Fluid-Solid Interaction Dynamics: Theory, Variational Principles, Numerical Methods and Applications [1 ed.] 9780128193525

Table of contents :
1. Introduction
2. Cartesian Tensor and Matrix Calculus
3. Fundamentals of Continuum Mechanics
4. Variational Principles of Linear FSI Systems
5. Solutions of Some Linear FSI Problems
6. Preliminaries of Waves
7. Finite Element Models for Linear FSI Problems
8. Mixed FE-BE Model for Linear Water-structure Interactions
9. Hydroelasticity Theory of Ship-water Interactions
10. Variational principles for nonlinear fluid-solid interactions
11. Mixed FE – CFD Method for Nonlinear Fluid-solid Interactions
12. Mixed Finite Element – Smoothed Particle Methods for Nonlinear Fluid-solid Interactions Appendices

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Fluid Solid Interaction Dynamics Theory, Variational Principles, Numerical Methods, and Applications

Fluid Solid Interaction Dynamics

Theory, Variational Principles, Numerical Methods, and Applications

Jing Tang Xing Fluid-Structure Interaction Group, School of Engineering Sciences, Faculty of Engineering & Physical Sciences, University of Southampton, Southampton, United Kingdom

Academic Press is an imprint of Elsevier 125 London Wall, London EC2Y 5AS, United Kingdom 525 B Street, Suite 1650, San Diego, CA 92101, United States 50 Hampshire Street, 5th Floor, Cambridge, MA 02139, United States The Boulevard, Langford Lane, Kidlington, Oxford OX5 1GB, United Kingdom Copyright © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved. No part of this publication may be reproduced or transmitted in any form or by any means, electronic or mechanical, including photocopying, recording, or any information storage and retrieval system, without permission in writing from the publisher. Details on how to seek permission, further information about the Publisher’s permissions policies and our arrangements with organizations such as the Copyright Clearance Center and the Copyright Licensing Agency, can be found at our website: www.elsevier.com/permissions. This book and the individual contributions contained in it are protected under copyright by the Publisher (other than as may be noted herein). Notices Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary. Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility. To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein. British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library Library of Congress Cataloging-in-Publication Data A catalog record for this book is available from the Library of Congress ISBN: 978-0-12-819352-5 For Information on all Academic Press publications visit our website at https://www.elsevier.com/books-and-journals

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Preface The story of this book goes back to the author’s studies as an MSc/PhD candidate in the department of engineering mechanics, Tsinghua University at Beijing China in 1978. At that time, investigations on fluid solid interaction (FSI) dynamics were motivated by widely intensive attention in world academic and engineering communities. While searching and reading references on FSI published in the world journals, he realized that the solution of practical FSI problems involving complex physical conditions must rely on some numerical simulations, so he chose numerical methods for structural and FSI dynamics as his research direction for his MSc and PhD theses. As the result of 6 years of study, the MSc thesis for the investigation of the mode synthesis approach based on variational principles and the PhD thesis on some of the theoretical and computational aspects of the finite element method (FEM) and substructure subdomain techniques for dynamic analysis of coupled FSI problems were completed and he passed his viva in 1980 and 1984, respectively. In these theses, the fundamental principles to construct the substructure methods were revealed, and some general theorems and variational principles for electrodynamics and FSIs were developed. Then these findings were used to create the corresponding displacement, equilibrium, and mixed types of finite element (FE) substructure models for numerical analysis of FSI problems. With his MSc/PhD degrees awarded, the author moved from the Institute of Structure and Strength, where he had worked as a research engineer in structure dynamic analysis and testing for about 10 years before his MSc study, and joined academia at Beijing University of Aeronautics and Astronautics (BUAA) in December of 1984. Later, the author was promoted to professor of theoretical and applied mechanics at BUAA. During that period, when dealing with a numerical simulation of an aircraft fuel tank wing interaction system, which aims to obtain its coupled natural vibration frequencies and modes for more accurate flutter analysis of aircraft, it was further realized that the occurrence of zero-energy modes in the displacement element model of fluids was a serious difficulty to overcome for FSI problems involving free surface motions in the lower frequency range. This motived the author to further tackle FSI problems with different free surface wave expressions. As a result, two variational principles were developed, which respectively adopt fluid pressure and velocity potential to replace fluid displacement as the variables describing fluid motions. These two principles have formed the basis for developing effective numerical models, especially the mixed pressure displacement type, for linear FSI analysis. When the author visited Brunel University, United Kingdom, in 1989 90, he met former Professor R.E.D. Bishop and Professor W.G. Price, known as the father of hydroelasticity. This visit started his joint research with Professor Price, which has been continued and further developed since his joining the Department of Ship Science at the University of Southampton in 1993. During more than 25 years of teaching and research in close collaborations with colleagues and students, the following developments on FSI analysis were made: (1) the variational principles for linear FSI problems were further modified to include more complex practical cases, such as the boundary conditions for surface tensions, gas liquid coupling interfaces, floating FSI interfaces, etc.; (2) variational principles for nonlinear FSI dynamics; (3) further development of the original mixed pressure displacement model for linear FSI problems to consider more complex boundary and coupling conditions and demonstrations dealing with many practical problems in maritime engineering, such as water very large floating structure (VLFS) interactions, liquefied natural gas (LNG) ship water couplings, and three-phase interaction for air water-shell systems, etc.; (4) development of the mixed FE boundary element (BE) model to deal with linear FSI problems involving infinite fluid or solid domains, so that the number of degrees of freedom for infinite domains can be largely reduced and the efficiency of computations increased; (5) creation of the mixed FE computational fluid dynamics (CFD) model, allowing nonlinear FSI problems to be simulated using available commercial solid FE and fluid CFD computer codes in association with a partitioned iteration approach; (6) proposal of the mixed FE smoothed particle method (SPM), which is convenient in simulating nonlinear FSI problems suffering from fluid solid separations and breaking waves, such as green water, and missiles moving into and out of water. These theoretical results and numerical methods, developed and practiced by the author and his colleagues, are presented in this book.

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Parts of chapters of the original draft of this book were used as lecture notes of linear and nonlinear numerical methods for FSI dynamics twice given at BUAA, for graduate students in the fields of mechanics and mathematics and three times at Zhejiang University for students in the field of ocean engineering. Considering the requirements of teaching and textbooks, when presenting each numerical method for a specified FSI problem, the author has always kept in mind that the main aim of a book is for readers, especially students and new researchers, to learn fundamental theoretical knowledge, essential techniques, and reliable methods, and then to apply their independent research ability to model and numerically solve practical complex FSI problems. Toward this end, for each numerical method described throughout the book, the author always tries to address the key steps to solve an engineering FSI problem (shown in Fig. 1.9). The first is to define the physical FSI problem of concern, from which the purpose, characteristics, and important and neglectable factors affecting the solution can be clarified to facilitate the formulation of a suitable mathematical model. For example, when dealing with a water structure interaction problem, the compressibility of water can be neglected for problems aiming to obtain the fluid loads on a structure caused by surface waves, but it has to be considered for a problem involving an underwater explosion wave problem. The second is to formulate the physical problem into a mathematical model consisting of a set of partial differential equations with boundary conditions defined in the continuum of the FSI system. This is completed by using assumptions to keep the essential factors and to neglect nonessential ones explored in the first step, based on the fundamental mechanical conservative laws governing the given FSI problem. The resultant formulation constructs a mathematical base to solve the problem. The third step is to convert the mathematical formulation derived in the second step into its numerical form for solving. To develop a practical and useful numerical scheme, the essential discretization approach, with its consistency, stability, and convergence performance, has to be considered. The last step is to illustrate and demonstrate the developed methods by the examples supported by available experiments or other reported results. To complete these basic tasks, this book provides the fundamental theories, methodologies, and results developed in different applications of FSI dynamics in a coherent format for readers to learn, understand, and use them to accomplish the four tasks to solve their physical FSI problems in engineering: (1) establish the fundamental principles, equations, and different types of boundary conditions in continuum dynamics, as well as their simplified forms, by introducing some assumptions, giving the necessary knowledge and approaches for readers to define their physical FSI problems and to formulate them in reasonable and effective mathematical models; (2) develop the variational principles for linear and nonlinear FSI systems, in which the conditions for free surface waves, surface tensions, floating FSI interfaces, and two-phase gas liquid coupling interfaces are included (these variational principles establish the basis for developing new numerical methods); (3) determine the related numerical discretization theories, approaches, their formulations and performance analysis, such as for the FE, BE, finite difference (FD), and SPM models described in each solution method for different types of FSI problems; (4) examine the simultaneous numerical solution approaches for linear FSI equations and the partitioned iteration approach for nonlinear problems, which enables the solid and the fluid equations for nonlinear FSI problems to be separately solved in time steps using commercial codes, and then arriving at convergence through coupling iteration until the time step of interest is reached. The book, to some extent, is of very practical use with its complete and comprehensive knowledge covering theories, numerical methods, and their solutions to deal with various FSI problems in engineering applications. No doubt that the number of pages in any book aiming to cover FSI dynamics will soon run out, as it is not possible—and actually there is no need—to give very detailed information on the related numerical methods that have been fully theoretically and practically demonstrated elsewhere. To amend this and to satisfy the further reading requirements for some readers, when describing related knowledge and numerical methods, the author always provides the references to world-influential books on the methods involved for more information if it is needed. Throughout the book, the author continually returns to some examples for each method and has tried to illustrate even the most abstract results in order to demonstrate the developed theory and approaches, as well as applications. The simpler examples can be solved by hand, and doing so can clearly enable readers, especially students, to better understand the related FSI mechanisms. The selected practical application examples include air liquid-shell three-phase interactions, LNG ship water sloshing, acoustic analysis of an air-building interaction system excited by human foot impacts, transient dynamic response of an airplane VLFS water interaction system excited by airplane landing impacts, turbulent flow body interactions, a structure dropping down on the water surface with breaking waves, etc. The numerical results are compared with available experiments or numerical data to demonstrate the accuracy of the various approaches and their value for engineering applications. Based on FSI analysis of integrated wave harvesting systems, it is revealed that the energy harvesting device acts as a damping mechanism in the system, which can keep the resonance of the linear system and the periodical oscillation, such as flutter, of the nonlinear system in their

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stable states. As a result, these two harmful phenomena—resonance and flutter—having to be avoided in traditional designs conversely are used to effectively harvest wave energies. The book consists of 12 chapters. The first chapter gives an introduction of FSI dynamics. After a general discussion on its position and characteristics in mechanics, some FSI problems in engineering are listed. Solution approaches for FSI problems include no coupling approximation, the quasicoupling approach, and integrated coupling methods. As a preliminary introduction, a simple example is given to show how to construct a numerical model in order to solve complex engineering problems. These include the Rayleigh Ritz method based on variational formulations, FE models, weighted residual approaches, and FD approximations, which are adopted to solve FSI problems in the book. Following this introduction, a short review of FSI dynamics is given, which covers the subdisciplines of the field, some historical events affecting the development and progress of research around the world, and important conferences, review papers, and books on FSI. This short review provides fundamental information and literature resources for readers who wish to engage in investigations of FSI problems in the future. Following this short review, the main aim of this book is given and explained in the last subsection of the chapter. In Chapter 2, Cartesian tensor and matrix calculus, some preliminary knowledge of Cartesian tensor analysis and matrix calculus, used throughout the book, is briefly discussed, and further reference books are listed. Readers who are familiar with tensor analysis and matrix calculus may not need to read this chapter. Chapter 3, Fundamentals of continuum mechanics, presents fundamental knowledge on continuum dynamics, which includes the different reference systems, deformation and stress analysis, conservative laws, constitutive equations, and various types of boundary conditions, as well as their corresponding simplifications by using some assumptions. This knowledge is necessary to formulate a FSI problem. Chapter 4, Variational principles of linear fluid solid interaction systems, presents the variational principles for linear FSI dynamic systems. Following a short introduction on the history of variational principles for linear FSI dynamics, it discusses the two new, further developed, variational principles by the author and his colleagues, in which the floating boundary conditions on the wet interface, the interaction conditions between two different fluids, and the surface tension conditions on the interfaces of liquids and gases are modeled. The first is a pressure acceleration form, an equilibrium or complementary energy form, in which the fluid pressure and the solid acceleration are taken as variational variables to describe the dynamics of the system. Considering the same conditions, the second example is its kinematic or potential energy form, in which the velocity potential of fluids and the displacement of solids are chosen as variables to describe the dynamic motions of the coupling system. After the detailed mathematical proofs on these two principal variational principles, some varieties are derived by replacing the involved variables or releasing some variational constraints by means of the Lagrangian multiplier approach. These modified variational formulations include the two mixed energy forms—a displacement pressure form and an acceleration velocity potential form—as well as two three-field forms: a displacement pressure velocity potential one and a displacement acceleration pressure one. To deal with FSI systems with damping, for which the integrated variational in real form cannot be derived, the virtual forms of two fundamental variational formulations are given, including the contributions from various types of damping in FSI systems. A comprehensive discussion on the complex form of the integral variational formulations based on adjoint variables for damped dynamic systems is also given in this chapter. As discussed in Chapter 1, Introduction, variational formulations provide a powerful approach to converting the governing equations formulating physical FSI problems in the continuum system into their corresponding numerical forms and then to construct effective numerical models or to find their approximate solutions for very complex engineering problems that are difficult to solve analytically. The variational principles presented in this chapter play an important role as the “bridges” used in Chapter 5, Solutions of some linear fluid solid interaction problems, to derive approximate solutions of some simple FSI problems and in Chapter 6, Preliminaries of waves, to develop FE models for linear FSI systems. Chapter 5, Solutions of some linear fluid solid interaction problems, deals with some simple FSI problems, whose theoretical solutions or the approximate solutions, based on the variational principles in association with the Rayleigh Ritz method, can be obtained. These selected FSI problems include one-dimensional (1D), 2D, and simplified 3D problems, such as axis-symmetrical problems and center-symmetrical problems. In Section 5.1.1, the dynamics of a 1D FSI system subjected to a pressure wave is discussed, in which the natural vibrations and the dynamic response are given. Sections 5.1.2 and 5.1.3 investigate 1D Sommerfeld systems to show that their natural vibrations rely on a complex eigenvalue problem and how to use the dry solid natural modes to seek the dynamic response of these systems. Section 5.1.4 discusses the effect of free surface waves based on Rayleigh Ritz approximations, while Section 5.1.5 studies the vibration of an FSI system with a floating boundary, in which the gravity potential needs to be considered. For 2D problems, Section 5.2.1 discusses the sloshing problem of a 2D incompressible water tank to further explain how the variable separation method works to find solutions. Section 5.2.2 investigates a 2D Sommerfeld system

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involving both free surface waves and compressible waves aiming to explain the new radiation parameter proposed by the author. Section 5.2.3 studies a dam-water system subjected to earthquake excitations, of which the solutions may be useful in dam design, while Section 5.2.4 investigates a beam water interaction system that is an often used as a model to tackle offshore structure water interactions. For 3D problems, an axis-symmetrical problem involving Bessel functions is discussed in Section 5.3.1, while a center-symmetrical problem on underwater noise reduction is solved in Section 5.3.2. It should be emphasized that, although the problems discussed in this chapter are simple, the solution approaches used in the examples are carefully selected to explain the methods and concepts mentioned in previous chapters, as well as to understand and reveal some FSI mechanisms more directly and obviously. For those researchers and students who are just starting to deal with FSI dynamics, these simple examples provide very handy practices. Chapter 6, Preliminaries of waves, presents some fundamental knowledge and concepts on waves. The concept of wave propagation is developed from the simplest model for 1D motion, from which d’Alembert’s solution, dispersive waves, and dissipation waves are introduced. Some nonlinear wave equations, such as Burgers, KdV, and Boussinesq equations with their solitary wave solutions, are discussed based on the Jacobian elliptic function expansion method. Other content concerns linear plane waves in order to provide knowledge on the method of variable separation, deep and shallow water waves, standing and traveling waves, as well as the wave field and its energy transmission. This knowledge can be used when dealing with wave structure interactions. Chapter 7, Finite element models for linear fluid structure interaction problems, presents the two FE models for linear FSI problems. The motions of solid structure are governed by the equations in linear elasticity theory, while the fluid motions satisfy a linear pressure wave equation with possible linear gravity wave conditions on the free surface, if it exists. Also, the surface tension can be considered if this is necessary. On the FSI boundaries, the conditions for the force equilibrium and the motion consistency, represented by the corresponding variables adopted in each model, are required. If a floating boundary is involved, the change of gravity potential caused by the flotation of the boundary is considered. The forms of conditions on the solid and fluid boundaries depend on the variables used in the two models, respectively. The first model is a displacement velocity potential model in which the displacement in the solid and the potential of velocity in the fluid are used as the variables to investigate the FSI. On the boundaries of the solid, the traction or the displacement condition can be defined, while the boundary conditions of the fluid are defined by the prescribed potential of velocity or its spatial derivatives (fluid velocity). The second model is a mixed one in which the displacement of the solid and the dynamic pressure of the fluid are chosen as the variables. Traction or acceleration is required on the boundary of solid, while on the fluid boundaries, the pressure or the acceleration is defined. To improve the efficiency of solving large complex FSI problems, the substructure subdomain methods are introduced. Since the presented numerical methods are based on FE models in both the solid and the fluid, any complex geometries of the problem can be modeled by choosing suitable elements, and the traditional FE procedures can be directly used in the calculations. For example, the subspace iteration program to solve the natural frequencies with the corresponding natural modes of FSI systems and the available time integral approaches can be adopted to solve their dynamic responses. The second displacement pressure model is more convenient to simulate many engineering problems involving defined pressure boundary conditions and obtaining dynamic pressures on structures, for example, the dynamic pressure on the tank wall caused by fluid sloshing, the sound pressure in structure acoustic volume interactions, the dynamic responses of FSI systems subject to various excitations: explosion pressure wave, dynamic impacts, earthquakes, human foot impacts, etc. Chapter 8, Mixed finite element boundary element model for linear water structure interactions, deals with linear structure water dynamic interactions based on a mixed FE BE model. The water is treated as an ideal incompressible fluid with its flow irrotational in the Euler coordinate system. Therefore its motions are governed by the theory of potential flows presented in Section 3.6.3. The structures are considered as elastic bodies satisfying the equations in the Lagrange coordinate system given in Chapter 3, Fundamentals of continuum mechanics. Following a description of the fundamentals of the BE method, an important water structure interaction problem is discussed: an integrated coupling system of a very large floating structure (VLFS) subject to aircraft landing impacts. The problem involves aircraft VLFS water interactions, for which a detailed mathematical description, the numerical solution method, as well as validation and examples are given. Chapter 9, Hydroelasticity theory of ship water interactions, discusses hydroelasticity theory. Section 9.1 presents the fundamentals of ship water interaction dynamics, including the definitions of the three reference coordinate systems, generalized nonlinear governing equations, the static equilibrium solution, and the steady motion solution of the system. The moving reference frame with the ship’s forward speed is used as a convenience in modeling the dynamics of moving ships. Section 9.2 summarizes some concepts on incident water waves. Following the knowledge described in Sections 9.1 and 9.2, Section 9.3 investigates the dynamic response of the linear integrated ship water interaction

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system excited by incident waves using the traditional approach in the hydroelasticity theory of ships. Since the mode summation approach is used in traditional hydroelasticity theory, which is not valid for nonlinear systems, only linear problems are discussed in this chapter, leaving nonlinear ones to be tackled in other related chapters for nonlinear FSI problems. Some differences between the mixed FE BE model for water structure interactions presented in Chapter 8, Mixed finite element boundary element model for linear water structure interactions, and the traditional hydroelasticity method are mentioned for readers to choose a suitable approach in solving their structure water interaction problems of interest. Chapter 10, Variational principles for nonlinear fluid solid interactions, presents the variational principles for nonlinear FSI dynamics developed by the author and his colleague. It is most important to find the main differences in deriving variations of a variational principle for nonlinear FSI compared with linear cases. For linear theory, we assume that the motions of the fluid and the solid are small, so that its original configuration is taken as our reference state, and there is no need to distinguish Lagrange and Euler coordinates, as well as the variations involved. For example, when we take the variation of a quantity defined in a fluid domain, we consider its boundary fixed at the original position and neglect the effect caused by boundary motion. Also, as used widely, we always freely exchange the order of time and space integrations in the linear variation process without any considerations. However, for nonlinear cases, these operations are no longer valid, since large motions cause boundary motions, which have to be considered in the variation process. After a short review of historical studies on the variations of nonlinear dynamical systems in Section 10.1, the fundamental concepts of the variational process valid for nonlinear FSI systems are detailed, as discussed in Section 10.2, for readers to learn these mathematical tools and to derive the variational principles for nonlinear systems. Based on this knowledge and these methods, the variational principles for nonlinear FSI systems are derived, and some selected application examples are presented in order to obtain their approximate solutions. Chapter 11, Mixed finite element computational fluid dynamics method for nonlinear fluid solid interactions, presents the mixed FE CFD method for nonlinear FSI systems, which has been developed and practiced by the author and his colleagues. In this mixed method, the well developed, both theoretically and numerically, FEMs in structural analysis and CFD for fluid flows are combined to deal with nonlinear FSI problems, the benefit of which is effectively using the available powerful commercial software for FEM and CFD to simulate complex nonlinear FSI problems in engineering. Generally, we assume that the structure undergoes large rigid motions as well as large elastic deformation, while the fluid flow is governed by nonlinear viscous or nonviscous field equations with nonlinear boundary conditions applied to the free surface and FSI interfaces. The updated Lagrangian description in FEM analysis of solids and the updated Arbitrary-Lagrangian Eulerian mesh description in CFD, discussed in Chapter 3, Fundamentals of continuum mechanics, are used to overcome the difficulty caused by large motions of coupling interfaces to solve the FSI equations in association with the partitioned iteration approach. The detailed numerical implication process and application examples are presented after the fundamental theory is described. Chapter 12, Mixed finite element smoothed particle methods for nonlinear fluid solid interactions, presents a mixed FE SPM, in which the solid structure is modeled by a powerful nonlinear FE model, while the fluid motions are modeled by the SPM, to deal with nonlinear FSI problems involving violent fluid flows, such as fluid solid separation and breaking waves. Since both FE and SPM are based on the Lagrange description of motion, this mixed approach easily traces fluid particle large motions occurring with breaking waves, compared with the FE CFD method given in Chapter 11, Mixed finite element computational fluid dynamics method for nonlinear fluid solid interactions. In order to for readers to fully understand the SPM involved in this mixed scheme, a briefing on the history, characteristics, developments with applications, and fundamental theory within the influential books are summarized in Sections 12.1 12.3 before discussing the proposed mixed FE SPM in Section 12.4. To demonstrate the proposed mixed method, selected examples include beam water interactions, the rigid and flexible wedges dropping on the water, flow-induced vibrations of a 2D cylinder, etc., whose results are compared with available experiments and other reports. Appendix A provides some numerical methods with its FORTRAN program to solve dynamic FE equations developed by author. The principles, solution processes, example, and comparison of five time integration methods often used in FEM are described. The computer code provides a generalized functional module for the five algorithms to be incorporated into any computer program to investigate complex dynamics problems in engineering. The book ends with an extensive list of more than 800 references. For the listed references, the author makes no claims for the completeness of the listed publications but has tried to include the bulk of the papers, monographs, and books that have proved useful to the author, his colleagues, and his students, but he recognizes that his bias probably makes this a rather eclectic selection, especially the papers by the author, his colleagues, and students in their collaborated researche on FSI problems.

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The book can be used as an undergraduate or graduate textbook or a comprehensive source for scientists, academics, researchers, and engineers, providing the state of the art on the theory, variational principles, numerical modeling, and applications of FSI dynamics. The author would like to thank his supervisors, former Professor Q.H. Du and former Professor Z.C. Zheng, as well as the teachers, especially Professor K.Z. Huang, in the department of engineering mechanics at Tsinghua University, who steered the author into engaging this research and who transferred his solid knowledge in mathematics and mechanics to the author, which has created the lifetime benefit of being able to deal with complex FSI problems. The author would like to deeply thank the Faculty of Engineering and Physical Sciences at the University of Southampton for awarding an Emeritus Professor position and for allowing the university’s office and facilities to be used in writing this book. The author’s colleagues in the Fluid Structure Interaction Research Group and the Institute of Sound and Vibration of the university, especially Professor W.G. Price, Fellow of Royal Society (FRS), deserve more thanks than anyone can give for their important help and support lasting more than 20 years of working together. The author extends his sincere thanks to his colleagues and friends who have helped him or jointly engaged in FSI research problems. In particular, the author thanks Professor R.A. Shenoi, who led a team including the author to manage the UK EPSRC and EU MASTRUC projects in which the involved FSI problems in marine engineering were further practiced and demonstrated; Professor G.E. Hearn, who helped to check the PhD thesis on aircraft VLFS water interactions completed by one of the author’s students; Dr. Y.P. Xiong, Dr. M. Tan, and Dr. K. Djidjeli, who have jointly completed FSI research projects, supervised graduate students, and published a number of FSI research papers with the author for many years. The author’s great thanks should be given to his former research assistant and PhD candidates for their dedication in FSI research, particularly Dr. Y. Chen for the mixed FE FD method, Dr. S. Zhao for beam water interactions, Dr. Z. Jin for aircraft VLFS water interactions, Dr. J. Yang for the nonlinear airfoil flutter system to harvest fluid energy, Dr. F. Sun and Dr. Z. Sun for the mixed FE SPM, and Dr. A. Javed for the flow-excited vibration with the mixed FD smoothed particle approach. The author also wishes to thank Dr. M. Toyoda at Ishikawajima-Harima Heavy Industries Co. Ltd. (IHI), Japan, for the joint research in the experiment on a water-spherical shell tank interaction vibration. Without the hard works of these RA and PhD students on the theoretical analysis and numerical simulations for the related FSI problems, this book could not have been presented in its current form. The author expresses his deep thanks to Professor Bohua Sun, member of the Academy of Science of South Africa, Cape Peninsula University of Technology, South Africa; Professor David Yang Gao, Alex Rubinov Professor of Mathematics, Federation University Australia; and Professor Pihua Wen, reader in Computational Mechanics, School of Engineering and Materials Science, Queen Mary University of London, for agreeing to let me nominate them as possible referees of this book, as required by the publisher when this book proposal was provided. Finally, the author would like to acknowledge the encouragement, advice, and gentle criticisms of editors, whose careful readings of the manuscripts enabled him to make corrections and improvements. At last, the author thanks his wife and children for their understanding, patience, and supports during the production of this addition to his family in the author’s retired life. Jing Tang Xing Southampton September 2018

Chapter 1

Introduction Chapter Outline 1.1 Fluid
solid interaction dynamics and its characteristics 1 1.2 Fluid
solid interaction problems in engineering 3 1.3 Solution approaches to fluid
solid interaction problems 6 1.3.1 Approximate solution with no fluid
solid interaction 6 1.3.2 Quasicoupling approximation method 6 1.3.3 Solution of integrated coupling fields 6 1.4 Approaches to deriving numerical equations 7 1.4.1 Problem and its governing equations 8 1.4.2 Analytical solution 8 1.4.3 Variational formulations and Rayleigh
Ritz method 13 1.4.4 Finite element method 16 1.4.5 Weighted residual methods 21

1.4.6 Finite difference method 1.5 Short historical review on fluid
solid interaction 1.5.1 Terms of fluid
solid interaction and its subdisciplines in literatures 1.5.2 Historical remarkable events and progress on fluid
solid interaction 1.5.3 World-recognized conferences 1.5.4 Influential review papers 1.5.5 Important books on fluid
solid interaction 1.6 Main aim and characteristics of this book 1.7 Suggestions how to choose some contents as lecture notes

23 24 24 26 27 33 38 40 41

This chapter presents an introduction to fluid
solid interaction (FSI) dynamics. After a general discussion of its position role, involved categories and characteristics in mechanics, some FSI problems in engineering are discussed, along with the related solution approaches, which include no coupling approximation, quasicoupling approach, and integrated coupling methods. As a preliminary introduction using a simple example, this chapter describes how to construct a numerical model in order to solve some complex engineering problems with no analytical solutions. These include the Rayleigh
Ritz method based on variational formulations, finite element (FE) models, weighted residual approaches, and finite difference (FD) approximations. These methods will be employed to solve the FSI problems investigated in this book. A short review on FSI dynamics is given, which includes the subdisciplines within FSI, some historical events affecting the development and progress of this research in the world, and the important conferences, review papers, and books on the subject. This short review provides the fundamental information and literature resources for those readers who wish to engage in investigations of FSI problems in the future. Following this short review, the main aim of the book is summarized in the last subsection.

1.1

Fluid
solid interaction dynamics and its characteristics

As we know, mechanics is the study of motion (or equilibrium) and the forces that cause such motion (or equilibrium). Mechanics is based on the concepts of time, space, force, energy, and matter. Knowledge of mechanics is needed in the study of all branches in physics, chemistry, biology, and engineering. In a given problem in mechanics, if inertial force is involved, the problem is considered a problem of dynamics; otherwise it is considered a static equilibrium problem. Fig. 1.1 shows the categories of mechanics according to the objects studied. Relative mechanics studies motion as it approaches the speed of light, where the measurement of absolute time and absolute space, used in Newtonian mechanics, is no longer valid, and a relative time-space measurement is adopted. In this book, we will not discuss problems involving the theory of relativity, only the ones in classical mechanics. As clearly indicated by its name, FSI dynamics is an interdisciplinary branch of mechanics that studies the dynamic behavior of a system consisting of both solids (structures) and fluids simultaneously. There are two key points when identifying whether a problem is an FSI problem. The first is that the system must include fluids and solids together,

Fluid
Solid Interaction Dynamics. DOI: https://doi.org/10.1016/B978-0-12-819352-5.00001-X © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved.

1

2

Fluid
Solid Interaction Dynamics

Relative mechanics Einstein's theory of relativity for microworld Classical mechanics Mechanics Mass point Newton's laws for macroworld

Rigid body Continuum

Solid Fluid

FIGURE 1.1 Mechanics and its categories according to the objects studied.

FIGURE 1.2 A solid dam
water interaction system.

and the second is that the fluid or the solid variables cannot be eliminated, so the problem has to be solved simultaneously. Fig. 1.2 shows a dam
water interaction system. A two-step approximate method is often used in engineering to solve this problem: (1) Assume that the solid dam is rigid in order to calculate the dynamic pressure added on the ^ (2) consider the elastic dam subjected to the dynamic pressure wet interface caused by the prescribed fluid pressure p; obtained in step 1 to calculate its deformation and stress distribution. In this approximate solution process, the first step is a pure dynamic problem in fluid mechanics, and the second step is a pure problem in solid mechanics. Therefore the equations governing the motions of fluid and solid are solved separately, but the simultaneous dynamic interaction between the dam and the water is neglected. The most important characteristic of FSI problems is “interaction”: how the fluid motions affect the force applied to the solid and at the same time how the solid motion affects the fluid flows. The governing equations for the fluid and for the solid are coupled to each other; they cannot be solved separately, only simultaneously. There are two large categories of FSI problems. In the first type, the fluid domain and the solid domain cannot be clearly defined, and there is no a clear interface between them, such as flows through a porous medium. To solve a problem of this type, the corresponding constitutive relationship specified for the problem is needed. In the second type of problem, there exists an obvious fluid
structure interaction interface where the FSI occurs. In this book, we focus our attention on the second type of problem. Fig. 1.3 shows the relationship of the forces in the second type of fluid
structure interaction problem. In the figure, the two large circles described by the dashed lines represent the fluid and the solid domains, respectively. The small circle at which the two dashed circles make contact denotes the FSI interface, through which the dynamic force of the fluid affects the motion of the solid and the solid motion affects the fluid flows. The motions of both the fluid and the solid on the interaction interface are unknown, and they can be known only after solving the total coupled equations governing the FSI problem. In general FSI cases, the motions of both fluid and solid are also affected by their inertia forces and the elastic forces, respectively, as denoted by the two rectangles at the centers of the two dashed circles. Depending on the aim of an investigated problem, you may focus your observation point on the fluid or the solid. Normally, researchers in fluid dynamics are mainly interested in the fluid flows, while researchers in solid mechanics are concerned with the deformation of the solid. In engineering, different approaches are adopted to derive the simplified coupling problems. For example, when studying water
structure interactions, the compressibility of the water could be neglected, so that we have the problem in terms of incompressible fluid
structure interactions. Similarly, considering the rigid motion, we may neglect the elastic deformation of the structure to investigate the

Introduction Chapter | 1

3

Compressible fluid dynamics

Fluid volume. deformation

Fluid inertia FSI probs Focusing on fluids

Incompressible fluid dynamics

Fluid inertia neglected

Fluid force Interface

FSI Solid motion

Solid inertia neglected

Rigid body problems FSI probs Focusing on solids Solid inertia

Solid elasticity

Flexible body dynamics FIGURE 1.3 The force relationships of FSI problems (Xing et al., 1997a).

fluid
rigid-body interaction. In some cases, we may neglect the inertia of either the fluid or of the solid to obtain the corresponding simplified problems. For a gas, neglecting its inertia (mass) implies considering it as a gas spring, which is often used in engineering. All of these types of simplified problems are denoted by the rectangles located on the two dashed circles in Fig. 1.3.

1.2

Fluid
solid interaction problems in engineering

Based on the amplitude of the relative motions between the fluid and the solid, as well as their characteristics when interacting, Zienkiewicz and Bettess (1978) categorize the three types of problems shown in Fig. 1.4: (A) FSI problems with large relative motions between the two phases; (B) short-period FSI problems with finite relative motions between the two phases; (C) long-period FSI problems with finite relative motions between the two phases.

4

Fluid
Solid Interaction Dynamics

(A)

(B)

(C)

FIGURE 1.4 Three types of fluid
structure interaction problems: (A) FSI problems with large relative motions between the two phases; (B) shortperiod FSI problems with finite relative motions between the two phases; (C) long-period FSI problems with finite relative motions between the two phases, (Zienkiewicz and Bettess, 1978).

θ Aerodynamics Aerodynamic Force A Elastic

Deformation

system

Inertial force I

θ

Mass inertia

FIGURE 1.5 The feedback loop in the flutter (Fung, 1955).

A typical type (A) problem is the flutter problem: air
structure interactions, for which the essential physical variables and process are explained in Fig. 1.5 by Fung (1955, 1969). In the figure, the airplane wing (structure) serves three main functions: producing aerodynamic force, inertia, and elastic deformation. The vibration of the wing produces the air lift A and the inertial force I, and the resultant force A 1 I causes the elastic deformation θ of the wing. This elastic deformation θ will produce a new resultant force A 1 I, creating a closed-loop feedback. If the amplitude of the wing vibration increases with time, a flutter results. The flutter, or aeroelasticity, problem is a very important fluid
structure interaction problem. In type (B) problems, the time period involved is short, and the amplitude of motion is finite. FSIs involving impacts and explosions have this characteristic, for which the compressibility of the fluid is more important. Type (C) problems involve a long-term FSI process with finite amplitude of motion. In engineering, there are huge problems with these characteristics: for example, offshore structures subjected to water waves or earthquake excitations, acoustic dynamic responses of an air volume
structure system, sloshing problems of liquid in tanks, and so on. For this type of problem, designers are concern with the dynamic responses of the system. Fig. 1.6 shows a few of the important fluid
structure interaction problems often encountered in engineering. Fig. 1.7 shows the imagined figure of a very large floating airport in Japan, which has been cited and discussed by many authors, such as Isobe (1999), Remmers et al. (1999), Watanabe et al. (2004), and Jin (2007). Fig. 1.8 gives a liquid
shell interaction test arrangement (Toyota et al., 2006; Xing et al., 2006). The review papers by Xing et al. (1997a) and Xing (2015, 2016) give a more detailed description of FSI problems in engineering, along with available references. Interested readers may refer to those papers for more information on this research topic.

Introduction Chapter | 1

5

flutters of airplanes, engine blades, suspension bridges, Problem A

wind electric line coupling vibrations, structure vibrations caused by outside flows, tube vibrations caused by inside flows, etc.

FSI Problem B

f-s vibrations by explorsions, ship slamming, piling in water, f-s vibrations by impacts

airplane landing on water/VLFS, fire system impacts/missile launch, etc.

dynamic responses of marine structures subjected

Problem C

by waves/earthquake excitations, acoustic volume–structure interaction vibrations, various noise vibrations: airplane, engine, cars, buildings, sloshing vibrations of various liquid tanks: chemical vessels, airplane/car fuel tanks, etc.

FIGURE 1.6 Some fluid
structure interaction problems in engineering.

FIGURE 1.7 A very large floating structure (VLFS): imagining a Mega floating airport used to validate experimental model in Japan (Isobe, 1999; Remmers et al., 1999; Watanabe et al., 2004; Jin, 2007).

FIGURE 1.8 A liquid
shell interaction system mounted on a vibration table for vibration tests (Toyota et al., 2006; Xing et al., 2006).

6

Fluid
Solid Interaction Dynamics

1.3

Solution approaches to fluid
solid interaction problems

The FSI problems discussed herein fall within the field of classical mechanics; therefore, the theory and methods developed in classical mechanics can be used directly to deal with them. FSIs involve fluid flows, solid motions, as well as their interactions. These problems are interdisciplinary in that the solution must use knowledge of both fluid mechanics and solid mechanics, or, more generally, knowledge of continuum mechanics. Due to the complexity of these problems, generally no analytical solutions can be derived, except in some very simplified cases. Scientists and engineers have to rely on numerical methods or experimental approaches to get to the solution of an FSI problem. For a full-scale product test, the costs are too high, so the numerical solution of the problem is more and more important, especially in the design stage when the product has not yet been produced. Therefore, in this book, our main focus is on numerical solution approaches for various FSI problems in engineering.

1.3.1

Approximate solution with no fluid
solid interaction

Historically, when numerical computation techniques were not well developed, designers could solve FSI problems only by using the following approach, which does not involve consideration of FSI. The computational process consists of two main steps: 1. Assume that the structure is rigid and fixed in space in order to solve a corresponding pure fluid problem and determine the forces on the wet structure interface produced by the fluid flows. 2. Solve the pure solid system subject to the forces applied on the wet interface obtained in step 1 in order to obtain the dynamic response of the structure. As discussed for the dam
water interaction system in Fig. 1.2, this approximate solution approach deals with an FSI problem as a pure fluid problem in fluid dynamics and as a pure solid problem in solid mechanics. This approximation is still used in the initial design stages of a product.

1.3.2

Quasicoupling approximation method

For some FSI problems, we may find an approximate relationship between the fluid dynamic force and the structure motion, so that the original FSI problem may be approximately solved by a semicoupling approximation approach using the following steps: 1. Investigate the fluid dynamic force produced by the solid structure with a set of given motions u to obtain the fluid dynamic force f(u) as a function of given solid motions. This force function may be an analytical function or discrete data obtained from the experiments. 2. Solve the solid equation subjected to the dynamic force obtained in step 1, that is, Lu 5 f ðuÞ;

(1.1)

where L denotes a differential operator in solid mechanics. In the history of airplane design, an aerofoil oscillating with a given harmonic motion in the airflow was investigated to derive the unsteady aerodynamic lift as a function of the given heavy pitch motions of the aerofoil. For marine structure
water interactions, the Morrison formulation is often used as the water force acting on the structure (Bishop and Price, 1979). Compared with the noncoupling approximation method described in Section 1.3.1, here the solid structure is not rigid and fixed in space but has a set of given motions. Therefore the first step solves a set of pure fluid problems, each of which one has a given solid motion, so that the function f ðuÞ can be obtained. Since for a real FSI problem, both the fluid and the solid motions are unknown and they cannot be solved separately, the force function f ðuÞ obtained in the condition of a given solid motion might not be valid for real coupling cases. For this reason, we call this method a semicoupling approach. If the obtained force function is also valid for the real coupling cases, even in some conditional range, this method will provide a real coupling solution.

1.3.3

Solution of integrated coupling fields

This method follows the principles in continuum mechanics to establish the governing equations describing both the solid and fluid motions as well as their boundary conditions, coupling conditions, and then numerically solve this set of coupling equations.

Introduction Chapter | 1

7

Affected by many complex factors Engineering problems

Based on assumptions and simplifications from experiments and observations Physical model

Experiment method

Based on principles of mechanics Analytical mathematical model Described by governing equations Partial differential equations Analytical method

River

Bridges to cross the rivers between analytical and numerical models

Numerical mathematical model

Solutions

Validations

Applications FIGURE 1.9 Flowchart to solve an engineering problem.

For a practical engineering problem, the flowchart shown in Fig. 1.9 should be followed to obtain its solution. Normally, an engineering problem is so complex that all of the involved practical factors cannot be considered in the analysis. Researchers need to analyze these practical factors, based on their experience and practical observations, in order to ignore some negligible factors and consider only the important factors in building a physical model. Based on this physical model, an experimental model can be constructed in order to obtain the experimental solution. Also, we can use the fundamental principles of physics to formulate the governing equations of the physical model and to derive the corresponding analytical model. In general cases, the governing equations consist of a set of partial differential equations from which the analytical solution can be obtained if this set of equations can be analytically solved. However, as previously mentioned, the analytical solution is normally impossible, and a numerical approach must be employed to solve the problem. A “river” flows between the analytical mathematical model and numerical model, so that a bridge is necessary to cross this river, that is, to transfer the analytical equations to numerical equations for the problem to be solved numerically.

1.4

Approaches to deriving numerical equations

In solid mechanics, finite element analysis (FEA) is a most powerful approach to simulate engineering problems, whereas in fluid dynamics the most popular approach is FD method. There are different approaches to derive the numerical equations from the continuous governing equations. Here, we discuss these approaches by a simple example.

8

Fluid
Solid Interaction Dynamics

fˆ O Fˆ (t)

x

L FIGURE 1.10 A uniform rod excited by the two dynamic forces.

1.4.1

Problem and its governing equations

Fig. 1.10 shows a uniform rod of length L, transverse section area B 5 1, and elastic modulus E, fixed at its left end O, ^ at its right end, and a uniform body force f^ per unit volume of the rod. We are required excited by a dynamic force FðtÞ to know the dynamic displacement u(x,t), the stress σ, and strain e of the rod excited by the forces. Using a fundamental knowledge of structural mechanics, we obtain the governing equations describing the rod motion: Dynamic equilibrium equation: @σ @2 u 1 f^B 5 ρB 2 ; @x @t

B

(1.2)

Stress
strain relationship: σ 5 Ee;

(1.3)

@u ; @x

(1.4)

Displacement
strain relationship: e5 Boundary conditions: uð0; tÞ 5 0; ^ BσðL; tÞ 5 FðtÞ:

Displacement:

(1.5)

Traction:

(1.6)

For this simple problem, we can derive an analytical solution. Substituting Eqs. (1.3) and (1.4) into Eqs. (1.2) and (1.6), we obtain the displacement solution equation EB

@2 u ^ @2 u 1 f B 5 ρB ; @x2 @t2

uð0; tÞ 5 0;

^ EB @u
5 FðtÞ; @x x5L

(1.7)

which is a second-order ordinary differential equation.

1.4.2

Analytical solution

1.4.2.1 Natural vibration If no external forces are applied to the system shown in Fig. 1.10, this system undergoes its natural vibration, which is ^ governed by the following equation derived from Eq. (1.7) by setting f^ 5 0 5 FðtÞ, that is, E

@2 u @2 u 5 ρ ; @x2 @t2

uð0; tÞ 5 0;

E @u
5 0: @x x5L

(1.8)

Eq. (1.8) can be solved analytically. Using a method of variable separation, we assume that uðx; tÞ 5 UðxÞTðtÞ;

(1.9)

Introduction Chapter | 1

9

from which, when substituted into Eq. (1.8), it follows that E

d2 U d2 T T 5 ρU ; dx2 dt2

Uð0ÞTðtÞ 5 0;

ET dU
5 0: dx x5L

(1.10)

Dividing both sides of the first equation in Eq. (1.10) by UT and considering the functions U and T involving only spatial variable x and time variable t, respectively, we obtain EUv T€ 5 5 2ω2 ; ρU T

(1.11)

where ðÞ0 5 dðÞ=dx, ð_Þ 5 dðÞ=dt, and ω is a real or complex constant to be determined. Based on Eq. (1.11), we have the following equations in the form of variable separation: T€ 1 ω2 T 5 0:

(1.12a)

ω2 U 5 0; E=ρ

Uv 1

Uð0Þ 5 0; U 0 ðLÞ 5 0:

(1.12b)

The solution of Eqs. (1.12a) and (1.12b) is in the form ω ω UðxÞ 5 a cos pffiffiffiffiffiffiffiffi x 1 b sin pffiffiffiffiffiffiffiffi x; E=ρ E=ρ

(1.13)

which, when the boundary conditions in Eqs. (1.12a) and (1.12b) are used, gives a 5 0;

ωL cos pffiffiffiffiffiffiffiffi 5 0: E=ρ

(1.14)

Finally, we obtain ð2n 2 1Þπ ωn 5 2L Un ðxÞ 5 sin

sffiffiffiffi E ; ρ

ð2n 2 1Þπ x; 2L

(1.15) n 5 1; 2; . . .:

The ωn is called as the nth natural frequency of the system, and the Un ðxÞ is the corresponding nth natural mode of the system. We can see that the natural frequencies and modes depend only on the distributions of the stiffness and mass of the system and that they are independent of any external forces, so that they are called the natural characteristics of the system. Eqs. (1.12a) and (1.12b) define an eigenvalue problem corresponding to the natural vibration of the system.

1.4.2.2 Orthogonal relationships of natural modes Assume that we have two different natural frequencies ωn ; ωm ; ðn 6¼ mÞ and the corresponding modes Un ðxÞ and Um ðxÞ, which must satisfy Eqs. (1.12a) and (1.12b), respectively, that is, Uvn 1

ω2n Un 5 0; E=ρ

Un ð0Þ 5 0; U 0n ðLÞ 5 0;

(1.16)

10

Fluid
Solid Interaction Dynamics

and Uvm 1

ω2m Um 5 0; E=ρ (1.17)

Um ð0Þ 5 0; U 0m ðLÞ 5 0:

Premultiplying the first equation in Eq. (1.16) by Um ðxÞ and taking an integration over the rod from 0 to L, we obtain  ðL  ðL ðL ω2n ω2 Um Uvn 1 Un dx 5 Um Uvn dx 1 Um n Un dx E=ρ E=ρ 0 0 0 ðL ðL   ω2 5 Um U 0n 0 2 U 0m U 0n dx 1 Um n Un dx E=ρ 0 0 (1.18)
L ð L ðL  0 0 
ω2n 0 0
Un dx 5 U m U n
2 U m U n dx 1 Um E=ρ 0 0 0 ðL ðL ω2 5 2 U 0m U 0n dx 1 Um n Un dx 5 0; E=ρ 0 0 in which, we have used an integration method by parts and the boundary conditions satisfied by the mode functions. In a similar process, starting from Eq. (1.17) and premultiplying both sides by mode Un ðxÞ and completing the related integration, we derive  ðL  ðL ðL ω2m ω2 0 0 Un Uvm 1 Um dx 5 2 U m U n dx 1 Um n Un dx 5 0: (1.19) E=ρ E=ρ 0 0 0 A subtraction between Eqs. (1.18) and (1.19) gives 

ω2m

Since ω2m 6¼ ω2n , we obtain

2 ω2n



ðL 0 ðL 0

ðL Um 0

1 Un dx 5 0: E=ρ

(1.20)

Um Un dx 5 0; (1.21) U 0m U 0n dx 5 0;

which is called the orthogonal relationships of natural modes. For a natural vibration in the form Un ðxÞ, its displacement, acceleration, strain, and stress can be derived by the following formulations: un 5 Un ðxÞTn ðtÞ;

u€n 5 Un ðxÞT€ n ðtÞ;

en 5 U 0n ðxÞTn ðtÞ;

σn 5 EU 0n ðxÞTn ðtÞ;

ρu€n 5 ρUn ðxÞT€ n ðtÞ;

(1.22)

so that the orthogonal relationships Eq. (1.21) can be rewritten in the following form: ðL ðL ðL Um Un dx 5 ½Um Tm ðtÞ½ρUn T€ n ðtÞdx 5 ½um ½ρu€n dx 5 0; 0

ðL 0

0

U 0m U 0n dx 5

ðL 0

0

½U 0m Tm ðtÞ½EU 0n ðxÞTn ðtÞdx 5

ðL

(1.23) ½em ½σn dx 5 0:

0

Physically, the first equation in Eq. (1.23) implies that the inertial force in the nth mode vibration does not do work in the displacement field with the mth mode form, while the second equation in Eq. (1.23) represents that the stress in the rod due to the nth mode vibration does not do work in the strain field produced by the mth mode vibration. As

Introduction Chapter | 1

11

we know, if a force is perpendicular to a displacement, that is, the force and the displacement are orthogonal, the force does not do work along the orthogonal displacement. The term “orthogonal” or “orthogonality” is used based on these physical explanations.

