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Equilibrium finite element formulations
 9781118926208, 111892620X, 9781118926215, 1118926218, 9781118925782

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Equilibrium Finite Element Formulations

Equilibrium Finite Element Formulations J. P. Moitinho de Almeida Department of Civil Engineering, Architecture and Georesources, Instituto Superior Técnico, University of Lisbon, Portugal

Edward A. W. Maunder College of Engineering, Mathematics and Physical Sciences, University of Exeter, United Kingdom

This edition first published 2017 © 2017 John Wiley & Sons Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom. For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com. The right of the author to be identified as the author of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988. All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. Limit of Liability/Disclaimer of Warranty: While the publisher and authors have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. It is sold on the understanding that the publisher is not engaged in rendering professional services and neither the publisher nor the author shall be liable for damages arising herefrom. If professional advice or other expert assistance is required, the services of a competent professional should be sought. Library of Congress Cataloging-in-Publication Data Names: Almeida, J. P. Moitinho de (Jose Paulo), 1958- author. | Maunder, E. A. W., author. Title: Equilibrium finite element formulations / J. P. Moitinho de Almeida, Edward A. W. Maunder. Description: Chichester, UK ; Hoboken, NJ : John Wiley & Sons, 2016. | Includes bibliographical references and index. Identifiers: LCCN 2016039762 (print) | LCCN 2016045455 (ebook) | ISBN 9781118424155 (cloth) | ISBN 9781118926215 (pdf ) | ISBN 9781118926208 (epub) Subjects: LCSH: Finite element method. | Equilibrium. | Structural analysis (Engineering)–Mathematics. Classification: LCC TA347.F5 A394 2016 (print) | LCC TA347.F5 (ebook) | DDC 518/.25–dc23 LC record available at https://lccn.loc.gov/2016039762 A catalogue record for this book is available from the British Library. Cover Design: Wiley Cover Credits: borzaya/Gettyimages Set in 10/12pt Warnock by SPi Global, Chennai, India

10 9 8 7 6 5 4 3 2 1

To • Bruce M. Irons • Baudouin M. Fraeijs de Veubeke • John C. de C. Henderson For their inspiration to think differently.

vii

Contents Preface xiii List of Symbols

xvii

1

Introduction 1

1.1 1.2 1.3 1.4 1.5 1.5.1 1.5.2 1.6

Prerequisites 1 What Is Meant by Equilibrium? Weak to Strong Forms 2 What Do We Gain From Strong Forms of Equilibrium? 3 What Paths Have Been Followed to Achieve Strong Forms of Equilibrium? 5 Industrial Perspectives 6 Simulation Governance 7 Equilibrium in Structural Design and Assessment 7 The Structure of the Book 8 References 9

2

Basic Concepts Illustrated by Simple Examples 11

2.1 2.2 2.2.1 2.2.2 2.2.3

Symmetric Bi-Material Strip 12 Kirchhoff Plate With a Line Load 16 Kinematically Admissible Solutions 16 Statically Admissible Solutions 19 Assessment of the Solutions Obtained 20 References 21

3

Equilibrium in Other Finite Element Formulations 22

3.1 3.2 3.3 3.4

Conforming Formulations and Nodal Equilibrium 22 Pian’s Hybrid Formulation 25 Mixed Stress Formulations 27 Variants of the Displacement Based Formulations With Stronger Forms of Equilibrium 28 Fraeijs de Veubeke’s Equilibrated Triangle 29 Triangular Equilibrium Elements for Plate Bending 30 Other Variants 31 Trefftz Formulations 32 Formulations Based on the Approximation of a Stress Potential 33 The Symmetric Bi-Material Strip Revisited 33 References 40

3.4.1 3.4.2 3.4.3 3.5 3.6 3.7

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Contents

4

Formulation of Hybrid Equilibrium Elements 43

4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.8.1 4.8.1.1 4.8.1.2 4.8.2 4.8.3 4.8.3.1 4.8.3.2 4.8.3.3 4.8.4 4.9 4.10 4.11

Approximation of the Stresses 43 Approximation of the Boundary Displacements 45 Assembling the Approximations 48 Enforcement of Equilibrium at the Boundaries of the Elements 48 Enforcement of Compatibility 51 Governing System 53 Existence and Uniqueness of the Solution 54 Elements for Specific Types of Problem 57 Continua in 2D 57 Exemplification of the Assembly Process 58 A Simple Numerical Example 60 Continua in 3D 62 Plate Bending 63 Reissner–Mindlin Theory 64 Kirchhoff Theory 65 Example 66 Potential Problems of Lower Order 66 The Case of Geometries With a Non-Linear Mapping 68 Compatibility Defaults 69 The Dimension of the System of Equations 70 References 71

5

Analysis of the Kinematic Stability of Hybrid Equilibrium Elements 73

5.1 5.2 5.2.1 5.2.2 5.2.3 5.2.3.1 5.2.3.2 5.2.3.3 5.2.3.4 5.2.3.5 5.2.3.6 5.2.3.7 5.2.3.8 5.2.4 5.3 5.3.1 5.3.1.1 5.3.1.2 5.3.2 5.3.2.1 5.3.2.2 5.3.3 5.3.3.1

Algebraic and Duality Concepts Related to Spurious Kinematic Modes Spurious Kinematic Modes in Models of 2D Continua 76 Single Triangular Elements 77 A Pair of Triangular Elements With a Common Interface 80 Star Patches of 2D Elements 82 Open Stars of Degree 0 84 Closed Stars of Degree 0 84 Open Stars of Degree 1 84 Closed Stars of Degree 1 85 Open Stars of Degree 2 85 Closed Stars of Degree 2 85 Examples of Unstable Closed Star Patches of Degree 2 86 Stars of Degree 3 or Higher 87 Observations for General 2D Meshes 87 Spurious Kinematic Modes in Models of 3D Continua 90 Single Tetrahedral Elements 90 Spurious Modes Associated With a Single Edge 92 Spurious Modes Associated With a Single Face 94 A Pair of Tetrahedral Elements 94 Primary Interface Spurious Modes 95 Pairs of Tetrahedral Elements With Coplanar Faces 96 Star Patches of Tetrahedral Elements 97 Edge-Centred Patches 98

73

Contents

5.3.3.2 5.4 5.4.1 5.4.2 5.4.3 5.4.3.1 5.4.3.2 5.4.3.3 5.4.3.4 5.4.4 5.5 5.6 5.7 5.7.1 5.7.2

Vertex-Centred Patches 98 Spurious Kinematic Modes in Models of Reissner–Mindlin Plates 99 A Single Triangular Reissner–Mindlin Element 100 A Pair of Reissner–Mindlin Elements 102 Star Patches of Reissner–Mindlin Elements 103 Open Stars of Degree 1 103 Closed Star Patches of Degree 1 103 Open Stars of Degree 2 103 Closed Star Patches of Degree 2 103 Observations for General Meshes of Reissner–Mindlin Elements 104 The Stability of Plates Modelled With Kirchhoff Elements 105 The Stability of Models for Potential Problems 106 How Do We Obtain a Stable Mesh for General Structural Models? 108 General Procedures 108 Macro-Elements 108 References 109

6

Practical Aspects of the Kinematic Stability of Hybrid Equilibrium Elements 111

6.1 6.1.1 6.1.2 6.2 6.3 6.4 6.5

Identification of Rigid Body and Spurious Kinematic Modes 111 Spurious Kinematic and Rigid Body Modes of an Element 112 Spurious Kinematic and Rigid Body Modes of a Mesh 113 Blocking the Spurious Modes 115 An Illustration of the Procedures to Remove Spurious Modes 116 How Do We Recognize Admissible Loads? 117 Quasi-Simplicial Hybrid Elements Created by Hierarchical Mesh Refinement 118 Non-Simplicial Hybrid Elements 120 A Cautionary Tale of ‘Near Misses’ 120 References 125

6.6 6.7

7.1 7.1.1 7.1.2 7.1.3 7.2 7.3 7.4

126 Potential Energy and Complementary Potential Energy 126 Existence and Uniqueness of Solutions 129 Properties of the Exact Solution 129 The Formal Relation Between Both Energies 130 Hybrid Complementary Potential Energy 131 Properties of the Generalized Complementary Energy 132 The Babuška–Brezzi Condition and Hybrid Equilibrium Elements 133 References 134

8

Recovery of Complementary Solutions 135

8.1 8.2 8.2.1 8.2.2 8.3 8.3.1

General Features of Partition of Unity Functions 136 Recovery of Compatibility From an Equilibrated Solution 138 Derivation of ũ E 140 An Illustration of the Technique 141 Recovery of Equilibrium From a Compatible Solution 143 Recovery From Star Patches: The General Case 144

7

A Variational Basis of the Hybrid Equilibrium Formulation

ix

x

Contents

8.3.2 8.3.3 8.3.3.1 8.3.3.2 8.3.3.3 8.3.3.4 8.4 8.4.1 8.4.2 8.5 8.5.1 8.5.2 8.5.2.1 8.5.2.2 8.5.3

Recovery From Star Patches: The Case of Linear Displacements 146 Element by Element Recovery of Equilibrium 150 Resolution of the Vertex Forces 150 Derivation of Statically Equivalent Codiffusive Tractions 153 Admissibility of the Derived Tractions 155 Derivation of the Element Stress Fields 156 Numerical Examples 157 Recovery of Compatibility From an Equilibrated Solution 157 Recovery of Equilibrium From a Compatible Solution 160 Extensions of the Recovery Procedures 163 Reissner–Mindlin Theory 163 Kirchhoff Theory 163 Recovery of Compatibility 163 Recovery of Equilibrium 164 Non-Simplicial Elements 164 References 164

9

166 Global Error Bounds 167 Revisiting the simple example 170 Estimation of the Error Distribution and Global Mesh Adaptation 177 The Convergence of the Simple Example 180 Obtaining Local Quantities of Interest 184 Bounding the Error of Local Outputs 187 Background 187 Bounds of the Error of Outputs Obtained From Complementary Solutions 187 Local Outputs for the Kirchhoff Plate With a Line Load 189 The Displacement at the Corner 190 The Average Displacement on the Loaded Side 192 The Average Displacement on the Free Side 193 Estimation of the Error Distribution and Mesh Adaptation for Local Quantities 194 Adaptivity for Multiple Loads and Multiple Outputs 195 References 196

9.1 9.1.1 9.2 9.2.2 9.3 9.4 9.4.1 9.4.2 9.5 9.5.1 9.5.2 9.5.3 9.6 9.7

Dual Analyses for Error Estimation & Adaptivity

10

Dynamic Analyses 199

10.1 10.2 10.3 10.4 10.5 10.6 10.6.1 10.6.2

Toupin’s Principle for Elastodynamics 200 Derivation of the Equilibrium Finite Element Equations 201 Analysis in the Frequency Domain 203 Analysis in the Time Domain 205 No Direct Bounds of the Eigenfrequencies? 206 Example 207 Eigenfrequencies 207 Forced Vibrations 209 References 211

Contents

11

Non-Linear Analyses 212

11.1 11.2 11.2.1 11.2.2 11.2.2.1 11.2.2.2 11.2.2.3

Elastic Contact 212 Material Non-Linearity 214 Non-Linear Elasticity 214 Elastoplastic Constitutive Relations 215 Direct Implementation 216 A Standard Return Mapping Implementation 217 A Return Mapping Implementation for Plasticity Defined in the Strain Space 218 Imposing the Yield Condition in a Weak Form 219 Limit Analysis 220 Introduction 220 General Statement of the Problem as a Mathematical Programme 220 Formulation (1) 221 Formulation (2) 221 Yield Constraints 222 Application of the Complementary (Dual) Programme 222 Implementation for Plate Bending Problems 222 Numerical Example 223 Geometric Non-Linearity 224 Weak Compatibility for Large Displacements With Small Strains 225 Equilibrium 227 Transformation of Boundary Displacement Parameters and Generalized Tractions 228 Governing System 229 Determination of the Rigid Body Displacements 229 Tangent Form of the Governing System 230 Variation of the Rigid Body Displacements 230 The Effect of a Variation in the Boundary Displacement Parameters on the Associated Transformations 231 Tangent Form of the Governing System for an Element 233 Large Displacements and Spurious Kinematic Modes 233 Numerical Example 234 References 235

11.2.2.4 11.3 11.3.1 11.3.2 11.3.2.1 11.3.2.2 11.3.2.3 11.3.2.4 11.3.3 11.3.4 11.4 11.4.1 11.4.2 11.4.3 11.4.4 11.4.5 11.4.6 11.4.6.1 11.4.6.2 11.4.6.3 11.4.7 11.4.7.1

A

Fundamental Equations of Structural Mechanics 237

A.1 A.1.1 A.1.2 A.1.3 A.1.4 A.1.4.1 A.1.4.2 A.2 A.2.1 A.2.2

The General Elastostatic Problem 237 Two Dimensional Elasticity 237 Three Dimensional Elasticity 238 Shear Stresses and Warping of a Beam Section 240 Plate Bending 245 Reissner–Mindlin Theory 247 Kirchhoff Theory 248 Compatibility of Strains 250 Integrability Conditions 250 Enforcement of the Kinematic Boundary Conditions 251

xi

xii

Contents

A.3

General Elastodynamic Problem References 252

B

Computer Programs for Equilibrium Finite Element Formulations 254

B.1 B.1.1 B.1.2 B.1.3 B.2 B.2.1 B.2.2 B.2.3 B.2.4 B.2.5 B.2.6 B.2.6.1 B.2.6.2

Auxiliary Programs 255 gmsh 255 The mche and mchf Classes 258 mtimesx 258 Structure of the Programs 259 Definition of the Mesh 259 Definition of the Material Properties and Boundary Conditions 260 Definition of the Approximation Functions 261 Enforcement of Boundary Conditions 263 Processing the Solutions 265 Code Snippets 265 Computing the Flexibility Matrix of an Element 265 The Equilibrium Matrix of a Side of a Plane Element 267 References 269 Subject Index 271

252

xiii

Preface This book is the result of many years of joint work and fun with equilibrium finite element formulations. Although they are often regarded as the ugly duckling of computational mechanics, we know that they have characteristics that are particularly attractive. It thus became our mission to spread the word that a ‘strongly equilibrated finite element’ is not a contradiction. It all started a long time ago, in the late 1960s, when Edward, then a PhD student in civil engineering at Imperial College, attended a lecture by Fraeijs de Veubeke.1 In retrospect he now has some questions that might have been interesting at the time, but he also admits that the recognition of the practical relevance of the equilibrium formulations for continua that were presented only really came later, while doing structural design of reinforced concrete using stress fields obtained from displacement based finite elements. Edward was also fortunate to have John Henderson as supervisor and later as a friend, colleague and father figure. John was a polymath with a firm belief in the benefits of a proper mathematical foundation to structural analysis, from vector spaces to algebraic topology. He encouraged the transfer of knowledge gained from the analysis of aircraft structures to the analysis of civil engineering structures, using concepts of static-kinematic duality in the pursuit of equilibrium via flexibility methods. Zé Paulo’s discovery of how to impose strong forms equilibrium, and as consequence the discovery of Fraeijs de Veubeke’s and John Henderson’s work, happened in 1985, also at Imperial College. His objective was to figure out the characteristics of the solutions of simple elastoplastic models which could enforce either equilibrium or compatibility, leading to interesting complementarities in the results. Bruce Iron’s book Techniques of Finite Elements, published in 1980, with its clear, imaginative and friendly style, focused on ‘enabling the understanding of mathematical and physical concepts, because effective trouble-shooting is best achieved with such harmony’, also had a very strong influence on both authors. The stars were thus aligned for equilibrium when we first met in the office of David Lloyd Smith at Imperial College, sometime in 1992.2 After many papers, projects and conferences, where stress fields and spurious kinematic modes were dissected to exhaustion, we decided to collect our ideas in a book. 1 Probably in the Department of Aeronautics, where he also attended memorable lectures given by Kelsey, of Argyris and Kelsey fame! 2 Following the publication of ‘An alternative approach to the formulation of hybrid equilibrium finite elements’.

xiv

Preface

The resulting text provides a comprehensive presentation of an equilibrium formulation of the finite element method, principally with application to 2D, 3D and plate flexure problems in structural mechanics, when strains can be assumed to be infinitesimally small. Equal weight is given to the construction of stress fields that strongly satisfy equilibrium within and between elements, as well as displacements on element boundaries that are ‘broken’ or discontinuous at vertices of 2D plate elements or edges of solid elements. We present up-to-date developments which enable dual analyses of models to be undertaken, either as a means of verification or as an alternative source of output that may be more directly useful to design engineers. The book starts with a simple introduction of the concepts involved, followed by a historical introduction of equilibrium in the context of finite element analysis, and comparison with other formulations. We discuss the details of the equilibrium formulation in the context of modelling linear elastic static and dynamic behaviour with particular emphasis on the associated problems of spurious kinematic modes. A more mathematical justification of the formulation is included, where we propose a relevant functional to be used in the variational analysis of the saddle point problem, and attempt to explain its significance in engineering terms to a non-mathematician. We then proceed to present methods to recover complementary conforming and equilibrating solutions from each other, and show how the dual nature of such solutions enables bounds to be enumerated on global or local quantities of interest. The text concludes by opening routes to extending the formulation in order to simulate various forms of non-linear behaviour. We make particular effort to explain the more mathematical concepts in straightforward terms which we hope will be understandable by the intended readership, namely senior undergraduates of engineering and applied mathematics, graduate researchers and practising engineers with an interest in verification, duality and safe structural design. The topics are illustrated with a range of numerical examples which have been carefully designed to be simple, but of just sufficient complexity to highlight particular features. The book contains two appendices: the first to summarize the fundamental equations of structural mechanics, and the second to serve as a companion to the computer programs that were developed in the course of writing the book. These programs are available on request to the first author and we intend to publish them, when they are more mature, under an open source licence. The time from initial thoughts, in 2009, to the formal proposal, submitted in April 2011 and accepted in March 2012, was almost as long as that to complete the text. As usual, longer than we had anticipated. We thank, first and foremost, our families for their patience and moral support. Our gratitude also extends to all the colleagues and friends who, directly or indirectly, and in many different ways, have helped us in developing our ideas before or during the writing of this book: in particular to Orlando Pereira and Pierre Beckers who read the manuscript, as well as to Angus Ramsay, Antonio Huerta, Bassam Izzuddin, Bill Harvey, Carlos Tiago, David Lloyd Smith, Eduardo Arantes e Oliveira, João Teixeira de Freitas, John Robinson, Luiz Fernando Martha, Pedro Diez, Philippe Bouillard and Pierre Ladevèze.

Preface

We do not forget to thank all the people in the editorial and production teams at Wiley, for their patience and advice throughout the preparation of this book. A special mention should be made regarding the helpful suggestions by the copy editor, Chris Cartwright. Our recognition obviously includes all those who we have forgotten – sorry for that. We hope you enjoy this book as much as we enjoyed writing it. Lisbon and Exeter, March 2016 Zé Paulo and Edward

xv

xvii

List of Symbols

The general meaning of the most commonly used symbols is given in this list. Other meanings may be specified in the text.

A  b or b𝛼 C 𝝌 or 𝜒𝛼𝛽  or  ⋆ D d◽ 𝚫 E 𝜺 or 𝜀𝛼𝛽 𝜖  f Γ 𝜸 or 𝛾𝛼 K k k𝛼𝛽 L𝛼 𝝀(𝝃) l Lk(V ) () m or m𝛼𝛽   n◽ N(V ) ‖◽‖ 𝜈

Nullspace of DT Area of a face (Generalized) body force vector or component Equilibrium/compatibility matrix for the mixed formulations Curvature vector or component Differential compatibility or equilibrium operator Equilibrium/compatibility matrix for the hybrid formulations Degree of a given polynomial approximation Rigid body displacement Young’s modulus (Generalized) strain vector or component Bound of the error of a pair of solutions Element flexibility matrix Material flexibility matrix Boundary of the problem Shear strain vector or component Element stiffness matrix Material stiffness matrix Signature function for an edge (𝛽 = + or −) or a face (𝛽 = 1 … 3) Area coordinate on a face Lagrange multiplier function Length of a side Link of vertex V Local output Distributed bending moment vector or component Mobility matrix Boundary normal operator Number of parameters of a given approximation Simplicial neighbourhood of vertex V Energy norm (may be defined for a displacement, strain, or a stress field) Poisson’s ratio

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List of Symbols

Domain of the problem Eigenfrequency Internal angle at a vertex of a triangle or dihedral angle at an edge of a tetrahedron 𝜑̄ cot 𝜑 Π Total potential energy Total complementary energy Πc Generalized complementary energy ΠGc Ψ Partition of unity function 𝝍 or 𝜓 𝛼 Interpolation matrix or function Distributed shear force vector or component q or q𝛼 𝜌 Mass density S Stress approximation matrix 𝝈 or 𝜎𝛼𝛽 (Generalized) stress vector or component ŝ Stress approximation parameters  Set (including vector space) of statically admissible stress fields T Kinetic energy density (Generalized) boundary traction vector or component t or t𝛼 Rotation vector or component 𝜽 or 𝜃𝛼  Set (including vector space) of side tractions U Strain energy Complementary strain energy Uc (Generalized) displacement vector or component u or u𝛼 Set (including vector space) of kinematically admissible displacement fields k 𝒗 Boundary displacement V Boundary displacement approximation matrix 𝒗̂ Boundary displacement approximation parameters  Set (including vector space) of boundary displacements V Work done by the applied forces Work done by the imposed displacements Vc W Strain energy density Complementary strain energy density Wc 𝑤 Transverse displacement of a plate 𝝃 Coordinate on the boundary of an element ◽e or ◽m Vector associated with element e or boundary entity m ◽e𝛼 or ◽m𝛼 Component of vector associated with element e or boundary entity m Ω 𝜔 𝜑

1

1 Introduction

1.1 Prerequisites A very concise description of this book is that it presents a methodology to predict and explain the distribution of forces and deflections that develop within a loaded structure.

For a layman who is unfamiliar with structural analysis, this description requires further explanations of many important points, namely: What is a structure? What are the forces within the structure? What are loads? Why and how do they get distributed? We will not try to address these questions. For their answers a basic book on structural analysis, for example, Coates et al. (1988); Hibbeler (2008) or Marti (2013) will provide the necessary knowledge on the concepts used to describe structural behaviour – equilibrium, compatibility and constitutive relations – as well as the variables involved – forces, displacements, stresses and strains. For all but the simplest problems, the mathematical equations used to describe the relations between these structural variables cannot be solved in a closed form. Of the various techniques that are used to obtain approximate solutions of these equations we will focus our attention on the application of a particular technique, the finite element method (FEM). Though it is possible to gain an understanding of FEM concepts solely from the information that will be presented in this text, it is more convenient to start with a basic book on finite element procedures, for example, Fish and Belytschko (2007). We will therefore assume that the reader has a basic knowledge of the problems of structural analysis, namely of the fundamental equations of solid mechanics and, at least, some understanding of the procedures involved in the application of the FEM, most probably using a conventional displacement based formulation. Such a reader, probably with an engineering background in the context of aeronautical, civil or mechanical engineering applications, given a title which includes ‘equilibrium’

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

2

Equilibrium F.E. Formulations

and ‘finite elements’ might rather wonder: ‘Why another book? The finite element method is well known and it provides solutions that satisfy equilibrium. Doesn’t it?’ The fact is that in most cases it doesn’t, since only an approximate form of equilibrium is achieved by displacement based finite element formulations. Our text presents a way to obtain solutions that are different from the ‘usual’ ones, because they exactly satisfy equilibrium. Nevertheless, since they normally omit the strict enforcement of compatibility conditions, it is not possible to say a priori which will be better. They just fail in different ways. We believe that exploiting the complementarity of the two approaches allows for an interpretation of the results that is more profound than what is possible with a single type of analysis, naturally providing the tools for the assessment of their quality. That, in the end, is our goal. Explaining in detail how equilibrated solutions can be obtained is just a step towards it.

1.2 What Is Meant by Equilibrium? Weak to Strong Forms We expect the reader to understand what is meant by a free body, and being in a state of equilibrium, that is, the forces and their moments sum to zero. However, although checks on equilibrium at the global or overall level of a structure, for example, as represented by its finite element model, are commonly undertaken, deeper investigations into local levels of equilibrium become more problematic. In FEM there are various shades of meaning, and perhaps expectation, when considering local equilibrium. The concept of a free body normally starts at the level of an infinitesimal element in a continuum (i.e. strong equilibrium between body forces and stresses), which is itself a mathematical abstraction – since we ignore the microscopic structure of the material. Then the concept moves to the level of a single finite element, and then it may move back to another mathematical abstraction – a node of an element, where we invoke the concept of nodal forces (i.e. corresponding to a weak form of equilibrium between statically equivalent forces). It is relevant here to note that the concept of a nodal force may not be explicitly mentioned in texts on finite elements, and we are aware of commercially available software where nodal forces are not available to the user, but only stress contours and tables of stresses at particular points! In practice some confusion exists, and engineers may be unaware of the ‘subtle’ distinctions between these different levels of equilibrium of free bodies, and their significance to the analysis of a finite element model. We frequently hear of engineers who look blank when advised that local equilibrium is usually violated – they appear to have a firm conviction that equilibrium is being satisfied in all necessary aspects. Their first response might be: ‘Does it matter if there are local violations?’ An appropriate reply might be: ‘It all depends on your needs and how well you know the distribution of the loads.’ This is a matter of judgement, but we would advise that engineers, when faced with many uncertainties, can proceed with more confidence knowing that their analysis provides complete equilibrium. Local violations can be regarded as residual loads that are equilibrated by the errors in stress, and such loads are made orthogonal to the displacements allowed by a conforming model. By refining the model, the solutions converge, even when residual loads persist.

Introduction

Our starting point is the fact that conventional finite element analyses ‘provide solutions that equilibrate the equivalent nodal forces’, where the adjective equivalent plays a central role that is often disregarded in the more basic introductions to the FE method.

Effectively there is equilibrium of equivalent nodal forces in the solutions provided by most FE programs. We will discuss in detail what that means and we will conclude that, in most cases, there are no nodal forces as such. Energetically consistent nodal forces are defined, which are required to produce the same work as the real forces and stresses for all displacements considered. But, in general, this is not sufficient to guarantee equilibrium in a strong, or pointwise, sense. This happens because only a finite subset of the possible displacements can be included in a given model and the solution space is generally infinite, therefore equilibrium is imposed on an average, or weak form. Generally the solutions provided by displacement based FE models do not enforce the equilibrium conditions at every point of the domain and/or its boundary.

Our objective is to present in this book a methodology whose models produce solutions that strongly verify all equilibrium conditions. As always there is a drawback for every new approach. In this case the gain in terms of equilibrium will imply a loss in terms of compatibility, which will only be imposed in a weak form. We will not pretend that these equilibrium formulations are always better than their displacement based counterparts, as each formulation locally enforces one set of conditions, while imposing a weak form of the other.

1.3 What Do We Gain From Strong Forms of Equilibrium? The complementary nature of these formulations is, in our opinion, the strongest reason for considering solutions obtained from both approaches. It does not matter which one is considered first, as the different approximate solutions that they produce are complementary, in the sense that they satisfy complementary equations in a strong and in a weak form.

As we will show, this complementarity can be used in a natural way to assess the quality of the solutions, and to drive a mesh adaptation process, deciding where it is important to have more, or fewer elements. From a practical point of view it is also relevant to point out that equilibrium solutions have the advantage of being immediately usable as a safe basis for design of ductile structures, when the Static Theorem of Limit Analysis can be invoked (fib, 2013; Marti, 2013; Nielsen and Hoang, 2010). In particular, equilibrium solutions give us a more rational way of accounting for stress concentrations, especially when they arise due to mathematical singularities where the structural geometry has been simplified, for example, at re-entrant corners.

3

4

Equilibrium F.E. Formulations

Displacement models tend to pollute local equilibria while seeking infinite values of stress, while equilibrium models continue to provide a statically admissible field in the neighbourhood of a mathematical singularity. Such a stress field represents a redistribution of stress near the singularity, and in this respect it is similar to the behaviour of the real structure, which may adapt itself by local yielding while maintaining stress equilibrium. On a historical note we recall that, before the introduction of computer techniques, methods requiring strong equilibrium as their starting point formed the basis for most structural design procedures. Considering the design of arches and their thrust lines, Figure 1.1, the analysis of statically indeterminate trusses and frames by Maxwell-Mohr methods, Figures 1.2 and 1.3, and also in variational methods such as the Trefftz method, we find that most ‘historical’ methods of analysis, in essence, sought to explain how the transmission of forces through the structure may be achieved. The reason for this can be explained by the fact that material resistance is intuitively related to the level of stress within the structure – displacements do not appear to be as important – leading to the application, implicit or explicit, of the Static Theorem of Limit Analysis, already mentioned. Furthermore, force, or flexibility, methods can lead to better conditioned systems of equations, and fewer of them (Argyris and Kelsey, 1960; Henderson and Maunder, 1969). These were critical factors when solutions were calculated with manual or semi-manual methods (Kurrer, 2008).

Figure 1.1 Robert Hooke (1676): ‘As hangs the flexible line, so, but inverted will stand the rigid arch’ (Heyman, 1982). The chain adapts its shape so that internal tensions balance the concentrated vertical forces, and transfer them to the ground/supports. The reflection of the shape of the chain is a thrust line, the trajectory of the compressive stress resultants. Source: Adapted from the hanging chain, Robert Hooke (1676), requoted from Heyman 1982.

Introduction

(a)

(b)

(c)

Figure 1.2 Statically indeterminate truss and a possible system of force transmission: (a) and (b) for external loads, from joints to ground; (c) for internal forces that are self-balanced.

Figure 1.3 Determination of a displacement of a hyperstatic beam using a statically admissible virtual moment distribution.

1.4 What Paths Have Been Followed to Achieve Strong Forms of Equilibrium? Generalizing the approach that is used for the analysis of frame structures using the force method, the obvious solution for the analysis of continua is to combine

5

6

Equilibrium F.E. Formulations

approximations that verify equilibrium in a strong sense so that the generalized relative displacements corresponding to the hyperstatic forces are zero. In the 1960s and 70s, as a consequence of the historical relevance given to equilibrium in technical culture, there was considerable effort devoted to obtaining such finite element methods (Argyris and Kelsey, 1960; Fraeijs de Veubeke, 1965; Gallagher, 1975; Robinson, 1973). This approach was practically abandoned, an exception being the work of Kaveh (2004, 2014), mainly due to the difficulty in setting up such approximations, and to the superior computational efficiency of direct stiffness assembly procedures (Przemieniecki, 1985). We will only briefly address this approach in this book. In the 1960s Fraeijs de Veubeke et al. proposed and developed finite element formulations providing solutions that verify equilibrium in the strong sense using two different approaches: to work with a stress potential, for example the Airy stress function, and to assemble elements where an internally equilibrated stress approximation is assumed in such a way that their boundary tractions are in equilibrium, that is, are codiffusive.

It appears that these equilibrium models did not find favour – maybe due to their relative complexity in implementation, and difficulties encountered in the application of boundary conditions. In any event, the belief in displacement models seems to have grown, together with the idea that the quality of solutions could be judged entirely by post-processing procedures such as those proposed by Zienkiewicz & Zhu in the late 1980s and early 1990s (Zienkiewicz and Zhu, 1987). However, the value of having complementary solutions for error estimation was still recognized, and a lot of research has been expended into a variety of ways to recover enhanced solutions from displacement models, for example, the superconvergent patch recovery (SPR) methods (Zienkiewicz and Zhu, 1987, 1992) and the error in constitutive relation from P. Ladeveze et al. (Ladevèze and Leguillon, 1983; Ladevèze and Maunder, 1996). These recovery methods, with the exception of the approach proposed by Ladevèze, only lead to a better approximation of strong equilibrium. Since the 1990s there has been a renewed interest and a renaissance in FEM, with hybrid formulations that directly enforce a strong form of equilibrium (Almeida and Freitas, 1991).

1.5 Industrial Perspectives Two different perspectives are particularly relevant when considering the industrial application of analyses based on equilibrium finite elements: • verification and validation in ‘simulation governance’ (Szabó and Actis, 2012); • the design and/or assessment of structures, explicitly or implicitly based on the Static Theorem of Limit Analysis. In this Section we briefly discuss how these aspects can be considered.

Introduction

1.5.1 Simulation Governance

With the ever wider reliance by industry on finite element analyses to justify compliance with codes of practice or statutory requirements, it is recognized that more formal verification and validation procedures should be adopted. General procedures have been defined in a rather pithy, but memorable way, to address two questions (Roache, 1998): i) ‘Am I solving the equations right?’ ii) ‘Am I solving the right equations?’ The ‘equations’ refer to the mathematical model which is assumed to describe the physical behaviour; in our case this corresponds to the equations of elasticity. Then (i) concerns verifying that the solution matches the mathematical model. It may fail to do so because of the intrinsic inaccuracy of the numerical model (for us the finite element approximations that are assumed), combined with other numerical aspects, for example, numerical instabilities and ill-conditioning that may be present. Point (ii) questions the validity of the mathematical model to adequately represent physical reality, for example, are potential non-linearities in behaviour properly accounted for; do the boundary conditions and loads reflect actual conditions; do the constitutive relations properly match those of the real materials? Thus two complementary approaches to finite element modelling that satisfy compatibility or equilibrium, and which can deliver opposite bounds to quantities of interest, are clearly advantageous, both for the peace of mind of the engineer, and as a means of providing evidence to satisfy statutory requirements on quality control. 1.5.2 Equilibrium in Structural Design and Assessment

Finite element models can evolve as the design of a structure or a device develops, but with different and evolving aims, particularly in the civil engineering context where structures tend to be ‘one-off’ as opposed to the mass produced artefacts of mechanical engineering. For the design of one-off civil engineering structures, high accuracy in the output of quantities of interest is not normally required, since the variability of materials, construction processes, and loading regimes (including the troublesome question of usually indeterminate residual stresses) do not generally justify this. However, a strong sense of equilibrium is important from the simplest initial models at early stages of design to the refined models necessary to justify the final design. Wittingly, or unwittingly, designers rely on what Heyman (1995) has termed ‘the master safe theorem’, that is, if any equilibrium state can be found, that is, one for which a set of internal forces is in equilibrium with the external loads, and, further, for which every internal portion of the structure satisfies a strength criterion, then the structure is safe. It is interesting to note that Wren must have had a rather similar basis for his designs: ‘The design must be regulated by the art of staticks, or invention of the centres of gravity, and the duly poising of all parts to equiponderate; without which, a fine design will fail and prove abortive. Hence I conclude, that all designs must, in the first place, be brought to this test, or rejected.’ (Addis, 2007). In pithier terms, Ed Wilson quotes: ‘equilibrium is essential, compatibility is optional’ and then emphasizes that stresses in conforming elements do not strongly satisfy equilibrium, so that mesh design needs to be considered in order to achieve acceptable levels of stress (Wilson, 2000).

7

8

Equilibrium F.E. Formulations

Of course these quotations leave open what is meant by ‘portion’ or ‘part’, but at the smallest practical level we can take this to mean infinitesimal parts of a continuum, and internal forces to mean stresses. The conventional conforming finite element model only enforces the weaker equilibrium of nodal forces, and so an element serves as a portion, but then we need a strength criterion! A useful and attractive feature of equilibrium finite elements is to transfer their interactions from the mathematical concept of nodal forces to the engineering concept of tractions on interfaces, backed up by fully equilibrated internal stress fields. These are immediately in a form amenable to comparison with strength criteria. It may be noted that, even when the non-elastic properties of the structural material do not strictly justify the use of the master safe theorem, equilibrium is a first line of defence! Concepts of equilibrium, and their use in design/assessment stages, are embodied in codes of practice for design, but with some warnings when conventional conforming finite element models are used, for example, Eurocodes EN 1990 (Basis of structural design), EN 1992 (Design of concrete structures), and fib Model Code for Concrete Structures 2010. • EN 1990 (1.5.6.2) Global analysis is defined as the ‘determination, in a structure, of a consistent set of either internal forces and moments, or stresses, that are in equilibrium with a particular defined set of actions on the structure, and depend on geometrical, structural and material properties.’ • EN 1992: brief reference to the use of finite element methods is made in Section 5 Structural analysis, 5.1.1 General requirements: ‘However, for certain particular elements, the methods of analysis used (e.g. finite element analysis) gives stresses, strains and displacements rather than internal forces and moments. Special methods are required to use these results to obtain appropriate verification.’ • The fib Model Code for Concrete Structures 2010 allows the use of the theory of plasticity in design, and this includes the ‘lower bound (static) theorem’. Verification of designs may be assisted by numerical simulations, including the finite element method: ‘In the case of the most widely used stiffness method, the shape of the displacement field is assumed and equilibrium is satisfied only in integral sense. The internal stresses are lower, compared with an exact solution. The approximations introduced by the finite element formulation only, can be a significant source of errors in numerical analysis.’ One of the most expensive and dramatic examples of a collapse attributed to the use of conforming models must be that of the Sleipner offshore oil production platform in 1991 (Rombach, 2011). A coarse mesh of solid elements led to a transverse shear stress resultant in a cell wall being underestimated by some 50% compared with later estimates of statically admissible internal forces. The latter forces were sufficient to cause local shear failure in a poorly detailed reinforced concrete wall, and consequently to trigger overall collapse during flotation trials.

1.6 The Structure of the Book After this Introduction we begin in Chapter 2 by illustrating the derivation of approximate solutions to some simple examples involving concepts of compatibility and/or equilibrium, without recourse to the use of finite elements. This is followed in Chapter 3

Introduction

by a summary of the main finite element formulations other than the hybrid equilibrium one, beginning with the more conventional compatible or conforming approach and then proceeding to variations which may involve stronger forms of equilibrium. Chapters 4 to 6 focus on the main topic of this book, that is, the hybrid equilibrium formulation, its elements and the particular features associated with their possible kinematic instabilities, known as the spurious kinematic modes and which may be likened to pseudo-mechanisms. The approach taken here is based on the duality between static and kinematic quantities, and after presenting general relations based on linear elastic behaviour, we consider specifics for 2D and 3D continua and plate bending. Chapter 7 places the formulations in the context of a variational basis and appropriate functionals corresponding to potential and complementary potential energies. This aspect is particularly important when considering the existence and convergence of solutions for typical saddle point problems. In Chapter 8 we consider the complementary nature of compatible and equilibrating finite element solutions, and present methods for recovering compatibility from equilibrium and vice-versa. This leads us to Chapter 9 which describes the roles of complementary solutions in obtaining error estimates of either solution, and explains how these estimates can be used to adapt a finite element mesh to best achieve desired outputs. In Chapter 10 we apply the hybrid equilibrium formulation to linear dynamic analyses by exploiting Toupin’s principle to account for inertia effects. Finally Chapter 11 takes a look beyond modelling linear behaviour, and provides some pointers towards analysing various forms of non-linearity with the hybrid approach. The book contains two appendices: A, with a resumé of the necessary fundamental equations of linear elasticity; and B, containing some insight and guidance on using the computer programs associated with this book. Please note that the quantities shown in numerical examples are without specific units; we assume that a particular system of units will always be consistent.

References Addis W 2007 Building: 3000 Years of Design Engineering and Construction. Phaidon London. Almeida JPM and Freitas JAT 1991 Alternative approach to the formulation of hybrid equilibrium finite elements. Computers & Structures 40(4), 1043–1047. Argyris JH and Kelsey S 1960 Energy Theorems and Structural Analysis. Butterworths. Coates RC, Coutie MG and Kong FK 1988 Structural Analysis 3 edn. Van Nost.Reinhold. fib 2013 fib Model Code for Concrete Structures 2010. Ernst und Sohn. Fish J and Belytschko T 2007 A First Course in Finite Elements. Wiley. Fraeijs de Veubeke BM 1965 Displacement and equilibrium models in the finite element method. In Stress Analysis (ed. Zienkiewicz OC and Holister GS). Wiley. Gallagher RH 1975 Finite Element Analysis: Fundamentals. Prentice-Hall. Henderson JCC and Maunder EAW 1969 A problem in applied topology: on the selection of cycles for the flexibility analysis of skeletal structures. IMA Journal of Applied Mathematics 5(2), 254–269. Heyman J 1982 The Masonry Arch. Ellis Horwood. Heyman J 1995 The Stone Skeleton: Structural Engineering of Masonry Architecture. Cambridge University Press.

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Hibbeler RC 2008 Structural Analysis 7 edn. Prentice Hall. Kaveh A 2004 Structural Mechanics: Graph and Matrix Methods. Research Studies Press. Kaveh A 2014 Computational Structural Analysis and Finite Element Methods. Springer. Kurrer KE 2008 The History of the Theory of Structures: from Arch Analysis to Computational Mechanics. Wiley-VCH. Ladevèze P and Leguillon D 1983 Error estimate procedure in the finite element method and applications. SIAM Journal on Numerical Analysis 20(3), 483–509. Ladevèze P and Maunder EAW 1996 A general method for recovering equilibrating element tractions. Computer Methods in Applied Mechanics and Engineering 137(2), 111–151. Marti P 2013 Theory of Structures: Fundamentals, Framed Structures, Plates and Shells. Wiley, Wilhelm Ernst & Sohn. Muttoni A, Schwartz J and Thürlimann B 1997 Design of Concrete Structures with Stress Fields. Springer. Nielsen MP and Hoang LC 2010 Limit Analysis and Concrete Plasticity. CRC press. Przemieniecki JS 1985 Theory of Matrix Structural Analysis. Dover Publications. Roache P 1998 Verification and Validation in Computational Science and Engineering. Hermosa Publishers. Robinson J 1973 Integrated Theory of Finite Element Methods. J. Wiley & Sons. Rombach G 2011 Finite-element Design of Concrete Structures: Practical Problems and Their Solutions. ICE. Schwer LE 2007 An overview of the PTC 60/V&V 10: Guide for verification and validation in computational solid mechanics. Engineering with Computers 23(4), 245–252. Schwer LE and Oberkampf WL 2007 Special Issue on Verification and Validation. Engineering with Computers 23(4), 243–244. Szabó B and Actis R 2012 Simulation governance: Technical requirements for mechanical design. Computer Methods in Applied Mechanics and Engineering 249–252, 158–168. Higher Order Finite Element and Isogeometric Methods. Wilson E 2000 Three Dimensional Static and Dynamic Analysis of Structures. Computers and Structures, Inc. Zienkiewicz OC and Zhu JZ 1987 A simple error estimator and adaptive procedure for practical engineering analysis. International Journal for Numerical Methods in Engineering 24, 337–357. Zienkiewicz OC and Zhu JZ 1992 The superconvergent patch recovery (SPR) and adaptive finite element refinement. Computer Methods in Applied Mechanics and Engineering 101, 207–224.

11

2 Basic Concepts Illustrated by Simple Examples

In this Chapter we illustrate concepts and demonstrate some of the principal features of conforming and equilibrating solutions through two simple examples which can be understood using mainly hand calculations and without involving finite elements explicitly! These solutions are generally obtained using the principles of minimum total potential or minimum complementary energy respectively, which are also recalled.

The first example is a symmetric bi-material strip under uni-axial tension, which is modelled exploiting symmetry. Although this appears to be a simple problem, it reveals the potential for complications due to singularities at the ends of the interface between the two materials (Szabo and Babuška, 1991). The problem is considered initially with either two kinematic variables for a conforming solution, or only one particular stress field for an equilibrating solution. The calculations are then very simple to carry out by hand, and the solutions illustrate important features. However, the extension of the static approach becomes more challenging since the use of polynomial stress fields requires the inclusion of terms of degree 3. Although the stress fields become quite complicated, the solution of the problem is still possible by hand, though we do not recommend it. The stress field obtained by minimizing the complementary strain energy is presented, and it is shown to be far superior to the conforming solution. A lesson might be: if you put more effort into thinking about ways of defining load paths, then rewards are to be found! A linear elastic reference solution for stresses and strains, obtained from a refined finite element model, is also included for comparison. Finally we complete this example by presenting bounds for the strain energy of the problem and note that they can be related to bounds for a specific average displacement. A second example concerns a rectangular plate bending problem governed by Kirchhoff plate theory, which applies to thin plates for which it is assumed that the transverse shear strains are zero. In this problem the plate is simply supported on two adjacent edges and loaded with a uniform line load on a third edge. We demonstrate conforming and equilibrating solutions obtained from up to three kinematic or static variables. Convergence of the two forms of solution is illustrated with respect to the strain energy.

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

12

Equilibrium F.E. Formulations

We provide a standard answer to the question ‘which is the best approach to solve a problem: equilibrium or compatibility?’: ‘It is better to look at both.’

2.1 Symmetric Bi-Material Strip A quadrant of the symmetric bi-material strip under uni-axial tension represented in Figure 2.1 is modelled as a plane stress problem, exploiting its symmetry, with the boundary conditions shown in the same figure, where symmetry is enforced by the sliding supports. For the homogeneous problem, materials in regions A and B have identical elastic properties, E and 𝜈. For a unit traction p this problem has the following exact solution: 1 𝜈 (2.1) 𝜎xx = 1, 𝜎yy = 𝜎xy = 0, ux = x , uy = −y . E E When the elastic properties are different a closed form analytical solution cannot be obtained, but very simple approximate solutions can be sought. We will start with two, inspired by the solution of the homogeneous problem: e e e = 1, 𝜎yy = 𝜎xy = 0. i) A statically admissible solution with 𝜎xx ii) A kinematically admissible solution with linear displacements, ucx = a1 x, ucy = a2 y.

The statically admissible solution is completely defined, without any degrees of freedom. We note that we can integrate the strains that result from 𝜎 e to obtain compatible displacement fields in each region, designated by 𝛼, u𝛼x = x∕E𝛼 , u𝛼y = −y 𝜈𝛼 ∕E𝛼 + c. The vertical displacement at the material interface can be made continuous by an appropriate choice of the constant c, but the horizontal component of these displacements is discontinuous along the interface between the materials. Therefore this is not the exact solution of the problem since the strains corresponding to these stresses are not integrable, that is, there isn’t a continuous displacement field leading to these strains. The kinematically admissible solution depends on two coefficients: a1 and a2 . The resulting elastic stresses are: p

p

A B A Figure 2.1 Bi-material strip and symmetry simplification.

y

p

A h

B x L

Basic Concepts

E𝛼 E c c , 𝜎yy,𝛼 = (a1 𝜈𝛼 + a2 ) 𝛼 2 , 𝜎xy,𝛼 = 0. 1 − 𝜈𝛼2 1 − 𝜈𝛼 It is important to realize that for the non-homogeneous case it is not possible to assign values to these two coefficients so that the resulting stresses satisfy all four stress related boundary conditions that are not automatically satisfied by this solution: horizontal stress at x = L equal to 1 for the two regions, vertical stress equal to zero at y = +h, and equal vertical stresses at the interface. Instead of arbitrarily deciding which conditions we want to enforce, a Rayleigh-Ritz approach (Pearson, 1959; Washizu, 1982) can be used to find the values of the coefficients that minimize the potential energy of the approximate solution, Π = U − V , and therefore best equilibrate the stresses, in an average sense. In the case when 𝜈A = 𝜈B = 𝜈, the strain energy of the solution, U, is c 𝜎xx,𝛼 = (a1 + a2 𝜈𝛼 )

𝜺

1 (𝜎 𝜀 + 𝜎yy 𝜀yy + 𝜎xy 𝛾xy ) dΩ 2 ∫Ω xx xx (EA + EB )(a21 + 2a1 a2 𝜈 + a22 ) = hL, 4 (1 − 𝜈 2 ) and the work done by the applied loads, V , is ∫Ω ∫0

∫Γt

𝝈(𝜺)T 𝜺 d𝜺 dΩ =

T t̄ u dΓ =

∫Γt

(t̄x ux + t̄y uy ) dΓ = a1 hL.

The total potential energy is obtained from these two functions of a1 and a2 . To minimize it we differentiate with respect to the two parameters, leading to a set of two linear equations, whose solution is 2 2𝜈 and a2 = − . a1 = EA + EB EA + EB The resulting stresses are 2 c 𝜎xx,A = EA , EA + EB

c 𝜎xx,B = EB

2 , EA + EB

c c 𝜎yy,𝛼 = 𝜎xy,𝛼 = 0.

This is not the exact solution because the boundary conditions on the horizontal stresses are not verified. However, for these particular conditions, the Rayleigh-Ritz method has chosen to impose zero vertical stresses throughout. It is important to recall c along the interface between the materials does not affect that the discontinuity of 𝜎xx equilibrium, because this stress component has zero projection on the interface. It is also possible to seek an equilibrated solution starting from stress approximations that have unknown parameters. Such a solution for this problem, which is not so simple to derive though it involves only a single parameter, is presented next. The verification of its admissibility is left as an exercise for the reader. ⎧1⎫ ⎧ (L − x)2 (L + 2x) ⎫ ⎪ b ⎪ ⎪ ⎪ 𝝈 A = ⎨0⎬ + ⎨−3(h − y)2 (L − 2x)⎬ 2 ; ⎪0⎪ ⎪ −6x(h − y)(L − x) ⎪ hL ⎭ ⎩ ⎭ ⎩ ⎧1⎫ ⎧ (L − x)2 (L + 2x) ⎫ ⎪ ⎪ ⎪3 ⎪ b 𝝈 B = ⎨0⎬ − ⎨ (h2 − 2y2 )(L − 2x)⎬ 2 . ⎪0⎪ ⎪ 2 6xy(L − x) ⎪ hL ⎩ ⎭ ⎩ ⎭

13

14

Equilibrium F.E. Formulations

The unknown parameter, b, is determined so as to minimize the complementary energy of the system, Πc = Uc − Vc . In this case there are no imposed displacements, so the work done by the reactions, Vc , is equal to zero, Vc =

∫Γu

t T ū dΓ = 0,

and, therefore, Πc is equal to the complementary strain energy of the stresses, Uc , 𝝈

∫Ω ∫ 0

𝝈 T 𝜺(𝝈) d𝝈dΩ

1 (𝜎 𝜀 + 𝜎yy 𝜀yy + 𝜎xy 𝛾xy ) dΩ 2 ∫Ω xx xx hL(EA + EB ) L2 (EB − EA ) = + b 2EA EB 2EA EB 7h4 (43EA + 3EB ) + 14h2 L2 (5EA + 11EB ) + 208L4 (EA + EB ) 2 + b. 1120EA EB hL

=

Differentiating with respect to the parameter, in a process similar to that applied to the compatible solution, we obtain b=

280hL3 (EA − EB ) . 7h4 (43EA + 3EB ) + 14h2 L2 (5EA + 11EB ) + 208L4 (EA + EB )

Figures 2.2 and 2.3 depict graphical representations of the stresses and strains of these solutions and of a very good approximation of the exact solution, for a geometry where L = 1 and h = 1, and material properties EA = 1, EB = 4 and 𝜈 = 0.25, obtained using a very refined finite element mesh. σxx

σyy

σxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

Figure 2.2 Bi-material strip: stress distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜎xx ; {−0.5 ∶ 0.5} for 𝜎yy ; {−0.5 ∶ 0.5} for 𝜎xy , with blue representing the minimum value, red the maximum and green the central value. (See plate section for colour representation of this figure).

Basic Concepts

εxx

εyy

γxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

Figure 2.3 Bi-material strip: strain distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜀xx ; {−0.5 ∶ 0.5} for 𝜀yy ; {−0.5 ∶ 0.5} for 𝛾xy . (See plate section for colour representation of this figure).

It is clear that neither of these simple solutions is exact, but the point to emphasize is that each of them, in spite of its failings, also has its virtues: • the compatible solution indicates that the stiffer material will bear higher stresses; • the simplest equilibrated solution already points at larger strains in the more flexible material; • the second equilibrated solution is the one that better captures the transfer of the horizontal force, uniform on the loaded face, into the stiffer material; this is a characteristic, quite often observed, of strongly equilibrated solutions, they provide better solutions, with fewer parameters, but the corresponding derivations are also considerably more complicated. Though these virtues may be exaggerated in this example, the really important aspect is that each of them is missed by the other solution.

Figure 2.4 presents a plot of the values of the strain energy of the solutions, assuming EA = 1, L = h = 1, 𝜈 = 0.25 and different values of EB . The most significant result provided by this example stands out: the strain energy of the exact solution is bounded by any two complementary approximate solutions. For EA = EB , as the exact solution is obtained, the bounds coincide. It is also noticeable in this figure that, for the particular values that were chosen, the energy obtained from an equilibrated solution is closer to the exact value. At least for the simplest equilibrated solution this should be regarded as a matter of chance. In fact, by adjusting the ratio L∕h, it is possible to argue in favour of the complementary conclusion. Clearly neither of these arguments is always valid.

15

Equilibrium F.E. Formulations

Compatible (2 dofs) Equilibrated (0 dofs) Equilibrated (1 dof) Exact

Strain energy

16

1

0.1 1 EB

0.1

10

Figure 2.4 Bi-material strip: strain energy of the solutions as a function of EB , for EA = 1.

Furthermore we note that the average horizontal displacement at the loaded edge is proportional to this energy, a subject that will be discussed in Chapter 9. We can, therefore, assess the quality of the value obtained for this displacement by checking the difference between the energy of two complementary solutions. A consequence of selecting the coefficients that lead to a form of minimization of a potential energy is that the resulting solutions always verify what is known as the Galerkin orthogonality condition. For a compatible solution this means that the work done by the unbalanced forces on any of the possible displacements is always zero. Conversely for an equilibrated solution, the work done by the lack of compatibility is also zero, for all possible self-balanced stress fields. A more formal definition of the Galerkin orthogonality condition will be given in Chapter 7.

2.2 Kirchhoff Plate With a Line Load In this example we consider the square plate in Figure 2.5, which is simply supported on two adjacent sides, and has a unit line load applied to one of the free sides. A direct (right handed) reference frame with the positive z axis pointing downwards, the usual engineering orientation for bending plates (Timoshenko and Woinowsky-Krieger, 1959), is used. 2.2.1 Kinematically Admissible Solutions

Kinematically admissible polynomial solutions for the transverse displacement, 𝑤(x, y), of degree d are constructed by combining functions aij xi yj , with i, j > 0 and i + j ≤ d. 2.00 E=1 v = 13

2.00 y

x

x

y

z

Figure 2.5 Simply supported plate of thickness h. Plan and isometric views.

Basic Concepts

For generic i and j the corresponding curvatures are: { 0 if i = 1 𝜒xx = −aij i(i − 1) xi−2 yj if i > 1 { 0 if j = 1 𝜒yy = i j−2 if j > 1 −aij j(j − 1) x y 𝜒xy = −2 aij i j xi−1 yj−1 The moment fields are derived using the constitutive relations ⎧m ⎫ ⎪ xx ⎪ ⎨myy ⎬ = Df ⎪mxy ⎪ ⎩ ⎭

⎡1 𝜈 0 ⎤ ⎧𝜒 ⎫ ⎢𝜈 1 0 ⎥ ⎪ xx ⎪ ⎢ ⎥ ⎨𝜒yy ⎬ , ⎢0 0 1 − 𝜈 ⎥ ⎪𝜒xy ⎪ ⎣ 2 ⎦⎩ ⎭

where Df =

E h3 12(1 − 𝜈 2 )

The simplest solution uses one function of degree 2: a11 x y. The bending curvatures are nil, so the strain energy only due to torsion is U = 8Df a211 ∕3. The work done by the applied forces is V = 4 a11 . Minimization of the total potential energy leads to a11 = 3∕(4Df ), with U = 1.5∕Df . This approximation, with zero bending moments and with constant twisting moment equal to −0.5, has zero normal bending moments and equivalent shear forces on all the sides and a concentrated Kirchhoff force at the free corner. It is therefore the exact solution when a force equal to 1 is applied at that free corner, but it is not the exact solution of the problem considered. Again we note that for the simplified deformed shape that was assumed, with a linear variation of the vertical displacement on the loaded side of length 2, the two load cases – unit line load and unit corner force – are energetically equivalent, that is, they produce the same work. Applying the same procedure to the three functions of degree 3, the following displacement and moment fields are obtained: Df 𝑤(x, y) = 0.0182186 x2 y − 0.097166 x y2 + 0.907895 x y; ⎧ ⎫ 0.0647773 x − 0.0364372 y ⎪ ⎪ 0.194332 x − 0.0121457 y m(x, y) = ⎨ ⎬. ⎪−0.0242915 x + 0.129555 y − 0.605263⎪ ⎩ ⎭ The strain energy of this solution is U = 1.62955∕Df . It can be observed that this is not the exact solution of the problem because the bending moments at the sides are not zero. To check boundary equilibrium we compare the forces and moments on the sides, obtained from the internal moments, with those that were prescribed. Note that the solution obtained would be exact if those moments and forces had been imposed instead. ̄ 0 = m(y) ≠ mxx (0, y) = −0.0364372 y

at x = 0;

̄ 0 = m(y) ≠ mxx (2, y) = 0.129555 − 0.0364372 y

at x = 2;

̄ 0 = m(x) ≠ myy (x, 0) = 0.194332 x

at y = 0;

̄ 0 = m(x) ≠ myy (x, 2) = 0.194332 x − 0.0242915

at y = 2;

17

18

Equilibrium F.E. Formulations

0 = F̄ ≠ −2mxy (2, 2) = 0.789474 𝜕mxy 𝜕mxx +2 = 0.323887 1 = r̄ (y) ≠ 𝜕x 𝜕y 𝜕myy 𝜕mxy 0 = r̄ (x) ≠ +2 = −0.0607287 𝜕y 𝜕x 𝜕 2 mxy 𝜕 2 mxx 𝜕 2 myy ̄ y) = + + 2 0 = p(x, 𝜕x2 𝜕y2 𝜕x 𝜕y

at x = 2, y = 2; at x = 2; at y = 2; for the whole domain.

All the derived loads on the sides and at the free corner are different from those that were prescribed, but the two sets remain energetically equivalent for each of the deformed shapes that were considered: xy, x2 y and xy2 . The verification of this equivalence is not obvious because all the boundary terms – normal moments, corner force and effective shear forces – produce work, even for the simplest bilinear deformed shape, 𝑤 = xy. For this shape, the displacements corresponding to the terms previously listed are the following: 𝜕𝑤 𝜕x 𝜕𝑤 𝜃x (y) = − 𝜕x 𝜕𝑤 𝜃y (x) = − 𝜕y 𝜕𝑤 𝜃y (x) = − 𝜕y 𝑤=4 𝜃x (y) = −

=y

at x = 0;

=y

at x = 2;

=x

at y = 0;

=x

at y = 2; at x = 2, y = 2;

𝑤 = 2y

at x = 2;

𝑤 = 2x

at y = 2.

This equivalence can be verified by comparing the work integrals: 2

∫0

1 × 2y dy = 4 = 2



2

(−0.0364372y) × (−y) dy +

∫0 2



∫0

∫0

(0.129555 − 0.0364372y) × (−y) dy

2

(0.194332x) × (−x) dx +

∫0

(0.194332x − 0.0242915) × (−x) dx

2

+ 0.789474 × 4 +

∫0

(0.323887) × (2y) dy

2

+

∫0

2

(−0.0607287) × (2x) dx +

∫0 ∫0

2

0 × xy dx dy.

Table 2.1 shows the values of the strain energy obtained for different approximation degrees, exhibiting convergence from below, as expected.

Basic Concepts

Table 2.1 Simply supported plate: Strain energy (U) of the compatible solutions, scaled by Df , as a function of the degree of the approximation (d) and of the number of variables (dof ). d/dof

2/1

3/3

4/6

5/10

10/45

20/190

U

1.50000

1.62955

1.66534

1.68785

1.68850

1.68851

UExact − U

0.18851

0.05895

0.02317

0.00066

0.00001

0.00000

2.2.2 Statically Admissible Solutions

Again the construction of a statically admissible solution is a more complex matter. We start by observing that a non-zero twisting moment field mxy is required, because solutions based solely on bending moments, mxx and myy , are not possible. The simplest equilibrated solution only has a twisting moment equal to (y − 2)∕2. Equilibrium is verified in the domain because the cross derivative of the linear twisting moment is zero. On the sides all bending moments are zero by definition, the effective shear force is zero for y = 2 because the twisting moment is constant and the line load at x = 2 is equilibrated by the effective shear force due to the variation of mxy . The corner force is zero because the corresponding twisting moment is also zero. This is not the exact solution because the curvatures corresponding to these moments are not compatible, that is, there isn’t a continuous displacement field that has these curvatures. The compatibility equations (A.15) express this condition, which in this case can be explained in simple terms by observing that the twisting curvature, which is proportional to the cross derivative of the displacement, is linear in y, while the curvature in y, proportional to the second derivative in y of the displacement, is nil. The first term requires that the displacement has a term in x y2 , which implies a non-zero curvature in y, contrary to what is required. Additional moments, of higher degree, can be considered. By imposing that they verify all the equilibrium conditions, a reduced set of alternatives is obtained. For example with quadratic moments the following moment field is obtained: ⎫ ⎧ 1 0 ⎧ 0 ⎫ ⎧ ⎫ ⎪− 4 (x − 2)x⎪ 1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ 1 (y − 2)y m=⎨ 0 ⎬+⎨ b1 + ⎨ (y − 2)y ⎬ b2 . ⎬ 4 y−2 ⎪ ⎪ ⎪ ⎪ 1 ⎪ 4 y−x ⎪ ⎩ 2 ⎭ ⎩− 4 (x − 2)(y − 1)⎭ ⎪ ⎪ 4 ⎭ ⎩ The values of the unknown parameters, b1 , b2 , are determined, just as for the bi-material example, so as to minimize the complementary energy of the system, leading to the following field of moments: ⎧ ⎫ 0.0224618 (x − 2) x ⎪ ⎪ −0.260557 (y − 2) y m=⎨ ⎬. ⎪x (0.238095 y − 0.215633) + 0.00134771 y − 0.52381⎪ ⎩ ⎭ The values of the strain energy for various approximation degrees are presented in Table 2.2, now exhibiting convergence from above.

19

20

Equilibrium F.E. Formulations

Table 2.2 Simply supported plate: Complementary strain energy (U c ) of the equilibrated solutions, scaled by Df , as a function of the degree of the approximation (d) and of the number of variables (dof ). d/dof

1/0

2/2

3/6

4/12

7/42

14/182

Uc

2.00000

1.73944

1.69271

1.68930

1.68855

1.68851

Uc − UExact

0.31149

0.05093

0.00420

0.00079

0.00040

0.00000

2.2.3 Assessment of the Solutions Obtained

The moment fields for some of these approximations are presented in Figure 2.6. The ‘exact’ solution is taken to correspond to the equilibrated solution of degree 14. We observe that the equilibrated solution captures quite well the distribution of twisting moments and recognizes the variation of myy in y, though it fails to represent its variation in x. Though the compatible solution has a non-zero value of the twisting moment at the free vertex, therefore indicating the presence of a corner force, it generally recognizes the correct sign for the twisting moment and is complementary to the equilibrated solution as regards identifying the variation of myy . The bending moment mxx , which is the most complicated field, but also has the lowest absolute value, is not captured by either of the first two approximations. The determination of equilibrated functions is considerably more involved, just as in the previous example. A plot of the convergence of the absolute error of the strain energy, as a function of the degree of the approximation functions used, is presented in Figure 2.7. It can be seen that the convergence rates achieved are similar for both approaches. mxx

myy

mxy

Compatible (3 dofs)

Equilibrated (2 dofs)

“Exact”

Figure 2.6 Simply supported plate: some moment distributions. The ranges of the colours are: {−0.1 ∶ 0.1} for mxx ; {−0.75 ∶ 0.75} for myy ; {−1 ∶ 1} for mxy . (See plate section for colour representation of this figure).

Basic Concepts

Figure 2.7 Simply supported plate: error of the strain energy of the solutions as a function of the degree of the approximation functions used.

1e+00 Compatible Equilibrated

Error in strain energy

1e-01 1e-02 1e-03 1e-04 1e-05 1e-06 2

5 Degree

10

12

Finally we note that in this example, interestingly, the displacement at the free vertex presents a very peculiar behaviour: we always obtain the exact value, both for the compatible and for the equilibrated solutions. This point will be further discussed in Section 9.5. For now it suffices to say that it happens because in this very particular model we know the exact solution for the auxiliary virtual problem that may be used to compute the displacement, the application of a unit vertical force at the free vertex. Consequently the integration of the curvatures obtained from either one of the compatible or one of the equilibrated solutions with the moments corresponding to the unit load always produces the exact displacement: 3∕Df .1

References Pearson CE 1959 Theoretical Elasticity vol. 6 of Harvard Monographs in Applied Science. Harvard University Press. Szabo B and Babuška I 1991 Finite Element Analysis. John Wiley & Sons. Timoshenko SP and Woinowsky-Krieger S 1959 Theory of Plates and Shells 2 edn. McGraw-Hill New York. Washizu K 1982 Variational Methods in Elasticity and Plasticity 3 edn. Pergamon Press.

1 This will become obvious in Section 9.5. In a nutshell, 𝜖̊ is zero, so that the bound of the error of the output is also zero.

21

22

3 Equilibrium in Other Finite Element Formulations The number of finite element species has grown considerably since their inception some 60 years ago, a situation that has evoked the image of a ‘zoo’ of elements and methods. Before presenting in detail the hybrid equilibrium formulation in Chapter 4, we now look at a variety of related species. In this context we focus our attention on specific applications of the finite element method to plane elasticity problems, as a vehicle to explain details that are relevant in the context of the equilibrium approaches that constitute the core of the present work. In order to provide some coherence between the presentations we try to use a similar approach for all types of formulation, by considering the following aspects: • What approximations are involved? • What variational principle is (explicitly or implicitly) used? • What conditions are being imposed? How? In particular, what levels of equilibrium are obtained from their solutions?

