamped-clamped 2D beams: numerical studies We consider a thin and narrow plate of length ˆl of 20 inches, width ˆb of 2.5 inches, and thickness tˆ of 0.1 inches (geometrically, it is a beam). With L0 = 10 inches the dimensionless plate is 2 × 0.05 × 0.01. We consider loads ˆ = 30×106 psi, E0 = 27.3×106 applied in the plane of the plate. We choose E ˆ = 10.5×106 psi, hence G = Gˆ = 0.3846; hence E = 1.0989; shear modulus G E0 and Poisson’s ratio ν = 0. Model problem 1: simply supported (SS) beam In this case we consider the plate to be “simply supported” as shown in Fig. 5.67. Points A and B are constrained in the y-direction but are free to move in the x-direction. On face AB of the plate σy = 10−6 and on face CD σy = −10−6 is applied causing deflection of the plate in the negative y-direction. At the center plane (EF ) the x-displacement (u) is constrained due to symmetry about x = 0.5ˆl. Since ˆb and tˆ are much smaller than ˆl, the deformation behavior is like a simply supported slender beam (shear deformation is not significant). The domain (l × b) 2 × 0.05 is modeled using a twenty element uniform discretization (ten elements along the length l and two elements along the width b) using nine-node p-version hierarchical plane stress elements with higher order global differentiability local approximation ¯ e ) scalar product space. Boundary conditions on the four boundin H k,p (Ω aries of the domain ABCD of the plate are also shown in Fig. 5.67. The nine-node elements are mapped into a square of two units with the origin of the natural coordinate system ξ, η at the center of the element. The element local approximation as well as all computations are performed using natural coordinate system ξ, η. The degrees of local approximation in ξ and η (pξ , pη ) are chosen to be equal p = pξ = pη and are chosen to be the same

341

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

for all dependent variables. Due to smoothness of the solution for this model problem both C 0 and C 1 solutions are expected to work well. We expect C 0 solutions to approach C 1 solutions upon convergence. In the numerical solutions presented here we choose k = 1; that is, local approximation of ¯ e ). class C 0 (Ω σy = −10−6 τxy = 0 u=0

σx = 0 τxy = 0 y

τxy = 0

C

E

D

A

F

B

0.05

0.01 x

2 v=0

σy = 10−6 τxy = 0

v=0

Figure 5.67: Schematic and boundary conditions used for a simply supported plate

A p-convergence study with p = pξ = pη = 3, 5, . . . shows that at p = 9 the residual functional I is of the order O(10−16 ), confirming that the equations are satisfied accurately in the pointwise sense. This is confirmed by ¯ e ) and their comparisons the similar studies with the solutions of class C 1 (Ω ¯ e ) studies. Thus, in the following we present results for p = pξ = with C 0 (Ω ¯ e ) for all dependent variables pη = 9 and local approximations of class C 0 (Ω using 20 element uniform discretization described earlier. Model problem 2: clamped-clamped (CC) beam This model problem consists of the same plate as used in model problem 1 but is considered clamped at the two ends (x = 0 and x = 2) as shown in Fig. 5.68. The boundary conditions on the boundaries AC and BD (clamped boundaries) are u = v = 0 (shown in Fig. 5.68). The details of dicretization, choice of p-levels, choice of order of space, etc. are the same for this model problem as described on the model problem 1. In this case ¯ e ) yields the residalso a p-convergence study for the solutions of class C 0 (Ω −16 ual functional I of the order of O(10 as is the case of model problem 1. ¯ e ) local apThus, for this model problem also p = pξ = pη = 9 and C 0 (Ω proximation for all dependent variables yield very accurate solutions, hence are used to compute the results presented here. Solutions for model problems 1 and 2 In the solutions presented here for model problems 1 and 2 the dimenˆ = 30 × 106 psi. Figure sionless modulus of elasticity E corresponds to E 5.69 shows a plot of v versus x at the centerline (y = 0.025) for the simply supported beam. The displacements v of the bottom and the top faces of

342

SELF-ADJOINT DIFFERENTIAL OPERATORS

σy = −10−6 τxy = 0 u=0

u=0 v=0 y

u=0 v=0

C

E

D

A

F

B

0.05

0.01 x

2 σy = 10−6 τxy = 0

Figure 5.68: Schematic and boundary conditions used for a clamped-clamped plate

the beam (y = 0.0 and y = 0.05) are virtually the same as the displacement v at the centerline (y = 0.025) as expected for a slender beam like what is used here (validating the assumption of inextensibility of transverse normals in a beam theory). Graphs of v versus x at the centerline of the clamped (CC) beam of Fig. 5.68 for the same values of σy on boundaries AB and CD and the same value of E are shown in Fig. 5.70. These are plotted using the same x, y scales as Fig. 5.69 so that displacements of the two can be compared. We observe substantially reduced v displacement in the case of   ∂v CC beam as expected. Rotation θz = 12 ∂u − versus x for SS and CC ∂y ∂x plates at the centerline are shown in Figs. 5.71 and 5.72. In both model problems θz is antisymmetric about x = 1.0. Much larger values of θz in case of SS plate (due to unconstrained θz at x = 0.0 and x = 2.0) compared to CC plate are clearly observed. Graph of τxy versus x at the centerline of the SS and CC plates are shown in Figs. 5.73 and 5.74. We observe that τxy is antisymmetric about x = 1.0 for both SS and CC plates. 0

Displacement v

−0.01

−0.02

−0.03 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.69: Displacement v at y = 0.025 versus distance x (simply supported plate)

343

5.3. TWO-DIMENSIONAL BOUNDARY VALUE PROBLEMS

0

Displacement v

−0.01

−0.02

−0.03 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.70: Displacement v at y = 0.025 versus distance x (clamped-clamped plate)

0.04

Rotation θz

0.02 0

−0.02 −0.04 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.71: Rotation θz at y = 0.025 versus distance x (simply supported plate)

0.04

Rotation θz

0.02 0

−0.02 −0.04 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.72: Rotation θz at y = 0.025 versus distance x (clamped-clamped beam)

344

SELF-ADJOINT DIFFERENTIAL OPERATORS

Shear Stress τxy

0.00004 0.00002 0 −0.00002 −0.00004 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.73: Shear stress τxy at y = 0.025 versus distance x (simply supported plate)

Shear Stress τxy

0.00004 0.00002 0 −0.00002 −0.00004 0

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 Distance x

2

Figure 5.74: Shear stress τxy at y = 0.025 versus distance x (clamped-clamped plate)

5.4 Three-dimensional boundary value problems 5.4.1 Three-dimensional boundary value problems in a single dependent variable Consider 3D steady-state Fourier heat conduction equation for an anisotropic medium with symmetric conductivity matrix:

−

∂qx ∂qy ∂qz − − + Q = 0, ∀x, y, z ∈ Ω ⊂ R3 ∂x ∂y ∂z

(5.423)

where (qx , qy , qz ) are the components of the heat flux vector q, which is related to the gradient of temperature ∇T by the Fourier heat conduction law, q = −k · ∇T :

345

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

∂T ∂T ∂T + kxy + kxz ∂x ∂y ∂z ∂T ∂T ∂T −qy = kyx + kyy + kyz ∂x ∂y ∂z ∂T ∂T ∂T + kzy + kzz −qz = kzx ∂x ∂y ∂z

−qx = kxx

(5.424)

¯ = Ω ∪ Γ, closure of Ω with kij = kji in which T is the temperature and Ω (i, j = x, y, z) and the boundary conditions T = T0 qx nx + qy ny + qz nz + β(T − T∞ ) + q = 0

on on

Γ2

(5.425)

Γ1

(5.426)

where (nx , ny , nz ) are the direction cosines of a unit exterior normal to boundary Γ1 , (qx , qy , qz ) are heat fluxes in x, y and z directions, respectively, β is a film coefficient, T∞ is the ambient temperature, and q is the applied heat flux normal to the boundary. If we substitute qx , qy and qz from (5.424) in (5.423), then we obtain a single second order partial differential equation in temperature T . One could show that for this differential equation (a) the differential operator is linear, (b) the adjoint A∗ of the differential operator is same as the operator, and thus (c) the Galerkin method with weak form will yield variationally consistent integral forms. In the following we consider finite element processes based on Galerkin method with weak form. 5.4.1.1 Galerkin method with weak form ¯ e of the discretization Ω ¯T = S Ω ¯ e with For an element e with domain Ω e

¯ e , we consider local approximation The of T over Ω

Z  ∂ (qx ) ∂x ¯e ¯e Ω Ω Z  ∂ ∂ + (qy ) + (qz ) v dx dy dz + Qv dx dy dz (5.427) ∂y ∂z

(A The + Q, v)Ω¯ e =

Z

(A The + Q)v dΩ = −

¯e Ω

346

SELF-ADJOINT DIFFERENTIAL OPERATORS

in which v = δφeh . Transferring one order of differentiation to v from each term in the integrand in (5.427) (A The + Q, v)Ω¯ e =

Z  ∂v e ∂v e ∂v e  (qx ) + (qy ) + (q ) dx dy dz ∂x ∂y ∂z z ¯e Ω I Z e e e − v(qx nx + qy ny + qz nz ) dΓ + Qv dΩ (5.428) Γe

¯e Ω

The concomitant hAThe + Q, viΓe in (5.428) is given by hAThe

I + Q, viΓe = −

Γe

v(qxe nx + qye ny + qze nz ) dΓ

Using concomitant (5.428) we can determine PVs, SVs, EBCs and NBCs; T ˜ 1 is the EBC and (q e nx + q e ny + is the PV and T = T˜ on some boundary Γ x y ˜ 2 is qze nz ) is the SV. Hence, (qxe nx + qye ny + qze nz ) = q˜ on some boundary Γ the NBC. As seen before, we note that the secondary variable in (5.428) is not known. If we let ˆ · qe ) qxe nx + qye ny + qze nz = qne (= n

(5.429)

in which qne is the flux normal to the element boundary Γe , then we can write (5.428) as (A The + Q, v)Ω¯ e =

Z  ∂v e ∂v e ∂v e  q + q + q dx dy dz ∂x x ∂y y ∂z z ¯e Ω I Z e − vqn dΓ + Qv dx dy dz Γe

(5.430)

¯e Ω

or (A The + Q, v)Ω¯ e = B e (The , v) − le (v)

(5.431)

in which B

e

Z  ∂v e ∂v e ∂v e  = q + q + q dx dy dz ∂x x ∂y y ∂z z ¯e Ω I Z le (v) = vqne dΓ − Qv dx dy dz

(The , v)

Γe

(5.432)

(5.433)

¯e Ω

(5.431) is the weak form of the boundary value problem (5.423) and (5.424).

347

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

5.4.1.2 Approximation space The boundary value problem contains up to second order derivatives of the temperature T and the test function v. ¯ T in the pointwise (i) Admissibility of Th is AT − f = 0 (f = −Q) in Ω ¯ e ); k = 3 is minimally conforming. sense requires, The ∈ Vh ⊂ H k,p (Ω For this choice, the integrals in the following are Riemann X X [B e (The , v) − le (v)] = 0 (AThe − f, v)Ω¯ e = (ATh − f, v)Ω¯ T = e

e

(5.434) ¯ e ), then all integrals are Riemann Thus, if we choose The of class C 2 (Ω in all forms in (5.434) and, hence we say that all three forms in (5.434) are precisely equivalent. ¯ e ), if T e ∈ Vh ⊂ (ii) Based on the weak form, if we choose The of class C 1 (Ω h 2,p e ¯ H (Ω ), then X X (ATh − f, v)Ω¯ T ⇔ (AThe − f, v)Ω¯ e ⇔ [B e (The , v) − le (v)] = 0 L L | {z } e e R

(5.435) − holds in the Riemann sense, Thus, for this choice e but all other integral forms in (5.435) are in the Lebesgue sense. ¯ e ), then P [B e (·, ·) − le (·)] only holds (iii) If we choose, The ∈ Vh ⊂ H 1,p (Ω e in the Lebesgue sense and the other two integral forms in (5.435) are meaningless P

[B e (·, ·)

le (·)]

5.4.1.3 Local approximation The Let The

=

n X

Ni (ξ, η, ζ) δie = [N ]{δ e }

(5.436)

i=1

¯ ξηζ . Then be the local approximation of T over an element e with domain Ω v = δThe = Nj (ξ, η, ζ), j = 1, 2, . . . , n

(5.437)

Ni (ξ, η, ζ) are the local approximation functions and δie are the nodal degrees of freedom. Substituting (5.436) and (5.437) into the weak form (5.431) Z  ∂Nj e ∂Nj e ∂Nj e  B e (The , Nj ) = q + q + q dx dy dz (5.438) ∂x x ∂y y ∂z z ¯e Ω Z I e e l (Nj ) = Nj qn dΓ − Q Nj dx dy dz (5.439) Γe

¯e Ω

348

SELF-ADJOINT DIFFERENTIAL OPERATORS

for i, j = 1, 2, . . . , n, where −qxe = kxx

n X ∂Ni i=1

−qye = kyx −qze = kzx

∂x

n X ∂Ni i=1 n X i=1

∂x

δie + kxy

n X ∂Ni i=1

δie + kyy

∂Ni e δ + kzy ∂x i

∂y

n X ∂Ni i=1 n X i=1

∂y

δie + kxz

n X ∂Ni i=1

δie + kyz

∂Ni e δ + kzz ∂y i

∂z

n X ∂Ni i=1 n X i=1

∂z

δie δie

(5.440)

∂Ni e δ ∂z i

Thus, we have B e (The , Nj ) = −[K e ]{δ e }, le (vNj ) = {P e } − {F e }

(5.441)

where (i, j = 1, 2, . . . , n) Z  ∂Nj ∂Nj ∂Nj  ∂Ni  e kxx + kxy + kxz Kij = ∂x ∂x ∂y ∂z ¯e Ω

∂Nj ∂Nj ∂Nj  ∂Ni  kyx + kyy + kyz ∂y ∂x ∂y ∂z  ∂Nj ∂Nj ∂Nj  ∂Ni  kzx + kzy + kzz dx dy dz + ∂z ∂x ∂y ∂z +

Pie =

I

(5.442)

qne Ni dΓ

(5.443)

QNi dx dy dz

(5.444)

Γe

Fie =

Z ¯e Ω

Remarks. ¯ e , i.e. (1) {P e } is yet to be determined, which requires the definition of Ω the domain of definition (geometry) of the element e. (2) Once the geometry is established, explicit form of φeh , the local approximation can be determined. (3) The approximation space Vh containing the local approximation functions Ni needs to be defined as well. ¯ e : Element geometry 5.4.1.4 Definition of Ω The element geometry in the physical coordinate space could be a tetrahedron with flat faces and straight edges or it could have curved faces and edges. The element geometry could also consist of a hexahedron with curved

349

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

faces and edges or flat faces and straight line edges. We consider hexahedron elements for illustrating the details. Fig. 5.75 (a) shows a twenty-seven-node three-dimensional hexahedron element in the physical coordinate space. The element faces and the edges are distorted. The element geometry in Fig. 5.75 (a) is mapped into a two-unit cube in Fig. 5.75 (b) with the origin of the coordinate system at the center of the element. Following the details in chapter 8, we have the following for mapping of points. x(ξ, η, ζ) = y(ξ, η, ζ) = z(ξ, η, ζ) =

27 X i=1 27 X i=1 27 X

¯i (ξ, η, ζ) xi N ¯i (ξ, η, ζ) yi N

(5.445)

¯i (ξ, η, ζ) zi N

i=1

η

ξ y

ζ x

z A 27-node element in x, y, z space

Element map in natural coordinate space ξ, η, ζ space

Figure 5.75: A 27-node three-dimensional hexahedron element

Mapping of length (dx, dy, dz) and (dξ, dη, dζ) is described by (can be derived following details similar to 2D case, see chapter 8)       ∂x ∂x ∂x ∂ξ ∂η ∂ζ dx dξ    ∂y ∂y ∂y  dy = [J] dη ; [J] =  (5.446) ∂ξ ∂η ∂ζ       ∂z ∂z ∂z dz dζ ∂ξ ∂η ∂ζ

[J] is the Jacobian of transformation and dx dy dz = |J| dξ dη dζ (can also be derived using details similar to 2D case, see chapter 8). We note that the ¯i (ξ, η, ζ) in (5.445) have the properties shape functions N ( 27 1, j = i X ¯ ¯i (ξj , ηj , ζj ) = N , Ni (ξ, η, ζ) = 1 (5.447) 0, j 6= i i=1

350

SELF-ADJOINT DIFFERENTIAL OPERATORS

¯i (ξj , ηj , ζj ) can For the nodal configurations used in Figs. 5.75 (a) and (b), N be easily derived using tensor product (see chapter 8) 5.4.1.5 Computations of element matrix [K e ] and vector {F e } ¯ ξηζ in the natural coordinate space ξ, η, ζ we Using the element map Ω ¯ ξη and (5.442) and (5.444) can transform integrals for [K e ] and {F e } over Ω can be written as e Kij

Z1 Z1 Z1  =

∂Nj ∂Nj ∂Nj  ∂Ni  kxx + kxy + kxz ∂x ∂x ∂y ∂z

−1 −1 −1

∂Nj ∂Nj ∂Nj  ∂Ni  kyx + kyy + kyz ∂y ∂x ∂y ∂z   ∂Nj ∂Nj ∂Nj  ∂Ni + kzx + kzy + kzz |J| dξ dη dζ ∂z ∂x ∂y ∂z +

(5.448)

Z1 Z1 Z1

Fie =

QNi dΩ

(5.449)

−1 −1 −1

The derivatives of the approximation functions Ni (ξ, η, ζ) with respect to x, y, z can be easily obtained using (see chapter 8)   ∂Ni     ∂x   ∂Ni

∂y     ∂Ni  

= [J T ]−1

∂z

  ∂Ni      ∂ξ  ∂Ni

∂η      ∂Ni 

(5.450)

∂ζ

Numerical values of [K e ] and {F e } can be obtained using gauss quadrature. 5.4.1.6 Details of secondary variable vector {P e } H Note that {P e } is defined by Γe vqne dΓ, v = Nj (ξ, η, ζ); j = 1, 2, . . . , n. ¯e = Thus, details of {P e } obviously require a clear definition of Γe in Ω Ωe ∪ Γe . For simplicity, we consider an eight-node hexahedron element in the x, y, z space (5.76. The element has flat faces and straight edges but the element shape may not have to be a prism. The element is mapped into a two-unit cube in ξ, η, ζ natural coordinate space. The boundary Γe of the element consists of six faces Γe =

6 [ i=1

Γei

351

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

η

φe8

φe7

8 φe5 6

5 φe4 y

ξ

3 φe3

1 ζ x

7

φe6

2

φe1

φe2

z Eight-node element in R3

Map of an eight-node element in ξ, η, ζ space

Figure 5.76: An eight-node three-dimensional hexahedron element

let us define Γei (i = 1, . . . , 6) as follows.

Γe1 has nodes: 1, 4, 8, 5 (ξ=−1)

(5.451)

Γe2 has nodes: 2, 3, 7, 6 (ξ=+1)

(5.452)

Γe3 has nodes: 1, 2, 6, 5 (η=−1)

(5.453)

Γe4 has nodes: 3, 4, 8, 7 (η=+1)

(5.454)

Γe5 has nodes: 1, 2, 3, 4 (ζ=−1)

(5.455)

Γe6 has nodes: 5, 6, 7, 8 (ζ=+1)

(5.456)

The integral over Γe can be written as the sum of the integrals over Γei ; i = 1, . . . , 6.

e

I

{P } = Γe

{N }T qne dΓ

=

6 Z X

{N }T qne dΓ =

i=1 Γe

6 X

P e |Γe i

(5.457)

i=1

i

We consider the integrals over Γei in (5.457). First, we note that {N } = {N1 , . . . , N8 }

(5.458)

352

SELF-ADJOINT DIFFERENTIAL OPERATORS

n o [N ]Γe1 = N1 , 0, 0, N4 , N5 , 0, 0, N8 ξ=−1 n o [N ]Γe2 = 0, N2 , N3 , 0, 0, N6 , N7 , 0 ξ=1 n o [N ]Γe3 = N1 , N2 , 0, 0, N5 , N6 , 0, 0 η=−1 n o [N ]Γe4 = 0, 0, N3 , N4 , 0, 0, N7 , N8 η=1 n o [N ]Γe5 = N1 , N2 , N3 , N4 , 0, 0, 0, 0 ζ=−1 n o [N ]Γe6 = 0, 0, 0, 0, N5 , N6 , N7 , N8

(5.459)

ζ=1

Using (5.459) for each of the integrals in (5.457). Consider Z n oT e T N1 , 0, 0, N4 , N5 , 0, 0, N8 {PΓe1 } = qne dΓ ξ=−1

(5.460)

Γe1

Let us define {P1e } as n o e T e e e e {PΓ1 } = P11 , 0, 0, P41 , P51 , 0, 0, P81 = {P1e }T

(5.461)

and similarly n o e e e e {PΓe2 }T = [0, P22 , P32 , 0, 0, P62 , P72 ,0 n o e e e e {PΓe3 }T = P13 , P23 , 0, 0, P53 , P63 , 0, 0 n o e e e e {PΓe4 }T = 0, 0, P34 , P44 , 0, 0, P74 , P84 n o e e e e {PΓe5 }T = P15 , P25 , P35 , P45 , 0, 0, 0, 0 n o e e e e {PΓe6 }T = 0, 0, 0, 0, P56 , P66 , P76 , P86

(5.462)

We note that the secondary variables for each of the six faces are perpendicular to the faces. Symbolically, we can write  e  6 P1   X .. e e {P } = {PΓei } = (5.463) .     i=1 P8e where Pie (i = 1, . . . , 8) are the sum of the secondary variables at the nodes of the elements in (5.462). We keep in mind that the sum of the secondary variables is symbolic as they are normal to the element faces.

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

353

Remarks. ¯ e ), then T e ∈ Vh ⊂ H 1,p (Ω ¯ e) (1) When The is of class C 0 (Ω h ¯ e ) local approximations (of any degree, i.e. p) permit inter-element (2) C 0 (Ω continuity of Th and its derivatives in the plane of the mating element faces, but the derivative of Th normal to the inter-element boundaries (faces) is discontinuous. This is exactly parallel to the 2D case except that in this case the inter-element boundaries are the surfaces. ¯ e ), the inter-element (3) When the local approximations are of class C 1 (Ω continuity of the first derivative of Th holds, but the second derivative of Th normal to the inter-element plane is discontinuous. (4) Element computation, assembly and solution follow standard procedure.

5.4.2 Three-dimensional boundary value problems in multivariables In this section we consider three dimensional boundary value problems in multivariables. The simplest examples of such boundary value problems are in 3D linear elasticity. Let u, v, w be the displacements of a point in the body Ω(x, y, z) in x, y and z directions. In the following we first derive the governing differential equations. Equations of equilibrium Consider force equilibrium in x, y and z directions: If τij ; i = x, y, z; j = x, y, z are the stresses at a point x, y, z such that τij = τji , then the force equilibrium in x, y and z directions yields ∂σx ∂τxy ∂τxz + + + fx = 0 ∂x ∂y ∂z ∂τxy ∂σy ∂τyz + + + fy = 0 ∀(x, y, z) ∈ Ω ⊂ R3 ∂x ∂y ∂z ∂τyz ∂τxz ∂σz + + + fz = 0 ∂x ∂y ∂z

(5.464)

in which fx , fy and fz are body forces per unit mass. Strain displacement relations (linear elasticity) ∂v ∂w ∂u , εy = , εx = ∂x ∂y ∂z ∂v ∂w ∂u ∂w ∂u ∂v = + , γzx = + , γxy = + ∂z ∂y ∂z ∂x ∂y ∂x εx =

γyz

(5.465)

354

SELF-ADJOINT DIFFERENTIAL OPERATORS

Constitutive equations: stress-strain relations Using generalized Hooke’s law, stresses can be expresses as a linear combination of strains or strains can be expressed as a linear combination of stresses. Expressing stresses in terms of strains we have (for orthotropic or isotropic material behavior). σx = D11 εx + D12 εy + D13 εz σx = D21 εx + D22 εy + D23 εz σx = D31 εx + D32 εy + D33 εz  ∂v ∂w  τyz = D44 + ∂z ∂y  ∂w ∂u  τxz = D55 + ∂x ∂z  ∂u ∂v  τxy = D66 + ∂y ∂x

(5.466)

Governing differential equations When the stresses from (5.466) are substituted into the equations of equilibrium (5.464) and (5.465) for strains, we obtain three partial differential equations in u, v and w that are second order in u, v and w. These equations constitute the mathematical model which is starting point in the finite element formulation. F (u, v, w) + fx = 0, G(u, v, w) + fy = 0, H(u, v, w) + fz = 0

(5.467)

Boundary conditions u = u0 , v = v0 , w = w0

on Γ2  σx nx + τxy ny + τxz nz = tˆx   τxy nx + σy ny + τyz nz = tˆy on Γ1   τxz nx + τyz ny + σz nz = tˆz

(5.468)

(5.469)

in which stresses are defined by (5.466) and the strains in the stresses by (5.465). Remarks. (1) One could show that the operator A is (a) linear (b) the adjoint A∗ of the operator is same as this operator A.

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

355

(c) and, hence, self-adjoint for this case (2) Thus, a variationally consistent finite element formulation of this boundary value problem is possible using Galerkin method with weak form. 5.4.2.1 Galerkin method with weak form Let w1 = δu, w2 = δv, w3 = δw be the test functions. Then for an ¯ e , we can write element e of the discretization with domain Ω Z
Φ1 = F (u, v, w) + fx w1 dx dy dz ¯e Ω

Z Φ2 =


G(u, v, w) + fy w2 dx dy dz

(5.470)

¯e Ω

Z Φ3 =


H(u, v, w) + fz w3 dx dy dz

¯e Ω

Substituting for F , G and H from (5.464) (in terms of stresses) and performing integration by parts yields Z  ∂w1 ∂w1  ∂w1 σx + τxy + τxz dx dy dz ∂x ∂y ∂z ¯e Ω I Z + w1 (σx nx + τxy ny + τxz nz ) dΓ + w1 fx dx dy dz = 0 (5.471)

Φ1 = −

Γe

¯e Ω

In which the concomitant hφ1 iΓe is given by I hφ1 iΓe = w1 (σx nx + τxy ny + τxz nz ) dΓ Γe

Z  ∂w2 ∂w2 ∂w2  Φ2 = − τxy + σy + τyz dx dy dz ∂x ∂y ∂z ¯e Ω I Z + w2 (τxy nx + σy ny + τyz nz ) dΓ + w2 fy dx dy dz = 0 (5.472) Γe

¯e Ω

In which the concomitant hφ2 iΓe is given by I hφ2 iΓe = w2 (τxy nx + σy ny + τyz nz ) dΓ Γe

356

SELF-ADJOINT DIFFERENTIAL OPERATORS

Z  ∂w3 ∂w3 ∂w3  τxz + τyz + σz dx dy dz ∂x ∂y ∂z ¯e Ω I Z + w3 (τxz nx + τyz ny + σz nz ) dΓ + w3 fz dx dy dz = 0 (5.473)

Φ3 = −

Γe

¯e Ω

In which the concomitant hφ3 iΓe is given by I hφ3 iΓe = w3 (τxz nx + τyz ny + σz nz ) dΓ Γe

We identify PVs, SVs, EBCs and NBCs from the concomitants in (5.471)– (5.473). PV: u, v, w   σx nx + τxy ny + τxz nz SV: τxy nx + σy ny + τzy nz   τxz nx + τyz ny + σz nz   ˜0 u = u ˜1 EBCs: v = v˜0 on some Γ   w =w ˜0  e  σx nx + τxy ny + τxz nz = q˜nx NBCs: τxy nx + σy ny + τzy nz = q˜ne y   τxz nx + τyz ny + σz nz = q˜ne z

(5.474)

˜2 on some Γ

Substituting NBCs from (5.474) into (5.471)–(5.473) and changing sign throughout yields Z  ∂w1 ∂w1 ∂w1  −Φ1 = σx + τxy + τxz dx dy dz ∂x ∂y ∂z e ¯ Ω I Z e − w1 qnx dΓ − w1 fx dx dy dz = 0 (5.475) Γe

−Φ2 =

¯e Ω

Z  ∂w2 ∂w2 ∂w2  τxy + σy + τyz dx dy dz ∂x ∂y ∂z ¯e Ω I Z e − w2 qny dΓ − w2 fy dx dy dz = 0 Γe

¯e Ω

(5.476)

5.4. THREE-DIMENSIONAL BOUNDARY VALUE PROBLEMS

−Φ3 =

Z  ∂w3 ∂w3 ∂w3  τxz + τyz + σz dx dy dz ∂x ∂y ∂z ¯e Ω I Z e − w3 qnz dΓ − w3 fz dx dy dz = 0 Γe

357

(5.477)

¯e Ω

Equations (5.475)–(5.477) represent the desired weak form. When the stresses are substituted into (5.475)–(5.477) we can write     −Φ1  B1e (u, v, w; w1 , w2 , w3 ) − l1e (w1 , w2 , w3 ) −Φ2 = B2e (u, v, w; w1 , w2 , w3 ) − l2e (w1 , w2 , w3 ) (5.478)    e  B3 (u, v, w; w1 , w2 , w3 ) − l3e (w1 , w2 , w3 ) −Φ3 = B e (u, v, w; w1 , w2 , w3 ) − le (w1 , w2 , w3 )

(5.479)

We note that (a) B e (·, ·) is bilinear and symmetric (b) le (·) is linear 5.4.2.2 Approximation spaces Since in this case the differential operator is second order (when the governing differential equations are expressed in terms of displacements) the discussion of the approximation spaces remains the same as in previous model problem (section 5.4.1) and is not repeated here. The governing differential equations contain second order derivatives of u, v and w and, hence, equal order, equal degree local approximations for u, v and w are a valid choice. 5.4.2.3 Local approximation If we let   {φeh }T = ueh vhe whe

(5.480)

¯ e (tetrahedron or hexahedron) we can write Then for a choice of Ω ueh = vhe =

n X i=1 n X

Ni (ξ, η, ζ) uei = [N ]{ue } Ni (ξ, η, ζ) vie = [N ]{v e }

i=1

whe =

n X i=1

Ni (ξ, η, ζ) wie = [N ]{we }

(5.481)

358

SELF-ADJOINT DIFFERENTIAL OPERATORS

and w1 = δueh = Nj (ξ, η, ζ), j = 1, 2, . . . , n w2 = δvhe = Nj (ξ, η, ζ), j = 1, 2, . . . , n w3 =

δwhe

(5.482)

= Nj (ξ, η, ζ), j = 1, 2, . . . , n

Element matrix [K e ] and vector {F e } By substituting (5.481) and (5.482) into (5.478), we can write   −Φ1  −Φ2 = [K e ]{δ e } − {P e } − {F e }   −Φ3

(5.483)

e of [K e ] and F e of {F e } can be easily obtained Explicit expressions for Kij i e and F e are obusing (5.475)–(5.477) and (5.466). Numerical values of Kij i tained using Gauss quadrature. Details of the secondary variables follow the same as for previous model problem in section 5.4.1.

5.5 Summary In this chapter, details of the finite element method for various 1D and 2D boundary value problems have been presented. Details of discretization, local approximation, integral forms, their variational consistency, higher degree as well as higher order local approximations, element equations, their assembly, the solutions of assembled equations, and post-processing are presented for each model problem considered in this chapter. Special emphasis is placed on GM/WF and LSP based on residual functional due to the fact that these methods of approximations yield VC integral forms, hence unconditionally stable finite element computational processes. Importance and benefit of higher degree as well as higher order global differentiability approximations is illustrated in the numerical studies. It is shown that if our objective is to satisfy the governing differential equations in the pointwise sense then the LSP based on the differential models with the highest orders of derivatives of the dependent variables and the local approximations in higher order spaces is most meritorious. Such computational process yield better accuracy than the GM/WF.

Problems 5.1 Consider a fully developed steady flow of a Newtonian fluid between two parallel plates. Let the distance between the plates be 2b. Let u and v be the velocity components in x and y directions.

359

PROBLEMS

v

y

u=0 u(y) 2b

u

x

u=0

Figure 5.77: Schematic of flow between parallel plates

For this simple flow u = u(y) and v = 0 and the governing differential equations reduce to the following: dp d2 u = µ 2 , −b ≤ x ≤ b (1) dx dy Since

∂p ∂y

= 0,

∂p ∂x

is constant (say f0 ), (1) reduces to µ

d2 u = f0 , µ is viscosity dy 2

(2)

with the following boundary conditions: u = 0 at y = ±b

(3)

(a) Construct the weak form of (2) using GM/WF. Give details of PV, SV, EBC and NBC as well as the nature of the resulting functionals. Establish VC or VIC of the integral form. (b) Consider a two-node element with linear approximation of u. Derive the element equations for a typical element e using the element map in the natural coordinate space.

Part I: Part Ia. Consider a four element uniform mesh for the entire domain (−b ≤ y ≤ b). Use the element equations derived in (b) to construct assembled equations in partitioned form and solve for the unknown nodal values of u. Compare your calculated solution with the analytical solution given by u=−

1 dp f0 (b2 − y 2 ), f0 = 2µ dx

(4)

Comment on your observations and findings. Plot graphs of theoretical solution u and computed solution uh versus y and compare. Part Ib. Because of symmetry it is only necessary to model half of the domain (0 ≤ y ≤ b). Construct a uniform discretization of the half domain (0 ≤ y ≤ b) using two two-node linear elements. Compare your calculated results with the analytical solution as well as the solution computed in Part Ia using the whole domain. Comment on your observations and findings. Plot graphs of theoretical solution u and computed solution uh versus y and compare.

360

SELF-ADJOINT DIFFERENTIAL OPERATORS

Part II: Consider the problem of steady state Couette flow between parallel plates. The bottom plate in stationary and the top plate in moving in its own plane with a constant velocity of u0 . The GDE for this case is also given by (2) but the boundary conditions are as follows (instead of (3)): y u0

2b

u(y) x

Figure 5.78: Schematic of Couette flow u(0) = 0, u(2b) = u0 = 1

(5)

The discretized equations derived in (b) for an element e are valid here also. Part IIa. Consider a non-uniform discretization of four linear elements beginning with y = 0 (h1 = 0.8b, h2 = 0.5b, h3 = 0.3b, and h4 = 0.4b). Compute numerical solutions for this discretization and compare them with the analytical solution given by u0 y 2b2 y  y − f0 1− (6) u(y) = 2b µ 2b 2b Plot graphs of theoretical solution u and computed solution uh versus y and compare the two. Discuss your findings. Part IIb. Consider a four-element uniform discretization. Compute a numerical solution for this discretization. Compare this solution with the theoretical solution (6) and the numerical solution in Part IIa. Discuss your results. Plot graphs of theoretical solution u and computed solution uh versus y and compare the two. Note: while performing numerical calculations you may choose the following values for various constants (if it is more convenient). b = 1, µ = 1, f0 = 1, u0 = 1 5.2 Consider an axial rod of length 12 units and cross-sectional areas A1 and A2 as shown in Fig. 5.79. The rod consists of two different materials of Young’s moduli E1 = 15 × 106 and E2 = 30 × 106 . It is subjected to an axial load P = 10, 000 at the free end. It is also subjected to axially distributed loads as shown in the figure.

h-version: (a) Discretize the stepped rod of Fig. 5.79 using two-node linear axial elements. Construct uniform meshes consisting of 2, 4, 6, 8, . . . elements and compute displacements, strains, stresses and the quadratic functional.

p-version: (b) Discretize the rod using only two three-node p-version axial elements. Compute results for p-levels of 2, 3, 4, 5, . . ..

361

REFERENCES FOR ADDITIONAL READING

600 



Axial load  f (x) for  0≤x≤6

1200 

 Axial load  f (x) for  6 ≤ x ≤ 12

y 1200

P

Fixed end 

 A1 = 4 6 E1 = 15 × 10  µ1 = 0.0



 A2 = 2 6 E1 = 30 × 10  µ2 = 0.0

6

6

Figure 5.79: Schematic of axial deformation of a rod

Results: (I) Plot a graph of the quadratic functional (I) versus degrees of freedom (dofs) for both h- and p-version approximations using regular or semi-log scales. (a) Comment on the dofs needed for convergence of I for h- and p-versions. (b) Comment on the rate of convergence of the quadratic functional I for h- and p-versions. (II) Plot graphs of: (a) displacement u versus axial distance x (b) strain εxx versus axial distance x (c) stress σxx versus axial distance x for both h- and p-models to observe their convergence as h, the element characteristic length, is reduced in the h-version with progressive discretization refinement for a fixed p-level (p = 1) and as p-level is increased in the p-version for fixed element characteristic length h. (i) Is the convergence of the quadratic functional I (for both h- and p-versions) monotonic, nonmonotonic or anything else? What is the expected behavior? (ii) Are the convergence of displacements, strains and stresses monotonic, nonmonotonic or something else? What is the expected behavior? (iii) Prepare a short write-up containing details of discretizations and description as well as discussion of results including the answers to the questions raised here. You may use FINESSE, ANSYS, or any other software for performing computations. GNUPlot or any other xy graphs display program may be helpful in making xy plots.

References for additional reading [1] B. Jiang. The Least-Squares Finite Element Method: Theory and Applications in Computational Fluid Dynamics and Electromagnetics. Springer, 1998.

362

REFERENCES FOR ADDITIONAL READING

[2] K. S. Surana, L. R. Anthoni, S. Allu, and J. N. Reddy. Strong and weak form of governing differential equations in least squares finite element processes in hpk framework. Int. J. Comp. Meth. in Eng. Sci. and Mech., 2008. [3] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 3(2):155–218, 2002. [4] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [5] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004.

6

Non-Self-Adjoint Differential Operators 6.1 Introduction In this chapter we consider boundary value problems described by nonself-adjoint differential operators. These could be single or multi-variable boundary value problems in single or multi-dimensional space. First, we review some basic properties of these differential operators and the properties of the integral forms resulting from various classical methods of approximation (presented in chapter 3). In particular, we look for whether the integral form resulting from a method of approximation for such operators is variationally consistent or variationally inconsistent. (1) The non-self-adjoint differential operators are linear but not symmetric. For such operators the adjoint A∗ of the operator A is not the same. (2) The properties of the integral forms (presented in chapter 3) resulting from various methods of approximation for non-self-adjoint differential operators are summarized in the following (a) GM, PGM and WRM yield VIC integral forms. (b) GM/WF also yields VIC integral forms. However, in this approach the contributions of the even order terms in the differential operator to the coefficient matrix becomes symmetric due to integration by parts. When these terms dominate the behavior of the solution, GM/WF is beneficial compared to GM, PGM and WRM. (c) LSM or LSP, when either used for GDEs containing highest order derivatives of the dependent variables or for those cast as systems of first order GDEs, yields VC integral forms. (d) Thus, for non-self-adjoint operators only GM/WF and LSP are worthy of consideration. (3) In GM/WF, the VIC integral forms remain VIC regardless of the remedies employed to alter them. In such cases, one must establish on a problem-by-problem basis when the computations will remain stable, i.e. the restrictions on h, p and k and the dimensionless parameters in the mathematical model. 363

364

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

(4) In finite element processes based on GM/WF one constructs an integral ¯ T . Weak form based on fundamental lemma over the discretization Ω form is constructed using integration by parts. Thus, if Aφ − f = 0 in ¯ e is the discretization of Ω, ¯ then we can ¯ T = ∪e Ω Ω is the BVP and if Ω write X X

(Aφh − f, v)Ω¯ T = Aφeh − f, v Ω¯ e = B e (φeh , v) − le (v) = 0 (6.1) e

e

¯ T and φe is the local apin which φh is the approximation of φ over Ω h e ¯ . For an element e with domain Ω ¯ e we construct proximation of φ over Ω the weak form (Aφeh − f, v)Ω¯ = B e (φeh , v) − le (v), v = δφeh

(6.2)

using the same procedure as described in chapter 5 for self-adjoint differential operators. In (6.2), we note that B e (·, ·) is bilinear but not symmetric and le (·) is linear. We use the local approximation φeh

=

n X

Ni δie = [N ]{δ e }

(6.3)

i=1

v = Nj , j = 1, . . . , n

with

(6.4)

to obtain B e (φeh , v) = [K e ]{δ e } e

e

(6.5)

e

l (v) = {F } + {P }

(6.6)

in which [K e ] is not symmetric, i.e. Kij = 6 Kji ; i, j = 1, . . . , n. Using (6.5) and (6.6) in (6.1), the assembly of the element equations follows standard procedure. (5) In the finite element processes based on minimization of residual func¯ T (noting that tional we construct the residual functional I(φh ) over Ω ¯ T and E e = Aφe − f over Ω ¯ e , then A is linear). Let E = Aφh − f over Ω h X X (6.7) I(φh ) = (E, E)Ω¯ T = Ie = (E e , E e )Ω¯ e e

e

The necessary conditions are given by X X X δI = 2(E, δE)Ω¯ T = δI e = 2 (E e , δE e )Ω¯ e = 2 {g e } = 2{g} = 0 e

e

e

(6.8) ¯ e , we construct For an element e with domain Ω {g e } = (E e , δE e )Ω¯ e = B e (φeh , v) − le (v)

(6.9)

365

6.2. 1D CONVECTION-DIFFUSION EQUATION

In the integral form (6.9) we note that B e (·, ·) is bilinear and symmetric and le (·) is linear. Upon substituting from (6.3) and (6.4) into (6.9) and noting that δE e = ANj ; j = 1, 2, . . . , n, we obtain B e (φeh , v) = [K e ]{δ e } e

e

l (v) = {F }

(6.10) (6.11)

in which e Kij = (A Ni , A Nj )Ω¯ e , i, j = 1, . . . , n

Fie = (fi , A Ni )Ω¯ e , i = 1, . . . , n

(6.12) (6.13)

e = K Clearly, the coefficients Kij ji ; i, j = 1, . . . , n. Assembly of the element equations (6.10) and (6.11) in (6.8) follows standard procedure.

In the following sections we consider specific model problems to derive finite element processes based on GM/WF and LSP and present some numerical studies.

6.2 1D convection-diffusion equation The steady-state one-dimensional convection-diffusion equation is representative of the energy equation encountered in more complex two- and three-dimensional flows. The steady-state one-dimensional convection-diffusion equation in the dimensionless form can be written as dφ 1 d2 φ − = 0 ∀x ∈ Ω = (0, 1) ⊂ R dx P e dx2

(6.14)

where φ is dimensionless temperature, P e is the Peclet number defined as P e = uL/R = 1/k, u is the velocity, L is the length of the domain and k is the diffusion coefficient. We shall consider the following boundary conditions: φ(0) = 1.0

and

φ(1) = 0.0

(6.15)

6.2.1 Analytical solution We note that φ(x) = c1 + c2 ex P e

(6.16)

satisfies (6.14). The constants c1 and c2 are evaluated using φ(0) = 1.0 and φ(1) = 0 and we have  eP e − ex P e  φ(x) = (6.17) eP e − 1

366

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

It is instructive to examine the derivatives of φ(x) in (6.17) of order n −(P e)n ex P e dn φ = dxn eP e − 1

(6.18)

−(P e)n eP e −(P e)n = eP e − 1 1 − 1/eP e

(6.19)

φ(n) (x) ≡ Therefore φ(n) (1) = From (6.19), we note that

lim φn (1) = −∞

P e→∞

(6.20)

However, for a finite value of P e, the derivatives of all orders of φ(x) at x = 1 and elsewhere remain bounded.

Remarks (1) For very low P e the convective term dφ dx and diffusion term ¯ play significant role over the entire domain Ω.

1 d2 φ P e dx2

both

(2) As P e increases the problem becomes convection dominated, i.e. with progressively increasing Peclet number, the diffusion becomes more and more isolated near x = 1.0. The solution φ remains one over the ma¯ except in the very small neighborhood near x = 1, generally jority of Ω O(1/P e) for large values of P e over which φ changes from 1 to 0. Hence, at P e = 106 , φ remains 1 from x = 0.0 to x = 1 − O(10−6 ) and changes from 1 to 0 over a length of O(10−6 ) near x = 1.0. Thus, in the physics of the problem, with increasing P e the solution φ is described by the con¯ However, vection dominated process for the majority of the domain Ω. when attempting a numerical simulation of such a process, it is clear that resolution of the isolated behavior of the diffusion near x = 1.0 must be correctly incorporated in the numerical process. (3) Based on the previous remark, we clearly see that accurate simulation of the isolated diffusion near x = 1.0 for high values of P e is of critical significance in solving this problem satisfactorily. (4) Figure 6.1 shows plots of the analytical solution φ(x) vs x for Peclet numbers of 10, 100 and 1000. Progressively increasing gradients of φ near x = 1.0 are rather obvious with increasing Peclet numbers.

367

6.2. 1D CONVECTION-DIFFUSION EQUATION

1.2 Pe=10 Pe=100 Pe=1000

1

φ

0.8

0.6

0.4

0.2

0 0

0.2

0.4

0.6

0.8

1

x

Figure 6.1: Theoretical solution φ versus x for P e = 10, 100 and 1000

(5) From the analytical solution (6.17) it is clear that ¯ φ(x) ∈ C ∞ (Ω)

(6.21)

(6) Another significant point to note is to clearly understand the minimum requirements on the approximation φh of φ in a numerical process to ensure that the physics is not disturbed in the numerical process. From (6.14) we note that ¯ (a) convection dφ dx must at least be continuous over Ω which implies that 1 T ¯ ) to ensure such behavior. φ ∈ C (Ω 2 ¯ which ne(b) diffusion P1e ddxφ2h must also at least be continuous over Ω 2 T ¯ . cessitates that φh must at least be of class C over Ω From (a) and (b) we note that in order for convection and diffusion both ¯ T , φh must at least be of class C 2 over Ω ¯ T . This to be continuous over Ω is the minimum requirement to ensure that the physics of convection and diffusion is not disturbed in the numerical process. (7) These points discussed above will be helpful in understanding the nature of various computed solutions in the subsequent sections and the restrictions on the local approximations.

6.2.2 The Galerkin method with weak form (GM/WF) We write (6.14) as Aφ − f = 0

∀x ∈ (0, 1) = Ω ⊂ R1

(6.22)

368

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

in which d 1 d2 , f =0 − dx P e dx2

A=

(6.23)

with boundary conditions φ(0) = 1 and φ(1) = 0

(6.24)

¯ e be the discretization of Ω ¯ and φh be the approximation ¯ T = ∪e Ω Let Ω T e ¯ and φ , the local approximation of φ over Ω ¯ e = [xe , xe+1 ] → of φ over Ω h ¯ ξ = [−1, 1]. We consider Ω

(Aφeh − f, v)Ω¯ e =

xZe+1

 dφe

h

dx

xe

−

1 d2 φeh  v dx, v = δφeh P e dx2

(6.25)

From (6.25) we obtain the weak form

(Aφeh −f, v)Ω¯ e

xZe+1

= xe

 dφe 1 dv dφeh  v h+ dx−v(xe ) P1e −v(xe+1 ) P2e (6.26) dx P e dx dx

where 1 dφeh P e dx xe 1 dφeh P2e = P e dx xe+1

P1e = −

(6.27) (6.28)

or (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(6.29)

in which B e (φeh , v) =

xZe+1 xe

 dφe 1 dv dφeh  dx v h+ dx P e dx dx

e

l (v) = v(xe ) P1e − v(xe+1 ) P2e

(6.30) (6.31)

Equations (6.29)–(6.31) are the weak form of (6.22). We note that (a) B e (·, ·) is the bilinear but not symmetric. (b) le (·) is linear.

369

6.2. 1D CONVECTION-DIFFUSION EQUATION

Approximation space The BVP contains up to second order derivative of the dependent variable φ but the weak form only contains up to first order derivatives of φeh and v. ¯ T in the pointwise sense requires (i) Admissibility of φh in Aφ − f = 0 in Ω ¯ e ) ; k = 3 is minimally conforming. For this choice φeh ∈ Vh ⊂ H k,p (Ω the integrals in the following are Riemann. X X
(Aφ−f, v)Ω¯ e = B e (φeh , v)−le (v) = 0 (6.32) (Aφh −f, v)Ω¯ T = e

e

¯ e ), then all integrals are Riemann That is, if we choose φeh of class C 2 (Ω in all forms in (6.32), hence all these forms in (6.32) are equivalent. ¯ e ), that is if (ii) Based on the weak form, if we choose φeh of class C 1 (Ω 2,p e e ¯ ), then the following holds. φh ∈ Vh ⊂ H (Ω X X B e (φeh , v) − le (v) (Aφh − f, v)Ω¯ T ⇔ (Aφeh − f, v)Ω¯ e ⇔ L

L

e

e

|

{z R

}

(6.33) − holds in For this choice of local approximation the Riemann sense, but all other integral forms in (6.33) hold only in the Lebesgue sense. ¯ e ), then P (B e (φe , v)−le (v)) only holds (iii) If we choose φeh ∈ Vh ⊂ H 1,p (Ω e h in the Lebesgue sense and the other two integral forms in (6.33) are meaningless. e e e (B (φh , v)

P

le (v))

Local approximation φeh Let φeh

=

n X

Ni (ξ) φei = [N ]{δ e }

(6.34)

i=1

¯e → Ω ¯ ξ . Then be the local approximation of φ over Ω v = δφeh = Nj (ξ), j = 1, . . . , n

(6.35)

in which Ni (ξ) are the local approximation functions (see chapter 8). Substituting from (6.34) and (6.35) into the weak form (6.29)–(6.31) B

e

(φeh , Nj )

xZe+1

Nj

= xe

n X dN

i e δi

i=1

dx



 n dNj 1 X dNi e  + δ dx dx P e dx i

(6.36)

i=1

le (Nj ) = Nj (xe ) P1e − Nj (xe+1 ) P2e , j = 1, 2, . . . , n

(6.37)

370

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

i, j = 1, 2, . . . , n, which can be written as B e (φeh , v) = [K e ]{δ e }  e  (P1 )ξ=−1         0     . e .. l (v) =       0      e  (P2 )ξ=+1

(6.38)

(6.39)

where e Kij

xZe+1



=

dNj 1 dNi dNj  + dx, i, j = 1, 2, . . . , n dx P e dx dx

Ni

xe

(6.40)

Consider C 0 local approximation with p = 1 That is, consider a two-node linear element. For this choice, we have 1 − ξ  1 + ξ  φeh (ξ) = φe1 + φe2 = N1 φe1 + N2 φe2 2 2 (6.41) v = Nj (ξ), j = 1, 2 and x(ξ) =

2 X

Ni (ξ) xi =

1 − ξ 

i=1

2

x1 +

1 + ξ  2

x2

he dξ, he = xe+1 − xe dξ 2 dNj dv 1 dNj 2 dNj = = = , j = 1, 2 dx dx J dξ he dξ dx = J dξ =

 dx 

dξ =

(6.42)

Additionally, dN1 1 dN2 1 =− , = dξ 2 dξ 2 e First, we write the matrix [K ] as

(6.43)

[K e ] = [K 1e ] + [K 2e ]

(6.44)

in which 1e Kij

Z1 Ni

=

dNj dξ dξ

−1 2e Kij =

Z1 −1

1 dNi dNj 2 2 dξ = P e dξ dξ he P e he

Z1 −1

(6.45) dNi dNj dξ dξ dξ

371

6.2. 1D CONVECTION-DIFFUSION EQUATION

Therefore we have 1e

Z1 "

[K ] = −1

dN1 dξ 1 N2 dN dξ

N1

2 [K ] = P e he 2e

dN2 dξ 2 N2 dN dξ

N1

Z1 " dN1 dN1 −1

dξ dξ dN2 dN1 dξ dξ

#

 1 −1 dξ = 2 −1

dN1 dξ dN2 dξ

dN2 dξ dN2 dξ

and e

{P } =



#

1 1



  1 1 −1 dξ = 1 P e he −1

P1e P2e

(6.46)

(6.47)

 (6.48)

Remarks (1) The matrix [K 1e ] is due to the convective term in the GDE. It is not symmetric. (2) The matrix [K 2e ] is purely due to the diffusion term in the GDE. It is symmetric, a benefit of the integration by parts for the terms with even order derivative, i.e. the diffusion term. (3) The matrix [K 1e ] due to convection is independent of he , the element length, i.e. convection is simulated correctly in the computations regardless of he . In other words, convection is independent of he . (4) The element matrix [K 2e ] due to diffusion is a function of he and P e. A change in P e influences the physics of diffusion and he influences how the physics of diffusion is simulated in the computations. Numerical studies Consider a four-element uniform discretization using two-node linear elements shown in Fig. 6.2. Using local node numbers 1 and 2 and (6.46) and (6.47), we can write the following for each of the four elements shown in Fig. 6.2.    ! e   e  1 −1 1 1 1 −1 φ1 P1 e (Aφh − f, v) = + − (6.49) φe2 P2e 2 −1 1 P e he −1 1 for e = 1, . . . , 4. After imposing inter-element continuity conditions on the nodal values of φ at the element nodes and then the assembly of the element matrices and vectors {P e } gives the following:
[K 1 ] + [K 2 ] {δ} = [K]{δ} = {P }

(6.50)

372

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

y

1

2

1

2

3

3

4

4

5 x

φ1

φ2

φ1 = 1.0 (BC) he =

φ3 1 4

φ4

= 0.25; e = 1, . . . , 4

φ5 φ5 = 0.0 (BC)

Figure 6.2: Four-element uniform discretization with boundary conditions

or  1 1 0 −2 2 − 1 1 0  2 2   0 − 1 0 2   0 − 21  0 0

0

0 0 1 2

0

0 − 12

      0 1 −1 0 0 0  φ1    0  −1   2 −1 0 0 φ2        1     0 + 0 −1 2 −1 0 φ3  P e he    0  1 0 −1 2 −1 φ4          2 0 0 0 −1 1 φ 5 1 2

  1 P   1      P21 + P12   2 3 = P2 + P1 (6.51)   3 4    P2 + P1     P24 Essential BCs are given by φ1 = 1.0, φ5 = 0.0

(6.52)

and the inter-element conditions on secondary variables are P11 = ? as φ1 = 1.0 P21 + P12 = 0 P22 + P13 = 0 P23 P24

+

P14

(6.53)

=0

= ? as φ5 = 0.0

We impose conditions (6.52) and (6.53) in (6.51) by modifying the rows and columns of the assembled [K] and {P }:

373

6.2. 1D CONVECTION-DIFFUSION EQUATION

 −     

1 2

1 − 0 0 0

0 1 P e he

0

2 P e he

1 2

0 0

1 P e he 2 1 1 P e he 2 − P e he 1 2 1 − 2 − P e he P e he 12

− 12 − P e1he 0 0

−

0

0

    φ1  1             φ2   0    φ3 = 0      φ4  0     − P e1he           φ 0 5 1 0 0 0

(6.54) We note that the assembled matrix [K] in (6.54) is a tri-diagonal matrix. For nodes i − 1, i, and i + 1 we can write 

−

1 1  1  2 1 − φi + − φi−1 + φi+1 = 0 2 P e he P e he 2 P e he

(6.55)

Pre-multiply (6.55) by P e he yields Pe h

e

2

 Pe h  e − 1 φi+1 + (2) φi − + 1 φi−1 = 0 2

(6.56)

This is a second order difference equation that can be solved exactly by letting φi = pi−1 (6.57) Therefore, (6.55) becomes Pe h

e

2

  Peh  e − 1 pi + (2) pi−1 + − − 1 pi−2 = 0 2

(6.58)

and dividing throughout by pi−2 yields p2

Pe h

 Pe h  e − 1 + (2) p − +1 =0 2

e

2

(6.59)

Equation (6.59) is quadratic in p and hence its roots p1 and p2 are −2 ± p=

q

(2)2 − 4 2

P e he 2 P e he 2


 −
−1

−1

or p1 = 1, p2 =

1+ 1−

P e he 2 P e he 2

P e he 2

+1


 (6.60)

(6.61)

The solution φi consists of a linear combination of pi−1 and pi−1 and can 1 2 be written as φi = C pi−1 + D pi−1 (6.62) 1 2

374

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

Substituting for p1 and p2 in (6.62) from (6.61) φi = C + D

1+ 1−

P e he 2 P e he 2

!i−1 (6.63)

in which, the constants C and D depend upon the boundary conditions, i.e. at i = 1 :

φi = φ1 = 1 ⇒ 1 = C + D

at i = N + 1 :

1+

φi = φN +1 = 0 ⇒ 0 = C + D

1−

P e he 2 P e he 2

!N

(6.64)

Solving for C and D from (6.64) 

1+ P e2he

N

1− P e2he

C=

 1−

1+ P e2he 1− P e2he

Therefore 

1+ P e2he

 1−

 1−

N

1− P e2he

φi =

1

N , D =

1+ P e2he 1− P e2he

N +



1+ P e2he

N

1− P e2he

1+ P e2he

i−1

1− P e2he

 1−

(6.65)

1+ P e2he

(6.66)

N

1− P e2he

From (6.66), we note that  Pe h   P e h i−1 1+ 2 e 1+ 2 e P e he is positive and thus is positive (a) If 2 < 1, then P e he P e he 1−

2

1−

2

for all values of i.  Pe h   P e h i−1 1+ 2 e 1+ 2 e (b) If P e2he > 1, then is negative and hence changes P e he P e he 1−

2

1−

2

sign with each successive i. This implies that the solution given by (6.66) oscillates from one node to the next, noting that C and D in (6.64) or (6.66) are constants. The theoretical solution for P e = 10, 100 and 1000 is shown in Fig. 6.1. The influence of the choice of P e2he on the computational process and the computed solution can also be shown through the finite element computations. Let us consider P e2he = 0.5, 1.5 and 2.5. The numerical studies for these three choices can be done in two ways: (i) We can choose a fixed P e and determine he values corresponding to P e2he = 0.5, 1.5 and 2.5 for which numerical studies can be performed. (ii) We can also choose a fixed he value and then determine P e values corresponding to P e2he = 0.5, 1.5 and 2.5. We

375

6.2. 1D CONVECTION-DIFFUSION EQUATION

present details here for the second choice. Consider a uniform discretization with he = 0.01 (100 element uniform mesh) with local approximations of class C 0 at p-level of one. For this value of he (fixed) and P e2he = 0.5, 1, 5 and 2.5 we obtain P e = 100, 300 and 500 for which we compute numerical solutions using the formulation based on GM/WF. In all three cases [K]{δ} − {F } − {P } = {0} is satisfied by the computed solution confirming that assembled coefficient matrix is not poorly conditioned for these choices. For P e2he = 0.5 (less than one), the computed solution is oscillation free (see Fig. 6.3). For P e2he = 1.5, the computed solution has spurious oscillation in the vicinity of x = 1.0. For P e2he = 2.5, the oscillations grow in magnitude and spread over larger domain. If we continue to increase P e, i.e. continue to increase P e he /2, the oscillation will continue to increase in magnitude and will continue to spread over larger domain until to a point at which the computations will cease due to near singular assembled coefficient matrix. 1.6 1.4 1.2

φh

1 0.8 0.6

—

C0(Ωe) ; p=1 , he=0.01

0.4

1

/2 Pehe=0.5 /2 Pehe=1.5 /2 Pehe=2.5

1 1

0.2

; Pe=100 ; Pe=300 ; Pe=500

0 0.9

0.91

0.92

0.93

0.94

0.95 0.96 Distance, x

Figure 6.3: Numerical solution for different



P e he 2



0.97

0.98

0.99

1

(The Galerkin method with weak

form: two node linear element)

Remarks (1) Consider a fixed discretization, p-level and k suitable for P e = P e1 . If the Peclet number is increased form P e1 to P e2 > P e1 and if the discretization, p-level and k are kept fixed, then it is clear that more localized behavior of diffusion at P e = P e2 will no longer be simulated correctly in the numerical process. Thus, what must happen in the numerical process is that the localized behavior of diffusion must diffuse

376

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

over a larger length. Progressively increasing Peclet number must continue to yield progressively more diffused solutions until it cannot diffuse any more. Such a solution will be a straight line connecting φ = 1 at 2 x = 0 to φ = 0 at x = 1. This in fact is the solution of BVP: ddxφ2 = 0 with φ(0) = 1 and φ(1) = 0 (φ(x) = (1 − x)). A consistent (i.e. VC) numerical process must behave in this manner. 1

0.8

0.6 —

φh

C1(Ωe) ; p=13 , he=0.2 Pe=100 Pe=200 Pe=300 Pe=400 Pe=500 Pe=1,000 Pe=10,000

0.4

0.2

0 0

0.25

0.5 Distance, x

0.75

1

Figure 6.4: Behavior of φh (x) with increasing P e for a fixed uniform discretization with characteristic length he of 0.2. VC integral form: minimization of residual functional

This aspect of the VC integral form can be illustrated numerically. Consider LSP for this model problem (see previous sections and sections 6.1, 6.2.3) without using auxiliary variable and auxiliary equation. This formulation yields VC integral form. We choose P e = 100, a five-element ¯e) p-version uniform discretization with local approximation in H 2,13 (Ω x 1 ¯ x ) at p-level of 13. For this space, i.e. local approximation of class C (Ω choice of he , p and k the numerically computed solution for P e = 100 is sufficiently close to the theoretical solution (see Fig. 6.4). Keeping he , p and k constant, we increase P e to 200, 300, 400, 500, 1,000 and 10,000 and compute numerical solutions (shown in Fig. 6.4) using VC LS formulation. Solutions for all P e numbers are smooth, i.e. oscillation free. With progressively increasing P e the computed solutions continue to diffuse and eventually become φh = (1 − x) for P e = 10, 000 as expected. Computations remain stable with positive definite coefficient matrices (see reference [1]).

377

6.2. 1D CONVECTION-DIFFUSION EQUATION

(2) Next, we examine the VIC integral forms from GM/WF constructed in this section. We note that the assembled equations after imposing boundary conditions would have the following form (assuming uniform discretization).   [K]{δ} = [K1 ] + P e1he [K2 ] {δ} = {f } (6.67) nonzero due to BC due to diffusion due to convection

For a fixed value of he , progressively increasing P e would result in smaller and smaller values of (1/P e he ) giving rise to more and more isolated diffusion and as a consequence the correct simulation of diffusion is no longer present in the computational process. In the limit, when P e → ∞, the term P e1he [K2 ] approaches a null matrix and (6.67) reduces to [K1 ]{δ} = {f } (6.68) in which [K1 ] is a non-symmetric matrix with zeros on the diagonal, hence [K1 ] is a singular matrix. Thus, computation of {δ} is not possible. We see that for a fixed discretization, VIC GM/WF progressively degrades with increasing P e and eventually becomes a degenerate process in which computation of the solution {δ} ceases. (3) From (6.68) it is quite clear that since [K1 ] is a non-symmetric matrix with zeros on the diagonal, a solution {δ} is possible from (6.67) if and only if P e1he [K2 ] makes contribution to [K1 ] that is significant enough to overcome the singular nature of [K1 ]. With increasing P e, contribution of P e1he [K2 ] to [K1 ] diminishes, i.e. [K] matrix which is well-conditioned for some low P e starts to become ill-conditioned and hence the appearance of spurious nodal oscillations in the solution and eventually becomes singular. (4) Based on (1)–(3) it is clear that the root causes of nodal oscillations in the solution are: (a) VIC integral form in GM/WF resulting in non-symmetric [K1 ] in (6.67). (b) The inability of a fixed discretization to simulate the physics of diffusion correctly for progressively increasing P e. (5) In the published literature this phenomenon described above is termed as negative diffusion in the Galerkin processes and hence the notion of addition of artificial diffusion through streamline upwinded PetrovGalerkin techniques such as SUPG, SUPG/DC, SUPG/DC/LS, and so on [2–8]. Arguments presented in (1)–(4) based on the mathematics and

378

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

physics clearly support what is concluded in (4) and hence the notion of ‘negative diffusion’ in the Galerkin processes has no basis at all and therefore all upwinding techniques to cure the spurious behaviors of the solution as possible remedies have no basis either [2–8]. In other words, upwinding methods in whatever form are completely without mathematical foundation and are totally unjustifiable from the point of view of the mathematics as well as the physics (see Surana et al. [1, 9–13]). (6) In conclusion, the Galerkin method for non-self-adjoint operators leads to VIC finite element processes which would only work satisfactorily with C 0 local approximations if appropriate discretization is considered in which physics of diffusion is simulated correctly in the numerical process. Such discretizations for the present model problem would result in the number of elements of the order of P e in R1 (for higher P e) which are impractical (i.e. 106 elements for P e = 106 in R1 ). (7) It is natural to ask, whether the solutions of classes higher than C 0 would result in any benefit in the GM/WF. This question is meritorious and is a subject of study in a published paper by Surana et al. [1] and also discussed in a later section.

6.2.3 Finite element formulation of convection-diffusion equation based on residual functional In this section we present a finite element formulation of the 1D steady state convection diffusion equation (6.14) based on the residual functional. of φ over a typical element e with domain Let φeh be the local approximation ¯ e of the spatial domain Ω ¯ = [0, 1]. Then ¯ e of the discretization Ω ¯T = S Ω Ω Ee =

dφeh 1 d2 φeh − dx P e dx2

¯e ∀x ∈ Ω

(6.69)

is the residual equation for an element e. The residual functional I e for an ¯ T are given element e and the residual functional I for the discretization Ω by I e = (E e , E e )Ω¯ e I=

M X i=1

Ie =

M X i=1

(6.70) (E e , E e )Ω¯ e

(6.71)

379

6.2. 1D CONVECTION-DIFFUSION EQUATION

Therefore δI =

M X

2(E e , δE e )Ω¯ e = 0

i=1 M X

δ2I = 2

(δE e , δE e )Ω¯ e > 0

(6.72)

(6.73)

i=1

Equation (6.73) implies that a solution obtained from (6.72) minimizes I in (6.71). Equations (6.71)–(6.73) confirm that the integral form (6.72) in the LSP is variationally consistent. Since (6.14) is a second order ODE, the ¯ e ) ; k ≥ 3, p ≥ 2k − 1 in which k = 3 is the local approximation φeh ∈ H k,p (Ω minimally conforming order of the approximation space. For this choice of ¯ T hold in the Riemann sense. k all integrals over Ω Details of element equations ¯ ξ = [−1, 1] be the map of Ω ¯ e in the natural coordinate space ξ and Let Ω let φeh (ξ) =

n X

Ni (ξ) φei = [N ]{δ e }

(6.74)

i=1

Substituting from (6.74) into (6.69) n n   1 d2 X d X e Ni (ξ) φei − N (ξ) φ Ee = i i dx P e dx2 i=1

which yields the following in matrix and vector notation: h dN i 1 h d2 N i e Ee = − {δ } dx P e dx2 n dN o 1 n d2 N o − {δE e } = dx P e dx2 We note that (6.72) can be written as (since E e is linear in φei ) M X

e

e

[K ]{δ } =

e−=1

M X 

(6.75)

i=1

e

e



e

(δE , δE )Ω¯ e {δ } =

e=1

M hX

(6.76) (6.77)

i (δE e , δE e )Ω¯ e {δ} = 0

e=1

(6.78) in which {δ} =

S

{δ e }. Therefore

e

[K e ] =

Z

{δE e }{δE e }T dx

(6.79)

¯e Ω

is explicitly defined by using (6.77). Numerical values are computed using Gauss quadrature. Numerical studies using this formulation are presented in later sections.

380

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

6.2.4 A finite element formulation of convection diffusion equation based on residual functional: first order system of equations A finite element formulation based on minimization of residual functional of the convection diffusion equation (6.14) containing up to second derivative of φ would require that the local approximation φeh of φ over an element at ¯ e ), i.e. φe should be at least of class C 2 . least belong to the space H 3,p (Ω h 2 If we permit inter-element discontinuity of ddxφ2 , then φeh of the class C 1 are admissible in (6.14). However, if we wish to use φeh of the class C 0 in the least squares processes then (6.14) must at least be recast as a system of first order differential equations using auxiliary variable. This form is sometimes also referred to as weak differential form or a first order system of the governing differential equation (6.14). If we let τ=

dφ , τ being an auxiliary variable dx

then, the GDE (6.14) can be written as  dφ 1 dτ  − =0  dx P e dx ∀x ∈ Ω = (0, 1) ⊂ R1 dφ   τ− =0 dx

(6.80)

(6.81)

The first order differential equations in φ and τ in (6.81) will permit C 0 ¯ e . The second equation local approximation for φ and τ over an element Ω in (6.81) is called auxiliary equation. ¯ e and Let φeh and τhe be local approximations of φ and τ over an element Ω e e 1,p e e e 0 ¯ ), i.e. let φ and τ be of class C . Furthermore let φh , τh ∈ Vh ⊂ H (Ω h h ¯ e ) is also admissible in choice of p = 1 (linear behavior of φeh and τhe over Ω e e (6.81). Upon substituting φh and τh in (6.81) we obtain, residuals functions E1e and E2e for an element e:  dφeh 1 dτhe  e  E1 = − dx P e dx ¯e ∀x ∈ Ω (6.82) e dφh  e e  E2 = τh − dx The residual functional I e for an element e can be constructed using I e = (E1e , E1e )Ω¯ e + (E2e , E2e )Ω¯ e

(6.83)

and for the whole discretization with M elements we have I=

M X e=1

Ie =

M X 2 X e=1

i=1

(Eie , Eie )Ω¯ e



(6.84)

381

6.2. 1D CONVECTION-DIFFUSION EQUATION

δI = 2

M X 2 X e=1

δ2I = 2

 (Eie , δEie )Ω¯ e = 0

(6.85)

(δEie , δEie )Ω¯ e > 0

(6.86)

i=1

M X 2 X e=1 i=1

Equations (6.86) imply that a solution {δ} from (6.85) minimizes I in (6.84). From (6.84)–(6.86) we confirm that the integral form (6.85) in the LSP is variationally consistent. Details of element equations Consider equal order, equal degree local approximations for φ and τ over e ¯ Ω . Let us choose a two-node linear element to illustrate the main steps in the process. Let φeh =

2 X

Ni φei = [N ]{φe } and τhe =

2 X

Ni τie = [N ]{τ e }

(6.87)

i=1

i=1

in which 1 + ξ  1 − ξ  and N2 (ξ) = N1 (ξ) = 2 2 1 − ξ  1 + ξ  x(ξ) = xe + xe+1 2 2

(6.88)

Therefore J=

dx xe+1 − xe he = = dξ 2 2

(6.89)

We note that (6.85) can be written as (since Eie are linear in φei and τie ) M X

e

e

[K ]{δ } =

e=1

M hX 2 X e=1

i

(δEie , δEie )

i=1

e

{δ } =

M X 2 hX

i (δEie , δEie ) {δ} = 0

e=1 i=1

(6.90) in which

e

{δ } =



   φe1         φe  

 {φe } 2 = , Nodal values of φ and τ for an element e e e {τ }  τ1         τ e   2 (6.91)

382

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

and ( δEie

=

∂Eie ∂{φe } ∂Eie ∂{τ e }

) =

 e ∂Ei     ∂φe1e      ∂E  ei   ∂φ2

e

∂Ei    ∂τ1e      e   ∂Ei  

, i = 1, 2

(6.92)

∂τ2e

therefore e

[K ] =

2 X

(δEie , δEie )Ω¯ e

(6.93)

i=1

The element coefficient matrix requires details of δEie in (6.92). Using (6.82) and (6.87) we can write E1e

2

2

i=1

i=1

  1 d X d X e e Ni (ξ) φi − Ni (ξ) τi = dx P e dx

E2e =

2 X i=1

2  d X Ni (ξ) φei Ni τie − dx

(6.94)

i=1

Details of δEie in (6.92) can be easily obtained using (6.94)  1 h dN i dN i , − = dx P e dx   h i dN e T {δE2 } = − , [N ] dx

(6.95)

h1 − ξ 1 + ξ i [N ] = [N1 (ξ) , N2 (ξ)] = , 2 2 h dN i h dN i dN 1 2 = , dx dx dx 1 h dN1 dN2 i 2 h 1 1i = , − , = J dξ dξ he 2 2

(6.96)

1 1 1 1 i , , , − he he P e he P e he h 1
1
i 1 1 e T {δE2 } = , − , 1−ξ , 1+ξ he he 2 2

(6.97)

{δE1e }T

h

in which

Therefore {δE1e }T =

h

−

383

6.2. 1D CONVECTION-DIFFUSION EQUATION

and we can write the following: (δE1e , δE1e )Ω¯ e

Z1 =

he dξ 2

{δE1e }{δE1e }T

−1

Z1

(δE2e , δE2e )Ω¯ e =

(6.98) he dξ 2

{δE2e }{δE2e }T

−1

which can be expressed as 1 (δE1e , δE1e )Ω¯ e

=

Z1 −1

1 −1 he 2 − 1 Pe 1 Pe

(δE2e , δE2e )Ω¯ e =

−1

− P1e

1 Pe

1

1 Pe

− P1e

1 Pe

1 P e2

− P1e2

− P1e

− P1e2

1 P e2

1 he 2

− h12

(1−ξ) 2he

(1+ξ) 2he

Z1

− h12

− h12

−(1−ξ) 2he

−(1+ξ) 2he

−1

(1−ξ) 2he

−(1−ξ) (1−ξ)2 2he 4

(1−ξ 2 ) 4

(1+ξ) 2he

−(1+ξ) 2he

(1−ξ 2 ) 4

(1+ξ)2 4

e

e

e

he dξ 2

(6.99)

he dξ 2

(6.100)

After performing integration in (6.99) and (6.100), we obtain the following element coefficient matrix [K e ] − h2e

1 2

− h2e

2 he

− 12

− 21

1 2

− 12

he 3

he 6

1 2

− 12

he 6

he 3

2 he

[K e ] =

1 2

+

1 P e he

0

0

-1

1

0

0

1

-1

-1

1

1

-1

1

-1

-1

1

(6.101)

and the element equations become {g e } = [K e ]{δ e }

(6.102)

in which {δ e } is defined by (6.91). Remarks (1) For a discretization with M elements the process of assembly and imposition of boundary conditions φ(0) = 1.0 and φ(1) = 0.0 follows the usual procedure.

384

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

(2) For a fixed discretization that works satisfactorily for a P e = P e1 using (6.101) as element matrix, if one progressively increases the P e, then for very large Peclet numbers, (6.101) would become − h2e

1 2

− h2e

2 he

− 12

− 12

1 2

− 21

he 3

he 6

1 2

− 21

he 6

he 3

2 he e

[K ]P e→∞ =

1 2

(6.103)

The element matrix [K e ] in (6.103) is obviously due to pure convection. We note [K e ] in (6.103) is symmetric with nonzero elements on the diagonal. Assembly of [K e ] in (6.103) and imposition of boundary conditions would yield a symmetric and positive definite matrix from which {δ} can be computed. Thus we see that LSP are non-degenerate processes. This property of the LSP is due to the variational consistency of the integral form.

Numerical solutions In this section we consider some special cases and compute solutions for them. Consider a two-element uniform mesh in which each element is a twonode linear element. This mesh is obviously deficient to simulate the solution of convection diffusion equations for P e → ∞. Nonetheless, we consider it to demonstrate the non-degenerate nature of LSP. 1

1

φ1 τ1 φ1 = 1.0

2 φ2 τ2

2

3 φ3 τ3 consider P e → ∞ φ3 = 0.0

h1 = 21 h2 = 21 x1 = 0.0 x2 = 0.5 x3 = 1.0 Figure 6.5: Two-element uniform discretization and boundary conditions

The element equations in this case would be (6.101) in which [K e ] is defined by (6.103). Substituting h1 = h2 = 12 we obtain the following element

385

6.2. 1D CONVECTION-DIFFUSION EQUATION

matrices:



4  −4 [K 1 ] = [K 2 ] =  0.5 0.5

−4 4 −0.5 −0.5

 0.5 −0.5  1 

0.5 −0.5 1 6 1 12

(6.104)

12 1 6

The degrees of freedom for the two elements are (after substituting interelement continuity conditions on φ and τ )     φ1  φ2            φ φ3 2 {δ 1 } = and {δ 2 } = (6.105) τ  τ       1   2  τ2 τ3 Assembled equations for the two element discretization are φ1

τ1

φ2

4

0.5

-4

0.5

1 6

-0.5

-4

-0.5

4+4

0.5

1 12

−0.5 +0.5

0

-4

-0.5

4

-0.5

0.5

1 12

-0.5

1 6

0 0

0

τ2 0.5 1 12 −0.5 +0.5 1 6

+

1 6

φ3

τ3

0

0

0

0

-4

0.5

-0.5

1 12

                                      



φ1    τ1 φ2

               

= {0}  τ2   

(6.106)

     φ3         τ3  

We note that assembled matrix [K] is symmetric (due to symmetry of [K e ]). After imposing boundary conditions φ1 = 1.0 and φ3 = 0.0, the reduced system of equations from (6.106) becomes   1    1 τ1  −0.5 12 0  −0.5  6     −0.5 8 0 0.5 φ2   4   1  = (6.107) 1   0 13 12 τ2  −0.5       12     1 1 0 0.5 12 τ3 0 6 A plot of the solution is shown in Fig. 6.6. We note that τ = dφ dx = −1 at all three nodes. The solution is in agreement with the explanation provided in previous remarks. Thus, we note that even with a ridiculous mesh and hopeless local approximation, the formulation based on minimization of residual functional does not degenerate and allows us to compute a solution satisfying the boundary conditions at x = 0.0 and x = 1.0. This property of the formulation based on minimization of residual functional is due to variational consistency of the integral form. This property also plays an extremely crucial role in designing adaptive processes.