1.4.2.3 Generalized mass and stiffness For the nth mode vibration, we define the corresponding generalized mass and stiffness as ðL ðL Kn Mn 5 Un ρBUn dx; Kn 5 U 0n EBU 0n dx; ω2n 5 : M n 0 0

(1.24)

Since the mode function in Eq. (1.15) can be multiplied by an arbitrary constant, we may choose this constant by obtaining a unit generalized mass Mn 5 1, so that the generalized stiffness Kn 5 ω2n .

1.4.2.4 General solution of free vibrations If the rod does not suffer any external forces except an initial disturbance at time t 5 0 with the initial displacement and velocity conditions _ 0Þ 5 V0 ðxÞ; uðx;

uðx; 0Þ 5 U0 ðxÞ;

(1.25)

the rod will undergo a vibration called free vibration. The general form of free vibration is the solution in Eq. (1.8). Based on the theory of ordinary differential equation, this solution takes the form uðx; tÞ 5

N X

Un ðxÞðcn cos ωn t 1 dn sin ωn tÞ;

(1.26)

n51

where the constants cn and dn can be determined by using the initial conditions in Eq. (1.25), so that we have U0 ðxÞ 5

N X

Un ðxÞcn ;

V0 ðxÞ 5

n51

N X

Un ðxÞωn dn :

(1.27)

n51

Using the orthogonal relationship in Eqs. (1.21) and (1.24), we obtain ðL ðL N X cn M n ; Um ðxÞU0 ðxÞdx 5 cn Um ðxÞUn ðxÞdx 5 ρB 0 0 n51 ðL ðL N X dn ωn Mn Um ðxÞV0 ðxÞdx 5 dn ωn Um ðxÞUn ðxÞdx 5 ; ρB 0 0 n51

(1.28)

and the constants can be obtained as ρB cn 5 Mn dn 5

ðL 0

ρB ωn Mn

Un ðxÞU0 ðxÞdx; ðL

(1.29) Un ðxÞV0 ðxÞdx:

0

1.4.2.5 General solution of forced vibrations The solution of Eq. (1.7) gives the vibrations of the rod excited by the external forces, which is called the forced vibration. We often use a mode summation method to derive the solution of this equation. Assume that the solution of Eq. (1.7) is in a mode summation form uðx; tÞ 5

N X

Un ðxÞqn ðtÞ;

(1.30)

n51

where qn is called a generalized coordinate corresponding to nth mode. Physically, Eq. (1.30) implies that the motion of the rod consists of the component of each mode. The set of mode function Un ðxÞ spans a complete

12

Fluid
Solid Interaction Dynamics

orthogonal function space, and qn ðtÞ is the coordinate for a motion, a point in this space. Substituting Eq. (1.30) into Eq. (1.7), we have EB

N X

Uvn ðxÞqn ðtÞ 1 f^B 5 ρB

N X

n51

N X

Un ðxÞq€n ðtÞ;

n51

Un ð0Þqn ðtÞ 5 0;

(1.31)

n51

EB

N X

^ U 0n ðLÞqn ðtÞ 5 FðtÞ:

n51

Premultiplying the first equation in Eq. (1.31) by Um ðxÞ and then taking the integration over the rod, we obtain ðL N ðL X Um ðxÞEBUvn ðxÞdxqn ðtÞ 1 Um ðxÞf^Bdx n51 0

5

N ðL X

0

(1.32)

Um ðxÞρBUn ðxÞdxq€n ðtÞ:

n51 0

Integrating the first term on the right-hand side of Eq. (1.32) by parts and using the boundary conditions in Eq. (1.31), we derive ðL N X Um ðxÞ EBUvn ðxÞqn ðtÞdx 0

n51

5

ð L ("

Um ðxÞ

0

N X

# EBU 0n ðxÞqn ðtÞ 0

2 U 0m ðxÞ

n51

"

) N X 0 EBU n ðxÞqn ðtÞ dx n51

#
L ð N
N L X X
5 Um ðxÞ EBU 0n ðxÞqn ðtÞ
2 U 0m ðxÞ EBU 0n ðxÞqn ðtÞdx
0 n51 n51 0 ðL N X ^ 2 U 0m ðxÞ EBU 0n ðxÞqn ðtÞdx; 5 Um ðLÞFðtÞ 0

(1.33)

n51

which is substituted into Eq. (1.32), and by using the orthogonal Eqs. (1.21) and (1.24), it is given that Mn q€n 1 Kn qn 5 F^ n ðtÞ; ðL ^ F^ n ðtÞ 5 Un ðxÞf^Bdx 1 Un ðLÞFðtÞ;

n 5 1; 2; . . .; N;

(1.34)

0

where F^ n ðtÞ is a generalized force for the nth mode. Each equation in Eq. (1.34) is independent, and its solution gives the generalized coordinate qn ðtÞ. If the external forces are the sinusoidal force of frequency Ω, such as f^ 5 f e jΩt ;

F^ 5 Fe jΩt ;

(1.35)

the generalized coordinate should take the form qn ðtÞ 5 Qn e jΩt ;

(1.36)

since the system is linear, and its dynamic response has the same frequency. It is necessary to mention that we have used a complex to represent the motion, which implies that the real part of this complex denotes the corresponding physical variable. When you calculate the work done by a complex force along a complex displacement, you must do the multiplication using their corresponding real parts. In the force conditions given by Eqs. (1.35) and (1.36), the solution of Eq. (1.34) can be obtained as ðL 2 Ω2 Mn Qn 1 Kn Qn 5 F n ; F n 5 Un ðxÞf Bdx 1 Un ðLÞF; 0

F n =Mn F n =ðMn ω2n Þ Qn 5 2 5 ; 1 2 η2 ω n 2 Ω2

η5

Ω ; ωn

(1.37)

where η denotes a frequency ratio. If the nth natural frequency is near the excitation frequency, the dynamic response of the nth mode will be very large. This is called a resonance and normally should be avoided for a safe engineering design.

Introduction Chapter | 1

1.4.3

13

Variational formulations and Rayleigh
Ritz method

Generally, the governing equations describing an FSI system are a set of partial differential equations that cannot be analytically solved. Variational formulations provide an integral description based on which a weak solution and numerical equation can be constructed. Here, as an explanation of the fundamentals of this method, we investigate the simple problem shown in Fig. 1.10.

1.4.3.1 Admissible displacement field The displacement solution of the problem shown in Fig. 1.10 must satisfy all of governing equations in Eqs. (1.2)
(1.7). An admissible displacement does not need to satisfy all the governing equations, but it is required to satisfy a part of the governing equations: displacement
strain relationship in Eq. (1.4) and displacement boundary condition in Eq. (1.5). This implies that any admissible displacement must be continuous and differentiable. Since admissible displacements do not satisfy the dynamic equilibrium equation in Eq. (1.2) and traction boundary condition in Eq. (1.6), there are many admissible displacement fields that construct an admissible displacement space. The solution of the problem must be included in this admissible displacement space. Variational formulation provides a means to seek the solution of the problem from all of the admissible displacements in this space.

1.4.3.2 Variational formulation The functional of potential energy of this problem is given by # ) ð t2 (ð L "  2 EB @u ρB 2 ^ u_ 2 Bf^u dx 2 FuðLÞ Π ½ u 5 2 dt: 2 @x 2 t1 0

(1.38)

The principle of potential energy states that in all admissible displacement fields satisfying the time terminal variation condition δuðt1 Þ 5 0 5 δuðt2 Þ, the solution satisfying the governing equations in Eqs. (1.2)
(1.7) makes the functional in Eq. (1.38) stationary. We prove this statement as follows. Taking the variation of the functional in Eq. (1.38), we derive    

ð t 2 ð L

@u @u ^ ^ _ u_ 2 Bf δu dx 2 FδuðLÞ dt: δΠ½u 5 EBδ 2 ρBuδ (1.39) @x @x 0 t1 Integration by parts gives ð t2 ð L t1 0

_ udxdt _ ρBuδ 5

ð t2 ð L

 € dxdt _ _Þ 2 uδu ρB ðuδu

t1 0


t2 ð t2 ð L

_
dx 2 € 5 ρBðuδuÞ ρBuδudxdt 0 t1 0 t1 ð t2 ð L € 52 ρBuδudxdt; t1 0 2 3    ð t2 ð L @u @u 4EBδ 5dxdt @x @x t1 0 2 3   ð t2 ð L 2 @ @u @ u 4EB δu 2 EB 2 δu5dxdt 5 @x @x @x t1 0 2 3 
L ð t2 ð L ð t2  2
@u @ u 4EB δu

dt 2 5 EB δu5dxdt @x @x2 t1 t1 0 0 2 3
ð t2 ð L ð t2 2

4EB @ u δu5dxdt; 5 EB @u @x
δuðLÞdt 2 @x2 t1 t 0 L 1 ðL

(1.40)

14

Fluid
Solid Interaction Dynamics

where the condition δuð0Þ 5 0 due to admissible displacements satisfying the displacement boundary condition in Eq. (1.5) and the time terminal variation conditions δuðt1 Þ 5 0 5 δuðt2 Þ having been introduced. Substituting Eq. (1.40) into Eq. (1.39), we obtain
  

ð t2 ð L  @2 u @u

^ δΠ½u 5 ðρBu€ 2 EB 2 2 Bf δudx 1 EB
2 F^ δuðLÞ dt: (1.41) @x @x L t1 0 Since the variation δu in the domain xAð0; LÞ and the variation δuðLÞ are arbitrary, the δΠ½u 5 0 derives the following variational stationary conditions:
@2 u @u

^ ^ € EB
5 F; EB 2 1 Bf 5 ρBu; (1.42) @x @x L which are the dynamic equilibrium equation and the traction boundary condition in the displacement form given in Eq. (1.7). The statement has been demonstrated.

1.4.3.3 Rayleigh
Ritz method Based on the variational formulation in Eq. (1.38), an approximate solution may be derived. The idea of Rayleigh
Ritz method is to find the best approximate solution from a subspace of the admissible space. Assume an approximate solution in the form uðx; tÞ 5 Φq;  Φ 5 φ1 ðxÞ φ2 ðxÞ ? φN ðxÞ ;  T q 5 q1 ðtÞ q2 ðtÞ ? qN ðtÞ ;

(1.43)

where φI ðxÞ (I 5 1, 2,. . ., N), stands for Ritz functions, continuous functions defined in the rod domain and satisfying the displacement boundary condition φI ð0Þ 5 0. The generalized coordinates vector qðtÞ, satisfying the time terminal variational condition δqðt1 Þ 5 0 5 δqðt2 Þ, is to be determined from the variational stationary condition δΠ½u 5 0. Substituting the Ritz expression in Eq. (1.43) into the functional in Eq. (1.38), we obtain  ð t2  1 T 1 T T q Kq 2 q_ Mq_ 2 q F dt; Π½q 5 2 t1 2 ðL ðL 0T 0 (1.44) K 5 Φ EBΦ dx; M 5 ΦT ρBΦdx; 0

F5

ðL

0

^ T ðLÞ: Bf^ΦT dx 1 FΦ

0

The variation of the functional in Eq. (1.44) gives  ð t2  1 T 1 δΠ½q 5 δ q Kq 2 q_ T Mq_ 2 qT F dt 2 2 t1 ð t2 5 ðδqT Kq 2 δq_ T Mq_ 2 δqT FÞdt 5

t1 ð t2

(1.45)

δqT ðKq 1 Mq€ 2 FÞdt;

t1

where the integration by parts

ð t2 t1

ð t2

 T ðδq Mq_ _Þ 2 δqT Mq€ dt t1
t2 ð t2
_

2 ðδqT MqÞdt € 5 ðδqT MqÞ

_ 5 δq_ Mqdt T

t1

t1

(1.46)

Introduction Chapter | 1

15

and time terminal variational condition δqðt1 Þ 5 0 5 δqðt2 Þ have been used. Finally, from δΠ½q 5 0, it follows Mq€ 1 Kq 5 F;

(1.47)

which is a matrix equation derived from Rayleigh
Ritz method. This equation can be solved by using computers. Obviously, if the chosen Ritz functions satisfy the orthogonal conditions ðL ðL φ0I Tφ0J dx 5 1; φTI φJ dx 5 1; I 6¼ J; (1.48) 0

0

the resultant matrices M and K will be diagonal, and Eq. (1.47) will be an N independent equation corresponding to each Ritz function.

1.4.3.4 Example 1.1 Choose one Ritz function φ1 ðxÞ to estimate the first natural frequency of the rod shown in Fig. 1.10. Solution: Choosing a Ritz function φ1 ðxÞ 5 x2 ;

φ01 ðxÞ 5 2x;

φ1 ð0Þ 5 0;

(1.49)

which is a continuous function in the rod domain and satisfies the fixed boundary condition at the left end. Using Eq. (1.44), we obtain ðL ðL 4EBL3 K11 5 φ01 EBφ01 dx 5 4EB x2 dx 5 ; 3 0 0 ðL ðL (1.50) ρBL5 4 M11 5 φ1 ρBφ1 dx 5 ρB x dx 5 ; 5 0 0 from which the first approximate natural frequency rffiffiffiffiffiffiffiffi sffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi sffiffiffiffi 20=3 E K11 20E ð1Þ ω1 5 5 5 : M11 3ρL2 L ρ

(1.51)

We may choose another Ritz function φ1 ðxÞ 5 sin

πx ; 2L

φ01 ðxÞ 5

π πx cos ; 2L 2L

φ1 ð0Þ 5 0;

(1.52)

and obtain ð π2 EB L 2 πx π2 EB dx 5 ; K11 5 cos 4L2 0 2L 8L 0 ðL ðL πx ρBL M11 5 φ1 ρBφ1 dx 5 ρB sin2 dx 5 ; 2L 2 0 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi sffiffiffiffi π2 EB=8L π E ð2Þ ω1 5 5 : ρBL=2 2L ρ ðL

φ01 EBφ01 dx 5

(1.53)

Compared with the analytical solution given in Eq. (1.15), we find that the approximate solution in Eq. (1.51) is larger than the analytical solution but that the approximate solution in Eq. (1.53) is same as the analytical solution, that is, ωð1Þ 1 . ω1 ;

ωð2Þ 1 5 ω1 :

(1.54)

Generally, Ritz functions constrain the deformation of the rod in the style expressed by the function form, which implies that a geometric constraint is added to the system. As a result, the stiffness of the system is increased, so that the corresponding approximate frequency is larger than the analytical one. The second Ritz function is chosen because it is the same as the first natural mode given in Eq. (1.15); therefore the second approximate solution is the same as the analytical one. The accuracy of a Ritz solution depends on the chosen Ritz function.

16

Fluid
Solid Interaction Dynamics

1.4.4

Finite element method

The Rayleigh
Ritz method provides a very useful approach to deriving approximate solutions of a dynamic system based on the principle of potential energy. We know that Ritz functions are defined in the full domain of the system. For simple geometric structures, such as rods, beams, or plates, these Ritz functions can be evaluated and chosen. However, for complex structures, like various airplanes, cars, or ships, these types of Ritz functions cannot be defined in the full domains of the structures, therefore essentially restricting the applications of the Ritz method. Before a finite element method (FEM) is created, in the structural matrix analysis of structures consisting of rods, beams, or plates, designers consider each part of the total structure as a member, for example, a rod member, a beam member, etc., and investigate the equilibrium of each member to generate the total matrix equilibrium equation to be solved by computers. In the history of the FEM, the original idea of FEM was as the combination of the structural analysis concept and the Ritz method, which may be explained as follows. Structural members are the parts of a complex structure, and they are produced in the production process. These members can be separated and identified by their connection positions, and they are called natural elements. FE creators took the concept of structural members to divide a continuous medium into many small parts, called manmade elements, and to study the equilibrium of each element used in structural analysis in order to generate the global equilibrium equation. For each element, the Ritz idea is used to define displacement functions, called interpolation functions, in order to study the element equilibrium. Since each element is located in a finite, small area, the Ritz function can be easily chosen, thus overcoming the difficulty of defining a Ritz function in the full domain. Therefore FEM was created by adopting the ideas of both the Ritz method and the structural matrix analysis method, thereby creating the Ritz functions in the full structure domain by the ones in small element domains. For an explanation, we study our example shown in Fig. 1.10 using FEM to see how the FE equation is generated.

1.4.4.1 Mesh As shown in Fig. 1.11, the rod is divided into two elements of length l 5 L=2 with the corresponding three nodes. The global node displacement vector is defined by  T U 5 U1 U2 U3 : (1.55)

1.4.4.2 Element analysis Fig. 1.12A shows a local coordinate system of a representative element (e) with its element variables for which the subindex and superindex indicate the corresponding node number and the element number, respectively. Fig. 1.12B shows the variables for a representative node I to be studied, where the superindices ðeR Þ and ðeL Þ denote the element numbers connected to the right and left sides of node I, respectively. The process of FEA for this rod is described as follows. Based on the Ritz method, we represent the displacement and x coordinate of element (e) in the following isoparametric element form: uðeÞ ðξ; tÞ 5 hI ðξÞuIðeÞ 1 hJ ðξÞuðeÞ J ; ðeÞ xðeÞ ðξÞ 5 hI ðξÞxðeÞ I 1 hJ ðξÞxJ ; 12ξ 11ξ hI ðξÞ 5 ; hJ ðξÞ 5 ; 2 2

(1.56)

Element 2 Length

Element 1 Length

fˆ

l = L/2

l = L/2

O Fˆ (t)

L Node 1

Node 2

Node 3

U1

U2

U3

FIGURE 1.11 Finite element model of the rod in Fig. 1.10.

x

Introduction Chapter | 1

17

Element e

(A)

o

I

f I(e)

J

u I(e)

f J(e )

ξ

u J(e )

FˆI (B)

f I( eR )

f I(eL ) Node I

FIGURE 1.12 (A) Equilibrium of element (e); (B) equilibrium of node I with an external force F^ I .

which can be rewritten in a matrix form uðeÞ ðξ; tÞ 5 HðeÞ uðeÞ ; xðeÞ ðξÞ 5 HðeÞ xðeÞ ;   T HðeÞ 5 hI ðξÞ hJ ðξÞ ; uðeÞ 5 uðeÞ ; uðeÞ I J  T xðeÞ 5 xðeÞ ; xðeÞ I J

(1.57)

from which it follows: uðeÞ ðξ; tÞ 5 HðeÞ uðeÞ ;

u_ðeÞ ðξ; tÞ 5 HðeÞ u_ ðeÞ ;

dxðeÞ dHðeÞ ðeÞ 1  5 x 5 21 dξ dξ 2

xðeÞ ðξÞ 5 HðeÞ xðeÞ ;

xðeÞ 2 xðeÞ l I 1 xðeÞ 5 J 5 ; 2 2

@uðeÞ dHðeÞ ðeÞ 1  21 1 uðeÞ ; 5 u 5 @ξ dξ 2 @uðeÞ @uðeÞ dξ 2dHðeÞ ðeÞ 1  5 21 5 u 5 @x @ξ dx ldξ l

(1.58) 1 uðeÞ :

Here the functions hI ðξÞ and hJ ðξÞ play the roles of Ritz functions, called interpolation functions. Since the length of the element, generally the space domain of an element, could be very small, some simple functions, such as linear functions, can provide a good approximation of the displacement distribution. This is the most valuable point in the FEM. These interpolation functions satisfy the conditions of Ritz functions: continuous functions in the domain of element and the displacement conditions at the nodes of the element boundary, that is, ξ 5 2 1; hI ð2 1Þ 5 1; hJ ð2 1Þ 5 0; xðeÞ ð2 1Þ 5 xðeÞ uðeÞ ð2 1Þ 5 uðeÞ I ; I ; ξ 5 1; hI ð1Þ 5 0; hJ ð1Þ 5 1; xðeÞ ð1Þ 5 xJðeÞ ; uðeÞ ð1Þ 5 uðeÞ J : For element (e), the functional in Eq. (1.38) is written in the integration over this element, that is, )  ð t2 (ð xðeÞ ðeÞ ðeÞ J @u EB @u ρB ðeÞ ðeÞ ðeÞ ðeÞ 2 u_ðeÞ ΠðeÞ ½u 5 u_ðeÞ 2 Bf^uðeÞ dx 2 uJ fJ 1 uI fI dt; 2 @x 2 @x t1 xðeÞ I

(1.59)

(1.60)

18

Fluid
Solid Interaction Dynamics

which, when Eq. (1.58) is substituted into it, gives ΠðeÞ ½uðeÞ  8 9 " # # ð t2 : A3 ; δij Bj 5 Bi ;

9 i51 > = i 5 2 5 Ai ; > ; i53

(2.6)

δij δjk 5 δik ; so that Eq. (2.4) can be rewritten in the form ds2 5 δij dxi dxj :

(2.7)

Cartesian tensor and matrix calculus Chapter | 2

2.1.4

45

Permutation symbol

The permutation symbol is defined by

8 1; > > < eijk 5 2 1 > > : 0;

ijk 5 123; 231; 312; ijk 5 213; 132; 321;

(2.8)

ijk 5 others:

Based on this definition, the permutation symbol has the following characteristics: 1. It vanishes whenever the values of any two indexes coincide. 2. It equals 1 when the subscripts permute like 1, 2, 3, and it equals 21 otherwise. 3. It is antisymmetrical to any two indexes, such as e123 5 2 e213 . Using this notation, we can calculate the determinant of a matrix aij by the following equation:   aij  5 erst ar1 as2 at3 5 erst a1r a2s a3t : The cross-multiplication of two vectors ai and bj can be calculated by    g1 g2 g3       ða 3 bÞ 5  a1 a2 a3  5 erst gr as bt ;   b b b  1 2 3 ða 3 bÞr 5 erst as bt : The volume of a parallelepiped having any three vectors ai , bj , and cj , as edged, is given by    c1 c2 c3      cUða 3 bÞ 5  a1 a2 a3  5 erst cr as bt :   b b b  1

From this equation, it follows that

2

(2.9)

(2.10)

(2.11)

3

  1 0 0        g1 U g2 3 g3 5  0 1 0  5 1 5 e123 ;   0 0 1   gi U gj 3 gk 5 eijk :

(2.12)

As an application, we can write the moment Ki , about the origin of the coordinate system, of a force fk acting at a point xj in the following tensor form:

2.1.5

2.1.6

Ki 5 eijk xj fk :

(2.13)

eijk eirs 5 δjr δks 2 δjs δkr

(2.14)

e 2 δ Identity

Differentiation of a function f(x1, x2, x3) df ðx1 ; x2 ; x3 Þ 5

@f dxi 5 f;i dxi @xi

(2.15)

46

Fluid
Solid Interaction Dynamics

2.1.7

Transformation of coordinates

As shown in Fig. 2.2, the coordinate system O 2 x~1 x~2 is obtained from O 2 x1 x2 by rotating an angle θ in the counterclockwise direction about origin O. For a point p, the coordinates under these two systems satisfy the following relationships: " # " #" # x1 cosθ 2sinθ x~1 5 ; x2 sinθ cosθ x~2 (2.16) " # " #" # x~1 cosθ sinθ x1 5 : x~2 2sinθ cosθ x2 These two equations can be represented as the following index forms: x~i 5 β ij xj ;

xi 5 β ji x~j ;

(2.17)

where β ij 5

  @x~i 5 cos x~i ; xj ; @xj (2.18)

   T @xi β ji 5 5 cos x~j ; xi 5 β ij : @x~j From Eqs. (2.16) and (2.17), we can obtain that  T  21 β ji 5 β ij 5 β ij ;

(2.19)

which implies that β ij is an orthogonal matrix. For an orthogonal matrix, we have βUβ21 5 βUβT 5 I; β il β jl 5 β li β lj 5

2.1.8

@x~i @xl @x~i 5 5 δij ; @xl @x~j @x~j

(2.20)

@x~l @xj @xj 5 5 δij : @xi @x~l @xi

Tensor

A set of quantities is called a scalar, a vector, or a tensor depending on how the components of the set are defined in the variables x1 , x2 , and x3 and on how they are transformed when the variables x1 , x2 , and x3 are changed to x~1 , x~2 , and x~3 .

p

x2

~ x2

~ x1

θ

o FIGURE 2.2 Rotation of coordinates.

x1

Cartesian tensor and matrix calculus Chapter | 2

47

Scalar

φ φ~ 5 φ

A single component Transformation A tensor of rank 0 Vector

Three components Transformation A tensor of rank 1

ai a~ i 5 β ij aj

Stress

τ ij τ~ ij 5 β ir β js τ rs

Nine components Transformation A tensor of rank 2

The important property of tensor fields is this: if a tensor equation. can be established in one coordinate system, then it must hold for all coordinate systems.

2.1.9

Quotient rule

Consider a set of functions Aði; j; kÞ; ði; j; k 5 1; 2; . . .; nÞ; for which we do not know whether it is a tensor. Suppose we know the nature of the product of Aði; j; kÞ with an arbitrary tensor; then we can determine whether the function Aði; j; kÞ is a tensor without applying the law of transformation directly. For example, let γ i be a vector, a tensor of rank 1, and the product Aði; j; kÞγ i 5

n X

AðI; j; kÞγ I 5 Ajk

(2.21)

I51

yields a tensor Ajk . Then we can demonstrate that Aði; j; kÞ is a tensor of the type Aijk . Proof: Since Ajk is a tensor, it is transformed into x~ coordinates as   A~ði; j; kÞγ~ i 5 A~jk 5 β jr β ks Ars 5 β jr β ks Aðm; r; sÞγ m : However, the tensor γ m 5 β im γ~ i , when substituted into Eq. (2.22), gives   A~ði; j; kÞ 2 β im β jr β ks Aðm; r; sÞ γ~ i 5 0:

(2.22)

(2.23)

Now, γ i is an arbitrary vector, so that the quantity within the bracket must vanish, and we obtain ~ j; kÞ 5 β im β jr β ks Aðm; r; sÞ; Aði;

(2.24)

which is precisely the law of transformation of the tensor Aijk . This demonstration can be generalized to higher-order tensors.

2.1.10

Index forms of some important variables

Using the index notation and defining an operator r 5 gi @=@xi , we can express the variables often used in continuum mechanics in their index forms in Table 2.1. The vector identity Δv 5 grad div v 2 curl curl v

(2.25a)

vi;jj 5 vj;ji 2 eijk ekst vt;sj ;

(2.25b)

can be expressed in the index form which is easily demonstrated by using the e 2 δ identity.

48

Fluid
Solid Interaction Dynamics

TABLE 2.1 The vector and index notations of some variables in mechanics. Name

Vector notation

Index notation

Its rank

Vector

v

vi

1

Dot, scalar, or inner product

λ 5 uUv

λ 5 ui vi

0

Cross or vector product

w5u3v

wi 5 eijk uj vk

1

Gradient of scalar field

grad φ 5 rφ

@φ=@xi 5 φ;i

1

Vector gradient

grad v 5 rv

@vi =@xj 5 vi;j

2

Divergence

div v 5 rUv

@vi =@xi 5 vi;i

0

Curl

curl v 5 r 3 v

eijk @vk =@xj 5 eijk vk;j

1

Laplacian

r2 v 5 rUrv 5 Δv

@2 vj =ð@xi @xi Þ 5 xj;ii

1

2.1.11

Two primary identities

For the application in Chapter 3, Fundamentals of continuum mechanics, to derive the deformation of continuums, we demonstrate the following two identities: Identity I: Assuming that B is an arbitrary tensor of rank 2 and that a; b; c are three arbitrary vectors, we have the following identity: ðBUaÞ 3 ðBUbÞUðBUcÞ 5 J B a 3 bUc;

(2.26)

in which J B 5 jBj, the determinant of the tensor B. Proof: Using Eqs. (2.9)
(2.12), we can obtain that       ðBUaÞ 3 ðBUbÞUðBUcÞ 5 Bij aj gi 3 Brs bs gr U Blm cm gl   5 Bij Brs Blm aj bs cm gi 3 gr Ugl 5 elir Bij Brs Blm aj bs cm

(2.27)

5 emjs J B aj bs cm 5 J B a 3 bUc: Identity II: With the same assumptions for the identity I, we have the second identity ðBUaÞ 3 ðBUbÞ 5 J B B2T Uða 3 bÞ: Proof: Based on the dot multiplication rules, we can rewrite Eq. (2.26) in the form  ðBUaÞ 3 ðBUbÞUðBUcÞ 5 BT U½ðBUaÞ 3 ðBUbÞ Uc 5 J B a 3 bUc;

(2.28)

(2.29)

in which the transpose of tensor B is introduced since the dot multiplication BUc is done by the second index of B. Now, because vector c is arbitrary, premultiplying the inverse tensor B2T on both sides of Eq. (2.29) gives Eq. (2.28).

2.2

Matrix calculus

Numerical equations are normally expressed in matrix form, based on which derivative of a matrix with respect to another vector is needed during the mathematical derivation. Therefore, the matrix derivative is a convenient notation for keeping track of partial derivatives while doing calculations. Here, we present the information on matrix calculus adopted in this book.

Cartesian tensor and matrix calculus Chapter | 2

2.2.1

49

Types of matrix derivatives

We use a letter, such as x or y, to denote a scalar, a bold letter to indicate a vector, and a bold capital letter for a matrix. The six types of derivatives that can be most neatly organized in matrix form are presented in Table 2.2. We could also consider the derivatives of a vector with respect to a matrix or any of the unfilled cells in Table 2.3. However, these derivatives are most naturally organized in tensor form, as discussed in Section 2.1, so they do not fit neatly into a matrix. Two competing notational conventions split the field of matrix calculus into two separate groups, which can be distinguished by whether they write the derivative of a scalar of size 1 with respect to a vector of size n as a 1 3 n row vector or as an n 3 1 column vector. The former is called as the numerator layout convention, or the Jacobian formulation, while the latter is referred to as the denominator layout convention, or the Hessian formulation. Generally, the derivative of a vector y with respect to a vector x may be denoted in the following formulation: Numerator layout:

dy ; dxT

Denominator layout:

dyT : dx

(2.30a)

From this definition, we obtain that 0 dx dxT 5I5 ; T dx dx

1T T dy @ A 5 dy ; T dx dx

(2.30b)

dxT dx 6¼ I 6¼ : dxT dx

Serious mistakes can result when combining results from different authors without carefully verifying that compatible notation has been used. Therefore great care should be taken to ensure notational consistency. TABLE 2.2 Types of matrix derivatives. Types

Scalar

Vector

Matrix @Y=@x

Scalar

@y=@x

@y=@x

Vector

@y=@x

@y=@x

Matrix

@y=@X

TABLE 2.3 Identities of vector by vector @y=@x. Derivatives

Numerator layout @y=@xT

Denominator layout @yT =@x

@a=@x

0

0

@x=@x

@x=@xT 5 I

@xT =@x 5 I

@ðAxÞ=@x

A

AT

@ðbuÞ=@x

b@u=@xT 1 u@b=@xT

b@uT =@x 1 @b=@xuT

@ðAuÞ=@x

A@u=@xT

@uT =@xAT

@½gðuÞ=@x

ð@g=@uT Þð@u=@xT Þ

ð@uT =@xÞð@gT =@uÞ

du

ð@u=@xT Þd x

d xT ð@uT =@xÞ

50

Fluid
Solid Interaction Dynamics

2.2.2

Derivatives with vectors

2.2.2.1 Vector by scalar The derivative of a vector  y 5 y1

y2

?

ym

T

(2.31a)

by a scalar x is written in numerator layout notation as @y  5 @y1 =@x @x

@y2 =@x

? @ym =@x

T

;

(2.31b)

which is known as the tangent vector of the vector y in vector calculus. For example, if x denotes the time t, Eq. (2.31b) gives the tangent vector of the curves generated by the position vector y.

2.2.2.2 Scalar by vector The derivative of a scalar y by a vector  x 5 x1

x2

? xn

T

(2.32a)

is written in denominator layout notation as a column vector, @y  5 @y=@x1 @x

@y=@x2

?

@y=@xn

T

;

(2.32b)

 @y=@xn :

(2.32c)

and as a row vector in numerator layout notation  @y 5 @y=@x1 T @x

@y=@x2

?

In vector calculus, Eq. (2.32b) gives the gradient vector of a scalar field yðxÞ.

2.2.2.3 Vector by vector The derivative of a vector function y in Eq. (2.31a) with respect to a vector x in Eq. (2.32a) is written in numerator layout notation as 3 2 @y1 =@x1 @y1 =@x2 ? @y1 =@xn 7 6 6 @y2 =@x1 @y2 =@x2 ? @y2 =@xn 7 @y 7 6 (2.33) 56 7; 7 @xT 6 ^ ^ & ^ 5 4 @ym =@x1

@ym =@x2

?

@ym =@xn

which is called the Jacobian matrix when the vector x is the position vector in a space.

2.2.3

Derivatives with matrices

2.2.3.1 Matrix by scalar The derivative of a matrix Y by a scalar x is known as the tangent matrix and is given in numerator layout notation by 3 2 @Y11 =@x @Y12 =@x ? @Y1n =@x 7 6 7 @Y 6 6 @Y21 =@x @Y22 =@x ? @Y2n =@x 7 (2.34) 56 7: 7 @x 6 ^ ^ & ^ 5 4 @Yn1 =@x @Yn2 =@x ? @Ynn =@x

Cartesian tensor and matrix calculus Chapter | 2

51

2.2.3.2 Scalar by matrix The derivative of a scalar function yðXÞ with respect to an m 3 n matrix X is given in numerator layout notation by 2 3 @y=@x11 @y=@x21 ? @y=@xm1 6 7 6 @y=@x12 @y=@x22 ? @y=@xm2 7 @y 6 7; (2.35) 5 7 @XT 6 ^ ^ & ^ 4 5 @y=@x1n @y=@x2n ? @y=@xmn of which each row is a 1 3 m vector. Important examples of scalar functions of matrices are the trace and determinant of a matrix.

2.2.3.3 Matrix by matrix In numerator layout, the derivative of an α 3 β matrix function matrix whose entries are an α 3 β matrix, i.e., 2 @F=@x11 @F=@x21 6 6 @F=@x12 @F=@x22 @F 6 T 56 @X ^ ^ 4 @F=@x1m

2.2.4

@F=@x2m

FðXÞ with respect to an n 3 m matrix X is an m 3 n ? ? & ?

@F=@xn1

3

7 @F=@xn2 7 7: 7 ^ 5 @F=@xnm

(2.36)

Identities

2.2.4.1 Vector by vector We assume that the vector a and the matrix A denote a constant vector and a constant matrix, respectively, and that bðxÞ is a scalar function of vector x, as well as two vector functions uðxÞ; g½uðxÞ. According to these derivative definitions, it is not difficult to demonstrate that the following identities are valid.

2.2.4.2 Scalar by vector Many scalar functions of vectors or matrix are often used in numerical equations, for example, the inner products of two vectors, the quadratic forms, and the norm of a vector. For those scalar functions, the derivative identities with respect to scalar shown in Table 2.4 are valid. It is not difficult to demonstrate the identities given in Tables 2.3 and 2.4 by using the definitions shown in Eqs. (2.30a) and (2.30b). For example,    T @ xT Ax xT A@x @ xT Ax 5 1 @xT @xT @xT (2.37)     xT A@x xT AT @x T T T T 5 1 5x A1A I5x A1A : @xT @xT Here, the first term denotes the derivative of column vector x, while the second term is the derivative of row vector xT that needs to transpose to x in order to use Eq. (2.30b).

2.3

Exercise problems

In order for readers, especially students, who are not familiar with tensor operations to learn and use this mathematical tool as used in this book, we present the following exercise problems and their solutions.

52

Fluid
Solid Interaction Dynamics

TABLE 2.4 Identities of scalar a by vector. Expressions of scalar a

Numerator layout @a=@xT

Denominator layout @aT =@x

Inner product a 5 uUv 5 uT v

uT @v=@xT 1vT @u=@xT

@uT =@xv 1@vT =@xu

Inner product a 5 xUx 5 xT x

2xT

2x

Inner product a 5 aUx 5 aT x

aT

a

Quadratic form a 5 uUAv 5 uT Av

uT A@v=@xT 1 vT AT @u=@xT

@uT =@xAv 1 @vT =@xAT u

Quadratic form a 5 xUAx 5 xT Ax

xT ðA 1 AT Þ

ðA 1 AT Þx

Quadratic form a 5 aT xxT b

xT ðabT 1 baT Þ

ðabT 1 baT Þx

Norm a 5 :x 2 a: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 ðx2aÞT ðx 2 aÞ

ðx2aÞT :x 2 a:

ðx 2 aÞ :x 2 a:

2.3.1

Problem 1: prove the following formulations

1. δii 5 3 Solution: This problem uses the summation convention and the definition of the Kronecker delta, so that δii 5 δ11 1 δ22 1 δ33 5 1 1 1 1 1 5 3:

(2.38a)

2. δij δij 5 3 Solution: Use the rule of Kronecker delta given by Eq. (2.6), which shows that, for any variable Ai with a subindex the same as the one of δij , the result of Ai δij equals the replacement of the same subindex i of the variable by another subindex j of δij . We then obtain δij δij 5 δii 5 δjj 5 3:

(2.38b)

3. eijk ejki 5 6 Solution: Using the properties of the permutation symbol for the shift of the subindex and the e 2 δ identity given in Eq. (2.14), we obtain eijk ejki 5 eijk eijk 5 δjj δkk 2 δjk δkj 5 δjj δkk 2 δjj 5 3 3 3 2 3 5 6:

(2.38c)

4. eijk Aj Ak 5 0 Solution: This problem involves the cross-multiplication of a vector A with itself, so that the result vanishes, i.e., eijk Aj Ak 5 ðA 3 AÞi 5 0:

(2.39a)

Cartesian tensor and matrix calculus Chapter | 2

53

However, here we wish to use the tensor operation to give the demonstration as follows: eijk Aj Ak 5 2 eikj Aj Ak 5 2 eikj Ak Aj 5 2 eijk Aj Ak ; 2eijk Aj Ak 5 0;

eijk Aj Ak 5 0;

(2.39b)

where the first equal sign results in a shift of subindexes j and k of the permutation symbol eijk , the second results from the exchange rule of multiplication Aj Ak , and the third is obtained by exchanging the repeating indexes jj and kk according to the summation convention rule. 5. δij δjk 5 δik Solution: This can be immediately proved by Eq. (2.6). Furthermore, the summation convention rule can be used to give the details, i.e.,  0; i 6¼ k; δij δjk 5 δi1 δ1k 1 δi2 δ2k 1 δi3 δ3k 5 5 δik : (2.40) 1; i 5 k; 6. eijk δjk 5 0 Solution: This can be proved by two methods. One is just to use Eq. (2.6) and the definition of the permutation symbol, by which the components with two same indexes vanish, i.e., eijk δjk 5 eijj 5 eikk 5 0:

(2.41a)

Another way is the same as the one used in Eq. (2.39b) and considering δjk 5 δkj , i.e., eijk δjk 5 2 eikj δjk 5 2 eijk δkj 5 2 eijk δjk ;

2.3.2

eijk δjk 5 0:

(2.41b)

Problem 2: prove the identity of three arbitrary vectors A 3 ðB 3 CÞ 5 ðAUCÞB 2 ðAUBÞC

Solution: Using Eq. (2.10) for the cross-multiplication of two vectors and e 2 δ identity given in Eq. (2.14), we can obtain the component i of the left-side vector:    A 3 ðB 3 CÞ i 5 eijk Aj eklm Bl Cm 5 δil δjm 2 δim δjl Aj Bl Cm 5 δil δjm Aj Bl Cm 2 δim δjl Aj Bl Cm  5 ðAm Cm ÞBi 2 ðAl Bl ÞCi 5 ðAUCÞB2ðAUBÞC i :

2.3.3

(2.42)

Problem 3: express the constitutive equation in a tensor form exx 5

1 ½σxx 2 νðσyy 1 σzz Þ; E

exy 5

ð1 1 νÞ σxy ; E

eyy 5

1 ½σyy 2 νðσzz 1 σxx Þ; E

eyz 5

ð1 1 νÞ σyz ; E

ezz 5

1 ½σzz 2 νðσxx 1 σyy Þ; E

ezx 5

ð1 1 νÞ σzx : E

Solution: We define Θ 5 σxx 1 σyy 1 σzz and use subindexes 1, 2, and 3 to represent x, y, and z, respectively, so that we can express the constitutive equation in tensor form: eij 5

ð1 1 νÞ ν σij 2 Θδij : E E

(2.43)

54

Fluid
Solid Interaction Dynamics

2.3.4

Problem 4: write the tensor equation in a coordinate (xyz) form  G ui;kk 1

1 @2 u i uk;ki 1 fi 5 ρ 2 1 2 2ν @t

Solution: This equation in tensor form represents three equations in the coordinate system. For index i taking values 1, 2, and 3 representing x, y, and z, respectively, we obtain the following three equations using the summation convention: 0 1 2 1 @Θ A 1 fx 5 ρ @ ux ; G@Δux 1 1 2 2ν @x @t2 0

1 2 1 @Θ A 1 fy 5 ρ @ uy ; G@Δuy 1 1 2 2ν @y @t2 0 G@Δuz 1

1 @ΘA @ uz 1 fz 5 ρ 2 ; 1 2 2ν @z @t

Θ 5 rUu 5

2.3.5 1. 2. 3. 4.

(2.44)

1 2

@ux @uy @uz 1 1 : @x @y @z

Problem 5: prove the following identities using index notations

curl curl v 5 grad div v 2 Δv divðr n rÞ 5 ðn 1 3Þr n curlðr n rÞ 5 0 Δðr n Þ 5 nðn 1 1Þr n22 Here, r 2 5 xj xj and r 5 xj gj . Other notations in the preceding equations are defined in Table 2.1. Solution: Using the derivative @r 2 2r@r 2xj @xj 5 5 5 2xj δji 5 2xi ; @xi @xi @xi @r xi 5 ; @xi r

we can derive the following solutions: ðcurl curl vÞi 5 ðr 3 r 3 vÞi 5 eijk ðr 3 vÞk;j   5 eijk eklm vm;lj 5 δil δjm 2 δim δjl vm;lj 5 vm;mi 2 vi;jj 5 ðrUvÞ;i 2 Δvi

(2.45a)

5 ðgrad div vÞi 2 ðΔvÞi : divðr n rÞ 5 5

@ðr n xi Þ nr n21 xi @r r n @xi 5 1 @xi @xi @xi nr n21 xi @r r n @xi nr n21 xi xi r n @xi 1 5 1 @xi @xi r @xi

5 nr n 1 3r n 5 ðn 1 3Þr n :

(2.45b)

Cartesian tensor and matrix calculus Chapter | 2

ðcurlðr n rÞÞi 5 eijk ðr n xk Þ;j 5 eijk 5 eijk nr n21

55

@r n @xk xk 1 eijk r n @xj @xj

xj xk 1 eijk r n δjk 5 nnr n22 eijk xj xk 1 eijj r n r

(2.45c)

5 0 1 0 5 0:

0 1 2 n   @ r @ @r @nr n21 A 5 @ nr n22 xi Δðr n Þ 5 5 @xi @xi @xi @xi @xi 5 nðn 2 2Þr n24 xi xi 1 nr n22

@xi @xi

(2.45d)

5 nðn 2 2Þr n22 1 3nr n22 5 nðn 1 1Þr n22 :

2.3.6

Problem 6: prove eijkaiajbk 5 0 for nonzero vectors a and b

Solution: Using Eq. (2.9) for the value of determinant, we obtain    a1 a1 b1      eijk ai aj bk 5  a2 a2 b2  5 0;   a a b  3 3 2 since the two column vectors in it are the same.

(2.46)

Chapter 3

Fundamentals of continuum mechanics Chapter Outline 3.1 Descriptions of the motion of a continuum 3.1.1 Material frame of reference 3.1.2 Spatial frame of reference 3.1.3 Arbitrary Lagrange Euler frame of reference 3.1.4 Updated Lagrangian system 3.1.5 Updated arbitrary Lagrange Euler system 3.2 Analysis of deformation 3.2.1 Displacement and strain 3.2.2 Velocity field and rate of deformation of fluids 3.3 Stress tensor 3.3.1 Cauchy’s stress 3.3.2 Piola
Kirchhoff stress 3.4 Constitutive equation 3.4.1 Solids 3.4.2 Fluids 3.5 Laws of conservation 3.5.1 Green theorem 3.5.2 Material derivatives of volume integral with mass density

57 57 59 60 63 63 64 64 69 70 70 71 72 72 72 73 73

3.5.3 Material derivatives of arbitrary integrands in a spatial system 3.5.4 Material derivatives of arbitrary integrands in the arbitrary Lagrange Euler system 3.5.5 General forms of the conservation laws 3.5.6 Jump condition and equation 3.5.7 Conservation of mass and the equation of continuity 3.5.8 Conservation of momentum and equations of motion 3.5.9 Conservation of energy and equation of energy 3.6 Navier
Stokes equations and boundary conditions 3.6.1 Displacement solution of solid mechanics 3.6.2 Velocity
pressure solution equations of fluid mechanics 3.6.3 Bernoulli equation and potential flows 3.6.4 Linear waves in fluids

75 76 77 81 82 83 87 92 92 94 97 99

74

To investigate fluid
solid interaction (FSI) dynamics, it is necessary to understand the equations describing the motions of fluids, solids, and their interactions. As discussed in Chapter 1, Introduction, both fluids and solids are of a continuum in which motions are governed by the fundamental equations of continuum mechanics. This chapter provides the fundamental knowledge used to formulate physical FSI problems in order to establish a set of partial differential equations with suitable boundary and FSI conditions. This knowledge includes the three coordinate reference systems [Lagrangian, Eulerian, and arbitrary Lagrange Euler (ALE)], describing motions in fluids and solids; deformation and stress analysis in the continuum; the conversion laws for deriving the governing equations of motions, as well as the various boundary conditions required in FSI problems. In considering a particular practical case, different assumptions and corresponding simplified formulations are given for use in the definition of the problem. For more detailed knowledge on continuum mechanics, refer to the world-influential books, such as Fung (1977), Truesdell (1966, 1991), Huang (1989), and Fung and Tong (2001).

3.1 3.1.1

Descriptions of the motion of a continuum Material frame of reference

Let a fixed coordinate system O 2 x1 x2 x3 be chosen, as shown in Fig. 3.1. A material particle located at a point ðx1 5 X1 ; x2 5 X2 ; x3 5 X3 Þ in time t 5 t0 is represented by the label ðX1 ; X2 ; X3 Þ. This label may be considered the name of this particle, and therefore it does not change while it moves in space. As time moves forward, the particle moves, and its location can be determined by the history of its time in motion in the form xi 5 xi ðXj ; tÞ;

ði 5 1; 2; 3; j 5 1; 2; 3Þ:

Fluid
Solid Interaction Dynamics. DOI: https://doi.org/10.1016/B978-0-12-819352-5.00003-3 © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved.

(3.1)

57

58

Fluid
Solid Interaction Dynamics

FIGURE 3.1 Material particle labeled by its original position coordinates.