3.1 Conforming Formulations and Nodal Equilibrium The most common finite element formulations are based on a compatible approximation of the displacement field within the elements, wherein a set of interpolation functions is used to describe each component of the displacement vector, ux and uy for plane elasticity problems (Szabo and Babuška, 2011; Zienkiewicz and Taylor, 2005). For a given element with n nodes, n interpolation functions, 𝜓i (x), with i = 1 … n, are defined, such that 𝜓i (x) has a unit value at node i and zero at the others. This implies that the weights of an arbitrary linear combination of these functions, for example, the ∑n f̂i s in f (x) = i=1 (𝜓i (x) f̂i ), are equal to the value of function f at node i. Therefore, if the same weight is used for the nodes that are shared between two elements the function will be continuous between these elements, at least at the nodes. In practice an appropriate choice of the number of nodes and of the shape of the elements is used, such that the resulting function is continuous everywhere.1

1 Problems that require C1 continuity, i.e. continuity of the approximation and of its derivatives, for example the bending of thin plates, as modelled by the Kirchhoff theory, demand more complex forms of interpolation. Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

Equilibrium in Other F.E. Formulations

Then the solution process just consists of determining the ‘best’ values of the unknown displacements at the free nodes, looking for the displacement field that is in some sense closest to the (generally unknown) exact one. The corresponding equations can be deduced by considering the minimization of the total potential energy amongst all possible approximations, which corresponds to imposing a weak form of equilibrium. It is assumed that these models are well known to the reader, but for completeness we recall that the element displacements are approximated as { } [ 1 ux 𝜓 = ue = uy 0

··· ···

𝜓n 0

0 𝜓1

··· ···

] ⎧ u1 ⎫ 0 ⎪ x⎪ ⋮ = 𝝍 e û e . 𝜓 n ⎨ n⎬ ⎪uy ⎪ ⎩ ⎭

The corresponding strains are immediately obtained from compatibility via (A.1), resulting, by application of the constitutive relation (A.3), in the approximation of the stresses in the elements: 𝜺e = ( 𝝍 e ) û e = Be û e

and

𝝈 e = k Be û e .

These stresses lead to the element stiffness matrix, Ke =

∫ Ωe

BTe k Be dΩ,

(3.1)

which, together with the nodal forces derived from the imposed body forces b̄ and boundary tractions t̄ , F̂ e =

∫ Ωe

𝝍 Te b̄ dΩ +

∫Γe

𝝍 Te t̄ dΓ,

(3.2)

are assembled, as expressed by operator A, in a global system of equations, from which the unknown nodal displacements are obtained: ̂ K û = F;

(3.3)

with K = A(K e ) and F̂ = A(F̂ e ). The solutions of this set of equations minimize the total potential energy of the system, which can be written as 1 ̂ ̂ = û T K û − û T F. Π(u) 2 Equation (3.3), involving forces, is obviously expressing a form of equilibrium. We seek to answer the following questions, in order to assess its nature: • How do we interpret this form of equilibrium? • How does it relate with the search for the ‘best’ value of the unknown displacements at the free nodes? In practical problems the nodal forces that appear in displacement based formulations are not real forces concentrated at the nodes, but equivalent nodal forces. The difference is subtle, but it is crucial when we seek to understand in detail how much equilibrium we have.

23

24

Equilibrium F.E. Formulations

Equivalent nodal forces, as defined in (3.2), replace continuous distributions of body ̄ and surface tractions, t̄ , with a set of discrete forces, as many as the number of forces, b, nodal displacements of the element, such that the work done by either of them, whatever the displacements, is the same: ( ) T T ̂F Te û e = ̄bT 𝝍 e dΩ + t̄ T 𝝍 e dΓ û e = b̄ ue dΩ + t̄ ue dΓ, ∀û e . ∫Ωe ∫Γe ∫ Ωe ∫Γe In other words, from the viewpoint of the element with its assumed displacements, it is not possible to distinguish the difference between a given body force, which is distributed inside the element, and its equivalent nodal forces, which are concentrated at the nodes. Understanding that this equivalence is valid for a given mesh, with given nodes, is critical. If the mesh is changed, for example by subdividing an element, the equivalent nodal forces will change. So what the compatible finite element model does is to accept this fact and restrict the requirement of equilibrium to what it is capable of: all its equivalent nodal forces must be in balance. This is imposed by (3.3), which equilibrates the nodal forces induced by the nodal displacements with the nodal forces equivalent to the applied loads. There are two main alternatives to explain the nature of the equivalent nodal forces that originate from the element stresses, as expressed by the stiffness matrix. The first is based on requiring that the nodal forces F̂ e,K , associated with imposing an arbitrary set of displacements, û e , are energetically equivalent to the strain energy associated with any given variation of the nodal displacements: T F̂ e,K 𝛿 û e = û Te

∫ Ωe

BTe k Be dΩ 𝛿 û e ,

∀𝛿 û e

=⇒

F̂ e,K =

∫ Ωe

BTe k Be dΩ û e .

Alternatively it is possible to obtain the equivalent nodal forces associated with the loads that equilibrate the stresses due to a given set of displacements: ( ) ̂F e,K = − 𝝍 Te  ⋆ (k Be ) dΩ + 𝝍 Te  Te k Be dΓ û e , (3.4) ∫ Ωe ∫Γe where the adjoint operator  ⋆ is defined in (A.2), the boundary operator  e is defined in (A.5), and the minus sign is used to obtain the body forces inside the element, so that they equilibrate the given stress distribution. Integration by parts of the first term in parentheses shows that the two alternative interpretations are equivalent. From the viewpoint of assessing equilibrium the second alternative allows for a better comprehension of the nature of the nodal forces associated with the stresses. Each column of K e corresponds to the equivalent nodal forces that must be applied to the element in order to impose a given unit displacement. These forces are obtained by projecting onto the nodes the internal body forces and the tractions on the surface of the element that occur when that unit displacement is imposed. When the question ‘how much equilibrium do we have’ is posed, we can just answer: a balance is required between the equivalent nodal forces associated with the stresses induced by the displacements, and those corresponding to the applied loads. The relevant point is that it is possible to prove that as the mesh is refined the corresponding solution converges, in a predictable way, to the exact solution of the continuous problem. T

Equilibrium in Other F.E. Formulations

̂ This is achieved by recognizing that K is a positive definite matrix, implying that Π(u) is convex. Then, when the solutions for two finite element meshes are considered, û H ̂ and û h , such that all possible solutions in û H are included in û h , the convexity of Π(u), implies that Π(û h ) ≤ Π(û H ). This leads directly to a proof of convergence (Hughes, 2000; Oliveira, 1968).

3.2 Pian’s Hybrid Formulation In Pian (1964) an alternative form for the derivation of element stiffness matrices was proposed, where ‘instead of a required continuous displacement function over the element, it is necessary only to write down the boundary displacements that will guarantee a complete displacement compatibility.’ A stress field is assumed inside each element, written as 𝝈e = P 𝜷

or, using the notation that we introduce in Chapter 4,

𝝈 e = Se ŝ e .

This field is required to be in equilibrium with zero body forces, that is,  ⋆ Se = 𝟎.2 Component-wise nodal interpolation schemes are not applicable to these functions, since more than one component of stress is generally associated to each parameter. Requiring, for example, the stationarity of the complementary energy inside each element, which is equivalent to a weak form of compatibility, results in ∫ Ωe

STe f Se dΩ ŝ e =  e ŝ e = DTe û e =

∫Γe

STe  e 𝝍 Γe dΓ û e ,

where the 𝝍 Γe s only need to be defined on the boundary of the element, that is, they can be a function of the side coordinates and a reference frame of the element does not need to be used. Furthermore, in order to have boundary displacements that ‘will guarantee a complete displacement compatibility’ it is sufficient to use functions that are continuous on the boundary of each element, wherein its nodes must lie. Therefore, internal nodes cannot be considered in this formulation. Then, since matrix  e is positive definite, the stress parameters, ŝ e , can be defined as a function of the nodal displacements: T ̂ e. ŝ e =  −1 e De u

The procedure to obtain the equivalent nodal forces corresponding to these stresses is similar to what is used for ‘normal’ compatible elements, resulting in: F̂ e,K =

T

∫Γe

T ̂ e = K e û e . 𝝍 TΓe  𝝈 e dΓ = De  −1 e De u

These element stiffness matrices are assembled into a global system, as for compatible displacement formulations, and balanced with the corresponding equivalent nodal forces.3 Given the solution of the system, in terms of nodal displacements, the stresses are obtained element-wise. 2 Details of the properties and the construction of the approximation functions in Se are given in Chapter 4 . 3 These nodal forces, immediately defined for applied boundary tractions, require a particular solution and its projection on the boundary when non-zero internal body forces are considered.

25

26

Equilibrium F.E. Formulations

This system of equations, defined in terms of nodal displacements, cannot be directly obtained from the total complementary energy of the model. Effectively, the complementary strain energy of a solution can be written as 1 ∑ T 1 ̂ ŝ e  e ŝ e = û T K u, UC (̂s) = 2 elements 2 but the term corresponding to the work done by the imposed displacements must be generalized, in order to include the traction discontinuities at the boundary of the elements. When only the boundary between two elements, a and b, is considered we have the following two additional terms to include in the total complementary energy: T

∫Γa

uTab  a 𝝈 a dΓ

and

T

∫Γb

uTab  b 𝝈 b dΓ.

Thus the contributions of the internal boundaries cancel each other when there is strict T T equilibrium between elements, that is,  a 𝝈 a +  b 𝝈 b = 𝟎, in the absence of tractions applied along the interface between a and b. This local detail of equilibrium is not generally satisfied, and hence the stationarity of the complementary potential energy, discussed in Section 7.1, cannot be invoked globally. The solution sought by this model corresponds instead to the stationarity of a hybrid complementary potential energy, which considers the following additional term: ∑ T ∑ T û e De ŝ e . uTΓe  Γe 𝝈 e dΓ = ∫ elements Γe elements Imposing continuity of the boundary displacements implies that in this formulation, the equivalent nodal forces at a vertex of an element collect contributions from the tractions on all sides that converge onto that vertex. Therefore, equilibrium, as imposed by (3.3), with the K e s obtained using this approach, will not lead to stresses that respect a strong form of equilibrium: in general side tractions do not balance, that is, they are not codiffusive. The number of parameters to include in the approximation of stresses and displacements, ns and n𝜓 , must respect the following condition ns ≥ n𝜓 − nrbm , where nrbm is the number of rigid body modes, which is 3 for plane elasticity problems. This condition requires that the number of stress parameters must be at least as large as the number of independent strain fields induced by the displacements. If this is not adhered to, the rank of the elemental stiffness matrix will be ns instead of n𝜓 − nrbm . As ns is increased, while n𝜓 is kept constant, an elemental stiffness matrix is obtained which, by describing more precisely the behaviour of the element when subject to the boundary displacements in 𝝍 Γe , can be said to be ‘better’. Nevertheless, the solutions thus obtained will not converge to the exact solution of the problem, unless it can be represented by the boundary displacements in 𝝍 Γe . In other words, if 𝝍 Γe is for example linear, the coefficients of the stiffness matrix will converge to the exact nodal forces obtained when linear boundary displacements are imposed. Unless the boundary displacements of all elements are linear in the exact solution, a model built using those stiffness matrices will not provide the exact stresses or displacements.

Equilibrium in Other F.E. Formulations

Fraeijs de Veubeke (1965) identified this situation as a ‘limitation principle’: If a net of finite elements is analysed by compatible displacement modes and the stresses left free to be determined by energy considerations, the best stresses are those associated with the strains derived from the displacements and the degrees of freedom in the displacement modes are governed by the ordinary principle of variation of displacements. In other words it is useless to look for a better solution by injecting additional degrees of freedom in the stresses, although the stresses obtained will not, as a rule, satisfy the detailed equilibrium conditions. We term such an approach as being one of ‘hybrid stress’ (Freitas et al., 1999), following Pian’s own classification of different finite element methods, first expressed in Pian and Tong (1969) and further detailed in Pian (1978, 1983). ‘Hybrid’ is used because it approximates a complementary field on the boundary of the elements, and ‘stress’ is used because these are the primary variables inside the elements.

3.3 Mixed Stress Formulations Stress approximations that are not constrained to verify internal equilibrium are much simpler to implement, but cannot be used for the hybrid stress element just discussed. They require an approximated displacement field that is also defined inside the elements. Then the nodal forces due to the approximated stresses can be written as ( ) T 𝝍 Te  Se dΓ − 𝝍 Te  ⋆ Se dΩ ŝ e = BT S dΩ ŝ e = C e ŝ e . F̂ e,K = ∫Γe ∫ Ωe ∫ Ωe e e The weak form of the compatibility condition (A.1), together with the constitutive relation (A.3), when weighted by the stress approximation functions becomes directly ∫ Ωe

STe f Se dΩ ŝ e =  e ŝ e = C Te û e =

∫Ωe

STe Be dΩ û e ,

Matrices C e , now obtained for the mixed model, and De of the hybrid model in Section 3.2, have a similar role of transforming stress parameters into nodal forces, while their dual transformations, using the transposed matrices, provide the generalized strains associated with given nodal displacements. Matrix De can be obtained from the definition of C e , provided a displacement field exists inside the element which corresponds to the assumed boundary ones; but the inverse is not true: C e cannot be obtained only from a boundary integral, unless the stresses in Se are equilibrated. The stiffness matrix of an element, given by T K e = C e  −1 e Ce ,

is used with the assembly operator A in a standard way and the nodal forces are obtained just as for displacement based formulations. In general a strong form of equilibrium is not obtained from these formulations, not even inside the elements. The reason for this is similar to that which was explained for the hybrid stress model: the weak form of equilibrium takes in the contributions from all

27

28

Equilibrium F.E. Formulations

the elements that share each nodal displacement, thus averaging to zero the lack of equilibrium, instead of imposing a strong form. The same conditions that must be verified by the hybrid stress element, regarding the number of parameters involved in the approximations, also apply here. It is interesting to check what happens in the case of the triangular plane element, with linear approximations, both for stresses and for displacements. We have in this case ns = 9 > 3 = 6 − 3 = n𝜓 − nrbm , so that we can expect K e to have rank n𝜓 − nrbm . The relevant equilibrium conditions are: • 2 for internal equilibrium, since the derivatives of the stresses are constant, with two components; • 4 for each side, since the boundary tractions are linear, also with two components. This results in: 2(internal equilibrium conditions) + 3(sides) × 4(traction parameters) = 14 conditions. This number is higher than n𝜓 , indicating that in general some of the conditions will not be imposed. In the quest for equilibrium it is possible to use a quadratic displacement approximation with linear stresses. In this case the condition between the number of stress parameters (9) and the number of deformation modes (9 = 6 × 2 − 3) is still verified, so that K e will have rank n𝜓 − nrbm , but as n𝜓 is still smaller than 14 it is not possible to enforce all the equilibrium conditions. If cubic displacement fields, with 2 × 10 parameters, are considered, the number of deformation modes becomes larger than the number of stress parameters, leading to an elementary stiffness matrix that is necessarily rank deficient. When several elements are assembled, the number of internal and inter element equilibrium conditions grows faster than the number of nodes. Therefore, for an assembled mesh, the number of equilibrium conditions becomes proportionally higher than the number of nodal forces. This is illustrated by considering a patch with two elements having linear stresses and quadratic displacements. The number of equilibrium conditions is: 2 elements × 2 (internal equilibrium conditions) + 5 (sides) × 4 (traction parameters) = 24, while the number of nodal forces is 18. The ratio between the number of equilibrium conditions and the number of nodal = 1.167 for an isolated element to 24 = 1.333 for the simforces changes from 14 12 18 plest patch. When there are several elements sharing a vertex this ratio grows even more.

3.4 Variants of the Displacement Based Formulations With Stronger Forms of Equilibrium There are displacement based finite element approaches which, directly or indirectly, seek equilibrium by trying to avoid the typical behaviour that is apparent in the presentations made in the previous sections, that is, that the strict enforcement of compatibility, for example by sharing a vertex node between adjacent elements, results in nodal forces that imply an average form of equilibrium.

Equilibrium in Other F.E. Formulations

We can say, roughly speaking, that a stronger form of equilibrium is achieved by assuming a displacement field that is not entirely compatible, which results in a larger number of nodal forces to equilibrate and, therefore, in a stronger form of equilibrium. This will be illustrated by a special case, of relevant historical significance, wherein a displacement based element leads to stress distributions that are strictly equilibrated: Fraeijs de Veubeke’s equilibrated triangle (Fraeijs de Veubeke, 1964),4 followed by a brief consideration of other similar approaches. 3.4.1 Fraeijs de Veubeke’s Equilibrated Triangle

This element can be formulated by assuming a linear displacement field which is interpolated by nodes located at the middle of the sides, as illustrated in Figure 3.1. By realizing that the strains are constant and equal to those of the shaded ‘skeleton element’, which is a conventional constant strain triangle (CST) element, we have that the compatibility matrix Be is the same for both elements (they have the same interpolation nodes and therefore the same 𝝍 e s), so that the stiffness matrix of Fraeijs de Veubeke’s element is equal to the stiffness matrix of the ‘skeleton’ CST, multiplied by 4. The corresponding strains and stresses are constant, so that equilibrium in the absence of body forces is automatically verified inside each element.5 The tractions on the sides are constant, implying that 3 (sides)×2 (traction parameters) = 6 conditions are necessary to impose equilibrium in an isolated element or, when interpreting the stiffness matrix as in (3.4), each nodal force only considers the resultant of tractions on its side. The analysis of an isolated element subject to arbitrary body forces is not necessarily well posed: • If at least three non-concurrent displacements are fixed, the solution in terms of nodal displacements and stresses is unique. • For an element that is not fixed, multiple solutions exist when the applied forces are in equilibrium, with a multiplicity corresponding to the rigid body movements of the element. Although a pure rigid body rotation is not strictly represented by uniform translations of the sides, nodal displacements conforming to a rotation do exist.6 The strains and the stresses are not affected by these motions and are therefore unchanged by them. When a new element is assembled onto an existing mesh, the number of new nodal forces is always equal to the number of new equilibrium conditions. This corresponds to Figure 3.1 Fraeijs de Veubeke’s equilibrated triangle.

4 In this text Fraeijs de Veubeke states that: ‘It is the transmission of stresses that is required and this can only be achieved by relaxing compatibility requirements.’ 5 It is possible to formulate the same element by assuming constant stresses, which are equilibrated. Its strains are integrable and correspond to a linear displacement field. 6 We term such movement a ‘pseudo-rigid body’ rotation.

29

30

Equilibrium F.E. Formulations

the number of new free sides, that is, the sides of the new element that are not connected to existing ones. Considering that for one element the number of nodal forces is equal to the number of equilibrium conditions, we have that this condition is extended to the assembled mesh. This means that on interior boundaries the tractions on adjacent elements balance each other, and on each external boundary the tractions are in equilibrium with the uniform applied traction that is equivalent to the nodal force being considered. Under these conditions, if a solution exists, the stress field is in equilibrium. The resulting displacement field does not have to be compatible, since the displacements of the vertices of adjacent elements are not constrained to be the same. Fraeijs de Veubeke (1964) noticed that ‘the possibility of rotation of one panel with respect to the adjacent one leads to peculiar difficulties’ and that ‘special care must therefore be exercised with respect to the pattern of subdivision into equilibrium models’. He further identified the conditions for which the assembled stiffness matrix is rank deficient. The identification of such situations is simple, by recognizing that they correspond to meshes of ‘skeleton elements’ that form mechanisms, as exemplified in Figure 3.2. This rank deficiency implies that some loadings, even when globally equilibrated, may be inadmissible, as exemplified in Figure 3.3. In Sander (1971) it is further noted that ‘the use of the first member of the family of quadrilateral membrane equilibrium elements7 is therefore always precarious’. The particular situation of assembling three or four triangles into a larger element, which is known to be stable under the appropriate conditions, was also studied by Fraeijs de Veubeke. All these situations, which may correspond to the occurrence of spurious kinematic modes in hybrid equilibrium elements, will be presented and discussed in Chapters 4 to 6. 3.4.2 Triangular Equilibrium Elements for Plate Bending

Finite elements for plate bending that strictly verify equilibrium were proposed around 1970, as explained in Allman (1971). Herrmann (1967) proposed a constant moment element based on using linear transverse displacement fields and a mixed variational principle, which imposed codiffusivity of the normal bending moments along the element sides.

Figure 3.2 Patches of Fraeijs de Veubeke’s equilibrated triangles. When the loading is restricted to the external sides, the patch on the left is stable, whereas the other two will not always achieve equilibrium. 7 Obtained by assembling four triangles with uniform stresses in each one.

Equilibrium in Other F.E. Formulations

p

p

p

2p

p 2p Figure 3.3 Admissible and inadmissible loadings for an unstable mesh of Fraeijs de Veubeke’s equilibrated triangles.

Morley (1967) proposed an equilibrium element with linear moments based on quadratic Southwell functions and a complementary energy principle using the underlying Kirchhoff theory (with more detail in Morley (1968)). In continuation, Morley (1971) described the constant moment equilibrium element based on the use of non-conforming quadratic functions for transverse displacements, and showed that the corresponding solutions for moments are equivalent to those produced by Herrmann (1967). Allman (1971) reformulated the same element using the complementary energy principle and he summarized and reviewed all these elements. The developments of equilibrium plate bending elements appear to have been driven largely by the needs of the aeronautical industry. Both Morley and Allman worked at the Royal Aircraft Establishment in Farnborough, UK. Simultaneously at the Laboratoire de Techniques Aéronautiques et Spatiales in Liège, Belgium, Fraeijs de Veubeke and Sander developed a family of triangular equilibrium elements. They also demonstrated that the elements could be based on the analogy between displacement functions and Southwell moment functions and their use in a complementary formulation (Fraeijs de Veubeke and Sander, 1968; Fraeijs de Veubeke and Zienkiewicz, 1967; Sander, 1971). 3.4.3 Other Variants

The most relevant procedures that have been used to achieve weaker forms of compatibility are based on the explicit or implicit use of discontinuous displacement approximations8 in hybrid or in mixed formulations. Either form can be used to explain Fraeijs de Veubeke’s equilibrated triangle,9 but other options exist, namely the application of reduced integration techniques, or the consideration of mixed formulations based on the approximation of ‘enhanced strain modes’. 8 Also referred to as ‘broken approximation spaces’. 9 The presentation in 3.4.1 assumes incompatible displacements that are linear inside each element, but the same results can be obtained by considering a hybrid formulation, as in 3.2, wherein the boundary displacements are discontinuous at the vertices and uniform on each side.

31

32

Equilibrium F.E. Formulations

The traction-based equilibrium elements in Wang and Zhong (2014) are a development of the former technique, applied to quadrilaterals with a broken stress field of degree 1. They provide exactly the same solutions as the macro-element of degree 1 presented in Maunder et al. (1996) and described in Section 5.7.2. When extended to higher approximation degrees, codiffusive tractions are obtained when the finite element model is built by considering that no nodes (also referred to as connectors) are located at the vertices of the element; that is, for an element with a stress approximation of degree ds , ds + 1 nodes are considered on each side (Fraeijs de Veubeke, 1973), whose behaviour can be explained by a generalization of the concept of a skeleton triangle. These elements are equivalent to the hybrid equilibrium elements discussed in Section 4.9 of Kempeneers (2006). In general the characterization of the solutions provided by each of these methods is not straightforward, as it requires a study of the solutions to assess convergence and stability. Traditionally this is achieved by means of a ‘patch test’ (Irons and Razzaque, 1972), wherein a patch of elements is subject to the boundary conditions corresponding to a known exact solution, which can be represented by the approximations assumed for the patch. The test is said to be passed successfully when the exact solution is recovered. It can, under certain circumstances, provide false results (both positive and negative) in terms of assessing the convergence of the solutions (Hartmann and Katz, 2004; Stummel, 1980; Wang, 2001). The Babuška-Brezzi condition (Boffi et al., 2013; Brezzi, 1974) provides a more rigorous criterion for convergence, but its application to a specific element is generally more complex than the almost straightforward application of the patch test.

3.5 Trefftz Formulations In this Section we consider finite element approaches that use approximation bases ‘locally satisfying all field equations’ (Jirousek, 1978). These bases are named after Erich Trefftz, who in 1926 proposed a numerical technique using similar approximations (Trefftz, 1926).10 Their derivation was initiated from the late 1970s by J. Jirousek, by assuming inside each element approximations based on solutions of the equations of linear elasticity, which are associated with both equilibrated stresses and compatible displacements, thus satisfying all field equations. The elements are then connected using a hybrid approach where either equilibrium of tractions or compatibility of displacements is imposed, leading to two complementary approaches. The resulting equations for the element focused on equilibrium are similar to those presented in Section 3.2 for Pian’s hybrid element where the assumed boundary displacements are continuous. The nomenclature used to distinguish these two alternatives may be confusing, since the approach based on trying to enforce inter element equilibrium approximates the displacements on the boundary, whereas the formulation associated with imposing compatibility approximates the tractions. Jirousek names the element according to the field that is approximated on the boundary (Jirousek and Zieli´nski, 1997): hybrid Trefftz 10 See Maunder (2003) for an English translation.

Equilibrium in Other F.E. Formulations

elements with displacement frame (HT-D) work upon a weak form of inter element equilibrium and hybrid Trefftz elements with traction frame (HT-T) explicitly impose a weak form of inter element compatibility. Following Freitas et al. (1999) we prefer to name the elements according to the field whose equations are paramount in the formulation. Under this convention the formulations imposing inter element equilibrium are termed ‘hybrid Trefftz stress’ and those imposing compatibility are called ‘hybrid Trefftz displacement’. Just as in the approaches previously presented, the vertex continuity of the approximation functions used in the element interfaces (for displacements or tractions) is optional. It is usual to refer to the continuous functions used in the boundary of the elements as ‘frame functions’. The consequences in terms of equilibrium of the solutions are then similar to what happens to Pian’s hybrid element. Both Pian’s hybrid element with linear stresses and Fraeijs de Veubeke’s equilibrated triangle can be considered particular cases of Trefftz elements, since the corresponding strains are always integrable.

3.6 Formulations Based on the Approximation of a Stress Potential The approximation of a stress potential seems to be, at first sight, the natural and simple way to obtain stress approximations that strictly enforce equilibrium. As presented in Chapter 4, equilibrated stresses in 2D can be obtained from the second derivatives of the scalar valued Airy function. Obtaining stresses from this function is analogous to obtaining curvatures from the transverse deflection of a thin plate. In a similar way, for plate bending problems, the moments are the first derivatives of the Southwell function, which is vectorial. For this problem obtaining the moments is similar to obtaining the strains from membrane displacements. Fraeijs de Veubeke and Zienkiewicz (1967) pointed out these analogies, indicating that thin plate elements can be used to obtain the Airy functions, whereas 2D elasticity formulations are appropriate to obtain the Southwell functions, thus resulting in a natural pair of dual solvers. Several finite element models developed from these ideas were proposed (Beckers, 1972; Harvey, 1983; Sarigul and Gallagher, 1989). In a nutshell, we can say that they face several difficulties: the codiffusivity of stresses obtained from the Airy function requires continuity of the first derivatives of the function (Charlwood, 1971); the decreased quality of the derivatives of the approximated fields; the more complex procedures required for the enforcement of kinematic boundary conditions. For 3D problems, equilibrated stresses can be obtained from the vector valued Maxwell or Morera stress functions. In this case there is an additional problem, since the equilibrated stresses obtained from a linearly independent basis defined for these functions are not themselves linearly independent.

3.7 The Symmetric Bi-Material Strip Revisited We present here a limited set of results obtained for the analysis of the bi-material strip example presented in Chapter 2, using some of the formulations discussed

33

34

Equilibrium F.E. Formulations

in this Chapter, as well as the hybrid equilibrium formulation, which is detailed in Chapter 4. They are intended to illustrate some of the properties of these models. A complete characterization would require a more comprehensive study, which is well beyond the scope of this brief presentation. We always use a coarse mesh with eight elements on a quarter of the strip, as indicated in Figure 3.5, assuming the same dimensions and properties that were considered for Figures 2.2 and 2.3. The following elements are considered: • Compatible elements with displacement approximations of degree d: Cd . For example C1 corresponds to the constant strain triangular element (CST), also known as T3. • Pian’s hybrid elements of varying degree. The notation used is Eds Cd𝜓 , where d◽ stands for the degree used for the approximation of field ◽. For example E3 C2 corresponds to an element with cubic stresses and quadratic displacements at its interfaces. • Trefftz elements with polynomial stress fields, considering either continuous or broken (discontinuous) interface displacement approximations, referred to as either Tds Cd𝜓 or Tds Bd𝑣 . For example T3 B2 corresponds to an element with cubic stresses and broken quadratic displacements at its interfaces;. • Hybrid equilibrium elements, using the notation Eds Bd𝑣 , so that E2 B2 is an element with quadratic stresses and quadratic interface displacements. Fraeijs de Veubeke’s equilibrated triangle is, in this notation, E0 B0 . A separate computer program was used to obtain the solution for the compatible elements. For the stress based formulations the code for ‘hybrid equilibrium’ elements was adapted according to the required conditions: • To enforce continuity of displacements at the vertices for the {E or T}◽ C◽ elements, additional constraints and variables were added to the governing system. • For Trefftz type elements, the stress modes associated with incompatible strains were removed from the approximation basis. The stresses obtained for cubic approximations are presented in Figure 3.4, where the three models that use continuous displacement fields are presented in the first three rows, while those that use broken displacement fields are presented in the remaining rows. This is followed in Figure 3.5 by a plot of the corresponding displacements. Lack of equilibrium is clearly apparent for the first three solutions, particularly by observing the values of the shear stresses on the boundary, the horizontal direct stress at the right hand side and the vertical direct stress at the interface between the two materials. Similarly the last three solutions show some lack of compatibility. Though the displacements in the first column of Figure 3.5 appear to be the same, this is in fact not the case. In particular the only solution truly associated with compatible displacements is the first one, as represented by also shading its domain. The other solutions in the first column do correspond to continuous boundary displacements, but they are not compatible because: • in the case of Pian’s Hybrid element the strains inside the elements, as obtained from the stresses via the constitutive relation, are not compatible, that is, there is no displacement field leading to these strains;

Equilibrium in Other F.E. Formulations

C3

Figure 3.4 Bi-material strip: stress distributions for the bi-material strip. Obtained from models based on cubic approximations on a mesh of eight elements, EB = 4EA . The ranges of the colours are: {−3.0 ∶ 3.0} for 𝜎xx ; {−0.5 ∶ 0.5} for 𝜎yy ; {−0.5 ∶ 0.5} for 𝜎xy . (See plate section for colour representation of this figure).

E3C3

T3C3

E3B3

T3 B3

T3 B2

σxx

σyy

σxy

35

36

Equilibrium F.E. Formulations

C3

E3B3

E3C3

T3 B 3

T3C3

T3 B 2

Figure 3.5 Bi-material strip: Deformed shapes of the bi-material strip. Obtained from models based on cubic approximations on a mesh of eight elements, EB = 4EA .

• for Trefftz elements the strains can be integrated, leading to element-wise displacement fields one degree higher than the stresses. Those displacements will match neither those of the boundary nor the adjacent elements. In Tables 3.1 to 3.4 we compare the values of some of the outputs at selected points, for the same mesh and the same formulations, as a function of the degree of the approximations. The reference values were computed on a mesh with a very high refinement at the right end of the interface, using approximations of degree 6. The corresponding stress plots are given in Figure 2.2. The relative errors in the strain energy in Table 3.5 are obtained from the difference between each solution and the reference value. Negative values indicate that the energy of the solution is smaller than the reference. The relative strain energies of the error are calculated from the energy of the stresses corresponding to the difference between the computed and the reference fields. A quick inspection of the values may lead to the impression that equilibrium is not verified for solutions Ed Bd and Td Bd in Tables 3.3 and 3.4, since neither is the vertical direct stress the same for the two materials, nor is the tangential stress zero. However, this is just a consequence of showing the average of the stress as measured at the two elements adjacent to the point. For example, when considering the vertical direct stress, the value for the two elements ‘inside’ the domain is equal, whereas the elements adjacent to the boundary have different values, thus producing a different average at each of the two materials.

Equilibrium in Other F.E. Formulations

Table 3.1 Bi-material strip: Average values of the horizontal and vertical displacements at the top right corner. Reference values are 𝛿x = 0.90668 and 𝛿y = −0.30984. d

Cd

Ed Cd

Td Cd

Ed Bd

Td B d

Td Bd−1

𝛿x

1

0.77408

0.77408

0.77408

0.83687

0.83687

0.25056

2

0.89123

0.90006

0.90007

0.91977

0.70388

0.84865

3

0.91290

0.90450

0.89993

0.91067

0.86946

0.89793

4

0.90778

0.91369

0.92513

0.90211

0.49911

0.94949

5

0.90631

0.90186

0.89129

0.90849

7.80363

0.87142

𝛿y

1

−0.20429

−0.20429

−0.20429

−0.42543

−0.42543

−0.07920

2

−0.30419

−0.29810

−0.29810

−0.29342

−0.51891

−0.24885

3

−0.31337

−0.31687

−0.31817

−0.31693

−0.41470

−0.30426

4

−0.31117

−0.31128

−0.30801

−0.32261

−0.33878

−0.31255

5

−0.31031

−0.31061

−0.30977

−0.30717

−10.21075

−0.30683

Table 3.2 Bi-material strip: Average values of the horizontal direct stress at the left hand end A B of the interface, for both materials. Reference values are 𝜎xx = 0.42003 and 𝜎xx = 1.58601. d

Cd

Ed Cd

Td Cd

Ed Bd

Td B d

Td Bd−1

A 𝜎xx

1

0.50255

0.50255

0.50255

0.52265

0.52265

0.06338

2

0.41081

0.42006

0.42224

0.37727

0.44320

0.49404

3

0.42035

0.41626

0.38576

0.44166

0.28826

0.39505

4

0.42288

0.42687

0.43363

0.39878

0.44481

0.45160

5

0.41735

0.41845

0.41479

0.44107

0.46426

0.39659

B 𝜎xx

1

1.50064

1.50064

1.50064

1.49414

1.49414

0.93662

2

1.64349

1.62272

1.62692

1.58917

1.65880

1.63292

3

1.51227

1.51390

1.51882

1.60514

1.79385

1.53990

4

1.64500

1.64617

1.65759

1.57483

1.55739

1.54171

5

1.53771

1.53577

1.52288

1.59626

1.61001

1.61234

It is observed that the solutions of C1 , E1 C1 and T1 C1 , as well as E1 B1 and T1 B1 , are equivalent. Effectively any equilibrated stress approximations of degree 1 is a Trefftz approximation, since it is associated with compatible strains, implying that E1 {C or B}◽ and T1 {C or B}◽ are essentially the same model.

37

38

Equilibrium F.E. Formulations

Table 3.3 Bi-material strip: Average values of the vertical direct stress at the left hand end of the interface, for both materials. Reference value is 0.12547. d

Cd

Ed Cd

Td Cd

Ed Bd

Td B d

Td Bd−1

A 𝜎yy

1

0.08177

0.08177

0.08177

0.22592

0.22592

0.11508

2

0.14031

0.12585

0.12610

0.05621

0.10477

−0.02320

3

0.12600

0.13175

0.13208

0.17602

0.06276

0.27788

4

0.11850

0.10903

0.09852

0.08050

0.16233

0.10394

5

0.13141

0.13998

0.14266

0.16715

0.17526

0.09053

B 𝜎yy

1

0.11434

0.11434

0.11434

0.26095

0.26095

0.13492

2

0.21860

0.11594

0.11686

0.03967

0.15067

0.46010

3

0.06328

0.17085

0.15859

0.18992

0.01194

−0.16473

4

0.16601

0.07364

0.09151

0.06643

0.13787

0.21117

5

0.09153

0.16921

0.15927

0.18056

0.15309

0.12672

Table 3.4 Bi-material strip: Average values of the tangential stress at the left hand end of the interface, for both materials. Reference value is 0. d

Cd

Ed Cd

Td Cd

Ed Bd

Td B d

Td Bd−1

A 𝜎xy

1

0.03576

0.03576

0.03576

0.03503

0.03503

−0.32215

2

0.04225

0.04209

0.04203

−0.01654

0.04591

−0.01791

3

−0.02063

−0.01858

−0.01090

0.01391

−0.05082

0.05050

4

0.01347

0.01244

−0.00805

−0.01407

−0.02445

−0.01506

5

−0.01163

−0.00350

0.00704

0.01341

−0.02217

−0.01578

B 𝜎xy

1

0.02938

0.02938

0.02938

0.03503

0.03503

0.87785

2

0.11433

0.10957

0.10981

−0.01654

0.04591

−0.01106

3

−0.10480

−0.09268

−0.08474

0.01391

−0.05082

0.02862

4

0.08155

0.06861

0.05086

−0.01407

−0.02445

0.00654

5

−0.06814

−0.05799

−0.05403

0.01341

−0.02217

−0.02475

The equivalence between C1 and {E or T}1 C1 is not so obvious. How is it possible that the latter solutions, which imply quadratic displacement fields within the elements, are equivalent to C1 ? We know that the problem of imposing linear displacements on a triangle has an exact solution, which corresponds to constant strains and stresses. Therefore,

Equilibrium in Other F.E. Formulations

Table 3.5 Bi-material strip: Relative error in strain energy and relative strain energy of the error. d

Cd

Ed Cd

Td Cd

Ed Bd

Td B d

Td Bd−1

Uc (𝝈 h ) − Uc (𝝈 ref ) U(𝝈 ref ) 1

−0.10267

−0.10267

−0.10267

0.02759

0.02759

−0.52956

2

−0.01911

−0.01306

−0.01306

0.00696

0.01978

−0.02281

3

−0.00549

−0.00250

−0.00192

0.00256

0.00882

−0.00190

4

−0.00210

−0.00071

0.00036

0.00126

0.00532

0.00229

5

−0.00099

−0.00025

0.00047

0.00069

0.00482

0.00195

Uc (𝝈 h − 𝝈 ref ) U(𝝈 ref ) 1

0.10267

0.10267

0.10267

0.02759

0.02759

0.54300

2

0.01911

0.01334

0.01333

0.00696

0.01978

0.03167

3

0.00549

0.00325

0.00314

0.00256

0.00882

0.00556

4

0.00210

0.00124

0.00118

0.00126

0.00532

0.00301

5

0.00099

0.00060

0.00070

0.00069

0.00482

0.00183

when subjected to continuous linear displacements on its sides, the hybrid triangular elements select the constant terms, out of the 7 stress approximation functions available. Actually all {E or T}◽ C1 elements show this behaviour.11 We also observe that monotonic convergence for pointwise values is not a rule, contrary to what might be expected. For example, non-monotonic values of the displacements at the top right corner are presented in Table 3.1 for both the compatible model Cd and the equilibrated model Ed Bd . Furthermore, the absolute values of the compatible components are sometimes greater than those of the equilibrated ones. This behaviour does not violate the rules governing the convergence of such solutions, which only concern integral quantities. Since the problem only has one horizontal uniformly distributed load on the right hand side and no imposed displacements, we can guarantee that the corresponding average horizontal displacement is larger for the equilibrated model than it is for the compatible one. And, furthermore, we can guarantee that the average of the horizontal displacement of the exact solution is located between these values. When it comes to other displacements or stresses no guarantees can be given. The strain energy of the compatible and equilibrated solutions, as well as their total energies, do converge monotonically, as can be observed in columns 1, 4 and 5 of the first block of Table 3.5, the error in strain energy, which for these columns is equal to the second block, the strain energy of the error. For the other columns (2, 3 and 6), which correspond to models that do not enforce a strong form of equilibrium on the element sides, the error in strain energy of the solution 11 This is not the case for hybrid quadrilateral elements with linear displacements, unless the values of the displacements at all four nodes happen to correspond to a linear field.

39

40

Equilibrium F.E. Formulations

does not have to converge monotonically and may thus be misleading. In these cases only a zero value of the strain energy of the error guarantees that the solution is exact. We have included a model where the degrees of the approximations are not the same, Td Bd−1 , because for higher degree approximations Td Bd is implicitly unstable. Effectively this element has 3 + 4d stress fields and 3 × 2 × (d + 1) displacement parameters, implying that for d > 0 the stiffness matrix of the element becomes singular, and spurious kinematic modes12 appear, as in Figure 3.5 and in Table 3.1. The number of these modes is reduced for the Td Bd−1 model, which has the same number of displacements as the Td Cd model, 3 × 2 × d. In this situation we get rid of at least some of the spurious modes, but we no longer have codiffusive tractions. Hence the oscillations in the last column in Table 3.5. The question in this context is: which model is better? The answer really depends on the characteristics that are required for the solution, and favourable points can be perceived from every solution. Certainly with some bias, we consider that the hybrid equilibrium models fare very well in this set of results.

References Allman DJ 1971 Triangular finite elements for plate bending with constant and linearly varying bending moments In High Speed Computing of Elastic Structures (ed. Fraeijs de Veubeke BM) University of Liège. Beckers P 1972 Les fonctions de tension dans la méthode des éléments finis Thèse de doctorat Université de Liège. Boffi D, Brezzi F and Fortin M 2013 Mixed Finite Element Methods and Applications. Springer. Brezzi F 1974 On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique 8(R2), 129–151. Charlwood RG 1971 Dual Formulations of Linear Elasticity Using Finite Elements Computer Aided Engineering: Proceedings of the Symposium Held at the University of Waterloo, May 11-13, 1971, p. 45–62 number 5 in SM Study University of Waterloo. Fraeijs de Veubeke BM 1964 Upper and lower bounds in matrix structural analysis. In AGARDograph 72: Matrix Methods of Stuctural Analysis, Pergamon Press London p. 165–201. Fraeijs de Veubeke BM 1965 Displacement and equilibrium models in the finite element method. In Stress Analysis (ed. Zienkiewicz OC and Holister GS). Wiley. Fraeijs de Veubeke BM 1973 Diffusive Equilibrium Models Lecture notes for the International Research Seminar of the Theory and Application of Finite Element Methods. University of Calgary. Fraeijs de Veubeke BM and Sander G 1968 An equilibrium model for plate bending. International Journal of Solids and Structures 4(4), 447–468. Fraeijs de Veubeke BM and Zienkiewicz OC 1967 Strain-energy bounds in finite-element analysis by slab analogy. The Journal of Strain Analysis for Engineering Design 2(4), 265–271. Freitas JAT, Almeida JPM and Pereira EMBR 1999 Non-conventional formulations for the finite element method. Computational Mechanics 23, 488–501. 12 Such modes will be defined and discussed in detail in Chapters 5 and 6.

Equilibrium in Other F.E. Formulations

Hartmann F and Katz C 2004 Structural Analysis with Finite Elements. Springer Science & Business Media. Harvey JW 1983 Dual analysis of plane stress problems by commonly based finite elements. International Journal for Numerical Methods in Engineering 19, 971–984. Herrmann LR 1967 Finite-element bending analysis for plates. Journal of the Engineering Mechanics Division, ASCE 93(5), 13–26. Hughes TJR 2000 The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications. Irons BM and Razzaque A 1972 Experience with the patch test for convergence of finite elements The Mathematical Foundations of the Finite Element Method with Applications to Partial Differential Equations Academic Press: New York p. 557–587. Jirousek J 1978 Basis for development of large finite elements locally satisfying all field equations. Computer Methods in Applied Mechanics and Engineering 14(1), 65–92. Jirousek J and Zieli´nski AP 1997 Survey of Trefftz-type element formulations. Computers and Structures 63(2), 225–242. Kempeneers M 2006 Eléments finis statiquement admissibles et estimation d’erreur par analyse duale Thèse de doctorat Université de Liège. Maunder EAW 2003 Trefftz in translation. Computer Assisted Mechanics and Engineering Sciences 10(4), 545–564. Maunder EAW, Almeida JPM and Ramsay ACA 1996 A general formulation of equilibrium macro-elements with control of spurious kinematic modes. International Journal for Numerical Methods in Engineering 39(18), 3175–3194. Morley LSD 1967 A triangular equilibrium element with linearly varying bending moments for plate bending problems. Journal of the Royal Aeronautical Society 71, 715. Morley LSD 1968 The triangular equilibrium element in the solution of plate bending problems. Aeronautical Quarterly 19, 149–169. Morley LSD 1971 The constant-moment plate-bending element. Journal of Strain Analysis 6(1), 20–24. Oliveira ERA 1968 Theoretical foundations of the finite element method. International Journal of Solids and Structures 4(10), 929–952. Pian THH 1964 Derivation of stiffness matrices based on assumed stress distribution. AIAA Journal 2, 1333–1336. Pian THH 1978 A historical note about ‘Hybrid Elements’. International Journal for Numerical Methods in Engineering 12(5), 891–892. Pian THH 1983 Reflections and remarks on hybrid and mixed finite element methods In Hybrid and Mixed Finite Element Methods (ed. Atluri SN, Gallagher RH and Zienkiewicz OC) John Wiley: New York p. 565–570. Pian THH and Tong P 1969 Basis of finite element methods for solid continua. International Journal for Numerical Methods in Engineering 1(1), 3–28. Sander G 1971 Application of the dual analysis principle In High Speed Computing of Elastic Structures (ed. Fraeijs de Veubeke BM) University of Liège. Sarigul N and Gallagher RH 1989 Assumed stress function finite element method: Two-dimensional elasticity. International Journal for Numerical Methods in Engineering 28, 1577–1598. Stummel F 1980 The limitations of the patch test. International Journal for Numerical Methods in Engineering 15(2), 177–188. Szabo B and Babuška I 2011 Introduction to Finite Element Analysis. John Wiley & Sons.

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Trefftz E 1926 Ein gegenstuck zum ritzschen verfahren. Proceedings of the 2nd International Congress of Applied Mechanics p. 131–137. Wang L and Zhong H 2014 A traction-based equilibrium finite element free from spurious kinematic modes for linear elasticity problems. International Journal for Numerical Methods in Engineering 99(10), 763–788. Wang M 2001 On the necessity and sufficiency of the patch test for convergence of nonconforming finite elements. SIAM Journal on Numerical Analysis 39(2), 363–384. Zienkiewicz OC and Taylor RL 2005 The Finite Element Method for Solid and Structural Mechanics. Butterworth-Heinemann.

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4 Formulation of Hybrid Equilibrium Elements In this Chapter we present a general hybrid finite element formulation which can provide approximations of the stress field that are strictly equilibrated (Almeida and Freitas, 1991, 1992). As explained in Chapter 3, elements are termed hybrid, following the nomenclature used by Pian (Pian, 1964, 1983; Pian and Tong, 1969), also used in Freitas et al. (1999), because a complementary displacement field is independently interpolated on the boundary. The stresses inside the elements are also approximated using functions that a priori satisfy equilibrium in the absence of body loads, as in the original hybrid elements, but by considering discontinuous boundary displacement approximations, the equilibrium conditions at element boundaries can be strictly enforced. Usually the enforcement of equilibrium leads to an indeterminate solution, with some parameters free. As is commonly done, their values are determined so that the resulting strains are as compatible as possible. We will present ways to achieve this, either by considering the explicit derivation of equivalent strains, or by making a generalized complementary energy functional of the system take a stationary value. For the sake of simplicity, details of the presentation will mostly be illustrated for two dimensional elasticity problems; nevertheless the formulation is presented in general terms. A strict enforcement of equilibrium can only be guaranteed under the assumptions of polynomial approximations and elements with straight sides/faces. They are recalled during the presentation whenever relevant, and their implications discussed. The case of curvilinear geometries is discussed at the end of the Chapter.

4.1 Approximation of the Stresses The basis used for the approximation of the stresses in each element is a linearly independent set of self-balanced stress fields, that is, each of these stress distributions verifies equation (A.2) in the absence of body loads. In general we require that these fields constitute a complete basis of a vector space, usually designated h , that is, they are able to represent all possible self-balanced distributions of a given polynomial of degree ds .

We collect the members of this basis in a matrix Se (x), which we usually denote simply as Se , whose components are functions that depend on the coordinates, x, of the point Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

44

Equilibrium F.E. Formulations

being considered, in the reference frame of the element. Each column corresponds to a self-balanced stress field, which can be derived directly or from a stress potential, for example, the Airy, Maxwell-Morera or Southwell functions, respectively for two and three dimensional elasticity problems and for plate bending problems. For example, for a linear approximation of the stresses in a two dimensional elasticity problem, we could potentially consider a basis with nine members corresponding to the constant, linear in x and linear in y terms for each stress component. But, as some of these stress distributions would not be self-balanced, we are required to combine some of the members of this basis, so that equilibrium is always verified. The dimension of the basis is thereby reduced from nine to seven. A possible choice for matrix Se (x) is: ⎡1 0 0 S e = ⎢0 1 0 ⎢ ⎣0 0 1

x y 0 0 −y 0

0 x 0

0⎤ y ⎥. ⎥ −x⎦

The projections of these stresses as boundary tractions on the unit square with the lower left corner at the origin are represented in Figure 4.1. In each element the stresses are approximated by a linear combination of the members of this basis, the weights being termed ŝ e , complemented with a particular solution, 𝝈 0e (x), which is an arbitrary solution of the non-homogeneous differential equilibrium

σxx = 1 σyy = 0 σxy = 0

σxx = 0 σyy = 1 σxy = 0

σxx = 0 σyy = 0 σxy = 1

σxx = x σyy = 0 σxy = –y

σxx = y σyy = 0 σxy = 0

σxx = 0 σyy = x σxy = 0

σxx = 0 σyy = y σxy = –x

y

x Figure 4.1 Projection of a basis of linear self-equilibrated stress distributions for two dimensional problems on the boundary of a square region with the lower left corner at the origin of the reference frame. Note that each distribution is in equilibrium, that is, it has zero resultant forces and moments, in the absence of body loads.

Formulation of Hybrid Equilibrium Elements

equation: 𝝈 e = Se ŝ e + 𝝈 0e ,

where  ⋆ 𝝈 0e + b̄ = 0.

(4.1)

As the approximation functions do not have interpolating properties, that is, they are not equal to unity at a particular node and zero at all others, it is not possible to interpret the meaning of the weights of the linear combination, the components of ŝ e , as nodal values of stress. Furthermore a single non-zero weight may imply that more than one stress component is non-zero.

The arbitrary nature of the particular solution for one element can be understood when two distinct solutions, 𝝈 10 and 𝝈 20 , are compared. The difference between them necessarily satisfies the homogeneous equilibrium equation, therefore that difference is included in Se , as long as it constitutes a complete basis of a degree no lower than that of 𝝈 0e . So the choice of 𝝈 10 or 𝝈 20 will result in different stress parameters, but in the same final stress representation, as long as ds0 ≤ ds , where ds0 is the polynomial degree of the particular solution. Throughout this book we always assume that this condition is verified. In a two dimensional problem with, for example, a constant body force in x equal to 1, we select two obvious particular solutions, 𝝈 10 = {−x, 0, 0} and 𝝈 20 = {0, 0, −y}. Clearly the difference between solutions 2 and 1 is equal to the fourth element of the stress approximation basis. Therefore the stress distribution represented by S ŝ 1 + 𝝈 10 is equal to S ŝ 2 + 𝝈 20 when the weight vectors are equal, except for the fourth component, which must satisfy ŝ14 = ŝ24 + 1.

4.2 Approximation of the Boundary Displacements The set of entities that constitute the boundary of the elements in a finite element mesh can naturally be divided into two subsets by considering whether they belong to the external boundary of the domain or not. We refer to those sets as the internal and the external boundary of the finite element mesh, furthermore differentiating the entities in the external boundary by considering whether they are subject to static or kinematic constraints. We thus say that an entity belonging to the boundary of a finite element can therefore belong to the internal boundary, Γi , to the static boundary, Γt , or to the kinematic boundary, Γu .1 The first point to consider when the question of approximating the boundary displacements is posed must be ‘on which entities of the boundary are approximations necessary?’ Before addressing this question we recognize that the boundary of an element is composed of sets of entities of lower dimension, which are always non-empty for polyhedral 1 Imposed element tractions or displacements always require that the boundary entity belongs to either Γt or Γu , even when it is inside the domain. This is not the case for imposed discontinuities either of tractions or displacements, both associated with jumps in element fields, which can be prescribed at Γi .

45

46

Equilibrium F.E. Formulations

and for polygonal elements. We assume for now that these elements may have an arbitrary configuration, with no restrictions on the number of entities that define it, or on its convexity. The boundary of a three dimensional element will be composed of faces, edges and vertices, whereas the boundary of two dimensional elements will only have edges and vertices. In the text we will refer to the edges of two dimensional elements as sides, in order to recognize their different nature. The question of defining on which parts of the boundary an approximation is necessary depends on the nature of the problem being considered, more precisely on the order of the differential operator involved. The reference frame on a boundary will vary according to its dimension, but we will represent it in general by 𝝃. In this Section we just state, in general terms, the properties of the functions used to approximate displacements on each entity of the boundary, avoiding the discussion of the details of specific problems, and focusing the presentation on the definitions and on the discussion of the characteristics of the functions used. Following the same general approach that was considered for stresses, the approximation of the displacement(s) on a given boundary entity m will be written as 𝒗m = V m 𝒗̂ m + 𝒗̄ m ,

(4.2)

where 𝒗̂ m is a vector of parameters defining the linear combination of the functions in V m , which constitute a basis for a vector space normally designated as h . All the other terms are functions of 𝝃, the term 𝒗̄ m being used to account for non-homogeneous prescribed kinematic boundary conditions. A displacement component at a boundary entity where kinematic conditions are imposed is accounted for by considering the corresponding members of V m to be zero. The precise description of the components of 𝒗m and the characteristics of each member of V m will be covered in later sections, in the context of specific types of element. For the approximation of displacements on faces we use a complete two dimensional polynomial basis of arbitrary degree, d𝑣 , referred to as V 2D , for each face.

This implies that for each component of the displacement on a face we have a total of (d𝑣 + 1)(d𝑣 + 2)∕2 degrees of freedom in the basis as indicated by the number of terms in a Pascal’s triangle. The simplest choice of functions uses monomials in the form si t j , with i, j ≥ 0 and i + j ≤ d𝑣 , so that 𝝃 = (s, t), with s and t being defined on the reference frame of the face, which dictates their range. For some geometries of the faces, in particular for triangular ones, the interpolation functions used in conventional two dimensional Lagrange type elements can be used. It is also possible to use interpolation functions that, rather than interpolating at the vertices and at equally spaced nodes, are based on the points of numerical integration. As we will later show, this can facilitate the process of numerically computing integrals on a face. d

One dimensional polynomials of degree d𝑣 , in V 1D𝑣 , are used to approximate each component of the displacements on each side.

Formulation of Hybrid Equilibrium Elements

In this case 𝝃 = (r) and the polynomial bases have dimension d𝑣 + 1. As for faces, we can use either simple monomials ri , with 0 ≤ i ≤ d𝑣 and the interval of r appropriately defined, or other interpolation functions, typically Lagrange polynomials either with the nodes uniformly distributed or coinciding with the Gauss-Legendre integration points. The approximation of a variable on a vertex, which we will consider for Kirchhoff plate elements, is done by considering a single degree of freedom associated with it, formally corresponding to V 0D . In this case d𝑣 is always 0 and the number of parameters is always 1.

This approach also allows us to consider components of the boundary where a given relation between displacements is imposed, for example, an inclined sliding support, by setting that relation in the approximation basis. For example in two dimensional elasticity, a sliding support allowing displacements at 45 degrees is modelled by using the same approximations, with the same weights, for ux and uy . { } [ ux 1 = uy 1

r r

r2 r2

] ⎧𝑣̂ 0 ⎫ · · · ⎪𝑣̂ 1 ⎪ · · · ⎨𝑣̂ 2 ⎬ ⎪ ⎪ ⎩⋮⎭

When relative displacements of internal sides or faces are prescribed, so that specific displacement discontinuities are imposed along boundaries between elements,2 both V m and 𝒗̄ m are non-zero. These approximations are taken independently for each entity. It is therefore normal, depending on the boundary from which the displacement is taken, to have different values at points that share boundary entities. For example, the lateral displacement at a vertex of a Kirchhoff plate mesh can be obtained from either the degrees of freedom of the vertex or from the value at the end of any side that is incident with it. Different values are an indication of lack of compatibility of the solution being considered. Complementary indications of lack of equilibrium occur in conventional displacement formulations when, for example, the normal moments on two elements that share a side are not equal. For flat faces and straight sides the geometry of the boundary is linear, as it is always possible to define a linear mapping from a reference frame on the boundary onto the reference frame of the element and vice-versa, that is, x(𝝃) and 𝝃(x) can be linear. In this situation the projection on the boundary of functions defined in the reference frame of the elements does not change their degree.

When this mapping is non-linear, the change of reference frame implies a change in the space spanned by the approximation functions used. The implications of this will be discussed at the end of the Chapter. 2 Such actions are used to obtain the generalized stress resultants dual to these discontinuities, as presented in Chapter 9

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48

Equilibrium F.E. Formulations

4.3 Assembling the Approximations The approximations just introduced define the corresponding fields on each element or boundary entity. When all the ne elements and nΓ boundary components of a mesh are considered, these approximations can be symbolically collected into two large vectors, resulting in the following definitions: ⎧ 𝝈 1 ⎫ ⎡S1 ⎪ 𝝈2 ⎪ ⎢ ⎨ ⋮ ⎬=⎢ ⎪ ⎪ ⎢ ⎩ 𝝈 ne ⎭ ⎣

S2

⎤ ⎧ ŝ 1 ⎫ ⎧ 𝝈 01 ⎫ ⎥ ⎪ ŝ 2 ⎪ ⎪ 𝝈 02 ⎪ ⎥⎨ ⋮ ⎬ + ⎨ ⋮ ⎬ ⎥⎪ ⎪ ⎪ ⎪ Sne ⎦ ⎩ŝ ne ⎭ ⎩𝝈 0n ⎭



e

and ⎧ 𝒗1 ⎫ ⎡V 1 ⎪ 𝒗2 ⎪ ⎢ ⎨ ⋮ ⎬=⎢ ⎪ ⎪ ⎢ ⎩𝒗nΓ ⎭ ⎣

V2



⎤ ⎧ 𝒗̂ 1 ⎫ ⎧ 𝒗̄ 1 ⎫ ⎥ ⎪ 𝒗̂ 2 ⎪ ⎪ 𝒗̄ 2 ⎪ ⎥⎨ ⋮ ⎬ + ⎨ ⋮ ⎬, ⎥⎪ ⎪ ⎪ ⎪ V nΓ ⎦ ⎩𝒗̂ nΓ ⎭ ⎩𝒗̄ nΓ ⎭

which we represent as 𝝈 = S ŝ + 𝝈 0

̄ 𝒗 = V 𝒗̂ + 𝒗.

and

The domain of each Se includes element e and its boundary, while the domain of each V m is the respective boundary entity, which is the intersection of the boundaries of its adjacent elements.

4.4 Enforcement of Equilibrium at the Boundaries of the Elements The requirement of equilibrium inside the elements is automatically satisfied by approximation (4.1). The enforcement of equilibrium at the boundaries is a more complex matter, which we discuss in this Section. In the kinematic boundary Γu of the domain, the freedom to impose an arbitrary reaction allows the boundary equilibrium condition (A.5) to be satisfied automatically. In the remaining boundaries of the elements of the mesh, which we identify as either the static boundary, Γt , or the internal boundary, Γi , that equation must be explicitly imposed. Towards this end we determine the contribution of element e to (A.5). Given the assumed stress approximation (4.1) this contribution can be written as: T

T

T

 e 𝝈 e =  e Se ŝ e +  e 𝝈 0e ≡ t e .

(4.3)

For a particular boundary entity of element e, provided it has a linear mapping (i.e. flat or straight), matrix  e consists of constant direction cosines. Then the projections on the boundary of the approximation matrix Se and of the initial stress vector 𝜎0e , that is, their trace (Reddy, 1986), Se (x(𝝃)) and 𝝈 0e (x(𝝃)), are polynomials of degree dsΓ , which is, at most, the same as the degree of the original functions in the reference frame of the element. Therefore the contribution of element e has the form of a polynomial that never has a degree higher than ds . It should be remembered that ds is not lower than ds0 .

Formulation of Hybrid Equilibrium Elements

When all the elements that are adjacent to a given boundary entity belonging to Γt ∑ or Γi are accounted for, the resulting polynomial form of t = t e must be equal to the applied tractions t̄ for strict equilibrium. In this case t̄ is termed admissible, and its form must be polynomial of degree dt , such that dt ≤ ds , where ds is the highest degree of the contributing elements. In order to enforce equilibrium at such a boundary we can start by explicitly writing down each polynomial boundary equilibrium equation, noting that the cost of doing this for a general case is considerable. Its decomposition in monomials implies that it is sufficient to equate each of the terms affecting each of the powers of the same degree of coordinate 𝝃, for example, si t j or ri . This implies that the polynomial equation of equilibrium on the boundary is equivalent to a set of as many linear equations as the dimension of the polynomial approximation space that is necessary to represent the traction component of highest degree. As an example, consider the projection on a vertical side of a two dimensional element, with x coordinate equal to 1, of the stress approximation exemplified in Section 4.1, without body loads, and with a horizontal traction equal to y. Mapping y directly onto r, the boundary equilibrium equations become: ŝ1 + ŝ4 + r ŝ5 = r; ŝ3 − r ŝ4 − ŝ7 = 0. As the highest degree present is 1, two linear functions in r are obtained, which are verified for every r if the following four equations are satisfied:3 ŝ1 + ŝ4 = 0; ŝ5 = 1; ŝ3 − ŝ7 = 0; −̂s4 = 0. We note that if the approximation is insufficient to represent the applied load, this approach allows us to directly detect the resulting inconsistency, expressed by an isolated constant on the right hand side which is required to be equal to zero. In the example, this could be induced by considering a horizontal load equal to y2 , which leads to the impossible equation ‘0 = 1’, corresponding to the monomial r2 . For a zero vertical traction the quadratic term would lead to the trivial equation ‘0 = 0’. The process that was presented, of identifying the equations that must be imposed on each boundary, is feasible for a general case but, as mentioned, it is excessively complex. As an alternative we can project each component of the tractions on a boundary entity, m, replacing the continuous equilibrium equation tm (𝝃) = t̄m (𝝃),

∀𝝃 ∈ Γm ⊂ Γt ∪ Γi ,

with a set of discrete forms, which we seek to make equivalent to the continuous one: ∫Γm

𝜓i (𝝃) t(𝝃) dΓ =

∫Γ m

𝜓i (𝝃) t̄(𝝃) dΓ,

∀ 𝜓i (𝝃).

3 The number of polynomial equilibrium equations (2) times the dimension of the highest one dimensional approximation space used (2) = 4.

49

50

Equilibrium F.E. Formulations

The discrete equilibrium equations can be interpreted as balancing energetically equivalent forces, where the weight functions 𝜓i are virtual displacement fields defined on the boundary – the equivalent force times the ‘unit’ displacement is equal to the work done by the traction with that displacement. The number of independent weight functions determines the number of equilibrium equations, and the enforcement of the discrete form implies that the continuous form is verified when tm (𝝃) and t̄m (𝝃) can be represented as a combination of the 𝜓i s. This is consistent with the reasoning that was presented when the explicit development into monomials was considered. Since an approximation of the boundary displacements is necessary when enforcing the compatibility conditions, it makes total sense to use the same set of functions in both contexts. Doing so is not strictly necessary, but ensures that the formulation is energetically consistent, and complies with the use of a Galerkin type approach. Then, the equation used to impose equilibrium at boundary entity m is: ) ( ∑ T T V Tm  e Se dΓ̂se + V Tm  e 𝝈 0e dΓ = V Tm t̄ m dΓ; (4.4) ∫ ∫ ∫ Γm Γm Γm e where the sum in e is extended to all the elements that are incident with boundary entity m. For the assembled stress and boundary displacement approximations, the boundary equilibrium equation for the mesh, which must be enforced at the static and at the internal boundary, can be written as: T

∫Γt,i

V T  S dΓ̂s +

T

∫Γt,i

V T  𝝈 0 dΓ =

∫Γt,i

V T t̄ dΓ

(4.5)

Equilibrium is imposed at Γt and at Γi using a weighted residual equation, which, for the polynomial basis considered, implies its local enforcement, provided dt ≤ ds ≤ d𝑣 .

The displacements are fixed in Γu , so there is no corresponding approximation of the boundary displacements. As mentioned in the beginning of this Section, the reactions take the values necessary to verify equilibrium: T

T

t =  S ŝ +  𝝈 0 ,

in Γu .

(4.6)

For the example that was previously used, on a side from y = 0 to y = 1, assuming [ ] r 1−r 0 0 V= , 0 0 r 1−r the discrete form of the boundary equilibrium equation becomes: 0.5 ŝ1 + 0.5 ŝ4 + 0.3333 ŝ5 = 0.3333; 0.5 ŝ1 + 0.5 ŝ4 + 0.1666 ŝ5 = 0.1666; 0.5 ŝ3 − 0.3333 ŝ4 − 0.5 ŝ7 = 0; 0.5 ŝ3 − 0.1666 ŝ4 − 0.5 ŝ7 = 0,

Formulation of Hybrid Equilibrium Elements

which is, as it should be, a linear combination of the equations obtained when the explicit development of the polynomials was considered. When d𝑣 is higher than dsΓ and dt , trivial equations ‘0 = 0’ will be introduced for each degree of freedom in this condition. This is not problematic from the point of view of the statement of the conditions, but may be inconvenient when an efficient solution of the system of equations is necessary. When the stress approximation is not able to represent the load, ds < dt , the inconsistency is automatically detected when d𝑣 > ds , as an impossible equation is obtained. But if d𝑣 ≤ ds only an approximate form of boundary equilibrium will be achieved.

4.5 Enforcement of Compatibility Stress distributions that verify the equations presented so far can be constructed so that they are statically admissible. In most situations, as observed in Chapter 2, these stress distributions are statically indeterminate, that is, there are more independent stress approximations in (4.1) than independent equilibrium constraints in (4.5). This was expressed in Sections 2.1 and 2.2.2 by the unknown coefficients b, b1 and b2 , which were determined using the principle of minimum complementary energy. This minimization of the complementary energy of the system seeks, amongst all statically admissible stress distributions, the one that has the ‘least incompatible’ strains (Washizu, 1982). This is imposed by requiring that, in a ‘certain average sense’, the strains obtained from the stresses, via the constitutive relations, are equal to the strains obtained from the displacements: ∫Ω

𝜰i f 𝝈 dΩ =

∫Ω

𝜰i u dΩ;

where the choice of the functions in 𝜰 determines what is meant by the ‘certain average sense’. Each row of 𝜰 , the row vector 𝜰i , can be interpreted as a stress field, which is multiplied by both strain fields to produce a strain energy density. Then the compatibility condition just presented requires that the strain energy obtained by projecting the strains onto the stress field 𝜰i must be the same, whether we compute it from the stresses or from the displacements. Though other choices are possible, we will always select to set 𝜰 = ST , that is, we make the rows of 𝜰 equal to the columns of S, the independent stress approximation functions, thus continuing to comply with a Galerkin approach, as in Section 4.4. Discrete generalized strains, ê , are obtained, for which the following work equivalence between the parameters and the corresponding continuous approximation is obtained: ∫Ω

𝜀T 𝝈 dΩ = ê T ŝ .

(4.7)

We observe that other options are possible, which would lead to non-symmetric governing systems, whose convergence properties are generally harder to assess.

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Equilibrium F.E. Formulations

When the stress approximation, S, is used for 𝜰 , the compatibility equation becomes: ê =

∫Ω

ST f S dΩ ŝ +

∫Ω

ST f 𝝈 0 dΩ =

∫Ω

ST u dΩ.