386

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1

φh

0.5

x 0.5

1

Figure 6.6: Plots of φh versus x, p = 1, 2 element mesh, first order system

Remarks (1) It is rather obvious that a single element discretization would also yield the same solution as shown above. (2) Even the use of higher degree C 0 local approximations (C 0 p-version) one could confirm the same behavior. (3) Since in this case P e = ∞, all discretizations, any number of degrees of freedom, and all orders of local approximation would yield the same solution as shown in Fig. 6.6. (4) The readers are reminded that the purpose of the above exercise is to demonstrate the non-degenerate nature of finite element processes based on minimization of the residual functional due to variational consistency of the integral form. (5) It is perhaps instructive to examine the analytical solutions of the BVP when P e = ∞. In this case we have dφ = 0, 0 < x < 1 (6.108) dx with φ(0) = 1 and φ(1) = 0 (6.109) We note that (6.108) is a first order ordinary differential equation. Nonetheless, we have two boundary conditions to be satisfied. Thus, this BVP has no solution. When attempting to compute a numerical solution of (6.108)–(6.109), one observes that the boundary conditions (6.109) must be satisfied (since they are imposed and thus no other choice). A solution φ(x) that satisfies both boundary conditions is of course φ(x) = 1 − x

(6.110)

387

6.2. 1D CONVECTION-DIFFUSION EQUATION

Obviously (6.110) is not the solution of (6.108)–(6.109), as it does not satisfy (6.108). Nonetheless, finite element processes based on minimization of residual functional will compute such a solution due to the fact that it satisfies boundary conditions and the residual minimization criterion. Due to solution (6.110), one finds that Z  I=

dφ dx

Z1

2

¯ Ω

dΩ =

(−1)2 dx = 1

(6.111)

0

which is what the least squares numerical process yields. (6) In the finite element processes based on minimization of residual functional one could indeed start with a rather coarse mesh and invoke hpkadaptive processes and arrive at an optimal combination (least dofs) to achieve the desired solution due to the fact that non-degenerate nature of the resulting algebraic system is always assured. (7) On the other hand, in GM/WF due to VIC integral form, the coarse meshes may result in degenerate computational processes and hence computations of the solution may not be possible and thus true adaptive processes may not be possible either. ¯ e ) spaces Numerical studies using local approximations in H k,p (Ω for k ≥ 2, p ≥ 2k − 1 In this section we present some numerical studies for P e = 100 and ¯ e ); k ≥ using graded discretizations and local approximations in H k,p (Ω 2, p ≥ 2k − 1 spaces [1]. In the finite element processes that are based on minimization of residual functional I, examination of the behavior of I versus degrees of freedom (dofs) is essential in determining its dependence ¯ e ) with k = 2, 3, . . ., and compute on h, p and k. We consider spaces H k,p (Ω sequences of solutions for increasing p-levels up to 19 in each space. We consider P e = 100 and 106 and use discretizations: 106

Element lengths for five-element graded discretization for P e = 100: { 0.6, 0.28, 0.08, 0.02, 0.02 } Element lengths for 10-element graded discretization for P e = 106 : {0.722215, 2.5e-1, 2.5e-2, 2.5e-3, 2.5e-4, 2.5e-5, 2.5e-6, 2.5e-6, 2.5e-6}

388

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1e+00 1e-02

p=5

Residual Functional I

1e-04 1e-06

p=9 k,p

1e-08

—e

H (Ω ) Spaces (Pe=100) p-convergence

1e-10

k=2 k=3 k=4 k=5 k=6 k=7 k-convergence

1e-12 1e-14 1e-16

p=13

p=17

1e-18 10

100 Degrees of Freedom

Figure 6.7: Minimization of residual functional (LSP), residual functional I versus dofs: P e = 100

1e+00 1e-02 Residual Functional I

p=7 1e-04 —

Hk,p(Ωe) Spaces (Pe=106)

1e-06

p=11

p-convergence 1e-08

k=2 k=3 k=4 k=5 k=6 k=7 k-convergence

1e-10 1e-12

p=15

1e-14 10

100 Degrees of Freedom

Figure 6.8: Minimization of residual functional (LSP), residual functional I versus dofs: P e = 106

Behavior of the residual functional Figures 6.7 and 6.8 show graphs of I versus dofs for P e = 100 and P e = 106 , respectively. We make the following remarks. (1) For a fixed k, the order of space and hence the global smoothness of order k − 1, the rate of convergence of I increases as p-level is increased. (2) As the order of the space k is increased from k to k + 1, lower values of I are achieved in the space of order k + 1 compared to the space of

6.2. 1D CONVECTION-DIFFUSION EQUATION

389

order k, regardless of the values of k, indicating improved solution in the space of order k + 1 compared to the space of order k. (3) While the rate of convergence of I (i.e. slope of I versus dofs) increases slightly with the change of the order of the space at lower p-levels, at higher p-levels (p ≥ 12), I versus dofs graphs are almost parallel to each other for all values of k, indicating same rate of convergence for all values of k. (4) For fixed p-level i.e., p = constant, and since the discretization is fixed, h is constant as well; one could study the k-process for the residual functional. In these cases h and p are fixed and only k, the order of ¯ e ) is changing, hence these represent k-process for the the space H k,p (Ω functional I. Dependence of I on the order of the space k, i.e. degree of global smoothness is quite obvious. At lower p-levels (3 − 5), the dependence of I on k is not as strong as it is for p ≥ 5. With increasing p-levels, the dependence of I on k becomes even stronger. This confirms ¯ e ) or in other words, the degree of that the order of the space k in H k,p (Ω global smoothness is undoubtedly an independent parameter in addition to h and p. (5) The k and pk-convergence of the error functional I (and likewise other quantities) is perhaps most illustrative of the influence of the order of space k on the convergence of I. The pk-convergence can be viewed in at least two different ways. (i) For increasing k as well as increasing p but p = 2k −1, i.e. minimum p-level for the order of the space k. In this case we observe much higher rate of convergence of pk-processes compared to ¯ e )). p-convergence of I in any of the spaces (i.e., any value of k in H k,p (Ω We also refer to this as the k-convergence. We note that increase in p with increasing k is inevitable, but in this case p is minimum. In the graphs this is represented as k-convergence. (ii) Perhaps, the most dramatic is the pk-convergence of the error or residual functional I in which p and k both change in such a way that the total degrees of freedom for the discretization do not vary much. Almost vertical lines show such behaviors for various combination of p and k in which total dofs do not change significantly but the I values decrease significantly. We observe exceptionally high slopes of such lines, even for very low total dofs. With increasing total dofs, slopes of these lines increase indicating an increase in the convergence rate of I. At close to 100 dofs, the pk-convergence graphs of I are almost vertical straight lines, indicating exponential convergence rate. It is worth noting that such behavior is the consequence of the fact that for fixed dofs, an increase in k would permit an increase in p-level, which in fact is responsible for the improvement i.e. decrease in the value of I.

390

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

(6) We observe that I continues to decrease in progressively higher order spaces for a given dofs, indicating improved performance of the LSP in higher order spaces. That is, the best approximation property of the LSP in E-norm yields progressively improved solutions in higher order spaces. Using the five and ten element discretizations, we present some numerical ¯ e ) and H 3,p (Ω ¯ e ) spaces. We present studies for P e = 100 and 106 in H 2,p (Ω φh versus x graphs for both values of P e numbers. ¯ e ) spaces Figures 6.9 (a) and (b) show plots of φh versus x in H 2,p (Ω for p = 3, 5, 7, . . .. Figures 6.10 (a) and (b) show graphs of φh versus x ¯ e ); p = 5, 7, 9, . . . spaces. Due to graded mesh and low P e (of in H 3,p (Ω 100) even for low p-levels the computed solutions are sufficiently accurate. ¯ e ) for p = Figures 6.11 (a) and (b) show the plots of φh versus x in H 2,p (Ω 3, 5, 7, 9, 18. ¯ e ) due In Figs. 6.11 (a), we observe the linear behavior of φh in H 2,3 (Ω 2,3 ¯ e ) are to the fact for this mesh at such low p-levels the solutions in H (Ω unable to account for the physics of diffusion in the numerical process. As the p-levels are increased, φh improves and at p = 18, φh is in excellent agreement with the theoretical solution. What appears to be a reasonable solution at p = 7 in Fig. 6.11 (a), is in error upon closer examination as shown in Fig. 6.11 (b). However, for p ≥ 9 all computed solutions are in excellent agreement with the theoretical solution (up to more than 5 or 6 ¯ e ); p = 5, 7, 9, 18 shown in Fig. 6.12 decimal places). Solutions in H 3,p (Ω ¯ e ), (a) and Fig. 6.12 (b) exhibit similar behaviors as the solutions in H 2,p (Ω i.e. for low values of p (≤ 7) the solutions are diffused but for p ≥ 9 sharp fronts are simulated accurately with excellent agreement with the theoretical solution. We remark that the computed solutions: (a) are always free of spurious oscillations and that the computational process in each case is nondegenerate due to variational consistency of LSP, (b) are always physical if judged based on how well chosen h, p and k allow the physics of the BVP to be incorporated in the numerical process, and (c) LSP permit much coarser discretizations than the variationally inconsistent Galerkin processes.

6.3 2D convection-diffusion equation The steady state 2D convection-diffusion equation is the 2D form of the energy equation in Eulerian description for an inviscid medium. The dimensionless form can be written as ∂φ ∂φ ∂  ∂φ ∂φ  ∂  ∂φ ∂φ  u +v − kxx + kxy − kyx + kyy −f =0 ∂x ∂y ∂x ∂x ∂y ∂y ∂x ∂y ∀x, y ∈ Ω = Ωx × Ωy = (0, 1) × (0, 1) ⊂ R2 (6.112)

391

6.3. 2D CONVECTION-DIFFUSION EQUATION

1

0.8

0.6 —

φh

H2,p(Ωe) Spaces (Pe=100) p-convergence 0.4

p=3 p=4 p=5 p=7 p=19

0.2

0 0

0.2

0.4

0.6

0.8

1

x

(a) Function value φh over entire domain 1

—

H2,p(Ωe) Spaces (Pe=100)

0.95

p-convergence p=3 p=4 p=5 p=7 p=19 theoretical

φh

0.9

0.85

0.8

0.75

0.7 0.7

0.75

0.8

0.85

0.9

0.95

1

x

(b) Function value φh : expanded view Figure 6.9: Minimization of residual functional (LSP); φh versus dofs: P e = 100, k = 2

in which u and v are known velocities in the x and y direction, φ is dimensionless temperature and kij (i, j = x, y) are the thermal conductivity coefficients. In the following, we consider more specific form of (6.112) for which analytical solution is possible.

392

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1

0.8

0.6 —

φh

H3,p(Ωe) Spaces (Pe=100) p-convergence 0.4

p=5 p=7 p=8 p=9 p=19

0.2

0 0

0.2

0.4

0.6

0.8

1

0.98

1

x

(a) Function value φh over entire domain 1

0.98

0.96 —

φh

H3,p(Ωe) Spaces (Pe=100) p-convergence 0.94

p=5 p=7 p=9 p=19 theoretical

0.92

0.9 0.9

0.92

0.94

0.96 x

(b) Function value φh : expanded view Figure 6.10: Minimization of residual functional (LSP); φh versus dofs: P e = 100, k = 3

Theoretical solution of the simplified form If we assume u = v = 1, kxx = kyy = to f = 0 then (6.112) can be written as

1 Pe

and kxy = kyx = 0 in addition

∂φ ∂φ 1  ∂ 2 φ ∂ 2 φ  + − + −f = 0, ∀x, y ∈ Ω = (0, 1)×(0, 1) ⊂ R2 (6.113) ∂x ∂y P e ∂x2 ∂y 2

393

6.3. 2D CONVECTION-DIFFUSION EQUATION

1

0.8

0.6 —

φh

H2,p(Ωe) Spaces (Pe=106) p-convergence 0.4

p=3 p=5 p=6 p=7 p=19

0.2

0 0

0.2

0.4

0.6

0.8

1

0.99998

1

x

(a) Function value φh over entire domain 1

—

H2,p(Ωe) Spaces (Pe=106)

φh

0.999

p-convergence p=7 p=9 p=13 p=19 theoretical

0.998

0.997

0.996 0.9999

0.99992

0.99994

0.99996 x

(b) Function value φh : expanded view Figure 6.11: Minimization of residual functional (LSP); φh versus dofs: P e = 106 , k = 2

We consider (6.113) with the following BCs (also shown in Fig. 6.13) and simplification. φ(1, y) = 0.0, 0 ≤ y ≤ 1 φ(x, 1) = 0.0, 0 ≤ x ≤ 1 1 − e(x−1)P e , 0≤x≤1 1 − e¯P e 1 − e(y−1)P e φ(0, y) = , 0≤y≤1 1 − e¯P e φ(x, 0) =

(6.114)

394

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1

0.8

0.6 —

φh

H3,p(Ωe) Spaces (Pe=106) p-convergence 0.4

p=5 p=7 p=8 p=9 p=19

0.2

0 0

0.2

0.4

0.6

0.8

1

0.99998

1

x

(a) Function value φh over entire domain 1

—

H3,p(Ωe) Spaces (Pe=106)

φh

0.999

p-convergence p=7 p=9 p=13 p=19 theoretical

0.998

0.997

0.996 0.9999

0.99992

0.99994

0.99996 x

(b) Function value φh : expanded view Figure 6.12: Minimization of residual functional (LSP); φh versus dofs: P e = 106 , k = 3

The theoretical solution of the BVP given by (6.113) with BCs (6.114) is

φ(x, y) =

(1 − e(x−1)P e )(1 − e(y−1)P e ) (1 − e−P e )(1 − e−P e )

(6.115)

395

6.3. 2D CONVECTION-DIFFUSION EQUATION

The differential operator A in (6.114) is ∂2  ∂ 1  ∂2 ∂ + + − ∂x ∂y P e ∂x2 ∂y 2

(6.116)

hence (6.114) can be written as Aφ = 0 as f = 0

(6.117)

The differential operator A is linear but not symmetric, hence A is a nonself-adjoint differential operator, thus the Galerkin method with weak form would yield VIC integral form. In the following we consider least squares finite formulations of (6.113) based on the residual functional as these yield VC integral forms. y

(0, 1)

φ(x, 1) = 0

(1, 1) Ω

φ(0, y) =

1−e(y−1)P e 1−e−P e

φ(1, y) = 0

x (0, 0)

φ(x, 0) =

1−e(x−1)P e 1−e−P e

(1, 0)

Figure 6.13: Domain of definition of the BVP (6.113) and boundary conditions

6.3.1 Least squares finite element formulation based on the residual functional Consider the non-simplified form of 2D convection-diffusion equation i.e. equation (6.112). Let φeh (x, y) be the local approximation of φ over a typical ¯ e of the discretization Ω ¯T = S Ω ¯ e of the spatial element e with domain Ω e ¯ = (0, 1) × (0, 1). The residual equation E e for an element e can domain Ω be written as ∂φe ∂φe ∂φe  ∂φe ∂  Ee = u h + v h − kxx h + kxy h ∂x ∂y ∂x ∂x ∂y ∂φeh ∂φeh  ∂  ¯e − kyx + kyy − f e ∀x, y ∈ Ω (6.118) ∂y ∂x ∂y

396

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

¯ e and Ω ¯ T can be written as The residuals I e and I for Ω I e = (E e , E e )Ω¯ e I=

M X

Ie =

M X

(6.119) (E e , E e )Ω¯ e

(6.120)

e=1

i=1

Therefore δI = 2

M X

(E e , δE e )Ω¯ e = 0

(6.121)

(δE e , δE e )Ω¯ e > 0

(6.122)

e=1

δ2I = 2

M X e=1

Since (6.113) is a second order ODE in φ, x and y, the local approxi¯ e ) ; k ≥ 3 and p = 2k − 1 in which k = 3 is the mation φeh (x, y) ∈ H k,p (Ω minimally conforming order of the approximation space. For this choice of ¯ T hold in the Riemann sense. k, all integrals over Ω Details element equations ¯ ξη = (−1, 1) × (−1, 1) be the map of an element e with domain Ω ¯e Let Ω in the natural coordinate space ξ, η and φeh (ξ, η)

=

n X

Ni (ξ, η) φei = [N ]{δ e }

(6.123)

i=1

¯ e (see chapter 8). Substituting (6.123) be the local approximation of φ over Ω into (6.118) n

n

i=1

i=1

  ∂ X ∂ X Ni φei + v Ni φei E =u ∂x ∂y e

n n X X ∂  ∂Ni e ∂Ni e  − kxx φi + kxy φ ∂x ∂x ∂y i i=1

i=1

n n X X ∂  ∂Ni e ∂Ni e  − kyx φi + kyy φ − fe ∂y ∂x ∂y i i=1 i=1  ∂N ∂N ∂ ∂Ni ∂Ni  i i δE e = u +v − + kxy kxx ∂x ∂y ∂x ∂x ∂y   ∂ ∂Ni ∂Ni − kyx + kyy , i = 1, 2, . . . , n ∂y ∂x ∂y

(6.124)

397

6.3. 2D CONVECTION-DIFFUSION EQUATION

We can also write these using the matrix and vector notation.  h h ∂N i h ∂N i ∂N i ∂  h ∂N i e E = u +v − kxx + kxy ∂x ∂y ∂x ∂x ∂y   h i h i ∂ ∂N ∂N {δ e } − f e − kyx + kyy ∂y ∂x ∂y n ∂N o n ∂N o n ∂N o ∂  n ∂N o {δE e } = u +v − kxx + kxy ∂x ∂y ∂x ∂x ∂y n ∂N o ∂  n ∂N o − kyx + kyy ∂y ∂x ∂y

(6.125)

Since E e is linear in φei , then E e = {δE e }T {δ e } − f e , hence (6.121) can be written as (for a discretization with M elements) M X

[K e ]{δ e } =

e=1

M X 

 (δE e , δE e )Ω¯ e {δ e }

e=1 M M hX i X e e = (δE , δE )Ω¯ e {δ} = (δE e , f e )Ω¯ e e=1

(6.126)

e=1

and therefore e

Z

e

e T

e

Z

{δE }{δE } dΩ and {F } =

[K ] = ¯e Ω

{δE e }f e dΩ

(6.127)

¯e Ω

is explicitly defined by using (6.125). Numerical values of the coefficients of [K e ] are computed using Gauss quadrature.

6.3.2 Least squares finite element formulation of (6.113) by recasting it as a system of first order PDEs In (6.113), we introduce auxiliary variables α and β defined by the following auxiliary equations  ∂φ  ∂φ + kxy α = − kxx ∂x ∂y (6.128)  ∂φ ∂φ  β = − kyx + kyy ∂x ∂y Negative sign in (6.128) are introduced to be consistent with definitions of heat vector [14]. Substituting from (6.128) into (6.113).  ∂φ ∂φ ∂α ∂β  +v + + −f =0  u   ∂x ∂y ∂x ∂y    ∂φ ∂φ α + kxx + kxy =0 ∀x, y ∈ Ω = (0, 1) × (0, 1) (6.129)  ∂x ∂y     ∂φ ∂φ  β + kyx + kyy =0  ∂x ∂y

398

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

¯ e , domain Let φeh , αhe , βhe be the local approximations of φ, α and β over Ω ¯ T . We consider φe , αe , β e ∈ of a typical element e of the discretization Ω h h h 1,p e ¯ ¯ e ) with equal degree of Vh ⊂ H (Ω ), local approximations of class C 0 (Ω approximation functions. Upon substituting the local approximations in (6.129) we obtain the following residual equations. ∂φeh ∂φe ∂αhe ∂β e +v h + + h − fe ∂x ∂y ∂x ∂y e e ∂φ ∂φ E2e = αhe + kxx h + kxy h ∂x ∂y e ∂φ ∂φe E3e = βhe + kyx h + kyy h ∂x ∂y E1e = u

       

¯e ∀x, y ∈ Ω

(6.130)

      

¯ e and Ω ¯ T can now be defined. The residual functional I e and I over Ω Ie =

I=

3 X i=1 M X

(Eie , Eie )Ω¯ e e

I =

e=1

(6.131)

M X 3 X e=1

(Eie , Eie )Ω¯ e



(6.132)

i=1

therefore δI = δ2I =

M X e=1 M X

e

δI = 2

M X 3 X

 (Eie , δEie )Ω¯ e = 0

e=1 i=1 M X 3 X

δ2I e = 2

e=1

e=1

 (δEie , δEie )Ω¯ e > 0

(6.133)

(6.134)

i=1

Details of element equations ¯e Consider unequal degree C 0 local approximation for φ, α and β over Ω for the sake of generality φeh

=

nφ X

Niφ (ξ, η) φei = [N φ ]{φe }

i=1

αhe =

nα X

Niα (ξ, η) αie = [N α ]{αe }

i=1 nβ

βhe =

X i=1

Niβ (ξ, η) βie = [N β ]{β e }

(6.135)

399

6.3. 2D CONVECTION-DIFFUSION EQUATION

Substituting from (6.135) into (6.130)

nφ

E1e

nφ

n

i=1

i=1

α  ∂ X φ e  ∂  X φ e ∂ X =u N φi + v N φi + N α αie ∂x ∂y ∂x

i=1

nβ

+

∂  X β e N βi − f e ∂y i=1

E2e =

nα X

N α αie + kxx

i=1 nβ

E3e =

X

nφ X ∂N φ i=1 nφ

N β βie + kyx

i=1

∂x

X ∂N φ i=1

∂x

φei + kxy

nφ X ∂N φ i=1 nφ

φei + kyy

∂y

X ∂N φ i=1

∂y

(6.136) φei φei

which can be written as h ∂N φ i h ∂N α i h ∂N β i h ∂N φ i e e e {φ } + u {φ } + {α } + {β e } − f e =u ∂x ∂y ∂x ∂y h ∂N φ i h ∂N φ i E2e = [N α ]{αe } + kxx {φe } + kxy {φe } (6.137) ∂x ∂y h ∂N φ i h ∂N φ i E3e = [N β ]{β e } + kyx {φe } + kyy {φe } ∂x ∂y

E1e

Let us define  e  {φ } [ e {δ } = {αe } and {δ} = {δ e }  e  e {β }

(6.138)

We can now consider {δE1e }, {δE2e } and {δE3e } which are given by

{δEie } =

 n e o ∂Ei     e   n ∂{φ e} o  ∂Ei

e

 n ∂{α e} o      ∂Eei  ∂{β }

, i = 1, 2, 3

(6.139)

400

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

and using (6.139) and (6.137), we obtain n φ o  n φo ∂N   u ∂x + v ∂N   ∂y    α e ∂N {δE1 } = n ∂xβ o       ∂N ∂y

{δE2e } =

{δE3e } =

n φo n φ o  ∂N ∂N   k + k xx xy   ∂x ∂y  

{N α }       {0} n o n o   ∂N φ ∂N φ   k + k yx yy   ∂x ∂y  

(6.140)

{0}

  

  

{N β }

We note that (6.133) can be written as M h X 3 X e=1

i  (δEie , δEie )Ω¯ e {δ e }

i=1

=

M X 3 hX e=1

(δEie , δEie )Ω¯ e

M nX 3 i o X {δ} = (δEie , f e )Ω¯ e (6.141) e=1

i=1

i=1

or M X

[K e ]{δ e } = [K]{δ} =

e=1

M X

{F e } = {F }

(6.142)

i=1

Therefore [K e ] =

3 X

(δEie , δEie )Ω¯ e =

i=1

{F e } =

3 Z X

3 Z X i=1 ¯ e Ω

{δEie }{δEie }T dΩ (6.143)

{δEie }f e dΩ

i=1 ¯ e Ω

are explicitly defined by using (6.139). Numerical values of the coefficients of [K e ] and {F e } are obtained using Gauss quadrature. In the following we consider a number of model problems using a slightly simplified form of (6.112). In all cases, numerical solutions are computed using least squares finite element formulation based on residual functional as presented above.

401

6.3. 2D CONVECTION-DIFFUSION EQUATION

6.3.3 Convection dominated thermal flow (advection skewed to a square domain) In this example [2, 5–7, 10] we consider advection skewed to a square domain with φ, dimensionless temperature, specified on two adjacent bound√ → − 2 2 aries with k a k = u + v = 1, that is, unidirectional flow skewed to the boundaries by angle θ and θ = 30◦ , 45◦ and 60◦ [see Fig. 6.14(a)]. The diffusivity coefficients are chosen such that the diffusivity matrix [K] = k[I] with k = 10−6 (i.e. kxx = kyy = 10−6 and kxy = kyx = 0) which results in a P´eclet number P e = 1/k = 106 . Thus, in this case, (6.112) reduces to (assuming f = 0) u

∂φ ∂φ 1  ∂2φ ∂2φ  +v − + 2 =0 ∂x ∂y P e ∂x2 ∂y

∀x, y ∈ Ω = (0, 1) × (0, 1) ⊂ R2

(6.144) For such high value of P´eclet number, the solution is essentially that of the pure advection case. We compare the numerical solutions computed here with those reported in the literature. The temperature φ is specified by the following boundary conditions (for sides AB and AD): φ = 100, x = 0, 0 ≤ y ≤ 1 ˆ φ = φ(x), y = 0, 0 ≤ x ≤ h1

(6.145)

φ = 50, y = 0, h1 ≤ x ≤ 1 ˆ where h1 is the length of the element along the x-axis located at A and φ(x) is a cubic function satisfying the following four conditions: ˆ φ(0) = 100 ,

ˆ 1 ) = 50 φ(h

,

∂ φˆ =0 , ∂x x=0

∂ φˆ =0 ∂x x=h1

(6.146)

¯ e ), hence we use the In this study, we consider solutions of class C 00 (Ω mathematical model (6.129) consisting of a system of first order PDEs. First, we consider a coarse 4 × 4 uniform discretization [see Fig. 6.14(b)] consisting of nine node p-version elements with boundary conditions defined by (6.145) and (6.146). The p-levels in the ξ and η directions are increased uniformly beginning with pξ = pη = 1. √ Figure 6.15 shows p-convergence of the residual functional I√( I versus degrees of freedom). At p-level of pξ = pη = 9, the value of I is of the order of O(10−3 ), i.e. the value of I is of the order of O(10−6 ) confirming good accuracy of the computed solution. Figure 6.16(a) shows a contour plot and 3D plot of φ over the square domain at p-level of p = 8 in which φ = 100 along AC and φ = 50 along EF . In the triangular region ACD, the temperature φ has a constant value of 100 and likewise in the region EBF , φ has a constant value of 50. Smooth and oscillation free computed solution

402

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

y

Zero flux boundary conditions on sides BC and CD

1 D

C kak = 1

Flow direction θ φ = 100

1

A

B

x

φ = 50

(a) Schematic of the BVP y

φ = 100 100 φ x h1

50

x

φ = 50

(b) Finite element discretization and boundary conditions

Figure 6.14: Schematic, boundary conditions, and finite element mesh for advection skewed to a square domain

is quite obvious from the 3D plot. Transition from φ = 50 to φ = 100 occurs in the region ACF E. Remarks (1) The first observation we make is that the computed solution (even for such a coarse discretization) is oscillation free. The present least squares formulation based on residual functional requires no addition of streamwise diffusion that is necessary in the Galerkin method with weak form. (2) We note that lines of constant φ are parallel to each other, that is, AC is parallel to EF and so is the case for all other contour lines between

403

6.3. 2D CONVECTION-DIFFUSION EQUATION

101 p=1 p=1 Uniform discretization

p=1

4x4 10x10 20x20

Square root of residual functional, √ I

100

10-1

10-2

p=9 10

p=8

-3

p=8

10-4 101

102

103 Degrees of freedom

104

105

Figure 6.15: Square root of the residual functional versus degrees of freedom for θ = 45◦ , ¯ e) solutions of class C 00 (Ω

AC and EF confirming the absence of crosswind diffusion. (3) The width h, given by h = h1 sin(θ), over which the function φ changes from 100 to 50 is a function of the element length h1 of the element located at x = 0, y = 0 and angle θ and it is not, in any way, related to the least squares formulation of the boundary value problem. (4) To illustrate the point discussed in (3), we consider two more refined uniform discretizations: 10 × 10 and 20 × 20. For both discretizations, p-levels are increased uniformly (pξ = pη = p) beginning with √ p = 2 and ending with p = 8. Figure 6.15 shows p-convergence of I ( I versus dofs) for all three discretizations (4 × √ 4, 10 × 10 and 20 × 20). We note that convergence rates (slopes of I versus dofs) are almost the same for the three discretizations. However, the coarsest discretization yields lowest values of I for a given number of degrees of freedom. This is in agreement with the expected behavior due to the smoothness of the solution φ, that is, for smooth solutions, coarse discretizations with higher p-levels result in the best performance. (5) Figures 6.16(b) and (c) show 3D and contour plots of φ for 10 × 10

404

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

φ

110 100 90 80 70 60 50 40

1

D

C

F

y

1 y

1 0

h1=0.25

0

x

0 A0

E h1=0.25

B x

1

(a) 4 × 4 uniform discretization

φ

110 100 90 80 70 60 50 40

1

D

C F

y

1 y

1 0

0

h1=0.10

x

E 0 A 0 h1=0.10

B x

1

(b) 10 × 10 uniform discretization

φ

110 100 90 80 70 60 50 40

1

D

C F

y

1 y

1 0

0 h1=0.05

x

E 0 A 0 h1=0.05

B x

1

(c) 20 × 20 uniform discretization ¯ e ) at pξ = pη = p = 8 Figure 6.16: Plots of φ for θ = 45◦ : solutions of class C 00 (Ω

and 20 × 20 discretizations respectively at p = 8. We note that the solution for 10 × 10 and 20 × 20 discretizations are also free of crosswind diffusion. The length h1 decreases as the discretization is refined and as a consequence, the width h over which φ changes from 100 to 50 also decreases.

405

6.3. 2D CONVECTION-DIFFUSION EQUATION

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

h1=0.25

0

0

x

0

h1=0.25

x

1

(a) 4 × 4 uniform discretization

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

0

h1=0.10

0 0 h1=0.10

x

x

1

(b) 10 × 10 uniform discretization

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

0 h1=0.05

x

0 0 h1=0.05

x

1

(c) 20 × 20 uniform discretization ¯ e ) at pξ = pη = p = 8 Figure 6.17: Plots of φ for θ = 30◦ : solutions of class C 00 (Ω

(6) Numerical solutions are computed for θ = 30◦ and 60◦ with the three uniform discretizations and uniform p-level. Figure 6.17 shows contour plots and 3D plots for θ = 30◦ for the three discretizations at p-level of pξ = pη = 8. Figure 6.18 shows similar plots for θ = 60◦ . All three studies confirm that for a fixed value of h (given by h =

406

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

h1=0.25

0

0

x

0

h1=0.25

x

1

(a) 4 × 4 uniform discretization

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

0

h1=0.10

0 0 h1=0.10

x

x

1

(b) 10 × 10 uniform discretization

φ

110 100 90 80 70 60 50 40

1

y

1 y

1 0

0 h1=0.05

x

0 0 h1=0.05

x

1

(c) 20 × 20 uniform discretization ¯ e ) at pξ = pη = p = 8 Figure 6.18: Plots of φ for θ = 60◦ : solutions of class C 00 (Ω

h1 sin(θ)), the computed solutions are free of crosswind diffusion. The results from the least squares finite element formulation computed and presented here are compared with Grygiel and Tanguy [15] and those reported by many researchers using streamline upwinded Petrov-Galerkin method (SUPG) and the method of characteristics. Discussions based on on these comparisons

6.3. 2D CONVECTION-DIFFUSION EQUATION

407

are presented in the following. Comparison with published works, discussion and remarks (1) Brookes and Hughes [2] have presented results for this problem for θ = 22.50◦ , 45◦ and 67.5◦ using 10 × 10 uniform mesh of bilinear elements employing four formulations: G (the Galerkin formulation), QU (quadrature method of adding diffusion, i.e., the convective terms are evaluated using one point quadrature), SU1 (one point quadrature for convective term in SUPG) and SU2 (2 × 2 quadrature for convective as well as diffusion term in SUPG). We make the following observations (a) The Galerkin formulation has wild oscillations in the computed solutions. The magnitude of the oscillations varies with angle θ. (b) The QU method adds excessive diffusion and smears the solution. Sharp front is destroyed. (c) The SU1 has oscillations similar to the Galerkin formulation for θ = 22.5◦ and 67.5◦ . For the case of θ = 45◦ , the solution is nodally exact. (d) The SU2 produces the best results out of the four techniques considered but some oscillations are still present. Hughes, Mallet and Mizukami [4] have also investigated the same problem using: SUPG (2 × 2 Gaussian quadrature rule for all terms for four node bilinear element), DC1 (SUPG with discontinuity capturing operator (τ1 = τ and τ2 = τ11 ), using four node bilinear), DC2 (SUPG with discontinuity capturing behavior (τ1 = τ and τ2 = max(0, τ11 − τ ), using four node bilinear element) and MH (Mizukami and Hughes use triangular elements with treatment similar to DC1 and DC2). Based on the results, we make the following comments. (a) The SUPG solution has oscillations. (b) The DC1 smoothes the oscillations while still maintaining the reasonable sharp front. (c) Discontinuity-capturing operator DC2 produces less diffusive solution than DC1. (d) In the case of the MH scheme, each triangular element is internally subdivided into more than one triangle with the diagonals of the triangles roughly oriented parallel to the internal layers. This scheme is monotonic and produces the sharpest fronts. (2) The SUPG based methods have significant amount of crosswind diffusion The amount of crosswind diffusion varies with the angle θ.

408

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

(3) Method of characteristics produces the best results for θ = 45◦ . The method of characteristics also has significant diffusion but it is less than the SUPG method. (4) The p-version LEFEF has no crosswind diffusion and solution is oscillation free for all values of θ. (5) Furthermore, the p-version least squares finite element formulation does not utilize upwinding, discontinuity capturing operator or any other such means. The p-version least squares finite element process is a straight forward formulation that utilizes the fact that: the presence of artificial diffusion in the solution of the LEFEF is a consequence of inadequate representation of the function behavior over each element. If this argument is valid, then adequate mesh with appropriate p-levels should be able to produce a good solution. The studies presented here confirm this.

6.3.4 Advection of a cosine hill in a rotating flow field In this numerical study [2, 5–7, 10] we consider almost pure advection (P e = 106 ) of a cosine hill in a rotating flow field over a unit square [see Fig. 6.19(a)]. The velocities u and v are given by the following. u = −y, v = x

(6.147)

We consider kxx = kyy = k = 10−6 and kxy = kyx = 0 which gives us P e = k1 = 106 and (6.112) reduces to (6.144). Due to such small value of diffusivity k, this problem is almost the same pure advection, i.e. −y

∂φ ∂φ 1  ∂ 2 φ ∂ 2 φ  +x − + = 0 ∀x, y ∈ Ω = (0, 1)×(0, 1) ⊂ R2 (6.148) ∂x ∂y P e ∂x2 ∂y 2

We set φ = 0 on all four boundaries of the domain of unit square. Along the line OA (internal boundary) φ is specified by a cosine hill. φ=


1 1 cos(4πy + π) + 1 , − ≤ y ≤ 0 2 2

(6.149)

We consider a 4×4 uniform discretization of nine node p-version elements shown in Fig. 6.19(b). The p-levels are increased uniformly in ξ and η. Figure 6.20 shows a √ graph of I versus dofs (p-convergence of I). At p-level of 5, the value of I is already of the order of O(10−5 ). Figure 6.21 shows the elevation of φ at p-level of 8 over the unit square computed using least squares finite element formulation.

409

6.3. 2D CONVECTION-DIFFUSION EQUATION 1 φ=0

y

φ=0

φ

Flow direction

x

O

φ=0 1

A O

A

φ=0

(a) Schematic of the BVP and boundary conditions

(b) An uniform discretization

Figure 6.19: Schematic, boundary conditions and finite element mesh for advection in a rotating flow field

10-1 p=1 p=2 Square root of residual functional, √ I

-2

10

10-3

p=4

10-4

p=6

p=8 -5

10

101

102 103 Degrees of freedom

104

Figure 6.20: Square root of the residual functional versus degrees of freedom, solutions ¯ e) of class C 00 (Ω

Figure 6.22 shows plots of φ versus y and φ versus x at x = 0 and y = 0 respectively (i.e. along the vertical and horizontal center lines of the unit square domain). These results can be compared with those reported by Hughes and Mallet [3] using various upwinding formulations based on the Galerkin method with weak form.

410

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1.2 1 0.8 0.6 φ 0.4 0.2 0 -0.2

0.5 y

0.5 x -0.5

¯ e ), p-level Figure 6.21: 3D Plot of φ: 4x4 uniform discretization; solutions of class C 00 (Ω of 8

1

1

0.8

0.8

0.6

0.6

φ

φ 0.4

0.4

0.2

0.2

0

0 -0.5

-0.25

0 y

0.25

0.5

(a) φ versus y at x = 0

-0.5

-0.25

0 x

0.25

0.5

(b) φ versus x at y = 0

¯ e ), Figure 6.22: Plots of φ along x = 0 and y = 0: 4x4 mesh; solutions of class C 00 (Ω p-level of 8

Comparison with published works, discussion and remarks [10] (1) For this problem, the Galerkin method with weak form produces reasonable results. Some oscillations are present in the computed solution near the boundaries of the domain. (2) The QU formulation fails drastically due to the presence of crosswind diffusion. (3) Both SU1 and SU2 give good results.

411

6.3. 2D CONVECTION-DIFFUSION EQUATION

(4) The p-version LSFEF produces excellent results for a rather coarse mesh without the need of upwinding or any other adjustments. From Fig. 6.21, the remarkable accuracy of the computed solution φ is worth noting. We note that the p-version LSFEF has no dispersion or diffusion. The cosine hill distribution specified along OA is maintained as it gets advected in the flow field.

6.3.5 Thermal boundary layer In this model problem [2,5–7,10], we consider a rectangular domain 1×0.5 with  x = 0.0, 0 ≤ y ≤ 0.5 φ = 1, y = 0.5, 0 ≤ x ≤ 1.0 (6.150) φ = 0, y = 0, 0 ≤ x ≤ 1.0 as the boundary conditions. The velocity components u and v are given by u = 2y, v = 0

(6.151)

No boundary conditions were imposed at the outflow (x = 1.0, 0 ≤ y ≤ 0.5). This model problem has been investigated by Franca et al. [8] for k = 7 × 10−4 which corresponds to P e = 1428.57. In the present study we consider P e of 100, 1428.57, 10, 000 and 106 . Figure 6.23 shows a schematic, and discretization using nine node p-version finite elements. We note that at x = y = 0, specification of φ is not unique. In the present study, we assume φ = 1.0 at x = y = 0 and allow it to vary in a cubic manner from 1 to 0 along the x-axis (0 ≤ x ≤ h1 ) for the element located at x = y = 0: ˆ φ = φ(x),

y

0.5

n ˆ ∂ φˆ ˆ ˆ 1 ) = 0, ∂ φ φ(0) = 1.0, φ(h = 0, = 0 (6.152) ∂x x=0 ∂x x=h1 φ=1

0.20603125 Velocities: u = 2y v=0

φ=1

0.20603125 0.075 0.01125 0.0016875

x

0 0

φ=0

(a) Schematic and boundary conditions

1.0

0.0225 0.15 h 1 = 0.003375

0.4120625

0.4120625

(b) Graded discretization (numbers indicate element lengths)

Figure 6.23: Schematic, boundary conditions and finite element mesh for thermal boundary layer problem

412

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

-1

10

p=1

Square root of residual functional, √ I

10-2

Pe = 100 Pe = 1,428.57 Pe = 10,000 Pe = 1,000,000

p=1 10-3

p=1 p=8

10-4

p=1

p=8

10-5

p=8 10-6 p=8

10-7 101

102 103 Degrees of freedom

104

Figure 6.24: Square root of the residual functional versus degrees of freedom, solutions ¯ e) of class C 00 (Ω

To accommodate φ in (6.152) we choose a rather small element at x = y = 0 as shown in Fig. 6.23(b). The p-levels in the ξ and η directions for all elements of the discretization are increased uniformly (pξ = pη = p). √ Figure 6.24 shows plots of I versus degrees of freedom (p-convergence of the residual functional I for P´eclet numbers of 100, 1428.57, 10, 000 and 106 . At p-level of 8, the residual functional I has a magnitude lower than of the order of O(10−4 ) confirming good accuracy of the computed solution. Plots of the computed solution φ at p-level of 8 versus y at x = 0.003375, 0.025875, 0.5 and 1.0 for P e = 100, 1428.57, 10, 000 and 106 are shown in Fig. 6.25. In addition, 3D plots of the computed solution φ over the rectangular domain 1 × 0.5 for all four values of P´eclet numbers are shown in Fig. 6.26. Remarks. (1) A 5 × 5 graded coarse discretization works well for the wide range of P´eclet numbers considered (100–106 ). (2) A small element size at x = y = 0 and the graded discretization is necessitated due to: (1) the non-unique nature of φ at x = 0 and (2) higher P´eclet numbers for which higher gradients of φ are present, i.e. sharper boundary layers.

413

6.4. SUMMARY

0.5

0.5 Pe: 100 1,428.57 10,000 106

0.3

0.2

At location, x=0.003375

0.1

0.3

0.2

At location, x=0.025875

0.1

0

0 0

0.2

0.4 0.6 Solution, φ

0.8

1

0

0.5

0.2

0.4 0.6 Solution, φ

0.8

1

0.8

1

0.5 Pe: 100 1,428.57 10,000 106

0.3

0.2

Pe: 100 1,428.57 10,000 106

0.4 Distance, y

0.4 Distance, y

Pe: 100 1,428.57 10,000 106

0.4 Distance, y

Distance, y

0.4

At location, x=0.5

0.1

0.3

0.2

At location, x=1.0

0.1

0

0 0

0.2

0.4 0.6 Solution, φ

0.8

1

0

0.2

0.4 0.6 Solution, φ

¯ e ), Figure 6.25: Plots of φ at different x locations: 5x5 mesh; solutions of class C 00 (Ω p-level of 8

(3) For P e = 1428.57, the computed solutions are in good agreement with the numerical solutions reported by Franca et al. [8]. (4) The numerical solution presented here are computed using a straight forward least squares finite element formulation based on a residual functional without any special treatments or use of upwinding methods. The method used here has excellent convergence characteristics.

6.4 Summary Through simple model problems we are able to demonstrate the most significant aspects of VIC and VC integral forms. In all applications with non-self-adjoint differential operators, the findings reported for the 1D and 2D convection-diffusion equations hold without exception. Another significant aspect to note here is that all upwinding methods in whatever form have no mathematical or physical basis and the solutions computed using such methods are in fact not the true solutions of the BVP. Higher order spaces play a significant role for such a BVP. The various issues and the research in this regard have been reported by Surana et al. [10]. Failure of GM/WF due to VIC integral form in very simply model problems is more

414

φ

NON-SELF-ADJOINT DIFFERENTIAL OPERATORS

1.2 1 0.8 0.6 0.4 0.2 0

φ

0.5

1.2 1 0.8 0.6 0.4 0.2 0

0.5 y

y

1

1

x

x

0

0

(a) Plots of φ for P e = 100 and 1, 428.57

φ

1.2 1 0.8 0.6 0.4 0.2 0

φ

0.5

1.2 1 0.8 0.6 0.4 0.2 0

0.5 y

1

y

1

x 0

x 0

(b) Plots of φ for P e = 10, 000 and 106 ¯ e ), Figure 6.26: 3D plots of φ for different values of P e: 5x5 mesh; solutions of class C 00 (Ω p-level of 8

than sufficient to alert to refrain from using GM/WF for BVPs in two and three dimensions in which the differential operators are non-self-adjoint. Importance of higher order spaces and higher degree local approximations is demonstrated. Finite element formulations based on the residual functional (LSP) are shown to yield VC integral forms regardless of whether the mathematical model consists of a higher order system of differential equations or a first order system of differential equations. VC integral forms yield unconditionally stable finite element computational processes, hence are completely free of upwinding processes. Use of higher order spaces in differential models containing the highest orders of the derivatives of the dependent variables is most prudent choice as this choice results in the computational processes with least number of degrees of freedom, hence most efficient computations.

Problems 6.1 Consider steady state one-dimensional convection-diffusion equation dφ 1 d2 φ − = 0, 0 < x < 1 = Ω dx P e dx2 φ(0) = 1, φ(1) = 0 This BVP represents dimensionless form of one-dimensional energy equation with unit velocity field. P e is called Peclet number and φ is dimensionless temperature.

415

PROBLEMS

(a) Construct weak form of the GDE over Ω using GM/WF. Give details of PV, SV, EBC and NBC as well as the nature of th resulting functionals. Establish VC or lack of it of the resulting integral form. ¯ e with linear geometry as well as linear local approx(b) Consider a two-node element Ω ¯ e . Give details of the weak form over Ω ¯ e . Derive discretized imation of φ over Ω e ¯ equations for an element with domain Ω using natural coordinate system. (c) Derive theoretical solution of the BVP. (d) Perform numerical studies for P e = 100 using element equations derived in (b). Consider progressively refined discretizations until you observe convergence of the numerical results. (1) Tabulate nodal values of φ obtained from finite element calculations and the theoretical values (at selected locations). (2) Also tabulate dφ obtained from F.E. studies and the theoretical values (at selected dx locations). (3) Plot graphs of φh versus x for all discretizations and compare with the theoretical solution φ (one single graph). h (4) Plot graphs of dφ versus x for all discretizations and compare with the theoretical dx dφ solution dx . h (5) Calculate L2 -norm of φh and dφ for all discretizations and compare with the dx corresponding L2 -norm obtained using the theoretical solution. Discuss the F.E. results and their behavior with progressively refined meshes. Comment on the accuracy of the F.E. solution compared to theoretical solutions using the graphs and the L2 -norms. 6.2 Consider one-dimensional steady state convection-diffusion equation in the dimensionless form. 1 d2 φ dφ − = 0, 0 < x < 1 = Ω dx P e dx2 φ(0) = 1, φ(1) = 0

(1)

The GDE (1) is referred to as the strong form of the GDE due to the fact that it contains highest order derivatives of the dependent variable φ. (a) Construct a least squares formulation of (1) over Ω using φh as global approximation of φ over Ω. Do not substitute approximation φh in the integrals. ¯ T of Ω ¯ in which Ω ¯ e is a two-node element. Derive the (b) Consider a discretization Ω e e ¯ ¯ e ) for the two-node element equations for LSP over Ω . Consider φh of class C 1 (Ω given by (for element map in ξ space: −1 ≤ ξ ≤ 1) (ξ 3 − ξ)  e  (1 + ξ) (ξ 3 − ξ)  e φ1 + − φ2 2 4 2 4  (ξ 3 − ξ) (ξ 2 − 1)  he  dφeh  + − 4 4 2 dx 1  (ξ 3 − ξ) (ξ 2 − 1)  he  dφeh  + + (2) 4 4 2 dx 2  e  e dφ dφ in which φe1 and φe2 are nodal values of φ at nodes 1 and 2 and dxh and dxh φeh (ξ) =

 (1 − ξ)

+

1

2

are the nodal values of derivate of φeh at nodes 1 and 2. he is the element length for ¯ e . Derive discretized equations of equilibrium for an element Ω ¯ e using local element Ω 1 ¯e approximation of class C (Ω ) given by (2) and linear geometry mapping between x and ξ spaces.

416

REFERENCES FOR ADDITIONAL READING

(c) Perform numerical studies for P e = 100 using element equations derived in (b). Consider five progressively refined discretizations. 2

h (1) Tabulate φh , dφ and ddxφ2h for all discretizations and their theoretical values. dx (2) Plot graphs of (for all discretizations) (i) φeh versus x and φ versus x h versus x and dφ versus x (ii) dφ dx dx2 d2 φh (iii) dx2 versus x and ddxφ2 versus x h (3) Calculate L2 -norms of φh and dφ for all discretizations and compare them with dx their theoretical values. Plot graphs of the L2 -norms versus dofs. (4) Compute least squares functional I for each discretization and plot a graph of I versus total dofs for each discretization on a single graph. What can you infer from these graphs?

Compare these solutions with GM/WF computed in 6.1. Provide a discussion of results and findings. [1, 9–13, 16–18]

References for additional reading [1] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [2] A. Brooks and T. J. R. Hughes. Streamline Upwinding/Petrov-Galerkin Formulation for Convection Dominated Flows with Particular Emphasis on the Incompressible Navier-Stokes Equations. Comp. Meth. in Appl. Mech. and Engg., 32:199–259, 1982. [3] T. J. R. Hughes and M. Mallet. A new Finite Element Formulation for Computational Fluid Dynamics; III. A Generalized Streamline Operator for Multidimensional Advection Diffusion Systems. Comp. Meth. in Appl. Mech. and Engg., 58:305–328, 1986a. [4] T. J. R. Hughes, M. Mallet, and A. Mizukami. A new Finite Element Formulation for Computational Fluid Dynamics; II. Beyond SUPG. Comp. Meth. in Appl. Mech. and Engg., 54:341–355, 1986b. [5] G. F. Carey and T. Plover. Variable Upwinding and Adaptive Mesh Refinement in Convection-Diffusion. Int. J. of Num. Meth. in Engg., 19:341–353, 1983. [6] J. Christie, D. F. Griffiths, and A. R. Mitchell. Finite Element Method for Second Order Differential Equations with Significant First Derivatives. Int. J. of Num. Meth. in Engg., 10:1389–1396, 1976. [7] J. Christie and A. R. Mitchell. Upwinding of Higher Order Galerkin Methods in Conduction-Convection Problems. Int. J. of Num. Meth. in Engg., 12:1764–1771, 1978. [8] L. P. Franca, S. L. frey, and T. J. R. Hughes. Stabilized Finite Element Methods: I. Application to the Advenction-Diffusion Model. Comp. Meth. in Appl. Mech. and Engg., 95(3):253–276, 1992. [9] D. Winterscheidt and K. S. Surana. p-Version Least Squares Finite Element Formulation for Convection-Diffusion Equation. International Journal of Numerical Methods in Engineering, 36:111–133, 1993. [10] K. S. Surana and J. S. Sandhu. Investigation of Diffusion in p-Version LSFE and STFSFE formulations. Computational Mechanics, 16(3):151–169, 1995. [11] K. S. Surana, O. Gupta, P. W. TenPas, and J. N. Reddy. h, p, k least squares finite element processes for 1d helmholtz equation. Int. J. Comp. Meth. in Eng. Sci. and Mech., 7(4):263–291, 2006.

REFERENCES FOR ADDITIONAL READING

417

[12] Surana, K. S., Mahanti, R. K. and Reddy, J. N. Galerkin/Least Squares Finite Element Processes for BVPs in h, p, k Mathematical Framework. International Journal of Computational Engineering Sciences and Mechanics, 8:439–462, 2007. [13] K. S. Surana, O. Gupta, and J. N. Reddy. Galerkin and least squares finite element processes for 2d helmholtz equation in h, p, k framework. Int. J. Comp. Meth. in Eng. Sci. and Mech., 8:341–361, 2007. [14] K. S. Surana. Advanced Mechanics of Continua. CRC/Taylor and Francis, 2015. [15] J. M. Grygiel and P. A. Tanguy. Finite Element Solution for Advection Dominated Thermal Flows. Comp. Meth. in Appl. Mech. and Engg., 93:277–289, 1991. [16] G. Hellwig. Differential Operators of Mathematical Physics. Addison-Wesley Publishing Co., 1967. [17] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964. [18] J. N. Reddy. An Introduction to the Finite Element Method. McGraw Hill Inc., New York, 3rd edition, 2006.

7

Non-Linear Differential Operators 7.1 Introduction In this chapter we consider finite element processes for BVPs described by non-linear differential operators. These BVPs could contain single or multi-dependent variables and could be in single or multi-dimensional spaces. First, we review the properties of these differential operators as well as the properties of the integral forms based on methods of approximation (presented in Chapter 3). (1) The non-linear differential operators are neither linear nor symmetric. (2) The properties of the integral forms (presented in Chapter 3) resulting from various methods of approximation for non-linear differential operators are summarized in the following. (a) GM, PGM, WRM and GM/WF all yield variationally inconsistent integral forms. (b) However, in GM/WF the contribution of the even order terms in the differential operator to the total integral form becomes symmetric due to transferring half of the order of differentiation from the dependent variable to the test function. When the even order terms in the GDEs dominate the solution behavior, GM/WF is meritorious over GM, PGM and WRM. (3) The LSM or LSP when used for GDEs containing highest orders of the derivatives of the dependent variables or for those cast as a system of first order PDEs using auxiliary equations can be made variationally consistent by: (a) neglecting second variation of the residuals in the second variation of the residual functional (b) solving the system of non-linear algebraic equations using Newton’s linear method (or Newton-Raphson method). Justification for (a) and importance of (b) have been discussed in Chapter 3 in connection with classical methods of approximation. 419

420

NON-LINEAR DIFFERENTIAL OPERATORS

(4) In GM/WF one constructs and integral form based on Fundamental Lemma of the calculus of variations, which is then converted to the discretization. For each element of the discretization weak form is constructed using by parts. Thus, if Aφ − f = 0 in Ω is the BVP S integration T e ¯ ¯ ¯ then we can write and if Ω = e Ω is the discretization of Ω, (Aφh − f, v)Ω¯ T =

X

(Aφeh − f, v)Ω¯ e =

e

X


B e (φeh , v) − le (v)

(7.1)

e

¯ T and φe is the local approxin which φh is the approximation of φ over Ω h ¯ e . For an element e with domain Ω ¯ e , we construct imation of φ over Ω the weak form (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v), v = δφeh

(7.2)

using the same procedure as described in Chapter 5 for self-adjoint operators. In (7.2), we note that (a) B e (φeh , v) is linear in v but not linear in φeh due to the fact that the differential operator A is non-linear. (b) B e (φeh , v) is obviously not symmetric (as bilinearity of B e (·, ·) is essential for symmetry). (c) le (·) is linear. We use the local approximation φeh =

n X

Ni δie = [N ]{δ e }

(7.3)

i=1

with v = δφeh = Nj , j = 1, . . . , n

(7.4)

in which δie are nodal degrees of freedom and Ni are local approximation functions to obtain B e (φeh ) = [K e ]{δ e } e

e

(7.5) e

l (v) = {P } + {F }

(7.6)

e 6= K e , i.e. [K e ] is not In (7.5), [K e ] is a function of {δ e } and Kij ji symmetric and {P e } is a vector of secondary variables resulting from the concomitant due to integration by parts. (5) In least squares finite element processes we construct residual functional ¯ T . Let E = Aφh − f over Ω ¯ T and E e = Aφe − f over Ω ¯ e, I(φh ) over Ω h then X X I(φh ) = (E, E)Ω¯ T = Ie = (E e , E e )Ω¯ e (7.7) e

e

421

7.1. INTRODUCTION

The necessary conditions are given by X X (E e , δE e )Ω¯ e δI e = 2 δI(φh ) = (E, δE)Ω¯ T = e

e

=2

X

e

{g } = 2{g} = 0

(7.8)

e

where {g} =

X

{g e } = 0

(7.9)

e

¯ e , we construct {g e } = (E e , δE e ) ¯ e and For an element e with domain Ω Ω then use (7.9) to obtain {g}. The necessaryScondition {g} = 0 must be used to calculate the nodal values {δ} = e {δ e } for the discretiza¯ T . Since the differential operator A is non-linear, we must find a tion Ω {δ} that satisfies (7.9) iteratively. Let {δ}0 be an assumed or starting solution, then  g({δ}0 ) 6= 0 (7.10) Let {δ} = {δ}0 + {∆δ}

(7.11)

be such that   g({δ}) = g({δ}0 + {∆δ}) = 0

(7.12) Expand g(·) in (7.12) in Taylor series about {δ}0 and retain only up to linear terms in {∆δ} (Newton’s linear method or Newton–Raphson method):   ∂{g} g({δ}0 + {∆δ}) = g({δ}0 ) + {∆δ} = 0 (7.13) ∂{δ} {δ}0 

Therefore

h ∂{g} i−1  {∆δ} = − g({δ}0 ) ∂{δ} {δ}0

(7.14)

We note that 1 ∂{g} = δ{g} = δ 2 I = (δE, δE)Ω¯ T + (E, δ 2 E)Ω¯ T ∂{δ} 2 We approximate

∂{g} ∂{δ}

(7.15)

in (7.15) by (see Chapter 3)

1 2 ∂{g} δ I= = δ{g} ≈ (δE, δE)Ω¯ T > 0 2 ∂{δ}

(7.16)

Hence, with this approximation we achieve variational consistency of the integral form. Therefore, (7.14) becomes h i−1  {∆δ} = − (δE, δE)Ω¯ T g({δ}0 ) (7.17) {δ}0

422

NON-LINEAR DIFFERENTIAL OPERATORS

and (δE, δE)Ω¯ T =

X

(δE e , δE e )Ω¯ e

(7.18)

e

and we use line search to determine {δ}, i.e. {δ} = {δ}0 + α{∆δ} with 0 < α ≤ 2 such that I({δ}) ≤ I({δ}0 ). We consider the iterative process converged when |gi | ≤ ∆ ; i = 1, . . . where ∆ is a preset tolerance for zero. Summary of steps: 1. Choose an assumed or starting solution {δ}0 P e P e e 2. I = I = (E , E )Ω¯ e e P e eP e 3. g = {g } = (E , δE e )Ω¯ e e e P 4. (δE, δE)Ω¯ T = (δE e , δE e )Ω¯ e e

5. {∆δ} = −[(δE, δE)Ω¯ T ]−1 {δ}0 {g({δ}0 )} 6. {δ} = {δ}0 + α{∆δ}; 0 ≤ α ≤ 2 such that I({δ}) ≤ I({δ}0 ) 7. If |gi | ≤ ∆, i = 1, . . . then the solution is converged. Otherwise set {δ}0 = {δ} and repeat steps 2 through 7. In the following we consider a specific model problem to present details of finite element processes based on GM/WF and LSP.

7.2 One dimensional Burgers equation The steady state 1D Burgers equation represents one dimensional form of momentum equation in viscous flows. The dimensionless form of 1D steady state Burgers equation is given by φ

dφ 1 d2 φ − = 0 ∀x ∈ (0, 1) = Ω ⊂ R1 dx Re dx2

(7.19)

We consider the following boundary conditions: φ(0) = 1,

φ(1) = 0

(7.20)

in which Re = ρuL µ is the Reynolds number where ρ, u, L and µ are reference density, velocity, length, and viscosity. Analytical solution of (7.19) and (7.20) is given by   φˆ Re(x−1) 1 − e φ = φˆ (7.21) 1 + eφˆ Re(x−1)

423

7.2. ONE DIMENSIONAL BURGERS EQUATION

in which φˆ is the solution of φˆ − 1 ˆ = e−φ Re φˆ + 1

(7.22)

We note that for Re > 5, φˆ in (7.22) is very close to 1.0. For Re < 5, we need to solve (7.22) to find φˆ in (7.21). The behavior of φ and its derivatives is similar to convection diffusion equation. Thus, for higher Re, the gradients of φ become localized near x = 1.0 with progressively increasing values. We ¯ note that the analytical solution is of class C ∞ (Ω).

7.2.1 The Galerkin method with weak form Let the Burgers equation be defined Aφ − f = 0 in Ω, where A = S ¯by 1 d2 e T ¯ such ¯ − Re dx2 and f = 0. Let Ω = e Ω be the discretization of Ω e ¯ that Ω = [xe , xe+1 ] with he = xe+1 − xe , the element length. Let φh be ¯ e with ¯ T and φe , the approximation of φ over Ω the approximation of φ in Ω h e v = δφh . Consider

d φ dx

(Aφeh

Z − f, v)Ω¯ e =

(Aφeh

xZe+1



− f )v dx =

φeh

xe

¯e Ω

dφeh 1 d2 φeh  − v dx (7.23) dx Re dx2

Using integration by parts for the second term in (7.23)    Z  dφe 1 dv dφeh  1 dφeh  xe+1 vφeh h + (7.24) (Aφeh − f, v)Ω¯ e = dx − v dx Re dx dx Re dx xe ¯e Ω

Let

 1 dφe  h P1e = − , Re dx xe

P2e =

 1 dφe  h Re dx xe+1

(7.25)

Therefore (Aφeh −f, v)Ω¯ e

=

Z 

vφeh

dφeh 1 dv dφeh  + dx−v(xe ) P1e −v(xe+1 ) P2e (7.26) dx Re dx dx

¯e Ω

or (Aφeh − f, v)Ω¯ e = B e (φeh , v) − le (v)

(7.27)

where B e (φeh , v) =

Z 

vφeh

dφeh 1 dv dφeh  + dx dx Re dx dx

(7.28)

¯e Ω

l (v) = v(xe ) P1e + v(xe+1 ) P2e e

(7.29)

424

NON-LINEAR DIFFERENTIAL OPERATORS

Equation (7.26) is the weak form of (7.19), the BVP being considered. (a) B e (φeh , v) is linear in v but is not linear in φeh . (b) B e (·, ·) is not symmetric. (c) le (v) is linear in v. Approximation space The BVP contains up to second order derivative of the dependent variable φ but, the weak form only contains up to the first order derivatives of φ and v. ¯ T in the pointwise sense requires (i) Admissibility of φh in Aφ − f = 0 in Ω k,p T ¯ φh ∈ Vh ⊂ H (Ω ); k ≥ 3 for which k = 3 is minimally conforming. For this choice the integrals in the following are Riemann. X X
(Aφh −f, v)Ω¯ T ⇔ (Aφeh −f, v)Ω¯ e ⇔ B e (φeh , v) − le (v) R

R

e

e

{z

|

R

}

(7.30) ¯ e ) (or higher) then, all integrals in That is, if we choose φeh of class C 2 (Ω (7.30) are Riemann and hence, all three forms in (7.30) are equivalent. ¯ e ), that is if (ii) Based on the weak form, if we choose φeh of class C 1 (Ω 2,p e e ¯ ), then the following holds. φh ∈ Vh ⊂ H (Ω X X
L L (Aφh − f, v)Ω¯ T

(Aφeh − f, v)Ω¯ e

B e (φeh , v) − l(v) L

L

e

e

|

{z R

} (7.31)

φeh ,

P

(B e (·, ·) − le (·))

For this choice of local approximation the sum e holds in the Riemann sense but all other integral forms in (7.31) are in the Lebesgue sense. ¯ e ) then P (B e (φe , v) − l(v)) only holds (iii) If we choose φeh ∈ Vh ⊂ H 1,p (Ω e h in the Lebesgue sense and the other two integral forms in (7.31) are meaningless. Local approximation φeh Let φeh =

n X

Ni (ξ) δie = [N ]{δ e }

(7.32)

i=1

¯e → Ω ¯ ξ , then be the local approximation of φ over Ω v = δφeh = Nj (ξ), j = 1, . . . , n

(7.33)

425

7.2. ONE DIMENSIONAL BURGERS EQUATION

in which Ni (ξ) are local approximation functions (see Chapter 8) and δie are nodal degrees of freedom. Substituting from (7.32) and (7.33) into the weak form (7.28) and (7.29) yields (with v = Nj ) B

e

(φeh , Nj )

=

Z h

Nj

n X

Ni δie

i=1

¯e Ω

n X dN

i

dx

i=1

δie



n

1 dNj X dNi e i + δ dx Re dx dx i i=1

(7.34) le (Nj ) = Nj (xe ) P1e + Nj (xe+1 P2e

(7.35)

B e (φeh , v) = [K e ]{δ e }  e  (P )   ξ=−1 1       0     . e .. l (v) =       0      e  (P2 )ξ=+1

(7.36)

or

(7.37)

e of [K e ] can be written in at least two possible ways. The coefficients Kij That is, we can write (from (7.34)) e Kij

Z 

=

1 dNi dNj  + dΩ, i, j = 1, . . . , n dx Re dx dx

(7.38)

dφeh 1 dNi dNj  + dΩ, i, j = 1, . . . , n dx Re dx dx

(7.39)

dNj Ni φeh

¯e Ω

or Z 

e Kij =

Ni Nj

¯e Ω e 6= K e . In (7.39), it appears that K e is In (7.38), we see clearly that Kij ij ij dφe

symmetric but dxh is a function of Ni . Thus, the integral of Ni Nj ¯ e cannot be ensured to be positive. Ω

dφeh dx

over

C 0 local approximations with p = 1 We consider a two-node linear element. For this choice we have φeh

=

1 − ξ  2

φe1

+

1 + ξ  2

v = δφeh = Nj (ξ), j = 1, 2

φe2

=

2 X

Ni φei = [N ]{δ e }

(7.40)

i=1

(7.41)

426

NON-LINEAR DIFFERENTIAL OPERATORS

and x(ξ) =

1 − ξ  2

xe1 +

1 + ξ  2

xe2 , J =

he = (xe2 − xe1 ) 2

(7.42)

he dξ dx = J dξ = 2

It is instructive to consider (7.34) for j = 1 and 2. For j = 1 then B e (φeh , v) becomes

B

e

(φeh , N1 )

=

Z1 h

1 − ξ  1 − ξ  e  1 + ξ  e  1 e 1 e  φ1 + φ2 − φ1 + φ2 2 2 2 2 2

−1

+


i 1  φ1 − φ2 dξ 2 Re he

(7.43)

and for j = 2 then B e (φeh , v) becomes

B e (φeh , N2 ) =

Z1 h

1 + ξ  1 − ξ  e  1 + ξ  e  1 e 1 e  φ1 + φ2 − φ1 + φ2 2 2 2 2 2

−1

+


i 1 −φ1 + φ2 dξ 2 Re he

(7.44)

which can be written as 1 B e (φeh , N1 ) = − (φe1 )2 + 3 1 B e (φeh , N2 ) = − (φe1 )2 − 6

1 e e φ φ + 3 1 2 1 e e φ φ + 3 1 2

1 e 2 1 (φ2 ) + (φe − φe2 ) 6 Re he 1 1 e 2 1 (φ2 ) + (−φe1 + φe2 ) 3 Re he

(7.45)

Writing (7.45) in the matrix form " B

e

(φeh , v)

=

− 13 φe1 − 16 φe1 − 13 φe2

1 3

φe1 + 16 φe2 1 3

φe2

#( ) φe1 φe2

1 + Re he

"

#( ) 1 −1 φe1 −1

1

φe2 (7.46)

or B e (φeh , v) =

h

 K 1e ({δ e }) +

1  2e i e K {δ } = [K e ]{δ e } Re he

(7.47)

If we consider a four-element uniform discretization with grid points 15 and function values at the grid points as φ1 , . . . , φ5 , then the assembled

427

7.2. ONE DIMENSIONAL BURGERS EQUATION

equations would be      φ1              1 1   0 0 φ  φ2 + φ3 0 2    3 6    1 1 1 1 φ3 0 − φ2 − φ3 φ3 + φ4 0   6 3 3 6       1 1 1 1    φ − φ − φ 0 φ4 + φ5  4  0 3 4   6 3 3 6        1 1 1   φ5   0 0 φ5 − φ4 − φ5 3 6 3        φ1     P11  1 -1    0 0 0              1     P    2 2     φ   -1 -1 0 2 0  2     +P1      2     P 2 = φ 3 -1 0 0 2 -1 3 +P1         3          P2     φ4  -1  -1 2 0 0     4     +P     1            4     1 -1 0 0 0 φ5 P2 

1 − φ1 3 1 1 − φ1 − φ2 6 3 0 0 0

+

1 Re he

1 1 φ1 + φ2 3 6

0

which can be written as h  K 1 ({δ}) +

0

0

1  2 i K {δ} = {P } Re he

(7.48)

(7.49)

Remarks (1) From the BCs, we note that φ1 = 1 and φ5 = 0. Hence, after imposing BCs in (7.48), the reduced [K 1 ] and [K 2 ] would be (3 × 3) matrices in which [K 1 ] would have zeros on the diagonals. (2) [K 1 ] is independent of he , that is, the non-linear convection part in [K] is not dependent on he , the discretization length. (3) For a fixed discretization (i.e. fixed he ), if one increases Reynolds number Re then for very large Re, Re1he [K 2 ] → [0], a null matrix and as a consequence [K] in (7.49) now only consists of [K 1 ] and hence becomes singular. (4) For a fixed discretization and for Reynolds numbers between 0 and ∞, the assembled [K] matrix progressively deteriorates for progressively increasing Reynolds number producing oscillations in the solution (in the same fashion as in case of convection diffusion equation) and eventually becomes singular for large values of Re. (5) Thus, we note that GM/WF leads to degeneracy of [K], a consequence of variationally inconsistent integral form.

428

NON-LINEAR DIFFERENTIAL OPERATORS

(6) The use of upwinding methods in this case can also be ruled out for the same reasons as given in the case of convection-diffusion equation (chapter 6). ¯ e ) spaces to demon(7) Many numerical studies have been presented in H k,p (Ω strate the various aspects discussed here (see references at the end of the chapter).

7.2.2 LSP based on residual functional Let φeh be the local approximation of φ over typical element e with domain e ¯ ¯T = S Ω ¯ e of the spatial domain Ω ¯ = [0, 1], then Ω of the discretization Ω e

dφeh 1 d2 φeh ¯e − ∀x ∈ Ω (7.50) dx Re dx2 is the residual equation for an element e. The residual functionals I e and I ¯ e and Ω ¯ T are given by for Ω E e = φeh

I e = (E e , E e )Ω¯ e I=

M X e=1

Ie =

M X

(7.51) (E e , E e )Ω¯ e

(7.52)

X

(7.53)

e=1

and δI = 2

M X e=1

δ2I ≈ 2

M X

(E e , δE e )Ω¯ e = 2

{g e } = 2{g} = 0

e

(δE e , δE e ) = 2δ{g} > 0

(7.54)

e=1

Since (7.50) is a second order ODE, the local approximation φeh ∈ Vh ⊂ ¯ e ) ; k ≥ 3, p ≥ 2k − 1 in which k = 3 is the minimally conforming H k,p (Ω ¯T order of the approximation space. For this choice of k all integrals over Ω hold in Riemann sense. Following the details of the LSP in the earlier section we have {∆δ} = −[(δE, δE)Ω¯ T ]−1 (7.55) {δ}0 {g({δ}0 )} We define [K e ] = (δE e , δE e )Ω¯ e

(7.56)

[K e ] = (δE e , δE e )Ω¯ e

(7.57)

in which The solution {δ0 } is an assumed or starting solution in the Newton’s linear method. A new updated solution {δ} is obtained using {δ} = {δ0 } + α{∆δ} Details on how determining α have already been discussed.

(7.58)

429

7.2. ONE DIMENSIONAL BURGERS EQUATION

Computational details ¯ ξ = [−1, 1] be the map of Ω ¯ e in the natural coordinate space ξ and Let Ω let the local approximation be given by φeh (ξ) =

n X

Ni (ξ) φei = [N ]{φe } = [N ]{δ e }

(7.59)

i=1

Substituting (7.59) in (7.50) Ee =

n X

Ni (ξ) φei

i=1

or

n n   d X  1 d2  X e N (ξ) φ Nj (ξ) φej − i i dx Re dx2 i=1

j=1


h dN i e  1 h d2 i e E e = [N ]{δ e } {δ } − {δ } dx Re dx2 h ∂N i   n dN o 1 n d2 N o e e e {δE } = {N } {δ } + [N ]{δ } − ∂x dx Re dx2

or {δE e } = {N }

(7.60)

n dN o ∂φeh 1 n d2 N o + φeh − ∂x dx Re dx2

(7.61)

(7.62) dφe

Thus [K e ] in (7.57) is defined. For an assumed solution {δ0 }, φeh , and dxh are known in (7.62), hence [K e ] is explicitly defined. Numerical values of the coefficients of [K e ] are calculated using Gauss quadrature.

7.2.3 LSP based on residual functional: first order system of equations Since the GDE for the BVP contains up to second order derivative of the dependent variable φ, the GDE must recast into a system of first order ODEs for C 0 local approximation to be admissible. Let τ=

dφ dx

(7.63)

then the GDE (7.19) can be written as φ

dφ 1 dτ − =0 dx Re dx dφ =0 τ− dx

  

∀x ∈ (0, 1) = Ω ⊂ R1

(7.64)

 

with BCs: φ(0) = 1 and φ(1) = 0. Let φeh and τhe be local approximations of φ ¯ e = [xe , xe+1 ] → Ω ¯ ξ = [−1, 1] an element e of the discretization and τ over Ω

430

NON-LINEAR DIFFERENTIAL OPERATORS

¯T = S Ω ¯ e . Then Ω e dφeh 1 dτhe − dx Re dx e dφ E2e = τhe − h dx E1e = φeh

  

¯e ∀x ∈ (0, 1) = Ω

(7.65)

 

The residual functional I e for an element e is given by I e = (E1e , E1e )Ω¯ e + (E2e , E2e )Ω¯ e

(7.66)

¯ T , we have For Ω I=

X

e

I =

2 XX

e

δI = 2

2 XX e

e

(Eie , Eie )Ω¯ e

(7.67)

i=1

X  (Eie , δEie )Ω¯ e = 2{g} = 2 {g e } = 0

(7.68)

e

i=1

or {g} =

X

{g e } = 0

(7.69)

e

Following the details of the LSP for nonlinear differential operators presented in the earlier section we have δ I∼ =2 2

2 XX e

(δEie , δEie )Ω¯ e = 2δ{g} > 0

(7.70)

1  2 −1  δ I {δ0 } g({δ0 }) 2

(7.71)

i=1

and {∆δ} = − We define

X 1 [K] = [δ 2 I] = [K e ] 2 e

in which [K e ] =

2 X

(δEie , δEie )Ω¯ e

(7.72)

(7.73)

i=1

Computational details Consider equal order, equal degree local approximations. We choose twonode C 0 linear local approximation for φeh and τhe . 1 − ξ 

φe1 +

1 + ξ 

φe2 = [N ]{φe } 2 2 1 + ξ  1 − ξ  τ1e + τ2e = [N ]{τ e } τhe = 2 2

φeh =

(7.74)

431

7.2. ONE DIMENSIONAL BURGERS EQUATION

and x(ξ) = Let

1 − ξ  2

xe +

1 + ξ  2

xe+1 , he = xe+1 − xe , J =

he 2

 e φ    e    1e   {φ } φ2 {δ e } = = , nodal dofs for element e e e {τ } τ     1e   τ2  e    ∂Eei    ∂φ1     n e o   e ∂Ei  ∂Ei       e e ∂{φ } ∂φ {δEie } = n ∂E e o =  ∂E2e  , i = 1, 2    ei   i   ∂τ1   ∂{τ e }     e  ∂E  ei  

(7.75)

(7.76)

(7.77)

∂τ2

in which E1e = φeh

dφe dφeh 1 dτhe − , E2e = τhe − h dx Re dx dx

(7.78)

Therefore ndN o dφe dφeh d(δφeh ) 1 d(δτhe ) 1 ndN o +φeh − = {N} h +φeh − dx dx Re dx dx dx Re dx n dN o e) d(δφ h = {N } − (7.79) {δE2e } = δτhe − dx dx Substituting for φeh , τhe and δφeh = Nj ; j = 1, 2 and δτhe = Nj ; j = 1, 2 in (7.79)  n o  e n o {N } dφh + φe dN   dN  h dx dx n dx e e o {δE1 } = , {δE2 } = (7.80)    {N }  − 1 dN {δE1e } = δφeh

Re

dx

Therefore e

[K ] =

Z1 

{δE1e }{δE1e }T + {δE2e }{δE2e }T

h

e

2

dξ

(7.81)

−1

We note that   1+ξ [N ] = 1−ξ 2 2 h dN i   = − 21 12 dξ h dN i 1 h dN i 2  1 −2 = = dx J dξ he

(7.82) (7.83) 1 2



(7.84)

We remark that [K e ] is calculated at {δ}0 , an assumed starting or guess dφe
solution, at which (φeh ){δ}0 and dxh {δ}0 appearing in (7.80) are known and hence are constants as far as the integrals are concerned.