Eq. (3.1) defines a single-valued, continuous, differentiable, and one-to-one transformation from a domain Ω0 ðXj Þ to a domain Ωf ðxj Þ with t as a parameter. The Jacobian of this transformation must not vanish in the domain Ω0 ðXj Þ, and therefore its reverse transformation exists: Xi 5 Xi ðxj ; tÞ;

ði 5 1; 2; 3; j 5 1; 2; 3Þ:

(3.2)

If Eq. (3.1) is known for every particle in the continuum, we know the history of motion of the entire body. Therefore this equation is said to be a material or Lagrange description for the motion of continuum. The velocity Vi and acceleration Wi of the particle Xj , respectively, are derived by the partial derivatives with respect to time t as follows:
@xi

Vi ðXj ; tÞ 5 ; (3.3) @t
Xj

@Vi

@2 x i
(3.4) Wi ðXj ; tÞ 5
Xj 5 2

: @t
@t Xj From Eq. (3.1), it follows that dxi 5

@xi dXj 5 xi; j dXj : @Xj

(3.5)

From Eq. (3.5), using Eq. (2.11), we determine that the volume dΩf , constructed by the three line elements dxðIÞ i corresponding to the three orthogonal material line elements dXiðIÞ 5 δiI dXi , (I 5 1,2,3), is calculated by ð2Þ ð3Þ ð1Þ ð2Þ ð3Þ dΩf 5 elmn dxð1Þ l dxm dxn 5 elmn xl; r xm; s xn; t dXr dXs dXt

5 elmn xl; 1 xm; 2 xn; 3 dX1ð1Þ dX2ð2Þ dX3ð3Þ


@xi

dX1 dX2 dX3 5 JdΩ0 ; 5

@Xj


(3.6)

dΩ0 5 dX1 dX2 dX3 : Here, the Jacobian of the motion governed by Eq. (3.1) takes the form




@x1 =@X1 @x1 =@X2 @x1 =@X3


@xi

J 5



5

@x2 =@X1 @x2 =@X2 @x2 =@X3

: @Xj
@x3 =@X1 @x3 =@X2 @x3 =@X3


(3.7a)

Fundamentals of continuum mechanics Chapter | 3

59

Its material derivative, that is, the time derivative with the material coordinate fixed, can be derived using the tensor expression of the Jacobian, that is, J 5 elmn xl; 1 xm; 2 xn; 3 , as follows:



@J
@t


5 elmn

@xl;1 @xm;2 @xn; 3 xm; 2 xn; 3 1 elmn xl; 1 xn; 3 1 elmn xl; 1 xm; 2 @t @t @t

5 elmn

@vl @vm @vn xm; 2 xn; 3 1 elmn xl; 1 xn; 3 1 elmn xl; 1 xm; 2 @X1 @X2 @X3

X

  @vr @vr @vr 5 ermn 1 elrn 1 elmr xl; 1 xm; 2 xn; 3 @xl @xm @xn

(3.7b)

5 vr; r elmn xl; 1 xm; 2 xn; 3 5 Jvr; r : Here we have used the equality relation ermn @vr elrn @vr elmr @vr 1 1 5 vr; r elmn ; @xl @xm @xn

(3.7c)

which is obviously valid for l 5 1; m 5 2; n 5 3, and, since it is antisymmetrical to any two indices of ðl; m; nÞ, it must be valid. Similarly, from Eq. (3.2), we have dΩm 5 dX1 dX2 dX3 5 X1; r X2; s X3; t dxr dxs dxt


@Xi


dx1 dx2 dx3 5 J 21 dΩf : 5
@x


(3.8)

j

3.1.2

Spatial frame of reference

It is not convenient to use material description to describe the flow of water; we do not seek to identify the origin location of every particle of water. Instead, we are generally interested in the instantaneous velocity field and its evolution with time. This leads to the spatial or Euler description traditionally used in hydrodynamics. In a spatial description, the motion of the continuum is described by the velocity vector field vi ðxj ; tÞ, which is a function of the location xj and time t. The velocity vi and acceleration wi of the particle are, respectively, given by
Dxi @xi

5 @t
; vi ðxj ; tÞ 5 x_i 5 Dt X @vi @vi dxj 1 5 vi; t 1 vi; j vj : wi 5 v_i ðxj ; tÞ 5 @t @xj dt

(3.9)

Here, a convenient terminology, the material derivative, is introduced: DðÞ @ðÞ @ðÞ 5 1 vj ð_Þ 5 : Dt @t @xj

(3.10)

A field variable Fðxj ; tÞ can be transformed into FðXj ; tÞ using Eq. (3.2). The material derivative of the variable Fðxj ; tÞ means the rate of change of the property FðXj ; tÞ of the particle Xj , so that

@F

@F

@F _ _ F 5
; F 5
1 vj : (3.11) @t X @t x @xj Physically, in the material description of the motion of a continuum, an observer focuses on each material point marked by coordinate Xj . In the spatial description, the observer focuses not on each material point but on each special point xj fixed in the space.

60

Fluid
Solid Interaction Dynamics

3.1.3

Arbitrary Lagrange Euler frame of reference

In addition to the spatial and material coordinate systems, a referential coordinate system called the referential or mixed coordinate is defined and independently prescribed as a function of space and time by zi 5 zi ðxj ; tÞ;

ði 5 1; 2; 3; j 5 1; 2; 3Þ;

(3.12a)

for which there exists an inverse transformation xi 5 xi ðzj ; tÞ; provided that, by analogy with Eqs. (3.7a
3.7c), the defined Jacobean


@xi
J~ 5



. 0: @zj Combining the mappings expressed in Eqs. (3.1) and (3.12a
3.12c), we may define a third mapping by


@zi
^ zi 5 zi ½xk ðXj ; tÞ; t 5 zi 5 zi ðXj ; tÞ; J 5



. 0; @Xj

(3.12b)

(3.12c)

(3.13)

which transforms the material region Ωm in the material coordinate system into a reference region Ωr in the referential system. The motions of a continuum can be investigated in this referential system. The velocity of a material point Xj in the referential system can be obtained by the time derivative of Eq. (3.13), that is,
@zi
v^i 5

; (3.14) @t X in which the material point is unchanged. This velocity in Eq. (3.14) is a vector defined in the referential coordinate system O 2 z1 z2 z3 . The referential coordinate system defined by Eqs. (3.12a
3.12c) provides a moving system of which the moving velocity in the spatial system can be obtained by the time partial derivative of Eq. (3.12b), that is,
@xi

v~i 5 ; (3.15) @t
z in which the referential point is fixed. Similar to the mathematical process in deriving Eqs. (3.7a
3.7c), we can obtain the following relationships for the transformation between spatial volume elements dΩf , material elements dΩm , and referential elements dΩz as well as the time derivatives of the two Jacobeans in Eqs. (3.12a
3.12c) and (3.13), respectively, as
~ v~j J@ @J~

~ dΩf 5 JdΩz ; 5 5 J~v~j; j ; (3.16)
@t z @xj
^ v^j @J^

J@ ^ dΩz 5 JdΩm ; 5 : (3.17)
@t X @zj In numerical analyses, the referential coordinate points can be treated as mesh points, so that Eq. (3.15) gives the mesh velocity, which may vary at different points. Comparing Eqs. (3.1) and (3.12a
3.12c), we can see that the referential coordinates zi , the name of a referential or mesh point, is the same as the Lagrangian coordinates Xi , which is being considered as the name of a material point. From Eq. (3.15), we may conclude that G G

G

v~i 5 0, implying the mesh point is fixed in the space, so that the referential system becomes the Euler system; v~i 5 vi , implying the mesh velocity equals the velocity of the material point, and the referential system moves with the material points, so that the referential system becomes the Lagrangian system; 0 6¼ v~i 6¼ vi , implying that the moving velocity of the mesh is independent, corresponding to a general referential system.

Since the referential system includes both Lagrangian and Euler descriptions as its special cases, the referential system is also called an arbitrary Lagrangian
Euler reference system.

Fundamentals of continuum mechanics Chapter | 3

Taking the material derivative of Eq. (3.12b), we obtain

Dxi @xi

@xi @zj

@xi 5 vi 5 1 5 v~i 1 v^j 5 v~i 1 ci ; @t
Z @zj @t
X Dt @zj @xi ci 5 v^j ; @zj

61

(3.18)

which physically implies that the velocity of a material point equals the mesh velocity v~i plus the relative velocity ci of the material point to the mesh point. The relative velocity ci is also called a convective velocity. As mentioned, geometrically, Eq. (3.14) gives the velocity vector v^j , defined in the referential system, and the relative velocity ci , defined in the spatial system. The second equation in Eq. (3.18) gives a vector transformation from the velocity v^j in the referential system into ci in the spatial system, following the vector transformation rule discussed in Chapter 2, Cartesian tensor and matrix calculus. Based on Eq. (3.18), for an arbitrary physical variable Fðzj ; tÞ defined in the referential system, we can obtain its material derivative as





D @F @zj

@F

1 @F @xi v^ Fðzj ; tÞ 5 @F

5 @F

1 5 @t X @t z @zj @t
X @t
z @xi @zj j Dt

(3.19a)
@F @F

@F
5
1 ci 5
1 cUgrad F: @t z @t z @xi Combining Eqs. (3.11) and (3.18), as well as the preceding equation, we obtain that


@F

5 @F

1 ðv 2 c Þ @F 5 @F

1 v~ @F ; i i i @t
z @t
x @t
x @xi @xi




@F

@F @F
@F @F
@t
5 @t
2 ðvi 2 ci Þ @x 5 @t
2 v~i @x ; i i x z z

(3.19b)

which give the formulations to transform the partial time derivatives between the spatial and referential systems. Physically, these equations imply that the time change rate of a physical field variable at a spatial point xi equals its time change rate at the corresponding mesh coordinate zi minus the one caused by the mesh motion. Eqs. (3.19a
3.19j) reflect a very important relationship in ALE description. The governing equations in the spatial system can be transformed into the corresponding forms in the ALE system, so that the problem can be solved in that system. The important characteristic of ALE description is that the moving velocity of the mesh points can be given according to practical computational requirements in order to maintain appropriate geometric shapes of meshes, as well as accurately to describe moving interfaces between fluids and solids. Now we have three coordinate systems, as shown in Fig. 3.2. Here, xi is a Euler coordinate standing for a point fixed in three-dimensional (3D) space. At time t 5 0, a material point (black dot) is located at a space position

FIGURE 3.2 Three coordinate systems: xi is the Euler special coordinates; Xj , the material coordinates to identify the material point (black dot); and zj , the referential coordinates (ALE or mesh coordinates) to mark the mesh point (circle). ALE, Arbitrary Lagrange Euler.

62

Fluid
Solid Interaction Dynamics

xm j 5 Xj , called a material coordinate, which is used to identify this material point, while a mesh point (circle) is located at a space point xzj 5 zj , called a referential coordinate or ALE coordinate, which is adopted to identify this mesh point. As time moves forward, the body moves to its new position at time t 5 τ, and the material point Xj , with its velocity vi
5 @xi =@tjX , moves to its new space position xi 5 xi ðXj ; τÞ, while the mesh point zj , with the mesh velocity v~i 5 @xi =@t
z , arrives at its new position xi 5 xi ðzj ; τÞ. We have now defined the three reference systems: the Lagrangian, Eulerian, and ALE, in which the acceleration wi of a material point, the material derivative of velocity vi , is respectively expressed as



@v @v i
Lagrangian: wi 5 ; w 5

;
@t X @t X



@v @v i
Eulerian: wi 5 1 vj vi; j ; w 5

1 vUrv; (3.19c) @t
x @t x



ALE: wi 5 @vi

1 cj vi; j ; w 5 @v

1 cUrv: @t z @t z These will be used in dynamic analysis for acceleration calculations. For further convenience in transforming the governing equations in the special coordinate system ðxi ; tÞ into the corresponding ones in the ALE system ðzi 5 ξ i ; t 5 τÞ, we derive the following fundamental relationships. For any function Fðxi ; tÞ, there exist the relationships between the various reference domains, such that Fðxi ; tÞ 5 F½xi ðξ j ; τÞ; τ 5 Fðξ j ; τÞ:

(3.19d)

Eq. (3.19b) becomes @F @F @F 5 1 v~i ; @τ @t @xi

@F @F @F ~ 5 2 v: @t @τ @xi

(3.19e)

Let Fðxi ; tÞ 5 Fðξ i ; τÞ 5 ξi , then @ξi @ξi ~ 5 v: @t @t

(3.19f)

The divergence of a vector field vi under the mesh coordinate system ξ i is represented by Thompson et al. (1999) in the form div v 5  from which, when letting v 5 Fðxi ; tÞ;

0;

~ i @ξ j =@xi Þ @ðJv ; ~ j J@ξ

(3.19g)

 0 , it follows that

j ~ @F @ðJF@ξ =@x1 Þ 5 ; ~ j @x1 J@ξ

(3.19h)

j ~ @F @ðJF@ξ =@xi Þ 5 ; ~ @xi J@ξ j

(3.19i)

~ j =@xi Þ @ðJ@ξ 5 0: ~ j J@ξ

(3.19j)

providing the more general relationship

which, by letting F 5 1, gives a special result

These equations provide the relationships that can transform the partial derivatives with respect to time t and space coordinates xi in the physical domain into the corresponding partial derivatives with respect to time τ 5 t and mesh coordinates ξ j , so that we can transform the governing equations established in the physical coordinate system into their counterparts in the mesh coordinate system.

Fundamentals of continuum mechanics Chapter | 3

63

FIGURE 3.3 Total and updated Lagrange coordinates. The black dots represent the same material point.

FIGURE 3.4 Total and updated ALE coordinates. The circles stand for the mesh points. ALE, Arbitrary Lagrange Euler.

3.1.4

Updated Lagrangian system

The total Lagrangian system adopts the original special position coordinates to identify the material points. As shown in Fig. 3.3, at time t 5 0 a representative material point P locating at a space point which is used as the Lagrange coordinates to identify this material point P to study the motion. In the total time period of the analysis, the Lagrange coordinates of point P remains unchanged. The total Lagrange description sometimes is not convenient for nonlinear analysis, so that an updated Lagrangian system is created. In the updated Lagrange description, the current position coordinates of a material point P at time as its material coordinate to identify the material point P to investigate it motion in the time period from to. In this short time period, the material coordinates will not change. When the numerical work in this time period from to is completed, at time the material point P moves to a new spatial position which is now chosen as a updated Lagrangian coordinates to identify same material point P in order to study the motion from time to. This updated process continues until the completion of computations.

3.1.5

Updated arbitrary Lagrange Euler system

The idea of the updated ALE coordinate system is similar to the updated Lagrange coordinate system. As shown in Fig. 3.4, at time t 5 0, a representative mesh point z, with its mesh velocity v~0i located at a space point x0i , is used along with the referential coordinates z0i 5 x0i to identify z in order to study its motion. In the total time period of the analysis, the referential coordinates of mesh point z for the total ALE description remain unchanged. In the updated ALE description, the current position coordinate xτi of mesh point z at time τ is used as its referential coordinate zτi 5 xτi to identify z in order to investigate the motion in the time period from τ to τ 1 Δτ. In this short time period, the mesh coordinates zτi 5 xτi will not change. When the numerical work in this time period from τ to τ 1 Δτ is completed, at time τ 1 Δτ the mesh point z moves to a new spatial position xτ1Δτ , which is now chosen as an i τ1Δτ τ1Δτ 5 xi to identify the point in order to study the motion from time τ 1 Δτ to updated mesh coordinate zi τ 1 2Δτ. This updated process continues until the computation is complete. As a result, the partial time derivative

64

Fluid
Solid Interaction Dynamics

FIGURE 3.5 Analysis of the deformation of a continuum body.


@ðÞ=@t
z , with the referential mesh coordinate zi fixed, is now replaced by @ðÞ=@tjx , which is more convenient in numerical analysis.

3.2 3.2.1

Analysis of deformation Displacement and strain

The material description of the motion of a continuum is typically used in solid mechanics, in which the displacement and strain in the body are considered as functions of the material point and time. As shown in Fig. 3.5, the displacement of the material point can be expressed in one of the following forms: Ui 5 xi ðXj ; tÞ 2 Xi ;

ui 5 xi 2 Xi ðxj ; tÞ:

(3.20)

The velocities and accelerations in Eqs. (3.3) and (3.4), as well as Eq. (3.9), are respectively obtained in the forms


@Ui

@Vi

@2 Ui

Vi ðXj ; tÞ 5 ; Wi ðXj ; tÞ 5 5 2
; (3.21) @t
Xj @t
Xj @t Xj vi ðxj ; tÞ 5

Dui ; Dt

wi ðxj ; tÞ 5

Dvi D 2 ui 5 : Dt Dt2

(3.22)

3.2.1.1 Strain tensors Assume that an infinitesimal line connecting the point Xj and a neighboring point Xj 1 dXj at a time t 5 0 moves to the line connecting the points xj and xj 1 dxj at time t. Using Eq. (3.5), the squares of this line element in the original and the current configurations are given, respectively, by 21 ds20 5 dXj dXj 5 Xm; i Xm; j dxi dxj 5 D21 ri Drj dxi dxj 5 cij dxi dxj ;

ds2 5 dxi dxi 5 xr; i xr; j dXi dXj 5 Dri Drj dXi dXj 5 Cij dXi dXj ; Dij 5

@xi @xr @xr ; Cij 5 Dri Drj 5 DTir Drj 5 ; Dir D21 rj 5 δij ; @Xj @Xi @Xj

D21 ij 5

@Xi @Xr @Xr 21 2T 21 ; cij 5 D21 : ri Drj 5 Dir Drj 5 @xj @xi @xj

(3.23)

Fundamentals of continuum mechanics Chapter | 3

65

In continuum mechanics, Dij 5 xi; j 5 @xi =@Xj is called the deformation gradient, and its reverse is D21 ij 5 Xi; j 5 @Xi =@xj . To explain the geometric structure of these tensors, we can write the deformation gradient in vector form D 5 DiJ gi GJ . The two base vectors GJ and gi are used to identify the differences for the Lagrange system and the spatial system, respectively. Since the first index i and the second index J of this tensor are related to the spatial and Lagrange systems, respectively, this tensor is called a two-point tensor. For convenience in the mathematical formulation, we will not use the capital index to distinguish the preceding different indices. However, it is necessary to remember that the second index of the deformation gradient tensor is related to the original configuration. The tensor C 5 DT UD is called the Green deformation tensor, which is a dot multiplication of the transpose of the deformation gradient and itself. Because the two indices relating to the spatial system vanish in the summation convention, the two indices of the resultant tensor belong to the Lagrange system, for which this tensor is defined. Similarly, the tensor c 5 D2T UD21 is for the spatial system and is called Cauchy deformation tensor. The difference between the squares of the length elements may be written as ds2 2 ds20 5 ðδmn xm; i xn; j 2 δij Þ dXi dXj 5 ðCij 2 δij Þ dXi dXj 5 2Eij dXi dXj ;

(3.24)

or ds2 2 ds20 5 ðδij 2 δmn Xm; i Xn; j Þ dxi dxj 5 ðδij 2 cij Þ dxi dxj 5 2eij dxi dxj ;

(3.25)

where δmn xm; i xn; j 2 δij Cij 2 δij 5 ; 2 2 δij 2 δmn Xm; i Xn; j δij 2 cij eij 5 5 : 2 2 Eij 5

(3.26)

Here, Eij was introduced by Green and St. Venant and is called Green’s strain tensor. The strain tensor eij was introduced by Cauchy for infinitesimal strains and by Almansi and Hamel for finite stains and is known as Almansi’s strain. The tensors Eij and eij are defined in the original nondeformed and deformed configurations, respectively. Therefore Eij is often referred to as Lagrangian and eij as Eulerian. Both strain tensors are symmetric. For a rigid motion of the body, Eqs. (3.24) and (3.25) vanish, and therefore all components of the Green and Almansi strain tensors are zero throughout the body. Using Eq. (3.20), we obtain that Ui; j 5 ui; j 5

@xi @Xi 2 5 xi; j 2 δij ; @Xj @Xj

@xi @Xi 2 5 δij 2 Xi; j ; @xj @xj

Dij 5 xi; j 5 Ui; j 1 δij ;

(3.27)

D21 ij 5 Xi; j 5 ui; j 1 δ ij :

(3.28)

The Ui;j is called a displacement gradient, which can be decomposed into Ui; j 5

1 1 ðUi; j 1 Uj; i Þ 1 ðUi; j 2 Uj; i Þ 5 Sij 2 Ωij ; 2 2

1 Sij 5 ðUi; j 1 Uj; i Þ; 2 1 Ωij 5 ðUj; i 2 Ui; j Þ 5 eijk Ωk ; 2

(3.29) Ωk 5

1 1 ekij Ωij 5 ekij Uj; i : 2 2

Here, Sij is symmetric and called the deformation tensor, and Ωij is antisymmetric called the rotation tensor. The Ωij has only three independent elements, and there exists its dual vector Ωi . The Green and Almansi strain tensors can be represented in the displacement form

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1 ½δmn ðUm; i 1 δmi Þ ðUn; j 1 δnj Þ 2 δij  2 1 5 ðUj; i 1 Ui; j 1 Um; i Um; j Þ; 2

(3.30)

1 ½δij 2 δmn ðδmi 2 um; i Þ ðδnj 2 un; j Þ 2 1 5 ðuj; i 1 ui; j 2 um; i um; j Þ: 2

(3.31a)

Eij 5

eij 5

In cases of small deformations of the body, the nonlinear terms in Eqs. (3.30), (3.31a), and (3.31b) can be neglected such that Eij 5

1 ðUj; i 1 Ui; j Þ; 2

eij 5

1 ðuj; i 1 ui; j Þ: 2

(3.31b)

3.2.1.2 Line element transformation and its stretch ratio Line element transformation The transformation equation between a line element dx in the deformation configuration and its nondeformed element dX in the original configuration is given by Eq. (3.5), which can be rewritten in the following forms using the deformation gradient defined in Eq. (3.23), that is, @xi dXj 5 Dij dXj ; @Xj @Xi dxj 5 D21 dXi 5 Xi; j dxj 5 ij dxj : @xj dxi 5 xi; j dXj 5

(3.32)

Stretch ratio Using Eq. (3.32), we can calculate the stretch ratio γ of a line element of length ds0 and the angle θ between two line elements ds and d^s as follows: ds γ5 5 ds0

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xn; i xn; j dXi dXj dxi dxi 5 ; ds0 ds0

dXi ð2Eij 1 δij Þ dXj 5 Ni ð2Eij 1 δij Þ Nj ; ds0 ds0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dXi DTin Dnj dXj γ2 5 5 Ni Cij Nj ; γ 5 Ni Cij Nj ; ds0 ds0 γ2 5

cos θ 5

dsUd^s dxi dx^i dXi xm; i x^m; j d X^ j Ni xm; i x^m; j N^ j 5 ; 5 5 ds ds^ ds0 γ γ^ ds^0 ds0 γ γ^ ds^0 γ γ^

(3.33a)

(3.33b)

where Ni 5 dXi =ds0 denotes a unit vector of the line element at time 0. Considering the three unit vectors NiJ (J 5 1,2,3) along the three coordinate directions, we have the three stretch ratios along three coordinate directions γJ 5

pffiffiffiffiffiffiffi CJJ ;

in which the index J does not follow the summation convention.

(3.33c)

Fundamentals of continuum mechanics Chapter | 3

67

^ constructed by two line elements before deformation to da 5 dx 3 dx^ . FIGURE 3.6 An area element dA 5 dX 3 d X,

If we define an engineering stretch ratio ΔN of a line element N as the ratio of the length increment over its original length, we have ΔN 5

jdxj 2 jdXj 5 γ N 2 1: jdXj

(3.33d)

3.2.1.3 Area element transformation and area ratio ^ constructed by two line elements before deformation is As shown in Fig. 3.6, an area element dA 5 dX 3 dX deformed to da 5 dx 3 dx^ due to the deformation of the body. Using the identity given by Eq. (2.26) in Chapter 2, Cartesian tensor and matrix calculus, we can derive the formulations governing the area element transformation and area ratio as follows. Area element transformation Using Eq. (3.32), we obtain the two line elements dx and dx^ , as well as the corresponding area ^ dx 5 DUdX; dx^ 5 DUdX; ^ da 5 dx 3 dx^ 5 ðDUdXÞ 3 ðDUdXÞ;

(3.34a)

which, when the identity in Eq. (2.28) is used, gives the area element transformation formulation ^ 5 JD2T UdA; da 5 JD2T UðdX 3 d XÞ @Xj dai 5 JD2T dAj : ij dAj 5 J @xi

(3.34b)

Area ratio The area ratio can also be calculated as

pffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi jdaj daUda ðJD2T UdAÞUðJD2T UdAÞ σðηÞ 5 5 5 jdAj jdAj jdAj pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi J dAUðD21 UD2T ÞUdA dA 5 J ηUC21 Uη; η 5 ; 5 jdAj jdAj

in which η denotes the unit normal vector of the area element dA.

(3.35a)

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As a special case, we consider the area element constructed by the two base vectors G1 and G2 of the Lagrange system; the unit normal of this area element is the base vector, therefore the vector η 5 ½0; 0; 1. For this special case, the area ratio from Eq. (3.35a) is qffiffiffiffiffiffiffiffi 21 : σðG3 Þ 5 J C33 (3.35b)

Unit normal vector of deformed area element ^ are two unit differential vector elements and perpendicular to each other, the vector dA 5 η If the vectors dX and dX represents a unit normal vector of the area before deformation. The unit normal vector of the corresponding deformation ^ can be obtained by area of element dA 5 dX 3 dX νi 5

dx 3 dx^ eijk dxj dx^k 5 : jdx 3 dx^ j jdx 3 d x^ j

(3.36a)

From Eqs. (3.1), (3.27), and (3.29), it follows that dxi 5 dXi 1 Sij dXj 1 eikj Ωk dXj ; d x^i 5 dX^ i 1 Sij dX^ j 1 eikj Ωk dX^ j ; dx 3 d x^ 5 η 1 Ω 3 η 1 Sjj η 2 SUη

(3.36b)

1 jSjS21 Uη 1 ðΩUSÞ 3 η 1 ðΩUηÞΩ; in which jSj 6¼ 0. This formulation, in combination with Eq. (3.36a), provides a representation of the unit normal on the surface of the structure under deformation in terms of the unit normal η on the surface before the deformation, the symmetrical tensor S, and the rotation vector Ω. In an infinitesimal theory of continuum mechanics, the elements of the tensor S and the vector Ω are assumed small, so that the products of terms are negligibly small and Eq. (3.36b) reduces to dx 3 dx^ 5 η 1 Ω 3 η 1 Sjj η 2 SUη; jdx 3 d x^ j2 5 1 1 2Sjj 2 2ηUSUη;

(3.36c)

and from Eq. (3.36a), ν 5 η 1 Ω 3 η 2 SUη 1 ðηUSUηÞη:

(3.36d)

^ Furthermore, if the structure is assumed rigid so that it experiences no strain, that is, S 5 0, the vectors dX and dX experience only a rigid rotation, so that the vectors dx and dx^ are also two unit vectors perpendicular to each other, and jdx 3 dx^ j 5 1. Therefore Eqs. (3.36b) and (3.36d) respectively reduce to dx 3 dx^ 5 ν 5 η 1 Ω 3 η 1 ðΩUηÞΩ; ν 5 η 1 Ω 3 η;

(3.36e)

of which the first equation gives a result involving the second-order quantity of rotation (Xing and Price, 2000), while the second one keeps only the first-order quantity of rotation, as given by Newman (1978). It should be mentioned that the results in Eqs. (3.36a
3.36d) are not valid for motion with large rigid rotation; the rotation tensor, as discussed in Chapter 2, Cartesian tensor and matrix calculus, is a nonlinear function of rotation angles and cannot be expressed by the small rotation vector Ω. To model the large rigid rotation motion of ships, the rotation tensor discussed in Section 10.6.4 must be used.

3.2.1.4 Volume element transformation and volume ratio ~ and dX ^ and located at point Xj in As shown in Fig. 3.7, a volume element constructed by three line vectors dX, dX, the original configuration at time t 5 0 moves to a new point xi , with the three corresponding line vectors dx, d x~ , and dx^ at time t 5 τ. We can derive the volume element transformation equation and volume ratio as follows.

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69

FIGURE 3.7 Volume element and volume ratio.

Volume element transformation From Eq. (3.5), it follows that @xi dXj 5 xi; j dXj 5 Dij dXj ; @Xj @x~i ~ d x~i 5 dX j 5 x~i; j dX~ j 5 Dij dX~ j ; @X~ j @x^i ^ d x^i 5 dX j 5 x^i; j dX^ j 5 Dij dX^ j : @X^ j

dxi 5

(3.37)

The volumes of the volume element in the nondeformed and deformed configurations can be calculated using Eq. (2.26), so that ~ 3 d XÞ ^ 5 eijk dXi d X~ j dX^ k ; dV 5 dXUðdX ~ 3 ðDUdXÞ ^ dv 5 dxUðdx~ 3 dx^ Þ 5 ðDUdXÞU½ðDUdXÞ 5 Jeijk dXi dX~ j d X^ k 5 JdV:

(3.38)

Volume ratio The volume ratio takes the following forms: dv 5 J; dV

dV 5 J 21 : dv

(3.39)

Furthermore, considering the mass conservation in the deformation, we have ρ0 dV 5 ρdv;

3.2.2

ρ0 5 Jρ;

ρ0 5 J: ρ

(3.40)

Velocity field and rate of deformation of fluids

The difference of the velocities at points xj and xj 1 dxj at the same time can be calculated by dvi 5

@vi dxj 5 vi; j dxj : @xj

(3.41)

We can express the partial derivative vi;j in the form 1 1 vi;j 5 ðvi; j 1 vj; i Þ 1 ðvi; j 2 vj;i Þ 5 Vij 2 ωij : 2 2

(3.42)

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FIGURE 3.8 Stress principle.

We may define the rate of deformation tensor Vij and the spin tensor ωij as 1 ðvi; j 1 vj; i Þ; 2 1 ωij 5 ðvj; i 2 vi; j Þ: 2 Vij 5

(3.43)

It is evident that tensor Vij is symmetric and ωij is antisymmetric. Hence the ωij has only three independent elements, and there exists a dual vector ωi , such that ωk 5

3.3 3.3.1

1 1 ekij ωij 5 ekij vj; i ; 2 2

ωij 5 eijk ωk :

(3.44)

Stress tensor Cauchy’s stress

To investigate the motion of a deformation body, we need to express the interaction between the material outside a closed surface and inside the interior, as shown in Fig. 3.8. This interaction can be divided into two types: one, due to the action of a distant type of force such as gravitation and the electromagnetic force, which can be expressed as force per unit mass and is called the body force; and the other due to the action across the closed surface, called the surface force. The well-accepted stress principle of Euler and Cauchy assumes that the interaction of the materials on the two sides of the surface element is momentless. Therefore this interaction can be expressed by the traction or stress vector Tjν on the surface with the outer normal vector ν j , defined as follows: ΔFj dFj 5 ; Δs-0 Δs ds

Tjν 5 lim

(3.45)

where ΔFj represents the interaction force acting on a small surface element of area Δs on the closed surface. If we consider a closed surface consisting of the six surfaces of an infinitesimal parallelepiped with its three outer normal vectors in the xi directions of the coordinate system, the superscript ν denoting the normal vector ν j can be replaced by superscript i, so that Eq. (3.45) defines a stress tensor in the form σij 5 Tji :

(3.46)

The stress tensor is symmetric due to momentlessness in the investigation of the equilibrium of the infinitesimal parallelepiped. If we know the stress tensor at a point in an elastic body, we can determine the traction Ti acting on any surface, with the outer normal vector ν j passing through this point, using Cauchy’s formula: Ti 5 σij ν j :

(3.47)

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71

FIGURE 3.9 An area element in the original nondeformed configuration (left) is moved to its current deformed configuration (right) showing the traction caused by the deformation.

3.3.2

Piola
Kirchhoff stress

Since the interaction between the material outside a closed surface and that in the interior happens in the deformation configuration, Cauchy’s stress σij in Eq. (3.46) and Cauchy’s formula given by Eq. (3.47) are defined in the current configuration of the continuum, and therefore these are convenient to use it in the Euler system. For convenience in investigating motion using the Lagrange description, two Piola
Kirchhoff stress tensors were introduced. To understand these two stress tensors, we need to consider the area element deformation shown in Fig. 3.9. In the original nondeformation configuration, its area and unit normal vector are dS and ηi , respectively. Due to the deformation, this element moves to the current position with its area ds and ν i , as well as a resultant traction dTi , caused by the deformation. This traction is calculated by Cauchy’s formula in Eq. (3.47) in the current configuration, that is, dTi 5 σij ν j ds:

(3.48)

In order to calculate this traction acting on the deformed area using the original nondeformed information, the first Piola
Kirchhoff stress tensor τ ir is defined by transforming the unit vector ν i and area ds in Eq. (3.48) into the original configuration using Eq. (3.34b), so that we have ν j ds 5 J

@Xr η dS; @xj r

dTi 5 τ ir ηr dS;

τ ir 5 Jσij

@Xr : @xj

(3.49)

For the first Piola
Kirchhoff stress tensor, the first and second indices involve the deformed and nondeformed configurations, respectively, and therefore it is also a two-point tensor. Using this tensor, we can calculate the traction dTi acting on an area ν j ds using the nondeformed area information. For convenience in analysis, we can transform the traction dTi defined in the current configuration to an imagined resultant traction dTi0 acting on the original nondeformed area in the original configuration using a vector transformation rule and then combining it with Eq. (3.49) to obtain @xi 0 @Xi dTj ; dTi0 5 dTj ; @Xj @xj @Xi dTi0 5 τ jr ηr dS 5 Σ ir ηr dS; @xj @Xi @Xi @Xr @xi Σ ir 5 τ jr 5 J σjs ; τ ij 5 Σ rj : @xj @xj @xs @Xr dTi 5

(3.50)

Here, Σ ij is defined as the second Piola
Kirchhoff tensor, which can be used to calculate the imaged traction on the original configuration using Cauchy’s formula. Also, the second Piola
Kirchhoff tensor Σ ij corresponds to the Green strain tensor Eij to calculate the work.

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3.4 3.4.1

Constitutive equation Solids

The constitutive equation provides a relationship between the Green strain Eij and the second Piola
Kirchhoff stress Σ ij . For a linear elastic body, this relationship is expressed as Σ ij 5 Cijkl Ekl ;

(3.51)

where Cijkl is a tensor of elastic constants. Since the stress tensor and the strain tensor are symmetric, the tensor Cijkl is symmetric with respect to the indices i and j, as well as k and l. Therefore, generally, it has a maximum of 36 independent constants. For an isotropic elastic solid, the tensor Cijkl takes the form Cijkl 5 λ δij δkl 1 μðδik δjl 1 δil δjk Þ;

(3.52)

where λ and μ are called the Lame constants, which can be represented by the elastic modulus E, the shear modulus G, and the Poisson’s ratio ν in the following forms: λ5

Eν ; ð1 1 νÞð1 2 2νÞ

μ 5 G;

E 5 2Gð1 1 νÞ:

(3.53)

The constitutive equation for linear isotropic elastic bodies becomes Σ ij 5 λ Ekk δij 1 2GEij :

3.4.2

(3.54)

Fluids

For nonviscous fluid, there is no shear stress, and the stress tensor takes the form σij 5 2 pδij ;

(3.55)

where p is a scalar called pressure. For a Newtonian fluid, the shear stress is linearly proportional to the rate of deformation. The stress
strain relationship is specified by σij 5 2 pδij 1 σ~ ij ;

σ~ ij 5 Dijkl Vkl :

(3.56)

Here, σ~ ij denotes the viscous stress tensor, and Dijkl is a tensor of viscous coefficients of the fluid. If the fluid is isotropic, Dijkl takes the form Dijkl 5 λδij δkl 1 μðδik δjl 1 δil δjk Þ;

(3.57)

which, when substituted into Eq. (3.56), gives that σij 5 2 pδij 1 λVkk δij 1 2μ Vij :

(3.58)

Therefore we obtain the summation of three normal stresses σjj 5 2 3p 1 ð3λ 1 2μÞVjj :

(3.59)

For Stokes fluid assumes that the mean normal stress σjj =3 is independent of the rate of dilation Vjj , so that 3λ 1 2μ 5 0;

(3.60)

and the constitutive equation becomes σij 5 2 pδij 1 2μ Vij 2

2μ Vkk δij : 3

(3.61)

In this case, only one material constant μ, called the coefficient of viscosity, sufficiently defines its property. For incompressible fluid, Vkk 5 0, and we have the constitutive equation σij 5 2 pδij 1 2μVij :

(3.62)

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73

The pressure term makes a fundamental difference between fluids and solids. To accommodate this new variable, it is often assumed that a kinetic equation of state exists that relates the pressure p, the mass density ρ, and the absolute temperature T: f ð p; ρ; TÞ 5 0:

(3.63)

For a constant temperature or if the effect of temperature is ignored, the kinetic state equation reduces to f ð p; ρÞ 5 0:

(3.64)

If the fluid is barotropic, this equation may expressed as dp 5 c2 ; dρ

(3.65)

where c is the speed of sound in the fluid. Therefore, for a barotropic fluid, there exists a potential function P of pressure, ðp dp ; (3.66a) Pð pÞ 5 p0 ρ which may be linearized as follows:

ðp

dp 5 P ð pÞ 5 p0 ρ

ðp



1 dρ dp 2 2 1? ρ ρ0 p0 0

 

ðx

p;i dxi : x0 ρ0

(3.66b)

Here, the subindex “0” denotes the variables at the reference position, such as p0 , may be the atmosphere on the free surface. From Eqs. (3.66a) and (3.66b), it follows that p; j p; j P; j 5 ; P; j  : (3.67) ρ ρ0

3.5

Laws of conservation

We now derive differential equations describing the motion of the continuum under specific boundary conditions. The formulation must obey the laws of conservation of mass, momentum, and energy. This section presents these laws in the forms suitable to describe the motion of the continuum in different cases.

3.5.1

Green theorem

Green theorem, also called Gauss’s theorem, provides the relationship between a volume integral and the corresponding surface integral. Consider a convex domain Ωf bounded by a surface Γf consisting of a finite number of parts whose outer normal ν i forms a continuous vector field. Such a domain is said to be regular. Let Fðxj Þ be a continuously differentiable function defined in the volume Ωf and on the surface Γf . The Green theorem confirms that ð ð F; i dΩf 5 Fν i dΓf : (3.68a) Ωf

Γf

Here, the function Fðxj Þ may be a scalar, vector, or tensor. In Fig. 3.10, we demonstrate the Green theorem as follows. Considering an integral of F;2 over the domain Ωf , we may complete this integral along a line segment parallel to axis o 2 x2 and obtain that ð ð ð F; 2 dΓf 5 F; 2 dx2 dx1 dx3 5 ðFR 2 FL Þ dx1 dx3 Ωf Γf ð Ωf 5 ðFR dx1 dx3 2 FL dx1 dx3 Þ (3.68b) ð Γf ð Fν 2 dΓf ; 5 ðFR ν 2 dΓ f 1 FL ν 2 dΓf Þ 5 Γf

Γf

where FR and FL represent the values of the function F on the right and left ends of the integrating line segment. Also, in the derivation process, we have considered the element dx1 dx3 as a projection of the surface area element dΓf on

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FIGURE 3.10 Demonstration of the Green theorem.

the plane x3 ox2 , so that ðdx1 dx3 ÞR 5 ν 2 dΓf and ðdx1 dx3 ÞL 5 2 ν 2 dΓf . Based on this demonstration, if we consider the volume integral with the cases of subindex i 5 1; 2 on the left-hand side in Eq. (3.68a), we can confirm that Eq. (3.68a) is valid.

3.5.2

Material derivatives of volume integral with mass density

We consider a volume integral IðtÞ of a continuously differentiable function Fðxj ; tÞ defined over a spatial domain Ωf ðxj ; tÞ occupied by a given set of material particles, ð IðtÞ 5 Fðxj ; tÞdΩf ; (3.69) Ωf

which is a function of time t. The rate of time change of IðtÞ, denoted by DI=Dt and called the material derivative of IðtÞ, is defined for a given set of material particles in the volume Ωf ðxj ; tÞ.

3.5.2.1 Spatial system We investigate the material derivative of the integration (3.69) over a spatial domain under the Eulerian reference system. In this integration, the integrand Fðxj ; tÞ is multiplied by mass density ρ, that is, ð D Fðxj ; tÞρdΩf : (3.70) Dt Ωf We know that the volume element dΩ0 in the material space becomes the volume element dΩf in the Euler space and that they satisfy the transformation given by Eq. (3.8), that is, dΩf 5 JdΩ0 :

(3.71)

Assume that the mass densities at the initial time and current time of the continuum are ρ0 and ρ, respectively, so that, due to mass conservation, we have ρdΩf 5 ρ0 dΩ0 ;

ρ0 5 ρJ;

(3.72)

which gives a relationship of the mass densities in the material and Euler spaces. Using the transformation in Eqs. (3.1) and (3.72), we can calculate the material derivative of the integral defined by Eq. (3.70) as follows ð ð D D Fðxj ; tÞρ dΩf 5 Fðxj ; tÞρ JdΩm Dt Ωf Dt Ωm ð ð ð (3.73a) D DF DF ρ0 dΩm 5 ρ dΩf ; 5 Fðxj ; tÞρ0 dΩm 5 Dt Ωm Ωm Dt Ωf Dt

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75

where the operation of material derivative D=Dt can move into the spatial integration over the original domain since the material domain Ωm is not changed with time t. Therefore Eq. (3.73a) shows that the operation of the material derivative of the integration in Eq. (3.70) and that of the spatial integration are commutative.

3.5.2.2 Arbitrary Lagrange Euler system Similarly, we can derive the material derivative of integration over a volume in the ALE system, in which the volume is moving in the space. Using Eq. (3.17) and considering mass conservation, we can obtain ð ð D D ^ ^ j ; tÞ^ρ JdΩ ^ m Fðzj ; tÞ^ρ dΩz 5 Fðz Dt Ωz Dt Ωm ð ð ð (3.73b) D DF^ DF^ ^ j ; tÞρ0 dΩm 5 ρ0 dΩm 5 5 Fðz ρ^ dΩz : Dt Ωm Ωm Dt Ωz Dt Here, we have considered mass conservation for the material derivative process, so that ρ0 5 ρ^ J^ is used, with a hat (^) to identify the variables defined in the ALE system.

3.5.3

Material derivatives of arbitrary integrands in a spatial system

We consider the material derivative of the integration in Eq. (3.69), with no mass density included in the integrand. In this case, the mass volume Ωf ðxj ; tÞ is moving due to mass motions. For a time change dt, the time change rate DI=Dt consists of two parts: one involves the rate of the function Fðxj ; tÞ while the volume is not changed, that is, @F=@t; and the other concerns the volume change ΔΩf ðxj ; tÞ while the function is not changed. As a result, we obtain ð ð DI @F 1 5 dΩf 1 lim F dΩf ; (3.74) dt-0 dt ΔΩ Dt Ωf @t f in which the integral defined in the volume change ΔΩf can be transferred into an integral over the surface Γf of the volume Ωf ðxj ; tÞ by considering the infinitesimal volume change shown in Fig. 3.11, that is, dΩf 5 x_j ν j dΓf dt 5 vj ν j dΓf dt and the integral ð ð ð 1 1 F dΩf 5 lim Fvj ν j dΓf dt 5 Fvj ν j dΓf : (3.75) lim dt-0 dt ΔΩ dt-0 dt Γ Γf f f Here vj and ν j represent the velocity of the point on the surface and the unit outer normal vector of the surface. Finally, the material derivative of IðtÞ takes the form

FIGURE 3.11 A material domain Ωf ðtÞ moves to a new position at time t 1 dt.

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DI 5 Dt

ð Ωf

@F dΩf 1 @t

ð Γf

Fvj ν j dΓf :

(3.76)

Using the Green theorem in Eqs. (3.68a) and (3.68b) and the definition of material derivative given by Eq. (3.10), we can further develop Eq. (3.76) into ð ð DI @F 5 dΩf 1 Fvj ν j dΓ f Dt Ωf @t Γf ð 5

Ωf

@F dΩf 1 @t

ð  5

Ωf

ð  5

Ωf

ð Ωf

ðFvj Þ; j dΩf

 @F @F @vj 1 vj 1F dΩf @t @xj @xj  DF @vj 1F dΩf Dt @xj

(3.77)

:

Eq. (3.77) implies that the operation of the material derivative and that of spatial integration are noncommutative in general. This important formulation will be used in the following sections to derive the laws of conservation under the spatial coordinate system.

3.5.4

Material derivatives of arbitrary integrands in the arbitrary Lagrange Euler system

Eq. (3.77) gives the material derivative of a volume integral in which the velocity vj of the volume motion is the velocity of the material particle. For nonlinear fluid
structure interactions, we often investigate the problem in the ALE system, for which the material derivative of a volume integral with moving boundaries is needed. There are the following two forms of this material derivative.

3.5.4.1 The form based on the spatial coordinates To derive this form, we consider an arbitrary volume Ωf ðzi ; tÞ of boundary Γf ðzi ; tÞ moving with a mesh velocity v~j given by Eq. (3.15) in the space, where the mesh coordinate zi identifies the volume and its boundaries, in order to investigate the time derivative dI=dt of the integration defined in Eq. (3.69). Here, the volume Ωf ðzi ; tÞ need not be a material volume with the same materials during its motion, but the mesh coordinate must be unchanged to mark the volume boundaries. As in the derivation of Eq. (3.76) and letting vj 5 v~j on the surface integration, we obtain the time derivative of a moving volume integral as follows:
ð ð dI

@F dΩ 5 1 F v~j ν j dΓf : (3.78a) f dt
z Ωf @t Γf Note that we use notation dI=dt to distinguish it from the material notation DI=Dt. In this equation, on the right-hand side, the first integration represents the time change rate with the volume not moving, and the second one denotes the additional change caused by the boundary motion with the mesh velocity v~j . Combining with Eq. (3.76), we can represent Eq. (3.78a) in a form involving the material derivative, that is,
ð dI

5 DI 2 Fðv 2 v~ Þν dΓ ; j j j f dt
z Dt Γf
ð DI dI

5
1 Fðvj 2 v~j Þν j dΓf ; (3.78b) Dt dt z Γf  
ð ð
@ Fðvj 2 v~j Þ DI 5@ FdΩf

1 dΩf : Dt @t Ωf @xj Ωf z This implies that the nonmaterial derivative dI=dt of the integration over a moving domain equals its material derivative reduced by the quantity, shown in the surface integration on the right-hand side of Eq. (3.78b), flowing out of the domain through the moving boundary.

Fundamentals of continuum mechanics Chapter | 3

77

The second and third equations in Eq. (3.78b) express the material derivative of a mesh volume integration in the spatial coordinate system. This expression provides convenience when formulating the numerical equations based on the updated ALE system discussed later in the chapter.

3.5.4.2 The form based on the referential coordinates We consider the volume integration defined in the referential coordinate system, that is, ð ^I 5 Fðz; ^ tÞdΩz ;

(3.79)

Ωz

where the volume Ωz of its boundary Γz and the integrand F^ are the functions of the referential coordinates and time. The moving velocity v~j of the boundary of the volume is given by Eq. (3.15), while the velocity v^j of the material point in this referential system is given by Eq. (3.14). The material derivative of the integration in Eq. (3.79) can be accomplished using Eqs. (3.14) and (3.17) to transfer the integration domain into the material one and then to complete the derivative operations, that is, ð
ð


DI^ @ @
^ ^ ^ 5 Fðz; tÞ dΩz
5 Fðz; tÞJ dΩ0

Dt @t Ωz @t Ω0 X X


 ð ð 

^ @ ^ ^ @Fðz; tÞ

Fðz; tÞJ^

dΩ0 5 F^ @J

1 J^ 5 dΩ0 (3.80) @t X @t
X Ω0 @t Ω0 X

ð  @v^j @F^

@F^ @zj

^ F^ 5 1
1 X JdΩ0 : @t @zj @t
@zj Ω0 z Now we transfer Eq. (3.80) back into the referential domain to obtain the form of the material derivative in the referential coordinate system:
ð ð
DI^ @ @ðF^ v^j Þ ^ 5 FdΩz

z 1 dΩz : (3.81) Dt @t Ωz Ωz @zj Using the Green theorem given in Section 3.5.1 to change the second volume integration into its surface integration, we can write Eq. (3.81) as
ð ð
DI^ @ ^ 5 (3.82) FdΩz

z 1 F^ v^j ν^ j dΓz : Dt @t Ωz Γz Combining Eqs. (3.77) and (3.81), we find that the two equations have similar mathematical structures but different material velocities defined in the spatial and referential systems, respectively. Eq. (3.81) provides a formulation to derive the law of conservation in the referential system.