(4.8)

The internal displacements, u, in the right hand side of (4.8) can be removed via an integration by parts. We start by recalling that, for one dimensional problems, the integration by parts of the product of one function and the derivative of another can be obtained considering that (u 𝑣)′ = u′ 𝑣 + u 𝑣′ =⇒ u 𝑣′ = (u 𝑣)′ − u′ 𝑣, therefore b

∫a

b

u 𝑣′ dx = [u 𝑣]ba −

∫a

u′ 𝑣 dx.

In (4.8) both the members of ST and u are components of a tensor field, while u is a vector field. The differential operator  transforms the vector field of the displacements into the tensor field of the strains and its adjoint  ⋆ transforms the tensor field of stresses into the vector field of body forces. For a vector 𝝉, with the components of a symmetric tensor field, and a vector field, 𝝊, the integration by parts rule is written, according to Gauss’s Theorem, as ∫Ω

𝝉 T 𝝊 dΩ =

∫Γ

𝝉 T  𝝊 dΓ −

∫Ω

( ⋆ 𝝉)T 𝝊 dΩ.

Applying this formula to the right hand side of compatibility condition (4.8) we get ∫Ω

ST u dΩ =

∫Γ

ST  u dΓ −

∫Ω

( ⋆ S)T u dΩ.

If the differential operator, , has second (or higher) derivatives, it is necessary to repeat the integration by parts until  ⋆ is operating on S. At the end of this process, as we will see in detail for Kirchhoff plates, provided different boundary approximations are considered, a similar final result is obtained. As the stress approximation functions are self-equilibrated, the last term on the right hand side is zero. Furthermore, we replace the displacements, which are only used on the boundary, by their approximation (4.2), leading to the following compatibility condition, where the possible influence of initial strains, 𝜀T , such as those induced by thermal effects, has also been included: ∫Ω

ST f S dΩ ŝ +

∫Ω

ST (f 𝝈 0 + 𝜀T ) dΩ = ∫Γt,i

ST  V dΓ 𝒗̂ +

∫Γu

ST  ū dΓ.

(4.9)

Compatibility is imposed in a weak form by equating the projections of the strains induced by the stresses with those induced by the boundary displacements. In general this weak form does not imply that a local enforcement of compatibility is possible.

Formulation of Hybrid Equilibrium Elements

The compatibility operator on the right hand side of (4.9), transforming boundary displacement parameters into equivalent strains, is the transpose of the equilibrium operator in (4.5), which transforms stress parameters into generalized boundary forces.

4.6 Governing System We use the following notation for the integrals introduced in the previous sections for element e: Flexibility matrix

e =

Equivalent initial strains

ê 0e =

∫ Ωe ∫ Ωe

STe f Se dΩ;

(4.10)

STe (f 𝝈 0e + 𝜀T ) dΩ −

Equilibrium matrix Equivalent initial tractions

Dme =

T

∫Γm

∫Γu

V m T  me Se dΓ;

STe  e ū dΓ;

(4.11) (4.12)

T V T t̄ dΓ; (4.13) t̂ 0me = − V Tm  me 𝝈 0e dΓ + ∫Γm ∫Γm m m

When all stress and boundary displacement parameters are collected into unique ̂ as presented in Section 4.3, these elemental stress and displacement vectors, ŝ and 𝒗, matrices and vectors can be assembled into global entities. Then the governing system corresponding to the model being developed can be written as: [ ]{ } { } ê ŝ − DT = ̂0 (4.14) 𝒗̂ t0 D 𝟎 Matrix  is block diagonal and D is block sparse. The non-zero entries in a row corresponding to boundary m are in the columns corresponding to the stress parameters of the elements that are adjacent to that boundary.4 Unlike the usual stiffness matrices, each non-zero entry in this system results from a single contribution, that is, there is no accumulation of values. Given the highly sparse nature of system (4.14), it is appropriate to solve its equations using general sparse solvers (Davis, 2006) and it is inadvisable, in general, to use solvers designed for dense or banded matrices. On the other hand, the block diagonal structure of  can be exploited, together with its positive-definitiveness, to eliminate the stress parameters from the system of equations, ŝ =  −1 DT 𝒗̂ −  −1 ê 0 , 4 The use of a Galerkin approach leads to a projection of the generalized stresses into tractions that is contragradient to the transformation of boundary displacements into generalized strains, expressed by D and DT respectively. This has been derived from static-kinematic duality (Freitas, 1989), but could also be obtained from the principle of virtual work.

53

54

Equilibrium F.E. Formulations

leading to a stiffness-like equilibrium equation, where the unknowns are the boundary displacement parameters. We can introduce a new operator, K , which corresponds to the Schur complement of the flexibility matrix: (K = (D −1 DT ))𝒗̂ = D −1 ê 0 + t̂ 0 .

(4.15)

This system is also sparse, thought not as much as (4.14), with a similar structure to that of the stiffness matrix of a displacement based finite element model. An efficient solution of this system is also possible, namely banded solvers can be used, provided an appropriate reordering of the displacement parameters is used and the assembly of the system of equations is performed element and boundary-wise. Instead of computing (D −1 DT ) globally, each elemental flexibility matrix must be individually inverted and T each term (Dme  −1 e Dne ) computed, before assembling each contribution into the global system.

4.7 Existence and Uniqueness of the Solution The matrix in systems of equations with the structure of (4.14), which is typical of hybrid and mixed finite element formulations, is not positive definite. This raises the questions of existence and uniqueness of the solution, which extend into the question of convergence of the formulation itself. All these points are characterized, from a mathematical point of view, by stability prerequisites, known as the Babuška-Brezzi condition (Boffi et al., 2013). In this Section we characterize the question of existence and uniqueness of the solution from an algebraic point of view, in preparation for Chapter 5, where we see how to predict a priori potential problems in an arbitrary mesh of hybrid equilibrium finite elements, and how to circumvent them. Potential issues related to the convergence of the method will be dealt with separately in Chapter 7. Matrix  is symmetric, by definition, and is positive definite when the columns of each Se are linearly independent and f is positive definite, which are requirements imposed a priori. The independence of the elements of a stress approximations basis, Se , is equivalent to ∫Ω STe Se dΩ being positive definite. The introduction of a positive definite kernel, f , in this e integral just corresponds to a change of the metric used in the projection.

In the special case of incompressible materials, where 𝜈 = 0.5, f is positive semi-definite, and so is  e . Each flexibility matrix is then rank deficient by one, since only isotropic stresses induce zero strains and there is only one such self-balanced stress distribution, the uniform one. Since this field is the same throughout the mesh, it corresponds to only one parameter, which is fixed by the boundary conditions. In the special case where all the external boundary is fixed it becomes indeterminate. The flexibility matrix associated with the other stress parameters is positive definite and the corresponding stresses are always uniquely defined. Therefore (4.14) depends of the rank of matrix D, which has dimen) rank (∑ of system ) (∑ the Γ e × = n n n sion 𝑣 × ns . Algebraically this corresponds to saying that the nΓ 𝑣 ne s nullspaces of K and DT are the same.

Formulation of Hybrid Equilibrium Elements

We start by assuming that: • the model is sufficiently supported, so as to exclude rigid body displacements; • if the material is incompressible its external boundary is not totally fixed, so as to exclude the indeterminate uniform isotropic stress field. Consider a vector 𝒗̂ 0 that belongs to the nullspace of K , that is, such that K 𝒗̂ 0 = 0. This implies that 𝒗̂ T0 K 𝒗̂ 0 = 0 = 𝒗̂ T0 D −1 DT 𝒗̂ 0 . When  is positive definite, this requires that DT 𝒗̂ 0 = 0. The case of an incompressible material can be dealt with by excluding the uniform isotropic stress state. It is also clear, from the definition of K , that if DT 𝒗̂ 0 = 0, then K 𝒗̂ 0 = 0.

There are, consequently, two possible situations regarding the solution of (4.14) or (4.15): i) The nullspace of DT is empty. Then D has full row rank, K is positive definite, and the solution for both 𝒗̂ and ŝ is unique. For this case to be possible it is necessary, but not sufficient, that n𝑣 ≤ ns ; ii) The nullspace of DT is not empty. Then D is row rank deficient, K is positive semi-definite and the equations need to be consistent for a solution to be possible. This requires that 𝒗̂ T0 (D −1 ê 0 + t̂ 0 ) = 0 for all 𝒗̂ 0 belonging to the nullspace of DT (or K ), that is, the generalized load has to be orthogonal to the nullspace. ̂ The difference When the equations are consistent, multiple solutions exist for 𝒗. between two distinct solutions, 𝒗̂ 1 and 𝒗̂ 2 , necessarily belongs to the nullspace of DT , because K 𝒗̂ 1 = K 𝒗̂ 2 = (D −1 ê 0 + t̂ 0 ), and then K (𝒗̂ 1 − 𝒗̂ 2 ) = 0, so that 𝒗̂ 1 − 𝒗̂ 2 belongs to the nullspace of DT and DT 𝒗̂ 1 = DT 𝒗̂ 2 . Therefore the different solutions are all mapped into the same vector by DT , and the solution for ŝ is unique. The first situation is the most desirable: When the nullspace of DT is empty the solution of (4.14) exists and is unique.

This should be the general goal, but it may not always be achieved.5 It might be considered that a non-empty nullspace is a failure of the formulation, but, as we shall see, it does not necessarily hinder the fulfilment of the objectives of the analysis. Nevertheless it is crucial to realize when the nullspace of DT is non-empty, and to interpret the consequences of such a situation. First we note that if n𝑣 > ns , the dimension of the nullspace is at least n𝑣 − ns . This means that if there are more equilibrium conditions, weighted by the functions approximating the 𝒗s, than static degrees of freedom, corresponding to the members of ŝ , the existence of a solution is in question, and if it exists it will not be unique. As, in practice, the problem is more complex than this, because global equilibrium also needs to be considered, as well as additional internal conditions, the detailed accounting of these dimensions will be postponed. 5 Guidance on how to achieve this is given in Section 5.7

55

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By considering that DT transforms boundary displacement parameters into generalized strains, we realize that the rigid body displacements, that is, those that imply movement without deformation, necessarily belong to the nullspace. But, as we are dealing with generalized strains (a form that is equivalent to the integrals of the stress approximation functions times the strains), there may exist other (non-rigid body) boundary displacements, which are termed spurious kinematic modes, that also induce zero strains, and are consequently part of the nullspace of DT . For now we will concentrate on the existence of rigid body movements, which can only occur when the mesh is not fixed. The consequences of having such a situation provides a simple interpretation of the conditions for the existence of a solution and of the difference between multiple solutions, which directly extends to the case of spurious kinematic modes. A structural model is not a mechanism when it has supports that block all possible rigid body movements. The number of necessary supports is three for a planar model, and six for a spatial one, but having this number of supports is not sufficient to guarantee proper support, because they may not be properly distributed, blocking the rigid body movement more than once in one direction, while allowing other movements to occur. If the system somehow lacks supports, the displacements of any of the possible rigid body movements introduce, by definition, zero strains. Therefore for a vector of displacement parameters describing such a movement, 𝒗̂ rbm , we have DT 𝒗̂ rbm = 0, that is, 𝒗̂ rbm belongs to the nullspace of DT , and the number of such independent vectors is equal to the number of possible rigid body movements. From the viewpoint of equilibrium, due to the lack of reactions, the static analysis of a model that is not properly supported is only possible when the loading that is applied to a model satisfies global equilibrium by itself. The conditions for this can be expressed as requiring that the applied loads produce no work under any rigid body movement, which is none other than the algebraic condition for consistency of the system: 𝒗̂ T0 (D −1 ê 0 + t̂ 0 ) = 0, ∀ 𝒗̂ 0 ∈ rbm , where rbm is the space of rigid body modes, which will be defined in Section 5.1. For a solution of (4.14) to exist, the loading must be admissible, that is, it must satisfy 𝒗̂ T0 (D −1 ê 0 + t̂ 0 ) = 0 for all 𝒗̂ 0 belonging to the nullspace of DT . This is an equilibrium requirement: in the second generalized row of (4.14) the projection of the stresses must be able to balance the projection of the applied forces, otherwise a solution does not exist. When the nullspace is formed only by the rigid body modes, this corresponds to requiring that the loading is self-equilibrated.

Furthermore, the solution in terms of displacements is undetermined, because if we add to a solution an arbitrary rigid body movement, or any other member of the nullspace of DT , without modifying the stresses, we continue to verify (4.14). ̂ is a solution of (4.14), then (̂s, 𝒗̂ + 𝒗̂ 0 ), where 𝒗̂ 0 is an arbitrary vector in the If (̂s, 𝒗) nullspace of DT , is also a solution of (4.14). This corresponds to saying that, when a solution exists, the stresses are unique, but the displacements are only defined up to a member of the nullspace. When the nullspace is formed only by the rigid body modes, this corresponds to saying that the solution is not modified by adding an arbitrary rigid body movement to it.

Formulation of Hybrid Equilibrium Elements

Though the origin of the spurious kinematic modes is more complex than the simple lack of support that was just interpreted, the consequences are quite similar: not all loadings are admissible and the resulting displacements are indeterminate. The details on their origin and on how to control them are the main topic of Chapter 5.

4.8 Elements for Specific Types of Problem In this Section we will discuss particular points related to the application of the hybrid equilibrium formulation to specific types of problem. 4.8.1 Continua in 2D

For two dimensional problems, plane stress or plane strain, the dimension of the approximation basis in (4.1) can be determined by considering that the space of a general bi-dimensional stress field of degree ds has dimension 3 × (ds + 1)(ds + 2)∕2. For this field to be self-equilibrated, its derivatives, which are of degree ds − 1, must be in equilibrium with zero body forces, thereby imposing 2 × ((ds − 1) + 1)((ds − 1) + 2)∕2 = ds (ds + 1) constraints on the general field. The dimension of e , that is, the number of parameters of the self-equilibrated stress field, is obtained by subtracting the number of constraints from the number of parameters of the general stress field. (ds + 1)(ds + 2) n2D − ds (ds + 1), s = 3× 2 (d + 1)(ds + 6) . (4.16) = s 2 The derivation of the components of an approximation basis can be made quite generally by the use of the Airy stress potential (Airy, 1863), Φ. Then the stresses are given by: ⎧ 𝜕2 ⎫ ⎪ ⎪ ⎧𝜎 ⎫ ⎪ 𝜕y2 ⎪ ⎪ xx ⎪ ⎪ 𝜕 2 ⎪ ⎨𝜎yy ⎬ = ⎨ ⎬ Φ = 𝔇Airy Φ. 2 ⎪𝜎xy ⎪ ⎪ 𝜕x ⎪ ⎩ ⎭ ⎪ 𝜕2 ⎪ ⎪− 𝜕x𝜕y ⎪ ⎩ ⎭ Therefore to obtain a complete self-balanced stress basis of degree ds we use a basis of degree ds + 2 for Φ, which has ((ds + 2) + 1)((ds + 2) + 2)∕2 = (ds + 3)(ds + 4)∕2 components. Noting that three components must be removed, as the constant and linear terms induce zero stresses, it can be verified that the number of components generated is equal to n2D s . As already mentioned there are multiple particular solutions. The most convenient approach is to set the tangential stress in 𝝈 0 equal to zero, and to set the normal stresses equal to minus the corresponding primitive of the body forces. In this case, for a general monomial body force, b, we have: ⎧ −𝛽 xi+1 yj ∕(i + 1) ⎫ ⎪ ⎪ x b= =⇒ 𝝈 0 = ⎨−𝛽y xk yl+1 ∕(l + 1)⎬ , ⎪ ⎪ 0 ⎭ ⎩ which can be extended to cover general polynomial body forces. {

𝛽x x i y j 𝛽y xk y l

}

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The boundary displacement is a vector with two components 𝒗 = {𝑣x , 𝑣y } defined on each side of the mesh. Where there are no kinematic boundary conditions imposed, the approximation of degree d𝑣 of 𝒗 on side Γm is written, as: [ { } d 𝑣x V 1D𝑣 = 𝒗m = 𝑣y m 𝟎

⎧ 𝑣̂ ⎫ ⎪ 1 ⎪ ]⎪ ⋮ ⎪ 𝟎 ⎪ 𝑣̂ d𝑣 +1 ⎪ ⎨ ⎬ = V m 𝒗̂ m . d V 1D𝑣 ⎪ 𝑣̂ d𝑣 +2 ⎪ ⎪ ⋮ ⎪ ⎪𝑣̂ 2(d +1) ⎪ ⎩ 𝑣 ⎭m

(4.17)

As already mentioned, kinematic boundary conditions are imposed by restricting V m : a fixed side requires a nil V m ; a horizontal sliding support requires the removal of the second block of V m ; and other conditions can be imposed by using appropriate combinations of the V 1D s. When we set d𝑣 equal to ds , which will be our general rule, strict boundary equilibrium is imposed by (4.5). We note that when d𝑣 < ds , these formulations may also converge to the exact solution. In this case the bounding properties and the monotonic nature of convergence cannot be guaranteed. When d𝑣 > ds it is guaranteed that matrix D will be rank deficient, but this condition may also occur when d𝑣 = ds . The conditions for this, and procedures to avoid it, are discussed in detail in Chapter 5, with a particular focus on triangular and tetrahedral (simplicial) elements. Though the formulation allows for the use of general elements, convex or non-convex, with an arbitrary number of sides, their usage is seldom of practical interest. An example to illustrate this point involves considering a mesh of hexagonal elements. The stress approximation on each element has (ds + 1)(ds + 6)∕2 components and on the boundary the displacement functions imply 6 × 2 × (d𝑣 + 1) constraints, which are reduced by three when the global equilibrium conditions (or the rigid body modes from the point of view of the displacements) are considered. Each element will then have (ds + 1)(ds + 6)∕2 − 6 × 2 × (d𝑣 + 1) − 3 unconstrained degrees of freedom. Setting d𝑣 equal to ds will lead to a negative number of degrees of freedom when ds = d𝑣 ≤ 18, meaning that arbitrary tractions of degree d𝑣 will only verify equilibrium under special constraints, stronger than global equilibrium.6 Only when d𝑣 is considerably smaller than ds may these elements eventually work. We note that the formulation allows for the application of more general particular solutions, for example, rational, trigonometric or exponential, which may correspond to the exact solution for particular boundary conditions, different from those being considered, which the model tries to correct with the complementary solution. In this case, as already mentioned, the strictly equilibrated properties of the solutions will, in general, be lost. 4.8.1.1 Exemplification of the Assembly Process

To exemplify the application of the Hybrid Equilibrium Formulation to a plane stress problem we use the model with two elements in Figure 4.2. 6 Actually the problem is more complex because hyperstatic modes (with zero projection on the boundary) may exist, further reducing the number of parameters available for equilibrating the applied tractions.

Formulation of Hybrid Equilibrium Elements

Figure 4.2 Mesh for a two dimensional problem.

e y b

a

x

1

2

d

α

c

The choice of the degree of the approximation of the stress fields within the elements, ds , leads to vectors ŝ 1 and ŝ 2 each with dimension n2D s , which are collected in the global vector of stress parameters ŝ . Regarding the approximation of the boundary approximations we must consider that the model has one free side, a, one internal side, b, two sliding sides, c and d, and one fixed side, e. As no displacements are imposed, 𝒗̄ i is always zero. For strict equilibrium d𝑣 must be equal to ds . The approximations on sides a and b are complete, with independent components along x and y, while on side e the displacements are all zero. On side c only the x displacement component is approximated, but on the inclined side, d, both components are non-zero, though they are not independent. The correct relation is achieved by setting { } [ ] 𝑣x cos(𝛼) d𝑣 = V 1D 𝒗̂ d . 𝑣y d sin(𝛼) The global boundary displacement vector collects the parameters of sides a to d, the first two of dimension 2 × n1D , the others with n1D components. The governing system takes the following form: ⎡− ⎢ 1 ⎢ 𝟎 ⎢ ⎢ Da1 ⎢ Db1 ⎢D ⎢ c1 ⎣ 𝟎

𝟎 − 2 𝟎 Db2 𝟎 Dd2

DTa1

DTb1

DTc1

𝟎

𝟎

DTb2

𝟎

DTd2

𝟎

⎤⎧ ⎥⎪ ⎥⎪ ⎥⎪ ⎥⎨ ⎥⎪ ⎥⎪ ⎥⎪ ⎦⎩

ŝ 1 ⎫ ⎧ ⎪ ⎪ ŝ 2 ⎪ ⎪ ⎪ 𝒗̂ a ⎪ ⎬=⎨ 𝒗̂ b ⎪ ⎪ 𝒗̂ c ⎪ ⎪ ⎪ 𝒗̂ d ⎪ ⎭ ⎩

ê 01 ⎫ ê 02 ⎪ ⎪ t̂ 0a ⎪ ⎬ t̂ 0b ⎪ t̂ 0c ⎪ t̂ 0d ⎪ ⎭

For the action in Figure 4.2 the right hand side of the governing system is zero, except for the horizontal components of t̂ 0a . The determination of the flexibility matrices by numerical integration is simpler than the determination of stiffness matrices in isoparametric displacement formulations, because there are no derivatives involved. The value of the S function matrix at each integration point needs to be determined, and the triple product ST f S times the integration weight is accumulated. The process for the determination of the equilibrium matrices is similar, with a one dimensional integration on the sides. It is crucial to specify an orientation for the sides, so that the integrals are performed in a consistent way, in particular regarding V (r). Finally it is important to point out that the origin of the reference frame that is chosen affects the numerical stability of the resulting governing system. In the absence of rounding errors this problem would not be relevant, but, as this is not the case, the precision of the integrals computed far away from the origin can be affected. The simple and efficient solution is to use a local reference frame for each element, parallel to the

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global one, for example, centred at the centroid of the element. A scaling of the coordinates used in matrix S, such that the element fits approximately inside a unit circle, can also improve conditioning. A similar approach can also be used for each side. With the mesh in Figure 4.2, irrespective of the degree ds , there is a single spurious kinematic mode associated with the right angle at the lower left corner. This mode is illustrated in Figure 4.3, together with an alternative mesh which is free from spurious modes. The existence of the spurious mode implies that D is rank deficient by one. The governing system is therefore singular, so only when the right hand side is consistent will a solution be possible, otherwise the solution of the system is impossible. The condition for consistency is, in this case, very simple. It just requires that the tangential tractions must correspond to a single state of stress at that corner, a condition that is not required for the alternative mesh. The solution has a multiplicity of one, the same as the rank deficiency of the system, which affects only the components of 𝒗 that are non-zero in Figure 4.3. The stresses are unique, as well as the displacements on the sides, b and d, that are not affected by the spurious mode. The general case for determining spurious kinematic modes and defining admissible loads will be addressed in Chapter 5. 4.8.1.2 A Simple Numerical Example

We illustrate the equilibrated nature of the solutions that are obtained with the hybrid equilibrium formulation by considering the model in Figure 4.4: a square cantilever plate of unit side, modelled using two elements with approximations of degree 5 for both stresses and boundary displacements. The three components of stress are presented in Figure 4.5. The equilibrated nature of these solutions can be appreciated by observing that 𝜎xx is zero on the right side, 𝜎yy is one at the bottom and zero at the top, 𝜎xy is zero throughout the static boundary, while 𝜎y′ y′ and 𝜎x′ y′ are continuous on the internal interface. Figure 4.6 shows the corresponding boundary tractions for each element and the stress trajectories7 inside them. As a direct consequence of the observations that were made regarding Figure 4.5, these tractions are in equilibrium. Finally, the boundary displacements are illustrated in Figure 4.7. On the left part of this figure the displacements are affected by a non-zero amplitude of the spurious kinematic

Figure 4.3 Two dimensional problem: Illustration of the spurious kinematic mode and an alternative stable mesh. 7 Stress trajectories are orthogonal families of curved lines whose tangents follow the directions of the principal stresses (Pereira and Almeida, 1994; Timoshenko and Goodier, 1951).

Formulation of Hybrid Equilibrium Elements

y E=1 ν = 0.15

x y'

x'

Figure 4.4 Square cantilever. Definition of the boundary conditions, material properties, mesh, reference frames and components of the stress tensor either in the (x, y) or in the (x ′ , y′ ) reference frame. Figure 4.5 Square cantilever. Distributions of the stress components in the two frames considered. The range of values for the colour representation are: {−5.0 ∶ 5.0} for 𝜎xx ; {−1.5 ∶ 1.5} for 𝜎yy ; {−1.5 ∶ 1.5} for 𝜎xy ; {−2 ∶ 2} for 𝜎x′ x′ , 𝜎y′ y′ and 𝜎x′ y′ ; with blue representing the minimum value, red the maximum and green the central value. (See plate section for colour representation of this figure).

Figure 4.6 Square cantilever. Element tractions and stress trajectories. The width of each trajectory varies proportionally to the corresponding principal stress, from −4 to 4, with blue representing the minimum value, red the maximum and green the central value. (See plate section for colour representation of this figure).

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Figure 4.7 Square cantilever. Boundary displacements, on the left with a non-zero spurious mode amplitude, on the right with the spurious mode blocked.

mode associated with the bottom right vertex. This amplitude is implicitly selected by the solver that was used. To obtain the ‘clean’ displacements on the right the technique that will be described in Section 6.2 is used to block the spurious mode. Notice that only the displacements of the bottom and right sides are different; the incompatible nature of this solution implies that the displacements are discontinuous at the vertices, even if only by a small amount. 4.8.2 Continua in 3D

For a three dimensional continuum, a polynomial of degree ds has (ds + 1)(ds + 2)(ds + 3)∕6 monomials, the dimension of Pascal’s pyramid of degree ds . Therefore, as the symmetric stress tensor has six independent components, the dimension of the complete approximation space of degree ds is 6 × (ds + 1)(ds + 2)(ds + 3)∕6. For this field to be self-equilibrated we impose 3 × ((ds − 1) + 1)((ds − 1) + 2)((ds − 1) + 3)∕6 = ds (ds + 1)(ds + 2)∕2 constraints on the general field. The dimension of the complete self-equilibrated stress approximation field of degree ds is, therefore: n3D s = (ds + 1)(ds + 2)(ds + 3) − ds (ds + 1)(ds + 2)∕2, 1 = (ds + 1)(ds + 2)(ds + 6). (4.18) 2 The Maxwell and the Morera stress potentials (Maxwell, 1866; Morera, 1892) may be used for the derivation of an approximation basis (Almeida and Pereira, 1996). Both these potentials are based on three independent potentials, Φi with i = 1, 2, 3, that is, a vector valued field, for which the stresses are given respectively by: ⎡ ⎢ ⋅ ⎢ 2 ⎧𝜎 ⎫ ⎢ 𝜕 ⎪ xx ⎪ ⎢⎢ 𝜕z2 ⎪𝜎yy ⎪ ⎢ 𝜕 2 ⎪ 𝜎zz ⎪ ⎢ ⎨𝜎 ⎬ = ⎢ 𝜕y2 ⎪ xy ⎪ ⎢ ⎪ 𝜎yz ⎪ ⎢ ⋅ ⎪𝜎zx ⎪ ⎢ 𝜕 2 ⎩ ⎭ ⎢− 𝜕y𝜕z ⎢ ⎢ ⋅ ⎣

𝜕2 𝜕z2 ⋅ 𝜕2 𝜕x2 ⋅ ⋅ −

𝜕2 𝜕x𝜕z

𝜕2 𝜕y2 𝜕2 𝜕x2

⎤ ⎥ ⎥ ⎥ ⎥ ⎥⎧ ⎫ ⎥ Φ ⋅ ⎥ ⎪ 1⎪ ⎨Φ2 ⎬ = 𝔇Maxwell 𝚽 𝜕 2 ⎥⎥ ⎪Φ3 ⎪ − ⎩ ⎭ 𝜕x𝜕y ⎥ ⎥ ⋅ ⎥ ⎥ ⋅ ⎥ ⎦

Formulation of Hybrid Equilibrium Elements

for the Maxwell stress potential; and ⎡ 𝜕2 ⎢ 𝜕y𝜕z ⎢ ⎢ ⋅ ⎧𝜎 ⎫ ⎢ xx ⎪ ⎪ ⎢ ⋅ ⎪𝜎yy ⎪ ⎢ ⎪ 𝜎zz ⎪ ⎢ 2 ⎨ 𝜎 ⎬ = ⎢− 1 𝜕 ⎪ xy ⎪ ⎢ 2 𝜕x𝜕z ⎪ 𝜎yz ⎪ ⎢ 2 ⎪𝜎zx ⎪ ⎢ 1 𝜕 ⎩ ⎭ ⎢ 2 𝜕x2 ⎢ 2 ⎢− 1 𝜕 ⎢ 2 𝜕x𝜕y ⎣

⋅ 𝜕2 𝜕x𝜕z ⋅ 1 𝜕2 2 𝜕y𝜕z 1 𝜕2 − 2 𝜕x𝜕y 1 𝜕2 2 𝜕y2 −

⎤ ⎥ ⎥ ⎥ ⋅ ⎥ 2 𝜕 ⎥ 𝜕x𝜕y ⎥ ⎧Φ1 ⎫ ⎥⎪ ⎪ 1 𝜕 2 ⎥ ⎨Φ2 ⎬ = 𝔇Morera 𝚽 2 𝜕z2 ⎥ ⎪Φ3 ⎪ ⎥⎩ ⎭ 1 𝜕2 ⎥ − 2 𝜕x𝜕z ⎥ ⎥ 1 𝜕2 ⎥ − 2 𝜕y𝜕z ⎥⎦ ⋅

for the Morera stress potential. To obtain a complete basis of degree ds , a basis for the potentials of degree ds + 2 is used, with 3 × (ds + 3)(ds + 4)(ds + 5)∕6 components. The difference between this number and the dimension of the self-equilibrated approximation space of degree ds previously deduced is due to the fact that many of these potentials generate zero stresses as well as linearly dependent components. If for example, Φ1 = z2 and Φ3 = x2 are used as Maxwell potentials, only one independent stress field is obtained, with constant 𝜎yy . A procedure to overcome this problem is proposed in Almeida and Pereira (1996), which consists basically in removing the zero stress modes and determining a linearly independent basis for the coefficients of the non-zero ones. An alternative procedure, which transforms a complete basis for the stresses into one which is self-equilibrated, is presented in Section B.2.3. The approximation of the boundary displacements is similar to the approach previously presented for two dimensional problems, with the displacement vector having three components and considering the indications given in Section 4.2 for the approximation of the displacements of faces. We note that it is particularly important to guarantee that the same reference frame is used for the faces of adjacent elements. A procedure to obtain a particular solution for a given body load is also similar to what is done in two dimensional problems, setting the tangential stresses to zero. When we want to consider elements that have faces with more than three vertices, for example, hexahedral elements, an additional complexity is introduced when these vertices are not coplanar. In this case matrix  is not constant on the face and the enforcement of equilibrium becomes more complex. This is a case of non-linear mapping of the boundary, similar to what is discussed in Section 4.9. 4.8.3 Plate Bending

The distinction between the two plate bending theories that we consider, Reissner– Mindlin and Kirchhoff, is basically of a kinematic nature, which has implications on some of the boundary conditions. But the determination of self-balanced moment distributions can be done the same way for both, by determining moments fields that satisfy the homogeneous form of (A.2), with  ⋆ =  ⋆M , as in Section A.1.4. The dimensions of a general moment field of degree ds and of a general bi-dimensional stress field are the same, 3 × (ds + 1)(ds + 2)∕2. To satisfy the differential equilibrium

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equation, the derivatives, which are of degree (ds − 2), must produce a zero load, which corresponds to imposing (ds − 1)ds ∕2 constraints. The number of parameters of the self-equilibrated moment field is, therefore: nM s = 3 × (ds + 1)(ds + 2)∕2 − (ds − 1)ds ∕2; = ds (ds + 5) + 3.

(4.19)

The Southwell two dimensional vector potential (Southwell, 1950) can be used to generate self-equilibrated moments: ⎧ ⎪ 0 ⎧m ⎫ ⎪ xx ⎪ ⎪ 𝜕 ⎪ ⎨myy ⎬ = ⎨ 𝜕x ⎪mxy ⎪ ⎪ ⎭ ⎪− 1 𝜕 ⎩ ⎪ 2 𝜕y ⎩

⎫ ⎪ ⎪{ } ⎪ Φx = 𝔇Southwell, m 𝚽; ⎬ Φ y 1 𝜕 ⎪ ⎪ − 2 𝜕x ⎪ ⎭ 𝜕 𝜕y 0

and the corresponding shear forces: 1 𝜕2 { } ⎧− ⎪ 2 𝜕y2 qx =⎨ 2 qy ⎪1 𝜕 ⎩ 2 𝜕y𝜕x

1 𝜕2 ⎫ { } 2 𝜕x𝜕y ⎪ Φx = 𝔇Southwell, q 𝚽. ⎬ 1 𝜕 2 ⎪ Φy − 2 𝜕x2 ⎭

When a complete base of degree ds + 1 is used for each Φ, only three linearly dependent moment fields are generated, because a constant Φx or Φy generates zero moments, and Φx = y generates the same moments as Φy = x. The resulting dimension, 2 × (ds + 2)(ds + 3)∕2 − 3, is equal to the number of parameters for the moment field, as previously determined. Note that only transverse body forces or pressures are considered in the particular solutions, excluding applied couples for simplicity. 4.8.3.1 Reissner–Mindlin Theory

A basis for the approximation of the generalized stresses is obtained from the Southwell potentials, as a (5 × nM s ) function matrix. The boundary displacements, which are independently defined on each side of the elements, comprise two components of rotation and a transverse translation. We consider different approximation degrees for the rotations and for the translation, as opposed to the most usual form of isoparametric displacement element for Reissner–Mindlin plates, which uses approximations of the same degree for both fields.

⎧𝑤⎫ ⎡V d𝑤 ⎪ ⎪ ⎢ 1D 𝒗m = ⎨𝜃x ⎬ = ⎢ 𝟎 ⎪ 𝜃y ⎪ ⎢ ⎩ ⎭m ⎣ 𝟎

𝟎 d V 1D𝜃 𝟎

𝑣̂ 1 ⎧ ⎪ ⋮ ⎪ 𝑣̂ d𝑤 +1 ⎪ ⎤ ⎪ 𝑣̂ d𝑤 +2 𝟎 ⎥⎪ ⋮ 𝟎 ⎥⎨ d𝜃 ⎥ ⎪ 𝑣 ̂ V 1D ⎦ d𝑤 +d𝜃 +2 ⎪ ⎪ 𝑣̂ d𝑤 +d𝜃 +3 ⎪ ⋮ ⎪ ⎩ 𝑣̂ d𝑤 +2 d𝜃 +3

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ = V m 𝒗̂ m . ⎪ ⎪ ⎪ ⎪ ⎪ ⎭m

(4.20)

Formulation of Hybrid Equilibrium Elements

These degrees are again selected so that each corresponding boundary equilibrium equation (4.5) is locally enforced. For moments of degree ds the shear forces are functions of degree ds − 1, therefore equilibrium is achieved if the rotations are approximated using functions of degree d𝜃 = ds and polynomials of degree d𝑤 = ds − 1 are used for the translations. We then have that n𝑣𝑤 = d𝑤 + 1 = ds and n𝑣𝜃 = d𝜃 + 1 = ds + 1. This seems unexpected because we tend to imagine plate deformations where the rotations are dominated by the gradients of the transverse displacement. However it must be appreciated that, on the one hand, such deformations are not mandatory for this theory, and on the other hand, these degrees do enable strict equilibrium to be enforced, although the consequent displacements are generally incompatible. When the approximated solutions converge, any displacement approximations that are more elaborate than they need be, tend to disappear. 4.8.3.2 Kirchhoff Theory

The approximation of the generalized stresses is obtained by removing the shear forces from the approximation used for Reissner–Mindlin plates. For the boundary displacements we must recognize that the boundary normal operator,   (A.11), is defined on distinct boundary entities: the sides and the vertices. This requires different approximations on the sides of the mesh and at its vertices. For the sides we use an approximation similar to that used for the Reissner–Mindlin plates, but with only a single component of rotation, normal to the side [ { } d V 1D𝑤 𝑤 𝒗m = = 𝜃n m 𝟎

⎧ 𝑣̂ 1 ⎪ ⋮ ]⎪ 𝟎 ⎪ 𝑣̂ d𝑤 +1 ⎨ d V 1D𝜃 ⎪ 𝑣̂ d𝑤 +2 ⎪ ⋮ ⎪ 𝑣̂ ⎩ d𝑤 +d𝜃 +2

while the approximation for each vertex is, { } [ ]{ } 𝒗m = 𝑤 m = 1 𝑣̂ 1 m = V m 𝒗̂ m .

⎫ ⎪ ⎪ ⎪ ⎬ = V m 𝒗̂ m , ⎪ ⎪ ⎪ ⎭m

(4.21)

(4.22)

The application of (A.11) allows us to derive the generalized boundary forces: normal moments and equivalent shear forces on the sides, and Kirchhoff forces at the vertices. This leads to an equilibrium matrix (4.12) which has some components associated with sides and others associated with vertices, recalling that for one element, at each vertex, two contributions need to be considered, one for each side adjacent to it. Just as for Reissner–Mindlin plates, in order to impose equilibrium locally when functions of degree ds are used to approximate the moments, the approximation of the normal rotations at the sides must be a function of the same degree, d𝜃 = ds , and the approximation of the transverse displacements must be one degree lower than the approximation used for the moments, d𝑤 = ds − 1. Although the compatibility operator is the transpose of the equilibrium one, this property is not so obvious. Its derivation is detailed in Section A.1.4. The hybrid equilibrium Kirchhoff plate elements of degrees ds = 0 or 1 correspond to the elements mentioned in Section 3.4.2.

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A

e

B

c

d

Figure 4.8 A finite element model of the plate problem.

2 b

1 C

a

D

4.8.3.3 Example

A finite element model for the analysis of the plate bending problem that was studied in Section 2.2 is presented in Figure 4.8. Now we once again have two triangular elements. Sides a and b are free, c is internal, while d and e are simply supported. Only transverse displacements may exist at the vertices, A, B and C are fixed and D is free. The dimension of the vector of stress parameters is always the same irrespective of the theory considered. When the Reissner–Mindlin theory is used, the dimension of vector 𝒗̂ depends on whether the simple supports are soft or hard, that is, whether the twisting rotations are ̂ has dimension (5 × 2 × n𝑣𝜃 + 3 × released or constrained respectively. Thus vector 𝒗, n𝑣𝑤 ) or (3 × 2 × n𝑣𝜃 + 2 × 1 × n𝑣𝜃 + 3 × n𝑣𝑤 ), which includes contributions from the five sides, two of them with zero transverse displacement. The resulting system is similar to that obtained for the two dimensional example. For the Kirchhoff theory the transverse displacement at the free vertex must be ̂ with dimension 5 × n𝑣𝜃 + 3 × n𝑣𝑤 + 1, includes contributions from included. Vector 𝒗, n the five sides and one vertex. On sides a, b and c the approximation includes both 𝜃n and 𝑤, but on d and e only 𝜃n is approximated. The governing system takes the following form, where the displacement at the free vertex is considered in the same way as the displacements of the sides: ⎡− 1 ⎢ 𝟎 ⎢ ⎢D ⎢ a1 ⎢ 𝟎 ⎢ Dc1 ⎢ Dd1 ⎢ ⎢ 𝟎 ⎣ DD1

𝟎 − 2 𝟎 Db2 Dc2 𝟎 De2 DD2

DTa1

𝟎

DTc1

DTd1

𝟎

𝟎

DTb2

DTc2

𝟎

DTe2

𝟎

DTD1 ⎤ ⎧ DTD2 ⎥⎥ ⎪ ⎪ ⎥⎪ ⎥⎪ ⎥⎨ ⎥⎪ ⎥⎪ ⎥⎪ ⎥⎪ ⎦⎩

ŝ 1 ⎫ ⎧ ŝ 2 ⎪ ⎪ ⎪ ⎪ 𝒗̂ a ⎪ ⎪ 𝒗̂ b ⎪ = ⎪ ⎬ ⎨ 𝒗̂ c ⎪ ⎪ 𝒗̂ d ⎪ ⎪ 𝒗̂ e ⎪ ⎪ ⎪ ⎪ 𝒗̂ D ⎭ ⎩

ê 01 ⎫ ê 02 ⎪ ⎪ t̂ 0a ⎪ t̂ 0b ⎪ ⎬ t̂ 0c ⎪ t̂ 0d ⎪ t̂ 0e ⎪ ⎪ ̂t 0D ⎭

The conditions for detecting the spurious kinematic modes that may be associated with these models are discussed in Chapter 5. This is only necessary for Reissner–Mindlin plate models, since the Hybrid Equilibrium Kirchoff plate models have no spurious kinematic modes. 4.8.4 Potential Problems of Lower Order

The formulation of complementary finite element formulations for potential problems governed by Poisson’s equation, which deals with scalar and vector variables

Formulation of Hybrid Equilibrium Elements

(Cannarozzi et al., 2000; Fraeijs de Veubeke and Hogge, 1972), is simpler than for solid mechanics problems, which normally operate with vector and tensor variables. For simplicity we use the heat conduction problem as exemplar, where the temperature plays the role of the displacements, the temperature gradient corresponds to the strains, the heat flux is analogous to the stresses, heat sources play the same role as applied forces and the conductivity constant replaces the elasticity coefficients. In conventional formulations the temperature field is approximated. In this problem the equivalent of the domain equilibrium conditions is the balance of the divergence of the heat flux with the heat sources. In particular, for zero domain sources this reduces to requiring a divergence free heat flux field and continuity of its normal components at the boundaries. Working with scalar balance conditions imposed on the heat flux as a vector field, implies that moment equilibrium is no longer relevant. In the context of structural mechanics, the determination of the shear stress in the cross-section of a prismatic beam due to shear and torsion is also a potential problem governed by equations similar to those of heat conduction. Equilibrium elements can, therefore, be used to obtain the stress distribution and the warping of a cross-section in such situations. In Section A.1.3 we present in detail the relevant equations, which are not widely available.

(i) Horizontal shear force.

(ii) Vertical shear force.

(iii) Twisting moment.

Figure 4.9 Illustrations of the shear stresses and warping functions on the cross-section of a prismatic bar caused by the action of different stress resultants. A mesh with 522 elements, with approximations of degree 4 was used to obtain these values.

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Typical results are exemplified in Figure 4.9. They consist of shear stress distributions inside the elements, which constitute a vector field, and warping displacements on the sides of the elements, which are considered as a scalar field. Inter element equilibrium is expressed by requiring the normal tractions on the sides of the elements to be the same on adjacent elements.

4.9 The Case of Geometries With a Non-Linear Mapping We now consider the difficulties created when applying the hybrid formulation to finite element models with more general curvilinear geometry. The approximation of the stresses on the reference frame of the boundary, S(𝝃) = S(x(𝝃)), is a polynomial of degree dsΓ = ds d𝜉 , where d𝜉 is the degree of x(𝝃). When d𝜉 is greater than 1, dsΓ is greater than ds , but the approximation is still a polynomial function. For other non-linear approximations of the boundary, for example, involving trigonometric functions, the approximation of the stresses on the reference frame of the boundary is not a polynomial function. To illustrate such a mapping, we consider a side, passing through nodes A, B and C, defined using quadratic Lagrange interpolation functions. of the nodes √The coordinates √ are A = (0; 0), B equal to one of B1 = (0.5; 0.5), B2 = (1 − 2∕2; 2∕2) or B3 = (0.6; 0.6) and C = (1; 1), as indicated in Figure 4.10. For simplicity the z coordinates are assumed constant. The corresponding mapping functions x(𝝃), where x = (x; y) and 𝝃 = r ∈ [−1; 1], are: AB1 C AB2 C AB3 C

(0.5 r + 0.5; (0.207107 r2 + 0.5 r + 0.292893; (−0.1 r2 + 0.5 r + 0.6;

0.5 r + 0.5); −0.207107 r2 + 0.5 r + 0.707107); −0.1 r2 + 0.5 r + 0.6).

We start by noticing that only the first mapping function is linear, although the geometry of the third case is also a straight line. The degree of dsΓ increases when d𝜉 > 1, irrespective of the geometry of the boundary. The mappings of the term x y, a function of degree 2 of x, which may represent one of the components of S(𝝃) = S(x(𝝃)), become: AB1 C AB2 C AB3 C

(0.5 r + 0.5)2 ; −0.0428932 r + 0.335786 r2 + 0.5 r + 0.207107; (−0.1 r2 + 0.5 r + 0.6)2 , 4

showing how the degree of the polynomial is only preserved for a linear mapping of the geometry. y

Figure 4.10 Possible configurations of the side ABC.

C B2 B3 B1

A

x

Formulation of Hybrid Equilibrium Elements

For non-linear mappings, the matrix of the normals,  e , is generally not constant – the third example just presented is an example of an exception to this rule. For a polynomial x(𝝃) the components of the normal may be polynomial functions, but in most cases they are fractions: a derivative of x(𝝃), divided by |J|, the ratio between lengths or areas defined in x and 𝝃, which is obtained from the Jacobian matrix. In these situations it is possible to scale the equilibrium condition by multiplying it with |J|, so that a polynomial form is recovered. The resulting tractions, when written in the boundary coordinate system, change accordingly. In any case using displacement approximations of the same degree that is used for the stresses is not sufficient to guarantee that equilibrium is strictly enforced by (4.5). When S(𝝃) and  are polynomials, t(𝝃) is also a polynomial. It is then possible to use polynomials of a higher degree for V , to strictly enforce equilibrium on the boundary, but this is generally inadvisable, as it may over-constrain the solution. When either S(𝝃) or  are not polynomial functions it is generally impossible to guarantee that equilibrium on the boundary is strictly enforced. The general advice is to accept the marginal loss of equilibrium that results from the use of weighting functions of lower degree than strictly necessary. When, within each element, the stress approximation is a function of coordinates in a curvilinear reference frame, it is still possible to define polynomial self-equilibrated bases. Their projection on the boundary is generally non-polynomial, but in practice the corresponding lack of codiffusivity is very small (Santos and Almeida, 2014). The use of such formulations has implications on the establishment of a solution basis that are beyond the scope of this book.

4.10 Compatibility Defaults The stresses obtained from the Hybrid Equilibrium model minimize the lack of compatibility of the corresponding strains. This concept is complementary to the lack of equilibrium of a stress field, as observed for example, in compatible models, which assesses how much the equilibrium conditions are violated, in the domain, by not verifying (A.2), and on the boundaries, by not verifying (A.5). The path to assess lack of compatibility is more complex because the corresponding equations, (A.1) and (A.4), are defined in terms of displacements, not strains, and its immediate application would require the integration of the given strains. The most practical approach considers separately: • whether the strain field is integrable, that is, if there is a displacement field that via (A.1) produces it; • whether the strains at the boundaries, internal or external, match those of the adjacent entities, those of the adjacent element or resulting from the prescribed displacements; • whether the displacement at a point is unique when it is obtained by integrating along different paths. These aspects of compatibility are enforced by the equations presented in Appendix A.2.

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4.11 The Dimension of the System of Equations A simple measure to compare the ‘cost’ of performing an analysis is the dimension of the system of equations that must be solved. Though it does not encompass all the details involved, the fact is that for very large systems the computational effort required to compute the matrices grows linearly with the number of elements, while the solution time does not. In order to make a fair comparison between the dimensions of the systems of equations of equilibrated and compatible models, we consider the dimensions of what we call the minimum global systems, nE and nC respectively, with the maximum possible number of variables eliminated at element level.8 For equilibrated models this implies considering only the parameters of the boundary displacement approximations, by locally solving each element system for the stress parameters, using (4.15). For the compatible models this implies solving the stiffness matrix of each element for its ‘bubble modes’. A more detailed study should consider other factors, namely the topological characteristics of the system, since for the equilibrated models each parameter is associated via an edge or a face with, at most two elements,9 while in the compatible models the nodes associated with a vertex, or an edge in 3D, normally involve more than two elements. nE = nedges × (ds + 1) × 2;

nC = (nedges × (du − 1) + nverts ) × 2.

The ratio between the number of edges and vertices in a planar mesh varies between 1 and 3, approaching 3 for large meshes of triangular elements, the ones where the dimension of the system is relevant. Setting nedges = 𝛼e𝑣 nverts we get nE = nverts × (ds + 1) × 𝛼e𝑣 × 2;

nC = nverts × ((du − 1) × 𝛼e𝑣 + 1) × 2.

and, therefore nE = nU

ds + 1 d +1 ≈ s for large meshes. 𝛼 −1 2 du − du − 𝑣 3 𝛼𝑣

This formula implies that for large meshes, when ds = du = d, as the degree is increased the ratio between the dimensions tends to one from above, that is, the dimension of the equilibrated system is always larger than the dimension of the compatible one. Even when we select ds = du − 1, in order to have approximations for the stresses of the same degree, the dimension of the equilibrated system remains larger, as shown in Table 4.1. In any case the dimensions are of the same order of magnitude. For three dimensional approximations of degree ds and du the dimension of the minimal global systems are, again without considering the effect of enforcing the boundary conditions, d2 + 3 ds + 2 nE = nfaces × s × 3; 2 ( ) du2 − 3 du + 2 C n = nfaces × + nedges × (du − 1) + nverts × 3. 2 8 This process is also termed condensation. 9 With the exception of the vertex displacements for Kirchhoff plates, where more than two elements may share each of those variables.

Formulation of Hybrid Equilibrium Elements

Table 4.1 Ratio between the dimension of the equilibrated and the compatible systems for two dimensional continua. du

1

2

3

4

5

ds = du

6.000

2.250

1.714

1.500

1.385

ds = du−1

3.000

1.500

1.286

1.200

1.154

Table 4.2 Ratio between the dimension of the equilibrated and the compatible systems for three dimensional continua. du

1

2

3

4

5

ds = du

36.000

9.000

4.444

3.103

2.495

ds = d u − 1

12.000

4.500

2.667

2.069

1.782

The ratio between the number of faces, edges and vertices in a solid mesh varies in a more complex way than for planar meshes. For large meshes of tetrahedral elements the ratio tends to nedges = 7 nverts and nfaces = 12 nverts (Remacle and Shephard, 2003). Setting nedges = 𝛼e𝑣 nverts and nfaces = 𝛼f 𝑣 nverts we get d2 + 3 ds + 2 nE = nverts × s × 𝛼f 𝑣 × 3; 2 ) ( 2 du − 3 du + 2 C n = nverts × 𝛼f 𝑣 + (du − 1) × 𝛼e𝑣 + 1 × 3. 2 and, therefore, for large meshes 6 ds2 + 18 ds + 12 nE ≈ . nC 6 du2 − 11 du + 6 The corresponding numbers are summarized in Table 4.2. The conclusion is similar to that obtained for two dimensional continua, now with a larger disadvantage for the equilibrated solutions. As shown in Figure 9.3 of Chapter 9, when the solutions are compared in terms of their global error, the disadvantage is not as significant as it seems at first sight.

References Airy GB 1863 On the strains in the interior of beams. Philosophical transactions of the Royal Society of London p. 49–79. Almeida JPM and Freitas JAT 1991 Alternative approach to the formulation of hybrid equilibrium finite elements. Computers & Structures 40(4), 1043–1047. Almeida JPM and Freitas JAT 1992 Continuity conditions for finite element analysis of solids. International Journal for Numerical Methods in Engineering 33(4), 845–853. Almeida JPM and Pereira OJBA 1996 A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids. International Journal for Numerical Methods in Engineering 39(16), 2789–2802.

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Boffi D, Brezzi F and Fortin M 2013 Mixed Finite Element Methods and Applications. Springer. Cannarozzi AA, Momanyi FX and Ubertini F 2000 A hybrid flux model for heat conduction problems. International Journal for Numerical Methods in Engineering 47, 1731–1749. Davis TA 2006 Direct Methods for Sparse Linear Systems. SIAM. Fraeijs de Veubeke BM and Hogge MA 1972 Dual analysis for heat conduction problems by finite elements. International Journal for Numerical Methods in Engineering 5(1), 65–82. Freitas JAT 1989 Duality and symmetry in mixed integral methods of elastostatics. International Journal for Numerical Methods in Engineering 28(5), 1161–1179. Freitas JAT, Almeida JPM and Pereira EMBR 1999 Non-conventional formulations for the finite element method. Computational Mechanics 23, 488–501. Maxwell JC 1866 On reciprocal diagrams in space, and their relation to Airy’s function of stress. Proceedings of the London Mathematical Society 1(1), 58–63. Morera G 1892 Soluzione generale delle equazioni indefinite dell’equilibrio di un corpo continuo. Rendiconti Accademia Nazionale dei Lincei(Ser. 5) 1, 137–141. Pereira OJBA and Almeida JPM 1994 Automatic drawing of stress trajectories in plane systems. Computers & Structures 53(2), 473–476. Pian THH 1964 Derivation of stiffness matrices based on assumed stress distribution. AIAA Journal 2, 1333–1336. Pian THH 1983 Reflections and remarks on hybrid and mixed finite element methods In Hybrid and Mixed Finite Element Methods (ed. Atluri SN, Gallagher RH and Zienkiewicz OC) John Wiley: New York p. 565–570. Pian THH and Tong P 1969 Basis of finite element methods for solid continua. International Journal for Numerical Methods in Engineering 1(1), 3–28. Reddy JN 1986 Applied Functional Analysis and Variational Methods in Engineering. McGraw-Hill New York. Remacle JF and Shephard MS 2003 An algorithm oriented mesh database. International Journal for Numerical Methods in Engineering 58(2), 349–374. Santos HAFA and Almeida JPM 2014 A family of Piola-Kirchhoff hybrid stress finite elements for two-dimensional linear elasticity. Finite Elements in Analysis and Design 85, 33–49. Southwell RV 1950 On the analogues relating flexure and extension of flat plates. The Quarterly Journal of Mechanics and Applied Mathematics 3(3), 257–270. Timoshenko SP and Goodier JN 1951 Theory of Elasticity. McGraw-Hill New York. Washizu K 1982 Variational Methods in Elasticity and Plasticity 3 edn. Pergamon Press.

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5 Analysis of the Kinematic Stability of Hybrid Equilibrium Elements In Chapter 4 we formulated the governing equations for an isolated hybrid element and for meshes of elements, and raised the question of existence and uniqueness of solutions. In this context the significance of spurious kinematic modes was realized. The relevance of the Babuška–Brezzi condition to the characterization of the associated solutions will be discussed in Chapter 7. In this Chapter we first recall the crucial link between spurious modes of displacement and admissible modes of traction as alternative, but complementary, aspects of kinematic stability. We then proceed to use this link to develop the descriptions of the spurious modes, which can be thought of as pseudo-mechanisms, for single triangular and tetrahedral (simplicial) hybrid elements. This is completed by considering the stability of assemblies of elements, either as star patches, or in more general meshes, identifying whether they are stable, or otherwise allow the propagation of spurious kinematic modes. We finally show that simplicial hybrid elements for modelling plates governed by Kirchhoff theory, or for modelling potential problems, are effectively stable and free from spurious modes. The stability of star patches is especially relevant in the contexts of characterizing macro-elements and recovering equilibrium using partition of unity methods (PUM) as discussed in Chapter 8. Knowing that a mesh is globally free from spurious modes is particularly important when selecting the numerical technique used to solve the system of equations, and it also guarantees that the rigid body displacements used in the corotational formulation presented in Chapter 11 are well determined. Guidelines to obtain a mesh that is free from spurious modes are summarized in Section 5.7, at the end of this Chapter. The presentation that follows relies heavily on Maunder and Almeida (2005, 2009) and particularly on Maunder et al. (2016). Our goal here is to provide detail on the basic aspects and on the physical interpretation of the most relevant results, particularly for 2D elements. Fuller details of the derivations are to be found in the aforementioned publications.

5.1 Algebraic and Duality Concepts Related to Spurious Kinematic Modes The primary variables that connect a hybrid equilibrium element to its neighbouring elements or an external boundary, are the boundary displacements. Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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These connections prompt two related questions: i) Apart from rigid body displacements, can all boundary displacements formed by the approximation functions in V be resisted by the elements, that is, do they all lead to non-zero work being done on the boundary and, in consequence, to non-zero strain energy being stored within the elements? If the answer is no, then those displacements that do not conform with rigid body movements and are not resisted by the elements are termed spurious kinematic modes, and they can be regarded in the same way as mechanisms at the structural level. A consequence of the existence of such a mechanism is of course that certain boundary tractions cannot be supported, that is, they are inadmissible. This leads to the second question: ii) Apart from global equilibrium, are there any other conditions to be satisfied for the prescribed tractions to be admissible? Clearly tractions should not excite a spurious mode, that is, admissible tractions should perform zero work with all spurious modes of displacement. Furthermore, for strict equilibrium in a local sense, tractions should not be of a higher degree than that of the approximation functions in S, and then for tractions t to be admissible they are such that there is a ŝ for which T t =  S ŝ . The answer to these questions, and the determination of any spurious kinematic modes and associated inadmissible tractions, is to be found within the characteristic matrix D for an assembled mesh of elements. This matrix contains the integrated work done by the assumed boundary displacements with the boundary tractions that equilibrate with the assumed internal stress fields. Thus, recalling Section 4.7, for a spurious kinematic or rigid body mode denoted by 𝒗̂ 0 : DT 𝒗̂ 0 = 𝟎 ⇒ ŝ =  −1 DT 𝒗̂ 0 = 𝟎,

and D −1 DT is singular

(5.1)

Denoting the dimensions of D by n𝑣 × ns , and its rank by nadm , we consider the nullspace of DT to be generated by the columns of a matrix A (Strang, 1988). These columns represent independent boundary displacements that do zero work with admissible tractions, that is, they are orthogonal to the tractions. They include the nrbm rigid body modes, as well as the nskm spurious kinematic modes when they exist. For structural applications nrbm equals 3 or 6 for models defined in a 2 or 3 dimensional space respectively, whereas it is always 1 for potential problems. Then matrix A has dimensions n𝑣 × nann where: nann = n𝑣 − nadm = nrbm + nskm Therefore the number and description of spurious kinematic modes for a mesh is entirely dependent on the rank and nullspace of DT . To obtain a convenient way to present and explain in general the relations between side displacements and tractions, we now introduce some further algebraic concepts concerning vector spaces (Shephard, 1966). The space h of side displacements introduced in Section 4.2 has as its dual the space h of side tractions with the work product as the associated linear functional. These spaces have dimension n𝑣 and are defined by complete polynomial distributions of degree d𝑣 . The admissible tractions form a subspace adm of h , and the nullspace of DT forms the subspace ann which is termed the annihilator of the subspace adm .

Analysis of the Kinematic Stability

We can also regard ann as the subspace which produces zero equivalent strains. Then the elements of the quotient space 𝜀 = h ∕ann are the cosets of displacements that produce non-zero equivalent strains. At this point we identify two further subspaces g ⊆ h and rbm ⊆ h relevant to a model, or a single element, without kinematic boundary conditions. g consists of all tractions satisfying the overall, or global, equilibrium conditions, that is, three conditions for plate models in 2D and six conditions for solid models. rbm is defined as the annihilator of g , and generally consists of the rigid body displacements of the sides/faces of a mesh, thus dim rbm = nrbm = 3 or 6, and clearly rbm ⊆ ann since admissible tractions must satisfy the global equilibrium conditions. However, when d𝑣 = 0, the rigid body rotational mode of displacement of a side is absent, and rbm needs to include discontinuous side displacements in order to enforce rotational equilibrium. Such modes of displacement conform with a rigid body rotation as far as the midpoints of the sides/faces are concerned, but the sides only undergo translations, as discussed in Section 3.4.1. However this ‘incomplete’ rotational mode is sufficient to enforce rotational equilibrium of the constant modes of traction. Although such modes of displacement are strictly speaking spurious kinematic ones, they are better classified as pseudo-rigid body modes. Finally we define the quotient space skm = ann ∕rbm whose elements are cosets of ann which consist of all displacements (𝑣skm + rbm ), where 𝑣skm is a spurious distribution of displacements, and independence of cosets implies that the corresponding spurious modes do not combine to form an element of rbm . We will represent an element of skm by a displacement vector 𝑣skm , but note that the side displacements are only uniquely defined to within an element of rbm . A further characteristic of spurious modes is their discontinuous nature at vertices of 2D elements or at edges of 3D elements. We also note that the left nullspace of DT is generated by the columns of a matrix T B , that is, DB = 𝟎, with dimensions nhyp × ns where nhyp = ns − nadm and nhyp is the number of self-stressing hyperstatic modes of internal stress, that is, stress fields that do not require any equilibrating tractions on the boundary. Hence ns − nhyp = nadm = n𝑣 − nann = n𝑣 − nrbm − nskm or nskm = (n𝑣 − nrbm ) − (ns − nhyp )

(5.2)

Expressions for nhyp have been established for simplicial continuum and plate bending elements (Fraeijs de Veubeke, 1973; Kempeneers et al., 2010; Maunder and Almeida, 2005) as functions of degree d. These expressions imply that the necessary and sufficient conditions for tractions to be admissible are that they satisfy the local rotational equilibrium conditions at vertices of triangular elements and along edges of tetrahedral elements, together with any independent global equilibrium conditions. Similar conditions apply to Reissner–Mindlin plate bending elements, but they are not sufficient for moment fields of degree 2. In that case an extra condition is required (Maunder and Almeida, 2005). These equilibrium conditions, including those of global equilibrium, lead directly to the annihilator AT and thereby to a basis for spurious modes. The definition of this matrix depends on the choice of the basis functions for the dual spaces of displacement and traction on a side or face of an element.

75

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Equilibrium F.E. Formulations

Clearly orthogonal bases would be convenient, and these are available, for example, in the form of Legendre polynomials for a side of a plate element, and their extension to a triangular face of a tetrahedral element in terms of Jacobi polynomials (Maunder et al., 2016; NIST, 2013). The topological and geometrical configurations that allow or block the propagation of spurious modes between elements can be determined by considering the compatibility of displacements due to the spurious modes of separate elements, or the transmission of admissible tractions. The former route will be followed in this Chapter as it is considered to be simpler and more direct, however details of the latter route for the 2D case can be found in Maunder and Almeida (2009).

5.2 Spurious Kinematic Modes in Models of 2D Continua We introduce this Section by considering the corner of an element where a horizontal and a vertical side meet, which can be assumed, for example, to correspond to the lower left corner of the problem in Chapter 4 (Figures 4.2 and 4.3). For this situation the interpretation of the two complementary facets of the spurious mode is particularly simple. • The tangential displacements on these sides only do work with the projection of the tangential stresses. If one of the sides moves in the direction of the tangential traction and the other moves in the opposite direction, these displacements produce zero work upon every self-balanced stress field, implying that the associated equivalent strains, defined in (4.8) and used in (4.9), are zero. From the viewpoint of the model the spurious mode displacements are similar to a rigid body movement, that is, in the sense of (4.8), they induce zero strains.

• The tangential tractions on these sides can only be in equilibrium if they have the same value and orientation at the corner, that is, when the tractions on each side are both directed towards or away from the corner. Otherwise the spurious mode displacements will do non-zero work with these tractions.1 The model, which assumes a continuous stress at the lower left corner, cannot equilibrate a distribution of tangential tractions that is discontinuous thereon.

The two aspects are naturally linked and it does not make sense to argue which one is ‘the first born’. What is important, when studying these models, is to understand the complementarity of these two aspects of a spurious mode. We can begin to generalize these aspects of spurious modes by considering a similar problem, which is oblique at the corner, instead of having a right angle. The generic triangular element used is illustrated in Figure 5.1, with corner 3 detailed in Figure 5.2. 1 Side tractions t are defined as forces per unit length per unit thickness of plate. Since only elements of constant thickness are considered, a unit thickness will generally be assumed.

Analysis of the Kinematic Stability

Figure 5.1 Notation used for a generic triangular element and definition of the side coordinates. The orthogonal reference frame used for side 3 is also illustrated. Note that this frame is different from the oblique frame used at each vertex.

–1 2

0 ξ1

1

n

s side 1

side 3 side 2

3 –1

0

1 ξ2

1

1

s1 1 sin ϕ

ts1

1 ts2

ϕ 1 sin ϕ

vs11

ξ1

s1 s2

side 1

ξ2

3 2

ts1

s2

side 2 2 ts2

3 vs22

Figure 5.2 Covariant (s1 , s2 ) and contravariant (̄s1 , s̄ 2 ) base vectors on the sides adjacent to corner 3. The illustrated tractions include all components, whereas the displacements are only those of a spurious mode for which 𝑣1s̄ = 0 = 𝑣2s̄ . 2

1

In this case it is easier to explain the consequences by using the oblique axes s1 and s2 parallel to the sides as the directions of covariant base vectors for the tractions.2 Now for the tractions to equilibrate in terms of moments with the internal stress field at a corner, and therefore be admissible, we must have ‘continuity’ in the form of the contravariant components of traction vectors, requiring that ts11 = ts22 at the corner. When we enforce this admissibility condition, the work product at the corner is written as ts12 𝑣1s̄2 + ts11 𝑣1s̄1 + ts11 𝑣2s̄2 + ts21 𝑣2s̄1 , which is zero when 𝑣1s̄2 = 0 = 𝑣2s̄1 ,

and 𝑣1s̄1 + 𝑣2s̄2 = 0.

This implies that the movements of the sides do zero work with all admissible tractions when their displacements are directed parallel to the contravariant base vectors s̄ 1 and s̄ 2 , implying that points on side 1 move along s̄ 1 , normal to side 2, and points on side 2 move along s̄ 2 , normal to side 1, as indicated in Figure 5.2. 5.2.1 Single Triangular Elements

In this Section we develop the functions that describe the distributions of displacements corresponding to the spurious kinematic modes of single triangular elements for 2D continua. 2 Using these axes, the traction vectors are the sum of the product of the contravariant components times the covariant base vectors, t = ts1 s1 + ts2 s2 . The boundary displacements are expressed in the complementary form, 𝒗 = 𝑣s̄1 s̄ 1 + 𝑣s̄2 s̄ 2 . By realizing that [s1 s2 ]T [̄s1 s̄ 2 ] is the identity matrix, we see that the work product t ⋅ 𝒗 at the corner is equal to ts1 𝑣s̄1 + ts2 𝑣s̄2 .