432

NON-LINEAR DIFFERENTIAL OPERATORS

Remarks (1) The formulation yields VC integral form with the approximation δ 2 I ∼ = 2(δE, δE)Ω¯ T and the use of Newton’s linear method to find {δ} that satisfies the necessary condition {g} = 0. (2) Computations of [K e ], and the solution procedure follows the standard procedure. ¯ e ) spaces using Numerical studies using approximations in H k,p (Ω LSP Since the differential operator is non-linear, GM/WF will yield VIC integral form. Surana et al. [1–3] have presented numerical studies for GM/WF ¯ e ) spaces. using local approximations in H k,p (Ω A few important points to note: (1) LSP is variationally consistent and hence the resulting computational processes are always non-degenerate regardless of the choice of h, p and k, and the dimensionless parameters controlling the physics. (2) In this formulation we utilize auxiliary variable to reduce Burgers equation into a system of two first order differential equa¯ e ) for the tions. (3) The minimally conforming space for LSP is H 3,p (Ω original differential equation without auxiliary variable, but if we permit 2 discontinuity of ddxφ2 at the inter-element boundaries, then it is possible to ¯ e ) spaces. Lack of convergence of such solutions will not be a use H 2,p (Ω surprise due to the fact that the local approximations in this space do not describe the physics of diffusion correctly in the computational process. Consider a system of first order ODEs for Burgers equation φ

dφ 1 dτ dφ − = 0, τ − =0 dx Re dx dx

(7.85)

Equation (7.85) is used in the majority of the published work. We make some observations here. (1) Consider a fixed discretization suitable for Re = Re1 , if the Reynolds number is increased from Re1 to Re2 > Re1 and if the discretization is kept fixed, then it is clear that more localized behavior of diffusion at Re = Re2 is no longer simulated correctly in the numerical process. Thus the more localized behavior of diffusion at Re = Re2 must diffuse over a larger length in the computational process. Progressively increasing Re must continue to yield progressively diffused solutions until the solution cannot diffuse anymore, one of the possible solutions will be a straight line connecting φ = 1 at x = 0 to φ = 0 at x = 1 which in fact is the 2 solution of the BVP ddxφ2 = 0 with φ(0) = 1 and φ(1) = 0.

7.2. ONE DIMENSIONAL BURGERS EQUATION

433

(2) It is instructive to examine the analytical solution of the resulting differential equation and its computed solution in case of 1D Burgers equation when Re = ∞. In this case we have dφ φ = 0, 0 < x < 1 (7.86) dx φ(0) = 1, φ(1) = 0 (7.87) We note that (7.86) is a first order non-linear differential equation, nonetheless we have two boundary conditions (7.87) to be satisfied by a solution of (7.86). Thus the BVP (7.86)–(7.87) has no unique solution. When attempting to compute a numerical solution of (7.86)–(7.87), one observed that boundary conditions (7.87) must be satisfied (since they are imposed and thus no other choice). A solution φh of lowest degree that satisfies the boundary conditions (7.87) is of course φh = 1 − x

(7.88)

Obviously, φ in (7.88) is not the solution of (7.86) and (7.87) because it does not satisfy (7.86). However, LSP will compute such a solution (if the local approximation is linear) due to the fact that it satisfies boundary conditions and LS minimization criterion, using (7.88) one finds that  Z  Z1 dφeh 2 1 I(φh ) = φh dΩ = [(−1)(1 − x)]2 dx = (7.89) dx 3 0

Ω

which is in fact what the LSP yields. This nonzero I(φh ) obviously confirms that the GDE (7.86) is not satisfied by φh = 1 − x. The theoretical solution of 1D Burgers equation at Re → ∞ is a step function located at x = 1.0. Since (7.86) is a non-linear differential equation and (7.86) and (7.87) do not have a unique solution. For example φh = 1 −

n X xi i=1

n

, n = 2, 3, . . .

(7.90)

satisfies the boundary conditions in (7.87). Each solution in (7.90), does not satisfy (7.86) and hence would yield a different value of the residual functional I. When solving (7.86) and (7.87) numerically, the nature of φh obviously depends upon h, p, and k. Nonetheless, the computational process in each case is non-degenerate. (3) The motivation for the above exercise is to demonstrate the non-degenerate nature of LSP due to variational consistency of the integral form. The computed solutions are always free of spurious oscillations and are always physical within the limitations due to the choices of h, p, k, and Re (see [1] for numerical studies with varying h, p and k).

434

NON-LINEAR DIFFERENTIAL OPERATORS

Graded discretizations: Re = 102 and Re = 106 (LSP)

In this section we consider numerical studies using the following graded discretizations for Re of 102 and 106 : Element lengths for the  5-element graded discretization for Re = 100: 0.779, 0.17, 0.017, 0.017, 0.017 Element lengths for the 13-element graded discretization for Re = 106 :  0.4944443, 0.4944443, 10−2 , 10−3 , 5 × 10−5 , 5 × 10 −5 , 5 × 10−6 , 5 × 10−6 , 10−6 , 10−7 , 10−7 , 10−7 , 10−7 These discretizations by no means are optimal, but care has been taken to ensure that these meshes permit resolution of the solution gradients. We ¯ e ) and H 3,p (Ω ¯ e ) spaces. At Re = 102 the soconsider solutions in H 2,p (Ω lution is relatively diffused; however, at Re = 106 , the diffusion is isolated over a length of O(10−6 ) near x = 1.0. First, we consider Re = 106 . So¯ e ) and H 3,p (Ω ¯ e) lutions are computed for p = 3 to p = 19 in both H 2,p (Ω spaces. Figures 7.1(a) and (b) show plots of computed solution φh versus ¯ e ) space. The solutions in H 2,p (Ω ¯ e ) space x at different p-levels in H 2,p (Ω progressively approach the theoretical solution as p-level is increased. At p = 19 the computed solution is in excellent agreement with the theoretical solution. Expanded plots of the solution shown in Figs. 7.1(b) reveal ¯ e) that the solutions are free of spurious oscillations. The solutions in H 3,p (Ω space for various p-levels are shown in Figs. 7.2(a) and (b). We observe the solutions to be diffused for p = 5, but for p ≥ 7 the computed solutions are in good agreement with the theoretical solution. Expanded plots of φh versus x are shown in Fig. 7.2(b). We note that the solutions at all p-levels are free of oscillations. i ¯ e ) and H 3,p (Ω ¯ e) Results for Re = 102 ( ddxφih versus x, i = 0, . . . , 3) H 2,p (Ω spaces are shown in Figs. 7.3(a) and (b) and 7.4(a) and (b). We observe minor oscillations in the solution at lower p-levels (< 6) near x = 1.0 in both spaces, indicating inadequacy of the mesh in resolving solution gradients at these p-levels. As the p-level is increased the solution and its gradients converge to the theoretical values. Oscillations in the expanded views completely disappear at higher p-levels. The purpose of presenting these results is to demonstrate that even though the solution is quite diffused at Re = 102 , but the inadequacy of the mesh and p-levels may results in oscillatory solutions which are eliminated in this case only by p-level increase. On the other hand oscillatory solutions due to VIC integral form result due to degeneracy of the computational process.

435

7.2. ONE DIMENSIONAL BURGERS EQUATION

1

0.8

0.6 —

φ

H2,p(Ωe) Spaces (Re=106) p-convergence 0.4

p=3 p=5 k=6 k=7 k=9 k=19

0.2

0 0

0.2

0.4

0.6

0.8

1

x

(a) Function value over entire domain 1.015

1.01 —

H2,p(Ωe) Spaces (Re=106)

φ

p-convergence 1.005

p=9 p=11 k=13 k=19 theoretical

1

0.99995

0.99996

0.99997

0.99998

0.99999

1

x

(b) Function value: expanded view Figure 7.1: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 106 , k=2

¯ e ) and k, Dependence of the solution on order k of space H k,p (Ω pk-convergence In this section we present numerical studies to demonstrate that the order ¯ e ) is an independent parameter in all finite element of space k in H k,p (Ω computations in addition to h and p. Dependence of the solution φh on

436

NON-LINEAR DIFFERENTIAL OPERATORS

1

0.8

0.6 —

φ

H3,p(Ωe) Spaces (Re=106) p-convergence 0.4

p=5 p=7 k=8 k=9 k=19

0.2

0 0

0.2

0.4

0.6

0.8

1

0.99999

1

x

(a) Function value over entire domain 1.015

1.01 —

H3,p(Ωe) Spaces (Re=106)

φ

p-convergence 1.005

p=9 p=11 k=13 k=19 theoretical

1

0.99995

0.99996

0.99997

0.99998 x

(b) Function value: expanded view Figure 7.2: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 106 , k=3

k in the Galerkin processes has already been demonstrated in the results presented by Surana et al. [1]. LSP are based on minimization of error or residual functional I, hence examination of the behavior of I versus degrees of freedom is essential in determining its dependence on h, p, and k. We ¯ e ) ; k = 2, 3, . . . spaces and compute sequences of solutions consider H k,p (Ω

437

7.2. ONE DIMENSIONAL BURGERS EQUATION

1

0.8

0.6 —

φ

H2,p(Ωe) Spaces (Re=100) p-convergence 0.4

p=3 p=4 k=5 k=7 k=19

0.2

0 0

0.2

0.4

0.6

0.8

1

x

(a) Function value over entire domain 1.02 1 H

0.98

2,p

—

(Ωe) Spaces (Re=100)

p-convergence 0.96

p=3 p=4 k=5 k=7 k=19 theoretical

φ

0.94 0.92 0.9 0.88 0.86 0.8

0.85

0.9 x

0.95

1

(b) Function value: expanded view Figure 7.3: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 100, k=2

for progressively increasing p-levels beginning with the lowest admissible plevel and increasing it up to 19 in each space. We consider Re = 102 and Re = 106 and use the discretizations given earlier.

438

NON-LINEAR DIFFERENTIAL OPERATORS

1

0.8

0.6 —

φ

H3,p(Ωe) Spaces (Re=100) p-convergence 0.4

p=5 p=7 k=8 k=9 k=19

0.2

0 0

0.2

0.4

0.6

0.8

1

x

(a) Function value over entire domain 1.02 1 H

0.98

3,p

—

(Ωe) Spaces (Re=100)

p-convergence 0.96

p=5 p=6 k=7 k=9 k=19 theoretical

φ

0.94 0.92 0.9 0.88 0.86 0.8

0.85

0.9 x

0.95

1

(b) Function value: expanded view Figure 7.4: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 100, k=3

Behavior residual functional I versus dof Figures 7.7 and 7.8 show graph of residual functional I versus degrees of freedom for Re = 102 and Re = 106 , respectively. We make the following remarks.

439

7.2. ONE DIMENSIONAL BURGERS EQUATION

1e+00

1e-02

Residual Functional, I

1e-04

1e-06

—

Hk,p(Ωe) Spaces (Re=100) p-convergence

1e-08

k=2 k=3 k=4 k=5 k=6 k=7 k-convergence

1e-10

1e-12

1e-14 10

100 Degrees of Freedom

Figure 7.5: Residual functional I versus dof: Re = 100

1e+00

Residual Functional, I

1e-02

—

H (Ωe) Spaces (Re=106) k,p

1e-04

p-convergence k=2 k=3 k=4 k=5 k=6 k=7 k-convergence

1e-06

1e-08

1e-10 10

100 Degrees of Freedom

Figure 7.6: Residual functional I versus dof: Re = 106

(1) For a fixed k, the order of space and hence the global smoothness, the rate of convergence of I increases as p-level is increased. (2) As the order of the space k is increased from k to k + 1 the computed values of I decrease in the space of order k + 1 compared to the space

440

NON-LINEAR DIFFERENTIAL OPERATORS

1

0.8

φ

0.6

0.4

0.2

Re=10 Re=100 Re=1000

0 0

0.2

0.4

0.6

0.8

1

x

Figure 7.7: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 100

1

0.999

φ

0.998

0.997

0.996

0.995 0.999

Re=1045 Re=106 Re=10

0.9992

0.9994

0.9996

0.9998

1

x

Figure 7.8: LSP based on residual functional; 1D Burgers equation; φ versus x: Re = 106

of order k, regardless of the values of k, indicating improved solution in the space of order k + 1 compared to the space of order k. (3) While the rate of convergence of I (i.e., slope of I versus dof graph) is increased slightly with the change in the order of the space at lower p-

7.2. ONE DIMENSIONAL BURGERS EQUATION

441

levels, at higher p-levels (p ≥ 12), I versus dof curves are almost parallel to each other for all values of k, indicating same rate of convergence for all values of k. (4) For fixed p-level (i.e., p = constant) and for the fixed discretization (i.e. constant h) we could study the k-convergence of the residual functional I (lines with dots). Along these lines, h and p are fixed and only k, ¯ e ) is changing, hence these lines represent the order of the space H k,p (Ω k-convergence of the functional I for LSP. Dependence of I on the order of the space k, that is, degree of global smoothness is quite obvious. At lower p-levels (3 − 5), the dependence of I on k is not as strong as it is for p > 5. With increasing p-levels, the dependence of I on k becomes even stronger. This confirms that the order of the space ¯ e ) or in other words, the degree of global smoothness is k in H k,p (Ω undoubtedly an independent parameter in addition to h and p. We observe similar dependence of all quantities on k (not shown here). (5) pk-convergence of the error functional I (and likewise other quantities) is perhaps most illustrative of the influence of the order of space k on the convergence rate of I. pk-convergence can be viewed in at least two different ways: (a) for increasing k as well as increasing p but p = 2k − 1, i.e., minimum p-level for the order of the space k. In this case we observe much higher rate of convergence of pk-processes compared to ¯ e )) p-convergence of I in any of the spaces (i.e, any value of k in H k,p (Ω (b) Perhaps, the most dramatic is the pk-convergence of the residual functional I in which p and k both change in such a way that the total degrees of freedom for the discretization do not vary much. Almost vertical lines show such behaviors for various combination of p and k in which total dof do not change significantly but I decreases substantially. We observe exceptionally high slopes of such lines, even for very low total dofs. With increasing total dofs, slopes of these lines increase showing an increase in the convergence rate of I. At close to 100 dofs, the pk-convergence graphs of I are almost vertical straight lines, indicating exponential convergence rate. It is worth noting that such behavior is the consequence of the fact that for fixed dofs, an increase in k would permit an increase in p-level, which in fact is responsible for the improvement in the value of I. (6) We observe that I continues to decrease in progressively higher order spaces for a given dofs, indicating improved performance of the LSP in higher order spaces. That is, the best approximation property of the LSP in E-norm gets progressively better in higher order spaces.

442

NON-LINEAR DIFFERENTIAL OPERATORS

7.3 Fully developed flow of Giesekus fluid between parallel plates (polymer flow) In this model problem we consider fully developed flow of an incompressible Giesekus fluid [4] between parallel plates. Figure 7.9 shows a schematic ˆ The origin of the flow. The plates are separated by a distance of 2H. of the xy-coordinate is located at the center of the plates and the positive x-direction is the direction of the flow. The flow is pressure driven i.e. ∂p ∂x (negative) is specified. We assume the fluid to be incompressible. We consider contravariant Cauchy stress tensor and Almansi strain tensors as conjugate measures of the stress and strain tensors in Eulerian description. This yields upper convected Giesekus constitutive model. The mathematical model describing the flow physics (for incompressible case with isothermal flow assumption) consists of x-momentum equation and the constitutive equations. The continuity equation in this case is satisfied identically. If we decompose the contravariant Cauchy stress tensor in equilibrium stress and deviatoric contravariant Cauchy stress tensor, then the equilibrium stress is mechanical pressure p and the deviatoric contravariant Cauchy stress tensor becomes a dependent variable in the constitutive theory.

y B BCs at B :

u=0, p=0 H Flow direction

x

A center line

BCs at A:

∂u =0 ∂y τxy = 0 p τxy

=0

(UCG1) (UCG2)

Figure 7.9: Schematic of 1D fully developed flow between parallel plates (half domain)

We begin with all quantities with their usual dimensions (units) in the development of the mathematical model and then non-dimensionalize them using the following. The quantities with the subscript zero are the reference

7.3. FLOW OF GIESEKUS FLUID BETWEEN PARALLEL PLATES

443

quantities. ηˆp ηˆs u ˆ x ˆ ηˆ ηˆs , ρ= , u= axial velocity , η=η= , ηs = , ηp = L0 ) η0 ) η0 η0 ρ0 u0   ρ0 y02 ; Characteristic kinetic energy  pˆ τˆ p= , τ= and p0 = τ0 = or  p0 τ0  µ0 u0 , Characteristic kinetic stress L0 (7.91) x=

We choose the large of the two for p0 (and τ0 ). This results in dimensionless form of the mathematical model given in the following: Momentum equations: In the absence of body forces  p  ∂p

 τ  ∂τ xy 0 =0 ρ0 u20 ∂x ρ0 u20 ∂y  p  ∂p  τ  ∂τ yy 0 0 − =0 ρ0 u20 ∂y ρ0 u20 ∂y 0

−

(7.92) (7.93)

Giesekus constitutive model: We consider the upper convected Giesekus constitutive model derived [4] in deviatoric Cauchy stress tensor τ . In this model, the first convected time derivative of τ , the deviatoric contravariant Cauchy stress tensor, is a dependent variable in the constitutive theory. Dimensionless form of the constitutive model is given by
∂u De  L0 τ0  2 τxx − 2De τxy −α (τxx + (τxy )2 = 0 ∂y η u0 η0
De  L0 τ0  2 τyy − α (τyy + (τxy )2 = 0 (7.94) η u0 η0 ∂u De  L0 τ0  ∂u L0 τ0 τxy − 2De τyy −α τxy (τxx + τyy ) = η τxy ∂y η u0 η0 u0 η0 ∂y Equations (7.92)–(7.94) constitute the complete mathematical model in dependent variables u, p, τxx , τyy and τxy for fully developed flow between ∂p is specified parallel plates. The flow is assumed to be pressure driven i.e. ∂x ∂p as input data. Since p0 = τ0 and ∂x is constant, from (7.92) we can determine τxy by integrating (7.92) with respect to y and using the BC: τxy = 0 at y = 0 (due to symmetry).  ∂p  τxy = y (7.95) ∂x

444

NON-LINEAR DIFFERENTIAL OPERATORS

A theoretical solution for the remaining dependent variables is not possible due to complexity of the constitutive model. We consider LSP for (7.92) ¯ e ) spaces; k ≥ 2, p ≥ 2k − 1. Theoretical solution for τxy (7.94) in H k,p (Ω given by (7.95) will serve as one of the checks on the validity and accuracy of computations. We consider equal order, equal degree local approximations for all dependent variables: u, p, τxx , τyy , τxy . With the choice p0 = τ0 , the quantities in the brackets in (7.92) and (7.93) can be factored and eliminated, but we do not do so in the followingSin order to maintain generality. ¯ e of Ω ¯ = (0, 1). Let ue , pe , (τyy )e ¯T = Ω Consider a discretization Ω h h h e

and (τxy )eh be local approximations of u, p, τxx , τyy and τxy . By substituting the local approximations in (7.92) - (7.94) and using the notations c1 =

p0 τ0 L0 τ0 1 u0 η0 , c2 = , c1 = and = 2 2 u0 η0 c3 L0 τ0 ρ0 u0 ρ0 u0

¯ e. we obtain the residual equations for an element e with domain Ω ∂(τxy )eh ∂peh − c2 ∂x ∂y e e ∂(τ ∂p yy )h E2e = c1 h − c2 ∂y ∂y E1e = c1


2
2  ∂ue De  E3e = (τxx )eh − 2De(τxy )eh h − α c3 (τxx eh + (τxy (7.96) ∂y η
2
2  De  E4e = (τyy )eh − α c3 (τyy )eh + (τxy )eh η   η ∂ue ∂ue De h c3 (τxy )eh (τxx )eh + (τyy )eh − E5e = (τxy )eh − 2De(τyy )eh h − α ∂y η c3 ∂y The residual functional I e for an element e is given by e

I =

5 X

(Eie , Eie )Ω¯ e

(7.97)

i=1

¯ T can be written as The functional I for the discretization Ω 5  X XX e I= I = (Eie , Eie )Ω¯ e e

e

(7.98)

i=1

Therefore δI =

X

δI ≈

X

δI e = 2

e

e

5 XX e

δI e = 2

i=1

(7.99)

e

i=1

5 XX e

 X (Eie , δEie )Ω¯ e = 2 {g e } = 2{g} = 0 

(δEie , δEie )Ω¯ e = 2δ{g} > 0

(7.100)

7.3. FLOW OF GIESEKUS FLUID BETWEEN PARALLEL PLATES

445

and  {∆δ} = −[δ 2 I]−1 {δ0 } g({δ0 })

(7.101)

We define X

[K e ]

(7.102)

(δEie , δEie )Ω¯ e

(7.103)

[K] = [δ 2 I] =

e

in which [K e ] =

5 X i=1

Computational details ¯ ξ = [−1, 1] be the map of Ω ¯ e in the natural coordinate space ξ and Let Ω let ueh

=

n X

Ni uei = [N ]{ue }

i=1

peh =

n X

Ni pei = [N ]{pe }

i=1

(τxx )eh = (τyy )eh =

n X i=1 n X

Ni (τxx )ei = [N ]{(τxx )e }

(7.104)

Ni (τyy )ei = [N ]{(τyy )e }

i=1

(τxy )eh =

n X

Ni (τxy )ei = [N ]{(τxy )e }

i=1

in which Ni = Ni (ξ); i = 1, 2, . . . , n. Substituting (7.104) in the residual ∂p equations and noting that ∂x is given i.e. known, we obtain h ∂N i  ∂p − c2 (τxy )e ∂x ∂y h ∂N i  h ∂N i  E2e = c1 {pe } − c2 (τyy )e ∂y ∂y      h ∂N i e  E3e = [N ] (τxx )e − 2De [N ] (τxy )e {u } ∂y 2   2  De   −α c3 [N ] (τxx )e + [N ] (τxy )e η (7.105) E1e = c1

446

NON-LINEAR DIFFERENTIAL OPERATORS

   2   2  De   E4e = [N ] (τyy )e − α c3 [N ] (τyy )e + [N ] (τxy )e η        e E5e = [N ] (τxy ) − 2De [N ] (τyy )e [N ]{ue }    η  De   −α [N ]{ue } c3 [N ] (τxx )e + [N ] (τyy )e − η c3 Let   T  T  T  {δ e }T = {ue }T , {pe }T (τxx )e (τyy )e (τxy )e

(7.106)

be the nodal dofs for an element e, then  "     T  ∂Eie T ∂Eie T ∂Eie ∂Eie e T , , , {δEi } = = ∂{δ e } ∂{ue } ∂{pe } ∂{(τxx )e }  T  T # ∂Eie ∂Eie , , i = 1, 2, . . . , 5 (7.107) ∂{(τyy )e } ∂{(τxy )e } Using (7.105)–(7.107) we can obtain    h ∂N i ∂E1e T T T T = {0} , {0} , {0} , {0} , −c2 ∂{δ e } ∂y     h i h h ∂N i e ∂E2 ∂N ∂N i T T T = {0} , c1 , {0} , −c2 , −c2 , {0} ∂{δ e } ∂y ∂y ∂y    h i ∂E3e e ∂N = −2De(τ ) , {0}T , xy h ∂{δ e } ∂y  αDe 2αDe e e e T (τxx )h [N ] − 2De uh [N ] − 2 c3 (τxy )h [N ], {0} [N ] − η α     ∂E4e 2αDe αDe e e T T T = {0} , {0} , {0} , − c (τ ) [N ], [N ]−2 c (τ ) [N ] 3 yy h 3 xy h ∂{δ e } η α    αDe ∂E5e η = −2De(τyy )eh [N ] − [N ], {0}T , − c3 (τxy )eh [N ], e ∂{δ } c3 η 
αDe αDe − 2De ueh [N ] − c3 (τxy )eh [N ], [N ] − c3 (τxx )eh + (τxx )eh [N ] α η (7.108) Therefore [K e ] =

Z

{δEie }{δEie }T dΩ

(7.109)

¯e Ω

Numerical values of the coefficients of [K e ] are calculated using Gauss quadrature with {δ0 }, a known starting solution which i assumed zero in the following.

447

7.3. FLOW OF GIESEKUS FLUID BETWEEN PARALLEL PLATES

Numerical studies We consider fully developed flow of an incompressible Giesekus fluid [4] with the following material coefficients that are constant. ρˆ = 800

kg ˆ = 0.006 s, α = 0.15 , ηˆs = 0.002 Pa.s, ηˆp = 1.426 Pa.s, λ m3

We choose ˆ = L0 = 0.003175 m, ρ0 = ρˆ = 800 kg , η0 = ηˆ = 1.426 Pa.s H m3 which gives H = L0 = 1, p0 = τ0 = ρu2o = 800u2o , Re = De =

ρL0 u0 = 1.7812u0 , η0

ˆ 0 λu = 18.89764u0 L0

and we also choose u0 = 0.5 m/sec for which Re = 0.8906 and De = 9.45. A good discretization of the spatial domain 0 ≤ y ≤ 1 is important in ensuring satisfactory convergence of the Newton’s linear method for the system of non-linear algebraic equations and good accuracy of the computed solutions. ∂p With progressively increasing ∂x , we expect a constant (approximately) velocity core or plug at the center of the flow. This suggests a highly biased finer discretization towards the walls. A four-element graded mesh with element length of 0.1, 0.15, 0.225, and 0.525 starting from the wall is used in the present study. The local approximations are p-version (3-node elements) in higher order spaces. Initial p-convergence studies with this discretization ¯ ex ) to be suffisuggest p = 11 and k = 2 (local approximations of class C 1 (Ω cient for good accuracy of results. For this choice of mesh, p-level and order o the space (k = 2), the residual or least squares functional values remain O(10−9 ) to O(10−31 ) indicating that the PDEs are satisfied accurately in the pointwise sense as the integrals are Riemann when the local approxima¯ e ). Newton’s linear method tions for u, p, τxx , τyy , and τxy are of class C 1 (Ω used for solving the non-linear algebraic equations converges in less that 10 iterations for all results presented in this section. ∂p In numerical studies we begin with ∂x = −0.1 for which a converged ∂p solution is obtained and then progressively increase it up to ∂x = −0.275 ∂p using a continuation procedure in which converged solutions at lower ∂x are used as initial (or starting) solution in the Newton’s linear method. ∂p Figure 7.10 shows graphs of velocity u versus y for different values of ∂x . ∂p ∂u Graphs of velocity gradient ∂y versus y for different values of ∂x are shown in Fig. 7.11. As expected with progressively increasing

∂p ∂x

(corresponding

448

NON-LINEAR DIFFERENTIAL OPERATORS

2 —

C1(Ωex) ; p=9

∂p/∂x values:

Distance, y

1.8

-0.1 -0.2 -0.25 -0.265 -0.275

1.6

1.4

1.2

1 0 0.25 0.5 0.75 1 1.25 1.5 1.75 2 2.25 2.5 2.75 3 3.25 3.5 3.75 4 Velocity, u

Figure 7.10: Velocity u versus distance y

2

1.8 —

Distance, y

C1(Ωex) ; p=9 1.6

1.4

∂p/∂x values: -0.1 -0.2 -0.25 -0.265 -0.275

1.2

1 -30 -28 -26 -24 -22 -20 -18 -16 -14 -12 -10 Velocity gradient, du/dy

Figure 7.11: Velocity gradient

du dy

-8

-6

-4

-2

0

versus distance y

to increasing flow rate) almost a constant velocity core is observed in the ¯ e ) local center of the flow. I values of O(10−9 ) or lower, and use of C 1 (Ω approximations ensure that the computed solutions satisfy GDEs accurately in the pointwise sense. Plots of τxx , τyy and τxy versus y for different values ∂p of ∂x are shown in Figs. 7.12 to 7.14. Computed τyy is in perfect agreement

449

7.3. FLOW OF GIESEKUS FLUID BETWEEN PARALLEL PLATES

with the theoretical solution (7.95). This model problem demonstrates the significance of VC integral forms in which regardless of the complexity of the mathematical model, the computations can be performed in a routine manner with good convergence characteristics and good accuracy. 2

1.8 —

Distance, y

C1(Ωex) ; p=9

∂p/∂x values:

1.6

-0.1 -0.2 -0.25 -0.265 -0.275

1.4

1.2

1 0

1

2

3

4

5 Stress, τxx

6

7

8

9

10

Figure 7.12: Stress component τxx versus distance y

2

1.8 1

—

Distance, y

C (Ωex) ; p=9 1.6

1.4

∂p/∂x values: -0.1 -0.2 -0.25 -0.265 -0.275

1.2

1 -0.12 -0.11 -0.1 -0.09 -0.08 -0.07 -0.06 -0.05 -0.04 -0.03 -0.02 -0.01 Stress, τyy

Figure 7.13: Stress component τyy versus distance y

0

450

NON-LINEAR DIFFERENTIAL OPERATORS

2

1.8 —

Distance, y

C1(Ωex) ; p=9

∂p/∂x values:

1.6

-0.1 -0.2 -0.25 -0.265 -0.275

1.4

1.2

1 -0.4

-0.35

-0.3

-0.25

-0.2 -0.15 Stress, τxy

-0.1

-0.05

0

Figure 7.14: Stress component τxy versus distance y

7.4 2D steady-state Navier–Stokes equations In this section we consider example problems that require numerical simulation of two-dimensional Navier-Stokes equations for isothermal, incompressible, Newtonian fluids. The dimensionless form of the two-dimensional steady state Navier–Stokes equations in xy frame (Eulerian description) consisting of continuity equation, momentum equations and the constitutive equations for isothermal, incompressible Newtonian fluids are given by (in the absence of sources and sinks) [5]   ∂u ∂v ρ + =0 ∂x ∂y     ∂u ∂u p0 ∂p τ0 ∂τxx ∂τxy ρ u +v + − + =0 (7.110) ∂x ∂y ∂x ∂y ρ0 u20 ∂x ρ0 u20     ∂τxy ∂τyy ∂v ∂v p0 ∂p τ0 ρ u +v + − + =0 2 2 ∂x ∂y ∂x ∂y ρ0 u0 ∂y ρ0 u0 where 

 η0 u0 ∂u τxx = 2η L0 τ0 ∂x   η0 u0 ∂v 2η τyy = L0 τ0 ∂y     η0 u0 ∂u ∂v τxy = η + L0 τ0 ∂y ∂x

∀x, y ∈ Ω ⊂ R2

(7.111)

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

451

in which u, v are velocity in x and y directions, p is mechanical pressure, τxx , τyy , τxy are components of deviatoric Cauchy stress tensor, ρ is density and η is viscosity (assumed constant), all are dimensionless including x, y. We have used x=

τˆij x ˆ yˆ ρˆ ηˆ pˆ , y= , ρ = , η = , τij = , p= L0 L0 ρ0 η0 τ0 p0

(7.112)

in which all quantities with ˆ (hat) have their usual dimensions and the quantities with subscript zero are reference quantities. We note that pˆ and τˆij both have units of LF2 (F is force and L is length), hence τ0 and p0 must not be chosen independent of each other. In fact, once we choose F0 and F0 L0 , τ0 and p0 both may be defined as L 2 . We use the following: τ0 = p0 = 0

η0 u0 2 0 max(ρ0 u20 , ηu L0 ). ρu0 is characteristic kinetic energy and L0 is characteristic viscous stress. For inertia dominated flows, generally ρ0 u20 > ηL0 u00 will hold. We note that when

τ0 p0 η0 u0 1 = 2 = 1 and = 2 L0 τ0 Re ρ0 u0 ρu0 η0 u0 τ0 p0 1 η0 u0 (2) p0 = τ0 = , = 2 = and =1 L0 Re L0 τ0 ρ0 u20 ρu0 (1) p0 = τ0 = ρ0 u20 ,

(7.113)

where Re is Reynolds number. Remarks (1) Equations (7.111) is a first order system of non-linear PDEs in dependent variables u, v, p, τxx τyy , τxy that naturally result in this form directly from the conservation laws and the constitutive theory. Equations (7.111) is a system of first order non-linear PDEs in u, v, p, τxx τyy , τxy . (2) In this particular case it is possible to substitute in stresses in the momentum equations. The resulting terms can be simplified using the following relations that can be obtained by differentiating the continuity equation with respect to x and y. ∂2v ∂2u =− 2 ∂x∂y ∂x 2 ∂ u ∂2v =− 2 ∂y∂x ∂y

(7.114)

The resulting continuity (unchanged) and momentum equations can be written as

452

NON-LINEAR DIFFERENTIAL OPERATORS

 ∂u ∂v =0 ρ + ∂x ∂y       ∂u ∂u p0 ∂p  η  ∂ 2 u ∂ 2 u ρ u + + − + 2 =0 ∂x ∂y Re ∂x2 ∂y ρ0 u20 ∂x      2    ∂v p0 ∂p ∂ v ∂v η ∂2v ρ u + =0 + − + ∂x ∂y Re ∂x2 ∂y 2 ρ0 u20 ∂y 

(7.115)

for all x, y ∈ Ω ⊂ R2 . Equations (7.115) are also a system of non-linear partial differential equations that contain first order derivatives of the mechanical pressure but up to second order derivatives of the velocities u and v. (3) Since both systems of PDEs ((7.111) and (7.115)) are systems of nonlinear PDEs, we only consider least squares finite element processes for both them as only the LSP in this case yields VIC integral forms.

7.4.1 LSP based on residual functional: first order system of PDEs of u, v, Let ueh , vhe , peh , (τxx )eh , (τyy )eh , (τxy )eh be the local approximations ¯T = S Ω ¯ e of the ¯ e of the discretization Ω p, τxx , τyy , τxy over an element Ω e ¯ ⊂ R2 . For generality, we consider unequal order unequal degree domain Ω local approximations. u

ueh

=

n X

Niu uei = [N u ] {ue }

i=1 v

vhe

=

peh = (τxx )eh = (τyy )eh = (τxy )eh =

n X i=1 np X i=1 τxx n X i=1 τyy n X i=1 τxy n X i=1

Niu vie = [N v ] {v e } Nip pei = [N p ] {pe } (7.116) Niτxx (τxx )ei = [N τxx ] {(τxx )e } τ

Ni yy (τyy )ei = [N τyy ] {(τyy )e } τ

Ni xy (τxy )ei = [N τxy ] {(τxy )e }

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

453

By replacing u, v, p, τxx , τyy , and τxy with their local approximation in (7.111) we obtain the residual equations.  e  ∂uh ∂vhe e E1 = ρ + ∂x ∂y   e e e ∂(τxx )eh ∂(τxy )h p0 τ0 e ∂uh e e ∂uh + − + vh + E2 = ρ uh ∂x ∂y ∂x ∂y ρ0 u20 ρ0 u20     e e e ∂(τxy )h ∂(τyy )eh p0 τ0 e e ∂vh e ∂vh E3 = ρ uh + − + vh + ∂x ∂y ∂x ∂y ρ0 u20 ρ0 u20 (7.117)   e ∂u η u 0 0 e e h 2η E4 = (τxx )h − L0 τ0 ∂x   ∂v e η0 u0 E5e = (τyy )eh − 2η h L0 τ0 ∂y     e ∂uh ∂vhe η0 u0 E6e = (τxy )eh − + η L0 τ0 ∂y ∂x Upon substituting for local approximations from (7.116) in (7.117) we can obtain explicit forms of the residual equations. Let us define nodal degrees of freedom {δ e } for an element e as   {δ e }T = {ue }T , {v e }T , {pe }T , {(τxx )e }T , {(τyy )e }T , {(τxy )e }T (7.118) Then {δEie }T

 =

    T ∂Eie T ∂Eie T ∂Eie , , , , ∂{v e } ∂{pe } ∂{(τxx )e } T  T  ∂Eie ∂Eie , , i = 1, 2, . . . , 6 (7.119) ∂{(τyy )e } ∂{(τxy )e }

∂Eie ∂{ue } 

T



and the element matrix [K e ] is given by   6 Z  X ∂Eie ∂Eie T [K ] = dΩ ∂{δ e } ∂{δ e } e

(7.120)

i=1 ¯ e Ω

Remaining details follow the standard procedure presented earlier in connection with other model problems and hence are omitted here. In this case least admissible order k of the approximation space is 2 if the integrals over ¯ T are to be in Riemann sense. However, k = 1 i.e. local approximaΩ ¯ e ) for all dependent variables are permissible if we accept tion of class C 0 (Ω ¯ T . In applications generally local apthe integrals in Lebesgue sense over Ω 0 e ¯ proximations of class C (Ω ) with same p-levels are used for all dependent variables.

454

NON-LINEAR DIFFERENTIAL OPERATORS

7.4.2 LSP based on residual functional: higher order systems of PDEs Let ueh , vhe , peh be the local approximations of u, v, and p over an element ¯T = S Ω ¯ e of Ω ¯ ⊂ R2 . In this case also we consider of the discretization Ω e unequal order, unequal degree local approximations for u, u and p.

¯e Ω

u

ueh

=

vhe = peh =

n X i=1 nv X i=1 np X

Niu uei = [N u ] {ue } Niu vie = [N v ] {v e }

(7.121)

Nip pei = [N p ] {pe }

i=1

By replacing u, v and p with their approximations in (7.115) we obtain residual equations   e ∂uh ∂vhe e + E1 = ρ ∂x ∂y    2 e e e ∂ uh ∂ 2 ueh p0 ∂peh τ0 e e ∂uh e ∂uh E2 = ρ uh + vh + − + (7.122) ∂x ∂y ρ0 u0 ∂x L0 ρ0 u0 ∂x2 ∂y 2  2 e   e e ∂ vh ∂ 2 vhe p0 ∂peh τ0 e ∂vh e e ∂vh + vh + − + E3 = ρ uh ∂x ∂y ρ0 u0 ∂y L0 ρ0 u0 ∂x2 ∂y 2 for all x, y ∈ Ω ⊂ R2 . Upon substituting for local approximation from (7.121) in (7.122) we can obtain explicit forms of the residual equations. Let us define nodal degrees of freedom {δ e } for an element e as   {δ e }T = {ue }T , {v e }T , {pe }T (7.123) Then {δEie }T =

"

∂Eie ∂{ue }

T     # ∂Eie T ∂Eie T , , , i = 1, 2, 3 ∂{v e } ∂{pe }

(7.124)

and the element coefficient matrix [K e ] is given by   3 Z  X ∂Eie ∂Eie T [K ] = dΩ ∂{δ e } ∂{δ e } e

(7.125)

i=1 ¯ e Ω

Remaining details follow the procedure presented in earlier examples. In this case k − 2 and k = 3 are minimum orders of approximation spaces for

455

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

peh and ueh , vhe for k ≥ 3 for all dependent variables would be admissible as well. Generally in applications in which the solutions are smooth k = 2 may ¯ T are in Lebesgue sense. suffice. For this choice of k the integrals over Ω In the following we present three numerical examples using the LS formulations presented in Sections 7.4.1 and 7.4.2. We consider numerical studies for the following model BVPs using Navier Stokes equations for incompressible isothermal flows. (a) (b) (c) (d)

Section 7.4.3: Section 7.4.4: Section 7.4.5: Section 7.4.6:

slider bearing, flow of a viscous lubricant a square lid-driven cavity asymmetric backward facing step flow past a circular cylinder

7.4.3 Slider bearing; flow of a viscous lubricant The problem has been studies in references [6, 7]. In reference [7], an approximated mathematical model for which the theoretical solution as well as finite element solutions are presented. Figure 7.15 shows a schematic of the slider bearing. The bearing consists of a sliding pad moving at a velocity u0 relative to the stationary pad inclined at a small angle to the moving pad. The page between the pads is filled with lubricant. The two open ends are assumed atmospheric pressure p0 (we assume this to be zero in the numerical studies). We consider the following properties of the fluid. ηˆ = 8 × 10−4 lb ft sec ft−2 , ρˆ = 50.0

lb ft3

and choose the following reference quantities lb , η0 = ηˆ = 8 × 10−4 lb ft sec ft−2 , ft3 ft L0 = 0.001 ft, v0 = 1.0 sec ρ0 = ρˆ = 50.0

With these choices of reference quantities η=

ˆ1 ˆ2 ˆ ηˆ h h L 0.36 = 1, h1 = = 0.8, h2 = = 0.4, L = = = 360 η0 L0 L0 L0 0.001

We choose ρˆ0 = τ0 = ρ0 v02 = 50 based on characteristic kinetic energy (CKE). With this choice p0 η0 v0 1 τ0 η 0 v0 η0 1 = = 1 and = = = = 2 2 2 L0 τ0 L0 ρ0 v0 Re 62.5 ρ0 v 0 ρ0 v 0 L0 ρ0 v0

456

NON-LINEAR DIFFERENTIAL OPERATORS

yˆ

uˆ = 0, vˆ = 0 pˆ = pˆ0 = 0 ˆ1 h

ˆ x) h(ˆ

ˆ2 h xˆ

uˆ = uˆ0 , vˆ = 0

pˆ = pˆ0 = 0

ˆ L ˆ = 0.36 ft, L

ˆ 1 = 8 × 10−4 ft, h

ˆ2 = h

ˆ1 h 2

= 4 × 10−4 ft,

uˆ0 = 30

ft sec

Figure 7.15: Schematic of slider bearing

Approximate theoretical solution Following references [6, 7], the flow can be approximated by (using the non-dimensionality form of the Navier–Stokes equations) ∂p 1 ∂2u η = Re ∂y 2 ∂x

(7.126)

where h − h  ∂p 1 6η u0  2h1 h2 H 2 1 = , h(x) = h1 + x, H = 1− (7.127) ˆ2 ∂x Re h h L h1 + h2 and  ∂p y   y h2 1− u = u0 − Re 2η ∂x h h 1 6η u0 L(H − 1 − h)(h − h2 ) p(ˆ x) = Re h2 (h21 − h22 ) η ∂u ∂p  h  η u0 τxy = = y− − Re ∂y ∂x 2 Re h

(7.128) (7.129) (7.130)

Finite element solution We present finite element solution of the Navier–Stokes equations using least squares finite element formulation presented in section 7.4.1 for a system of first order equations. Figure 7.16 shows a 5 × 3 uniform discretization of the dimensionless domain using nine-node p-version two-dimensional elements (see chapter 8 for details). We consider local approximations of ¯ e ) with equal degree interpolations. Even though the integrals class C 0 (Ω

457

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

for this choice of approximation space are in Lebesgue sense but due to the smoothness of the solution, we expect accurate computed solutions upon ¯ e ) equal degree interconvergence. Computations are performed using C 0 (Ω polations with p-level of 1, 2, 3, . . .. y

u = 0, v = 0 p=0 0.8 0.4 x p=0

u0 = 30, v = 0 360

Figure 7.16: Computational domain and 5 × 3 uniform mesh of nine-node p-version elements

Figure 7.17 shows a plot of the square root of the residual functional I versus degrees of freedom for progressively increasing p-levels. With increasing p-levels progressively reduce values of I confirm improved accuracy of the solution. Figure 7.18 shows plots of pressure p versus x at y = 0 (at the moving plate) for various p-levels and comparison with the theoretical solution of (7.126) shows similar plots for shear stress τxy at y = 0. We note that the theoretical solution of the approximate mathematical model underestimates pressure distribution but overestimates the shear stress when compared with the converged solution of the actual Navier–Stokes equations without approximation presented here.

7.4.4 A square lid-driven cavity In this model problem we consider a dimensionless unit square lid-driven cavity, a schematic of which is shown in Figure 7.20(a). The boundary conditions for dimensionless velocities u, v and pressure p are also shown in Fig. 7.20(a). Figures 7.20(b) and (c) two graded finite element discretizations using 36 nine-node p-version finite elements. The main difference in the two discretizations is the size of the elements adjacent to the four boundaries of the cavity. In discretization A the elements are 0.1 units where in discretization B they are 0.05 units. The physical size of the cavity is 3 cm × 3 cm and the fluid properties used are: ρˆ = 998.2 kg/m3 , ηˆ = 1.002 × 10−3 P a

458

NON-LINEAR DIFFERENTIAL OPERATORS

100

Square root of residual functional, √I

p=1

10-1 0

—e

C (Ω ) p=3 -2

10

p=5 10-3

p=9 p=7

10-4 100

1000 Degrees of freedom

10000

Figure 7.17: Square root of the residual functional versus dofs 300 p=1 p=2 p=3 p=4,5,...,9 Approx. Theoretical

280 260 240

—

C0(Ωe) ; LSP

220

Pressure, p

200 180 160 140 120 100 80 60 40 20 0 0

40

80

120

160

200

240

280

320

Distance, x

Figure 7.18: Pressure solution along the moving plate (y = 0)

We consider the following reference values: ρ0 = ρˆ =998.2 kg/m3 , η0 = ηˆ = 1.002 × 10−3 P a, L0 = 0.03 m, υ0 = 0.03346 m/s

360

459

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

0 —

C0(Ωe) ; LSP -0.2

p=1 p=2 p=3 p=4,5,...,9 Approx. Theoretical

Stress, τxy

-0.4

-0.6

-0.8

-1

-1.2

-1.4 0

40

80

120

160 200 Distance, x

240

280

320

360

Figure 7.19: Shear stress distribution along the moving plate (y = 0) y A

B

u=1

p=0

v=0

y

C D

u=0 v=0

u=0 v=0

u=0 v=0

x

(a) Schematic

y

hd

0.15 0.15

0.05 0.15

0.30

0.30

0.30

0.30

0.15 0.15

x

hd

0.15 0.05

x

(c) Mesh B

(b) Mesh A

Figure 7.20: Driven cavity problem schematic and the finite element discretizations

With this choice of reference values, we have ρ = 1, η = 1, Re =

ρ0 L0 υ0 = 1000 η0

The velocity of the lid is assumed to vary from zero at the vertical walls to one in a continuous and differentiable manner over a length of hd representing the characteristic lengths of the elements at the top two corners. Using the conditions ∂u u = 0, = 0 at xA = 0 and x = xB = hd ∂x we can obtain a cubic distribution of u over 0 ≤ x ≤ hd that is continuous and differentiable. Likewise, using u = 0,

∂u = 0 at x = xC = 1 − hd and x = xD = 1 ∂x

460

NON-LINEAR DIFFERENTIAL OPERATORS

we can obtain another cubic distribution of u over 1 − hd ≤ x ≤ 1 that is continuous and differentiable. In the computations of the numerical solutions we consider the mathematical model consisting of a first order system of PDEs (7.111) as well as the higher order system of equations (7.115). We consider local approxi¯ e ) and C 1 (Ω ¯ e ) with equal degree interpolations for mations of classes C 0 (Ω all variables. p-levels in ξ and η directions are chosen same (pξ = pη = p) and are uniformly increased from 3 to 9. Numerical solutions are computed for both mesh A and mesh B. Newton’s linear method with line search is considered converged when |gi | ≤ ∆ ; i = √ 1, 2, . . . in which ∆ is a present tolerance. Figure 7.21 shows graphs of I versus degrees of freedom for both mathematical models with mesh A and mesh B when τ0 = p0 = ρ0 υ02 (CKE) is used to non-dimensionalize pressure and stresses. Similar graphs for po = τ0 = η0 υ0 /L0 (CVS) are shown in Fig. 7.22. From Figs. 7.21 and 7.22 we note that C 1 solutions yield lower values of I compared to C√0 solutions for a given dofs confirming better accuracy of C 1 solutions. I in Fig. 7.21 is lower than in Fig. 7.22 for the same degrees of freedom suggesting that CKE approach is superior to CVS approach for non-dimensionalizing √ pressure and stresses. Mesh B yields lower I compared to mesh A (for the same dofs), hence more accurate computed solutions.

Square root of residual functional, √I

100

—

Mesh A (Higher order PDEs) : C11(Ω—ee) Mesh B (Higher order PDEs) : C0(Ω—e) Mesh A : C0(Ω— ) p=3 Mesh B : C (Ωe)

10-1 p=3 10-2

p=9

10-3 p=9 10-4 100

1000 10000 Degrees of freedom

100000

Figure 7.21: Square root of the residual functional versus degrees of freedom (p0 = τ0 = ρ0 υ02 ); CKE

√ Convergence rates of I vs dof √ for solutions of class C 0 and C 1 are comparable. From these graphs of I in Figs. 7.21 and 7.22, the most

461

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

Square root of residual functional, √I

10

0 1

—

Mesh A (Higher order PDEs) : C1(Ω—e) Mesh B (Higher order PDEs) : C0(Ω—e) Mesh A : C (Ω—e) Mesh B : C0(Ωe) p=3 p=3

10-1

10-2

p=9

p=9

10-3 100

1000 10000 Degrees of freedom

100000

Figure 7.22: Square root of the residual functional versus degrees of freedom 0 υ0 (p0 = τ0 = ηL ); CVS 0

accurate results are obtained using mesh B for the mathematical model consisting of the first order system of PDEs with local approximation of class ¯ e ) and using CKE to non-dimensionalize pressure and stresses, hence C 0 (Ω is used here to present the computed results. These are also compared with mesh A. Figures 7.23 and 7.24 have plots of velocities u and v versus distance at vertical and horizontal centerlines of the cavity for different p-levels using mesh A. Similar plots for mesh B are shown in Figs. 7.25 and 7.26. At lower plevels (p = 3), computed solutions using meshes A and B differ as expected due to coarse meshes and low p-levels, but beyond p = 3 both meshes yield almost same results. At p = 5 the computed solutions are almost converged. The solutions for these meshes are also reported in references [8, 9] and are compared with the solutions of reference [10] with good agreement.

7.4.5 Asymmetric backward facing step In this model problem we consider 2:3 asymmetric backward facing step shown in Fig. 7.27(a) in dimensionless form. The fluid is water with the following properties: ρˆ = 998.2

kg , ηˆ = 1.002 × 10−3 P a m3

462

NON-LINEAR DIFFERENTIAL OPERATORS

1

Distance, y

0.75

0.5

—

Mesh A : C0(Ωe) p=3 p=5 p=7 p=9

0.25

0 -0.5

0 Velocity, u at x=0.5

0.5

1

¯ e) Figure 7.23: Velocity u at x = 0.5 for mesh A using local approximation of class C 0 (Ω (CKE as reference pressure and stress) 0.75

Velocity, v at y=0.5

0.5

0.25

0 —

Mesh A : C0(Ωe) p=3 p=5 p=7 p=9

-0.25

-0.5

-0.75 0

0.25

0.5

0.75

1

Distance, x

¯ e) Figure 7.24: Velocity v at y = 0.5 for mesh A using local approximation of class C 0 (Ω (CKE as reference pressure and stress)

We choose the following reference values: ρ0 = ρˆ = 998.2

kg , η0 = ηˆ = 1.002 × 10−3 P a, L0 = 0.015 m m3

Then ρ = 1, n = 1, Re =

ρ0 L0 u0 = 14943.1137 u0 η0

463

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

1

Distance, y

0.75

0.5

—

Mesh B : C0(Ωe) p=3 p=5 p=7 p=9

0.25

0 -0.5

0 Velocity, u at x=0.5

0.5

1

¯ e) Figure 7.25: Velocity u at x = 0.5 for mesh A using local approximation of class C 0 (Ω (CKE as reference pressure and stress)

0.75

Velocity, v at y=0.5

0.5

0.25

0 —

Mesh B : C0(Ωe) p=3 p=5 p=7 p=9

-0.25

-0.5

-0.75 0

0.25

0.5

0.75

1

Distance, x

¯ e) Figure 7.26: Velocity v at y = 0.5 for mesh A using local approximation of class C 0 (Ω (CKE as reference pressure and stress)

For u0 = 0.004885 we have Re = 73 and for u0 = 0.00153247 we obtain Re = 229. The experimental measurements for this model problem for these two Reynolds numbers have been reported by Denham and Patrick [11]. The numerical simulations for Re = 73 and Re = 229 of the experiment in [11] have also been given in references [9] using the discretizations shown in Figs. 7.27(b) and (c). In the numerical simulations presented in refer-

464

NON-LINEAR DIFFERENTIAL OPERATORS 4/3

28 u=0, v=0

2

y

τxx = 0 v=0

u=0, v =0 x

u=0, v=0

(a) Schematic x=4

x=8

y

x

(b) A 7 element discretization x=4

x=8

y 0.2 x 0.2

(c) A 20 element discretization 0.2 0.8

x=4

x=8

y

0.8 0.2 x 0.2

(d) A 32 element discretization

Figure 7.27: Backward facing step problem and finite element discretizations

ences [9] using LSP based on residual functional using first order system of PDEs, the experimental inlet velocities of reference [9] were used so that the numerically computed solutions could be compared with experimental measurements. Published numerical solutions in references [9] show extremely good agreement with the experimental measurements even for such coarse discretizations of Figs. 7.27(b) and (c). Reference [9] shows that even the coarsest possible mesh of Fig. 7.27(b) produces reasonable computed solutions. In reference [9] numerically computed solutions for discretization of Fig. 7.27(c) are compared with the experimental results of reference [11]. In the numerical solutions presented here we employ a slightly more re¯ e ) p-version hierarchical elements fined discretization of 32 nine-node C 0 (Ω shown in Fig. 7.27(d). We use LSP based on residual functional for the first order system of PDEs (7.111). Same p-levels are considered for all depen-

465

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

10-1

—e

C0(Ω )

Square root of residual functional, √I

p=3 p=9 10-2

CKE Re=73 CVS Re=73 CKE Re=229 CVS Re=229

p=3

10-3 p=9

10-4 1000

10000

100000

Degrees of freedom

Figure 7.28: Square root of the residual functional versus degrees of freedom

dent variable in ξ and η direction. p-levels are uniformly increased for all elements of the discretization from 3 to 9. Fully developed parabolic u velocity (dimensionless) with peak value of 1.5 is specified at the inlet for both √ Reynolds numbers. Figure 7.28 shows graphs of I versus dof for Re = 73 and Re = 229 when CKE and CVS are used to non-dimensionalize pressures and deviatoric Cauchy stresses. As seen in Fig. 7.28, I is O(10−6 ) or lower at p = 9 confirming extremely good accuracy of the computed solutions. Newton’s method with line search converges in less than 10 iterations for all computations √ using |gi | ≤ O(10−6 ) ; i = 1, 2, . . . as convergence criteria. Lower values of I for a given dofs for CKE compared to CVS indicate better accuracy of the computed solutions when using CKE to nondimensionalize pressure and deviatoric stresses. Thus, in the following we present the computed solutions using CKE at p-levels of nine as these are converged solutions. Plots of velocity u vs y at x = 0.0, 0.8, 2.0 and 28.0 or Reynolds numbers of 73 and 229 at p = 9 are shown in Figs. 7.29–7.30. As expected at x = 0, onset of expansion, the axial velocity u is parabolic for both Reynolds numbers (same as at the inlet). Negative u velocity in vicinity of lower boundary for y < −0.25 (approximately) confirms recirculation in the expansion corner. At x = 28.0 we observe perfect fully-developed parabolic velocity profiles for both Reynolds numbers.

466

NON-LINEAR DIFFERENTIAL OPERATORS

2 1.75 —

1.5

C0(Ωe) ; p=9

1.25

CKE, Re=73 CKE, Re=229

Distance, y

1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -0.2

0

0.2

0.4

0.6 0.8 Velocity, u

1

1.2

1.4

1.6

Figure 7.29: Velocity u at x = 0.0 (step location) using local approximation of class ¯ e) C 0 (Ω

2 1.75 0

—

1.5

C (Ωe) ; p=9

1.25

CKE, Re=73 CKE, Re=229

Distance, y

1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -0.2

0

0.2

0.4

0.6 0.8 Velocity, u

1

1.2

1.4

1.6

¯ e) Figure 7.30: Velocity u at x = 0.8 using local approximation of class C 0 (Ω

Figures 7.33 and 7.34 show contour plots of velocity u for Re = 73 and Re = 229. Contour plots of velocity v are shown in Figs. 7.35 and 7.36. We clearly observe larger recirculation zone for Re = 229, as expected. The contour plots confirm that the length of 28 dimensionless units beyond expansion point is sufficient for the flow to become fully developed.

467

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS

2 1.75 —

1.5

C0(Ωe) ; p=9

1.25

CKE, Re=73 CKE, Re=229

Distance, y

1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -0.2

0

0.2

0.4

0.6 0.8 Velocity, u

1

1.2

1.4

1.6

¯ e) Figure 7.31: Velocity u at x = 2 (step location) using local approximation of class C 0 (Ω 2 1.75 0

—

1.5

C (Ωe) ; p=9

1.25

CKE, Re=73 CKE, Re=229

Distance, y

1 0.75 0.5 0.25 0 -0.25 -0.5 -0.75 -1 -0.2

0

0.2

0.4

0.6 0.8 Velocity, u

1

1.2

1.4

1.6

¯ e) Figure 7.32: Velocity u at x = 28 using local approximation of class C 0 (Ω

7.4.6 Flow past a circular cylinder Figure 7.37 shows a schematic of the flow domain. Figure 7.38 shows boundary conditions on the four boundaries of the domain and the boundary of the cylinder. Since L0 = 1 and the dimensionless radius of the cylinder is 1 we have

468

NON-LINEAR DIFFERENTIAL OPERATORS

¯ e) ; p = 9 Figure 7.33: Velocity u for Re = 73 (CKE), C 0 (Ω

¯ e) ; p = 9 Figure 7.34: Velocity u for Re = 229 (CKE), C 0 (Ω

¯ e) ; p = 9 Figure 7.35: Velocity v for Re = 73 (CKE), C 0 (Ω

rˆ = 1 meter. Flow domain consists of length L, height h and the cylinder is symmetrically located in the flow domain. Figure 7.39 (a) shows a 1300 element graded discretization for (x×y) = (−30.5, 30.5)×(−21.5, 21.5). The cylinder is located at the center of the domain. Figure 7.39(b) shows another discretization in which the discretization of Fig. 7.39(a) remains unchanged but additional length has been added to the right of the cylinder. This discretization has 1400 elements.

p=1 p=2 p=3 p=4,5,...,9 Approx. Theoretical

Stress, τxy

-0.4

-0.6

469

7.4. 2D STEADY-STATE NAVIER-STOKES EQUATIONS -0.8

-1

-1.2

-1.4 0

40

80

120

160 200 Distance, x

240

280

320

360

¯ e) ; p = 9 Figure 7.36: Velocity v for Re = 229 (CKE), C 0 (Ω ρ = 1, u = uf s , v = 0, T = 1

v=0 ρ=1 ufus f= s 1

∂v/∂x = 0 r

v=0

rˆ

ˆ =0 ∂v/∂y h

∂T /∂x = 0

T =1 u = 0, v = 0, ∂T /∂n = 0

∂u/∂x = 0

ˆ L ρ = 1, u = uf s , v = 0, T = 1

(a)Figure Boundary Conditions compressible flow a circular cylinder 7.37: forSchematic ofpast Flow over circular

cylinder

u = uf s , v = 0

v=0

uf s = 1

∂v/∂x = 0

r

v=0

∂v/∂y = 0 u = 0, v = 0, ∂T /∂n = 0

∂u/∂x = 0

u = uf s , v = 0

Figure 7.38: Boundary conditions for flowflow past circular cylinder (incompressible flow) (b) Boundary Conditions for incompressible past a a circular cylinder Figure 1: Boundary conditions for flow past a circular cylinder (compressible flow)

We consider water at NTP with the following properties: ρ0 = ρˆ = 997.78 kg/m3 , µ0 = µ ˆ = 9.774 × 10−4 Pa · s, L0 = 1 m 2

Therefore ˆ and Re = ρ = 1, µ = 1, L = L

997.78 u0 9, 774 × 10−4

470

NON-LINEAR DIFFERENTIAL OPERATORS

25 20 15 10 5 0 −5 −10 −15 −20 −25 −40

−30

−20

−10

0

10

20

30

40

(a) Mesh of 1300 elements 25 20 15 10 5 0 −5 −10 −15 −20 −25 −40

−20

0

20

40

60

80

(b) Mesh of 1400 elements

Figure 7.39: Discretizations for the two different domain lengths for flow past a cylinder

We choose solution of class C 22 at p-level of 5 for ρ, u, v in the mathematical model given by equations (7.115), higher order system. We choose u0 such that Re =10, 20, 30, 40, 60, 80, 100, 200 and 500 are obtained and uf s = 1 in all cases. For Re = 10 no initial or starting solution is used (other than null), thus Stokes flow becomes the starting solution at the end for first iteration in Newton’s linear method. The convergence of the Newton’s method is achieved in 5 iterations with |gi |max = 0.332 × 10−6 and I = 0.213 × 10−4 . This solution at Re = 10 is used as initial solution for Re = 20 (new u0 ) to obtain a converged solution for Re = 20. This continuation process is continued until Re = 500 is reached for both 30.5 and 60.5 lengths of the domain past the cylinder. Table 7.1 gives details of Re, I, |gi |max and number of iterations. Contour maps of u velocity for Re = 10, 100 and 500 for lengths of 30.5 and 60.5 past the cylinder are shown in Figs. 7.40–7.42. We observe the

471

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

Table 7.1: Values of Re, I, g for incompressible flow calculations Re 10 20 40 60 80 100 200 500

I 0.213 × 10−4 0.332 × 10−4 0.687 × 10−4 0.141 × 10−3 0.206 × 10−3 0.254 × 10−3 0.749 × 10−3 0.204 × 10−2

|gi |max 0.332 × 10−6 0.645 × 10−6 0.159 × 10−7 0.196 × 10−6 0.598 × 10−6 0.208 × 10−6 0.510 × 10−6 0.932 × 10−6

no. of iterations 5 5 6 6 7 8 9 25

following. (a) For low Re (up to 200), the lengths of 30.5 and 60.5 yield same results due to the lack of influence of outflow boundary on the flow physics in the near field past the cylinder. (b) At Re = 500, the length of 30.5 is obviously not sufficient whereas length of 60.5 is still sufficient to capture the flow physics behind the cylinder.

Figure 7.40: Comparison of velocity u for Re = 10 of lengths of 30.5 and 60.5 behind the cylinder (incompressible flow)

7.5 2D compressible Newtonian fluid flow Following reference [5], the conservation and balance laws: conservation of mass, balance of momenta and the first law of thermodynamics yield continuity, momenta and energy equations. For stationary process (BVP) the dimensionless form of conservation and balance laws can be written as

472

NON-LINEAR DIFFERENTIAL OPERATORS

Figure 7.41: Comparison of velocity u for Re = 100 of lengths of 30.5 and 60.5 behind the cylinder (incompressible flow)

Figure 7.42: Comparison of velocity u for Re = 500 of lengths of 30.5 and 60.5 behind the cylinder (incompressible flow)

(in the absence of sources and sinks and using ideal gas law): ∂(ρu) ∂(ρv) + =0 (7.131) ∂x ∂y ∂u ∂u  p0  ∂p  τ0  τxx τxy  ρu + ρv + − + =0 (7.132) ∂x ∂y ∂x ∂y ρ0 v02 ∂x ρ0 v02 ∂v ∂v  p0  ∂p  τ0  ρu + ρv + − ∂x ∂y ρ0 v02 ∂y ρ0 v02 (7.133) ρcv  ∂T ∂T  1  ∂qx ∂qy  u +v + + Ec ∂x ∂y Re Br ∂x ∂y  p   ∂u ∂v    1 ∂vi 0 2 p + − 2ηD + λ(D ) =0 (7.134) − ij kk ∂x ∂y Re ∂xj ρ0 v02 η v  0 0 τij = (2ηDij + λδij Dkk ) , i, j = 1, 2 (7.135) L0 τ0 (7.136)

473

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

where ∂T ∂T , qy = −k ∂x ∂y ∂vj  1  ∂vi + , i, j = 1, 2 Dij = 2 ∂xj ∂xi R p T  0 0 0 p(ρ, T ) = RρT p0 qx = −k

(7.137) (7.138) (7.139)

In deriving (7.131)–(7.139), we have used the following definition of the dimensionless variables ˆ ˆ vˆi ρˆ ηˆ λ L , vi = , ρ= , η= , λ= L= L0 v0 ρ0 η0 η0 ˆ ˆ τˆij pˆ T k (7.140) τij = , p= , T = , k= τ0 p0 T0 k0 tˆ L0 cˆv t = , t0 = , cv = , p0 = τ0 = ρv02 (CKE) t0 v0 cv0 In (7.140), the quantities with hat notation are with their usual dimensions and those with subscript zero are their reference values. Re, Br and Ec are defined as Re =

η0 v02 v02 v 0 ρ0 L 0 , Br = , Ec = η0 k0 T0 cv0 T0

In addition we have used the following for specific internal energy e in the energy equation e = cv T (7.141) in which we assume constant cv . Equations (7.131)–(7.137) is a system of time first order PDEs in nine dependent variables ρ, u, v, τij , qx , qy , T . By substituting the Cauchy deviatoric stresses τij from (7.135) and the heat vectors qx , qy from (7.137) into (7.134), these can be reduced into a system of four PDEs in four dependent variables ρ, u, v, T . ∂(ρvi ) =0 ∂xi

(7.142)

∂vi  p0  ∂p 1 ∂ + − (2ηDIJ + λδij Dkk ) = 0 (i = 1, 2) (7.143) 2 ∂xj ρ0 v0 ∂xi Re ∂xj ρ cv  ∂T ∂T  k  ∂2T ∂2T  u +v − + Ec ∂x ∂y Re Br ∂x2 ∂u2  p  ∂u ∂v    1 ∂vi 0 2 − + − 2ηD + λ(D =0 (7.144) ij kk Re ∂xj ρ0 v02 ∂x ∂y ∂p ∂p  ∂p  ∂p  ∂T  − + =0 (7.145) ∂xi ∂ρ ∂xi ∂T ∂xi

ρvj

474

NON-LINEAR DIFFERENTIAL OPERATORS

In (7.142)–(7.145), x1 and x2 refer to x and y, likewise v1 and v2 imply u ∂p ∂ρ and v, velocities in x and y directions. In (7.145), ∂ρ and ∂T are deterministic from the equation of state (ideal gas law (7.139)). Equations (7.142)–(7.144) is a system of four PDEs in four dependent variables ρ, u, v, T . These contain up to second order derivatives of velocities u, v and temperature T . A variationally consistent least squares finite element formulation of (7.142)–(7.144) is constructed using residual functional (similar to one presented in section 7.4. Details are straight forward. Since the PDEs contain up to second order derivatives of u, v and T , the approximation space V ¯ e ), k ≥ 3 for the integrals to be Riemann for this must satisfy V ⊂ H k,p (Ω k,p e ¯ but V ⊂ H (Ω ), k = 2 when the integrals are Lebesgue i.e. solutions of class C 2 correspond to minimally conforming spaces but solutions of class C 1 are admissible if the integrals can be accepted in the Lebesgue sense. Using equal order equal degree interpolation for all dependent variables we can consider ρ, u, v, T to be of class C 1 or C 2 . In the following we present numerical solutions of two boundary value problems using the VC least squares finite element formulation based on residual functional using PDEs (7.142)–(7.144). (a) Carter’s plate: Laminar flow over a flat plate at Mach 1,3 and 5 is considered. The medium is air and the transport properties are assumed to be constant with ideal gas law. (b) Flow over a circular cylinder: Flow over a circular cylinder with varying trailing lengths is considered. Medium is air with constant transport properties at NTP and ideal gas law. ¯ e ) as well For both model problems we consider solutions for class C 1 (Ω 2 e ¯ as solutions of class C (Ω ) at various p-levels. These two model problems have also been investigated in references [12, 13].

7.5.1 Carter’s plate In this study we consider flow of compressible fluid (air) over a stationary impermeable plate at various mach numbers. Plate is two meters in length. At the left end of the plate a constant velocity field is imposed. The objective is to compute the flow features in a 2 m × 0.01 m domain over the plate. Figure 7.43 shows a schematic of the Carter’s plate. Boundary conditions and non-dimensional computational domain L × h are shown in Fig. 7.44. The following properties are used for air at NTP. ρ0 = 1.12254 cv0 = 717.0

kg , η0 = 0.1983 × 10−4 Pa · s, T0 = 410.52 K m3

J ˆ = 286.9965 J , k0 = 0.028854 N , R kg · K kg · K kg · K

475

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

ˆ = 0.01m h

uˆf s

ˆ = 2m L

Figure 7.43: Schematic of Carter’s plate y ρ = 1 , ∂ρ/∂x = 0 , T = 1 , ∂T /∂x = 0 , ∂T /∂y = 0 D

C

ρ = 1, T = 1, u = uf s , v = 0 h = 0.01

uˆf s

∂T /∂x = 0

∆y A

u = 0, v = 0, ∂T /∂y = 0

B

L=2

x

Figure 7.44: Carter’s plate Boundary conditions

We choose the following values for the reference quantities: ˆ cv0 = cˆv , v0 = 343.25 m L0 = 1.0m, ρ0 = ρˆ, µ0 = µ ˆ, k0 = k, s J kg T0 = 410.52 K, R0 = 286.99 , τ0 = p0 = ρo v02 = 0.14438 × 106 kg · K m·s With these reference quantities we have the following dimensionless quantities and dimensionless parameters: ρ = 1.0, µ = 1.0, k = 1.0, cv = 1.0, R = 1.0, γ = 1.4

476

NON-LINEAR DIFFERENTIAL OPERATORS

Re = 0.212 × 106 , Br = 0.19724, Ec = 0.40027 ˆ = 0.01 m). Specification ˆ = 2 m) and h = 0.01 (h We choose L = 2 (L of the dependent variables on the boundaries is crucial to ensure that the BVP is well posed but not overly constrained. That is, we only define what is permissible based on physics and that to only basic quantities and thus avoiding redundant descriptions through BCs. At the inlet(no heat transfer along and across inlet AD): ρ = 1,

T =1 ⇒

∂T ∂T = 0, =0 ∂y ∂x

At the plate (no heat transfer normal to the plate): u = 0,

v = 0,

∂T =0 ∂y

At the top boundary CD (no heat transfer normal to the boundary CD): ρ=1 ⇒

∂p = 0; T = 1 ⇒ ∂x

∂T ∂T = 0; =0 ∂x ∂y

At the outflow boundary BC (no heat transfer across the boundary BC): ∂T =0 ∂x Fully developed flow conditions on BC are inappropriate to use if the length L is not sufficient. To avoid over specification of BCs on CD, we avoid defining u and v and/or their gradients. We present numerical studies for free stream velocity u ˆf s of Mach 1, 2, 3 and 5. The domain (2 × 0.01) is discretized using a (70 × 30) graded mesh in which the element size at the lower left corner is 10−5 . We choose us to be the speed of sound. Solutions for all Mach numbers are of class C 11 at p = 3 in space and time. 7.5.1.1 Mach 1 flow If we choose u0 = us then uf s = 1. The other reference quantities are chosen as shown above. This choice of u0 gives rise to ReM 1 , EcM 1 , BrM 1 , i.e. Re, Ec and Br at Mach 1. The choice of initial solutions is critical as well. We choose ρ = 1, T = 1, v = 0 and u = 1 as initial solution. At the leading edge, the velocity distribution of 0 to uf s is applied over a distance of ∆y in the element located at the leading edge in a continuous and differentiable manner (shown in Fig. 7.44). Newton’s linear method with line search converged in 35 iterations with |gi |max = 0.862 × 10−6 and

477

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

I = 0.325 × 10−2 . The largest value of I is from the element located at the leading edge. Other than the single element at the leading edge, I values elsewhere are much smaller than O(10−2 ).

Density, ρ

Figures 7.45–7.47 show contour plots of density, velocity and temperature over (x, y) domain of (0 , 2) × (0 , 0.005). From the figures we observe that boundary layer may not be fully developed, hence the reason for not imposing fully developed flow conditions at the outflow. Progressive development of the boundary layer is clearly observed. Temperature rise due to viscous dissipation at the plate with progressively higher values for increasing x are clearly seen in Fig. 7.47. Increase in density beyond 1 (initial state) indicates existence of mild shock.

Velocity, u

Figure 7.45: Contour of the density ρ for Mach 1.0 flow

Figure 7.46: Contour of the velocity u for Mach 1.0 flow

478

Temperature, T

NON-LINEAR DIFFERENTIAL OPERATORS

Figure 7.47: Contour of the temperature T for Mach 1.0 flow

7.5.1.2 General consideration for higher Mach number flows For calculating flow at Mach 2, 3 and 5 a direct calculation similar to that described for Mach 1 resulted in lack of convergence of the iterative solution method for solving non-linear algebraic equations. This is largely due to inadequate starting or initial solution for Newton’s linear method for solving non-linear algebraic equations i.e., the choice of free stream values as initial solution (as used for Mach 1 flow) is not in close proximity of the actual solution sought from Newton’s linear method upon convergence. Thus, a continuation procedure becomes essential to use. For example we could use Mach 1 solution as a starting or initial solution for Mach 2. For illustration purposes consider Mach 2 flow. For Mach 2 flow, u0 = 2us gives uf s = 1 (same as in case of Mach 1). But (Re)M 2 , (Ec)M 2 , (Br)M 2 will be different compared to Mach 1 flow. With this choice, the boundary and free stream values of u, ρ, T and v in Mach 2 solution are same as those for Mach 1 flow but new Re, Br and Ec for Mach 2 flow are characteristic dimensionless parameters of the flow at Mach 2. Thus we can proceed as follows: Mach 2 flow u0 = 2us , uf s = 1 All other reference values same as those for Mach 1 Starting solution from Mach 1 calculations Mach 3 flow

u0 = 3us , uf s = 1

(a) Starting solution from Mach 1 flow (b) Starting solution from Mach 2 flow Mach 5 flow

u0 = 5us , uf s = 1

479

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

(a) Starting solution from Mach 1 flow (b) Starting solution from Mach 2 flow (c) Starting solution from Mach 3 flow 7.5.1.3 Mach 2 flow

Density, ρ

Using the continuation from Mach 1 described above with u0 = 2us , uf s = 1 (same as Mach 1 flow), the Newton’s linear method converges in 17 iteration with |gi |max = 0.883 × 10−6 and I = 0.105 × 10−2 . The contour plots of ρ, u, T are shown in Figs. 7.48–7.50 over the same (x, y) domain as used for Mach 1 flow. Thinner boundary layer and increased temperature values at the plate are clearly observed compared to Mach 1 flow. Larger increase in density (1.05) compared to Mach 1 (1.02) is also observed.

Velocity, u

Figure 7.48: Contour of the density ρ for Mach 2.0 flow

Figure 7.49: Contour of the velocity u for Mach 2.0 flow

480

Temperature, T

NON-LINEAR DIFFERENTIAL OPERATORS

Figure 7.50: Contour of the temperature T for Mach 2.0 flow

7.5.1.4 Mach 3 flow In this case Mach 1 solution as well as Mach 2 solutions were used as starting or initial solutions in two separate studies. The final converged solutions at Mach 3 were identical from either of these two as initial solutions. Using Mach 1 solution as initial solutions, Newton’s linear method converges in 17 iterations with |gi |max = 0.254 × 10−6 and I = 0.716 × 10−3 .

Density, ρ

Contour plots of ρ, u, T are shown in Figs. 7.51–7.53. Thinner boundary layer and increased temperature compared to Mach 2 flow are quite clear.

Figure 7.51: Contour of the density ρ for Mach 3.0 flow

481

Velocity, u

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

Temperature, T

Figure 7.52: Contour of the velocity u for Mach 3.0 flow

Figure 7.53: Contour of the temperature T for Mach 3.0 flow

7.5.1.5 Mach 5 flow For Mach 5 flow, the initial or starting solutions were used from Mach 1 as well as Mach 2 and Mach 3 for three different studies (with u0 = 5us and uf s = 1 ). All three studies converged to the identically same solution. We present results obtained using the Mach 1 solution as the starting or initial solution. Contour of ρ, u and T are shown in Figs. 7.54–7.56. The Newton’s linear method converges in 29 iterations with |gi |max = 0.473 × 10−6 and I = 0.425 × 10−3 . Contour plots of ρ, u, T are shown in Figs. 7.54–7.56. We observe similar trend of diminishing boundary layer thickness with increasing temperature values at the plate. Density increase in this case is largest (1.1), higher than Mach 1, 2 and 3 flows indicating slightly stronger shock.