3.5.5

General forms of the conservation laws

Let Fðxj ; tÞ, rðxj ; tÞ, and qðxj ; tÞ be a physical property per unit mass of the continuum in a domain Ωf , the property generated per unit mass per unit time by a source, and the flux of property per unit time through a unit area of the boundary Γf of the volume, respectively. In continuum mechanics, we normally define the positive direction of the flux vector qðxj ; tÞ pointing from the inside to the outside of the domain, so that the property through the boundary into the domain will be negative. Based on the universal laws governing the conservation of the mass, momentum, and energy of the system, we can obtain the following two general forms of the conservation laws. One is in an integration and the other in a differential form. The integration form is normally used to build numerical formulations based on the corresponding volume interactions, such as the finite volume method for computational fluid dynamics, while the differential equations are always used in finite difference formulations. These general forms of conservation laws will be used in the following sections to derive the governing equations of the continuum by defining the corresponding physical properties Fðxj ; tÞ, rðxj ; tÞ, and qðxj ; tÞ.

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3.5.5.1 Spatial system Integration form The integration form of the generalized conservation laws under the spatial coordinate system is given by ð ð ð ð D DF ρ dΩf 5 rρ dΩf 2 qUν dΓf ; Fρ dΩf 5 Dt Ωf Ωf Dt Ωf Γf

(3.83a)

in which Eqs. (3.73a) and (3.73b) are introduced. This integration equation physically represents that the material derivative of the property Fðxj ; tÞ in the domain Ωf equals the summation of its generation rate by the source rðxj ; tÞ and the input rate of the property through the boundary by the flux 2qðxj ; tÞ. Using Eq. (3.77) in the left-side material derivative and the Green theorem in preceding integration, we can write it in the form  ð

ð ð @ðρFÞ 1 ðρFvÞUr dΩf 5 rρdΩf 2 qUrdΩf ; @t Ωf Ωf Ωf

ð  ð ð (3.83b) @ ρFdΩf 1 ðρFv 1 qÞUrdΩf 5 rρdΩf ; @t Ωf Ωf Ωf x Or, using the middle integration in Eq. (3.83a), we have  ð  ð ð @F 1 vUrF ρdΩf 5 rρdΩf 2 qUrdΩf : @t Ωf Ωf Ωf

(3.83c)

Here, we have two generalized integration forms of conservation laws in which the one in Eq. (3.83c) is simplified when the equation ð ð D D1 ρdΩf 5 0 ρdΩf 5 (3.83d) Dt Ωf Ωf Dt is valid for any spatial domain, which, as explained later, is the mass conservation equation in continuum mechanics. Differential form From Eqs. (3.83b) and (3.83c) and considering an arbitrary integration domain, we respectively have the following two differential forms of generalized conservation laws: @ðρFÞ 1 ðρFv 1 qÞUr 5 ρr; @t   @F ρ 1 vUrF 5 rρ 2 qUr; @t

(3.83e) (3.83f)

since the integration domain could be infinitely small.

3.5.5.2 Lagrange system The generalized conservation laws in the preceding spatial coordinate system can be transformed into the Lagrange system. Integration form Using Eq. (3.40), (3.49), and (3.50), we can transfer Eq. (3.83a) as follows: ð ð ð ð DF DF @Xj ρJdΩ0 5 ρ0 dΩ0 5 rρ0 dΩ0 2 qi Jη dΓ0 @xi j Ω0 Dt Ω0 Dt Ω0 Γ0 ð ð @Xj ρ @Xj 5 rρ0 dΩ0 2 q0j; j dΩ0 ; q0j 5 qi J 5 qi 0 : @xi ρ@xi Ω0 Ω0

(3.84a)

Here, ηj denotes the unit normal vector of the area element dΓ0 , and the related variables are the ones in the original Lagrange configuration, although the index 0 is neglected in order to identify them. The q0j represents the flux vector in the Lagrange system, of which the volume ratio is involved since the mass density changes.

Fundamentals of continuum mechanics Chapter | 3

79

Differential form Considering that the integration volume could be arbitrary small, Eq. (3.84a) gives its differential form as
DF @F

ρ0 5 ρ0
5 rρ0 2 q0j; j : Dt @t X

(3.84b)

3.5.5.3 Arbitrary Lagrange Euler system Integration form The generalized forms of conservation laws in the ALE system can be derived using the material derivative of a volume interaction over the referential or ALE system given by Eqs. (3.81), (3.82), and (3.73b). They are, respectively, in the forms
ð ð ð
@
^ ^ ^ ρ^ F dΩz
1 ð^ρF^v 1 qÞUrdΩ r^ ρ^ dΩz ; z5 @t Ωz Ωz Ωz z
ð ð ð (3.85a)
@ ρ^ F^ dΩz

1 ð^ρF^ v^j 1 q^j Þν^ j dΓz 5 r^ ρ^ dΩz ; @t Ωz Γz Ωz z and ð Ωz

!
ð ð @F^

^ ρ^ dΩz 5 r^ ρ^ dΩz 2 qUrdΩ ^ ^ 1 v Ur F z: @t
z Ωz Ωz

(3.85b)

It should be noted that the variables and the geometrical volume with its surface are all defined in the referential system, so that the hat (^) over the variables is used to distinguish this purpose. Differential form From the two equations in Eq. (3.85a), we respectively obtain the two differential forms of generalized conservation laws in the ALE system:
^
@ð^ρFÞ
1 ð^ρF^ ^ v 1 qÞUr ^ 5 ρ^ r^ ; (3.85c) @t
z !
@F^

^ ^ 5 r^ ρ^ : ρ^ 1 v^ UrF 1 qUr (3.85d) @t
z Comparing Eqs. (3.85c) and (3.85d) with Eqs. (3.83e) and (3.83f), we have found that the structures of the generalized conservative laws in the Eulerian and ALE systems are in similar form and need only to use the corresponding variables. For instance, the notation v denotes the velocity of a material point relative to the spatial point fixed in the space, while v^ is the velocity of the same material point relative to the moving mesh coordinates defined by Eq. (3.14).

3.5.5.4 Updated arbitrary Lagrange Euler system The generalized forms of conservation laws in the updated ALE system can be derived based on the material derivative of a moving volume given by Eq. (3.78b), in which the current spatial coordinates are taken as the referential coordinates so that the variables and their derivatives, as well as the geometrical volume, are referred to the current spatial configuration. From Eq. (3.78b), we obtain the conservation laws as follows. Integration form @ @t

ð Ωf


ð ð

ρFdΩf
1 ½ρFðv 2 v~ Þ 1 qUrdΩf 5 rρdΩf : x

Ωf

(3.86a)

Ωf

Here, the current spatial domain Ωf is used as the updated ALE domain, with the moving velocity v~ of boundary and the partial time derivative with respect to the referenced coordinate z being x.

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Considering the mass conservation condition in Eqs. (3.83d), Eq. (3.86a) can be written as  ð

ð ð @F 1 ðv 2 v~ ÞUrF ρdΩf 5 rρdΩf 2 qUrdΩf : Ωf @t Ωf Ωf

(3.86b)

Differential form Similar to the driving process for the spatial system, from the integration forms in Eqs. (3.86a) and (3.86b
3.86d), we obtain the corresponding differential forms for the updated ALE system:  
@ ðvj 2 v~j ÞρF @ðρFÞ

1 1 qUr 5 ρr (3.86c) @xj @t
x and ρ


 @F

@F ~ 1 ðv 2 v Þ 1 qUr 5 ρr: j j @t
x @xj

(3.86d)

3.5.5.5 Updated Lagrange system As discussed in Section 3.1.4, the updated Lagrange system takes the spatial coordinate in the current configuration domain Ωf as the updated material coordinate to identify the material particle at time t for investigation of motion in the time period from t to t 1 Δt. The generalized conservation laws given in Eqs. (3.83a
3.83f) for the spatial system, when changing the partial time derivative in the spatial system to the material derivative, can be transferred into the updated Lagrange forms. Integration form Since the current spatial coordinate xi is chosen as the updated material coordinate Xi to study the motion of the continuum from t to t 1 Δt, in this form, the material derivative in Eqs. (3.83a
3.83f) is calculated by
DF Fðxi ; t 1 ΔtÞ 2 Fðxi ; tÞ @F

5 lim 5
; (3.87a) Δt-0 Dt Δt @t x which, when substituted into Eqs. (3.83b) and (3.83c), gives the two integration forms for the updated Lagrange system:
ð ð ð @ðρFÞ

dΩ 5 rρdΩ 2 qUνdΓf ; (3.87b) f f
Ωf @t Ωf Γf x
ð ð ð @F

ρ dΩ 5 rρdΩ 2 qUrdΩf : (3.87c) f f @t
x Ωf Ωf Ωf

Differential form Eqs. (3.87c) and (3.87d), respectively, have their differential forms:
@ðρFÞ

1 qUr 5 ρr; @t
x
@F

ρ 1 qUr 5 ρr; @t
x

(3.87d) (3.87e)

which can also be obtained by setting vj 5 v~j and replacing the coordinate z with its material counterpart in Eqs. (3.86c) and (3.86d). These updated Lagrange forms of governing equations will be used for the smooth particle method used to simulate nonlinear fluid flows.

Fundamentals of continuum mechanics Chapter | 3

81

FIGURE 3.12 Small volume around a jump interface.

3.5.6

Jump condition and equation

In continuum mechanics, there always exists a moving surface through which some physical properties are not continuous. For example, in the two sides of an interface between a liquid and a gas in a coupling system, the mass densities of the continuum are different. The shock wave surface in gas dynamics is also a typical example of this phenomenon. As shown in Fig. 3.12, a small volume VðtÞ on the two sides of an interface of area sðtÞ with its moving velocity v~i is considered. Since this small domain moves in the space, Eq. (3.78b) is used to represent the material derivative in Eq. (3.76) in order to derive a general equation on jump conditions. In this case, the general conservation law for this volume is ð ð ð ð d Fρdv 5 rρdv 2 qUνds 2 ρFðv 2 v~ ÞUνds: (3.88a) dt V V s s If we let the thickness of this thin layer h-0, the volume integral on the left-hand side in Eq. (3.88a) vanishes, but the volume integral on the right-hand side may not vanish if there is a source r on the jump interface. This volume integration can be expressed by a surface integral on the jump interface, that is, ð ð ð rρdv 5 2rρhds 5 rd ds; (3.88b) V

s

s

d

where r represents the source rate per unit area on the jump interface. As a result, Eq. (3.88a) gives the equation of jump condition ½½ρFðv 2 v~ ÞUν 1 ½½qUν 2 rd 5 0;

(3.88c)

where the jump value of the property through the jump interface is defined as ½½ðÞ 5 ðÞ1 2 ðÞ2 :

(3.88d)

For example, on the interaction interface between the liquid and gas, the condition of the mass conservation can be obtained by letting F 5 1, rd 5 0, and q 5 0 in Eq. (3.88c), that is, ½½ρðv 2 v~ ÞUν 5 0; ½ρðvj 2 v~j Þν j 1 2 ½ρðvj 2 v~j Þν j 2 5 0:

(3.89a)

If the jump interface is a surface fixed in the space, its moving velocity v~j 5 0, and Eq. (3.89a) becomes ½ρvj ν j 1 2 ½ρvj ν j 2 5 0:

(3.89b)

This represents that the mass flowing through the fixed interface is the same even though the mass densities on both sides are different. If the jump interface is a material surface, vj 5 v~j , Eq. (3.89a) is automatically satisfied.

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Also, we can obtain the jump condition of momentum conservation by letting F 5 v, rd 5 0, and q 5 2σ in Eq. (3.88c), that is, ½½ρvðv 2 v~ ÞUν 2 ½½σUν 5 0; ½ρvi ðvj 2 v~j Þν j 2σij ν j 1 5 ½ρvi ðvj 2 v~j Þν j 2σij ν j 2 :

(3.90)

½ρvi vj ν j 2σij ν j 1 5 ½ρvi vj ν j 2σij ν j 2 ; v~j 5 0; fixed surface; ½σij ν j 1 5 ½σij ν j 2 ; vj 5 v~j ; material surface:

(3.91)

This equation has its special forms:

3.5.7

Conservation of mass and the equation of continuity

The law of mass conservation requires that the mass M contained in a domain Ωf closed by its surface Γf with outer normal vector ν i is not changed. Let F 5 1, r 5 0, and q 5 0 in Eqs. (3.83a
3.87e), we can derive the following equations governed by the mass conservation law.

3.5.7.1 Spatial system Integration form From Eqs. (3.83a) and (3.83b), we obtain the integration form of mass conservation law under the spatial coordinate system: ð D ρdΩf 5 0; (3.92a) Dt Ωf  ð

@ρ 1 ðvj ρÞ;j dΩf 5 0: (3.92b) Ωf @t Differential system Eq. (3.83e) gives the differential form for mass conservation: ρ; t 1 ðρvj Þ; j 5 0;

Dρ 1 ρvj; j 5 0: Dt

(3.92c)

For an incompressible fluid, the mass density of the fluid is a constant, and Eq. (3.92c) is simplified to vj; j 5 0:

(3.92d)

3.5.7.2 Lagrange system Integration form Using Eq. (3.84a), we derive the Lagrange form of the mass conservation law:
ð ð ð D Dρ0 @ρ0

dΩ0 5 ρ dΩ0 5
dΩ0 5 0: Dt Ω0 0 Ω0 Dt Ω0 @t X Differential form It is easy to derive the differential form corresponding to Eq. (3.93a), that is,
Dρ0 @ρ0

5 5 0: Dt @t
X

(3.93a)

(3.93b)

3.5.7.3 Arbitrary Lagrange Euler system From equations given by Eqs. (3.85a
3.85d) and by setting F 5 1, r 5 0, and q 5 0, we obtain the forms of mass conservation in the ALE system.

Fundamentals of continuum mechanics Chapter | 3

83

Integration form D Dt

ð

ð Ωz

ρ^ dΩz 5

Ωz


 @^ρ

1 ð^ρ v^j Þ; j dΩz 5 0: @t
z

(3.94a)

Differential form

@^ρ

@^ρ

@^ρ D^ρ ^ 1 ρ^ v^j; j 5 0: 1 ð^ ρ v Þ 5 1 v^j 1 ρ^ v^j; j 5 j ;j @t
z @t
z @zj Df

(3.94b)

3.5.7.4 Updated arbitrary Lagrange Euler system The mass conservation laws for the updated ALE system can be directly obtained using Eqs. (3.86a) and (3.86c). Integration form ð

Ωf


 @ρ

@½ρ ðvj 2 v~j Þ 1 dΩf 5 0: @t
z @xj

(3.95a)


@ρ

@½ρ ðvj 2 v~j Þ 1 5 0:
@t z @xj

(3.95b)

Differential form

3.5.7.5 Updated Lagrange system Similarly, we can use Eqs. (3.87b) and (3.87d) to derive the integration and differential forms, respectively, of the mass conversation law in the updated Lagrange system as follows. Integration form
@ρ


dΩf 5 0: Ωf @t x

(3.96a)


@ρ

5 0; @t
x

(3.96b)

ð

Differential form

which can also be obtained by setting vj 5 v~j and replacing the coordinate z by its material counterpart respectively in Eqs. (3.95a) and (3.95b).

3.5.8

Conservation of momentum and equations of motion

The principle of momentum is based on Newton’s second law from which Euler’s two motion laws can be derived. The first Euler motion law requires that the material derivative of the linear momentum of a body is equal to the resultant force of all applied forces, while the second law confirms that the material derivative of the angular momentum, that is, momentum moment, of the body equals the resultant moment of all applied forces. Following the first Euler motion law, we can let F 5 v, r 5 f, and q 5 2σ in Eqs. (3.83a
3.83f), (3.84a), and (3.84b) in order to derive the equation governing the conservation of linear momentum of the continuum. However, to derive the equation for the conservation of angular momentum of the continuum, we need to introduce a couple-stress tensor m and a body moment vector M, of which the former represents the interaction moment between the two adjacent parts in the continuum, and the second denotes the body moment acting per unit mass of the continuum. For micropolar

84

Fluid
Solid Interaction Dynamics

materials, these two additional quantities must be considered. The detailed theories on this type of material, as well as its mathematical descriptions, can be found in many classic publications, for example, Eringen (1964, 1966), Eringen and Suhubi (1964), Suhubi and Eringen (1964), and Iesan (1967, 1969). For materials with no couple-stresses and where body moments exist, the equation governing the conservation of angular momentum is automatically valid based the law for linear momentum conservation, so that the equation of angular momentum conservation is not necessary. This book in its current version does not involve macropolar materials in the following chapters; we present only the equation on angular momentum conservation law in the spatial system but omit its corresponding forms in the other coordinate systems. If this omitted material is needed in further research, readers may follow the approach discussed in this book to transform the equation in the spatial system into the forms in the related reference systems.

3.5.8.1 Spatial system Integration form Form Eqs. (3.83b) and (3.83c), we obtain the following two forms of the linear momentum conservation law in the spatial system:  ð ð

D @ðρvÞ 1 ðρvvÞUr dΩf vρdΩf 5 Dt Ωf @t Ωf (3.97a) ð ð 5 fρ dΩf 1 σUνdΓf ; Ωf

ð Ωf

Γf

 @v 1 vUrv ρdΩf @t Ωf ð ð 5 fρ dΩf 1 σUνdΓf :

Dv ρdΩf 5 Dt

ð 

Ωf

(3.97b)

Γf

Here, Eq. (3.97b) is simplified when Eq. (3.83d)—that is, the equation of mass conservation—is introduced. Differential form Using the Green theorem to change the surface integrations in Eqs. (3.97a) and (3.97b) into the corresponding volume integrations and considering the integration domain as arbitrary, we can derive their differential forms, respectively, as @ðρvÞ 1 ðρvvÞUr 5 σUr 1 ρf 5 0; @t   Dv @v ρ 5 1 vUrv ρ 5 σUr 1 ρf 5 0: Dt @t

(3.97c) (3.97d)

These two equations can be rewritten in their component forms: @ðρvi Þ 1 ðρvi vj Þ; j 5 σij; j 1 ρfi ; @t

 Dvi @vi 5ρ 1 vj vi; j 5 σij; j 1 ρfi : ρ Dt @t

(3.97e) (3.97f)

Angular momentum conservation Cross-multiplying Eq. (3.97f) by the position vector xk , we obtain ρerki xk

Dvi Dðerki xk vi Þ 5ρ 5 erki xk σij; j 1 ρerki xk fi ; Dt Dt

(3.98)

in which the equation erki Dxk =Dtvi 5 erki vk vi 5 0 is used. Eq. (3.98) physically represents the equilibrium of angular momentum of the body in which no couple-stresses and body moments are considered. Therefore, if Eq. (3.97f) is valid, Eq. (3.98) is automatically valid for continuums without couple-stresses and body moments involved.

Fundamentals of continuum mechanics Chapter | 3

85

Including the couple-stress tensor m and the body moment vector M, if they are required, Eq. (3.98) becomes ρ

Dðerki xk vi Þ 5 mrj; j 1 erki xk σij; j 1 ρerki xk fi 1 ρMr : Dt

(3.99)

This is a differential form of angular momentum conservation law for the continuum with couple-stresses and body moments considered. Integrating Eq. (3.99) over the spatial volume and using the Green theorem, we have its integration form: ð ð Dðerki xk vi Þ ρdΩf 5 ðmrj; j 1 erki xk σij; j 1 ρerki xk fi 1 ρMr Þ dΩf Dt Ωf Ω ð f ð (3.100a) 5 ðerki xk fi 1 Mr ÞρdΩf 1 ðmrj 1 erki xk σij Þν j dΓf ; Ωf

Γf

which can be rewritten in the corresponding vector form ð ð D Dðx 3 vÞ ρdΩf ðx 3 vρÞ dΩf 5 Dt Ωf Dt Ωf ð ð 5 ðx 3 f 1 MÞρ dΩf 1 ðm 1 xUσÞUνdΓf : Ωf

(3.100b)

Γf

Physically, x denotes the position vector of a point in the space, x 3 vρ is the moment of momentum per unit volume of the body, x 3 fρ represents the moment of the body force, while Mρ is the body moment per unit volume, and mUν and xUσUν are the traction moment and the traction force per unit area on the surface.

3.5.8.2 Lagrange system Integration form Based on the transformations given by Eqs. (3.49) and (3.50), we can transfer the area element νdΓf and Cauchy’s stress σ in the surface integration over the spatial configuration in Eqs. (3.97a) and (3.97b) into the corresponding area element ηdΓ0 and the first Piola
Kirchhoff stress τ for the Lagrange configuration by the following mathematical process represented by the component form: ð ð ð @Xj σir ν r dΓf 5 σir J ηj dΓ0 5 τ ij ηj dΓ0 ; (3.101a) @xr Γf Γ0 Γ0 which, when Eqs. (3.95a), (3.95b), and (3.6) are introduced, transfers Eq. (3.97a) into the Lagrange system form: ð ð ð D vρ dΩ0 5 fρ0 dΩ0 1 τUη dΓ0 ; (3.101b) Dt Ω0 0 Ω0 Γ0 which is further derived into


 ð  ð @v

0 ρ0
2 τUr dΩ0 5 fρ0 dΩ0 : @t X Ω0 Ω0

(3.101c)

Differential form From Eq. (3.101c), we can obtain the differential form of momentum conservation in the Lagrange system: ρ0

@2 Ui 5 τ ij; j 1 ρ0 fi ; @t2

ρ0

@2 Ui 5 ðxi; r Σ rj Þ; j 1 ρ0 fi 5 ½ðδir 1Ui; r ÞΣ rj ; j 1 ρ0 fi : @t2

(3.101d)

Here, we replace the material time derivative of the velocity in Eq. (3.101c) by the acceleration and the first Piola
Kirchhoff stress τ by the second one Σ, defined in Eq. (3.50).

86

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3.5.8.3 Arbitrary Lagrange Euler system Integration form The integration form of the momentum conservation law under the ALE system can be derived using Eqs. (3.85a
3.85d). They respectively are
 ð

ð ð @ð^ρ v^ Þ

^ ^ ν^ dΓz ; 1 ð^ρ v^ v^ ÞUr dΩz 5 f ρ^ dΩz 1 σU (3.102a) @t
z Ωz Ωz Γz  ð 
ð ð @ v^

^ ^ ν^ dΓz : (3.102b)
1 v^ Urv^ ρ^ dΩz 5 fρ dΩz 1 σU Ωz @t z Ωz Γz Differential form Using the Green theorem in Eqs. (3.102a) and (3.102b), we obtain the two differential forms of generalized conservation laws for the ALE system:
@ð^ρ v^ Þ

^ 1 ð^ρ v^ v^ ÞUr 5 ρ^ f^ 1 σUr; (3.102c) @t
z 
 D^v @ v^

^ ^ ^ 5 ρ^ ρ^ 1 v Ur v 5 f^ ρ^ 1 σUr: (3.102d) Dt @t
z

3.5.8.4 Updated arbitrary Lagrange Euler system Integration form The momentum conservation law in the updated ALE system can be derived using the generalized forms of the conservation laws given in Eqs. (3.86a) and (3.86b) using same replacements: F 5 v, r 5 f, and q 5 2σ. We obtain the following results:
 ð

ð ð @ðρ vÞ

@½ρvðvj 2 v~j Þ 1 dΩf 5 fρ dΩf 1 σUν dΓf ; (3.103a) @t
z @xj Ωf Ωf Γf  ð
ð ð @vi

@vi ρ 1 ðvj 2 v~j Þ dΩf 5 fi ρ dΩf 1 σij ν j dΓf : (3.103b) @t
z @xj Ωf Ωf Γf Here, the current spatial domain Ωf is used as the updated ALE domain, so that the moving velocity v~ of boundary and the partial time derivative with referenced coordinate z fixed are involved. Differential form From Eqs. (3.103a) and (3.103b), we obtain the corresponding differential forms for the updated ALE system:
@ðρvÞ

@½ρvðvj 2 v~j Þ 1 5 σUr 1 ρf; (3.103c) @t
z @xj


 @vi

@vi ~ ρ 1 ðv 2 v Þ 5 σij; j 1 ρfi : (3.103d) j j @t
z @xj

3.5.8.5 Updated Lagrange system Integration form The momentum conservation law in the updated Lagrange system is obtained using Eqs. (3.87a
3.87e). We have the following integration and differential forms:
 ð

ð ð @ðρvÞ

(3.104a) dΩf 5 fρ dΩf 1 σUν dΓf ; @t
x Ωf Ωf Γf

Fundamentals of continuum mechanics Chapter | 3


ð ð @v
ρ

dΩf 5 fρ dΩf 1 σUr dΩf : Ωf @t x Ωf Ωf

87

ð

(3.104b)

Differential form Eqs. (3.104a) and (3.104b), respectively, give their corresponding differential forms:
@ðρvÞ

5 fρ 1 σUr; @t
x
@v
ρ

5 σUr 1 fρ 5 0: @t x

(3.104c) (3.104d)

In this updated Lagrange system, the time derivative is given by Eq. (3.87a) with the fixed current spatial coordinate, which now is referred to as the updated Lagrange coordinate.

3.5.9

Conservation of energy and equation of energy

Before deriving the general energy conservation equation in continuum mechanics, we may consider a special case in which only mechanical energy is of interest without any other energy, such as thermal or electric or magnetic. In this case, the mechanical energy conservation equation is merely the first integral of the equation of motion, so that it is not an independent equation. This equation can be obtained by multiplying the equation of momentum conservation given in Section 3.5.8. For example, considering Eq. (3.97f) premultiplied by the velocity vi , we obtain 2 3   @ðv v =2Þ Dðvi vi =2Þ i i 1 vj vi2vi 5 5 ρ ρ4 (3.105a) @t Dt ;j 5 vi σij; j 1 ρ vi fi 5 ðvi σij Þ; j 1 ρvi fi 2 vi; j σij : This equation, when Eq. (3.43) is introduced, is rewritten as ρ

Dðvi vi =2Þ 5 2 Vij σij 1 ðvi σij Þ; j 1 ρ vi f ; Dt

ρ

Dðe~ 1 vi vi =2Þ 5 2 q~j; j 1 q^f ; Dt

ρ

De~ 5 Vij σij ; dt

(3.105b)

q~j 5 2 vi σij ;

q^f 5 ρ vi fi :

~ respectively, represent the kinetic energy-density and the internal mechanical energy-density per Here, vi vi =2 and e, unit mass of the continuum; q~j is defined as the energy-flow density vector with its positive direction along the outside normal vector direction, and q^f denotes the input power done by the body force fi . Integrating the preceding equation over the spatial volume and using the Green theorem, we obtain its integration form: ð ð   Dðvi vi =2Þ dΩf 5 ðvi σij Þ; j 2 Vij σij 1 ρ vi fi dΩf ρ Dt Ωf Ωf ð ð 5 vi σij ν j dΓf 1 ðρ vi fi 2 Vij σij Þ dΩf ; ð Ωf

ρ

Dðe~ 1 vi vi =2Þ dΩf 5 Dt

ð

Γf

Ωf

Ωf

ð2 q~j; j 1 q^f Þ dΩf

ð

52

ð Γf

q~j ν j dΓf 1

Ωf

q^f dΩf :

(3.105c)

88

Fluid
Solid Interaction Dynamics

These two equations are the energy-flow balance equation in its differential and integration forms developed by Xing and Price (1999). Physically, Eq. (3.105c) implies that the time change rate of the kinetic and internal energies equals the input power that includes that done by the body force and the input energy flow through the boundary of the volume. However, if thermal process is significant, then the equation of energy becomes an independent equation to be satisfied. This equation is derived by the law of conservation of energy. Letting F 5 e 1 vi vi =2, r 5 vi fi 1 θ, and q 5 hj 2 vi σij in the generalized conservation laws given by Eqs. (3.83a
3.83c), we can derive the energy conservation laws as follows. Here, e is the internal energy per unit mass, θ is a heat generation rate (i.e., the heat energy produced per unit mass per second), and hj is the heat flux vector denoting the heat passing a unit area per second.

3.5.9.1 Spatial system Integration form From Eqs. (3.83a
3.83c), we can obtain the following two integration forms of the energy conservation law: D Dt

ð

@½ρðe 1 vUv=2Þ 1 ½ρðe 1 vUv=2ÞvUr dΩf ðe 1 vUv=2Þρ dΩf 5 @t Ωf Ωf ð ð 5 ðvUf 1 θÞρ dΩf 2 ðh 2 vUσÞUνdΓf ð Ωf ð Γf 5 ðvUf 1 θÞρ dΩf 2 ðh 2 vUσÞUrdΩf ; ð 

Ωf

ð Ωf

Ωf

@ðe 1 vUv=2Þ 1 vUrðe 1 vUv=2Þ ρdΩf @t Ω ð f ð 5 ðvUf 1 θÞρ dΩf 2 ðh 2 vUσÞUνdΓf ð Ωf ð Γf 5 ðvUf 1 θÞρ dΩf 2 ðh 2 vUσÞUrdΩf :

Dðe 1 vUv=2Þ ρdΩf 5 Dt

(3.106a)

ð 

Ωf

Ωf

When Eq. (3.105c) is introduced, the preceding equation can be reduced to ð ð  De @e ρdΩf 5 1 vUre ρ dΩf @t Ωf Dt Ω ð f ð 5 ðVij σij 1 θρÞ dΩf 2 hUνdΓf ð Ωf ð Γf 5 ðVij σij 1 θρÞ dΩf 2 hUrdΩf : Ωf

(3.106b)

(3.106c)

Ωf

Differential form The differential forms of the energy conservation law derived from Eqs. (3.106a
3.106c) are given in the following three component forms:  @½ρðe 1 vi vi =2Þ  1 ρðe1vi vi =2Þvj 1ðhj 2vi σij Þ ; j 2 ρðvi fi 1 θÞ 5 0; @t

(3.106d)

@ðe 1 vi vi =2Þ 1 ρvj ðe1vi vi =2Þ; j 1 hj; j 2 ðvi σij Þ; j 2 ρðvi fi 1 θÞ 5 0; @t

(3.106e)

ρ

Fundamentals of continuum mechanics Chapter | 3

ρ

De @e 5 ρ 1 ρvj e; j 5 Vij σij 2 hj; j 1 ρθ: Dt @t

89

(3.106f)

Since the heat flux vector and the internal thermal energy are involved in this thermal process, we need to introduce the Fourier’s law of heat conduction with a heat conduct coefficient κ and the caloric equation of state, respectively, as follows: hj 5 κT; j ;

(3.107)

e 5 eðT; ρÞ;

(3.108)

to obtain a set of closed equations for a thermal fluid problem. In this closed set of equations, there are 22 equations: one equation of continuity, three equations of momentum, one energy equation, six constitutive equations, one kinetic equation of state, three Fourier’s equation of heat conduction, one caloric equation of state, and six velocitydeformation rate equations. The 22 unknown variables consist of six stress components σij , three velocities vj , three heat flux components hj , one pressure p, one mass density ρ, one internal energy e, one temperature T, and six deformation components Vij .

3.5.9.2 Lagrange system Integration form Based on Eqs. (3.49) and (3.50), the corresponding flux vector q0j in Eq. (3.84a) takes the following form for the energy conservation law: @Xj 5 h0j 2 ηr Σ rj ; @xi @Xr ηr 5 v j ; @xj

q0j 5 ðhi 2 vr σri ÞJ h0j 5 hi J

@Xj ; @xi

(3.109a)

in which, to avoid any confusion, the index 0 is used to identify the variables in the Lagrange system. Now, replacing the variables in Eq. (3.84a) by F 5 e 1 vi vi =2, r 5 vi fi 1 θ, and q 5 hj 2 vi σij , as well as noting the result in Eq. (3.109a), we obtain the energy conservation equations for the Lagrange system as D Dt

ð

ð

Ω0

Dðe 1 vUv=2Þ ρ0 dΩ0 Dt ð ð 5 ðvUf 1 θÞρ0 dΩ0 2 ðh0 2 v0 UΣÞUηdΓ0 ;

ðe 1 vUv=2Þρ0 dΩ0 5

Ω0

Ω0

(3.109b)

Γ0

in which some of variables appearing in the dot products are not marked by index 0 because the scalar product is not changed during the coordinate transformation. However, we should remember that these variables are also the ones in the Lagrange system. Considering Eq. (3.105c) valid for mechanical processes in which no thermal process is involved and noting that the scalar Vij σij dΩf 5 Vij0 Σij dΩ0 , we can obtain its Lagrange form: ð Ω0

ρ0

DðvUv=2Þ dΩ0 5 Dt

ð

ð Γ0

v0 UΣUη dΓ0 1

Ω0

ðvUfρ0 2 Vij0 Σ ij Þ dΩ0 ;

which, when combining with Eq. (3.109b), gives the simplified energy conservation equation: ð ð D De ρ0 dΩ0 eρ0 dΩ0 5 Dt Ω0 Ω0 Dt ð ð 0 5 ðθρ0 1 Vij Σ ij ÞdΩ0 2 h0j ηj dΓ0 : Ω0

Γ0

(3.109c)

(3.109d)

90

Fluid
Solid Interaction Dynamics

Differential form From Eqs. (3.109a
3.109c), using the Green theorem to transfer the surface integrations into the corresponding volume forms and considering an arbitrarily small integration volume, we obtain their differential forms, respectively, as
@ðe1vi vi =2Þ

0 0 ρ0 (3.109e)
1 ðhj 2vi Σij Þ; j 5 ðvi fi 1 θÞρ0 ; @t X
@ðvi vi =2Þ

0 0 ρ0 (3.109f)
2 ðvi Σ ij Þ; j 5 vi fi ρ0 2 Vij Σij ; @t X
@e
ρ0

1 h0j; j 5 θρ0 1 Vij0 Σij : (3.109g) @t X

3.5.9.3 Arbitrary Lagrange Euler system The energy conservation equation for the ALE system can be directly obtained from Eqs. (3.85a
3.85d) using the same replacements—F 5 e 1 vi vi =2, r 5 vi fi 1 θ, and q 5 hj 2 vi σij —as those adopted in Section 3.5.9.1, so that we have the following results. Integration form From Eqs. (3.85a) and (3.85b), we obtain 8 9
ð < =
^ ^ vU^v=2Þ
@½^ρðe1 ^ ^ 1 ½^ ρ ð e 1 v U^ v =2Þ^ v Ur dΩ
; z @t Ωz : z ð ð ^ ρ dΩz 2 ðh^ 2 v^ UσÞUν ^ 5 ð^vUf^ 1 θÞ^ dΓz ; ð

Ω

Γ

z z 2 3
^ v^ U^v=2Þ

4@ðe1 5
1 v^ Urðe^ 1 v^ U^v=2Þ ρ^ dΩz @t Ωz z ð ð ^ ρ dΩz 2 ðh^ 2 v^ UσÞUνdΓ ^ 5 ð^vUf^ 1 θÞ^ z:

Ωz

Ωz

(3.110c)

Γz

which, when substituted into Eq. (3.110b), reduces Eq. (3.110b) into  ð 
ð ð @e^

^ ^ νdΓ ^ ^ z: ^ ^ ^ 1 v Ur e ρ dΩ 5 ð θ^ ρ 1 V σ ^ ÞdΩ 2 hU z ij ij z @t
z

(3.110b)

Γz

We can also transfer Eq. (3.105c) into the one in the ALE system, that is, 2 3
ð
4@ð^vU^v=2Þ
1 v^ Urð^vU^v=2Þ5ρ^ dΩz
@t Ωz z ð ð ^ ^ ^ ν^ dΓz ; 5 ð^vUf ρ^ 2 V ij σ^ ij Þ dΩz 1 v^ UσU

Ωz

(3.110a)

Ωz

(3.110d)

Γz

Differential form Using the Green theorem in these equations, we obtain the following differential forms of energy conservation laws for the ALE system
^ v^ U^v=2Þ

@½^ρðe1 ^ ^ ^ ^ (3.110e)
1 ½^ρðe^ 1 v^ U^v=2Þ^vUr 5 ð^vUf 1 θÞ^ρ 2 ðh 2 v^ UσÞUr; @t z 2 3

^ v^ U^v=2Þ
^ ^ 4@ðe1 5 ^ ^ (3.110f)
1 v^ Urðe^ 1 v^ U^v=2Þ ρ^ 5 ð^vUf 1 θÞ^ρ 2 ðh 2 v^ UσÞUr; @t z


 @ð^vU^v=2Þ

^ ^ ^ (3.110g)
1 v^ Urð^vU^v=2Þ ρ^ 5 v^ Uf ρ^ 2 V ij σ^ ij 1 ð^vUσÞUr; @t z

Fundamentals of continuum mechanics Chapter | 3




 @e^

^ ρ 1 V^ ij σ^ ij 2 hUr: ^ ^ ^ 1 v Ur e ρ^ 5 θ^ @t
z

91

(3.110h)

3.5.9.4 Updated arbitrary Lagrange Euler system The energy conservation laws in the ALE system can be derived using Eqs. (3.86a
3.86d) or its spatial forms given in Section 3.5.9.1 to replace the time partial derivatives as well as the fixed spatial coordinates by the ones with fixed ALE coordinates. Therefore we have their corresponding forms for the updated ALE system as follows. Integration form Taking F 5 e 1 vi vi =2, r 5 vi fi 1 θ, and q 5 hj 2 vi σij in Eqs. (3.86a) and (3.86b), we obtain
ð  @½ρðvj 2 v~j Þðe 1 vi vi =2Þ @½ρðe1vi vi =2Þ

dΩf
1 @xj @t Ωf z ð ð 5 ðvi fi 1 θÞρ dΩf 2 ðhj 2 vi σij Þν j dΓf ; Ωf

2

Γf

3
ð
4@ðe1vi vi =2Þ
1 ðvj 2 v~j Þðe1vi vi =2Þ;j 5ρ dΩf
@t Ωf z ð ð 5 ðvi fi 1 θÞρ dΩf 2 ðhj 2 vi σij Þν j dΓf : Ωf

which, when substituted into Eq. (3.111b), gives 2 3
ð ð ð
4@e
1 ðvj 2 v~j Þe; j 5ρdΩf 5 ðθρ 1 Vij σij ÞdΩf 2 hj ν j dΓf : @t
Ωf

z

(3.111b)

Γf

Also, we have the corresponding form of the mechanical energy conservation Eq. (3.105c): 2 3
ð ð
  @ðv v =2Þ
5 ~ ρ4 i i 1 ðv 2 v Þðv v =2Þ dΩ 5 ðvi σij Þ; j 2 Vij σij 1 ρvi fi dΩf ; j j i i f ;j
@t Ωf Ωf z

Ωf

(3.111a)

(3.111c)

(3.111d)

Γf

In the preceding equations for the updated ALE system, the current spatial domain Ωf is used as the updated ALE domain, so that the moving velocity v~ of boundary and the partial time derivative with referenced coordinate z fixed are involved. Differential form From the integration forms in Eqs. (3.111a
3.111d) with application of the Green theorem to transfer the surface integrations to the form for the volume, we obtain the corresponding differential forms for the updated ALE system
@½ρðvj 2 v~j Þðe 1 vi vi =2Þ @½ρðe1vi vi =2Þ

5 ðvi fi 1 θÞρ 2 ðhj 2vi σij Þ; j ; (3.111e)
1 @t @xj z " #
@ðe1vi vi =2Þ

(3.111f)
1 ðvj 2 v~j Þðe1vi vi =2Þ; j ρ 5 ðvi fi 1 θÞρ 2 ðhj 2vi σij Þ; j ; @t "

z

#
@ðvi vi =2Þ

ρ
1 ðvj 2 v~j Þðvi vi =2Þ; j 5 ðvi σij Þ; j 2 Vij σij 1 ρvi fi ; @t z


 @e

~ 1 ðv 2 v Þe j j ; j ρ 5 θρ 1 Vij σ ij 2 hj; j : @t
z

(3.111g) (3.111h)

92

Fluid
Solid Interaction Dynamics

3.5.9.5 Update Lagrange system Based on Eqs. (3.87a
3.87e), we obtain the energy-flow conservation laws for updated Lagrange system as follows. Integration form Substituting F 5 e 1 vi vi =2, r 5 vi fi 1 θ, and q 5 hj 2 vi σij into Eqs. (3.87b) and (3.87c) gives the two integration forms for the updated Lagrange system: ð Ωf


ð ð @½ρðe1vi vi =2Þ

d Ω 5 ðv f 1 θÞρ dΩ 2 ðhj 2 vi σij Þν j dΓf ; f i i f
@t Ω Γ x f

ð Ωf


ð ð @ðe1vi vi =2Þ

ρ d Ω 5 ðv f 1 θÞρ dΩ 2 ðhj 2vi σij Þ; j dΩf : f i i f
@t Ω Ω x f

(3.112a)

f

(3.112b)

f

We can also write the mechanical energy conservation Eq. (3.105c) in its updated Lagrange form:
ð ð ρ@ðvi vi =2Þ

d Ω 5 v σ ν dΓ 1 ð ρvi fi 2 Vij σij Þ dΩf ; f i ij j f
@t Ωf Γf Ωf x

ð

(3.112c)

which combines with Eq. (3.112b) to give the following simplified form ð


ð ð @e

ρ
dΩf 5 ðθρ 1 Vij σij Þ dΩf 2 hj; j dΩf : Ωf @t x Ωf Ωf

(3.112d)

Differential form Eqs. (3.112a
3.112d), respectively, correspond to their differential forms:
@½ρ ðe1vi vi =2Þ


5 ðvi fi 1 θÞρ 2 ðhj 2vi σij Þ; j ; @t x
@ ðe1vi vi =2Þ

ρ
5 ðvi fi 1 θÞρ 2 ðhj 2vi σij Þ; j ; @t x
@ ðvi vi =2Þ

ρ
5 ρvi fi 2 Vij σij 1 ðvi σij Þ; j ; @t x
@e
ρ

5 θρ 1 Vij σij 2 hj; j : @t x

3.6

(3.112e)

(3.112f)

(3.112g)

(3.112h)

Navier
Stokes equations and boundary conditions

We have obtained the fundamental equations to formulate a broad range of problems in continuum mechanics. Here, the important displacement solution for solids and the velocity solution for fluids are presented.

3.6.1

Displacement solution of solid mechanics

In solid mechanics, the Lagrange or the material coordinate system O 2 X1 X2 X3 is often used to describe the motion of a solid body. A displacement solution method chooses the displacement Ui of a solid as a main variable to be solved. The corresponding Green strain Eij and the second Piola
Kirchhoff stress can be calculated based on the obtained displacement. All of the variables are considered the functions of the material coordinates Xi and time t. To understand how this method works, we formulate the motion of a solid body occupying a domain Ωs of its boundary S 5 SU , ST with a unit outer normal vector ηi by the following governing equations.

Fundamentals of continuum mechanics Chapter | 3

93

3.6.1.1 Governing equations Equation of motion Eq. (3.101d) in Ωs : ρ0

  @2 Ui 5 ðxi; r Σ rj Þ; j 1 ρ0 fi 5 ðδir 1Ui; r ÞΣ rj ; j 1 ρ0 fi : 2 @t

(3.113)

Displacement
strain relationship Eq. (3.30) in Ωs : Eij 5

1 ðUj; i 1 Ui; j 1 Um; i Um; j Þ: 2

(3.114)

Constitutive Eq. (3.54) in Ωs : Σij 5 λEkk δij 1 2GEij :

(3.115)

Displacement on SU : Ui 5 U^ i ;

(3.116)

Traction on ST : ðδir 1 Ui; r ÞΣrj ηj 5 T^ i :

(3.117)

Boundary conditions:

3.6.1.2 Displacement solution equation Substituting Eq. (3.114) into Eq. (3.115), we obtain   1 Σ ij 5 λ Uk; k 1 Um; k Um; k δij 1 GðUj;i 1 Ui; j 1 Um;i Um; j Þ; 2

(3.118)

which, when further substituted into Eqs. (3.113) and (3.117), gives the displacement solution equation of solids ρ0

 @2 U i  5 ðδir 1Ui; r ÞΣ rj ; j 1 ρ0 fi ; @t2

Ui 5 U^ i ; ðδir 1 Ui; r ÞΣrj ηj 5 T^ i ;   1 Σ rj 5 λ Uk; k 1 Um; k Um; k δrj 1 GðUj; r 1 Ur; j 1 Um; r Um; j Þ: 2

(3.119)

3.6.1.3 Linear waves in solids For linear cases where the nonlinear terms in Eq. (3.119) are neglected, the well-known Navier’s equation is derived as GUi; jj 1 ðλ 1 GÞUj; ji 1 ρ0 fi 5 ρ0

@2 Ui ; @t2 (3.120)

Ui 5 U^ i ; ½λUk; k δij 1 GðUj; i 1 Ui; j Þηj 5 T^ i :

Dilatational wave in solids Taking a partial derivative of Eq. (3.120) with respect to Xi , and neglecting the body force fi , we obtain a dilatational wave equation in solids: ðλ 1 2GÞΘ; jj 5 ρ0

@2 Θ ; @t2

(3.121)

94

Fluid
Solid Interaction Dynamics

where Θ 5 Ui;i is the divergence of the displacement and represents the volume strain of the body. The velocity of the dilatational wave is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi λ 1 2G Cd 5 : (3.122) ρ0 Shear wave in solids Similarly, taking a cross multiplication of Eq. (3.120) by the gradient operator r, that is, multiplying its both sides by ersi @=@Xs , and noting ersi Uj; jis 5 0, we obtain a shear wave equation in solids: GΨr; jj 5 ρ0

@ 2 Ψr ; @t2

(3.123)

where Ψr 5 ersi Ui; s is the rotation vector of the displacement field and involves the shear deformation of the body. The velocity of shear wave is sffiffiffiffiffi G Cs 5 : (3.124) ρ0

3.6.2

Velocity
pressure solution equations of fluid mechanics

In fluid mechanics, the Euler or the spatial coordinate system O 2 x1 x2 x3 is often used to describe the motion of fluids. Corresponding to the displacement solution method in solid mechanics, the velocity vi and the pressure p of fluid are often chosen as the two fundamental variables to seek the solution of the problem. The corresponding deformation rate tensor Vij and Cauchy’s stress σij can be calculated based on the velocity and pressure obtained. All of the variables are considered the functions of the spatial coordinate xi and time t. Assuming that no thermal process is involved, we use this approach to formulate the motion of a Newtonian fluid occupied a domain Ωf of its boundary Γ 5 Γf , Γv with a unit outer normal vector ν i by the following governing equations.