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Equilibrium F.E. Formulations

For a single element e we could denote the space of side tractions by he , but where the restriction to an element is clear, we will omit the superscript e. As with side displacements in (4.17), we can express boundary tractions on side m as a vector with two contravariant components: ] [ { } d ts 1 𝟎 ̂ h1D𝑣 tm. = tm = d ts 2 m 𝟎 h1D𝑣 d

If we select a basis h1D𝑣 that is dual to the basis used to approximate the covariant d side displacements, V 1D𝑣 , then the work product between tractions and displacements is expressed by the scalar product of the vectors of traction and displacement parameters. Considering all the sides of an element leads to the total work being: 3 ∑

T t̂ e 𝒗̂ e =

t̂ Tm 𝒗̂ m

m=1

Now, in order to construct a spurious kinematic mode, we need to find the displacement parameters in 𝒗̂ e for which the work product is zero, for all all admissible tractions t̂ e . The admissibility conditions for tractions only involve the rotational equilibrium equations at the three corners, which for corner 3, as previously noted, can be written as: ts11 |𝜉1 =1 = ts22 |𝜉2 =−1 ; ⏟⏟⏟ ⏟⏟⏟ From side 1

From side 2

where 𝜉i denotes the non-dimensional coordinate for a side, as indicated in Figure 5.1. In terms of the traction parameters, this equation becomes: −h1 (1) t̂s11 1 − h2 (1) t̂s11 2 − … + h1 (−1) t̂

2 s2 1

+ h2 (−1) t̂

2 s2 2

+ … = 0,

d

where hi is the ith component of h1D𝑣 . Then the vector 𝒗̂ e which verifies the zero work condition for all tractions satisfying this homogeneous equation must have the form: } { T } { { } 𝟎 𝟎 −h (1) , 𝒗̂ 3 = . , 𝒗̂ 2 = 𝒗̂ 1 = 𝟎 hT (−1) 𝟎 In order to obtain explicit expressions for these displacements in a simple way, it is convenient to use the dual bases for a side of length l, in terms of the Legendre polynomials Pn (𝜉i ), with n = 0 to d𝑣 , for example, hn+1 = Pn (𝜉i ), Since h(1) =



1

1

d

V 1D𝑣

n+1

1

= h⋆n+1 = Pn⋆ (𝜉i ), with Pn⋆ (𝜉i ) =

1





and h(−1) =



1

2 n+1 Pn (𝜉i ). l

−1

1

−1

⌋ … ,

we can express the non-zero covariant components of side displacements as: 𝑣1s̄1 = −

n=d ∑ n=0

Pn⋆ (𝜉1 ) ≡ −

k+ ; l1

and 𝑣2s̄2 =

n=d ∑ n=0

(−1)n Pn⋆ (𝜉2 ) ≡

k− . l2

(5.3)

Note that these explicit expressions for a spurious kinematic mode are independent of the basis chosen for h , and two other similar spurious modes are defined associated with the other two corners of the element.

Analysis of the Kinematic Stability

Since the contravariant base vectors s̄ 1 and s̄ 2 , after scaling by − sin 𝜑3 , coincide with the unit outward normal vectors n2 and n1 to sides 2 and 1 respectively, we conclude that:

(5.4) where k + and k − , defined in (5.3), are termed the 2D signature functions of sides 1 and 2, incident with corner 3, and  denotes the area of the element. The displacements described in (5.4) are taken to define the unit spurious mode associated with the corner. The unit spurious modes of one corner are illustrated in Figure 5.3 when d𝑣 = 0 to 4. It should be noted that the points on the sides are uniformly spaced in the original state, and they indicate relative stretching in the displaced state. Two other modes associated with the other corners are similarly defined for each degree by using cyclic symmetry. The dimensions of the relevant vector spaces of an element are dependent on its degree d = ds = d𝑣 , and these are recalled here from Section 4.8.1: dim h = n𝑣 = 6(d + 1);

dim h = ns =

(d + 1)(d + 6) . 2

Then, since nrbm = 3, and nhyp = 0 when d ≤ 3, nhyp =

(d − 2)(d − 3) , when d > 3, 2

we have nann = nrbm + nskm = 3, 5 or 6 when d = 0, 1 or ≥ 2 respectively.

(5.5)

Equation (5.5) tells us that there are no more spurious modes other than those associated with the three corners, but for degrees less than two, dependencies must exist. When d = 0 we can superimpose the unit spurious mode of corner 3 with the spurious mode of corner 2 with amplitude −1, and then all three sides translate with displacement 𝒗2 as shown in Figure 5.4a. Hence any two of the spurious modes can be combined to produce side displacements that conform with a rigid body translation of the element, and consequently only one of the spurious modes is independent of the rigid body modes. This one may be taken as the unit mode associated with corner 1 say, and then it is effectively a pseudo-rigid body rotation about the midpoint of side 1. When d = 1, the three unit spurious modes can be combined to produce side displacements that conform with an anticlockwise rotation 𝜔 = 3∕ of the element about its centroid as shown in Figure 5.4b. This property implies that the number of spurious modes independent of the rigid body modes is only two, as given by (5.5).

Figure 5.3 Illustration of the spurious kinematic modes associated with the top corner for d𝑣 = 0 to 4.

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80

Equilibrium F.E. Formulations 1

υ

3

3 υ1 ⊥ side 2 υ2 ⊥ side 1

υ1 ⊥ side 3 1

ω υ2 F 1

2

υ3 ⊥ side 1 (a)

3 A

υ1 E C G υ3

2

υ

υ3

3 CE A 3 CF A 3 CG A

2

(b)

Figure 5.4 Illustrations of dependent spurious modes when d = 0 or 1.

5.2.2 A Pair of Triangular Elements With a Common Interface

The spurious modes belonging to an individual element can propagate to neighbouring elements when these modes are compatible at their interfaces. This possibility is now discussed for a pair of elements labelled A and B as illustrated in Figure 5.5. Displacements of one side, for example side 1, associated with arbitrary spurious modes at each incident vertex is given by (5.6): ( ) ( ) k+ k− 𝒗1 = p3 (5.6) n2 − p2 n3 l1 sin 𝜑3 l1 sin 𝜑2 where p2 and p3 denote the amplitudes of the spurious modes associated with vertices 2 and 3 respectively. Then the distributions of relative side displacements, in the tangential or normal directions, are expressed by (d + 1) dimensional vectors. Using the basis functions defined in 5.2.1, the components of tangential displacement of side 1 of element A are given by: { 𝒗̂ A1 s

= 𝔼d

pA3 pA2

}

⎡1 ⎢1 where 𝔼d = ⎢ ⋮ ⎢ ⎣1

1 ⎤ −1 ⎥ ⋮ ⎥ ⎥ (−1)d ⎦

and the components of normal displacement are given by: { A} p3 A ̂𝒗A1 n = ℂd 𝔻c pA2

A 1

ϕ3B Θ Δ

T

ϕ2A

(5.8)

Figure 5.5 A pair of triangular elements, with a rigid body displacement of element B.

V ϕ3A

(5.7)

B ϕ1B S

2

Analysis of the Kinematic Stability

where

[ −1 ℂd = 𝔼d 0

0 1

] and

𝔻Ac

[ A 𝜑̄ = 3 0

] 0 , 𝜑̄ A2

where 𝜑̄ ej ≡ cot 𝜑ej at corner j of element e. At the interface between elements A and B, illustrated in Figure 5.5, compatibility of displacements requires 𝒗̂ A1 = 𝒗̂ B2 , when the internal numbering of element vertices and sides is anticlockwise with vertex i opposite to side i for each element. However, when defining the relative displacements at an interface of length l between a pair of elements, we need to match the rigid body modes of side displacements. This can be done by allowing element B to move as a rigid, or pseudo-rigid, body as shown in Figure 5.5. This leads to the compatibility equation (5.9), using (5.7) and (5.8). [

𝔼d −ℂd 𝔻Bc

𝔼d ℂd 𝔻Ac

where

{ pA =

pA3 pA2

]{

pA pB

}

⎧Δ ⎫ ⎪ s⎪ = l ℝ ⎨Δn ⎬ ⎪Θ⎪ ⎩ ⎭

} ,

and pB =

B

(5.9)

{ B} p3 , pB1

and the 2(d + 1) × 3 matrix:

ℝB =

[ 1

0

] 0

0

1

0

⎡1 ⎢0 or ⎢ ⎢0 ⎢ ⎣0

0 0 1 0

⎡1 0 ⎢ 0 ⎤ ⎢ 𝟎d × 3 0 ⎥ ⎥ or ⎢⎢ 0 1 0 ⎥ ⎢0 0 ⎥ ⎢ l∕6⎦ ⎢ 𝟎(d−1) × 3 ⎣

0 ⎤ ⎥ ⎥ ⎥ 0 ⎥, l∕6⎥ ⎥ ⎥ ⎦

when degree d = 0, 1, or > 1. The solutions to (5.9), which depend on the shapes of the elements and on their degree, determine the forms of the spurious kinematic modes that can be transmitted. We will refer to such modes as being malignant. The following cases where propagation of spurious modes can occur are considered, as the degree d is increased from zero: When d = 0, the ‘spurious modes’ defined by pA are reclassified as pseudo-rigid body rotations, which are free to propagate.

When d = 1, (pB1 − pB3 ) = (pA3 − pA2 ). Physically this condition implies that the two sides of the interface have equal extensions. Thus any spurious mode from element A can propagate to element B, and hence there are two independent solutions for pB , for example when pB1 = 0 or pB3 = 0.

When d = 2, pB = −pA , and (𝜑̄ A3 + 𝜑̄ B3 ) pA3 = (𝜑̄ A2 + 𝜑̄ B1 ) pA2 .

81

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Equilibrium F.E. Formulations

In this case a simple geometric interpretation of the compatibility condition exists, since, with reference to Figure 5.5 we obtain the relations: (A + B )TV ⋅ VS (A + B )TS ⋅ VS ; (𝜑̄ A2 + 𝜑̄ B1 ) = . A B 2( ⋅  ) 2(A ⋅ B ) These relations imply that: (𝜑̄ A3 + 𝜑̄ B3 ) =

(5.10)

TV ⋅ pA3 = TS ⋅ pA2 .

(5.11)

Hence, in order for the spurious modes of element A to propagate to element B, the modal amplitudes associated with corners 2 and 3 must be such that they satisfy (5.11). Both amplitudes are non-zero, except when either TV or TS is zero, a situation that arises when two sides of the adjacent elements are colinear. This occurs when (𝜑A2 + 𝜑B1 ) = 𝜋, or (𝜑A3 + 𝜑B3 ) = 𝜋. We term such configurations degenerate.

In such cases the spurious modes need to satisfy (5.12): pA3 is arbitrary,

pA2 = 0,

pA3

pA2

= 0,

is arbitrary,

when TV = 0; when TS = 0.

(5.12)

When d ≥ 3, pB = −pA and ℂd [𝔻Ac + 𝔻Bc ] pA = 𝟎.

Thus spurious kinematic modes cannot propagate unless rank [𝔻Ac + 𝔻Bc ] < 2, which can only occur in the degenerate cases. As when d = 2, the malignant spurious mode is defined by (5.12). Typical modes are illustrated in Figure 5.6. 5.2.3 Star Patches of 2D Elements

We now consider assemblies of simplicial elements in the form of star patches, which for 2D meshes imply that the elements share a common ‘star’ vertex V . The stability, or freedom from spurious kinematic modes, of such an assembly depends on the topological and geometric configurations as well as the degree of the elements. It is useful here to invoke terminology and concepts associated with simplicial complexes, the mathematical entities that embody the topological and geometrical properties of finite element meshes (Munkres, 2000). The properties of a single 2D triangular element are described in terms of simplexes. The 2D space occupied by an element comprises the union of its corner points, as 0-simplexes or vertices, its sides, as 1-simplexes or edges that include their end vertices, and a 2-simplex which includes the edges and vertices as well as the interior points. In this hierarchical structure, a simplex includes all its simplexes of lower dimension. A star patch of elements then defines a simplicial subcomplex as the simplicial neighbourhood N(V ) of vertex V , and the link Lk(V ), which is the subset of simplexes of N(V ) that excludes V . In this case Lk(V ) is a 1-dimensional complex. Star patches of elements are described as open or closed depending on whether the corresponding link Lk(V ) forms an open path (chain), or a closed circuit. Such patches are illustrated in Figure 5.7.

Analysis of the Kinematic Stability

V 3

3

2

B A 1

1

2

V

3

3

2

B A 1

1

2 3

V 3 2

B 1

A 2

1

Figure 5.6 Illustration of the spurious kinematic modes of degree 0, 1 and 2 that can be transmitted between a pair of elements. Non-degenerate and degenerate case, the latter is illustrated when either the rigid body movement of the common interface or the external sides are blocked.

V

V

Figure 5.7 Simplicial neighbourhoods of vertices, and their links.

In particular we will derive the kinematic properties of open and closed star patches as special cases of chains and circuits of elements. In the absence of degenerate geometrical arrangements, it transpires that meshes are stable when d > 2, are unstable when d < 2, and are stable unless the geometric configurations satisfy certain conditions when d = 2. For each degree, ranging from 0 to 3 and higher, we begin by considering open stars of elements whose kinematic properties are based on an extension of those of a pair of elements, followed by the analysis of closed stars.

83

84

Equilibrium F.E. Formulations

5.2.3.1 Open Stars of Degree 0

The smallest open star is a single element, and as already noted in Section 5.2.1, this has only one independent spurious mode which can be described by translations of the sides to conform with a pseudo-rigid body rotation of the rigid lamina shown in Figure 5.8a. The lamina or ‘skeleton triangle’, as first proposed by Fraeijs de Veubeke (1964), connects the midpoints of the sides, and the centre of rotation can be at an arbitrary location. This representation of a spurious mode is extended to an assembly of elements by pin-jointing the corresponding laminae to form skeletal models as illustrated in Figure 5.8b and 5.8c. The pseudo-mechanisms within a skeletal model define the spurious modes of the finite element model. The single spurious mode of each element can occur within an open patch, so such a patch contains n independent spurious modes where n is the number of elements. 5.2.3.2 Closed Stars of Degree 0

The skeletal model of a closed star patch behaves, from the kinematic point of view, as a closed linkage that is unstable when it has more than three elements. The example in Figure 5.8c is clearly unstable as a 7-bar linkage. Only the case of three elements forms a stable macro-element of degree 0. 5.2.3.3 Open Stars of Degree 1

The compatibility condition for a pair of elements derived in Section 5.2.2 requires equal extensions of each side of an interface, for example, the extension of side 1 of element A is (pA3 − pA2 )∕l. Since each element has two independent spurious modes, each time an element is added when building a chain of elements, one of the new spurious mode parameters is used to enforce compatibility, and one parameter remains free. A chain of n elements thus has one kinematic degree of freedom per interface plus one at each end of a chain. Figure 5.9a illustrates an open star with a set of relative values of spurious mode amplitudes that define a local mode at the interface between elements A and B. In this case the spurious mode only involves deformations of the interface and of the two adjacent sides in Lk(V ). Figure 5.9b illustrates the particular case of a chain of three elements whose interfaces form the diagonals of a quadrilateral with one side missing. The amplitudes shown imply that the sides of Lk(V ) are undeformed and can remain in their original positions. Closure of this patch with the missing element is an important case when characterizing the behaviour of closed stars.

V

(a)

V

(b)

Figure 5.8 Star patches of elements of degree 0.

(c)

Analysis of the Kinematic Stability

1 0

B

–1

A

0

0 0 1

00 V

0

–1

1

0 0

V 0 (b)

(a)

Figure 5.9 Open star patches of elements of degree 1.

5.2.3.4 Closed Stars of Degree 1

The nature of the mode for a pair of elements A and B in Figure 5.9a involves deformations of the interface and the two sides belonging to Lk(V ), while the other two sides move as rigid bodies. This implies that a closed star of n elements would allow one mode per interface if three closure constraints of the rigid body displacements were not imposed. Hence we normally have (n − 3) spurious modes for a closed star. In practice, in order to avoid spurious modes with this degree, we can resort to macro-elements in the form of triangular closed stars with n = 3, or quadrilateral closed stars with n = 4 and subdivision by diagonals. In the latter case, one spurious mode still exists, but it is restricted to the internal sides of the macro-element, as can be observed from the second example in Figure 5.9b, by realizing that when a fourth element is added to close the star, the spurious mode associated with vertex V will have amplitude −1, whereas the modes associated with the vertices on the link will have amplitude 0. Therefore this mode is not excited by tractions applied to the external sides of the macro-element. 5.2.3.5 Open Stars of Degree 2

When the degree d = 2, a chain of elements normally has three spurious modes: one mode continues to propagate from one element to the adjacent element, plus one mode per end vertex. The amplitudes of the corner modes involved in the propagation depend on the local geometry, that is, the intersections of successive diagonals. This is illustrated in Figure 5.10a where the amplitudes of the corner modes at the star vertex V alternate between +1 and −1, and the amplitudes of the modes at the vertices on Lk(V ), for example, at vertex S, must satisfy pA2 = −TV ∕TS = −pB1 . In a degenerate case where the intersection point T is positioned on Lk(V ), as in Figure 5.10b, the spurious mode cannot propagate throughout the patch, and it becomes confined to the pair of elements A and B with relative amplitudes as shown. 5.2.3.6 Closed Stars of Degree 2

When degree d = 2, the spurious mode that propagates around a closed star must of necessity satisfy the relative values established in Section 5.2.2 for successive pairs of adjacent elements to ensure that the deformational components of these modes are compatible. Additional conditions are necessary to ensure a proper matching of the modes at the closure of the patch: • compatibility of the deformational components; • compatibility of the rigid body modes of the sides.

85

86

Equilibrium F.E. Formulations

A

S

B T

p2A pB 1

B

T

1 –1

0 –1

–1 1

A

0

0 0

1

V

V

(a)

(b)

Figure 5.10 Open star patches of elements of degree 2.

Since the amplitudes pi3 need to have the same magnitude but alternate signs, the first requirement alone tells us that a closed star patch with an odd number of elements is stable and free from spurious modes, apart from degenerate cases. However, having an even number of elements is not a sufficient condition for allowing propagation, because the rigid body modes have to match. The compatibility conditions for these rigid body modes can be expressed (Maunder et al., 2016) by: n ∑

(−1)i Wi = 0;

and

i=1

n ∑

(−1)i Wi (li ni ) = 𝟎.

(5.13)

i=1

where, with reference to (5.10), Wi =

𝜑̄ A3 + 𝜑̄ B3 li2

=

A + B TV 2 A B VS

(5.14)

for a typical interface between elements A and B, where li = VS, and ni is the unit vector normal to the interface with a direction agreeing with an anticlockwise rotation about V . 5.2.3.7 Examples of Unstable Closed Star Patches of Degree 2

These extra compatibility conditions are satisfied, and hence malignant spurious modes can be excited, when some form of symmetry exists in the geometrical arrangements, as illustrated in Figures 5.11a to 5.11d. The vertices in the links of polygons a to d lie on a circle with centre at the star vertex V . Polygons a and b are regular, and polygons c and d have cyclic symmetry in the sense that in c 𝜑13 = 𝜑53 , 𝜑33 = 𝜑73 , 𝜑23 = 𝜑43 = 𝜑63 = 𝜑83 , and the relative values of Wi equal either W1 or W2 are clearly balanced; and in d 𝜑13 = 𝜑33 = 𝜑53 , 𝜑23 = 𝜑43 = 𝜑63 , and all values of Wi are equal. Thus (5.13) are satisfied for both cases c and d. Polygon e is a quadrilateral with internal diagonals so that Wi = 0 for each vertex in Lk(V ), and the spurious mode does not include displacements of the sides in the link unlike patches a to d. Polygon f is a degenerate hexagon in the form of a quadrilateral with four internal sides defined by the diagonals, and two degenerate vertices 1 and 2 in Lk(V ). In this case, all the conditions are satisfied for a spurious mode to exist. However, in order for a spurious mode to have compatible deformations of the sides, it is necessary that all the corner mode amplitudes be zero with the exception of the corners at vertices 1 and 2. Hence a spurious mode cannot propagate around the patch. The degenerate vertices give rise to two independent local spurious modes that each involve displacements of the three sides joined to these vertices.

Analysis of the Kinematic Stability

–W2

W2

V

ϕ83

ϕ43 ϕ13 ϕ2 ϕ3 3 3

–W1 W1 (a)

ϕ63 ϕ5 V 3 ϕ43 ϕ13 3 ϕ2 ϕ3 3

–W1

6 ϕ73 ϕ3 ϕ53

V

V

W1

(b)

(c)

W2

–W2

2 V

(d)

(e)

1

V

(f)

Figure 5.11 Examples of unstable closed star patches of degree 2.

5.2.3.8 Stars of Degree 3 or Higher

In this case no spurious mode can propagate from one element to the adjacent element in a chain of elements. Thus normally there are only the two spurious modes associated with the end vertices of an open star. The exception again occurs with degenerate vertices where local spurious modes occur. No spurious modes exist except for the degenerate cases, as illustrated in Figures 5.11e and 5.11f. 5.2.4 Observations for General 2D Meshes

We can classify spurious modes according to their origin, that is, they may be ‘topologically inherent’ and essentially dependent on the connections between elements as well as the degree – these modes are inherent to a system of elements and their existence is independent of changes in geometry; otherwise they can be termed ‘geometrically induced’ when their existence depends not only on the topology and degree, but also on the particular alignments of the sides of elements. The simplest example of a topological mode occurs where a vertex belongs to a single element and the adjacent sides are unrestrained. This is the only topological mode for elements of degree greater than 1. A stable mesh for degree d < 2 generally requires the use of macro-elements assembled from three simplicial elements. For degree d > 2 all meshes are stable with the localized exceptions as described for a closed star patch in Section 5.2.3. A mesh of elements of degree d = 2 is susceptible to spurious modes induced by its geometry. Provided a mesh contains one stable star patch, the complete mesh is stable with the exception of the local pathological cases of the quadrilateral star patch with internal diagonals, as illustrated in Figure 5.11e, and degenerate vertices on the boundary.

87

88

Equilibrium F.E. Formulations

Unstable star patches, such as those illustrated in Figures 5.10 and 5.11, may be stabilized by: • relocating one or more vertices to destroy the symmetry; • subdividing one element into a macro-element; or • constraining one side to prevent change of its shape, for example, by external buttressing with an element for either an open or a closed star patch. The open star patch considered in Figure 5.10a may have its potential spurious modes blocked by the addition of another element of degree 2, for example, as shown shaded in Figure 5.12a by the element joining the vertices Q, R and S. The block is effective if and only if the lines joining VR, SP and QU are not concurrent. This geometric condition can be explained by the use of Ceva’s theorem for a 2-simplex (Coxeter and Greitzer, 1967). By considering the compatibility conditions required for the transmission of a spurious mode between the element joining vertices V , S and Q and the three adjacent elements, we would require, according to (5.11), the following relations between the corner mode amplitudes a, b, and c shown in Figure 5.12a: TV × a = TS × b;

and NS × b = NQ × c;

⎡ TV =⇒ ⎢ 0 ⎢ ⎣−MV

−TS NS 0

0 ⎤ −NQ⎥ ⎥ MQ ⎦

and MQ × c = MV × a

⎧a⎫ ⎧0⎫ ⎪ ⎪ ⎪ ⎪ ⎨b⎬ = ⎨0⎬ ⎪ c ⎪ ⎪0⎪ ⎩ ⎭ ⎩ ⎭

A solution is only possible when the determinant of the 3 × 3 matrix is zero, and this requires: TV × NS × MQ − TS × NQ × MV = 0. According to Ceva’s theorem for a 2-simplex, this requirement is only met when the lines VN, SM, and QT are concurrent, as illustrated in Figure 5.12b. Thus blocking of the transmission of a spurious mode in the configuration of elements in Figure 5.12a is effective provided these lines are not concurrent. Examples of meshes of degree 2 that do not contain a stable star patch are those formed by tessellations based on a triangular element of constant shape arranged into R N –c

–b

b –b

M

U

S N

T

a –a –a

c

Q –c

R

U

S

Q

V

T M

V

P

P

(a)

(b)

Figure 5.12 An external element may buttress a star patch.

Analysis of the Kinematic Stability

Figure 5.13 A hexagonal star patch extracted from a tessellation.

ϕ3 ϕ1

e

l1 ϕ2

overlapping hexagonal star patches, like the one illustrated in Figure 5.13. We demonstrate that a malignant spurious mode can be transmitted throughout such a mesh when it has no kinematic boundary constraints, by considering first a typical element e on its own. If unit spurious modes exist at each of its vertices, then the tangential and normal displacements of side 1 can be resolved from the vector function in (5.6) to give: 𝑣1s =

2 (P + 5 P2 ); l1 0

and 𝑣1n =

1 ((𝜑̄ 2 − 𝜑̄ 3 )(P0 + 5 P2 ) − 3 (𝜑̄ 2 + 𝜑̄ 3 ) P1 ). l1

If the element is rotated as a rigid body about its centroid through an anticlockwise angle 𝜃, then the corresponding displacements of side 1 are described by: 𝑣1s (𝜃) =

2𝜃 P0 ; 3 l1

and 𝑣1n (𝜃) =

𝜃 ((𝜑̄ 2 − 𝜑̄ 3 ) P0 − 3 (𝜑̄ 2 + 𝜑̄ 3 ) P1 ). 3 l1

Hence it follows that when a rotation 𝜃 = −3∕ of the element (i.e. clockwise) is superimposed on the displacements due to the spurious modes, we are left with the following quadratic mode of displacement: { 1} } { 5P2 𝑣s 2 ; (5.15) = 𝑣1n (𝜑̄ 2 − 𝜑̄ 3 ) l1 and similar forms of displacement of the other sides are obtained using cyclic symmetry. These displacements exclude rigid body modes of the sides, and they are compatible with those from adjacent elements so long as the spurious modes’ parameters and the rotations of the elements alternate in sign. It appears that the sides joining pairs of vertices, where equal angles are subtended in an element, translate as rigid bodies in the tangential directions. This is not the case because although the lengths of the sides do not change, the quadratic displacement fields induce longitudinal strains in them. Thus the tessellation in Figure 5.13 can be extended indefinitely, and a single spurious mode exists with each element having mode amplitudes equal to ±1. Such a spurious mode can occur around openings in the mesh providing there are an even number of elements bordering an opening so as to maintain the alternation in sign. However, the blocking of such a mode is a simple matter, for example, by the use of a local subdivision, or by constraining one side to be rigid. The corresponding displacements are illustrated in Figure 5.14 for a mesh of isosceles triangular elements where it can be seen that the displacements are compatible so long as the spurious mode parameters and the rigid body rotations alternate in sign. In the special case of a mesh of equilateral elements the displacements normal to the sides all become zero. This pattern of the spurious displacements is malignant, and is reminiscent of the plane-filling patterns designed by Escher (Piller et al., 2015).

89

90

Equilibrium F.E. Formulations

Figure 5.14 A uniform mesh of isosceles elements, with its spurious mode of degree 2.

Clearly tractions or body forces applied to an element having a resultant couple will be inadmissible when an adjacent element receives an equal and opposite couple which implies a self-balanced system of loads.

5.3 Spurious Kinematic Modes in Models of 3D Continua We now extend our ideas on spurious modes and admissible tractions to solid elements, and initially consider these entities in the neighbourhood of an edge where two faces in different planes meet, as shown in Figure 5.15. Local equilibrium conditions along the edge can be expressed again in terms of contravariant components of traction with oblique reference axes, as indicated in Figure 5.16. The traction admissibility conditions require ts11 = ts22 along an edge, where tsii denotes the contravariant tangential component on face i in the direction normal to that edge.3 Then the zero work property of spurious kinematic modes associated with the edge implies that the displacements of the adjacent edges are directed parallel to the contravariant base vectors as shown in Figure 5.16, where s̄ 1 = −n2 ∕ sin 𝜑 and s̄ 2 = −n1 ∕ sin 𝜑. Thus the displacements of points in face 1 are normal to face 2 and those of face 2 are normal to face 1. In a similar way to that used for 2D elements in Section 5.2, we develop the functions which describe the typical spurious kinematic modes associated with an edge of a tetrahedral element in Section 5.3.1. Then, in Section 5.3.2, we present the compatibility conditions required for propagation of spurious modes between a pair of elements with a triangular interface. The results from Section 5.3.2 are used in Section 5.3.3 to consider the kinematic stability of star patches of elements centred on a common edge or a common vertex. 5.3.1 Single Tetrahedral Elements e The space of spurious modes of an element, skm , has its dimension defined by (5.2), and this is obtained by considering the terms on the right hand side.

3 The other components of the traction in the oblique frame are not relevant for the equilibrium condition at this edge.

Analysis of the Kinematic Stability

n2

z

z

23

y

3

x

2 n2

n1 Face 2

x

y

Face 2

Face 1

1

1 1

3 3

1

1

n1

n2

3

n1

Face 2

Face 1

Face 2

3

1 Face 1

n2

2

y

z

x

2

2 2 1

y z

x

Figure 5.15 Notation used for the neighbourhood of an edge of a tetrahedral element presented in different views, including three first angle orthographic projections. Figure 5.16 Oblique reference frame for a normal section along an edge where 𝜑 is the dihedral angle; at a face the direction of the displacement associated with a spurious kinematic mode of an edge is parallel to the normal of the opposite face.

23  s3  s3 n2

s2

ϕ

s1

n1

v2 n1 Face 2

Face 1

v1 n2

s2 s1

The displacements of a face of a tetrahedron are described by complete polynomials of degree d, and this implies that: (d + 1)(d + 2) 2 is the dimension of a bi-dimensional scalar field of degree d, and the space of rigid and/or pseudo-rigid body displacements has dimension nrbm = 6. Internal stress fields are described by polynomials of the same degree, and these are complete within the constraints set by equilibrating with zero body forces. In this case dim  = n𝑣 = 4 × 3 × nb , where nb =

91

92

Equilibrium F.E. Formulations

the dimension of the space of stress fields is recalled from (4.18) and the dimension of the subspace of hyperstatic stress fields is given by Kempeneers et al. (2010): (d + 1)(d + 2)(d + 6) , 2 (d − 3)(d − 2)(d + 2) = 0 when d ≤ 3, nhyp = for d > 3 2

dim S = ns = nhyp and thus

nann = n𝑣 − ns + nhyp = 6, 15 or 6 (d + 2) when d = 0, 1 or ≥ 2, respectively. (5.16) The number of independent spurious kinematic modes is then defined by nskm = (nann − nrbm ) = 0, 9 or 6 (d + 1) when d = 0, 1 or ≥ 2, respectively. (5.17) This number indicates that (d + 1) spurious modes can be associated with each edge of an element, but they are not independent unless d ≥ 2. 5.3.1.1 Spurious Modes Associated With a Single Edge

As in Section 5.2.1 we can derive the spurious mode displacements from the property of being orthogonal to the admissible tractions. Instead of repeating the full derivation, we just invoke the similarity between Figures 5.2 and 5.16 to state that a spurious mode of the faces adjacent to the edge must satisfy the following condition, 𝑣1s̄1 + 𝑣2s̄2 = 0, with all the other components equal to zero. Convenient orthogonal functions derived from the Jacobi polynomials over a triangular domain (Maunder et al., 2016; NIST, 2013) are used to define a basis for the contravariant traction components.4 These functions can be ordered in a hierarchical sequence hnm of increasing degree n = 0 to d, and for each value of n, index m varies from 0 to n. Furthermore the functions can be expressed so that along one selected edge of the face, hnm = Pm (𝜉), where Pm is the Legendre polynomial of degree m. Similar independent bases could be defined using cyclic symmetry to exchange one selected edge for another. n To define the spurious modes we use hn⋆ m , the dual basis of hm : hn⋆ m =

(2m + 1)(n + 1) n hm . face

4 They are defined as:

hnm (L1 , L2 , L3 )

( = (1 − L1 )

m

0,2m+1 Pn−m (1

−2

0,0 L1 ) Pm

L3 − L 2 1 − L1

) ;

where L1 , L2 , L3 are the area coordinates and Pn𝛼,𝛽 (◽) is a Jacobi polynomial. These functions, as presented, are applicable for edge 1, which corresponds to L1 = 0. The functions for edges 2 and 3 require the area coordinates to be exchanged to comply with cyclic symmetry, that is, they are reordered to (L2 , L3 , L1 ) and (L3 , L1 , L2 ) respectively.

Analysis of the Kinematic Stability

While in the 2D case the condition at the vertex involves a single scalar, which leads to one spurious mode, we now have (d + 1) conditions at an edge, corresponding to the Legendre polynomials. This naturally leads to (d + 1) spurious modes. For edge e adjacent to faces 1 and 2, these spurious modes then take the form: 𝒗1m =

e km n, 1 sin 𝜑 2

𝒗2m = −

e km n 2 sin 𝜑 1

(5.18)

e where km is termed the signature function of mode m, which varies from 0 to d. This mode number matches the index m in hn⋆ m , and n1 and n2 are the unit outward normals to faces 1 and 2 with areas 1 and 2 respectively, as illustrated in Figure 5.15. e is simply defined in terms of the dual functions hn⋆ The signature function km m : e km = face

d ∑

hn⋆ m .

(5.19)

n=m

Since the dual functions for a face are inversely proportional to its area face , this area e , so ensuring its independence appears as the factor applied to the summation for km from the size or shape of the face. The signature functions when degree d = 2 and 5 are presented in Figure 5.17 for the edge shown in bold. It can be observed that along this edge the functions are always the Legendre polynomials, nevertheless over the face the functions vary in a different way, dependent upon d. Figure 5.18 illustrates for a tetrahedron the three spurious modes associated with one edge, obtained from (5.18), when degree d = 2.

(a) Quadratic approximation (d = 2).

(b) Quintic approximation (d = 5). 1 , for different approximation degrees. Figure 5.17 Face signature functions, km

93

94

Equilibrium F.E. Formulations

Figure 5.18 The three spurious modes associated with the edge shown in bold, when d = 2.

In a similar way, a total of 6(d + 1) spurious modes are defined for a tetrahedral element of degree d. It should be noted that in the case of degree d = 0, the signature function and therefore the mode displacements at the faces are constant, as defined in (5.20). 1 1 (5.20) n , 𝒗2 = − n 𝒗10 = 1 sin 𝜑 2 0 2 sin 𝜑 1 As with the 2D case in Section 5.2.1, the displacements of their centroids conform with a rotation of the element. In this case the axis of rotation crosses the tetrahedron through the centroid of two faces, as detailed in Maunder et al. (2016). Thus the six constant spurious modes corresponding to the six edges form the pseudo-rigid body rotations, of which only three are independent. In the case of degree d = 1, there are two linear spurious modes per edge. Out of the total of 12 modes, only nine are independent since they can be combined to conform to the three independent rigid body modes of rotation. This is a generalization of the 2D case, illustrated in Figure 5.4b. 5.3.1.2 Spurious Modes Associated With a Single Face

Now we express the spurious mode displacements of face i, with area i , as an arbitrary combination of the modes associated with all three of its edges: ( d ) 3 e ∑ aem km 1 ∑ ± ne , (5.21) 𝒗i = i e=1 sin 𝜑e m=0 where index e is now used to denote a particular edge with the amplitude of its mode m denoted by aem , and ne denotes the unit outward normal vector to the other face adjacent to edge e. The sign to be used for edge e depends on its orientation, given by the unit edge vector t e and the direction of increasing coordinate 𝜉, as compared with the face vector ni × si , where si denotes the unit covariant base vector associated with edge e and face i, as illustrated in Figure 5.16. If the orientation is the same, then the face is type 1 and the sign is plus, otherwise it is type 2 and the sign is minus. In Figure 5.19 all the edges of the shaded face have negative orientations and hence the signs to be used in (5.21) are negative. 5.3.2 A Pair of Tetrahedral Elements

A pair of elements A and B have the interface shown shaded in Figure 5.19, and retaining the sequence 1 to 3 for the edges of the interface, its displacements can be expressed for each element: ( d ) ( d ) e Be e ∑ aAe 1 ∑ 1 ∑ ∑ am km m km A − ne , and 𝒗B = nBe 𝒗A = A B  e  sin 𝜑 sin 𝜑 e e m=0 e m=0

Analysis of the Kinematic Stability

3

3

n1

n2 Edge 2

A

Q

Edge 1

A

T

P

2 1

n3

Edge 3

1

R

ω

B 2

Figure 5.19 Local numbering of vertices and edges of a face/interface.

where the unit outward normals nAe and nBe relate to the other faces of A and B that are adjacent to edge e, and  is the area of the interface. Note that the expression for elements A and B carry different signs since the edge orientations imply that the interface is of type 2 for element A, and of type 1 for element B as in (5.18). Compatibility at the interface requires 𝒗A = 𝒗B , and solutions for the mode amplitude B Be vectors aA = {aAe m } and a = {am } have been derived for a general degree d. The numbers of spurious modes that can be transmitted between a pair of elements depend upon d, and these are presented in Table 5.1 for the non-degenerate cases. For low degrees, that is, d = 0 or 1, the displacements of the interface are constant or linear respectively, and thus no deformations involving normal displacements occur. It should be noted that, for degree d = 0, the malignant modes are more properly described as pseudo-rigid body ones. In particular: When degree d = 0, all displacements of an interface conform with rigid body translations, and the 3 pseudo-rigid body modes associated with the 3 constant edge modes can be transmitted between a pair of elements.

and When degree d = 1, the 2 spurious modes per edge of the interface can be transmitted provided aB = −aA . The in-plane tangential displacements correspond to rigid body modes or states of constant strain of the interface. 5.3.2.1 Primary Interface Spurious Modes

It has been shown that for all degrees d, one of the compatible solutions can be expressed in the following form when the mode amplitudes associated with edge 1 are zero: Table 5.1 Number of malignant modes transmitted between a pair of elements of general degree (Maunder et al., 2016). d

0

1

2

3

≥4

number of malignant modes

3

6

6

5

3

95

96

Equilibrium F.E. Formulations

𝕜 (L ) 𝒗A = d 1 

(

)

1 1 − n2 + n3 TQ sin 𝜑A2 TR sin 𝜑A3

;

(5.22)

where TQ and TR are the perpendicular distances between T, the point where the line joining vertices A and B intersects the interface, and the edges of the interface as illustrated in Figure 5.19, and ( ) j=n n=d ∑ ∑ 𝕜d (L1 ) = (n + 1) ((2j + 1)Pj (𝜉)) , when L1 = 0.5 (1 + 𝜉). n=0

j=0

Since L1 = 0 on edge 1 and L1 = 1 at vertex 1, the distribution of interface displacements varies with distance from edge 1 according to a combination of Legendre polynomials, as illustrated in Figure 5.20 up to degree 4, but does not vary along lines parallel to this edge. Two other independent solutions are generated using cyclic symmetry, with zero amplitudes associated with edge 2 or edge 3. The three corresponding spurious modes of a pair of elements are termed their primary interface modes. The displacement field of the interface defined in (5.22) for element A is illustrated in Figure 5.21 for degree d = 3. This figure also includes the displacements of the other two faces due to the spurious mode amplitudes of the edges of the interface. It should be noted that the displacements of all three faces incident with vertex 1 are based on the same function 𝕜3 (L1 ), differing only in direction and magnitude. The displacements of faces 1A2 and 1A3 are normal to the interface, and the 4th face 2A3 remains stationary. There are three such primary malignant spurious modes for each interface, irrespective of the degree, and each mode corresponds to zero amplitudes for one of its edges. The one illustrated in Figure 5.21 has zero amplitudes for the edge 2 − 3 (shown in bold). 5.3.2.2 Pairs of Tetrahedral Elements With Coplanar Faces

Two configurations are possible, which are illustrated in Figure 5.22: Case (a): Two faces that are adjacent to an edge of the interface are coplanar. Then the intersection point T in Figure 5.19 is located on this edge, and consequently 𝜑̄ Ae + 𝜑̄ Be = 0. Case (b): T occurs at a vertex of the interface. Then we have two pairs of coplanar faces. For degrees 0 or 1, there are no changes to be considered since the corresponding spurious modes are unaffected by changes in dihedral angles. However, for higher degrees we note two different situations: 1.00

𝕜d (L)

0.00

d= d= d= d= d=

Figure 5.20 Normalized functions 𝕜d (L), L ∈ [0, 1].

0 1 2 3 4 L

Analysis of the Kinematic Stability

Figure 5.21 A primary spurious mode of degree d = 3 which can be transmitted to a neighbouring element via the face containing point T, as defined in Figure 5.19.

A

3

1

T

2

B

3

T

A

B

3

A

1T

1 2 Case (a)

2 Case (b)

Figure 5.22 Examples of the possible cases of pairs of tetrahedral elements with coplanar faces.

Case (a): The malignant spurious modes include the primary mode with zero amplitudes on the edge containing T plus the (d + 1) spurious modes associated with this edge alone. They are similar, respectively, to those in Figures 5.21 and 5.18, with T on the bold edge. For the particular degree d = 2, two further malignant modes can exist, and their details can be seen in Maunder et al. (2016). Case (b): The malignant spurious modes just consist of the 2 × (d + 1) modes associated with the 2 edges of the interface incident with T. Only modes similar to those in Figure 5.18 exist.

5.3.3 Star Patches of Tetrahedral Elements

We focus on the propagation of the primary spurious modes, which follow the same pattern for all degrees, and are the only ones when d ≥ 4. With tetrahedral elements, star patches exist in two types of configuration: • edge-centred patches where elements share a common edge; • vertex-centred patches where elements share a common vertex. In Maunder et al. (2016) it is conjectured that the spurious kinematic properties of a vertex-centred patch are governed by those of its constituent edge-centred patches, based on the edges incident with the vertex. Accordingly we first address the kinematic properties of edge-centred patches, before considering vertex-centred patches.

97

98

Equilibrium F.E. Formulations

A 1

B

Q

3

T

P

R 2

A 1

T

23

B

Figure 5.23 Patch of tetrahedral elements centred on edge 2–3.

5.3.3.1 Edge-Centred Patches

The patch in Figure 5.23 is shown with a typical interface shaded, together with the intersection point T and points P, Q and R on edges 1, 2 and 3. The link of the common edge, Lk(E), can be defined (Giblin, 1977) as the edges of the patch which are not incident with the common edge E. Hence when Lk(E) forms a closed path, the patch is closed, otherwise it is open. The primary interface mode for elements A and B, as defined in (5.22), can be activated with: i) Zero mode amplitudes a1m for the common edge 1. In this case the spurious mode is localized to the interface and the faces adjacent to edges 2 and 3, and propagation does not occur any further. If a pair of faces adjacent to edge 2 or 3 are coplanar, then the local mode is replaced by (d + 1) spurious modes associated with the corresponding edge as discussed in Section 5.3.2. ii) Zero mode amplitudes aem for edge 2 or 3 after redefining the mode to account for cyclic symmetry. Then the spurious mode propagates via adjacent elements around the patch when the amplitudes alternate in sign and are appropriately scaled to suit changes in the relative positions of the intersection points T. Hence propagation is only possible in an open patch, or a closed patch with an even number of elements. Special situations again occur when some faces are coplanar and point T is situated on an edge or a vertex of one or more interfaces. 5.3.3.2 Vertex-Centred Patches

The prediction of malignant spurious modes in vertex-centred patches is more problematic than in the 2D case, and we restrict ourselves here to pointing out some of their features. For this purpose, we select the three examples illustrated in Figure 5.24. As in Section 5.2.4, we classify the modes as ‘topologically inherent’ or ‘geometrically induced’. In this context, it is convenient to use further topological terminology: The terms link-face, link-edge and link-vertex denote the entities belonging to the Lk(V ), where V denotes the common vertex. The valency of a link-vertex is the number of link-edges incident with it. The following hypothesis has been proposed in Maunder et al. (2016) as a guide to identifying the spurious modes in a vertex-centred star patch which are topologically inherent. This has been verified by numerous numerical tests, which also confirm the independence of their number from the location of the vertex.

Analysis of the Kinematic Stability

7

7

4 8 1

5

V 3

8

6 3

5

V

4

6

V

4

2

2 2 (a)

1

(b)

3

1

(c)

Figure 5.24 Star patches of tetrahedral elements.

A basis for skm of a vertex-centred star patch of tetrahedral elements is defined as the collection of the spurious modes of each of the subpatches centred on the edges incident with the vertex. In defining these modes, the faces of a sub-patch that connect with elements outside it are assumed to have zero displacements, that is, to be fixed. Four elements assembled as a macro-tetrahedron The patch in Figure 5.24a contains four

subpatches centred on the edges joining the link vertices to V , and each one contains three elements. Since this is an odd number, a spurious mode cannot be propagated around its edge. The stability of this macro-element, predicted by Fraeijs de Veubeke (1973), was numerically verified in Almeida and Pereira (1996) and Kempeneers et al. (2003), with a proof in Kempeneers et al. (2010). Twelve elements assembled as a macro-hexahedron-like domain The patches in Figures 5.24b

and 5.24c differ in the values of the valencies of their link vertices. In Figure 5.24b there are four link vertices which have valency 4, and hence each edge joining them to V allows a spurious mode to exist. On the other hand, the assembly shown in Figure 5.24c has the property that all the link vertices have an odd valency. Consequently there are no topologically inherent spurious modes. However, these conclusions change when any pair of link-faces adjacent to an edge are coplanar. Thus for example, when the mesh is formed from a regular hexahedron with flat sides, then the only spurious modes are those associated with the edges forming the diagonals of the sides. The total number of these modes is 6(d + 1).

5.4 Spurious Kinematic Modes in Models of Reissner–Mindlin Plates In this Section we consider the number and descriptions of spurious kinematic modes in the context of modelling plate bending problems with triangular hybrid equilibrium elements governed by Reissner–Mindlin theory. In this type of element, shear deformation through the thickness of the plate is accounted for by first order shear deformation theory (Mindlin, 1951; Reissner, 1945). In this theory, the through thickness normals remain straight but they are free to rotate about an axis lying in the midplane of the plate independently of the displacements of the midplane.

99

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Equilibrium F.E. Formulations

The stress resultants for these elements include bending and torsional moments together with transverse shear forces. The side displacements include rotations of the normals to the midplane, about normal and tangential axes in the midplane, and transverse displacements normal to the midplane. The corresponding dual tractions consist of torsional and bending moments, and transverse shear forces. The determination of spurious modes for this type of element follows a similar approach to that in Section 5.2, and begins by considering the admissibility conditions for moment tractions at the corner of a plate as illustrated in Figure 5.25. For moment tractions to be admissible, its covariant components at a corner must satisfy m1s̄2 + m2s̄1 = 0. Then the zero work equation for all such moments leads to the spurious modes of side rotation involving only the contravariant components 𝜃s12 of side 1 and 𝜃s21 of side 2. We continue in this Section with defining the spurious modes of a single element, and then consider a pair of elements before addressing more general assemblies. In Section 5.5 we examine plates governed by the classical Kirchhoff theory where shear deformation is neglected and the through thickness normals rotate to conform with the local rotations of the midplane. 5.4.1 A Single Triangular Reissner–Mindlin Element

The dimensions of the spaces of stress resultants and side displacements for a single element are given by Maunder and Almeida (2005): dim h = n𝑣 = 3 (3 d + 2),

dim h = ns = (d2 + 5 d + 3);

nrbm = 1 when d = 1, and nrbm = 3 when d > 1; nhyp = 0 when d ≤ 3, nhyp = (d2 − 4 d + 3) when d > 3. When d = 0, an element would have constant moment fields with zero transverse shear forces, and consequently elements based on Reissner–Mindlin theory are made redundant. In any case, as can be shown, assemblies of such elements into star patches are always unstable, and are mentioned here only for completeness. Hence, they will not be considered further. When d = 1, the sides of an element can only displace in a transverse direction by constant amounts. Hence the only real rigid body mode of displacement that is included within the space of displacements of an element is that of a uniform transverse translation.

s1 1 sin ϕ

s2

side 1

θs1 2

ξ1

s1 s2

1 sin ϕ

ms1 1

ms1 2

3

ϕ

ξ2

ms2 1

side 2 ms2 2

3

θs2 1

θs1 1

θs2 2

Figure 5.25 Oblique components of moment and rotation on sides adjacent to a corner.

Analysis of the Kinematic Stability

Figure 5.26 A pseudo-rigid body rotation 𝜽 for degree d = 1. 1

2

h 2 θ

3

δ

Using (5.2), the number of independent spurious kinematic modes is given by: nskm = 5 for d = 1 and nskm = 3 when d > 2, but nskm = 4 when d = 2. (5.23) As with other elements, we obtain the description of spurious kinematic modes from the conditions for side tractions to be admissible. Using the conditions that apply to the covariant components of moment at each corner, the associated unit spurious mode takes a similar form to (5.4) with displacement vector 𝒗 replaced by a vector 𝜽 representing rotation of a transverse normal, that is, ( ( ) ) k+ k− (5.24) s2 and 𝜽2 = s 𝜽1 = l1 sin 𝜑3 l2 sin 𝜑3 1 where s1 and s2 are the covariant base vectors, which have the direction of the sides, as illustrated in Figure 5.25. It should be noted that the three spurious modes so defined are independent for all degrees. We may consider the displaced forms in Figure 5.3 to represent the displacements of the top edges of the sides of the plate relative to its midplane, which remains stationary. Hence we cannot combine these modes so as to conform with a rigid body movement of the element, unlike the membrane cases discussed in Section 5.2.1, when d = 0 or 1. When d = 1, two additional pseudo-rigid body modes of rotation exist, due to the incompleteness of the displacement space to represent the true rigid body modes. The additional modes are described by side displacements that conform with rigid body rotations of an element, with the exception of the linear components of transverse translation, since these are excluded from the displacement space of the element. All tractions belonging to h , and hence those excluding linear distributions of transverse shear, do zero work with these displacements when they satisfy overall equilibrium. This is a default requirement of admissibility. An example of such a mode of displacement is illustrated in Figure 5.26, where all fibres in the sides are displaced to conform with a rigid body rotation 𝜽 about side 1. This rotation requires linear translations along sides 2 and 3 as indicated in the Figure, with value 0 along side 1 and value h × 𝜃 at corner 1, when corner 1 is distance h from side 1. The displacements belonging to h allow all the fibres to rotate by 𝜽, but only allow sides 2 and 3 to translate by 𝛿 = 0.5 h × 𝜃. However, when d = 2, it is no longer sufficient for tractions to be admissible that they only satisfy the corner conditions. One extra condition is required, and this involves shear force as well as moment tractions on all three sides. The details of its derivation are

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given in Maunder and Almeida (2005). It makes use of the internal equilibrium equation between shear forces and moment derivatives. The displacements of the unit mode are expressed in (5.25) for side 1 as typical. ⎫ ⎧ ⎧𝑤 ⎫ −0.75 l1 P1 ⎪ ⎪ 1⎪ 1 ⎪ ⎨𝜃1n ⎬ = l ⎨((𝜑̄ 3 − 𝜑̄ 2 )P0 + 3 (𝜑̄ 3 + 𝜑̄ 2 ) P1 − 2.5 (𝜑̄ 3 − 𝜑̄ 2 ) P2 )⎬ 1 ⎪ ⎪ ⎪ 𝜃1s ⎪ (−2 P0 + 5 P2 ) ⎭ ⎩ ⎩ ⎭

(5.25)

The values for other sides are obtained using cyclic symmetry. In (5.25), 𝜃1n and 𝜃1s denote rotations about the tangential axis s and about the inward normal axis −n respectively. 5.4.2 A Pair of Reissner–Mindlin Elements

The number and form of malignant spurious modes in a pair of elements generally depends on their degree d. From the results in Maunder et al. (2016) concerning the compatibility of spurious displacements at an interface, there are essentially three separate cases to consider, corresponding to d = 1, 2 or ≥ 3. When degree d = 1, three independent spurious modes are associated with the vertices of an element and they do not involve transverse displacements. The amplitudes of the spurious modes of element B are dependent on those of adjacent element A, as illustrated in Figure 5.5, replacing p’s with r’s, according to (5.26).

[ (𝜑̄ A3 − 𝜑̄ B1 ) 1 rB = B (𝜑̄ 1 + 𝜑̄ B3 ) (𝜑̄ A3 + 𝜑̄ B3 )

] (𝜑̄ A2 + 𝜑̄ B1 ) A r . (𝜑̄ A2 − 𝜑̄ B3 )

(5.26)

where the corner mode amplitude vectors are { } { } r3A r3B A B and r . = r = r2A r1B When degree d = 2, an element has a 4th spurious mode as defined in (5.25). The amplitude vectors are augmented with extra components r4A and r4B to account for this.

Compatibility of spurious mode displacements now requires r B = −r A and, provided the pair of elements contains no degenerate vertices, the relative values of the mode amplitudes are determined by: ⎧ TS ⎫ { A} r4A ⎪1 + 3 TV ⎪ r3 = r2A TV ⎬ 4 ⎨ ⎪ ⎪1 + 3 ⎩ TS ⎭

(5.27)

Analysis of the Kinematic Stability

When degree d ≥ 3, there are no malignant modes unless the geometrical configuration is degenerate, for example, TS = 0, or TV = 0. 5.4.3 Star Patches of Reissner–Mindlin Elements

We discuss here the question of stability of open and closed star patches of plate bending elements, in a similar way to that for the 2D continuum case in Section 5.2.3. We recognize again that when d ≥ 3, meshes are generally stable with the only exceptions being for degenerate geometries. 5.4.3.1 Open Stars of Degree 1

In this case we have three independent corner spurious modes per element. Both corner modes at an interface of one element propagate to an adjacent element in a unique way as defined by (5.26). Propagation is continued around an open star patch of n elements, having V as a common vertex, with one extra spurious mode introduced in each element. Thus we obtain a total of (3 + (n − 1)) = (n + 2) independent spurious modes in a patch. 5.4.3.2 Closed Star Patches of Degree 1

For this case we consider first an open star patch with n elements similar to that illustrated in Figure 5.7, but with the two sides of the elements at the end of the chain coincident but unconnected. This open patch has (n + 2) independent spurious modes. Closure of the unconnected sides requires five compatibility constraints to match linear rotations and constant transverse displacements. Then this implies that (n + 2 − 5) = (n − 3) independent spurious modes remain, apart from the two pseudo-rigid body rotations as already described for an element in Section 5.4.1. 5.4.3.3 Open Stars of Degree 2

In order for a single mode to propagate via the interfaces throughout an open star patch, the mode amplitudes must satisfy r B = −r A as well as (5.27) at each interface. With reference to Figure 5.10, these conditions can only be satisfied in general when the ratio TS∕TV is constant for all interfaces. This puts severe constraints on the configurations that allow this to occur, for example, a patch of similar isosceles triangular elements. In the degenerate case, where for example, TSi = 0, the single spurious mode cannot propagate throughout a patch, but it becomes localized at an interface with r3e = 0 = r4e for the elements adjacent to the interface. 5.4.3.4 Closed Star Patches of Degree 2

As in the 2D case, patches of elements with degree 2 require particular consideration. In addition to the compatibility conditions for open star patches, it is necessary that the relative rigid body displacements for each element be compatible within a closed star. These displacements are derived in Maunder et al. (2016), and they lead to the additional form of compatibility conditions in (5.28): ∑ (−1)i Wi li t i = 𝟎 (5.28) i

where Wi and li have the same meaning as in Equations (5.13), and t i is the unit vector directed along interface i, from V .

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The geometric conditions that allow spurious modes to propagate within a closed star patch of Reissner–Mindlin elements are thus more onerous than those for membrane elements. Some examples of stable and unstable configurations of closed star patches having various forms of symmetry are illustrated in Maunder and Almeida (2009) and Maunder et al. (2016). Two examples of closed star patches are considered in Figure 5.27 to illustrate the use of the compatibility conditions in determining the stability or otherwise of the patches. Figure 5.27a shows a hexagonal patch assembled from a set of six similar isosceles triangular elements with area . At each interface Ti V ∕Ti Si = 1, and Wi li t i = V⃗Ti ∕. It is clear by inspection of this Figure that −V T1 + V T2 − V T3 = 0, and hence (5.28) is satisfied for the whole patch. Consequently a single spurious mode can propagate within the patch and it is unstable, as is a similar patch of 2D elements. We thus see that the spurious mode that is propagated within this star patch involves rigid body rotations of the interfaces that are of the same magnitude but alternate in sign. The semi-regular octagonal patch in Figure 5.27b has its link vertices positioned on a circle with centre at V , it is symmetrical about horizontal and vertical centrelines, and the corner angles 𝜑23 and 𝜑43 are not equal. This patch has sufficient symmetry to enable a single spurious mode to propagate in 2D, but it is clear that the more onerous requirement for a constant ratio Ti V ∕Ti Si at each interface is not satisfied. Consequently this patch is a stable configuration for Reissner–Mindlin plates. On the other hand, a regular octagonal configuration would be unstable, since it satisfies all the conditions to allow a spurious mode to propagate. 5.4.4 Observations for General Meshes of Reissner–Mindlin Elements

The general observations for 2D meshes in Section 5.2.4 apply to meshes of Reissner– Mindlin plate elements. However, the geometric conditions that allow spurious modes to propagate are more onerous than in the 2D case. In other words, meshes of Reissner– Mindlin elements tend to be more stable than their 2D configurations. The buttressing effect of an additional element when degree d = 2, as shown in Figure 5.12, still holds, but now this effect still applies for the concurrent configuration in Figure 5.12b provided the points M, N and T are not at the midpoints of the sides. The tessellation based on a uniform element of degree 2 remains unstable, with a single spurious mode that can propagate throughout a hexagonal patch, as illustrated S2

S1

S2

T1

T2 S3

ϕ 23

S3

T3 V

T2

T1

T3 V

ϕ 43

(a)

S1

(b)

Figure 5.27 Hexagonal and octagonal patches with d = 2.

Analysis of the Kinematic Stability

in Figure 5.13, or throughout an extended mesh. This mode can again be defined by considering the superposition of all four unit spurious modes of an element. For example the corresponding displacements of side 1 are: ⎧𝑤 ⎫ ⎧ ⎫ −l1 P1 ⎪ 1⎪ ⎪ 3 ⎪ 𝜃 ⎨ 1n ⎬ = 4 l ⎨10 (𝜑̄ 2 − 𝜑̄ 3 ) P2 ⎬ 1 ⎪ ⎪ 𝜃1s ⎪ ⎪ 20 P2 ⎩ ⎭ ⎩ ⎭ These displacements match those of an adjacent element when the signs of its unit spurious modes are changed. Thus propagation continues to rely on the mode amplitudes having the same values but alternating in sign for adjacent elements. When this is not possible, the corresponding mesh is stable.

5.5 The Stability of Plates Modelled With Kirchhoff Elements The kinematic freedoms in Kirchhoff elements are different from those in Reissner– Mindlin elements as a consequence of the extra assumption that shear deformations are zero. Thus on a side of an element, rotations 𝜃n remain free but rotations 𝜃s are constrained to equal the rotations of the midplane along a side, that is, 𝜃s = − 𝜕𝑤 . This 𝜕s assumption implies that we can take 𝜃n and 𝑤 as the independent kinematic freedoms associated with a side. To obtain dual pairs consistent with the derivation of generalized boundary tractions summarized in (A.11) we define the independent kinematic variables as distributions of 𝜃n and 𝑤 along the sides plus discrete values of 𝑤 at the corners of an element, with the distributions of 𝑤 on the sides being one degree lower than that of 𝜃n (Almeida and Maunder, 2013). With these definitions we can apply (5.2) to obtain the number of spurious kinematic modes, with n𝑣 = 2 × 3 (d + 1), and ns = (d + 2)(d + 3) − 3 since the internal moment and shear fields are the same as for the Reissner–Mindlin plate element in Section 5.4. The hyperstatic fields of moment and shear require zero (values of normal bending ) 𝜕m moment mn and the equivalent Kirchhoff shear force rn = qn + 𝜕sns , as derived in Section A.1.4, after replacing local coordinate r with tangential coordinate s, together with zero corner forces. In order to determine nhyp it is convenient to turn again to the Southwell vector potential U of degree (d + 1) to generate the moment and shear fields. For the hyperstatic fields, it is necessary for U to satisfy the following conditions on the sides of the element: mn =

𝜕Us 𝜕2 U = 0, rn = − 2 n = 0. 𝜕s 𝜕s

We assume that U is a continuous function, and then these conditions imply that if U were to represent a planar displacement field, the boundary displacements would conform with a rigid body motion. Furthermore, the stress fields and tractions are unchanged by the addition of a linear potential field 𝜹U which represents a rigid body motion of the complete element. Hence, without loss of generality, we can assume that U = 0 around the boundary of the element.

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w3

w3

h (a)

(b)

Figure 5.28 Pseudo-rigid body modes for a Kirchhoff element when degree d = 0 or 1. The initial position of the sides is marked in light grey and the arrow represents the unit vertex displacement.

It can be readily shown that polynomials of degree up to (d + 1) that are complete, but have zero value on the perimeter of a triangular domain, generate a space of dimension 1 d(d − 1). 2 Thus the two components of the vector function U of degree (d + 1) generate d(d − 1) hyperstatic moment and shear fields, that is, nhyp = d(d − 1), and consequently nann = n𝑣 − ns + nhyp = 3 for all degrees of a triangular element. When d = 0, there are no true rigid body modes within h , but three pseudo-rigid body modes exist. A typical mode is shown in Figure 5.28a, where the side rotations and vertex translations are given by: cos 𝜑2 cos 𝜑1 1 , 𝜃2n = − , 𝑤1 = 𝑤2 = 0, 𝑤3 = −1. 𝜃3n = , 𝜃1n = − h h h This can be regarded as a pseudo-rigid body rotation about side 3. The zero work condition with admissible tractions is thus equivalent to enforcing global moment equilibrium about an axis parallel to side 3, and thus two of such spurious modes are independent. The summation of all three such spurious modes involves only constant transverse displacements at the 3 corners, and thus zero work with this pseudo-rigid translation mode enforces transverse equilibrium of the corner forces. When d = 1, the translational rigid body mode exists within h . A typical pseudo-rigid body mode is shown in Figure 5.28b, where the side displacements are similar to those when d = 0 but in addition sides 1 and 2 are translated by −0.5. As when d = 0, the zero work condition with admissible tractions is thus equivalent to enforcing global moment equilibrium about an axis parallel to side 3. Two such independent pseudo-rigid body modes exist, corresponding to moment equilibrium. When d ≥ 2, the number of rigid body modes equals 3 and nskm = 0. Since zero work with the pseudo-rigid body modes imposes no extra conditions on the tractions beyond those of global overall equilibrium, we can expect all such tractions to be admissible, and no spurious modes should exist in a mesh of triangular elements of any degree.

5.6 The Stability of Models for Potential Problems When we consider the 2D potential problem, for example, typified by the heat conduction problem, side displacements of a hybrid element are replaced by a scalar potential, for example, temperature T, and the internal stress resultant tensor is replaced by a vector, for example, heat flux q. The hybrid equilibrium element aims to satisfy strongly the balance of fluxes, for example, div q = 0. Independent temperature distributions are defined along element sides, and internally we define balanced vector flux fields.

Analysis of the Kinematic Stability

The work product on the boundary becomes qn ⋅ T where qn is the outward normal component of flux. Thus a spurious mode of boundary temperature T would generate no internal flux, that is, it would satisfy: ∮𝜕Ω

qn ⋅ T ds = 0,

for all q. For a triangular element of degree d, we have: n𝑣 = 3 (d + 1),

nrbm = 1.

We can define flux fields using a scalar polynomial heat flux potential field Φ of degree (d + 1) so that: 𝜕Φ 𝜕Φ qx = , qy = − . 𝜕y 𝜕x The space of complete heat flux potential fields up to degree (d + 1) has dimension 1 (d + 2)(d + 3), and thus ns = 12 (d + 2)(d + 3) − 1 since a constant heat flux potential 2 produces zero flux. Then, invoking Equation (5.2) again: d (1 − d) + nhyp . 2 = 0 along the sides of the element, that is, Φ is conHyperstatic flux fields require 𝜕Φ 𝜕s stant along the sides, and again without loss of generality Φ may be assumed to be zero thereon. The situation is now similar to the case of the plate bending element governed by Kirchhoff theory in SubSection 5.5, and the dimension of the hyperstatic space is d (d − 1). This value confirms the conclusion that no spurious modes exist in triangular 2 elements for potential problems. For 3D potential problems, we could also deduce from consideration of a 3D flux potential 𝚽 that the tetrahedral element has no spurious modes. However, a more direct, although equivalent, argument begins by establishing the number of divergence free flux fields within a tetrahedral domain.5 . The polynomial spaces of scalar fields of degree d have dimensions: 1 1 d d ) = (d + 1) (d + 2) in 2D, dim(P3D ) = (d + 1) (d + 2) (d + 3) in 3D. dim (P2D 2 6 d ), and The total number of independent vector flux fields of degree d in 3D is 3 dim(P3D d−1 the number of independent constraints to enforce zero divergence is dim(P3D ). Therefore, 1 d d−1 ) − dim (P3D ) = (d + 1) (d + 2) (2d + 9). ns = 3 dim (P3D 6 In order to find the hyperstatic fields within a tetrahedral domain, we need to enforce d ) conzero normal flux over each of its four faces. For each face this involves dim(P2D straints. However, when the flux is already divergence free, one of the normal flux conditions must be redundant, and hence: 1 d ) + 1 = d(d − 1) (2d + 5). nhyp = ns − 4 dim (P2D 6 nskm =

5 This argument could also have been used for the 2D case.

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But we have that the space of scalar (e.g. temperature) fields over the four faces has d ), and there exists a single constant temperature field that is dimension n𝑣 = 4 dim(P2D equivalent to a rigid body mode. Consequently: d ) − 1 = (n𝑣 − nrbm ) =⇒ nskm = 0. (ns − nhyp ) = 4 dim(P2D

5.7 How Do We Obtain a Stable Mesh for General Structural Models? In this Chapter we have analysed the sources of kinematic instability in a given mesh in the form of spurious kinematic modes and the configurations that allow their propagation. In other words, we have determined the conditions which must apply to prescribed loads so that they are admissible. It may be inconvenient to have such restrictions on general patterns of load, and in that case it is important to have advice on how to obtain meshes that are stable for all loads. The fundamentals for this guidance are now presented. 5.7.1 General Procedures

We have already noted in previous sections that simplicial meshes for Kirchhoff plate bending and potential problems are inherently stable for any configuration. In the field of structural mechanics, simplicial meshes for 2D continua and plate bending governed by Reissner–Mindlin theory are fully stabilized when using elements of degree d ≥ 3, and avoiding pathological configurations, that is, those that include quadrilateral closed star patches with internal sides aligned with diagonals, and triangular open star patches with their vertices connected to a pair of collinear sides of a static boundary. However, in these cases, we also need to be aware of problems that may be caused by configurations which we term ‘near misses’, that is, those that, as will be discussed and illustrated in Section 6.7, are in close proximity to an unstable form. It is then still possible to obtain solutions, but their quality may become polluted by unrealistic relative stiffnesses. The safer alternative for these problems, and also for simplicial meshes for 3D continua, is to invoke the concept of macro-elements (otherwise known as ‘super-elements’). 5.7.2 Macro-Elements

Macro-elements can be considered as special closed star patches of ‘primitive’ triangular or tetrahedral elements. The equations for these patches can be pre-assembled and condensed, so that in practice each macro-element is treated as a single element. The resulting system of equations is smaller than it would have been if the degrees of freedom had not been condensed. When every macro-element is free from spurious modes the resulting system is necessarily stable. If spurious modes are present, and they can be made internal to the macro-element, then the resulting governing system is also stable. As we have noted in Sections 5.2.4, 5.3.3 and 5.4.4, individual triangular or tetrahedral macro-elements are stable for all degrees, and hence meshes of such elements are free from spurious modes.