NON-LINEAR DIFFERENTIAL OPERATORS

Density, ρ

482

Velocity, u

Figure 7.54: Contour of the density ρ for Mach 5.0 flow

Temperature, T

Figure 7.55: Contour of the velocity u for Mach 5.0 flow

Figure 7.56: Contour of the temperature T for Mach 5.0 flow

Comparison of results for Mach 1, 2, 3 and 5 flows A comparison of the results for mach 1, 2, 3, and 5 is presented in Figs. 7.57–7.61. Figure 7.57 shows plots of velocity u at the outflow bound-

483

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW

ary for Mach 1, 2, 3 and 5 flows. An exploded view of these is also shown in Fig. 7.58. We clearly see thinning boundary layer for increasing Mach numbers. Free stream value of u = uf s = 1 for y > 0.004 for all Mach numbers shows that h-value of 0.01 chosen in numerical studies is sufficiently large. Plots of ρ, T and v at the outflow for Mach 1, 2, 3 and 5 are shown in Figs. 7.59–7.61. Progressively decreasing density at the plate for progressively increasing Mach number flows is clearly observed. Free stream density of one is clearly seen for y > 0.004, conforming adequate choice of h. From Fig. 7.60 we observe progressively increasing temperature values at the plate for progressively increasing Mach number flows and a free stream temperature of 1 for y > 0.004. Plots of v at the outflow boundary confirm that v of the order of 10−5 implies very close to fully developed flow. Slightly larger values of density beyond 1.0 indicate existence of milder shocks. 0.01

Mach Mach Mach Mach

0.009

1 2 3 5

0.008

Distance, y

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0

0.2

0.4

0.6

0.8

1

1.2

Velocity, u

Figure 7.57: Velocity u at the outflow for Mach 1, 2, 3 and 5 flows

7.5.2 Mach 1 flow past a circular cylinder We consider same geometry, discretizations as used for example 7.4.6 in case of incompressible isothermal flow of a Newtonian fluid considered here. We consider flow of air (at NTP) past a circular cylinder at Mach 1 with the following properties of the medium. ρˆ = 1.12254

kg , µ ˆ = 0.1983 × 10−4 Pa · s, T0 = 410.52 K m3

J ˆ = 286.9965 J , kˆ = 0.028854 N , R kg · K kg · K kg · K and we choose the following reference quantities and their values. cˆv = 717.0

ˆ cv0 = cˆv L0 = 1.0m, ρ0 = ρˆ, µ0 = µ ˆ, k0 = k,

484

NON-LINEAR DIFFERENTIAL OPERATORS 0.002

Mach Mach Mach Mach

0.0018

1 2 3 5

Distance, y

0.0016

0.0014

0.0012

0.001

0.0008

0.0006 0.8

0.85

0.9

0.95

1

1.05

Velocity, u

Figure 7.58: Exploded view of velocity u for Mach 1, 2, 3 and 5 flows 0.01

Mach Mach Mach Mach

0.009

1 2 3 5

0.008

Distance, y

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

Density, ρ

Figure 7.59: Density ρ at the outflow for Mach 1, 2, 3 and 5 flows

u0 = 343.25

m J , T0 = 410.52 K, R0 = 286.99 s kg · K

τ0 = p0 = ρ0 u20 = 0.14438 × 106

kg m · s2

With these reference quantities we have the following dimensionless quantities and the dimensional parameters. ρ = 1.0, µ = 0, k = 1.0, cv = 1.0, R = 1.0 Re = 0.212 × 106 , Br = 0.19724, Ec = 0.40027

485

7.5. 2D COMPRESSIBLE NEWTONIAN FLUID FLOW 0.01 Mach Mach Mach Mach

0.009

1 2 3 5

0.008

Distance, y

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0

0.5

1

1.5

2

2.5

3

3.5

4

Temperature, T

Figure 7.60: Temperature T at the outflow for Mach 1, 2, 3 and 5 flows 0.01 0.009

Mach Mach Mach Mach

1 2 3 5

0.008

Distance, y

0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 -0.0001

-8e-05

-6e-05

-4e-05

-2e-05

0

2e-05

Velocity, v

Figure 7.61: Velocity v at the outflow for Mach 1, 2, 3 and 5 flows

A schematic with BCs is shown in Fig. 7.62. We choose rˆ = 1 m, hence r = rˆ/L0 = 1. The length from the inlet to the center of the cylinder is 6 units. From the studies performed for Carter’s plate we find that a good starting or initial solution is quite essential when solving BVPs for compressible flow. We choose the incompressible flow solution (for same geometry and same discretization) as starting solution for compressible flow. When using a length of 60.5 behind the cylinder, the use of incompressible flow results in non-converged Newton’s linear method. Experiments with length of 30.5 behind the cylinder show convergence of the Newton’s linear method. We

0

40

80

120

160 200 Distance, x

486

240

280

320

360

NON-LINEAR DIFFERENTIAL OPERATORS

ρ = 1, u = uf s , v = 0, T = 1

v=0 ρ=1

∂v/∂x = 0

uf s = 1

∂v/∂y = 0

r

v=0

∂T /∂x = 0

T =1 u = 0, v = 0, ∂T /∂n = 0

∂u/∂x = 0

ρ = 1, u = uf s , v = 0, T = 1 (a) Boundary Conditions for compressible flow past a circular cylinder

Figure 7.62: Boundary conditions for flow past a circular cylinder (compressible flow) u = uf s , v = 0

realize that this length is not sufficient for the flow to be fully developed at the outflow boundary. We present results to demonstrate that the VCLSP has no difficulty in computing numerical solution. We consider solutions v=0 ¯ e ) with p-level of 5. Using the converged solution from fs = 1 of uclass C 22 (Ω ∂v/∂x =the 0 r v=0 incompressible flow at Re = 500, the Newton’s linear method converges ∂v/∂y = 0in −4 and I = 0.206 × 10−2 and only few 50 iterations with |gi |max =u0.613 10/∂n = 0, v =× 0, ∂T =0 ∂u/∂x = 0 elements around. Contour plots of u, v, ρ and T are shown in Figs. 7.63–7.66. High values of I indicate the GDEs are not satisfied accurately by the computed solution due to inadequate h and p and the length past the cylinder. Nonetheless the computational process works well the sense that with continuation u = uin f s, v = 0 in Re, for the given h and p, the length past the cylinder and the outflow (b) Boundary Conditions for incompressible flow past a circular cylinder boundary conditions a solution satisfying discretized form of the GDEs is found (|gi | ≥ 10−6 ). With h, p refinement, adequate length past the cylinder Figure 1: Boundary conditions for flow past a circular cylinder (compressible flow) and appropriate outflow conditions we see no issues in resolving the correct physics with the laminar flow assumption.

7.6 Summary

2

Numerical solutions of a variety of model problems in R1 and R2 containing nonlinear differential operators have been considered. It has been shown in chapter 3 that for such operators, all methods of approximations leading to integral forms yield VIC integral forms. It was established that the integral forms resulting from the residual functional (LSP) can be made VC by small adjustments in the solution process for nonlinear systems of algebraic equations. It is established that by neglecting the second variation of the

487

Velocity, u

7.6. SUMMARY

Velocity, v

Figure 7.63: Contour plot of velocity u for compressible flow past a cylinder of lengths of 30.5

Density, ρ

Figure 7.64: Contour plot of velocity v for compressible flow past a cylinder of lengths of 30.5

Figure 7.65: Contour plot of density ρ for compressible flow past a cylinder of lengths of 30.5

residuals in the second variation of the residual functional and by employing Newton’s linear method for obtaining numerical solutions of the nonlinear algebraic equations, the integral forms become VC, hence the resulting coefficient matrices in the algebraic systems remain unconditionally positive definite. The rationale and validity of this approach was also established in chapter 3. All numerical studies presented in this chapter utilize the ap-

488

Temperature, T

NON-LINEAR DIFFERENTIAL OPERATORS

Figure 7.66: Contour plot of temperature T for compressible flow past a cylinder of lengths of 30.5

proach (see chapter 3 for more details). Hence, regardless of the choice of p, p and k and the parameters in the mathematical models (like P e, Re, De etc.), the computational process used here remains unconditionally stable. As discussed in chapter 3, the modified least squares formulation utilized here is free of upwinding methods. The residual functions and the residual functionals used in the formulation correspond to actual nonlinear differential operators. With appropriate choices of p and k, this method can incorporate desired features of the theoretical solution in the computational process. Adaptive computations based on residual functional and h, p refinements provide a mechanism to achieve true convergence to the theoretical solutions. By maintaining integrals in the Riemann sense (appropriate choice of k) and by ensuring that the residual functionals for the whole discretization approach zero, we ensure that the computed numerical solutions indeed satisfy the governing differential equations in the pointwise sense, hence these solutions are indeed same as the theoretical solutions. [1–3, 7–39]

Problems 7.1 Consider the one-dimensional steady state Burgers equation: 1 d2 φ ∂φ =0 − dx Re dx2 φ(0) = 1, φ(1) = 0 φ

0 2 we uniformly contract he such that he = 2 in ξ coordinate space. The mapping so described is obviously linear. Other possibilities exist as well (discussed later).

8.1.1 Mapping of points Regardless of the precise nature of the mapping, in the abstract sense we could describe it by x = x(ξ) (8.1) Such mapping can be easily constructed by using, for example, functions in ξ space and position coordinates in x space. If we consider the line segment being defined by the position coordinates of its two ends then we could write 1 + ξ  1 − ξ  xe + xe+1 . (8.2) x = x(ξ) = 2 2
One could verify that such a mapping is a linear stretch and that 1−ξ 2
and 1+ξ indeed are functions corresponding to points 1 and 2 in ξ space 2 which are a map of xe and xe+1 in x-space. Other possibilities exist, but for now it suffices to consider this abstract form of the mapping defined by (8.1) for developing the general theory. Form (8.1) we note that such a ¯ ξ we could determine mapping only maps points, i.e. given a location ξ ∗ ∈ Ω ∗ ∗ e ¯ . Mapping (8.1) must be one-to-one and the corresponding x = x(ξ ) ∈ Ω onto, i.e. the inverse of the mapping ξ = ξ(x) must exist and must be unique. For our purpose here, (8.1) suffices.

8.1.2 Mapping of lengths Elemental length dx, dξ in x, ξ spaces can be related using dx = in which we define J = Obviously we have

dx dξ ,

dx dξ dξ

(8.3)

called the Jacobian of mapping or transformation. xZe+1

he =

Z1

dx = xe

−1

dx dξ = dξ

Z1 J dξ −1

(8.4)

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

495

¯e 8.1.3 Behavior of dependent variable φ over Ω ¯ e . Instead of Our interest of course is to describe the behavior of φ over Ω ¯ ξ using interpolation doing so, we can also describe the behavior of φ over Ω theory, φ = φ(ξ) (8.5) Since for a ξ, say ξ ∗ , the corresponding x, say x∗ , is given by x(ξ ∗ ) using (8.1) and using (8.5) we in fact have φ(ξ ∗ ) and thus the behavior of φ over ¯ e . This approach is advantageous in that the interpolation theory only Ω ¯ ξ and not for each Ω ¯ e (e = 1, 2, . . . , N ). needs to be developed for Ω Obtaining

di φ ; dxi

i = 1, . . .

Since φ = φ(ξ) and x = x(ξ) we have dφ(ξ) dφ(ξ) dx dφ(ξ) = = J dξ dx dξ dx

(8.6)

Therefore

dφ(ξ) 1 dφ(ξ) = dx J dξ It is straight forward to show that di φ(ξ) 1 di φ(ξ) = dxi (J)i dξ i

(8.7)

(8.8)

For example, when the mapping is defined by (8.2) we have J=

dx 1 he = (xe+1 − xe ) = dξ 2 2

Other forms of mapping in one dimension will be discussed in later sections. In any case, once x = x(ξ) is defined, everything else is defined.

¯ ξ = [−1, 1] 8.2 Elements of interpolation theory over Ω ¯ ξ = [−1, 1] in the natural coordinate space Consider the element map Ω ξ. In this section we consider various alternatives to establish an analytical expression for the behavior of a function f = f (ξ), ξ ∈ [−1, 1].

8.2.1 A polynomial approximation in one dimension Let f (ξ) have the values f (ξ0 ) = f0 , f (ξ1 ) = f1 , . . . , f (ξn ) = fn at n + 1 points ξ0 , ξ1 , . . . , ξn ∈ [−1, 1]. then we can describe f (ξ) as a polynomial of degree n that passes through these n + 1 points f (ξ) = a0 + a1 ξ + a2 ξ 2 + · · · + an ξ n

(8.9)

496

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

or f (ξ) = [1

ξ

ξ2

···

  a0       a1   ξn] . ..         an

(8.10)

Upon substituting f (ξ) = fi and ξ = ξi ; i = 0, 1, . . . , n in (a), we obtain the following set of linear simultaneous algebraic equations among ai (i = 0, 1, . . . , n) from which ai (i = 0, 1, . . . , n) can be calculated:      1 ξ0 ξ02 . . . ξ0n   a0     f0    a1  1 ξ1 ξ 2 . . . ξ n      f1   1 1  = (8.11)  .. .. .. . . ..  .. .. . . . . .   .   .          1 ξn ξn2 . . . ξnn an fn or [C]{a} = {f }

(8.12)

{a} = [C]−1 {f }

(8.13)

Therefore Substituting from (8.13) into (8.10) for {a} we obtain   f0        f1  2 n −1 f (ξ) = [1 ξ ξ · · · ξ ][C] ..  .       fn or f (ξ) = [L0 (ξ)

L1 (ξ)

or f (ξ) =

n X

···

  f0       f1   Ln (ξ)] . ..         fn

Li (ξ)fi

(8.14)

(8.15)

(8.16)

i=0

in which Li (ξ) are basis functions or local approximation functions corresponding to locations ξi ; i = 0, 1, . . . , n and f (ξ) is called local approxi¯ ξ . Local approximation is a linear combination of the mation of f over Ω basis functions and the function values fi (constants) at ξi ; i = 0, 1, . . . , n generally called degrees of freedom.

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

497

Remarks. (1) In order for {a} and, hence, f (ξ) to be unique, it is obvious that the matrix [C] should not be singular, i.e. det [C] 6= 0. (2) If ξi (i = 0, 1, . . . , n) are distinct then det [C] 6= 0 is assured and, hence, the uniqueness of {a} is assured as well. (3) Such approach is error prone due to matrix inversion and is computationally not very efficient. As the number of points increases, we need to invert increasingly large matrices. Choices of ξi are obviously crucial as well. This process described in the above section can be accomplished more efficiently using Lagrange interpolating polynomials (described in the following in section 8.2.2) in which the inverse of the matrices is avoided.

8.2.2 Lagrange interpolating polynomials in one dimension ¯ ξ = [−1, 1] be the domain (i.e. map of an element in the natural Let Ω coordinate system ξ). Let (ξi , f (ξi )) (i = 0, 1, . . . , n) be the pair of points Theorem 8.1. There exists a unique polynomial φ(ξ) or degree not exceeding n called the Lagrange interpolating polynomial such that φ(ξi ) = f (ξi ), i = 0, 1, . . . , n

(8.17)

Proof. The existence of the polynomial φ can be proved if we can establish the existence of polynomials Lk ; k = 0, 1, . . . , n with the properties (i) Each polynomial Lk is a polynomial of degree at most n. ( 1 j=i (ii) Li (ξj ) = , i = 0, 1, . . . , n 0 j 6= i Assuming the existence of these polynomials, we can write φ(ξ) =

n X

fk Lk (ξ)

(8.18)

k=0

where fk = f (ξk ). (iii) We also note that n X i=0

Li (ξ) = 1

(8.19)

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Remarks. (1) Lk (ξ) are polynomials of degree at most n. (2) φ(ξ) is a linear combination of Lk (ξ) and fk , hence φ(ξ) is a polynomial of degree at most n. (3) φ(ξm ) = fm = f (ξm ), since Lk (ξm ) = 0 for k 6= m and Lk (ξm ) = 1 for k=m. (4) Lk (ξ) are called Lagrange interpolating polynomials. The Lagrange interpolating polynomials Lk (ξ) can be obtained using n Y ξ − ξm
, k = 0, 1, . . . , n Lk (ξ) = ξk − ξm

(8.20)

m=0 m6=k

The polynomials Lk (ξ) have the desired properties as described earlier and, hence, we can write f (ξ) = φ(ξ) =

n X

fi Li (ξ), fi = f (ξi )

(8.21)

i=0

¯ e = [−1, 1]). ConExample 8.1 (Quadratic Lagrange polynomials over Ω sider the points (ξ1 , f (ξ1 )),(ξ2 , f (ξ2 )) and (ξ3 , f (ξ3 )) where (ξ1 , ξ2 , ξ3 ) are (−1, 0, 1). Then we can express f (ξ) by φ(ξ) given by f (ξ) = φ(ξ) = L1 (ξ)f1 + L2 (ξ)f2 + L3 (ξ)f3

∀ξ ∈ [ξ1 , ξ3 ] = [−1, 1]

So all we need to do is to establish L1 (ξ), L2 (ξ) and L3 (ξ). We do so in the following. Let 3  Y ξ − ξm  , k = 1, 2, 3 Lk (ξ) = ξk − ξm m=1 m6=k

Thus, we have k=1:

3  Y ξ − ξm  ξ(ξ − 1) (ξ − ξ2 )(ξ − ξ3 ) L1 (ξ) = = = ξ1 − ξm (ξ1 − ξ2 )(ξ1 − ξ3 ) 2 m=1 m6=1

k=2:

L2 (ξ) =

3  Y ξ − ξm  (ξ − ξ1 )(ξ − ξ3 ) = = (1 − ξ 2 ) ξ2 − ξm (ξ2 − ξ1 )(ξ2 − ξ3 )

m=1 m6=2

k=3:

3  Y ξ − ξm  (ξ − ξ1 )(ξ − ξ2 ) ξ(ξ + 1) L3 (ξ) = = = ξ3 − ξm (ξ3 − ξ1 )(ξ3 − ξ2 ) 2 m=1 m6=3

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

499

Remarks. ( 1 m=k (1) Lk (ξm ) = 0 m= 6 k (2)

P3

k=1 Lk (ξ)

=

ξ(ξ−1) 2

, k = 1, 2, 3. + (1 − ξ 2 ) +

ξ(ξ+1) 2

=1

(3) Plots of Lk (ξ) as a function of ξ for ξ ∈ [−1, 1] are shown in Fig. 8.2. L2 (ξ)

L1 (ξ)

L3 (ξ)

1

1

1

2

3

ξ=0

ξ=1

ξ ξ = −1

Figure 8.2: Plots of Li (ξ) (i = 1, 2, 3) for ξ ∈ [−1, 1]

(4) We have f (ξ) = φ(ξ) = L1 (ξ)f (ξ1 ) + L2 (ξ)f (ξ2 ) + L3 (ξ)f (ξ3 ) or f (ξ) = φ(ξ) =

3 X

Li (ξ)f (ξi ) =

i=1

3 X

Li (ξ)fi

i=1

where f1 , f2 , f3 are the values of the function at the nodes or points 1, 2, 3 located at ξ = −1, 0, 1. In this case, the element would be a three node element with values of the function at the nodes as unknowns and φ(ξ) is quadratic over [−1, 1]. (5) We note that Li (ξ) (i = 1, 2, 3) are quadratic basis functions corresponding to nodes 1, 2, 3. (6) Henceforth, we refer to f (ξ) = φ(ξ) interpolation as local approximation ¯ ξ . This is due to the fact that if Ω ¯ξ of the dependent variable φ over Ω e ¯ is the map of an element Ω , then φ(ξ) is purely local to this element without regard to the neighboring elements. ¯ e = [−1, 1]). Consider Example 8.2 (Cubic Lagrange polynomials over Ω the points (ξi , f (ξi )), (i = 1, . . . , 4) where ξ1 , . . . , ξ3 ) are (−1, − 31 , 13 , 1). Then we can express f (ξ) by φ(ξ) given by f (ξ) = φ(ξ) =

4 X i=1

Li (ξ)f (ξi ) =

4 X i=1

Li (ξ)fi

∀ξ ∈ [ξ1 , ξ4 ] = [−1, 1]

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

In this case, also, we need to establish Lk (ξ). Recall that Lk (ξ) =

4  Y ξ − ξm  , k = 1, 2, 3, 4 ξk − ξm

m=1 m6=k

or L1 (ξ) =

4  Y ξ − ξm  (ξ − ξ2 )(ξ − ξ3 )(ξ − ξ4 ) = ξ1 − ξm (ξ1 − ξ2 )(ξ1 − ξ3 )(ξ1 − ξ4 )

m=1 m6=1

1 1 9 (1 − ξ)( + ξ)( − ξ) 16 3 3 4   Y ξ − ξm (ξ − ξ1 )(ξ − ξ3 )(ξ − ξ4 ) L2 (ξ) = = ξ2 − ξm (ξ2 − ξ1 )(ξ2 − ξ3 )(ξ2 − ξ4 ) =−

m=1 m6=2

27 1 (1 + ξ)(1 − ξ)( − ξ) 16 3 4   Y ξ − ξm (ξ − ξ1 )(ξ − ξ2 )(ξ − ξ4 ) L3 (ξ) = = ξ3 − ξm (ξ3 − ξ1 )(ξ3 − ξ2 )(ξ3 − ξ4 ) =

m=1 m6=3

27 1 (1 + ξ)(1 − ξ)( + ξ) 16 3 4   Y ξ − ξm (ξ − ξ1 )(ξ − ξ2 )(ξ − ξ3 ) L4 (ξ) = = ξ4 − ξm (ξ4 − ξ1 )(ξ4 − ξ2 )(ξ4 − ξ3 ) =

m=1 m6=4

=−

9 1 1 ( + ξ)( − ξ)(1 + ξ) 16 3 3

Remarks. ( 1 m=k (1) Lk (ξm ) = 0 m 6= k P4 (2) k=1 Lk (ξ) = 1

, k = 1, 2, 3, 4.

Plots of Lk (ξ) as a function of ξ for ξ ∈ [−1, 1] are shown in Fig. 8.3. ¯ ξ = [−1, 1] (3) As the degree of the polynomial increases over the domain Ω so do the number of nodes or points, that is, a linear polynomials needs two nodes or points, a quadratic behavior requires three nodes or points, ¯ ξ would need four nodes or points and likewise a cubic polynomial over Ω between [−1, 1] in the ξ coordinate space and so on. ¯ ξ with 2, 3, 4, . . . nodes in the ξ coordinate system could (4) These domains Ω be viewed as maps of the 1D finite elements from the physical space x to the natural coordinate space ξ.

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

501

(5) At each node only the function value is required to establish the ana¯ ξ . Thus, when such elements are lytical behavior of the function over Ω used in the discretization, we only have inter-subdomain continuity of the function, i.e. all such interpolation may consist of higher degree but are of class C 0 and, hence, will exhibit inter-subdomain discontinuity of the first derivatives of the function and the derivatives of order higher than one are not defined at the inter-element node points. ¯ ξ as follows (6) In summary, we can list 1D C 0 interpolation of f over Ω f (ξ) =

n X

Li (ξ) fi

i=1

where n = 1, 2, 3, . . . for the linear, quadratic, and cubic cases. Li (ξ) are called Lagrange functions, local approximation functions, or basis functions (see Fig. 8.4). L1 (ξ)

L2 (ξ)

1

1

1

2

−1

− 13

3

4

1 3

1

ξ

1

2

−1

− 31

3

4

1 3

1

ξ

L4 (ξ)

L3 (ξ)

1

1

1

2

−1

− 13

3

4

1 3

1

ξ

1

2

−1

− 13

3

4

1 3

1

ξ

Figure 8.3: Plots of Li (ξ); i = 1, 2, 3 for ξ ∈ [−1, 1]

8.2.3 p-version hierarchical functions in one dimension ¯ ξ = [−1, 1] as the domain of interpolation (referred to as elConsider Ω ement or finite element map from the physical coordinate space x to ξ). ¯ ξ is referred to as element interpolation or Behavior of a function f over Ω ¯ ξ . The subject local approximation as it is local to the element domain Ω of element interpolation or local approximation is crucial as it establishes how the function or dependent variable behaves over the element domain.

502

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Nodal configuration 1

2

-1

1

1

2

3

-1

0

1

ξ

ξ

Basis functions L1 (ξ) =

1−ξ 2

L2 (ξ) =

1+ξ 2

L1 (ξ) =

ξ(ξ−1) 2

L2 (ξ) = 1 − ξ 2 L3 (ξ) =

Type C0

C0

ξ(ξ+1) 2

9 1 ( 3 + ξ)( 31 − ξ)(1 − ξ) L1 (ξ) = − 16

1

2

3

-1

− 31

1 3

4 1

ξ

L2 (ξ) = L3 (ξ) = L4 (ξ) =

1 27 16 (1 + ξ)(1 − ξ)( 3 − ξ) 27 1 16 (1 + ξ)(1 − ξ)( 3 + ξ) 9 1 ( 3 + ξ)( 31 − ξ)(1 + ξ) − 16

C0

Figure 8.4: Linear, quadratic, and cubic Lagrange interpolation functions in terms of the natural coordinate

It is well established that higher degree local interpolations as opposed to lower degree yield higher convergence rates and have many other beneficial properties. The simplest manner in which the higher degree local interpolations can be constructed is by using Lagrange interpolation functions. ¯ ξ = [−1, 1]. Figure 8.5 shows a Consider the one dimensional case in which Ω two-node linear, three-node quadratic and four-node cubic Lagrange configurations in the natural coordinate space ξ for which the basis functions (or local approximation functions or Lagrange interpolation functions) can be easily constructed using Lagrange interpolation (as shown in section 8.2.2). Remarks. (1) From Figure 8.5, we note that for linear local approximations or Lagrange functions the degree of the polynomials is one (p = 1). For quadratic local approximations the degree of the polynomial is two (p = 2) and for cubic local approximations the degree of the polynomial is three (p = 3). Thus, this process does provide a mechanism for increasing the degree of local approximation, i.e. p-level. (2) As we increase the p-level (i.e. degree of local approximation) additional ¯ e ), which is nodes need to be added to the existing elements (domain Ω an undesirable feature. (3) If we examine the basis functions (Lagrange interpolation functions) for the linear and quadratic case, we note that Lagrange functions L1 (ξ) = 1−ξ 1+ξ 2 and L2 (ξ) = 2 for linear approximation do not explicitly appear

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

Nodal configurations 1

2

ξ

-1 (linear: p = 1) 1

f (ξ) =

2 P

Li (ξ) fi

i=1

1

2

3

-1

0

1

ξ

3 P

Li (ξ) fi

2

3

4

-1

− 13

1 3

1

(cubic: p = 3)

f (ξ) =

4 P

1−ξ 2

L2 (ξ) =

1+ξ 2

L1 (ξ) =

ξ(ξ−1) 2

L2 (ξ) = 1 − ξ 2

Type C0

C0

ξ(ξ+1) 2

9 1 L1 (ξ) = − 16 ( 3 + ξ)( 31 − ξ)(1 − ξ)

i=1

1

L1 (ξ) =

L3 (ξ) =

(quadratic: p = 2)

f (ξ) =

Basis functions

503

ξ

L2 (ξ) = L3 (ξ) = L4 (ξ) =

27 1 16 (1 + ξ)(1 − ξ)( 3 − ξ) 1 27 16 (1 + ξ)(1 − ξ)( 3 + ξ) 9 1 ( 3 + ξ)( 31 − ξ)(1 + ξ) − 16

C0

Li (ξ) fi

i=1

¯ ξ = [−1, 1] Figure 8.5: 1D Lagrange basis functions Ω

in the Lagrange functions for the quadratic or the cubic local approximations. Yet, we know that a quadratic local approximation is quite ¯ ξ . In other words, the linear capable of linear local approximations over Ω behavior is implicitly embedded in the quadratic behavior but the linear basis functions do not explicitly appear in the basis functions for the quadratic case. This property of quadratic local approximation to contain linear approximation is known as an ‘embedding property’. Thus, in general we can say that when 1D local approximations are constructed using Lagrange interpolation functions, the lower degree basis functions are implicitly embedded in the higher degree basis functions. Due to the fact that the embedding property of these basis functions is not explicit, when computations are performed for progressively increasing p-levels (degree of the polynomial), we cannot take advantage of the computations performed at the lower p-level when performing computations for higher p-levels. That is, when computing at p-level of (p + 1), computations performed at p-level of p are of no use i.e. cannot be used. (4) In order to remedy the shortcomings described in (3), we need to accomplish the following. (a) We should be able to increase the p-level, i.e. degree of local approximation, without adding additional nodes. With this feature the discretizations and the nodes will remain fixed for all p-levels, which is advantageous.

504

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

(b) The embedding property of the higher degree local basis functions needs to be explicit rather than implicit. That is, when we examine the basis functions for the quadratic local approximation, the linear basis functions and the corresponding nodal degrees of freedom must appear explicitly in the quadratic case. In other words, a linear local approximation must be an explicit subset of a quadratic local approximation and likewise a quadratic local approximation must be a subset of a cubic approximation and so on. In general, a local approximation of degree p must be a complete and explicit subset of the local approximation of degree (p + 1). This property will be referred to as the hierarchical property of the local approximations. Derivation of C 0 p-version hierarchical basis functions Recall that the local approximations derived using Lagrange interpolation functions do posses the embedding property, but the embedding property is implicit. The purpose of this derivation is to make the embedding property explicit and at the same time eliminate the need for additional nodes for increasing p-levels. We present details of the development in the following. Let (ξi , f (ξi )) = (ξ, fi ) (i = 1, . . . , n) be the node locations and function values at n nodes of an n-node Lagrange element. Then f (ξ) =

n X

Li (ξ)fi

(8.22)

i=1

in which Li (ξ) are Lagrange interpolation functions or polynomials. Consider a quadratic case, that is n = 3 and substitute Li (ξ) (i = 1, 2, 3) in (8.22) 1 1 f (ξ) = − ξ(1 − ξ)f1 + (1 − ξ 2 )f2 + ξ(1 + ξ)f3 2 2

(8.23)

By rearranging terms in (8.23) we can write  ξ2 − 1  1 1 f (ξ) = (1 − ξ)f1 + (1 + ξ)f3 + (f1 − 2f2 + f3 ) 2 2 2

(8.24)

Differentiating (8.23) or (8.24)twice with respect to ξ we obtain d2 f = f1 − 2f2 + f3 dξ 2

(8.25)

Note that for the three node quadratic element, node 2 is located at ξ = 0, so that (8.25) can be written as  d2 f  dξ 2

ξ=0

= f1 − 2f2 + f3

(8.26)

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

505

Substituting back into (8.24) we obtain  ξ 2 − 1  d2 f  1 1 f (ξ) = (1 − ξ)f1 + (1 + ξ)f3 + 2 2 2 dξ 2 ξ=0

(8.27)

where f1 = f |ξ=−1 and f3 = f |ξ=+1 . Recall that for the linear approximation of f (two-node element) we have f (ξ) =

1 − ξ  2

f1 +

1 + ξ  2

f2

(8.28)

where f1 = f |ξ=−1 and f2 = f |ξ=+1 . Comparing (8.28) with (8.24) we note that the linear approximation is an explicit subset of the quadratic approximation in the sense that the linear approximation functions and the nodal dofs in (8.28) appear explicitly in the quadratic approximation (8.27). Thus, when using (8.27) and (8.28) for linear and quadratic approximations of f , when we increase the p-level form 1 to 2 we only need to add the  ξ 2 −1 d2 f term to the linear approximation to obtain a quadratic ap2 dξ 2 ξ=0

proximation. Hence, the linear approximation is explicitly embedded in the quadratic approximation. In other words, the linear and quadratic approximations described by (8.27) and (8.28) are hierarchical. ¯ ξ = [−1, 1] for a Next, we consider a cubic Lagrange interpolation over Ω ξ ¯ four-node element in Ω , and we can write f (ξ) =

4 X

Li (ξ)fi

(8.29)

i=1

or  1  1    1  9 27  f (ξ) = − 1−ξ +ξ − ξ f1 + 1+ξ 1−ξ − ξ f2 16 3 3 16 3   1   1   9 1 27  + 1+ξ 1−ξ + ξ f3 − +ξ − ξ 1 + ξ f4 (8.30) 16 3 16 3 3 If we differentiate f (ξ) with respect to ξ twice and thrice and evaluate these at ξ = 0 and then substitute these into (8.30), we obtain  ξ 2 − 1  d2 f   ξ 3 − ξ  d3 f  1 1 + f (ξ) = (1 − ξ)f1 + (1 + ξ)f3 + 2 2 2! dξ 2 ξ=0 3! dξ 3 ξ=0 (8.31) Comparing the cubic approximation (8.31) with the quadratic approximation (8.27), we note  3that  the cubic approximation can be obtained by adding the ξ 3 −ξ d f term to the quadratic approximation. Another point to 3! dξ 3 ξ=0

note is that the addition of this term does not require a new node, i.e. for the cubic approximation the same three-node configuration used for the

506

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

quadratic case suffices. Clearly, the quadratic approximation is a complete subset of the cubic approximation, i.e. has an explicit embedding property and hence is hierarchical. The linear, quadratic and cubic approximations described by (8.28), (8.24) and (8.31) are hierarchical C 0 p-version approximations which require only a three-node element configuration in which the nodes are located at ξ = −1, ξ = 0 and ξ = 1. Thus, we note that using the procedure described here the two-node, three-node and four-node Lagrange elements have been reduced to an equivalent three-node configuration for the linear, quadratic and cubic approximations described by (8.28), (8.27) and (8.31) (see Fig. 8.6). In general, if we consider pξ , the degree of Lagrange interpo¯ ξ and follow a similar procedure lating polynomials over (pξ + 1), nodes in Ω as described above, then we find that this can also be reduced to a threenode configuration (with nodes at ξ = −1, ξ = 0 and ξ = 1) for which we can write the following hierarchical p-version approximation for f (ξ). 1

2

-1

1

1

2

-1

3

0

ξ

ξ

1

1

2

3

4

-1

− 13

1 3

1

         

ξ

can be reduced to

        

1

2

3

-1

0

1

ξ

3-node p-version element

.. . Lagrange elements Figure 8.6: Lagrange to Hierarchical interpolation functions

pξ

X ξ i − a  di f  1 1 f (ξ) = (1 − ξ)f1 + (1 + ξ)f3 + 2 2 i! dξ i ξ=0

(8.32)

i=2

or f (ξ) = N11ξ (ξ)f |ξ=−1 + N31ξ (ξ)f |ξ=+1 +

pξ X

N2iξ (f,ξi )ξ=0

(8.33)

i=2

where N11ξ =

1 + ξ   ξi − a  , N31ξ = , N2iξ = 2 2 i!  di f  di f (f,ξi )ξ=0 = = dξ i ξ=0 dξ i node 2

1 − ξ 

(8.34)

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

507

and ( 1, a= ξ,

i is even i is odd

Equations (8.32) and (8.33) represents hierarchical p-version C 0 approxima¯ ξ = [−1, 1] in which the element nodal configuration is a tion of f over Ω three-node configuration with nodes at ξ = −1, 0 and +1 and remains fixed regardless of p-level. The p-version hierarchical approximation of C 0 type described by (8.32) or (8.33) can be more conveniently expressed in tabular form in terms of the basis functions and nodal variable operators as opposed to nodal variables, as shown in Table 8.1. Nodal variable operators assigned to specific nodes act on the dependent variables to produce nodal degrees of freedom. Table 8.1: C 0 basis functions and nodal variable operators 1D hierarchical p-version basis functions (C 0 type)

1

2

3

−1

0

1

1 (1 2

1 (1 2

− ξ)

p-level

1D hierarchical p-version nodal variable operators (C 0 type)

ξ

+ ξ)

1

2

3

−1

0

1

1

ξ2 −1 2! ξ3 −ξ 3! ξ4 −1 4!

1

1 d2 dξ2 d3 dξ3 d4 dξ4

2 3 4 .. .

.. . p ξ ξ −a pξ !

ξ

.. . p

d ξ p dξ ξ

pξ

8.2.4 Higher order global differentiability approximations in one dimension: p-version Let x = x(ξ) define the map of a three-node (equally spaced) element ¯ e ) to ξ-space (Ω ¯ ξ ). Let xi (i = 1, 2, 3) be the coordinates of from x-space (Ω the nodes of the element in x-space. Then, x(ξ) =

1 − ξ  2

x1 +

1 + ξ  2

x3

(8.35)

defines the mapping of the points from ξ-space to x-space. As usual, we have J=

dx he = dξ 2

(8.36)

508

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

where he is the element length in x-space. Let φ be the quantity of interest to ¯e ¯e be interpolated S ¯ e over Ω . Ω is the domain of an element e of the discretization T ¯ ¯ e by φe . Ω = e Ω . In finite element processes we approximate φ over Ω h S T e ¯ is given by φh = Thereby they approximation of φh over Ω e φh . The global differentiability of φh is naturally dependent on the differentiability S achievable by e φeh and is controlled by the inter-element boundaries. Thus, in this section we consider development of φeh , that is, approximations of φ ¯ e that are capable of yielding global differentiability of orders 1, . . . , i over Ω of φh . For the sake of simplicity we drop the subscript h and superscript e ¯ e ), C 2 (Ω ¯ e ) and in the following. We consider approximations of type C 1 (Ω i e ¯ also of C (Ω ). ¯ e) 8.2.4.1 Local approximation of class C 1 (Ω In the following we switch from f to φ, as in finite element processes we have commonly used φ for the dependent variable. ¯ξ The one dimensional C 0 p-version hierarchical interpolation of φ over Ω (a three-node element) can be written as (for the 3-node configuration shown in Fig. 8.7) φ(ξ) =

1 − ξ  2

φ1 +

where

1 + ξ  2

p  i X ξ − a  di φ φ2 + i! dξ i ξ=0

(8.37)

i=2

( 1, i is even a= ξ, i is odd η ξ 1 ξ = −1

2

3 2 ξ=0

ξ=1

Figure 8.7: A C 0 , p-version hierarchical interpolation for the 3-node configuration

Here, φ1 and φ2 are the nodal values of φ at the end nodes 1 and 2. Differentiating φ(ξ) with respect to ξ, we obtain p  i−1 X iξ dφ 1 1 −a ¯  di φ = − φ1 + φ2 + dξ 2 2 i! dξ i ξ=0 i=2

(8.38)

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

where

( 0, a ¯= 1,

Evaluating these by

dφ dξ

dφ1 dξ

509

i is even i is odd

at ξ = −1 (node 1) and at ξ = +1 (node 2) and denoting

and

dφ2 dξ ,

we obtain

dφ dφ1 1 1 d2 φ 1 d3 φ = = − φ1 + φ2 − + dξ ξ=−1 dξ 2 2 dξ 2 ξ=0 3 dξ 3 ξ=0 p  X i(−1)i−1 − a ¯  di φ + i! dξ i ξ=0

(8.39)

i=4

dφ2 1 1 d2 φ 1 d3 φ dφ = = − φ1 + φ2 + + dξ ξ=1 dξ 2 2 dξ 2 ξ=0 3 dξ 3 ξ=0 p  X i(1)i−1 − a ¯  di φ + i! dξ i ξ=0

(8.40)

i=4

Solving for



d2 φ dξ 2 ξ=0

and



d3 φ dξ 3 ξ=0

from (8.39) and (8.40) and substituting

these into (8.37), we obtain φ(ξ) =

 (ξ 3 − ξ) (ξ 2 − 1)  dφ (ξ 3 − ξ)  1 φ1 + − 2 4 4 4 dξ  (1 + ξ) (ξ 3 − ξ)   (ξ 3 − ξ) (ξ 2 − 1)  dφ 2 + − φ2 + + 2 4 4 4 dξ p  (ξ i − 1) − 1 (ξ 2 − 1)  di φ X 2 + i! dξ i

 (1 − ξ)

+

ξ=0

i=4,6,...

+

p X i=5,7,...

 (ξ i − ξ) −


i−1 2

i!

(ξ 3 − ξ)  di φ (8.41) dξ i ξ=0

The interpolation defined in (8.41) for φ(ξ) ensures inter-element continuity of both φ and dφ dξ in the natural coordinate space ξ. However, in the solution of differential equations using the finite element method, we require interelement continuity of φ and dφ dx . This can be accomplished easily by noting that dφ 1 dφ = (8.42) dx J dξ Hence, dφ1 dφ1 dφ2 dφ2 =J and =J (8.43) dξ dx dξ dx

510

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

If the element is a straight line with equally spaced nodes, then J = h2e where he is the length of element e. Thus, the interpolation (8.41) can be written as φ(ξ) =

 (ξ 3 − ξ) (ξ 2 − 1)  dφ (ξ 3 − ξ)  1 φ1 + − J 2 4 4 4 dx  (1 + ξ) (ξ 3 − ξ)   (ξ 3 − ξ) (ξ 2 − 1)  dφ 2 + − φ2 + + J 2 4 4 4 dx p  (ξ i − 1) − 1 (ξ 2 − 1)  di φ X 2 + i! dξ i ξ=0 i=4,6,... p  (ξ i − ξ) − i−1
(ξ 3 − ξ)  di φ X 2 (8.44) + i! dξ i ξ=0

 (1 − ξ)

+

i=5,7,...

or φ(ξ) =

N10 (ξ)φ1

+

dφ1 N11 (ξ) dx

+

N20 (ξ)φ2

+

dφ2 N21 (ξ) dx

+

p X

N3i (ξ)

i=4

di φ dξ i ξ=0 (8.45)

where (ξ 3 − ξ)  2 4  (ξ 3 − ξ) (ξ 2 − 1)  N11 (ξ) = − J 4 4  (1 + ξ) (ξ 3 − ξ)  − N20 (ξ) = 2 4  (ξ 3 − ξ) (ξ 2 − 1)  + J N21 (ξ) = 4 4  (ξi −1)− 1 (ξ2 −1)  2  i is even i! N3i (ξ) =  (ξi −ξ)− i−1 (ξ3 −ξ)   2 i is odd i! N10 (ξ) =

 (1 − ξ)

+

(8.46)

(i = 4, 5, . . . , p)

We can illustrate these graphically in terms of nodal approximation functions and the nodal variable operators as follows (see Fig. 8.8). The interpolation functions defined in (8.44) are unique and satisfy interelement continuity of both φ and dφ dx . To establish uniqueness, it suffices to consider a special case of (8.44) in which the terms corresponding to i = 4, 5, . . . , p are absent. In particular, we need to show that (1) the dimension of the space of φ(ξ) is the same as the total degrees of freedom for the element and (2) that if all the degrees of freedom are identically zero then ¯ ξ . With i = 4, 5, . . . , p absent in (8.44) we have a complete φ(ξ) = 0 ∀ξ ∈ Ω cubic polynomial in ξ with basis functions or monomials 1, ξ, ξ 2 and ξ 3 which

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

η 1

511

η

3

1

2

3

2

ξ

ξ

N10 (ξ)

N20 (ξ)

1

N11 (ξ)

N21 (ξ)

d dx

1 d dx

di i dξ

N3i (ξ) i = 4, 5, . . . , p

ξ=0

i = 4, 5, . . . , p

¯ e ) basis functions C 1 (Ω

Nodal variable operators ¯ e ) interpolations C 1 (Ω

for

Figure 8.8: C 1 basis functions and nodal variable operators

is a complete set of linearly independent monomials. Hence, the dimension of the space of φ(ξ) is four (equal to the number of basis functions). The dφ2 1 degrees of freedom for this element are φ1 , dφ dx , φ2 and dx . If φ1 = 0, dφ1 dφ2 dx = 0, φ2 = 0 and dx = 0 and if φ(ξ) is a cubic polynomial in ξ, then ¯ ξ . Alternatively, since φ(ξ) = N 1 φ1 +N 1 dφ1 +N 0 φ2 +N 1 dφ1 φ(ξ) = 0 ∀ξ ∈ Ω 0 1 dx 2 2 dx ¯ ξ if all of the degrees of freedom at the nodes are zero. then φ(ξ) = 0 ∀ξ ∈ Ω The inter-element continuity follows from the fact that both φ and dφ dx are nodal degrees of freedom at the end nodes of the element. ¯ e ): 8.2.4.2 Interpolations or local approximations of class C 2 (Ω ¯ e ) basis Following the procedure similar to that used for deriving C 1 (Ω 2 e ¯ ) type p-version interpolation functions (or functions, we can derive C (Ω ¯ e ) be given by (C 0 basis functions) in one dimension. Let φ(ξ) of class C 0 (Ω p-version hierarchical for the three node configuration). φ(ξ) =

1 − ξ  2

φ1 +

 ξ 2 − 1  d2 φ  ξ 3 − ξ  d3 φ φ2 + + 2 2! dξ 2 ξ=0 3! dξ 3 ξ=0 p  ξ 5 − ξ  d5 φ X di φ i + + N3 (ξ) i (8.47) 5! dξ 5 dξ

1 + ξ 

 ξ 4 − 1  d4 φ + 4! dξ 4 ξ=0

ξ=0

i=6

ξ=0

Hence  2   3 4 3ξ − 1 d3 φ ξ dφ 1 1 d2 φ d φ = − φ1 + φ2 + ξ 2 + + 3 dξ 2 2 dξ ξ=0 3! dξ ξ=0 6 dξ 4 ξ=0  4  p X 5ξ − 1 d5 φ dN3i di φ + + (8.48) 5! dξ 5 ξ=0 dξ dξ i ξ=0 i=1

512

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

 2 4  3 5 d2 φ d φ d φ d2 φ d3 φ ξ ξ = + ξ + + dξ 2 dξ 2 ξ=0 dξ 3 ξ=0 2 dξ 4 ξ=0 6 dξ 5 ξ=0 p X d2 N3i (ξ) di φ + dξ 2 dξ i ξ=0

(8.49)

i=6

We use (8.48) and (8.49) to evaluate then define

dφ dξ

and

d2 φ dξ 2

at ξ = −1 and ξ = 1 and

 p   P dN3i   di φ    dξ ξ=−1 dξ i ξ=0   dφ   d2 φ      i=6 1  p         2       i P dξ dξ i dN       d φ 3 3φ         dφ 2 d       i dξ dξ φ1 3 ξ=1 ξ=0 dξ dξ i=6 + P = [B] + [A] d4 φ (8.50)  p  2 i d2 φ1 φ2 d N3       di φ     dξ 2  dξ 4        2 i dξ      d2 φ2    d5 φ    ξ=−1 dξ ξ=0  i=6     p  2 i dξ 2   dξ 5 ξ=0 P iφ d N   d 3     2 i dξ dξ i=6

ξ=1

ξ=0

where dφ2 dφ dφ dφ1 , = = dξ dξ ξ=−1 dξ dξ ξ=1 d2 φ1 d2 φ d2 φ2 d2 φ = , = dξ 2 dξ 2 ξ=−1 dξ 2 dξ 2 ξ=1

(8.51)

and  1  −1 31 − 61 30  1 1 1 1 3 6 30  [A] =   1 −1 1 − 1  , 2 6 1 1 21 16

 1 −2 − 1 2 [B] =   0 0

1 2 1 2

(8.52)

0 0

From (8.50) we obtain  p   P dN3i   di φ    dξ ξ=−1 dξ i ξ=0   dφ     d2 φ    i=6 1         p  2       i P dξ dξ i dN       d φ 3 3φ         dφ 2 d       i dξ dξ φ 3 ξ=1 ξ=0 dξ 1 −1 −1 −1 dξ i=6 2φ = [A] −[A] [B] −[A]   4 p d 1 d φ P d2 N3i φ2      di φ     dξ 2  dξ 4        2 i dξ dξ 2       5  d φ2    d φ ξ=−1 ξ=0 i=6   2     p dξ   dξ 5 ξ=0 2 i P iφ d N   d 3     i dξ 2 dξ i=6

ξ=1

ξ=0

(8.53)

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

513

Next, we rewrite (8.47) in the form



φ1 φ(ξ) = [a(ξ)] φ2

 +

 d2 φ    dξ2       d3 φ  

3 [b(ξ)] ddξ4 φ     dξ5 4    d φ  5 dξ ξ=0

+

p X

N3i

i=6

di φ dξ i ξ=0

(8.54)

where [a(ξ)] =

 1−ξ

[b(ξ)] =

h

2

1+ξ 2



ξ 2 −1 ξ 3 −ξ ξ 4 −1 ξ 5 −ξ 2! 3! 4! 5!

(8.55)

i

Substituting from (8.50) into (8.54), we obtain



φ1 φ(ξ) = [a(ξ)] φ2 



 p    P dN3i   di φ    dξ ξ=−1 dξ i ξ=0    dφ     i=6  1        p       i P dξ i dN     d φ 3         dφ2   i dξ dξ   φ1 ξ=1 ξ=0 dξ i=6  2φ + [b(ξ)][A]−1  − [B] −   p d  21   P d2 N3i iφ φ   d 2      dξ    2 i dξ dξ 2       d φ     ξ=−1 ξ=0 2 i=6  dξ2      p  2 i     P i d N3   d φ     dξ 2 dξ i i=6

+

ξ=1 p X

N3i

i=6

ξ=0

di φ (8.56) dξ i ξ=0

Using dφ dφ =J dξ dx

and

2 d2 φ 2d φ = J dξ 2 dx2

(8.57)

we have h

dφ1 dφ2 d2 φ1 d2 φ2 dξ dξ dξ 2 dξ 2

iT

h iT dφ2 2 d2 φ1 J 2 d2 φ2 1 = J dφ J J dx dx dx2 dx2

(8.58)

514

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Substituting from (8.58) into (8.56), we obtain  φ(ξ) = [a(ξ)]

φ1 φ2 



 p    P dN3i   di φ    dξ ξ=−1 dξ i ξ=0         i=6     dφ1    P p  J      i dN3   dx  di φ         dφ     2 i dξ  J dx  φ1 ξ=1 dξ ξ=0 −1  i=6 + [b(ξ)][A]  − [B] − P   p  2 i d2 φ1 2  i φ2 d N3   d φ 2    J dx      2 i dξ dξ    2 d2 φ2      ξ=−1 ξ=0  i=6  J dx2        p     P d2 N3i   di φ     dξ 2 dξ i ξ=1

i=6

+

p X i=6

ξ=0

di φ

i N3 i dξ

(8.59) ξ=0

From (8.59) we can write  dφ  J dx1         J dφ2   h i φ 1 dx 2φ φ(ξ) = [a(ξ)] − [b(ξ)][A]−1 [B] + [b(ξ)][A]−1 d φ2  J 2 21     2φ   2 ddx 2 J dx2   p   i P dN3   di φ      i=6 dξ ξ=−1 dξi ξ=0         p    i P i dN3   d φ   p  X  i dξ ξ=1 dξ i ξ=0 −1 i d φ i=6 (8.60) − [b(ξ)][A] N3 i  + p  2 i P d N3   dξ ξ=0 di φ   i=6  dξ 2 ξ=−1 dξ i ξ=0      i=6       p   P d2 N3i   di φ     dξ 2 dξ i i=6

ξ=1

ξ=0

Finally, from (8.60) we can write dφ1 ˆ21 (ξ) dφ2 +N dx dx p 2 2 X d φ1 d φ2 di φ 2 2 i ˆ ˆ ˆ + N1 (ξ) 2 + N2 (ξ) 2 + N3 (ξ) (8.61) dx dx dξ ξ=0

ˆ10 (ξ)φ1 + N ˆ20 (ξ)φ2 + N ˆ11 (ξ) φ(ξ) = N

i=6

¯ e ) interpolation or approximation of φ over Ω ¯ e. This is the desired C 2 (Ω These are illustrated for the three-node element in Fig. 8.9. ¯ e ) type interpolation functions in (8.61) are We can show that the C 2 (Ω d2 φ unique and satisfy the inter-element continuity of φ, dφ dx and dx2 . To show the ¯ e ), it suffices to consider a special case of (8.61) in uniqueness of φ ∈ C 2 (Ω

¯ ξ = [−1, 1] 8.2. ELEMENTS OF INTERPOLATION THEORY OVER Ω

η 1

515

η

3

1

2

3

2

ξ

ξ

ˆ 0 (ξ) N 1

ˆ 0 (ξ) N 2

1

1

ˆ 1 (ξ) N 1

ˆ 1 (ξ) N 2

d dx

d dx

ˆ 2 (ξ) N 1

ˆ 2 (ξ) N 2

d2 dx2

d2 dx2



di dξ i

ˆ i (ξ) N 3 i = 6, 7, . . . , p ¯ e ) basis functions C 2 (Ω

ξ=0

i = 6, 7, . . . , p Nodal variable operators ¯ e ) interpolations C 2 (Ω

for

¯e Figure 8.9: C 2 basis functions and nodal variable operators in Ω

which the terms corresponding to i = 6, . . . , p are absent. We need to show that (1) the dimension of the space of φ(ξ) is the same as the total number of degrees of freedom for the element, and (2) if all degrees of freedom for ¯ ξ . With i = 6, . . . , p absent, the element are zero then φ(ξ) = 0 ∀ξ ∈ Ω we have a complete fifth degree polynomial in ξ for φ(ξ) with monomials [1 ξ ξ 2 ξ 3 ξ 4 ξ 5 ]; hence, the dimension of the space of φ(ξ) is six (equal to the dofs or the number of monomials). The dofs for this element are 2 dφ2 d2 φ2 1 d φ1 φ1 , dφ dx , dx2 and φ2 , dx , dx2 at the two end nodes of the element. Thus, the dimension of the space is equal to the total number of degrees of freedom ¯ ξ . This for the element. If all dofs are zero, then obviously φ(ξ) = 0 ∀ξ ∈ Ω 2 e ¯ is obvious from (8.61). Thus, C (Ω ) interpolation functions in (8.61) are unique. The C 2 inter-element continuity is inherent in φ(ξ) due to the fact d2 φ that φ, dφ dx and dx2 are degrees of freedom at the end nodes of the element. ¯ e) 8.2.4.3 Local approximations of class C i (Ω ¯ e ) and Following the procedure presented for interpolation of types C 1 (Ω ¯ e ), we can derive the following for C i (Ω ¯ e ) type interpolation functions C 2 (Ω in one dimension that ensure continuity of the interpolation of φ of order i ¯ ξ , we can write ¯ T of Ω. ¯ For φ(ξ) ∈ C i (Ω ¯ e ) ∀ξ ∈ Ω over the discretization Ω d2 φ1 di φ1 dφ1 + N 21 (ξ) 2 + · · · + N i1 (ξ) i φ(ξ) = N 01 (ξ)φ1 + N 11 (ξ) dx dx dx e e e e 2 dφ2 d φ2 di φ2 + N 02 (ξ)φ1 + N 12 (ξ) + N 22 (ξ) 2 + · · · + N i2 (ξ) i dx dx dx e e e e pξ j X d φ j + N 3 (ξ) j (8.62) dξ ξ=0 e j=2(i+1)

516

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Proofs of uniqueness and the inter-element continuity follow directly by in¯ e ) type interpolation. ¯ e ) and C 2 (Ω duction using the proofs given for C 1 (Ω Details of (8.62) can be illustrated for the three node configuration shown in Fig. 8.10. η 1

3

η 1

2

3

2

ξ j

j

N (ξ)

N (ξ)

e1

e2

ξ dj dxj

dj dxj



dk dξ k ξ=0

N k (ξ) e3

j = 0, 1, 2, . . . , i

j = 0, 1, . . . , i

k = 2(i + 1), . . . , p

k = 2(i + 1), . . . , p

¯ e ) basis functions C i (Ω

Nodal variable operators ¯ e ) interpolations C i (Ω

for

¯e Figure 8.10: C i basis functions and nodal variable operators in Ω

8.3 Mapping in two dimensions: quadrilateral elements In two dimensions, the element geometry may be of triangular shape or of quadrilateral shape with straight or curved sides. Such distorted irregular shapes present difficulty when performing integration over the area of the element for calculating coefficients of the element matrices and vectors. Another difficulty that is much more significant is the construction of local approximation functions for quadrilateral and irregular shapes. Both of these difficulties can be overcome by mapping the element physical domains from (x, y)-space into a natural coordinate space (ξ, η)-space. In the following, we first consider quadrilateral shapes with curved boundaries. To illustrate the mapping of points, lines, and areas between the (x, y) and (ξ, η) systems, ¯ e in the (x, y) coordinate (physical) space, consider a four-sided element Ω as shown in Fig. 8.11. Let (xi , yi ) (i = 1, . . . , n) be the coordinates of the nodes located on the boundary (and possibly in the interior) of the element. ¯ e into a square of two units, Ω ¯ ξη , with the origin of the Consider a map of Ω ¯ ξη . coordinate system (ξ, η) located at the center of Ω Mapping of points: Regardless of the precise nature of the mapping, in the abstract sense one could describe this mapping by x = x(ξ, η), y = y(ξ, η)

(8.63)

517

8.3. MAPPING IN TWO DIMENSIONS: QUADRILATERAL ELEMENTS

η

7

4

9

8 y

2

1

7

5

6

2

8

3

5

6

9

1

4

2

ξ

3

2

x ¯ e in xy space Ω

Element map in the natural coordinate space ξη

Figure 8.11: Mapping of points from a physical element to points in a square of two units in natural coordinates

and if the mapping is one-to-one and onto, then the inverse of the mapping exists and is unique and we could also write ξ = ξ(x, y), η = η(x, y)

(8.64)

In the present case the form (8.63) is preferable over (8.64). In (8.63), we note that (a) The mapping (8.63) is explicit in ξ and η but (8.64) is implicit in x and y; that is, given ξ ∗ and η ∗ in (ξ, η)-space, the mapping allows explicit determination of their map (x∗ , y ∗ ) in the (x, y)-space. (b) However, given (x∗ , y ∗ ) in (x, y)-space, determination of their map (ξ ∗ , η ∗ ) in (ξ, η)-space is not explicit but requires solution of a system of simultaneous equations in ξ and η. (c) The form of the mapping (8.63) only allows mapping of points from (ξ, η)-space to (x, y)-space. This mapping has no concept of lengths or areas. Explicit form of the mapping in (8.63) is not too difficult to establish. ¯i (ξ, η) be the basis functions in the natural coordinate space (ξ, η) such Let N that ( j=i ¯i (ξj , ηj ) = 1, N , i = 1, . . . , m (8.65) 0, j 6= i and

m X i=1

¯i (ξ, η) = 1 N

(8.66)

518

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Then m X

x=

¯i (ξ, η)xi N

i=1

y=

m X

(8.67) ¯i (ξ, η)yi N

i=1

¯ e in the natural coordinate space ξ, η. accomplishes the desired mapping of Ω The element map in the natural coordinate space is also generally referred ¯ m or Ω ¯ ξη . to as the master element denoted by Ω Mapping of lengths In this section we establish a relationship between line lengths dx and dy along the x and y axes and lengths dξ and dη along the ξ and η axes (see Fig. 8.12). Since, x = x(ξ, η) and y = y(ξ, η) we can write ∂x ∂x dξ + dη ∂ξ ∂η ∂y ∂y dy = dξ + dη ∂ξ ∂η

dx =

or in matrix form   " ∂x dx ∂ξ = ∂y dy

∂x ∂η ∂y ∂ξ ∂η

in which

#

dξ dη





x x = ξ η yξ yη







x x [J] = ξ η yξ yη

dξ dη

(8.68)





dξ = [J] dη

 (8.69)

(8.70)

is called the Jacobian of transformation. For the transformation (8.63) to η

dy dx ξ dη y

dξ x ¯e Domain Ω

¯ ξ or Ω ¯m Domain Ω

Figure 8.12: Mapping of lengths from a physical element to lengths in a square of two units in natural coordinates

519

8.3. MAPPING IN TWO DIMENSIONS: QUADRILATERAL ELEMENTS

be one-to-one and onto det [J] > 0

¯ m or Ω ¯ ξη ∀(ξ, η) ∈ Ω

Clearly, (8.70) describes the relationship between the elemental lengths in the two coordinate systems. Derivatives of x and y with respect to ξ and η needed in [J] can be easily obtained using (8.67). Mapping of areas Consider elemental length dx, dy along x and y axes forming an area dΩ = dx dy in the x, y-space. Likewise consider length dξ, dη along ξ and η forming an area dΩm = dξ dη. In this section we establish a relationship between dΩ and dΩm . Let ~i and ~j be the unit vectors along x and y axes and ~eξ and ~eη be the unit vectors along ξ and η axes (see Fig. 8.13). η

dy dx

~eη

ξ

~eξ dη y

dξ

~j ~i

x

¯ ξ or Ω ¯m Domain Ω

¯e Domain Ω

Figure 8.13: Mapping of areas from a physical element to areas in a square of two units in natural coordinates

Then the cross product of the vectors dx~i and dy ~j would yield a vector perpendicular to the plane containing vectors dx~i and dy ~j and the magnitude of this vector represents the area formed by the two vectors, dΩ. Thus, dx~i × dy ~j = dx dy ~i × ~j = dx dy ~k (8.71) but ∂x ∂x dξ ~eξ + dη ~eη ∂ξ ∂η ∂y ∂y dy ~j = dξ ~eξ + dη ~eη ∂ξ ∂η dx~i =

(8.72)

Substituting from (8.72) into the left side of (8.71) we obtain dx dy ~k =



∂x ∂x ∂y ∂y dξ ~eξ + dη ~eη × dξ ~eξ + dη ~eη ∂ξ ∂η ∂ξ ∂η

(8.73)

520

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

expanding the right side of (8.73) gives ∂x ∂y ∂x ∂y dx dy ~k = dξ dξ ~eξ × ~eξ + dξ dη ~eη × ~eξ ∂ξ ∂ξ ∂η ∂ξ ∂x ∂y ∂x ∂y + dξ dη ~eξ × ~eη + dη dη ~eη × ~eη ∂ξ ∂η ∂η ∂η

(8.74)

Noting that ~eξ × ~eξ = 0 = ~eη × ~eη ~eξ × ~eη = ~eζ = ~k

(8.75)

~eη × ~eξ = −~eζ = −~k we obtain the following from (8.75)   ∂x ∂y ∂x ∂y ~ dx dy k = − dξ dη ~k ∂ξ ∂η ∂η ∂ξ Hence

 dx dy =

∂x ∂y ∂x ∂y − ∂ξ ∂η ∂η ∂ξ

but det [J] = |J| =

(8.76)

 dξ dη

∂x ∂y ∂x ∂y − ∂ξ ∂η ∂η ∂ξ

(8.77)

(8.78)

Hence dx dy = |J| dξ dη

(8.79)

dΩe = |J| dΩm = |J| dΩξη

(8.80)

or Equation (8.80) gives the desired relationship between the elemental areas in the two coordinate spaces.

¯ m : quadrilateral el8.4 Local approximation over Ω ements ¯ ξη , then we can write Let φ(ξ, η) is the local approximation of φ over Ω n X

Ni (ξ, η) δie

(8.81)

φ(ξ, η) = [N (ξ, η)]{δ e }

(8.82)

φ(ξ, η) =

i=1

or where Ni (ξ, η) are the local approximation functions corresponding to the nodes of the element and δie are nodal degrees of freedom. Ni (ξ, η) are also

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

521

referred to as basis functions for the element e. Explicit details of Ni (ξ, η) (i = 1, . . . , n), n being the total degrees of freedom and the corresponding δie depend upon many considerations. (1) The first important issue is the choice of the nodal configuration for the element. That is, how many nodes and their locations. (2) Means of constructing Ni (ξ, η) for: (a) (b) (c) (d)

C 0 local approximation of higher degree C 0 p-version hierarchical local approximations C i,j (i, j ≥ 1) for rectangular family elements C i,j (i, j ≥ 1) for elements with distorted shapes in the x, y-space

(3) In the development of (2), the choices of δie , nodal dofs are of course crucial and require careful considerations. (4) The developments in (2) and (3) could be based on: (a) Lagrange interpolation functions, (b) Legendre polynomials or (c) Chebyshev polynomials. The specific details presented in the following are based on Lagrange polynomials. Their extensions to the other two types are straight forward and the details can be found in the cited references. Obtaining derivatives of φ(ξ, η) with respect to x and y With the local approximation of φ defined by (8.82) in which φ = φ(ξ, η), obtaining derivatives of φ(ξ, η) with respect to x and y needed in the integral form is not direct and needs to be considered. First, we note that when φ(ξ, η) is defined by (8.82) we can write n

∂φ X ∂Ni (ξ, η) e = δi ∂x ∂x i=1

∂φ = ∂y

n X ∂Ni (ξ, η) i=1

∂y

(8.83) δie

∂φ ∂Ni ∂Ni Thus, obtaining ∂φ ∂x and ∂y implies establishing ∂x and ∂y (i = 1, . . . , n). Once again, since Ni (i = 1, . . . , n) are functions of ξ and η and x = x(ξ, η), y = y(ξ, η), we can proceed as follows.

∂Ni ∂Ni ∂x ∂Ni ∂y = + ∂ξ ∂x ∂ξ ∂y ∂ξ ∂Ni ∂Ni ∂x ∂Ni ∂y = + ∂η ∂x ∂η ∂y ∂η

(8.84)

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

arranging (8.84) into matrix form (

∂Ni ∂ξ ∂Ni ∂η

)

" =

∂x ∂ξ ∂x ∂η

∂y ∂ξ ∂y ∂η

#(

∂Ni ∂x ∂Ni ∂y

)

( = [J]T

∂Ni ∂x ∂Ni ∂y

) (8.85)

Therefore, (

∂Ni ∂x ∂Ni ∂y

)

( T −1

= [J ]

∂Ni ∂ξ ∂Ni ∂η

) i = 1, . . . , n

;

(8.86)

Using (8.86), derivatives of Ni with respect to ξ and η can be transformed into the derivatives of Ni with respect to x and y and, hence, the derivatives of φ with respect to x and y in (8.84) are defined.

¯ ξη : polynomial 8.4.1 C 00 local approximations over Ω approach ¯ m of type C 00 , i.e across First, we consider interpolation φ(ξ, η) of φ over Ω the inter subdomain boundaries only φ is continuous but the derivatives of φ normal to the inter-element boundaries in the physical domain may be ¯ m , a bilinear behavior of φ (in most discontinuous. For the master element Ω cases this would be lowest degree admissible behavior) would require, φ(ξ, η) = c1 + c2 ξ + c3 η + c4 ξη

(8.87)

Evaluation of constants ci ; i = 1, . . . , 4 naturally requires four conditions, and for C 00 behavior of φ we can choose φei = φ(ξi , ηi ), i = 1, . . . , 4

(8.88)

in which (ξ, η) are the coordinates of the four corner nodes, as shown in Fig. 8.14. η (ξ4, η4) 4

(ξ3, η3) 3 ξ

1 (ξ1, η1)

2 (ξ2, η2)

Figure 8.14: A four-node square of two units in the natural coordinates

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

523

Using (8.87) and (8.88) we obtain    c1  1 ξ1       c2 1 ξ2 =  1 ξ3  c3     c4 1 ξ4

η1 η2 η3 η4

−1  e  ξ1 η 1 φ     1e   ξ2 η 2  φ2 ˆ e}  = [C]{φ e ξ3 η 3   φ  3     ξ4 η 4 φe4

(8.89)

Substituting from (8.89) into (8.87) we obtain   ˆ e} φ(ξ, η) = 1 ξ η ξη [C]{φ

(8.90)

or φ(ξ, η) = [N ]{φe } = [N ]{δ e } =

4 X

Ni (ξ, η)φei

(8.91)

i=1

in which

( 1, Ni (ξj , ηj ) = 0,

j=i , i = 1, . . . , 4 j 6= i

(8.92)

and 4 X

Ni (ξ, η) = 1

(8.93)

i=1

This approximation of φ is referred to as bilinear Lagrange interpolation. ¯ ξη Higher degree of approximation of φ over Ω Let us consider a quadratic (complete second degree in ξ and η) approx¯ m . This requires an expression similar to imation of φ in ξ and η over Ω (8.87) but with a complete second degree polynomial in ξ and η and hence a decision on which terms to consider. Secondly, in order to evaluate the constants ci , we need to know the locations of the nodes over the master ¯ m . In case of quadratic behavior of φ over Ω ¯ m , it may not be very element Ω difficult to decide on these two aspects, but as the degree of approximation or local interpolation is increased the decision on these two aspects is not straight forward. The decision on which ξ and η terms to consider for φ and where the ¯ m can be facilitated by using what is called Pascal’s nodes are located over Ω rectangle. The family of local interpolations generated in this approach are Lagrange family of local interpolations. First, we explain how to interpret the information in Fig. 8.15. (a) The locations of terms in the rectangular arrangement are the locations ¯ m configuration. of the nodes in Ω

524

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Rectangular array 1

ξ

ξ2

ξ3

ξ4

η

ξη

ξ2η

ξ3η

ξ4η

η2

ξη 2

ξ 2η2

ξ 3η2

ξ 4η2

η3

ξη 3

ξ 2η3

ξ 3η3

ξ 4η3

η4

ξη 4

ξ 2η4

ξ 3η4

ξ 4η4

Rectangular elements of various degree local approximations Nodal configuration

Monomials to be used

Degree of approximation p of φ

constant; p = 0

1

ξ

1

bilinear; p = 1 η

ξη

1

ξ

ξ2

η

ξη

ξ2η

η2

ξη 2

ξ 2η2

1

ξ

ξ2

ξ3

η

ξη

ξ2η

ξ3η

η2

ξη 2

ξ 2η2

ξ 3η2

η3

ξη 3

ξ 2η3

ξ 3η3

bi-quadratic; p = 2

bi-cubic; p = 3

Figure 8.15: Pascal’s rectangle and Lagrange family of interpolations

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

525

(b) The terms shown in the rectangle are the monomials to be used in the expression for φ(ξ, η). (c) For example, in the case of bilinear local approximation one would consider only up to linear terms in ξ and η, which would yield a four-node element with terms 1, ξ, η and ξη in the expression for φ(ξ, η). ¯ m , one would (d) In the case of bi-quadratic (complete) behavior of φ over Ω consider up to quadratic monomials in ξ and η, which would yield a ninenode element with terms 1, ξ, ξ 2 , η, ξη, ξ 2 η, η 2 , ξη 2 , ξ 2 η 2 in the expansion for φ(ξ, η). (e) Likewise, for a bi-cubic element we would have a sixteen-node element for which the corresponding monomials in the interpolation for φ(ξ, η) are shown in Fig. 8.15. (f) Once we know the monomials to be used in the expansion for φ(ξ, η) and the locations of the nodes, it is a rather simple matter (following the same procedure as used for the bilinear case) to derive φ(ξ, η) =

n X

Ni (ξ, η) φei

i=1

= [N ]{φe } = [N ]{δ e }

(8.94)

where n is the number of nodes. Remarks. (1) The local interpolations of various degrees derived above are referred to as C 0 Lagrange family of interpolations in which function value at the nodes are the nodal degrees of freedom. (2) A serious drawback of this approach is that as the degree of local approximation is increased, so does the size of the matrix to be inverted in deriving Ni (ξ, η). (3) The most fundamental shortcoming of this approximation is that as the degree of approximation increases, the number of nodes increase dramatically, a very undesirable feature from the point of view of constructing discretizations for various p-levels. (4) In the following section 8.4.2 we correct some of these shortcoming.

8.4.2 C 00 Lagrange type local approximation using tensor product Recall that in the case of 1D local approximations in ξ for −1 ≤ ξ < 1, the basis functions or local approximations can be easily generated using Lagrange interpolation functions which allows us to bypass the inversion of

526

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

the matrices. For an element with l equally spaced nodes in −1 ≤ ξ < 1, we could write φ(ξ) as l X φ(ξ) = Ni (ξ) φei (8.95) i=1

in which

 l  Y ξ − ξm , k = 1, . . . , l Nk (ξ) = Lk (ξ) = ξk − ξm

(8.96)

m=1 m6=k

with

( 1, Ni (ξj ) = 0,

and

l X

j=i , i = 1, . . . , l j 6= i

(8.97)

Ni (ξ) = 1

(8.98)

i=1

In this, the degree of approximation is (l − 1), that is, p = l − 1 and the local approximations are of type C 00 due to the fact that only the function values are the unknowns or the degrees of freedom at the nodes. To understand the basic concept of tensor product, consider a four-node ¯ m and a two-node linear element in ξ natural coordinate bilinear element in Ω ξ system with N1 (ξ) and N2ξ (ξ) as the basis functions and also a two-node linear element in η natural coordinate system with N1η (η) and N2η (η) as the basis functions. Then the four-node bilinear element can be thought of as a map of the traces of the two-node linear configuration in η traversing along the two-node linear configuration in ξ. This is illustrated in Fig. 8.16 and Table 8.2. η (−1, 1) 4

η

(1, 1)

(+1) N2η

2η

(−1) N1η

1η 1ξ

3 ξ

1 (−1, −1)

2 (1, −1)

¯m (a) Element map Ω

N1ξ (−1)

2ξ

(b)

N2ξ (+1)

ξ

Figure 8.16: Derivation of four-node bilinear element as a tensor product of two-node line elements

The information expressed in Table 8.2 can also be constructed in the following manner. If we declare N1ξ and N2ξ as column vectors and N1η and

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

527

Table 8.2: Correspondence between the basis functions of the bilinear element and the 1D linear element 2D configuration

Product of 1D basis functions

Node

Basis function

1

N1 (ξ, η)

=

N1ξ N1η

2

N2 (ξ, η)

=

N2ξ N1η

3

N3 (ξ, η)

=

N2ξ N2η

4

N4 (ξ, η)

=

N1ξ N2η

N2η as a row matrix (with corresponding ξ and η coordinates) then a tensor product of these two would be a simple product of these two ( )  N1ξ : ξ = −1  n N1 : η = −1 N2η : η = +1 ξ N2 : ξ = +1   ξ η   N1 N1 N1ξ N2η N1 (ξ, η) N4 (ξ, η) (−1,1) (−1,−1) (8.99) = N2ξ N1η N1ξ N1η N2 (ξ, η) N3 (ξ, η) (1,−1) (1,1)

Remarks. (1) When taking the tensor product, we always keep track of ξ η coordinates associated with the 1D functions so that the results of the tensor product can be appropriately associated with the corresponding nodes of the 2D configuration. (2) Whether we use this classical definition of the tensor product in (8.99) or tabular form given earlier, the end result is the same. Example 8.3 (Explicit expressions for the standard Lagrange basis func¯ m ). In the case of the four-node tions for the four-node bilinear element in Ω bilinear element N1ξ =

1−ξ 1+ξ 1−η 1+η , N2ξ = , N1η = , N2η = 2 2 2 2

(8.100)

and hence (8.99) would yield (

1−ξ 2 1+ξ 2

) : ξ = −1  1−η : ξ = +1

2

: η = −1

1+η 2

 : η = +1

 1−ξ 1−η 1+η  ( 2 )( 2 ) ( 1−ξ 2 )( 2 )   (−1,1) N1 (ξ, η) N4 (ξ, η)  (−1,−1)  =  1+ξ 1−η (8.101) = 1+η N2 (ξ, η) N3 (ξ, η) ( 2 )( 2 ) ( 1+ξ 2 )( 2 ) (1,−1)

(1,1)

528

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Therefore, we hve   1−η 1−ξ N1 (ξ, η) = 2 2    1+ξ 1−η N2 (ξ, η) = 2 2    1+ξ 1+η N3 (ξ, η) = 2 2    1+η 1−ξ N4 (ξ, η) = 2 2 

(8.102)

Obviously the basis functions in (8.102) are the correct local approximation functions for the four-node bilinear element as they satisfy the properties ( 1, j = i Ni (ξj , ηj ) = 0, j 6= i and

4 X

Ni (ξ, η) = 1

i=1

Example 8.4 (Explicit expressions for the standard Lagrange basis functions ¯ ξη ). Consider a nine-node (determined using for the bi-quadratic element in Ω ¯ m . The basis functions (or local Pascal’s rectangle) bi-quadratic element in Ω approximation functions) for this element can be generated by taking the tensor product of 1D quadratic basis functions for three-node configurations in ξ and η directions (see Fig. 8.17): i n oh η N3η N2η N1 = N1ξ : ξ = −1 N2ξ : ξ = 0 N3ξ : ξ = +1 (η=−1) (η=0) (η=+1)     N1ξ N1η N1ξ N2η N1ξ N3η (−1,−1)  (−1,0) (−1,1) N1 (ξ, η) N8 (ξ, η) N7 (ξ, η)     ξ η ξ η ξ η    N2 N1 N2 N2 N2 N3  = N (ξ, η) N (ξ, η) N (ξ, η)  (8.103) 9 6  (0,−1) (0,0) (0,1)   2     ξ η ξ η ξ η N3 (ξ, η) N4 (ξ, η) N5 (ξ, η) N N N N N N 3

1

3

2

(1,−1) (1,0)

3

3

(1,1)

Explicit expressions for Niξ and Niη (i = 1, 2, 3) are ξ(ξ − 1) , 2 N2ξ = 1 − ξ 2 , ξ(ξ + 1) N3ξ = , 2

N1ξ =

η(η − 1) 2 η N2 = 1 − η 2 η(η + 1) N3η = 2 N1η =

(8.104)

529

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

Hence Ni (ξ, η) (i = 1, . . . , 9) can be determined explicitly using (8.103) and (8.104). η (−1, 1)

(0, 1) 7

(−1, 0)

8 1

(−1, −1)

η

(1, 1)

(1) N3η

3η

(0) N2η

2η

(−1) N1η

1η

5

6 (0, 0) 9

4 2

(0, −1)

(1, 0) ξ

3 (1, −1)

¯m (a) Element map Ω

1ξ

2ξ

3ξ

N1ξ (−1)

N2ξ (0)

N3ξ (1)

ξ

(b)

Figure 8.17: Derivation of nine-node bi-quadratic element as a tensor product of threenode line elements

Higher degree local basis functions of complete degrees in ξ and η can be easily derived using this procedure. It is rather clear that the tensor product procedure avoids inverting matrices to determine constants in the polynomial expansions of φ(ξ, η). In this approach of constructing the local approximations one could increase the degree of the polynomial (p-level) in ξ and η directions as desired. These local approximations could be referred to as p-version. The two main drawbacks of this approach are: (i) that an increase in p-level requires a new nodal configuration for the element, i.e. essentially new geometric description of additional nodes and (ii) and secondly the local approximations lack hierarchical property, i.e. the approximations for lower p-levels are not an explicit subset of those at higher p-levels. These two drawbacks can be corrected by deriving the C 00 p-version hierarchical local approximation functions presented in the next section for quadrilateral elements based on Lagrange polynomials.