3.6.2.1 Governing equations Equation of motion in Eq. (3.97f): Dvi 5 σij; j 1 ρfi : Dt

(3.125)

ρ; t 1 ðρvj Þ; j 5 0:

(3.126)

ρ Equation of continuity in Eq. (3.92c):

Geometrical relationship in Eq. (3.43): Vij 5

1 ðvi; j 1 vj; i Þ; 2

Ωij 5

1 ðvj; i 2 vi; j Þ: 2

(3.127)

Constitutive Eq. (3.61) σij 5 2 pδij 1 λVkk δij 1 2μ Vij :

(3.128)

f ð p; ρÞ 5 0:

(3.129)

Kinetic state Eq. (3.64):

Boundary conditions: Solid boundary On the interface between the fluid and a solid body with prescribed motions v^i , if the fluid is considered to be a viscous fluid, a no-slip condition is called for, which requires the velocity of the fluid to equal the velocity of the solid. If the

Fundamentals of continuum mechanics Chapter | 3

95

fluid is assumed to be nonviscous fluid, the fluid can freely move along the tangent direction τ i of the solid, which implies that no tangent force is applied to the fluid. However, the normal velocity in the normal direction ν i of the fluid domain must be equal to the normal velocity of the solid. Therefore we have the following boundary conditions on the surface sv of the solid body. Viscous fluid: Nonviscous fluid:

no-slip condition; normal velocity;

Tangent force:

vi 5 v^i ; vi ν i 5 v^i ν i ;

σij τ j 5 0:

(3.130a) (3.130b) (3.130c)

Free surface On the free surface of a liquid, the particles of the liquid are always on the surface, assuming the liquid and gas cannot be mixed. Therefore the free surface is a material surface for which the material derivative vanishes. This defines a kinematic boundary condition D ðx3 2 hÞ 5 0; Dt

x3 5 hðx1 ; x2 ; tÞ;

(3.131a)

where the second equation defines the elevation of free surface if we take x3 5 0 as the plane of the undisturbed free surface, and the x3 axis is positive upward. Since the position of the free surface is unknown, an additional dynamic boundary condition is imposed. If the surface tension is neglected, the pressure p of the fluid must equal a given pressure p0 , such as the atmospheric pressure. However, if the surface tension is considered, a certain amount of energy is required to create an interface between two fluids. A surface tension coefficient, that is, the energy per unit area with dimensions ½forceUlength=length2 5 force=length is denoted by χ. As a result, an equilibrium condition of membrane forces acting on an element of surface must be satisfied, which is called the Laplace law (Morand and Ohayon, 1995; Ibrahim, 2005a). Therefore we have the following boundary conditions on the free surface Γf : No surface tension: p 5 p0 ;   1 1 Surface tension: nlg χ 1 5 p 2 p0 ; R1 R2

(3.131b) nlg 5 1; 2;

(3.131c)

where nlg 5 1; 2 implies the cases of a single surface or double surfaces involving surface tension, respectively. For a soap bubble, there are two surfaces between air and soap liquid, and therefore the coefficient is nlg 5 2. Here, as shown in Fig. 3.13, R1 and R2 represent the radii of curvature of a surface element in x1 and x2 directions, respectively. Geometrically, Fig. 3.13 shows the equilibrium of an infinitesimal element on the free surface considering the surface tension in order to derive Eq. (3.131c). From the wave height in Eq. (3.131a),

we determine the slopes of the surface in two coordinate directions, tan θ1 5 @h=@x1  θ1 and tan θ2 5 @h=@x2  θ2 , respectively. Assuming the two coordinate directions are the principal directions of curvatures, we have the principal curvatures







κ1 5 @θ1 =@x1 5 @2 h=@x21 and κ2 5 @θ2 =@x2 5 @2 h=@x22 , respectively. Therefore the two principal radii of curvature of the surface are R1 5 2 1=κ1 and R2 5 2 1=κ2 . Here, the negative notations are introduced because the two curvatures are negative for the surface deformation shown in Fig. 3.13. The two infinitesimal angles can be calculated by dθ2 5 2 κ2 dx2 , and another infinitesimal angle in x1 direction is dθ1 5 2 κ1 dx1 , which is not shown in the figure. Considering the equilibrium of the infinitesimal element, we have  

dθ2 dθ1 1 dx2 sin ðp 2 p0 Þ dx1 dx2 5 2nlg χ dx1 sin 5 χnlg dx1 dθ2 1 dx2 dθ1 2 2 (3.131d)  2   

@ h @2 h 1 1 5 2 nlg χ κ1 1 κ2 dx1 dx2 5 2 nlg χ 1 dx dx 5 n χ 1 dx dx ; 1 2 lg 1 2 R1 R2 @x21 @x22 which, when divided by dx1 dx2 , gives Eq. (3.131c).

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FIGURE 3.13 The equilibrium of an infinitesimal element on the free surface subjected to the atmosphere pressure p0 , the liquid pressure p, and the surface tension.

3.6.2.2 Navier
Stokes equation To derive the Navier
Stokes equation, we substitute Eq. (3.127) into Eq. (3.128) to obtain

σij 5 2 pδij 1 λvk; k δij 1 μ vi; j 1 vj; i ;

(3.132)

which, when further substituted into Eq. (3.125), gives ρ



  Dvi 5 ρfi 2 p; i 1 λvk; k ; i 1 μ vi; j 1vj; i Þ ; j Dt

(3.133)

5 ρfi 2 p; i 1 ðλ 1 μÞvk; ki 1 μvi; jj : This equation, in association with the equation of continuity Eq. (3.126) and the state Eq. (3.129), establishes a set of equations involving three variables: velocity vi , pressure p, and mass density ρ, which combines with the suitable boundary conditions to solve a fluid problem in a constant temperature. If the temperature is not constant, an energy equation is required.

3.6.2.3 Different cases of fluids Stokes fluid For the Stokes fluid, λ 5 2 2μ=3, Eq. (3.133) is written as Dvi 1 υ 5 fi 2 p; i 1 vk; ki 1 υvi; jj ; Dt ρ 3

(3.134)

where υ 5 μ=ρ is the kinematic viscosity.

Perfect fluids Perfect fluids have no viscosity so that υ 5 μ=ρ 5 0; Eq. (3.134) is simplified to Dvi 1 5 fi 2 p;i : ρ Dt

(3.135)

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97

Incompressible fluids For incompressible fluids vj;j 5 0, the Navier’s Eq. (3.134) is simplified to Dvi 1 5 fi 2 p; i 1 υvi; jj : ρ Dt

(3.136a)

The nondimensional form of this equation with body force neglected ( fi 5 0) has been very often used in the computational fluid dynamics (CFD) of incompressible fluids. To derive the nondimensional equation, a reference length L for the space coordinate, a time scale T for the time, a velocity scale V for the velocity field, and ρV 2 for the pressure are chosen, from which, when keeping the same notations for all variables but now considered as nondimensionalized, we obtain @vi 1 vi; jj ; 5 2 p;i 1 @t Re

Re 5

ρVL VL 5 : μ υ

(3.136b)

Here, we have used Eq. (3.10) for the material derivative and consider the measure scales VT 5 L for simplification of the resultant equation. Re, called the Reynolds number, is a very important nondimensional parameter in fluid dynamics and physically represents the ratio between the inertial force and the viscous force. For very small values of this number, that is, for strongly viscous-dominated flows, the convection terms (p;i ) can be neglected with respect to the viscous terms. Using the vector identity Eqs. (2.24) and (3.42) for incompressible fluid, we obtain vi; jj 5 2 eilm emst vt; sl 5 2 2eilm ωm; l ;

(3.137)

from which Eqs. (3.136a) and (3.136b) become Dvi 1 5 fi 2 p; i 2 2υeilm ωm;l ; Dt ρ

(3.138)

which is an equation involving the vortex vector ωm . For irrotational flows ωm 5 0, Eq. (3.138) reduces to Eq. (3.135) for perfect fluids. However, this does not imply that the motion of perfect fluids must be irrotational since in Eq. (3.138) υ 5 0 cannot guarantee a zero vector ωm included in the material derivative term on the left-hand side of this equation.

3.6.3

Bernoulli equation and potential flows

3.6.3.1 Bernoulli equation Using Eqs. (3.10) and (3.43), the material derivative Dvi =Dt is further expressed in the form Dvi Dt

@vi @vi 1 vj vi; j 5 1 vj ðvi; j 2 vj; i Þ 1 vj vj; i @t @t ;

@vi 1 2vj ωji 1 12 vj vj ;i 5 @t 5

which, when substituted into Eq. (3.135) for perfect fluids, gives   @vi 1 1 1 2vj ωji 1 vj vj 5 fi 2 p; i : @t 2 ρ ;i

(3.139)

(3.140)

Furthermore, we assume that the fluid is barotropic, satisfying Eq. (3.67), and that the body force is a potential force, such as gravity, which can be obtained by the gradient of a potential Ψ, that is, f j 5 2 Ψ; j :

(3.141)

Substituting Eqs. (3.141) and (3.67) into Eq. (3.140), we have

@vi 2 Ψ1P1 12 vj vj ;i 5 1 2vj ωji ; @t

@vi vi 1 2vj ωji vi : 2 Ψ1P1 12 vj vj ;i vi 5 @t

(3.142)

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Since vj ωji vi 5 0, for a stable flow in which the velocity field is not changed with time so that vi;t 5 0, we obtain   1 2 Ψ1P1 vj vj vi 5 0; (3.143) 2 ;i which physically represents that the gradient rðΨ 1 P 1 vj vj =2Þ is orthogonal to the velocity vector, and therefore the total potential function does not change along a flow line, that is, Ψ1P1

1 vj vj 5 const; 2

(3.144)

which is the Bernoulli equation for stable barotropic flows of a perfect fluid subject to a potential body force.

3.6.3.2 Potential flows Furthermore, if the flow is irrotational, the spin tensor vanishes, ωji 5 0, and there exists a potential of velocity φðxj ; tÞ that satisfies grad φ 5 v;

vj 5 φ ; j :

(3.145a)

From Eqs. (3.140) and (3.142), we obtain the equation governing potential flows of a perfect barotropic fluid subject to a potential body force as   @φ; i 1 5 φ; ti : (3.145b) 2 Ψ1P1 φ; j φ; j 5 2 @t ;i This equation is rewritten as

  1 2 Ψ1P1 φ; j φ; j 1φ; t 5 0; 2 ;i

(3.146)

1 φ φ 1 φ; t 5 const: 2 ;j ;j

(3.147)

therefore Ψ1P1

3.6.3.3 Incompressible potential flows If the fluid is incompressible, for which the mass density ρ is a constant, and the body force is the gravity, we have the potential of pressure function in Eq. (3.67) and the potential of body force, respectively, as p 2 p0 P5 ; Ψ 5 g xi δi3 ; (3.148) ρ where g is the gravitational acceleration and x3 axis is defined as positive upward. As a result, Eq. (3.147) becomes g xi δi3 1

p 2 p0 1 1 φ; j φ; j 1 φ; t 5 const: ρ 2

(3.149)

We may choose the position of undisturbed free surface x3 5 0 as our reference point with a zero potential. At this undisturbed reference point, the potential of velocity vanishes, and the pressure p 5 p0 is the undisturbed pressure on the free surface; therefore the right-hand side constant vanishes, and Eq. (3.149) becomes g xi δi3 1

p 2 p0 1 1 φ; j φ; j 1 φ; t 5 0: ρ 2

(3.150)

For the static case with φ 5 0, we have g xi δi3 1

p 2 p0 5 0; ρ

p 5 p0 2 ρ g xi δi3 :

(3.151)

This gives the static pressure at a point ðx3 # 0Þ in the fluid, which equals a summation of the pressure on the free surface and the fluid pressure (2ρ g xi δi3 ) caused by gravity.

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99

3.6.3.4 Free surface conditions for incompressible potential flows Based on the potential φ of velocity, the boundary conditions in Eqs. (3.131a) and (3.131b) can be represented in the form of the velocity potential. Substituting these boundary conditions on the free surface into Eq. (3.150), we obtain   1 1 1 φ φ 1 φ; t ; gh 1 φ; j φ; j 1 φ; t 5 0; h 5 2 (3.152) 2 g 2 ;j ;j which, when substituted into Eq. (3.131a), gives   Dx3 Dh 21 D 1 5 5 φ φ 1 φ; t ; Dt Dt g Dt 2 ; j ; j

(3.153)

that is, an alternative boundary condition for the velocity potential on the free surface x3 5 h. Using the material derivative formulation, we can write Eq. (3.153) in the form φ; tt 1 2φ; j φ; jt 1

1 ðφ φ Þ φ 1 gφ; j δ3j 5 0: 2 ;j ;j ;i ;i

(3.154)

If the nonlinear terms in this equation are neglected, a linear condition is obtained φ; tt 1 gφ; j δ3j 5 0:

(3.155)

Since Eq. (3.148) is valid for the linearized potential function of pressure defined by Eq. (3.66b), the linearized condition in Eq. (3.155) is also valid for compressible potential flows.

3.6.4

Linear waves in fluids

3.6.4.1 Wave equations Considering linear approximation and neglecting the body force, we may write the equation of motion given by Eq. (3.135) and the equation of continuity in Eq. (3.126) as @vi 1 1 p; i 5 0; ρ0 @t

(3.156)

ρ; t 1 ρ0 vj; j 5 0:

(3.157)

Here, ρ0 is the mass density at the initial time. Using the condition in Eq. (3.65) of barotropic flows and the irrotational condition in Eqs. (3.145a) and (3.145b), we obtain p; i 5

dp ρ 5 c2 ρ ; i ; dρ ; i

vi; t 5 φ; it ;

vj; j 5 φ; jj ;

which, when substituted into Eqs. (3.156
3.157), gives     @φ @φ 1p 5 ρ0 1c2 ρ 5 0; ρ0 @t @t ;i ;i ρ; t 1 ρ0 φ; jj 5 0:

(3.158)

(3.159) (3.160)

Because any time function added into the potential of velocity ϕ does not affect the velocity, without losing any generality, we can define p 5 2 ρ0

@φ ; @t

ρ0

@φ 1 c2 ρ 5 0: @t

(3.161)

Taking a derivative of the second equation in Eq. (3.161) with respect to time t and combining it with Eq. (3.160), we obtain @2 φ 5 c2 φ; jj : @t2

(3.162)

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This is the wave equation in fluids in the form of potential of velocity. Similarly, we can obtain its density and pressure forms as follows: @2 ρ 5 c2 ρ; jj ; @t2

@2 p 5 c2 p; jj : @t2

(3.163)

3.6.4.2 Radiation conditions Sommerfeld condition Sommerfeld’s original proof (Sommerfeld, 1912) of the uniqueness theorem of the radiation solution ϕ 5 φe2iωt of the wave Eq. (3.162) defined in a full infinite 3D space assumed an additional condition   @φ ω 2 iκφ 5 0; κ 5 : (3.164a) lim r r-N @r c pffiffiffiffiffiffiffiffi Here, the quantity r stands for the distance from any fixed point in the space r 5 0; i 5 2 1, κ represents the ratio of the circular frequency ω of the stimulation, and c is the speed of the wave in a full infinite 3D space. This condition is called the general condition of radiation (Sommerfeld, 1949). The fact that this condition is superfluous has been rigorously proven by Rellich (1943) even for the case of an arbitrary number of dimensions h where the radiation condition reads   ðh21Þ=r @φ lim r 2 iκφ 5 0: (3.164b) r-N @r Courant and Hilbert (1962) further discussed and demonstrated the characterization of outward energy radiation of wave radiation problems satisfying the Sommerfeld condition in the form given by Magnus and Oberhettinger (1949):
 
2 ðð
@φ

lim 2iκφ

dSr 5 0; (3.164c)
r-N @r r5jx2x0 j where dSr denotes the surface element of a large sphere Sr of radius r about a fixed point x0 . As mentioned by Courant and Hilbert (1962), this integration form is less demanding than Eq. (3.164b), yet it suffices for the characterization of outward radiation. Sommerfeld’s condition physically represents the case that waves produced and radiated from the sources must scatter to infinity with the velocity c along r direction and that no waves may be radiated from infinity into the prescribed singularities of the field. Similarly, for a wave coming from infinity, the absorption condition defined by Sommerfeld (1949) has a form for an arbitrary number of dimensions h:   @φ 1 iκφ 5 0: (3.164d) lim r ðh21Þ=2 r-N @r Radiation condition involving both free surface and pressure waves The Sommerfeld condition was defined for the infinite 3D space in which only one radiation wave of the speed of wave c is involved. For a problem not defined in a full infinite space, the boundary conditions of a finite domain have to be involved. For example, investigating a water-wave problem entailing both the pressure wave in water and the surface wave on the free surface, we have two different wave speeds, one for the pressure wave and the other for the free surface wave. For this case, the original Sommerfeld condition given in Eqs. (3.164a
3.164d) cannot be used, since the resultant radiation wave speed is physically affected by both waves. To deal with this type of practical problems, Xing (2007, 2008) developed the following radiation condition: 0 1 @φ limr-N r ðh21Þ=2 @ 2 ikφA 5 0; @r (3.165a) ω 2iωt k 5 ; ϕ 5 φe ; ξ

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101

in which an integrated resultant speed ξ of the radiation wave is introduced. This resultant wave speed generally is different from the speed of the pressure wave in water or the surface wave on the free surface. Therefore we choose notations k and ξ, depending on the problem, to represent the integrated speed of the wave in the radiation condition. The variable ξ is a positive real number to satisfy the case of ξ 5 c for the full infinite domain problem discussed by Sommerfeld. Considering natural vibrations ϕ 5 φe2iωt , Eq. (3.165a) is written in an equivalent form.   @ϕ 1 @ϕ 1 lim r ðh21Þ=2 5 0: (3.165b) r-N @r ξ @t Similarly, for a wave coming from infinity, the corresponding absorption condition is defined by   @φ lim r ðh21Þ=2 1 ikφ 5 0: r-N @r

(3.165c)

Sommerfeld’s condition has been widely adopted to investigate incompressible water-wave radiation problems; for example, see Newman (1978), Bishop and Price (1979), and Eatock Taylor (1981). In the acoustic field, Gaunaurd and Brill (1984) presented a study of the resonance scattering problem of an infinitely long axisymmetric elastic cylinder excited by an incident plane wave. The far-field water pressure was expressed by the Hankel function to manage a wave condition at an infinite boundary. The characteristic matrix of the system was presented. In a comprehensive critical review paper by Tang and Fan (2004), the mechanisms of sound scattering and radiation of a submerged elastic structure
acoustic interaction system excited by incident wave were further discussed. In the conclusion, it is clearly mentioned that the roots of the characteristic equation of the studied structure
acoustic coupling system are complex. Moreover, Filippi (1999) used a Fourier transformation method to derive the dynamic response of a one-dimensional vibro-acoustic system excited by an external force and subject to the imposed Sommerfeld radiation condition at infinity. However, the main objective in these publications mentioned was to study the dynamic responses of the system affected by external excitations or vibration sources; they were not intended to explore the natural behavior of the system in which no external excitations are applied, although the characteristic equation of the individual studied system was involved; see, for example, Gaunaurd and Brill (1984), Filippi (1999), and Tang and Fan (2004). As is well known (see, e.g., Courant and Hilbert, 1962; Thomson, 1988), the natural vibration of a dynamic system is defined by an eigenvalue problem of the corresponding idealized system with no material damping assumed and no external forces. From the defined eigenvalue problem, the real natural frequencies and modes of the system are derived or calculated using finite element methods (Bathe, 1996; Zienkiewicz and Taylor, 1989, 1991). For example, Morand and Ohayon (1995) presented some detailed methods for numerical modeling of linear natural vibration analysis of elastic structures coupled to internal fluids. Xing and Price (1991), as well as Xing et al. (1996), proposed a mixed finite element substructure
subdomain method to simulate natural vibrations and dynamic responses of various linear fluid
structure interaction problems. A fluid
structure interaction system subject to Sommerfeld’s condition is defined as a Sommerfeld system in Xing’s papers (Xing, 2006, 2007, 2008). Following the definition of natural vibration of a system, the natural dynamic characteristics of a Sommerfeld system with no excitations, such as incident waves or vibration sources (Xing, 2006, 2007, 2008), were revealed. In Chapter 5, Solutions of some linear fluid
solid interaction problems, we will investigate some examples of the natural vibrations of fluid
structure interaction problems subject to the previous radiation conditions.

Chapter 4

Variational principles of linear fluid
solid interaction systems Chapter Outline 4.1 Short review on historic background 4.2 Fluid
solid interaction problems and interaction conditions 4.2.1 Geometric and dynamic conditions on material interfaces 4.2.2 Interactions on nonfloating fluid
solid interaction interface 4.2.3 Interaction on floating fluid
solid interaction interface 4.2.4 Interactions on air
liquid interface 4.2.5 Conditions of surface
tension interactions 4.2.6 Boundary conditions on infinity and moving structures 4.3 A complementary energy model: pressure
acceleration form 4.3.1 Governing equations 4.3.2 Variational formulation 4.4 A potential
energy model: displacement
velocity potential form 4.4.1 Governing equations 4.4.2 Variational formulation

104 105 107 109 109 111 112 115 116 116 118 120 120 122

4.5 Mixed energy models: displacement
pressure and acceleration
velocity potential forms 4.5.1 Displacement
pressure form 4.5.2 Acceleration
velocity potential form 4.6 Three field variational formulations 4.6.1 Displacement
pressure
velocity potential form 4.6.2 Displacement
acceleration
pressure form 4.7 Formulations with displacement potential or pressure impulse as a variable 4.7.1 Displacement
potential form 4.7.2 Pressure impulse form 4.8 Variational formulations for dissipative systems 4.8.1 Damping types 4.8.2 Virtual variational formulations 4.8.3 Complex variational formulation 4.9 Variational formulations for pipes conveying fluid 4.9.1 Description and assumptions of the problem 4.9.2 Variational formulation 4.9.3 Variational stationary conditions for governing equations 4.9.4 Natural vibrations and first approximate frequency

124 125 126 127 127 129 130 130 132 133 134 136 137 138 139 140 141 142

In this chapter, we present the variational principles for linear fluid
solid interaction (FSI) dynamic systems. Following a short introduction on the history of variational principles for linear FSI dynamics, we describe the mathematical equations and the corresponding boundary and FSI conditions for various FSI problems, which is a basis for investigating their variational formulations. We mainly discuss the two further developed variational principles, included in which are the floating boundary effect on the wet interface, the interaction conditions between two different fluids, and the surface tension conditions on the interfaces of liquids and gases. The first is a pressure
acceleration form, an equilibrium or complementary energy form, in which the fluid pressure and the solid acceleration are taken as variational variables to describe the dynamics of the system and its stationary conditions, given the governing equations of the system. The second one is a velocity potential
displacement model, a kinematical or potential energy form, in which the velocity potential of fluids and the displacement of solids are chosen as variables to describe the dynamic motions of the coupling system and in which its stationary conditions are the governing equations of the system in the velocity potential
displacement style. After the detailed mathematical proofs of these two variational principles, we further discuss some varieties of variational principles for FSI systems that are derived by replacing the involved variables or by releasing some variational constraints by means of the Lagrangian multiplier approach based on the two fundamental variational principles. These modified variational formulations include the two mixed energy forms—displacement
pressure and acceleration
velocity potential—as well as two three-field forms—displacement
pressure
velocity potential and displacement
acceleration
pressure. Fluid
Solid Interaction Dynamics. DOI: https://doi.org/10.1016/B978-0-12-819352-5.00004-5 © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved.

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In dynamic response analysis, the damping of the FSI system, generated by the materials of solids and fluids or by radiation conditions for infinite fluid domains, should be considered. Since the damping involves the first-order time derivatives of variational variables, no corresponding integral variational formulations are generated in real forms, so we present the virtual forms of the two fundamental variational formulations to include the contributions from various types of damping in FSI systems. A useful discussion on the integral variational formulations based on adjoint variables for damped dynamic systems using complex forms is also given. As discussed in Chapter 1, Introduction, variational formulations provide a powerful approach to constructing numerical models or to finding their approximate solutions for very complex engineering problems that are difficult to solve analytically. The variational principles presented in this chapter will be used in Chapter 5, Solutions of some linear fluid
solid interaction problems, to derive approximate solutions of some simple FSI problems and in Chapter 6, Preliminaries of waves, to develop some finite element (FE) models for FSI problems.

4.1

Short review on historic background

The Hamilton principle in analytical dynamics is the most well-known, being extensively described in many textbooks (see, for example, Morse and Feshbach, 1953; Whittaker, 1917; Yourgrau and Mandelstam, 1956/1968; Gantmacher, 1960). This variational principle adopts the displacement of the system and its time derivative, the velocity, as the variables to construct the kinetic and potential energy to derive the dynamic equations governing the motions of system. It is considered a potential form of variational principles for dynamics. Many theoretical development and modifications to Hamilton’s principle have been reported. For example, Toupin (1952) and Crandall (1957) developed complementary principles to describe the dynamic systems using momentum as the variable that satisfies the dynamic equilibrium equations, whereas Bailey (1981) discussed its more precise statement. In fluid dynamics, pressure has been used (Bateman, 1929, 1944) as the Lagrangian in a variational principle to derive the equations of motion in an inviscid, incompressible fluid, whereas the potential of velocity has been used as the Lagrangian in a developed variational principle for a fluid with a free surface in order to invoke the equations of motion with the nonlinear boundary condition on the free surface (Luke, 1967). The variational formulation based on Hamilton’s principle is regarded as the most powerful tool to solve liquid sloshing problems; for example, Lawrence et al. (1958) provided a variational solution of fuel sloshing modes, and Moiseev (1964), as well as Moiseev and Petrov (1964), developed the theory of oscillations of liquid-containing bodies based on Hamilton’s principle. Earlier references and historical developments on sloshing dynamics involving variational principles in fluid dynamics may be found in Serrin (1959) and Ibrahim (2005a). For the solid dynamics, Green and Zerna (1954) introduced Hamilton’s principle into the theory of elastodynamics. With the development of computational techniques, variational principles are at the center of many advances in the analysis of the dynamic behavior of elastic bodies. Xing (1981) systematically investigated the variational principles for elastodynamics and their applications to reveal the theoretical basis for establishing dynamic substructure models and time element approaches (Xing and Zheng, 1983, 1985, 1986, 1987). Oden and Reddy (1983) provided an excellent description of the variational principle in theoretical mechanics, but they constrain the variations of both the displacement and momentum to vanish at the time instants t1 and t2 . These unreasonable constraints have been relaxed by Xing (1984, 1986a,b,c) and by Xing and Zheng (1986) in their development of variational principles to derive solutions to elastodynamic problems subject to four types of initial and final time instant conditions. These theoretical ideas were further developed to describe the dynamics of generalized conservative holonomic systems, so that a more general variational principle framework was constructed that covered many previously quoted results (Xing and Price, 1992). Earlier contributions on Hamilton’s principle dealing with FSI problems concern a very frequent and effective physical model called the acoustoelastic model, according to which a structure with a linear elastic behavior interacts with an acoustic fluid in such a way that both systems undergo small displacements. The importance of this model has led to the development of many variational formulations. Depending on the type of variables used for the structure and for the fluid, these variational formulations can be divided into three major categories: (1) displacement or potential form, in which the displacement for the structure and the displacement or the potential of velocity for the fluid are used (Gladwell and Zimmermann, 1966; Gladwell, 1966; Gladwell and Mason, 1971; Tabarrok, 1978; Thompson, 1982, 1983; Xing, 1984, 1986a,b,c, 1988); (2) equilibrium or complementary form, where the force-like quantities, the structure stress or acceleration, and the fluid pressure are chosen as variational variables (Gladwell and Zimmermann, 1966; Xing, 1984, 1986a,b,c, 1988; Xing and Price, 1991; Xing et al., 1996); (3) mixed energy form, in which the structure displacement is combined with a force-like quantity for the fluid, or, in a reverse case, the structure acceleration combined with the fluid potential velocity is adopted as the variational variable (Dowell et al., 1977; Tabarrok, 1978; Xing, 1984, 1986a,b). In these research publications, the following developments should be further emphasized. Gladwell and Zimmermann (1966) first presented the two variational formulations in which the potential one uses the displacements of the acoustic field as the

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solid interaction systems Chapter | 4

105

variable, and the other is its complementary one using a force-like variable, of which the result was further developed by Gladwell (1966) to include radiation damping in the acoustic fields by using complex adjoint variables (Morse and Feshbach, 1953). Later, Tabarrok (1978) discussed in more detail the dual variational formulations for acoustostructural vibrations, in which the potential form adopts both displacements of the gas and the plate as variables, whereas the complementary version uses the impulse of gas pressure in association with the impulse momentum of the plate. Moreover, Xing (1984) in his PhD thesis investigated these three types of variational principles for FSI dynamic systems and their applications in establishing three types of FE numerical models: displacement, equilibrium, and mixed-element models (Xing, 1986a,b,c; Xing et al., 1986), in which the micropolar theory (Eringen, 1964, 1966) is adopted in order to introduce a penalty coefficient to suppress the zero energy modes. There are four symmetrization approaches, one of which is the same as the one by Irons (1970), as well as the corresponding approximate approaches to tackling FSI problems in engineering. These approaches were further discussed (Xing and Price, 1991; Xing et al., 1996) for the mixed FE model derived from the acceleration
pressure variational principle by Xing (1988) in order to deal with more practical FSI dynamic problems, such as free surface waves and transient dynamic excitations. Three field formulations were initially concerned in order to modify the nonsymmetrical form of numerical equations derived from mixed variational formulations with solid displacement and fluid pressure as variables. To this aim, Morand and Ohayon (1979), based on their investigations of the variational formulations for the elastoacoustic problem (Morand and Ohayon, 1976), developed a three-field symmetric formulation by introducing a fluid displacement potential into the mixed displacement
pressure variational form. Further discussion on this model was given by Ohayon and Valid (1981, 1984) and by Vullierme and Ohayon (1984), and more on this topic can be found in Morand and Ohayon (1995). Following the idea of fluid displacement potential and introducing sloshing motions, seismic, and body forces excitations into the equations, Liu and Uras (1988) derived a more general three-field mixed variational formulation by using Lagrangian multiplier approach, and they used their formulation to generate symmetric FE equations. Meanwhile, Olson and Bathe (1983, 1985) proposed a direct symmetric coupled formulation by introducing static fluid pressure into the potential variational formulation, with the solid displacement and the fluid potential of velocity as variables, to generate a three-field variational formulation. A further worthy contribution to three-filed formulation was given by Kanarachos and Antoniadis (1988), who introduced solid acceleration as the third variable in the mixed variational formulation and developed a new three-field symmetrical variational form with solid displacement and acceleration, as well as fluid pressure, as variables. In their paper, the authors gave some discussion on the free surface condition in Lagrangian description represented by a wave height variable z, which is not the pressure form in the original displacement
pressure variational formulation. The first and second variational principles described in the following sections were originally developed by Xing (1984, 1986a,b,c, 1988) to investigate FSI dynamic problems involving linear free surface waves. Since that time, there were no interaction problems involving two different fluids, and the corresponding interaction conditions on the interface between two different fluids was not revealed. Until 2006, when a thin elastic spherical tank
air
water interaction problem was tackled (Xing et al., 2006, 2007a; Tan et al., 2006; Toyota et al., 2006; Xiong et al., 2006a,b), it was realized that this interaction condition is different from one on the interface between two domains of the same fluid due to the different mass densities of the two fluids. This new revealed condition is further theoretically confirmed in the paper by Xing et al. (2009a,b). With more applications of the developed variational principles and application practices to the ship
water interaction simulations, we also realized that the wet interface between the ship and the water is a floating FSI interface, whose motions affect the gravitational potential of the system. So a new form of FSI boundary condition, including floating effects, was defined to represent this physical phenomenon. Moreover, when investigating the nonlinear low-frequency waves of a dragon washbasin (Zhou et al., 2013), it was found that, in some cases, free surface tension may play an important role, and so its effect also needs to be included in the corresponding variational principles. These new ideas have now been formulated in modified variational principles (Xing, 2015a, 2016). Therefore in this chapter, the descripted variational principles are the new versions, including the new contributions just mentioned above: (1) the boundary condition between two difference fluids; (2) the FSI boundary condition covering floating effects; (3) the surface tension contribution. These more generalized variational formulations will be beneficial in investigating more practical FSI problems in engineering.

4.2

Fluid
solid interaction problems and interaction conditions

Now we restrict our focus to investigate linear fluid
structure interaction problems in which the disturbances of both fluids and solids are negligibly small. The initial configuration of the system is therefore chosen as the reference position to measure the dynamic disturbances of the system, and there is no need to distinguish the Euler coordinates o 2 x1 x2 x3 for fluids and the corresponding Lagrange coordinates o 2 X1 X2 X3 for solids. For this type of problem, we assume that the fluid of mass density ρf is nonviscous compressible and that its motions are irrotational,

106

Fluid
Solid Interaction Dynamics

so that it is governed by the wave equations in the form of pressure p or velocity potential φ formulated in Section 3.6.4 of Chapter 3, Fundamentals of continuum mechanics. The solid structures are exactly governed by the linear elasticity theory with small motions of displacements Ui , velocity Vi 5 Ui;t , and acceleration Wi 5 Ui;tt ; therefore we do not need to distinguish different strain and stress tensors for nonlinear analysis with finite displacement fields; only Cauchy’s stress σij and linear strain Eij in the structure are adopted. Here, for illustration purposes, we give some example figures drawn from practical engineering problems. All of these similar problems can be formulated by the variational formulations discussed in this chapter. Fig. 4.1 shows a dam
water interaction system in which the solid, of mass density ρs and elastic tensor Cijkl , occupies a domain Ωs of boundary ST , Sw , Σ with its outside unit normal vector ηi . The wet interface Σ does not float with the free surface motion of fluid in the domain Ωf of boundary Γf , Γw =Γv , Γp =Γφ =Γv , Σ with its outside unit normal vector ν i . The right-hand-side boundary of the water can be treated as a different boundary depending on the numerical models. For the pressure p model, it can be considered a Γp boundary on which a pressure wave p^ or the gradient of pressure, boundary acceleration w^ i , is given, which may be caused by explosions in the water or earthquakes. For the fluid model using the potential of velocity φ as a variable, this boundary could be a velocity boundary Γv with a ^ On the base boundary denoted given velocity v^i or a velocity potential boundary Γφ with a given velocity potential φ. ^ ^ by Su or Sw , the displacement U i or acceleration W i of the solid can be given, and on the boundary denoted by Γv or Γw for the fluid, the velocity v^i or acceleration w^ i may be defined. Fig. 4.2 shows a ship
water interaction system in which the same notations as used in Fig. 4.1 are applied to identify the domains of fluid and solid as well as their boundaries, except the floating wet interface is now denoted by Σf to distinguish that this interface floats with the free surface motion; two infinite boundaries are marked by Γ2N and ΓN . In general cases, the ship is moving on the water with a constant velocity V^ along x2 direction, and it is in its steadystate equilibrium. With an oncoming wave from boundary Γ2N , where the dynamic pressure, or velocity, or potential of velocity of the water can be defined, the ship will be disturbed from its steady-state equilibrium. Due to the damping effects of the water, the disturbances cannot be transmitted onto the right infinite boundary ΓN , which is called a nondisturbed boundary on which the dynamic pressure and velocity would be zero. Fig. 4.3 shows an air
liquid
shell interaction system in which there exists an air
liquid interaction interface ΓðLAÞ through which the mass densities of the two fluids are different. This system could be considered in modeling the dynamic response of chemical liquid tanks subjected to earthquake excitations.

ηi

Tˆi

ST

νi

Γf

x3

x2

o

x1

ΩS Ωf

Σ Fˆi

Γp , pˆ / Γv , vˆ / Γ , φˆ

fˆi

φ

Γw / Γv

S w / Su

FIGURE 4.1 Dam
water interaction system in which the wet interface Σ does not float with the free surface wave.

ST

x3

νi

pˆ 0

ηi Tˆi

Fˆi

o Ωs

x1

x2 Σf

Γ−∞

Γ∞

s

fˆi

Ωf

Γw / Γv

FIGURE 4.2 Ship
water interaction system using the same notation as shown in Fig. 4.1.

Γf

Variational principles of linear fluid
solid interaction systems Chapter | 4

ηi

107

Tˆi

ST

Σ

νi

Fˆi

Air

Γ Liquid

( LA )

Ωs Ωf

Earthquake excitation

Sw

FIGURE 4.3 Air
liquid
structure interaction system of which there exists an air
liquid interaction interface ΓðLAÞ through which the mass densities of the two fluids are different.

4.2.1

Geometric and dynamic conditions on material interfaces

The examples shown in Figs. 4.1
4.3 involve four possible dynamic interactions: (1) the FSI on the wet interface Σ that does not float with the free surface motion; (2) the FSI on the floating wet interface Σf that does float with the free surface motion; (3) the air
liquid interaction on the interface ΓðLAÞ on the two sides, where the mass densities of the fluids are different; and (4) if the surface tension cannot be neglected, a possible surface tension
liquid interaction on the free surface Γf or the air
liquid interface ΓðLAÞ . More generally, the surface tension on the air
solid or the liquid
solid interface, which is neglected here, may also need to be considered. Interested readers may refer to the more detailed discussions in the book and the original publications by Morand and Ohayon (1995) and Ibrahim (2005a,b). As shown in Figs. 4.1
4.3, we chose a Cartesian coordinate system o 2 x1 x2 x3 in which its origin o is fixed on the free surface in the steady-state equilibrium. For the cases in Figs. 4.1 and 4.3, this steady-state is the static equilibrium state of the integrated system. For the system in Fig. 4.2, this state implies that the ship moves with a constant velocity V^ along x2 direction in which all the forces, such as its propulsion, water resistance force, gravitation, etc., are in a stable stationary equilibrium state, which is considered unchanged as observed by people standing on the ship. The gravitational acceleration is along the negative direction of the coordinate axis o 2 x3 , so that the coordinate x3 on the free surface defines the wave height. To develop the generalized governing equations describing the linear fluid
structure interaction dynamics, the interaction boundary conditions on these types of boundaries have to be investigated using the theory of continuum mechanics given in Chapter 3, Fundamentals of continuum mechanics. Since the four interaction interfaces are considered material surfaces, on the two sides of which the materials cannot mix with each other, the following geometrical and dynamic conditions have to be satisfied.

4.2.1.1 Dynamic condition For linear FSI problems with small disturbances from the reference state, the potential function of pressure P is approximated by Eq. (3.66b), and the Bernoulli equation in Eq. (3.147) is linearized to the form p 2 p0 gxi δi3 1 1 φ;t 5 0; (4.1) ρf where a zero constant on the right-hand side of Eq. (3.147) is chosen, and the subindex 0 indicates the pressure value on the free surface. Considering the static equilibrium or steady-state equilibrium in the reference state, the potential of velocity can be represented as
0; statice quilibrium φ 5 φ^ 5  (4.2) φ; steady-state equilibrium; and the Bernoulli equation in Eq. (4.1) becomes gx^i δi3 1

p^ 2 p0 1 φ^ ;t 5 0; ρf

(4.3)

108

Fluid
Solid Interaction Dynamics

where the hats “^” mark the variables in the static or steady-state equilibrium. Subtracting Eq. (4.3) from Eq. (4.1), we obtain gðxi 2 x^i Þδi3 1

p~ 1 φ~ ;t 5 0; ρf

(4.4)

^ φ~ 5 φ 2 φ;

(4.5)

where ^ p~ 5 p 2 p;

denoting the disturbance of the dynamic pressure and the velocity potential relative to the reference equilibrium state, respectively. To simplify the notation, we neglect the wave (B) over the variables and write it directly in the form p 1 φ;t 5 0; ρf p ghðx1 ; x2 ; tÞ 1 1 φ;t 5 0; ρf

gðxi 2 x^i Þδi3 1

ðxi 2 x^i Þδi3 5 hðx1 ; x2 ; tÞ;

(4.6)

and consider the dynamic pressure p, the potential of velocity φ, and the floating height h, all measured from the reference equilibrium state. Taking the first partial derivative of Eq. (4.6) with respect to xi and noting φ;3 5 v3  h;t for linear approximation, we obtain p;i p;i p;i 1 φ;it 5 gδi3 1 1 vi;t 5 gδi3 1 1 wi 5 0; ρf ρf ρf p;i ν i p;i ν i p;3 g1 1 φ;it ν i 5 g 1 1 wi ν i 5 g 1 1 h;tt 5 0: ρf ρf ρf

gδi3 1

(4.7a)

Furthermore, taking the partial derivative of the second equation in Eq. (4.7a) with respect to x3 , noting h;3 5 x3;3 5 1, we have p;33 5 0; ρf

(4.7b)

which implies that, in linear approximations, the dynamic pressure around the floating surface in the vertical direction is a linear function of the floating height h. It is important to remember that both xi and x^i are space coordinates, so that xi 5 x^i implies the same space point inside the fluid, at which Eq. (4.6) becomes p 1 φ;t 5 0; ρf

(4.8)

which provides a relationship between the pressure and the potential of velocity. Taking the derivative of Eq. (4.8) with respect to t and xi , we respectively obtain p;t 1 φ;tt 5 0; ρf p;i p;i p;i 1 φ;it 5 1 vi;t 5 1 wi 5 0; ρf ρf ρf

p;i 5 2 ρf wi ;

(4.9a)

in which φ;i 5 vi and wi  vi;t are introduced. The formulations given by Eqs. (4.8) and (4.9a) are valid everywhere in the fluid domain for linear approximations. Since, for linearized approximation, the pressure is a linear function of the height h, Eq. (4.9a) gives p;3 5

@p 5 2 ρf w3 5 constant; @x3

@w3 5 0: @xi

(4.9b)

Variational principles of linear fluid
solid interaction systems Chapter | 4

109

4.2.1.2 Geometrical condition For linear problems, the geometrical condition on the floating material boundary in Eq. (3.131a) reduces to Dx3 5 v3 5 φ;3 5 φ;i ν i 5 h;t ; Dt

(4.10a)

where φ;i ν i 5 φ;3 and ν i 5 δi3 are introduced. On the original horizontal floating interference interface, the curvature of the surface vanishes, so that the surface tension component in the vertical direction also vanishes. Therefore the pressure in Eq. (3.131c) takes the value p 5 p0 5 0: As a result, Eq. (4.6) gives h52

φ;t ; g

(4.10b)

from which, when combining with Eq. (4.10a), it follows that φ;i ν i 5 2

φ;tt : g

(4.11a)

The application of Eq. (4.8) can replace the velocity potential in Eq. (4.11a) by the pressure and gives p;tt p;i ν i 5 2 : g

(4.11b)

Using the derived formulations for the geometric and dynamic conditions, we construct the interaction conditions on the discussed interfaces as follows.

4.2.2

Interactions on nonfloating fluid
solid interaction interface

The original position on a nonfloating FSI interface Σ is considered the reference position in investigating its geometrical constraint and dynamic equilibrium. This interface is a material interface, so the conditions given by Eq. (3.91) apply, requiring the motions of the fluid and the solid to be consistent and the dynamic force of the fluid to be balanced by the traction force on the solid. Based on Eqs. (4.8), (4.9a), and (4.9b), we have the following interaction conditions for nonviscous FSI problems.

4.2.2.1 Velocity potential
displacement model Normal velocity consistence:

φ;i ν i 5 2 Ui;t ηi :

Normal traction equlibrium: ηi σij ηj 5 2 p 5 ρf φ;t ; Tangent force vanishing: τ i σij ηj 5 0:

(4.12a) (4.12b)

4.2.2.2 Pressure
acceleration model Normal acceleration consistence:

p;i ν i 5 W i ηi : ρf

Normal traction equlibrium: ηi σij ηj 5 2 p; Tangent force vanishing: τ i σij ηj 5 0:

4.2.3

(4.13a)

(4.13b)

Interaction on floating fluid
solid interaction interface

On a floating FSI boundary Σf , x^3 stands for the static position of the floating wet interface, such as the wet interface between a ship bottom and the water; the dynamic interaction pressure is not known. In this case, Eqs. (4.6), (4.7a), and (4.7b) should be used to define the interaction conditions. On the FSI floating interface, the floating height and acceleration of the water should equal those of the solid, respectively, that is, h 5 Ui ν i 5 2 Ui ηi ;

h;t 5 Vi ν i 5 2 Vi ηi ;

h;tt 5 Wi ν i 5 2 Wi ηi ;

(4.14)

110

Fluid
Solid Interaction Dynamics

from which, with Eqs. (4.6), (4.7a), (4.7b), (4.11a), and (4.11b), we can define the following interaction conditions on the floating FSI interface.

4.2.3.1 Velocity potential
displacement model Normal velocity consistence:

φ;i ν i 5 2 Ui;t ηi :

(4.15a)

Normal traction: ηi σij ηj 5 2 p 5 ρf ðφ;t 2 gδi3 ν i Uj ηj Þ 5 ρf ½φ;t 2 ðgUηÞðUUηÞ; Tangent force:

(4.15b)

τ i σij ηj 5 0:

4.2.3.2 Pressure
acceleration model Normal acceleration consistence:

p;i ν i 5 Wi ηi 2 gδi3 ν i ρf 5 WUη 1 gUν 5 ðW 2 gÞUη:

Normal traction equlibrium: ηi σij ηj 5 2 p; Tangent force vanishing: τ i σij ηj 5 0:

(4.16a)

(4.16b)

Here, Eq. (4.16a) is derived using Eq. (3.147a) by taking the partial derivative with respect to xi and then multiplying the result by ν i , considering the normal acceleration φ;it ν i 5 2 Wi ηi as well as the effect of gravitational acceleration g, only along the x3 5 δi3 xi direction. The boundary conditions given in Eqs. (4.15b) and (4.16a) cover the case as shown in Fig. 4.4, where the FSI interface Σf is an inclined surface relative to the vertical direction, that is, the gravitational acceleration direction. In this case, a part of gravitation potential is included. From Fig. 4.4, it is not difficult to understand the gravity vector represented by gi 5 2 gδi3 ;

2 gδi3 ν i 5 gUν 5 g cos α;

(4.16c)

where α denotes the angle between the gravity vector g and the unit vector ν pointing from the fluid to the solid on the FSI interface. The three special cases are as follows: α 5 π; cos α 5 2 1; horizontal Σf with solid above; π α 5 ; cos α 5 0; vertical Σf ; no floating effect; 2 α 5 0;

cos α 5 1;

horizontal Σf with fluid above:

v

α Fluid

Solid Σf

g FIGURE 4.4 Interaction condition on FSI interface. FSI, Fluid
solid interaction.

(4.16d)

Variational principles of linear fluid
solid interaction systems Chapter | 4

4.2.4

111

Interactions on air
liquid interface

As mentioned, we assume that the gas and the liquid cannot be mixed and that an obvious material interface between the gas and the liquid exists. As demonstrated by Eq. (3.91) for jump conditions, on this material interface, the normal velocities perpendicular to the interface are the same, and the dynamic pressures of the fluids satisfy the dynamic equilibrium. We now derive the appropriate forms of these conditions for different numerical models based on Eqs. (4.6)
(4.11b).

4.2.4.1 Two synthesized formulations As shown in Fig. 4.3, this interface is denoted by ΓðLAÞ , under which is the heavy liquid, marked by superindex L, and over which is the light air marked by A. In this notation, the variables with superindex A are the variables of the air, and the ones with superindex L are for the variables of the liquid. We define the unit normal vector of this interface as pointing from the liquid to the air, as shown in Fig. 4.3, so that ν i 5 δi3 , since this air
liquid interface is always in the horizontal plane. We can synthesize the geometric and dynamic conditions presented in Section 4.2.1 to derive two formulations, respectively, for the pressure and velocity potential model. Pressure form Using Eqs. (4.9a), (4.9b), (4.11a), and (4.11b) to replace wi and p;i ν i in Eqs. (4.7a) and (4.7b), we obtain the equation for the pressure model: p;tt ν 3 p;i ν i g5 1 : (4.17a) gρf ρf Velocity
potential form Taking the time derivative of Eq. (4.6), we obtain the equation for the velocity potential model: ρf φ;i ν i 1

ρf φ;tt p;t 1 5 0: g g

(4.17b)

Eqs. (4.17a) and (4.17b) provide the formulations to derive the interaction conditions on the air
liquid interface, for which the superindexes L and A are the variables of the liquid and the air, respectively. Because the original interface is ðAÞ a horizontal plane, the unit outside the normals satisfies ν ðLÞ i 5 δi3 5 2 ν i :

4.2.4.2 Velocity potential
displacement model For the velocity potential
displacement model, the consistent condition requires the velocity potentials of the air and the liquid to be same. To satisfy the dynamic equilibrium, the dynamic pressures of the air and the liquid are also the ðAÞ same at any time on the interface, which implies the time derivative pðLÞ ;t 5 p;t . Therefore using Eq. (4.17b), subtracting its form for the air from that for the liquid, we have the following interaction conditions: Velocity potential consistence:

φðLÞ 5 φðAÞ :

(4.18a)

Time derivative of pressure: ðLÞ ðLÞ ρðLÞ f φ;i ν i 1

ðLÞ ðLÞ ρðLÞ f φ;tt ν 3

g

ðAÞ ðAÞ 1 ρðAÞ f φ;i ν i 1

ðAÞ ðAÞ ρðAÞ f φ;tt ν 3

g

5 0:

(4.18b)

Based on Eqs. (4.18a) and (4.18b), we have the following two special cases. Same fluid ðAÞ ðAÞ For an interface inside the same fluid, the mass density ρðLÞ and the second-order time derivative t φðLÞ ;tt 5 φ;tt , so f 5 ρf that this equation reduces to ðLÞ ðAÞ ðAÞ φðLÞ ;i ν i 1 φ;i ν i 5 0;

(4.18c)

112

Fluid
Solid Interaction Dynamics

which physically represents that the velocity on the interface in the same fluid is consistent. As a result, Eq. (4.18b) ðAÞ actually is a geometric condition, while Eq. (4.18a) implies a pressure equilibrium condition since it gives φðLÞ ;t 5 φ;t , ðLÞ ðAÞ ðLÞ ðAÞ so that 2p 5 ρf φ;t 5 ρf φ;t 5 2 p , as confirmed by Eq. (4.8). Free surface of water If the air is the atmosphere, the velocity potential of air φðAÞ 5 0, that is, the one defined for the static equilibrium state, so that Eq. (4.18a) is no longer needed, and Eq. (4.18b) is reduced to ðLÞ φðLÞ ;i ν i 1

ðLÞ φðLÞ ;tt ν 3 5 0: g

(4.18d)

This is just the condition for free surface shown in Eqs. (4.11a) and (4.11b) or Eq. (3.155).