Analysis of the Kinematic Stability

Quadrilateral macro-elements of degree d = 1 are always affected by spurious modes, while for d > 1, this is so only in the case when the internal vertex lies at the intersection of the diagonals of the quadrilateral. As explained in Maunder et al. (1996), these spurious modes can be made internal to the patch or removed altogether, by a proper selection of the position of the internal vertex. The performance of quadrilateral macro-elements is similar to that of triangular macro-elements, unless the loads that are defined inside the macro-element are discontinuous. In that case, ‘near miss’ situations, such as those described in Section 6.7, may occur. Hexahedral macro-elements may also be defined by assembling tetrahedral elements (Almeida and Pereira, 1996; Pereira, 2008), but no general configurations have been found such that all spurious modes are internal. In order to recover the stress field, the condensation needs to be reversed. This is a process that is local to the macro-element. Alternative condensation schemes can be used: • By eliminating the degrees of freedom associated with the internal interfaces 𝒗̂ i , some stress parameters are made dependent, resulting in a broken space of statically admissible stress fields inside the macro-element, together with an appropriate particular solution 𝝈 0 . Then the variables of the macro-element are the remaining stress parameters and the displacement parameters of the external interfaces, 𝒗̂ e . • By eliminating all the internal parameters, of stress and displacement, a stiffness matrix that directly relates the external displacement parameters 𝒗̂ e with the external traction parameters t̂ e is obtained. The second option is generally preferred (Maunder et al., 1996), not only because it results in a smaller number of variables, but also because the elimination of the stress parameters, already presented in (4.15), is a straightforward process. There is some evidence that the convergence rate of the solutions obtained with macro-elements of degree d is similar to that obtained from the same mesh with primitive elements of degree d + 1 (Johnson and Mercier, 1978; Pereira, 1996).

References Almeida JPM and Maunder EAW 2013 A general degree hybrid equilibrium finite element for Kirchhoff plates. International Journal for Numerical Methods in Engineering 94(4), 331–354. Almeida JPM and Pereira OJBA 1996 A set of hybrid equilibrium finite element models for the analysis of three-dimensional solids. International Journal for Numerical Methods in Engineering 39(16), 2789–2802. Coxeter HSM and Greitzer SL 1967 Geometry Revisited. Math. Assoc. Amer., Washington DC. Fraeijs de Veubeke BM 1964 Upper and lower bounds in matrix structural analysis. In AGARDograph 72: Matrix Methods of Stuctural Analysis, Pergamon Press London p. 165–201. Fraeijs de Veubeke BM 1973 Diffusive Equilibrium Models Lecture notes for the International Research Seminar of the Theory and Application of Finite Element Methods. University of Calgary. Giblin PJ 1977 Graphs, Surfaces and Homology. Chapman and Hall.

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Johnson C and Mercier B 1978 Some equilibrium finite element methods for two-dimensional elasticity problems. Numerische Mathematik 30(1), 103–116. Kempeneers M, Beckers P, Almeida JPM and Pereira OJBA 2003 Modèles équilibre pour l’analyse duale. Revue européenne des éléments finis 12(6), 737–760. Kempeneers M, Debongnie JF and Beckers P 2010 Pure equilibrium tetrahedral finite elements for global error estimation by dual analysis. International Journal for Numerical Methods in Engineering 81(4), 513–536. Maunder EAW, Almeida JPM and Pereira OJBA 2016 The stability of stars of simplicial hybrid equilibrium finite elements for solid mechanics. International Journal for Numerical Methods in Engineering 107(8), 633–668. Maunder EAW, Almeida JPM and Ramsay ACA 1996 A general formulation of equilibrium macro-elements with control of spurious kinematic modes. International Journal for Numerical Methods in Engineering 39(18), 3175–3194. Maunder EAW and Almeida JPM 2005 A triangular hybrid equilibrium plate element of general degree. International Journal for Numerical Methods in Engineering 63(3), 315–350. Maunder EAW and Almeida JPM 2009 The stability of stars of triangular equilibrium plate elements. International Journal for Numerical Methods in Engineering 77(7), 922–968. Mindlin RD 1951 Influence of rotary inertia and shear on flexural motions of isotropic elastic plates. Journal of Applied Mechanics 18, 31–38. Munkres JR 2000 Topology. 2nd ed. Upper Saddle River, NJ: Prentice Hall. NIST 2013 Digital Library of Mathematical Functions http://dlmf.nist.gov/, Release 1.0.6 of 2013-05-06. Pereira OJBA 1996 Utilização de elementos finitos de equilíbrio em refinamento adaptativo PhD thesis Universidade Técnica de Lisboa. Pereira OJBA 2008 Hybrid equilibrium hexahedral elements and super-elements. Communications in Numerical Methods in Engineering 24, 157–165. Piller M, Elliott P and Peterse F 2015 The Amazing World of MC Escher. National Galleries of Scotland. Reissner E 1945 The effect of transverse shear deformation on the bending of elastic plates. Journal of Applied Mechanics 12, 69–77. Shephard GC 1966 Vector Spaces of Finite Dimension. Oliver and Boyd. Strang G 1988 Linear Algebra and its Applications. Harcourt Brace Jovanovich.

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6 Practical Aspects of the Kinematic Stability of Hybrid Equilibrium Elements In Chapter 5 we presented an algebraic procedure for the identification of the spurious kinematic modes of simplicial elements, either isolated or assembled into star patches. This procedure is entirely based on topological and geometric properties, which enabled us to determine when spurious modes can propagate, and to define the form of the modes. Limited conclusions were made concerning the stability of general meshes. Recognizing that a variety of practical issues can arise from the existence of spurious modes, we address in this Chapter the following particular aspects: • the identification of stable meshes and, in the case of unstable ones, the determination and removal of the spurious kinematic and rigid body modes; • the characterization of the admissibility of the loads in unstable meshes; • the spurious modes potentially introduced by a hierarchical mesh refinement; • the spurious modes of non-simplicial elements; • the behaviour of mesh configurations that are in close proximity to being unstable.

6.1 Identification of Rigid Body and Spurious Kinematic Modes In this Section we discuss techniques to identify the spurious kinematic and rigid body modes in a solution when loading is admissible and yet the former modes exist. This identification is problematic since spurious modes can exist with arbitrary values of their parameters, which effectively pollute the displacements in a solution. Such techniques are crucial when the rigid body displacements are directly sought, as for example in the recovery of compatibility, to be discussed in Section 8.2, or for the large displacement analysis presented in Section 11.4. They are also relevant when we seek to constrain the spurious kinematic modes, modifying the global system so that it is non-singular. The objective being that in the solution of this system the amplitude of the spurious modes is zero. As a consequence the direct representation of the boundary displacements is not polluted by these modes. Spurious modes are not important when only the stresses are sought since the stress distribution is not affected by them. This also implies that the determination of a local displacement by the application of admissible virtual loads, as discussed in Section 9.3, is similarly not affected because it only depends on solutions in terms of their stress fields. Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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Equilibrium F.E. Formulations

6.1.1 Spurious Kinematic and Rigid Body Modes of an Element

When an isolated element is considered, the nullspace of DTe , which has dimensions ne𝑣 × neann , represents boundary displacements that are combinations of rigid body and spurious kinematic modes. Identifying the eventual existence of a spurious mode within a solution would allow for its removal, so that the rigid body displacement parameters could then be easily determined. In this context it is relevant to introduce a metric into the displacement space he by, for example, defining the inner product ⟨𝑣i , 𝑣j ⟩ = 𝒗̂ Ti 𝒗̂ j using appropriately scaled Legendre polynomials as a basis. Then, given the singular value decomposition (SVD) of matrix De (Golub and Van Loan, 2013; Strang, 1988) De = De ⌈𝚺De ⌋TDe ,

(6.1)

the nullspace of DTe is comprised of the left singular vectors comprising the columns of De that correspond to the zero singular values in 𝚺De . These columns form submatrix 0De and they serve as a basis for  eann , and the columns of its complement 1De form a basis for  e𝜀 , the space of straining modes: [ ] De = 1De 0De . In terms of an inner product space, these two subspaces when obtained from an SVD are orthogonal complements, giving a unique decomposition of an arbitrary displacement 𝑣 ∈ he . Each of the columns of the nullspace corresponds to a combination of rigid and spurious modes, and there is no unique way to separate them because the spurious modes do not form a direct complement of the rigid body modes within the nullspace.1 Likewise, the straining modes do not form a direct complement of the nullspace of DTe , until we define an inner product. In other words, one particular spurious mode in a solution, normally involving several elements, does not have to be orthogonal to the rigid body modes. This is explained by realizing that in most cases the elements adjust their position as a function of the amplitude of that spurious mode. The rigid body mode amplitude is obtained when the amplitude of the spurious mode in the solution is zero. Otherwise a polluted value is obtained. The rigid body displacement of an element is written as ueR = U eR 𝚫e , where 𝚫e represents the rigid body displacement parameters, typically translations and/or rotations, while U eR is a matrix of functions describing the displacements induced by these parameters. The corresponding boundary displacement parameters, 𝒗̂ eR , can be expressed as 𝒗̂ eR =  eR 𝚫e .

(6.2)

There are (at least) three different ways to construct this operator: • by directly setting the coefficient (i, j) of  eR equal to the value of 𝑣̂ ei when Δej is equal to one; 1 If it were a direct complement, the intersection of the two subspaces would be empty, so that each element of the nullspace could be written uniquely as the sum of a rigid body mode and a spurious kinematic mode. This is not the case because when we add a rigid mode to a spurious mode we still have a spurious mode.

Practical Aspects of Kinematic Stability

• by imposing on each side a weak form of the compatibility equation for the trace of U eR on the sides, V e 𝒗̂ e = U eR |Γe 𝚫e , which when weighted by V Te has a unique solution that is equivalent to the strong form, so that  eR = −1 e

∮Γe

V Te U eR dΓ,

where e = ∮Γe V Te V e dΓ; • by recognizing that within the nullspace of DTe only the rigid body motions lead to continuous boundary displacements, so that imposing continuity of the displacements defined by 0De directly produces a definition of  eR . The first approach is simple to implement, when a nodal approximation of the boundary displacements is used, whereas the second one, although more complex is more generally applicable. They both consider a 𝚫e that is selected a priori, unlike the third option in which the physical meaning of the rigid body displacement parameters is not explicitly set. When  eR is known, the rigid body displacement parameters can be obtained from the boundary displacement parameters, 𝒗̂ e , by solving  TeR ̄ e  eR 𝚫e =  TeR ̄ e 𝒗̂ e ,

(6.3)

where ̄ e can be equal to either the identity matrix or to e , in order to minimize either ∥ 𝒗̂ e − 𝒗̂ eR ∥2 or ∮Γe (𝒗e − 𝒗eR )2 dΓ. The problem with the values thus obtained is that they may be polluted by the presence of a spurious kinematic mode in 𝒗̂ e . In order to filter it out, it is necessary to know how that mode globally affects the mesh, or a relevant part of it, so that its effect can be removed from the displacement parameters. Procedures to globally characterize the spurious modes will be discussed in the following Section. Assuming for the moment that the boundary displacement parameters describing the spurious modes affecting the element are given by 𝒗̂ eS = Z e ẑ , with global parameters ẑ known, we can obtain the filtered boundary displacement parameters, which are not affected by the amplitudes of those spurious modes, 𝒗̂ eF = 𝒗̂ e − Z e ẑ ,

(6.4)

leading to unpolluted rigid body displacement parameters. It should be noted that, for a given solution, the parameters of the spurious kinematic modes, ẑ , are global and represent modes over a complete mesh, not just a single element. 6.1.2 Spurious Kinematic and Rigid Body Modes of a Mesh

The determination of the spurious modes, Z, or of their parameters, ẑ , cannot be done for an isolated element, that is, without connection to other elements or supports, since rigid body displacements of an element can be arbitrarily combined into the spurious modes of a mesh as described by the columns of Z. This is the fundamental difference between A, the nullspace of DT , and Z, because in the former matrix the rigid body displacements of the mesh are also included. When the model is properly supported, A and Z are equivalent.

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However, for an element we can still exploit the decomposition from the left singular e vectors in (6.1) of he into ann and its orthogonal complement 𝜀e . We already mentioned in Section 5.1 the possibility of determining the spurious modes of a finite element mesh from the nullspace of the global matrix DT . The definition just given of 0De , which is determined at element level, allows for an equivalent, but more compact form of determining this nullspace, by explicitly removing the space of straining modes, 1De , of each element. Then we have for each element boundary displacements that are driven only by its annihilating parameters: 𝒗̂ e = 0De ẑ eann . Assembling the compatibility equations for all sides of the mesh that do not belong to the static boundary2 leads to a global system of equations, C ẑ ann = 𝟎. This problem is easier to solve because the number of columns of C is smaller than that of D, since neann < nes . The nullspace of C defines ann for the mesh. On a properly supported structure3 the nullspace directly defines the spurious kinematic modes of the mesh when they exist. Hence when the nullspace is empty, the mesh is free from such modes and we can establish that the finite element model is fully stable. Since the calculation of the nullspace using the algebraic procedures presented in Sections 5.2 to 5.4 may not be viable for an arbitrary mesh, it is generally necessary to use a numerical approach. The most numerically stable technique to determine the nullspace of DT or C is via its SVD, which is a rather expensive procedure for a large model.

Alternatively, we can consider the possibility of working with star patches of elements. In particular, for 2D elasticity with approximations of degree greater than 2, we have shown in Section 5.2.3 that spurious modes are always associated with pathological configurations of the star patch of a vertex. In these modes the link of a closed star can normally be considered to be fixed, so that Z is null everywhere except for the sides that are adjacent to the vertex being considered, that is, the internal sides of the star patch. Exceptions occur in the degenerate cases, for example, when adjacent sides of a link are colinear. The spurious mode may therefore be determined from the patch, by considering that the sides of the link are fixed. This can readily be done at the level of a star patch using the results from Section 5.2. Alternatively, as for the global problem, this can be done numerically either by processing the assembled D matrix of the patch or the assembled annihilators, 0De , of the elements into matrix C. The former process should be clear without further explanation, but the latter deserves additional detailing. Each column of the annihilator of each element describes independent sets of boundary displacements that lead to zero equivalent strains. The parameters associated with free sides (that is, for open stars the sides incident with V and with the ends of the link, which are on the static boundary) are not constrained. Compatibility conditions have to be imposed on the remaining set of sides: the displacement parameters associated with sides on the link and with internal sides that belong to the kinematic boundary have to be set to zero, as well as the relative displacements between adjacent elements, always on internal sides. 2 Because on the static boundary the displacements are free. 3 i.e. a structure where the supports prevent the rigid body displacements of the structure as a whole.

Practical Aspects of Kinematic Stability

The nullspace of either matrix, DT or C, corresponds directly to the non-zero components of Z associated exclusively with the patch. The description of Z obtained from this patch-wise approach will provide a complete description of the spurious modes for plane elasticity problems with elements of degree d > 2. However, for lower degrees there are cases in which the spurious kinematic modes propagate beyond a single patch, as shown for example in Figure 5.14. The procedure just described, by fixing the link of each patch, blocks such spurious modes and is therefore unable to detect them. It may be possible to obtain an indication of these situations by determining 0D for a patch with a free link, and by realizing that spurious modes which involve sides connected to V and sides on the link are an indication that at least one of them is propagating beyond the patch. To determine how much these modes can eventually propagate may require the consideration of an arbitrary number of patches, which in the limit extends to the whole mesh. In order to avoid such situations it is recommended that for elements of degree d ≤ 2 special care should be taken in designing the mesh, so as to avoid global spurious modes, as discussed in Section 5.7. It is relevant to recall that, as mentioned in 5.2.4, for d = 2 global spurious modes are blocked provided the mesh contains one stable star patch. For 2D meshes, as already noted in Section 5.2 and illustrated in Figures 5.11 and 5.14, the propagation of spurious modes for degree 2 requires some form of symmetry. Consequently an arbitrary unstructured mesh is unlikely to allow global spurious modes. For degree 1, the spurious modes of closed stars normally involve displacements of the link, so that fixing the link will block them. Hence, in general, the determination of global spurious modes will require a global analysis. Once Z is known, we can determine ẑ from the boundary displacement parameters, by requiring the filtered displacements to be orthogonal to the spurious modes in Z,

̂ ẑ = (Z T Z)−1 Z T 𝒗. and use these values in (6.4).

A number of relevant problems have been raised in this Section, and some of them remain unresolved as yet. We hope that this discussion serves to point directions for future research.

6.2 Blocking the Spurious Modes If Z is known we can add fictitious springs with an arbitrary positive definite stiffness matrix k s , associated with the spurious modes, which leads to a non-zero lower right block, Z k s Z T , of rank nskm , in (4.14). k s will normally be assumed to be diagonal, but that is not a requirement. The equilibrium equations in (4.14) then become D ŝ + Z k s Z T 𝒗̂ = t̂ 0 . By definition Z T D is zero, as well as Z T t̂ 0 , provided the loads are admissible. The equilibrium equations projected onto the spurious modes then become

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Z T Z k s Z T 𝒗̂ = 𝟎. Since the modes are independent, Z T Z is positive definite,4 implying that the boundary displacement parameters verify Z T 𝒗̂ = 𝟎. This states that they are orthogonal to the spurious modes and are, therefore, free from their influence. Furthermore, the modified matrix in (4.14) is now non-singular. This can be perceived by realizing that the rank deficiency of the original matrix is exactly compensated by the new block. The boundary displacements obtained from the solution of the modified system are the same as the filtered boundary displacements, 𝒗̂ F , in the previous Section. They both satisfy (4.14) and are orthogonal to the spurious modes. It should be noted that when either Z is not properly computed, or the loading is inadmissible, the application of this blocking technique will lead to erroneous results. A non-zero value of Z T 𝒗̂ is always an indication of an error.

6.3 An Illustration of the Procedures to Remove Spurious Modes In the previous sections we discussed two alternative approaches to obtain solutions free from the effect of spurious kinematic modes. We now illustrate the result of their application to a simple example. We consider a square, for convenience rotated 45∘ , with all external sides fixed, subject to a uniform horizontal traction acting on its horizontal diagonal, illustrated in Figure 6.1. A mesh with four elements is used, with the internal vertex at the centre of the square. For elements of degree greater than 0 this model has one spurious mode, which is not excited by the prescribed load. In Figure 6.2 we show the spurious mode and both polluted and clean boundary displacements, when approximations of degree 3 are used.5 y

p = 1.00

2.00

x E = 1.00 v = 0.15

2.00

Figure 6.1 Rotated square: Definition of the problem and finite element mesh. 4 It is the identity matrix when Z is obtained from an SVD. 5 It should be realized that although the displaced boundaries corresponding to the spurious modes are straight, the tangential displacements of the sides are not linear, and that there is an overlap of two sides, horizontal or vertical, depending on the arbitrary sign of the spurious mode amplitude.

Practical Aspects of Kinematic Stability

(a)

(b)

(c)

Figure 6.2 Rotated square with approximations of degree 3: Spurious mode, polluted and clean boundary displacements. Notice that the displacements in part (c) are not actually continuous at the vertices, although they appear to be so.

Regarding the polluted boundary displacements, it is crucial to realize that in Figure 6.2b the use of a different solver, or even a different ordering of the variables, will normally lead to different polluted displacements. Concerning the clean boundary displacements, as explained in Section 6.2, the same result is obtained by either filtering the direct solution of (4.14), or by blocking the spurious modes.

6.4 How Do We Recognize Admissible Loads? The loads for hybrid equilibrium models of 2D or 3D continua, in force driven problems, can be prescribed as body forces applied within elements, and/or side/face tractions. In displacement driven problems we can prescribe any initial strains and/or side/face displacements. In the latter case an equilibrated solution is always possible, unlike what happens with ‘standard’ displacement models which cannot find a solution for problems with discontinuous displacements. We must note, nevertheless, that whenever the imposed boundary displacements correspond only to a spurious mode, the equilibrated solution will have zero stresses – a poor approximation which could still be used to bound the solution. Admissible body forces must of necessity be described by element-wise polynomials of degree not greater than (d − 1), and the degree of either the projection on the boundary of the initial stresses or of the applied tractions cannot be larger than the degree of the projection of S on the boundary. Body forces then require to be locally equilibrated by particular solutions 𝝈 0e of degree T ≯ d in each element. The self-equilibrated tractions,  S ŝ , are admissible by default, T but the component  𝝈 0 may not be admissible. This only happens for modes that involve the sides of more than one star, which can only happen for d ≤ 2. An example of this situation is the mesh in Figure 5.14, or even more simply an isolated pair of adjacent elements. When globally equilibrated body forces that induce alternating moments are considered on each element of these meshes they will excite the spurious mode. Since the boundary tractions of the particular solution are inadmissible we term such body forces inadmissible.

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This situation requires that all sides affected by this mode are unconstrained, consequently it seldom happens in practice. The only case where it must always be considered is the procedure for recovery of equilibrium in star patches, presented in Chapter 8, where small patches of unconstrained elements are used. In all cases this type of inadmissibility can be obviated by using stress approximations of degree 3 or higher. All side/face tractions are admissible in the absence of spurious modes, but when such modes are present, care needs to be taken to avoid an ill-posed problem in which spurious modes would be excited. The general rule is to verify that no work is done for any of the spurious modes. If all the spurious modes in Z are known, as discussed in Section 6.1.2, this verification is immediate and guaranteed. On the other hand, when Z is not known, the potential inadmissibilities are normally associated with discontinuous loads. Then the characterization of admissibility only needs to consider the star patch of each of the vertices where the discontinuities occur. The directions of displacements due to local spurious modes are immediately recognizable for special geometric configurations, for example, those pathological situations illustrated in Figure 6.3 for 2D elements, where we can consider either patch ABC, a pair of elements that have colinear sides AV and VC on a free boundary, or patch ABCD, where four elements form a quadrilateral patch, with their internal sides aligned with the diagonals of the quadrilateral. For both patches, tractions on sides AV and VC are admissible provided the traction components normal to side BV are continuous at the vertex V . Therefore the loading in part (a) is admissible, while that in part (b) is not.

6.5 Quasi-Simplicial Hybrid Elements Created by Hierarchical Mesh Refinement In order to refine a mesh in a non-uniform way, whatever the criterion leading to that operation, it is possible either to remesh the whole domain or to subdivide selected elements in a hierarchical manner. In the former case the stability of the new mesh must be established ab initio, while for the latter the stability conditions can be inferred from the properties of the original mesh, as discussed in this Section. We first note that the topological nature of a hierarchically refined mesh of simplicial elements is such that some of its elements have more than three sides or four faces. This D

A

V

D

C

B

A

V

C

B (a)

(b)

Figure 6.3 Admissible and inadmissible discontinuous tractions on a patch of 2D elements where a spurious kinematic mode is present.

Practical Aspects of Kinematic Stability

Figure 6.4 Hierarchical refinement of a mesh of triangular elements. A C B

is illustrated in Figure 6.4 for 2D, where the number of vertices of some of the ‘triangles’ exceeds three. It is possible to organize the vertices so that those in excess lie on the sides defined by the original vertices. We term such elements as quasi-simplicial. In the hybrid equilibrium finite element model, the stresses are independently approximated inside each element in the standard way, while the approximations of the boundary displacements can be based either on the initial undivided sides/faces, or on the smaller entities created by the subdivision. In order to impose codiffusivity of tractions, the latter option must be followed. It has been observed (Pereira et al., 1999) that this approach to mesh refinement does not lead to any additional spurious modes being invoked. In this Section we address this observation and provide a proof that for two dimensional problems this is always true. However for low degree elements it is possible for a hierarchical refinement to introduce additional spurious modes, which are not excited by a load that is admissible for the initial mesh. We start with an initial mesh, with elements of degree d, for which the load is known to be admissible. This implies that there is at least one stress field 𝝈 0 throughout the mesh which verifies equilibrium. A hierarchical refinement of the mesh using elements of the same or of a higher degree retains 𝝈 0 as a possible stress field in the refined mesh. Internal equilibrium is automatically guaranteed, since the body forces are not changed. Along the original interfaces, which are also present in the refined mesh, the tractions will remain the same, while at the new interfaces we have zero tractions since this stress field is continuous. Hence at least 𝝈 0 equilibrates the loads on the refined mesh. In the case of a non-uniform refinement, extra spurious modes will exist in ‘degenerate’ elements (i.e. the ones that are not equally subdivided along the sides). As an example, consider element C in Figure 6.4 where only one side is subdivided into two. The number of stress fields is unchanged, but the number of independent displacement parameters increases by 2 × (d + 1), which is, for the isolated element, the number of extra spurious modes. However, when d > 2, they will not be able to propagate into an adjacent refined element that is itself stable by virtue of its degree. On the other hand, if we use a low degree, d = 1 say, then extra spurious modes can propagate, but, as previously stated, these will not be excited by the admissible loads. Of course such cases would become more complicated if we use different degrees in different elements. This procedure is effective because in 2D the subdivision of a triangle produces four similar triangles and therefore the quality of the refined mesh is unchanged. However such similarity does not occur in 3D, because the subdivision of a tetrahedron into eight tetrahedra with the same volume is not unique and generally results in tetrahedra with different shapes.

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In any case, although the arguments regarding admissibility of the loading still apply, the question of whether the number of spurious modes is unchanged is more complicated and has not yet been addressed.

6.6 Non-Simplicial Hybrid Elements If we consider a general quadrilateral element for modelling 2D problems, a popular shape in practice for conforming elements, and use the same approximation functions as for the simplicial element, then we can expect the number of spurious modes to increase above three for a single element. Referring to (5.2), it is evident that n𝑣 increases as the number of sides increases, nrbm and ns remain unchanged, and nhyp tends to decrease as more constraints are imposed on the stress fields in h . Hence nskm must increase, and numerical evidence in Ramsay et al. (1997) reports that nskm = 12 for degree d between values of 4 and 10 inclusive. The recognition of admissible tractions becomes more problematic, since more conditions need to be considered over and above the ‘continuity’ ones at the corners as specified in Section 5.2. We may try to reduce nskm by increasing the degree ds of the stress fields, or decreasing the degree d𝑣 of the side displacement fields. However, although stability may be achieved, side tractions will generally lose the property of being completely codiffusive, which is a central aim of hybrid equilibrium models. This can also be used to stabilize simplicial meshes, although at the cost of no longer enforcing complete codiffusivity (Fraeijs de Veubeke, 1973). It is important to notice that when d𝑣 = 0 is used with ds > 0, the solutions do not converge because the equilibrated stresses produce work with the pseudo-rigid body rotations. Nevertheless, as discussed in Section 4.7, there are exceptions to this which can be managed by a judicious selection of stress and displacement subspaces from the complete spaces h and h for a given degree d. An example is to be found in Almeida and Freitas (1992), where a square hybrid element is used with an incomplete quadratic stress subspace h of dimension 8, and a complete linear space h of dimension 16, despite the existence of spurious modes at the element level. Alternatively, a weaker form of equilibrium may be organized by using stress fields in h of degree ds ≥ 1, and displacement fields in h which correspond to degree d𝑣 = 0 for tangential components and degree d𝑣 = 1 for normal components. In this case the resultant forces and moments will be codiffusive, even when the tractions are not.

6.7 A Cautionary Tale of ‘Near Misses’ When spurious modes are induced by the geometrical configuration of a mesh, certain patterns of loading become inadmissible. If such loads are of interest, then it might be assumed that it is only necessary to shift the positions of the vertices by small amounts in order to stabilize the mesh and obtain a solution. While in theory such changes, or ‘near misses’, should result in a stable mesh, in practice the solutions themselves can exhibit their own form of instability, for example, small changes in position can lead to significant changes in strain energy and displacements. The relevance of this, with reference to displacement elements, was pointed out in Irons and Ahmad (1980) ‘... we

Practical Aspects of Kinematic Stability

V

Figure 6.5 Shallow 3-pinned arch. H k1

k2

L

L

h

must beware also of the near-mechanisms. Too little is known to be at all confident. We should take note of the fact that mathematicians are very noncommital over questions of convergence with fine mesh, in the presence of spurious mechanisms.’ The introduction of a spurious mode implies a corresponding extra hyperstatic stress mode according to (5.2), and although this should disappear if a perturbation is made to avoid a spurious mode, a similar non-hyperstatic stress field may exist which can be activated by certain loads in elastic solutions, showing a very flexible mode of deflection similar to a spurious mode. Such a form of instability can be immediately recognized, and very simply demonstrated, for a physical structure which exhibits similar characteristics: a 3-pinned polygonal shallow arch modelled with two bar elements having the same lengths but different stiffnesses, k1 and k2 , as illustrated in Figure 6.5. This structure is generally statically determinate, and if we assume a linear behaviour, the vertical and horizontal deflections at the apex 𝛿𝑣 and 𝛿h due to force V tend to the expressions in (6.5), when the rise h is small compared with the span 2L: ) ( )2 )( ) ( ( V 1 1 V 1 L L 1 . (6.5) 𝛿𝑣 → + and 𝛿h → − + 4 k1 k2 h 4 k1 k2 h On the other hand the deflections due to force H tend to the expressions in (6.6), also for small values of h: )( ) ) ( ( H 1 H 1 1 L 1 𝛿𝑣 → and 𝛿h → + ; (6.6) − + 4 k1 k2 h 4 k1 k2 H when h = 0. (k1 + k2 ) Thus if h > 0, then the arch is statically determinate, and the vertical deflection due to V tends to a very large value as h is reduced, inducing a geometrically non-linear behaviour, which we are not considering here. If the relative stiffnesses of the bars are different, the horizontal deflection due to V and the vertical deflection due to H also tend to large values, though not as quickly as the vertical deflection due to V . On the other hand, the horizontal deflection due to H remains constant for all non-zero values of h. However, when h = 0, the arch becomes a mechanism as regards force V , and a hyperstatic structure as regards force H, with internal forces dependent on the relative stiffnesses of the bars. If the bars have equal stiffness k, then due to force H, 𝛿𝑣 = 0 and 𝛿h = 0.5 H∕k. If the bars have unequal stiffness, then the internal forces in the bars and the value of 𝛿h change abruptly compared to the values when h > 0. V becomes inadmissible when h = 0, but H remains admissible. However we see a sudden increase in horizontal stiffness as the hyperstatic stress field is activated. but 𝛿𝑣 = 0 and 𝛿h =

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Hence we can see that when the geometry of the structure is in close proximity to being a mechanism, the stiffness, measured for a load which would excite the mechanism, becomes very low and is sensitive to the precise position of the node at the apex. Furthermore, the internal forces and deflections can also be very dependent on this position when the stiffness properties are not symmetric. This type of behaviour, which is dependent on the form of the structure, is precisely similar to that which can occur in a mesh of hybrid equilibrium elements. However, in the latter case, it is a consequence of the mathematical layout of the elements of a mesh when its geometric properties are in close proximity to those which induce spurious modes. Figure 6.6 shows a version of the problem used in Section 6.3, modified to illustrate this behaviour: with a uniform traction, either horizontal or vertical, acting on its horizontal diagonal, going from the left vertex up to an arbitrary point, P. In order to guarantee that a strictly equilibrated solution can be obtained using element-wise polynomial approximations, the point where the load ends has to coincide with a vertex of the mesh. For the simple meshes with four and five elements this is either V or, for the case of the load acting on the whole diagonal, the right vertex of the square. When V is located at the centre of the plate, the corresponding mesh with four elements is the same as in Section 6.3, with one spurious kinematic mode, already shown in Figure 6.2a. As explained in the first part of this Section, this mode is excited by any horizontal traction that is discontinuous at V ; in the present case this means that P coincides with the centre of the square. On the other hand, the mesh with five elements is always stable. The variation of the strain energy of the solution obtained with hybrid equilibrium and with compatible elements of degrees 1, 3 and 6, as a function of the horizontal position of V , is presented in Figures 6.7, 6.8 and 6.9 for three distributions of tractions. It is clear from these plots that although the convergence from above/below is strictly verified, the quality of the equilibrated solutions obtained from the meshes with four elements that are ‘close’ to being unstable is, to put it mildly, poor when the discontinuous horizontal load is considered, thus exposing the problem of having a model that is ‘barely stable’ together with an action that excites its weakness. However, it is also clear that for the other load cases, which are admissible even when P or V is central, the equilibrated solutions are quite good. y

p = 1.00

2.00

x

P

V

V

E = 1.00 v = 0.15

2.00

Figure 6.6 Rotated square: definition of the problem and simple finite element meshes with 4 and 5 elements.

Practical Aspects of Kinematic Stability

Energy

0.40

Equilibrated 4 Elements 5 Elements

1

3

6

6

3

1

1

0.20

Compatible 4 Elements 1 0.00 –1.00

0.00

1.00

Position

Figure 6.7 Horizontal load applied to the left of P; strain energy of the solution for the simple meshes, as a function of the position of V, using elements of degrees 1, 3 and 6.

Energy

0.20

1

Equilibrated 4 Elements 5 Elements

0.10

1 Compatible 4 Elements

1 0.00 –1.00

0.00

Position

1.00

Figure 6.8 Vertical load applied to the left of P; strain energy of the solution for the simple meshes, as a function of the position of V, using elements of degrees 1, 3 and 6.

On a marginal note we can say that these results also illustrate the well known fact that compatible elements with linear displacement approximations misbehave when distorted. Other than that, compatible models appear to be immune to problems in every situation tested.6 6 The compatible model would fail, for example, at representing non-homogeneous kinematic boundary conditions that are incompatible, for example, when the displacements of the supports are discontinuous at a vertex.

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0.40

Energy

124

Equilibrated 4 Elements 5 Elements

0.20

1

0.00 –1.00

Compatible 4 Elements

0.00

Position

1.00

Figure 6.9 Horizontal load applied across the whole diagonal of the plate; strain energy of the solution for the simple meshes, as a function of the position of V, using elements of degrees 1, 3 and 6.

When vertex V is located exactly at the intersection of the diagonals and the load is admissible, a reduction in the strain energy of the solution may exist only for that position. This is due to the presence of an additional hyperstatic stress field, which can be exploited by the model to improve compatibility. The possible existence of equilibrium models for which the results ‘explode’ is clearly a negative point that affects such models. Generally it can be avoided by using finite element meshes that are not close to having spurious kinematic modes that can be excited by the applied loads. Spurious modes are problematic either when they are real, or when they would become real for a small change in the geometry of the mesh. This is not a concern when the loading is continuous, since such modes cannot then be excited. However, it is still necessary to guarantee that the technique used to solve the system of equations can deal with a singular matrix.7 For arbitrary loads, for example in dynamical problems wherein by virtue of the inertial effects ‘the structure creates the load’, the ‘near misses’ tend to be exaggeratedly excited. In such cases the influence of the ‘barely stable’ and thus ‘extra flexible’ modes may lead to erroneous results.8 These modes can be removed either by subdividing the critically located elements or by globally refining the mesh. The good news is that when a uniform refinement is considered, the refinement of a stable (even if barely) mesh with four elements always converges to the exact solution at the same rate, independent of the position of V , as shown in Figure 6.10. As V gets closer to the centre of the plate, the error of the solution obtained for the initial mesh is larger, but as the mesh is refined, convergence is guaranteed. Note that in each stage of refinement an element is subdivided into four new elements which create new nodes but retain 7 It is always a good idea to verify that the loading is admissible and that a proper solution has been obtained by checking that ∥ Ax − b ∥≈ 𝟎. 8 The model is very flexible, as shown in Chapter 10, leading to very low eigenfrequencies.

Practical Aspects of Kinematic Stability

Load Partial

Partial

Partial

Total

Position of V Equilibrated Compatible @0.1 Degree 1 Degree 3 Degree 6 @0.01 Degree 1 Degree 3 Degree 6 @0 Degree 1 Degree 3 Degree 6 @0.01 Degree 1 Degree 3 Degree 6

k = 1.95 k = 2.00 k = 2.00

k = 1.76 k = 2.01 k = 2.00 1e+00

k = 1.96 k = 2.00 k = 2.00

k = 1.75 k = 2.01 k = 2.00

k = 1.89 k = 2.00 k = 2.00

k = 1.77 1e – 04 k = 2.01 k = 2.00

k = 2.07 k = 3.13 k = 3.13

k = 1.98 k = 3.13 k = 3.13 1e – 08

Degree 1

Degree 3

Degree 6

0 1 2 3 4 5

0 1 2 3 4 5

0 1 2 3 4 5

Figure 6.10 Rotated square: action of a partial horizontal load. Convergence curves in terms of the strain energy, and corresponding convergence rates, k, for a uniform refinement of the simple finite element meshes. The initial mesh has 4 elements, except for the ‘Partial @0’ solution, where P coincides with the diagonal; in this case the initial mesh has five elements.

the old ones. Consequently, the spurious mode that is excited by the discontinuous load when V is central is retained at all stages, but it only affects the central star patch.

References Almeida JPM and Freitas JAT 1992 Continuity conditions for finite element analysis of solids. International Journal for Numerical Methods in Engineering 33(4), 845–853. Fraeijs de Veubeke BM 1973 Diffusive Equilibrium Models. Lecture notes for the International Research Seminar of the Theory and Application of Finite Element Methods. University of Calgary. Golub GH and Van Loan CF 2013 Matrix Computations. 4th edition, John Hopkins University Press. Irons BM and Ahmad S 1980 Techniques of Finite Elements. Ellis Horwood, Chichester. Pereira OJBA, Almeida JPM and Maunder EAW 1999 Adaptive methods for hybrid equilibrium finite element models. Computer Methods in Applied Mechanics and Engineering 176(1-4), 19–39. Ramsay ACA, Almeida JPM and Maunder EAW 1997 Curious convergence with hypostatic hybrid equilibrium models. Communications in Numerical Methods in Engineering 13(7), 541–552. Strang G 1988 Linear Algebra and its Applications. Harcourt Brace Jovanovich.

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7 A Variational Basis of the Hybrid Equilibrium Formulation In this text we follow an approach based on using the concepts of equilibrium, compatibility and the assumed constitutive relation as the starting point for the presentation of the formulations being considered. Nevertheless, it would be unreasonable not to look at the problem from the viewpoint of energy methods. In particular because it must be recognized that their more abstract nature, together with the generality of the tools of functional analysis, provide a simpler way to obtain a clear answer to the questions of whether and how the solutions converge, as well as an alternative explanation to the questions of existence and uniqueness of the solution for a given model. The role of this Chapter is to present such an approach. The mathematical language may be more abstract, but physical interpretations will always be provided. We start by revisiting the classical results related to the potential and the complementary potential energy, before presenting the generalized complementary principle that is related to the hybrid equilibrium formulation, and studying its properties. Elastic, but not necessarily linear, materials and infinitesimal displacements are assumed.

7.1 Potential Energy and Complementary Potential Energy The potential energy of a mechanical system, already introduced in Chapter 2, is a functional of the displacement field u (Washizu, 1982): Π(u) = U(𝜺(u)) − V (u); 𝜺(u)

U(𝜺(u)) = =

∫Ω ∫𝜺T ∫Ω

𝝈(𝝐)T d𝝐 dΩ;

V (u) =

∫Ω

T b̄ u dΩ +

∫Γt

T t̄ u dΓ;

W (𝜺(u)) dΩ;

where W is the strain energy density, U is the total strain energy and V is the work done by the applied forces. Similarly, the complementary potential energy of a mechanical system is a functional of the stress field 𝝈: Πc (𝝈) = Uc (𝝈) − Vc (𝝈); 𝝈

Uc (𝝈) = =

∫Ω ∫0 ∫Ω

𝜺(𝝇)T d𝝇 dΩ;

Vc (𝝈) =

T

∫Γu

( 𝝈)T ū dΓ;

Wc (𝝈) dΩ;

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

A Variational Basis

where Wc is the complementary strain energy density, Uc is the complementary strain energy and Vc is the work done by the imposed displacements. For an arbitrary strain field, 𝜺, and for its associated stress, 𝝈(𝜺), the sum of the strain energy densities W (𝜺) + Wc (𝝈(𝜺)) is always equal to 𝜺T 𝝈(𝜺). Similarly, for an arbitrary stress field, 𝝈, and for its associated strain, 𝜺(𝝈), we have that W (𝜺(𝝈)) + Wc (𝝈) is always equal to 𝝈 T 𝜺(𝝈). For a linear elastic constitutive relation, which we consider in most of this text, the strain energy densities take the following form: 1 1 Wc (𝝈) = 𝝈 T f 𝝈 + 𝝈 T 𝜺T . W (𝜺(u)) = (𝜺 − 𝜺T )T k (𝜺 − 𝜺T ); 2 2 This implies, as stated, that W (𝜺) + Wc (k 𝜺) = 𝜺T k(𝜺 − 𝜺T ),

and Wc (𝝈) + W (f 𝝈) = 𝝈 T (f 𝝈 + 𝜺T ).

Both potential energies can be written using the same compact notation for the functional: 1 Ξ(g) = a(g, g) − p(g); 2 where a(., .) is a bilinear form,  ×  → ℜ, and p(.) is a linear form,  → ℜ. The following definitions are obtained: ak (u, 𝒗) = pk (u) =

𝜺(𝒗)

∫Ω ∫0 ∫Ω

𝝈(𝜺(u))T d𝝐 dΩ;

T b̄ u dΩ +

∫Γt

T t̄ u dΓ;

1 a (u, u) − pk (u) 2 k for the Potential Energy functional, and Π(u) =

as (𝝈, 𝝉) = ps (𝝈) =

𝝉

∫Ω ∫0

𝜺(𝝈)T d𝝇 dΩ; T

∫Γu

( 𝝈)T ū dΓ;

(7.1) (7.2) (7.3)

(7.4) (7.5)

1 (7.6) a (𝝈, 𝝈) − ps (𝝈) 2 s for the complementary potential energy functional. The point(s) corresponding to the stationarity conditions of such a functional, satisfy the following condition (e.g. Hughes (2000)): Πc (𝝈) =

a(g, h) = p(h),

∀h ∈  .

To apply the stationarity condition to the potential energy functional we introduce two sets of kinematically admissible displacement fields: • k , whose members satisfy the continuity conditions necessary for the existence of an integrable strain field and the boundary conditions on Γu . • k0 , whose members are also associated to integrable strains, and, furthermore, verify homogeneous boundary conditions on Γu . This set constitutes a vector space because any linear combination of its members also belongs to the set.

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Equilibrium F.E. Formulations

The stationarity of the potential energy, for any kinematically admissible variation of the displacement field, which is necessarily a member of k0 , corresponds to the weak form of the equilibrium conditions of the system. Find u ∈ k such that: ak (u, 𝒗) = pk (𝒗) ∫Ω

𝝈(𝜺(u))T ( 𝒗) dΩ =

∫Ω

∀𝒗 ∈ k0

T b̄ 𝒗 dΩ +

∫Γt

T t̄ 𝒗 dΓ

∀𝒗 ∈ k0

(7.7)

Integration by parts of (7.7), where the term on Γu cancels because 𝒗 belongs to k0 , shows that this equation implies both domain and boundary equilibrium: 1 − ( ⋆ 𝝈(𝜺(u)))T 𝒗 dΩ = b̄ 𝒗 dΩ − ( 𝝈(𝜺(u)))T 𝒗 dΓ ∫Ω ∫Ω ∫Γi ∪Γt T

+

∫Γt

T

T t̄ 𝒗 dΓ

∀𝒗 ∈ k0

Similarly, to apply the stationarity condition to the complementary potential energy functional, we introduce two sets of statically admissible stress fields: • s , whose members equilibrate the prescribed body forces, and prescribed tractions on Γt . • s0 , whose members are in equilibrium with zero body forces and tractions on Γt . This set constitutes a vector space because any linear combination of its members also belongs to the set. It defines the hyperstatic stress distributions of the model. The stationarity of the complementary potential energy, for any statically admissible stress variation, corresponds directly to the weak form of the compatibility conditions of the system. Find 𝝈 ∈ s such that: as (𝝈, 𝝉) = ps (𝝉)

∀𝝉 ∈ s0 T

∫Ω

ū T  𝝉 dΓ 𝜺(𝝈)T 𝝉 dΩ = − 𝜺TT 𝝉 dΩ + ∫Ω ∫Γu

∀𝝉 ∈ s0

(7.8)

This equation imposes the condition that the equivalent strains obtained from 𝝈, the left hand term, must be equal to the equivalent strains due to the initial strains and to the imposed displacements, the right hand term. Alternatively this equation may be obtained by integrating by parts the weak form of the compatibility equation (A.1), where we omit, for simplicity, the initial strains, and where the term on Γt cancels, since 𝝉 belongs to s0 , as does the term in Γi , because for an equilibrated stress field the projection of the stresses on any internal interface is,

1 In the resulting equation internal boundaries, Γi , are introduced to account for potentially discontinuous stress fields, due to either discontinuous material properties or discontinuous strains.

A Variational Basis

by definition, balanced: ∫Ω that is,

𝜺(𝝈)T 𝝉 dΩ =

∫Ω

(u)T 𝝉 dΩ,

∀𝝉 ∈ s0 ;

𝜺(𝝈)T 𝝉 dΩ = − uT ( ⋆ 𝝉) dΩ + uT  𝝉 dΓ, ∫Ω ⏟⏟⏟ ∫Γ T

∫Ω

∀𝝉 ∈ s0 ;

=0

then

∫Ω

𝜺(𝝈)T 𝝉 dΩ =

T

∫Γu

ū T  𝝉 dΓ,

∀𝝉 ∈ s0 .

Equations (7.7) and (7.8) express what we have termed in Chapter 2 the Galerkin orthogonality conditions. 7.1.1 Existence and Uniqueness of Solutions

Knowing the conditions for the existence of solutions, and whether they are unique or not, is a crucial point. This was implicitly addressed in previous chapters, because of the existence of spurious kinematic modes, and is now presented for solutions based purely on displacements. If the quadratic form a(g, g) is positive definite, that is, a(g, g) > 0, ∀g ≠ 0, the stationary point of Ξ is unique and is a minimum. In case it is strictly positive semi-definite, that is, a(g, g) ≥ 0, ∀g and there is (at least) one g ≠ 0 for which a(g, g) = 0, the problem may have no stationary points. If it has, they are multiple, all corresponding to a minimum of Ξ(g). When defined in terms of the strains, the quadratic form ak (., .) is positive definite. When the displacements are used, it must be considered that the space of kinematically admissible displacements may include rigid body movements, for which ak (urbm , urbm ) = 0, although urbm ≠ 0. This situation allows us to illustrate the condition for the existence and uniqueness of a minimum of the functional. A solution exists provided pk (urbm ) = 0, that is, the loading must not excite the rigid body movement(s) or, in other words, the loading must be self-equilibrated in the direction of the possible rigid body movement(s). Otherwise the work done by the applied forces changes with a rigid body movement, while the strain energy does not, and a stationary point does not exist. If the existence condition is satisfied, the solution is multiple because adding an arbitrary linear combination of the possible rigid body movements to a solution does not change its energy, both in terms of the strain energy, which by definition is not changed by a rigid body movement, and of the work done by the applied forces, which is also unchanged because of the existence condition. In practice this situation is generally avoided by excluding the rigid body movements from the space of kinematically admissible displacements, that is, by requiring that the structural model is properly supported. 7.1.2 Properties of the Exact Solution

By expanding the definitions of the functionals we verify that, for the exact solution, the total energies sum to zero, that is, Π(u) + Πc (𝝈) = 0: Π(u) + Πc (𝝈) = U(𝜺(u)) − V (u) + Uc (𝝈) − Vc (𝝈);

129

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Equilibrium F.E. Formulations

=

∫Ω

=

∫Ω

=

∫Ω = 0.

𝜺(u)T 𝝈 dΩ − 𝜺(u)T 𝝈 dΩ + 𝜺(u)T 𝝈 dΩ −

∫Ω ∫Ω ∫Ω

T b̄ u dΩ −

T

∫Γ

( 𝝈)T u dΓ;

( ⋆ 𝝈)T u dΩ − 𝝈 T (Du) dΩ +

T

∫Γ

( 𝝈)T u dΓ; T

∫Γ

( 𝝈)T u dΓ −

T

∫Γ

( 𝝈)T u dΓ;

Therefore, for the exact solution, the potential energy is equal to the negative of the complementary potential energy. For force driven problems, where the imposed displacements and the initial strains are zero, the complementary potential energy is equal to the complementary strain energy, which depends only on the stress. In this situation, provided the complementary strain energy density is a positive definite functional (Uc (𝝈) > 0, ∀𝝈 ≠ 0), the exact solution corresponds to a minimum of the complementary potential energy. Similarly, for displacement driven problems, where the body forces and the applied tractions are zero, the potential energy is equal to the strain energy. Provided the strain energy density is a positive functional (U(𝜺) > 0, ∀𝜺 ≠ 0), the exact solution corresponds to a minimum of the potential energy. For mixed problems, where both forces and displacements are imposed, no assumptions can be made, a priori, regarding the bounding properties of the solutions. 7.1.3 The Formal Relation Between Both Energies

The potential energy and the complementary potential energy are dual functionals in the sense of the Legendre–Fenchel transformation (Boffi et al., 2013; Ciarlet et al., 2011). Formally, given a convex functional G(𝑣) which maps 𝑣 ∈  onto ℜ, and a space  ′ , dual of , its Legendre transformation G⋆ (𝑣⋆ ), with 𝑣⋆ ∈  ′ , is given by: ) ( G⋆ (𝑣⋆ ) = sup ⟨𝑣, 𝑣⋆ ⟩ − G(𝑣) . 𝑣∈

Since the inclusion of a particular solution does not affect the outcome, for our problem we assume that  corresponds to k0 (𝑣 becomes u), whereas  ′ corresponds to s0 (𝑣⋆ becomes 𝝈). We start with the primal functional defined as G(u) = Π(u). When we define the projection between the dual spaces to correspond to the product between strains and stresses: ⟨u, 𝝈⟩ =

∫Ω

(u)T 𝝈 dΩ =

∫Ω

𝜺T 𝝈 dΩ;

and substitute in the definition of the Legendre transformation: ) ( G⋆ (𝝈) = sup 𝜺T 𝝈 dΩ − Π(u) , u∈ 0 ∫Ω k

we verify after an integration by parts that G⋆ (𝝈) = ΠC (𝝈). It is also possible to follow the opposite path, from complementary to potential energy. In this case the projection between dual spaces involves  ⋆ , which transforms stresses into body forces, so that we have a product between forces and displacements.

A Variational Basis

7.2 Hybrid Complementary Potential Energy The approximation of stresses used in the hybrid equilibrium formulation ensures a priori that the relevant conditions are verified inside each element, but does not consider equilibrium between elements and on Γt , which needs to be imposed a posteriori, so that the resulting stress field becomes a member of s . Actually, such an approximation forces the stress field inside each element to belong to a finite dimensional subspace of the space of all stresses that verify equilibrium without body forces or imposed tractions on its boundary. A stress field that verifies equilibrium internally and at every boundary is said to be codiffusive. It is always worth repeating that codiffusivity is not equivalent to continuity. Along an arbitrary given interface, without locally applied tractions, the former only implies continuity of the projection of the stress tensor on that interface, while the latter implies that all components of the tensor must be continuous. This implies that discontinuous stress fields may be codiffusive. Since the stress approximations verify equilibrium, and are continuous inside each element, in order to apply the principle of minimum complementary energy it is necessary to enforce codiffusivity where it is lacking: at the boundary between elements, Γi , and at the static boundary Γt . This is achieved by introducing Lagrange multiplier functions, 𝝀(𝝃), at these interfaces and adding their inner products with the conditions to be imposed to the complementary energy functional, resulting in a new generalized complementary functional: ΠGc (𝝈, 𝝀) = Uc (𝝈) − Vc (𝝈) ±

𝝀T ( 𝝈 − t̄ ) dΓ. T

∮Γe ∖Γu

(7.9)

Its stationary point(s) verify a weak form of compatibility, as for the original functional, and at least a weak form of codiffusivity. On a generic internal boundary, i, joining two elements, e and m, the condition being T T imposed can also be written as  e,i 𝝈 e −  e,i 𝝈 m − t̄ = 0, because the outward normal to element m is always equal and opposite to the normal of the adjacent element, e. The Lagrange multiplier functions, 𝝀(𝝃), represent vector quantities at each interface between elements, Γi , or external boundary with Neumann boundary conditions, Γt . Since the inner product of these quantities with the boundary tractions is an energy, they can be interpreted as generalized boundary displacements, 𝒗. Although the sign of the Lagrange multiplier term is arbitrary, we use the negative sign so that all terms on the boundary have the same sense, corresponding to minus the work done by boundary tractions. We set this work to zero, for arbitrary displacements, in order to minimize the lack of equilibrium. Formally there are no constraints on what are admissible 𝝀s or their variations. Any (infinite) basis that spans all possible boundary displacements can, in principle, be used. Nevertheless, in order to guarantee that a finite number of variations of these displacements locally enforces codiffusivity for the projection of the approximated stresses, which are also obtained from a basis with a finite dimension, the problem must be approached with special care. The tractions are continuous on each boundary entity, 2 and because of the discontinuity of the normal they are, in general, discontinuous between them. Therefore the option 2 On each extremity for 1D elements, on each side for 2D elements and on each face for 3D elements.

131

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Equilibrium F.E. Formulations

of selecting variations that are continuous on each interface, but can be discontinuous between them, is consistent with this behaviour, leading to equilibrium equations that are independent for each boundary. Furthermore, under appropriate conditions, as we have seen in the derivation of the hybrid equilibrium formulation, these independent equations allow us to achieve the goal of defining strictly codiffusive stress fields.

7.3 Properties of the Generalized Complementary Energy The generalized functional in (7.9) can be written as 1 a (𝝈, 𝝈) − bs (𝝈, 𝒗) − ps (𝝈) + qs (𝒗), 2 s where 𝝈 and 𝒗 belong, respectively, to: ΠGc (𝝈, 𝒗) =

• the set of element-wise statically admissible stresses, se , that equilibrate the applied body forces inside each element; • the set of kinematically admissible boundary displacements, kb , that satisfy the boundary conditions on Γu , and the additional forms are given by: bs (𝝈, 𝒗) =

T

∫Γe ∖Γu

𝒗T  𝝈 dΓ and qs (𝒗) =

∮Γe ∖Γu

𝒗T t̄ dΓ.

We start by noting that this functional corresponds to a saddle point problem, that is, its stationary points do not correspond to a maximum or to a minimum. Effectively, though as (𝝈, 𝝈) is positive definite, the term bs (𝝈, 𝒗) has no defined sign, therefore if at a stationary point a given variation (𝛿𝝈, 𝛿𝒗) increases the value of the functional, then the variations (−𝛿𝝈, 𝛿𝒗) and (𝛿𝝈, −𝛿𝒗) will necessarily decrease it. In the sets of functions introduced, se is a superset of s and kb is a superset of the projection of k onto the boundaries of the finite element mesh, its trace. These functions can have stronger discontinuities, since the stresses are not required to be codiffusive between elements, and the displacements are not continuous between boundaries. The sets corresponding to such functions are said to be broken. For the definition of the stationarity conditions of the generalized functional, the subsets of se and kb corresponding respectively to zero body forces, that is, to self-equilibrated stresses, s0e , and to homogeneous boundary displacements on Γu , k0b , are introduced. These sets form broken vector spaces. The stationarity conditions for this problem are then: Find (𝝈, 𝒗) ∈ se × kb , such that: { as (𝝇, 𝝈) − bs (𝝇, 𝒗) = ps (𝝇), ∀𝝇 ∈ s0e ; (7.10) −bs (𝝈, 𝒘) = −qs (𝒘), ∀𝒘 ∈ k0b , where as (𝝇, 𝝈) is the work done by the strains corresponding to 𝝇 with 𝝈; bs (𝝇, 𝒗) is the work done by the boundary displacements 𝒗 with the trace of 𝝇; ps (𝝇) is the work done by the imposed displacements with the trace of 𝝇; and qs (𝒗) is the work done by the boundary displacements 𝒗 with the applied tractions.

A Variational Basis

The first set of equations imposes a weak form of compatibility, which corresponds basically to the element-wise application of the stationarity of the potential complementary energy, with the displacements of the element boundaries contributing to the equivalent generalized strain. In these equations the elements that share a boundary are connected via the boundary displacements. To understand why the variations are considered on s0e , and not on se , a point that might have been considered earlier, we could just invoke the integration by parts following (7.8), where  ⋆ 𝝇 is required to be zero. The corresponding physical interpretation is that the work of any equilibrated stresses with the strains corresponds to the integral of the displacements weighted by the body forces that correspond to those stresses. For zero body forces the domain displacements are removed from the equation, resulting in an equation where all the contributions to an average displacement that is known to be nil are required to balance. We note, nevertheless, that similar equations, with stresses that correspond to non-zero body forces, are used to obtain average displacements. The second set of equations imposes a weak form of equilibrium on the element boundaries, which, when the projection of the stresses on the boundary plus the applied tractions and the displacements are both polynomial functions of degrees dt and d𝑣 , respectively, corresponds to the strong form of equilibrium, provided dt ≤ d𝑣 . From the viewpoint of kinematics it is obvious that the displacements are fixed on Γu , thus justifying the choice of k0b for the test functions. The complementary reasoning is, in our opinion, easier to understand: since the reactions on Γu always balance the existing tractions, the corresponding equations do not need to be considered.

7.4 The Babuška–Brezzi Condition and Hybrid Equilibrium Elements The Babuška–Brezzi condition defines in a very general way the conditions for a solution of a saddle point problem to exist, whether that solution is unique and how a given approximation technique will converge to it (Boffi et al., 2013). Without going into details we will try to explain in simple terms how it applies to our formulation, the first point being to recognize that the use of the generalized complementary energy functional leads to a saddle point problem, the principal feature of which is that the solutions correspond to stationary points, where the functional is neither a minimum or a maximum. The genesis of this idea is partly related to equilibrium elements, namely via Brezzi (1974), where the strongly diffusive elements of Fraeijs de Veubeke are often referred to as important examples. The most noted inequality that is considered in this condition, when adapted to our formulation becomes: inf

0b 𝝎∈k0b ∕ann

sup

𝝇∈s0e

bs (𝝇, 𝝎) ≥ 𝛽 > 0, ||𝝇|| ||||𝝎∥𝜀

(7.11)

where we must note that our solution is sought in the product of two sets of functions se × kb , whereas in equation (7.11) it is required that the functions must belong to two vector spaces. This is achieved by considering particular solutions that satisfy the non-homogeneous terms, which when included in (7.10), will induce additional terms on the right hand sides.

133

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Equilibrium F.E. Formulations

In a nutshell, inequality (7.11) requires that it is always possible to have positive work done by the projection on the boundary of the stresses with the boundary displacements. Following Boffi et al. (2013) (Theorems 4.2.3 and 4.2.4) we know that the condition in (7.11) has to be supplemented by the following properties: • as must be coercive, that is, as (𝝇, 𝝇) ≥ 𝛼 ∥ 𝝇∥2 , with 𝛼 > 0; • for a solution to exist either ‘Im B = Q∗ ’, or ‘Im B ≠ Q∗ but g ∈ Im B’. For our models the first property is verified, as discussed in 4.7. The second one, which still needs to be explained, may be easily read in terms of our concepts of spurious kinematic modes and admissibility of the loads. In our terms Q∗ corresponds to the space of boundary tractions h , dual to h , while ‘Im B’ corresponds to the projection of the internal stresses on the boundary, that is, to the admissible tractions. It is immediately recognizable that requiring ‘Im B = Q∗ ’, that is, the equivalence of those two spaces, corresponds to requiring the absence of spurious kinematic modes. The alternative condition ‘Im B ≠ Q∗ ’, with ‘g ∈ Im B’, just corresponds to stating that in the presence of spurious kinematic modes, the loads must belong to the space of admissible tractions (Im B) for a solution to exist. In this case the solution for displacements is not unique, because of the possible inclusion of arbitrary elements of ann . Coefficient 𝛽 in (7.11), together with a similar 𝛼 obtained from as are used to determine the convergence rates of the approximate solutions.

References Boffi D, Brezzi F and Fortin M 2013 Mixed Finite Element Methods and Applications. Springer. Brezzi F 1974 On the existence, uniqueness and approximation of saddle-point problems arising from Lagrangian multipliers. ESAIM: Mathematical Modelling and Numerical Analysis-Modélisation Mathématique et Analyse Numérique 8(R2), 129–151. Ciarlet PG, Geymonat G and Krasucki F 2011 Legendre-Fenchel duality in elasticity. Comptes Rendus de l’Académie des Sciences, Paris, Série 1 349(9 – 10), 597–602. Hughes TJR 2000 The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Dover Publications. Reddy JN 1986 Applied Functional Analysis and Variational Methods in Engineering. McGraw-Hill New York. Washizu K 1982 Variational Methods in Elasticity and Plasticity 3 edn. Pergamon Press.

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8 Recovery of Complementary Solutions It has long been recognized that, in the absence of exact solutions, knowledge of feasible complementary solutions, that is, one with conforming strains and one with statically admissible stresses, enables judgements to be made about the quality of either one of them. A pair of complementary solutions can always be obtained from independent global analyses of conforming and equilibrating finite element models. However, this may involve considerable computational effort. Much research has been focused on formulating procedures to recover complementary solutions more efficiently, for example, Ladevèze et al. (1991) presented an element-wise procedure to recover an equilibrating stress field from a global compatible solution, and Pereira et al. (1999) proposed a dual form of procedure to recover a compatible displacement field from a global equilibrating solution based on a hybrid equilibrium model. In this Chapter we will consider procedures, based on the analysis of localized subdomains in the form of star patches of elements, to recover one form of solution from its complement. The essence of these recovery procedures lies in the use of ‘partition of unity’ functions that are associated with the subdomains, whose extents are limited to where the functions are non-zero. The functions act as weighting functions for fields of displacements, strains, stresses and loads so that these quantities become localized distributions that are only non-zero within star patches. The summation of each quantity over all overlapping patches recovers the total quantity. Basic ideas are introduced and discussed in the context of 1D and 2D continua and simplicial elements, though the concepts can be extended to 3D continua (Almeida and Maunder, 2010), as well as plates, and to other non-simplicial forms of element albeit with extra complexity. Reference should be made to Almeida and Maunder (2009) and to Maunder and Almeida (2012) for detailed presentations of the recovery of equilibrium from compatible solutions, which include the case of 3D continua. In this Chapter we will again refer to topological entities relevant to simplicial elements, for example, the link of a star, and we recall that such entities are defined in Section 5.2.3. In Section 8.1 we define partition of unity functions and summarize their general features. In particular we explain the derivation and significance of quantities which we term fictitious. In Section 8.2, we derive a kinematically admissible continuous displacement field uC from a hybrid solution consisting of an equilibrated stress field 𝝈 E , and a displacement field 𝒗E at the element interfaces and the external boundaries, which is Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

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generally incompatible at the vertices. In Section 8.3, we derive a statically admissible stress field 𝝈 E from a compatible solution with a displacement field uC .

8.1 General Features of Partition of Unity Functions A set of partition of unity (PU) functions consists of one function Ψi associated with each vertex i of a mesh. Each function is C 0 continuous, zero outside the closure of star patch i, and has unit value at vertex i. These functions sum to unity at every point, and as a consequence, the first derivatives of the functions in any particular direction sum to zero at any point, that is, these derivatives form a partition of nullity. We will use the most obvious choice of function for simplicial elements: the union of the linear interpolation/shape functions associated with a vertex and its adjoining elements. This function is illustrated in Figure 8.1 for a closed star patch. The function in element e will be denoted by Ψei . We apply these PU functions as weighting functions to a pair of global displacement and stress fields (u, 𝝈). By doing so we obtain partitioned fields, each of them restricted to a localized subdomain defined by its star patch. We want to use the derivatives of these partitioned quantities so that their sum satisfies, at least in a weak form, the original equation of compatibility or equilibrium. To do so it becomes necessary to include terms which, because they are not present in the real problem, we call fictitious. These fictitious quantities, which can be seen as a by-product of the PU of the solutions, necessarily cancel out when the local solutions are summed. This is illustrated for a uniform 1-dimensional domain with the aid of Figure 8.2. In this illustration we assume that the axial displacement u or stress 𝜎 varies linearly over the patch associated with vertex i. Then the corresponding derivative represented by the axial strain 𝜀, or the stress gradient, is constant. The latter is the negative of the equilibrating body force b. These quantities are shown in Figures 8.2a and 8.2b, respectively. When the partition of unity function Ψi , shown in Figure 8.2c, is applied to them, we obtain piecewise quadratic or piecewise linear fields as in Figure 8.2d or 8.2e. We now show that the weighted derivatives are not equal to the derivatives of the weighted functions. Let us first consider the strain fields 𝜀i = Ψi 𝜀(u) and 𝜀(Ψi u) as defined by: dui dΨi d du dΨi 𝜀(Ψi u) = = (Ψi u) = Ψi + u = Ψi 𝜀(u) + u, dx dx dx dx dx that is, dΨi 𝜀i = Ψi 𝜀(u) = 𝜀(Ψi u) − u (8.1) dx Figure 8.1 A partition of unity function over a closed star patch.

i

Recovery of Complementary Solutions

u or σ

a

a – gh

ε or –b

a + gh

g

g h

h

h i (a) Assumed field over a patch.

g h

i

(b) First derivative of the assumed field. Ψi 1 h

h i (c) Partition of unity function for the patch. ui = Ψiu or σi = Ψiσ

a

g

h i (d) Partition of the assumed field. h

a h

–g

a h

h i (e) Partition of the first derivative. h

dσi dui or dx dx

+g

a h

– ha + g i

εi = Ψiε or –bi = –Ψib

– ha – g

(f) Derivative of the partitioned field.

–g

a h

dΨi dΨi σ u or dx dx

– ha

– ha – g

i (g) Fictitious derivatives.

Figure 8.2 A partition of unity function applied to a 1-dimensional patch.