8.4.3 C 00 p-version hierarchical local approximations based on Lagrange polynomials Consider quadrilateral elements with four corner nodes, four mid side nodes and a node at the center of the element. The element shape and sides can be distorted in the (x, y)-space, as shown in Fig. 8.18(a). Consider a map of the element in the (ξ, η)-space into a square of two units with the origin of the coordinate system located at the center of the element, as shown in Fig. 8.18(b). The C 00 p-version hierarchical local approximations for the element of Fig. 8.18(b) can be derived by considering 1D p-version

530

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

hierarchical local approximation in ξ and η for the three-node element and then taking their tensor product. ¯ m or Ω ¯ ξη . Consider Let φ be the field variable to be interpolation over Ω a three node element in ξ-coordinate space as shown in Fig. 8.18(c). Then for this element we can write φ(ξ) = N11ξ (ξ)(φ)ξ=−1 + N31ξ (ξ)(φ)ξ=1 +

pξ X

N2iξ (φ,ξi )ξ=0

(8.105)

i=2 i

where φ,ξi = ∂∂ξφi and pξ is the highest degree of the polynomial in ξ. Similarly, for the three-node configuration of Fig. 8.18(c) in η direction we can write φ(η) =

N11η (η)(φ)η=−1

+

N31η (η)(φ)η=1

+

pη X

N2iη (φ,ηi )η=0

(8.106)

i=2

where φ,ηj =

∂j φ ∂η j

and pη is the highest degree of the polynomial in η. η

4

9

8 y

2

1

7

5

6

7

2

8

3

9

1

4

2

ξ

3

2

x

(a)

5

6

A quadrilateral element in xy-space

(b) A nine-node quadrilateral element map in ξη-space

η

3

2 1 ξ 1

2

3

(c) 1D three-node elements in ξ and η spaces

Figure 8.18: Mapping of nine-node bi-quadratic element in (x, y)-space into (ξ, η)-space

531

8.4. LOCAL APPROXIMATION OVER QUADRILATERAL ELEMENTS

The 1D local approximations given by (8.105) and (8.106) can be expressed more conveniently in terms of 1D functions in ξ and η and the corresponding nodal variable operators. Keeping in mind that nodal variable operators act on the dependent variables(s) to produce dofs at the corresponding nodes. Figure 8.19 shows the arrangement of the 1D functions in ξ and η and Fig. 8.20 show the corresponding nodal variable operators. The approximation functions and the nodal variable operators for the nine-node 2D element of Fig. 8.18(b) can be constructed by simply taking the tensor products of the 1D C 0 p-version hierarchical approximation functions and the corresponding nodal variable operators. The resulting C 00 2D p-version hierarchical approximation functions and the nodal variable operators are shown in Fig. 8.20. Since the 1D C 0 p version functions and the nodal variable operators used in deriving the 2D C 00 approximation are hierarchical, the hierarchical nature of the 2D approximations is preserved. Thus, for the nine-node C 00 p-version hierarchical element the dependent variable φ can ¯ m or Ω ¯ ξη using be interpolated over Ω φ(ξ, η) = [N (ξ, η)]{δ e }

(8.107)

where node 2 node 4 3 " node 1 z }| { node }| { z}|{ z}|{ z p 1 1pη ξ 11 11 21 31 12 13 [N (ξ, η)] = N1 , [N2 , N2 , . . . , N2 ], N3 , [N4 , N4 , . . . , N4 ], node 6

node 8 7 }| { node }| { z}|{ z z}|{ z p 1 1pη ξ 11 11 21 31 12 13 N5 , [N6 , N6 , . . . , N6 ], N7 , [N8 , N8 , . . . , N8 ],

node 5

node 9 z }| { pξ 3 p ξ2 23 33 22 32 [N9 , N9 , . . . , N9 ], [N9 , N9 , . . . , N9 ], node 9 z }| { # p p 2p 3p . . . , [N9 η , N9 η , . . . , N9 ξ η ] (8.108)

and node 2

}| { "node 1 z 3  dpξ φ  i node z}|{ h d2 φ   d3 φ  z}|{ {δ e }T = φ1 , , ,..., , φ3 , dξ 2 2 dξ 3 2 dξ pξ 2 node 4

z }| { 5 h d2 φ   d3 φ   dpη φ  i node z}|{ , ,..., , φ5 , dη 2 4 dη 3 4 dη pη 4

532

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY node 6

}| { z 7  dpξ φ  i node h d2 φ   d3 φ  z}|{ , ,..., , φ7 , dξ 2 6 dξ 3 6 dξ pξ 6 node 8

z }| {  dpη φ  i h d2 φ   d3 φ  , ,..., , dη 2 8 dη 3 8 dη pη 8 node 9

{ zh 4   5 }|  dpξ +2 φ  i d φ d φ , ,..., , dη 2 dξ 2 9 dη 2 dξ 3 9 dη 2 dξ pξ 9 node 9

}| { z  dpξ +3 φ  i h d5 φ   d6 φ  , ,..., , dη 3 dξ 2 9 dη 3 dξ 3 9 dη 3 dξ pξ 9 node 9

z }| { # h dpη +2 φ   dpη +3 φ   dpη +pξ φ  i ..., , ,..., dη pη dξ 2 9 dη pη dξ 3 9 dη pη dξ pξ 9

(8.109)

For a chosen p-level in ξ and η directions (i.e. pξ and pη ), explicit expressions for the approximation functions [N ] and the nodal dofs {δ e } are easily obtained. We remark that the above development is based on Lagrange polynomials due to the fact that 1D C 0 p-version hierarchical functions in ξ and η directions were based on Lagrange polynomials. Following the approach presented here, one could easily derive 2D C 00 p-version hierarchical local approximations based on Legendre and Chebyshev polynomials.

¯ e ) p-version local approximations: 8.5 2D C ij (Ω Rectangular family of elements ¯ e ) type local approximation funtions In this section we derive 2D C ij (Ω in xy coordinate space which possess interelement continuity of orders i and j in x and y directions with complete polynomials of orders pξ and pη and present proofs of necessary and sufficient conditions. Consider a nine-node p-version rectangular element in x, y space with sides parallel to x and y axes [Fig. 8.22(a)] and its map in natural coordinate space ¯ e ) type interpolations for the element shown [Fig. 8.22(b)]. We derive C ij (Ω in Fig. 8.22(a) by using tensor products of one dimensional C i , C j interpolations in x and y directions [Fig. 8.22(c)] for the three-node configurations shown in Fig. 8.22(a). Let pξ be the degree of interpolation in ξ directions (parallel to x-axis and pointing in the same direction) and i be order of continuity in the x direction, then for the three-node 1D configuration of Fig. 8.22(c) in ξ direction we

¯ E ) P -VERSION LOCAL APPROXIMATIONS 8.5. 2D C IJ (Ω

533

can write interpolations of type C i in x direction as follows for a dependent variable φ: ∂ i φ1 ∂φ1 ∂ 2 φ1 i + · · · + N (ξ) + N 2 (ξ) ∂x ∂x2 ∂xi e1 e1 e1 e1 2 ∂φ2 ∂ φ2 ∂ i φ2 i + N 0 (ξ) φ2 + N 1 (ξ) + N 2 (ξ) + · · · + N (ξ) ∂x ∂x2 ∂xi e2 e2 e2 e2 pξ X ∂ i1 φ 3 ¯ ξ and ∀x = x(ξ) ∈ Ω ¯e N i1 (ξ) + ∀ξ ∈ Ω x ∂ξ i1 e3

φ(ξ) = N 0 (ξ) φ1 + N 1 (ξ)

(8.110)

i1 =2(i+1)

η p = pη

pη = 3

pη = 2

pη = 1

N31η =

(pη )η

N2 ( b=

1; η;

=

η pη −b pη !

η 3 −η 3!

N23η =

N22η =

1+η 2

η 2 −1 2

3

(η = 1)

2

(η = 0)

1

(η = −1)

pη is even pη is odd N11η =

(ξ = −1)

(ξ = 0)

1−η 2

(ξ = 1) ξ

1 N11ξ =

2

3 N31ξ =

1−ξ 2

pξ = 1

N22ξ =

ξ 2 −1 2

pξ = 2

N23ξ =

ξ 3 −ξ 3!

pξ = 3

(pξ )ξ

N2 ( a=

1+ξ 2

1; ξ;

=

ξ

pξ −a

pξ !

p = pξ

pξ is even pξ is odd

Figure 8.19: 1D, C 0 hierarchical functions in ξ and η directions.

534

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

η p = pη

pη = 3

pη = 2

pη = 1

1

∂3 ∂η 3

∂ pη ∂η pη

∂2 ∂η 2

1

(ξ = −1)

(ξ = 0)

3

(η = 1)

2

(η = 0)

1

(η = −1)

(ξ = 1) ξ

1

2

1

3 1

pξ = 1

∂2 ∂ξ 2

pξ = 2

∂3 ∂ξ 3

pξ = 3

∂ pξ ∂ξ pξ

p = pξ

Figure 8.20: 1D, C 0 hierarchical nodal variable operators in ξ and η directions.

2

1 ∂ φ1 in which (φ1 , φ2 ) are function values at nodes 1 and 2 and ( ∂φ ∂x , ∂x2 ) are first and second derivatives of φ with respect to x at node one. Likewise, 2 2 ∂ φ2 ( ∂φ ∂x , ∂x2 ) are the first and second derivatives of φ with respect to x at node 2, and so on.

Let pη be the degree of interpolation in η directions and j be the order of continuity in the y-direction then for the three-node 1D configuration of Fig. 8.22(c) in η direction, we can write the following for the dependent variable φ:

¯ E ) P -VERSION LOCAL APPROXIMATIONS 8.5. 2D C IJ (Ω

535

η N6i1 (ξ, η) = (N2iξ N31η )

N711 (ξ, η) = (N11ξ N31η )

7

N81j (ξ, η) = (N11ξ N2jη )

6

5

9

4

8

j = 2, . . . , pη

N511 (ξ, η) = (N31ξ N31η )

i = 2, . . . , pξ

ξ N9ij (ξ, η)

=

(N2iξ

N41j (ξ, η) = (N31ξ N2jη )

N2jη )

j = 2, . . . , pη

i = 2, . . . , pξ j = 2, . . . , pη

N111 (ξ, η) = (N11ξ N11η )

3

2

1

N311 (ξ, η) = (N31ξ N11η )

N2i1 (ξ, η) = (N2iξ N11η ) i = 2, . . . , pξ (a) Approximation functions

η ∂i ∂ξ i

(1)




(1)

i = 2, . . . , pξ

7

∂j ∂η j




6

5

9

4

8

ξ ∂ i+j ∂η j ∂ξ i

j = 2, . . . , pη

∂j ∂η j




i = 2, . . . , pξ j = 2, . . . , pη

1 (1)

j = 2, . . . , pη

3

2 ∂i ∂ξ i







(1)

i = 2, . . . , pξ (b) Nodal variable operators non-hierarchical nodes hierarchical nodes

Figure 8.21: p-version approximation functions and nodal variable operators for a ninenode 2D element

536

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

η

6

7

8

1

y

5

6

7

4

9

3

2

4

9

8

5

ξ

2

1

3

x (a)

A nine-node rectangular element in ¯ e) xy space (Ω

(b)

Map of element of (a) in natural co¯ m) ordinate space ξη (Ω

η

2

η=1

3

η=0

1

η = −1 ξ = −1

ξ=0

ξ=1

3

2

ξ 1 (c)

1D three-node configurations in ξ and η directions

Figure 8.22: 2D nine-node element and 1D three-node configurations in ξ and η directions

∂φ1 ∂ 2 φ1 ∂ j φ1 j + N 2 (η) + · · · + N (η) ∂y ∂y 2 ∂y j e1 e1 e1 e1 2 ∂ φ2 ∂φ2 ∂ j φ2 j + N 0 (η) φ2 + N 1 (η) + N 2 (η) + · · · + N (η) ∂y ∂y 2 ∂y j e2 e2 e2 e2 pη X ∂ j1 φ 3 ¯ e (8.111) ¯ η and ∀y = y(η) ∈ Ω + ∀η ∈ Ω N j1 (η) y ∂η j1 e3

φ(ξ) = N 0 (η) φ1 + N 1 (η)

j1 =2(j+1)

¯ E ) P -VERSION LOCAL APPROXIMATIONS 8.5. 2D C IJ (Ω η

η

N l1 (η) e2

2

∂ l1 ∂y l1

2

3

l1 = 0, 1, . . . , j ∂ l2 l2 = 2(j + 1), . . . , pη ∂y l2

3

1

∂ l1 ∂y l1

l1 = 0, 1, . . . , j l2 = 2(j + 1), . . . , pη

N l2 (η) e3

N l1 (η) e1

537

1

3

2

k1

k2

k1

1 1

3

2

ξ

ξ k1

∂ ∂xk1

k2

∂ k1 ∂xk2

k1 = 0, 1, . . . , i

∂ ∂xk2 k1 = 0, 1, . . . , i

k2 = 2(i + 1), . . . , pξ

k2 = 2(i + 1), . . . , pξ

N (ξ)

N (ξ)

e1

N (ξ)

e3

e2

(a) C i , C j 1D basis functions

(b) C i , C j 1D nodal variable operators

Figure 8.23: 1D higher order continuity basis functions and nodal variable operators

By taking tensor product of 1D functions and the nodal variable opera¯ e) tors in (8.110) and (8.111) (shown in Fig. 8.23) we can generate 2D C ij (Ω type interpolations for φ = φ(ξ, η) (i.e. interpolation functions and the nodal ¯ e ) 2D approximation functions and variable operators). The resulting C ij (Ω the nodal variable operators are shown in Fig. 8.24 and we can write φ(ξ, η) = [N (ξ, η)]{δ e }

¯m ∀ξ η ∈ Ω

and

¯e (x, y) ∈ Ω

(8.112)

In the following, we present a theorem and its proof that ensures necessary ¯ e ). and sufficient conditions for interpolation (8.112) to be of type C ij (Ω N k1 (ξ) N l1 (η) e1

e3

e2

e1

e3

9 k2

e1

e1

4

N k1 (ξ) N l2 (η) e3

8

e1

1 N k1 (ξ) N l1 (η) e2

e1

(a) Basis functions

∂ k1+l1 ∂y l1 ∂xk1 5

4

∂ k2+l2 ∂y l2 ∂xk2

3

N k2 (ξ) N l1 (η)

6

9

∂ k1+l2 ∂y l2 ∂xk1

e3

2 e3

∂ k2+l1 ∂y l1 ∂xk2

∂ k1+l1 ∂y l1 ∂xk1 7

e2

l2

N (ξ) N (η) 1

e2

5

8 e3

N k1 (ξ) N l1 (η)

e2

6

7

N k1 (ξ) N l2 (η)

N k1 (ξ) N l1 (η)

N k2 (ξ) N l1 (η)

e2

2

3

∂ k2+l1 ∂y l1 ∂xk2

∂ k1+l1 ∂y l1 ∂xk1

(b) Nodal variable operators

k1 = 0, 1, . . . , i

l1 = 0, 1, . . . , j

k2 = 2(i + 1), . . . , pξ

l2 = 2(j + 1), . . . , pη

∂ k1+l2 ∂y l2 ∂xk1

¯ e ) basis functions and nodal variable operators Figure 8.24: 2D C ij (Ω

∂ k1+l1 ∂y l1 ∂xk1

538

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

¯ e )∀ξ ∈ Ω ¯ ξ or ∀x = x(ξ) ∈ Ω ¯ e of degree pξ and Theorem 8.2. If φ(ξ) ∈ C i (Ω x x ¯ η or ∀y = y(η) ∈ Ω ¯ e of degree pη are one dimensional ¯ ey )∀η ∈ Ω φ(η) ∈ C j (Ω y interpolations for the three-node 1D configurations of Fig. 8.22 (c), then the tensor product of generating 2D interpolations for φ(ξ, η) by (8.112) ensures ¯ e )∀x(ξ), y(η) ∈ Ω ¯ e for necessary and sufficient conditions for φ(ξ, η) ∈ C ij (Ω the nine node element of Fig. 8.22(a). The proof of this theorem is constructed by induction and is presented in the following sections.

¯ e ) with p-levels 8.5.1 2D interpolations of type C 11 (Ω of pξ and pη ¯ e ) interpolations. (a) Proof of the uniqueness of C 11 (Ω It suffices to consider pξ = 3 and pη = 3 in which case there are no degree of freedom at the midside nodes and the center node. We need to show (i) That the dimension of the space of φ(ξ, η) is same as the total degrees of freedom for the element. (ii) If all degrees of freedom are identically zero, then φ(ξ, η) = 0 ∀ξ, η ∈ ¯ m. Ω For a nine-node element with complete bicubic interpolations in ξ and η, the total number of basis functions required is sixteen (based on Pascal rectangle). Hence, the dimension of the approximation space of φ(ξ, η) is sixteen. For pξ = pη = 3 (bicubic) the element has no degrees of ∂φ freedom at the midside nodes and at the center node and φ, ∂φ ∂x , ∂y and ∂2φ ∂y∂x

(defined as set S) as degrees of freedom at each of the four corner nodes giving rise to a total of sixteen degrees of freedom for the element. Hence the dimension of the space of φ(ξ, η) is equal to the total degrees of freedom. If all degrees of freedom are zero, then it follows directly from ¯ m . Thus C 11 (Ω ¯ e) the expression for φ(ξ, η) that φ(ξ, η) = 0 ∀ξ, η ∈ Ω interpolations generated using tensor product in (8.112) are unique. ¯ e ) inter-element continuity. (b) Proof of C 11 (Ω In this case also we can consider pξ = pη = 3 without loss of generality. Consider a three-element discretization in which the element one has elements two and three as neighbors with mating boundaries with ¯ 14 and L14 (L ¯ 14 being closed set, i.e. it includes nodes common sides L ¯ 12 and L12 1 and 4; L14 being open set) parallel to y axis and sides L ¯ (L12 being closed set, that is, it includes nodes 1 and 2; L12 being open set) parallel to x axis. Then, we need to show that the elements of the

¯ E ) P -VERSION LOCAL APPROXIMATIONS 8.5. 2D C IJ (Ω

Table 8.3: Degrees of freedom, order of polynomials for φ, ¯ 14 its derivatives ∀y ∈ L14 and ∀y ∈ L Degrees of freedom at nodes 1 and 4 ∂φ ∂y
∂φ ∂ ∂y ∂x

φ, ∂φ , ∂x

Third degree polynomial ¯ 14 ∀y ∈ L φ ∂φ ∂x

∂φ ∂x
∂φ ∂ ∂x ∂y

φ,

∂φ , ∂x

∂iφ ; i = 1, 2, 3 ∂y i
∂φ ∂i ; i = 1, 2, 3 ∂y i ∂x

∂φ , ∂y

∂φ ∂y

Uniqueness ∀x ∈ L12 ∂iφ ; i = 1, 2, 3 ∂xi
∂φ ∂i ; i = 1, 2, 3 ∂xi ∂y

φ, ∂φ , ∂y

and uniqueness of φ and

Uniqueness ¯ 14 ∀y ∈ L φ,

φ,

Third degree polynomial ¯ 12 ∀y ∈ L φ

∂φ ∂x

Uniqueness ∀y ∈ L14

Table 8.4: Degrees of freedom, order of polynomials for φ, ¯ 12 its derivatives ∀y ∈ L12 and ∀y ∈ L Degrees of freedom at nodes 1 and 2

539

∂φ ∂y

∂φ ∂ , ∂x ∂y

∂φ ∂y


: set S1

∂φ : set S2 ∂x

and uniqueness of φ and

Uniqueness ¯ 12 ∀x ∈ L φ,

∂φ ∂x

∂φ ∂ , ∂y ∂x


: set T1

∂φ : set T2 ∂y

∂φ ∂2φ ¯ ¯ set S, that is, φ, ∂φ ∂x , ∂y and ∂y∂x , are unique ∀y ∈ L14 and ∀x ∈ L12 . The proof is best presented by referring to Tables 8.3 and 8.4. First, consider ¯ 14 (Table 8.3). With φ and ∂φ as degrees of freedom at nodes 1 and L ∂y ¯ 14 ensuring uniqueness 4, φ = φ(y) is a cubic polynomial in y ∀y ∈ L ∂iφ of φ, ∂yi , i = 1, 2, 3 ∀y ∈ L14 (row 1, column 3 in Table 8.3). However ∂φ ¯ only φ and ∂φ ∂y (set S1 ) are unique ∀y ∈ L14 due to the fact φ and ∂y are the only dofs at nodes
1 and 4. From row 2 of Table 8.3, we note that ∂φ ∂ ∂φ with ∂x and ∂y ∂x (set S2 ) as degrees of freedom at nodes 1 and 4, ∂φ ∂φ ¯ ∂x is also a
polynomial of degree three in y ∀y ∈ L14 ensuring that ∂x i ∂φ ∂ and ∂y i ∂x ; i = 1, 2, 3 ∀y ∈ L14 (row 2, column 3) are unique. Thus, ¯ 14 . Hence, this interpolation is of type set S = S1 ∪ S2 is unique ∀y ∈ L 11 e ¯ ¯ ¯ 12 (Table 8.4). With φ and ∂φ C (Ω ) ∀y ∈ L14 . Next we consider L ∂x
∂ ∂φ (set T1 ) and ∂φ , (set T ) as degrees of freedom at nodes 1 and 2 ∂y ∂x ∂y ¯ 2, φ and ∂φ ∂y are polynomials of degree three in x ∀x ∈ L12 ensuring
i ∂ i ∂φ uniqueness of φ, ∂∂xφi ; i = 1, 2, 3 and ∂φ ∂y , ∂xi ∂y ; i = 1, 2, 3 ∀x ∈ L12 , ∂φ ∂ ∂φ
¯ 12 due to the however only φ, ∂φ are unique ∀x ∈ L ∂x and ∂y , ∂x ∂y fact that these are nodal degrees of freedom at nodes 1 and 2. Hence, ¯ 12 . Thus the the elements of the set S = T1 ∪ T2 are unique ∀x ∈ L 11 e ¯ ¯ interpolation is of type C (Ω ) ∀x ∈ L12 . Since the interpolation is ¯ e ) ∀x ∈ L ¯ 12 and also of type C 11 (Ω ¯ e ) ∀y ∈ L ¯ 14 , the of type C 11 (Ω 11 e e ¯ ¯ interpolation is of type C (Ω ) ∀x, y ∈ Ω . This completes the proof.

540

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

¯ e ) with p-levels 8.5.2 2D interpolations of type C 22 (Ω of pξ and pη ¯ e ) interpolations. (a) Proof of the uniqueness of C 22 (Ω It suffices to consider pξ = 5 and pη = 5 in which case there are no degree of freedom at the midside nodes and the center node of the nine-node element. We need to show that: (i) the dimension of the space is same as the total degrees of freedom for the element, and (ii) if all degrees of freedom are identically zero, then φ(ξ, η) = 0 ∀ξ, η ∈ ¯ m. Ω For a nine-node p-version element with complete fifth degree polynomial in ξ and η for local approximation, the total number of basis functions required is 36 (based on Pascal rectangle). Hence, the dimension of the approximation space of φ(ξ, η) is 36. This local approximation has φ,
∂φ ∂ 2 φ ∂φ ∂ ∂φ
∂ ∂ 2 φ
∂ 2 φ ∂ 2 ∂φ
∂2 ∂2φ ∂x , ∂x2 , ∂y , ∂y ∂x , ∂y ∂x2 , ∂y 2 , ∂y 2 ∂x and ∂y 2 ∂x2 as degrees of freedom at each of the four corner nodes (defined as set S) and no degrees of freedom at the mid side and center nodes, giving a total of 36 degrees of freedom for the element. Hence, the dimension of the approximation space for φ(ξ, η) is equal to the total number of degrees of freedom for the element. If all degrees of freedom are zero, then it follows directly ¯ m . Thus, the from the expression for φ(ξ, η) that φ(ξ, η) = 0 ∀ξ, η ∈ Ω 22 e ¯ C (Ω ) interpolations of p-levels of pξ and pη in ξ and η directions are unique. ¯ e ) interelement continuity. (b) Proof of C 22 (Ω Without loss of generality we can consider pξ = pη = 5. Consider a three-element discretization in which the element one has elements two and three as neighbors that is with common boundaries (see Fig. 8.26) ¯ 14 and L14 parallel to y axis and sides L ¯ 12 and with common sides L L12 parallel to x axis. Then we need to show that the elements of the ¯ 14 and ∀x ∈ L ¯ 12 . For the proof we refer to set S are unique ∀y ∈ L ¯ 14 Tables 8.5 and 8.6 corresponding to the interelement boundaries L ∂φ ∂2 ¯ and L12 . With φ, ∂y and ∂y2 as degrees of freedom at nodes 1 and 4, ¯ 14 ensuring uniqueness φ = φ(y) is a fifth degree polynomial in y ∀y ∈ L ∂iφ of φ, ∂yi (i = 1, 2, . . . , 5) ∀y ∈ L14 (row 1, column 3 of Table 8.5), how∂2φ ¯ ever only φ, ∂φ ∂y and ∂y 2 (set S1 ) are unique ∀y ∈ L14 due to the fact that only these are the degrees of freedom at nodes 1 and 4. Similarly, from Table 8.6, we can conclude that elements of sets S2 and S3 are unique ¯ 14 . Thus, the set S is unique ∀y ∈ L ¯ 14 and the interpolation is of ∀y ∈ L

¯ E ) P -VERSION LOCAL APPROXIMATIONS 8.5. 2D C IJ (Ω

Table 8.5: Degrees of freedom, order of polynomials for φ, ¯ 14 of φ and its derivatives ∀y ∈ L14 and ∀y ∈ L Degrees of freedom at nodes 1 and 4

Fifth degree polynomial ¯ 14 ∀y ∈ L

541 ∂iφ ; ∂xi

i = 1, 2 and uniqueness

Uniqueness ∀y ∈ L14

Uniqueness ¯ 14 ∀y ∈ L

i

φ, ∂φ ∂ , ∂x ∂y ∂2φ ∂ , ∂x2 ∂y

∂φ ∂ 2 φ , ∂y ∂y 2 ∂φ ∂x




∂2φ ∂x2

,




,

∂2 ∂y 2 ∂2 ∂y 2

φ ∂φ ∂x

∂φ ∂x




∂2φ ∂x2




∂2φ ∂x2

φ, ∂∂yφi i = 1, . . . , 5
∂φ ∂φ ∂ i , ∂x ∂y i ∂x i = 1, . . . , 5
∂2φ ∂2φ ∂i , ∂x2 ∂y i ∂x2 i = 1, . . . , 5

Table 8.6: Degrees of freedom, order of polynomials for φ, ¯ 12 of φ and its derivatives ∀x ∈ L12 and ∀x ∈ L Degrees of freedom at nodes 1 and 2

Fifth degree polynomial ¯ 12 ∀y ∈ L

φ,

∂φ ∂ 2 φ , ∂y ∂y 2

∂φ ∂ , ∂x ∂y ∂2φ ∂ , ∂x2 ∂y

∂iφ ; ∂y i





∂φ

: set S1 2

∂ , ∂y 2 : set S2
∂2 ∂2φ , ∂y2 ∂x2 : set S3 ∂x

∂φ ∂x





∂2φ ∂x2





i = 1, 2 and uniqueness

Uniqueness ∀y ∈ L12

Uniqueness ¯ 12 ∀y ∈ L

i

φ, ∂φ ∂ , ∂y ∂x ∂2φ ∂ , ∂y 2 ∂x

∂φ ∂ 2 φ , ∂x ∂x2 ∂φ ∂y


,

∂2φ ∂y 2




,

∂2 ∂x2 ∂2 ∂x2

φ ∂φ ∂y

∂φ ∂y




∂2φ ∂y 2




∂2φ ∂y 2

φ, ∂∂xφi i = 1, . . . , 5
∂φ ∂ i ∂φ , ∂y ∂xi ∂y i = 1, . . . , 5
∂2φ ∂i ∂2φ , ∂y 2 ∂xi ∂y 2 i = 1, . . . , 5

φ,

∂φ ∂ 2 φ , ∂x ∂x2

∂φ ∂ , ∂y ∂x ∂2φ ∂ , ∂y 2 ∂x




; set S1


∂φ ∂φ ∂2 , ∂x 2 ∂y ∂y ; set S2
∂ 2 ∂ 2 φ

∂2φ , ∂x2 ∂y2 ∂y 2 ; set S3

¯ e ) ∀y ∈ L ¯ 14 . Next we consider L ¯ 12 (Table 8.6). With φ, ∂φ type C 22 (Ω ∂x 2 and ∂∂xφ2 as nodal dofs at nodes 1 and 3, φ(x) is a polynomial of degree ¯ 12 ensuring uniqueness of φ, ∂ i φi ; i = 1, 2, . . . , 5 ∀x ∈ L12 five in x ∀x ∈ L ∂x

2

∂ φ (row 1, column 3 of Table 8.6), however only φ, ∂φ ∂x and ∂x2 (set T1 ) are ¯ 12 (row degrees of freedom at nodes 1 and 2 and hence unique ∀x ∈ L 1, column 2 of Table 8.6). Similarly for rows 2 and 3 of Table 8.6, we ¯ 12 . thus, conclude that elements of sets T2 and T3 are unique ∀x ∈ L ¯ 12 . Thus the elements of the set S = T1 ∪ T2 ∪ T3 are unique ∀x ∈ L 22 e ¯ ¯ the interpolation is of type C (Ω ) ∀x ∈ L12 . Since the interpolation ¯ 12 and also of type C 22 (Ω¯e ) ∀x ∈ L ¯ 14 , the is of type C 22 (Ω¯e ) ∀x ∈ L 22 e ¯ . This completes the proof. interpolation is of type C (Ω¯e ) ∀(x, y) ∈ Ω

542

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

¯ e ) interpolations of p-levels pξ and pη 8.5.3 2D C ij (Ω A proof of the necessary and sufficient conditions for the general case in which the interpolations are of orders i and j in x and y directions and are of polynomial degrees pξ and pη in ξ and η directions follows by induction. This completes the proof of the theorem. Remarks. (1) Since 1D approximation functions and the nodal variable operators used in the tensor product are only valid when ξ and η are parallel to x and y ¯ e ) local approximations generated by using axes. The family of 2D C ij (Ω them are only valid when the 2D elements in xy space are rectangular with their sides parallel to x and y axes with the additional restriction that ξ and η axes must be pointing in the same directions as x and y axes. This obviously limits their usefulness in practical applications requiring distorted element shapes in the xy space. ¯ e ) element with pξ = (2) The local approximation functions for a 2D C 11 (Ω pη = 3 will only have basis functions and dofs at the corner nodes. If p-levels are increased in ξ and η beyond three (could be unequal), then additional dofs will appear at the mid side and center nodes of the nine ¯ e ) local node element. We note that pξ = pη = 3 is minimum for C 11 (Ω approximation. ¯ e ) local approximation, then pξ = pη = 5 is (3) If we consider a 2D C 22 (Ω the minimum p-levels needed. For this local approximation, the basis functions and the dofs will only appear at the corner nodes. If p-levels in ξ and η increased beyond five, then additional dofs will appear at the mid side and the center node of the nine-node element. (4) The need for higher order global differentiability local approximations for distorted quadrilateral 2D elements is rather obvious. We consider this in the next section.

¯ e ) approximations for quadrilateral 8.6 2D C ij (Ω elements: higher order global differentiability approximations (HGDA) In this section we present development of higher order global differentiability local approximations for two dimensional quadrilateral elements of distorted geometries in xy space. The distorted quadrilateral elements in physical coordinate space xy are mapped into a master element in ξη natural coordinate space in a two unit square with the origin at the center of the element. For the master element, 2D C 00 p-version hierarchical local

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

543

2

∂φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y

¯ 14 or L14 L

4

3

1

2

¯ 12 or L12 L

2

1

2

∂φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y

2

∂φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y

3 y

x

Figure 8.25: A C 11 element one with neighboring C 11 elements two and three

2

2

2

∂φ ∂ φ ∂ φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y , ∂x2 , ∂y 2 ,
∂2 ∂2φ
2
∂ φ ∂ ∂2φ ∂ ∂y ∂x2 , ∂x ∂y 2 , ∂x2 ∂y 2

¯ 14 or L14 L 4

3

1

2

¯ 12 or L12 L 2

2

1 2

2

2

2

∂φ ∂ φ ∂ φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y , ∂x2 , ∂y 2 ,
∂ ∂2φ
∂2 ∂2φ
∂ ∂2φ ∂y ∂x2 , ∂x ∂y 2 , ∂x2 ∂y 2

2

∂φ ∂ φ ∂ φ ∂ φ φ, ∂φ ∂x , ∂y , ∂x∂y , ∂x2 , ∂y 2 ,
∂2 ∂2φ
2
∂ ∂2φ ∂ ∂ φ ∂y ∂x2 , ∂x ∂y 2 , ∂x2 ∂y 2 y

3

x

Figure 8.26: A C 22 element one with neighboring C 22 elements two and three

approximations are considered. The degrees of freedom and the approximation functions from the mid-side nodes and/or center node are borrowed to derive desired derivative degrees of freedom at the corner nodes in the ξη space for various higher order global differentiability approximations in

544

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

η

4

9

8 y

2

1

7

5

6

7

2

6

8

3

9

1

4

2

ξ

3

2

x

(a)

5

(b) Map of the element of (a) in natural coordinate space ξη

Nine-node 2D distorted quadrilateral in xy-space η ∂iφ ∂ξ i

φ

7 ∂j φ ∂η j

8

9

5 4

i+j

∂ φ ∂η j ∂ξ i

1 φ

6

φ

2 ∂iφ ∂ξ i

∂j φ ∂η j

ξ

3 φ

i = 2, 3, . . . , pξ j = 2, 3, . . . , pη

(c) Dofs for C 00 p-version hierarchical element with p-levels pξ and pη directions

Figure 8.27: 2D distorted p-version hierarchical element with p-levels pξ and pη in ξ and η directions

the ξη space. These derivative degrees of freedom at the corner nodes in ξη space are then transformed from the natural coordinate space (ξ, η) to the physical coordinate space (x, y) using Jacobians of transformations to obtain the desired higher order global differentiability local approximations in the xy coordinate space. Pascal rectangle is used to establish a systematic procedure for the selection of degrees of freedom and the corresponding approximation functions from C 00 p-version hierarchical element for the global differentiability of any desired order in xy space. The higher order global differentiability local approximations for distorted geometries cannot be derived using the tensor product approach presented earlier in this chapter and also utilized in reference [1]. Here we present derivations of the higher order global differentiability local approx-

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

545

imations for distorted quadrilateral elements using a completely different approach. Ahmadi, Surana and Reddy [2] developed basic strategy for distorted quadrilateral elements using Lagrange monomials. If the elements in xy space are distorted, we could possibly consider an alternative. The distorted element from xy physical coordinate space is first mapped in a two unit square in ξη natural coordinate space. We can consider 1D higher order global differentiability approximations in ξ, η directions. A tensor product of these 1D approximations would yield higher order differentiability approximations in ξ, η space. The requirement of higher order global differentiability in xy space necessitates that the derivative degrees of freedom at the corner nodes be transformed from ξη space to xy space. ∂ ∂ ∂2 For example, in case of C 11 HGDA, ∂ξ , ∂η , ∂ξ∂η need to be transformed to ∂ ∂2 ∂ ∂x , ∂y , ∂x∂y

∂2 ∂2 ∂3 ∂4 ∂ ∂ ∂2 ∂3 ∂ξ , ∂η , ∂ξ 2 , ∂η 2 ∂ξ∂η , ∂ξ 2 ∂η , ∂ξ∂η 2 , ∂ξ 2 ∂η 2 ∂ ∂2 ∂3 ∂ ∂2 ∂2 ∂3 ∂4 , ∂y , ∂x need to be transformed into ∂x 2 , ∂y 2 ∂x∂y , ∂x2 ∂y , ∂x∂y 2 , ∂x2 ∂y 2 . Due to the fact that degrees of freedom (dofs) in ξη space for C ij higher order

and for C 22 HGDA,

approximations are not a complete set, this transformation is not possible. ∂ ∂ ∂ ∂ In case of C 11 , ∂ξ , ∂η can be transformed into ∂x , ∂y but there is no feasible ∂2 ∂2 22 ∂ξ∂η into ∂x∂y . In case of C , we can transform ∂ ∂ ∂2 ∂2 ∂2 ∂2 ∂2 ∂2 ∂x , ∂y and ∂ξ 2 , ∂η 2 , ∂ξ∂η to ∂x2 , ∂y 2 , ∂x∂y . However, we cannot 3 4 ∂3 , ∂ , ∂ into their counterparts in xy space. Similar ∂ξ 2 ∂η ∂ξ∂η 2 ∂ξ 2 ∂η 2

resolution for transforming ∂ ∂ ∂ξ , ∂η

to

transform situation exists for orders higher than two as well. Thus, the derivation of HGDA for 2D distorted elements in xy space requires a fundamentally different approach. Guidelines In deriving the desired HGDA for 2D distorted elements of quadrilateral family in xy space, we use following guidelines.

(a) The distorted element geometry is mapped from xy physical coordinate space into ξη natural coordinate space, a two-unit square (see Fig. 8.27(a) and (b)). The origin of the ξ, η coordinate system is located at the center of the map in ξη space and we have the following for the mapping of points,   X   n x ¯i (ξ, η) xi = N y yi

(8.113)

i=1

¯i (ξ, η) In which, (xi , yi ) are the Cartesian coordinates of the nodes and N are the shape functions. We could use eight node configuration (i.e. n=8) with serendipity functions (see later section for details) for this

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

purpose, or standard Lagrange family tensor product basis or shape functions. (b) If possible, we would like to consider C 00 p-version hierarchical local approximation as a starting point in the derivation of 2D HGDA due to the fact that this would permit increase in p-levels without rediscretization. Figure 8.27 (c) shows nodal degrees of freedom for a standard C 00 p-version hierarchic element in which ϕ is the field variable being interpolated. The degrees of freedom at the corner nodes of this element consist of only function values. (c) Different degrees of freedom are needed at the corner nodes than those for the 2D HGDA generated using tensor product. This is due to the fact that dofs in tensor product 2D HGDA do not transform from xy to ξη or vice versa. Obviously the choices of the dofs at the corner nodes are dictated by the transformations of the derivatives between xy and ξη spaces. (d) The degrees of freedom for 2D HGDA element of distorted shape are chosen such that they can be transformed using standard Jacobians of transformation from natural coordinate space to physical coordinate space. The choices of nodal operators (or dofs) at the corner nodes listed in Table 8.7 for C 11 , C 22 and C 33 HGDA satisfy this requirement. We note that for C 11 , the derivative operators are a complete set of first order operators. For C 22 HGDA, the set for C 11 is augmented by a complete second order set and so on. Table 8.7: Choices of nodal operators at the corner nodes for C ij 2D distorted quadrilateral elements in xy space

Type of HGDA

Nodal operators at the corner nodes

C 11

∂ ∂ 1, ∂x , ∂y

C 33

2 2 ∂2 , ∂ , ∂ ∂x2 ∂y∂x ∂y 2 3 3 3 ∂2 ∂2 ∂3 , ∂ , ∂ , ∂ ∂y∂x , ∂y 2 , ∂x3 ∂y∂x2 ∂y 2 ∂x ∂y 3

∂ ∂ 1, ∂x , ∂y ,

C 22 ∂ ∂ 1, ∂x , ∂y ,

∂2 , ∂x2

(e) Since C 00 p-version hierarchical approximations are used as a starting point and since in these local approximations only the function values are the degrees of freedom at the corner nodes, we must establish some rules and a systematic procedure that allow us to borrow the desired number of degrees of freedom from C 00 p-version hierarchical approximations to generate the desired dofs at the corner nodes of the 2D HGDA element of distorted geometry in xy space.

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

547

Transformation matrices In this section we present details of the transformation matrices essential to derive 2D HGDA for distorted quadrilateral geometries in xy space. These transformations permit transformations of the derivative degrees of freedom from ξη space to xy space. Figure 8.27 (c) shows nodal degrees of freedom for C 00 p-version hierarchical element in which ϕ is dependent variable. From (8.113), we obtain the following for mapping of lengths in (ξ, η) and (x, y) spaces,     dx dξ = [J] (8.114) dy dη where " [J] =

∂x ∂ξ ∂y ∂ξ

∂x ∂η ∂y ∂η

# =

h

xy ξη

i

  xξ xη = yξ yη

(8.115)

Using the C 00 p-version hierarchical approximations for a nine-node p-version hierarchical element [see Fig. 8.116(c)], the field variable φ can be approximated using φ(ξ, η) = [N (ξ, η)]{δ e } (8.116) in which [N (ξ, η)] is a row matrix of C 00 p-version hierarchical local approximations and {δ e } are the corresponding nodal dofs (arranged in some suitable fashion). We define h iT ∂φ ∂φ {φ}ξη = (8.117) 1 ∂ξ ∂η iT h ∂φ ∂φ (8.118) {φ}xy 1 = ∂x ∂y h 2 iT ∂ φ ∂2φ ∂2φ {φ}ξη = (8.119) 2 2 2 ∂ξ ∂η∂ξ ∂η h 2 i T ∂ φ ∂2φ ∂2φ (8.120) {φ}xy 2 = ∂x2 ∂y∂x ∂y 2 iT h 3 ∂ φ ∂3φ ∂3φ ∂3φ {φ}ξη = (8.121) 3 2 2 3 3 ∂ξ ∂η∂ξ ∂η ∂ξ ∂η h 3 i T ∂ φ ∂3φ ∂3φ ∂3φ {φ}xy (8.122) 3 = ∂x3 ∂y∂x2 ∂y 2 ∂x ∂y 3 where T denotes transpose. We note that (8.117) and (8.118) are a complete set of first order derivatives. Equations (8.119) and (8.120) are a complete set of second order derivatives, and (8.121) and (8.122) are a complete set of third order derivatives in ξη and xy coordinate spaces. In this manner, we can define {φ}ξη i and xy {φ}i as complete sets of the derivatives of order i in (ξ, η) and (x, y) spaces.

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Next we define the rules of transformation between the sets of different order derivatives in (ξ, η) and (x, y) spaces. xy {φ}ξη i = [Ji ]{φ}i

(8.123)

Obviously, 

 xξ yξ [J1 ] = = [J]T xη yη

(8.124)

Using chain rule of differentiation, we can determine the transformation matrices for higher order derivatives of the dependent variable. We will use the following notations for higher order derivatives of physical coordinate x with respect to natural coordinates ξ and η  ∂x i ∂ix ∂ i+j x ∂x , xiξ = , xξ i = , x = (8.125) xξ = i j ξ η ∂ξ ∂ξ ∂ξ i ∂ξ i ∂η j and

 ∂x i ∂x ∂ix , xiη = , xη i = (8.126) ∂η ∂η ∂η i Similarly, we have the following notations for derivatives of physical coordinate y with respect to natural coordinates ξ and η.  ∂y i ∂y ∂iy ∂ i+j y yξ = , yξi = , y ξ i = i , yξ i η j = i j (8.127) ∂ξ ∂ξ ∂ξ ∂ξ ∂η xη =

and

 i ∂y ∂y ∂iy i yη = , yη = , yη i = (8.128) ∂η ∂η ∂η i Following these notations, the transformation matrices for the second order derivatives are as follows: h i xy ξη xy −1 1 {φ}2 = [J2 ] {φ}2 − [J2 ]{φ}1 (8.129)     2xξ yξ yξ2 xξ2 yξ2 x2ξ     1  xξη yξη  (8.130) [J ] = [J2 ] =  x x x y + x y y y η η η η ξ ξ ξ ξ 2     2 2 x η 2 yη 2 xη 2xη yη yη Similarly, transformation matrices for the third order derivatives are as follows: h i xy xy −1 1 2 {φ}xy {φ}ξη (8.131) 3 = [J3 ] 3 − [J3 ]{φ}2 − [J3 ]{φ}1   x3 3x2ξ yξ 3xξ yξ2 yξ3   ξ x2 x 2x x y + x2 y 2x y y + y 2 x y y 2  ξ η ξ  ξ η η ξ ξ ξ η ξ η η ξ (8.132) [J3 ] =  2  xη xξ 2xη xξ yη + x2η yξ 2xη yη yξ + yη2 xξ yξ yη2    x3η 3x2η yη 3xη yη2 yη3

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω



3xξ xξ2

3xξ2 yξ + 3yξ2 xξ

549



3yξ yξ2

  x x 2 + 2x x x 2 y + 2x y + x y 2 + 2x y y y 2 + 2y y  η η η η ξ ξη ξη ξ ξ ξη ξ ξη ξ ξ ξ ξ   [J31 ] =   xξ xη2 + 2xη xξη xξ yη2 + 2xξη yη + xη2 yξ + 2xη yξη yη2 yξ + 2yη yξη    3xη xη2

3xη2 yη + 3yη2 xη

3yη yη2

(8.133) 



x 3 yξ3  ξ  x 2 y 2   ξ η ξ η 2 [J3 ] =   xξη2 yξη2   

(8.134)

xη3 yη3

From (8.123), (8.129) and (8.131) we can write the following general expression, i h ξη xy xy xy i−1 1 2 −1 (8.135) {φ} − [J ]{φ} − [J ]{φ} − . . . − [J ]{φ} {φ}xy = [J ] i i i 1 i i−1 i−2 i i

8.6.1 C 11 HGDA for 2D distorted quadrilateral elements in xy space In order to show specific details of the development, we consider C 11 HGDA. Figure 8.28 (a) shows the dofs at the corner nodes of C 11 HGDA element (subscript indicates differentiation). Comparing Figure 8.28 (a) with C 00 p-version element of Fig. 8.27 (c), we note that the element of Fig. 8.28 (a) requires φx and φy as additional dofs at each of the four corner nodes, η η φ, φx , φy

7 8 1 φ, φx , φy

∂iφ ∂ξ i

φ, φx , φy

6

5

9

4

7 ξ

2

3

∂j φ ∂η j

6 9

8

4

∂ l+m φ ∂η l ∂ξ m

1 φ, φx , φy

(a) Nodal dofs at the corner nodes of a 2D C 11 HGDA element

5

2

∂j φ ∂η j

3

∂iφ ∂ξ i

i = 4, 5, . . . , pξ l = 2, 3, . . . , pξ

j = 4, 5, . . . , pη m = 2, 3, . . . , pη

(b) Dofs at the hierarchical nodes of 2D C 11 HGDA element

Figure 8.28: Nodal dofs for a 2D C 11 HGDA distorted quadrilateral element

ξ

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

a total of eight dofs for the four corner nodes. We need to borrow eight dofs and the corresponding C 00 p-version approximation functions to generate these dofs and the corresponding approximation functions for the 2D C 11 HGDA element. This would obviously result in reduction of dofs at the hierarchical nodes of the 2D HGDA element. In doing so we must follow a systematic procedure. For this case, the choice of dofs from C 00 element is rather straightforward. We borrow two dofs which are associated with the lowest p-levels (i.e. p = 2 and 3) and the corresponding approximation functions from each of the four mid-side nodes. This implies that the degrees 2 3 2 3 of freedom ∂∂ξφ2 , ∂∂ξφ3 from nodes 2, 6 and ∂∂ηφ2 , ∂∂ηφ3 from nodes 4 and 8 and the corresponding approximation functions are borrowed. These dofs must be eliminated to generate the derivative dofs at the corner nodes of 2D C 11 HGDA element [see Fig. 8.28(a)]. Figure 8.28(b) shows the dofs at the hierarchical nodes of the 2D HGDA element. The remaining details for the derivation of the 2D C 11 HGDA element can be obtained from the general derivation given in a later section. We note that the dofs removed from the mid side nodes of the C 00 pversion element correspond to p-levels 2 and 3 and hence consistent with the tensor product C 11 element. The first degree of freedom at the mid side nodes of the C 11 HGDA element corresponds to p-level of 4. For C 11 HGDA element, we do not need to borrow any dofs from the center node of C 00 p-version element.

8.6.2 C 22 HGDA for 2D distorted quadrilateral elements in xy space We consider 2D C 22 HGDA for distorted quadrilateral elements in xy space. Figure 8.29 (a) shows dofs at the corner nodes of the element. Comparing this with C 00 HGDA of Fig. 8.27(c), we note that φx , φy , φx2 , φxy and φy2 are additional dofs at each of the four corner nodes, i.e. a total of twenty. We need to borrow twenty dofs and the corresponding C 00 p-version approximation functions from 2D C 00 p-version approximation to generate these dofs and the corresponding approximation functions for the 2D C 22 HGDA element. This would obviously result in reduction of dofs at the hierarchical nodes of the 2D HGDA element. For this case, the choice of dofs from C 00 element is not as straight forward as those for C 11 HGDA element. We borrow four dofs which are associated with the p-levels 2, 3, 4 and 5 and the corresponding approximation functions from each of the four mid-side nodes to maintain conformity with C 22 tensor product element. This implies 2 3 4 5 2 3 that the degrees of freedom ∂∂ξφ2 , ∂∂ξφ3 , ∂∂ξφ4 , ∂∂ξφ5 from nodes 2, 6 and ∂∂ηφ2 , ∂∂ηφ3 , ∂4φ ∂5φ , ∂η 4 ∂η 5

from nodes 4 and 8 and the corresponding approximation functions are borrowed. This would result in a total of 16 degrees of freedom.

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

∂6φ ∂6φ ∂η 2 ∂ξ 4 ∂η 4 ∂ξ 2 ∂ l+m φ ∂η l ∂ξ m

η η

φ, φx , φy

φ, φx , φy

∂iφ ∂ξ i

φx2 , φxy , φy2

φx2 , φxy , φy2

7

6

5

9

4

7

8 1

ξ

2

3

∂j φ ∂η j

6

8

4

φ, φx , φy

φ, φx , φy

φx2 , φxy , φy2

φx2 , φxy , φy2

(a) Nodal dofs at the corner nodes of a 2D C 22 HGDA element

2

l = 2, 3, . . . , pξ m = 2, 3, . . . , pη l+m≥7

5

9 1

551

∂j φ ∂η j

ξ

3

i

∂φ ∂ξ i

i = 6, 7, . . . , pξ

j = 6, 7, . . . , pη

(b) Dofs at the hierarchical nodes of 2D C 22 HGDA element

Figure 8.29: Nodal dofs for a 2D C 22 HGDA distorted quadrilateral element

The remaining four degrees of freedom are borrowed from the center node (node 9). The degrees of freedom that are borrowed from center node of the C 00 tensor product element are borrowed in such a way that the dofs corresponding to lower p-levels are selected before those corresponding to higher p-levels. Figure 8.30 shows the dofs generated at the center node of a C 00 element corresponding to p-levels pξ (in ξ direction) and pη (in η direction). The dofs illustrated with a circle are selected in deriving a C 22 HGDA element. They correspond to (pξ , pη ) pairs of (2, 2), (3, 2), (2, 3) and (3, 3). Degree of freedom corresponding to p-level pair of (3, 3) is chosen over (4, 2) or (2, 4) to ensure symmetry with respect to pξ and pη . Symmetry in the degrees of freedom pairs is maintained to preserve the symmetry of finite element solutions for symmetric discretizations. These dofs must be eliminated from C 00 p-version approximations to generate the derivative dofs at the corner nodes required for 2D C 11 HGDA element [Fig. 8.29(a)]. Figure 8.29(b) shows the dofs at the hierarchical nodes of the 2D HGDA element.

8.6.3 C 33 HGDA for 2D distorted quadrilateral elements in xy space We consider 2D C 33 HGDA for distorted quadrilateral elements in xy space. Compared to a C 00 HGDA of Fig. 8.27(c), we note that φx , φy , φx2 , φxy , φy2 , φx3 , φx2 y , φxy2 and φy3 are additional dofs at each of the four corner nodes, a total of thirty six dofs. Hence, we need to borrow thirty six dofs from the hierarchical nodes of C 00 p-version element, keeping in mind that the remaining dofs at the mid side nodes must begin with p-level of eight. This is to ensure that C 33 HGDA is in conformity with C 33 tensor

552

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

pξ pη 1

2

3

4

5

6

η 

η 

∂3 ∂η

η 

∂4 ∂η

3

4

∂3  ξ ∂ξ 

5

∂4  ξ ∂ξ

ξ 2 η2

ξ3η2

ξ 4η2

ξ 2η3

ξ3η3

ξ 4η3

ξ 2η4

ξ3η4

6

∂5  ξ ∂ξ

ξ5η2

∂6  ξ ∂ξ

ξ 6 η2

ξ5η3

ξ6η3

ξ 4 η4

ξ5η4

ξ6η4

∂5  ∂η

ξ 2η5

ξ3η5

ξ 4η5

ξ5η5

ξ6η5

∂6 ∂η

ξ 2η6

ξ3η6

ξ 4η6

ξ5η6

ξ6η6

η 

η 

∂2 ∂η

2

∂2  ξ ∂ξ

Additional dofs from the center node of C00 p-version element for C22 HGDA element Additional dofs from the center node of C00 p-version element for C33 HGDA element

Figure 8.30: Dofs at the center node of a C 00 p-version hierarchical element

product element. This allows us to borrow twenty four dofs (corresponding to p-levels of 2, 3, 4, 5, 6 and 7) from each of the mid side nodes (nodes 2, 4, 6, 8) of Fig. 8.27(c), making a total of twenty four. The remaining twelve dofs needed to generate the dofs at the corner nodes of C 33 HGDA element must come from the center node (node 9). From Fig. 8.30, the dofs illustrated with circle and square are the dofs selected from the center node of C 00 tensor product element in deriving a C 33 HGDA element. The additional dofs correspond to (pξ , pη ) pairs of (2, 2), (3, 2), (2, 3), (3, 3), (2, 4), (2, 5), (3, 4), (3, 5), (4, 2), (4, 3), (5, 2) and (5, 3). These degrees of freedom are chosen in such a way that symmetry is ensured with respect to pξ and pη . With the discussion of the concepts relating to the transformation matrices and the selection of the dofs, the general derivation of the C ij approximations for distorted quadrilateral elements is now presented in the following.

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

553

8.6.4 Derivation of C ij approximations for distorted quadrilateral elements In this section, we describe a general methodology which utilizes C 00 p-version hierarchical interpolation functions as a starting point (described in earlier sections) to generate desired higher order global differentiability approximations for distorted quadrilateral elements. Since the approximation functions are functions of natural coordinates ξ, η, Ni = Ni (ξ, η), the desired derivative degrees of freedom need to be generated first in ξη space and then transformed into xy space as described above for C 11 , C 22 , and C 33 HGDA. The transformations described earlier assist us in transforming the desired derivative degrees of freedom from ξη coordinate space to xy coordinate space. The dofs {δ e } of a nine node C 00 p-version hierarchical element are separated into those corresponding to corner nodes (denoted by co), mid-side nodes (m) and center node (c) as follows: e e e ϕ(ξ, η) = [a]{δco }r1 + [b]{δmc }el + [c]{δm }r2 + [d]{δce }r3

(8.136)

where the subscript r1 denotes the degrees of freedom retained at the corner nodes. Subscript el corresponds to the degrees of freedom borrowed from the mid-side nodes and the center node that are to be eliminated to incorporate the new derivative degrees of freedom at the corner nodes of a C ij HGDA element. The subscripts r2 and r3 denote the remaining degrees of freedom (to be retained) from mid-side nodes and center node (after borrowing the required degrees of freedom). [a], [b], [c] and [d] are row matrices containing C 00 p-version local approximations corresponding to the dofs in the r1 , el, r2 , r3 sets respectively. e } consists of dofs from mid-side nodes For a C 11 HGDA element, {δmc el e } ={δ e } , only since we do not need any dofs from the center node, i.e. {δmc el m el which consists of the following:   2 ∂ φ ∂ 3 φ ∂ 2 φ ∂ 3 φ ∂ 2 φ ∂ 3 φ ∂ 2 φ ∂ 3 φ T e {δm }e = , , , , , , , ∂ξ 2 2 ∂ξ 3 2 ∂η 2 4 ∂η 3 4 ∂ξ 2 6 ∂ξ 3 6 ∂η 2 8 ∂η 3 8 (8.137) The subscript in (8.137) indicates node number. For classes higher than e } will consist of dofs from mid-side nodes as well as center node C 11 , {δmc el as described in sections 8.6.2 and 8.6.3. Let the desired new derivative dofs at the corner nodes of a C ij HGDA 11 HGDA element, these dofs element be denoted by {δ e }xy n . In case of a C will consist of complete set of first order derivatives of the dependent variable evaluated at the four corner nodes, i.e. n oT {δ e }ξη = φ | , φ | , φ | , φ | , φ | , φ | , φ | , φ | (8.138) η 1 η 3 η 5 η 7 ξ 1 ξ 3 ξ 5 ξ 7 n

554

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

where subscripts ξ and η on φ denote differentiations, φξ |1 =

∂φ ∂ξ |node 1

=

∂φ ∂ξ |(ξ=−1,η=−1 ).

For classes higher than C 11 , the new derivative dofs at the corner nodes will be augmented with the complete sets of derivatives up to the class being derived. Differentiating (8.136) with respect to ξ and η and evaluating the resulting expression at each of the four corner nodes, we get e e e e {δ e }ξη n = [A]{δco }r1 + [B]{δmc }el + [C]{δm }r2 + [D]{δc }r3

(8.139)

e } in (8.139), Solving for the degrees of freedom to be eliminated i.e. {δmc el we get e −1 e −1 e −1 e {δmc }el = [B]−1 {δ e }ξη n − [B] [A]{δco }r1 − [B] [C]{δm }r2 − [B] [D]{δc }r3 (8.140) Substituting Jacobian of transformation from (8.135) into the above equation, we can transform the new derivative dofs from ξη space to xy space. Equation (8.140) can thus be written as e −1 e −1 e −1 e {δmc }el = [B]−1 [Ji ]{δ e }xy n −[B] [A]{δco }r1 −[B] [C]{δm }r2 −[B] [D]{δc }r3 (8.141) e Now, substituting {δmc }el from (8.141) into (8.136),

 −1 e φ(ξ, η) = + [b] [B]−1 [Ji ]{δ e }xy n − [B] [A]{δco }r1  −1 e −1 e e − [B] [C]{δm }r2 − [B] [D]{δc }r3 + [c]{δm }r2 + [d]{δce }r3 e [a]{δco }r1

(8.142)

Collecting terms in the (8.142), we get the final form of the C ij HGDA local approximations as follows:
e φ(ξ, η) = [a] − [b][B]−1 [A] {δco }r1 + [b][B]−1 [Ji ]{δ e }xy n

−1 e −1 + [c] − [b][B] [C] {δm }r2 + [d] − [b][B] [D] {δce }r3

(8.143)

8.6.5 Limitations of 2D C 11 global differentiability local approximations for distorted quadrilateral elements In the proposed framework, 2D C ij global differentiability local approximations are derived by borrowing appropriate degrees of freedom and the corresponding approximation functions from the hierarchical nodes of C 00 element. In (8.143), [a], [c] and [d] contain C 00 local approximations which are retained at corner, mid-side and center nodes whereas [b] contains C 00 local approximation functions which are borrowed from mid-side and center nodes. [A], [B], [C] and [D] are matrices containing derivatives of C 00

¯ E ) APPROXIMATIONS FOR QUADRILATERAL ELEMENTS 8.6. 2D C IJ (Ω

555

approximations collected in [a], [b], [c] and [d] with respect to ξ and η evaluated at the corner nodes. The approximation functions for the C 11 distorted element at the corner, mid-side and center nodes (which are retained) are obtained by modifying the corresponding functions for the C 00 element by [b][B]−1 [A], [b][B]−1 [C] and [b][B]−1 [D] respectively. In case of 2D C 11 HGDA element, the new derivative degrees of freedom introduced at the corner nodes are first order derivatives with respect to x and y. The nature of the C 00 local approximation functions and the coordinates of the corner nodes (ξ and η coordinates are either +1 or −1) always result in all of the coefficients of [D] matrix to be zero regardless of the p-level. The coefficients of matrices [A], [B], [C] however are not all zero. This results in the approximation functions at the center node of C 11 distorted element to be exactly same as those corresponding to C 00 element (since [b][B]−1 [D] is a row matrix containing all zeros). When we derive approximation functions for C 22 and higher order elements, the derivative degree of freedoms introduced at the corner nodes include mixed derivatives with respect to x and y. The mixed derivatives of the C 00 approximation functions in [d] (center node) evaluated at the corner nodes are not all zero and hence coefficients in [D] matrix are not all zero. This possibly results in loss of complete basis for C 11 HGDA elements. This is currently under investigation. An alternative way to generate C 11 HGDA element is being considered. Remarks. 1. The derivation presented above is general and is independent of the nature of the C 00 interpolation functions. Hence, the row matrices [a], [b] , [c] and [d] can contain approximation functions of any kind (for example, Lagrange, Legendre or Chebyshev functions). The different choices of C 00 approximation functions would yield the corresponding C ij HGDA elements. This could be important from the point of view of conditioning number of the resulting matrices in applications. 2. The matrices [A], [B], [C], [D] contain derivatives of C 00 approximation functions with respect to ξ and η evaluated at the corner nodes. They can be precomputed once and used to generate approximation functions of any order C ij ; i, j ≥ 2 element. This is important from the point of view of efficiency of computations. 3. The approximation functions that are borrowed from the mid-side nodes and center node of C 00 elements should be such that: (i) lowest degree admissible functions (corresponding to a lower p-level) are selected first (ii) and a symmetric pattern maintained in their selection so that symmetric discretizations for symmetric behaviors would yield numerical solutions that are also symmetric.

556

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

8.7 Interpolation theory for 2D triangular elements: basis functions of class C 00 based on Lagrange interpolation 8.7.1 Langrange family C 00 basis functions based on Pascal triangle Following a procedure similar to that for quadrilateral elements using Pascal’s rectangle, local approximations of class C 00 can be established for triangular elements as well. However, in this case the mapping of the element from the physical space x, y or x, y, z to a natural coordinate space is somewhat different than for the quadrilateral shapes. First, let us consider a triangular subdomain or element with straight sides in the physical coordi¯ e would require a decision nate space. A local approximation φ(x, y) over Ω on (just like quadrilateral elements): (a) The choice of the monomial terms in x, y and xy etc. to be used in the ¯ e based on complete linear, linear combination to describe φ(x, y) over Ω quadratic, cubic, etc. polynomials in x and y. (b) The choice of the locations of the nodes on the boundary of the element as well as its interior. This decision can be greatly facilitated by considering what is known as Pascal’s triangle. Pascal’s triangle is a systematic arrangement of monomials of various degrees in x and y and their interaction such that the resulting polynomials of any fixed degree in x and y can be readily constructed. Along one side of the triangle we have increasing powers of x starting with zero while on the other side of the triangle we have increasing powers of y also starting with zero. The horizontal lines connecting the corresponding like power terms in x and y provides the interaction terms between x and y. Using the Pascal’s triangle local approximation functions of any degree can be generated for the triangular elements. First, we note the following: (a) The location of the terms in the triangular arrangement are the locations ¯ e. of the nodes for the triangular element Ω (b) Also, the terms themselves are the functions to be used in the linear combination for φ(x, y). (c) Based on these two criteria, elements with constant, bilinear, bi-quadratic ¯ e with 1, 3, 6 and 10 nodes (see Fig. 8.31) and bi-cubic behavior of φ over Ω can be easily generated. Example 8.5. Consider the six-node bi-quadratic element in Fig. 8.32, which shows the configuration of nodes as well as the monomials to be con-

557

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS Pascal’s triangle

Degree of the polynomial

1

0

1

1

3

2

6

3

10

4

15

5

21

6

28

y

x x

2

xy

x3 x4 x5 x6

xy x4 y

y3

xy 2

x2 y 3

x5 y

y

2

xy

x3 y 2

3

y

3

x2 y 3

x4 y 2 x3 y 3

y

4

xy 4

x2 y 4

y5

xy 5

y6

Element with nodes

Number of terms in the polynomial

Figure 8.31: Pascal’s triangle for the C 00 Lagrange family of triangular elements.

¯ e . For the local approxsidered in the polynomial representation of φ over Ω ¯ e , we can immediately write imation of φ over Ω φ(x, y) = c1 + c2 x + c3 y + c4 xy + c5 x2 + c6 y 2

(8.144)

Let (xi , yi ); i = 1, . . . , 6 be the coordinates of the nodes of the element and let φei (i = 1, . . . , 6) be the function values of φ at these nodes. Then  e  6 φ1   X   .. −1 e 2 2 ˆ φ(x, y) = 1 x y xy x y [C] = [N (x, y)]{φ } = Ni (x, y)φei .   e i=1 φ6 (8.145) in which   1 x1 y1 x1 y1 x21 y12 . . . . . . ˆ = [C] (8.146)  .. .. .. .. .. ..  1 x6 y6 x6 y6 x26 y62

6 4

1

1

5 2 Nodes

y

x 3

x2

xy

y2

Monomials

Figure 8.32: Six-node triangular element

Once again, we could verify that ( 1, Ni (xj , yj ) = 0,

j=i , i = 1, . . . , 6 j 6= i

(8.147)

558

and

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

6 X

Ni (x, y) = 1

(8.148)

i=1

Remarks. (1) The shortcomings of this approach are similar to those for the quadrilateral elements based on Pascal’s rectangle. With increasing p-levels, number of nodes per element increase requiring a new discretization. Establishing the local approximation functions requires inverse of progressively increasing size coefficient matrices with progressively increasing p-levels. (2) The basis functions derived using this approach are clearly of C 0 type (as function values are the only nodal dofs) of Lagrange family. (3) It is significant to note that regardless of the shortcomings of this approach, Pascal’s triangle provides a systematic means of the nodal configurations and the selection of monomials based on p-levels.

8.7.2 Lagrange family C 00 basis functions based on area coordinates The concept of area coordinates is the most natural way to derive basis functions for triangular elements. Consider a three node triangular element with straight sides shown in Fig. 8.33. Let the sides opposite to node i be labelled side i for i = 1, 2, 3. Consider a point inside the element and connect it with straight lines to the nodes of the element. This divides the area A of the triangle into three triangular areas A1 , A2 and A3 , noting that area Ai is along the side i, i = 1, 2, 3. Let Li be the ratio of AAi (i = 1, 2, 3). Li =

Ai , i = 1, 2, 3 A

(8.149)

using A = A1 + A2 + A3

(8.150)

and dividing both sides by A and then using (8.149) 1 = L1 + L2 + L3

(8.151)

Li (i = 1, 2, 3) are called area coordinates or natural coordinates for the triangular region of area A. The properties of L1 , L2 and L3 are important to note. (i) Li has a value of 1 at node i and has a value of zero on side i and thus has a value of zero at the other nodes. That is, L1 is one at node 1, zero on side 1 and, hence zero at nodes 2 and 3.

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS

559

3 Side 2

Side 1 A1

A2 A3

y

2

Side 3

1 x

Figure 8.33: Triangular coordinates

(ii) Their sum is obviously one at any point within the element ((8.151)). The area coordinates Li can be used to define the Cartesian coordinates of ¯ e . If (xi , yi ) are the Cartesian coordinates of the nodes of the any point in Ω element, then we can write x = L1 x1 + L2 x2 + L3 x3 y = L1 y1 + L2 y2 + L3 y3 using (8.151) and (8.152)        1 1 1 L1  L1  1 x = x1 x2 x3  L2 = [C] L2       L3 L3 y y1 y2 y3 Therefore

      L1   1  N1 (x, y) L2 = [C]−1 x = N2 (x, y)       N3 (x, y) L3 y

(8.152)

(8.153)

(8.154)

We could easily confirm that Ni (x, y) (i = 1, 2, 3) are same as those using Pascal triangle. We note that det[C] = A2 , hence [C]−1 in (8.154) is unique. Thus, for the triangular element with three nodes (see Fig. 8.33) we can use L1 , L2 , L3 as basis functions instead of Ni (x, y) (i = 1, 2, 3). In the next section, we extend this concept of area coordinate to basis functions of higher p-level.

8.7.3 Higher degree C 00 basis functions using area coordinates Consider the triangular elements with straight sides shown in Fig. 8.34 with vertex nodes 1,2 and 3. First, we explain how the figures in Fig. 8.34 are constructed. Consider Fig. 8.34(a). Draw m equally spaced lines parallel to side 1 labelled 0, 1, . . . , m, ‘0’ being side 1. Similarly we also draw m equally

560

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

3

3

Side 2

Side 1

2

1 m m−1 p

1 0

0

1

(a) Equally spaced m lines parallel to side 1 3 m

2 1 p

m−1 m

(b) Equally spaced m lines parallel to side 2 3 p q

m−1

pqr

r

r

1 0 1

1

2

Side 3

(c) Equally spaced m lines parallel to side 3

2

(d) Identification of a point based on lines parallel to sides 1 to 3

Figure 8.34: Identification of element nodes based on area coordinates

spaced lines parallel to sides 2 and 3 [see Fig. 8.34(b) and (c)], sides 2 and 3 being marked ‘0’. If we consider typical lines p, q and r parallel to sides 1, 2 and 3, then their intersection marked pqr [Fig. 8.34(d)] defines a point within the element. The interaction of these parallel lines with the sides 1, 2 and 3 and amongst themselves define the locations of the nodes. These are in agreement with the Pascal triangle. Consider triangle in Fig. 8.34(a). We note that L1 = 0 on side 1 and has a value 1 at node 1. We can setup Lagrange type interpolations to define functions N0 (L1 ), N1 (L1 ), . . . , Nm (L1 ) corresponding to the (m + 1) parallel lines. We define Ni (L1 ) by Ni (L1 ) =

i  Y mL1 − j + 1  j=1

j

, for j ≥ 1; i = 1, 2, . . . , m

(8.155)

= 0, for i = 0 Similarly in the other two directions we setup Ni (L2 ) and Ni (L3 ) Ni (L2 ) =

i  Y mL2 − j + 1  j=1

j

= 0, for i = 0

, for j ≥ 1; i = 1, 2, . . . , m

(8.156)

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS

Ni (L3 ) =

i  Y mL3 − j + 1 

j

j=1

, for j ≥ 1; i = 1, 2, . . . , m

561

(8.157)

= 0, for i = 0 The two dimensional basis function for a node located at p, q, r i.e. Npqr (L1 , L2 , L3 ) can now be defined Npqr (L1 , L2 , L3 ) = Np (L1 ) Nq (L2 ) Nr (L3 )

(8.158)

Np (L1 ), Nq (L2 ) and Nr (L3 ) are defined by (8.155) to (8.157). We consider some examples in the following. Three-node triangular element (p-level of one) In this case m = 1 and we have the pqr values at the nodes shown in Fig. 8.35, and N0 (L1 ) = 1 N1 (L1 ) = L1 , m = 1, j = 1, i = 1 in (8.155) N0 (L2 ) = 1 N1 (L2 ) = L2 , m = 1, j = 1, i = 1 in (8.156) N0 (L3 ) = 1 N1 (L3 ) = L3 , m = 1, j = 1, i = 1 in (8.157)

(8.159)

(8.160) (8.161)

(pqr) = (001) 3 Side 1

Side 2

2

1 (pqr) = (100)

Side 3

(pqr) = (010)

Figure 8.35: pqr values for three node element

The basis functions Ni (L1 , L2 , L3 ) (i = 1, 2, 3) for the three-node triangular element are now obtained using the following: N1 (L1 , L2 , L3 ) = N

= N1 (L1 ) N0 (L2 ) N0 (L3 ) = L1

(8.162)

N2 (L1 , L2 , L3 ) = N

= N0 (L1 ) N1 (L2 ) N0 (L3 ) = L2

(8.163)

N3 (L1 , L2 , L3 ) = N

= N0 (L1 ) N0 (L2 ) N1 (L3 ) = L3

(8.164)

pqr (100)

pqr (010)

pqr (001)

562

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Six-node triangular element (p-level of two) Figure 8.36 shows a six-node triangular element for p = 2 in x and y (based on Pascal’s triangle) and pqr values at the nodes of the element. We generate Ni (L1 ), Ni (L2 ) and Ni (L3 ) functions using (8.155) to (8.157). In this case m = 2. N0 (L1 ) = 1 N1 (L1 ) =

1  Y 2L1 − j + 1 

j

j=1

N2 (L1 ) =

(8.165)

2  Y 2L1 − j + 1 

j

j=1

= 2L1

= L1 (2L1 − 1)

3 (002) 5 6

4

(101)

(011) 2

1 1 (200)

(110)

3 (020) 2

vertex nodes for L1, L2, and L3 lines and locations of 1, 2, . . . , 6 nodes of the six-node triangular element

Figure 8.36: pqr values at the nodes of a six-node quadratic triangular element

Similarly, N0 (L2 ) = 1 N1 (L2 ) =

1  Y 2L2 − j + 1  j=1

N2 (L2 ) =

j

2  Y 2L2 − j + 1  j=1

j

= 2L2 (8.166) = L2 (2L1 − 1)

N0 (L3 ) = 1 N1 (L3 ) =

1  Y 2L3 − j + 1  j=1

N2 (L3 ) =

j

2  Y 2L3 − j + 1  j=1

j

= 2L3 (8.167) = L3 (2L3 − 1)

563

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS

The basis functions Ni (L1 , L2 , L3 ) (i = 1, 2, . . . , 6) for the six-node quadratic element can be generated using the following: pqr (L1 , L2 , L3 ) (200)

= N2 (L1 ) N0 (L2 ) N0 (L3 ) = L1 (2L1 − 1)

N2 (L1 , L2 , L3 ) = N

pqr (L1 , L2 , L3 ) (111)

= N1 (L1 ) N1 (L2 ) N0 (L3 ) = 4L1 L2 )

N3 (L1 , L2 , L3 ) = N

pqr (L1 , L2 , L3 ) (020)

= N0 (L1 ) N2 (L2 ) N3 (0) = L2 (2L1 − 1)

pqr (L1 , L2 , L3 ) (011)

= N0 (L1 ) N1 (L2 ) N1 (L3 ) = 4L2 L3

N5 (L1 , L2 , L3 ) = N

pqr (L1 , L2 , L3 ) (002)

= N0 (L1 ) N0 (L2 ) N2 (L3 ) = L3 (2L3 − 1)

N6 (L1 , L2 , L3 ) = N

= N1 (L1 ) N2 (0) N3 (1) = 4L1 L3

N1 (L1 , L2 , L3 ) = N

N4 (L1 , L2 , L3 ) = N

pqr (L1 , L2 , L3 ) (101)

(8.168) Ten-node triangular element (p-level of three) Figure 8.37 shows a ten-node cubic triangular element (p-level of three) in x and y (see Pascal triangle also) and pqr values at the nodes of the element. We generate Ni (L1 ), Ni (L2 ) and Ni (L3 ) functions using (8.155) to (8.157). In this case m = 3. 3 7

(003) 6

8 (102)

(012) 10

9 (201) 1

5 (021)

(111)

1 (300)

2 (210)

3 (120)

4 (030)

2

vertex nodes for L1, L2, and L3 lines 1, 2, . . . , 10 nodes for p-level of three

Figure 8.37: pqr values at the nodes of a ten-node cubic triangular element

We have N0 (L1 ) = 1 N1 (L1 ) =

1  Y 3L1 − j + 1  j=1

j

= 3L1

564

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

N2 (L1 ) =

2  Y 3L1 − j + 1 

j

j=1

N3 (L1 ) =

3  Y 3L1 − j + 1 

j

j=1

3 = L1 (3L1 − 1) 2 =

L1 (3L1 − 1)(3L1 − 2) 2

(8.169)

Similarly, N0 (L2 ) = 1 N1 (L2 ) = 3L2 3 N2 (L2 ) = L2 (3L2 − 1) 2 L2 N3 (L2 ) = (3L2 − 1)(3L2 − 2) 2

(8.170)

N0 (L3 ) = 1 N1 (L3 ) = 3L3 3 N2 (L3 ) = L3 (3L3 − 1) 2 L3 N3 (L3 ) = (3L3 − 1)(3L3 − 2) 2

(8.171)

The basis functions Ni (L1 , L2 , L3 ) (i = 1, 2, . . . , 10) for the ten-node cubic triangular element (see Fig. 8.37) can be generated using the following: N1 (L1 , L2 , L3 ) = N

pqr (L1 , L2 , L3 ) (300)

= N3 (L1 ) N0 (L2 ) N0 (L3 )

L1 (3L1 − 1)(3L1 − 2) 2 N2 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N2 (L1 ) N1 (L2 ) N0 (L3 ) =

(210)

9 = L1 L2 (3L1 − 1) 2 N3 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N1 (L1 ) N2 (L2 ) N0 (L3 ) (120)

9 = L1 L2 (3L2 − 1) 2 N4 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N0 (L1 ) N3 (L2 ) N0 (L3 ) (030)

=

L2 (3L2 − 1)(3L2 − 2) 2

8.7. INTERPOLATION THEORY FOR 2D TRIANGULAR ELEMENTS

pqr (L1 , L2 , L3 ) (021)

N5 (L1 , L2 , L3 ) = N

565

= N0 (L1 ) N2 (L2 ) N1 (L3 )

9 = L2 L3 (3L2 − 1) 2 N6 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N1 (L1 ) N2 (L2 ) N0 (L3 ) (012)

9 = L2 L3 (3L3 − 1) 2 N7 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N0 (L1 ) N0 (L2 ) N3 (L3 ) (003)

L3 = (3L3 − 1)(3L3 − 2) 2 N8 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N1 (L1 ) N0 (L2 ) N2 (L3 ) (102)

9 = L1 L3 (3L3 − 1) 2 N9 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N2 (L1 ) N0 (L2 ) N1 (L3 ) (201)

9 = L1 L3 (3L1 − 1) 2 N10 (L1 , L2 , L3 ) = N pqr (L1 , L2 , L3 ) = N0 (L1 ) N0 (L2 ) N3 (L3 ) (111)

= 27L1 L2 L3

(8.172)

Using the procedure used for linear, quadratic and cubic triangular elements, higher degree C 00 local approximation functions can be derived easily. Remarks. (1) As in case of quadrilateral family of 2D elements based on Lagrange functions derived using tensor product, these triangular element have similar short comings with increasing p-level the number of nodes per element increase requiring a new discretization. Secondly the approximation functions do not have hierarchical property. (2) In applications distorted triangular elements are essential, hence we need to establish a mechanism to map the distorted triangular element into a regular master element that has straight sides. (3) Many other details of the derivatives of Ni (L1 , L2 , L3 ) with respect to x, y as needed in the integral forms and the details of integration over the element area also need to be considered. (4) The developments needed in (1)–(3) are facilitated if we consider Legendre polynomials instead of Lagrange polynomials in the derivations of the basis functions.