4.2.4.3 Pressure
acceleration model For the pressure
acceleration model, the dynamic equilibrium requires the pressures of the air and the liquid to be balanced, and the consistence condition is derived from Eq. (4.17a), so that we have the following results: Pressure equlibrium: pðLÞ 5 pðAÞ : 0 1 0 1 ðLÞ ðLÞ ðAÞ ðAÞ 1 @ ðLÞ ðLÞ p;tt ν 3 A 1 @ ðAÞ ðAÞ p;tt ν 3 A Normal acceleration consistence: ðLÞ p;i ν i 1 1 ðAÞ p;i ν i 1 5 0: g g ρf ρf

(4.19a) (4.19b)

Similarly, we also have the two following special cases. Same fluid ðAÞ ðLÞ ðAÞ For an interface inside the same fluid, the mass density ρðLÞ f 5 ρf and the second-order time derivative t p;tt 5 p;tt , and this equation reduces to ðLÞ ðAÞ ðAÞ pðLÞ ;i ν i 1 p;i ν i 5 0;

(4.19c)

which physically that represents the normal acceleration on the interface in the same fluid is consistent. Free surface of water For the atmosphere, the dynamic pressure pðAÞ 5 0, defined for the static equilibrium state, so that Eq. (4.19a) is no longer needed and Eq. (4.19b) is reduced to ðLÞ pðLÞ ;i ν i 1

ðLÞ pðLÞ ;tt ν 3 5 0; g

(4.19d)

which is the condition on the free surface in pressure form.

4.2.5

Conditions of surface
tension interactions

4.2.5.1 Synthesized equations on surface tension interface If the surface tension is considered, the pressure boundary condition Eq. (3.131c) must be satisfied. Replacing the dynamic pressure p in Eq. (4.6) by the pressure difference p 2 p0 in Eq. (3.131c), and using Eq. (3.131d), we obtain gh 2

nlg χðκ1 1 κ2 Þ nlg χ 1 φ;t 5 gh 2 h;II 1 φ;t 5 0; ρf ρf

h;II 5

2 X @2 h @2 h @ 2 h 5 2 1 2: @xI @xI @x1 @x2 I51

(4.20)

Here, a summation convention for a two-dimensional case with subindex I 5 1; 2 has been introduced, which will be used in the following derivations.

Variational principles of linear fluid
solid interaction systems Chapter | 4

113

Velocity potential form Taking the time derivative of Eq. (4.20) and noting for linearized approximation, h;t 5 v3 5 φ;3 5 φ;i ν i ;

(4.21a)

we obtain gϕ;i ν i 2

nlg χ φ ν i 1 φ;tt 5 0: ρf ;IIi

For incompressible fluids, φ;jj 5 0; so that Eq. (4.21b) reduces to   nlg χ @2 φ gφ;i ν i 1 ν i 1 φ;tt 5 0: ρf @x23 ;i

(4.21b)

(4.21c)

Pressure form In a process similar to Eq. (4.8) to derive Eqs. (4.11a) and (4.11b), we can derive the pressure forms of Eqs. (4.21b) and (4.21,c) as follows: nlg χ gp;i ν i 2 p;IIi ν i 1 p;tt 5 0; (4.22a) ρf for compressible fluids, and   nlg χ @2 p gp;i ν i 1 ν i 1 p;tt 5 0; ρf @x23 ;i

(4.22b)

for incompressible fluids. Surface membrane acceleration form Using Eqs. (4.9a) and (4.9b) to replace the pressure gradient p;j in the middle term involving the surface tension in Eqs. (4.22a) and (4.22,b) by the surface acceleration wj , we respectively obtain the corresponding equations on the surface in the acceleration forms: gp;i ν i 1 nlg χwi;II ν i 1 p;tt 5 0;

(4.23a)

and  gp;i ν i 2 nlg χ

 @2 w i ν i 1 p;tt 5 0: @x23

(4.23b)

4.2.5.2 Surface tension interaction on air
liquid interface We assume that the surface tension on the air
liquid interface ΓðLAÞ is not neglected, so the interaction conditions derived in Section 4.2.4 have to be modified to include the effect of the surface tension. Due to the single surface tension on the air
liquid interface, nlg 5 1: Due to the effect of surface tension, the pressures on the two sides of the interface are different, and the velocity potentials may also be different. However, the equilibrium equation Eq. (4.21a) including the surface tension effect is valid for both the liquid and the air domains. We imagine a middle layer in the tension surface to separate the liquid and the gas and only half the tension to be balanced by the pressure from the liquid and gas, respectively, so that we have ðLÞ φðLÞ ;i ν i 2 ðAÞ ϕðAÞ ;i ν i 2

χ 2ρðLÞ f g χ 2ρðAÞ f g

ðLÞ φðLÞ ;IIi ν i 1

1 ðLÞ ðLÞ φ ν 5 0; g ;tt 3

(4.24)

ðAÞ φðAÞ ;IIi ν i 1

1 ðAÞ ðAÞ φ ν 5 0; g ;tt 3

(4.25)

which provides the two dynamic equilibrium conditions in the form of potential of velocity on the surface
tension interface.

114

Fluid
Solid Interaction Dynamics

x3

h( x1 , x2 , t ) l

x1

x2 Γ

nI

dx1dx2

FIGURE 4.5 Integration approximation of tension
surface energy.

In process similar to Eq. (4.8) to derive Eqs. (4.11a) and (4.11b), we can transfer Eqs. (4.24) and (4.25) into their pressure forms: 1 ðLÞ ðLÞ χ ðLÞ ðLÞ pðLÞ pðLÞ ;i ν i 1 p;tt ν 3 2 ;IIi ν i 5 0; g 2ρðLÞ g f ðAÞ pðAÞ ;i ν i 1

1 ðAÞ ðAÞ χ ðAÞ p ν 2 ðAÞ pðAÞ ;IIi ν i 5 0: g ;tt 3 2ρf g

(4.26)

4.2.5.3 Energy integration of surface tension In order to include the surface tension in the functionals, it is necessary to determine its energy form. In Fig. 4.5, Γ of its boundary l denotes a surface located in the horizontal plane in the static equilibrium state, and the outside unit normal of the boundary l is represented by nI ; ðI 5 1; 2Þ. Due to a disturbance, this surface floats to a height h, which has the potential energy (Soedel, 1981; Morand and Ohayon, 1995; Ibrahim, 2005a,b). ð 1 H½h 5 χh;I h;I dΓ: (4.27) Γ2 This is a functional of the surface height, so that we have to transfer it to either the velocity potential or the pressure as follows. Velocity
potential form Eq. (4.10b) can be used to replace the floating height h in Eq. (4.27) by the velocity potential φ, so that we obtain the functional ð t2 ð 1 Hχ ½φ 5 dt χφ;It φ;It dx1 dx2 : (4.28a) 2 2g t1 Γ The variation of this functional is derived as ð ð χ t2 δHχ ½φ 5 2 dt δφ;It φ;It dΓ g t1 Γ ð ð χ t2 5 2 dt ½ðδφ;t φ;It Þ;I 2 δφ;t φ;IIt dΓ g t1 Γ ð ð" ð ð χ t2 χ t2 5 2 dt ðδφ;t φ;It ÞnI dl 2 2 dt δφ;t φ;IIt dΓ g t1 l g t1 Γ ð t2 ð χ 5 2 2 dt ½ðδφ φ;IIt Þ;t 2 δφ φ;IItt dΓ g t1 Γ # ð ð 2 χ t2 5 dt δφ φ;II3 dΓ: g t1 Γ

(4.28b)

Here, in the mathematical derivation process, we have used the Green theorem in 2D case to replace the surface integration on Γ to a curve-line integration along the boundary l, as shown in Fig. 4.5. We also neglect the surface tension between the fluid and the solid and require φ;It 5 0 to be along the boundary l, which physically implies that the

Variational principles of linear fluid
solid interaction systems Chapter | 4

115

time change rates of the slope of the surface of the velocity
potential function along the two horizontal directions xI vanish. Further, the variational constraints δφðt1 Þ 5 0 5 δφðt2 Þ are added. Pressure form For the functional of the pressure
acceleration form, the acceleration energy of dimension N m=s2 is involved. Therefore we take the second-order time derivative of Eqs. (4.28a) and (4.28b) to obtain ð @2 @ χ H ½φ 5 φ φ dΓ χ @t2 @t Γ g2 ;It ;Itt ð χ 5 ðφ;Itt φ;Itt 1 φ;It φ;Ittt ÞdΓ (4.29) 2 Γg ð χ 5 φ φ dΓ; 2 ;Itt ;Itt Γg in which φ;Ittt 5 2 gφ;3tI 5 2 gw3;I 5 0 is used according to Eqs. (4.9b) and (4.11a). Using Eq. (4.8) to replace the velocity potential by the pressure and multiplying 2 times the time integration for Eq. (4.29), we can construct a functional of surface tension in the pressure form ð t2 ð χ Πχ ½p 5 dt p p dΓ; (4.30a) 2 ρ2 ;It ;It 2g t1 Γ f which has the dimension N m/s, the power of work. As with the assumptions to derive Eq. (4.28b), we neglect the surface tension between the fluid and the solid and require p;It 5 0 along the boundary l, which physically implies that the time change rates of the slope of the surface of pressure function along the two horizontal directions xI vanish, as well as the variational constraints. We add δpðt1 Þ 5 0 5 δpðt2 Þ. In these assumptions for linear approximations, the variation of the functional in Eq. (4.30a) is derived as ð t2 ð χ δΠχ ½p 5 dt δp;It p;It dΓ 2 ρ2 g t1 Γ f ð ð χ t2 5 2 2 dt ½ðδp;t p;It Þ;I 2 δp;t p;IIt dΓ g ρf t1 Γ ð ð ð ð χ t2 χ t2 5 2 2 dt ðδp;t p;It ÞnI dl 2 2 2 dt δp;t p;IIt dΓ (4.30b) g ρf t1 l g ρf t1 Γ ð ð χ t2 5 2 2 dt ½ðδpp;IIt Þ;t 2 δpp;IItt dΓ g ρf t1 Γ ð ð χ t2 5 2 dt δpp;II3 dΓ: gρf t1 Γ The expressions of the functionals involving the surface tension energy, given in Eqs. (4.28a) and (4.30a), provide a base to introduce the surface tension effect into the variational formulations developed by Xing (1988), as discussed in the two Sections 4.3 and 4.4.

4.2.6

Boundary conditions on infinity and moving structures

4.2.6.1 Conditions on infinity boundaries Γ6 N For the transient dynamic analysis of marine structures, an “infinite” water domain may be involved. It is more difficult to model infinite domain using a FE approach, although some researchers have employed infinite elements. In this numerical modeling, a water boundary sufficiently far from structures is considered a boundary of the problem. For the pressure model, on this boundary, we can consider that the dynamic pressure vanishes ðp^ 5 0Þ or that the dynamic pressure gradient ðw^ i 5 0Þ vanishes. For the velocity
potential model, on this boundary, we can consider that the velocity potential vanishes ðφ^ 5 0Þ or that the gradient of velocity potential, that is, the fluid velocity ðv^i 5 0Þ, vanishes.

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Fluid
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These boundary conditions are adopted based on the following physical facts: (1) due to the damping of the system, any dynamic disturbance of water caused by vibrations of a structure cannot be transmitted to infinity; (2) in transient dynamic analysis, such as impact analysis, the important dynamic loads happen during a very short, finite time period in which the disturbance can travel only a finite distance. Therefore in practical cases, there exists a finite boundary where the pressure and velocity of the water are undisturbed in the time period of interest. To determine a suitably far boundary, the multiplication of the speed of sound in the fluid within the given time period provides a reference distance. The chosen boundary may be 2
3 times the reference distance.

4.2.6.2 Boundary condition caused by a moving structure To analyze the transient dynamic responses of a moving marine structure, such as a ship, we are not interested in the dynamics of the steady-state motion of the structure. As discussed in detail by Newman (1978), in the steady-state motion of ships, a ship is considered a rigid body and moves with a mean constant forward velocity V^ in calm water, which produces a steady flow field. The solution of the steady-state problem is itself of interest, particularly with regard to the calculation of wave resistance in calm water, and it has been well investigated (Newman, 1978). We are mainly concerned with extra dynamic forces caused by unsteady disturbances added to the steady-state motion. We assume that the velocity of the structure is constant. To investigate the unsteady dynamics added into the steady motion, it is convenient to use the moving system that, due to the constant velocity of the structure, is an inertial system and therefore all the governing equations given in this section are valid. Under this moving inertial system, for steady-state motion, the rigid motion of the ship relative to the moving system vanishes, and the relative velocity of the water on ^ the negative ship velocity, which determines the steady-state motion of the undisturbed infinite boundary equals 2 V, the ship-water system. The dynamic motions of interest caused by the extra disturbance relative to the steady-state can be solved in a manner of the static ship in linear analysis for which the superposition principle is valid.

4.3

A complementary energy model: pressure
acceleration form

Now we discuss a new version of variational formulation in a complementary energy model, that is, the pressure
acceleration form for linear structure
compressible fluid dynamic interactions based on a linearized theory. The new defined boundary conditions include interfaces between two different fluids, with FSI floating boundaries and boundaries with surface
tension effect considered. The fluid is treated as an ideal compressible fluid with its flow irrotational, so that its behavior is governed by an acoustic equation in the pressure form under the Euler coordinate system. The structures are considered a linear elastic body satisfying the equations for linear elasticity in the Lagrange coordinate system. The theory described in this section can be used to deal with the problems involving pressure waves in fluids, for example, air
structure interactions and structure
water interaction systems subject to a pressure wave caused by an earthquake or explosion in the water. Based on the principles in continuum mechanics, as well as the conditions derived in Section 4.2, the governing equations for generalized FSI systems can be given as follows.

4.3.1

Governing equations

Using the notations and the assumptions shown in Figs. 4.1
4.3 and explained in Section 4.2, we can list the governing equations of the FSI systems herein. For a generalized theory, we imagine an FSI system in which there exists a nonfloating FSI boundary Σ, a floating FSI boundary Σf , a free surface Γf , and a liquid
air floating interface ΓðLAÞ , with surface tension considered for the last two interfaces Γf and ΓðLAÞ .

4.3.1.1 Solid structure Dynamic equation σij;j 1 F^ i 5 ρs Wi ;

ðXi ; tÞAΩs 3 ðt1 ; t2 Þ

(4.31)

Strain-displacement Eij 5

1 ðUi;j 1 Uj;i Þ; 2

ðXi ; tÞAΩs 3 ðt1 ; t2 Þ

(4.32)

Variational principles of linear fluid
solid interaction systems Chapter | 4

117

Constitutive equation σij 5 Cijkl Ekl ;

ðXi ; tÞAΩs 3 ðt1 ; t2 Þ

(4.33)

and we have Vi 5 Ui;t ;

Wi 5 Vi;t ;

Vij 5 Eij;t 5

1 ðVi;j 1 Vj;i Þ: 2

(4.34)

Boundary conditions Acceleration: Reaction:

Wi 5 W^ i ; σij ηj 5 T^ i ;

ðXi ; tÞASw 3 ½t1 ; t2 

(4.35)

ðXi ; tÞAST 3 ½t1 ; t2 

(4.36)

4.3.1.2 Fluid subdomains ðβ 5 L or AÞ We use the superindex β to distinguish the liquid ðβ 5 LÞ and the air ðβ 5 AÞ. Therefore the following equations with the superindex β are valid for both the liquid and the air. Dynamic equation 2 ðβÞ ðβÞ pðβÞ ;tt 5 ðc Þ p;ii ;

ðxi ; tÞAΩðβÞ f 3 ðt1 ; t2 Þ

(4.37)

Boundary conditions On the prescribed pressure boundary Γp , the dynamic pressure of the fluid equals the prescribed pressure, that is, pðβÞ 5 p^ðβÞ ;

ðxi ; tÞAΓðβÞ p 3 ½t1 ; t2 :

(4.38)

On the prescribed acceleration boundary Γw , the normal acceleration should satisfy the following condition: ðβÞ ðβÞ ðβÞ ðβÞ ^ i νi ; pðβÞ ;i ν i 5 2 ρf w

ðxi ; tÞAΓðβÞ w 3 ½t1 ; t2 :

(4.39)

4.3.1.3 Conditions on interaction interfaces Fluid
solid interaction interface Σ On the FSI interfaces discussed in Section 4.2.2, the gravity potential effect may need to be considered depending the angle α between the gravity vector gi and the outside unit normal vector ν i of the fluid shown in Fig. 4.4. As presented in Eq. (4.16c), with the different values of angle α, the corresponding FSI boundary could be a full or part floating boundary or a nonfloating one where no gravity effect exists. The defined gravitational acceleration vector in Eq. (4.16c), that is,  T gi 5 2 gδi3 ; g 5 0 0 2g ; (4.40) enables these difference cases to be represented in one equation. The boundary conditions in Eqs. (4.13a), (4.13b), (4.16a)
(4.16d), as well as the general case shown in Fig. 4.4, can be denoted in the following forms: Kinematic:

W i ηi 1 gi ν i 5

p;i ν i ; ρf

ðxi ; tÞA

X

3 ½t1 ; t2 :

(4.41a)

Equilibrium: Normal traction: ηi σij ηj 5 2 p: XðαÞ Tangent force: τ i σij ηj 5 0; ðxi ; tÞA 3 ½t1 ; t2 :

(4.41b) (4.41c)

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Fluid
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As shown in Fig. 4.5 and discussed for Eqs. (4.16c) and (4.16,d), we have gi ν i 5 g cos α 5 2 gδi3 ν i 5 2 gν 3 ;

(4.41d)

for which ν 3 5 1ðα 5 πÞ, implying that the FSI interface is the surface, a full  floating FSI boundary shown in Fig. 4.2 with the boundary condition given by Eq. (4.16a), while ν 3 5 0 α 5 6 π=2 corresponds to the nonfloating FSI boundary Σ shown in Fig. 4.1 with the boundary condition in Eq. (4.13a). The generalized equation in Eq. (4.41a) also includes the partial effect of gravity in the case of 0 , jν 3 j , 1 when the FSI interfaces are not in the vertical and horizontal planes but in an inclined plane, such as shown in Fig. 4.4. Gas
liquid interaction interfaces ΓðLAÞ f

We define ΓðLAÞ 5 Γf , ΓðLAÞ and a parameter λ to distinguish the free surface and the air
liquid interface. From f Eq. (4.26), it follows ðLÞ pðLÞ ;i ν i 1

1 ðLÞ ðLÞ χ ðLÞ p ν 2 ðLÞ pðLÞ ;IIi ν i 5 0; g ;tt 3 λρf g

ðAÞ pðAÞ ;i ν i 1

1 ðAÞ ðAÞ χ ðAÞ p;tt ν 3 2 ðAÞ pðAÞ ;IIi ν i 5 0; g λρf g


λ5

1; 2;

xi AΓf ; xi AΓðLAÞ ;

(4.42a)

ðxi ; tÞAΓðLAÞ 3 ½t1 ; t2 : f

ðAÞ The notations ν ðLÞ 3 and ν 3 denote the x3 component of the unit outward normal vector on the interface of the fluid domain and the gas domain, respectively. As mentioned, due to the gravity, the interfaces between two different fluids must be in the horizontal plane in the static equilibrium state. The heavier fluid is located beneath this interface. Therefore using the chosen coordinate system shown in Figs. 4.1
4.3, for the heavier or lighter fluid, we have that ðlightÞ ðLAÞ ν ðheavyÞ 5 ν ðLÞ 5 ν ðAÞ 5 Γf , the dynamic 3 3 5 1 and ν 3 3 5 2 1. As a special case, if this interface is a free surface Γf ðAÞ pressure p 5 p0 5 0, and Eq. (4.42a) reduces to the one for the free surface

1 ðLÞ ðLÞ χ ðLÞ ðLÞ ðLÞ pðLÞ ;i ν i 1 p;tt ν 3 2 ðLÞ p;IIi ν i 5 0; g ρf g

ðxi ; tÞAΓf 3 ½t1 ; t2 ;

(4.42b)

therefore Eq. (4.42a) covers two cases. Conditions on infinity boundaries Γ6 N As discussed in Section 4.7.1, we treat the infinite boundary as the undisturbed boundary on which the dynamic pressure vanishes ðp^ 5 0Þ, which is formulated by Eq. (4.38) or the dynamic pressure gradient ðw^ i 5 0Þ vanishing in Eq. (4.39). Therefore these infinite boundaries as special cases are included in Eqs. (4.38) and (4.39).

4.3.2

Variational formulation

The variational principles (Xing, 1988; Xing and Price, 1991; Xing et al., 2009a,b) are further modified to include the interactions between two different fluids and surface tension effects. In this functional, the dynamic pressure p in the fluid and the acceleration Wi 5 Ui;tt in the solid are chosen as the arguments, providing a foundation to establish a mixed pressure
displacement FE model for linear FSI problems. We find that the following variational formulation can be derived, that is, Πsf ½p; Wi  5 Πs ½Wi  1 Πf ½p 1 Πint ½p; Wi ; where Πs ½Wi 

(4.43a)

8 0 9 1 ð t2 > ; n 5 1; > > > 2κ > > > > > < j ð1Þ (8.120h) GðxÞ 5 4 H0 ðκjxjÞ; n 5 2; > > > > > > > ejκjxj > > n 5 3: > : 4πjxj ; Here, H0ð1Þ is the first Hankel function of order 0; see Karman and Biot (1940), in which Eq. (8.120g) in the spherical coordinate system becomes a Bessel equation and its solution method is given. For an inhomogeneous Helmholtz equation r2 PðxÞ 1 κ2 PðxÞ 5 2 f ðxÞ; its solution can be obtained by a convolution PðxÞ 5 ðG 3 f ÞðxÞ 5

ð Ωf

Gðx; x Þf ðx Þdx :

(8.121a)

(8.121b)

Here, Gðx; x Þ is the Green function with a source point at x , which can be obtained through replacing x by x 2 x in Eq. (8.120h).

8.4.2

Green identity

For the Helmholtz equation, the Green identity in Eq. (8.16a) is also valid, which can be demonstrated as follows. For the source point at x , Eq. (8.120g) becomes r2 Gðx; x Þ 1 κ2 Gðx; x Þ 5 2 Δðx 2 x Þ:

(8.122a)

Premultiplying a function P satisfying Eq. (8.120f) and then integrating the resultant equation over the fluid domain Ωf , we obtain

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Fluid
Solid Interaction Dynamics

ð Ωf

  PG; ii ðx; x Þ 1 κ2 PGðx; x Þ dΩðxÞ 5 2

ð Ωf

PΔðx 2 x ÞdΩðxÞ 5 2 Pðx Þ:

Using the Green theorem, we obtain ð ð ð   PG; ii ðx; x Þ dΩðxÞ 5 2 P; i G; i dΩðxÞ 1 PG; i ðx; x Þν i dΓ ðxÞ; Ωf

Ωf

Σ

(8.122b)

(8.122c)

where the boundary condition on the infinity boundary in Eq. (8.120g) has been introduced. Combining Eqs. (8.122b) and (8.122c), we have ð ð ð κ2 PGdΩ 2 P; i G; i dΩ 1 PG; i ν i dΓ 5 2 Pðx Þ: (8.122d) Ωf

Ωf

Σ

Similarly, premultiplying the first equation in Eq. (8.120f) by the Green function Gðx; x Þ and then integrating the resultant equation over the fluid domain Ωf , we obtain ð ð ð 2 κ PGdΩ 2 P; i G; i dΩ 1 P; i Gν i dΓ 5 0; (8.122e) Ωf

Ωf

Σ

from which, when Eq. (8.122d) is subtracted, the Green identity based on the Helmholtz equation follows, that is, ð ½P; i Gðx; x Þ 2 PG; i ðx; x Þν i dΓ 5 Pðx Þ: (8.122f) Σ

8.4.3

Acoustic radiation in infinite fluid domain

Fig. 8.27 shows a structure occupying a domain Ωs with an outside unit normal η on its boundary Σ, submerged in an infinite fluid domain Ωf of outside unit normal ν 5 2η on Σ. A Cartesian coordinate system O 2 x1 x2 x3 is fixed at the mass center O of the body. The structure is considered a linear isotropic elastic body of mass density ρs and elastic constant tensor Cijkl , and the outside fluid is assumed to be compressible and inviscid, of mass density ρf and the speed of sound c. A harmonic force T^ i 5 T~ i e2jωt is applied at a point x^i of the body, causing the vibrations of the structure. This vibration through the FSI interface Σ transfers into the outside fluid and radiates into its infinity. We will use the Green function in Eq. (8.120h) to solve this radiation problem.

8.4.3.1 Governing equations of the problem The governing equations of this FSI radiation problem are as follows. For the solid structure, its linear dynamic equations are given in Section 4.3.1.1, from which the displacement equation can be derived in the form
 (8.123) Cijkl uk;lj 1 T^ i Δ xj 2 x^j 5 ρs ui; tt : The motion of the fluid satisfies the linear wave equation in terms of dynamic pressure p shown in Eq. (4.37) with its infinity boundary condition, that is,

FIGURE 8.27 Acoustic radiation of a body submerged in an infinite fluid domain.

Mixed finite element
boundary element model for linear water
structure interactions Chapter | 8

p; ii 5

1 p; tt ; c2

p is an outgoing wave; when x-N:

323

(8.124)

On the FSI interface Σ, the kinematic and equilibrium conditions in Eq. (4.40) are required. When neglecting the gravity effect, these are ui; tt ν i 5 2

1 p; i ν i ; ρf

ν i Cijkl uk;l ν j 5 2p;

τ i Cijkl uk; l ν j 5 0:

(8.125)

Here, vi and τ i, respectively, denote the unit normal vector and tangent vector on the fluid boundary pointing from inside to outside the fluid domain. The delta function Δðxj 2 x^j Þ is used to model the excitation force T^ i at point x^j in the solid body. Since the system is linear, the dynamic response of the system will be a harmonic in the same frequency as that of the excitation force. As a result, we can assume that ui 5 Ui ðxÞe2jωt ;

p 5 PðxÞe2jωt ;

(8.126a)

from which, when this is substituted into Eqs. (8.121)
(8.123), it follows Cijkl Uk;lj 1 T~ i Δðx 2 x^ Þ 5 2ρs ω2 Ui ;

xAΩs ;

(8.126b)

2ω2 P 5 2 κ2 P; xAΩf ; c2 1 ω2 Ui ν i 5 P; i ν i ; ν i Cijkl Uk;l ν j 5 2P; ρf τ i Cijkl Uk; l ν j 5 0; xAΣ: P; ii 5

(8.126c)

(8.126d)

8.4.3.2 Solution approach Mode summation for solid structure Based on the mode summation approach, we assume that the displacement of the solid is represented by Ui 5 ΦðiÞ q;  q 5 q1 q2

? qN

T

;

 ΦðiÞ 5 φ1

φ2

? φN

ðiÞ

;

(8.127a)

where ΦðiÞ denotes the matrix of the solid mode functions, in which the element φðiÞ I representing the Ith mode of the structure, which satisfies the following orthogonal relationships: ð
2 ðkÞ 2 ^I ; ΦðiÞT ; j Cijkl Φ;l dΩ 5 Λ 5 diag ω ð

Ωs

Ωs

(8.127b) ΦðiÞT ρs ΦðiÞ dΩ 5 IN 3 N :

Here, ω^ I is the Ith natural frequency of the structure, which may include the buoyance stiffness effect, and IN 3 N is a unit matrix of order N equaling the number of retaining modes of structure. Using the mode transformation, we can transfer Eqs. (8.126b) and (8.126d) into the corresponding mode form. Substituting Eq. (8.127a) into Eq. (8.126b), we obtain ~ ^ Þ 5 2ρs ω2 ΦðiÞ q; Cijkl ΦðkÞ ;lj q 1 T i Δðx 2 x

xAΩs :

(8.128a)

Premultiplying Eq. (8.128a) by ΦðiÞT and then integrating the resultant equation as well as using the Green theorem, Eq. (8.126d), and the orthogonal relationships in Eq. (8.127b), we have ðΛ2 2 ω2 IÞq 2 F~ 5 2P; ð ΦðiÞT ν i PdΣ 5 P; F~ 5 T^ i ΦðiÞT ð^xÞ; Σ

and the first equation in Eq. (8.126d) becomes

(8.128b)

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Fluid
Solid Interaction Dynamics

ω2 ΦðiÞ ν i q 5

1 P; i ν i : ρf

(8.128c)

Mode functions of fluid corresponding the Ith solid mode

We assume that, for a mode function φðiÞ I of a solid structure, there is a corresponding mode function ψI of the fluid pressure, which can be obtained by means of the Green identity in Eq. (8.122f). Physically, this mode function of the fluid is the one produced by the solid mode motion on the FSI interface Σ, the vibration sources of the Green function. Therefore, substituting the boundary conditions of Eq. (8.126d) into Eq. (8.122f), we can obtain the pressure mode function ψI as follows:  ð

ðkÞ   (8.129a) ψ I ð x Þ 5 ðxÞν G ð x; x Þ 1 ν C φ ðxÞν G ð x; x Þν ρf ω2 φðiÞ i r rjkl I;l j ;i i dΣðxÞ: I Σ

Define the fluid mode matrix as

 Ψ 5 ψ1

ψ2

?

 ψN ;

which, when further combining with Eq. (8.129a), gives ð h i   ~  Þ 1 Ψðx Þ; Ψðx Þ 5 ρf ω2 ΦðiÞ ðxÞν i Gðx; x Þ 1 ν r Crjkl ΦðkÞ ðxÞν G ðx; x Þν dΣðxÞ 5 ω2 Ψðx j ; i i ;l Σ ð ~ Þ 5 Ψ ~ R ðx Þ 1 jΨ ~ I ðx Þ 5 ρ ΦðiÞ ðxÞν i Gðx; x ÞdΣðxÞ; Ψðx f ðΣ h i R I  Ψðx Þ 5 Ψ ðx Þ 1 jΨ ðx Þ 5 ν r Crjkl ΦðkÞ ;l ðxÞν j G; i ðx; x Þν i dΣðxÞ:

(8.129b)

(8.129c)

Σ

Due to the complex Green function in Eq. (8.120h), the fluid mode matrices in Eq. (8.129c) are complex, of which the real and imaginary parts are identified by the up-indexes R and I, respectively. When x -x on the boundary, which is assumed to be smooth, the surface integration should exclude the point x, so that Eq. (8.129c) becomes ~  Þ 1 Ψðx Þ; cðx ÞΨðx Þ 5 ω2 Ψðx

(8.129d)



where, cðx Þ is same as given in Eq. (8.17b). Mode summation of the pressure and fluid
solid interaction equation The complex fluid pressure field can now be represented by a mode summation in the form   ~  Þ 1 Ψðx Þ q; x in Ωf ; Pðx Þ 5 ω2 Ψðx :   ~  Þ 1 Ψðx Þ q; x on Σ: Pðx Þ 5 c21 ðx Þ ω2 Ψðx

(8.129e)

Substituting the pressure on the boundary into Eq. (8.128b), we obtain the pressure vector P 5 ω2 mq 1 kq;

ð

m 5 m 1 jm 5 R

I

ð k 5 kR 1 jkI 5

Σ

Σ

~ ΦðiÞT ν i c21 ðxÞΨðxÞdΣ;

(8.129e)

ΦðiÞT ν i c21 ðxÞΨðxÞdΣ;

and the dynamic equation of the FSI system in the form ~ ½ðΛ2 2 ω2 IÞ 1 ðkR 1 ω2 mR Þ 1 jðkI 1 ω2 mI Þq 5 F:

(8.129f)

In this equation, ðkR 1 ω2 mR Þ denotes the added mass/stiffness to the solid from the fluid, and the imaginary part involves the added damping from the acoustic radiation, which loses the vibration energy. The solution q can be

Mixed finite element
boundary element model for linear water
structure interactions Chapter | 8

325

obtained from Eq. (8.129f); the displacement field of the solid and the pressure field of the fluid can be obtained from Eqs. (8.127a) and (8.129e), respectively. It should be noted that if the theoretical mode functions are not available and we have to use the numerical modes derived by FE analysis, the integrations involved in the method must be numerically completed.

8.4.3.3 Example 8.5: Acoustic radiation of an underwater small ball Here, we discuss an example shown in Fig. 8.28, where an underwater small ball of radius r0 is moored at the deep seabed and subjected to a radial excitation force fe2jωt uniformly applied on the spherical surface of radius r 5 r^ # r0 . Neglecting the effect of the mooring cable, we wish to estimate the acoustic pressure caused by its central symmetrical radial vibration in the mode φðrÞ 5 r. Vibration of elastic ball The components of the mode φðrÞ and the excitation force in the coordinate O 2 x1 x2 x3 are φðiÞ 5 φðrÞ

xi 5 xi ; r

φðiÞ ; j 5 δij ;

T~ i 5 fxi r^;

νi 5 2

xi : r

(8.130a)

Assume that the displacement of the ball and the pressure in the fluid are expressed as ui 5 qφðiÞ e2jωt ;

Ui 5 qφðiÞ ;

p 5 qψe2jωt ;

(8.130b)

from which, when substituted into Eqs. (8.126a)
(8.126d), it follows Cijkl φðkÞ lj q 1 f x^i r^Δðx 2 x^i Þ 5 2ρs ω2 qφðiÞ ;

xAΩs :

(8.130c)

Premultiplying this equation by ϕðiÞ and integrating over the Ωs , as well as using the Green theorem, we obtain ð ~ P~ 5 φðiÞ ν i PdΣ; ^ 2 P; ðK 2 ω2 MÞq 5 f ϕðrÞ Σ ð ð ð r0 4πρS r05 M 5 ρS φðiÞ φðiÞ dΩ 5 ρS xi xi dΩ 5 4ρS πr 4 dr 5 ; 5 Ωs Ωs 0 ð ð (8.130d) ðiÞ ðkÞ K 5 φ; j Cijkl φ;l dΩ 5 δij Cijkl δkl dΩ Ω Ωs ð r0s 4πð9λ 1 4μÞr03 5 4πr 2 ð9λ 1 4μÞdr 5 ; 3 0 where Eq. (3.52) for isotropic material has been introduced to replace the tensor of elastic constant Cijll .

FIGURE 8.28 Underwater structure moored in the infinite deep sea.

326

Fluid
Solid Interaction Dynamics

Pressure mode The pressure mode corresponding to the solid mode φðiÞ can be obtained by using Eq. (8.129c), that is,  ð

ðkÞ  2 ðiÞ   ψðx Þ 5 ρf ω φ ðxÞν i Gðx; x Þ 1 ν r Crjkl φ;l ðxÞν j G; i ðx; x Þν i dΣðxÞ:

(8.131a)

Σ

Considering that the ball is so much smaller than the wide deep sea, we can approximately imagine the ball being a point with its FSI boundary Σ as a small spherical surface Γ ε about point O in order to complete the surface integration in preceding equation, so that we obtain ð h i ψðx Þ 5 ρf ω2 rGðε; x Þ 1 ð3λ 1 2μÞG;r ðε; x Þ dΓ Γε 2 3 (8.131b) 2 jκjx 2εj jκjx 2εj jκjx 2εj 4πε e jκe e 4 54πε2 ; 5 ρf ω2 ε 1 ð3λ 1 2μÞ 1 4πjx 2 εj 4πjx 2 εj 4πjx 2εj2 from which, when ε-0, it follows 

ψðx Þ 5 ð3λ 1 2μÞ

ejκjx j : jx j2

(8.131c)

Here, in the same way as to derive Eq. (8.130d), Eq. (3.52) has been used, that is, ν r Crjkl φðkÞ ;l ν j 5 Cijkl

xi xj xi xj δkl 5 Cijll 2 r2 r

5 ½λδij δll 1 μðδil δjl 1 δil δjl Þ 5 ð3λ 1 2μÞδij

xi xj r2

(8.131d)

xi xj xj xj 5 ð3λ 1 2μÞ 2 5 ð3λ 1 2μÞ: r2 r

This equation gives the fluid pressure mode corresponding to the displacement mode in Eq. (8.130a). The pressure field in the fluid is given by pðr; tÞ 5 ψðrÞqe2jωt 5 ð3λ 1 2μÞq

ejðκr2ωtÞ ; r2

ejκr PðrÞ 5 ð3λ 1 2μÞq 2 ; r

(8.132)

in which we have replaced the distance jx j from a field point x to the origin O of the coordinate system by r.

Fluid
solid interaction radiation equation Now, substituting the pressure mode in Eq. (8.132) into the first equation in Eq. (8.130d), we obtain ^ ½ðK 2 ω2 MÞ 1 ðkR 1 jkI Þq 5 f ϕð^rÞ 5 f r; ð jκr e P~ 5 rð3λ 1 2μÞq 2 dΣ 5 ðkR 1 jkI Þq; r Σ

(8.133)

kR 5 4πð3λ 1 2μÞr0 cosðκr0 Þ; kI 5 4πð3λ 1 2μÞr0 sinðκr0 Þ: From this equation, the generalized coordinate q can be solved, and then the dynamic displacement and the radiation fluid pressure can be obtained by Eq. (8.130b).

Mixed finite element
boundary element model for linear water
structure interactions Chapter | 8

327

FIGURE 8.29 Acoustic radiation problem of a structure in quarter water and subjected to a harmonic force excitation.

8.4.4

Generalized acoustic radiation problems

We investigate the generalized acoustic radiation problem described in Fig. 8.29, in which a structure with its mass center oð1Þ at point oð1Þ ðxð1Þ oi Þ in a Cartesian coordinate system O 2 x1 x2 x3 is in infinite quarter deep water and is ð1Þ ð1Þ subjected to an excitation force Ti 5 T^ i e2jωt at point x^ð1Þ in the system oð1Þ 2 xð1Þ j 1 x2 x2 . We intend to solve this FSI problem to get the radiation water pressure in the water.

8.4.4.1 Governing equations ð1Þ ð1Þ In the system oð1Þ 2 xð1Þ 1 x2 x2 , the displacement of the structure satisfies the equation

ð1Þ Cijkl uk;lj 1 Ti Δ xð1Þ 5 ρs ui; tt ; xð1Þ j 2 x^j i AΩs ;

(8.134)

and the dynamic pressure of the water is governed by a wave equation p; ii 5

1 p; tt ; c2

xð1Þ i AΩf :

(8.135)

On the fixed water boundary Γ b , the gradient of pressure vanishes, that is, p; i ν i 5 0;

; xð1Þ i AΓ b

(8.136a)

and on the free surface Γ f , there are two cases: considering or not considering free surface wave disturbance, for which the boundary conditions are No surface wave: Surface wave:

p 5 0; p; i ν i 5 2

1 @2 p : g @t2

(8.136b)

For the infinite boundary Γ N , the Sommerfeld condition is required, implying a wave with finite amplitude extending to infinity. On the FSI interface Σ, the kinematic and equilibrium conditions in Eq. (4.40) are required. When neglecting the gravity effect, these are ui; tt ν i 5 2

1 p; i ν i ; ρf

ν i Cijkl uk;l ν j 5 2p;

τ i Cijkl uk;l ν j 5 0:

(8.137)

8.4.4.2 Image method We have learned in Section 8.4.3 that the radiation problem in an infinite water domain can be solved using the Green function. Therefore, in the infinite domain, the solution of the structure (J 5 1) in Fig. 8.29 can be obtained as

328

Fluid
Solid Interaction Dynamics

FIGURE 8.30 Imaged system in infinite water from the quarter water radiation problem in Fig. 8.29.

ðJÞ uðJÞ xðJÞ e2jωt ; i 5 Ui j

pðJÞ 5 PðJÞ xðJÞ e2jωt ; j

(8.138a)

PðJÞ 5 ΨðxðJÞ ÞqðJÞ :

(8.138b)

in which UiðJÞ 5 Φi ðxðJÞ ÞqðJÞ ;

It is more important to know that the mode function matrix Φi is dimensionless and depends only on its geometrical and physical parameters, as well as the local coordinate arrangement, but it does not involve any external forces, so that ðJÞ ðJÞ for the four bodies J in its local system oðJÞ 2 xðJÞ 1 x2 x2 , (J 5 1, 2, 3, 4), shown in Fig. 8.30, the function form of Φi is the same. The corresponding pressure mode function matrix Ψ is obtained, using the Green identity in Eq. (8.129c), by means of the solid mode function matrix and the Green function in the infinite domain; therefore the form of pressure mode matrix Ψ for the four bodies is also the same. The solutions in Eq. (8.132) satisfy the governing equations in Eqs. (8.134) and (8.135) and the FSI condition in Eq. (8.137), as well as the infinite condition on Γ N , but they do not satisfy the boundary condition in Eq. (8.136a) on the fixed wall Γ b and the one in Eq. (8.136b) on the free surface Γ f . We intend to use the image method to obtain the solution based on Eq. (8.138b) for an infinite water domain. Fig. 8.30 shows the imaged infinite system from the quarter water radiation problem in Fig. 8.29. This imaged system is considered in the infinite water domain with no free surface and fixed wall considered. However, it is symmetrical to the horizontal plane x1 Ox2 and to the vertical plane x1 Ox3 . The resultant dynamic pressure PðXi Þ at any point Xi in the infinite water domain is the summation of the four produced pressures by the four body vibrations, so that PðXi Þ 5

4 X J51

4 X

ðJÞ ðJÞ ðJÞ PðJÞ xðJÞ Ψ x ; x ; x 5 qðJÞ ; i 1 2 3

(8.139)

J51

in which, the transformation between the global coordinate Xi in the reference system O 2 x1 x2 x3 and the local ðJÞ ðJÞ ðJÞ ðJÞ coordinate xðJÞ i in the local system o 2 x1 x2 x2 , (J 5 1, 2, 3, 4), is given by

Mixed finite element
boundary element model for linear water
structure interactions Chapter | 8

ðJÞ Xi 5 xðJÞ oi 1 xi :

329

(8.140)

Based on Eqs. (8.139) and (8.140), we obtain @ðÞ @ðÞ 5 ðJÞ ; @Xi @xi

(8.141a)

from which, it follows that the gradient of the resultant pressure field at a point on any surface of the unit normal vector ν j in the global system is given by P; j ðXi Þν j 5

4 X

4

X ðJÞ ðJÞ ðJÞ ðJÞ PðJÞ ν ðJÞ Ψ; j xðJÞ ν ðJÞ ; j xi j 5 j q ; 1 ; x2 ; x3

J51

(8.141b)

J51

ðJÞ ðJÞ ðJÞ ðJÞ where ν ðJÞ j denotes the corresponding unit normal vector of the global surface in the local system o 2 x1 x2 x2 .

Relationships valid for the symmetrical image system Since the image system is symmetrical to the horizontal plane x1 Ox2 and to the vertical plane x1 Ox3 , for a point respectively located on these two symmetrical planes, the following relationships for the geometry, the pressure mode, and its gradient are valid. For a point AðXiA Þ located on the horizontal symmetrical plane x1 Ox2 ð2Þ ν ð1Þ 3 5 2 ν3 ;

ð1Þ ð2Þ ð2Þ ν ð1Þ 1 5 ν2 5 0 5 ν1 5 ν2 ;

ð4Þ ν ð3Þ 3 5 2 ν3 ;

ð3Þ ð4Þ ð4Þ ν ð3Þ 1 5 ν2 5 0 5 ν1 5 ν2 ;

xð1ÞA 5 xð2ÞA 1 1 ;

xð1ÞA 5 xð2ÞA 2 2 ;

xð1ÞA 5 2xð2ÞA 3 3 ;

xð3ÞA 5 xð4ÞA 1 1 ;

xð3ÞA 5 xð4ÞA 2 2 ;

xð3ÞA 5 2xð4ÞA 3 3 ;





ð1ÞA ð1ÞA ð2ÞA ð2ÞA Ψ xð1ÞA 5 Ψ xð2ÞA ; 1 ; x2 ; x3 1 ; x2 ; x3



ð3ÞA ð3ÞA ð4ÞA ð4ÞA Ψ xð3ÞA 5 Ψ xð4ÞA ; 1 ; x2 ; x3 1 ; x2 ; x3



ð1ÞA ð1ÞA ð2ÞA ð2ÞA Ψ;3 xð1ÞA 5 2 Ψ;3 xð2ÞA ; 1 ; x2 ; x3 1 ; x2 ; x3



ð3ÞA ð3ÞA ð4ÞA ð4ÞA Ψ;3 xð3ÞA 5 2 Ψ;3 xð4ÞA : 1 ; x2 ; x3 1 ; x2 ; x3

(8.142a)

(8.142b)

For a point BðXiB Þ located on the vertical symmetrical plane x1 Ox3 ð1Þ ν ð3Þ 2 5 2ν 2 ;

ð3Þ ð1Þ ð1Þ ν ð3Þ 1 5 ν3 5 0 5 ν1 5 ν3 ;

ð2Þ ν ð4Þ 2 5 2ν 2 ;

ð4Þ ð2Þ ð2Þ ν ð4Þ 1 5 ν3 5 0 5 ν1 5 ν3 ;

xð3ÞB 5 xð1ÞB 1 1 ;

xð3ÞB 5 2xð1ÞB 2 2 ;

xð3ÞB 5 xð1ÞB 3 3 ;

xð4ÞB 5 xð2ÞB 1 1 ;

xð4ÞB 5 2xð2ÞB 2 2 ;

xð4ÞB 5 xð2ÞB 3 3 ;





ð3ÞB ð3ÞA ð1ÞB ð1ÞB ð1ÞB Ψ xð3ÞB ; x ; x 5 Ψ x ; x ; x ; 1 2 3 1 2 3



ð4ÞB ð4ÞB ð2ÞB ð2ÞB Ψ xð4ÞB 5 Ψ xð2ÞB ; 1 ; x2 ; x3 1 ; x2 ; x3



ð3ÞB ð3ÞA ð1ÞB ð1ÞB Ψ;2 xð3ÞB 5 2Ψ;2 xð1ÞB ; 1 ; x2 ; x3 1 ; x2 ; x3



ð4ÞB ð4ÞB ð2ÞB ð2ÞB Ψ;2 xð4ÞB 5 2Ψ;2 xð2ÞB : 1 ; x2 ; x3 1 ; x2 ; x3

(8.143a)

(8.143b)

330

Fluid
Solid Interaction Dynamics

Radiation pressure for the quarter water with no free surface wave To satisfy the no surface wave condition in Eq. (8.136b), the resultant pressure at any point AðX1A ; X2A ; 0Þ should vanish, that is, 4


 X ðJÞA ðJÞA P XiA 5 Ψ xðJÞA qðJÞ 5 0: 1 ; x2 ; x3

(8.144a)

J51

Based on the relationships for points on the horizontal plane given by Eqs. (8.142a) and (8.142b), we obtain ~ qð1Þ 5 2qð2Þ 5 q;

^ qð3Þ 5 2qð4Þ 5 q:

(8.144b)

Also, to satisfy the fixed wall condition in Eq. (8.136a), the gradient of the resultant pressure at any point BðX1B ; 0; X3B Þ on the vertical symmetrical plane Γ b should vanish, that is, 4

X
 ðJÞB ðJÞB P; j XiB ν j 5 Ψ; j xðJÞB ; x ; x qðJÞ 5 0; ν ðJÞB j 1 2 3

(8.145a)

J51

which, when using the relationships in Eqs. (8.143a) and (8.143b), requires qð3Þ 5 qð1Þ ;

qð4Þ 5 qð2Þ ;

qð3Þ 5 qð1Þ ;

qð4Þ 5 qð2Þ 5 2 qð1Þ :

(8.145b)

which implies ^ q~ 5 q;

(8.146)

Finally, the radiation pressure at any point Xi for the quarter water problem with no free surface wave case can be obtained by Eq. (8.139) with the generalized coordinate vector in Eq. (8.146), that is, PðXi Þ 5 Ψ ðXi Þqð1Þ ; Ψ ðXi Þ 5 Ψ xð1Þ 2 Ψ xð2Þ 1 Ψ xð3Þ 2 Ψ xð4Þ : i i i i

(8.147)

Here, Ψ ðXi Þ is a composite pressure mode matrix, which provides the pressure mode matrix satisfying the wave equation and the fluid boundary conditions. The vector qð1Þ can be derived from the governing equation of the solid given in Eq. (8.134) with the FSI conditions in Eq. (8.137) by using the mode summation forms for the solid displacement and the fluid pressure: Uið1Þ 5 Φi ðxð1Þ Þqð1Þ ;

Pð1Þ 5 Ψ ðXð1Þ Þqð1Þ ;

(8.148)

in the same process as presented in Section 8.4.2.2. In this calculation, the coordinates of the point Xið1Þ in the global system on the FSI interface Σ have to be transformed into the corresponding ones for using Eq. (8.147). We do not repeat the description of this process. Radiation pressure for the quarter water with free surface wave In this case, we choose the composite pressure in the form PðXi Þ 5

4 X



ðJÞ ðJÞ ð21ÞJ11 Ψ xðJÞ ; x ; x qðJÞ : 1 2 3

(8.149)

J51

In this composition, the fixed wall condition requires 4

X
 ðJÞB ðJÞB P; j XiB ν j 5 ð21ÞJ11 Ψ; j xðJÞB ν ðJÞB qðJÞ 5 0; j 1 ; x2 ; x3

(8.150a)

J51

from which, considering Eqs. (8.143a) and (8.143b), it follows







ð3ÞB ð3ÞA ð1ÞB ð1ÞB ð1ÞB ð2ÞB ð2ÞB ð2ÞB ð4ÞB ð4ÞB ð4ÞB ð3Þ ð1Þ ð2Þ ; x ; x 1 Ψ x ; x ; x 2 Ψ x ; x ; x 2 Ψ x ; x ; x Ψ;2 xð3ÞB q q q qð4Þ ;2 ;2 ;2 1 2 3 1 2 3 1 2 3 1 2 3




(8.150b)   ð1ÞB ð1ÞB ð2ÞB ð2ÞB ð2ÞB
ð4Þ ð1Þ ð3Þ ð2Þ 1 Ψ 5 0; 5 Ψ;2 xð1ÞB ; x ; x 2 q x ; x ; x 2 q q q ;2 1 2 3 1 2 3

Mixed finite element
boundary element model for linear water
structure interactions Chapter | 8

331

which requires ~ qð3Þ 5 qð1Þ 5 q;

^ qð4Þ 5 qð2Þ 5 q:

(8.150c)

To satisfy the free surface wave condition in Eq. (8.136b), on the free surface plane x1 Ox2 , the pressure mode function must satisfy P; i ν i 5

ω2 P: g

(8.151a)

Substituting Eq. (8.150a), as well as using Eqs. (8.142a) and (8.142b) and (8.150c), into Eq. (8.151a), we obtain h



i ð1ÞA ð1ÞA ð3ÞA ð3ÞA ^ Ψ;3 xð1ÞA 1 Ψ;3 xð3ÞA ðq~ 1 qÞ 1 ; x2 ; x3 1 ; x2 ; x3 5



i ω2 h ð1ÞA ð1ÞA ð1ÞA ð3ÞA ð3ÞA ^ Ψ x1 ; x2 ; x3 ; x ; x 1 Ψ xð3ÞA ðq~ 2 qÞ; 1 2 3 g

(8.151b)

which is further denoted by 2

3 2 3
A  ω2
A 
A  ω2
A  ~ ;3 x 1 ~ ;3 x 2 Ψ ~ x 5q~ 5 4Ψ ~ x 5q; 4Ψ ^ Ψ i i i i g g




 ~ xA 5 Ψ xð1ÞA ; xð1ÞA ; xð1ÞA 1 Ψ xð3ÞA ; xð3ÞA ; xð3ÞA : Ψ i 1 2 3 1 2 3

(8.151c)

Since each pressure mode in the pressure mode matrix Ψ is obtained without considering the condition in Eq. (8.151a) on the free surface, the coefficient matrix for the vector q~ should be invertible for the solution to ~ After this, the reduce to one generalized coordinate vector and to remain the only independent one, such as q: rest of the work is the same as that in the case of no free surface wave, which has been discussed based on Eq. (8.148).