The strain fields for this example are plotted in Figures 8.2e to 8.2g. They confirm the need to subtract the fictitious strains in 8.2g from those in 8.2f in order to obtain the correct partitioned strain field 𝜀i in 8.2e. In other words, when we fix the link, the strains in the patch require the fictitious strains in 8.2g to supplement the partitioned ∑ strains in 8.2e. The total strain field is recovered in the sum 𝜀 = i 𝜀i . Similarly let us consider the equilibrating body force fields bi = Ψi b (𝜎) and b (Ψi 𝜎) as defined by: b (Ψi 𝜎) = −

d𝜎i dΨi d d𝜎 dΨi = − (Ψi 𝜎) = −Ψi − 𝜎 = Ψi b(𝜎) − 𝜎 dx dx dx dx dx

that is, dΨi 𝜎 (8.2) dx Figures 8.2e and 8.2f now represent the distributions of stress gradients which define the negative of the equilibrating body forces. They confirm the need to add the fictitious body forces in 8.2g to those equilibrated with 𝜎i implied in 8.2f, in order to obtain the correct partitioned body forces in 8.2e. The body forces implied in 8.2f reflect the effect bi = Ψi b (𝜎) = b (Ψi 𝜎) +

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Equilibrium F.E. Formulations

of weighting the patch stresses 𝜎i in order to equilibrate with zero tractions on the link. It should thus be appreciated that these body forces are self-balanced. The total body ∑ forces are recovered in the sum b = i bi . These concepts of fictitious strains and body forces, as expressed in (8.1) and (8.2), are extended to star patches of elements in 2D or 3D models in (8.3) and (8.4): i) When the strain field in a star patch is derived from weighted displacements: 𝜺(Ψi u) = (Ψi u) = Ψi u + (Ψi )u = Ψi 𝜺(u) + (Ψi )u, that is, 𝜺i = Ψi 𝜺(u) = 𝜺(Ψi u) − (Ψi )u.

(8.3)

Since the fictitious strains, (Ψi )u, sum to zero over all patches, the total strains are correctly recovered by summing either 𝜺(Ψi u) or Ψi 𝜺(u). ii) When the equilibrating body force field in a star patch is derived from weighted stresses: b(Ψi 𝝈) = − ⋆ (Ψi 𝝈) = −Ψi  ⋆ 𝝈 − ( ⋆ Ψi )𝝈 = Ψi b(𝝈) − ( ⋆ Ψi )𝝈, that is, bi = Ψi b (𝝈) = b (Ψi 𝝈) + ( ⋆ Ψi )𝝈

(8.4)

Since the fictitious body forces, ( ⋆ Ψi )𝝈, sum to zero over all patches, the total body forces are correctly recovered by summing either b (Ψi 𝝈) or Ψi b (𝝈). Thus the body forces b (Ψi 𝝈) form a self-balanced load on a patch which implies that the problem of analysing the patch for stresses is a well-posed one, and the summation of the patch stresses must equilibrate with the total loads.

8.2 Recovery of Compatibility From an Equilibrated Solution In this Section we consider the problem of finding a compatible displacement field, uC , from an equilibrated finite element solution, defined in terms of stresses and boundary displacements. This compatible displacement field will be defined using the approximation basis of a conforming finite element model with the same configuration as the hybrid model, but we will restrict further analyses to subdomains where non-zero PU functions Ψi are applied, thus avoiding the necessity of solving the global problem again. We start by defining the displacement field in each patch as ui = 𝝍 i û i , with corresponding strain fields 𝜺̄ i = Bi û i , where Bi = 𝝍 i , matrix 𝝍 i contains the shape functions, and vector û i contains the displacement parameters for a patch, generally associated with nodal displacements. In this context a node may refer to a vertex, a point on a side of an element, or a whole side in a hierarchical formulation. Then we proceed to determine ui as if it corresponds to the partitioned displacement field Ψi uC , and independently analyse each star patch with zero displacements at its link. The stiffness matrix of a patch is Ki =

∫Ωi

BTi kBi dΩ,

Recovery of Complementary Solutions

and using (8.3) we can express: K i û i =

∫Ωi

BTi k 𝜺̄ i dΩ =

∫ Ωi

BTi kΨi 𝜺C dΩ +

∫ Ωi

BTi k 𝜺̃ i dΩ

where 𝜺C = uC

𝜺̃ i = (Ψi )uC ,

and

After integrating by parts, we get: ∫ Ωi

BTi kΨi 𝜺C dΩ =

∫ Ωi

BTi Ψi 𝝈 C dΩ =

∫ Ωi

𝝍 Ti b(Ψi 𝝈 C )dΩ +

∫Γi

𝝍 Ti Ψi t C dΓ,

and using (8.4) we have: ∫ Ωi

𝝍 Ti b(Ψi 𝝈 C )dΩ =

∫ Ωi

𝝍 Ti Ψi b (𝝈 C )dΩ −

∫Ωi

𝝍 Ti ( ⋆ Ψi )𝝈 C dΩ.

Thus the stiffness equations to be solved for a patch, which enforce the weak equilibrium conditions, take the form: K i û i = ĝ i , where ĝ i =

∫ Ωi

̄ 𝝍 Ti Ψi bdΩ +

∫Γi

𝝍 Ti Ψi t̄ dΓ −

∫ Ωi

𝝍 Ti ( ⋆ Ψi )𝝈 E dΩ +

∫ Ωi

BTi k(Ψi )ũ E dΩ,

after substitution of 𝝈 E for 𝝈 C , and ũ E for uC in the right hand side. The first two integrals represent the partitioned body and boundary traction loads on the patch, and the other two integrals represent the fictitious loads and initial strains. The initial strain term depends on a discontinuous element-wise displacement field ũ E which is derived from stresses 𝝈 E and side displacements 𝒗. The derivation is detailed in Section 8.2.1. This term has an essential role in ensuring that the vertex of a star patch reaches its ‘correct’ position when its link has zero displacements. The total displacement of vertex i is defined solely by û i , since there are no contributions to this displacement from the analyses of other star patches. The boundary conditions to be imposed on a patch may consist of prescribed tractions or displacements weighted by the appropriate PU function. However, for sides on the link of a patch Ψi = 0, and we apply homogeneous conditions for displacements, with at least one side being fixed. Thus K i is non-singular since the homogeneous Dirichlet boundary conditions on all or part of the link prevent rigid body movements. The boundary conditions are illustrated in Figure 8.3 for patches whose vertices form an internal vertex (8.3a), or a vertex on the boundary of the complete model (8.3b); the links are indicated by bold solid lines, and are assumed to be fixed. The application of the partition of unity to a non-homogeneous ū implies that the degree of the approximation used for 𝜓i must be greater than the degree used to ̄ describe u. The solutions from all the patch models sum to give the conforming displacements: ∑ ∑ uC = i uCi = i 𝝍 i û i .

139

140

Equilibrium F.E. Formulations

ui = 0

ui = 0

i

i

ui = Ψiu or ti = Ψit (a)

(b)

Figure 8.3 Star patches in the recovery of compatibility.

8.2.1 Derivation of ũ E

We now describe a method to derive element-wise displacement fields from the hybrid equilibrium solution. Although these fields are not compatible at element interfaces, they reflect the shapes and positions of the elements well enough for the purposes of quantifying the fictitious initial strains. The shape of an element is derived from the prolongation condition applied to the strains from a compatible solution and the equilibrating solution, that is, ∫ Ωe

BTe k 𝜺C dΩ =

∫ Ωe

BTe k 𝜺E dΩ,

(8.5)

or K e û e = ĝ e ,

(8.6)

where Ke =

∫ Ωe

BTe k Be dΩ

and

ĝ e =

∫ Ωe

BTe 𝝈 E dΩ.

In (8.6), K e is singular, and the solution û e is only unique to within a rigid body displacement. Thus we can express û e = û ep + e 𝚫e where û ep are particular displacement parameters which represent the deformed shape of the element, and vector 𝚫e represents an arbitrary rigid body movement. With reference to the notation used in 6.1.1, we have that e = U eR |nodes . The position of an element, as embodied in vector 𝚫e , is adjusted to minimize the lack of fit between boundary displacements defined by ue = 𝝍 e û e , and those defined by 𝒗e = V e 𝒗̂ e from the hybrid solution. For the best results, the latter displacements should have had any contributions from spurious kinematic modes of the mesh filtered out, either using (6.4) or by blocking them, as described in Section 6.2. This is illustrated by the example in Section 8.4.1. Thus we seek to minimize ∮Γ e

𝜹Te 𝜹e dΓ, where 𝜹e = {𝝍 e û e − V e 𝒗̂ e } = {ue − 𝒗e },

with 𝚫e as the variable vector. This leads to Equation (8.7) { } ] [ T e M e e 𝚫e = Te N e 𝒗̂ e − M e û ep

(8.7)

Recovery of Complementary Solutions

Figure 8.4 The initial position of an element, its boundary displacements and the element-wise displacement field. 𝚫e is chosen to minimize the integral on the boundary of the square of ue − 𝒗e .

ve

ue – ve ue

Δe

where Me =

∮Γ e

𝝍 Te 𝝍 e dΓ and N e =

∮Γ e

𝝍 Te V e dΓ.

Finally the element-wise displacements for element e are given by ũ E = 𝝍 e {û ep + e 𝚫e }. This solution is illustrated diagrammatically in Figure 8.4. 8.2.2 An Illustration of the Technique

We consider a simple 1D example to illustrate the analysis in detail for a typical patch. A uniform bar has its left hand end fixed, and its right hand end is subjected to a tensile force P. The use of linear hybrid elements leads to the constant stress 𝝈 E , and a linear continuous displacement field ũ E as shown in Figure 8.5a. The equilibrium solution is of course the exact one in this case, nevertheless this simple example serves to illustrate the analysis of a typical star patch. Figure 8.5b shows the patch PU function and its first derivative. From the nature of the stress field we would expect linear displacements to be recovered, and consequently we should work with at least quadratic displacement conforming models for a star patch. We will assume hierarchic shape functions for the patch as shown in Figure 8.5c, together with their first derivatives, or axial strains. The patch stiffness matrix is derived as follows: ⌊ ⌋ ⌊ ⌋ d𝜓i d𝜓j d𝜓k 𝜀̄ i = Bi û i = û i , ui = 𝝍 i û i = 𝜓i 𝜓j 𝜓k û i , ds ds ds and 0 0 ⎤ ⎡ 2 EA ⎢ 0 16∕3 0 ⎥. Ki = BTi [EA]Bi ds = ∫ Ωi ⎥ h ⎢ 0 0 16∕3 ⎦ ⎣ The patch load vector is derived in this case (in the absence of prescribed body forces b̄ and concentrated tractions t̄ ) from just the two integrals involving fictitious loads in their integrands: [ ] E dΨ i) Fictitious body forces equal to − ds i 𝜎 E = − 𝜎h , along element 1 of the patch, on the [ ] E dΨ left of vertex i, and to − ds i 𝜎 E = 𝜎h , along element 2 of the patch, on the right of vertex i. [ ] dΨ ii) Fictitious initial strains ds i ũ E .

141

142

Equilibrium F.E. Formulations

h

h P

i–1

P

ui+1

ui

ui–1

i+1

i

(a)

i–1

1 h

Ψi

1

i+1

i

i–1

– 1h

dΨi ds

i+1

i

(b) 1

ψi

1 4 h

1 h

dψi ds



ψj

ψk

dψj ds

4 h

– h4

1 h

1 dψk ds

– h4

(c)

Figure 8.5 A 1-dimensional example. The mesh is uniform, with all elements of length h.

Thus, noting the simple representation of the derivatives of the integrands in Figures 8.5b and 8.5c: ( ĝ i = − 𝝍 Ti ∫ Ωi

dΨi ds

(

) 𝜎 E A ds +

∫Ωi

BTi E

⎧ ⎪ ⎪ ( ) dΨi ⎪ T E = − 𝝍i 𝜎 A ds + EA ⎨ ∫ Ωi ∫ Ωi ⎪ ds ⎪ ⎪ ⎩

dΨi ds

) ũ E A ds,

⎫ ⎪ ⎪( ) ⎪ dΨi E ⎬ ds ũ ds, ds ⎪ d𝝍 k ⎪ ⎪ ds ⎭ d𝝍 i ds d𝝍 j

⎧ 0 ⎫ ⎧ (ū 1 + ū 2 ) ⎫ ⎪ ⎪ ⎪ EA ⎪ −4 𝛿∕3 ⎬ . = 𝜎 E A ⎨ −2∕3 ⎬ + ⎨ h ⎪ ⎪ ⎪ ⎪ ⎩ 2∕3 ⎭ ⎩ 4 𝛿∕3 ⎭ Since the variation of ũ E is linear, the displacement in element 1 varies from ū 1 − 𝛿 at vertex (i − 1) to ū 1 + 𝛿 at vertex i, and the displacement in element 2 varies from ū 2 − 𝛿 at vertex i to ū 2 + 𝛿 at vertex (i + 1), where ū 1 and ū 2 denote the average displacements of elements 1 and 2 respectively. The extension of each element is given by 2 𝛿, and 𝜎 E = 2 𝛿E∕h, where h is the length of each element.

Recovery of Complementary Solutions

Figure 8.6 Plot of axial displacements of patch i.

– δ 2

i–1

u1 + u2 2 i

δ 2

i+1

Solving K i û i = ĝ i gives the displacement parameters: ⎧ (ū 1 + ū 2 ) ⎫ ⎪ 1⎪ û i = ⎨ −𝛿 ⎬. 2⎪ ⎪ 𝛿 ⎭ ⎩ The resulting displacements are shown diagrammatically in Figure 8.6: It should be noted that: • The displacement of the star patch vertex i has the correct value after just this analysis, since the solutions from other patches are zero at this vertex. • The quadratic components of the displacement fields will cancel out when displacements from adjacent patches are summed together, leaving the total displacement field linear. • For this example, elements based only on linear shape functions would produce the same total solution. If, in this example, only the linear shape function 𝝍 i is used, the patch equations reduce to ) ( EA 1 2 EA û i = (ū + ū 2 ). h h The quadratic components of displacement are absent, and the displacement parameter for vertex i is unchanged.

8.3 Recovery of Equilibrium From a Compatible Solution In this Section, we assume that we already have a compatible solution consisting of a kinematically admissible displacement field uC together with a conforming strain field 𝜺C = uC and corresponding stress field 𝝈 C = k 𝜺C . This stress field satisfies the weak form of equilibrium as expressed in (8.8) for all displacement fields u ∈ h , the space of conforming displacement fields of the finite element model. −

T

∫Ω

𝝈 C u dΩ +

∫Ω

T b̄ u dΩ +

∫Γ

t T u dΓ = 𝟎,

(8.8)

where, in the last integrand, t is prescribed on Γt and u is prescribed on Γu . We now seek a strongly equilibrated stress field 𝝈 E , that is, one satisfying  ⋆ 𝝈 E + T b̄ = 𝟎, and  𝝈 E = t̄ . Unless we have the exact solution, this stress field provides an incompatible strain field 𝜺E = f 𝝈 E . In the following we detail such a recovery process by way of a hybrid equilibrium finite element model.

143

144

Equilibrium F.E. Formulations

8.3.1 Recovery From Star Patches: The General Case

We again restrict further analyses to subdomains consisting of star patches. Here we consider the stress field 𝝈 Ei = Ψi 𝝈 E = Si ŝ i + 𝝈 i0 , where 𝝈 i0 is a particular solution which equilibrates with the patch loads. These loads consist of body forces as defined from (8.4), that is, b̄ i ≡ b (Ψi 𝝈 E ) = Ψi b̄ − ( ⋆ Ψi )𝝈 E , as well as the partitioned prescribed tractions t̄ i = Ψi t̄ . Since 𝝈 E is not yet known, it will be convenient to substitute the stress field 𝝈 C in place of 𝝈 E in the term representing the fictitious body forces, since ∑ i) these forces will cancel out in the summation i ( ⋆ Ψi )𝝈 C , and, as will be shown in the following, ii) the patch loads (b̄ i , t̄ i ) remain self-balancing so that the problem of analysing the patch is well posed. Property (ii) is valid when the loads are orthogonal to all rigid body displacements, that is, ∫ Ωi

T b̄ i u dΩ +

∫Γi

T t̄ i u dΓ = 𝟎, for all rigid body displacements u.

The first integrand is expanded to T ̄ T u − (( ⋆ Ψi )𝝈 C )T u, b̄ i u = (Ψi b) T T T = b̄ (Ψi u) − 𝝈 C (Ψi u) + 𝝈 C Ψi (u }.

For rigid body displacements we have u = 𝟎, and the orthogonality condition becomes: −

T

∫ Ωi

𝝈 C (Ψi u) dΩ +

∫Ωi

T b̄ (Ψi u) dΩ +

∫Γi

T t̄ (Ψi u) dΓ = 𝟎,

(8.9)

which is the weak equilibrium condition imposed on the stress field of the conforming model when Ψi u ∈ h . If the conforming model is based on linear displacement fields, then Ψi u ∉ h when u represents a rigid body rotation, and, therefore, the loads are only self-balanced in the translational sense. Further consideration thus needs to be given to rotational equilibrium (Maunder and Almeida, 2012). This is discussed in Subsection 8.3.2, where a corrective stress field is introduced. If the conforming model is based on complete quadratic or higher degree displacement fields, then self-balanced loads for a star patch are guaranteed without further correction. We enforce equilibrium at the internal sides of the patch using the side shape functions contained within V i assumed for the mesh of hybrid elements (as in (4.5)): T

∫Γi

V Ti  i 𝝈 Ei dΓ =

∫Γi

V Ti (Ψi t̄ ) dΓ

Recovery of Complementary Solutions

Weak compatibility conditions within a patch are imposed as follows, making use of (8.3) and taking 𝜺Ei = Ψi (uC ): ∫ Ωi

STi 𝜺Ei dΩ = =

∫Ωi ∫Ωi

STi Ψi uC dΩ ( ( )) STi  Ψi uC dΩ −

∫ Ωi

STi (Ψi )uC dΩ

where 𝜺Ei = f 𝝈 Ei . Integration by parts leads to: ∫ Ωi

( ( )) STi  Ψi uC dΩ =

∫ Γi

( ) STi  i Ψi uC dΓ

Now we replace the side displacements Ψi uC by the hybrid displacements V i 𝒗̂ i to obtain the patch equations in the usual form from a hybrid formulation, as in (4.14): ]{ } { } [ ê ŝ i − i DTi = ̂i 𝒗̂ i ti Di 𝟎 where i = ê i =

∫ Ωi ∫ Ωi

STi f Si dΩ, STi f 𝝈 i0 dΩ +

Di = ∫Ωi

T

∫Γi

V Ti  i Si dΓ,

STi (DΨi )uC dΩ,

and t̂ i =

∫Γi

V Ti (Ψi t̄ ) dΓ −

T

∫Γi

V Ti  i 𝝈 i0 dΓ.

We should note that the hybrid model for the patch may be ill-posed, due not to the unbalanced property of the loads (this has already been addressed), but to their inadmissibility. The nature of the load distribution is discontinuous at element interfaces – so possibilities can exist for exciting spurious kinematic modes if they exist in the hybrid model of a star patch. We are of course free to construct whatever hybrid model we like, to fit with the geometrical arrangement, so models can be designed to be free of spurious modes, apart perhaps from the ‘pathological’ cases mentioned in Chapter 5, but even these can be avoided by the use of appropriate macro-elements as stabilizers. The total equilibrating stress field is then given by: ∑ ∑( ) 𝝈 Ei = Si ŝ i + 𝝈 i0 . 𝝈E = i

i

On a patch we require the degree of 𝝈 i0 to match that of the partitioned body forces (whether fictitious or resulting from the prescribed ones), and the degree of Si to match the degree of the partitioned tractions. The link part of the patch boundary is generally subjected to homogeneous Neumann boundary conditions, unless Dirichlet boundary conditions apply there. In that case the analysis of the patch could impose either type of homogeneous condition on the link. ̄ and it does not However, when a side in a patch has prescribed displacements u, belong to the link, we have two options as to how to account for them in a patch anal̄ and the ysis. One is to apply the displacements in the weighted form, that is, ū i = Ψi u, other option is to derive reactive tractions which are statically equivalent to the reactive

145

146

Equilibrium F.E. Formulations

ti = 0

ti = 0

i

i ui = Ψiu or ti = Ψit

(a)

(b)

Figure 8.7 Star patches in the recovery of equilibrium

forces, and apply these as if they were prescribed tractions. Patch boundary conditions are illustrated in Figure 8.7 for the cases of a patch with an interior vertex (8.7a), and a patch with a vertex on the external boundary of the model (8.7b). The link is again indicated by the bold solid lines. 8.3.2 Recovery From Star Patches: The Case of Linear Displacements

When the space h is based on element-wise linear displacements, (Ψi u) ∈ h implies that a rigid body displacement u must be a constant translation without rotation. Then the system of loads on patch i, (b̄ i , t̄ i ) is not generally in rotational equilibrium, and it has a resultant moment M [i] about vertex i. The contribution to this moment from element e in the patch is given by: M e[i] =

∫ Ωe

r ei × b̄ ei dΩ + i

∫Γe

r ei × t̄ ei dΓ

(8.10)

i

where r ei denotes the position vector of the infinitesimal domain dΩ or dΓ in element e relative to vertex i. In general ∑ M e[i] ≠ 𝟎, M [i] = e

however, as shown in Maunder and Almeida (2012), the contributions from one element to the moments around all its vertices, have the useful property of summing to zero: ∑ M e[i] = 𝟎. i

We make use of this property to determine a ‘corrective’ stress field 𝝈̃ such that the ̃ do lead to self-balanced loads on star patches. fictitious body forces based on (𝝈 C + 𝝈) The fictitious body forces from 𝝈̃ should only produce vertex moments to counterbalance those from 𝝈 C while producing zero additional vertex forces. We can find an appropriate 𝝈̃ from analysing another global finite element problem based on mixed elements with displacement parameters that are dual/conjugate to vertex forces and moments. At first sight such an analysis may appear to involve more computational effort than that required from a dual analysis! However, regarding the vertex moments as representing the loads, we see that the total load is the superposition of load cases, each one consisting of a set of self-balanced moments acting on the vertices of a single element k.

Recovery of Complementary Solutions

∑ Denoting the corresponding stress field by 𝝈̃ {k} , we obtain 𝝈̃ = k 𝝈̃ {k} . Although 𝝈̃ {k} is a global field, we may restrict its non-zero values to within a subdomain Ω{k} local to element k. This would save computational effort without violating the essential cor̃ This idea of localization is prompted by the principle of rective property required of 𝝈. Saint Venant in elasticity, that is, stresses remote from the application of a load are independent of the distribution of the load. Hence when the local loads are self-balanced, we expect the stresses remote from the load to tend towards zero. Although it is not strictly necessary that the strains due to 𝝈̃ be compatible, as they would be in an exact elastic solution, nevertheless, the quality of the end result (𝝈 E ) ̃ and the compatibility of will depend to some extent on that of the corrective stress 𝝈, its associated strain field. Consequently, a reasonable approximation to 𝝈̃ {k} should be obtained by carrying out an elastic analysis of a neighbourhood patch Ω{k} subjected to a set of three vertex moments M k[i] associated with element k, which will be termed the kernel element of the patch. The extent of a neighbourhood patch is a matter of choice. At the simplest level, this patch could just consist of element k by itself. However, the choice affects the quality of the recovered equilibrating stress field, and a compromise needs to be made between this quality and the computational effort involved. Figure 8.8 illustrates a range of neighbourhood patches for an element k, generated with successive bands of elements around it. The analyses of the neighbourhood patches exploit a mixed formulation, in which both displacement and stress fields are assumed within the domains of the elements, as presented in Section 3.3. In the present context, these fields are assumed as follows: i) Stress fields 𝝈̃ e = S̃ e s̃e , the basis for S̃ e may be quite general, and it is not necessary to limit it to stress fields that are statically admissible with zero body forces. ii) Displacement fields { } ̃ ̃ e 𝜹e , ̃ue = U 𝜽̃ e ̃ e is based on linear shape functions to interpolate from vertex displacements where U as translational degrees of freedom, 𝜹̃ ei , and quadratic shape functions to conform with vertex rotations as drilling degrees of freedom, 𝜃̃ei . These degrees of freedom ̃ e . In the 2D case the disare conjugate to vertex forces F̃ ei and vertex moments M i placements are defined in a similar way to those for Allman’s original conforming triangular element with drilling degrees of freedom (Allman, 1984; Tian and Yagawa, 2007), that is, ũ ei = Ψei 𝜹̃ ei + (𝜃̃ei Ψei ) 𝚲 × r ei

(8.11)

k

Figure 8.8 Neighbourhoods of element k

147

148

Equilibrium F.E. Formulations

Figure 8.9 Rotational displacements over a closed star patch in 2D.

∼ θi i

define displacements due to a vertex translation or rotation respectively, where 𝚲 denotes the unit vector normal to the plane of the element, with a positive direction conforming with the third axis of a right handed set of reference axes. In this case, the compatible displacements in a star patch of elements due to rotation of the vertex are as illustrated in Figure 8.9, but the value of the drilling rotation is twice that defined by Allman (1984). The corresponding strain field is 𝜺̃ ei = B𝛿ei 𝜹ei + B𝜃ei 𝜃ei where B𝛿ei = Ψei

and

B𝜃ei = (Ψei )𝚲 × r ei .

In the mixed formulation, we define generalized vertex forces and moments in terms of the stress fields: ̃ 𝛿e s̃e , F̃ ei = D i

̃ 𝛿e = where D i

̃ 𝜃e s̃, ̃e =D M i i

̃ 𝜃e = where D i

∫ Ωe ∫ Ωe

(Ψei )T S̃ e dΩ

and

(𝚲 × r ei )T (Ψei )T S̃ e dΩ,

and enforce weak compatibility between the strains due to the stress field 𝝈̃ e and any initial strain fields 𝜺̃ eT , with those due to the displacement fields, that is, ∫ Ωe

T S̃ e (𝜺 (𝝈̃ e ) + 𝜺̃ eT ) dΩ =

∫ Ωe

T S̃ e 𝜺 (ũ e ) dΩ,

̃ 𝛿e 𝜹̃ e + D ̃ 𝜃e 𝜽̃ e , ⇒ ̃ e s̃e + ẽ e0 = D T

T

where 𝛿

̃ e =

∫Ωe

T S̃ e f S̃ e dΩ, ẽ e0 =

∫Ωe

T S̃ e 𝜺̃ eT

̃e ⎡D 1 ⎢ 𝛿 𝛿 ̃e = ⎢D ̃ dΩ, D ⎢ e2 ⎢ ̃𝛿 ⎣ De 3

̃ 𝜃e ⎤ ⎡D 1 ⎥ ⎢ 𝜃 𝜃 ⎥ , and D ̃e = ⎢D ̃ ⎥ ⎢ e2 ⎥ ⎢ ̃𝜃 ⎦ ⎣ De3

⎤ ⎥ ⎥. ⎥ ⎥ ⎦

Recovery of Complementary Solutions

The complete system of equations for an element becomes: ⎡ −̃ e ⎢ ⎢ D ̃𝛿 ⎢ e 𝜃 ⎢ D ⎣ ̃e

T T ̃ 𝛿e D ̃ 𝜃e ⎤ ⎧ s̃ ⎫ ⎧ ẽ ⎫ D ⎥ ⎪ e ⎪ ⎪ e0 ⎪ 𝟎 𝟎 ⎥ ⎨ 𝜹̃ e ⎬ = ⎨ F̃ e ⎬ ⎥⎪ ⎪ ⎪ ̃ ̃ ⎪ 𝟎 𝟎 ⎥⎦ ⎩ 𝜽e ⎭ ⎩ M e ⎭

(8.12)

We should note that this mixed element has at least one spurious kinematic mode that occurs when its vertices have equal rotations 𝜃̃e . The corresponding displacement field is then given by: ∑ 𝜃̃ei Ψei (𝚲 × r ei ) ũ e = i∑ = 𝜃̃e 𝚲 × {Ψei r ei } i { ( ) }} { ∑ ∑ = 𝜃̃e 𝚲 × = 𝟎, Ψei r e − 𝚲 × Ψei r i ∑

i



i

since r ei = {r e − r i }, i Ψei = 1 and i Ψei r i = r e , where r e and r i denote the position vectors of a point within element e, and vertex i, respectively, relative to the global origin of a reference frame. This mode is not the same as a rigid body rotation since it results in zero displacements as well as zero strains. In fact, of the nine strain fields due to the unit displacement parameters, only five are independent: three constant strain fields due to vertex translations, and two linear strain fields due to vertex rotations. Consequently if S̃ e includes T T ̃ 𝛿e D ̃ 𝜃e ] is five. all nine linearly independent linear stress fields, then the rank of [D This follows by considering the strain field 𝜺̃ due to an arbitrary combination of the five displacement parameters that produce strain (with the other displacement parameters zero). 𝜺̃ is a non-zero linear strain field, and the scalar work product ∫Ω 𝝈 T 𝜺̃ dΩ cannot e be zero for all linear stress fields (for example, including the linear stress field 𝝈 that has the same values as 𝜺̃ ). Thus the rank cannot be less than five. In Maunder and Almeida (2012) we conclude that when polynomial stress fields are used that are complete in linear terms, stress fields of higher degree are not excited. Hence the complete linear basis is sufficient, and a single element has no more than one spurious mode. Furthermore for such a mode to propagate through a mesh, the rotations of all vertices need to have the same value. It is therefore sufficient for one vertex to have a prescribed rotation, for example, a zero value, for the spurious kinematic mode to remain dormant. In the analyses of neighbourhood patches, these mixed elements replace the conforming elements in the finite element models, and we identify ̃ e −−−−→ M ̃e , and M F̃ ei −−−−→ F̃ e[i]{k} i [i]{k} where subscripts [i] and {k} refer to vertex i within the neighbourhood patch associated with element k. Then we enforce: ∑ ∑ ∑ ̃ e = 0, ∀i ∉ Ωk ; ̃ e = −Mk , ∀i ∈ Ωk , M M F̃ e[i]{k} = 𝟎; [i]{k} [i]{k} [i] e

e

e

and assume zero initial generalized strains ẽ e0 . The approximations introduced in 𝝈̃ {k} by restricting analyses to neighbourhood ̃ patches, become evident in the incompatibilities of the strains corresponding to 𝝈.

149

150

Equilibrium F.E. Formulations

While these incompatibilities do not affect the corrective equilibrating property ̃ they do indicate a non-optimal solution, the strain energy of which could be of 𝝈, further reduced. This can be achieved by adopting an iterative strategy where residual strains from earlier analyses of neighbourhood patches become the initial strains in future analyses. Further details of these strategies are discussed in Maunder and Almeida (2012), where we also consider alternative definitions of the extents of the neighbourhoods of a kernel element, the simplest definition being that of element k by itself. 8.3.3 Element by Element Recovery of Equilibrium

It is also possible to recover an equilibrated solution in a procedure which analyses each element rather than each overlapping star patch. Furthermore, the existence of this procedure is not dependent on the degree of the displacement field. The essential step consists of deriving codiffusive tractions on the sides of each element, which equilibrate with its prescribed loads. This is achieved by first resolving the nodal forces determined at each vertex of the conforming elements into equal and opposite components acting at each interface. When we identify the partition of unity Ψei with the linear shape function associated with vertex i in a hierarchical formulation of element e, equivalent vertex forces F̂ ei are defined by: F̂ ei =

∫Ωe

( ⋆ Ψei )𝝈 C dΩ −

∫Ωe

Ψei b̄ dΩ −

∫Γe

Ψei t̄ dΓ.

(8.13)

F̂ ei can also be expressed as: F̂ ei = − b̄ ei dΩ − t̄ ei dΓ, ∫ Ωe ∫Γe

(8.14)

with b̄ ei = Ψei b̄ − ( ⋆ Ψei )𝝈 C

and

t̄ ei = Ψei t̄ .

It is shown next, with reference to Figures 8.10 and 8.12, that each force F̂ ei at vertex i of element e can be replaced by statically equivalent codiffusive tractions acting on the two sides of the element adjacent to the vertex. An element subjected to these tractions plus the loads represented by (b̄ ei , t̄ ei ) is then in translational, but not necessarily rotational, equilibrium. The moment of these forces about vertex i is here denoted by Mei , and its value is defined as in (8.10). ∑ In general Mei ≠ 0, but i Mei = 0, and the tractions obtained by replacing F̂ ei for all vertices of an element do lead to a well-posed problem for a single element. Then, apart from the question of admissibility when spurious kinematic modes exist, each element can be analysed to obtain another equilibrated solution 𝝈 Ee so that 𝝈 E = ∪ 𝝈 Ee . This is the essence of the method originally proposed by Ladevèze et al. (1991), and further developed by Ladevèze and Maunder (1996), Ladevèze and Rougeot (1997), and Ladevèze and Pelle (2005). 8.3.3.1 Resolution of the Vertex Forces

We use the weak equilibrium condition imposed in the analysis of the conforming ∑ model, that is, e F̂ ei = 𝟎 for the set of nodal forces at a vertex defined by its adjacent elements.

Recovery of Complementary Solutions

(a) A closed star patch of elements sharing a common vertex i

(b) A closed force polygon as a Maxwell diagram with an internal pole point

(c) Exploded view of the Maxwell diagram including the pair of forces Fˆ [em]i into which each vertex force is decomposed

(d) Exploded view of the star patch including the components of the vertex forces Fˆ ei for each element

Figure 8.10 Decomposition of vertex forces in a closed star patch.

Each force F̂ ei can be resolved into two components that are associated with the two sides of the connected element. This resolution can always be done so that the forces obtained for each side of an interface are equal in magnitude and opposite in direction. The existence of such a decomposition can be simply explained by exploiting a Maxwell diagram for a set of equilibrating forces.

151

152

Equilibrium F.E. Formulations

In a Maxwell diagram the set of equilibrating forces are represented graphically by a set of vectors that form a closed polygon. If these vectors are constructed in the order of the corresponding elements identified by traversing around the link of the star patch, then a pole point P defines an appropriate resolution of each force. This is illustrated for a closed star patch in the four part Figure 8.10. Thus we have for example F̂ 1i = F̂ [15]i + F̂ [12]i ; F̂ 2i = F̂ [21]i + F̂ [23]i ;

and

F̂ [12]i = −F̂ [21]i

and codiffusive components F̂ [12]i and F̂ [21]i are assigned to the ends of the interface between elements 1 and 2, and so on for the remainder of the patch. For a closed patch the decomposition is not unique since it depends on the position of the pole point P, and this has 2 degrees of freedom. We can however form target forces P̂ [em]i for interface components such as F̂ [em]i , in terms of the average tractions acting between a pair of elements, by defining: P̂ [em]i =

∫Γ[em]

0.5 Ψei (t C[em] − t C[me] ) dΓ

T

(8.15)

T

where t C[em] =  [em] 𝝈 Ce , and t C[me] =  [me] 𝝈 Cm denote the tractions which equilibrate with the stresses obtained from the compatible solution in the adjacent elements. We then seek to locate P so as to ∑ ‖ ‖2 minimize 𝑤[em]i ‖F̂ [em]i − P̂ [em]i ‖ , (8.16) ‖ ‖ e

where 𝑤[em]i is a weighting factor. The solution to this problem has a convenient graphical interpretation (Ladevèze and Maunder, 1996) when we place the target vectors on the Maxwell diagram so that the tail end of the vector P̂ [em]i coincides with the vertex of the force polygon where forces F̂ mi and F̂ ei intersect, as illustrated in Figure 8.11 with P̂ [54]i . Then the position of P corresponds to the centroid of a set of masses 𝑤[em]i placed at the tip ends of the target vectors. However for an open patch, P is constrained in position in order to avoid imposing tractions on the sides which belong to a free boundary. So for example, if the star patch in Figure 8.12a contains only the elements 1 to 4, then the force polygon in Figure 8.12b has only four forces and P lies at the intersection of forces F̂ 1i and F̂ 4i . Figures 8.12c and 8.12d illustrate the decomposition of the vertex forces in a similar way as for the closed star patch. Fˆ5i

Figure 8.11 Maxwell diagram with target force vectors.

Pˆ[54]i w[54]i

Fˆ4i

Recovery of Complementary Solutions

P Fˆ1i 1

2 3

i

Fˆ2i

4 (a) An open star patch of elements sharing a common vertex i.

Fˆ4i

Fˆ3i

(b) A closed force polygon as a Maxwell diagram with a pole point P .

P Fˆ1i Fˆ4i Fˆ[21]i Fˆ[23]i Fˆ2i

Fˆ[34]i Fˆ[32]i Fˆ3i

(c) Exploded view of the Maxwell diagram including the pair of forces Fˆ[em] into which i each vertex force is decomposed.

(d) Exploded view of the star patch including the components of the vertex forces Fˆei for each element.

Figure 8.12 Decomposition of vertex forces in an open star patch.

8.3.3.2 Derivation of Statically Equivalent Codiffusive Tractions

A general way to derive codiffusive tractions was initially proposed by Ladevèze et al. (1991), and this required that the definition of 𝝈 E in each element should satisfy the prolongation conditions, that is, for each element e: ∫ Ωe

BTe 𝝈 C dΩ =

∫ Ωe

BTe 𝝈 E dΩ.

(8.17)

153

154

Equilibrium F.E. Formulations

The prolongation condition ensures that the work done by the original conforming and the recovered equilibrated stress fields with any conforming strain field is the same. These conditions were later relaxed, but we will now explain the implications of the conditions when implemented with a hierarchical formulation. Compliance with (8.17) requires, via integration by parts: ∫ Ωe

BTe 𝝈 C dΩ = =

∫ Ωe ∫ Ωe

BTe 𝝈 E dΩ 𝝍 Te b̄ dΩ +

∮𝜕Ωe

𝝍 Te t dΓ

T

where t =  e 𝝈 E , that is, ∮𝜕Ωe

𝝍 Te t dΓ =

∫ Ωe

BTe 𝝈 C dΩ −

∫ Ωe

𝝍 Te b̄ dΩ ≡ F̂ e

(8.18)

where F̂ e denotes the vector of conjugate forces corresponding to the displacement parameters of the conforming model. Thus the tractions satisfy the consistency relations with these forces acting on the elements. For elements of general degree, we can exploit the hierarchical structure of the displacement space h . Thus consider he as the direct sum: he = Ve ⊕ Se ⊕ Ie

(8.19)

where the subspaces denoted by subscripts V , S and I are based on shape functions corresponding to vertex nodes, sides and the interior respectively. The shape functions for Ve are linear in a hierarchical formulation, and they are identical to the partition of unity functions Ψei for vertex node i of element e. Various alternative polynomial basis functions for the subspaces Se and Ie can be used provided they are zero at each vertex, or on the boundary of the element respectively. Thus, from (8.18): i) When the shape function ∈ Ve , we satisfy (8.18) for vertex i by imposing for each interface between elements e and m the following conditions on tractions t(≡ t E[em] ): ∫Γ[em]

Ψi t E[em]

dΓ = F̂ [em]i

(8.20)

where F̂ [em]i is the resolved component of F̂ ei applied to the interface. ii) When the shape function ∈ Se , and is associated with side mode j, the right hand ̂ [em] acting on element e at the side of (8.18) is represented by a generalized force G j interface between elements e and m. This force is counter-balanced by an equal and opposite generalized force on element m, and the prolongation condition takes the form: ∫Γ[em]

̂ [em] 𝝍 T[em]j t E[em] dΓ = G j

(8.21)

iii) When the shape function ∈ Ie , it is associated with an internal mode with zero displacements on all sides of the element. In this case (8.18) has no influence on t. Hence we can satisfy the prolongation condition by deriving consistent distributions of traction independently on each side by making use of the resolved vertex forces and

Recovery of Complementary Solutions

1 (P 2 0

k

+ P1)

1 (P 2 0

– P 1)

1 (P 2 0

+ P1)

1 (P 2 0

– P1)

1 (P 3 2

– P 0)

Fˆ [em]k

ˆ [em] G j

j Fˆ [em]i

i 1 (P 2 0

+ 3P1)

1 (P 2 0

– 3P1)

Fˆ [em]i 1 (P + 3P 1 2 0

+ 5P2)

1 (P – 3P 1 2 0

+ 5P2)

15 P 2 2

Figure 8.13 Hierarchical shape functions and their dual traction functions for d = 1 or 2.

the generalized side forces. As an example we consider the hierarchical formulation described by Akin (1994), for which the basis of Se is formed from shape functions defined on one side by an (Pn − Pn−2 ) for degree n ≥ 2, where Pn and Pn−2 are Legendre polynomials of degree n and (n − 2) respectively and the coefficient an is chosen for convenience. When d = 1 or 2, the shape functions from Ve and Se associated with one side are illustrated in the top part of Figure 8.13, and their dual traction functions are illustrated in the bottom part. The dual functions in Figure 8.13 define traction distributions on a side consistent with unit resultant forces, denoted by for example, F̂ [em]i and F̂ [em]k at the end vertices, and, in the case of d = 2, a unit generalized force at the side denoted by for example, ̂ [em] . G j 8.3.3.3 Admissibility of the Derived Tractions

The simplest mesh of hybrid elements to replace a single conforming element would consist of just a single element. In general the tractions derived from forces F̂ [em]i and ̂ [em] may violate the admissibility conditions at the vertices, that is, the requirement of G i continuity of the tangential contravariant traction component as discussed in Section 5.2, and thereby excite spurious kinematic modes of the element. One way to avoid this problem is to use stable macro-elements, as discussed in Chapter 5, for which there are no restrictions on the tractions apart from the need for overall equilibrium. However, an alternative approach would be to introduce self-balancing variations of the tractions so as to satisfy the admissibility conditions, although in doing so we will in general violate the prolongation conditions. Although the latter conditions are not strictly necessary for obtaining bounded error estimates, it is an open question as to which conditions will lead to the most useful equilibrating solution. For a closed star, the admissibility conditions at the internal vertex can be uniquely satisfied by varying only the tangential tractions on the interfaces. With reference to Figure 8.14b, a self-balanced quadratic distribution of tangential traction is defined by

155

156

Equilibrium F.E. Formulations

1 a1

1

1

a5

r1

r2

–0.5

r5

a2 r3

r4

– 0.25

a4

a3 –1

1 P1

(a)

P2

1 (P 2 1

+ P2)

(b)

Figure 8.14 Variation of tangential tractions in a closed star patch.

combining the Legendre polynomials of degree 1 and 2, for example, 0.5(P1 + P2 ). This combination has unit value at the internal vertex of a star patch and zero value at the vertex in the link. These distributions at the interfaces are combined so as to negate the residual terms in the admissibility conditions for the vertices of the elements which connect to the internal vertex of the star patch as indicated in Figure 8.14a. The combination is defined by the solution to the following equations: ⎡ ⎢ ⎢ ⎢ ⎢ ⎣

1 1 0 0 0

0 1 1 0 0

0 0 1 ⋱ 0

0 1 0 0 0 0 ⋱ 0 1 1

⎤⎧ ⎥⎪ ⎥⎪ ⎥⎨ ⎥⎪ ⎦⎪ ⎩

a1 a2 a3 ⋮ an

⎫ ⎧ ⎪ ⎪ ⎪ ⎪ ⎬ = −⎨ ⎪ ⎪ ⎪ ⎪ ⎭ ⎩

r1 r2 r3 ⋮ rn

⎫ ⎪ ⎪ ⎬ ⎪ ⎪ ⎭

(8.22)

where aj and rj denote the multipliers of the unit interface tangential traction distributions and the residual values of the admissibility equations of the elements respectively. Since the square matrix on the left hand side is non-singular, the solution is unique. For an open star, variations in both tangential and normal components of traction on the interfaces are generally necessary, and then alternative variations are possible to achieve admissibility. 8.3.3.4 Derivation of the Element Stress Fields

The analysis of a triangular subdomain delineated by a conforming element, to obtain 𝝈 Ee , can be undertaken once the question of the admissibility of tractions has been resolved, along with the design of an appropriate mesh of hybrid equilibrium elements. The mesh may consist of just one element, one macro-element with three triangular subdivisions, or a more complex configuration. The analysis proceeds in a similar way to that for a mesh or for a star patch using (4.14), with non-homogeneous Neumann or Dirichlet boundary conditions, as appropriate. Then the total equilibrating stress field 𝝈 E = ∪ 𝝈 Ee . Although the recovery procedures in Section 8.3.3 lead to statically admissible stress fields, and hence they guarantee an upper bound to global error, minimal attention to compatibility implies that this bound can be excessively high. For further work to ameliorate such bounds, the reader is referred to Ladevèze and Pelle (2005). However, the procedures do provide equilibrating codiffusive tractions between elements, and thus also between substructures corresponding to any partition of the conforming model. This can be a useful tool for the design of complex structural forms.

Recovery of Complementary Solutions

tmax = 3 y y

x

E=1 v = 0.15

1.00

y′

x′

1.00

Figure 8.15 Square plate. Definition of the boundary conditions, material properties, mesh, reference frames and components of the stress tensor either in the (x, y) or in the (x ′ , y′ ) reference frame.

8.4 Numerical Examples To illustrate solutions obtained from some of the procedures described in this Chapter we use a very simple example, similar to the one in Almeida and Maunder (2009): a square plate where the top side is free and the other sides have sliding supports, as represented in Figure 8.15. For now we only consider the simple mesh with four elements and a central vertex, and the action of a linearly varying normal traction on the the top face, defined in that figure. Results for other actions and for more refined meshes will be given in Chapter 9. Quadratic approximation bases are used for the displacements of the compatible model, as well as for the stresses and boundary displacements of hybrid equilibrium models. This model has one spurious mode, which only affects the internal sides. 8.4.1 Recovery of Compatibility From an Equilibrated Solution

The recovery of compatibility process, described in Section 8.2, is illustrated in Figures 8.16 to 8.20. The starting point of this process is a boundary displacement field obtained from the equilibrated finite element solution, as in Figure 8.16, where the effect of the spurious mode has been removed as explained in Section 6.2, and Figure 8.17, where the influence of the spurious mode is clearly visible. The strains corresponding to the equilibrated

Figure 8.16 Boundary displacements of the equilibrated solution in which the spurious mode has been removed. Notice how the displacements at the vertices, amplified five times in the detail, are not exactly continuous.

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158

Equilibrium F.E. Formulations

Figure 8.17 Boundary displacements of an equilibrated solution affected by the spurious mode. In this figure the detailed displacements are shown with the same scale as the whole mesh.

Figure 8.18 The first row shows the element-wise displacement fields obtained from the boundary displacements in Figures 8.16 and 8.17, that is, unaffected and affected by the spurious mode. In the second row we show the displacement field corresponding to the difference between these two solutions, which is a rigid body motion for each element. Different scale factors are used for each plot, in order to better visualize them.

stresses fail to locally satisfy compatibility, both on the boundaries and inside the elements. While the latter point cannot be graphically verified, the first one is observable by checking that the displacements are not continuous at the vertices. From these boundary displacements, using the procedure in Section 8.2.1, we obtain the element-wise displacement fields, ũ E , presented in the first row of Figure 8.18. It is again observed that, even in the absence of spurious modes, these displacements

Recovery of Complementary Solutions

Figure 8.19 Compatible displacements on the star patches, recovered from the equilibrated solution that is free from spurious modes.

Figure 8.20 Compatible displacements: as recovered from the equilibrated solution; as obtained from the compatible model; and their difference. The maximum amplitudes of the displacements are 2.23924 and 2.27220. The maximum amplitude of the difference, which does not occur where the displacement is maximum, is 0.043216. Different scales are applied to the arrows in order to make the small differences visible.

are discontinuous between elements and do not satisfy the kinematic boundary conditions. The different nature of these boundary and element-wise displacement fields is evident by comparing their values at common points, as detailed in Figures 8.16 and 8.18: • The boundary displacements of the equilibrated solution are defined on the sides, so that we have the same displacement for two adjacent elements, except at the vertices, where we have as many displacements as there are incident sides.

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Equilibrium F.E. Formulations

• The element-wise displacement fields imply that along an internal side a different displacement is defined for each element and that at a vertex there are as many displacements as there are incident elements. Starting from the solution without the effect of the spurious mode, the compatible solutions for each of the five star patches considered in the recovery of compatibility process are presented in Figure 8.19, where the vertex and the link of each star are indicated. On each link the displacements are zero, so that a compatible solution is obtained by recovery when we sum those five solutions. This is presented in Figure 8.20, together with a compatible solution, obtained from a compatible model, as well as their difference. All the fields in this Figure are compatible and are described by the same approximation basis. A compatible solution would also be obtained by combining the element-wise displacements affected by the spurious mode. The quality of the resulting solution is considerably worse, but it would nevertheless be compatible. 8.4.2 Recovery of Equilibrium From a Compatible Solution

Figures 8.21 to 8.23 illustrate the general recovery of equilibrium process, described in Section 8.3.1. The cases of linear displacements and the element by element recovery schemes are illustrated in the references given in Sections 8.3.2 and 8.3.3. In these figures we represent the components of the stress tensor either in the (x, y) or in the (x′ , y′ ) reference frame, according to the scheme in Figure 8.15. The magnitude of components in the colour scale are symmetric, as given in Table 8.1, so that the green colour always corresponds to a zero value. The starting point of the recovery of equilibrium process is the stress field in Figure 8.21, corresponding to the compatible finite element solution, which fails to locally satisfy equilibrium, either on the boundaries or inside the elements. While the latter point cannot be graphically verified, the first one is observable by checking that

Figure 8.21 Stresses obtained from the compatible solution. (See plate section for colour representation of this figure).

Recovery of Complementary Solutions

Figure 8.22 Equilibrated stresses on the star patches, recovered from the compatible solution. (See plate section for colour representation of this figure).

𝜎yy is not varying as prescribed on the top face, 𝜎xy is not zero on the external sides and the rotated components of the stress tensor acting on the internal boundaries are unequal. The application of the star patch recovery process leads to the five stress distributions in Figure 8.22, which are organized according to the position of the vertex of the star. Each of these distributions locally verifies equilibrium with the partition of unity of the applied loading together with the fictitious body forces. Effectively we have continuity of the rotated components of the stress tensor acting on the internal boundaries (equal to zero on the link of each star patch), the shear stresses on the sliding supports are always zero and 𝜎yy has quadratic variations on the top face, corresponding to the product of the applied normal traction with the partition of unity functions. The equilibrated solution obtained by recovery, the sum of the five solutions in Figure 8.22, is presented in Figure 8.23a, followed by the equilibrated solution in Figure 8.23b, obtained from the hybrid equilibrium model.

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Equilibrium F.E. Formulations

(a)

(b)

Figure 8.23 (a) Equilibrated stresses recovered from the compatible model (b) Stresses obtained from the equilibrated model. (See plate section for colour representation of this figure). Table 8.1 Magnitude of the stress components in Figures 8.21 to 8.23. 𝜎xx

𝜎yy

𝜎xy

𝜎x ′ x ′

𝜎y ′ y ′

𝜎x ′ y ′

Minimum

−1.6

−3.0

−0.5

−2.0

−2.0

−1.2

Maximum

1.6

3.0

0.5

2.0

2.0

1.2

Recovery of Complementary Solutions

Two relevant points must be observed with respect to the sum that leads to the recovered solution: the fictitious body forces that were present in each of the star patches of Figure 8.22 disappear since they sum to zero, and the partitioned loads sum to the original values. Equilibrium between elements is observed because the tractions on the free sides of the link of each star are always zero and inter element equilibrium was verified inside each star. Although Figures 8.23a and 8.23b represent two equilibrated solutions, which use the same basis for stress approximation, it is apparent that they are not equal. The global solution selects the stress field that corresponds to the strains that are the ‘least incompatible’. It should finally be noted that the loads on each patch are admissible, and the potential spurious modes are not activated by them. However, with quadratic approximations the question of admissibility can be problematic, as pointed out in Section 6.4. The occurrence of inadmissible fictitious body forces in the recovery process will be exemplified in Section 9.1, where this problem is considered with meshes obtained by uniform mesh refinement.

8.5 Extensions of the Recovery Procedures As pointed out at the beginning of this Chapter, the recovery concepts can be applied to 3D solid continua, and should be applicable to plate bending problems, as well as potential problems. Although the examples in the previous sections only illustrate 1D or 2D problems, the corresponding formulations are not dependent on the dimension of the problem; only the details will vary when the dimension changes. Plate bending problems require special consideration according to which plate theory is used. We will briefly mention the two usual cases: 8.5.1 Reissner–Mindlin Theory

In this case the differential operators  M and  ⋆M remain 1st order but include constant terms as detailed in Appendix A.1.4. The recovery procedures follow similar lines to those for 2D or 3D continua, and the PU functions can retain the same piecewise linear forms. Recovery of equilibrium from a compatible solution again requires analyses of star patches with self-balanced loads. When the displacement space  k is based on polynomials of degree 1 for translations, we again have a lack of rotational equilibrium of the loads on a star patch, which requires correcting. The contributions of an element to a vertex moment, which represents a lack of rotational equilibrium of a star patch, can be defined so that they sum to zero for the three vertices of that element. Hence similar procedures can be used to determine a corrective stress field as in the 2D case. 8.5.2 Kirchhoff Theory

In this case the differential operators  K and  ⋆K become 2nd order as detailed in Appendix A.1.4, and we now consider briefly each route of recovery. 8.5.2.1 Recovery of Compatibility

We can directly obtain element-wise displacement fields, ũ E , as defined in Section 8.2.1, consisting only of transverse displacements. The rigid body movements are obtained by

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minimizing the contour integral of (ue − 𝒗e )2 , in which the integral incorporates terms from the difference of transverse displacements and normal rotations on the boundary, as well as vertex displacements. These displacements are not compatible between elements, but will generally provide a good approximation of the displacement field. For the star patch recovery process the derivation of the fictitious body forces and strains becomes more complicated. Since  K =  1  2 is a second order operator, the corresponding derivations involve performing the derivatives in two stages. The corresponding expressions are not presented here. In any case the analyses to obtain fully conforming Kirchhoff elements require shape functions of at least degree five, as, for example, in the TUBA family of compatible Kirchhoff elements (Argyris et al., 1968; Bell, 1969). 8.5.2.2 Recovery of Equilibrium

Starting from a compatible solution, for example generated from TUBA elements of degree ≥ 5, it is possible to use the differential operators for the Kirchhoff plate to obtain self-balanced loads on each star patch. As second derivatives are involved, the resulting expressions are again more complicated than for 2D or 3D continua. They will be the subject of further research. 8.5.3 Non-Simplicial Elements

We note that all these applications tend to be problematic when non-simplicial elements are involved, since within each element the partition of unity functions Ψi are no longer linear and their derivatives Ψi and  ⋆ Ψi are no longer constant. Further complications arise when, for example, the elements in a conforming model are mapped with curvilinear coordinates with non-constant Jacobians, and we wish to recover equilibrium. Currently this kind of problem would not occur when recovering compatible strains from a hybrid equilibrium model, since in this case the models are generally defined with simplicial elements having straight sides.

References Akin JE 1994 Finite Elements for Analysis and Design. Academic Press. Allman DJ 1984 A compatible triangular element including vertex rotations for plane elasticity analysis. Computers & Structures 19(1-2), 1–8. Almeida JPM and Maunder EAW 2009 Recovery of equilibrium on star patches using a partition of unity technique. International Journal for Numerical Methods in Engineering 79, 1493–1516. Almeida JPM and Maunder EAW 2010 Recovery of Equilibrium on Three-Dimensional Star Patches and its Application in the Determination of Solution Bounds for Large Scale Problems. In ECCM 2010 IV European Conference on Computational Mechanics (ed. Allix O and Wriggers P). Argyris JH, Fried I and Scharpf DW 1968 The TUBA family of plate elements for the matrix displacement method. The Aeronautical Journal of the Royal Aeronautical Society 72(692), 701–709. Bell K 1969 A refined triangular plate bending finite element. International Journal for Numerical Methods in Engineering 1(1), 101–122.

Recovery of Complementary Solutions

Ladevèze P and Maunder EAW 1996 A general method for recovering equilibrating element tractions. Computer Methods in Applied Mechanics and Engineering 137(2), 111–151. Ladevèze P and Pelle JP 2005 Mastering Calculations in Linear and Nonlinear Mechanics. Springer, New York. Ladevèze P and Rougeot P 1997 New advances on a posteriori error on constitutive relation in f.e. analysis. Computer Methods in Applied Mechanics and Engineering 150(1), 239–249. Ladevèze P, Pelle JP and Rougeot P 1991 Error estimation and mesh optimization for classical finite elements. Engineering Computations 8(1), 69–80. Maunder EAW and Almeida JPM 2012 Recovery of equilibrium on star patches from conforming finite elements with a linear basis. International Journal for Numerical Methods in Engineering 89(12), 1497–1526. Pereira OJBA, Almeida JPM and Maunder EAW 1999 Adaptive methods for hybrid equilibrium finite element models. Computer Methods in Applied Mechanics and Engineering 176(1-4), 19–39. Tian R and Yagawa G 2007 Allman’s triangle, rotational DOF and partition of unity. International Journal for Numerical Methods in Engineering 69(4), 837–858.

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9 Dual Analyses for Error Estimation & Adaptivity The idea of using approximate solutions of a complementary nature to obtain bounds of the exact value from such approximations can be traced back to the ‘Measurement of a Circle’ by Archimedes (Heath, 1897), who used the perimeter of inscribed and of circumscribed polygons to obtain accurate bounds of the ‘ratio of the circumference to the diameter’. In the context of solid mechanics the work by Trefftz (1926) is usually identified as the first application of two solutions of a complementary nature for the determination of bounds of a given output, specifically the torsional stiffness of a square and of an angle section.1 The modern framework for the derivation of solution bounds in elasticity is based on the work by Prager and Synge (1947), which ‘applies to any elastic body possessing a positive definite strain energy function, quadratic in the components of stress’ to obtain ‘approximate solutions of elastic boundary value problems with errors which are calculable’. Prager’s approach has the limitation of only being applicable to either force or displacement driven problems, that is, when either set of boundary conditions is homogeneous. Furthermore, because these errors correspond to a global measure of the difference between solutions, they cannot be used to directly assess the quality of selected quantities, which, in terms of error estimation, are referred to as local outputs. This was made possible by the extensions proposed by Greenberg (1948) and Washizu (1953), who derived expressions that can be applied to bound the error of local outputs obtained from such approximate solutions. This approach was applied by Fraeijs de Veubeke (1961), at the Université of Liège, who extended it to the context of the finite element method (Fraeijs de Veubeke, 1965). Another relevant contribution is the work by Ladevèze. In his thesis (Ladevèze, 1975) he introduces ‘une nouvelle notion d’erreur, que nous appellerons erreur en relation de comportement relative à un couple admissible’.2 This concept of considering that for a pair of complementary solutions the error is ‘focused’ in the constitutive relation was subsequently exploited by him and his co-workers for many different problems (Ladevèze and Pelle, 2005), in parallel with the development of techniques for the recovery of statically admissible solutions, some of which have been discussed in Chapter 8. 1 for translation and comments see Maunder (2003). 2 A new concept of error, which we will term error in constitutive relation associated with an admissible pair. Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

Dual Analyses for Error Estimation

The constraint associated with requiring either force or displacement driven problems was removed by Debongnie (Debongnie, 1983; Debongnie et al., 1995), who showed ‘that dual analysis, in the sense of an error measure, remains valid with a very slight modification in the case of general boundary conditions’. The application of these concepts to finite element solutions, in the form presented in Almeida and Pereira (2006), enables obtaining tighter bounds of local outputs, equivalent to those that are obtained from the parallelogram inequality, applied by Babuška and Miller (1984) and Prudhomme and Oden (1999) for the estimation of the error in local outputs. More general formulae, applicable to any pair of equilibrated/compatible solutions are given in Almeida (2013). Similar results are given in Wang and Zhong (2015). The Chapter starts by laying out the basis of the procedures used to obtain bounds of the global energy error, whose characteristics are exemplified by its application to the simple problem already used in Chapters 4 and 8. From the expressions leading to these bounds, the relative importance of the error in the elements is assessed and this information is used to set the basis for an adaptive process. It is then shown that local outputs and their bounds can be obtained using a procedure that is very similar to the one used to obtain bounds of the error in the global energy. A goal oriented adaptive process is then presented, which aims to control the error in a given local output, or in a selected combination of local outputs. All the examples are given for two dimensional continua, but the formulae are generally applicable to any of the problems considered in this book.

9.1 Global Error Bounds In this Section we demonstrate how a bound of the strain energy of the error of a solution can be obtained for a problem with general boundary conditions. The derivation is inspired by the presentation by Debongnie (1983), using the notation and results from Chapter 7, but we note that Equations (9.1) and (9.2) can also be obtained by integrating by parts the definitions of the energies involved and using the properties of the exact solution (Oden et al., 1989). Any arbitrary kinematically admissible displacement field, u⋆ , can be written as a function of the exact solution u, u⋆ = u + eku . The error, eku , is a compatible displacement field with zero imposed displacements and with corresponding strains ek𝜀 . The total potential energy of this arbitrary approximate solution is given by (7.3): 1 a (u + eku , u + eku ) − pk (u + eku ) 2 k 1 1 = ak (u, u) + ak (u, eku ) + ak (eku , eku ) − pk (u) − pk (eku ) 2 2 1 = Π(u) + (ak (u, eku ) − pk (eku )) + ak (eku , eku ) 2 Because u is the exact solution, the term that is linear in the error vanishes for any admissible variation of displacements (see (7.7), where only the term in Γu is not zero for the exact solution). Therefore Π(u⋆ ) =

2(Π(u⋆ ) − Π(u)) = ak (eku , eku ) =∥ eku ∥2E ≡∥ ek𝜀 ∥2E .

(9.1)

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Equilibrium F.E. Formulations

This non-negative quantity can be interpreted as a squared norm, which corresponds to the strain energy of the error in displacements. It can be used as a measure of the error of u⋆ . Similarly, any arbitrary statically admissible stress field, 𝝈 ⋆ , can be written as a function of the exact solution 𝝈, 𝝈 ⋆ = 𝝈 + es𝜎 . The error, es𝜎 , is a self-balanced or equilibrated stress field, otherwise known as hyperstatic, that is, it is in equilibrium with zero body forces and zero applied tractions. The total complementary potential energy of 𝝈 ⋆ is 1 Πc (𝝈 ⋆ ) = as (𝝈 + es𝜎 , 𝝈 + es𝜎 ) − ps (𝝈 + es𝜎 ) 2 1 1 = as (𝝈, 𝝈) + as (𝝈, es𝜎 ) + as (es𝜎 , es𝜎 ) − ps (𝝈) − ps (es𝜎 ) 2 2 1 s = Πc (𝝈) + (as (𝝈, e𝜎 ) − ps (es𝜎 )) + as (es𝜎 , es𝜎 ) 2 Because 𝝈 is the exact solution, the linear term in es𝜎 also vanishes, on account of (7.8). Therefore 2(Πc (𝝈 ⋆ ) − Πc (𝝈)) = as (es𝜎 , es𝜎 ) = ||es𝜎 ||2Ec .

(9.2)

This is a measure of the error of 𝝈 ⋆ , corresponding to the complementary strain energy of the error in stresses. We note that, for the linear elastic case, ||e𝜎 ||Ec = ||f e𝜎 ||E and ||ke𝜀 ||Ec = ||e𝜀 ||E . Because for the exact solution, as shown in 7.1.2, Π(u) + Πc (𝝈) = 0, by adding (9.1) and (9.2) we obtain that ||ek𝜀 ||2E + ||es𝜎 ||2Ec = 2(Π(u⋆ ) + Πc (𝝈 ⋆ )) = 𝜖 2 ≥ 0.

(9.3)

The following results are then immediate: ||eku ||2E ≤ 𝜖 2 ;

(9.4)

||es𝜎 ||2Ec ≤ 𝜖 2 .

(9.5)

By defining ek𝜎 = 𝝈(u⋆ ) − 𝝈, and es𝜀 = 𝜺(𝝈 ⋆ ) − 𝜺, it is possible to write an error of the compatible solution as as (ek𝜎 , ek𝜎 ) and an error of the equilibrated solution as ak (es𝜀 , es𝜀 ). For the linear elastic case all these measures are equivalent. A very important result that follows regards the orthogonality, in the sense of the strain energy products, of the two errors, either in stress, es𝜎 and ek𝜎 , or in strain, es𝜀 and ek𝜀 : ak (es𝜀 , ek𝜀 ) = 0;

(9.6)

as (es𝜎 , ek𝜎 )

(9.7)

= 0.

This is demonstrated, as follows, by considering the integration by parts of either term. Starting from the error of the compatible solution leads to the work of the error in displacements, which is not zero, acting upon the error in body forces and applied tractions of the equilibrated solution, which is zero by definition. When we start from the error of the equilibrated solution, the self-equilibrated stress field is integrated by parts, and the result corresponds to the work done by the error in stresses and the error in the reactions, which are not zero, acting upon the lack of

Dual Analyses for Error Estimation

compatibility and the error on the imposed displacements of the compatible solution, which are also zero by definition.3 Denoting 𝝈(u⋆ ) by 𝝈 k and 𝝈 ⋆ by 𝝈 s , we then have that: as (𝝈 k − 𝝈 s , 𝝈 k − 𝝈 s ) = as ((𝝈 k − 𝝈) − (𝝈 s − 𝝈), (𝝈 k − 𝝈) − (𝝈 s − 𝝈)) = as (ek𝜎 − es𝜎 , ek𝜎 − es𝜎 ) = as (ek𝜎 , ek𝜎 ) + as (es𝜎 , es𝜎 ) − 2 as (ek𝜎 , es𝜎 ); = as (ek𝜎 , ek𝜎 ) + as (es𝜎 , es𝜎 ) ≡ 𝜖s2 ;

(9.8)

similarly, denoting 𝜺(u⋆ ) by 𝜺k and 𝜺(𝝈 ⋆ ) by 𝜺s : ak (𝜺k − 𝜺s , 𝜺k − 𝜺s ) = ak (ek𝜀 − es𝜀 , ek𝜀 − es𝜀 ) = ak (ek𝜀 , ek𝜀 ) + ak (es𝜀 , es𝜀 ) − 2 ak (ek𝜀 , es𝜀 ); = ak (ek𝜀 , ek𝜀 ) + ak (es𝜀 , es𝜀 ) ≡ 𝜖k2 .

(9.9)

These results relate the (complementary) strain energies of the difference between a pair of complementary solutions, which is a computable quantity, with the sum of the strain energies of the errors of these solutions. An immediate consequence is that 𝜖, the error bound in (9.4) and (9.5), is a function of 𝜺s and 𝜺k , which can be obtained as a sum of elemental contributions. The strain energy of the error of either solution is bounded by the strain energy of the difference between the stresses or the strains of the pair of complementary solutions.