566

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

8.8 1D and 2D local approximations based on Legendre polynomials 8.8.1 Legendre polynomials Legendre polynomials1 are given by Rodriguez formula [3] Pn (ξ) =

 1 dn  2 (ξ − 1)n ; ξ ∈ [−1, 1]; n = 0, 1, 2, . . . n n n!2 dξ

(8.173)

For example, we have P0 (ξ) = 1 P1 (ξ) = ξ 3ξ 2 − 1 2 5ξ 3 − 3 P3 (ξ) = 2 35ξ 4 − 30ξ 2 + 3 P4 (ξ) = 8

P2 (ξ) =

(8.174)

These polynomials satisfy Legendre differential equation.
0 (1 − ξ 2 )Pn0 (ξ) + n(n + 1)Pn (ξ) = 0, ξ ∈ (−1, 1) The Legendre polynomials have the following orthogonal property  Z1  2 , for m = n Pn (ξ) Pm (ξ) dξ = 2n + 1  0, for m 6= n −1

(8.175)

(8.176)

The Legendre polynomials can also be represented by a recursive relation. P0 (ξ) = 1 P1 (ξ) = ξ

(8.177)


1 (2i + 1)ξ Pi (ξ) − iPi−1 (ξ) , i = 1, 2, . . . Pi+1 (ξ) = i+1 The Legendre polynomials can be used to define 1D p-version hierarchical approximation functions. 1

Adrien-Marie Legendre (1752–1833) was a French mathematician, who made numerous contributions to mathematics, such as the Legendre polynomials and Legendre transformation. Legendre polynomials are solutions to Legendre’s differential equation: h i 2 d d P (ξ) + n(n + 1)Pn (ξ) = 0. (1 − ξ ) dξ dξ n

567

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS

8.8.2 1D p-version C 0 hierarchical approximation functions (Legendre polynomials) Babuska and Szabo [4] have shown that for the three-node 1D configuration of Fig. 8.38 the C 0 p-version hierarchical interpolation for a dependent variable φ can be written as (for an element e) φeh (ξ) = N1 (ξ) φe1 + N2 (ξ) φe2 +

pξ X

N3i (ξ) δie

(8.178)

i=2

in which 1−ξ 2 1+ξ N2 (ξ) = 2

N1 (ξ) =

Ni (ξ) = p

(8.179) 1

2(2i − 1)


Pi (ξ) − Pi−2 (ξ) , i = 2, 3, . . .

φe1 and φe2 are function values at nodes 1 and 2 and δie ; i = 2, . . . are the nodal degrees of freedom at the hierarchical node 3 corresponding to plevels of 2, 3, . . .. We can show that these approximation functions satisfy the desired properties. Based on the C 00 p-version hierarchical derivation i presented earlier using Lagrange polynomials, we can designate δie = ∂∂ξφi (i = 2, 3, . . .). 1

3

2

-1

0

1

ξ

Figure 8.38: 1D three-node configuration

8.8.3 2D p-version C 00 hierarchical interpolation functions for quadrilateral elements (Legendre polynomials) If we consider a none-node element configuration in ξη space (in a two unit square), then using the 1D functions and nodal variable operators defined by (8.178) in ξ and η directions and by taking their tensor products, we can construct 2D p-version C 00 hierarchical local approximations or interpolations for the nine-node element. This procedure is identical to what has been used for Lagrange family of 2D C 00 p-version hierarchical local approximations based on tensor product. Details are straight forward.

568

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

8.8.4 2D C ij p-version interpolations functions for quadrilateral elements (Legendre polynomials) The derivation of these interpolation functions and nodal dofs follows exactly same procedure as used for C ij 2D p-version interpolations based on Lagrange family except that the corresponding C 00 2D p-version hierarchical interpolation functions of Lagrange type are replaced by those based on Legendre type. The details are straight forward.

8.8.5 2D C 00 p-version interpolation functions for triangular elements (Legendre polynomials) We consider a seven-node (three vertex, three mid side and a center node) distorted triangular element in xy space [see Fig. 8.39(a)]. The distorted element of Fig. 8.39 is mapped into ξη space in a two-unit equilateral triangle [Fig. 8.39(b)]. The midside and the center nodes are hierarchical nodes whereas the corner nodes are non-hierarchical nodes. The origin of the ξη coordinate system is located at node 2 of the equilateral triangle. The mapping of points can be defined using     X n ¯ x x ¯ = Ni (L1 , L2 , L3 ) i y yi

(8.180)

i=1

¯i (L1 , L2 , L3 ) are standard shape functions based on area coordiin which N nates derived using seven-node configuration of Fig. 8.39(b). We could also use six-node configuration with parabolic shape functions derived earlier. η

5

5

6 y

7

2

1

4

6

3

x

(a)

A seven-node distorted triangular element in xy space

4

7

1

3

2 1

ξ

1

(b) A seven-node master element in ξη space

Figure 8.39: A seven-node distorted triangular element in xy space and its map in the master element in ξη

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS

569

The area coordinates L1 , L2 , L3 can be related to the orthogonal natural coordinates ξ, η through the relations introduced by Szabo [4] 1 1−ξ− 2 1 L2 = 1+ξ− 2 η L3 = √ 3 L1 =

η  √ 3 η  √ 3

(8.181)

Relations (8.181) can be used to convert the interpolation functions given in (8.180) from L1 , L2 , L3 to natural coordinates ξ, η.     X n ¯ x x ¯ = Ni (ξ, η) i y yi

(8.182)

i=1

We now present C 00 p-version hierarchical interpolations over the element of Fig. 8.39(b). Following Babuska and Szabo [4], we can consider sevennode triangular element in the ξη space (Fig. 8.40). The nodal degrees of freedom for pξ = pη = p in ξ and η are also shown in Fig. 8.40: 1. Interpolation functions for vertex nodes N1 = L1 N3 = L2

(8.183)

N5 = L3 L1 + L2 + L3 = 1 2. Interpolation functions for mid-side nodes

(p − 1) mid-side interpolation functions are defined in terms of Legendre polynomials (Pi ) at each of the three mid-side nodes (2, 4 and 6), thus giving a total of 3(p − 1) interpolation functions  N2i = L1 L2 ψi (L2 − L1 ) 

N4i = L2 L3 ψi (L3 − L2 ) N6i

= L1 L3 ψi (L1 − L3 )

, i = 2, 3, . . . , p

(8.184)

 

where ψi (·) is defined by  ψi (α) =

 Pi (α) − Pi−2 (α) 4 p , i = 2, 3, . . . , p (1 − α2 ) 2(2i − 1)

(8.185)

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

η

5 φ δni ; i = 1, 2, . . . , (p−1)(p−2) 2 j δs2 ;

6 j = 2, . . . , p

φ 1

7

4 j δs1 ; j = 2, . . . , p

3 φ 2 j δs3 ; j = 2, . . . , p

ξ

Figure 8.40: Degrees of freedom for C 00 p-version hierarchical triangular element

in which α assumes a value of (L2 − L1 ), (L3 − L2 ), (L1 − L3 ) for mid-side interpolation functions corresponding to nodes 2, 4 and 6 respectively. First few terms of ψi (α) are √ ψ2 (α) = − 6 √ ψ3 (α) = − 10α (8.186) r 7 (5α2 − 1) and so on ψ4 (α) = − 8 3. Internal interpolation function From the Pascal triangle the total number of interpolation functions corresponding to p-level of p are 12 (p + 1)(p + 2) for completeness. From the sum of the interpolation functions for the vertex and the mid-side nodes, we require 21 (p − 1)(p − 2) additional interpolation functions for completeness. These are defined at the internal node (node 7). these are non-zero only in the interior of the triangular domain and vanish on all three sides of the triangular element. The internal approximation functions associated with node 7 (center node) of the master element for p ≥ 3 are as follows: N7j (L1 , L2 , L3 ) = L1 L2 L3 Pp−i−2 (L2 − L1 )Pi−1 (2L3 − 1) where i = 1, 2, . . . , p − 2 1 j = 1, 2, . . . , (p − 1)(p − 2) 2

(8.187)

For example, for a p-level of 3, we only have one internal interpolation

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS

571

function. N71 (L1 , L2 , L3 ) = L1 L2 L3

(8.188)

For p-level of 4, we have the following three internal interpolation functions. N71 (L1 , L2 , L3 ) = L1 L2 L3 N72 (L1 , L2 , L3 ) = L1 L2 L3 P1 (L2 − L1 ) N73 (L1 , L2 , L3 )

(8.189)

= L1 L2 L3 P1 (2L3 − 1)

For p-level of 9 the triangular C 00 p-version element contains 55 interpolation functions: 3 at vertex nodes, 24 at mid-side nodes and 28 internal. We transform the C 0 p-version interpolation functions in L1 , L2 , L3 to ξη space using (8.181). Remarks. 1. The integration of the coefficients of the element matrix and vectors encountered in finite element processes needs to be considered (see Appendix A). 2. The degrees of freedom at the mid-side nodes in fact are the tangential derivatives of various orders.

8.8.6 2D C ij interpolation functions for triangular elements (Legendre polynomials) In deriving higher order continuity interpolations in xy space for triangular elements, we use C 0 p-version interpolations based on Legendre family using area coordinates as starting point. We follow some guidelines. Selection of the derivative degrees of freedom at the vertices of a 2D HGDA triangular element is dictated by the transformation rules for the derivatives of various orders between xy and ξη spaces. The following choices of nodal operators (or dofs) at the vertices listed in Table 2.1 for C 11 , C 22 and C 33 HGDA satisfy the requirements. We note that for C 11 HGDA element, the derivative operators at the vertex nodes are a complete set of first order operators. For C 22 HGDA, the set of C 11 is augmented by a complete second order set of derivatives and so on. This selection of degrees of freedom is consistent with the framework developed for 2D distorted quadrilateral elements presented in a previous section and reference [2]. Since C 00 p-version hierarchical approximations are used as a starting point, which have only function value as a degree of freedom at the vertex nodes, we must establish some rules that allow us to borrow some dofs from C 00 p-version hierarchical approximations to generate the desired dofs at the vertices of the 2D HGDA for the distorted triangular element.

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

The details presented in for quadrilateral elements for transforming derivatives of various orders between ξη and xy spaces also holds for 2D distorted triangular elements between ξη and xy spaces and hence not repeated. The only difference being that we need to use the geometry mapping equations (8.180) in their construction. Table 8.8: Choices of dofs at the corner nodes for C ij 2D distorted triangular elements in xy space

Type of HGDA C 11

Nodal Operators at the corner nodes ∂ ∂ 1, ∂x , ∂y

C 33

2 2 ∂2 , ∂ , ∂ ∂x2 ∂y∂x ∂y 2 3 3 3 ∂2 ∂2 ∂3 , ∂ , ∂ , ∂ ∂y∂x , ∂y 2 , ∂x3 ∂y∂x2 ∂y 2 ∂x ∂y 3

∂ ∂ 1, ∂x , ∂y ,

C 22 ∂ ∂ , ∂y , 1, ∂x

∂2 , ∂x2

C 11 HGDA for 2D distorted triangular elements in xy space In this section we present specific details for C 11 HGDA element. Figure 8.41 (a) shows the dofs at the vertices of C 11 HGDA element (subscript indicates differentiation). Comparing Fig. 8.41 (a) with C 00 p-version element of Fig. 8.40, we note that the C 11 element requires φx and φy as additional dofs at each of the three vertices i.e. a total of six dofs for the three vertices. We borrow six dofs and the corresponding C 00 p-version approximation functions to generate the desired derivative dofs and the corresponding approximation functions for the 2D C 11 HGDA element. This would obviously result in reduction of dofs at the hierarchical nodes of the C 0 p-version element. In doing so we must follow a systematic procedure. For all HGDA element, the choice of dofs from C 00 element is rather straightforward. We borrow dofs corresponding to p-levels of 2 and 3 from mid-side nodes 2, 4 and 6. These dofs must be eliminated from C 00 pversion approximations to generate the derivative dofs at the vertices of 2D C 11 HGDA element as shown in Fig. 8.41 (a). Figure 8.41 (b) shows the dofs at the hierarchical nodes of the 2D HGDA element. The first degree of freedom at the mid-side nodes of the C 11 HGDA element corresponds to p-level of 4. For C 11 HGDA element, we do not need to borrow any dofs from the internal node of C 00 p-version element. C 22 HGDA for 2D distorted triangular elements in xy space Here we consider 2D C 22 HGDA for distorted triangular elements in xy space. Figure 8.42 (a) shows dofs at the corner nodes of the element. Comparing this with C 00 p-version element of Fig. 8.40, we note that φx ,

573

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS

η

5 6

φ, φx , φy 1

η

5

φ, φx , φy

7

2

4

6 j δs2

ξ 3 φ, φx , φy

1

7

4

δij

j δs3

j 2 δs1

3

ξ

j = 4, . . . , p (a) Nodal dofs at the corner nodes of a 2D C 11 HGDA element

i = 1, 2, . . . , (p−1)(p−2) 2 (b) Nodal dofs at the hierarchical nodes of a 2D C 11 HGDA element

Figure 8.41: Nodal dofs for C 11 2D distorted triangular element

φy , φx2 , φxy and φy2 are additional dofs at each of the three vertex nodes, a total of fifteen. Hence, we need to borrow fifteen dofs from the hierarchical nodes of C 00 element shown in Fig. 8.40, keeping in mind that the remaining dofs at the mid-side nodes of C 00 element must begin with p-level of six. This is due to the fact that a quintic polynomial describes a C 22 approximation in 1D. This allows us to borrow four dofs (corresponding to p-levels of 2, 3, 4 and 5) from each of the mid-side nodes of Fig. 8.40, making a total of twelve. The remaining three dofs needed to generate the dofs at the corner nodes of C 22 HGDA element must come from the internal node. The degrees of freedom are borrowed from internal node of the C 00 pversion element in such a way that the dofs corresponding to a lower p-level are selected before those corresponding to higher p-levels. Figure 8.42 (b) shows the dofs at the hierarchical nodes of the 2D HGDA element. With the discussion of the concepts relating to the selection of the dofs for 11 C and C 22 HGDA, we now present a derivation of the C ij approximations for distorted triangular elements. C ij HGDA for 2D distorted triangular elements in xy space We propose a new methodology (parallel to that used for 2D quadrilateral elements in the previous section and reference [2]) which utilizes C 00 p-version hierarchical interpolation functions as a starting point and generates desired order global differentiability approximations for distorted triangular elements. Since the approximation functions are functions of natural coordinates ξ, η, i.e. Ni = Ni (ξ, η), the desired derivative degrees of freedom can be easily generated first in ξη space and then transformed into xy space.

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

η φ, φx , φy φx2 , φxy , φy2

5 6

φ, φx , φy 1 φx2 , φxy , φy2

η

7

2

5

4

6 j δs2

ξ 3 φ, φx , φy φx2 , φxy , φy2

(a) Nodal dofs at the corner nodes of a 2D C 22 HGDA element

1

7

4

δij j 2 δs1

j δs3

3

ξ

j = 6, . . . , p i = 3, 4, . . . , (p−1)(p−2) 2 (b) Nodal dofs at the hierarchical nodes of a 2D C 22 HGDA element

Figure 8.42: Nodal dofs for C 22 2D distorted triangular element

The transformation matrices similar to those presented for quadrilateral elements assist us in transforming the desired derivative degrees of freedom from ξη to xy space. Using the C 00 p-version hierarchical approximations for a seven node element (see Fig. 8.40), the field variable φ can be approximated as, φeh (ξ, η) = [N (ξ, η)]{δ e }

(8.190)

in which [N (ξ, η)] is a row matrix of C 00 p-version hierarchical local approximations and {δ e } are the corresponding nodal dofs (arranged in some suitable fashion). The dofs in {δ e } of (8.190) are grouped into those corresponding to vertex or corner nodes (denoted by co), mid-side nodes (denoted by m) and internal node (denoted by i) as follows: e e e φeh (ξ, η) = [a]{δco }r1 + [b]{δmi }el + [c]{δm }r2 + [d]{δie }r3

(8.191)

where a subscript r1 denotes the degrees of freedom retained from corner nodes. Subscript el corresponds to the degrees of freedom borrowed from the mid-side nodes and the internal node that are to be eliminated to derive the new derivative degrees of freedom at the corner nodes of a C ij HGDA element. Subscripts r2 and r3 denote the degrees of freedom remaining at the mid-side nodes and internal node (after borrowing the required degrees of freedom), and [a], [b], [c] and [d] are vectors containing C 00 p-version local approximations corresponding to the dofs in the r1 , el, r2 and r3 sets respectively. e } consists of dofs from mid-side nodes For a C 11 HGDA element, {δmi el e } = {δ e } , only, since we do not need any from the internal node, {δmi el m el

8.8. 1D AND 2D APPROXIMATIONS BASED ON LEGENDRE POLYNOMIALS

575

which would contain the following dofs: n oT e {δm }e = δ 2 |2 , δ 3 |2 , δ 2 |4 , δ 3 |4 , δ 2 |6 , δ 3 |6

(8.192)

e } will consist of dofs from mid-side nodes For classes higher than C 11 , {δmi el as well as internal node. Let the desired new derivative dofs at the corner nodes of a C ij HGDA 11 HGDA element, these dofs in element be denoted by {δ e }xy n . In case of a C ξη space will consist of complete set of first order derivatives of the dependent variable with respect to ξ and η evaluated at the three corner nodes,

n oT {δ e }ξη n = φξ |1 , φη |1 , φξ |3 , φη |3 , φξ |5 , φη |5

(8.193)

∂φ where subscript denotes differentiation i.e. φξ |1 = ∂φ ∂ξ |node 1 = ∂ξ |ξ=−1,η=0 . For classes higher than C 11 , the new derivative dofs constituting complete sets are added at the corner nodes. Differentiating (8.191) with respect to ξ and η and evaluating the resulting expression at each of the three corner nodes, we get e e e e {δ e }ξη n = [A]{δco }r1 + [B]{δmi }el + [C]{δm }r2 + [D]{δi }r3

(8.194)

e } in (8.194), we Solving for degrees of freedom to be eliminated i.e. {δmi el get e −1 e −1 e −1 e {δmi }el = [B]−1 {δ e }ξη n − [B] [A]{δco }r1 − [B] [C]{δm }r2 − [B] [D]{δi }r3 (8.195) Substituting Jacobian of transformation [Ji ] into the above equation, we can transform the new derivative dofs from ξη space to xy space. Equation (8.195) can thus be written as e −1 e −1 e −1 e {δmi }el = [B]−1 [Ji ]{δ e }xy n −[B] [A]{δco }r1 −[B] [C]{δm }r2 −[B] [D]{δi }r3 (8.196) e } from (8.196) into (8.191) Now, substituting {δmi el e −1 e φeh (ξ, η) = [a]{δco }r1 + [b] [B]−1 [Ji ]{δ e }xy n − [B] [A]{δco }r1
e e − [B]−1 [C]{δm }r2 − [B]−1 [D]{δie }r3 + [c]{δm }r2 + [d]{δie }r3

(8.197)

Collecting terms in the (8.197), we get the final form of the C ij HGDA local approximations as follows:
e φeh (ξ, η) = [a] − [b][B]−1 [A] {δco }r1 + [b][B]−1 [Ji ]{δ e }xy n

e + [c] − [b][B]−1 [C] {δm }r2 + [d] − [b][B]−1 [D] {δie }r3

(8.198)

576

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Limitations of 2D C 11 global differentiability local approximations for distorted triangular elements In the proposed framework, 2D C ij global differentiability local approximations are derived by borrowing appropriate degrees of freedom and the corresponding approximation functions from the hierarchical nodes of C 00 element. In (8.198), [a], [c] and [d] contain C 00 local approximations which are retained at corner, mid-side and internal nodes whereas [b] contains C 00 local approximations functions which are borrowed from mid-side and internal nodes. [A], [B], [C] and [D] are matrices containing derivatives of C 00 approximations with respect to ξ and η collected in [a], [b], [c] and [d] evaluated at the corner nodes. The approximation functions for the C 11 distorted element at the corner, mid-side and internal nodes (which are retained) are obtained by modifying the corresponding functions for the C 00 element by [b][B]−1 [A], [b][B]−1 [C] and [b][B]−1 [D] respectively. In case of 2D C 11 HGDA element, the new derivative degrees of freedom introduced at the corner nodes are first order derivatives with respect to x and y. The nature of the C 00 local approximation functions and the √ coordinates of the corner nodes (ξ and η coordinates are either +1, 0, 3 or −1) always result in all the coefficients of [D] matrix to be zero regardless of the p-level. The coefficients of matrices [A], [B], [C] however are not all zero. This results in the approximation functions at the internal node of C 11 distorted element to be exactly same as those corresponding to C 00 element (since [b][B]−1 [D] is a row matrix containing all zeros). As a consequence, it appears that we have an incomplete C 11 HGDA distorted element, which may result in inaccurate behaviors for coarser discretizations. When we derive approximation functions for C 22 and higher order elements, the derivative degree of freedoms introduced at the corner nodes include mixed derivatives with respect to x and y. The mixed derivatives of the C 00 approximation functions in [d] (internal node) evaluated at the corner nodes are not all zero and hence coefficients in [D] matrix are not all zero. Alternate ways of deriving C 11 local approximations is under investigation. Remarks. (1) The derivation presented here is general. Hence, C 00 p-version hierarchical local approximation can be Legendre, Lagrange or Chebyshev. The choice of these functions changes the matrices [a], [b], [c] and [d] accordingly, but the details of the development remains unaffected. (2) The matrices [A], [B], [C] and [D] contain derivatives of C 00 p-version hierarchical approximations with respect to ξ and η evaluated at the vertex nodes. These can be precomputed once and stored and hence can be reused. This results in substantial computational efficiency.

577

8.9. 1D AND 2D INTERPOLATIONS BASED ON CHEBYSHEV POLYNOMIALS

(3) The C 00 p-version functions borrowed from the mid-side nodes and the internal node must be so chosen that (i) lowest degree admissible functions (corresponding to lower p-levels) are selected first before those corresponding to the higher p-levels (ii) the symmetric pattern is maintained in their choice so that symmetry of computed solutions for symmetric discretization is preserved. (4) Mapping of the distorted triangular element in ξη coordinate space and the transformation of the C 00 approximation functions from L1 , L2 , L3 area coordinates to ξη is essential. The derivative degrees of freedom generated at the vertex nodes in ξη space can be easily transformed into the corresponding derivative degrees of freedom in the xy space needed for C ij HGDA elements.

8.9 1D and 2D interpolations based on Chebyshev polynomials 8.9.1 Chebyshev polynomials Chebyshev polynomials satisfy the Chebyshev differential equation [3] The polynomials Ti ; i = 0, 1, 2, . . . are given by T0 = 1 T1 = ξ

(8.199)

Ti+1 = 2ξ Ti − Ti−1 , i = 1, 2, . . .

8.9.2 1D C 0 p-version hierarchical interpolations based on Chebyshev polynomials For nodes 1 and 2 (non-hierarchical) we have the standard functions 1−ξ 2 1+ξ N2 (ξ) = 2 N1 (ξ) =

(8.200)

at node 3, we introduce the interpolation functions N3i (ξ) defined by ( Ti − 1 ; if i is even i for i = 2, 3, . . . , pξ (8.201) N3 (ξ) = Ti − ξ ; if i is odd If φ is the dependent variable, then approximation of φ(ξ), φeh (ξ), is given by pξ X φeh (ξ) = N1 φe1 + N2 φe2 + N3i δie (8.202) i=2

578

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

We can choose

δie

=



∂iφ ; ∂ξ i ξ=0

i = 2, 3, . . . , pξ . As an example if pξ = 2 then

we have N32 = 2ξ 2 − 2

(8.203)

If pξ = 3, then we have N32 = 2ξ 2 − 2 N33 = 4ξ 3 − 4ξ

(8.204)

etc.

We note that the functions in (8.203) and (8.204) are indeed zero at ξ = ±1. Thus (8.202) with (8.200) and (8.201) is the desired 1D C 0 p-version hierarchical interpolation based on Chebyshev polynomials. 1

3

2

−1

0

1

ξ

Figure 8.43: 1D three-node p-version element in natural coordinate space

8.9.3 2D p-version C 00 hierarchical interpolation functions for quadrilateral elements (Chebyshev polynomials) If we consider nine-node element configuration in ξη space (in a two-unit square), then by using the 1D functions and nodal variable operators defined by (8.202) in ξ and η directions and by taking their tensor products we can construct 2D p-version C 00 hierarchical local approximation (or interpolations) for the nine-node element. The procedure is identical to what has been used for Lagrange family 2D C 00 p-version hierarchical local approximations based on tensor product. Details are straight forward.

8.9.4 2D C ij p-version interpolation functions for quadrilateral elements (Chebyshev polynomials) The derivation of these interpolation functions and nodal dofs follows exactly the same procedure as used for C ij 2D p-version interpolations based on Lagrange family except that the corresponding C 00 2D p-version hierarchical interpolation functions of Lagrange type are replaced by those based on Chebyshev type. The details are straight forward.

8.10 Serendipity family of C 00 interpolations over square subdomains “Serendipity” means discovery by chance. Thus, this family of elements has very little theoretical or mathematical basis other than the fact that in

8.10. SERENDIPITY FAMILY OF C 00 INTERPOLATIONS

579

generating approximation functions for these elements we only utilize the two fundamental properties of the approximation functions, ( 1, j=i Ni (ξj , ηj ) = (i = 1, . . . , m) (8.205) 0, j 6= i and

m X

Ni (ξ, η) = 1

(8.206)

i=1

(a) The main motivation in generating these basis functions is to possibly eliminate some or many of the internal nodes that appear in generating the interpolations using tensor product for family of higher degree interpolation functions. (b) For example, in the case of a bi-quadratic local approximation requiring a nine-node element, the corresponding serendipity element will contain eight boundary nodes, as shown in Fig. 8.44. η

7

8

η

9

1

4

2

7

5

6

3

Nine-node Lagrange bi-quadratic element

ξ

5

6

8 4

1

2

ξ

3

Eight-node serendipity element

Figure 8.44: Nine-node Lagrange and eight-node serendipity elements

(c) In the case of a bi-cubic element requiring 16-nodes with four internal nodes, the corresponding serendipity element will contain 12 boundary nodes (see Fig. 8.45) (d) While in the case of bi-quadratic and bi-cubic local approximations it was possible to eliminate the internal nodes and thus serendipity elements were possible. This may not always be possible for higher degree local approximations than three.

8.10.1 Method of deriving serendipity interpolation functions We use the two basic properties that the approximation functions must satisfy (stated by (8.205) and (8.206)). Let us consider a four-node bilinear

580

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY η

η

ξ

ξ

16-node bi-cubic element

12-node cubic serendipity element

Figure 8.45: Sixteen-node Lagrange and twelve-node serendipity elements

element. In this case, obviously, non-serendipity and serendipity approximations are identical. Nonetheless, we derive the approximation functions for this element using the approach used for serendipity basis functions. 1−η =0

η 4

1−ξ =0

3 ξ

1+ξ =0

1

2 1+η =0

Figure 8.46: Derivation of 2D bilinear serendipity element

(a) First, we note that the four sides of the elements are described by the equations of the straight lines as shown in the figure. Consider node 1. N1 (ξ, η) is one at node 1 and zero at nodes 2,3 and 4. Hence, equations of the straight lines connecting nodes 2 and 3 and nodes 3 and 4 can be used to derive N1 (ξ, η). That is, N1 (ξ, η) = c1 (1 − ξ)(1 − η)

(8.207)

in which c1 is a constant. But N1 (−1, −1) = 1, hence using (8.207) we get N1 (−1, −1) = 1 = c1 (1 − (−1))(1 − (−1))

⇒

c1 =

1 4

(8.208)

Thus, we have 1 N1 (ξ, η) = (1 − ξ)(1 − η) 4

(8.209)

8.10. SERENDIPITY FAMILY OF C 00 INTERPOLATIONS

581

which is the correct approximation function for node 1 of the bilinear element. Similarly, for node 2, 3 and 4 we can write N2 (ξ, η) = c2 (1 + ξ)(1 − η) N3 (ξ, η) = c3 (1 + ξ)(1 + η)

(8.210)

N4 (ξ, η) = c4 (1 − ξ)(1 + η) But N2 (1, −1) = 1

⇒

N3 (1, 1) = 1

⇒

N4 (−1, 1) = 1

⇒

1 4 1 c3 = 4 1 c4 = 4 c2 =

(8.211)

Thus, from (8.210) and (8.211) we obtain 1 N2 (ξ, η) = (1 + ξ)(1 − η) 4 1 N3 (ξ, η) = (1 + ξ)(1 + η) 4 1 N4 (ξ, η) = (1 − ξ)(1 + η) 4

(8.212)

(8.209) and (8.212) are the correct approximation functions for the four-node bilinear element. (b) In the above derivations we have only utilized the property (8.205), hence we must show that the interpolation functions in (8.209) and (8.212) satisfy (8.206). In this case, obviously they do. However, this may not always be the case. Eight-node serendipity element: Consider node 1 first. We have N1 (ξ, η)|(−1,−1) = 1 and zero at all the remaining nodes. Hence, for node 1 we can write N1 (ξ, η) = c1 (1 − ξ)(1 − η)(1 + ξ + η) Since N1 (ξ, η)|(−1,−1) = 1 we obtain

⇒

c1 = −

1 4

1 N1 (ξ, η) = − (1 − ξ)(1 − η)(1 + ξ + η) 4

(8.213)

(8.214)

(8.215)

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

1−η =0

η 7 1+ξ+η =0

6

5

8

4

1−ξ =0 ξ

1+ξ =0 1

2 1+η =0

3

Figure 8.47: Derivation of 2D bilinear serendipity element: node 1

1+ξ−η =0

1−ξ−η =0 5

1−η =0

1+ξ =0

7

1−ξ =0

1+ξ =0

3 1−ξ+η =0

for node 3

1+η =0

for node 5

1+η =0

for node 7

Figure 8.48: Derivation of 2D bi-quadratic serendipity element: nodes 3, 5, and 7

For nodes 3, 5 and 7 one may use the equations of the lines indicated in Fig. 8.48 and the conditions similar to (8.214) for N2 , N3 and N4 . For the mid side nodes, the product of the equations of straight lines not containing the mid side nodes provide the needed expressions and we have 1 N1 = (1 − ξ)(1 − η)(−1 − ξ − η) 4 1 N2 = (1 − ξ 2 )(1 − η) 2 1 N3 = (1 + ξ)(1 − η)(−1 + ξ − η) 4 1 N8 = (1 − ξ)(1 − η 2 ) 4 1 N4 = (1ξ )(1 − η 2 ) 2 1 N7 = (1 − ξ)(1 + η)(−1 − ξ + η) 4 1 N6 = (1 − ξ 2 )(1 + η) 2 1 N5 = (1 + ξ)(1 + η)(−1 + ξ + η) 4

8.10. SERENDIPITY FAMILY OF C 00 INTERPOLATIONS

In this case also we must show that

8 P

583

Ni (ξ, η) = 1, which holds.

i=1

Twelve-node serendipity elements: Using procedures similar to the four-node bilinear and eight-node biquadratic element (see Fig. 8.49) we can also derive the interpolation functions for the twelve-node serendipity element. η

9

10

11 12

7

8

5

6

ξ

1

2

3

4

Figure 8.49: Derivation of 2D bi-cubic serendipity element

N1 = N2 = N3 = N4 = N5 = N6 = N7 = N8 = N9 = N10 = N11 = N12 =

1 (1 − ξ)(1 − η)[−10 + 9(ξ 2 + η 2 )] 32 9 (1 − ξ 2 )(1 − η)(1 − 3ξ) 32 9 (1 − ξ 2 )(1 − η)(1 − 3ξ) 32 1 (1 + ξ)(1 − η)[−10 + 9(ξ 2 + η 2 )] 32 9 (1 − ξ)(1 − η 2 )(1 − 3η) 32 9 (1 + ξ)(1 − η 2 )(1 − 3η) 32 9 (1 − ξ)(1 − η 2 )(1 + 3η) 32 1 (1 + ξ)(1 − η 2 )(1 + 3η) 32 1 (1 − ξ)(1 + η)[−10 + 9(ξ 2 + η 2 )] 32 9 (1 − ξ 2 )(1 + η)(1 − 3ξ) 32 9 (1 − ξ 2 )(1 + η)(1 + 3ξ) 32 1 (1 + ξ)(1 + η)[−10 + 9(ξ 2 + η 2 )] 32

584

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Remarks. (1) Serendipity interpolations are obviously incomplete polynomials in ξ and η, hence have poorer local approximation compared to the local approximations based on Pascal rectangle. (2) There is no particular theoretical basis for deriving them. (3) In view of p-version hierarchical elements presented in the previous sections, serendipity elements are precluded and are of no practical significance.

8.11 Interpolation functions for 3D elements For BVPs in R3 the domain of definition of the BVP is a subspace of i.e. a volume. Discretizations of such domain naturally leads to 3D subdomains, i.e. 3D elements. In this section we consider hexahedron and tetrahedral families of subdomains (i.e. elements). R3 ,

8.11.1 Hexahedron elements Figure 8.50 shows hexahedron elements with distorted faces and edges and their maps in the natural coordinate space ξ, η, ζ in a two unit cube with the origin of the coordinate system located at the center of the element. 8.11.1.1 Mapping of points In the abstract sense, the mapping of points between ξ, η, ζ and x, y, zspaces is defined by x = x(ξ, η, ζ) y = y(ξ, η, ζ)

(8.216)

z = z(ξ, η, ζ) The inverse of the mapping is given by ξ = ξ(x, y, z) η = η(x, y, z)

(8.217)

ζ = ζ(x, y, z) As in the case of 2D elements, (8.216) is preferable over (8.217). The explicit ¯i (ξ, η, ζ) form of the mapping defined by (8.216) can be established. Let N ¯ ξηζ be basis functions in the natural coordinate space such that ∀ξ, η, ζ ∈ Ω m ¯ we have or Ω ( ¯i (ξj , ηj , ζj ) = 1, j = i N (8.218) 0, j 6= i

585

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

η

ξ y

ζ x

z Eight-node element in x, y, z space

Eight-node element in ξ, η, ζ space η

ξ y

ζ x

z 27-node element in ξ, η, ζ space

27-node element in x, y, z space

Figure 8.50: Hexahedral 3D elements

n ¯ X

¯i (ξ, η, ζ) = 1 N

(8.219)

i=1

¯ m is the map of Ω ¯ e in ξ, η, ζ space, i.e. mapping (8.216) is such in which Ω ¯m → Ω ¯ e or by (8.217), Ω ¯e → Ω ¯ m . Using N ¯i (ξ, η, ζ) we can write, that Ω

x = x(ξ, η, ζ) =

n ¯ X

¯i (ξ, η, ζ) xi N

i=1

y = y(ξ, η, ζ) = z = z(ξ, η, ζ) =

n ¯ X i=1 n ¯ X

¯i (ξ, η, ζ) yi N

(8.220)

¯i (ξ, η, ζ) zi N

i=1

in which (xi , yi , zi ) are coordinates of node i in x, y, z space. Equation (8.220) map a point (ξ ∗ , η ∗ , ζ ∗ ) into (x∗ , y ∗ , z ∗ ).

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

8.11.1.2 Mapping of lengths In this section we establish a relationship between length dx, dy, dz in x, y, z space and dξ, dη, dζ in ξ, η, ζ space. Based on (8.216), we can write ∂x ∂x ∂x dξ + dη + dζ ∂ξ ∂η ∂ζ ∂y ∂y ∂y dy = dξ + dη + dζ ∂ξ ∂η ∂ζ ∂z ∂z ∂z dz = dξ + dη + dζ ∂ξ ∂η ∂ζ

(8.221)

    dx dξ  dy [J] dη     dz dζ

(8.222)

dx =

or

where

 [J] =

∂x ∂ξ  ∂y   ∂ξ ∂z ∂ξ

∂x ∂η ∂y ∂η ∂z ∂η



∂x ∂ζ  ∂y  ∂ζ  ∂z ∂ζ

(8.223)

is called the Jacobian of transformation. For the mapping to be unique, i.e. one-to-one and onto, det [J] > 0

¯m ∀(ξ, η, ζ) ∈ Ω

(8.224)

must hold. Equations (8.222) is the desired relationship for mapping of lengths. The elements of [J] can easily be obtained using (8.220) in which ¯i (ξ, η, ζ) and (xi , yi , zi ) are known. N 8.11.1.3 Mapping of volumes In this section we derive a relationship that relates elemental volume dx dy dz in x, y, z space to the elemental volume dξ dη dζ in ξ η ζ space. Let ~i, ~j, ~k be the unit vectors in x, y, z space and let ~eξ , ~eη , ~eζ be unit vectors in ξ, η, ζ space, then we can write ∂x ∂x ∂x dξ ~eξ + dη ~eη + dζ ~eζ ∂ξ ∂η ∂ζ ∂y ∂y ∂y dy ~j = dξ ~eξ + dη ~eη + dζ ~eζ ∂ξ ∂η ∂ζ ∂z ∂z ∂z dz ~k = dξ ~eξ + dη ~eη + dζ ~eζ ∂ξ ∂η ∂ζ

(8.225)

dx~i · (dy ~j × dz ~k) = dx~i · dy dz ~i = dx dy dz

(8.226)

dx~i =

We note that

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

587

Substituting for dx~i, dy ~j, dz ~k from (8.225) into (8.226) and using properties of the dot product and cross products of the unit vectors in x, y, z and ξ, η, ζ spaces, we obtain dx dy dz = det[J] dξ dη dζ (8.227) Based on (8.227), det[J] > 0 must hold otherwise the volume in xyz space could be negative or zero if det[J] = 0. 8.11.1.4 Obtaining derivatives of φeh (ξ, η, ζ) with respect to x, y, z With the local approximation φeh (ξ, η, ζ) of φ defined by (8.232), and assuming that Ni (ξ, η, ζ) are known functions, obtaining derivatives of φeh (ξ, η, ζ) with respect to x, y, z needed in the integral form is not directly possible due to the fact that Ni (·) are functions of ξ, η and ζ. First, we note that when φeh (ξ, η, ζ) is defined by (8.232) we have n

∂φeh X ∂Ni e = δ ∂x ∂x i i=1 n

∂φeh X ∂Ni e = δ ∂y ∂y i

(8.228)

i=1 n

∂φeh X ∂Ni e = δ ∂z ∂z i i=1

∂φe ∂φe ∂φe ∂Ni ∂Ni i Thus, obtaining ∂xh , ∂yh and ∂zh implies establishing ∂N ∂x , ∂y and ∂z ; i 1, . . . , n. Since Ni = Ni (ξ, η, ζ) and x = x(ξ, η, ζ), y = y(ξ, η, ζ) and z

= =

z(ξ, η, ζ), we proceed as follows.

or

 ∂Ni ∂Ni ∂x ∂Ni ∂y ∂Ni ∂z   = + +  ∂ξ ∂x ∂ξ ∂y ∂ξ ∂z ∂ξ    ∂Ni ∂x ∂Ni ∂y ∂Ni ∂z  ∂Ni = + + , i = 1, . . . , n ∂η ∂x ∂η ∂y ∂η ∂z ∂η     ∂Ni ∂Ni ∂x ∂Ni ∂y ∂Ni ∂z    = + + ∂ζ ∂x ∂ζ ∂y ∂ζ ∂z ∂ζ     ∂Ni  ∂Ni        ∂x    ∂ξ  T ∂Ni ∂Ni = [J] ∂η  ∂y  , i = 1, . . . , n       ∂Ni    ∂Ni  ∂ζ

Therefore

  ∂Ni     ∂x   ∂Ni ∂y     ∂Ni   ∂z

(8.229)

(8.230)

∂z

h i−1 = [J]T

  ∂Ni     ∂ξ   ∂Ni

∂η     ∂Ni   ∂ζ

, i = 1, . . . , n

(8.231)

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BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

Using (8.231) derivatives of Ni (ξ, η, ζ) with respect to ξ, η and ζ can be transformed into the derivatives of Ni (ξ, η, ζ) with respect to x, y, z and, hence, the derivatives of φeh (ξ, η, ζ) with respect to x, y, z in (8.228) are defined.

8.11.2 Local approximation for a dependent variable ¯m φ over Ω 8.11.2.1 Hexahedron elements ¯ m . Then symboliLet φeh (ξ, η, ζ) be the local approximation of φ over Ω cally we can write φeh (ξ, η, ζ) =

n X

Ni (ξ, η, ζ) δie = [N ]{δ e }

(8.232)

i=1

where Ni (ξ, η, ζ) are the local approximation functions or interpolation functions corresponding to the nodes of the element and δie are the nodal degrees of freedom. Functions Ni (ξ, η, ζ) are also referred to as basis functions. Explicit forms of Ni (ξ, η, ζ) depend upon many considerations (similar to these for 2D elements). (1) The first important issue is the choice of nodal configuration for the element. That is, how many nodes and their locations. (2) Means of constructing Ni (ξ, η, ζ) for (a) (b) (c) (d)

C 000 local approximations of higher degree C 000 p-version hierarchical local approximations C ijk , i, j, k ≥ 1 local approximations for prism family C ijk , i, j, k ≥ 1 local approximations for distorted geometries in R3

(3) In the developments in (2), the choices of δie , nodal dofs are crucial and require careful considerations. (4) The developments in (2) and (3) could be based on (i) Lagrange interpolation functions (ii) Legendre polynomials or (iii) Chebyshev polynomials. The specific details presented in the following are based on Lagrange polynomials. The extensions to the other two types are straight forward. We consider details in the following ¯ e ) polynomial approximations: linear approximation C 000 (Ω ¯ e ), i.e. across the interWe consider local approximations of type C 000 (Ω element boundaries φ is continuous but the derivative of φ normal to the inter-element boundaries may be discontinuous in the physical domain. For

589

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

¯ e ), local approximation function values at the nodes suffice as degrees C 000 (Ω of freedom. Consider a two-unit cube in ξ, η, ζ coordinate space. For this geometry the least possible nodal configuration is eight nodes located at the corner points. This element in the physical space x, y, z could have distorted shape with flat faces and straight edges (see Fig. 8.51). η

φe8

φe7

8 φe5

φe6

6

5 φe4 y

1 ζ x

φe1

7 ξ

3 φe3 2 φe2

z Eight-node element in R3

Map of an eight-node element in ξ, η, ζ space

Figure 8.51: Eight-node trilinear element in 3D

In the polynomial approach, we consider φ(ξ, η, ζ) as a linear combination of monomials in ξ, η ζ. Noting that we have eight function values at the element nodes as dofs, φ(ξ, η, ζ) can be a polynomial in ξ, η and ζ containing up to eight constants, φ(ξ, η, ζ) = c1 + c2 ξ + c3 η + c4 ζ + c5 ξη + c6 ηζ + c7 ξζ + c8 ξηζ

(8.233)

Equation (8.233) is a complete linear polynomial in ξ, η and ζ. The constants c1 , . . . , c8 in (8.233) are evaluated using φei = φ(ξi , ηi , ζi ), i = 1, . . . , 8

(8.234)

in which (ξi , ηi , ζi ) are coordinates of the eight nodes in ξ, η, ζ space. first, we rewrite (8.233) as   c1          c2  φ(ξ, η, ζ) = 1 ξ η ζ ξη ηζ ξζ ξηζ ..  .       c8

(8.235)

Substituting from (8.234) into (8.233) and solving for ci ; i = 1, . . . , 8 in terms

590

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

of φei (i = 1, . . . , 8)    c1  1 ξ1     c2   1 ξ2  .. =  .. ..   .  . .      1 ξ8 c8

−1  e  . . . ξ 1 η 1 ζ1 φ1      φe   . . . ξ 2 η 2 ζ2  2  ˆ e} = [C]{φ ..  .. ..  . .   .     e  η8 ζ8 . . . ξ8 η8 ζ8 φ8

η1 η2 .. .

ζ1 ζ2 .. .

Substituting from (8.236) into (8.235),   ˆ e} φ(ξ, η, ζ) = 1 ξ η ζ . . . ξηζ [C]{φ or φ(ξ, η, ζ) =

8 X

Ni (ξ, η, ζ) φei

(8.236)

(8.237)

(8.238)

i=1

in which

and

( 1, j = i Ni (ξj , ηj , ζj ) = 0, j = 6 i 8 X

Ni (ξ, η, ζ) = 1

(8.239)

(8.240)

i=1

The approximation φeh (ξ, η, ζ) in (8.238) is linear in ξ, η and ζ and is ¯ e ) and is based on linear Lagrange interpolation functions of class C 000 (Ω (Ni (ξ, η, ζ)), p-level of one in ξ, η and ζ. ¯m 8.11.2.2 Higher degree approximations of φ over Ω Based on the derivation of linear approximation presented in the previous section, it is clear that if we wish φeh (ξ, η, ζ) to be a quadratic approximation in ξ, η and ζ then in (8.233) we need additional monomials in ξ, η and ζ and we also need to decide on the locations of the additional nodes (as many as the number of additional terms included in (8.233)). For progressively increasing degree (p-level) of approximation in ξ, η and ζ, the decision on the choices of additional monomials and locations of the additional nodes can be facilitated by considering Pascal’s prism. The family of interpolations generated using this approach is Lagrange type. Consider Fig. 8.52. First, we explain how to interpret the information in this figure. (1) There are three independent coordinates: ξ, η and ζ. For each coordinate we have monomials of orders 0, 1, . . . , pξ , pη and pζ . (2) First, we construct Pascal’s rectangle in ξ and η, similar to what was done for 2D quadrilateral elements. Locations of the terms are the location

591

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS 1

ξ

ξ2

ξ3

ξ4

η

ξη

ξ2η

ξ3η

ξ4η

η2

ξη 2

ξ 2η2

ξ 3η2

ξ 4η2

η3

ξη 3

ξ 2η3

ξ 3η3

ξ 4η3

η4

ξη 4

ξ 2η4

ξ 3η4

ξ 4η4

ζ

ζ2

ζ

3

ζ4

(a) Pascal prism Nodal configuration

Monomials to be used

Degree of approximation p of φ

ξ

1

p=1

ζξ

ζ η

ξη

ζη

ζξη 1 η ζξ

ζ1 ζη ζ 21

2

η2

ζξη

ζξ

ξ

ξ2

ξη

ξ2η

ξη 2

ξ 2η2

2

ζξ 2 η

p=2

2 2

ζ ξ

ζ ξ ζη 2

ζξη 2

ζ 2η

ζ 2 ξη

ζ 2ξ2η

ζ 2η2

ζ 2 ξη 2

ζ 2ξ 2η2

ζξ 2 η 2

(b) nodal configurations and monomial terms Figure 8.52: Pascal’s prism, nodal configurations and monomial terms

of the nodes for the 2D case and the terms themselves are to be used in the linear combination like the one in (8.233). (3) We let this configuration traverse along the ζ direction to the desired

592

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

p-level. As it encounters a term in ζ it produces a trace of nodes (same as in ξη) and each term in ξη configuration gets multiplied by the corresponding ζ term. Thus, for ζ = 1 the configuration in ξη as shown in the figure holds. At location ζ all terms in ξη configuration get multiplied with ζ. Similarly, at location ζ 2 we multiply the terms in ξη configuration by ζ 2 and so on. (4) Thus, in this scheme, one chooses the desired p-levels in ξ, η and ζ. This controls up to what terms we need to consider in ξ, η and ζ. (5) For example, if pξ = pη = pζ = 2, then in ξ, η and ζ we have 1, ξ, ξ 2 , 1, η, η 2 and 1, ζ, ζ 2 monomials and the result is a 27-node hexahedron element shown in Fig. 8.52 (b). (6) Likewise, for cubic φeh (ξ, η, ζ) in ξ, η and ζ we would need a 64-node element for which the approximation will be generated using the monomials 1, ξ, ξ 2 , ξ 3 , 1, η, η 2 , η 3 and 1, ζ, ζ 2 , ζ 3 . (7) Once we know the monomials to be used in the linear combination for φ(ξ, η, ζ) and the locations of the nodes, it is straight forward to derive (following the same procedure as shown for pξ = pη = pζ = 1), φeh (ξ, η, ζ)

=

n X

Ni (ξ, η, ζ) φei = [N ]{φe } = [N ]{δ e }

(8.241)

i=1

where n is the number of nodes (same as the total number of degrees of freedom due to the fact that each node only has function value as degree of freedom). Here Ni (ξ, η, ζ) are Lagrange interpolation functions. Remarks. (1) The local approximations discussed here are of class C 000 based on Lagrange family of interpolation functions. (2) A serious drawback of this approach is that as the degree of local approximation increases, so does the size of the matrix to be inverted in deriving Ni (ξ, η, ζ). (3) Another serious shortcoming of this approach is that as the degree of local approximation increases, the number of nodes for an element increases dramatically. This necessitates a new discretization for each p-level change. In the following sections we correct both of these drawbacks. 8.11.2.3 C 000 Lagrange type local approximations using tensor product Recall that in the case of 1D local approximations in ξ for −1 ≤ ξ ≤ 1, the basis functions or local approximation functions can be easily generated

593

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS η

η Nm

m

N2η

2

N1η

1 1

2

n

N1ξ

N2ξ

Nnξ

1

ξ N1ζ

2 N2ζ q Nqζ ζ

Figure 8.53: 1D Lagrange interpolation functions in ξ, η and ζ

η N1ξ N2η =

1−ξ 2




1+η 2




N2ξ N2η =

1+ξ 2




1+η 2




1−η 2




ξ

N1ξ N2η =

1−ξ 2




1−η 2




N2ξ N1η =

1+ξ 2




1 N1ζ =

1−ζ 2

2 N2ζ =

1+ζ 2

Figure 8.54: Tensor product of 1D functions in ξ, η with 1D functions in ζ for pξ = pη = pζ = 1

using Lagrange interpolation functions which allows us to bypass inversion of matrices. For a 1D element with n equally spaced nodes in −1 ≤ ξ ≤ 1

594

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY η 7

8 N8

N7

6

5 N7

N6 ξ

4

N4

3 N3

1

ζ

2 N2

N1

Figure 8.55: 3D element with pξ = pη = pζ = 1

η η N2ξ Nm

ξ η Nn−1 Nm

η N1ξ Nm

η Nnξ Nm

η Nnξ Nm−1

η N1ξ Nm−1

ξ

Nnξ N2η

N1ξ N2η

N1ξ N1η

1

N2ξ N1η

ξ Nn−1 N1η

N1ξ N1η

N1ζ

2 N2ζ q Nqζ ζ

Figure 8.56: Tensor product in ξ, η and 1D functions in ζ

we could write φeh (ξ) as

φeh (ξ) =

n X i=1

Niξ (ξ) φei

(8.242)

595

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

η

N2η =

1+η 2

N1η =

1−η 2

2

1 ξ N1ξ =

N1ζ =

2 N2ζ =

2

1

1

1−ξ 2

N2ξ =

1+ξ 2

1−ζ 2

1+ζ 2

ζ Figure 8.57: 1D approximation functions in ξ, η and ζ for pξ = pη = pζ = 1

in which Nkξ (ξ) = Lk (ξ) =

n Y ξ − ξm
, k = 1, . . . , n ξk − ξm

(8.243)

m=1 m6=k

with ( 1, j = i Niξ (ξj ) = 0, j = 6 i

(8.244)

and n X

Niξ (ξ) = 1

(8.245)

i=1

In this, the degree of approximation is pξ = n − 1 and the local approximations are of C 0 type due to the fact that only the function values are the unknowns at the end nodes (at ξ = ±1). The concept of tensor product used for 2D quadrilateral elements described earlier can be easily extended to the 3D case. consider 1D Lagrange type interpolation in ξ, η and ζ with p-levels of pξ = n − 1, pη = m − 1, pζ = q − 1. Here, n, m, q are the number of nodes for ξ, η and ζ 1D local approx-

596

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

imations. Then the following holds in ξ, η and ζ. φeh (ξ)

=

n X

Niξ (ξ) φei

i=1

φeh (η) = φeh (ζ) =

m X i=1 q X

Niξ (η) φei

(8.246)

Niξ (ζ) φei

i=1

Schematically, we can show these in Fig. 8.53. Following the procedure described for 2D elements, we first take the tensor product of 1D functions in ξ and η. This gives us (n×m) configuration of nodes in ξ, η and the corresponding 2D approximation functions and the 1D functions in ζ remain as they were in Fig. 8.53. Now we take the tensor product of (n × m) configuration in ξ, η with the 1D q-node configuration in ζ. This will yield an element with (n×m×q) nodes with explicit expressions of the approximation functions for each node. As an example, if pξ = pη = pη = 1 (i.e., n = m = q = 2) then the 1D Lagrange interpolation functions in ξ, η and ζ to be considerred in the tensor product to generate eight-node linear hexahedron C 000 element are shown in Fig. 8.55. Tensor product of 1D functions in ξ and η gives 2D functions in ξη but leaving 1D functions in ζ as they are in Fig. 8.56. Tensor product of 2D ξη interpolation functions of Fig. 8.56 with 1D interpolation functions in ζ gives the eight-node hexahedron linear 3D element of Fig. 8.57. The explicit expressions for the nodal interpolation functions are given in the following with pξ = pη = pζ = 1:  1 − ζ  1 − ξ  1 − η  N4 (ξ, η, ζ) = N1ζ N1ξ N1η = 2 2 2  1 − ζ  1 + ξ  1 − η  ζ ξ η N3 (ξ, η, ζ) = N1 N2 N1 = 2 2 2  1 − ζ  1 + ξ  1 + η  N7 (ξ, η, ζ) = N1ζ N2ξ N2η = 2 2 2  1 − ζ  1 − ξ  1 + η  ζ ξ η N8 (ξ, η, ζ) = N1 N1 N2 = 2 2 2 (8.247)  1 + ζ  1 − ξ  1 − η  N1 (ξ, η, ζ) = N2ζ N1ξ N1η = 2 2 2  1 + ζ  1 + ξ  1 − η  ζ ξ η N2 (ξ, η, ζ) = N2 N2 N1 = 2 2 2  1 + ζ  1 + ξ  1 + η  ζ ξ η N6 (ξ, η, ζ) = N2 N2 N2 = 2 2 2  1 + ζ  1 − ξ  1 + η  N5 (ξ, η, ζ) = N2ζ N1ξ N2η = 2 2 2

597

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

Using this approach we can generate interpolation or approximation functions of degrees (n − 1), (m − 1) and (q − 1) in ξ, η and ζ directions. Tensor product avoids inversion of matrices as required in the polynomial approach. Still, there are two drawbacks in this approach: (i) increase in p-level requires a new nodal configuration, i.e. essentially a new geometric description of additional nodes and (ii) the local approximations lack hierarchical property. These drawbacks are corrected in the next section by considering C 000 pversion hierarchical local approximations.

8.11.2.4 C 000 p-version 3D hierarchical local approximations: using tensor product Consider a three-node p-version hierarchical local approximations in ξ, η and ζ directions and the corresponding nodal variable operators shown in Figs. 8.58 and 8.59. As in the previous section, we first take the tensor products of 1D approximation functions and the nodal variable operators in ξ and η. This yields 2D nodal configuration, the corresponding approximation functions and the nodal variable operator in ξ, η space as shown in Figs. 8.60 and 8.61. The 1D functions and nodal variable operators in ζ remain the same as in Figs. 8.58 and 8.59. η

N31η

3

N2jη

2

(j = 2, . . . , pη ) N31η

1

1

2

3

N11ξ

N2iξ (i = 2, . . . , pξ )

N31ξ

N11ζ

ξ 1

2 N31ζ 3

N2kζ

(k = 2, . . . , pζ )

ζ

Figure 8.58: C 000 1D p-version hierarchical approximation functions

The tensor product of basis functions in ξ, η configuration in Fig. 8.58 with 1D basis functions in ζ gives approximation functions for 27-node pversion hierarchical hexahedron element. The corresponding nodal variable operators are obtained by similar processes for 2D nodal variable operators in ξ, η configuration and 1D nodal variable operators in ζ in Fig. 8.58. The details are straight forward and are left as an exercise. We note that 1D

598

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY η

1

3

∂j ∂η j (j = 2, . . . , pη )

2

1

1

1

2

1

∂i ∂ξ i (i = 2, . . . , pξ )

3 ξ

1 1 2

1

∂k ∂ζ k (k = 2, . . . , pζ )

3 1 ζ

Figure 8.59: C 0 1D p-version hierarchical nodal variable operators η N11ξ

N31η

N11ξ N2jη j = 2, . . . , pη

N11ξ N11η N11ζ

N2iξ N31η 1ξ 1η i = 2, . . . , pξ N3 N3

ξ N2iξ N2jη i = 2, . . . , pξ j = 2, . . . , pη N2iξ N11η i = 2, . . . , pξ

N31ξ

N2jη j = 2, . . . , pη

N31ξ N11η

N2iζ k = 2, . . . , pζ N31ζ ζ

Figure 8.60: C 0 1D p-version hierarchical approximation functions in ξ and C 00 2D p-version hierarchical approximation functions in ξ, η

p-version hierarchical basis functions in ξ, η and ζ are given by 1 − ξ  1 + ξ  N11ξ = , N31ξ = 2 2 (  ξi − a  1, if i is even N2iξ = , a= i! ξ, if i is odd

(8.248)

599

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

η 1

∂i ∂ξ i

i = 2, . . . , pξ 1

∂j ∂η j

ξ

j = 2, . . . , pη

i = 2, . . . , pξ j = 2, . . . , pη 1 1

∂j ∂η j

∂ i+j ∂ξ i∂η j

∂i ∂ξ i

j = 2, . . . , pη

1

i = 2, . . . , pξ

∂k ∂ζ k

k = 2, . . . , pζ 1 ζ

Figure 8.61: C 0 1D p-version hierarchical nodal variable operators

1 + η  , N31η = 2 2 (  ηj − b  1, if j is even = , b= j! η, if j is odd

(8.249)

1 + ζ  , N31ζ = 2 2 (  ζk − c  1, if k is even = , c= k! ζ, if k is odd

(8.250)

N11η = N2jη

N11ζ = N2kζ

1 − η 

1 − ζ 

Thus, the approximation functions for the 27-node p-version hierarchical hexahedron element are completely defined. The nodal degrees of freedom are obtained by letting the nodal variable operators act on the dependent variable(s). ¯ e ) p-version local approximations: Hexahedron 8.11.2.5 3D C ijk (Ω elements ¯ e ) p-version family of interpolations in xyz space which possess 3D C ijk (Ω interelement continuity of orders i, j and k in xyz directions with complete polynomials of orders pξ , pη and pζ are considered in this section. We consider a twenty seven node hexahedron element (two-unit cube) in ξηζ space

600

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

with sides parallel to xyz axes and pointing in the same direction as xyz axes. We consider map of this element in xyz space, a prism where sides are also parallel to xyz axes and pointing in the same direction. By using tensor products of C i , C j and C k one-dimensional interpolations in xyz directions, the local approximation functions and nodal variable operators for the 27 node prism family of hexahedron can be derived in a straight forward manner. These elements have same limitations and restrictions as discussed in section 8.5 for similar local approximations for 2D case in xy space. We remark that these interpolation functions can be of Lagrange, Legendre or Chebyshev type by choosing 1D interpolations based on these. ¯ e ) p-version interpolations for 8.11.2.6 3D C ijk (Ω distorted hexahedron elements: 27 node element In deriving the local approximations for this family of hexahedron elements we can follow the procedure similar to that used for 2D elements (section 8.6). The decision of the choice of dofs at the corner is straight forward due to the fact that these must constitute a transformable set between ξηζ and xyz space. However, from which nodes to borrow the needed degrees of freedom is not as straight forward but can be facilitated by examining Pascal prism and ensuring that lower order admissible terms are chosen first. These interpolations can also be of Lagrange, Legendre, or Chebyshev type depending upon the choice of 1D interpolations. 8.11.2.7 Interpolation theory for 3D tetrahedron ¯ e) elements: basis functions of class C 000 (Ω based on Lagrange interpolations Parallel to Pascal prism, in case of tetrahedron elements we have Pascal pyramid (see Fig. 8.62). The locations of the terms are the locations of the nodes and the terms themselves are to be used in the linear combination for the local approximations. For example a linear tetrahedron will contain four vertex nodes with the local approximation (for a dependent variable φ) φeh (x, y, z) = c1 + c2 x + c3 y + c4 z

(8.251)

Using Cartesian coordinates of the vertex nodes (xi , yi , zi ) are the function values φ(xi , yi , zi ) = φei (i = 1, 2, . . . , 4). In (8.251), it is straight forward to establish 4 X φeh (x, y, z) = Ni (x, y, z) φei (8.252) i=1

601

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

1 x

y z

x2 x

x3 2

xz xz 2

xy

2

zy

z2 x2 y z3

y2 y3

xy 2 z2y

zy 2

Figure 8.62: Pascal pyramid

in which Ni (x, y, z) are the interpolation functions with the properties ( 1, if j = i Ni (xj , yj , zj ) = 0, if j 6= i (8.253) 4 X Ni (x, y, z) = 1 i=1

¯ e) Using this procedure higher degree local approximations of class C 000 (Ω can be constructed for the tetrahedron elements. As in case of hexahedron elements based on Pascal prism, here also with increasing degree of local approximations the number of nodes for an element increase and we require inverse of progressively increasing size coefficient matrices to determine Ni (x, y, z). 8.11.2.8 Lagrange family C 000 interpolations based on volume coordinates Parallel to the area coordinates for 2D triangular elements, in case of tetrahedron elements we introduce the concept of volume coordinates. Consider a four-node tetrahedron element shown in Fig. 8.63. Consider a point P in the interior of the element. Connect point P with the vertices of the tetrahedron by straight lines. By doing so we divide the volume V into four volumes V1 , V2 , V3 and V4 . Let the side opposite to node i be side i with volume Vi .

602

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

1

P 2

4 3

Figure 8.63: Tetrahedron element with volume coordinates

Let Vi , i = 1, 2, . . . , 4 V 4 X with Li = 1 L1 =

(8.254)

i=1

be the volume coordinates. Clearly Li is equal to 1 at node i but zero on side i. The volume coordinates can be used to define the Cartesian coordinates of any point on the element. If (xi , yi , zi ) are the Cartesian coordinates of the nodes of the element, then x = L1 x1 + L2 x2 + L3 x3 + L4 x4 y = L1 y1 + L2 y2 + L3 y3 + L4 y4

(8.255)

z = L1 z1 + L2 z2 + L3 z3 + L4 z4 Using (8.255) and the second equation of (8.254), we can write         L1  L1  1 1 1 1 1                  x x1 x2 x3 x4  L2 L2 = = = [C] L  L  y   y1 y2 y3 y4            3   3  z z1 z2 z 3 z4 L4 L4 Therefore

      L1  1 N1 (x, y, z)                L2 x N2 (x, y, z) −1 = [C] = L  y  N (x, y, z)       3       3  L4 z N4 (x, y, z)

(8.256)

(8.257)

We can show that Ni (x, y, z) here in (8.257) are same as these using Pascal pyramid. We note that det[C] is related to the volume of tetrahedron hence

8.11. INTERPOLATION FUNCTIONS FOR 3D ELEMENTS

603

det[C] > 0 holds implying that Ni (x, y, z) in (8.257) are unique. Thus for the four-node tetrahedron we can use L1 , L2 , L3 , L4 as interpolation functions instead of Ni (x, y, z); i = 1, 2, . . . , 4. 8.11.2.9 Higher degree C 000 basis functions using volume coordinates Consider the four-node tetrahedron element shown in Fig. 8.63 with vertex nodes 1, 2, 3 and 4. We draw m equally spaced planes parallel to side 1 labelled 0, 1, . . . , p . . . , m, 0 being label for side 1. Similarly, we also draw m equally spaced planes parallel to sides 2, 3 and 4, sides 2, 3 and 4 being marked ‘0’ and q, r, s being intermediate locations of the planes parallel to sides 2, 3, and 4. If we consider typical planes p, q, r and s (parallel to sides 1, 2, 3 and 4) then their intersection pqrs defines a point within the tetrahedron element. The intersection of these planes with sides 1, 2, 3 and 4 and amongst themselves define the location of the nodes in agreement with the Pascal pyramid. Consider side 1 and planes parallel to it. We note that L1 is one at node 1 and zero on side 1, hence we can setup Lagrange type interpolations to define functions N0 (L1 ), N1 (L1 ), . . . , Nm (L1 ) corresponding to (m + 1) parallel planes. We can define Ni (L1 ) by Ni (L1 ) =

i  Y mL1 − j + 1  j=1

=0

j

, for j ≥ 1; i = 1, 2, . . . , m

(8.258)

for i = 0

Similarly in the other three directions, we setup Ni (L2 , Ni (L3 ) and Ni (L4 ) Ni (L2 ) =

i  Y mL2 − j + 1  j=1

=0 Ni (L3 ) =

=0

j

=0

(8.259)

, for j ≥ 1; i = 1, 2, . . . , m

(8.260)

for i = 0

i  Y mL4 − j + 1  j=1

, for j ≥ 1; i = 1, 2, . . . , m

for i = 0

i  Y mL3 − j + 1  j=1

Ni (L4 ) =

j

j

, for j ≥ 1; i = 1, 2, . . . , m

(8.261)

for i = 0

The 3D interpolations or basis functions for a node located at p, q, r, s i.e. Npqrs (L1 , L2 , L3 , L4 ) can now be defined using Npqrs (L1 , L2 , L3 , L4 ) = Np (L1 ) Nq (L2 ) Nr (L3 ) Ns (L4 )

(8.262)

604

BASIC ELEMENTS OF MAPPING AND INTERPOLATION THEORY

in which Np , Nq , Nr and Ns are defined by (8.258) to (8.261). We consider some examples in the following. 8.11.2.10 Four-node linear tetrahedron element (p-level of one) In this case m = 1 and we have p, q, r, s have values of 0 and 1. N0 (L1 ) = 1 N1 (L1 ) = L1 , m = 1, j = 1, i = 1

in (8.258)

N0 (L2 ) = 1 N1 (L2 ) = L2 , m = 1, j = 1, i = 1

in (8.259)

N0 (L3 ) = 1 N1 (L3 ) = L3 , m = 1, j = 1, i = 1

in (8.260)

N0 (L4 ) = 1 N1 (L4 ) = L4 , m = 1, j = 1, i = 1

in (8.261)

(8.263)

(8.264)

(8.265)

(8.266)

The basis or interpolation functions Ni (L1 , L2 , L3 , L4 ) (i = 1, 2, . . . , 4) for the four-node tetrahedron can now be constructed using the following. N1 (L1 , L2 , L3 , L4 ) = N

= N1 (L1 ) N0 (L2 ) N0 (L3 ) N0 (L4 ) = L1

N2 (L1 , L2 , L3 , L4 ) = N

= N0 (L1 ) N1 (L2 ) N0 (L3 ) N0 (L4 ) = L2

N3 (L1 , L2 , L3 , L4 ) = N

= N0 (L1 ) N0 (L2 ) N1 (L3 ) N0 (L4 ) = L3

N4 (L1 , L2 , L3 , L4 ) = N

= N0 (L1 ) N0 (L2 ) N0 (L3 ) N1 (L4 ) = L4

pqrs (1000) pqrs (0100) pqrs (0010) pqrs (0001)

(8.267)

8.11.2.11 A ten-node tetrahedron element (p-level of 2) In this case m = 2 in each of the three directions L1 , L2 , L3 and L4 . The element will contain 10 nodes. Using (8.263) to (8.266) with m = 2 we establish Ni (Lj ) (i = 0, 1, 2) for j = 1, 2, 3 and 4 and then use (8.267) to determine Ni (L1 , L2 , L3 , L4 ) for i = 1, 2, . . . , 10.

8.12 Summary Basic elements of mapping and the interpolation theory are presented in this chapter. The finite elements in physical spaces R1 , R2 , and R3 are mapped into natural coordinate spaces ξ; ξ, η; and ξ, η, ζ in predefined geometric configurations. Mapping of points, lengths, areas, and volumes are established. The interpolation theories in R1 , R2 , and R3 are presented using natural coordinate space with fixed element geometries using Lagrange

605

PROBLEMS

interpolation; tensor product and Pascal’s triangle, pyramid, and rectangle. For triangular and tetrahedron families of elements, area and volume coordinates are utilized. Interpolation theories, hence local approximations are also presented using Legendre and Chebyshev polynomials. In all cases p-version hierarchical local approximations of class C 0 as well as of higher classes in R1 are established first and then utilized through tensor product or otherwise to derive C 0 and higher order continuity local approximations for regular and distorted geometries (in physical space) in R2 and R3 .

Problems 8.1 Figure 8.64(a) shows a three-node parabolic one-dimensional element in one-dimensional Cartesian coordinate space.

η y

x1 = 2.0

x2 = 3.0

x3 = 6.0

1

2

3 ξ

x

1

2

3

2 (a) Schematic in xy space

(b) Schematic in ξη space

Figure 8.64: Element map in physical space xy and natural coordinate space ξη

Figure 8.64(b) shows a map of the element in the natural coordinate space. The element approximation functions are defined in the book in the natural coordinate system. (1) (2) (3) (4)

Write equations describing the element mapping between xy and ξη spaces. Derive an expression for the determinant of the Jacobian. Calculate the value of the determinant of the Jacobian at the element nodes. Derive expressions for the derivatives of the approximation functions with respect to x. (5) Calculate the derivatives of the approximation functions with respect to x at the element nodes. (6) If the element of Fig. 8.64(a) was used in stress analysis, can you comment on the nature of the strain and stress at the element nodes. 8.2 Consider two-dimensional finite elements shown in Fig. 8.65 (a), (b) and (c). The Cartesian coordinates of the nodes are given. The elements are mapped into ξ, η natural coordinate space into a two-unit square. (I) Determine the Jacobian matrix of transformation and its determinant for each element. Calculate and tabulate the value of the determinant of the Jacobian at the nodes of the element. (II) Calculate the derivatives of the approximation function with respect to x and y for (ξ,η) (ξ,η) node 3 (i.e. ∂N3∂x and ∂N3∂y ) for each of the three elements shown above. 8.3 Consider a two-dimensional eight-node finite element shown in Fig. 8.66. The Cartesian coordinates of the nodes of the element are given in Fig. 8.66. The element is mapped

REFERENCES FOR ADDITIONAL READING

606 y

(6.5,7)

y

y

(5,6)

3 4

3

4

(10,6) 5

6

7 3

10

5 1

2

8 3

2

1

x

10

5

(a)

(b)

4 3 1.5

2

1

x

3

x

3 0.5 (c)

Figure 8.65: 2D elements in xy space

into natural coordinate space ξ, η into a two-unit square with the origin of the ξη coordinate system at the center of the element. y

(5,6) 7

(10,6) 5

6

3 8 1 2 3

4 3 1.5

1 1 3

x

3 0.5

Figure 8.66: A 2D element in xy space

(a) Write a computer program (or calculate otherwise) to determine the Cartesian coordinates of the points midway between the nodes. Tabulate the x, y coordinates of these points. Plot the sides at the element in xy space by taking more intermediate points. (b) Determine the area of the element using Gaussian quadrature. Select and use the minimum number of quadrature points in ξ and η directions to calculate the area exactly. Show that increasing the order of the quadrature does not affect the area. (c) Determine the locations of the quadrature points (used in (b)) in the Cartesian space. Provide a table of these points and their locations in xy space. Also mark their locations on the plot generated in part (a). Provide program listing, results, tables and plots along with a write-up on the equations used as part of the report. Also provide a discussion of your results. [1–11]

References for additional reading [1] K. S. Surana, S. R. Petti, A. R. Ahmadi, and J. N. Reddy. On p-version hierarchical interpolation functions for higher-order continuity finite element models. Int. J. Comp. Eng. Sci., 2(4):653–673, 2001.

REFERENCES FOR ADDITIONAL READING

607

[2] A. R. Ahmadi, K. S. Surana, Maduri, A. Romkes, and J. N. Reddy. Higher order global differentiability local approximations for 2d distorted quadrilateral elements. Int. J. Comp. Meth. in Eng. Sci. and Mech., 10:1–19, 2009. [3] I. S. Sokolnikoff and R. M. Redheffer. Mathematics of Physics and Modern Engineering. McGraw-Hill, 2nd edition, 1966. [4] B. A. Szabo and I. Babuska. Finite Element Analysis. Wiley-Interscience, 1991. [5] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-self-adjoint operators in BVP. Int. J. Comp. Eng. Sci., 4(4):737–812, 2003. [6] K. S. Surana, A. R. Ahmadi, and J. N. Reddy. The k-version of finite element method for non-linear operators in BVP. Int. J. Comp. Eng. Sci., 5(1):133–207, 2004. [7] K. S. Surana, S. Allu, and J. N. Reddy. The k-version of finite element method for initial value problems: Mathematical and computational framework. Int. J. Comp. Eng. Sci., 8(3):123–136, 2007. [8] G. Strang. Variational crimes in the finite element method. In A. K. Aziz, editor, The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations, pages 689–710. Academic Press, 1972. [9] J. Patera and F. T. Pettman. Isoparametric hermite elements. Int. J. Num. Meth. Eng., 37:3489–3519, 1994. [10] D. W. Wang, I. N. Katz, and B. A. Szabo. Implementation of c1 triangular element based on the p-version of the finite element method (structural analysis computer code). Comp. Struct., 19(3):381–392, 2011. [11] R.H. Gallagher. Finite Element Analysis: Fundamentals. Prentice Hall, New Jersey, 1975.

9

Linear Elasticity using the Principle of Minimum Total Potential Energy 9.1 Introduction In chapters 3 and 5 we had considered finite element processes based on GM/WF and least squares methods for BVPs described by self-adjoint differential operators. The approach in GM/WF is based on fundamental Lemma whereas the LSPs are based on residual functional. In this chapter we summarize GM/WF for BVPs described by self-adjoint differential operators and present an alternate approach in which the details of the finite element processes can be derived directly without using the fundamental Lemma and integration by parts. While the mathematical basis for the approach presented here remains the same as GM/WF, but the approach presented in this chapter is perhaps more appealing as it deals directly with the physics. First, we revisit some basic concepts (chapters 3 and 5). Consider a BVP Aφ − f = 0 in Ω

(9.1)

in which A is a self-adjoint differential operator, which contains only even order derivatives of the dependent variables (we consider homogeneous boundary conditions for the sake of simplicity). More specifically A only needs to be linear and its adjoint A∗ needs to be same as A. In classical GM we consider (Aφn − f, v)Ω¯ = 0 (9.2) ¯ and v = δφn . In (9.2), for the in which φn is an approximation of φ over Ω even order derivative terms in φn we transfer half of the differentiation from φn to the test function v. Simplification of the concomitant using BCs and regrouping terms gives the weak form (Aφn − f, v)Ω¯ = B(φn , v) − l(v) = 0

(9.3)

In (9.3), B(φn , v) is bilinear and symmetric, B(φn , v) = B(v, φn ), and l(v) is linear. Hence, there exists a functional I(φn ) 1 I(φn ) = B(φn , φn ) − l(φn ) = 0 (9.4) 2 609

610

ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

such that the first variation of I(φn ) gives the weak form δI(φn ) = B(φn , v) − l(v) = 0

(9.5)

δ 2 I = B(v, v) > 0

(9.6)

and is a unique extremum principle and hence the weak form is a variationally consistent integral form. Inequality (9.6) implies that φn obtained from δI(φn ) = 0 minimizes I(φn ) in (9.4). In case of linear elasticity and linear structural mechanics, I(φn ) represents the total potential energy of the deforming body in which 21 B(φn , φn ) is the strain energy or elastic energy stored by the body and l(φn ) is the potential energy of loads. Thus minimization of I(φn ) is in fact the well known principle of minimization of total potential energy. Stated simply: If a deformed elastic body is in equilibrium then its total potential energy is minimum. Thus, if we could construct the total potential energy of a deformed body (integral over the volume of the body) then, its minimization will yield the integral form defining stable equilibrium of the deformed body. This indeed is the weak form, which in this case is variationally consistent integral form hence will yield unconditionally stable computational processes.