8.4.4.3 Example 8.6: A quarter radiation Now assume that the body in Fig. 8.30 is the small ball in Example 8.5, and we use the image method to solve the related radiation problem in a quarter domain. The pressure mode function is given by Eq. (8.131c), from which the four pressure functions for the system in Fig. 8.30 are ðJÞ ejκjr j ψðJÞ ðr ðJÞ Þ 5 ð3λ 1 2μÞ 2 ; r ðJÞ

J 5 1; 2; 3; 4:

(8.152)

Case for no free surface wave From Eq. (8.147), the composite pressure is PðrÞ 5 ψ ðRÞq;

R 5 r0ðJÞ 1 r ðJÞ ;

(8.153) ψ ðRÞ 5 ψð1Þ ðr ð1Þ Þ 2 ψð2Þ ðr ð2Þ Þ 1 ψð3Þ ðr ð3Þ Þ 2 ψð4Þ ðr ð4Þ Þ; where R denotes the position vector of a field point in the global coordinate system O 2 x1 x2 x3 , which equals the vector summation of the global position vector r0ðJÞ of the origin oðJÞ and the local position vector r ðJÞ in the local system ðJÞ ðJÞ oðJÞ 2 xðJÞ 1 x2 x2 . Therefore the composite pressure mode function 4 jκ R2r0ðJÞ X e ψ ðRÞ 5 ð3λ 1 2μÞ ð21ÞJ11 2 ; ðJÞ (8.154) J51 R2r0 ~ PðRÞ 5 ψ ðRÞq:

332

Fluid
Solid Interaction Dynamics

Case for free surface wave From Eq. (8.151c), we have

2




4ψ~ ;3 R

 A

3 2 3 2


  ω ω ~ A 5 ^ 2 ψ~ RA 5q~ 5 4ψ~ ;3 RA 1 ψ R q; g g 2

~ A Þ 5 ψð1Þ ðr ð1ÞA Þ 1 ψð3Þ ðr ð3ÞA Þ ψðR 2 3 jκ RA 2r0ð1Þ jκ RA 2r0ð3Þ e 6e 7 5 ð3λ 1 2μÞ4 2 1 5; A ð1Þ A ð3Þ 2 R 2r0 R 2r0 from which, when Eq. (8.150c) is introduced, it follows

~ AÞ q~ 5 ψ~ ;3 ðRA Þ2 ωg ψðR 2

^ qð3Þ 5 qð1Þ 5 aðRA Þq;

21

3 2 ω ~ A Þ5q^ 5 aðRA Þq; 4ψ~ ;3 ðRA Þ 1 ^ ψðR g

(8.155)

2

(8.156)

^ qð4Þ 5 qð2Þ 5 q:

Finally from Eq. (8.149), we obtain the composite pressure mode function as follows:   ψ ðRÞ 5 aðRA Þψðr ð1Þ Þ 2 ψðr ð2Þ Þ 1 aðRA Þψðr ð3Þ Þ 2 ψðr ð4Þ Þ ; jκ R2r0ðJÞ e  ^ ψðr ðJÞ Þ 5 ð3λ 1 2μÞ 2 ; PðRÞ 5 ψ ðRÞq: ðJÞ R2r 0

(8.157)

Chapter 9

Hydroelasticity theory of ship
water interactions Chapter Outline 9.1 Fundamentals for ship
water interactions 9.1.1 Frames of reference 9.1.2 Governing equations 9.1.3 Equations for static equilibrium state 9.1.4 Equations for steady motion 9.2 Incident waves 9.2.1 Equation of incident water waves

334 334 336 339 340 341 341

9.2.2 Linear plane gravity waves 9.2.3 Frequency of wave encounter 9.3 Linear hydroelasticity theory 9.3.1 Linearized governing equations 9.3.2 Equations in the modal space 9.3.3 Numerical solutions 9.3.4 Examples

342 343 343 343 344 351 358

Historically, the earliest concern on fluid
solid interaction (FSI) problems is with airplane design, and the related welldeveloped FSI theory is termed aeroelasticity, in which the air is considered a compressible fluid, and the main aim of analysis is to determine the critical speed of airplane flutter (see, e.g., Fung, 1955, 1969; Bisplinghoff et al., 1955; Bisplinghoff and Ashley, 1962). For ship design, the water is traditionally treated as an incompressible fluid with its flows irrotational, and the term hydroelasticity in naval engineering first appeared in the literature (Heller and Abramson, 1959). In offshore and maritime engineering, highly complex fluid
structure interaction mechanisms are encountered between the seaway and structure, as described in the theories relating to waves, resistance, and propulsion, sea keeping, maneuvering, wave loads, and structural responses (see Havelock, 1928, 1932, 1955; Wehausen and Denis, 1950; Kinsman, 1965; Wehausen, 1963, 1971, 1973, 1978; Mei, 1978, 1983; Ogilvie, 1977; Newman, 1977, 1978, 1994; Bishop and Price, 1979; Bishop et al., 1986; Wu, 1984, 1990; Wu and Eatock Taylor, 1987a,b; Isaacson and Cheung, 1992; Isaacson and Joseph, 1993; Duan, 1995; Lewis, 1988; Lloyd, 1989; Faltinsen, 1990; Baltrop and Adams, 1991; Breslin and Andersen, 1994; Price and Tan, 1992; Tan, 1994; Farmer et al., 1994; Baar, 1986). Ships generally move with a mean forward velocity, and their oscillatory motions in waves are superposed upon a steady flow field. Traditionally, a ship is regarded as an unrestrained rigid body with 6 degrees of freedom, and the unsteady motions of the ship and the waves are assumed to be of small amplitude. One of the principal problems encountered is the solution of the steady-state case, particularly with regard to the calculation of wave resistance in calm water; see, for example, Wehausen and Denis (1950), Wehausen and Laitone (1960), Wehausen (1973), and Baar (1986). The ship
wave interaction case is considered separately as the superposition of two problems. Namely, a radiation problem, where the ship undergoes prescribed oscillatory motion in otherwise calm water, and a diffraction problem, where incident waves act upon the ship in its equilibrium position. Interaction between these two first-order radiation and diffraction problems is of second order in the oscillatory amplitudes and is therefore neglected. This topic was reviewed by Wehausen (1971), and early numerical solutions were described by Mei (1978). The linear problem of ship motions in waves is solved by a superposition of the steady- and unsteady-state cases. Interaction between the steady and oscillatory flow fields complicates the more general problems, which were discussed by Ogilvie (1977) and Newman (1977, 1978) and which presented in detail the theory of ship motion and formulated hydrodynamic forces of oceangoing rigid ships. Bishop and Price (1979) gave up the rigid ship assumption and developed a two-dimensional (2D) linear hydroelasticity theory, based on superposition methods for the incident, diffraction, and radiation potentials, including the vibration modes of the structure, in order to deal with beam-like flexible ships interacting with the water. From their contribution, the term “hydroelasticity” for incompressible water
ship interactions has been widely used as same as the term “aeroelasticity” for compressible air
plane interactions. This linear hydroelasticity theory has been further developed into a more general three-dimensional (3D) case by Wu (1984), Price and Wu (1985), and Bishop et al. (1986) for Fluid
Solid Interaction Dynamics. DOI: https://doi.org/10.1016/B978-0-12-819352-5.00009-4 © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved.

333

334

Fluid
Solid Interaction Dynamics

ship
water interaction dynamical analysis, for which the related investigations have been completed successfully and used in the engineering design of ships (Wu, 1984, 1990). Recently, in a special issue of Journal of Engineering for the Maritime Environment edited by Temarel (2009), two review papers on hydroelasticity were given (see Wu and Cui, 2006, 2009; Hirdaris and Temarel, 2009) in which more historical publications, developments, and future trends were described. Interested readers may refer to these sources for more information. Compared with the mixed finite element
boundary element (FE
BE) model for water
structure interactions presented in Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, one of essential difference is using different Green functions. In Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, the fundamental solutions of the Laplace equation defined in a 2D or 3D infinite water domain, which do not need to satisfy the boundary conditions on the free surface of water, are used in Green’s third identity to generate the BE equation defined on the free surface and wet interface of the structure. However, in traditional hydroelasticity, the fundamental solutions of the Laplace equation defined in a half-infinite water domain, with the corresponding free surface boundary conditions satisfied, so that the velocity potential of the water interacting with a surface-piercing structure in motion, can be expressed as a singularity integral equation only on the water
structure interaction interface, with the structure motions included in the integral equation as a source density; these approaches are called singularity distribution methods (Brard, 1948, 1972). Another main difference is that in hydroelasticity formulation, two pure fluid problems—(1) the diffracted potential in which the solid is fixed and (2) the mode-radiated potential in which the solid motion is given as a natural mode shape of ships—may be required to get to the solution of the problem. However, in the FE
BE method, the fluid potential on the wet interface is denoted by a summation form of all BEs at each of which the value of the potential is a variable to be solved by the integrated water
structure interaction equations. This chapter discusses this hydroelasticity theory. Section 9.1 presents the fundamentals of ship
water interaction dynamics, which includes the definitions of the three reference coordinate systems, the generalized nonlinear governing equations, the static equilibrium solution, and the steady motion solution of the system. The moving reference frame with the ship forward speed is used herein for convenience in modeling the dynamics of moving ships. Section 9.2 summarizes some concepts on incident water waves. Following the knowledge described in Sections 9.1 and 9.2, Section 9.3 investigates the dynamic response of the integrated ship
water interaction system excited by incident waves using the traditional approach in the hydroelasticity theory of ships. As the publications indicate, in the traditional hydroelasticity theory of ships, the mode superposition principle is based on linear elastic structures by which the ship motion and the velocity potential of water are represented by the corresponding mode summation formulation. However, for nonlinear water
ship interactions, the mode summation principle is no longer theoretically valid because the deformation/motion of the ship is nonlinear, so that this chapter discusses the linear hydroelasticity theory based on the linearized governing equations given in prior two subsections, and we leave the nonlinear water
ship interactions to be investigated in later chapters where more complex numerical processes are involved.

9.1

Fundamentals for ship
water interactions

For ship
water dynamic interactions in the hydroelasticity theory, the water is treated as an ideal incompressible fluid with its flow irrotational in the Euler coordinate system. Therefore its motions are governed by the theory of potential flows presented in Section 3.6.3. The structures are considered elastic bodies satisfying the equations in the Lagrange coordinate system given in Chapter 3, Fundamentals of continuum mechanics.

9.1.1

Frames of reference

In order to predict the motions of a ship traveling with forward speed V^ and subject to waves as well as various dynamic loads, the ship may be regarded as an elastic body floating on the water, as shown in Fig. 9.1. Here, o0 2 x01 x02 x03 is a coordinate system fixed at a point o0 on the calm free surface of the water in the space, and axis o0 2 x03 is vertical and positive upward, that is, along the negative direction of gravitational acceleration. Another reference frame o 2 x1 x2 x3 is a set of axes moving with forward speed V^ in x01 direction and remaining parallel to o0 2 x01 x02 x03 . Therefore there exists the following transformation between the systems o0 2 x01 x02 x03 and o 2 x1 x2 x3 : ^ i1 ; v0i 5 x_0i 5 x_i 1 Vδ ^ i1 5 vi 1 Vδ ^ i1 ; x0i 5 x~ 0i 1 xi 1 Vtδ @ðÞ @ðÞ @xi @x0i ^ i1 ; 5 2 Vδ 5 ; @xi @t @t @x0i

(9.1)

Hydroelasticity theory of ship
water interactions Chapter | 9

335

FIGURE 9.1 Three coordinate systems for ship
water interactions.

where x~0i denotes the coordinates of the origin o in the system o0 2 x01 x02 x03 at time t 5 0: For convenience, without losing any generality, we assume x~0i 5 0: The frame of reference O 2 X1 X2 X3 is a material coordinate system fixed in the structure at O such that it coincides with o 2 x1 x2 x3 in the absence of any disturbance. The origins o and O are located at convenient positions in the body, usually on the line formed by the intersection of a longitudinal plane of symmetry and the calm water surface. The water is considered as an idealized fluid with its motion irrotational described by a total potential Φ0 ðx0i ; tÞ of velocity that is a function of spatial coordinates x0i and time t. By using the transformation in Eq. (9.1), the total potential may be represented in the moving frame of axes as the form ^ i1 ; tÞ  Φðxi ; tÞ; Φ0 ðx0i ; tÞ 5 Φ0 ðxi 1 Vtδ 0 1 ^ i1 @Φ ^ j1 @ @Φ0 @Φ Vδ @ Vδ AΦðxi ; tÞ; 2 5 5@ 2 @t @t @xi @t @xj

(9.2)

@Φ @Φ 5 ; @xi @x0i 0

as defined by the Lorentz transformation. Here, the last equation in Eq. (9.2) physically implies that the absolute velocity of a water particle at a physical point x0i in the space, which is also marked by the coordinate xi in the moving system, is not changed with the coordinates to identify it because it is the same water particle. Therefore the potential of velocity Φðxi ; tÞ and its partial derivative @Φ=@xi describe the absolute motion with the corresponding velocity of the water, although it is represented as a function of the moving coordinate xi and time t. For convenience, we may represent the potential of velocity as ^ i δi1 1 φðxi ; tÞ; Φðxi ; tÞ 5 Vx @Φ ^ i1 1 @φ ; 5 Vδ @xi @xi

(9.3)

where φ and its partial derivative @φ=@xi denote the velocity potential and the velocity of water relative to the moving system o 2 x1 x2 x3 . For the steady motion of the ship in calm water, the velocity field of the water relative to the moving system is not ^ i Þ independent of time t. changed with time, and the relative potential velocity takes the form φ 5 φðx As shown in Fig. 9.2, the ship occupies a material domain Ωs of its boundary S 5 Σ , ST with a unit outer normal vector ηi , and it has an interface Σ contacting the water occupying a spatial domain Ωf of its boundary Γ 5 Γ b , Γ 2N , Γ N , Γ f , Σ with a unit outer normal vector ν i . The ship is subjected to a body force F^ i per unit mass and traction T^ i per unit area on the surface ST , and the water has its body force f^i 5 2 gδi3 , the gravitational force,

336

Fluid
Solid Interaction Dynamics

FIGURE 9.2 Ship
water interaction system.

and is subject to an incident wave of its potential φ0 ðx0i ; tÞ of velocity. The ship motion relative to the moving set of axes o 2 x1 x2 x3 is described in the Lagrange coordinate system, and therefore its displacement Ui , Green strain Eij , and second Piolar
Kirchhoof stress Σ ij are considered the functions of the material coordinates Xi and time t. The displacement Ui0 , velocity Vi0 , and acceleration Wi0 , in the fixed system o0 2 x01 x02 x03 , of a material particle of the ship can be calculated as Ui0 ðXj ; tÞ 5 x0i ðXj ; tÞ 2 Xi ^ i1 2 Xi 5 Ui ðXj ; tÞ 1 Vtδ ^ i1 ; 5 xi ðXj ; tÞ 1 Vtδ 0 0 _ _ ^ ^ i1 ; Vi ðXj ; tÞ 5 U i ðXj ; tÞ 5 U i ðXj ; tÞ 1 Vδi1 5 Vi ðXj ; tÞ 1 Vδ 0 0 Wi ðXj ; tÞ5 U€ i ðXj ; tÞ 5 U€ i ðXj ; tÞ 5 Wi ðXj ; tÞ:

9.1.2

(9.4)

Governing equations

From the knowledge presented in Chapter 3, Fundamentals of continuum mechanics, the governing equations describing the dynamics of ship
water interaction systems are presented as follows.

9.1.2.1 Ship structure The deformation of the ship is investigated under the material coordinate system. The material coordinates of a particle are defined by its original coordinates in the fixed system at time t 5 0. Since it is chosen that x~0i 5 0 in Eq. (9.1), at time t 5 0, the moving system coincides with the fixed system. Therefore the material coordinates of a particle of the ship are the same for both the fixed and moving coordinate systems, and the equations for the ship structure are valid in both systems. Equation of motion in Ωs @2 U i 5 ½ðδir 1Ui; r ÞΣ rj ; j 1 ρ0 F^ i ; @t2   1 Σ ij 5 λ Uk; k 1 Um; k Um;k δij 1 GðUj; i 1 Ui; j 1 Um; i Um; j Þ: 2 ρ0

(9.5a) (9.5b)

Traction boundary condition on ST ðδir 1 Ui; r ÞΣ rj ηj 5 T^ i :

(9.5c)

For linear cases where the nonlinear terms in Eqs. (9.5a) and (9.5b) are neglected, we have @ Ui GUi; jj 1 ðλ 1 GÞUj; ji 1 ρ0 F^ i 5 ρ0 2 ; @t Σ ij ηj 5 T^ i ; Σ ij 5 λUk; k δij 1 GðUj; i 1 Ui; j Þ 5 σij : 2

(9.6a) (9.6b)

Hydroelasticity theory of ship
water interactions Chapter | 9

337

9.1.2.2 Water The spatial coordinates are used to describe the water flows. The spatial coordinates and the related time derivatives in the moving systems are different from the ones in the fixed system. The governing equations for the water flows are first derived in the fixed system, which then are transformed into the corresponding ones under the moving system o 2 x1 x2 x3 by using the transformations in Eqs. (9.1) and (9.2). For example, from those equations, it follows that ^ i; j δi1 1 φ; j 5 Vδ ^ ij δi1 1 φ; j 5 Vδ ^ j1 1 φ; j ; Φ0; j 5 Φ; j 5 Vx 0 ^ j1 Φ; j 5 φ; t 2 Vδ ^ j1 ðVx ^ i δi1 1φÞ; j Φ; t 5 Φ; t 2 Vδ 2 ^ j1 φ; j ; 5 φ; t 2 V^ 2 Vδ

(9.7)

which we use in the following derivations. Bernoulli equation in Ωf Fixed system: gx0i δi3 1 Moving system: gxi δi3 1

gxi δi3 1

p 2 p0 1 1 Φ0; j Φ0; j 1 Φ0; t 5 0; ρ 2

(9.8a)

p 2 p0 ^ j1 Φ; j 1 1 Φ; j Φ; j 1 Φ; t 5 0; 2 Vδ ρ 2

(9.8b)

2 p 2 p0 V^ 1 2 1 φ; j φ; j 1 φ; t 5 0: 2 ρ 2

(9.8c)

Fixed system: Φ0; jj 5 0;

(9.9a)

Equation of continuity in Ωf

Moving system:

Φ; jj 5 φ; jj 5 0:

(9.9b)

Boundary conditions On the seabed Γ b : The normal velocity of the water vanishes v^i 5 0, from which it follows Fixed system: Φ0; j ν j 5 0;

(9.10a)

^ j1 1 φ; j Þν j 5 0; Moving system: Φ; j ν j 5 ðVδ ^ 1 1 φ; j ν j 5 φ; j ν j 5 0; Φ; j ν j 5 Vν

if

^ ν\V:

(9.10b) (9.10c)

Physically, Eq. (9.10c) implies that the seabed is parallel to the ship’s moving direction, so that its component of normal vector ν 1 5 0: In practical numerical simulations, we always use Eq. (9.10c) to neglect the effect of seabed geometry. On the free surface Γ f : The kinematic condition and dynamic condition should be satisfied. These two equations now become D 0 ðx 2 hÞ 5 0; x03 5 hðx01 ; x02 ; tÞ; Dt 3 p 5 p0 : Using the Bernoulli equation in Eq. (9.8a) and the preceding two conditions, we have   1 0 0 1 1 0 0 0 0 Φ Φ 1 Φ; t ; gh 1 Φ; j Φ; j 1 Φ; t 5 0; h 5 2 2 g 2 ;j ;j

(9.11a) (9.11b)

(9.12a)

which, when substituted into Eq. (9.11a), gives

  Dx03 Dh 21 D 1 0 0 5 5 Φ; j Φ; j 1 Φ0; t ; Dt Dt g Dt 2

(9.12b)

that is, an alternative boundary condition for the velocity potential on the free surface x03 5 h, or Φ0; tt 1 2Φ0; j Φ0; jt 1

1 0 0 ðΦ Φ Þ Φ0 1 gΦ0; j δ3j 5 0: 2 ;j ;j ;i ;i

(9.12c)

338

Fluid
Solid Interaction Dynamics

If the nonlinear terms in this equation are neglected, a linear condition is obtained: Φ0; tt 1 gΦ0; j δ3j 5 0:

(9.13)

Eqs. (9.12a) and (9.12c) are transformed into the following forms in the moving system by using Eqs. (9.2) and (9.7):   1 1 ^ Φ; j Φ; j 1 Φ;t 2 Vδj1 Φ; j ; h52 (9.14a) g 2   2 V^ 1 1 2 φ;j φ;j 1 φ; t ; h5 (9.14b) g 2 2g 1 ^ j1 Φ; jt 1 V^ 2 δj1 δl1 Φ; jl 2 2Φ; j Vδ ^ l1 Φ; jl 5 0; ðΦ; j Φ; j Þ; i Φ; i 1 gΦ; j δ3j 2 2Vδ 2 1 2 φ; tt 1 2φ; j φ; jt 1 ðφ; j φ; j Þ; i φ; i 1 V^ δi1 δj1 φ; ij 1 gφ; j δ3j 5 0; 2

Φ; tt 1 2Φ; j Φ:jt 1

(9.14c) (9.14d)

which have the linearized forms 1 ^ j1 Φ; j Þ; h 5 2 ðΦ; t 2 Vδ g

(9.15a)

2 V^ 1 h5 2 φ; t ; 2g g

(9.15b)

^ j1 Φ; jt 1 V^ δj1 δl1 Φ; jl 1 gΦ; j δ3j 5 0; Φ; tt 2 2Vδ

(9.15c)

φ; tt 1 V^ δi1 δj1 φ; ij 1 gφ; j δ3j 5 0:

(9.15d)

2

2

On the far-field boundary Γ 6 N One of the following suitable conditions may be imposed: 1. An undisturbed condition physically implies that the disturbance associated with the ship cannot reach infinity due to physical damping in the water. Therefore the velocity of the water equals the velocity of the incident wave, that is, Fixed system: Φ0; j 5 φ0:j ; Moving system: Φ; j 5 φ0; j ;

(9.16a) ^ φ0 ðxi ; tÞ 5 φ0 ðxi 1 δi1 VtÞ;

^ j1 1 φ; j 5 φ0; j : Vδ

(9.16b) (9.16c)

Here, φ0 ðxi ; tÞ represents the incident wave described in the moving system. If no incident wave applies, Eqs. (9.16a) and (9.16b) represent the velocity on the far-field boundary vanishes, which means the water is in its static equilibrium state. However, Eq. (9.16c) represents that the relative velocity of ^ j1 , which implies that people standing on the ship see the water moving at the the water at the far-field equals 2 Vδ ^ j1 . speed 2 Vδ 2. A radiation condition assumes that the energy flux of waves associated with the disturbance of the ship is directed away from the ship into infinity, as discussed in Section 3.6.4.2 (Sommerfeld, 1912, 1949; Xing, 2006, 2007, 2008). Interaction conditions On the current deformed wet interface Σ, the normal velocity of the ship is equal to that of the adjacent fluid. The appropriate boundary condition is Fixed system:

Vi0 ηi 5 Φ0; i ηi 5 2 Φ0; i ν i ;

^ i1 1 Ui; t Þηi 5 2 Φ;i ν i ; Moving system: ðVδ Ui; t ηi 5 2 φ; i ν i :

(9.17a) (9.17b) (9.17c)

Hydroelasticity theory of ship
water interactions Chapter | 9

339

The fluid pressure, as a traction, is applied on the wet surface; therefore another force equilibrium equation similar to Eq. (9.5c) is required, that is, ðδir 1 Ui; r ÞΣ rj ηj 5 2 pδij σηj :

(9.17d)

Here, σ is the area ratio as given by Eq. (3.35). For a large deformation, a unit area on the wet surface of the ship before deformation is different from that after deformation. The fluid pressure is defined as the pressure per unit area after deformation, and therefore this area ratio is introduced for the force equilibrium on the wet surface. This equation can be also represented in the current reference frame form: σij ν j 5 2 pν i ;

(9.17e)

in which Cauchy’s stress σij in the ship and the unit outer normal ν j on the wet interface of the water domain are used. Compared with the unit vector ηi defined in the material system, the vector ν j changes due to the motion. This vector can be calculated based on Eq. (3.36).

9.1.3

Equations for static equilibrium state

In the static equilibrium state, the moving speed of the ship vanishes, that is, V^ 5 0, and the fixed and the moving systems are the same. The potential velocity Φ0 of the water and its wave height h on the free surface Γ f are assumed to be zero. For the total system, the external forces include the gravitational force, which is the body force per unit mass of the ship and water F^ i 5 2 gδi3 5 f^i3 , and the atmospheric pressure p0 applied on the free surface of the water and the ship. The displacement of the ship caused by these static forces is denoted by Uis , which can be assumed to be small. The equations governing this static equilibrium state, denoted by a superscript “s”, is derived from the ones given in Section 9.1.2. Static equilibrium equation of the ship σsij; j 2 ρ0 gδi3 5 0:

(9.18a)

σsij ηj 5 2 p0 ηi :

(9.18b)

Traction boundary condition on ST

Bernoulli equation for water gx0i δi3 1

ps 2 p0 5 0: ρ

(9.18c)

Equilibrium equation on the wet interface Σ σsij ηj 5 2 ps ηi :

(9.18d)

Taking the integration of Eq. (9.18a) over the volume Ωs of the ship and using the Green theorem in Eq. (3.68), we obtain ð ð ð ðσsij; j 2 ρ0 gδi3 ÞdΩ 5 σsij ηj dS 2 ρ0 gδi3 dΩ 5 0; (9.19a) Ωs

ST , Σ

Ωs

which, when Eqs. (9.17) and (9.9a) are used, is further reduced to ð ð ð ð ð s 0 2 p0 ηi dS 2 p ηi dS 5 2 p0 ηi dS 1 ρgx3 ηi dS 5 ρ0 gδi3 dΩ: ST

Σ

ST , Σ

Σ

(9.19b)

Ωs

Ð Since the atmospheric pressure p0 is a constant, the integration ST , Σ p0 ηi dS 5 0 on the closed surface of the ship, and we obtain ð ð ρgx03 ηi dS 5 ρ0 gδi3 dΩ: (9.19c) Σ

Ωs

The integration on the right-hand side of Eq. (9.19c) represents the total gravity of the ship, and the one on the lefthand side equals the buoyant force of the water, so that this equation is a global equilibrium equation of a static ship floating on the calm water surface.

340

Fluid
Solid Interaction Dynamics

Taking a cross-multiplication of Eq. (9.19a) by the position vector x0i , we obtain the moment equilibrium equation of the internal and external forces of the body about the origin of the system: ð ð 0 s ersi xs ðσij; j 2 ρ0 gδi3 ÞdΩ 5 ersi ½ðx0s σsij Þ; j 2 δsj σsij 2 ρ0 gx0s δi3 dΩ 5 0: (9.20a) Ωs

Ωs

Using the Green theorem in Eq. (3.68) and noting σsij 5 σsji , as well as the tensor operation rules presented in Chapter 2, Cartesian tensor and matrix calculus, we have the following integrations: ð ð ersi ðx0s σsij Þ; j dΩ 5 ersi x0s σsij ηj dS Ωs ST , Σ ð ð 5 2 ersi x0s p0 ηi dS 2 ersi x0s ps ηi dS S Σ ðT ð 5 2 ersi x0s p0 ηi dS 2 ersi x0s ðp0 2 ρ0 gx0j δj3 Þηi dS S Σ ð T ð 0 52 ersi xs p0 ηi dS 1 ersi x0s ρ0 gx03 ηi dS; (9.20b) ST , Σ Σ ð ð ð ersi ðx0s σsij Þ; j dΩ 5 ersi δsj σsij dΩ 1 ersi x0s σsij; j dΩ Ωs Ωs Ωs ð ð ersi δsj σsij dΩ 5 ersi σsis dΩ 5 0; Ωs Ωs ð ð ð ersi x0s p0 ηi dS 5 ersi ðx0s p0 Þ; i dΩ 5 p0 ersi δsi dΩ 5 0: ST , Σ

Therefore from Eq. (9.19a) it follows

Ωs

Ωs

ð Ωs

ð ersi ρ0 gx0s δi3 dΩ 5

Σ

ersi x0s ρ0 gx03 ηi dS:

(9.20c)

This is the equilibrium equation of the moments of the gravitational force of the ship and the buoyant force about the origin of the coordinate system.

9.1.4

Equations for steady motion

For the steady motion, marked by a hat (^), of the ship in calm water with its constant velocity V^ in the x2 direction and with its displacement U^ i relative to the moving system independent of time t, its relative velocity and acceleration vanish. The velocity field of the water relative to the moving system is not changed with time, and the relative velocity ^ i Þ and the wave height hðx ^ 1 ; x2 Þ are independent of time t. Furthermore, we may assume that the motion potential φ 5 φðx and deformation of the ship in the steady state are small, so that the linearized approximation for the structure is considered. Based on these conditions and the governing equations presented in Section 9.1.2, we obtain the equations describing the steady motion of the water
ship interaction in the moving system as follows. Equation of ship motion σ^ ij; j 2 ρ0 gδi3 5 0:

(9.21)

σ^ ij 5 λU^ k; k δij 1 GðU^ j; i 1 U^ i; j Þ:

(9.22)

σ^ ij ηj 5 2 p0 ηi :

(9.23)

2 V^ 1 p^ 2 p0 gxi δi3 1 2 1 φ^ ; j φ^ ; j 5 0: 2 ρ 2

(9.24)

Stress-displacement relation

Traction condition on ST

Bernoulli equation

Hydroelasticity theory of ship
water interactions Chapter | 9

341

Equation of continuity φ^ ; jj 5 0:

(9.25)

^ j1 1 φ^ ; j Þν j 5 0; On sea bed Γ b : ðVδ

(9.26a)

Boundary conditions

1 ^2 ^ ^ ðV 2 φ; j φ; j Þ; h^ 5 2g

(9.26b)

1 ^ ^ 2 ðφ; j φ; j Þ; i φ^ ; i 1 V^ δi1 δj1 φ^ ; ij 1 gφ^ ; j δ3j 5 0: 2

(9.26c)

On free surface Γ f :

On the far-field boundary:

^ j1 : Γ 6 N : φ^ ; j 5 2 Vδ

(9.26d)

Interaction condition on wet interface Σ Relative velocity:

φ^ ; i ηi 5 U^ i; t ηi 5 0;

^ i: Pressure: σ^ ij ηj 5 2 pη

(9.27a) (9.27b)

Subtracting Eq. (9.18c) from Eq. (9.24), we obtain gðx3 2 x03 Þ 1

2 V^ 1 p^ 2 ps 2 1 φ^ ; j φ^ ; j 5 0: 2 ρ 2

Therefore, at a space point of x3 2 x03 5 0, the pressure change is ρ 2 p^ 2 ps 5 ðV^ 2 φ^ ; j φ^ ; j Þ: 2

(9.28a)

(9.28b)

^ so On the free surface, the pressure equals the atmosphere p^ 5 p0 5 ps , and the wave height x3 2 x03 5 h^ 2 0 5 h, we have ρ 2 ρgh^ 5 ðV^ 2 φ^ ; j φ^ ; j Þh^: (9.28c) 2 We find that the fluid pressure change is caused by the free surface wave produced by the ship traveling with a con^ The extra fluid pressure p^ 2 ps is applied on the wet interface, and it is balanced by the engine forces, stant velocity V. so that the ship keeps a steady motion.

9.2 9.2.1

Incident waves Equation of incident water waves

In Chapter 6, Preliminaries of waves, the fundamentals of water waves are given, from which some solutions of water waves with their characteristics, the velocity and pressure fields, as well as energy transmission, can be obtained. To study the dynamic response of marine structures to water waves, it is very often the practice to choose an incident wave as the prescribed excitation. The velocity potential φ0 ðx0i ; tÞ of the incident wave must satisfy the governing equations for the water domain without the ship floating on its surface. Letting Φ0 5 φ0 5 φ0 in the equations for the water given in Section 9.1.2, we obtain the governing equations satisfied by the incident wave as follows. Bernoulli equation gx0i δi3 1

p0 2 p0 1 1 φ0; j φ0; j 1 φ0; t 5 0: ρ 2

(9.29)

Equation of continuity φ0; jj 5 0:

(9.30)

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Fluid
Solid Interaction Dynamics

Boundary conditions On the seabed Γb : φ0; j ν j 5 0; On the free surface Γf :

φ0; tt 1 2φ0; j φ0; jt 1

gh0 1

1 0 0 φ φ 1 φ0; t 5 0; 2 ;j ;j

1 0 0 ðφ φ Þ φ0 1 gφ0; j δ3j 5 0: 2 ;j ;j ;i ;i

(9.31) (9.32a)

(9.32b)

For the linearized theory in which the nonlinear terms are neglected, Eqs. (9.29), (9.32a), and (9.32b) become, respectively, p0 2 p0 1 φ0; t 5 0; ρ

gx0i δi3 1

gh0 1 φ0; t 5 0; φ0; tt 1 gφ0; j δ3j 5 0;

9.2.2

in water;

on x03 5 0; on x03 5 0:

(9.33a) (9.33b) (9.33c)

Linear plane gravity waves

As discussed in Section 6.3.2, for linear plane gravity waves in the plane x1 2 o 2 x3 , the Laplace Eq. (9.30), the rigid seabed condition Eq. (9.31) at a constant depth x03 5 2 H, and the free surface condition Eq. (9.33c) are satisfied by the solution φ0 ðx01 ; x03 ; tÞ 5 A cosh κðx03 1 HÞ sinðκx01 2 ω0 tÞ;

(9.34)

ω20 5 gκ tanh κH:

(9.35)

where the frequency of the wave

By using Eq. (9.33b), the corresponding wave elevation is 1 h0 5 2 φ0; t 5 a cosðκx01 2 ω0 tÞ; g

a5

ωA cosh κH: g

(9.36)

For this linear plane gravity wave, the wavenumber κ, wavelength λ0 , frequency ω0 , period T0 , as well as its speed c, are related by the following expressions: κ5 The wave speed can be further written as

2π ; λ0

ω0 5

2π ; T0

c5

λ0 ω0 5 : T0 κ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi gλ0 2πH c5 tanh : λ0 2π

As shown in Section 6.3.2.3, from shallow water waves, H=λ0 -0, so the wave speed reduces to pffiffiffiffiffiffiffi c 5 gH ;

(9.37)

(9.38)

(9.39)

which is independent of the wavelength. On the other hand, for deep water waves, H=λ0 -N, and the wave speed is rffiffiffiffiffiffiffiffi gλ0 c5 ; ω20 5 gκ; (9.40) 2π which is a dispersion equation showing that waves on deep water are propagated with speeds depending on the wavelength.

Hydroelasticity theory of ship
water interactions Chapter | 9

9.2.3

343

Frequency of wave encounter

~ called the heading angle, between the ship motion direction x01 (also direction As shown in Fig. 9.1, there is an angle χ, x1 ) and the direction x~1 of the incident wave φ0 . Assume that a sinusoidal sea wave has its wave profile as h0 5 a cosðκx~1 2 ω0 tÞ: The coordinate x~1 may be transformed into coordinates in the o

0

2 x01 x02 x03

(9.41) system by the transformation

^ ~ x~1 5 x01 cos χ~ 1 x03 sin χ~ 5 ðx1 1 VtÞcos χ~ 1 x3 sin χ;

(9.42)

where Eq. (9.1) is used. The wave elevation in Eq. (9.41) becomes ^ h0 5 a cos ½κðx1 1 VtÞcos χ~ 1 κx3 sin χ~ 2 ω0 t 5 a cosðκx1 cos χ~ 1 κx3 sin χ~ 2 ωe tÞ;

(9.43a)

in which the frequency of encounter of the ship is defined as ^ cos χ: ~ ωe 5 ω0 2 Vκ

(9.43b)

Physically, the frequency of encounter of the ship represents the frequency seen by people standing on the moving ship. If the ship is static, V^ 5 0, and the frequency of encounter is the same as the frequency of the wave relative to the fixed system. When the incident wave is along the ship’s forward direction ðχ~ 5 0Þ, the frequency of encounter is minimum, whereas if the wave takes the negative direction of the ship’s motion ðχ~ 5 πÞ, it is maximum. There are extensive solutions of Laplace’s equation and the linearized free surface condition Eq. (9.33c), described especially by Wehausen and Laitone (1960). Of particular importance is the plane-progressive wave of constant amplitude a and sinusoidal profile, for which the velocity potential in deep water is given by a complex form: φ0 ðx0i ; tÞ 5

    jga exp κ x03 2 jx01 cos χ~ 2 jx02 sin χ~ 1 jω0 t ; ω0

(9.44a)

which is expressed in the moving reference frame as φ0 ðxi ; tÞ 5

    jga exp κ x3 2 jx1 cos χ~ 2 jx2 sin χ~ 1 jωe t : ω0

(9.44b)

Here, κ is the wavenumber shown in Eq. (9.37), ω0 is the radian frequency in the space-fixed reference frame, χ~ denotes the angle of wave propagation relative to the x01 axis as shown in Fig. 9.1, and the frequency ωe of encounter is defined by Eq. (9.43b). In these complex expressions, their real parts denote the physical incident waves.

9.3

Linear hydroelasticity theory

Here, we investigate linear ship
water interactions using the hydroelasticity approach. The studied interaction system is shown in Fig. 9.2, of which the interaction dynamics under the defined moving/material reference coordinate systems is governed by the linearized governing equations derived by neglecting the nonlinear terms of the nonlinear equations presented in Section 9.1.2. For our convenience in discussion, we adopt the generalized constitutive Eq. (3.51) with the tensor Cijkl of elastic constant, as well as the geometric relation Eq. (3.31b) to replace Eq. (9.51), in order to summarize these equations as follows.

9.3.1

Linearized governing equations

9.3.1.1 Ship structure The ship displacement Ui ðXj ; tÞ is a function of material coordinate Xj and time, which satisfies the dynamic equations @2 Ui 5 σij; j 1 ρ0 F^ i ; @t2 σij 5 Cijkl Uk; l ; ρ0

and the boundary condition

in Ωs ;

(9.45a)

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Fluid
Solid Interaction Dynamics

ηj σij 5 T^ i ;

on ST ;

(9.45b)

where the body force F^ i per unit mass normally is the gravity of ship, and the traction T^ i per unit area may be some external excitation forces on the ship’s dry surface, such as those caused by possible mechanical impacts, winds, etc.

9.3.1.2 Water The water is considered an incompressible fluid with irrotational motions, so that its velocity potential Φðxi ; tÞ satisfies Eqs. (9.8)
(9.18), of which the involved ones described by the moving reference system are relisted here. Bernoulli equation in Ωf gxi δi3 1

p 2 p0 ^ j1 Φ; j 1 1 Φ; j Φ; j 1 Φ; t 5 0: 2 Vδ ρ 2

(9.46)

^ as shown by Eq. (9.7), so that Here, the term Φ; j Φ; j =2 includes the first-order quantities caused by the ship speed V, it cannot be neglected but retained until further analysis. Equation of continuity in Ωf Φ; jj 5 0:

(9.47)

Boundary conditions On the seabed Γ b : The normal velocity of the water vanishes v^i 5 0, Φ; j ν j 5 0:

(9.48a)

On the free surface Γ f : The kinematic condition and dynamic condition should be satisfied: 1 ^ j1 Φ; j Þ; h 5 2 ðΦ; t 2 Vδ g

(9.48b)

^ j1 Φ; jt 1 V^ 2 δj1 δl1 Φ; jl 1 gΦ; j δ3j 5 0: Φ; tt 2 2Vδ

(9.48c)

On the far-field boundary Γ 6 N : An undisturbed condition Eq. (9.16b) is required, that is, Φ; j 5 φ0; j ;

^ tÞ: φ0 ðxi ; tÞ 5 φ0 ðxi 1 δi1 Vt;

(9.48d)

9.3.1.3 Interaction interface On the wet interface Σ, the normal velocity of the ship equals that of the adjacent fluid, and the water pressure is balanced by the solid stress, that is,

9.3.2

^ i1 1 Ui; t Þηi 5 2 Φ0; i ν i 5 2 Φ; i ν i ; ðVδ

(9.49a)

σij ν j 5 2 pν i :

(9.49b)

Equations in the modal space

9.3.2.1 Mode summation method for ship motions Natural frequencies and modes with orthogonality The ship is considered a linear elastic body with its motion assumed small; therefore its motions can be fully formulated by the mode summation form: Ui ðXj ; tÞ 5 Yi Q;  Yi 5 Y1 Y2 ?

 YN ;

 Q 5 q1

q2

? qN

T

;

(9.50)

where Yi denotes a mode matrix of the dry free
free ship, of which a representative mode YI is a function of material coordinates Xi , and Q is the corresponding generalized coordinate vector with its element qI being a time function relating to the mode YI . The subindex N denotes the retained mode number to describe the ship motions in approximate form. Normally, these modes include the 6 rigid modes of the ship and the N 2 6 elastic modes in the lower frequency range.

Hydroelasticity theory of ship
water interactions Chapter | 9

345

As discussed in Section 1.4.2.1, these modes and corresponding frequencies are the natural characteristics of ship structures, which are fully determined by the distributions of mass and stiffness of ship structure and do not involve any external forces. The natural modes and frequencies of the dry ship can be obtained by solving the eigenvalue problem defined by Eqs. (9.45a) and (9.45b) with F^ i 5 0 5 T^ i and Eq. (9.49b) with p 5 0, that is, @2 Ui 5 Cijkl Uk; lj ; inΩs ; @t2 ηj Cijkl Uk; l 5 0; on ST , Σ 5 S: ρ0

(9.51)

The natural modes and frequencies can be obtained by the following three approaches: (1) FE analysis, which is a generalized and powerful method to obtain them for every complex structure; (2) analytical forms for some engineering structure members, such as rods, shafts, beams, and plats, of which the related analytical solutions are available; and (3) the experimental method, which measures the required natural characteristics by experiment, such as for aircraft designs; ground vibration resonance tests of full-size aircrafts are still used to obtain the natural characteristics of a new aircraft for flutter. The natural modes and frequencies satisfy the orthogonal relationships demonstrated as follows. Assume that YI ; ωI and YJ ; ωJ represent two difference modes with their frequencies, respectively, so that these two modes, respectively, satisfy the equations UiI 2 2 ρ0 ωI YIi

5 YIi ejΩI t ; 5 Cijkl YIk; lj ;

ηj Cijkl YIk;l 5 0;

in Ωs ;

(9.52a)

on ST , Σ 5 S;

and UiJ 5 YJi ejΩJ t ; 2 ρ0 ω2J YJi 5 Cijkl YJk; lj ; ηj Cijkl YJk;l 5 0;

in Ωs ;

(9.52b)

on ST , Σ 5 S:

Multiplying the middle equation in Eq. (9.52a) by the mode YJi and then integrating the resultant equation over the ship domain Ωs , as well as using the Green theorem given in Section 3.5.1, we obtain ð ð ð ð ð 2 ρ0 ω2I YIi YJi dΩ 5 YJi Cijkl YIk;lj dΩ 5 YJi Cijkl YIk;l ηj dS 2 YJi;j Cijkl YIk;l dΩ 5 2 YJi;j Cijkl YIk;l dΩ; (9.53a) Ωs

Ωs

Ωs

Ωs

Ωs

where the boundary condition in Eq. (9.52a) is introduced. In a similar procedure but multiplying Eq. (9.52b) by YIi , we obtain ð ð 2 ρ0 ω2J YJi YIi dΩ 5 2 YIi;j Cijkl YJk;l dΩ: (9.53b) Ωs

Ωs

Noting the symmetry of elastic constant tensor, that is, Cijkl 5 Cklij , and the summation convention of tensors given in Chapter 2, Cartesian tensor and matrix calculus, we can demonstrate that the right-hand sides of Eqs. (9.53a) and (9.53b) are equal. Therefore a subtraction of Eq. (9.53b) from Eq. (9.53a) gives ð 2 2 ðωJ 2 ωI Þ ρ0 YIi YJi dΩ 5 0; (9.54) Ωs

so that, for two different modes I and J with different frequencies, we have the following orthogonal relationships: ( ð 0; I 6¼ J; ρ0 YIi YJi dΩ 5 MII ; I5J; Ωs ( (9.55) ð KII 0; I 6¼ J; 2 YIi;j Cijkl YJk;l 5 ωI 5 : KII ; I 5 j; MII Ωs Here, KII and MII are called the generalized stiffness and mass of the mode I of the ship. If the orthogonality is done for a unit generalized mass MII 5 1; the generalized stiffness KII 5 ω2I , which is often used in dynamic analysis. For a dry free
free ship, there are six rigid modes with frequencies of zero. These six modes span a subspace of rigid motion with six independent base vectors that can be normalized to satisfy Eq. (9.55), in which the generalized

346

Fluid
Solid Interaction Dynamics

stiffness KII 5 0 and the generalized masses for the sway, surge, and heave modes are the total mass of the ship, while those for the pitch, roll, and yaw modes are the inertial moments of ship rotations about three body axes, respectively. In this consideration, we can say that the natural modes of the ship, including the rigid modes of zero frequencies, satisfy the orthogonal relationship given by Eq. (9.55), which will be used in our dynamic response analysis. Mode equation of ship motions Substituting Eq. (9.50) into Eqs. (9.45a), (9.45b), (9.49a), and (9.49b), we obtain € 5 Cijkl Yk;lj Q 1 ρ F^ i ; ρ0 Y i Q 0 ηj Cijkl Yk;l Q 5 T^ i ;

(9.56)

ηj Cijkl Yk;l Q 5 2 pδij ηj ; _ 5 φ;i η : ηi Yi Q i

Premultiplying Eq. (9.56) by YTi and then integrating the resultant equation over the ship domain Ωs , using the Green theorem given in Section 3.5.1 as well as the orthogonality Eq. (9.55), we obtain the following results: ð ð € € € ρ0 YTi Yi QdΩ 5 MQ; M 5 ρ0 YTi Yi QdΩ 5 I 5 diagð1Þ; ð

Ωs

Ωs

Ωs

ð YTi Cijkl Yk;lj QdΩ 5 ð

S

5 ST

ð FT 5 ð K5

ST

Ωs

ð

ηj YTi Cijkl Yk;l QdS 2 YTi T^ i dS 2

ð Σ

Ωs

YTi;j Cijkl Yk;l QdΩ ð

YTi ηi pdS 2

5 FT 1 P 2 KQ; ð YTi T^ i dS; P 5 2 YTi ηi pdS; Σ

ð Fb 5

Ωs

Ωs

YTi;j Cijkl Yk;l QdΩ

(9.57)

YTi ρ0 F^ i dΩ;

YTi;j Cijkl Yk;l dΩ 5 Λ 5 diagðω2I Þ:

From this the mode equation of ship motions can be written in the form € 1 KQ 5 Fb 1 FT 1 P; MQ

€ 1 ΛQ 5 Fb 1 FT 1 P: or IQ

(9.58)

Here, Fb ;FT , and P represent the mode force vectors caused by the body force, traction force, and the water pressure, respectively.