The error of the stress field obtained by averaging a kinematically and a statically admissible solution is ea𝜎 = (𝝈 k𝜎 + 𝝈 s𝜎 )∕2 − 𝝈 = (ek𝜎 + es𝜎 )∕2 and a similar quantity is defined for the error in strains. We then have that 1 ak (ea𝜀 , ea𝜀 ) = ak (ek𝜀 + es𝜀 , ek𝜀 + es𝜀 ); 4 1 = (ak (ek𝜀 , ek𝜀 ) + ak (es𝜀 , es𝜀 ) + 2 ak (ek𝜀 , es𝜀 )); 4 𝜖2 1 = (ak (ek𝜀 , ek𝜀 ) + ak (es𝜀 , es𝜀 )) = k . 4 4 Similarly we obtain that as (ea𝜎 , ea𝜎 ) =

𝜖2 1 (as (ek𝜎 , ek𝜎 ) + as (es𝜎 , es𝜎 )) = s . 4 4

The exact strain energy of the error of the average stress field is equal to the sum of the energies of the error of the complementary solutions, divided by four.

3 An alternative definition can be obtained from the error of a stress potential (typically the Airy stress function for 2D, the Maxwell or the Morera stress functions for 3D, or the Southwell functions for bending plates), multiplied by the compatibility defaults, which are zero for the compatible solution.

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tymax = 3

Force driven

tmax y =3

Displacement driven

umax y = 1.4381

Mixed

umax y = 1.4381

Figure 9.1 Different problems considered for the computation of errors.

9.1.1 Revisiting the simple example

The characteristics of the error for uniform meshes are illustrated here for the simple problem that was presented in Chapter 8. Three different types of action are now considered (Figure 9.1): • a force driven problem – a distributed vertical force, with a linear variation of tractions at the top; • a displacement driven problem – an imposed vertical displacement, with a linear variation at the bottom;4 • a mixed problem, corresponding to the combination of the other two. The numerical values of these actions were selected so that the strain energy of the force driven problem is close to one, and either total energy of the mixed problem is close to zero. Reference values were obtained for a non-uniform mesh with 1524 elements, using approximation functions of degree 7, for which the difference between complementary solutions is smaller than the precision used. Energy values are given in Tables 9.1 to 9.4 for the initial mesh of four elements, and then for uniformly refined meshes obtained by dividing every element into four. Both equilibrated and compatible solutions, global and locally recovered solutions, are considered, for approximation degrees from 1 to 3. It is also indicated whether the strain energy of the solution is larger, smaller or numerically equal to the reference solution. Some recovered solutions are absent from these tables. In 9.2 this happens in the first set of results when we try to recover equilibrium from a finite element solution with linear displacements, in which case the equilibrium of moments in the patch is not verified without correcting the lack of rotational equilibrium that arises for the star patches, as discussed in Section 8.3.2. For the second set of missing values in that table, the recovery procedure fails for the larger meshes because the self-balanced body forces on the stars are inadmissible for certain patches of elements of degree 2. Hence they excite the single spurious mode discussed and illustrated for a tessellation in Section 5.2.4. In this case a solution cannot be found for these patches. However, in the case of the mesh with four elements all stars are connected to a support, which blocks the spurious mode mentioned in that Section, while the internal pathological mode is not excited, so that a solution exists. 4 The corresponding solution is singular, since we are imposing a non-zero shear strain at the two lower corners of the domain, while requiring a zero shear stress on both adjacent sides.

Dual Analyses for Error Estimation

Table 9.1 Energies of equilibrium finite element solutions. Force driven

Displacement driven

Mixed

Elements

ds

Uc

Vc

Uc

Vc

Uc

Vc

4

1

1.34969 >

0.00000

0.11490


1.42821

16

1

1.33861 >

0.00000

0.11744
𝜀Y , then assuming that the strain is equal to 𝜎∕E leads to a plastic potential, Φ𝜀 , that violates the yield condition. A non-zero plastic stress must be considered in order to obtain an admissible stress that does not violate the yield condition. The strain is directly derived from the elastic stress, 𝜎 + 𝜎P 𝜎 . 𝜀= E = E E The plastic stress is then obtained by solving ( ) ) )( ( 𝜎 + 𝜎P E 𝜎 . 𝜀Y − , 𝜎P = 0 =⇒ 𝜎P = E 1 − ′ Φ𝜀 E E E The equivalence between Equations (11.4) and (11.5) is demonstrated by realizing that 𝜺P = f 𝝈 P . 11.2.2.4 Imposing the Yield Condition in a Weak Form

An alternative to the ‘discrete’ enforcement of the yield condition at the integration points is to consider an explicit weak form of the yield condition. We illustrate this point for the case when the plasticity conditions are expressed in the stress space, but its extension to plasticity in the strain space is straightforward. Instead of Φ ≤ 0, ∀x, we require that ∫ Ωe

ΛΦ dΩ ≤ 0,

∀Λ such that Λ(x) ≥ 0, ∀x ∈ Ωe ,

so that when all possible Λs are used this condition becomes equivalent to the local enforcement of the yield condition. In the numerical model a finite set of Λs is used for each element, Λie , leading to the definition of generalized plastic potentials, which impose the yield condition in a weak form: ̂ ie = Φ

∫Ωe

Λie Φ dΩ ≤ 0.

The corresponding plastic multipliers 𝛾̂ei must be non-negative, and complementarity is ̂ ie 𝛾̂ei = 0. The plastic multipliers cause plastic strains directly imposed by requiring that Φ within the element which are given by ∑ 𝜕Φ 𝜺P = (̂𝛾ei Λie ) . 𝜕𝝈 i

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220

Equilibrium F.E. Formulations

The strict imposition of these conditions in the incremental process causes additional difficulties, because the increments of the plastic strain vary within the element not only because of Λie , but also on account of the variation of 𝜕Φ∕𝜕𝝈, which must be accordingly combined for every point. This corresponds to another weak form of the conditions of the continuous problem, since plastic strains may occur at points where the plastic potential is not zero. In practice what seems to be a ‘simple and clean’ approach becomes excessively complex, because in addition, the explicit definition of the integrals is in general infeasible and a numerical integration is required. It is then preferable to resort to the option of working directly with the plasticity conditions at the integration points.

11.3 Limit Analysis 11.3.1 Introduction

The procedures for non-linear analysis outlined in Section 11.2.2 cannot be directly applied when material non-linear behaviour exhibits perfect plasticity, that is, without work hardening, Φ𝜎 = 𝜎 − 𝜎Y , and plastic deformation can continue under constant stress. However this material property is exploited in limit analysis where the capacity of a structure to support load, that is, the collapse load, is determined without the need to perform a non-linear incremental form of analysis. In addition to using this material property, limit analysis normally assumes that displacements are small, which implies that equilibrium equations can be based on the undeformed geometry, and the normality rule applies to plastic strains at yield. With these assumptions the upper and lower bound theorems apply (Drucker et al., 1952; Kamenjarzh, 1996; Nielsen and Hoang, 2011), so that bounds on the collapse load can be determined by using kinematically or statically admissible fields of displacement or stress respectively. In this Section we discuss the application of hybrid equilibrium elements to obtain lower bound solutions. 11.3.2 General Statement of the Problem as a Mathematical Programme

Let 𝝈 = 𝜆 𝝈 0 + 𝝈 hyp , where 𝝈 0 is statically admissible with the applied loads, b̄ and t̄ , and 𝝈 hyp is any hyperstatic stress field. Then the limit analysis problem is to maximize the load multiplier 𝜆 when subject to the stress constraints Φ𝜎 (𝝈(x)) ≤ 0 for all points x in the domain of the problem. In the usual terminology of a mathematical programme, 𝜆 is the objective function, the hyperstatic stress fields, 𝝈 hyp , are the variables and the yield conditions form the constraints. When this is implemented using a hybrid equilibrium finite element model, we can formulate the programme by either of the following approaches: 1) Determining a particular stress field 𝝈 0e for each element and a basis Se as in (4.1), and introducing local equilibrium equations that enforce codiffusive tractions at each interface, simultaneously with the yield constraints. In this case hyperstatic stress fields are not explicitly constructed. 2) Determining a particular stress field 𝝈 0 for the complete mesh, together with an explicit basis for the hyperstatic fields, belonging to the space s0 .

Non-Linear Analyses

11.3.2.1 Formulation (1)

The equilibrium equations for each side/face can be enforced during the optimization with static variables ŝ in a separate set of equality constraints of the form: D̂s = 𝜆t̂ 0 , where, as in (4.13), we balance the tractions prescribed on boundary sides with those derived by projection from 𝝈 0e . Then 𝝈 = Ŝs + 𝜆𝝈 0e . Clearly these equilibrium equations need to be consistent in order to obtain a solution, that is, the traction parameters in t̂ 0 must be admissible and therefore orthogonal to any spurious mode of the mesh. 11.3.2.2 Formulation (2)

This approach starts by considering that a particular stress field can be obtained from prescribed loads, b̄ and t̄ , that is, corresponding to a unit load multiplier, in two stages. In the first stage we define element-wise 𝝈 0e that satisfy  ⋆ 𝝈 0e + b̄ = 𝟎, as in Formulation (1), and then derive the corresponding contribution to t̂ 0 using (4.13). In stage 2 we seek a solution to: D̂s0 = t̂ 0 , and then a globally equilibrated particular solution, 𝝈 0 = Ŝs0 + 𝝈 0e . Although t̂ 0 is uniquely determined from (4.13), a solution for ŝ 0 is generally not unique, and in order to exist the prescribed loading must be admissible when spurious modes are possible. A basis for the hyperstatic fields can be obtained from the nullspace B of D, as introduced in Section 5.1, by taking: 𝝈 hyp = Ŝshyp ,

with

̂ ŝ hyp = B h,

where the nhyp dimensional vector ĥ collects the hyperstatic variable parameters. The use of the hyperstatic stress fields offers the advantage of a much smaller number of static variables when compared with Formulation (1), but requires the determination of the nullspace B. This task can be simplified when the elements are free of spurious modes, as can be arranged, for example, by using stable macro-elements. In this case, all tractions up to degree d are admissible, and we can exploit the concept of load paths selected from a dual graph of the mesh. This graph represents the topological connectivity of the elements, and it consists of a node in each element and an edge joining a pair of nodes when an interface exists between the corresponding elements (Armstrong, 1979; Maunder and Savage, 1994; Maunder, 1987). Hyperstatic stress fields can be derived from element tractions associated with (a) resultant forces and moments from constant and/or linear modes of traction identified with independent closed paths, for example, a circuit around a vertex of a closed star patch; and (b) higher degree self-balanced modes of traction identified at interfaces between pairs of elements. Single elements can themselves contain hyperstatic stress fields, for example, 2D triangular elements of degree d > 3. The utilization of the approximation basis for tractions based on Legendre polynomials is particularly useful in this context. A particular stress field can also be derived from tractions obtained from load paths associated with a tree type subgraph of the dual graph when it is rooted at a side/face on Γu .

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11.3.2.3 Yield Constraints

In the finite element model, the pointwise material yield conditions now take the form: Φ𝜎 (𝝈(xi )) ≤ 0,

for a predetermined finite set of material points with coordinates xi .

As yet the authors are unaware of a way to enforce such constraints for all intervening points unless stress approximation functions are element-wise constant or linear, as noted in Section 11.2.2. However for higher degree stress fields, a possible approximate method consists of first enforcing these constraints at a selected finite set of control points within each element leading to a collapse multiplier 𝜆0 . This factor may be excessive since yield may be violated at other points not included in the set. It is then necessary to search the corresponding stress field for additional point(s), x̄ i , where the yield condition may be violated, that is, Φ𝜎 (𝝈(̄xi )) > 0. Searching can be done over each element by the use of a fine grid of additional points. Where Φ𝜎 (𝝈(̄xi )) > 0 a scale factor 𝛼i < 1 is determined so that Φ𝜎 (𝛼i 𝝈(̄xi )) = 0 and the minimum value of 𝛼i is sought. The collapse multiplier is then reduced to 𝜆1 = min(𝛼i )𝜆0 . Due to the convex nature of the yield function, we can then assert that 𝜆1 is a safe lower bound collapse multiplier, provided the grid is sufficiently fine. Solutions may be determined in a linear programme (LP) after appropriate linearization of the yield surfaces, or in a non-linear programme, for example, a second order cone programme (SOCP) (Alizadeh and Goldfarb, 2003). Either programme offers a choice of formulation in primal or dual forms. 11.3.2.4 Application of the Complementary (Dual) Programme

The statement of the problem has been expressed so far in terms of the primal form of the mathematical programme. Alternatively, the programme can be expressed in its dual form where the objective function is to minimize the plastic work of the yielding material subject to: • the weak compatibility conditions of the kinematic parameters that are conjugate (dual) to the static parameters of the primal programme; and • verifying a plastic potential condition Φ ≥ 0. The two forms of programme differ in terms of the relative numbers of variable parameters and their constraints, and these numbers can affect the computational efficiency of the associated numerical techniques. However it should be noted that the static parameters obtained from non-optimal primal solutions are always safe from the point of view of structural design or assessment based on limit analysis, whereas solutions from the dual programme are only safe when the solution has been optimized. 11.3.3 Implementation for Plate Bending Problems

Plate bending problems may use yield conditions defined by the von Mises or the square (for example, Nielsen and Hoang (2011)) yield criteria, for moments in metallic plates or reinforced concrete slabs, respectively. If yield is governed solely by the moment field, then Kirchhoff plate elements are appropriate (for example, Maunder and Ramsay (2012)); otherwise Reissner–Mindlin elements should be used to allow for a yield condition that may involve transverse shear forces.

Non-Linear Analyses

11.3.4 Numerical Example

We consider the analysis of the square plate with side length 2, and simply supported on two adjacent sides, previously considered in Sections 2.2, 4.8.3 and 9.5, now subject to a uniformly distributed load, p. Nielsen’s square yield criterion is assumed for an isotropic homogeneous plate with equal sagging and hogging plastic yield moments mY . A simple yield line pattern, similar to that in the left hand part of Figure 11.5, with 2 parameters, gives an upper bound of the limit load equal to 1.37544 mY . Quadratic and cubic moment approximations are used, subject to the square yield criterion. For every element the yield condition is imposed locally at each control point of a triangular grid.3 An adapted version of the code by Passos (2011) was used, which basically maximizes the load parameter, subject to the (linear) equality constraints that impose equilibrium and to the (quadratic and convex) inequality constraints that enforce the yield condition at each point. Note that this code corresponds to a very straightforward implementation and has limitations when dealing with large problems. Table 11.1 lists the values of the limit load obtained for the different approximations and for the meshes represented in Figure 11.4, when a unit plastic moment is considered. For the mesh with 412 elements, the identification of the points that reached yield and the bending moment diagrams mxx and mxy for the ultimate load are presented in Figure 11.5. Table 11.1 Limit loads for quadratic stress approximations and Nielsen’s yield criterion. Number of elements

ds = d𝑣 = 2

ds = d 𝑣 = 3

6 points

15 points

45 points

10 points

45 points

91 points

26

1.41183

1.37363

1.37118

1.39520

1.37252

1.37224

66

1.39840

1.37318

1.37242

1.38150

1.37289

1.37284

242

1.38686

1.37315

1.37291

1.37732

1.37311

1.37306

412

1.38434

1.37316

1.37296

1.37562

1.37306

1.37302

1.37544 is an upper bound of the limit load. Note that, due to the characteristics of the SOCP solver, numerical errors may be present in solutions with a higher number of elements and/or control points.

Figure 11.4 Meshes used for limit analysis of the square plate simply supported on two adjacent sides, subject to a uniformly distributed load.

3 With 6, 10, 15, 45 or 91 points, corresponding to 3, 4, 5, 9 or 13 points in each direction.

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Equilibrium F.E. Formulations

(a)

(b)

(c)

Figure 11.5 (a) Location of the points where plastic strains develop for the ultimate load on the mesh with 412 elements, with a cubic approximation of the moments and 91 control points; distribution of moments mxx in (b) and mxy in (c) for quadratic moments and 45 control points. (See plate section for colour representation of this figure).

The results in Table 11.1 generally follow a pattern: • For a given finite element mesh and degree of approximation, increasing the number of control points corresponds to a decrease of the limit load. This happens because there are fewer regions inside each element where the yield condition may be violated, and the value obtained tends to a lower bound of the limit load. • Increasing the number of elements corresponds to a decrease of the limit load when the number of control points is ‘small’ and to an increase of the limit load when this number is ‘large’. In the first case the stronger enforcement of the yield conditions associated with having more elements and thus more control points dominates. The increase of the limit load when the yield condition is enforced with a very high density of points corresponds to a convergence from below to the exact limit load. Note that this convergence is not guaranteed because the enforcement of the yield condition is not strictly observed everywhere.

11.4 Geometric Non-Linearity In this Section we present an approach to including the effect of large displacements in the hybrid equilibrium finite element formulation. This technique, which can be classified as a corotational formulation, is similar to those used for displacement based elements by, for example, Wempner (1969), Horrigmoe and Bergan (1976), Belytschko and Hsieh (1973), Crisfield and Moita (1996), Felippa and Haugen (2005), Izzuddin (2005) and De Borst et al. (2012). A similar implementation for quadrilateral hybrid equilibrium Reissner–Mindlin flat shell elements has been proposed by Maunder and Izzuddin (2014), where each quadrilateral is treated as a macro-element consisting of four triangles, in which the force stress resultants are linear and the moments are quadratic, while the side translations are linear and the rotations are quadratic. The corotational formulations become simple and almost immediate by assuming that the strains are small, so that when the large rigid body displacements of each element are ‘removed’ the strain state of each element can be accurately characterized using the equations of the linear problem. The small strain hypothesis also implies that equilibrium inside each element can be enforced in a globally rotated, but undeformed element, and that when equilibrium

Non-Linear Analyses

between adjacent elements is considered the deformation of the interfaces can be neglected. In the following, we restrict our attention to models of 2D continua, but similar approaches should be applicable to 3D and plate bending. 11.4.1 Weak Compatibility for Large Displacements With Small Strains

Following the previously cited authors, the displacement field of element e can be decomposed in two parts, a rigid body motion, ueR and a strain inducing movement, ue𝜀 . For simplicity of notation, index e will be assumed in the following when discussing a generic element. It is also assumed that every element in a mesh has its initial axes in the same directions, but with different local origins. We then have: u = uR + u𝜀 . The corotational approach allows arbitrarily large values for uR , while u𝜀 is taken to be small, so that the strains can be assumed to be proportional to these displacements. In 2D problems an element may have three rigid body displacements, two translations (uTx and uTy ) and a rotation (𝜃R ). These components are collected, as in Section 8.2.1, in vector 𝚫R . The rigid body displacement of an arbitrary point with the initial position vector r = {x, y}, is then defined by the non-linear equations: { } uTx − y sin 𝜃R − x (1 − cos 𝜃R ) uR (𝚫R ) = . (11.6) uTy + x sin 𝜃R − y (1 − cos 𝜃R ) This expression shows that the rigid body translations correspond to the displacement of the origin of the reference frame, uT , which change if a different origin is considered. However, the rotation remains unchanged, provided the same uR is considered. ̆ y̆ ), The rigid body rotation naturally defines a reference frame local to the element, (x, which is illustrated in Figure 11.6. The corresponding transformation of the components ̆ is expressed by a rotation matrix T R , of an arbitrary vector, a to a, [ ] cos 𝜃R sin 𝜃R ă = T R a = a, − sin 𝜃R cos 𝜃R and vice-versa by the transposed matrix T TR . When an element is subjected to a rigid body rotation 𝜃R , the initial position r associated with the arbitrary point is changed to r 𝜃 , maintaining the same coordinates in uε

Figure 11.6 Large displacements of a body, with small strains. The undeformed configuration marked with the dashed line considers just the effect of the rigid body translation, uT , while the dotted line correspond to the effect of uR = u T + u 𝜃 .

u

uθ uT y ˘

uT y x

x ˘

θR

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Equilibrium F.E. Formulations

the local frame, r̆ 𝜃 = r. In the initial frame we have therefore that r 𝜃 = T TR r, so that the displacement induced by this rotation becomes: u𝜃 = r 𝜃 − r = T TR r − r. The strain inducing displacement, assumed to be small, also illustrated in Figure 11.6, is u𝜀 = u − uT − u𝜃 . In the local frame this is expressed as ŭ 𝜀 = T R (u − uT + r) − r, where we recall that r̆ 𝜃 = r. These displacements can be used in the weak form of compatibility (4.8), as defined in the local frame, that is, using the strains in the local frame, weighted by a local ̆ which will be defined in the next self-equilibrated stress approximation function, S, Section. The ensuing integration by parts is not affected by the magnitude of the displacements involved.4 This results in the following expression for the generalized strains, as obtained from the displacements of the sides and from the rigid body parameters: ê̆ 𝜀 =

∫Ω̆

=

∮Γ̆

=

∮Γ̆

=

∮Γ̆

T ̆ ̆ ŭ 𝜀 dΩ S̆ 

T ̆ T R (u − uT + r) dΓ̆ − S̆  T ̆ T R (u + r) dΓ̆ − S̆  T ̆ ̂̆ − V̆ dΓ̆ 𝒘 S̆ 

∮Γ̆

∮Γ̆

∮Γ̆

T ̆ ̆ r dΓ; S̆ 

T ̆ ̆ r dΓ; S̆ 

T ̆ ̆ r dΓ, S̆ 

where we recognize that an arbitrary rigid body translation produces zero work on a self-balanced stress field. The concept used in Chapter 4, of replacing the displacements on the boundary by their approximations, is generalized so as to include the ̆ = initial position vector of each point on the boundary, that is, 𝒘 = u + r = 𝒗 + r, or 𝒘 T R (𝒗 + r) = 𝒗̆ + T R r. Recall that, as illustrated in Figure 11.7 and further detailed in Figure 11.8, r is the initial position vector of a point (in the undeformed configuration), while 𝒘 = u + r is its final position (in the deformed configuration). The approximation functions in V̆ can be used to represent both displacements 𝒗̆ and the position vector r on the boundary of the rotated element. For straight sides an exact representation of r just requires linear approximation functions. Then r = V̆ r̂ (which could also be expressed as r̆ = V̆ r̂̆ ), with )−1 ( T T V̆ V̆ dΓ V̆ r dΓ. r̂ = ∮Γe ∮Γe The constitutive relation is unchanged from the case of small displacements. When initial strains and initial stresses are assumed to be zero, the generalized strains for each 4 If u𝜀 is not small this strain measure is not correct.

Non-Linear Analyses

Figure 11.7 Large displacements of a body, with small strains. The position vectors of a point in the initial, rotated and final configurations, r, r 𝜃 and 𝒘.

u



w

θR

r

r

Figure 11.8 Large displacements of a body, with small strains. When the rigid body translation is removed, the strain inducing displacement, u𝜀 , is equal to the difference between the position vector of the deformed configuration, 𝒘, and the rotated undeformed position vector, r 𝜃 .

uε uθ w

rθ r θR

element, induced by the stresses, are: ê̆ 𝜎 =

∫Ω̆ e

T S̆ f̆ S̆ dΩ ŝ̆ .

At element level the compatibility between the generalized strains induced by the stresses, ê̆ 𝜎 , and by the boundary displacements, ê̆ 𝜀 , can then be rewritten as: ∫Ω̆ e

T S̆ f̆ S̆ dΩ ŝ̆ =

∮Γ̆ e

T ̆ ̂̆ − V̆ dΓ 𝒘 S̆ 

∮Γ̆ e

T ̆ V̆ dΓ̂r , S̆ 

or, ̂̆ = D ̆ 𝒘 ̆ r̂ . −̆ ŝ̆ + D T

T

(11.7)

where the right hand term corresponds to the constant part of the generalized strain associated with the rigid body rotation, u𝜃 . ̆ which is defined in the local frame of the element, corresponds to matrix D Matrix D, in (4.12). 11.4.2 Equilibrium

When u𝜀 is small, equilibrium inside each element can be written in the local frame, so that it is verified by the following generalization of (4.1): 𝝈̆ = S̆ ŝ̆ + 𝝈̆ 0 .

(11.8)

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Equilibrium F.E. Formulations

The approximation functions in S, defined in the original position of the element, can be exactly the same as in the rotated frame of the element, directly replacing (x, y) by ̆ y̆ ). A particular solution 𝝈̆ 0 could also be considered to balance the body forces trans(x, ̄ however we will also assume, for simplicity, that formed to the local frame, b̄̆ = T R b, there are no body forces. Equilibrium between elements has to be written in terms of the generalized tractions in the global frame. It is nevertheless more convenient, for an isolated element, e, to begin by expressing these generalized tractions in its local frame. Each element contribution, as in (4.4), is ∫Γ e

T ̆ T V̆  S̆ dΓŝ̆ ,

or ̆ ŝ̆ = t̂̆ e , D

(11.9)

where the explicit reference to an element is made because, in the assembled t̂̆ , these elemental contributions on a side must be summed and balanced with the prescribed boundary tractions. 11.4.3 Transformation of Boundary Displacement Parameters and Generalized Tractions

̆ that is, V = V̆ , When the same basis is used to approximate each component of 𝒗 and 𝒗, the transformation matrix in an element, T R , can be generalized so that ̂ 𝒗̆ = T R V 𝒗̂ = V̆ T R𝑣 𝒗; this implies that 𝒗̂̆ = T R𝑣 𝒗̂

and

𝒗̂ = T TR𝑣 𝒗̂̆

(11.10)

that is, matrix T R𝑣 transforms boundary displacement parameters defined in the global frame into their counterparts in the local frame and its transpose performs the inverse transformation. Likewise, ̂̆ ̂̆ = T 𝒘 ̂ ̂ = T T 𝒘. (11.11) and 𝒘 𝒘 R𝑣

R𝑣

̂ = 𝒗̂ + r̂ , we also have that for each element Since 𝒘 ̂̆ = 𝒗̂̆ + T r̂ . 𝒘 R𝑣

(11.12)

Furthermore, if the same approximation functions are used for each component of V d and V̆ , that is, the same V 1D𝑣 as in (4.17), then T R𝑣 is obtained as the assembly of four diagonal matrices, by realizing that: [ ] d V 1D𝑣 𝟎 = TR d 𝟎 V 1D𝑣 [ ][ ] d ⌈TR11 ⌋(d𝑣 +1)×(d𝑣 +1) ⌈TR12 ⌋(d𝑣 +1)×(d𝑣 +1) V 1D𝑣 𝟎 ; d ⌈TR21 ⌋(d𝑣 +1)×(d𝑣 +1) ⌈TR22 ⌋(d𝑣 +1)×(d𝑣 +1) 𝟎 V 𝑣 1D

so that T R𝑣

[ T I = R11 (d𝑣 +1) TR21 I (d𝑣 +1)

] TR12 I (d𝑣 +1) . TR22 I (d𝑣 +1)

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Similarly the transformation of generalized tractions between reference frames is: t̂̆ = T R𝑣 t̂

and

t̂ = T TR𝑣 t̂̆

(11.13)

11.4.4 Governing System

Transforming to a global frame, and assembling the expressions just presented for all ̂̆ variables in (11.7) are not the same for elements and sides, requires realizing that the 𝒘 elements that share a side. By using (11.12), this change of variable is accounted for by considering that: ̂̆ = D ̆ T 𝒗̂̆ + D ̆ T T R r̂ . ̆ T𝒘 D 𝑣 The following system is then obtained: [ ]{ } { } ̆ TTR ̂̆ ̆ T (̂r − T R r̂ ) −̆ D s D 𝑣 𝑣 = , ̆ 𝒗̂ T TR D 𝟎 t̂ 0

(11.14)

𝑣

where, in the absence of body forces and initial strains, the terms on the right hand side represent the generalized strain associated with the rigid body rotations5 and the prescribed tractions. This system is non-linear because T R𝑣 is a non-linear function of the rigid body rotation of each element, which in turn depends on its boundary displacements, as will be explained in Section 11.4.5. ̂ and from the corresponding 𝜃R s, both sides of (11.14) For a given set of values (ŝ̆ , 𝒗), can be directly computed. The difference between these two vectors is the residual corresponding to these values, and this will be zero for a solution of the system: ]{ } } [ { } { T ̆ TTR ̆ (̂r − T R r̂ ) RC −̆ D ŝ̆ D 𝑣 𝑣 . − = R= T ̆ RE 𝒗̂ TR D 𝟎 t̂ 0 𝑣

In practice an iterative procedure that eliminates the residual at each load step is applied incrementally, to obtain a set of solutions for a given load history. 11.4.5 Determination of the Rigid Body Displacements

In Section 6.1.1 the rigid body displacements are determined assuming that the rotation component in 𝚫R is small, so that sin 𝜃R ≈ 𝜃R and cos 𝜃R ≈ 1. That assumption is not valid for the present problem which becomes non-linear because  eR , in (6.2), depends on 𝜃R . Given the values of 𝒗̂ and an initial guess of the rigid body displacements, 𝚫0R , typically obtained from the previous load step, a standard Newton iteration can be used to determine an improved approximation, by considering the tangent approximation of the rigid body displacements, loosely expressed as: uR = u0R +

2 𝜕uR 1 𝜕 uR 2 𝛿𝚫R + 𝛿𝚫R + higher order terms, 𝜕𝚫R 2 𝜕𝚫2R

which, by considering that the second derivatives of the translations is equal to zero and discarding the higher order terms, becomes: 1 T 2 uR = u0R + u1Δ R 𝛿𝚫R − T R r 𝛿𝜃R , 2 5 Which is zero when the rigid body rotation is zero.

229

230

Equilibrium F.E. Formulations

where u0R is obtained by using 𝚫0R in (11.6) and ] [ [ 𝜕uR 1 0 −x sin 𝜃R0 − y cos 𝜃R0 1Δ = I2 = uR = 0 0 𝜕𝚫R 0 1 x cos 𝜃 − y sin 𝜃 R

T ] . T 1𝜃 R r

R

The difference at each point on the boundary between the rigid body movement and the boundary displacements becomes: 1 T 2 ̂ + u1Δ q = (u0R − V 𝒗) R 𝛿𝚫R − T R r 𝛿𝜃R ; 2 1 T 2 = q0 + u1Δ R 𝛿𝚫R − T R r 𝛿𝜃R . 2 The variation of the rigid body displacements that minimizes ∮Γ qT q dΓ is now obtained by solving: M𝛿𝚫R + R𝚫 = 0;

(11.15)

where M=

⎛ ⎡0 0 ⎜u1ΔT u1Δ − ⎢0 0 ∮Γ ⎜ R R ⎢ ⎣0 0 ⎝

R𝚫 =

∮Γ

0 ⎤⎞ 0 ⎥⎟ dΓ, ⎥⎟ T r T R q0 ⎦⎠

and T

0 u1Δ R q dΓ.

Then 𝚫0R + 𝛿𝚫R becomes the new approximation of 𝚫0R , and the iterative process conT q0 becomes approximately zero. tinues until the integral of u1Δ R Since the integrals in (11.15) only involve the sine and cosine of 𝜃R , the coordinates of each point on the sides and the functions used to approximate the boundary displacements, it is possible to obtain closed form expressions for them. They are functions of sin 𝜃R , cos 𝜃R , the geometric properties of the sides of the element (the perimeter, the first and the second moments), and integrals involving the approximation functions. These expressions are simplified by using a reference frame positioned at the centroid of the sides of the element, together with Legendre polynomials as the approximation functions. 11.4.6 Tangent Form of the Governing System

The application of a Newton procedure for the determination of a solution of (11.14) requires the determination of its tangent matrix which describes the linear approxî From this mation of the variation induced by a change in the solution vector, (𝛿 ŝ̆ , 𝛿 𝒗). tangent matrix it is possible to determine a value for this variation such that the residual ̂ in (11.14) would cancel if the linear approximation corresponding to (ŝ̆ 0 + 𝛿 ŝ̆ , 𝒗̂ 0 + 𝛿 𝒗) was exact. 11.4.6.1 Variation of the Rigid Body Displacements

The first point to consider is that a change in the displacement parameters will generally lead to a change of the rigid body rotation, which directly affects T R𝑣 . We now express

Non-Linear Analyses

the approximation of the difference between the rigid body movement and the boundary displacements as: 1 T 2 ̂ + u1Δ ̂ q = (u0R − V 𝒗) R 𝛿𝚫R − T R r 𝛿𝜃R − V 𝛿 𝒗; 2 1 T 2 ̂ = q0 + u1Δ R 𝛿𝚫R − T R r 𝛿𝜃R − V 𝛿 𝒗. 2 The generalization of (11.15), so as to consider the variation of the boundary displacement parameters becomes: M𝛿𝚫R + R𝚫 −

̂ u1Δ R V dΓ 𝛿 𝒗 = 0. T

∮Γ

R𝚫 is zero when the rigid body displacements corresponding to the initial solution (ŝ̆ 0 , 𝒗̂ 0 ) are used, so that we can directly obtain the first order approximation of the variation of the rigid body displacements, as a function of the change in the boundary displacement parameters: 𝛿𝚫R = M −1

̂ ̂ u1Δ R V dΓ 𝛿 𝒗 = Q 𝛿 𝒗. T

∮Γ

(11.16)

The variation of the rigid body rotation is obtained from Q𝜃 in the last row of matrix Q. 11.4.6.2 The Effect of a Variation in the Boundary Displacement Parameters on the Associated Transformations

The effect of a variation of the boundary displacement parameters is twofold, corresponding to the changes in the two terms of the products in either (11.10) or (11.11). ̂ which are In any case, since the position vector is constant, the variations of 𝒗̂ and 𝒘, measured in the global frame, are the same: ̂ 𝛿 𝒗̂ = 𝛿 𝒘. ̂̆ Its tangent form thus involves the Equation (11.7) is non-linear only because of 𝒘. ̂̆ which is loosely expressed as: variation 𝛿 𝒘, 𝜕T R𝑣 ̂̆ = T 0 𝛿 𝒘 ̂ 𝒘 ̂ 0 + higher order terms; 𝛿𝒘 𝛿𝒘 R𝑣 ̂ + ̂ 𝜕𝒘 𝜕T R𝑣 ̂ 𝒗̂ 0 + r̂ ) + higher order terms. = T 0R𝑣 𝛿 𝒗̂ + 𝛿 𝒗( 𝜕 𝒗̂ Using a matrix form in this derivation may be misleading since a third order tensor is involved. We opt, therefore, to make the following deductions using indicial notation. 𝛿 𝑤̂̆ i = TR0 𝛿 𝑤̂ j + 𝑣ij

𝜕TR𝑣

ij

𝜕 𝑤̂ k

𝛿 𝑤̂ k 𝑤̂ 0j + higher order terms.

̂ can be The derivatives of components of T R𝑣 with respect to components of 𝒘 expressed in terms of derivatives with respect to 𝜃R , together with 𝜕𝜃R 𝜕𝜃 = R = Q𝜃 m , 𝜕 𝑤̂ m 𝜕 𝑣̂ m which is a direct consequence of (11.16). Then 𝜕TR𝑣 𝜕TR𝑣 ij ij 𝛿 𝑤̂ k 𝑤̂ 0j = Q𝜃k 𝛿 𝑤̂ k 𝑤̂ 0j = TR1𝜃𝑣 𝑤̂ 0j Q𝜃k 𝛿 𝑤̂ k . ij ̂ 𝜕𝜃R 𝜕 𝑤k

231

232

Equilibrium F.E. Formulations

Thus, neglecting the higher order terms: 𝛿 𝑤̂̆ i = (TR0 + TR1𝜃𝑣 𝑤̂ 0k Q𝜃j )𝛿 𝑤̂ j , 𝑣ij

ik

which can now be written in matrix form as ̂̆ = T 𝛿 𝒘 ̂ = T 𝒗̂ 𝛿 𝒗, ̂ 𝛿𝒘 𝒗̂

(11.17)

where ̂ 0 + r̂ )Q𝜃 . ̂ 0 Q𝜃 = T 0R𝑣 + T 1𝜃 T 𝒗̂ = T 0R𝑣 + T 1𝜃 R𝑣 𝒘 R𝑣 ( 𝒗 We also need to quantify the effect of a variation 𝛿 t̂̆ on the global generalized boundary ̂ which, as tractions 𝛿 t̂ , so that equilibrium can be imposed for an arbitrary change of 𝛿 𝒗, ̂ mentioned, is equivalent to a change of 𝛿 𝒘. In order to maintain equilibrium when the solution changes, we start by equating the work done by both generalized tractions on an arbitrary variation of the boundary displacements: T T ̂̆ = t̂̆ T T 𝛿 𝒘 ̂ = t̂ 𝛿 𝒘, ̂ t̂̆ 𝛿 𝒘 𝒗̂

thus we have6 t̂ = T T𝒗̂ t̂̆ . Then, again in indicial notation, ( ) 0 𝜕 TR0 + TR1𝜃𝑣 𝑤̂ 0k Q𝜃j 𝛿 𝑤̂ l t̂̆i . 𝛿 t̂j = T𝑣0̂ 𝛿 t̂̆i + 𝑣ij ij ik 𝜕 𝑤̂ l

(11.18)

As for the variation of the global displacement parameters, the second term can be simplified,7 and the higher order terms dropped, leading to ) ( 0 TR1𝜃𝑣 Q𝜃l + TR1𝜃𝑣 𝛿kl Q𝜃j + TR2𝜃𝑣 𝑤̂ 0k Q𝜃j Q𝜃l 𝛿 𝑤̂ l t̂̆i , ij

ik

ik

which can be written as ( ) 0 0 0 TR1𝜃𝑣 t̂̆i Q𝜃l + Q𝜃j t̂̆i TR1𝜃𝑣 + Q𝜃j t̂̆i TR2𝜃𝑣 𝑤̂ 0k Q𝜃l 𝛿 𝑤̂ l . ij

il

Observing that 𝛿 t̂ = T 𝒗0̂

T

T

= T 𝒗0̂

T

= T 𝒗0̂

TR2𝜃 𝑣ij

ik

−TR0 , 𝑣ij

= Equation (11.18) becomes, in matrix form, ( ) T T T 0 0 ̂̆ Q + QT t̂̆ 0 T 1𝜃 − QT t̂̆ 0 T 0 𝒘 ̂ ̂ 𝛿 t̂̆ + T 1𝜃 Q t 𝜃 𝜃 𝛿 𝒘; 𝜃 𝜃 R𝑣 R𝑣 R𝑣 ) ( 0 ̂ ̂ 0 𝛿 𝒘; 𝛿 t̂̆ + H t̂̆ , 𝒘 ( ) 0 ̂ 𝛿 t̂̆ + H t̂̆ , 𝒗̂ 0 + r̂ 𝛿 𝒗,

(11.19)

where matrix H is symmetric.

6 Note that unlike the transformations in (11.13), in this equating of virtual work, we are accounting for changes in displacements which also affect the directions of the local axes. 7 Notice that 𝜕 𝑤̂ 0k ∕𝜕 𝑤̂ l = 𝛿kl , the Kronecker delta.

Non-Linear Analyses

11.4.6.3 Tangent Form of the Governing System for an Element

The governing system of element e provides the contribution to the variation of the residuals, due to a given change of the variables of the element. When written in the local reference frame, this system is linear, and is expressed as: [ ]{ } { } ̆T 𝛿 ŝ̆ −𝛿 R̆ C −̆ D = . ̂̆ ̆ −𝛿 R̆ E e 𝛿𝒘 D 𝟎 e e

In order to assemble the global system two steps are necessary: • Transform the variation of the local displacements into global ones, which are the ̂ by using (11.17). same in terms of 𝛿 𝒗, • Transform the variation of generalized boundary tractions, from the local frame to the global one, via (11.19). The elemental contribution to the tangent form of the governing system appropriate for assembly is obtained from } { [ ]{ } ̂̆ + 𝛿 ê̆ ̂̆ ̆ TT0 −𝛿 e 𝛿 s −̆ D 𝜎 𝜀 𝒗̂ = , T T 0 ̆ 𝛿 𝒗̂ e ̂ T 𝒗0̂ 𝛿 t̂̆ = 𝛿 t̂ − (H(t̂̆ , 𝒗̂ 0 + r̂ ) 𝛿 𝒗) T 𝒗0̂ D 𝟎 e

e

which, in the absence of body forces and imposed displacements, becomes [ ]{ } { } ̆ TT0 −̆ D −𝛿 ê̆ 𝜎 + 𝛿 ê̆ 𝜀 𝛿 ŝ̆ 𝒗̂ = , T ̆ H(t̂̆ 0 , 𝒗̂ 0 + r̂ ) 𝛿 𝒗̂ e 𝛿 t̂ T0 D e 𝒗̂

(11.20)

e

These elemental equations are assembled in a standard way, considering that for the right hand side the sum of all increments of the incompatible generalized strains for the elements must balance the residual in compatibility RC , and the sum of all increments to the generalized tractions for the sides must balance RE , the residual in equilibrium. 11.4.7 Large Displacements and Spurious Kinematic Modes

An unstable mesh, that is, one where potential spurious modes exist, may present difficulties to non-linear analyses because of large displacements: • The determination of the current element-wise rigid body displacements may be polluted by the presence of spurious modes, as already discussed in Section 6.1. Note here that the displacements control the non-linear analysis, so their pollution becomes more problematic than in the linear case! • Loads which are conservative (i.e. remain fixed in direction and are transported with an element) and are initially admissible may become inadmissible after finite changes in geometry. Alternatively, changes to the configuration of a stable mesh may tend to transform it to being an unstable mesh. As has already been noted in Section 6.7, the behaviour of a mesh can still be problematic even when the configuration is in close proximity to being unstable. The most reliable procedure therefore, in the present context, would appear to be to use a mesh which is based on simplicial macro-elements, which are inherently stable for all configurations.

233

234

Equilibrium F.E. Formulations

11.4.7.1 Numerical Example

To illustrate the application of this technique we present the results for the analysis of a straight prismatic beam with length 10 and depth 1, as a plane stress problem. The side elevation has the shape of a parallelogram having vertical ends and long sides with a small upward gradient of 1 in 100. One end is fixed and a uniform horizontal pressure is applied to the other end. The material properties are 200 for Young’s modulus and 0.15 for Poisson’s ratio. The buckling load for the perfect beam model, assuming consistent units, is: 𝜆=

1 𝜋 2 200 12 𝜋 2 EI = ≈ 0.411. (2 L)2 (2 × 10)2

y

λ

1.00 0.10

x

10.00

Figure 11.9 Axially loaded beam with imperfection. Initial geometry and finite element mesh. λ λ = 1.0

1

0.8 0.6 0.4 δH δV

0.2 0

0

2

4

6

8

10

δ

12

λ = 0.4

λ = 0.1

Figure 11.10 Axially loaded beam with imperfection. Horizontal (𝛿H ) and vertical (𝛿V ) displacement curves of the point at the centre of the free end. The vertical direct Cartesian stress (𝜎yy ) for different load factors is also represented, with the elements subjected to their rigid body displacements. The ranges of the contours are {−0.025 ∶ 0.025} for 𝜆 = 0.1; {−0.5 ∶ 0.5} for 𝜆 = 0.4; and {−35 ∶ 35} for 𝜆 = 1. (See plate section for colour representation of this figure).

Non-Linear Analyses

The mesh with 62 macro-elements (186 triangular elements) presented in Figure 11.9 was used, with cubic stress and boundary displacement approximations, to obtain the the results presented in Figure 11.10: the load displacement curves for the point8 at the centre of the free end, and the contour plots of the vertical direct Cartesian stress (𝜎yy ) for selected values of the load. In agreement with the hypothesis of the corotational formulation, the stress contours in the deformed configurations are plotted by assuming that each element only undergoes a rigid body displacement. It is clear that, although the model captures the different phases of the non-linear response, the strains at post-buckling levels of load become significant and can no longer be regarded as infinitesimal, for example, the maximum absolute value of 𝜀yy is equal to 0.0025 for 𝜆 = 0.4, but becomes 0.175 for 𝜆 = 1.0. Although the underlying assumption of ‘small strains’ is then not valid for the purposes of predicting actual behaviour, nevertheless the numerical procedures of the analysis are verified. This leads to a ‘deformed configuration of undeformed elements’ that implies significant displacement incompatibilities. Such incompatibilities affect the strict enforcement of equilibrium because the outward normals for adjacent elements may differ.

References Alizadeh F and Goldfarb D 2003 Second-order cone programming. Mathematical Programming 95(1), 3–51. Armstrong MA 1979 Basic Topology. McGraw-Hill. Baker J, Horne MR and Heyman J 1965 The Steel Skeleton (Vol II: Plastic Behaviour & Design). Cambridge University Press. Belytschko T and Hsieh BJ 1973 Non-linear transient finite element analysis with convected co-ordinates. International Journal for Numerical Methods in Engineering 7(3), 255–271. Cook RD et al. 2002 Concepts and Applications of Finite Element Analysis. John Wiley & Sons. Crisfield MA and Moita GF 1996 A unified co-rotational framework for solids, shells and beams. International Journal of Solids and Structures 33(20), 2969–2992. De Borst R, Crisfield MA, Remmers JJ and Verhoosel CV 2012 Nonlinear Finite Element Analysis of Solids and Structures. John Wiley & Sons. de Souza Neto EA, Peric D and Owen DRJ 2011 Computational Methods for Plasticity: Theory and Applications. John Wiley & Sons. Drucker DC, Prager W and Greenberg HT 1952 Extended limit design theorems for continuous media. Quarterly of Applied Mathematics 9(4), 381–389. Felippa CA and Haugen B 2005 A unified formulation of small-strain corotational finite elements: I. Theory. Computer Methods in Applied Mechanics and Engineering 194(21), 2285–2335. Hoffman J 1992 Numerical methods for scientists and engineers. McGraw-Hill, New York. Horrigmoe G and Bergan PG 1976 Incremental variational principles and finite element models for nonlinear problems. Computer Methods in Applied Mechanics and Engineering 7(2), 201–217. 8 Actually the averages of the displacements at the ends of the edges that converge onto that point, with coordinates (10; 0.6).

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Izzuddin BA 2005 An enhanced co-rotational approach for large displacement analysis of plates. International Journal for Numerical Methods in Engineering 64(10), 1350–1374. Kamenjarzh JA 1996 Limit Analysis of Solids and Structures. CRC press. Khan AS and Huang S 1995 Continuum Theory of Plasticity. John Wiley & Sons. Kuss F and Lebon F 2009 Stress based finite element methods for solving contact problems: Comparisons between various solution methods. Advances in Engineering Software 40(8), 697–706. Lorentz GG 2012 Bernstein Polynomials. American Mathematical Society. Maunder E and Savage G 1994 A graph-theoretic model for finite elements with side variables. Civil Engineering Systems 11(2), 111–141. Maunder EAW 1987 On Stress-Based Equilibrium Elements and a Flexibility Method for the Analysis of Thin Plated Structures. In The Mathematics of Finite Elements and Applications VI (ed. Whiteman JR), p. 261–269. Academic Press. Maunder EAW and Izzuddin BA 2014 An Equilibrium Finite Element Model for Shells with Large Displacements. In 22nd UK Conference of the Association of Computational Mechanics in Engineering (ed. Javadi A and Hussein MS). University of Exeter. Maunder EAW and Ramsay ACA 2012 Equilibrium models for lower bound limit analyses of reinforced concrete slabs. Computers & Structures 108, 100–109. Naghdi PM and Trapp JA 1975a On the nature of normality of plastic strain rate and convexity of yield surfaces in plasticity. Journal of Applied Mechanics 42(1), 61–66. Naghdi PM and Trapp JA 1975b Restrictions on constitutive equations of finitely deformed elastic-plastic materials. The Quarterly Journal of Mechanics and Applied Mathematics 28(1), 25–46. Nielsen MP and Hoang LC 2011 Limit Analysis and Concrete Plasticity. CRC press. Passos O 2011 Análise Plástica de Lajes. Cálculo de minorantes e majorantes da carga de colapso Master’s thesis Instituto Superior Técnico, Technical University of Lisbon Portugal. Taylor MA, Wingate BA and Vincent RE 2000 An algorithm for computing Fekete points in the triangle. SIAM Journal on Numerical Analysis 38(5), 1707–1720. Washizu K 1982 Variational Methods in Elasticity and Plasticity 3 edn. Pergamon Press. Wempner G 1969 Finite elements, finite rotations and small strains of flexible shells. International Journal of Solids and Structures 5(2), 117–153.

k

σxx

σyy

σxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

k

Figure 2.2 Bi-material strip: stress distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜎xx ; {−0.5 ∶ 0.5} for 𝜎yy ; {−0.5 ∶ 0.5} for 𝜎xy , with blue representing the minimum value, red the maximum and green the central value.

εxx

εyy

γxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

Figure 2.3 Bi-material strip: strain distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜀xx ; {−0.5 ∶ 0.5} for 𝜀yy ; {−0.5 ∶ 0.5} for 𝛾xy .

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

k

k

k

mxx

myy

mxy

Compatible (3 dofs)

k

Equilibrated (2 dofs)

“Exact”

Figure 2.6 Simply supported plate: some moment distributions. The ranges of the colours are: {−0.1 ∶ 0.1} for mxx ; {−0.75 ∶ 0.75} for myy ; {−1 ∶ 1} for mxy .

k

k

k

C3

E3C3

T3C3

E3B3

k

k

T3 B3

T3 B2

σxx

σyy

σxy

Figure 3.4 Bi-material strip: stress distributions for the bi-material strip. Obtained from models based on cubic approximations on a mesh of eight elements, EB = 4EA . The ranges of the colours are: {−3.0 ∶ 3.0} for 𝜎;xx ; {−0.5 ∶ 0.5} for 𝜎yy ; {−0.5 ∶ 0.5} for 𝜎xy .

k

k

Figure 4.5 Square cantilever. Distributions of the stress components in the two frames considered. The range of values for the colour representation are: {−5.0 ∶ 5.0} for 𝜎xx {−1.5 ∶ 1.5} for 𝜎yy {−1.5 ∶ 1.5} for 𝜎xy {−2 ∶ 2} for 𝜎x′ x′′ , 𝜎y′ y′ and 𝜎x′ y′ ; with blue representing the minimum value, red the maximum and green the central value.

k

k

Figure 4.6 Square cantilever. Element tractions and stress trajectories. The width of each trajectory varies proportionally to the corresponding principal stress, from −4 to 4, with blue representing the minimum value, red the maximum and green the central value.

k

k

Figure 8.21 Stresses obtained from the compatible solution.

k

k

Figure 8.22 Equilibrated stresses on the star patches, recovered from the compatible solution.

k

k

(a) Equilibrated stresses recovered from the compatible model.

k

k

(b) Stresses obtained from the equilibrated model.

Figure 8.23 (a) Equilibrated stresses recovered from the compatible model (b) Stresses obtained from the equilibrated model.

k

k

k

k

(a)

(b)

(c)

Figure 11.5 (a) Location of the points where plastic strains develop for the ultimate load on the mesh with 412 elements, with a cubic approximation of the moments and 91 control points; distribution of moments mxx in (b) and mxy in (c) for quadratic moments and 45 control points.

k

k

λ λ = 1.0

1

0.8 0.6 0.4 δH δV

0.2

k

0

0

2

4

6

8

10

δ

12

λ = 0.4

λ = 0.1

Figure 11.10 Axially loaded beam with imperfection. Horizontal (𝛿H ) and vertical (𝛿V ) displacement curves of the point at the centre of the free end. The vertical direct Cartesian stress (𝜎yy ) for different load factors is also represented, with the elements subjected to their rigid body displacements. The ranges of the contours are {−0.025 ∶ 0.025} for 𝜆 = 0.1; {−0.5 ∶ 0.5} for 𝜆 = 0.4; and {−35 ∶ 35} for 𝜆 = 1.

k

k

k

σxx

σyy

σxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

k

Figure 2.2 Bi-material strip: stress distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜎xx ; {−0.5 ∶ 0.5} for 𝜎yy ; {−0.5 ∶ 0.5} for 𝜎xy , with blue representing the minimum value, red the maximum and green the central value.

εxx

εyy

γxy

Compatible (2 dofs)

Equilibrated (0 dofs)

Equilibrated (1 dof)

Exact

Figure 2.3 Bi-material strip: strain distributions obtained for EB = 4EA . The values for the colours are: {−3.0 ∶ 3.0} for 𝜀xx ; {−0.5 ∶ 0.5} for 𝜀yy ; {−0.5 ∶ 0.5} for 𝛾xy .

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

k

k

237

A Fundamental Equations of Structural Mechanics A.1 The General Elastostatic Problem We consider a general elastostatic problem governed by the following equations: Internal Compatibility Internal Equilibrium

𝜺 =  u;  𝝈 + b̄ = 0;

in Ω

(A.1)

in Ω

(A.2)

in Ω

(A.3)

u = ū

on Γu

(A.4)

 𝝈 = t̄

on Γt

(A.5)



𝝈 = k (𝜺 − 𝜺T ) Elasticity

or 𝜺 = f 𝝈 + 𝜺T

Kinematic Boundary Conditions Static Boundary Conditions

T

It is assumed that, unless another reference frame is explicitly mentioned, the (generalized) displacements, forces, strains and stresses are defined in a Cartesian reference frame. A.1.1 Two Dimensional Elasticity

For two dimensional elasticity problems we use the sign convention illustrated in Figure A.1 for the projection of the positive stresses on an infinitesimal square. The components of the symmetric, 2 × 2, stress and strain tensors (Wunderlich and Pilkey, 2014), are collected in vectors 𝝈 and 𝜺 𝝈 = {𝜎xx , 𝜎yy , 𝜎xy } and 𝜺 = {𝜀xx , 𝜀yy , 𝛾xy }, where we note that 𝛾xy = 2 𝜀xy , that is, the third component of the strain vector, 𝛾xy , is twice the shear component of the strain tensor, 𝜀xy , so that the tensor product of the strains times the stresses is equal to the inner product 𝜺T 𝝈, which represents a work density. The compatibility differential operator, mapping strains from displacements, is ⎡𝜕 ⎢ 𝜕x ⎢ =⎢ 0 ⎢𝜕 ⎢ ⎣ 𝜕y

0⎤ ⎥ 𝜕 ⎥ 𝜕y ⎥ . 𝜕 ⎥⎥ 𝜕x ⎦

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

238

Equilibrium F.E. Formulations

σyy

Figure A.1 Positive Cartesian components of stress in 2D.

σxy y

σxx σxx σxy σyy x

For this problem the adjoint operator  ⋆ , mapping body forces from stresses is the transpose of . The elasticity operators k and f are given, for an isotropic medium, by E k= 1 − 𝜈2

f =

⎡1 1⎢ −𝜈 E⎢0 ⎣

⎡1 𝜈 ⎢𝜈 1 ⎢ ⎢0 0 ⎣ −𝜈 1 0

0 ⎤ 0 ⎥, 1 − 𝜈 ⎥⎥ 2 ⎦ 0 ⎤ 0 ⎥ ⎥ 2(1 + 𝜈)⎦

and ⎡1 − 𝜈 ⎢ 𝜈 E k= (1 + 𝜈)(1 − 2𝜈) ⎢⎢ 0 ⎣ ⎡ 1 − 𝜈2 1⎢ −𝜈(1 + 𝜈) f = E⎢ 0 ⎣

𝜈 1−𝜈

−𝜈(1 + 𝜈) 1 − 𝜈2 0

0

0 ⎤ 0 ⎥, 1 − 2𝜈 ⎥⎥ 2 ⎦

0 ⎤ 0 ⎥ ⎥ 2(1 + 𝜈)⎦

for plane stress and plane strain problems, respectively. Finally the boundary operator  is given by the normal components corresponding to the derivatives in : ⎡nx  = ⎢0 ⎢ ⎣ny

0⎤ ny ⎥ . ⎥ nx ⎦

A.1.2 Three Dimensional Elasticity

For three dimensional elasticity problems we use the sign convention illustrated in Figure A.2 for the projection of the stress components on an infinitesimal cube. The components of the symmetric, 3 × 3, stress and strain tensors (Wunderlich and Pilkey, 2014) are also collected in two vectors, 𝝈 = {𝜎xx , 𝜎yy , 𝜎zz , 𝜎xy , 𝜎yz , 𝜎zx } and 𝜺 = {𝜀xx , 𝜀yy , 𝜀zz , 𝛾xy , 𝛾yz , 𝛾zx },

Fundamental Equations

σyy

Figure A.2 Positive Cartesian components of stress in 3D.

σxy

σyz

y

σxy

σyz σzz

σzx

σzx

σxx

x

z

where the vector component 𝛾𝛼𝛽 is equal to two times the corresponding strain, so that the tensor product of the strains times the stresses is equal to the inner product 𝜺T 𝝈. The compatibility differential operator, mapping strains from displacements, is ⎡𝜕 ⎢ 𝜕x ⎢0 ⎢ ⎢ ⎢0 =⎢ 𝜕 ⎢ ⎢ 𝜕y ⎢0 ⎢ ⎢ ⎢𝜕 ⎣ 𝜕z

0 𝜕 𝜕y 0 𝜕 𝜕x 𝜕 𝜕z 0

0⎤ ⎥ 0⎥ ⎥ 𝜕 ⎥ ⎥ 𝜕z ⎥ . 0⎥ ⎥ 𝜕 ⎥ 𝜕y ⎥ ⎥ 𝜕 ⎥ 𝜕x ⎦

For this problem the adjoint operator  ⋆ , mapping body forces from stresses is the transpose of . The elasticity operators k and f are given, for an isotropic medium, by ⎡1 − 𝜈 ⎢ 𝜈 ⎢ 𝜈 ⎢ E ⎢ 0 k= (1 + 𝜈)(1 − 2𝜈) ⎢ ⎢ 0 ⎢ ⎢ 0 ⎣ ⎡1 ⎢−𝜈 ⎢ 1 ⎢−𝜈 f = E⎢0 ⎢0 ⎢ ⎣0

−𝜈 1 −𝜈 0 0 0

−𝜈 −𝜈 1 0 0 0

𝜈 1−𝜈 𝜈

𝜈 𝜈 1−𝜈

0

0

0

0

0 0 0 1 − 2𝜈 2 0

0

0

0

0 0 0 2(1 + 𝜈) 0 0

0 0 0 0 2(1 + 𝜈) 0

0 ⎤ 0 ⎥ ⎥ 0 ⎥ 0 ⎥ 0 ⎥ ⎥ 2(1 + 𝜈)⎦

0 0 0 0 1 − 2𝜈 2 0

⎤ ⎥ ⎥ ⎥ 0 ⎥, ⎥ 0 ⎥ ⎥ 1 − 2𝜈 ⎥ 2 ⎦ 0 0 0

239

240

Equilibrium F.E. Formulations

Figure A.3 Reference frame used for the beam and shear stresses.

 z

x y

Finally the boundary operator  is given by the normal components corresponding to the derivatives in : ⎡nx 0 0 ⎤ ⎢ 0 ny 0 ⎥ ⎥ ⎢ 0 0 nz ⎥  =⎢ . ⎢ny nx 0 ⎥ ⎢ 0 nz ny ⎥ ⎥ ⎢ ⎣nz 0 nx ⎦ A.1.3 Shear Stresses and Warping of a Beam Section

This problem, discussed and illustrated in Chapter 4, is solved for a homogeneous and isotropic section of a prismatic beam, considering only the shear stresses due to the action of shear forces and a twisting moment. The kinematic hypotheses usually admitted in Timoshenko’s theory for prismatic beams are considered, which include the assumption that longitudinal warping displacements 𝑤 of the cross-section are free and constant along the axis.1 For a cross-section associated with the reference frame with origin at , indicated in Figure A.3, the displacements are expressed by: ux (x, y, z) = ux (z) − y 𝜃z (z); uy (x, y, z) = uy (z) + x 𝜃z (z); uz (x, y, z) = uz (z) + y 𝜃x (z) − x 𝜃y (z) + 𝑤(x, y), where 𝜃x (z), 𝜃y (z) and 𝜃z (z) denote rigid body rotations of the cross-section. We will focus our study on one term in the definition of the displacements, the warping of the cross-section, 𝑤, which is associated with shear strains and stresses. The corresponding equilibrium equation leads to the definition of the shear forces and of the twisting moment, while the corresponding shear strains are induced by shearing and twisting of the cross-section. The object of this Section is to present a numerical procedure to compute these relations, which generally cannot be expressed in closed form, as direct functions of the mechanical and geometrical properties of the cross-section. 1 This is the simplest model possible. More complex assumptions include consideration of the effects introduced by gradients of the axial deformation and of the curvatures, the distortion of the cross-section induced by Poisson’s ratio effect, as well as non-uniform warping.

Fundamental Equations

Since warping is assumed constant along the axis, the longitudinal strains corresponding to the assumed displacements are d𝜃y duz d𝜃 +y x −x ; dz dz dz = 𝜀zz + y 𝜒yy − x 𝜒xx .

𝜀zz =

Together with the elastic constitutive relations and the global equilibrium of the cross-section, these expressions lead directly to a linear relationship between some of the generalized forces (normal force and bending moments) and the corresponding generalized strains (axial strain at the origin and curvatures), which can be expressed in a flexibility matrix. If the origin of the reference frame is located at the centroid of the cross-section and the axes are principal, relative to the moments and products of area of the section, this matrix is diagonal, but this is not the case in general. The assumptions on the displacements are such that the stresses in the plane of the cross-section, 𝜎xx , 𝜎yy and 𝜎xy , are zero. Consequently, from the viewpoint of equilibrium, the only relevant equation corresponds to the component of (A.2) along z, the axis of the beam. 𝜕𝜎xz 𝜕𝜎yz 𝜕𝜎zz ̄ + + + bz = 0. 𝜕x 𝜕y 𝜕z The normal strains are linear in the section; therefore, the term associated with the normal stresses is also linear. Body forces may be present, to represent distributed couples. Their variation is consistently assumed to be linear, so that the last two terms in the previous equation can be replaced by a linear function, which represents an, as yet unknown, equivalent body force: [ 𝜕𝜎zz ̄ + bz = 𝛽0 + 𝛽x x + 𝛽y y = 1 𝜕z

x

] ⎡𝛽0 ⎤ T y ⎢𝛽x ⎥ = b̃ 𝜷. ⎢ ⎥ ⎣ 𝛽y ⎦

It has to be noted that, in this case, there is no second order tensor that needs to be represented as a vector of its components. The relevant components of shear stresses and strains involved are two dimensional vectors, the shear traction on the face and the shear strain, which can be written as 𝝈 = {𝜎xz , 𝜎yz } and 𝜺 = {𝛾xz , 𝛾yz }. The assumed displacement field implies the following expressions for the strains: dux − 𝜃y − y dz  duy + 𝜃x + x 𝛾yz = dz

𝛾xz =

d𝜃z 𝜕𝑤 𝜕𝑤 + = 𝛾̄x − y 𝛼 + ; dz 𝜕x 𝜕x d𝜃z 𝜕𝑤 𝜕𝑤 + = 𝛾̄y + x 𝛼 + , dz 𝜕y 𝜕y

(A.6) (A.7)

where 𝛾̄x = dux ∕dz − 𝜃y and 𝛾̄y = duy ∕dz + 𝜃x , are the reference shear deformations of the cross-section, as measured at the origin, and 𝛼 = d𝜃z ∕dz, is the rate of twist. These are the generalized strains of the cross-section, which are collected in vector  = {̄𝛾x , 𝛾̄y , 𝛼}. The displacement vector, which has only one unknown component, 𝑤, may therefore be regarded as a scalar. Then the compatibility differential operator, mapping strains

241

242

Equilibrium F.E. Formulations

from the warping displacement, is ⎡𝜕 ⎤ ⎢ ⎥  = ⎢ 𝜕x 𝜕 ⎥. ⎢ ⎥ ⎣ 𝜕y ⎦ The adjoint operator  ⋆ , mapping body forces from stresses, is, as expected, the transpose of , as can be realized from the equation of equilibrium in the direction of the longitudinal axis of the beam. The constitutive relations just involve the shear modulus G, which we define as a function of E and 𝜈. The elasticity operators, k and f , are [ ] [ ] 2(1 + 𝜈) 1 0 1 0 E and f = . k= 0 1 2(1 + 𝜈) 0 1 E The boundary operator  is [ ] n  = x . ny Homogeneous static boundary conditions are assumed, which require that shear stresses normal to the external boundaries, 𝜎zn , are zero. This guarantees that the complementary shear stress, 𝜎nz , acting longitudinally on the external faces of the beam, is zero. On the internal boundaries this condition implies equilibrium of the complementary shear stresses. The definitions (A.6) and (A.7) of the shear strains imply that a uniform warping of the cross-section is a rigid body mode, corresponding to a longitudinal translation of the bar. It can be set to zero at an arbitrary point. Similarly, linear warping displacements induce constant shear strains, which are already accounted for by the reference shear deformations of the cross-section. We choose to remove these three dependent modes by forcing the average of 𝑤 and its ̃ introduced first order moments to be zero. Their integrals are expressed using vector b, in the linear approximation of the equivalent body forces: ⎡1⎤ ⎢x⎥ 𝑤 dΩ = b̃ 𝑤 dΩ = 𝟎, ∫Ω ⎢ ⎥ ∫Ω ⎣y⎦ noting that when these conditions are verified for a given reference frame, they will be verified for any other. This ensures that 𝑤 is orthogonal to the reference shear deformations and to the average axial strain. Since the hybrid equilibrium model does not approximate the displacements inside the elements, this condition has to be imposed using an energetically equivalent procedure. ̃ By determining This is achieved by interpreting the weighting factors as body forces b. ̄ which equilibrate the three unit equivalent a 2 × 3 matrix of particular stress fields, 𝝈, body forces, we have that  ⋆ 𝝈̄ + b̃ = 𝟎. Then ̄ T 𝑤 dΩ; − b̃ 𝑤 dΩ = 𝟎 = ( ⋆ 𝝈) ∫Ω ∫Ω =

∫Γ

𝝈̄ T  𝑤 dΓ −

∫Ω

𝝈̄ T (𝑤) dΩ;

Fundamental Equations

𝟎=

∫Γ

𝝈̄ T  𝑤 dΓ −

∫Ω

̄ dΩ. 𝝈̄ T (f 𝝈 − E)

(A.8)

The matrix Ē transforms generalized strains into the strains at a point, as already defined in (A.6) and (A.7): [ ] 1 0 −y Ē = . 0 1 x There are multiple sets of stress fields that correspond to the desired body forces, for example, 𝝈̄ 1 and 𝝈̄ 2 . The difference between them has to be a self-equilibrated distribution satisfying  ⋆ (𝝈̄ 1 − 𝝈̄ 2 ) = 0. The first equation leading to (A.8) shows that this difference does not affect the enforcement of the required conditions. ̄ is a particular solution that equilibrates the equivalent body We also note that 𝝈 0 = 𝝈𝜷 T ̃ force b 𝜷. The generalized stresses on the section are given by: ⎡1 ⎡Vx ⎤ ⎢0  = ⎢ Vy ⎥ = ⎢ ⎥ ∫Ω ⎢ ⎣−y ⎣T ⎦

[ ] 0⎤ [ ] 𝜎 T 𝜎zx 1⎥ zx dΩ = Ē dΩ. 𝜎zy ∫Ω ⎥ 𝜎 x⎦ zy

(A.9)

The governing system in (4.14) is used as a starting point, and involves selecting the generalized forces as the independent variables. This implies that ê 0 depends on the unknown generalized strains, , and also on the unknown coefficients, 𝜷, which define the equivalent body force, while t̂ 0 depends only on 𝜷: ê 0e = e0e  + e𝛽0 𝜷; e

t̂ 0me = t 𝛽0 𝜷; me

where

e𝛽0 = e

∫Ωe

STe f 𝝈̄ dΩ;

t 𝛽0 = − V Tm  me 𝝈̄ dΓ; me ∫Γm T

e0e = − STe Ē dΩ. ∫ Ωe With respect to (4.14) there are two additional sets of constraints: one cancelling the average longitudinal displacements (A.8) and the other imposing equilibrium (A.9). We also consider two additional sets of variables: the generalized strains,  and the generalized stresses . Upon discretization, (A.8) becomes: − 𝝈̄ T f S dΩ̂s + 𝝈̄ T  V dΓ𝒗̂ + 𝝈̄ T Ē dΩ  − 𝝈̄ T f 𝝈̄ dΩ 𝜷 = 𝟎, ∫Ω ∫Γ ∫Ω ∫Ω which we write as −e𝛽0 ŝ − t 𝛽0 𝒗̂ + e𝛽 0  −  𝛽 𝜷 = 𝟎. T

T

The discrete form of (A.9) becomes: ∫Ω

T Ē S dΩ̂s +

∫Ω

T Ē 𝝈̄ dΩ 𝜷 = ,

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244

Equilibrium F.E. Formulations

−e0 ŝ + e𝛽 0 𝜷 = ; T

T

leading, finally, to the following system of equations: ⎡ − ⎢ ⎢ D ⎢ ⎢−e T ⎢ 0 ⎢−e𝛽 T ⎣ 0

DT

−e0









−t 𝛽0

e𝛽 0

T

−e𝛽0 ⎤ ⎥ ⎧ ŝ ⎫ ⎧ 𝟎 ⎫ −t 𝛽0 ⎥ ⎪ 𝒗̂ ⎪ ⎪ 𝟎 ⎪ ⎥ ⎨ ⎬ = ⎨ ⎬ . T I ⎥  e𝛽 ⎪ 3⎪ 0 ⎥⎪ ⎪ ⎩𝜷 ⎭ ⎩ 𝟎 ⎭ − 𝛽 ⎥⎦

(A.10)

The solution of this system, which has full rank, leads to the definition of the stress, displacement, generalized strains and equivalent body force parameters, as a function of the generalized forces. It can be expressed as a matrix with three columns, one for each generalized force. The values for  represent the matrix of the constitutive relation for the section, in flexibility format, as measured at , the origin of the reference frame:  ⎧𝛾̄  ⎫ ⎡xx ⎢ ⎪ x ⎪ ⎢  ⎨𝛾̄y ⎬ = ⎢ yx ⎪𝛼⎪ ⎢  ⎩ ⎭ ⎣Tx

 xy  yy  Ty

 ⎤ xT ⎧ ⎫ ⎥ Vx ⎪  ⎥⎪ yT ⎨ Vy ⎬ . ⎥   ⎥ ⎪T ⎪ TT ⎭ ⎩ ⎦

When a translated reference frame, centred at point (x , y ), is used, the twisting moment and the reference shear deformations must be corrected, while the other variables are invariant: 𝛾̄x = 𝛾̄x − 𝛼 y ;

Vx = Vx = Vx ;

𝛾̄y = 𝛾̄y + 𝛼 x ;

Vy = Vy = Vy ;

𝛼 = 𝛼 = 𝛼;

T  = T  − V x y  + V y x .