9.2 New notation To maintain consistency with the notation used in linear elasticity, structural mechanics, and energy, we introduce the following new notation. We first consider classical methods of approximation. Let the quadratic functional I(φ) representing total potential energy of a deformed body in equi¯ we librium be denoted by Π(φ). Then, for approximation φn of φ over Ω have I(φn ) = Π(φn ). Let Π1 (φn ) and Π2 (φn ) be the strain energy stored by the deformed body and the potential energy of loads respectively for an ¯ Then approximation φn of φ over Ω. 1 Π1 (φn ) = B(φn , φn ) 2 Π2 (φn ) = l(φn )

(9.7)

Hence, based on (9.4) and (9.7) we can write Π(φn ) = Π1 (φn ) − Π2 (φn )

(9.8)

First variation of Π(φn ) set to zero, δΠ(φn ) = 0, gives the necessary conditions from which we determine φn and δ 2 Π(φn ) > 0 is the extremum principle which when unique ensures that φn obtained from δΠ(φn ) = 0 in fact minimizes Π(φn ).

611

9.3. APPROACH

9.3 Approach Based on the details presented above, we can proceed as follows. (1) Derive an expression for the total potential energy Π of the deformed body directly from the physics of deformation. This establishes expressions for Π1 and Π2 . (2) Minimization of the total potential energy Π would yield the necessary conditions, namely, the equations of equilibrium for the deformed body from which we determine the solution. ¯ T = S Ωe . (3) First, (1) and (2) need to be converted to the discretization Ω e Consider ¯T Π = Π1 − Π2 for Ω (9.9) and Π=

X

Πe

¯e for Πe over Ω

(9.10)

e

minimization of Π implies δΠ = δ

X

Πe =

X

e

δΠe = 0

(9.11)

e

Equation (9.11) describes equilibrium of the whole discretizion of the ¯ (i.e. Ω ¯ T ). In (9.11), we only need to consider δΠe for an domain Ω ¯ e . This would yield the desired discretized element e with domain Ω equations for an element e that can be assembled or summed to obtain ¯ T . We present the discretized equations for the whole discretization Ω ¯e details of the discretized equations for an element e with domain Ω e e using Π and δΠ . The total potential energy is Πe = Πe1 − Πe2

(9.12)

9.4 Element equations 9.4.1 Local approximation of the displacement field Let {φeh } = {ueh } or



ueh vhe

 or

 e  uh  ve  he  wh

(9.13)

be the local approximation of the displacement field for ∀x or x, y or x, y, z ∈ ¯e → Ω ¯ m and hence, ∀ξ or ξ, η or ξ, η, ζ ∈ Ω ¯ m. Ω {φeh } = [N ]{δ e }

(9.14)

in which [N ] is the local approximation function matrix and {δ e } are nodal degrees of freedom.

612

ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

9.4.2 Stresses and strains Assuming small deformation, small strain (linear elasticity), the strains at a point can be expressed in terms of the gradients of the displacements u, v and w in x, y and z directions. Expressing strain tensor as a vector (Voigt’s notation), we can write strains as functions of the gradients of displacements n  u v w o {ε} = ε xyz

(9.15)

and by substituting from (9.14) into (9.15), we can express the strains in terms of nodal degrees of freedom {δ e }. {ε} = [B]{δ e }

(9.16)

Let {σ0 } be initial stress in the body before the application of external loads and also let {ε0 } be the initial strain. Then the total stress field in the body before deformation is given by the expression −[D]{ε0 } − {σ0 }

(9.17)

in which [D] is the matrix of material constants relating strains to stresses. The stresses due to deformation, {ε}, are given by (a suitable constitutive law) {σ} = [D]{ε} (9.18) Hence, the total stress is given by {σ}total = {σ} − [D]{ε0 } − {σ0 } = [D]{ε} − [D]{ε0 } − {σ0 }

(9.19)

Equation (9.18) is generalized Hooke’s law. [D] is a symmetric matrix containing material coefficients.

9.4.3 Strain energy Πe1 and potential energy of loads Πe2 If Πe1 is total strain energy or elastic energy stored in an element e with ¯ e , then domain Ω Z 1 Πe1 = {ε}T {σ} dΩ (9.20) 2 ¯e Ω

Substituting for {σ} from (9.18) into (9.20), Z 1 e Π1 = {ε}T [D]{ε} dΩ 2 ¯e Ω

(9.21)

613

9.4. ELEMENT EQUATIONS

Substituting for {ε} from (9.16) into (9.21) Πe1 =

1 2

Z

{δ e }T [B]T [D][B]{δ e } dΩ

(9.22)

¯e Ω

¯ e . When the elasNext we consider potential energy of loads Πe2 over Ω tic body undergoes deformation, the points of application of external loads experience motion. The body may be subjected to body forces such as acceleration and gravitational loads and/or pressure loads on some boundaries of it. The work is done by all of these disturbances acting on the volume ¯ e . When an element is isolated from the discretization Ω ¯ T , it must be in Ω equilibrium (free body diagram) under the action of external loads and the internal forces on its boundaries exerted by the remaining discretization (cut principle of Cauchy). Let us introduce the following notation: {P e } be a vector of secondary variables at the element nodes (i.e. internal nodal forces). {p} be a vector of distributed loads (say pressure) in x, y, z direction per ¯ e. unit length or area on a boundary Γe∗ of an element e with domain Ω {f b } be a vector of distributed body forces (due to gravity or acceleration and/or centrifugal forces) per unit volume. Then, the potential energy of loads can be written as Πe2 = {δ e }T {P e } +

Z

{φeh }T {p} dΓ +

Γe∗

Z

{φeh }T {f b } dΩ

¯e Ω

{ε}T [D]{ε0 } dΩ +

+

Z

¯e Ω

Z

{ε}T {σ0 } dΩ

(9.23)

¯e Ω

Substituting for {ε} from (9.16) and {φeh } from (9.14) in (9.23) Πe2 ={δ e }T {P e } −

Z

{δ e }T [N ]T {p} dΓ +

Γe∗

Z

e T

T

¯e Ω

{δ e }T [N ]T {f b } dΩ

¯e Ω

Z

{δ } [B] [D]{ε0 } dΩ +

+

Z

e T

(9.24) T

{δ } [B] {σ0 } dΩ ¯e Ω

This is the final expression for the potential energy of loads.

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

9.4.4 Total potential energy Πe for an element e Using (9.22) and (9.24) and noting that Πe = Πe1 − Πe2 , we can write Z Z 1 Πe = {δ e }T [B]T [D][B]{δ e } dΩ − {δ e }T {P e } − {δ e }T [N ]T {p} dΓ 2 ¯e Γe∗ Ω Z Z Z e T T b e T T − {δ } [N ] {f } dΩ − {δ } [B] [D]{ε0 } dΩ − {δ e }T [B]T {σ0 } dΩ ¯e Ω

¯e Ω

¯e Ω

(9.25) We note that {δ e } is independent of coordinates and hence can be taken outside of the integrals. Z i 1 e h e Π = {δ } [B]T [D][B] dΩ {δ e } − {δ e }T {P e } 2 ¯e Ω Z Z   − {δ e }T [N ]T {p} dΓ − {δ e }T [N ]T {f b } dΩ (9.26) Γe∗ e T

− {δ }

Z

¯e Ω



T

e T

[B] [D]{ε0 } dΩ − {δ }

Z

[B]T {σ0 } dΩ



¯e Ω

¯e Ω

We introduce the following notation: Z e [K ] = [B]T [D][B] dΩ

(9.27)

¯e Ω e

Z

{F }p =

[N ]T {p} dΓ

Γe∗

{F e }b =

Z

[N ]T {f b } dΩ

¯e Ω e

{F }ε0 =

Z

(9.28)

[B]T [D]{0 } dΩ

{F e }σ0 = [B]T {σ0 } dΩ in which [K e ] is called the element stiffness matrix and {F e }p , {F e }b , {F e }ε0 and {F e }σ0 are the equivalent nodal load vectors due to: pressure loading on the element faces, body forces, initial strain and initial stress. Substituting from (9.27) and (9.27) into (9.26) Πe = {δ e }T [K e ]{δ e } − {δ e }T {P e } − {δ e }T {F e }p − {δ e }T {F e }b − {δ e }T {F e }ε0 − {δ e }T {F e }σ0

(9.29)

615

9.4. ELEMENT EQUATIONS

We note that Πe = Πe {δ e }




(9.30)

and Π=

X



Πe {δ e } = Π {δ}

(9.31)

e

where {δ} = ∪e {δ e } Therefore (Π)min ⇒ or

(9.32)

∂Π =0 ∂{δ}

(9.33)

X ∂Πe  ∂Π =0 = ∂{δ} ∂{δ e } e From (9.29), we can obtain

(9.34)

∂Πe ∂{δ e }

∂Πe = [K e ]{δ e } − {P e } − {F e }p − {F e }b − {F e }ε0 − {F e }σ0 ∂{δ e }

(9.35)

Substituting from (9.35) into (9.34)  X ∂Π = [K e ]{δ e }−{P e }−{F e }p −{F e }b −{F e }ε0 −{F e }σ0 = 0 (9.36) ∂{δ} e or X

 X X X X X [K e ]{δ e } = {P e } + {F e }p + {F e }b + {F e }ε0 + {F e }σ0

e

e

e

e

e

e

(9.37) or

hX

i

K e {δ} = {P } + {F }p + {F }b + {F }ε0 + {F }σ0

(9.38)

[K]{δ} = {P } + {F }p + {F }b + {F }ε0 + {F }σ0

(9.39)

e

or ¯ T ) represent assembly in which e [K e ] = [K]. Sums over e (elements of Ω of element equations symbolically. Forces on the right side of (9.39) are due to assembly of element equivalent nodal loads and secondary variables. P

Remarks. (1) The derivation presented in this chapter is based on minimization of total potential energy of an elastic body in equilibrium under the action of external disturbances. Since the differential operator is self-adjoint, the total potential energy is in fact quadratic functional I(φ).

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

(2) This direct approach of constructing the total potential energy functional using the physics of deformation is quite general and is perhaps more appealing in applications such as linear elasticity and structural mechanics. (3) Various types of 1D, 2D and 3D elements in linear solid mechanics (i.e. linear elasticity), can be easily derived using the general derivation presented above. We outline the main steps in the following. (a) Identify displacement components. (b) Identify stress and strain components. ¯ e, (c) Establish the nature of local approximations over a subdomain Ω i.e [N ], the approximation function matrix and the dofs {δ e }. (d) Express strains in terms of the derivatives of displacements and substitute local approximations to derive [B] matrix. (e) Identify constitutive behavior [D] expressing material behavior relating strains to stresses. (f) Express initial strains. If thermal strains, then express them in terms of temperature difference with respect to stress free temperature and the coefficients of thermal expansion. (g) All other external loadings such as body forces (acceleration and centrifugal loads) etc. can also be expressed as a function of basis functions and subdomain or element volume or mass. With (a)–(g), all matrices and vectors in the element equations are defined except secondary variables for the elements as well as in assembled equations. (4) Using the approach outlined above it is straight forwards to derive the details of the finite element equations for applications such as (a) (b) (c) (d) (e) (f) (g) (h) (i) (j) (k) (l)

plane stress plane strain axi-symmetric deformation 2D beam bending axi-symmetric shells 3D membranes plate bending 3D solids 3D thick shells 3D shells 3D beam bending 1D, 2D and 3D axial rods, spars and others

617

9.5. FINITE ELEMENT FORMULATION FOR 2D LINEAR ELASTICITY

What one needs to recognize is that the generic nature of the basic derivation remains the same in all cases. What changes from one type of element (i.e. physics) to another are the specific contents of various matrices and vectors.

9.5 Finite element formulation for 2D plane stress linear elasticity To illustrate details of various matrices and vectors in the direct approach described above we consider plane stress linear elasticity as an example. In plane stress behavior a material particle in the current (deformed) configuration experiences displacements u and v in x, y directions of the o − xy fixed frame. We consider a distorted quadrilateral element in x, y space and its map m ¯ ¯ ξη in the ξ, η natural coordinate space in the two-unit square (see Ω or Ω Fig. 9.1). 7

5

5

6

6

7 9 8

v

η

4

2

8

9

¯e Ω

y

2

4

ξ

¯ m or Ω ¯ ξ,η Ω

3

1

1 x u

2

3

2

Element geometry in xy space

Element map in the natural coordinate space ξη

Figure 9.1: Distorted 2D quadrilateral element and its map in ξ, η space

Geometry: The mapping can be described using   X   n ¯ x x ¯ = Ni (ξ, η) i y yi i=1 ( 1, if j = i ¯i (ξj , ηj ) = with N 0, if j 6= i and

n ¯ X i=1

¯i (ξ, η) = 1 N

(9.40)

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

where n ¯ could be nine if the element shape is distorted or four (in which case, it is a four nodeh quadrilateral element) if the sides are straight. Additionally, i xy we have [J] = ξη .

¯ e or Ω ¯ ξη 9.5.1 Local approximation of u and v over Ω Consider C 0 local approximations of displacements u and v based on Lagrange family of interpolation functions, hence the nodal dofs are values of u and v at the element nodes (it could be p-version hierarchical also). Equal degree and equal order local approximations for u and v is valid because the GDEs for the plane stress deformation contain same order derivatives of u ¯ e, and v. Let ueh , vhe be local approximations of u and v over an element Ω then  e X  e n uh u = Ni (ξ, η) ei = [N ]{δ e } (9.41) vhe vi i=1

in which for a typical node i, uei , vie are the degrees of freedom. The submatrix [Ni ] for node i can be written as   Ni (ξ, η) 0 [Ni ] = 0 Ni (ξ, η)

(9.42)

with {δie } =

 e ui vie

(9.43)

9.5.2 Stresses, strains and constitutive equations Using Voigt’s notation, the strain and stress, as vectors {ε} and {σ} can be written as follows. Also, the constitutive theory for stresses can be expressed in terms of strains:      ∂u    εx   ∂x   ∂v {ε} = εy = ∂y      γxy  ∂u + ∂v   ∂y ∂x    σx  {σ} = σy = [D]{ε}   τxy

(9.44)

(9.45)

If we consider homogeneous and isotropic material then global x, y direction suffice for defining stresses, strains and the material matrix [D]. It is

9.5. FINITE ELEMENT FORMULATION FOR 2D LINEAR ELASTICITY

619

straight forward to express strains in terms of stresses, modulus of elasticity E and Poisson’s ratio ν    1    − Eν 0  εx   σx  E 0  = σy εy =  − Eν E1 (9.46)     2(1+ν) γxy τxy 0 0 E Inverse of (9.46) gives    1 ν  σx  E  σy = ν 1   (1 − ν 2 ) τxy 0 0

  0  εx  0  εy = [D]{ε}  1−ν  γxy 2

(9.47)

9.5.3 [B] matrix relating strains to nodal degrees of freedom Using local approximation (9.41) for u and v ∂ueh X ∂Ni e ∂ueh X ∂Ni e = u , = u ∂x ∂x i ∂y ∂y i X ∂Ni X ∂Ni ∂vhe ∂vhe = vie , = ve ∂x ∂x ∂y ∂y i

(9.48)

Substituting (9.58) into (9.44) {ε} = [B]{δ e } in which [Bi ] for a node i is given by 

∂Ni ∂x

 [Bi ] =   0

∂Ni ∂y

0

(9.49)



∂Ni  ∂y  ∂Ni ∂x



(9.50)

9.5.4 Element stiffness matrix [K e ] e

Z

[K ] =

[B]T [D][B] dΩ

(9.51)

¯e Ω

in which [B] and [D] are defined by (9.51) and (9.47). We note that dΩ = t(ξ, η) det[J] dξ dη

(9.52)

t(ξ, η), thickness at a point ξ, η can be defined by t(ξ, η) =

n ¯ X i=1

¯ (ξ, η) ti N

(9.53)

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

e ] of [K e ] for nodes i and j can where ti are nodal thicknesses. A typical [Kij be written as Z1 Z1 e [Kij ] = [Bi ]T [D][Bj ] t(ξ, η) det[J] dξ dη, i, j = 1, . . . , n (9.54) −1 −1

or e [Kij ]=

Z1 Z1 −1 −1

T    ∂Nj D11 D12 0 ∂x      0 ∂Ni   ∂y  D12 D22 0   0  ∂Nj ∂Ni ∂Ni 0 0 D33 ∂y ∂x ∂y 

∂Ni ∂x

0

0



∂Nj   ∂y  ∂Nj ∂x

t(ξ, η) det[J] dξ dη

(9.55) in which n is the number of nodes. Numerical values of [K e ] are obtained using Gauss quadrature.

9.5.5 Transformations from (ξ, η) to (x, y) space Using x = x(ξ, η), y = y(ξ, η) defining mapping between ξη and xy spaces we can write the following for mapping of lengths between the two spaces     dξ dx (9.56) = [J] dη dy where " [J] =

∂x ∂ξ ∂y ∂ξ

∂x ∂η ∂y ∂η

# (9.57)

in which n ¯ n ¯ ¯i ¯i ∂x X ∂ N ∂x X ∂ N = xi , = xi ∂ξ ∂ξ ∂η ∂η i=1

∂y = ∂ξ

i=1

n ¯ X ¯i ∂N

n ¯ X ¯i ∂N

i=1

i=1

∂y yi , = ∂ξ ∂η

∂η

(9.58) yi

and (

∂Ni ∂x ∂Ni ∂y

)

( T −1

= [J ]

∂Ni ∂ξ ∂Ni ∂η

) , i = 1, . . . , n

(9.59)

9.5.6 Body forces Body forces in solid and structural mechanics consist of acceleration or gravity loads and centrifugal forces in spinning objects. Recall that  b Z e T fx {F }b = [N ] dΩ (9.60) fyb ¯ Ω

9.5. FINITE ELEMENT FORMULATION FOR 2D LINEAR ELASTICITY

621

in which Gx and Gy are body forces percent volume in x and y directions. In ξ, η space, equation (9.60) can be written as e

Z1 Z1

{F }b = −1 −1

 b fx [N ] det[J] t(ξ, η) dξ dη fyb T

(9.61)

Let gx and gy be accelerations in x and y directions and wz be the angular velocity about the z-axis, then  b     fx ρ gx + ρ x wz2 ρ(gx + x wz2 ) = = (9.62) fyb ρ gy + ρ y wz2 ρ(gy + y wz2 ) in which ρ is mass density of the material. For a typical node i {Fie }b =

( ) Fxei Fyei

b

  Z1 Z1  ρ(gx + x wz2 ) Ni (ξ, η) 0 t(ξ, η) det[J] dξ dη (9.63) = 0 Ni (ξ, η) ρ(gy + y wz2 ) −1 −1

9.5.7 Initial strains (thermal loads) Let α be the coefficient of thermal expansion and Tref be the stress free temperature, then     α(T (ξ, η) − Tref ) εx0  {ε0 } = α(T (ξ, η) − Tref ) = εy0 (9.64)     0 0 in which εx0 and εy0 are thermal strains in the x and y directions. The temperature T (ξ, η) may be defined as X ˆ (ξ, η) Tie T (ξ, η) = N (9.65) in which Tie are nodal temperatures for an element e. Therefore   Z εx0  {F e }ε0 = [B]T [D] εy0 dΩ   0 ¯e Ω or {F e }ε0

  εx0  = [B]T [D] εy0 t(ξ, η) det[J] dξ dη   0 −1 −1

(9.66)

Z1 Z1

(9.67)

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

For a typical node i, we have {Fie }ε0 =



Fxei Fyei

 ε0

  εx0  = [Bi ]T [D] εy0 t(ξ, η) det[J] dξ dη   0 −1 −1 Z1 Z1

(9.68)

9.5.8 Equivalent nodal loads {F e }p due to pressure acting normal to the element faces The distributed pressure loads are assumed to act normal to the faces of the element as shown in Fig. 9.2. A quadrilateral element has four faces F1 , F2 , F3 , F4 corresponding to ξ = −1, ξ = 1, η = −1 and η = 1. Therefore, the details of the computations of {F e }p must be considered for all four faces. If Γe∗ is a face or side of the element, then Z e {F }p = [N ]T {p} dΓ (9.69) Γe∗

Let

    px l {p} = = x p(ξ, η) py ly

(9.70)

in which lx , ly are the direction cosines of the unit exterior normal to the boundary Γe∗ on which pressure p(ξ, η) is acting. We need to consider Γe∗ to be Γei ; i = 1, . . . , 4, that is faces Fi ; i = i = 1, . . . , 4. As an illustration consider face F2 of the element e. On this face, ξ = +1 is constant. A vector ~t2 tangent to the face F2 is given by   ∂x   ∂η   ~t2 = ∂y (9.71)   ∂η   0 Exterior normal to the face F2 (i.e. ~n2 ) can be obtained by taking cross e

product of ~t2 with ~k, a unit vector in the z direction   i j k  ∂x   ∂η  ∂y ~n2 = ∂x 0 = − ∂y ∂η  ∂η ∂η  e 0 0 1  0 

(9.72)

Therefore     ∂x    lx  ∂η  1 ∂y ~n2 = e = q = − ∂η  ly 
||~n2 || ∂y
2  ∂x 2   + ∂η lz ∂η e 0 ~n2

(9.73)

623

9.5. FINITE ELEMENT FORMULATION FOR 2D LINEAR ELASTICITY

and dΓ =

p (dx)2 + (dy)2

(9.74)

dξ +

∂x 2  ∂y ∂y 2 dη + dξ + dη ∂η ∂ξ ∂η

(9.75)

or s dΓ =

 ∂x ∂ξ

For face F2 (ξ = 1), we have dΓ =

s

∂x 2  ∂y 2 + ∂η ∂η

 dη

(9.76)

Substituting from (9.73) and (9.76) into (9.69) for face F2 and defining {F e }Fp 2 as equivalent load vector due to pressure on face F2 , we obtain {F e }Fp 2

Z1 =

[N ]Tξ=1 q

−1


∂x 2 ∂η

∂y ∂η − ∂x ∂η

(

1 +

∂y
2 ∂η

)s

 ∂x 2 ∂η

+

 ∂y 2 ∂η

p(ξ, η) dη (9.77)

or {F e }Fp 2

Z1 =

∂y ∂η − ∂x ∂η

( [N ]Tξ=1

−1

) p(ξ, η) dη

(9.78)

p(ξ, η) is known on face F2 (applied pressure loading). We note that positive pressure acts in the direction of the unit exterior normal to the face (i.e. away from the face). Expressions similar to (9.78) can also be derived for faces F1 , F3 and F4 . Final expressions for all four faces are given in the Table 9.1. η p(ξ, η)

6

7

5

6

7

F4 (Γe4 )

5

η=1

t~2

η = 1 face ξ = −1 face 8

9

n~2

4 ξ = 1 face

8

9 ξ = −1

ξ=1

F1 (Γe2 ) y

3 2 η = −1 face

1 x

Element in xy space

ξ 4 F2 (Γe2 )

η = −1

1

2 F3 (Γe3 )

3

Element map in ξ, η space (Element faces F1 , . . . , F4 )

Figure 9.2: Pressure loads acting normal to the element faces in physical coordinate space and the element map in natural coordinate space

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ELASTICITY USING THE PRINCIPLE OF MINIMUM TOTAL POTENTIAL ENERGY

Table 9.1: Expressions for equivalent nodal forces {F e }p for faces Fi ; i = 1, 2, 3, 4

Face F1

F2

F3

F4

{F e }Fp 1

{F e }p ( ∂y ) R1 − ∂η = [N ]Tξ=−1 p(ξ, η) dη ∂x −1

{F e }Fp 2 =

{F e }Fp 3 =

R1 −1

R1 −1

{F e }Fp 4 =

∂η ∂y ∂η − ∂x ∂η

( [N ]Tξ=1

∂y ∂ξ − ∂x ∂ξ

( [N ]Tη=−1

R1 −1

[N ]Tη=1

) p(ξ, η) dη

)

( ∂y ) − ∂ξ ∂x ∂ξ

p(ξ, η) dη

p(ξ, η) dη

9.6 Summary In this chapter, derivations of the details of finite element formulations have been presented for linear structural and linear solid mechanics using the principle of minimum potential energy. It is shown that minimization of total potential energy is identical to Galerkin method with weak form for selfadjoint differential operators (A is linear and A∗ = A) that are encountered in linear structural and linear solid mechanics. In this approach, we directly write a statement of total potential energy from physical consideration and thus avoid using differential mathematical models as well as integral forms derived using GM/WF. In section 9.4, a general formulation is presented for various matrices and vectors resulting from this approach that can be used for any application in linear structural and linear solid mechanics (listed at the end of section 9.4). To illustrate how to utilize the general formulation of section 9.4 for specific applications, details of the finite element formulation for a plane stress problem are presented in section 9.5.

10

Linear and Nonlinear Solid Mechanics using the Principle of Virtual Displacements

10.1 Introduction In the study of continuum mechanics [1,2] and variational methods [3–5], we often use principle of virtual work to derive Hamilton’s principle in Lagrangian description and Euler-Lagrange equations in Lagrangian as well as Eulerian descriptions which in fact are the momentum equations in the respective descriptions. In solid mechanics we refer to these as equations of equilibrium. The principle of virtual work is applicable to linear as well as non-linear processes. This is rather obvious from the resulting momentum equations that hold for all processes that are linear as well as non-linear. We have seen in the earlier chapters in this book that once we have a mathematical description of a physical process in terms of a differential model, i.e. a system of partial differential equations, the formulations of the finite element computational process using methods of approximation, discretizations and local approximations is rather straight forward. Many of the details and intricacies in this approach for non-linear differential operators describing reversible processes encountered in solid mechanics can be avoided if we use principle of virtual work to derive details of the finite element processes. In doing this we utilize the principle of virtual work in deriving the finite element formulation at a much earlier stage, long before the differential mathematical models are extracted from the equations of virtual work. This approach eliminates the use of PDEs and hence construction of integral form from the PDEs in the mathematical models as the statement of virtual work itself is an integral statement. Thus, PDEs are not required when using principle of virtual work. If we pursue the principle of virtual work to derive differential models then the result is momentum equations or equations of equilibrium. These equations are a statement of force balance and hence cannot differentiate between reversible and irreversible processes. 625

626

LINEAR AND NONLINEAR SOLID MECHANICS

As we know, in a reversible process the rate of mechanical work on a volume of matter does not result in entropy production. Whereas in an irreversible process the rate of mechanical work results in rate of entropy production which in turn influences specific internal energy. To describe this physics we need energy equation and entropy inequality. This is beyond the scope of virtual work. Thus, within the thermodynamic framework consisting of conservation and balance laws, the principle of virtual work only addresses momentum equations hence can only address reversible processes as the irreversibility requires other balance laws. More specifically, in this approach strictly speaking we can only consider restricted class of solid matter such as elastic solids. The thermoelastic solids with and without memory cannot be considered within the framework of principle of virtual work as such solids have mechanisms of dissipation, hence entropy production due to mechanical rate of work must be accounted for. The elastic behavior can be linear or nonlinear. Furthermore, since the momentum equations resulting from the principle of virtual work are valid for linear as well as nonlinear processes as long as additional physics of rate of entropy is not considered, this allows us to consider elastoplastic deformation using principle of virtual work so long as we only consider material nonlinearly due to plasticity without accounting for rate of entropy production. This is rather popular and straight forward approach to incorporate elastoplastic deformation in finite element processes using principle of virtual work.

10.2 Principle of virtual displacements Definition 10.1. When a deformed solid in equilibrium is subjected to virtual displacements then the virtual work done by these virtual displacements in moving through the actual forces is zero. This is known as the principle of virtual displacements (see Reddy [1, 6, 7]). Thus, virtual displacements are admissible displacements such that due to applications of these the equilibrium of the deformed body is not disturbed. This allows us to use principle of virtual work to derive equations describing the stable equilibrium state of the deformed solids. In this chapter, we use principle of virtual work to derive details of the finite element computational process for finite deformation and finite strain solid mechanics applications in which the deformation process is reversible. When solid continua are disturbed this results in kinetic energy, strain energy due to deformation, strains, and stresses, and work done by body forces and the surface tractions. In this chapter we only consider stationary processes in which kinetic energy can be neglected, i.e. the virtual work due to the inertial forces can be neglected. In most of the derivations presented here we only consider Lagrangian description.

10.3. VIRTUAL WORK STATEMENTS

627

10.3 Virtual work statements Let V ({x}, t) be the volume of the solid matter in the reference configuration in which {x} are the coordinates of the material points at time t = 0. Let V¯ be the deformed volume at time t (current configuration) with deformed material point locations {¯ x} = {x} + {u({x}, t)}; {u({x}, t)} being displacements of the material point at location {x} in the reference configurations. Consider the work done by all forces on the volume V over certain virtual displacements {δu} defined as a function of the deformed position {¯ x}. Since ¯ ¯ the volume of matter V with surface {A} is in equilibrium under the action of body forces and surface forces, the virtual work due to these forces must equilibrate with the virtual work due to internal stress field. Let   u {φ} = v   w

(10.1)

in which u, v and w are displacements of a material point at location {x} in the reference configuration. Let {δφ} represent the variation of {φ} i.e. virtual displacements at material point {x}.    δu  {δφ} = δv   δw

(10.2)

Consider the virtual work done by the body forces, distributed loads on the surface of the body and the internal stress field. Then we can write the following in Lagrangian description using principle of virtual work for finite deformation. Z Z Z  ¯ T b T dA {δφ} ρ{F } dV + {δφ} {¯ p} dA = δ{ε}T {σ} dV (10.3) dA A V V in which {σ} and {ε} are conjugate stress and strain tensors. We consider {σ} to be the second Piola-Kirchhoff stress tensor and {ε}, the Green’s strain tensor. The second Piola-Kirchhoff stress measure is valid for finite deformation [2]. In (10.3), the first term on the left side is the virtual work due to body forces. The second term is the virtual work due to distributed pressure loads and the right side is the virtual work of the internal stress field. Traditionally in the principle of virtual work we continue further with (10.3) to derive the associated Euler-Lagrange equations (in this case in the absence of inertial effects), but here instead we begin derivation of the details of finite element processes using (10.3).

628

LINEAR AND NONLINEAR SOLID MECHANICS

¯ T be the discretization of V¯ such that Let Ω [ ¯e ¯T = Ω Ω

(10.4)

e

¯ e is the domain of an element e such that Ω ¯ e = Ωe ∪ Γe , closure of in which Ω e e domain Ω with closed boundary (or surface) Γ . Note that henceforth an over bar on Ω or Ωe does not refer to the current or deformed configuration but rather closures of Ω and Ωe . Likewise, we consider AT , discretization of surface or area A bounding volume V [ Γe (10.5) AT = e

in which Γe is the surface or boundary of Ωe and [  [  ¯ T = ΩT ∪ AT = Γe Ω Ωe ∪

(10.6)

e

e

¯T We can write (10.1) over Ω Z

{δφh }T ρ{F b } dV +

¯T Ω

Z

{δφ}T

 dA¯  dA

Z {¯ p} dA =

δ{ε}T {σ} dV

(10.7)

¯T Ω

AT

or XZ e ¯e Ω

{δφeh }T ρ{F b } dV

+

XZ e

{δφeh }T

Γe

 dA¯  dA

{¯ p} dA =

XZ

δ{ε}T {σ} dV

e ¯e Ω

(10.8) ¯ T and {φe } is local ap{φh } is approximation of {φ} over discretization Ω h ¯ e. proximation of {φ} over Ω ¯e Consider local approximation {φeh } over Ω  e  uh  {φeh } = vhe = [N ]{δ e }  e wh

and {δφeh } = [N ]δ{δ e }

(10.9)

in which {φeh } is the local approximation of displacements {φ}T = [u v w] ¯ e . In (10.35), [N ] and {δ e } are defined by over Ω    e  [N ] [0] [0]  {u }   e  e e   [0] [N ] [0] [N ] =   ; {δ } =  {v e}  e {w } [0] [0] [N ] e

(10.10)

629

10.3. VIRTUAL WORK STATEMENTS

[N ] is the local approximation matrix and [N ] are local approximation funce

tions for ueh , vhe and whe . {ue }, {v e } and {we } are nodal degrees of freedom for ueh , vhe and whe . {δ e } are the total degrees of freedom for element e in the local approximation (10.9). We also need δ{ε} i.e. variation of Green’s strain in (10.7) or (10.8). The Green’s strain {ε} can be written as

  εxx        εyy        εzz {ε} = = 2εyz          2ε    zx   2εxy

∂ueh   ∂x   e ∂vh    ∂y   e  ∂wh  ∂z e e ∂vh ∂wh    ∂z + ∂y     e e    ∂w ∂u h h  +     ∂x ∂z   ∂ue    h + ∂vhe  ∂y ∂x



           

+

∂ueh
2 ∂v e
2 + ∂xh + ∂x ∂ueh
2 ∂v e
2 + ∂yh + ∂y e
2 ∂ueh
2 ∂vh + + 2 ∂z ∂z e ∂v e ∂ueh ∂ueh ∂vh  h   ∂y ∂z + ∂y ∂z +   e ∂v e e ∂ue  ∂v ∂u h h h h    ∂x ∂z + ∂x ∂z +  e e e   ∂uh ∂uh + ∂vh ∂vhe + ∂x ∂y ∂x ∂y

  1   2     1    2  1

e
2 ∂wh   ∂x    e
2  ∂wh    ∂y   e
∂wh 2   ∂z e ∂wh ∂w   ∂y ∂z    e ∂wh ∂w    ∂x ∂z   e ∂wh ∂w   ∂x ∂y



(10.11) or {ε} = {εl } + {εn }

(10.12)

 ∂ueh ∂vhe ∂whe , , {θx } = ∂x ∂x ∂x  e  ∂uh ∂vhe ∂whe T {θy } = , , ∂y ∂y ∂y  e  ∂uh ∂vhe ∂whe T {θz } = , , ∂z ∂z ∂z

(10.13)

  {θ}T = {θx }T , {θy }T , {θz }T

(10.14)

Let T



and

Using (10.14) we can write 1 {ε} = {εl } + {εn } = [H]{θ} + [Aθ ]{θ} 2

(10.15)

in which  1 0  0 [H] =  0  0 0

0 0 0 0 0 1

0 0 0 0 1 0

0 0 0 0 0 1

0 1 0 0 0 0

0 0 0 1 0 0

0 0 0 0 1 0

0 0 0 1 0 0

 0 0  1  0  0 0

(10.16)

630

and

LINEAR AND NONLINEAR SOLID MECHANICS

 {θx }T   {0}T    {0}T [Aθ ] =   {0}T   {θz }T  {θy }T

{0}T {0}T



 {θy }T {0}T    {0}T {θz }T   {θz }T {θy }T    T T {0} {θx }   {θx }T {0}T

In (10.15) {εl } and {εn } are linear and nonlinear strains. Consider {εl } = [H]{θ} We define

   e}  [Gx ]  {u      e e  {θ} = [G]{δ } =  [G ] {v }  y     [Gz ] {we }

(10.17)

(10.18)

(10.19)

in which  ∂[N ] e  ∂x  [Gx ] =  [0] [0]  ∂[N ] e  ∂y [Gy ] =   [0] [0]  ∂[N ] e  ∂z [Gz ] =   [0] [0]

[0] ∂[N ] e ∂x

[0] [0] ∂[N ] e ∂y

[0] [0] ∂[N ] e ∂z

[0]

[0]



 [0]   ∂[N ] e ∂x  [0]  [0]   ∂[N ] e ∂y  [0]  [0]   ∂[N ] e ∂z

(10.20)

(10.21)

(10.22)

Substituting from (10.19) into (10.18) {εl } = [H][G]{δ e } = [B l ]{δ e } Clearly [B l ] in (10.23) is given by  ∂[N ]  [0] [0] e   ∂x ∂[N ]  [0]  [0] e   ∂y  ∂[N ]  l  [B ] =  [0] [0] ∂ze    ∂[N ] ]  e [0] ∂[N e ∂x   ∂z ∂[N ] ∂[N ] [0] e ∂xe ∂y

(10.23)

(10.24)

631

10.3. VIRTUAL WORK STATEMENTS

Using (10.19) in (10.15), {εn } can be written as 1 1 1 {εn } = [Aθ ]{θ} = [Aθ ][G]{δ e } = [B n ]{δ e } 2 2 2

(10.25)

where [B n ] = [Aθ ][G]. Therefore we have the following for {ε}   1 {ε} = {εl } + {εn } = [B l ] + [B n ] {δ e } 2

(10.26)

Using (10.19) we can write δ{θ} = [G]δ{δ e }

(10.27)

δ{εl } = [B l ]δ{δ e } 1 1 δ{εn } = δ[Aθ ]{θ} + [Aθ ]δ{θ} 2 2 1 1 = [Aθ ]δ{θ} + [Aθ ]δ{θ} 2 2 = [Aθ ]δ{θ} = [Aθ ][G]δ{δ e } = [B n ]δ{δ e }

(10.28)

and

(10.29) (10.30) (10.31)

Therefore   δ{ε} = δ{εl } + δ{εn } = [B l ] + [B n ] δ{δ e }

(10.32)

δ{ε} = [B]δ{δ e } ; [B] = [B l ] + [B n ]

(10.33)

or Equations (10.26) and (10.32) are fundamental relationships. Now since we ¯ T resulting from the principle have all the variations, we return to (10.8) for Ω of virtual work. Substituting from (10.9) and (10.33) in (10.8)  dA¯  XZ XZ e T T b δ{δ } [N ] ρ{F } dV + δ{δ e }T [N ]T {¯ p} dA dA e ¯e e e Γ Ω XZ = δ{δ e }T [B]T {σ} dV (10.34) e ¯e Ω

or X

e T

Z

δ{δ }

e

T

b

Z

[N ] ρ{F } dV +

T

[N ]

 dA¯  dA

Z {¯ p} dA −

Γe

¯e Ω

T



[B] {σ} dV

=0

¯e Ω

(10.35) or δ{δ}T

XZ e

¯e Ω

[N ]T ρ{F b } dV +

Z Γe

[N ]T

 dA¯  dA

Z {¯ p} dA −

[B]T {σ} dV

 =0

¯e Ω

(10.36)

632

LINEAR AND NONLINEAR SOLID MECHANICS

where δ{δ} =

[

δ{δ e }

(10.37)

e

¯ T . Since {δ} and {δ} are the total degrees of freedom for discretization Ω δ{δ} in (10.36) are arbitrary, the following must hold.  ¯ XZ XZ XZ T dA T b [N ] [N ] ρ{F } dV + {¯ p} dA = [B]T {σ} dV dA e e e Γe

¯e Ω

¯e Ω

(10.38) Let {Rbe }

Z =

[N ]T ρ{F b } dV

(10.39)

¯e Ω

{Rpe } =

Z

[N ]T

 dA¯  dA

{¯ p} dA

(10.40)

Γe e

Z

{P } =

[B]T {σ} dV

(10.41)

¯e Ω

where {Rbe }, {Rpe } are nodal load vectors due to body forces and surface loads and {P e } is the vector of internal forces due to stress field. Using (10.39) - (10.41) in (10.38) we can write X X X {Rbe } + {Rpe } = {P e } (10.42) e

e

e

or {Rb } + {Rp } = {P }

(10.43)

{R} = {P }

(10.44)

{ψ} = {R} − {P }

(10.45)

or or For nonlinear problems (such as finite deformations, plasticity) both {R} and {P } can be functions of {δ}, hence we must find a solution {δ} that satisfies the residual vector {ψ} = {0} iteratively. It is preferable to use the following form for (10.45). XZ ψ = {R} − [B]T {σ} dV (10.46) e ¯e Ω

¯ T , we need only to determine {Re }, {Rpe } and Thus, for an element e of Ω b {P e } defined by (10.39) - (10.41). {Rbe } and {Rpe } pose no particular problem as these are loads, hence are vectors containing numbers. We consider details of {P e } in (10.41) in the following.

633

10.3. VIRTUAL WORK STATEMENTS

10.3.1 Stiffness matrix Consider {P e } in (10.41). Let us assume that the total stress {σ} is related to the total strain {ε} through {σ} = [Ds ]{ε}

(10.47)

Substituting for {ε} from (10.25)   1 {σ} = [Ds ] [B l ] + [B n ] {δ e }. 2 Substituting from (10.34) and (10.48) in (10.41) Z  l T   1 e {P } = [B ] + [B n ] [Ds ] [B l ] + [B n ] {δ e } dV 2

(10.48)

(10.49)

¯e Ω

or e

Z



{P } =

l

n

[B ] + [B ]

T

  l 1 n [Ds ] [B ] + [B ] dV {δ e } 2

(10.50)

¯e Ω

or {P e } = [Kse ]{δ e }

(10.51)

[Kse ]

is called the element secant stiffness matrix first derived by Oden [8, 9]. [Kse ] is non-symmetric even if [Ds ] is symmetric. Use of [Kse ] requires non-symmetric algebraic equations solver and it is restrictive due to the stress-strain relationship (10.47). Instead of using the total stress-total strain relationship, we can consider an incremental form δ{σ} = [DT ]δ{ε}

(10.52)

Use of (10.52) in (10.41) requires that we derive an incremental form of (10.41), i.e we need to consider δ{P e }. Z  e T δ{P } = δ [B] {σ} dV (10.53) ¯e Ω

or e

δ{P } =

Z 

 δ[B]T {σ} + [B]T δ{σ} dV

(10.54)

¯e Ω

where Z

Z

T

[B] δ{σ} dV = ¯e Ω

Z

T

[B] [DT ] δ{ε} dV = ¯e Ω

Z = ¯e Ω

[B]T [DT ][B] δ{δ e } dV

¯e Ω



[B]T [DT ][B] dV δ{δ e } = [K e ] δ{δ e }

(10.55)

634

LINEAR AND NONLINEAR SOLID MECHANICS

and Z

Z

T

δ[B] {σ} dV = ¯e Ω

  δ [B l ]T + [B n ]T {σ} dV

¯e Ω

Z =

δ[B n ]T {σ} dV

¯e Ω

Z =

 T δ [Aθ ][G] {σ} dV

¯e Ω

Z =

[GT ] δ[Aθ ]T {σ} dV

(10.56)

¯e Ω

We can show that δ[Aθ ]T {σ} = [S] δ{θ} = [S][G] δ{δ e }

(10.57)

  σxx [I] σxy [I] σxz [I]    [S] =  σ [I] σ [I] yy yz   σzz [I]

(10.58)

where

Therefore Z Z T δ[B] {σ} dV = [G]T [S][G] δ{δ e } dV ¯e Ω

¯e Ω

"Z =

# T

[G] [S][G] dV δ{δ e } = [Kσe ] δ{δ e }

(10.59)

¯e Ω

Substituting from (10.55) and (10.59) into (10.54)   δ{P e } = [K e ] + [Kσe ] δ{δ e }

(10.60)

Furthermore e

Z

[K ] =

   l T [B ] + [B n ] [DT ] [B l ] + [B n ] dV

(10.61)

¯e Ω

= [Kle ] + [Kne ] in which [Kle ]

Z = ¯e Ω

[B l ]T [DT ][B l ] dV

(10.62)

(10.63)

635

10.4. SOLUTION METHOD

and [Kne ]

Z 

l T

n

n T

l

n T

n



[B ] [DT ][B ] + [B ] [DT ][B ] + [B ] [DT ][B ] dV

=

(10.64)

¯e Ω

[Kle ] is the usual small displacement stiffness matrix. [Kne ] is termed as initial displacement matrix by Marcal [10]. Using (10.62) in (10.60) we finally have   e e e e δ{P } = [Kl ] + [Kn ] + [Kσ ] δ{δ e } = [KTe ] δ{δ e } (10.65) Matrix [Kσe ] is influence of the stress field on the element stiffness. The derivation given above for finite deformation and finite strain has also been presented in references [6, 11–18]. [KTe ] is called element tangent stiffness matrix. [KTe ] is obviously symmetric if [DT ] is symmetric.

10.4 Solution method We note that we are faced with finding a solution {δ}, nodal degrees of ¯ T that satisfies freedom for discretization Ω    ψ({δ}) = R({δ}) − P ({δ}) = 0 (10.66) in which ψ(·) is a nonlinear function of {δ}. Thus, we must use (10.66) to find a solution {δ} iteratively. At this stage many strategies are possible. In the following we only discuss Newton’s linear method or Newton–Raphson method. Let {δ0 } be a known solution, a guess or generally a null vector or established using linear solution, then  ψ ({δ0 }) 6= {0} (10.67) Let {∆δ} be correction to {δ0 } such that  ψ({δ0 } + {∆δ}) = 0

(10.68)

Expand ψ(·) in (10.68) in Taylor series about {δ0 } and retain only up to linear terms in {∆δ} (Newton’s linear method).   ∂{ψ} ∼ ψ({δ0 } + {∆δ}) = ψ({δ0 }) + {∆δ} = 0 (10.69) ∂{δ} {δ0 } or 

∂{ψ} {∆δ} = − ∂{δ}

−1 {δ0 }

 ψ({δ0 })

(10.70)

636

LINEAR AND NONLINEAR SOLID MECHANICS

New solution {δ} is obtained using {δ} = {δ0 } + {∆δ}

(10.71)

{ψ} = {R} − {P }

(10.72)

Recall that If we assume that {R} is not a function of {δ} i.e the loads are conservative (maintain their direction during deformation process), then X ∂{ψ} ∂{P } [KTe ] =− = −[KT ] = − ∂{δ} ∂{δ} e Using (10.73) in (10.70) and then (10.70) in (10.71) X 
o  e  −1 n ψ {δ0 } {δ} = {δ0 } − KT e

(10.73)

(10.74)

{δ0 }


Convergence of the iterative process is checked by examining ψ {δ} for proximity to null vector.

10.4.1 Summary of solution procedure 1. Assume {δ0 }. 2. Calculate [KTe ], {Rbe }, {Rpe }, {P e }. 3. Assemble {R} =

X

{Rbe } + {Rpe }

and {P } =

X

{P e }

e

e

4. Form {ψ} = {R} − {P } 5. Check for convergence (a) (b) (c) (d)

|ψi | ≤ ∆1 , i = 1, . . . , n∗ , n∗ = total number of degrees of freedom ∆1 : preset tolerance for zero If converged: stop If not converged then follow steps 6-8

6. Calculate {∆δ} and {δ} using X   e  −1 n
o {∆δ} = − KT ψ {δ0 } e

{δ0 }

{δ} = {δ0 } + {∆δ}

10.5. FINITE ELEMENT FORMULATION FOR 2D SOLID CONTINUA

637

7. Calculate [KTe ], {Rbe }, {Rpe }, {P e } using {δ}. 8. Form {ψ} = {R} − {P } Set {δ0 } to {δ} and go to step 5.

10.5 Finite element formulation for 2D solid continua In this section we present details of the finite element formulation for isotropic, homogeneous 2D continua (plane stress or plane strain) in Lagrangian description for finite deformation. We follow the general derivation presented in earlier sections that holds in R1 , R2 and R3 and for any kinematic description. In this section we are seeking to establish specific forms of [N ], [B l ], [B n ], hence [B], {θ}, [Aθ ], [G], [Kle ], [KneS], [Kσe ], hence [KTe ] and [S] ¯ e a two-dimensional ¯T = for 2D continua. Consider a discretization Ω e Ω of e ¯ Let a typical element e with domain Ω ¯ = Ωe S Γe be a nine-node domain Ω. p-version element with distorted faces in xy space [see Fig. 10.1(a)]. A map ¯ ξη of this element (master element) in the natural coordinate space ξη in Ω ¯ ξη , master element a two-unit square is shown in Fig. 10.1(b). Mapping of Ω ¯ e in xy space can be described using the following (or any in ξη space into Ω other suitable form):

x(ξ, η) = y(ξ, η) =

9 X i=1 9 X

¯i (ξ, η) xi N (10.75) ¯i (ξ, η) yi N

i=1

¯i (ξ, η) are Lagrange family 2D shape functions of class C 00 (Ω ¯ e ). Let ue , v e N h h ξη ¯ be the local approximations of u and v over Ω : ueh = [N (ξ, η)]{ue } vhe

e

= [N (ξ, η)]{v e } e

(10.76)

638

LINEAR AND NONLINEAR SOLID MECHANICS η ¯e Ω

7

5

6

¯ ξη Ω

5

6

7

4

9

2

8

9

4

8 y

3

2 1

1

3

2

x

(a)

ξ

2 ¯ e in the natural coordi(b) Map of Ω ¯ ξη ) nate space ξ, η (i.e. Ω

Quadrilateral element in xyspace

¯ e and its map Ω ¯ ξη in natural coordinate space Figure 10.1: A quadrilateral element Ω ξ, η

where [N ] is (1 × n) approximation function matrix in which n = (pξ + e

1)(pn + 1); pξ and pη being p-levels in ξ and η directions. {ue } and {v e } are degrees of freedom for ueh and vhe . Equations (10.76) can be combined in matrix form as " #   "[N ] [0] # [N ] [0] {ue }  uh e δ e (2n×1) = e = e = [N ]{δ e } {φh } = [0] [N ] {v e } vh [0] [N ] e

(2×2n)

e

(10.77) where



e

{δ } =

 {ue } {v e }

(10.78)

We note that {δφeh } = [N ]δ{δ e }. Let ( e ) ∂uh     ∂xe            ∂vh   {θx } [Gx ] {ue } ∂x e {θ} = = ( ∂ue ) = [G][δ ] = h {θy } [Gy ] {v e }     ∂ye       ∂vh  

(10.79)

∂y

in which

 [Gx ] =

∂[N ]  ∂xe

[0]

   

[0] ∂[N ] e ∂x ∂ueh



 and



   

[Gy ] =

∂[N ]  ∂ye

[0]

[0] ∂[N ] e ∂y

 (10.80)



  e 2  e 2  ∂uh ∂v 1     + ∂xh   2 ∂x         

  ∂x  εxx  e ∂vh {ε} = εyy + 1 = ∂y      2  2εxy   ∂ueh + ∂vhe      ∂y ∂x

∂ueh ∂x

2

∂ueh ∂ueh ∂x ∂y

+

+

e ∂vh ∂y

e ∂v e ∂vh h ∂x ∂y

2

    

(10.81)

10.5. FINITE ELEMENT FORMULATION FOR 2D SOLID CONTINUA

639

or {ε} = {εl } + {εn }

(10.82)

{εl } = [H]{θ} = [H][G]{δ e } = [B l ]{δ e }

(10.83)

  1000 [H] = 0 0 0 1 0110

(10.84)

  ∂[N ] [0] e  ∂x ∂[N ]  l  [B ] =   0 ∂ye  ∂[N ] ∂[N ] e ∂xe ∂y

(10.85)

  {θx }T {0}T [Aθ ] =  {0}T {θy }T  {θy }T {θx }T

(10.86)

in which

Clearly

Let

Then, following the derivation in section 10.3 we can write 1 1 1 {εn } = [Aθ ]{θ} = [Aθ ][G]{δ e } = [B n ]{δ e } 2 2 2

(10.87)

[B n ] = [Aθ ][G]

(10.88)

δ{εl } = [B l ]δ{δ e }

(10.89)

where

n

n

e

δ{ε } = [B ]δ{δ }

(10.90)

  δ{ε} = [B]δ{δ e } = [B l ] + [B n ] δ{δ e }

(10.91)

Therefore Consider incremented form of the constitutive equations δ{σ} = [DT ] δ{ε} or

      D11 D12 0 σxx   εx  δ σyy = D12 D22 0  δ εy , γxy = 2εxy     σxy 0 0 D33 γxy

(10.92)

(10.93)

If we only consider linear elastic material, then Dij for plane stress and plane strain are given by [1, 2]

640

LINEAR AND NONLINEAR SOLID MECHANICS

Plane Stress: D11 = D22 =

E 1 − ν2

νE 1 − ν2 E = =G 2(1 + ν)

D12 = D33

(10.94)

Plane Strain: D11 = D22 =

E(1 − ν) (1 + ν)(1 − 2ν)

ν D11 1−ν E = =G 2(1 + ν)

D12 = D33

(10.95)

In which E is modulus of elasticity, ν is Poisson’s ratio, and G is shear modulus. Now we can determine [KTe ], tangent stiffness matrix Z e [KT ] = [B]T [DT ][B] dV + [Kσe ] (10.96) ¯e Ω

or [KTe ] =

Z



   [B l ]T + [B n ]T [DT ] [B l ] + [B n ] dV + [Kσe ]

(10.97)

¯e Ω

We use the element map in the natural coordinate system ξ, η [see Fig. 10.2(b)]. See chapter 8 for details. dV = t(ξ, η) det[J] dξ dη  " ∂x ∂x #  x, y ∂ξ ∂η [J] = = ∂y ∂y ξ, η ∂ξ ∂η ( ∂N i ) ( ∂N i ) e e ∂x ∂ξ = [J T ]−1 ∂N , i = 1, 2, . . . , n ∂N i i e e ∂y ∂η

(10.98) (10.99)

(10.100)

t(ξ, η) is thickness of the plate for plane strain case. In case of plane strain t(ξ, η) = 1. Now we can write [KTe ] in (10.96) as [KTe ] =

Z1 Z1



   [B l ]T + [B n ]T [DT ] [B l ] + [B n ] t(ξ, η) det[J] dξ dη + [Kσe ]

−1 −1

(10.101)

10.6. FINITE ELEMENT FORMULATION FOR 3D SOLID CONTINUA

641

[B l , [B n ] and [DT ] are defined in (10.85), (10.88) and (10.94), (10.95). We keep in mind that [KTe ] is to be calculated at {δ0e }, assumed solution, thus [Aθ ] used in (10.101) is defined. Calculations of load vectors due to body forces and conservative surface loads is rather straight forward. We also note that [KTe ] can be written as [KTe ] = [Kle ] + [Kne ] + [Kσe ]

(10.102)

in which [Kle ] =

Z1

[B l ]T [DT ][B l ] t(ξ, η) det[J] dξ dη

(10.103)

−1

[Kne ]

Z1 = [B l ]T [DT ][B n ]+[B n ]T [DT ][B l ] −1

+[B n ]T [DT ][B n ] t(ξ, η) det[J] dξ dη [Kσe ]

Z1 =

[G]T [S][G] t(ξ, η) det[J] dξ dη

(10.104) (10.105)

−1

and 

 σxx [I2 ] σxy [I2 ] [S] = σxy [I2 ] σyy [I2 ]



,

10 [I2 ] = 01

 (10.106)

10.6 Finite element formulation for 3D solid continua The details of the finite element formulation presented in section 10.4 clearly holds for 3D solid continua for finite deformation as well. Based on the choice of element shape, i.e. hexahedron, tetrahedron, etc, we can address the remaining details. Consider a 27-node distorted hexahedron element in x, y, z space [see Fig. 10.2(a)] mapped into a two-unit cube in ξ, η, ζ natural coordinate space with the origin of ξ, η, ζ coordinate system at the center of the element [see Fig. 10.2(b)]. Let  e   e  uh   {u }  {φeh } = vhe = [N (ξ, η, ζ)]{δ e } = [N ] {v e } ; {δφeh } = [N ]δ{δ e }  e  e e  e wh {we } (10.107) ¯ e with its map in be the local approximations of u, v, w over an element Ω natural coordinate space. See chapter 8 for details of [N (ξ, η, ζ)] functions. e

642

LINEAR AND NONLINEAR SOLID MECHANICS

Following the details of chapter 8 we have the following mapping of points between element mapped in x, y, z and ξ, η, ζ spaces. x(ξ, η, ζ) = y(ξ, η, ζ) =

27 X i=1 27 X

¯i (ξ, η, ζ) xi N ¯i (ξ, η, ζ) yi N

(10.108)

i=1

z(ξ, η, ζ) =

27 X

¯i (ξ, η, ζ) zi N

i=1

¯i (ξ, η, ζ) are Lagrange family of 3D shape functions, and [N i ] in where N e [G], [N ] can be any suitable choice such as p-version hierarchical. Details of e [Aθ ], [B l ], [B n ] etc. have already been presented in section 10.3. Calculation of element tangent matrix [KTe ] requires constitutive equations. Consider incremental form of the constitutive relations. δ{σ} = [DT ]δ{ε}

or δ{ε} = [CT ]δ{σ}

(10.109) η

ξ y

ζ x

z (b) Element map in natural coordinate space in ξ, η, ζ

(a) A 27-node element in x, y, z space

Figure 10.2: A 27-node three-dimensional hexahedron element

The components of {ε} and {σ} are arranged as in (10.11). The material matrix [CT ] for isotropic, homogeneous elastic matter can be written as   C11 C12 C13 0 0 0 C21 C22 C23 0 0 0    C31 C32 C33 0 0 0   [CT ] =  (10.110)  0 0 0 C44 0 0     0 0 0 0 C55 0  0

0

0

0

0 C44

10.6. FINITE ELEMENT FORMULATION FOR 3D SOLID CONTINUA

643

in which C21 = C12 , C31 = C13 , C32 = C23 and 1 E −ν = E

C11 = C22 = C33 = C12 = C13 = C23

(10.111)

and [DT ] = [CT ]−1 . The element tangent stiffness matrix [KTe ] is given by [KTe ] =

Z

[B]T [DT ][B] dV + [Kσ ]

(10.112)

¯e Ω

Considering the element map in ξ, η, ζ coordinate space dV = |J| dξ dη dζ and    xξ xη xζ x, y, z =  yξ yη yζ  [J] = ξ, η, ζ zξ zη zζ 

(10.113)

In (10.113) the subscripts ξ, η, ζ indicate differentiations with respect to these and since Ni = Ni (ξ, η, ζ) ; i = 1, 2, . . . , n we have  ∂N   ∂N  i i    e e      ∂x     ∂ξ ∂N i ∂N i T −1 = [J ] , i = 1, 2, . . . , n e e ∂y  ∂η   ∂N     i    ∂N i  e e ∂z ∂ζ

(10.114)

Finally, we have the following for the tangent stiffness matrix for element e, [KTe ] =

Z1 Z1 Z1

[B]T [DT ][B] |J| dξ dη dζ + [Kσ ]

(10.115)

−1 −1 −1

in which [B] = [B l ] + [B n ]

(10.116)

Also if we substitute (10.116) in (10.115), we obtain [KTe ] = [Kle ] + [Kne ] + [Kσe ]

(10.117)

The matrices in (10.116) and (10.117) are defined in section 10.3. Using the

644

LINEAR AND NONLINEAR SOLID MECHANICS

element map in ξ, η, ζ coordinate space we can define these as [Kle ]

Z1 Z1 Z1 =

[B l ]T [DT ][B l ] |J| dξ dη dζ

(10.118)

−1 −1 −1

[Kne ] =

Z1 Z1 Z1


[B l ]T [DT ][B n ] + [B n ]T [DT ][B l ] + [B n ]T [DT ][B n ] |J| dξ dη dζ

−1 −1 −1

(10.119) [Kσe ] =

Z1

Z1

Z1

[G]T [S][G] |J| dξ dη dζ

(10.120)

−1 −1 −1

(10.121) in which

  σxx [I] σxy [I] σxz [I] [S] = σxy [I] σyy [I] σyz [I] σxz [I] σyz [I] σzz [I]

(10.122)

where [I] is a (3 × 3) identity matrix.

10.7 Axisymmetric solid finite elements These finite element formulations can be used to study axisymmetric deformation in bodies of revolution due to axisymmetric loads. The deformation field in such cases is independent of the circumferential coordinate θ. In the following we choose x to be radial direction r and y to be axial direction z of a (r, θ, z) cylindrical coordinate system. Figure 10.3 shows a nine-node p-version element in x, y space. Let u, v be the displacements in ¯ e or P (ξ, η) in Ω ¯ ξη . In the following we x, y directions at a point P (x, y) in Ω consider finite element formulation of axisymmetric finite elements for finite deformation in Lagrangian description. Mapping of geometry from x, y to ξ, η space is as usual.   X   9 x xi ¯ = N (ξ, η) (10.123) y yi i=1

¯ (ξ, η) are standard 2D Lagrange interpolation functions. Consider an eleN ¯ e = Ωe ∪Γe in which Γe is the boundary of the element. ment e with domain Ω The local approximation {φeh } of the displacement field {φ}T = [u v] over ¯ e (or Ω ¯ ξη ) can be written using p-version hierarchical local approximation Ω (see chapter 8).  e  e    e  uh {u } [N ] [0] {u } e e {φh } = = [N ]{δ } = [N ] = e (10.124) e e vh [0] [N ] {v e } e e {v } e

645

10.7. AXISYMMETRIC SOLID FINITE ELEMENTS

and δ{φeh } = [N ]δ{δ e } e [N ] is the approximation function matrix and [N i ] are local approximation e e functions (N i (ξ, η); i = 1, 2, . . . , n) used in defining sub-matrices of [N ]. e e The Jacobian of mapping is given by [J] =

h

x,y ξ,η

i



x x = ξ η yξ yη

 (10.125)

and ( ∂N i ) e ∂x ∂N i e ∂y

= [J T ]

( ∂N i ) e ∂ξ −1 ∂N i e ∂η

, i = 1, 2, . . . , n

η

y

(10.126)

v

(axial z) P (ξ, η)

u P (x, y) ξ

¯e Ω

x (radial r)

Figure 10.3: A nine-node p-version element

The Green’s strain tensor {ε} can be written as     1  ∂ue
2 ∂ueh
2  h ∂ueh  +       2  ∂x ∂y    ∂x      εxx      e
2 e
2  e ∂v ∂v       ∂vh h h      1 + εyy 2 ∂x ∂y ∂y {ε} = = ∂ue + = {εl } + {εn } e e e e e ∂vh ∂uh ∂vh ∂uh ∂vh h γ       xy +    ∂x + ∂y  ∂y            ∂x ∂x e ∂y  2    εθ ueh u      1 h   x 2

x

(10.127) in which εθ is the circumferential strain. Let {σ}T = [σxx , σyy , σxy , σθ ] be the second Piola-Kirchhoff stress tensor. We consider incremental stressstrain relations between {σ} and {ε} δ{σ} = [DT ]δ{ε}

(10.128)

646

LINEAR AND NONLINEAR SOLID MECHANICS

where

 D11 D21 [DT ] =   0 D41

D12 D22 0 D22

0 0 D33 0

 D14 D24   0  D44

in which D21 = D12 , D41 = D14 , D42 = D24 .  ν 1 1−ν 0   E(1 − ν) 1 0  [DT ] = 1−2ν (1 + ν)(1 − 2ν)   sym. 2(1−ν) Let

(10.129)



ν 1−ν  ν  1−ν 

(10.130)

 0  1

  ( ∂ue ) ( ∂ue )   {θx } h h ∂x ∂y {θ} = {θey } ; {θx } = ∂ve ; {θy } = ∂v e h h    uh  ∂x ∂y

(10.131)

x

and

  [Gx ] {θ} = [G]{δ e } = [Gy ] {δ e } [G3 ]

(10.132)

in which  [Gx ] =

∂[N ]  ∂xe

[0]

 [Gy ] = [G3 ] =

∂[N ]  ∂ye

h

[0] ∂[N ] e ∂x

[0]

[0]

∂[N ] e ∂y

1 x [N ]

i [0]

 

(10.133)

 

(10.134) (10.135)

e Using (10.127) and (10.132) we can write {εl } as {εl } = [H]{θ} = [H][G]{δ e } = [B l ]{δ e }   10000 0 0 0 1 0  [H] =  0 1 1 0 0 00001

(10.136)

(10.137)

Clearly [B l ] is given by  ∂[N ]  [0] e  ∂x ∂[N ]   [0] e  ∂y [B l ] =   ∂[N ] ∂[N ]   ∂ye ∂xe  1 x [N ] [0] e

(10.138)

647

10.7. AXISYMMETRIC SOLID FINITE ELEMENTS

Let   {θx }T {0}T 0 [Aθ ] =  {0}T {θy }T 0  ueh 0 0 x

(10.139)

Following the derivation in section 10.3 we can write 1 1 1 {εn } = [Aθ ]{θ} = [Aθ ][G]{δ e } = [B n ]{δ e } 2 2 2

(10.140)

where [B n ] = [Aθ ][G]

(10.141)

and δ{εl } = [B l ] δ{δ e } n

n

e

(10.142)

δ{ε } = [B ] δ{δ }

(10.143)

  δ{ε} = [B] δ{δ e } = [B l ] + [B n ] δ{δ e }

(10.144)

Therefore

Element tangent stiffness matrix [KTe ] is given by [KTe ]

Z =

[B]T [DT ][B] dV + [Kσe ]

(10.145)

¯e Ω

We note that for axisymmetric solids dV = 2πx dx dy = 2πx|J| dξ dη

(10.146)

where 

   x, y xξ xη [J] = = yξ yη ξ, η ( ∂N i ) e ∂x ∂N i e ∂y

( ∂N i ) e ∂ξ = [J T ]−1 ∂N ; i e ∂η

i = 1, 2, . . . , n

(10.147)

(10.148)

If we use [B] = [B l ] + [B n ] in (10.145), then we can write [KTe ] = [Kle ] + [Kne ] + [Kσe ]

(10.149)

648

LINEAR AND NONLINEAR SOLID MECHANICS

in which [Kle ]

Z1 Z1 =

[B l ]T [DT ][B l ] 2π x(ξ, η) |J| dξ dη

(10.150)

−1 −1

[Kne ]

Z1 Z1 = [B l ]T [DT ][B n ]+[B n ]T [DT ][B l ] −1−1


+[B n ]T [DT ][B n ] 2π x(ξ, η) |J| dξ dη [Kσe ] =

Z1

Z1

(10.151)

[G]T [S][G] 2π x(ξ, η) |J| dξ dη

(10.152)

    σxx [I2 ] σxy [I2 ] 0 10   [S] = σxy [I2 ] σyy [I2 ] 0  ; [I2 ] = 01 0 0 σθ

(10.153)

−1 −1

in which

10.8 Summary The finite element formulations are derived in this chapter using principle of virtual work. These formulations are valid for solid matter undergoing finite deformation and finite rotations. Incremental form of the constitutive theory between second Piola-Kirchhoff stress and the Green’s strain tensor yields symmetric tangent stiffness matrix when Newton’s linear method is employed for obtaining solutions of non-linear algebraic equations. Even though the derivation is presented for elastic material, its extension to plasticity is quite straight forward. This eventually results in modification of [DT ]. In case of small deformation {ε} is replaced by {εl } and the second Piola-Kirchhoff stress is simply Cauchy stress, hence [KTe ] only consists of [Kle ] if the influence of stress field on the stiffness defined by [Kσe ] is neglected as done in linear elasticity. References [6, 11–17] provide additional material for further reading on various finite element formulations such as beams, axisymmetric shells, curved shells, etc. for finite deformation, finite rotations and finite strains. These references also contain many model problems and their numerical solutions using the finite element formulations presented in them that are parallel to the ones considered here. The finite element formulations that incorporate large deformation, finite rotations and finite strains and are based on second Piola-Kirchhoff stress and Green’s strain tensor with incremental elastic constitutive equations (like the ones presented here) are generally referred to as geometrically nonlinear finite element formulations in the published literature.

REFERENCES FOR ADDITIONAL READING

649

References for additional reading [1] J. N. Reddy. An Introduction to Continuum Mechanics. Cambridge University Press, 2nd edition, 2013. [2] K. S. Surana. Advanced Mechanics of Continua. CRC/Taylor and Francis, 2015. [3] M. Gelfand and S. V. Fomin. Calculus of Variations. Dover Publications, 2000. [4] G. Mikhlin. Variational Methods in Mathematical Physics. Pergamon Press, 1964. [5] J. N. Reddy. Applied Functional Analysis and Variational Methods in Engineering. McGraw Hill Company, 1986. [6] J. N. Reddy. An Introduction to Nonlinear Finite Element Analysis. Oxford University Press, 2nd edition, 2015. [7] J. N. Reddy. Energy Principles and Variational Methods in Applied Mechanics. John Wiley, New York, (3rd in print) 2nd edition, 2002. [8] J. T. Oden. Numerical solution of nonlinear plasticity problems. J. Struct. Div., 93:235–255, 1967. [9] J. T. Oden. Finite plane stress of incommpressible elastic solids by finite element method. The Aero. Quart., 19:254–264, 1968. [10] P. V. Marcal. The effect of initial displacements on problems of large deflection and stability. Technical Report ARPA E54, Brown University, November 1967. [11] K. S. Surana. Geometrically non-linear formulations for the axi-symmetric shell elements. Int. J. Num. Meth. Eng., 18:447–502, 1982. [12] K. S. Surana. Geometrically non-linear formulations for the axi-symmetric transition finite elements. Comp. Struct., 17(2):243–255, 1983. [13] K. S. Surana. Geometrically non-linear formulations for the three-dimensional solidshell transition finite elements. Comp. Struct., 15(5):549–566, 1982. [14] K. S. Surana. Geometrically non-linear formulations for the two-dimensional curved beam elements. Comp. Struct., 17(1):105–114, 1983. [15] K. S. Surana. Geometrically non-linear formulations for the curved shell elements. Int. J. Num. Meth. Eng., 19:581–615, 1983. [16] K. S. Surana. Geometrically non-linear formulations with large rotations for finite elements with rotational degrees of freedom. Comp. Struct., 23(2):279–289, 1986. [17] K. S. Surana. Geometrically non-linear formulations for the three-dimensional curved beam elements with large rotations. Int. J. Num. Meth. Eng., 1987. [18] K. S. Surana and R. M. Sorem. Three dimensional curved beam elements with large translations and rotations. Presented at 12th Canadian Congress of Applied Mechanics, May 1989.

11

Additional Topics in Linear Structural Mechanics 11.1 Introduction In this chapter we consider various finite element formulations for linear structural mechanics in R1 , R2 , and R3 . The mathematical descriptions for structural members are momentum equations in Lagrangian description, often referred to as equations of equilibrium as these are statement of force balance in R1 , R2 , and R3 . The constitutive theories describing relations between Cauchy stress tensor and infinitesimal strain are generally based on generalized Hooke’s law. In case of linear structural mechanics the differential operators appearing in the mathematical descriptions are always self-adjoint, hence Galerkin method with weak form, principle of minimum potential energy, or Ritz method including principle of virtual work are all ideally suited to derive finite element formulations for such applications. The outcomes from each one of these methods are identically the same in terms of the resulting finite element formulations and the associated element equations. Generally it is a matter of choice and preference as to which method one chooses. In this chapter we explore applications of some of these methods in deriving finite element formulations i.e. element equations for structural elements in R1 , R2 , and R3 and consider some simple model problems.

11.2 1D axial spar or rod element in R1 (1D space) In this section we consider finite element formulations for purely one dimensional deformation of bars or rods in R1 . The resulting finite elements are called bar, rod, or spar elements in R1 . The governing differential equations describing stationary 1D deformation is the momentum equation in the axial direction. Considering spatial direction x, the following BVP describes the axial deformation of a bar or a rod in R1 in the absence of body forces:   du(x) d E − f (x) = 0 (11.1) dx dx In which E(x) is modulus of elasticity, u(x) is displacement, and f (x) is applied external load along the length of the rod. 651

652

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

Consider a 1D rod of length L subdivided into four two-node elements [see Fig. 11.1(a)]. The element equations can be derived using GM/WF, minimization of total potential energy (quadratic functional), or principle of virtual work, the final outcome is the same. Consider an element e [see Fig. 11.1(b)] showing nodal displacements ue1 and ue2 , nodal internal forces (secondary variables) P1e and P2e at local node numbers 1 and 2. y

1

2

1

3

2

4

3

4

5

P

x

f (x) (a) Finite element discretization of 1D rod

y

P1e

ue1 1 xe

e

ue2 2 P2e xe+1

f (x) he

x

(b) A two node 1D element Figure 11.1: Discretization of a 1D rod and a typical 2 node element e

We derive the element equations using principle of minimum potential energy (or minimization of quadratic functional). In the derivations we consider a local coordinate system x ¯, y¯ with its origin at node 1 and the x ¯ axis pointing from node 1 to node 2 (see Fig. 11.2). y

y¯ e x

P¯1e

u¯e1

u¯e2

1 f (¯ x) he

2 P¯2e

x¯

Figure 11.2: Element local coordinate system x ¯y¯ and nodal variables

Figure 11.2 shows an element e with local node numbers 1 and 2 with nodal displacements u ¯e1 and u ¯e2 and internal forces P¯1e and P¯2e . The local approximation u ¯eh (¯ x) describing u ¯ at a point x ¯ in the element can be written

11.2. 1D AXIAL SPAR OR ROD ELEMENT IN R1 (1D SPACE)

as u ¯eh =

2 X

¯i (¯ ¯] N x)¯ uei = [N



i=1

ue1 ue2

653

 (11.2)

¯i (¯ where N x) (i = 1, 2) are nodal local approximation functions and are given by (see Chapter 8)   x ¯ x ¯ ¯2 (¯ ¯ , N x) = (11.3) N1 (¯ x) = 1 − he he We note that

¯1 (¯ ¯1 (¯ N x1 ) = 1, N x2 ) = 0 ¯2 (¯ ¯2 (¯ N x1 ) = 0, N x2 ) = 1

That is

( 1, j = i ¯i (¯ N xj ) = 0, j = 6 i

, i, j = 1, 2

(11.4)

(11.5)

¯ ] is local approximation function matrix. and [N

11.2.1 Stresses and strains For purely one dimensional deformation of the element we have εx¯x¯ =

d¯ u , σx¯x¯ = Eεx¯x¯ d¯ x

(11.6)

In matrix and vector notation 

   d¯ u d ¯ u} {¯ ε} = {εx¯x¯ } = = {¯ u} = [L]{¯ d¯ x d¯ x ¯ ε} ; [D] ¯ = [E] {¯ σ } = {σx¯x¯ } = [D]{¯

(11.7)

where E is modulus of elasticity.

11.2.2 Total potential energy: Πe The total potential energy Πe of an element e is the sum of strain energy and the potential energy of loads, i.e. Πe = Πe1 + Πe2

(11.8)

where Πe1 is the strain energy and Πe2 is the potential energy of loads: Πe1

1 = 2

Z ¯e Ω

1 {¯ ε} {¯ σ } dΩ = 2 T

Zhe {¯ ε}T {¯ σ }A d¯ x 0

(11.9)

654

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

¯ e = [0, he ], the domain in which A is area of the cross-section of the rod and Ω of an element e, Πe2 = work done by P¯1e , P¯2e , and f (¯ x) or Πe2

=

−¯ ue1 P¯1e

−

u ¯e2 P¯2e

Zhe u ¯(¯ x)f (¯ x) d¯ x

−

(11.10)

0

Using (11.9) and (11.10) in (11.8) 1 Π = 2 e

Zhe Zhe T e ¯e e ¯e ¯(¯ x)f (¯ x) d¯ x {¯ ε} {¯ σ }A d¯ x−u ¯ 1 P1 − u ¯ 2 P2 − u

(11.11)

0

0

Substituting for {¯ ε} and {¯ σ } from (11.7) and replacing u ¯(¯ x) with u ¯eh (local approximation for element e) 1 Π = 2 e

Zhe 

d¯ ueh d¯ x

T

 E

d¯ ueh d¯ x



A d¯ x−u ¯e1 P¯1e − u ¯e2 P¯2e −

0

Zhe u ¯(¯ x)f (¯ x) d¯ x (11.12) 0

x) from (11.3) Substituting u ¯eh (¯ 1 Π = 2 e

  e T  ¯   e  Zhe  ¯ dN1 dN¯2 dN1 dN¯2 u ¯1 u ¯1 E A d¯ x e u ¯2 u ¯e2 d¯ x d¯ x d¯ x d¯ x 0

−

u ¯e1 P¯1e

−

u ¯e2 P¯2e

Zhe ¯1 u ¯2 u ¯e1 + N ¯e2 )f (¯ x) d¯ x (11.13) − (N 0

or  ¯ ¯ ¯1 dN ¯2 dN1 dN1 dN Zhe  d¯ 1 x d¯ x d¯ x d¯ x ¯e1 u ¯e2 ]  Πe = [u  ¯2 dN ¯1 dN ¯2 dN ¯2 2 dN 0 d¯ x d¯ x d¯ x d¯ x −

u ¯e1 P¯1e

−

u ¯e2 P¯2e

   



u ¯e1 u ¯e2

 EA d¯ x

Zhe ¯1 u ¯2 u − (N ¯e1 + N ¯e2 )f (¯ x) d¯ x (11.14) 0

11.2. 1D AXIAL SPAR OR ROD ELEMENT IN R1 (1D SPACE)

655

or  1 ¯e1 u ¯e2 ]  Πe = [ u 2

Zhe 0

¯1 ¯1 dN dN  d¯ x  x d¯  ¯ ¯ dN2 dN1 d¯ x d¯ x 

 ¯1 dN ¯2  dN  e  u ¯1 d¯ x d¯ x   EA d¯ x   u ¯e2 ¯ ¯ dN2 dN2 d¯ x d¯ x Zhe e ¯e e ¯e ¯1 u ¯2 u ¯e1 + N ¯e2 )f (¯ x) d¯ x (11.15) −u ¯ 1 P1 − u ¯ 2 P2 − (N 0

Since Πe = Πe (¯ ue1 , u ¯e2 ), minimization of total potential energy implies ∂Πe =0 ∂u ¯e1

and

∂Πe =0 ∂u ¯e2

(11.16)

Using (11.16) and (11.15) and writing the results in the matrix and vector form, we obtain   ¯ ¯ ¯1 dN ¯2   dN1 dN1 dN  e Zhe   u  x d¯ ¯1 x d¯ x d¯ x   d¯  EA  d¯  x    ¯ ¯ u ¯e2 ¯2 dN ¯2 dN2 dN1 dN 0 d¯ x d¯ x d¯ x d¯ x   he R    ¯1 (¯   x)f (¯ x) d¯ x  N   ¯e  P 0 = he (11.17) + ¯1e R P2     ¯  x)f (¯ x) d¯ x  N2 (¯  0

Equations (11.17) are the force equilibrium equations for element e. By ¯2 ¯1 1 dN 1 dN ¯1 (¯ ¯2 (¯ = − , = into (11.17) and substituting N x), N x) and d¯ x he d¯ x he carrying out the integration, we obtain the following equations for element e (assuming E and A are constant over the length of the element and are Ee and Ae ):   e   e  e Ae Ee 1 −1 u ¯1 P¯ f¯ = ¯1e + ¯1e (11.18) e −1 1 f2 P2 u ¯2 he We note that the local coordinate system x ¯, y¯ is due to pure translation of the x, y coordinate system hence ue1 , ue2 , P1e , P2e , f1e , f2e in the x, y coordinate system are same as u ¯e1 , u ¯e2 , P¯1e , P¯2e , f¯1e , f¯2e in the x ¯, y¯ coordinate system, hence (11.18) in the x, y coordinate system (fixed global cartesian frame) can be written as   e   e  e Ae Ee 1 −1 u1 f1 P1 = + (11.19) e e −1 1 u f P2e he 2 2 This is same as (5.62) in (5.59) using Galerkin method with weak form. Numerical studies using (11.19) for a rod fixed at one end are presented in section 5.2 (example 5.2.1), hence are not repeated here.

656

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

11.3 1D axial spar or rod element in R2 (2D space) In this section we consider 1D axial rod or spar elements in R2 or in 2D space. Figure 11.3 shows a one dimensional rod or spar element in two dimensional space x, y. y

x¯ 2

y¯

u ¯e2 P¯2e f¯e

e

2

1 v

θ u ¯e1 P¯1e f¯e 1

x u Figure 11.3: 1D spar in R2 (2D space)

In the local coordinate system x ¯, y¯ the element of Fig. 11.3 is purely axial (following section 11.2), hence we can write the following: ¯ e ]{¯ [K ue } = {f¯e } + {P¯ e }

(11.20)

or 

¯e K ¯e K 11 12 ¯e K ¯e K 21 22



u ¯e1 u ¯e2



 =

f¯1e f¯2e



 +

P¯1e P¯2e

 (11.21)

or   e   e  e Ae Ee 1 −1 u ¯1 f¯ P¯ = ¯1e + ¯1e e f2 −1 1 u ¯2 P2 he

(11.22)

11.3.1 Coordinate transformation Consider x ¯, y¯ coordinate system obtained by rotating x, y coordinate system through an angle θ (see Fig. 11.4). From simple geometry, coordinate x, y of a point P¯ can be expressed in terms of its coordinates x ¯, y¯ in the x ¯y¯-frame.

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2

657

y y¯ P (¯ x, y¯) y ¯ sin θ

x¯ ¯ cos θ θ y y ¯ x ¯ θ x ¯ cos θ

x ¯ sin θ x

Figure 11.4: Coordinate transformation

x=x ¯ cos θ − y¯ sin θ y=x ¯ sin θ + y¯ cos θ

(11.23)

In matrix and vector form we can write (11.23) as        x cos θ − sin θ x ¯ x ¯ = = [Re ] y sin θ cos θ y¯ y¯ or

(11.24)     x ¯ x e T = [R ] y¯ y

We note that [Re ]−1 = [Re ]T since [Re ] is orthogonal. Using (11.24), we can transform displacements and forces from x ¯y¯-frame to xy-frame.     u u ¯ e = [R ] v v¯     fx f¯ e = [R ] ¯x fy fy

(11.25)

Let ue1 , v1e and ue2 , v2e be the displacements of nodes 1 and 2 in the x, y e , P e and P e , P e be secondary variables at coordinate system. Let P1x 1y 2x 2y e , f e and f e , f e be the forces at nodes 1 and 2 in the xy-frame and f1x 1y 2x 2y nodes 1 and 2 due to f (x) acting along the length of the element. If (x1 , y1 ) and (x2 , y2 ) are the coordinates of the element nodes in the x, y coordinate system, then p he = (x2 − x1 )2 + (y2 − y1 )2 (11.26)

658

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

If l, m are the direction cosines of the line segment constituting element e i.e. x ¯ axis, then l = cos θ =

x2 − x1 y2 − y1 , m = sin θ = he he

Thus for an element e we can write [Re ] as   y2 − y1 x2 − x1     − cos θ − sin θ l −m   [Re ] = = =  y2 h−e y1 x2 −hex1  sin θ cos θ m l he he

(11.27)

(11.28)

Hence, using global coordinates (x1 , y1 ) and (x2 , y2 ) of the two nodes of the element the coordinate transformation matrix [Re ] is defined. Using [Re ] and (11.25) we can transform the displacements and the forces at nodes 1 and 2 from x ¯, y¯ local coordinate system to x, y global coordinate system. At node 1:  e    e l u ¯1 u1 e e u ¯e1 = {te }¯ ue1 = = [R ] {δ1 } = m 0 v1e  e   e   P1x l P¯1 e e P¯1e = {te }P¯1e {P1 } = = [R ] = (11.29) e P1y m 0  e   e   f1x l f¯1 e e = [R ] {f1 } = = f¯1e = {te }f¯1e e f1y 0 m Similarly, at node 2: 

ue2 v2e



= {te }¯ ue2  e  P2x {P2e } = = {te }P¯2e e P2y  e  f2x e {f2 } = = {te }f¯2e e f2y {δ2e }

=

(11.30)

From (11.29) and (11.30) we can write the following using the property that [te ]T [te ] = 1: u ¯e1 = {te }T {δ1e }, P¯1e = {te }T {P1e }, f¯1e = {te }T {f1e } u ¯e2 = {te }T {δ2e }, P¯2e = {te }T {P2e }, f¯2e = {te }T {f2e }

(11.31)

Returning to the element equations (11.21) (in symbolic form) in x ¯y¯ coordinate system, we can write e ¯e u ¯ e ¯e = f¯e + P¯ e K 11 ¯1 + K12 u 2 1 1 e e e e e ¯ ¯ ¯ ¯ K21 u ¯1 + K22 u ¯2 = f2 + P2e

(11.32)

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2

659

Substituting for {¯ ue1 } and {¯ ue2 } from (11.31) in (11.32) e e ¯ 11 ¯ 12 K {te }T {δ1e } + K {te }T {δ2e } = f¯1e + P¯1e e e ¯ 21 ¯ 22 K {te }T {δ1e } + K {te }T {δ2e } = f¯2e + P¯2e

(11.33)

Premultiply (11.33) by {te } ¯ e {te }T {δ e } + {te }K ¯ e {te }T {δ e } = {te }f¯e + {te }P¯ e {te }K 11 1 12 2 1 1 e ¯e e T e e ¯e e T e e ¯e e ¯e {t }K21 {t } {δ1 } + {t }K22 {t } {δ2 } = {t }f2 + {t }P2

(11.34)

Using (11.29) and (11.30) for the secondary variables in (11.34) and combining both equations in (11.34) in the matrix and vector form,  e e e T e e e T e   e   e  ¯ {t } {t }K ¯ {t } {t }K {δ1 } {f1 } {P1 } 11 12 = + (11.35) e e e T e e} e} ¯ ¯ e {te }T {t }K21 {t } {t }K {δ {f {P2e } 22 2 2 or [K e ]{δ e } = {f e } + {P e }

(11.36)

where [K e ] is the stiffness matrix of a 1D spar in xy-space in R2 , {δ e }T = [{δ1e }T {δ2e }T ] = [ue1 v1e ue2 v2e ] is a vector of nodal displacements in global xy coordinate system, {f e } is a vector of equivalent nodal forces, {P e } is a vector of secondary variables (internal nodal forces) at the element nodes, and [K e ] is often called global element stiffnes matrix: e e e e f1y f2x f2y ] {f e }T = [{f1e }T {f2e }T ] = [{te }T f¯1e {te }T f¯2e ] = [f1x

likewise e e e e {P e }T = [{P1e }T {P2e }T ] = [{te }T P¯1e {te }T P¯2e ] = [P1x P1y P2x P2y ]

Remarks. (1) The two dimensional spar elements in their own local coordinate system x ¯, y¯ can only experience purely axial deformation and loads. (2) When such elements are members of two dimensional pin connected structures, then due to application of external loads these members rotate relative to each other such that in the final equilibrium state the loads in the members are purely axial.

11.3.2 A two member truss Example 11.1. Consider a two member truss shown in Fig. 11.5. All members are pin connected. Grid points 1 and 2 are constrained from translations in x and y directions but the members connected to them are free to rotate

660

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

and the grid point 3 is free to move in x and y directions. The members at grid point 3 are free to rotate relative to each other. A load of 10 tons is applied at grid point 3 in the negative y-direction. The areas of cross-section A1 and A2 of the members are 2 and 4 square inches and the modulus of elasticity E of both members is 30 × 106 psi. The dimensions of the truss are also shown in Fig. 11.5(a). y y¯1 2

8’

1

A1

x ¯2

3

8’

x

A2

x ¯1

y¯2

8’

10 tons

2

1

(a) Schematic of the two member truss

(b) Local coordinate systems

Figure 11.5: A two member truss and member local coordinate systems

Figure 11.5(b) shows local coordinate systems x ¯1 , y¯1 and x ¯2 , y¯2 of the two members of the truss. Figure 11.6 shows the displacements and the forces at the two nodes of each member in their local coordinate system as well as in the xy global coordinate system. Our objective is to determine deflection u3 and v3 of grid point 3, reactions at grid points 1 and 2, and the axial forces in the two members of the truss. Figure 11.6(a) shows the two members of the truss with nodal displacements and forces in their own local coordinate systems (¯ x1 , y¯1 ) and (¯ x2 , y¯2 ). Figure 11.6(b) shows the forces and displacements at the nodes of the members in global xy coordinate system. 11.3.2.1 Computations The coordinates of the grid points are

√ (x1 , y1 ) = (0, 4), (x2 , y2 ) = (0, −4), (x3 , y3 ) = ( 48, 0)

The direction cosines l1 , m1 of element 1 and l2 , m2 of element 2 are √ √ x3 − x2 48 − 0 48 y3 − y2 0−4 1 l1 = = = , m1 = = =− h1 8 8 h1 8 2 (11.37) √ √ x3 − x1 48 − 0 48 y3 − y1 0+4 1 l2 = = = , m2 = = = h2 8 8 h2 8 2

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2

y¯1

661

1 1 v2 , P2y , f2y 1 1 u2 , P2x , f2x

1

1

u ¯11 P¯11

1 1 u3 , P3x , f3x

x ¯1 u ¯12 P¯21 x ¯2 u ¯22 P¯22

2

y¯2

1 1 v3 , P3y , f3y

2 2 v3 , P3y , f3y 2 2 u3 , P3x , f3x 2 2 v1 , P1y , f1y

u ¯21 P¯ 2

2

1

2 2 , f1x u1 , P1x

(a) Displacements and forces in the element local coordinate systems

(b) Displacements and forces in x, y global coordinate system

Figure 11.6: Element local and global displacements and forces

Therefore {t1 } =



l1 m1

 =

√   48  8

 −1  2

, {t2 } =



l2 m2

 =

√   48  8



1 2

(11.38)



A1 E1 A2 E2 = 7.5 × 106 lb /f t , = 15 × 106 lb /f t (11.39) h1 h2 Element equations for elements 1 and 2 in their own local coordinate systems:  1  1 A1 E1 ¯ 1 u ¯1 P¯ = ¯11 [K ] 1 u ¯2 P2 h1 (11.40)  2  2 A2 E2 ¯ 2 u ¯1 P¯1 [K ] = u ¯22 P¯22 h2 ¯ 1 ] and [K ¯ 2 ] from (11.18) Using (11.39) and [K   1  1 1 −1 u ¯1 P¯ 6 7.5 × 10 = ¯11 ; 1 −1 1 u ¯2 P2   2  2 1 −1 u ¯1 P¯ 15 × 106 = ¯12 ; 2 −1 1 u ¯2 P2

element 1 (11.41) element 2

The element equations in the global coordinate system can be obtained using ¯e = K ¯ e = 1 and K ¯e = K ¯ e = −1 for e = 1 and 2, the (11.35). Since K 11 22 12 21

662

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

element equations (11.35) reduce to    1   P2x   u2    1 1 T     P1   1 1 T {t }{t } −{t }{t } v2 6 2y 7.5 × 10 = , element 1 (11.42) −{t1 }{t1 }T {t1 }{t1 }T  u   P1     3    3x  1 v3 P3y and

15 × 106



   2  P  u1            P1x 2  2 2 T 2 2 T {t }{t } −{t }{t } v1 1y = , element 2 (11.43) 2 u3  −{t2 }{t2 }T {t2 }{t2 }T  P3x          2  v3 P3y

 {t1 }{t1 }T =  −

3 4 √

3 4

√

3 4

− 1 4





 , {t2 }{t2 }T =

3  √4 3 4

√

3 4

1 4

 

(11.44)

Using (11.44) in (11.42) and (11.43) we have 

3 4 √

 − 3  4 6 7.5 × 10  3  −4  √ 3 4

√

−

3 4

1 4 √ 3 4 − 14

√



3  4  √  u2  3 1   − 4   v2  4 √  3  u3   − 43   4   v3  √ − 43 14

− 43

 1  P      P2x 1  2y = ; 1  P3x    1   P3y

element 1 (11.45)

and 

15 ×

3  √4  3  4 106   −3  4  √ − 43

√

3 4 1 4 √

3 4 − 14

−

√

−

3 4

3 4 √ 3 4

√



3  4  u  1    − 14    v1  √  3   u3   4    v3  1 4

− 43 −

 2     P1x   2  P1y = , element 2 2 P3x      2  P3y

(11.46)

Let {δ}T = [u1 v1 u2 v2 u3 v3 ] be the arrangement of the degrees of freedom at grid points 1, 2, and 3, then the element equations (11.45) and (11.46) can be assembled into the following system of linear simultaneous

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2

equations in {δ}. 

663

 √ .. . − 23 − 23   √    .. 3 1 1   u1   0 0 . − −    2 2 2    v1   √ . √      3 . 3 3 3   0 − . −   0   u 4 4 4 4   2   √ √ 6  . v 7.5 × 10  3  2  0 − 43 14 .. − 14 4   0 ...      .     . . . . . . . . . . . . ..   . . . . . .  u 3       √ √ √ √  3 3 .. 3 3 3 3 v 3   − 2 − 23 − 43 . + − + 4 4 2 4 2   √ √ √ √ 3 3 3 3 1 1 1 .. 1 − 2 −2 −4 . − 4 + 2 4 4 + 2   2   P1x         2   P   1y       1   P   2x 1 = (11.47) P2y         ...      1 2   P3x + P3x       P1 + P2   3 2 √ 3 2

√

3 2

0

0

3y

3y

Boundary Conditions: (a) Essential BCs u1 = 0, v1 = 0 u2 = 0, v2 = 0

(11.48)

(b) Natural BCs 1 1 2 2 P1x , P1y , P2x , P2y : unknown 1 2 P3x + P3x = 0 : no external load at grid point 3 in x direction 1 2 P3y + P3y = −20000lbs : external load

(11.49) Partition equations (11.47). Let {δ1 }T = [u1 v1 u2 v2 ], {δ2 }T = [u3 v3 ] 1 1 2 2 1 2 1 2 {P1 }T = [P1x P1y P2x P2y ], {P2 }T = [P3x + P3x , P3y + P3y ]

Then, (11.47) can be written as      [K11 ] [K12 ] {δ1 } {P1 } = [K21 ] [K22 ] {δ2 } {P2 }

(11.50)

(11.51)

664

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

Using (11.48) and (11.49) in (11.50) and then using (11.50) in (11.47) we obtain [K11 ]{δ1 } + [K12 ]{δ2 } = {P1 } (11.52) [K21 ]{δ1 } + [K22 ]{δ2 } = {P2 } In which (from (11.47)) 

3  √2  3 6 2

[K11 ] = 7.5 × 10  0  0

√

3 2 1 2

0

0

3 4 √

0

0 −

3 4

0

(11.53)





  √ − 3 0   2 6 √ , [K12 ] = 7.5 × 10  3 3  − 4  − 4 √ 1 4



√

√

3 4

√



3 2  − 21   √  3  4  − 41

− 32 −



− 32 − 23 − 34 43  √ [K21 ] = [K12 ]T = 7.5 × 106  √3 − 2 − 12 43 − 41  √ √  3 3 3 3 + − + 4 2 4 2 6 √  √ [K22 ] = 7.5 × 10 1 1 − 43 + 23 4 + 2 (11.54) From (11.53) {δ2 } = [K22 ]−1 {P2 } − [K21 ]{δ1 }




(11.55)

Therefore    −1      0 0 2.25 0.433023 u3 −6 (11.56) − = 7.5 × 10 0 −20000 0.433023 0.75 v3       0.0092376” u3 0.00076980 (11.57) = ∴ = −0.048” −0.0040 v3 and  1  P      P1x 1  1y {P1 } = = [K11 ]{δ1 } + [K12 ]{δ2 } = {0} + [K12 ]{δ12 } 2 P2x      2  P2y   0.00076980 {P1 } = [K12 ] −0.0040 Using [K12 ] from (11.56), we obtain  1    P 17320.5     1x   P1      10000.0 1y {P1 } = = 2 P    −17320.5      2x     2 P2y 10000.0

(11.58)

(11.59)

(11.60)

11.3. 1D AXIAL SPAR OR ROD ELEMENT IN R2

665

where {P1 } are the reactions in x and y directions at nodes 1 and 2. The computed results can be compactly represented graphically as shown in Fig. 11.7. y

10000.0 lbs

17320.5 lbs (u3 , v3 ) = (0.0092376”, −0.048”)

1

x

2 20000 lbs 17320.5 lbs

10000.0 lbs Figure 11.7: Reactions, loading, and displacements (example 11.1)

11.3.2.2 Post-processing First we convert grid point displacements to element nodal displacements in the element local coordinate systems. Element 1: #   "√  48 1 u3 0.0092376” 1 1 T u ¯2 = {t } = − (11.61) v3 −0.048” 8 2 u ¯12 = 0.032” x ¯ x ¯ ¯1 (¯ ¯2 (¯ ¯ (¯ u1h (¯ x) = N x)¯ u11 + N x)¯ u12 , N x) = = he 96   x ¯ 0.001 u1h (¯ x) = (0.032”) = x ¯ 96 3 ∂u1h (¯ x) 0.001 = = 0.0003333 (axial strain) ∂x ¯ 3 = Eε1x¯x¯ = 30 × 106 (0.0003333) = 10000 psi (axial stress)

ε1x¯x¯ = σx1¯x¯

(11.62)

(11.63)

Element 2: u ¯22

2 T

= {t }



u3 v3

"√

 =

48 1 8 2

#

0.0092376” −0.048”

 (11.64)

666

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

u ¯22 = −0.016” x ¯ x ¯ ¯ (¯ N x) = = he 96

¯1 (¯ ¯2 (¯ u2h (¯ x) = N x)¯ u21 + N x)¯ u22 ; u2h (¯ x)



x ¯ = (−0.016”) = 96

−0.001 6

 x ¯

(11.65)

∂u2h (¯ x) −0.001 = = −0.0001666 (axial strain) ∂x ¯ 6 = Eε2x¯x¯ = 30 × 106 (−0.0001666) = −5000 psi (axial stress)

ε2x¯x¯ = σx2¯x¯

(11.66)

11.4 1D axial spar or rod element in R3 (3D space) The transformations derived in section 11.3 for 1D spars in R2 (2D space) can be extended for 1D spars in R3 or 3D space. Figure 11.8 shows a 1D spar element in its own local coordinate system x ¯, y¯, z¯ in R3 (i.e. xyz-space). The element nodal displacements, equivalent nodal forces, and the secondary variables in the element local coordinate system (¯ x, y¯, z¯) are shown in Fig. 11.8. In the element local coordinate system the element equations remain the same as derived for 1D space: 

¯e ¯e K K 11 12 ¯e ¯e K K 21 22



u ¯e1 u ¯e2



 =

f¯1e f¯2e



 +

P¯1e P¯2e

 (11.67)

Figure 11.9 shows displacements, equivalent nodal loads, and the secondary variables at the nodes of element e in the global x, y, z coordinate system. (x2 , y2 , z2 ) 2

e

y¯

x ¯ u ¯e2 P¯2e f¯e 2

1

(x1 , y1 , z1 )

y

u ¯e1 P¯1e

v

z¯

u

f¯1e

x

w z

Figure 11.8: 1D spar element in R3

11.4. 1D AXIAL SPAR OR ROD ELEMENT IN R3 (3D SPACE)

667

e e v2e , f2y , P2y

e e v1e , f1y , P1y

2

e e ue2 , f2x , P2x

e e e w2e , f2z , P2z

1 e e ue1 , f1x , P1x

y e e w1e , f1z , P1z

x

z

Figure 11.9: Nodal forces and displacements in the global x, y, z coordinate system

Using the coordinates (x1 , y1 , z1 ) and (x2 , y2 , z2 ) of nodes 1 and 2 of the element e, its length he , and its direction cosines l, m, and n in the xyz-frame can be determined. p he = (x2 − x1 )2 + (y2 − y1 )2 + (x2 − x1 )2 (11.68) x2 − x1 y2 − y1 z 2 − z1 l= , m= , n= he he he Let    l  e {t } = m   n

(11.69)

  x y = {te }¯ x   z

(11.70)

Then

At node 1:  e  e   u1   f1x  e e e e e e {δ1 } = v1 = {t }¯ u1 , {f1 } = f1y = {te }f¯1e ,  e  e  w1 f1z  e   P1x  e = {te }P¯1e {P1e } = P1y  e  P1z

(11.71)

668

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

At node 2:  e  e   u2   f2x  e e e e e e {δ2 } = v2 = {t }¯ u2 , {f2 } = f2y = {te }f¯2e ,  e  e  w2 f2z  e   P2x  e = {te }P¯2e {P2e } = P2y  e  P2z

(11.72)

From (11.71) and (11.72) we obtain u ¯e1 = {te }T {δ1e }, f¯1e = {te }T {f1e }, P¯1e = {te }T {P1e } u ¯e2 = {te }T {δ2e }, f¯2e = {te }T {f2e }, P¯2e = {te }T {P2e }

(11.73)

Expanding (11.67) e e e e ¯ 11 ¯ 12 K u ¯1 + K u ¯2 = f¯1e + P¯1e e e e e ¯ 21 ¯ 22 K u ¯1 + K u ¯2 = f¯2e + P¯2e

(11.74)

Substituting for u ¯e1 and u ¯e2 from (11.73) and pre-multiplying both equations by {te } and writing the resulting equations in the matrix and vector form  e e e T e e e T e   e e  e e ¯ {t } {t }K ¯ {t } {t }K {t }P¯1 {t }f¯1 {δ1 } 11 12 (11.75) + = e e e e e e T e e e T ¯ ¯ ¯ {te }P¯2e {t }f2 {δ2 } {t }K21 {t } {t }K22 {t } or [K e ]{δ e } = {f e } + {P e } [K e ] is the global stiffness matrix of 1D spar in R3 and   {δ e }T = {δ1e }T {δ2e }T = [ue1 v1e w1e ue2 v2e w2e ]   e e e e e e {f e }T = {f1e }T {f2e }T = [f1x f1y f1z f2x f2y f2z ]   e e e e e e P1y P1z P2x P2y P2z ] {P e }T = {P1e }T {P2e }T = [P1x

(11.76)

(11.77)

are nodal displacements, nodal loads, and the nodal secondary variables (internal nodal forces) at the two nodes of element e.

11.5 The Euler–Bernoulli beam element The Euler-Bernoulli beam theory is based on the assumption that the plane cross-sections perpendicular to the beam axis remain planar and perpendicular to the axis after deformation. The schematic in Fig. 11.10 shows a cantilever beam of length L along x-axis subjected to distributed load q(x) along its length and a moment ML and shear force FL at x = L. The end of the beam is completely clamped.

669

11.5. THE EULER–BERNOULLI BEAM ELEMENT z w

q(x)

FL ML

u

x

x=L x=0

Figure 11.10: Schematic of a beam along x-axis

Let w be the transverse displacement of the beam. The differential equation describing the bending behavior of the beam is given by  2  d2 d w b 2 = q(x) ∀x ∈ (0, L) ⊂ R1 (11.78) 2 dx dx This is a fourth order ordinary differential equation in displacement w, hence requires four boundary conditions. For the schematic shown in Fig. 11.10, the boundary conditions would be dw w(0) = 0, =0 dx x=0 (11.79)  2   2  d d w d w = ML , = FL b 2 b 2 dx dx dx x=L x=L In (11.78) b = EI; E is modulus of elasticity and I is bending moment of inertia of the beam cross-section. Figure 11.11 shows an m element discretization of the center line of the beam using two node beam elements. z FL

q(x)

ML x 1

e

2

x=0

m

q(x) 1

2

xe

xe+1

e he

Figure 11.11: Discretization and typical two node beam element

670

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

¯ e = [xe , xe+1 ] and of length he = Isolate an element e with domain Ω xe+1 − xe with local node numbers 1 and 2. We derive element equations for this beam element using the differential equation (11.78) and the Galerkin method with weak form as in this case the differential operator  2  d2 d A= 2 b 2 (11.80) dx dx is linear and we can also show that A∗ = A i.e. the adjoint of the operator is same as the operator, hence GM/WF will yield VC integral form. This ensures that the coefficient matrix in the element algebraic equations will be symmetric.

11.5.1 Derivation of the element equations (GM/WF) Let v = δw be the test function, then the following integral form is valid based on fundamental lemma of calculus of variations. xZe+1

(Aw − q, v)Ω¯ e = xe

d2 dx2

 2   d w b 2 − q v dx dx

(11.81)

We transfer two orders of differentiation from w to v using integration by parts xZe+1

(Aw − q, v)Ω¯ e

 d2 v d2 w b − qv dx = dx2 dx2 xe   2   xe+1  2   xe+1 d w d d w dv b 2 v + − b 2 dx dx dx dx xe xe

From the boundary terms (concomitant) we conclude that w,  d dx

2 b ddxw2

2 b ddxw2

dw dx

(11.82)

are primary

and are secondary variables, hence w = w ˆ and variables, dw ˆ dx = θ on some boundary Γ are essential boundary conditions and the secondary variables taking fixed values on some boundary Γ are natural boundary conditions. Γ of course consists of the end points of the element. Let us introduce the following notations for the secondary variables:  2   2  d w d d w d e b 2 = Q1 ; − b 2 = Qe2 ; shear forces dx dx dx dx xe xe+1 d2 w d2 w e b 2 = M1 ; −b 2 = M2e ; moments dx xe dx xe+1 (11.83)

11.5. THE EULER–BERNOULLI BEAM ELEMENT

671

Expanding boundary terms in (11.82) and then using (11.83) xZe+1 d2 w d2 v qv dx (Aw − q, v)Ω¯ e = b 2 2 dx − dx dx xe xe     dv dv e e e M e (11.84) − v(xe )Q1 − v(xe+1 )Q2 − − M − − dx xe 1 dx xe+1 2 xZe+1

or (Aw − q, v)Ω¯ e = B e (w, v) − le (v) where

xZe+1

e

B (w, v) =

b xe

e

d2 w d2 v dx dx2 dx2

(11.85)

(11.86)

xZe+1

qv dx + v(xe )Qe1 + v(xe+1 )Qe2

l (v) = xe

 +

   dv dv e − M + − Me dx xe 1 dx xe+1 2

(11.87)

B e (·, ·) is bilinear and symmetric and le (·) is linear, a direct consequence of linearity of A and the fact that A∗ = A. The quadratic functional representing the total potential energy can be obtained using 1 (11.88) I e (w) = B e (w, v) − le (w) 2 in which the first term represents elastic strain energy and the second term is the potential energy of loads i.e. work done by f (x) and also due to secondary variables at the nodes of the element.

11.5.2 Local approximation We note that the differential equation describing the bending behavior of a beam is a fourth order differential equation in displacement w. Let whe be ¯ e of the descretization the local approximation of w over an element domain Ω ¯ T = ∪Ω ¯ e , then the approximation wh of w over Ω ¯ T is given by wh = ∪we . Ω e e h T ¯ Consider the integral form over Ω based on fundamental lemma used as a starting step in the GM/WF:  Z Z  2  2  d d wh (Awh − q)v dΩ = b − q v dΩ = 0 (11.89) dx2 dx2 ¯T Ω

¯T Ω

672

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

This integral form is valid if the integrand is continuous. This requires that ¯T ) wh ∈Vh ⊂ H 5 (Ω f ∈H 1 v ∈V ⊂ H

(11.90) 1

¯ T and hence over Ω ¯ e. That is, wh and hence whe must be of class C 4 over Ω This is generally not entertained in the conventionally used finite element formulations for bending of beams even though this is precisely what needs to be done. Here we only present the approach used currently. The integral ¯ e implies that for the discretization Ω ¯ T the following form (11.85) over Ω holds:
P e e (Awh − q, v)Ω¯ T = (11.91) B (wh , v) − le (v) = 0 ; v = δwhe e

¯ T , wh must be at least of class i.e. for (Awh − q)v to be continuous over Ω ¯ T ), i.e. wh ∈ Vh ⊂ H 3 (Ω ¯ T ). However, if we accept integral over Ω ¯ T in C 2 (Ω e 1 T ¯ and hence the Lebesgue sense, then wh and hence wh of class C over Ω e ¯ over Ω can be used in the integral form resulting from the GM/WF. This is what is done commonly. We present details of this approach in the following. ¯ T ). This requires that we establish we such Consider wh of class C 1 (Ω h ¯ T ). We refer to such local approximation we that wh = ∪whe is of class C 1 (Ω h e ¯ T ). In simple terms this requires of class C 1 as it yields wh of class C 1 (Ω that at the inter-element boundaries w and − dw dx = θ (rotation) must be continuous. Thus, if we choose w and θ as nodal degrees of freedom at each of the two nodes of the element then the resulting local approximation ¯ T ). Two degrees of freedom at each node of the would yield wh of class C 1 (Ω element (total of four degrees of freedom for the element) suggest that we can begin with whe = c0 + c1 x + c2 x2 + c3 x3 (11.92) The constants c0 , c1 , c2 , and c3 can be evaluated using   dwhe e e e wh (xe ) = w1 , θ1 = − , dx xe   dwhe e e e wh (xe+1 ) = w2 , θ2 = − dx xe+1

(11.93)

Using (11.92) and (11.93) we obtain w1e = c0 + c1 xe + c2 x2e + c3 x3e θ1e = −c1 − 2c2 xe − 3c3 x2e w2e = c0 + c1 xe+1 + c2 x2e+1 + c3 x3e+1 θ2e = −c1 − 2c2 xe+1 − 3c3 x2e+1

(11.94)

673

11.5. THE EULER–BERNOULLI BEAM ELEMENT

or  e  w  1    e1    θ1 0 = e  w 1     e2   θ2 0

  xe x2e x3e c0       −1 −2xe −3x2e   c1 2 3 xe+1 xe+1 xe+1   c    2  2 −1 −2xe+1 −3xe+1 c3

(11.95)

Solving for ci ; i = 0, 1, . . . , 3 from (11.95) and substituting in (11.92), we obtain whe = N1w (x)w1e + N1θ (x)θ1e + N2w (x)w2e + N2θ (x)θ2e

(11.96)

in which    x − xe 3 x − xe 2 +2 =1−3 he he 2  x − xe N1θ (x) = −(x − xe ) 1 − he  2   x − xe x − xe 3 w N2 (x) = 3 −2 he he !  2 x − xe x − xe θ N2 (x) = −(x − xe ) − he he

N1w (x)



(11.97)

These are called Hermite cubic interpolation functions or simply C 1 local approximation functions. The local approximation functions can also be expressed in an element local coordinate system. Consider a local coordinate system x ¯ with its origin at node 1 and x ¯ axis pointing from node 1 to node 2. The x and x ¯ coordinates are related by x ¯ = x − xe , which when used in (11.97), yields the local approximation functions in the local coordinate system: N1w (¯ x)

 =1−3

x ¯ he

2

 +2

x ¯ he

  x ¯ 2 = −¯ x 1− he  2  3 x ¯ x ¯ w N2 (¯ x) = 3 −2 he he !  2 x ¯ x ¯ θ N2 (¯ x) = −¯ x − he he

3

N1θ (¯ x)

(11.98)

From (11.97) we note that the local approximation functions have the fol-

674

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

lowing properties: N1w (xe )

N1w (xe+1 )

= 1,

= 0,

N2w (xe ) = 0, N2w (xe+1 ) = 1, N1θ (xe )

N1θ (xe+1 )

= 0,

N2θ (xe ) = 0;

= 0,

N2θ (xe+1 ) = 0,

dN1w dN1w = 0, =0 dx xe dx xe+1 dN2w dN2w =0 = 0, dx xe dx xe+1 dN1θ dN1θ =0 − = 1, dx xe dx xe+1 dN2θ dN2θ = 0, − =1 dx xe dx xe+1

(11.99)

These properties of the local approximation functions ensure that condition (11.93) are satisfied by whe . Substituting local approximation whe in (11.84) and using (11.100) v = δwhe = N1w , N2w , N1θ , and N2θ we obtain (using (11.84)) (Awhe − q, v)Ω¯ T = [K e ]{δ e } − {F e } − {P e }

(11.101)

If we let N1w = N1 , N1θ = N2 , N2w = N3 , and N2θ = N4 , then v = δwhe = Nj ; j = 1, 2, . . . , 4 and [K e ] can be represented more compactly as (using be for b) e

xZe+1

be

[K ] = xe

d2 Ni d2 Nj dx ; dx2 dx2

i, j = 1, 2, . . . , 4

e T

{δ } = [w1e θ1e w2e θ2e ] xZe+1 e Fi = f (x)Ni (x) dx

(11.102)

xe e T

{P } = [Qe1 M1e Qe2 M2e ] If we choose q(x) = qe , a constant value over element e, then we obtain   6 −3he −6 −3he 2be −3he 2h2e 3he h2e   [K e ] = 3  he  −6 3he 6 3he  −3he h2e 3he 2h2e (11.103)   6     qe he  −he  e {F } = 6  12      he in which be = Ee Ie ; I being bending moment of inertia.

11.6. EULER-BERNOULLI FRAME ELEMENTS IN R2

675

11.6 Euler-Bernoulli frame elements in R2 In this section we consider finite element formulations of structural members in R2 that can be used to study frames in which the structural members are welded or reveted to each other. In such applications the member can support axial load as well as bending behavior. Thus, the finite element formulation for such structural members for linear elastic behavior can be derived by superposition of a rod element in R2 and a pure bending element (of section 11.5) in R2 . w2e ue2

2

z¯

e

w1e z

x¯

1

θ¯2e β

¯e ue1 θ1 x

Figure 11.12: A frame element in R2

Let (x1 , z1 ), (x2 , z2 ) be the coordinates of the two nodes of the element in x, z frame, then the direction cosines of the line segment 1-2 are l=

x2 − x1 z2 − z1 = cos β ; n= = sin β he he      x ¯ cos β − sin β x = z¯ sin β cos β z

(11.104)

or      x ¯ cos β sin β x = z¯ − sin β cos β z

(11.105)

     ¯ cos β 0 sin β  x  x y¯ =  0 1 0  y     z¯ − sin β 0 cos β z

(11.106)

and

For bending we can write  e    ¯  cos β sin β 0  ue  u w ¯ e = − sin β cos β 0 we  ¯e   e θ 0 0 1 θ

(11.107)

676

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

Thus,  e   e  u ¯ cos β sin β 0 0 0 0 u1     1  e            w ¯1  − sin β cos β 0 0 0 0  w1e          ¯e    e  θ1  θ1 0 0 1 0 0 0   = ¯e2  0 0 cos β sin β 0 ue2  u     0       e e           w ¯ 0 0 0 − sin β cos β 0 w    ¯e2    e2   θ2 0 0 0 0 0 1 θ2

(11.108)

or {δ¯e } = [T e ]{δ e }

(11.109)

For axial deformation (1D rod in R2 ) we have  e  1 0 −1 0  ¯1   u e w Ae Ee  0 0 0 0   ¯1 = {F¯ e } + {P¯ e } u ¯e  he −1 0 1 0    2e   00 00 w ¯2 

(11.110)

After introducing θ¯1 and θ¯2 in (11.110) we cam write the following for axial rods.   e   fe he   ¯ e   1 0 0 −1 0 0  ¯1  P1x   2   u         e  0 0 0 0 0 0        0 w ¯ 0      1               e ¯   Ae Ee  0 0 0 0 0 0 θ1 0 0 + = (11.111) e fe he  u ¯e2  P¯2x     he  −1 0 0 1 0 0        2         0 0 0 0 0 0       0    0  w ¯e        ¯e2       000 000 θ2 0 0 The beam element equation (11.101) can also be written as  0 0 0 6  2Ee Ie  0 −3he 3 he  0 0 0 −6 0 −3he

0 −3he 2h2e 0 3he h2e

0 0 0 −6 0 3he 0 0 0 6 0 3he

  0  e    0 u ¯ 0      1      qe he      e e ¯       −3he   w ¯ Q 2     1 1      ¯e   ¯ e   qe h2e h2e  θ M 1 1 12  = + e 0  u ¯ 0 0      2          e qe he    ¯e       w ¯ Q 3he     ¯e2     ¯ 2e      qe2h2e θ2 2h2e M2 12

        

(11.112)

       

A frame element in R2 is simply superposition of (11.111) and (11.112). Since both (11.111) and (11.112) have same degrees of freedom their superposition is simply their addition and we have the following for the frame element in the element local coordinate system x ¯z¯: ¯ e ]{δ¯e } = {F¯ e } + {P¯ e } [K

(11.113)

677

11.7. THE TIMOSHENKO BEAM ELEMENTS

¯ e ], {δ¯e }, {F¯ e }, and {P¯ e } are given by in which [K 

Ae h2e 2Ie

 0   2E I  0 e e e ¯ [K ] =  2 h3e − Ae he  2Ie  0 0

0 6 −3he 0 −6 −3he

2

0 − A2Ie hee −3he 0 2h2e 0 Ae h2e 0 2Ie 3he 0 2 he 0

0 −6 3he 0 6 3he

 0 −3he    h2e   0   3he 

(11.114)

2h2e

{δ¯e }T = [¯ ue1 w ¯1e θ¯1e u ¯e2 w ¯2e θ¯2e ] e ¯e ¯ e ¯e ¯e ¯ e {P¯ e }T = [P¯1x Q1 M1 P2x Q2 M2 ] (11.115) 1 1 1 1 1 1 e T 3 3 {F¯ } = [ fe he qe he − qe he fe he − qe he qe he ] 2 2 12 2 2 12 Using {δ¯e } = [T e ]{δ e } in (11.113) and premultiplying (11.113) by [T e ]T , we obtain  eT e e e ¯ ][T ] {δ } = [T e ]T {F¯ e } + [T e ]T {P¯ e } [T ] [K (11.116) or [K e ]{δ e } = {F e } + {P e }

(11.117)

Equations (11.117) are the element equations for the frame element of Fig. 11.12 in the global coordinate system xz. Explicit expressions for [K e ] can be obtained.

11.7 The Timoshenko beam elements The Euler-Bernoulli beam theory is based on the assumption that the plane sections remain planar and normal to the longitudinal axis of the beam after deformation. A consequence of this assumption is that transverse shear strain εxz is zero. This is obviously non-physical in non-slender beams. If we assume that the plane sections remain planar but not necessarily normal to the longitudianl axis of the beam, then the shear strain εxz is not zero. For this case the rotation of cross-sections (transverse normal planes to the longitudinal axis) is not equal to − dw dx . The beam bending theory based on this assumption is a shear deformation theory most commonly referred to as Timoshenko beam theory. In this theory since the rotation of the cross-section about y-axis is not described by − dw dx , we need to denote this by an independent parameter ψ(x). The balance of linear momenta for the stationary case i.e. equations of equilibrium are given by    d dw GAks ψ + +q =0 (11.118) dx dx

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ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

d dx

    dw dψ EA − GAks ψ + =0 dx dx

(11.119)

In which G is shear modulus and ks is shear correction factor to correct the constant shear stress across the cross-section in this theory compared to actual parabolic distribution. The Euler-Bernoulli beam theory
is recovered dw from (11.118) and (11.119) by substituting GAks ψ + dx from (11.119) into (11.118) and by letting ψ be − dw dx . From (11.118) and (11.119) we observe that both contain up to second order derivatives of w and ψ thus GM/WF appears to be a viable option. Alternatively, since we are considering thermoelastic behavior, a reversible process, the strain energy suggests existence of a quadratic functional, the total potential energy, all of which suggest GM/WF for constructing finite element formulations for (11.118) and (11.119).

11.7.1 Element equations: GM/WF Let p1 and p2 be the test functions representing variations of w and ψ ¯ e , the domain [xe , xe+1 ] of an element i.e. p1 = δw and p2 = δψ, then for Ω e, we can write (let GAks = α) xZe+1 xe xZe+1

− xe



d dx

    dw α ψ+ + q p1 dx = 0 dx

(11.120)

    dψ dw EI −α ψ+ p2 dx = 0 dx dx

(11.121)

d − dx

Integrating first term in (11.120) and (11.121) by parts once xZe+1 xe

      xe+1 dp1 dw dw α ψ+ − qp1 dx − p1 α ψ + = 0 (11.122) dx dx dx xe

xZe+1 xe

     dψ dw dψ xe+1 dp2 EI +α ψ+ p2 dx − p2 EI = 0 (11.123) dx dx dx dx xe

xZe+1

xZe+1

xZe+1

xe

xe

xe

dp1 dw α dx + dx dx

xZe+1

EI xe

dp1 α ψ dx = dx

dp2 dψ + αp2 ψ dx dx

(11.124)   xe+1 dw dψ αp2 dx = p2 EI (11.125) dx dx xe

xZe+1

 dx +

xe

   xe+1 dw qp1 dx + p1 α ψ + dx xe

679

11.7. THE TIMOSHENKO BEAM ELEMENTS

Let    dw = v1e α ψ+ dx  xe   dw − α ψ+ = v2e dx xe+1 dψ = M1e EI dx xe dψ − EI = M2e dx xe+1

(11.126)

Using (11.126) in (11.124) and (11.125) xZe+1

xZe+1

xZe+1

xe

xe

xe

dp1 dw α dx + dx dx

dp1 α ψ dx = dx

qp1 dx + p1 (xe )v1e + p1 (xe+1 )v2e (11.127)

xZe+1 xe

dp2 dψ + αp2 ψ EI dx dx

xZe+1

 dx +

αp2 xe

dw dx = p2 (xe )M1e + p2 (xe+1 )M2e dx

(11.128) e ¯ be local approximations for w and ψ over Ω = [xe , xe+1 ]. and Let Since ψ is derivative of w with respect to x, it suggests that if whe is an algebraic polynomial of degree p then ψhe must be an algebraic polynomial of degree p − 1 so that whe and ψhe are consistent. The integral forms in (11.127) ¯ e ) if the integrals over Ω ¯ T are and (11.128) require whe and ψhe of class C 1 (Ω ¯ T , then we and ψ e of to be Riemann. If we accept Lebesgue integrals over Ω h h 0 e ¯ can be of class C (Ω ). We choose this option here. Let whe

ψhe

whe = ψhe

=

nw P

Niw wie = [N w ]{we }

i=1 nψ P

i=1

(11.129) Niψ ψie

ψ

e

= [N ]{ψ }

in which Niw (x) and Niψ (x) are local approximation functions for whe and ψhe and wie and ψie are nodal degrees of freedom for whe and ψhe : p1 = δwhe = Njw (x), j = 1, 2, . . . , nw p2 = δψhe = Njψ (x), j = 1, 2, . . . , nψ

(11.130)

680

ADDITIONAL TOPICS IN LINEAR STRUCTURAL MECHANICS

Substituting (11.129) and (11.130) into (11.127) and (11.128) xZe+1 xe

dNjw α dx

xZe+1  n  nψ  w dN w dNjw P P ψ e e i N ψ dx α wi dx + dx i=1 i i i=1 dx xe

xZe+1

qNjw dx + Njw (xe )v1e + Njw (xe+1 )v2e , j = 1, 2, . . . , nw (11.131)

= xe

xZe+1

EI

dNjψ

xe

dx

xZe+1

αNjψ

+ xe nw P 11 j=1 nw P 21 j=1

or

nψ P dNiψ e ψi i=1 dx

!

 nψ ψ P

+ αNj

Niψ ψie

! dx

i=1

n  w dN w P e i wi dx = Njψ (xe )M1e + Njψ (xe+1 )M2e (11.132) dx i=1

e e Kij wj +

e e Kij wj +

nψ P 12 j=1 nψ P 22 j=1

e e Kij ψj = 1 Feie + 1 Pie , i = 1, 2, . . . , nw

(11.133)

e e Kij ψj = 2 Feie + 2 Pie , i = 1, 2, . . . , nψ

(11.134)

 11 e 12 e   e  ( 1 e )  1 e  { P } [ K ][ K ] {w } { Fe } = + 21 e 22 e e {2 P e } [ K ][ K ] {ψ } {2 Fee }

(11.135)

or [K e ]{δ e } = {Fee } + {P e }

(11.136)

in which 11

12

21

22

e Kij =

e Kij =

e Kij =

e Kij

xZe+1

α xe xZe+1

α xe xZe+1

dNiw dNjw dx ; dx dx dNiw ψ N dx ; dx j

= xe

i = 1, 2, . . . , nw j = 1, 2, . . . , nψ (11.137)

dNjw αNiψ dx ; dx

xe xZe+1

i, j = 1, 2, . . . , nw

ψ

i = 1, 2, . . . , nψ j = 1, 2, . . . , nw

dNiψ dNj EI + αNiψ Njψ dx dx

! dx, i, j = 1, 2, . . . , nψ

11.8. FINITE ELEMENT FORMULATIONS IN R2 AND R3

681

{δ e }T = [{we }T {ψ e }T ] xZe+1 1 ee qNiw dx ; i = 1, 2, . . . , nw Fi = xe 2

Feie = 0 ;

(11.138) i = 1, 2, . . . , nψ

{1 P e }T = [V1e 0 0 . . . 0 V2e ] {2 P e }T = [M1e 0 0 . . . 0 M2e ] Numerical values of the coefficients of [k e ] and the vector {F˜ e } can be calculated using Gauss quadrature.

11.8 Finite element formulations in R2 and R3 Finite element formulations for plane stress, plane strain, and axisymmetric solids in R2 and deformation in R3 have already been presented in: (1) chapter 5 using differential mathematical models and GM/WF, LSP and (2) chapter 9 using principle of minimum potential energy in which the total potential energy statement is derived directly using physics of deformation as opposed to differential model.

11.9 Summary The main objective of the finite element formulations presented in this chapter for linear structural mechanics is to illustrate the concept of local coordinate system, 1D elements in 2D and 3D space, and to present some representative finite element formulations for bending of beams. The formulations presented in chapters 5 and 9 for linear solid and structural mechanics are not repeated here for the sake of brevity.

12

Convergence, Error Estimation, and Adaptivity

12.1 Introduction In this chapter basic concepts of the convergence of a computed finite element solution, rates of convergence, a priori and a posteriori error estimations, computations of error in the finite element solutions, adaptive and self-adaptive finite element processes are considered. There are three main sources of error in the finite element computational processes that are based on variationally consistent integral forms: due to approximation of boundaries, due to inadequate precision in computation and faulty or inadequate algorithms, and due to local approximations of the theoretical solution. The first two sources can be almost completely eliminated while the error due to local approximations remains intrinsic in all finite element computations. In this chapter and elsewhere in the book the error implies error due to local approximation. In finite element computation processes as more degrees of freedom are added to a discretization the accuracy of the computed solution improves when the theoretical solution of the BVP is analytic or regular (i.e. smooth) and when the integral forms are VC. The characteristic length h of the discretization, degree of local approximations p, and the order k of the approximation space (that defines the global smoothness of the approximation over the whole discretization of order k − 1) are three independent parameters in all finite element computational processes, thus hpk framework. Parameters h, p, k control the achievable accuracy of a finite element solution. We define convergence of a finite element solution as the process through which the computed solution approaches theoretical solution as more degrees of freedom are added to the discretization. Our objective of course is to approach the theoretical solution for least number of degrees of freedom in the discretization. A finite element process that requires the addition of fewest degrees of freedom in some progression is said to have the fastest convergence rate. We discuss details of the convergence and convergence rates of the finite element processes for BVPs in this chapter. A priori error estimates 683

684

CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

are estimates of the error established prior to the computations that reveal specific functional dependence of the error in the finite element solution measured in some norm based on h, p, and k so that prudent judgments can be made regarding their choices to achieve the fastest convergence rate. A posteriori error estimates are those that are derived or determined using the currently computed solution and are often used to guide how and where more degrees of freedom need to be added to improve the accuracy of the current solution. Adaptivity or adaptive finite element processes help in determination of specific changes in h, p, and k using current solution in specific regions of the discretization for improving accuracy of the computed solution. A self-adaptive finite element process initiates computation with some discretization (h, p, k) and continuously modifies h, p, and k through a built-in or intrinsic mechanism in the most prudent manner (least degrees of freedom) to achieve desired accuracy. We consider specific details of various aspects of convergence, convergence rates, errors, and adaptivity in the following sections. We only consider those finite element processes in which the integral forms are variationally consistent (VC), hence ensuring symmetric, positivedefinite coefficient matrices in the resulting algebraic systems, thus unconditionally stable computational processes. Variational consistency of the integral form requires that for self-adjoint differential operators we consider GM/WF or LSP based on residual functional and for non-self adjoint differential operators (linear but A∗ 6= A) and nonlinear differential operators we only consider LSP based on residual functional. VC integral forms are of paramount importance in determining the convergence rates of the finite element processes for all three classes of differential operators (self adjoint, non-self adjoint, and non-linear), as shown in later sections of this chapter.

12.2 h-, p-, k-versions of FEM and their convergence Consider a BVP Aφ − f = 0 in Ω (12.1) ¯ be closure of Ω such that Ω ¯ = Ω ∪ Γ, Γ being closed boundary of Ω. Let Ω T ¯ ¯ Let Ω be discretization of Ω such that ¯T = S Ω ¯e Ω (12.2) e

¯ T . Let φh be approximation ¯ e being a finite element e of the discretization Ω Ω ¯ T and φe be local approximation of φ over an element e with of φ over Ω h e ¯ domain Ω such that S φh = φeh (12.3) e

12.2. H-, P -, K-VERSIONS OF FEM AND THEIR

CONVERGENCE

685

¯ e , then we Let he be the characteristic length of an element e with domain Ω ¯ T by define characteristic length h of the discretization Ω h = max(he ) e

(12.4)

¯ e , then we define Let pe be the degree of local approximations of φeh over Ω T ¯ by p, the p-level for the discretization Ω p = min(pe ) e

(12.5)

12.2.1 h-version of FEM and h-convergence When additional degrees of freedom are added to a discretization by refining the discretization i.e. subdividing existing elements thereby reducing h but holding p and k constant, then we have h-version of the finite element method. If we define error in some norm and monitor reduction in this error norm as a function of the deegrees of freedom, then we have h-convergence of the finite element method. The h-refinement and the corresponding hconvergence may be of the following types. Uniform h-refinement When the mesh refinement is done uniformly over the discretization (i.e. h is reduced to h/2 and then to h/4 and so on) we have uniform h-convergence). Uniform h-refinement may be wasteful as in this process we will be subdividing all elements in the discretization, hence adding additional degrees of freedom in portions of the discretization where the computed solution is already sufficiently converged, hence has desired accuracy. This is obviously wasteful. Selective h-refinement In this process uniform mesh refinement is done selectively in the desired portions of the discretization. This obviously requires additional information to determine locations of such regions. Element-wise error indicators, either estimated or computed, are needed to accomplish this. This is discussed further in later sections. Graded h-refinement When element error indicators are either not available or are not reliable, this approach is useful. In this method we consider a sequence of discretizations in which the element characteristic lengths are in some fixed geometric ratio. Smaller element sizes are intentionally biased towards the known locations of high solution gradients. The increase in the number of elements in this progressive sequence of discretizations can be by one or more at a time as deemed necessary.

686

CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

12.2.2 p-version of FEM and p-convergence When additional degrees of freedom are added to a discretization by increasing the degree of approximation pe of the elements and thereby increasing p for the whole discretization but holding h and k constant, then we have p-version of the finite element method. If we define error in some norm and monitor reduction in this error norm as a function of the degrees of freedom then we have p-convergence of the finite element method. The p-refinement and the corresponding p-convergence may be of the following types. Uniform p-refinement When the degree of local approximation (p-level) is increased uniformly for each element of the discretization we have uniform p-refinement. As in case of uniform h-refinement, this also can be wasteful due to addition of extra degrees of freedom in the portions of the discretization where the solution is sufficiently converged or has desired accuracy. Selective p-refinement In this process p-level increase is initiated selectively in the desired portions of the discretization. This obviously requires additional information to determine locations of such regions. Element-wise error indicators either estimated or computed are necessary to accomplish this. In the elements or regions requiring additional degrees of freedom the choice of selective hor p- refinement also requires a criterion through which the choice can be automated. In selective p-refinement we may encounter interelement boundaries with dissimilar p-levels. Such boundaries require special consideration to ensure that the solution behavior remains physical at these boundaries. As a general rule the additional degrees of freedom at a common boundary due to the higher p-level element must be constrained such that the common boundary behavior is in conformity as dictated by the lower p-level element.

12.2.3 hp-version of FEM and hp-convergence In this process h and p are changed simultaneously in the entire discretization, leading to uniform hp-refinement, or selectively in the desired portions of the subdomains, resulting in selective hp-refinement. Changes in h and p simultaneously in the selected desired portions of the discretization requires prudent indicators, either estimated or computed, that guide the hp-refinement process.

12.2. H-, P -, K-VERSIONS OF FEM AND THEIR

CONVERGENCE

687

12.2.4 k-version of FEM and k-convergence Surana et al. [1–4] have shown the order k of the approximation space to be an independent parameter in all finite element computational processes in addition to h and p, hence k-version of finite element method in addition to h- and p-versions. The order k of the approximation space ensures global differentiability of order k − 1 over the whole discretization. The appropriate choice of k is essential in ensuring that (1) the desired physics is preserved in the computational process and (2) the integrals are Riemann in the entire finite element process so that the equivalence of BVP with the integral form is preserved and the errors in the calculated solution can be computed correctly without knowledge of the theoretical solution. We elaborate more on some of these aspects in the following. If the differential operator contains highest order derivatives of the dependent variables of orders 2m, then the approximation of the solutions of the BVP must at least be of class C 2m (i.e. of global differentiability of order 2m in order for this approximation to be admissible in the BVP in the pointwise sense). This requires that order k of the approximation space must at least be 2m+1; that is, k = 2m+1 is minimally conforming order of the approximation space. Clearly the order k of the minimally conforming space is determined by the highest order of the derivatives of the dependent variable(s) in the BVP. When k ≥ 2m + 1 all integrals over the discretization ¯ T remain Riemann. When k = 2m, the integrals over Ω ¯ T are in Lebesgue Ω ¯T sense and the corresponding approximation φh of the solution φ over Ω is not admissible in the BVP Aφ − f = 0 in the pointwise sense. When ¯ T is not admissible at all in k ≤ 2m − 1, the approximation φh of φ over Ω the BVP. Choosing k > 2m + 1 may be beneficial if the theoretical solution φ of the BVP is of higher order global differentiability than 2m as this choice incorporates higher order global differentiability aspects of φ in the computational process. ¯T Let us define residual function E over Ω E(φh ) = Aφh − f

¯T Ω

over

¯T ) ∀φh ∈ Vh ⊂ H(Ω

(12.6)

or E=

P e P Aφh − f = E e e

e

¯ e) ∀φeh ∈ Vh ⊂ H(Ω

(12.7)

¯T and the residual functional I(φh ) over Ω I(φh ) = (E, E)Ω¯ T

(12.8)

or I(φh ) =

P e

(E e , E e )Ω¯ e

(12.9)

688

CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

where E e = Aφeh − f

(12.10)

For theoretical solution φ of Aφ − f = 0 we have I(φ) = (E, E)Ω¯ = 0

(12.11)

as E(φ) = Aφ − f = 0

over

¯ Ω

(12.12)

¯ T are Consider k ≥ 2m + 1. For this choice of k all integrals over Ω Riemann, hence when I(φh ) → 0, then E(φh ) → 0 and E e (φeh ) → 0 in the ¯ T or Ω). ¯ That is pointwise sense (i.e. φh → φ in the pointwise sense over Ω the converged finite element solution φh is same as the theoretical solution in all aspects up to the derivative of order k − 1. Thus proximity of I(φh ) to zero is a measure of the error in the computed solution φh . This is obviously deterministic without the knowledge of the theoretical solution of the BVP. Furthermore, since P I(φh ) = I e (φeh ) (12.13) e

is a measure of error in the approximation φeh for an element e. Thus can be computed for each element using current solution and can ¯e be used for refinement (adaptivity). Residual functional values I e over Ω provide built-in measure of error in the computed solution for each element of the discretization and adaptivity. Choice of h-, p-, or hp-refinement is not clear at this stage but will be considered in more details in a following section. In any case, the elements with values of I e greater than some average ¯ T become candidates for addition of more degrees of threshold value for Ω freedom. I e (φeh ) I e (φeh )

Remarks. (a) Significance of the choice of minimally conforming k or greater than minimally conforming (≥ 2m + 1) is quite clear. Without this choice details and conclusions in section 12.2.4 do not hold in the precise sense. (b) When k ≥ 2m + 1, I and I e are measures of error in the computed ¯ T or Ω ¯ e as I and I e are zero for the theoretical solution. solution over Ω (c) When k ≥ 2m + 1, computations of I e serve as error indicators and provide a rational mechanism for adaptivity that is built into the com¯ T where I e are greater than a putational process in specific regions of Ω ¯T . threshold value chosen for Ω (d) When k ≥ 2m + 1 and when I → 0 (≤ 10−8 ), φh → φ in the pointwise sense. That is the numerically computed solution is undoubtedly same as theoretical solution within the limitations of chosen k. This claim is

12.3. CONVERGENCE AND CONVERGENCE RATE

689

only possible within hpk framework. Without higher values of k (k > 1), that is, with just hp framework with solutions of class C 0 this claim can not be made. (e) The most significant aspect of k and the choice k ≥ 2m + 1 is that the physics of the BVP that is intrinsic in the mathematical description is preserved in the computational process. (f) The choice k ≥ 2m + 1 permits design of finite element computational processes in which desired features of the theoretical solution can be incorporated in the computational process with absolute assurance of unconditional stability if the integral forms are variationally consistent. (g) In k-convergence we keep h and p constant and progressively increase k. As explained in earlier chapters for fixed h and p, increasing k results in decrease in the degrees of freedom. Thus, for fixed degrees of freedom increase in k permits increase in p so that the same number of degrees of freedom are maintained. This results in substantial improvement in the solution accuracy (see chapters 5, 6, and 7). (h) Equations (12.6) – (12.12) should not be misunderstood to imply that these only hold for finite element processes based on residual functional. This is not the case. Computations of E e and I e only require spaces in which k ≥ 2m + 1, but the integral form could be from any desired methods of approximation.

12.3 Convergence and convergence rate Convergence of a finite element solution implies behavior of the error in the finite element solution (measured in some norm) as a function of the degrees of freedom or the characteristic length of the discretization. When the theoretical solution is known, the error in the finite element solution in some norm (L2 -norm, H 1 -norm, etc.) can be computed and therefore we can study its behavior as a function of the degrees of freedom. When the theoretical solution is not known, perhaps estimating the error in some norm in the computed solution is a viable option. However, we shall see in a later section that this option only works in a restricted range of the behavior of error norm versus dofs. The third option is that if we are using minimally conforming spaces Vh ⊂ H then residual functional I(φh ) can be ¯T computed precisely as for minimally conforming spaces all integrals over Ω are Riemann. Proximity of I(φh ) to zero is a measure of error due to the fact that when φh ' φ, I(φh ) ' I(φ) = 0. Thus, I(φh ) is in fact error ¯ T . This option can always be used for any measure in the solution φh over Ω applications as it does not require theoretical solution but necessitates the approximation φh to be in a space of order k ≥ 2m + 1. In what follows we

690

CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

¯ T , hence convergence of the comcan use I(φh ) as a measure of error over Ω puted solution φh to φ implies studying I(φh ) versus dofs as more degrees of freedom are added to the discretization. When I(φh ) ≤ ∆, a predetermined tolerance of computed zero, we consider the finite element p solution φh to be converged p to the theoretical solution φ. We consider I(φh )√versus dofs or √ I = (E, E) = ||E||L2 , L2 -norm of residual E. We study I versus dofs using log-log scale, or more precisely we study log||E||L2 versus log(dof s). log||E||L2 and log(dofs) or log-log scale are necessary as the range of I could be O(101 ) - O(10−20 ) and the range of dof could be O(101 ) - O(106 ) or higher.

12.3.1 Convergence behavior of computations The material presented in this section is based on ||E||L2 versus dof behavior, but the same concepts hold true for any other measure of error norm (i.e. ||E||L2 can be replaced with any other error norm without affecting the basic behavior of the convergence graph). A typical convergence behav√ ior of log( I) or log(||E||L2 ) versus log(dof ) is shown in Fig. 12.1. This graph is generated using 1D convection-diffusion equation (a second order ODE) with P e = 1000 and least squares finite element formulation based on residual functional. The progressively graded discretizations are generated beginning with two elements using a constant geometric ratio of 1.5. The smallest element is located at x = 1.0. k = 3 is used as it corresponds to the minimally conforming space. Minimum p-level of 5 (needed for k = 3) is considered for each progressively refined discretization. From Fig. 12.1 √ we observe five distinct zones. In each one of these zones the behavior of I versus dofs is unique and distinct. Pre-asymptotic range (AB): The range AB is called pre-asymptotic range. In this range as we move from location A toward location B additional degrees of freedom are added to the discretization but there is virtually no measurable reduction in the L2 -norm of E. The accuracy of the computed solution in this range is very poor (due to ||E||L2 of the O(1)) due to poor √ accuracy of the solution φh , hence I and the I e values for the elements are poor as well, therefore these can not be used to guide any form of adaptive refinement process. A posteriori error estimations in this range are not possible either as these require some regularity in the computed solution which is absent in φh in range AB. Thus, in this range adaptive processes are not possible as reliable indicators (either estimated or computed) based on φh are not possible.

691

12.3. CONVERGENCE AND CONVERGENCE RATE

0.5 AB : Pre-asymptotic range A

B

BC : Onset of asymptotic range

0

CD : Asymptotic range DE : Onset of post-asymptotic range

C

-0.5 2

log(√I = || E ||L )

EF : Post-asymptotic range -1

-1.5

-2 D -2.5 E

F

-3 1.4

1.5

1.6

1.7

1.8 log(dofs)

1.9

2

2.1

2.2

Figure 12.1: Typical convergence behavior of a finite element solution

Onset of asymptotic range (BC): The range BC is called onset of asymptotic range. In this range addition of degrees of freedom to the discretization results in measurable reduction in ||E||L2 reflecting progressive improvement in accuracy of the computed solution φh from B to C. In this range I e values or any other possible element error indicators are more accurate than range AB. In this range adaptive processes in h, p, or hp can be utilized keeping in mind that as we move closer to C, the values of I e (or other indicators) for the elements of the discretization become more accurate, hence can be more effective in the adaptive process. Asymptotic range (CD): In this range as more dofs are added to the discretization the improvement (reduction) in ||E||L2 is most significant. This range on log-log scale is nearly linear, hence constant slope. Adaptive refinements in this range are most effective in reducing ||E||L2 . We observe that between C and D there are several orders of magnitude reduction in the value of ||E||L2 . Slope of the error norm versus dof graph in this range is called the asymptotic convergence rate of the finite element solution. Onset of post-asymptotic range (DE): This range is almost reverse of the onset of asymptotic range. In this range reduction in ||E||L2 progressively diminishes with the addition of degrees of freedom to the discretization indicating that substantial achievable reduction in ||E||L2 has taken place up to point D. Computations in this range result in waste of significant resources (dofs) with very little gain in the objective of reducing ||E||L2 .

692

CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

Post-asymptotic range (EF ): In this range in spite of the addition of dofs to the discretization no measurable reduction is observed in ||E||L2 . This is generally due to the fact that within the accuracy of the computations (i.e. the word size on the computer we have reached a limit), hence the accuracy remains limited to the same number of decimal places in ||E||L2 regardless of the increase in dofs.

12.3.2 Convergence rates In an abstract sense the convergence rate of a finite element computational process is the rate at which the computed solution φh is approaching the theoretical solution φ as more degrees of freedom are added to the discretization through refining h or increasing p or changing k. That is it is the rate at which the error norm is approaching zero as more degrees of freedom are added. Thus, a measure of convergence rate of the finite element solution √ could be the slope of I (or ||E||L2 ) versus dof behavior. Since dofs can be added through h, p, and k, the convergence rate of a finite element solution can be a function of h, p, k, and the smoothness of the theoretical solution at this stage of the discussion. In range AB, the slope is almost zero. From B to C the slope increases as more dofs are added to the discretization thereby progressively increasing convergence rate from B to C. From C to D, the asymptotic range, the slope of ||E||L2 versus dofs is almost constant and the reduction in ||E||L2 is most significant as more dofs are added. Thus, in the asymptotic range the convergence rate is the highest (due to highest slope of log(||E||L2 ) versus log(dof )) and is constant. In the onset of post-asymptotic range DE the convergence rate decreases and eventually becomes almost zero in the postasymptotic range EF . Remarks. (i) Behavior of ||E||L2 versus dofs shown in Fig. 12.1 is typical of other error norms as well, hence the discussion and conclusions related to Fig. 12.1 are applicable in the convergence behavior study using any other desired error norm. (ii) Pre-asymptotic range AB, onset of post-asymptotic range DE, and post-asymptotic range EF should be avoided as in these ranges solution accuracy improvement is poor. (iii) In range AB I e values (or other measures) are not accurate enough to guide an adaptive process of any kind. (iv) Adaptive processes (h, p, k) can be initiated in the range BC as I e values in this range are reasonable measure of error. Adaptive processes

12.4. ERROR ESTIMATION AND ERROR COMPUTATION

693

become more and more effective when we initiate them as we approach from B to C, i.e. closer to C. In the range BC the slope of ||E||L2 versus dof increases from B to C indicating improving convergence rate and eventually achieves the highest convergence rate value at C which remains almost constant in the asymptotic range CD. (v) A priori and a posteriori error estimates are only valid in the asymptotic range due to the fact it is only in this range that computed φh has desired regularity and the convergence rate is the highest, hence worth estimating a priori. The error estimates (a priori and a posteriori) can neither be derived accurately nor can be used meaningfully in regions other than BC.

12.4 Error estimation and error computation There are two types of error estimations generally considered: a priori error estimation and a posteriori error estimation. A priori error estimation refers to establishing dependence of some error norm on h, p, k, and the regularity of the theoretical solution before the computations are performed so that we have knowledge of the precise nature of the functional dependence of error norm on h, p, k, and the regularity of the theoretical solution. A posteriori error estimation refers to error estimates derived using a computed solution with specific choices of h, p, and k. The sole purpose of a posteriori error estimation is to use current finite element solution to derive element indicators that can perhaps be used to guide an adaptive process. Both of the error estimations require some regularity of the computed solution which only exists in the asymptotic range (range CD, Fig. 12.1). This is a very significant restriction on the use of these estimates. For example, a priori error estimate can not be used to predict convergence rate in the ranges AB, BC, DE, and EF as this is specifically derived using the regularity of φh that only exists in the asymptotic range. Likewise a posteriori estimate can not be used for adaptivity in any ranges except CD. Another point to note is that a posteriori error estimates are generally derived such that they quantify the weakness(es) in the finite element global approximation φh = ∪φeh . Their derivations are largely based on C 0 local e approximations which result in interelement discontinuity of the first derivatives normal to the interelement boundaries. This may be quantified by establishing bounds that can be used for adaptivity. However if we use φeh of class C 1 thereby φh of class C 1 , then such bounds are meaningless. In k-version of finite element methods enabling higher order global differentiability approximations, majority of the a posteriori error estimates based on interelement discontinuity of the derivatives are not meaningful. With the use of higher order approximations φh , the integrals can be maintained Rie-

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CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

mann, ||E||L2 and ||E e ||L2 are true measures of the error in the finite element ¯ T and Ω ¯ e and can indeed be used in adaptive processes. These solution for Ω aspects are discussed in more details in later sections.

12.5 A priori error estimation: derivation of error estimate and convergence rates The published literature on a priori error estimation is almost exclusively for BVPs described by self adjoint differential operators in which the integral form, hence the finite element formulation, is constructed using Galerkin method with weak form. The finite element solution using GM/WF for BVPs described by self adjoint differential operators has best approximation property in B-norm. This property has been viewed as essential in deriving the a priori error estimates. Thus, we begin with boundary value problems described by self adjoint differential operators.

12.5.1 Galerkin method with weak form (GM/WF): self-adjoint operators In this section we revisit main steps of GM/WF for self-adjoint operators. Let Aφ − f = 0

in

Ω

(12.14)

be a boundary value problem in which the differential operator A is symmetric and its adjoint A∗ = A (i.e. the differential operator A is self adjoint). Based on fundamental lemma (chapter 2) we can write the following integral form: Z Z Z (Aφ − f, v)Ω¯ = (Aφ − f )v dΩ = (Aφ)v dΩ − f v dΩ = 0 (12.15) ¯ Ω

¯ Ω

¯ Ω

in which v = 0 on Γ∗ if φ = φ0 (given) on Γ∗ . v is called test function, hence v = δφ is admissible in (12.15). When v = δφ in (12.15), the integral form (12.15) is called integral form in Galerkin method. Since A is self adjoint, the BVP (12.14) only contains even order derivatives of φ. We transfer half of the differentiation from φ to v using integration by parts in the first term in (12.15) and collect those terms that contain both φ and v and define them collectively as B(φ, v) and those that contain only v and define them as l(v), hence we can write the following. B(φ, v) = l(v)

(12.16)

12.5. A PRIORI ERROR ESTIMATION

695

Each term in B(φ, v) contains both φ and v but more importantly the orders of derivatives of φ and v in each term is same (i.e. B(φ, v) is symmetric), thus B(φ, v) = B(v, φ) (12.17) and since A is linear, B(φ, v) is bilinear in φ, v and l(v) is linear in v. Hence in this case quadratic functional I(φ) is possible and is given by 1 I(φ) = B(φ, φ) − l(φ) 2

(12.18)

The integral form (12.16) is called weak form of (12.14). Due to the fact that (12.15) is integral form in Galerkin method, the weak form (12.16) is called integral form in Galerkin method with weak form (GM/WF). The quadratic functional I(φ) has physical significance as explained in chapters 2 and 5. If (12.14) represents a BVP associated with linear elasticity in solid mechanics, then 12 B(φ, φ) is strain energy, l(φ) is potential energy of loads and I(φ) is the total potential energy of the system described by (12.14). Theorem 12.1. The weak form B(φh , v) = l(v) resulting from GM/WF for self adjoint differential operator A in Aφ − f = 0 in which B(·, ·) is symmetric is variationally consistent. Proof. Variational consistency of the weak form B(φh , v) = l(v) requires that there exist a functional I(φh ) such that δI(φh ) = 0 gives the weak form, the Euler’s equation resulting from δI(φh ) = 0 is the BVP, and δ 2 I(φh ) yields unique extremum principle. Following section 12.5.1 the existence of the functional I(φh ) is by construction (equation (12.18)) 1 I(φh ) = B(φh , φh ) − l(φh ) 2 If I(φh ) is differentiable in φh , then δI(φh ) = 0 is a necessary condition for an extremum of I(φh ). Using δφh = v (due to GM/WF), 1 1 δI(φh ) = B(v, φh ) + B(φh , v) − l(v) = 0 2 2 Since B(·, ·) is symmetric, we obtain δI(φh ) = B(φh , v) − v) = 0 or B(φh , v) = l(v), the weak form The unique extremum principle (or sufficient condition) is given by
δ 2 I(φh ) = δ B(φh , v) − l(v) = B(v, v) > 0 ∀v ∈ Vh ⊂ H

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CONVERGENCE, ERROR ESTIMATION, AND ADAPTIVITY

Hence, a unique extremum principle. To show that the Euler’s equation resulting from the weak form is in fact the BVP, we just have to transfer differentiation back to φ (or φh ) from v in the weak form using integration by parts. This is rather straightforward. Thus, the weak form B(φh , v) resulting from the GM/WF is variationally consistent. δ 2 I(φh ) = B(v, v) > 0 implies that a φh from the weak form minimizes I(v), ∀v ∈ Vh , I(φh ) ≤ I(v) ∀v ∈ Vh

Theorem 12.2. Let Aφ − f = 0 be a BVP in which A is self adjoint and let B(φh , v) = l(v) be weak form resulting from GM/WF in which B(φh , v) = B(v, φh ) and φh , v ∈ Vh ⊂ H, then φh has best approximation property in B(·, ·)-norm. That is, if e = φ − φh , φ ∈ H being theoretical solution, then (a)

B(e, v) = 0 ∀v ∈ V

(b)

B(e, e) ≤ B(φ − w, φ − w), ∀w ∈ V

Proof. (a) B(φh , v) = l(v) B(φ, v1 ) = l(v1 ), v1 ∈ H Choosing v1 = v ∈ V ⊂ H B(φ, v) = l(v) Hence, B(φ − φh , v) = 0 or B(e, v) = 0 This implies that no element of V is a better approximation of φ than φh , the solution for the weak form when measured in B(·, ·) as e is B(·, ·)orthogonal to every element v of V . This is called the best approximation property of GM/WF for self adjoint operators. (b) For any v ∈ Vh B(e + v, e + v) = B(e, e) + 2B(e, v) + B(v, v) But B(e, v) = 0, hence B(e + v, e + v) = B(e, e) + B(v, v)

697

12.5. A PRIORI ERROR ESTIMATION

Since B(v, v) > 0 we have B(e, e) ≤ B(e + v, e + v) e + v = φ − φh + v = φ − (φh − v) φh , v ∈ V ;

hence φh − v = w ∈ Vh

Thus, B(e, e) ≤ B(φ − w, φ − w) ∀w ∈ Vh ⊂ H or B(φ − φh , φ − φh ) ≤ B(φ − w, φ − w) or ||φ − φh ||B ≤ ||φ − w||B That is, error in φh in B-norm is the lowest compared to any other solution w. This completes the proofs of (a) and (b).

12.5.2 GM/WF for non-self adjoint and non-linear operators Theorem 12.3. Let Aφ − f = 0 in Ω be a BVP in which A is a non-self adjoint differential operator. Let B(φ, v) − l(v) = 0 be all possible weak forms. Then all such integral forms are variationally inconsistent. Proof. Let there exist a functional I(φ) such that δI(φ) = 0 yield the weak form B(φ, v) − l(v) = 0. Since A is non-self adjoint, B(φ, v) is bilinear but not symmetric (i.e. B(φ, v) 6= B(v, φ)), hence δ 2 I(φ) = δ(B(φ, v) − l(v)) = B(δφ, v)    >0 = B(v, v) = 0 ∀v ∈ V   0 ∀v ∈ Vh

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12.5. A PRIORI ERROR ESTIMATION

Hence, the integral form resulting from δI(φh ) = 0 is variationally consistent. Theorem 12.6. The integral form in least-squares method based on residual functional is variationally consistent when the BVP is described by non-self adjoint operator. Proof. Since non-self adjoint operators are linear the proof of this theorem is same as that for self adjoint operators (Theorem 12.5) which are also linear.

12.5.4 Least-squares method based on residual functional for non-linear operators Theorem 12.7. Let Aφ − f = 0 in Ω be a boundary value problem in which A is a non-linear differential operator. Let φh be approximation of φ in ¯ T = ∪Ω ¯ e , discretization of Ω ¯ and let Aφh − f = E be the residual function Ω e ¯ Then the integral form resulting from the first variation of the residual in Ω. ∼ (δE , δE) functional I(φh ) = (E, E) set to zero is VC provided δ 2 I(φh ) = and the system of non-linear algebraic equations resulting from δI(φh ) = 0 are solved using Newton-Raphson or Newton’s linear method. Proof. Since A is non-linear, E is a non-linear function of φh , hence δE is a function of φh . I(φh ) = (E, E) = (Aφh − f, Aφh − f ) ;

existence of I(φh )

If I(φh ) is differentiable in φh , then δI(φh ) = 2(E, δE) = 2g(φh ) = 0 Hence, g(φh ) = 0 is a necessary condition. Since δE = δ(Aφh − f ) = δA(φh ) + Av, g(φh ) = (Aφh − f, δA(φh ) + Av) = 0 or
Aφh , Av + δA(φh ) = (f, δA(φh ) + Av) or B(φh , v) = l(v) Also   > 0 δ 2 I(φh ) = 2(δE, δE) + 2(E, δ 2 E) = 0  