This defines the transformation of the generalized forces and strains from the origin to an arbitrary point: ⎡ 1 T =⎢ 0 ⎢  ⎣−y

0 1 x

0⎤ 0⎥ ; ⎥ 1⎦

 = T T ;

  = T  .

The application of these transformations to the flexibility matrix of the cross-section leads to the determination of the shear centre. When the cross-section is acted upon by a twisting moment, the reference shear strains measured at that point are zero. Likewise, the rate of twist caused by an arbitrary shear force acting through that point is also zero. The relevant terms of the matrix are:     = Tx = xT − TT y ; xT     yT = Ty = yT + TT x .

Setting these terms to zero we obtain directly the coordinates of the shear centre.  It is noted that in general, the term xy is not equal to zero. This implies that to obtain a diagonal flexibility matrix, a rotation of the reference axes is necessary. These principal axes in shear are, in an arbitrary section, unrelated to the principal axes in bending.

Fundamental Equations

Additionally, the values of 𝜷 provide information about the distribution of body forces that lead to each of the generalized forces, which can be expressed as: V

Vx ⎧𝛽 ⎫ ⎡𝛽0 ⎪ 0 ⎪ ⎢⎢ Vx ⎨𝛽x ⎬ = ⎢𝛽x ⎪ 𝛽y ⎪ ⎢ V ⎩ ⎭ ⎣𝛽 x y

𝛽0 y V

𝛽x y V

𝛽y y

𝛽0T ⎤ ⎧ ⎫ ⎥ ⎪Vx ⎪ T⎥ V 𝛽x ⎨ y ⎬ . ⎥⎪ ⎪ T T⎥ ⎩ ⎭ 𝛽y ⎦

The last column of this matrix is always zero, since the problem of torsion, as formulated, involves no longitudinal stresses or applied forces, and the columns corresponding to Vx and Vy can be used to determine the position where both the corresponding linear distributions are zero, that is, the centroid of the cross-section (xG , yG ): V

V

V

𝛽0 x + 𝛽x x xG + 𝛽y x yG = 0;

V

V

V

𝛽0 y + 𝛽x y xG + 𝛽y y yG = 0.

Actually, these values of the 𝛽s are also related to the moments and products of area of the cross-section. This can be seen by considering that in the absence of applied couples about the transverse axes x and y, b̄ z = 0, and the equivalent body load is equal to the derivative of 𝜎zz . Since this stress can be defined in terms of the bending moments and of the moments and products of area of the cross-section, its derivative depends on the shear forces (the derivatives of the bending moments in this case) and on the same section properties. Considering the moments and products of area defined at the centroid,2 this derivative is written as: ((Vx Ixx + Vy Ixy )(x − xG ) + (Vx Ixy + Vy Iyy )(y − yG )) 𝜕𝜎zz . = 2 𝜕z Ixx Iyy − Ixy Equating this expression to the definition of the equivalent body forces as a function of the 𝜷s, for each of its monomials, leads to the following system of non-linear equations for the moments and product of area: V

V

2 𝛽0 y (Ixx Iyy − Ixy ) = −Ixy xG − Iyy yG ;

V

2 𝛽x y (Ixx Iyy − Ixy ) = Ixy ;

V

2 𝛽y y (Ixx Iyy − Ixy ) = Iyy ;

2 𝛽0 x (Ixx Iyy − Ixy ) = −Ixx xG − Ixy yG ;

V

2 𝛽x x (Ixx Iyy − Ixy ) = Ixx ;

V

2 𝛽y x (Ixx Iyy − Ixy ) = Ixy ;

V

V

From the last set of four equations, the pair for Ixy implies that 𝛽x y = 𝛽y x , and three equations with three unknowns remain. Their non-trivial solution is: V

Ixx =

V

V

V

𝛽x x 𝛽y y − (𝛽x y )2

;

Iyy =

V

𝛽y y

V

𝛽x x

V

V

V

𝛽x x 𝛽y y − (𝛽x y )2

;

Ixy = −

𝛽x y V

V

V

𝛽x x 𝛽y y − (𝛽x y )2

.

Given this solution, the first two equations can be used as an alternative way to determine the position of the centroid. A.1.4 Plate Bending

The internal equilibrium and compatibility equations for plate bending can be written in general terms by initially considering that the plate transverse displacements and the 2 The convention is that Ixy = − ∫ xy dΩ

245

246

Equilibrium F.E. Formulations

x y

z

mxx

θx

mxy

θy myy mxy

qx

mnn q n

mns

θn

θs

qy

Figure A.4 Positive Cartesian stress resultants for plate bending, acting on an infinitesimal element and projected onto a general boundary. The positive plate rotations are also illustrated.

rotations of the ‘normal fibres’, which are assumed to remain straight, are independent. These equations are written in terms of generalized stress and strain vectors that collect both the components of the symmetric, 2 × 2, moment or curvature tensors and of the shear force or transverse shear strain vectors: 𝝈 = {mxx , myy , mxy , qx , qy } and 𝜺 = {𝜒xx , 𝜒yy , 𝜒xy , 𝛾x , 𝛾y }. We use the reference frame and sign convention in Wunderlich and Pilkey (2014), where moment m𝛼𝛽 results from the integral on the thickness of the plate of the stress 𝜎𝛼𝛽 times the distance from the midplane and the shear force q𝛼 is the integral on the thickness of the plate of the stress 𝜎z𝛼 . The projections on an infinitesimal square of the resulting vectors are illustrated in Figure A.4, together with their projection on a general boundary. The displacement vector becomes u = {𝑤, 𝜃x , 𝜃y }, where the positive sense for the rotations is defined in the same figure. The compatibility operator becomes ⎡0 ⎢ ⎢0 ⎢ ⎢ =⎢ 0 ⎢ ⎢𝜕 ⎢ 𝜕x ⎢𝜕 ⎢ ⎣ 𝜕y

𝜕 𝜕x 0 𝜕 𝜕y 1 0

0⎤ ⎥ 𝜕 ⎥ 𝜕y ⎥ 𝜕 ⎥⎥ . 𝜕x ⎥ ⎥ 0⎥ ⎥ 1⎥ ⎦

For this problem, the equilibrium operator,  ⋆ , is no longer the transpose of . The terms with the first order operator just transpose, but the constant terms change sign, resulting in: ⎡0 ⎢ ⎢𝜕 ⋆ = ⎢ ⎢ 𝜕x ⎢ ⎢0 ⎣

0

0

0

𝜕 𝜕y

𝜕 𝜕y

𝜕 𝜕x

𝜕 𝜕x −1 0

𝜕⎤ 𝜕y ⎥ ⎥ 0 ⎥. ⎥ ⎥ −1⎥ ⎦

Fundamental Equations

We note that the enforcement of equilibrium of moments implies that the shear forces may be considered as dependent variables. In the absence of applied couples this dependency is expressed by: 𝜕 { } ⎡ ⎢ 𝜕x qx =⎢ qy ⎢0 ⎣

0 𝜕 𝜕y

𝜕 ⎤⎧ m ⎫ 𝜕y ⎥ ⎪ xx ⎪ myy ⎬ 𝜕 ⎥⎥ ⎨ ⎪mxy ⎪ ⎭ 𝜕x ⎦ ⎩

In this case the two equations in (A.2) can be condensed into one, as a function only of the moments and of the transverse pressure load. The corresponding equilibrium operator is ] [ 2 𝜕2 𝜕2 𝜕 2  ⋆M = . 𝜕x2 𝜕y2 𝜕x 𝜕y A.1.4.1 Reissner–Mindlin Theory

For Reissner–Mindlin theory no further assumptions are made regarding the transverse shear deformations, implying that all components of u can vary independently. The constitutive relation operators k and f are, for an isotropic medium, 𝜈h2 ⎡ h2 ⎢1 − 𝜈 1 − 𝜈 ⎢ 2 h2 ⎢ 𝜈h Eh ⎢1 − 𝜈 1 − 𝜈 k= 12(1 + 𝜈) ⎢ 0 0 ⎢ ⎢ 0 0 ⎢ 0 0 ⎣ 𝜈 ⎡ 1 − 2 0 ⎢ h2 h ⎢ 𝜈 1 0 ⎢− 2 h2 12 ⎢ h 1 +𝜈 f = 0 2 2 Eh ⎢⎢ 0 h ⎢ 0 0 0 ⎢ ⎢ 0 0 0 ⎣

⎤ 0 0⎥ ⎥ 0 0⎥ ⎥, 0 0⎥⎥ 5 0⎥ ⎥ 0 5⎦

0 0 h2 2 0 0 0 0

0 1+𝜈 5 0

0 ⎤ ⎥ ⎥ 0 ⎥ ⎥ 0 ⎥. ⎥ 0 ⎥ ⎥ 1 + 𝜈⎥ 5 ⎦

These equations implicitly involve the bending modulus of the plate, Df , used in Section 2.2, where h is the thickness of the plate: Df =

E h3 . 12(1 − 𝜈 2 )

The boundary operator becomes ⎡0 ⎢0 ⎢  = ⎢0 ⎢nx ⎢n ⎣ y

nx 0 ny 0 0

0⎤ ny ⎥ ⎥ nx ⎥ ; 0⎥ 0 ⎥⎦

247

248

Equilibrium F.E. Formulations

its application in (A.5) implies that equilibrium on a given boundary is imposed on the projection on that boundary of the shear forces, qn = nx qx + ny qy , and of the moment, which has Cartesian components mx = nx mxx + ny mxy and my = ny myy + nx mxy . Kinematic admissibility requires C0 continuity of transverse displacements and rotations. A.1.4.2 Kirchhoff Theory

In Kirchhoff theory it is assumed that the transverse shear strains are zero. This implies that the rotations can be written as a function of the plate transverse displacement and only the curvatures are independent. The generalized stress and strain vectors simplify to 𝝈 = {mxx , myy , mxy } and 𝜺 = {𝜒xx , 𝜒yy , 𝜒xy }, while the displacement vector becomes u = {𝑤}. The compatibility operator becomes 2 ⎡ 𝜕 ⎤ ⎡𝜕 ⎢ 𝜕x2 ⎥ ⎢ 𝜕x ⎢ 𝜕2 ⎥ ⎥ = − ⎢⎢ 0 K = − ⎢ ⎢ 𝜕y2 ⎥ ⎢𝜕 ⎢ 𝜕2 ⎥ ⎢ ⎥ ⎢2 ⎣ 𝜕y ⎣ 𝜕x 𝜕y ⎦

0⎤ ⎥⎡ 𝜕 ⎤ 𝜕 ⎥ ⎢ 𝜕x ⎥ 𝜕y ⎥ ⎢ 𝜕 ⎥ = − 2  1 . 𝜕 ⎥⎥ ⎢⎣ 𝜕y ⎥⎦ 𝜕x ⎦

The adjoint of  K , the equilibrium operator  ⋆K , is minus the transpose of  K , and then  ⋆K ≡  ⋆M . The elasticity operators correspond to the first three rows and columns of those matrices that are used for the Reissner–Mindlin plates. Defining the boundary operators, and interpreting them is complex, because the kinematic assumptions made have implications on the definition of consistent static boundary conditions, in a ‘not so obvious’ way, which challenged some of the brightest minds of the early 19th century, until solved by Kirchhoff (1850). From a kinematic point of view the initial, but naive, idea might be to prescribe one displacement and two rotations along a given side, which defines a local Cartesian reference frame from its tangent s and outward normal n. However, this is not possible, because when the transverse displacement 𝑤 is prescribed the corresponding rotation 𝜃s is also known, or if that rotation is prescribed the displacement is also known to within a constant; that is, only the displacement and the rotation perpendicular to the side, 𝜃n are independent variables. The corresponding boundary condition can be written as: [ 1 0

0 nx

] { } ] ⎧𝑤⎫ [ 1 { } 0 ⎪ ⎪ 𝑤̄ 𝜕 𝜕 𝜃 u = = . z −nx − ny ny ⎨ x ⎬ 𝜃̄n ⎪ 𝜃y ⎪ 𝜕x 𝜕y ⎩ ⎭

The ‘obvious’, but wrong, way to proceed for the static boundary conditions would be to prescribe the projection of the moments and of the shear forces, illustrated in Figure A.4, imposing three conditions, one more than we can prescribe from a kinematic perspective.

Fundamental Equations

This inconsistency is corrected by considering an energetically equivalent force that embodies the relation between the transverse displacement of the plate and the tangential rotation. For a general displacement, on a boundary that is not necessarily straight,3 the work done by the the moments and forces on the boundary is r2

∫r1

(mnn 𝜃n + mns 𝜃s ) dr,

but the Kirchhoff hypothesis implies that 𝜃s = −𝜕𝑤∕𝜕r (Timoshenko and Woinowsky -Krieger, 1959; Wempner, 1973). The work then becomes r2

∫r1

(mnn 𝜃n − mns

𝜕𝑤 + qn 𝑤) dr. 𝜕r

Integrating by parts (details in Section 4.5) we obtain r2

∫r1

(mnn 𝜃n + (qn +

𝜕mns r )𝑤) dr − [mns 𝑤]r21 . 𝜕r

This provides the definition of the energetically equivalent force, which consistently combines the effect of the shear force and of the twisting moment on the boundary, usu𝜕mns ally designated as the equivalent shear force, rn = qn + . Furthermore, this result 𝜕r indicates that the twisting moment at the extremities of each side must also be considered as vertex forces RV when enforcing equilibrium. An interpretation of the physical meaning of this transformation was provided by Kelvin and Tait (Kelvin/Thomson and Tait, 1883; Love, 1892; Southwell, 1936). The nature of the normal operator  is changed by this projection, which now includes derivatives and components of the normal. This can be understood by realizing that the integration by parts of a term that includes  K (or  ⋆K ) must be done in two steps, one for  1 , the other for  2 , resulting in terms that combine  i s and  j s. We observe that from the viewpoint of the hybrid equilibrium formulation we need to define an operator that, for a given straight side, transforms the internal moments into the normal moment and the equivalent shear force on that side. Using the tensor transformation rules and the derivatives along r as functions of the Cartesian derivatives, we obtain the following definitions, which make distinct projections for sides and for vertices: } ⎡ { n2y n2x mnn 𝜕 𝜕 𝜕 𝜕 =⎢ 2 −nx n2y + ny (1 + n2x ) rn ⎢nx (1 + n2y ) − nx ny 𝜕x 𝜕y 𝜕x 𝜕y ⎣ ⎤ ⎧mxx ⎫ 2nx ny ⎪ ⎪ 𝜕 𝜕⎥ m for side entities; ny (1 − n2x + n2y ) + nx (1 + n2x − n2y ) ⎥ ⎨ yy ⎬ 𝜕x 𝜕y ⎦ ⎪mxy ⎪ ⎭ ⎩ {

}

[

RV = −nx ny

nx ny

n2x



n2y

⎫ ⎧ ] ⎪mxx ⎪ ⎨myy ⎬ for vertex entities; ⎪mxy ⎪ ⎭ ⎩

3 The local coordinate r, introduced in Section 4.2, is used to indicate this.

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which are written symbolically as T

t =   𝝈.

(A.11)

The equation given for RV is the contribution from one side of an element to the equivalent Kirchhoff shear force concentrated at the initial vertex of one side. For the other vertex the contribution changes sign. Note that, when the corner has two orthogonal sides, the classical result is obtained of a Kirchhoff corner force equal to twice the twisting moment defined with reference to the axis of the sides. However, for other geometries the result is different. Kinematic admissibility requires C1 continuity of transverse displacements.

A.2 Compatibility of Strains Normally compatibility is considered using the displacements as the starting point. In this case, if the required continuity conditions and the kinematic boundary conditions are satisfied, the displacement field is compatible. When only a strain field is defined, usually obtained from a stress field, the question of assessing its compatibility is more complicated. Before trying to integrate the displacements it is advisable to determine whether this is possible and, if it is, to determine whether it is possible to satisfy all the kinematic boundary conditions. A.2.1 Integrability Conditions

The differential equations that must be verified imply that the strain field is compatible, so they are strictly speaking compatibility conditions. To differentiate them from the conditions on the displacements they are frequently referred to as ‘strain compatibility’ or as ‘integrability’ conditions. For two dimensional solid mechanics problems, three strain components are derived from two fields of displacement components. This implies that there is one condition between the strain components that must be verified for a corresponding displacement field to exist. This is obtained from the second derivatives of the strains, as 𝜕 2 𝜀xx 𝜕 2 𝜀yy 𝜕 2 𝛾xy + − = 0. (A.12) 𝜕y2 𝜕x2 𝜕x𝜕y For three dimensional solid mechanics problems, the reasoning is similar: six components of strain and three independent components of displacement imply three compatibility conditions. Normally six equations are presented (Love, 1892), these being the ones that are different out of the 81 obtained from the application of the summation convention using indicial notation in the form 𝜕 2 𝜀ij 𝜕xk 𝜕xl

+

𝜕 2 𝜀jl 𝜕 2 𝜀kl 𝜕 2 𝜀ik − − = 0. 𝜕xi 𝜕xj 𝜕xj 𝜕xl 𝜕xi 𝜕xk

(A.13)

But it should be noted that these six conditions are linearly dependent (Wunderlich and Pilkey, 2014), and can be reduced to three independent equations. In Reissner–Mindlin plates there are five strain components and three displacements. The two corresponding integrability conditions are: 2 𝜕𝜒xx 𝜕𝜒xy 1 𝜕 2 𝛾x 1 𝜕 𝛾y = 0; − − + 𝜕y 𝜕x 2 𝜕x𝜕y 2 𝜕x2

(A.14a)

Fundamental Equations

𝜕𝜒yy 𝜕x



𝜕𝜒xy 𝜕y

+

2 2 1 𝜕 𝛾x 1 𝜕 𝛾y − = 0. 2 𝜕y2 2 𝜕x𝜕y

(A.14b)

Kirchhoff plates, with three strains and one displacement, also require two conditions. By setting the 𝛾s to zero the equations for Reissner–Mindlin plates can be adapted as: 𝜕𝜒xx 𝜕𝜒xy − = 0; (A.15a) 𝜕y 𝜕x 𝜕𝜒yy 𝜕𝜒xy − = 0. (A.15b) 𝜕x 𝜕y Finally we note that for potential problems, where the gradient of the scalar field, g = 𝛁 f , plays the role of the strains, the integrability conditions are written for problems in two and three dimensions,4 as: 𝜕gy 𝜕gx = ; 𝜕y 𝜕x 𝜕gy 𝜕gx = , 𝜕y 𝜕x

𝜕gy 𝜕z

=

𝜕gz , 𝜕y

𝜕gz 𝜕g = x. 𝜕x 𝜕z

A.2.2 Enforcement of the Kinematic Boundary Conditions

The verification of the equations previously presented ensures that there is a displacement field that originates from the strains being considered. It is also possible to predict whether the displacements that will be obtained from given strains can match the prescribed kinematic boundary conditions or the displacements obtained for the adjacent elements. Instead of detailing every possible situation, which tends to become rather complicated, we will just point to the main aspects that must be considered when this facet of the compatibility of a strain field is being assessed. The first aspect to consider is whether the traces of the strains match at each boundary entity. In continuum problems this simply corresponds to requiring that the curvatures and the elongations of a side/face must match the values that are computed from the adjacent entity, either a kinematic boundary or another element (Pereira et al., 1999). This guarantees that by adding a rigid body displacement, the solutions can be made to match. Furthermore it is necessary to impose compatibility at the boundary of each interface,5 which was not considered in the aforementioned paper. This requires that the shear strains at the vertices/edges must be such that the sum of the shear induced rotations6 around the vertex/edge is zero (Beckers, 1972). The displacements must be adjusted for every boundary entity, but only one rigid body correction is available per element. If the kinematic boundary is simply connected, that is, topologically equivalent to a sphere or to a circle, this is automatically verified, because the continuity of the imposed displacements at the connections between entities (edges or vertices) ensures uniqueness of the solution. 4 In this case, as for the 3D solid mechanics problems, with linearly dependent conditions. 5 The endpoints of the sides for 2D and the edges of the faces for 3D. 6 Actually the sum of the jumps in 𝜀ns .

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Equilibrium F.E. Formulations

When the kinematic boundary is not simply connected, additional conditions must be imposed. A very simple 1D example is sufficient to illustrate this situation. An arbitrary (finite and piecewise continuous) strain field in a bar is always integrable. The integration constant, which corresponds to a rigid body displacement, can be used to adjust the value at one end, but if this bar has fixed values at both ends, we can only say that the strains are compatible when the relative displacement obtained by integration of the strains matches the difference between the imposed values at the ends. This is generalized for more complex problems by requiring that the relative displacements, as computed from the strains along arbitrary paths joining distinct parts of the kinematic boundary, match the differences between the corresponding imposed values.

A.3 General Elastodynamic Problem When dynamic forces are considered, (A.2) is rewritten as: Internal Equilibrium

 ⋆ 𝝈 + b̄ − 𝝆 ü = 0;

in Ω

(A.16)

The mass density matrix 𝝆 corresponds directly to the mass density scalar, 𝜌, for two and three dimensional elasticity, but this is not the case for plate bending where the moments are generalized stresses, otherwise termed stress resultants, that is, integrals of stresses. Then the mass density matrix is: h∕2

𝝆=

∫−h∕2

0 0 ⎤ ⎡𝜌(z) ⎢ 0 0 ⎥ dz, z2 𝜌(z) ⎥ ⎢ 2 0 z 𝜌(z)⎦ ⎣ 0

where the terms including z2 appear because horizontal displacements/accelerations are proportional to z, as well as the dynamic couple they induce. Often this rotational inertia is disregarded and only the term corresponding to the transverse displacement, 𝑤, is considered. Otherwise the rotational inertia forces must be considered as distributed couples.

References Beckers P 1972 Les fonctions de tension dans la méthode des éléments finis Thèse de doctorat Université de Liège. Kelvin/Thomson W and Tait PG 1883 Treatise on Natural Philosophy. Cambridge University Press. Kirchhoff GR 1850 Ueber das Gleichgewicht und die Bewegung einer elastichen Scheibe. Journal fur die Reine und Angewandte Mathematik 40, 51–58. Love AEH 1892 A Treatise on the Mathematical Theory of Elasticity. Cambridge University Press. Pereira OJBA, Almeida JPM and Maunder EAW 1999 Adaptive methods for hybrid equilibrium finite element models. Computer Methods in Applied Mechanics and Engineering 176(1-4), 19–39. Southwell RV 1936 An Introduction to the Theory of Elasticity for Engineers and Physicists. Oxford University Press.

Fundamental Equations

Timoshenko SP and Woinowsky-Krieger S 1959 Theory of Plates and Shells 2 edn. McGraw-Hill New York. Wempner GA 1973 Mechanics of Solids with Applications to Thin Bodies. McGraw-Hill. Wunderlich W and Pilkey WD 2014 Mechanics of Structures: Variational and Computational Methods. CRC press.

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B Computer Programs for Equilibrium Finite Element Formulations This Appendix is written in order to: • provide insight on how the theory was implemented; • illustrate the possibilities of the programs that were developed; • serve as a minimal ‘User Guide’ for these programs. A reader, seeking ideas for the development of his/her own work, should appreciate the following points: • The programs are written in the MATLAB programming language (The MathWorks Inc., 2013). • We have deliberately avoided developing a general purpose application. Each program is focused on a specific type of problem. • These programs are not concerned with user interface aspects. The definition of the meshes and boundary conditions, as well as the post-processing of the solutions is done using text files that are processed by other programs. • In order to achieve this goal our implementation relies mainly on the use of gmsh, (Geuzaine and Remacle, 2009).1 Although it is possible to prepare examples and to interpret results without using it, this is not recommended. • We use two MATLAB classes (mche/mchf), which facilitate the task of determining topological relations in a general finite element mesh, as well as the mtimesx2 routines for ‘Fast Matrix Multiply with Multi-Dimensional Support’, developed by James Tursa, for an efficient implementation of the numerical integration schemes. • The programs developed for this book are available on request to the first author. They will eventually be published under an open source licence, but they remain under copyright. This appendix contains some points about the auxiliary programs that are used, followed by an introductory explanation of how the equations of the hybrid equilibrium models are obtained.

1 Homepage at http://gmsh.info/ A three-dimensional finite element mesh generator with built-in pre- and post-processing facilities. 2 Files and instructions at http://www.mathworks.com/matlabcentral/fileexchange/25977-mtimesx-fastmatrix-multiply-with-multi-dimensional-support Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

Computer code

B.1 Auxiliary Programs B.1.1 gmsh

The use of gmsh for mesh generation and for visualization of results is recommended. Its manual, together with many examples and tutorials, are provided with the program. It is out of the scope of this brief introduction to describe the use of the program. The relevant point is that from the definition of a geometry (2D or 3D), gmsh will generate a mesh, which is saved in a ‘mesh file’ (normally with the extension ‘.msh’). We will only use the text (ASCII) format, exemplified in Figures B.1 and B.2. Each mesh file includes at least three sections: $MeshFormat; $Nodes; and $Elements. The first two are self explanatory. We just note that our 2D programs assume that the mesh is in a plane with constant z, by disregarding the value of this coordinate. Observing Figure B.1, the initial value of 8 in the $Elements file section, which describes the elements of the mesh, may seem surprising for a mesh with 3 triangles. This happens because the five edges on the external boundary of the domain are also considered elements, since they were explicitly declared as ‘Physical groups’, identified as number 8 for the lower and for the leftmost sides, and number 7 for the others. The format of the lines describing each element is always: number, type, number_of_tags, tags, vertices

The basic element types, relevant for our applications, are described in Table B.1. The number of tags is not fixed in the definition of the format, but gmsh always creates $MeshFormat 2.2 0 8 $EndMeshFormat $Nodes 5 1 1.3 -0.5 0 2 -1.1 -0.9 0 3 -0.3 0.5 0 4 1 0.6 0 5 0.1 -0.7 0 $EndNodes $Elements 8 1 1 2 8 1 2 5 2 1 2 8 1 5 1 3 1 2 7 2 1 4 4 1 2 7 3 4 3 5 1 2 8 4 3 2 6 2 2 9 6 5 1 4 7 2 2 9 6 5 4 3 8 2 2 9 6 2 5 3 $EndElements

4

4

3

7 5

6

8

2

1

3

5

2

1

Figure B.1 Contents of a simple bidimensional mesh file (Simple2D.msh) and the corresponding mesh.

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Equilibrium F.E. Formulations

$MeshFormat 2.2 0 8 $EndMeshFormat $Nodes 7 1 1.3 -0.5 0.0 2 -1.1 -0.9 0.0 3 -0.3 0.5 0.0 4 1.0 0.6 1.0 5 0.1 -0.7 0.0 6 -0.05 -0.15 0.5 7 0.6075 -0.2198 0.3440 $EndNodes $Elements 7 1 2 2 18 8 4 3 6 2 2 2 18 8 3 2 6 3 4 2 17 16 4 6 3 7 4 4 2 17 16 3 6 5 7 5 4 2 17 16 6 3 5 2 6 4 2 17 16 1 3 5 7 7 4 2 17 16 1 4 3 7 $EndElements

4 3 6

7

2 5 1 Figure B.2 Contents of a simple tridimensional mesh file (Simple3D.msh) and the corresponding mesh.

Table B.1 Main element types in gmsh type

Description

1

Line

number of vertices 2

2

1

2

3

Triangle

3 2

1

4

3

Tetrahedron

4 2

1

15

Node

1

4

1

Computer code

$MeshFormat 2.2 0 8 $EndMeshFormat $NodeData 1 "Scalar node data for Simple2D" 1 0.0 3 0 1 5 1 0 2 4 3 0 4 4 5 0 $EndNodeData Scalar node data for Simple2D 0

2

4

Figure B.3 A scalar field defined on the mesh of Figure B.1, using a $NodeData file section.

files with two tags for each element. The first indicates the physical group to which the element belongs and the second the geometric entity from which it was created. See, for example, that in Figure B.1 elements 1, 2 and 5 belong to group 8, while elements 1 and 2 were generated from the line 1. In 2D problems the physical group tag of the edges (1D elements)3 is used to assign boundary conditions, defined in a ‘.prob’ file, which will be described later. The physical group of the faces (2D elements) is used to assign material properties, for example, elastic constants and body loads. For 3D problems the boundary conditions are defined for the faces (2D elements) and the material properties are assigned from the physical group of the cells (3D elements). For the example in Figure B.2, boundary conditions can only be explicitly assigned to the two uppermost faces of the solid. The number of vertices depends on the type of element, as described in Table B.1, which only lists the elements recognized by our programs. The optional file sections: $NodeData; $ElementData; and $ElementNodeData are used to store outputs for postprocessing. A simple file is presented in Figure B.3, where the initial statements define the data being presented: one literal label (the description of the data, used in the scale); one real value (0.0, assumed to be a time reference of this solution) and three integer values (0, assumed to be the time step index; 1, the number of components of the solution at each node; and an indication that there are 5 nodal values). The last 5 lines are again self explanatory: the number of each node and the corresponding nodal value. Note that the number of components of the solution can also be 3, for vectors, and 9, for tensors. 3 and of the nodes (0D elements), in the case of Kirchhoff plates.

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Equilibrium F.E. Formulations

A $ElementNodeData file section is more powerful, but also more complex. Using it requires an $InterpolationScheme file section, which defines the basis functions used, in terms of the local coordinates of the element. It should be noticed that the nodal definition of the solution, used in a $NodeData file section, naturally leads to a continuous field, whereas the element-wise solution, in a $ElementNodeData file section, may represent a discontinuous field. B.1.2 The mche and mchf Classes

The mche and mchf classes implement the index-based ‘Modified Compact Half-Edge’ and ‘Modified Compact Half-Face’ data structures described in Lage et al. (2016), which allow efficient determination of the topological relations between entities (volumes, faces, edges or vertices) of a mesh, at a computational cost that is practically independent of the dimension of the mesh. In the case of our finite elements it is necessary to relate the boundary of the elements (faces/sides) to the corresponding global entities. We therefore use these classes to determine the relations between faces and edges in 2D meshes and, for 3D meshes, between volumes and faces. As an example, the following statements create a member of the che_gmsh class, load the ‘.msh’ file of Figure B.1 and assign two lists, where the faces and edges adjacent to vertex 5 are stored: m = che_gmsh(); m.load('Simple2D.msh'); [faces, edges] = m.get_ce_v(5); Unlike gmsh, the mche class defines all the entities involved, implying that in this case there are three faces and four edges adjacent to vertex 5. The values in the faces and edges lists are, respectively, {1, 2, 3} and {1, 3, 5, 6}. The corresponding numbers in gmsh are obtained from the following commands, which transform the numbers used for the topological entities into those used by gmsh: gmsh_faces = m.elems_faces(faces); gmsh_edges = m.elems_edges(edges); which yield, respectively, {6, 7, 8}, and {2, 0, 0, 1} (as a sparse vector). B.1.3 mtimesx

The mtimesx routines enable efficient multiplication of multi-dimensional arrays. The fundamental idea is that when the number of dimensions of the arrays is greater than 2, a ‘normal’ matrix multiplication is performed on the first two indices for each of the remaining dimensions, which must be equal for both arrays.4 In our applications this is normally followed by the computation of the sum for the higher indices. For example, % % C D %

A B = = D

has dimensions {n1 , n2 , nP } has dimensions {n2 , n3 , nP } ∑ mtimesx(A, B); % cijk = m aimk bmjk ∑ sum(C, 3); % dij = k cijk has dimensions {n1 , n3 }

4 One of the dimensions can also be 1 or missing: in that case that component of the array behaves as a scalar.

Computer code

is equivalent to D = zeros(size(A,1), size(B,2)); % (n1 , n3 ) % Could also check the values of % n2 , using (size(A,2)==size(B,1)), % and nP , using (size(A,3)==size(B,3)). for k = 1:size(A,3) D = D + A(:,:,k)*B(:,:,k); end We use it to perform the matrix multiplications involved in the numerical computation of element matrices, illustrated here for an elementary flexibility matrix, defined in (4.10). When we assume uniform material properties within each element, this formula becomes: e =

∫Ωe

STe

f Se dΩ ≈

GP ∑

STe (xi ) f Se (xi ) 𝑤i ∥ Ji ∥ .

i=1

We assume that the following arrays are defined: S - the stress approximation matrices, computed at each Gauss point, represented by an array with dimensions 3 × ns × GP; f - the material flexibility matrix, which has dimensions 3 × 3; GwdJ - the product of the Gauss weights by the determinant of the Jacobian - a vector with GP elements. % compute STe ( f Se ) at each Gauss point (third index) aux = mtimesx(S, 't', mtimesx(f, S)); % each Gauss weight is multiplied by the triple product % at the corresponding point aux = mtimesx(reshape(GwdJ, [ 1 1 size(GwdJ,2) ]), aux); % sum the products at all Gauss points Flex = sum(aux, 3);

B.2 Structure of the Programs B.2.1 Definition of the Mesh

The definition of the mesh is entirely left to either the mche or to the mchf classes. The resulting object manages the housekeeping of all the necessary information, which can be obtained either by directly accessing the class members, or by using one of the ‘get_xx_y’ functions. For the application of the hybrid formulation to 2D problems it is critical to identify the edges and the vertices of a face, when setting up and assembling the elementary matrices. This information is directly obtained from either ‘get_ev_c’ or ‘get_cv_e’, depending on whether we are finding the edges adjacent to an element or the elements adjacent to an edge. The vertices are used to obtain the geometry, while the edges provide the connectivity. Likewise for 3D problems functions ‘get_fev_c’ and ‘get_cev_f’ are crucial for the hybrid formulations. We note that in this case the information on the edges is not used.

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Equilibrium F.E. Formulations

Figure B.4 Problem definition for the mesh of of Figure B.1.

B.2.2 Definition of the Material Properties and Boundary Conditions

An auxiliary function load_problem is used, as the name indicates, to load the properties of a given problem from a text file. This file, which by default uses the extension ‘.prob’, has a simple structure, consisting of a sequence of ‘Sections’, that provide the data for the entities that match a given tag. Each section is identified by a header line, which has an asterisk, the type of section and the definition of the data needed by that program for an entity of that type. The lines preceding the first section are disregarded and should be used for comments. We use Points, Edges, Faces and Vols sections in our problem files, but the function allows for the definition of sections with arbitrary names.5 What information each section provides depends on the problem being considered. For a plane problem Points and Edges may have the definition of the boundary conditions, while the Faces will have information on the characteristics of the domain (e.g. body forces and material properties) and the Vols section will not be used. The absence of a section indicates that default values should be assumed. This format, which uses the text scanning facilities in MATLAB (The MathWorks Inc., 2013), is very flexible. We refer to MATLAB’s documentation to explain the ‘magic’ involved in expressions such as %d %d %q, which is used to read two integers and one string in quotes. In the example of Figure B.4 we consider, for the mesh of Figure B.1, that the sides with tag 8 are fixed in both directions and that on the sides with tag 7 a load of {0, −1} is applied. The elements derived from the face with tag 9 have a Young’s modulus of 1, a Poisson’s ratio of 0.15 and zero body forces. A typical usage might be: prob = load_problem('Simple2D.prob'); ... % Define properties of element elem (in mche numbering) 5 Beware of the risk of defining sections with mistyped names.

Computer code

tag = m.ele_tags{m.elems_faces(elem)}(1); which_tag = find(prob.Faces.tags == tag); if (isempty(which_tag)) % assign default values ... else % vals{1} and vals{2} are vectors Young = prob.Faces.vals{1}(which_tag); Poisson = prob.Faces.vals{2}(which_tag); % vals{3} is a cell. We use a {} to access the string load = prob.Faces.vals{3}{which_tag}; end B.2.3 Definition of the Approximation Functions

The numerical calculation of the element matrices and the evaluation of fields for post-processing require the availability of procedures that allow for the computation of the approximation functions, in terms of coordinates that can be expressed either in the global or in a local (elemental) reference frame. For the stresses we base the approximation of each component on a complete basis of a given degree, defined in the global (x, y) or (x, y, z) frame, which is constrained to produce self-balanced stresses. Although it is possible to use self-balanced stress approximations that are based on local coordinates, we do not follow this approach, avoiding the additional complexities involved in their derivation. For the case of a linear self-equilibrated stress approximation in two dimensional elasticity, used to illustrate (4.1), a complete linear basis is 𝝓1 = [1 x y] and a general stress approximation is ⎡𝝓1 0 0 ⎤ 𝝈̃ e = ⎢ 0 𝝓1 0 ⎥ s̃̂ e = S̃ e s̃̂ e . ⎥ ⎢ ⎣ 0 0 𝝓1 ⎦ This approximation is not self equilibrated, since [ ] 0 1 0 0 0 0 0 0 1 ̂ ⋆̃ ̂ s̃  Se s̃ e = 0 0 0 0 0 1 0 1 0 e is not nil for arbitrary values of s̃̂ e , but by applying a reduction of the basis, so that s̃̂e2 = −s̃̂e9 and s̃̂e6 = −s̃̂e8 we achieve this goal. Such a reduction of basis may be represented, for example, by ⎡1 ⎢0 ⎢ ⎢0 ⎢0 s̃̂ e = ⎢0 ⎢ ⎢0 ⎢0 ⎢0 ⎢0 ⎣

0 1 0 0 0 0 0 0 −1

0 0 0 0 1 0 0 1 0 0 0 0 0 0 0 0 0 0

0 0 0 0 1 0 0 0 0

0 0 0 0 0 1 0 −1 0

0⎤ 0⎥ ⎥ 0⎥ 0⎥ 0⎥ ŝ e = T ŝ e , ⎥ 0⎥ 1⎥ 0⎥ 0⎥⎦

that is, Se = S̃ e T.

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This matrix can be automatically determined for an arbitrary polynomial basis by explicitly determining the transformation that zeroes the body forces, that is, a T, such that (⋆ S̃ e ) T = 𝟎. ⋆ S̃ e is, in the general case, a function of the coordinates, which we require to be zero everywhere. This is achieved by expressing each component of the body forces as a combination of its monomial coefficients, that are then required to be identically zero. This ‘identification of the coefficients of the monomials’ transforms a matrix of polynomials, ⋆ S̃ e , into a matrix of constants, which we term 𝔹e , with a number of rows equal to the number of rows of ⋆ multiplied by the dimension of the polynomial basis that represents the body forces. Matrix T corresponds to a basis for the null space of 𝔹e , found, for example, by using a singular value decomposition: ] [ 1T ] [ [ ] ⌈𝚺𝔹 ⌋ 𝔹 𝔹e = 1𝔹 0𝔹 . T ⌈𝟎⌋ 0𝔹 In this format matrix T is equal to 0𝔹 , and a particular solution for a given body load can be determined from 1𝔹 . ̄ in vector be , with the same format as Writing the coefficients of the body loads, b, 𝔹e , the coefficients that define a particular solution, as a function of S̃ e , are obtained by solving 1𝔹 ⌈𝚺𝔹 ⌋1𝔹 s̃̂ e0 + be = 𝟎, T

s̃̂ e0 = −1𝔹 ⌈𝚺𝔹 ⌋−1 1𝔹 be . T

=⇒

This approach has the inconvenience of requiring the determination of vector be for each type of load considered, as well as some parsing of the expressions defining general loads, in order to express them as combinations of the load types. A more general, and formally simpler, alternative procedure is used instead. Equilibrium is enforced, in a weighted residual sense, between the applied body loads and the loads induced by the stress approximation, which can be expressed by (⋆ S̃ e )1𝔹 . The fundamental idea is that by using that basis as weighting functions, we guarantee that the resulting system is not singular. The coefficients defining the particular solution are thus obtained by solving: 1𝔹 (⋆ S̃ e )T (⋆ S̃ e )1𝔹 dΩ s̃̂ e0 + T

∫ Ωe

1𝔹 (⋆ S̃ e )T b̄ dΩ = 𝟎. T

∫ Ωe

In this context it is important to recognize that whenever b̄ cannot be represented by ⋆ S̃ e this procedure cannot produce a particular solution that locally verifies equilibrium. This can be an advantage as it produces the ‘most’ quasi-equilibrated solution in such a situation, for example for higher degree or non-polynomial body forces, but the potential loss of local equilibrium must always be pointed out. This condition can be identified by checking whether ∫ Ωe

T b̄ (⋆ S̃ e )1𝔹 dΩ s̃̂ e0 +

∫Ωe

T b̄ b̄ dΩ

is equal to zero. For example, when a quadratic basis is used in a 2D elasticity problem, 𝝓2 has six components and, therefore, S̃ e has 3 rows and 18 columns. ⋆ S̃ e is a linear polynomial,

Computer code

with 2 rows and 18 columns, which can be expressed as a function of the members of 𝝓1 . The coefficients of these polynomials can be arranged in a matrix of constants, ̄ times the 3 members of 𝝓1 ) with 6 rows (2 components of the vector of body forces, b, and 18 columns, the null space of which corresponds to 0𝔹 , that is to matrix T, with dimension 18 × 12, while 1𝔹 has dimension 18 × 6. Since this matrix is independent of the geometry of the element, it only needs to be determined once for each approximation basis, and recorded for all subsequent operations. In the example programs provided, this task is performed by a Mathematica notebook which writes the files with MATLAB functions for the computation of the stress at a set of points. This notebook also computes the definitions needed for the graphical representation of the stresses as an $ElementNodeData section of gmsh: the transformation between global and element coordinates; and the $InterpolationScheme scheme for the approximation basis used. The utilization of these functions in a MATLAB program requires a call to the initialization code, for example, DS = DefStress2D(degree), which creates a structure with the function that computes the basis at a set of points, the function that expresses the coordinate transform for a given geometry, and the T matrix. The computation of the self-equilibrated stress basis at a set of points is then achieved by a call such as s = CalcStress2D(xy, DS), which results in a multi-dimensional array s that, for a quadratic approximation and for an xy with 6 points, has dimensions 3 × 12 × 6. We notice that it is more common to derive the self-equilibrated stress approximations from a stress generation function, for example the Airy, Maxwell–Morera or Southwell functions. We opted instead to implement an alternative that is, in our opinion, more general and numerically efficient, since it only requires the computation of 𝝓n at each point and the multiplication of the resulting S̃ e by T. Computing 𝝓n only once saves operations, particularly for higher degree polynomials, which also require a larger number of integration points. Performing the multiplication of S̃ e and T, without explicitly assembling either, only multiplying 𝝓n by the corresponding part of T, which is stored as a sparse array, is the other ingredient for an efficient implementation. The definition of the approximation of the boundary displacements is more straightforward. Since we always use linear mappings of the element boundaries, a basis defined in terms of a local set of coordinates spawns the same approximation space as a Cartesian basis defined in terms of the geometry of the boundary. Therefore each V m matrix in (4.2) is written as function of a local coordinate basis, typically 𝜉 ∈ [−1, 1] for sides and 𝜉 ∈ [0, 1], 𝜂 ∈ [0, 1 − 𝜉] for triangular faces. The code in function_V1D and function_V2D provides bases defined by the monomials of the local coordinates, but the application of, for example, Lagrange or Legendre interpolation polynomials, would be equivalent. B.2.4 Enforcement of Boundary Conditions

As already stated, in equilibrium formulations the essential boundary conditions are equilibrium related, while the compatibility conditions correspond to the natural boundary conditions. We consider the general case, where both conditions can be applied on the same boundary, necessarily in different directions.

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When the hybrid form is used it operates on the essential boundary conditions, leading to the enforcement of equilibrium on the vertices/sides/faces of the elements. On those boundary entities where the displacements are imposed the explicit enforcement of equilibrium is not necessary, on account of the reactions. This leads to two alternative options for the enforcement of the kinematic boundary conditions: • to explicitly remove the boundary displacements corresponding to the vertices/ sides/faces that are fixed; • to consider all possible boundary displacements to be free, and then enforce their prescribed values by using additional constraints; in this case the associated Lagrange multipliers are related to the reactions. The first option leads to a smaller governing system, but is more complex in terms of identifying the numbering of the variables and of considering either non-homogeneous boundary displacements or inclined supports. The governing system corresponding to the second option is larger and has a non-zero pattern that is more complex. We opted to implement the second option because it is simpler, it easily allows working with non-homogeneous boundary displacements and inclined supports, and because MATLAB’s handling of sparse arrays is not sensitive to the pattern of the matrices. Γ ̄ where On an interface m, belonging to Γu , the compatibility condition is 𝒗mu = u, Γu 𝒗m = T m 𝒗m , collects the components of the displacement that are fixed along side m, in ̄ The transformation matrix T m will, for example, the reference frame used to define u. correspond to an identity matrix for fixed sides, and to the vector normal to the side for a sliding support. We impose this compatibility condition in a weak form, using as weighting functions Γ a basis, V mu , suitable to approximate 𝒗 on Γu : Γ

∫Γm

(V mu )T T m V dΓ 𝒗̂ m =

Γ

∫Γm Γ

(V mu )T ū dΓ

or

C m 𝒗̂ m = 𝒗̂̄ 0m .

It is assumed, in general, that V mu is equal to V for sides where all displacements are fixed. Otherwise only some of the components of V are normally used. In this case an optimal approximation of ū is obtained. Γ Nevertheless this is not a requirement. If the degree of V mu is lower than the degree of the projection of V on Γu , a weaker form of compatibility is imposed. The quality of the solution will, in general, be worse, but an equilibrated solution can be determined, as long as the structure is sufficiently fixed. Γ On the other hand, setting the degree of V mu higher than the degree of the projection of V should be avoided, since it introduces dependent constraints. The transpose of matrix C m is also introduced in the equilibrium equation corresponding to interface m, where it is multiplied by a set of variables, r̂ m , which Γ define the distribution of the reactions on the side: r = V mu r̂ m . We recall that ū being equal to 𝒗 is a compatibility condition, which is imposed Γ weakly in equilibrium formulations. The definitions of ū and of V mu do not affect the enforcement of equilibrium.

Computer code

B.2.5 Processing the Solutions

After solving the governing system, each program writes a text file describing the computed solution at selected points and two .msh files, which are used to visualize the stress distributions and the displacements of the element boundaries. Such results can be directly visualized with gmsh. Typical screenshots are provided with the program. A thorough explanation of the process involved in computing the solutions for gmsh is not given here, but it is relevant to explain the basic steps involved. In each element, the components of a solution are approximated using a combination of the basis functions 𝝓. This basis is defined in the global reference frame, x.6 Since gmsh uses functions that are defined in terms of the coordinates of a master element, 𝝃, it is necessary to transform 𝝓(x) into 𝝓(𝝃). This task is performed by a function, DS.handleC, which performs the transformation from the basis in x, used for analysis, into the basis in 𝝃, used for the visualization. This is a square matrix, with the same dimension as 𝝓, whose coefficients depend on the geometry of the element. weights = zeros(mesh.nc, DS.nB, 3); for elem = 1:mesh.nc [ ̃, verts ] = mesh.get_ev_c(elem); coords = mesh.coords(mesh.nodes_verts(verts),1:2) -... ones(3,1)*Center(:,elem)'; T = DS.handleC(coords(:,1), coords(:,2))'; for k = 1:3 weights(elem,:, k) =... T*(DS.T{k}*x((elem-1)*nS+(1:nS)) +... DS.T0{k}*s0(elem,:)'); end end B.2.6 Code Snippets

In this Section we present and discuss some code snippets for typical operations. The first example corresponds to the computation of the flexibility matrix in two dimensional problems. Its implementation is similar for all the problems considered: • • • •

define material flexibility; compute the coordinates of the Gauss points and the Jacobian; compute the equilibrated stress approximation basis at the Gauss points; perform the triple product, using mtimesx, as explained in B.1.3.

B.2.6.1 Computing the Flexibility Matrix of an Element

flex = [ 1 -nu 0; -nu 1 0; 0 0 2*(1+nu)]/Elast; % Plane stress % Get global coordinates of element. % Use a frame at its centre 6 We normally introduce a translation, so that the origin coincides with the centre of the element, but no scaling or rotations are used.

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Equilibrium F.E. Formulations

elem_coords = ... mesh.coords(mesh.nodes_verts(cell_verts), 1:2); Center(:,elem) = sum(elem_coords)/3; elem_coords = elem_coords - ones(3,1)*Center(:,elem)'; % Determinant of the Jacobian of the element dJac = det(DPhi*elem_coords); % Transform the Gauss points into coordinates coordGP = Phi*elem_coords; % Stresses at the Gauss points StressGP = CalcStress2D(coordGP, DS); GWdJ = reshape(NI2D.GW*dJac, [ 1 1 NI2D.npts ]); Flex = sum(mtimesx(GWdJ, mtimesx(StressGP, 't',... mtimesx(-flex,StressGP))), 3); The determination of the equilibrium matrix requires an integration on the boundary of the element, which involves a basis defined on the boundary, V , a basis defined on the element, S, and an operator that depends on the outward normal, N. While this is immediate for vertices, the definition of the reference frames used for edges and faces requires special care, because the same boundary approximation functions must be used for both adjacent elements. It is not advisable to use reference frames that are based on the elements, since they require, at least for one of the elements adjacent to the boundary being considered, an additional change of coordinates, as illustrated in Figure B.5. This transformation may be obtained by considering, in 2D, the orientation and origin of the reference frames. For 3D meshes the situation is more complex, since there are multiple options for the direction of the axes on the boundary.

tB



B

A

tA Figure B.5 For the shared side we have that, in general, tA ≠ tB ≠ tΓ . A change of coordinates is necessary in order to guarantee that, for example, tA = fAB (tB ). We opt to define all functions in terms of tΓ , from which the coordinates in the frame of the element are obtained.

Computer code

To avoid this complexity, the code we present opts to define the integration points, the approximation functions and the normals from the geometry and topology of the boundary entity. The half-edge and half-face data structures store for each boundary entity the reference to its adjacent elements, always one or two. Consistent values are obtained by setting  A =  Γ for the first member of this list and  B = − Γ for the other one, when it exists. The coordinates in the frame of the element are directly obtained from the coordinates in the frame of the boundary entity and from the coordinates of the vertices of the boundary entity. B.2.6.2 The Equilibrium Matrix of a Side of a Plane Element

[ elems_edge, verts_edge ] = mesh.get_cv_e(cell_edges(edge)); % Check orientation of cell_edge relative to the edge entity if (elems_edge(1) == elem) % First element in adjacency list pm = 1; else pm = -1; end % GPs of the edge, in the frame of the element, 3 × 2 vert_coords = mesh.coords(... mesh.nodes_verts(verts_edge),[1 2]) -... ones(2,1)*Center(:, elem)'; % Phi has dimension GPΓ × 3 coordGP = Phi_side*vert_coords; StressGP = CalcStress2D(coordGP, DS); dxdy = pm*(vert_coords(2,:) - vert_coords(1,:)); % The normal is not divided by the length Normal = [ dxdy(2) 0 -dxdy(1); 0 -dxdy(1) dxdy(2) ]; TSide = mtimesx(Normal, StressGP); vtside = mtimesx(VG, 't', TSide); % Not multiplied by the length (compensates for the normal) GWdJ = reshape(NI1D.GW, [ 1 1 NI1D.npts ]); Dside = sum(mtimesx(vtside, GWdJ),3); For solid elements the determination of the equilibrium matrix is quite similar, with the normal defined by the outer product of the vectors defining two edges of the triangular face being considered. In the case of bending plates, in particular for Kirchhoff ’s theory, the expressions are more elaborate, as given in Section A.1.4. The following code snippet computes the contribution of side edge of element elem to the D matrices of that side and of its vertices. % Both the stresses (moments) and their derivatives % have to be obtained

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Equilibrium F.E. Formulations

StressGP = CalcStressKT(coordGP, DS); [ DStressGPx, DStressGPy ] = CalcDStressKT(coordGP, DS); dxdy = pm*(vert_coords(2,:) − vert_coords(1,:)); % % % %

mn = mx * nx ̂ 2 + my * ny ̂ 2 + 2 * mxy * nx* ny; rn = dmxdx*(nx + nx*ny ̂ 2) - dmxdy*nx ̂ 2*ny − ... dmydx*nx*ny ̂ 2 + dmydy*(ny + nx ̂ 2*ny) + ... dmxydx*(2*ny ̂ 3) + dmxydy*(2*nx ̂ 3);

% The normal is divided by the length % (the alternative would be to scale the derivatives) lside = norm(dxdy); dxdy = dxdy/lside; nx = dxdy(2); ny = -dxdy(1); % For the normal moment Normal = pm*[ nx ̂ 2, ny ̂ 2, 2*nx*ny ]; % For the equivalent shear force NormalDx = [ (nx+nx*ny ̂ 2), -(nx*ny ̂ 2), (2*ny ̂ 3) ]; NormalDy = [ -(nx ̂ 2*ny), (ny+nx ̂ 2*ny), (2*nx ̂ 3) ]; % For the corner forces NormalMxy = [ (-nx*ny), (nx*ny), (nx ̂ 2-ny ̂ 2) ]; % The edge terms TSide = [ mtimesx(Normal, StressGP);... mtimesx(NormalDx, DStressGPx)... + mtimesx(NormalDy, DStressGPy) ]; vtside = mtimesx(VG, 't', TSide); GWdJ = lside*reshape(NI1D.GW, [ 1 1 NI1D.npts ]); Dside = sum(mtimesx(vtside, GWdJ),3); % The vertex terms. Note that % StressV = CalcStressKT(elem_coords, DS); % is computed outside the loop on the sides verts = [ edge (mod(edge,3)+1) ]; mvs = mtimesx(NormalMxy, StressV(:,:,verts)); % +mnt on one side, -mnt on the other % This value accumulates for all sides of the element Dv(verts,:) = Dv(verts,:) +... [1 0; 0 -1]*reshape(mvs, [nS 2])';

Computer code

References Geuzaine C and Remacle JF 2009 Gmsh: A 3-D finite element mesh generator with built-in pre- and post-processing facilities. International Journal for Numerical Methods in Engineering 79(11), 1309–1331. Lage M, Martha LF, Moitinho de Almeida JP and Lopes H 2015 IBHM: index-based data structures for 2D and 3D hybrid meshes. Engineering with Computers p. 1–18. doi:10.1007/s00366-015-0395-0. The MathWorks Inc. 2013 MATLAB Version 8.1.0 (R2013a), Natick, Massachusetts.

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Subject Index 2D continua 22, 57, 237 3D continua 62, 238

a Acceleration 199 Adaptivity 166 Adjoint load 187 Adjoint operator 24, 238, 246 Admissibility conditions 155 Admissible loading 56, 101, 115 Body forces 117 Tractions 74, 134 Airy stress potential see Stress potential Annihilator 74, 114 Arc-length method 215 Asymptotic convergence 178 Average displacement 185 Average stress 185

b Babuška–Brezzi condition 32, 54, 133 Bending plates see Plates Bernstein polynomials 214 Bi-material strip 11, 33 Bilinear form 127 Blocking technique see Spurious kinematic modes Boundary Displacements 45 Tractions 44 Velocities 202 Boundary operator 24, 48, 238, 247 Broken approximation spaces 31, 132 Buttressing 88, 104

c Clean boundary displacements 117 Closed star patch 82, 98, 103, 152 Codes of practice 8 Codiffusive 6, 131, 150 Collapse multiplier 222 Compatibility 22, 25, 51, 81, 95, 102, 250 Defaults 69 Complementary Energy 11, 25, 31, 51, 126 Strain energy 11, 20, 26, 127 Complementary (dual) programme 222 Complementary solutions 13, 135, 187 Computational cost 70, 184 Conditioning (numerical) 60 Conforming formulations 22 Constant strain triangle 29, 34 Contravariant 77, 90 Convergence 24, 39, 58, 125, 134, 178 Convergence rate 125, 182, 195 Coplanar faces 96 Corotational formulation 224 Corrective stress field 146 Coset 75 Covariant 77 Curvilinear geometry 68

d Damping 206 Degenerate configuration 82 Design, Structural 7, 156 Dihedral angle 96 Dimension of the system 70 Dirac delta function 184

Equilibrium Finite Element Formulations, First Edition. J. P. Moitinho de Almeida and Edward A. W. Maunder. © 2017 John Wiley & Sons Ltd. Published 2017 by John Wiley & Sons Ltd.

272

Subject Index

Displacement driven problem 130, 170 Drilling degrees of freedom 147 Dual analyses 166 Dual graph see Graph Dual load 187 Dynamic equilibrium 199

e Edge 46 Edge-centred patches 97 Effectivity 190 Eigenproblem 203 Elastodynamics 200 Equivalent nodal force see Nodal force Equivalent shear force 65, 249 Error estimation 6, 166 Error in the constitutive relation 6, 166 Error of the solutions 168 Existence and uniqueness 54, 129 Extractor load 187

f Face 46 Fictitious body forces 138, 144 Fictitious springs 115 Fictitious strains 137 Flexibility matrix 53, 244 Force driven problem 130, 170 Force method 6 Forced vibrations 209–211 Fraeijs de Veubeke’s equilibrated triangle 29, 33 Free body 2 Frequency domain 203

g Galerkin approach 51 Galerkin orthogonality 16, 129, 188 Gauss–Legendre integration 47 Generalized complementary functional 131 Generalized displacement, stress see Local output Global error bounds 167 Graph, dual 221

h Hardening, strain 216 Harmonic excitation 203 Heat conduction 67, 106 Hexagonal elements 58 Hexagonal patch 104 Hexahedral elements 63 Hexahedral macro-element 109 Hierarchical formulation 138, 150 Hierarchical refinement 119 Hybrid complementary potential energy 26, 131 Hybrid stress element 27 Hybrid Trefftz displacement 33 Hybrid Trefftz stress 33 Hyperelastic materials 214 Hyperstatic flux field 107 Hyperstatic stress field 75, 92, 105, 121, 128, 168, 188, 220

i Inadmissible loads Inertial 204 Static 117 Incompressible material 55 Initial strains 52 Inner product space 112 Integrability conditions 250 Internal boundary 45 Interpolation functions 22

j Jacobi polynomials 76, 92 Jacobian matrix 69

k Kernel element 147 Kinematic boundary 45 Kinematic stability 73 Kinematically admissible 12, 16, 128, 138, 167 Kinetic energy density 200 Kirchhoff corner force 65, 250 Kirchhoff plate element 65, 105

Subject Index

l Lagrange multiplier 131 Lagrange polynomials 47 Large displacements 224 Legendre polynomials 78, 92 Legendre–Fenchel transformation Limit analysis 3, 220 Limitation principle 27 Linear form 127 Linear mapping 68 Linear programme 222 Link 82, 98, 146 Load paths 11, 221 Local error bounds 185 Local output 184, 195 Displacement 184 Multiple 195 Stress 184 Lower and upper bounds of eigenfrequencies 207 Lower bound theorem 8, 220

130

m Macro-element 87, 99, 108 Malignant spurious mode 81, 95, 103 Mass density matrix 252 Master safe theorem 7 Maxwell diagram 151 Maxwell stress potential see Stress potential Maxwell–Mohr methods 4 Mechanism, pseudo 73, 121 Membranes see 2D continua Mesh adaptivity 178, 194 Mixed formulation 27, 147 Mixed problem 130, 170 Mobility matrix 202 Moment potential see Stress potential Morera stress potential see Stress potential Multiple loads 195 Multiple outputs see Local output Multiple solutions 55

n Neighbourhood patch 147 Newmark time integration 205 Newton–Raphson iteration 214, 229 Nodal force 2, 8, 23, 150 Non-linear analyses 212

Non-linear elasticity 214 Non-linear mapping 68 Non-simplicial elements 120, 164 Normality rule 220 Nullspace 55, 74, 112 Numerical stability 59

o Open star patch 82, 98, 103, 153 Optimal mesh 178 Orthogonal complement 112 Orthogonality 115, 168

p Partition of nullity 136 Partition of unity 135, 150 Pascal’s pyramid 62 Pascal’s triangle 46 Patch test 32 Pathological (dynamic) mode 207 Pian’s hybrid element 25 Plane strain 57, 238 Plane stress 58, 234, 238, 266 Plastic potential 215 Plasticity 8, 215 Plates 30 Kirchhoff 16, 47, 65, 163, 189, 248 Reissner–Mindlin 64, 99, 163, 247 Poisson’s equation 67, 106 Polluted boundary displacements 117 Positive definite 25, 53, 116, 129 Potential energy 13, 23, 126 Potential problems 67, 106 Primary interface spurious modes 95 Principle of Saint Venant 147 Prolongation condition 140, 153 Pseudo-rigid body rotation 29, 75, 94, 101, 106

q Quadratic eigenvalue problem 207 Quadratic form 129 Quadrilateral macro-element 109 Quasi-simplicial 119 Quotient space 75

r Rayleigh damping 206 Rayleigh–Ritz method 13

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274

Subject Index

Recovery methods 6, 135 Recovery of compatibility 138, 163 Recovery of equilibrium 143, 164 Reissner–Mindlin elements see Plates Residual 229 Response curve, dynamic 211 Return mapping 217 Rigid body displacements 29, 56, 140, 229 Rigid body modes 26, 56, 111

s Saddle point problem 132 Schur complement 54 Schwartz inequality 188 Second order cone programme 213, 222 Shear centre 244 Shear stress 67, 240 Side 46 Signature functions On faces 93 On sides 79 Simplexes 82 Simplicial neighbourhood 82 Simulation governance 6 Singular value decomposition 112, 262 Singularity 4, 11, 179 Skeleton elements 30, 84 Sleipner platform 8 Sliding support 47, 213 Small strains 225 Southwell moment potential see Stress potential Spurious kinematic modes 56, 73, 111, 134, 145, 149, 204, 233 Blocking technique 115 Geometrically induced 87, 98 Topologically inherent 87, 98 Star patch 82, 114, 136 Static (eigen) modes 206 Static boundary 45 Static condensation 203 Static Theorem 3 Statically admissible stress field 12, 19, 128, 156, 168, 216 Statically indeterminate stress field 51 Stiffness matrix 23, 138

Strain energy 13, 126, 200 Complementary, see Complementary strain energy Stress intensity factors 184 Stress potential 6, 33 Airy 57 Maxwell 63 Morera 63 Southwell 31, 64, 105 Superconvergent patch recovery 6 Surface tractions, see Boundary tractions

t Tangent form 230 Temperature 67 Tessellation 89, 105 Tetrahedral elements 90 Tetrahedral macro-element 108 Time domain 205 Timoshenko’s beam theory 240 Torsion, twisting 67, 240 Toupin’s principle 200 Trefftz formulations 32 Triangular macro-element 108 TUBA family of plate elements 164

u Unilateral contact 212 Upper bound theorem 220

v Valency 98 Velocity 200 Verification 6 Vertex 46 Vertex force 148 Vertex-centred patch 97 Virtual actions 184, 187, 196

w Warping displacements Weak compatibility 3 Weak equilibrium 2 Weighted residual 50

y Yield condition 222 Yield line 223

67, 240