9.3.2.2 Water domain Potential of velocity The total potential of velocity may be represented in the form ^ i Þ 1 Φðxi ; tÞ; Φðxi ; tÞ 5 Φðx

(9.59a)

^ Φ denote velocity potentials for the steady motion of the structure in calm water and the unsteady forced where Φ; motion in wave, respectively. As discussed in Section 9.1.4, when there is only a steady motion, the potential Φ^ is ^ i Þ; ^ 5 Vx ^ i δi1 1 φðx Φ

(9.59b)

^ i Þ is the relative velocity potential, and the velocity of the steady flow relative to the moving equilibrium where φðx frame of reference is v^i 5

@φ^ ^ ^ 5 @Φ=@x i 2 Vδi1 : @xi

(9.59c)

The unsteady component of the velocity potential Φðxi ; tÞ must include contributions from the distortions of the structure in the water, as well as the incident and diffracted wave fields, so that it is expressed as

Hydroelasticity theory of ship
water interactions Chapter | 9

Φðxi ; tÞ 5 φ0 ðxi ; tÞ 1 φD ðxi ; tÞ 1

N X

R

φI ðxi ; tÞ:

347

(9.60a)

I51

The quantity φ0 ðxi ; tÞ denotes the incident wave potential described in the moving frame of reference, which is given by the transformation Eq. (9.48d), that is, ^ i1 ; tÞ; φ0 ðxi ; tÞ 5 φ0 ðxi 1 Vtδ

(9.60b) R φI

denote the from the prescribed incident wave function φ0 ðx0i ; tÞ in the fixed frame of reference. The quantities φD ; diffracted wave potential and radiation potential arising from the response of the flexible structure. Physically, the diffraction potential is the velocity potential around a fixed rigid structure subject to waves, while the radiation potential is the one caused by structural oscillations. Based on the linear theory of the problem, we shall postulate the existence of a series of potentials R φI ; ðI 5 1; 2; 3; . . .; NÞ, each corresponding to one of the natural modes of the dry structure and hence to one of the generalized coordinates, qI ðtÞ. Thus, as in Eq. (9.50), the radiation potentials can be written in mode space form: R

R

φI ðxi ; tÞ 5 φI ðxi ÞqI ðtÞ;

(9.60c)

from which, when the first N natural modes of the dry structure are retained, Eq. (9.60a) can be expressed in matrix form: Φ 5 φ0 ðxi ; tÞ 1 φD ðxi ; tÞ 1 φR Q; h i φR 5 φR1 φR2 ? φRN :

(9.60d)

Here, φR denotes a matrix of radiation modes, and the generalized coordinate vector Q has been defined by Eq. (9.50). Since the system is linear, the generalized coordinate vector Q and the unsteady potential for the sinusoidal wave excitation formulated by Eq. (9.44b), respectively, take the oscillations with the same frequency ωe in the forms Q 5 Qejωe t ;

h i Φðxi ; tÞ 5 Φðxi Þejωe t 5 φ0 ðxi Þ 1 φD ðxi Þ 1 φR Q e jωe t :

Bernoulli equation Substituting Eqs. (9.59a) and (9.59b) into Bernoulli Eq. (9.46), we obtain its nonlinear form:

1 p 2 p0 1 2 gxi δi3 1 1 v^j Φ;j 1 v^j v^j 2 V^ 1 Φ;j Φ;j 1 Φ;t 5 0; ρ 2 2 and its linearized form: gxi δi3 1

p 2 p0 1 2 1 v^j Φ;j 1 v^j v^j 2 V^ 1 Φ;t 5 0: ρ 2

(9.60e)

(9.61a)

(9.61b)

The Bernoulli equation gives the relationship between the fluid pressure and the potential velocity at point xi at time t. For the static equilibrium state and the steady motion cases, Eq. (9.61b) reduces to Eqs. (9.18c) and (9.24), respectively. Boundary conditions Based on Eqs. (9.59a)
(9.59c) expressing the total potential of velocity in the two parts, the steady one ^ i Þ and the unsteady forced one Φ, and noting the steady potential Φ^ satisfying the boundary conditions ^ 5 Vx ^ i δi1 1 φðx Φ of the fluid, we can derive the boundary conditions satisfied by the unsteady potential as follows: Φ; j ν j 5 0;

on Γ b ;

^ j1 Φ; jt 1 V^ δj1 δl1 Φ; jl 1 gΦ; j δ3j 5 0; Φ; tt 2 2Vδ 2

Φ; j 5 φ0; j ;

on Γ 6 N :

on Γ f ;

(9.62)

348

Fluid
Solid Interaction Dynamics

9.3.2.3 Interaction interface Kinematic condition When Eq. (9.59a) is used, the kinematic consistency condition in Eq. (9.49a) on the wet interface Σ now takes the form ^ ;i ηi 1 Φ;i ηi ; ^ i2 1 Ui;t Þηi 5 2 ðΦ^ ;i 1 Φ;i Þν i 5 Φ ðVδ

(9.63a)

from which, when Eq. (9.59b) is introduced, it follows ^ ;i 2 Vδ ^ i1 Þηi 5 ðUi;t 2 φ^ ;i Þηi 5 Φ;i ηi : Ui;t ηi 2 ðΦ

(9.63b)

^ i Þ is the velocity potential of the steady flow relative to the moving reference Here, as defined by Eq. (9.59b), φðx ^ frame, so that its derivative φ;i gives the velocity of the steady flow relative to the moving coordinate system. Therefore Eq. (9.61b) implies that the normal velocity of the unsteady flow on the current wet interface must be consistent with the structure velocity relative to the steady flow. Eq. (9.61b) is defined on the current wet interface, on which a material point Xi moves to the current position: xi 5 Xi 1 Ui ;

(9.64a)

so that the normal vector ηi on the wet interface, according to Eqs. (3.36e) and (3.29), becomes a current vector ηci 5 ηi 1 eijk Ωj ηk ;

Ωj 5

1 ejkl Ul;k ; 2

(9.64b)

and the velocity of steady flow φ^ ;i 5 v^i has its current value v^ci 5 v^i 1 Uj v^i;j :

(9.64c)

Substituting Eqs. (9.64b) and (9.64c) into Eq. (9.61b), we obtain its linearized form: Φ;i ηi 5 ½Ui;t 2 ðv^i 1 Uj v^i;j Þðηi 1 eijk Ωj ηk Þ 5 ðUi;t 1 eijk Ωj v^k 2 Uj v^i;j Þηi ; or

@Φ _ 1 Ω 3 v^ 2 ðUUrÞ^vUη; 5 ½U @η

(9.64d)

where the second-order quantities have been neglected, and an equality 2 v^i eijk Ωj ηk 5 eijk Ωj v^k ηi and the steady condition v^i ηi 5 0 given by Eq. (9.27a) have been introduced. If we assume that the ship is rigid, its strain given by Eq. (3.31.b) vanishes, that is, 1 ðUi;j 1 Uj;i Þ 5 0; 2

(9.65a)

1 21 eijk ejrs Us;r v^k 5 ðδir δks 2 δis δkr ÞUs;r v^k 2 2 21 ðUs;i v^s 2 Ui;s v^s Þ 5 2 1 5 Ui;s v^s 2 ðUs;i 1 Ui;s Þv^s 5 Ui;s v^s 2 Eis v^s 5 Ui;s v^s ; 2 Ω 3 v^ 5 ð^vUrÞU;

(9.65b)

Eij 5 from which and from Eq. (2.14), it follows eijk Ωj v^k 5

or

so that for rigid ships, Eq. (9.64d) can be expressed in the form @Φ _ 1 ð^vUrÞU 2 ðUUrÞ^vUη 5 ½U _ 1 r 3 ðU 3 v^ ÞUη; 5 ½U @η

(9.65c)

in which a vector identity has been introduced. This boundary condition for rigid ships was derived by Timman and Newman (1962) to account in a consistent manner for the interaction between steady and oscillatory flow fields.

Hydroelasticity theory of ship
water interactions Chapter | 9

349

Fluid pressure on wet interface Here, the Bernoulli Eq. (9.61b) is used to determine the hydrodynamic force applied on the wet interface. As defined in Section 9.1.1, the origins of both the fixed and the moving coordinate systems are chosen on the calm free surface, so that a point xi on the wet interface is expressed as follows: x i 5 Xi ; ^ i1 1 Xi 1 U^ i ; xi 5 Vδ ^ i1 1 Xi 1 Ui ; xi 5 Vδ

for static equlibrium state; for steady motion state; for oscilation state;

(9.66a)

where U^ i and Ui denote the displacements of the ship in the steady motion and in the oscillation motion, respectively. Respectively substituting the four expressions in Eqs. (9.64a)
(9.64d) into the Bernoulli Eqs. (9.18c), (9.24), and (9.61b), we obtain ps 2 p0 5 0; xi AΣ s ; gXi δi3 1 (9.66b) ρ p^ 2 p0 v^2 2 V^ 1 5 0; gðXi 1 U^ i Þδi3 1 ρ 2 2

gðXi 1 Ui Þδi3 1

xi AΣ^;

2 p 2 p0 v^2 2 V^ 1 v^j Φ; j 1 1 Φ; t 5 0; ρ 2

(9.66c) xi AΣ:

(9.66d)

Here, Σs , Σ^, and Σ denote the wet interfaces in the static equilibrium state, in steady motion, and in oscillatory motion, respectively. Since the fluid velocity v^i in the steady motion is finite, the term v^2 on the current wet interface is derived by the following differential approximation based on the static wet interface, that is: v^2Σ 5 v^2 1 Ui ðv^2 Þ;i ;

xi AΣ s ;

(9.66e)

from which, Eq. (9.66d) can be rewritten as p 2 p0 v^2 2 V^ 1 v^j Φ; j 1 1 Φ; t 1 Ui ðv^2 Þ;i 5 0; ρ 2 2

gðXi 1 Ui Þδi3 1

xi AΣ s ;

(9.66f)

on the original static wet interface, in which an the last additional term on the left-hand side is added. The fluid pressure applied on the wet surface of the ship can be derived from Eq. (9.66f) as p 5 p0 1 ps 1 pd ; ps 5 2 ρgXi δi3 ;

xi AΣ s ;

2 pst 5 2 ρðv^2 2 V^ Þ=2;   pd 5 2 ρ gUi δi3 1 v^j Φ;j 1 Φ;t 1 Ui ðv^2 Þ;i ;

(9.66g)

in which p0 , ps , pst , and pd represent the atmosphere pressure, static water pressure, steady motion pressure, and dynamic pressure caused by the ship oscillations. Since the atmosphere pressure is also applied on the dry surface of the ship, it is often set as p0 5 0 in ship dynamics analysis. The pressure ps and the term pst are ^ i1 , functions of special coordinates only but independent of time t. Approximately, if we assume v^i 5 2 Vδ which neglects the variation of fluid velocity in the steady motion around the ship structure, the dynamic pressure reduces to   ^ ;1 1 Φ;t ; xi AΣs : pd 5 2 ρ gUi δi3 2 VΦ (9.66h)

9.3.2.4 Generalized fluid forces Based on Eqs. (9.50) and (9.60e), the hydrodynamic pressure given by Eq. (9.66g) can be expressed in the form pd 5 2 ρ½gYi Qδi3 1 v^j ðφ0;j 1 φD;j 1 φR;j QÞ 1 jωe ðφ0 1 φD 1 φR QÞ 1 Yi Qðv^2 Þ;i ejωe t ; which, when substituted into Eq. (9.57), gives the generalized fluid force vector

(9.67)

350

Fluid
Solid Interaction Dynamics

P 5 PS 1 PST 1 PRS 1 PW 1 PR ; ð PS 5 ρgYTi ηi Xj δj3 dS; Σ

ð PST 5

ρYTi ηi ðv^2 2 V^ Þ ; 2dS Σ 2

PRS 5 2 kQ; k 5 kS 1 kST ; ð ð kS 5 2 ρgYTi ηi Yj δj3 dS; kST 5 2 ρYTj ηj Yi ðv^2 Þ;i dS; Σ Σ ð PW 5 ejωe t ρYTi ηi ½v^j ðφ0;j 1 φD;j Þ 1 jωe ðφ0 1 φD ÞdS; Σ ð   PR 5 ejωe t ρYTi ηi v^j φR;j 1 jωe φR QdS:

(9.68)

Σ

Here, PS ; PST ; PRS ; PW , and PR denote the vectors of hydrostatic force, steady motion force, restoring buoyancy force, wave exciting force, and radiation force, respectively. The matrix k is an added buoyancy stiffness matrix. Historically, in the wave exciting force, the component involving the incident wave function φ0 has become known as the Froude
Krylov exciting force (Froude, 1861; Kriloff, 1896). In the hydroelasticity theory of ships, the radiation force is normally divided into the following two parts: PR 5 PRM 1 PRD ; € PRM 5 ω2e ejωe t mR Q 5 2 mR Q; _ PRD 5 2 jωe ejωe t dR Q 5 2 dR Q; where

ð

  ρYTi ηi v^j φR;j 1 jωe φR dS; Σ ð   21 dR 5 2 ωe Im ρYTi ηi v^j φR;j 1 jωe φR dS mR 5 ω22 e Re

(9.69a)

(9.69b)

Σ

are called the added mass and damping matrices from the water to the ship owing to radiation oscillations.

9.3.2.5 Generalized equation of motion Substituting Eqs. (9.68), (9.69a), and (9.69b) into Eqs. (9.60a)
(9.60e), we obtain the generalized equation of the ship motion in the mode space, that is, € 1 ðD 1 dR ÞQ _ 1 ðK 1 kÞQ 5 Fb 1 FT 1 PS 1 PST 1 PW ; ðM 1 mR ÞQ

(9.70)

in which the damping matrix of structure D has been added. This damping matrix can be obtained by experiment (Xing, 1975) or be given by D 5 αM 1 βK (see, e.g., Bathe, 1996) where the parameter α and β are also determined by experiment. Equation of static equilibrium For the static equilibrium state of the structure with the gravity body force Fb and the surface traction force FT 5 0; there is a static solution QS satisfying the equation ðK 1 kS ÞQS 5 Fb 1 PS ;

(9.71a)

from which, the static deformation of the structure can be obtained. Equation of steady motion Furthermore, for the flexible structure with steady traveling in calm water, there exists a steady-state solution QST with frequency ωe 5 0 satisfying the equation

Hydroelasticity theory of ship
water interactions Chapter | 9

ðK 1 kÞQST 5 Fb 1 PS 1 PST ;

351

(9.71b)

which, as shown previously by Bishop and Price (1979), gives a modal description of structural distortion in still water, trim, sinkage, etc. Equation of flexible structure subject waves The generalized linear equation of motion for a freely floating structure that is moving or stationary in waves, after extracting from Eqs. (9.72a)
(9.72g) the portion accounting for the steady-state solution in Eq. (9.71b), is written in the form € W 1 ðD 1 dR ÞQ _ W 1 ðK 1 kÞQW 5 PW : ðM 1 mR ÞQ

(9.71c)

The generalized equations in Eqs. (9.71a)
(9.71d) are matrix equations of order N 3 N, of which, if the mass orthogonal modes of structure are adopted, the generalize mass and stiffness matrices of structure are diagonal ones, as given by Eq. (9.57), that is, M 5 I 5 diagð1Þ;

K 5 Λ 5 diagðω2I Þ:

(9.71d)

Furthermore, for three-dimensional floating structures, the first six natural frequencies are zero with the corresponding rigid modes, while for 2D cases, the first three natural frequencies are zero. The matrix k is the contribution of the water buoyancy, which does not vanish even for rigid floating structures, but it will be negligibly small for fixed structures with no floating wet boundaries.

9.3.3

Numerical solutions

As mentioned at the beginning of this chapter, compared with the mixed FE
BE model for water
structure interactions presented in Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, the essential difference for the hydroelasticity analysis of ships is using different Green functions. In Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, the Green functions given by Eqs. (8.6) and (8.7) satisfy the Laplacian equations defined in an infinite 2D or 3D domain, respectively, while the Green functions normally used in hydroelasticity analysis are further required to satisfy the boundary condition on the free surface. Therefore these Green functions are called free surface Green functions, discussed as follows.

9.3.3.1 Free surface Green functions Equations for free surface Green functions A free surface Green function in the form Gðxj ; ξj Þejωt is required to satisfy the equation G; jj 5 Δðxj 2 ξj Þ 5 Δðx1 2 ξ1 ÞΔðx2 2 ξ2 ÞΔðx3 2 ξ3 Þ; G; j δj3 5 0; xj δj3 5 2 H; ω2 G; j δj3 5 κG; xj δj3 5 0; κ 5 ; g pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi G is finite; r 5 ðxj 2 ξj Þðxj 2 ξj Þ- 1 N:

(9.72a)

Here, κ is a wavenumber for infinite-depth water, the second and third equations represent the seabed and free surface conditions, respectively, while the last condition is called the Sommerfeld radiation condition discussed in Chapter 5, Solutions of some linear fluid
solid interaction problems. If we consider the moving speed of the structure, the free surface condition on x3 5 0 in Eq. (9.72a) should be replaced by the one in Eq. (9.15d), that is, 2 2ω2 G 1 V^ δj1 δl1 G;jl 1 gG;j δ3j 5 0;

on x3 5 0:

(9.72b)

This gives a Green function physically corresponding to a source of oscillatory strength, moving constant velocity ^ Havelock (1928) derived this Green function in the following method. He considered the two source points, respecV. tively, at point ξi and its mirror image point ξ~ i with respect to the free surface plane x3 5 0, so that the dynamic pressure p on the free surface plane vanishes, since the disturbances caused by the two source points cancel each other out.

352

Fluid
Solid Interaction Dynamics

Therefore, the linear Bernoulli equation, when neglecting the nonlinear term and constant mass density in Eqs. (9.79a) and (9.79b), gives @G ~ 5 p0 ; 1 gh 1 μG @t

(9.72c)

where μ~ denotes a viscous coefficient of the water. Taking the derivative with respect to time t, we obtain @2 G @h @G 5 0: 1 g 1 μ~ @t2 @t @t

(9.72d)

Considering the velocity potential caused by the moving sources disturbance is a wave with speed V^ in the x2 direction ^ i1 Þ; Gðxi ; tÞ 5 Gðx0i 2 Vtδ

(9.72e)

and for linearized cases @h=@t 5 @G=@x3 , we can transfer Eq. (9.72d) into the form @2 G g @G μ @G 1 2 2 5 0; 2 @x1 V^ @x1 V^ @x3 to be satisfied at x3 5 0. The solution of Eq. (9.72f) is given in the form 2 3 1 4 1 2 1 1r~ 1 Ψðxi ; ξi Þ5; Gðxi ; ξ i Þ 5 4π r r 2 5 ðxi 2 ξi Þðxi 2 ξi Þ; r~2 5 ðxi 2 ξ~ i Þðxi 2 ξ~ i Þ; ξ~ 1 5 ξ1 ; ξ~ 2 5 ξ2 ; ξ~ 3 5 2 ξ 3 :

(9.72f)

(9.72g)

Here the function Ψ satisfies Laplacian equation Ψ ;jj 5 0 everywhere; see the detailed explanation and discussion by Brard (1972). Two-dimensional case For 2D flow governed by Eq. (9.72a) in the x1 2 x3 plane, we can use the separation of variables, as discussed in Section 5.2.3.2, to obtain a set of eigenfunctions: Zn ðx3 Þ 5 cos λn ðx3 1 HÞ=Z^n , (n 5 0, 1, 2, . . .), of which λn are the roots of the equation κ tanðλn HÞ 5 2 ; (9.73a) λn with λ0 being purely imaginary and λn ðn $ 1Þ being purely positive real numbers as, respectively, shown by Figs. 5.11 and 5.10. Z^n is chosen so that the eigenfunctions are orthogonal, that is, ð0 Zm Zn dx3 5 δmn ; (9.73b) 2H

which gives

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi cosðλn HÞ sinðλn HÞ 1 λn H Z^n 5 : 2λn

(9.73c)

Using the eigenfunctions expansion method, the Green function and the Delta function in Eqs. (9.72a)
(9.72g) can, respectively, take the forms Gðxj ; ξj Þ 5

N X

Xn ðx1 ÞZn ðx3 Þ;

n50

Δðx3 2 ξ 3 Þ 5

N X n50

(9.74) Zn ðξ3 ÞZn ðx3 Þ:

Hydroelasticity theory of ship
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353

From these expressions, when substituted into the first equation in Eqs. (9.72a)
(9.72g) for the 2D case, it follows that  2 X N N X @ 2 2 λ Xn ðx1 ÞZn ðx3 Þ 5 Δðx1 2 ξ1 Þ Zn ðξ3 ÞZn ðx3 Þ: (9.75) n 2 @x2 n50 n50 Premultiplying this equation by the eigenfunction Zn ðx3 Þ, then integrating it with respect to x3 from 2 H to 0 and using the orthogonal relationship in Eq. (9.73b), we obtain  2  @ 2 2 λn Xn ðx1 Þ 5 Δðx1 2 ξ1 ÞZn ðξ3 Þ; (9.76a) @x22 for which the solution can be obtained by an initial value problem (Karman and Biot, 1940). Integrating Eq. (9.76a) with respect to x1 from ξ12 to ξ11 , we obtain ξ dXn  21 2 λ2n Xn ðx1 ÞΔξ1 5 Zn ðξ 3 Þ; (9.76b) dx1 ξ22 x1 Aðξ12 ; ξ 11 Þ;

Δξ1 5 ξ 11 2 ξ12 ;

from which, when Δξ1 -0, it follows dXn dXn dXn ðξ11 Þ 2 ðξ12 Þ 5 2 ðξ Þ 5 Zn ðξ3 Þ; (9.76c) dx1 dx1 dx1 1       where dXn =dx1 ðξ11 Þ 2 dXn =dx1 ðξ12 Þ 5 2 dXn =dx1 ðξ 1 Þ is used. Therefore the solution of Eq. (9.76a) can be obtained by solving the following initial value problem: 0 1 2 @ 2 @ 2 λn AXn ðx1 Þ 5 0; @x21 (9.76d) dXn Zn ðξ3 Þ : ðξ Þ 5 2 dx1 1 The solution, which is nondivergent when x1 -N, is given by Xn ðx1 Þ 5

2 Zn ðξ3 Þ 2λn jx1 2ξ1 j e ; 2λn

(9.76e)

so that the Green function in Eqs. (9.76a)
(9.76h) is obtained as e2λn jx1 2ξ1 j Zn ðξ3 ÞZn ðx3 Þ 2λn n50 N X e2λn jx1 2ξ1 j 5 2 cos λn ðξ3 1 HÞcos λn ðx3 1 HÞ 2 2λn Z^n n50

Gðxj ; ξ j Þ 5

5

N X

N X n50

2

2

(9.76f)

e2λn jx1 2ξ1 j cos λn ðξ 3 1 HÞcos λn ðx3 1 HÞ : cosðλn HÞ sinðλn HÞ 1 λn H

Using Eq. (9.73a), we obtain κ sinðλn HÞ 5 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; κ2 1 λ2n

λn cosðλn HÞ 5 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; κ2 1 λ2n

(9.76g)

from which, when this is substituted into Eq. (9.76f), it follows that the Green function can be rewritten in the form qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi N X κ2 1 λ2n e2λn jx1 2ξ1 j cos λn ðξ3 1 HÞ cos λn ðx3 1 HÞ G5 : (9.76h) κλn 2 ðκ2 1 λ2n Þλn H n50

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As a special case of Eqs. (9.76f) and (9.76h), if the source point (ξ1 ; ξ3 ) is on the free surface, that is, ξ3 5 0, we have the free surface Green function in the form G5

N X

2

n50

or G5

e2λn jx1 2ξ1 j cosðλn HÞ cos λn ðx3 1 HÞ cosðλn HÞsinðλn HÞ 1 λn H

N X

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi κ2 1 λ2n e2λn jx1 2ξ1 j cosðλn HÞ cos λn ðx3 1 HÞ

n50

κλn 2 ðκ2 1 λ2n Þλn H

(9.77a)

:

(9.77b)

In the two papers by John (1949, 1950), the forms of free surface Green functions and many important results on the existence and uniqueness of solutions to the water-wave problem were described in detail. Source points and multiple solutions that satisfy the linearized free surface condition are described systematically by Wehausen and Laitone (1960), in which an integration form of the free surface Green function with the source point at the origin on the free surface is formulated by ðN 1 eλx3 cosðλx1 Þ Gðx1 ; x3 Þ 5 2 lim dλ : (9.77c) 2π μ-0 1 0 λ 2 ðω2jμÞ2 =g As discussed by Newman’s important review paper (Newman, 1978), the parameter μ can be interpreted as a Rayleigh viscosity coefficient, representing a fictitious dissipation that suppressed incoming waves at infinity. Since the imaginary part of the pole λ 5 ðω2jμÞ2 =g is negative, μ may be set to zero if the contour of integration is deformed to pass above the pole at λ 5 ω2 =g  κ. Some discussions on ocean surface waves can also be read in Mei (1983). Three-dimensional case The 3D Green function that satisfied the free surface condition in Eq. (9.72b) was derived initially by Haskind (1946a, b) and subsequently by Brard (1948). The solution is described by Wehausen and Laitone (1960), Lighthill (1967), and Newman (1978). For the source point at the origin (ξi 5 0) of the coordinate system on the free surface, this Green function, under the reference coordinate system defined by Fig. 9.1, is given by ðN ð 2π 21 e½λx3 1jλðx1 cos ϑ1x2 sin ϑÞ Gðxi Þ 5 2 lim λdλ dϑ ; (9.78) ^ cos ϑÞ2 =g 8π μ-0 1 0 λ 2 ðω2jμ1 Vλ 0 in which the integration angle ϑ starts from x1 axis in the right-hand system. In Newman’s review paper (Newman, 1978), some more detailed discussion on this function with its various approximations is given; interested readers may refer to it.

9.3.3.2 Green identity and the solution of ship excited by waves ^ i1 tÞe jωe t , which causes vibrations We assume that the incident wave in Eq. (9.16b) is given by φ0 ðxi Þe jωe t 5 φ0 ðxi 1 Vδ of this linear system in the same frequency ω, so that the velocity potential in Eq. (9.60a) can be expressed as   Φðxi ; tÞ 5 φðxi ; tÞ 5 φ0 ðxi Þ 1 φD ðxi Þ 1 φðxi Þ ejωe t ; (9.79a) in which φðxi Þejωe t denotes the radiated velocity potential, of which the index “R” in Eq. (9.60a) is neglected for convenience in the following math. The diffraction potential can be obtained by solving equations for a rigid ship with no disturbance, as in Section 9.1.4; the incident wave potential is given in Section 9.2, so that the main aim is to determine the radiated one, which satisfies the following equations: φ; jj 5 0;

xi AΩf ;

φ; j ν j 5 0;

xi AΓ N , Γ b ;

2 ω2e φ 1 V^ φ; 11 1 gφ; j δ3j 5 0; 2

jωe U i ηi 5 2 φ; i ν i ;

xi AΣ:

xi AΓ f ;

(9.79b)

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Here, U i ðxr ; tÞ 5 U i ðxr Þejωe t denotes a disturbed displacement of the structure, and we have used the boundary conditions on the water boundary, such as on the free surface, Eq. (9.15d). We can determine this disturbed velocity potential as follows. Green third identity Using the similar derivation shown in Section 8.1.2 for Eqs. (9.79b) and (9.72a), we can obtain the Green third identity: ð   φðxi Þ 5 Gðxi ; ξi Þφ; j ðξi Þ 2 φðξ i ÞG; j ðxi ; ξi Þ ν j dΓ : (9.80a) Γ

for the ship
water interaction system under the moving coordinate system shown in Fig. 9.2. Here, G denotes the Green function in Eq. (9.72g) with Ψ 5 G chosen and surface integration over the boundary Γ 5 Γ b , Γ f , Γ N , Σ of the water domain. If point xi moves onto the boundary, Eq. (9.80a) turns to ð   cφðxi Þ 5 Gðxi ; ξi Þφ; j ðξ i Þ 2 φðξi ÞG; j ðxi ; ξi Þ ν j dΓ ; (9.80b) Γ

where c 5 0:5 for a smooth boundary, as shown in Eq. (8.17b). On the fixed boundary Γ b and the far-field boundary Γ N , both the Green function and the velocity potential satisfy the same boundary condition, so that the integration on Γ b , Γ N vanishes. On the free surface Γ f ; (x3 5 0 5 ξ3 ), ν 5 ð0; 0; 1Þ; the functions φ and G satisfy Eq. (9.72f) (μ 5 0), and therefore the integration on Γ f in Eq. (9.80b) becomes 2ð

V^ 2 Gðxi ; ξi Þφ;11 1 φðξi ÞG;11 dΓ g Γf 2ð n     o V^ 5 2Gðxi ; ξ i Þφ;1 ;1 1 φðξi ÞG;1 ;1 dΓ (9.80c) g Γf 2ð   V^ Gðxi ; ξi Þφ;1 2 φðξi ÞG;1 ν 1 dl: 5 g Cf Here, the contour C with its positive direction shown in Fig. 9.3 is the intersection curve of the structure’s outer surface and the mean calm water surface (x3 5 0), which is the interboundary of the integrated free surface area. In the last step of mathematical derivation, we have used the Green theorem to transfer the integration on the Γ f plane to its boundary C, considering its outer boundary at infinity with a vanishing integrated value. The positive line element dl in Eq. (9.80c) follows the right-hand rule in the coordinate system, so that it is along the negative direction of the contour C. As a result, Eq. (9.80a) reduces to ð   φðxi Þ 5 Gðxi ; ξi Þφ;j ðξi Þ 2 G;j ðxi ; ξi Þφðξ i Þ ν j dΓ Σ

1

2ð   V^ Gðxi ; ξi Þφ;1 2 φðξi ÞG;1 ν 1 dl: g Cf

(9.80d)

FIGURE 9.3 Free surface of water domain outside the ship, the positive direction of the contour C on the internal boundary of the water, and the direction of line element dl.

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Fluid
Solid Interaction Dynamics

Physically, φ;1 ν 1 is the component on axis o 2 x1 of the fluid velocity on the wet interface, so that the wet interface condition in Eq. (9.79b) confirms φ;j ν j 5 U j;t ν j 5 jωU j ν j 5 Qðξ i Þ; φ;1 ν 1 5 ðU j;t ν j ÞνUν 1 g1 5 Qðξ i Þðν j gj ÞUν 1 g1 5 Qðξi Þν 21 ;

(9.80e)

from which, when substituted into Eq. (9.80d), it follows φðxi Þ 5 RU ðxi Þ 1 Rp ðxi Þ; ð 2ð V^ RU ðxi Þ 5 Gðxi ; ξ i ÞQðξi ÞdΓ 1 Gðxi ; ξ i ÞQðξi Þν 21 dl; g Cf Σ ð 2ð V^ Rp ðxi Þ 5 2 φðξi ÞG;j ðxi ; ξi Þν j dΓ 2 φðξi ÞG;1 ν 1 dl: g Cf Σ

(9.80f)

For a point xi , the dynamic pressure in linearized Eq. (9.8c) can be obtained as p 5 2 ρφ;t 5 2 jωe ρφ; so that 2j Rp ðxi Þ 5 ωe ρ

(ð

) 2ð V^ pðξ i ÞG;j ðxi ; ξi Þν j dΓ 1 pðξ ÞG;1 ν 1 dl : g Cf i Σ

(9.80g)

(9.80h)

Physically, Eq. (9.80f) is explained as follows. On considering the fluid in the domain Ωf as a free-body diagram, its motion is excited by the solid displacement Ui and the pressure p on the wet interface Σ , Cf ; therefore RU ðxi Þ and Rp ðxi Þ represent the fluid velocity potentials contributed by the displacement and pressure on the FSI interface, respectively. Historically, as discussed by Brard (1972), the line integral along the waterline in Eq. (9.80g) is generally omitted by the authors, who attempt to represent the hull by singularity distributions over its surface. The necessity of such a line, however, was recognized by Wehausen (1963) and by Peters and Stoker (1957). Also, the possibility of using a line integral along the waterline was mentioned by Kotik and Morgan (1969) in order to express the difference between the wave resistances yielded at zero Froude number by a source distribution and by a normal doublet distribution over the hull. An analogous remark was expressed but not used by Breslin (1965). A very important report by Gotman (2002) presented very detailed calculation results of ship resistances based on different integration formulations proposed by many authors, which were compared with the corresponding results of experiments. In this outstanding report, nearly all historical publications are listed and discussed by the author, so that interested readers may refer to this publication for more details on the problem discussed here. Considering that the plane of symmetry of the structure is o 2 x1 x3 and that the normal vector on the wet interface satisfies ν 1 ðξ 1 ; ξ 2 ; ξ 3 Þ 5 ν 1 ðξ 1 ; 2 ξ2 ; ξ3 Þ; ν 2 ðξ 1 ; ξ 2 ; ξ 3 Þ 5 2 ν 2 ðξ1 ; 2 ξ2 ; ξ3 Þ; (9.81a) ν 3 ðξ 1 ; ξ 2 ; ξ 3 Þ 5 ν 3 ðξ 1 ; 2 ξ2 ; ξ3 Þ; Bishop et al. (1986) and Wu et al. (1993) define the following composite potential functions: 6

5 Gðξ1 ; ξ2 ; ξ3 Þ 6 Gðξ1 ; 2 ξ 2 ; ξ 3 Þ;

6

5 φðξ1 ; ξ2 ; ξ3 Þ 6 φðξ1 ; 2 ξ 2 ; ξ 3 Þ;

G φ

(9.81b)

Q 6 5 Qðξ1 ; ξ2 ; ξ3 Þ 6 Qðξ1 ; 2 ξ 2 ; ξ 3 Þ; in which the notations “ 1 ” and “ 2 ” define the symmetrical and the antisymmetrical functions to the plane o 2 x1 x3 , respectively. They represent the velocity potential in the form ð 2 ð V^ 1 6 6 6 φ ðxi Þ 5 G ðxi ; ξi ÞQ 6 ðξi ÞdΓ 1 G ðxi ; ξ i ÞQ 6 ðξi Þν 21 dl; (9.81c) 4π Σ 4πg Cf

Hydroelasticity theory of ship
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357

in which a constant factor 1=4π, as used in Eq. (9.47g), is included, and the term Rp ðxi Þ in Eq. (9.80g) is excluded, although it may not vanish.

9.3.3.3 Solution of the ship
water interaction in hydroelasticity Generalized coordinate equation of structure From Eq. (9.50), the disturbance motion QW in Eq. (9.71c) for the structure can be denoted as Ui ðXj ; tÞ 5 Yi QW ejωe t ;   Y i 5 Y 1 Y2 ? Y N ;  T Q W 5 q1 q2 ? qN W ;

(9.82a)

which, when substituted into Eq. (9.71c) and PW 5 PW ejωe t is considered, gives ½ 2 ω2e ðM 1 mR Þ 1 jωe ðD 1 dR Þ 1 ðK 1 kÞQW 5 PW ;

(9.82b)

½ 2 ω2e ðI 1 mR Þ 1 jωe ðD 1 dR Þ 1 ðΛ 1 kÞQW 5 PW ;

(9.82c)

or

for using the mass orthogonal modes of structure as shown in Eq. (9.71d). Matrix equation for the radiated velocity potential The radiated velocity potential in Eq. (9.80f), caused by the disturbance motion of structure denoted by generalized coordinate QW in Eq. (9.71c), can be written, as defined in Eq. (9.60e), in mode summation form: h i φðxi Þ 5 φR QW ; φR 5 φR1 φR2 ? φRN : (9.83a) Also, based on Eq. (9.50), we can rewrite the source Qðξi Þ in Eq. (9.80e) in matrix form: Qðξ i Þ5 jων i Yi QW ;  Y i 5 Y 1 Y2 ? Y N ;  T Q W 5 q1 q2 ? qN W : Substituting Eqs. (9.83a) and (9.83b) into Eq. (9.80f), we obtain   R φ ðxi Þ 5 jωe RU ðxi Þ 1 Rp ðxi Þ QW ; where the neglected index “R” is added, and the matrices 2 3 ð ^2 ð V RU ðxi Þ 5 jωe 4 Gν j Yj dΓ 1 Gν j Yj ν 21 dl5; g Cf Σ 2 3 ð ^2 ð V Rp ðxi Þ 5 2 4 φR G;j ν j dΓ 1 φ G;1 ν 1 dl5: g Cf R Σ

(9.83b)

(9.83c)

(9.83d)

Solution process of ship
water interaction The following process can be used to complete the calculation of the ship
water interaction analysis using the hydroelasticity approach: 1. Conduct a mode analysis of the dry free
free ship by using a FE program, from which the first N natural modes Yi and frequencies Λ can be obtained. These N natural modes, of which the first six are rigid modes of zero frequency, enable the construction of a subspace to describe motions of structure. 2. Run a pure fluid simulation to obtain the diffraction potential, φD ðxi Þ; xi AΣ, of the fluid, in which the structure is considered rigid and fixed in the space, so that there is no disturbance motion of structure.

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Fluid
Solid Interaction Dynamics

3. Run a pure fluid simulation to obtain the radiation potential ϕRI ðxi Þ; xi AΣ, excited by the mode I (I 5 1, 2,. . ., N), of the structure. Since the motion of the structure is given, the analysis is a pure fluid problem. 4. Solve Eqs. (9.82a)
(9.82c) to obtain the generalized coordinate QW , which is substituted into Eqs. (9.83a)
(9.83d) to derive the radiated potential. The ship
water interaction problem is solved. In this process, in general, no theoretical solutions for steps 2 and 3 are available, so numerical works have to be used. The idea is to divide the wet interface into many elements, as in Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, of which for each element k the velocity potential φk can be assumed constant, since the element is sufficiently small. Then, based on a Green formulation such as Eq. (9.80f), establish a numerical equation to derive the variable φk .

9.3.3.4 Comparison with finite element
boundary element method The approach described in Section 9.3.3.3 is based on a mode summation method, as well as the solutions of two types of pour fluid problems in steps 2 and 3 to solve water
ship interaction problems. The mixed FE
BE method given in Chapter 8, Mixed finite element
boundary element model for linear water
structure interactions, provides another approach to solving the same problem. In this case, the wet interface is still divided into many elements, and for each element, the total velocity potential, both radiated and diffracted, may be expressed by a linear function φ 5 Hφ to allow its derivative with respect to the existing coordinates in order to derive the fluid dynamic pressure by means of Eq. (9.67), which is to be used in the interpolation matrix and the node potential vector of the element. Based on these approximations of the total potential on the wet interface, we can establish a set of coupling equations using Eqs. (9.57) and (9.80b). In this set of equations, the variable vectors are the solid generalized coordinate vector Q and the global node potential vector φG for the wet interface. The detailed math process is similar to the FE and BE methods, and it is ignored in this book.

9.3.4

Examples

To illustrate the applications of the approach discussed in Section 9.3.3 as well as to show more detail on the method, we investigate the following simple examples. These examples can be completed quite simply by hand in order to show more clearly the detailed method in each process.

9.3.4.1 Dynamic response of a two-dimensional beam-like ship to an incident wave As shown in Fig. 9.4, a 2D uniform beam-like ship is traveling in a velocity V^ along x1 direction. The length, width, mass density per unit length, and bending stiffness are denoted by L, B 5 1, ρS , and EJ, respectively. Assume that the central line of the beam is axis o 2 x1 on the free surface and that the origin o of the coordinate system is located at the middle point of the beam. We also neglect the beam height and consider that the wet interface Σ is its central line, so that its outside normal vector ν 3 5 1; ν 1 5 0; and on the left and right ends of the beam, ν 1 5 2 1 and ν 1 51 1, respectively. Now we intend to find the dynamic response of the ship
water interaction system excited by an incident ^ i1 tÞejωt . wave φ0 ðxi Þejωt 5 φ0 ðxi 1 Vδ

FIGURE 9.4 Two-dimensional beam-like ship traveling on the water.

Hydroelasticity theory of ship
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359

Radiated potential of beam-like ship To use the hydroelasticity approach described in Section 9.3.3, we need to know the radiated potentials corresponding to the natural modes of the beam. Given this aim, we give the beam motion in its Ith natural mode form, U 3 ðx1 Þejωt 5 YI ðx1 Þejωt , to seek the corresponding radiated velocity potential ϕRI ðx1 Þ; x1 AΣ. Using Eq. (9.80e), we obtain Qðξi Þ 5 jωYI ðx1 Þν 3 5 jωYI ðx1 Þ;

(9.84a)

from which, when this is substituted into Eq. (9.80f), it follows ð   jωGðxi ; ξ1 ÞYI ðξ 1 Þ 2 φðξ 1 ÞG;3 ðxi ; ξ1 Þ dΓ 1 φC ðxi Þ φðxi Þ 5 Σ

5

ð L=2 2L=2



 jωGðxi ; ξ1 ÞYI ðξ 1 Þ 2 φðξ 1 ÞG;3 ðxi ; ξ1 Þ dξ 1 1 φC ðxi Þ;

2 3         2 jωV^ 4 L L 2L 2L 5 φC ðxi Þ 5 G xi ; YI 1 G xi ; YI 2 2 2 2 g

(9.84b)

2 3         2 V^ 4 L L 2L 2L 5 G xi ; φ φ : 2 2 G;1 xi ; g 2 2 2 2 We divide the wet interface into Nf elements, of which the length of each element is ΔL 5 L=Nf . A representative element Σ l is located from one point in the domain x1 5 2 L=2 1 ðl 2 1ÞΔL to another point x1 5 2 L=2 1 lΔL, and its center is at xl1 5 2 L=2 1 ðl 2 1=2ÞΔL. For a point xk1 AΣ, we define ð ð Gkl 5 Gðxk1 ; ξ1 ÞYI ðξ 1 Þdξ1 ; H kl 5 G;3 ðxk1 ; ξ1 Þdξ1 ; Σl

Σl

        L 2L k k 2L k k L GL 5 G xi ; YI ; G2L 5 G xi ; YI ; 2 2 2 2     k k 2L k k L HL 5 G;1 xi ; ; H2L 5 G;1 xi ; ; 2 2

(9.84c)

and approximate the function φðx1 Þ; x1 AΣ l as a constant, that is, ϕl 5 φðxl1 Þ;

ϕ0 5 φð2 0:5LÞ;

ϕNf 11 5 φð0:5LÞ;

(9.84d)

from which Eq. (9.84b) can be represented by a summation form cφðxk1 Þ 5 ϕk 5 δkl ϕl 0 1 2 2 Nf X ^ V V^ k GkC A 2 H kl ϕl 2 ðHLk ϕ1 2 H2L 5 jω@ Gkl 1 ϕNf Þ; g g l51

(9.84e)

k GkC 5 GkL 1 G2L :

Finally, we obtain the following algebraic equation: 2 2 Nf

X V^ k ~ kl V^ kl k cδkl 1 H~ ϕl 5 jωGk ; Gk 5 Gkl 1 GC ; H 5 H kl 1 ðHLk δl1 2 H2L δlNf Þ; g g l51

(9.84f)

where c 5 1/2, as given in Eq. (9.80b), the solution of which gives the required radiated potential in vector form:  T (9.84g) ϕRI 5 ϕ1 ϕ2 ? ϕNf : Its derivative with respect to x1 can be obtained by a difference formulation: h iT ϕl21 2 ϕl21 : ϕl;1 5 ϕRI;1 5 ϕ1;1 ϕ2;1 ? ϕN;1f ; 2ΔL

(9.84h)

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Fluid
Solid Interaction Dynamics

The total radiation mode vector and its derivative matrices are obtained as    ϕR 5 ϕR1 ϕR2 ? ϕRN ; ϕR;1 5 ϕR1;1 ϕR2;1 ?

 ϕRN;1 :

(9.84i)

Diffracted potential of beam-like ship To determine the diffracted potential of the beam, we assume that the beam is fixed and its motion U 3 ðx1 Þ 5 0 in order to seek the velocity potential φD ðxi Þejωt ; xi AΣ excited by the incident wave. In this case, we replace the potential φðxi Þ in Eq. (9.80f) by the total potential φ0 ðxi Þ 1 φD ðxi Þ to obtain ð   φ0 ðxi Þ 1 φD ðxi Þ 5 2 φ0 ðξ1 Þ 1 φD ðξ 1 Þ G;3 ðxi ; ξ 1 ÞdΓ 2 φD ðxi Þ; Σ 2 3         2 (9.85a) ^ V 4 L L 2L 2L 5 : φD ðxi Þ 5 G xi ; φ 2 G;1 xi ; φ 2 2 2 2 g In a similar process as used to derive Eq. (9.84g), we can obtain

 kl  kl cδkl 1 H~ ϕlD 5 2 cδkl 1 H~ ϕl0 ;     ϕl0 5 φ0 xl1 ; ϕlD 5 φD xl1 ; xl1 AΣl : From this equation we can obtain the diffracted potential on the wet interface in vector form:  T φD 5 ϕ1D ϕ2D ? ϕNDf ; of which the derivative matrix with respect to x1 is  φD;1 5 ϕ1D;1 ϕ2D;1

?

N

f ϕD;1

T

; ϕlD;1 5

l21 ϕl11 D 2 ϕD : 2ΔL

(9.85b)

(9.85c)

(9.85d)

Dynamic response of interaction system The free
free beam’s theoretical natural modes and frequencies are available, as given by Eq. (8.87), from which, with Eqs. (9.57) and (9.58), we can represent the beam motion in mode summation form: ð € 1 ΛQ 5 YT pdΓ ; IQ (9.86a) 3 Σ

where the beam traction force T^ i 5 0 and the vector ηi 5 2 ν i 5 2 v3 on the wet interface are introduced. Also, we approximately consider that the steady fluid velocity is given by ^ i1 ; v^i 5 2 Vδ

2 v^2 5 V^ ;

ðv^2 Þ;i 5 0;

(9.86b)

so that the dynamic force in Eq. (9.67) and the related matrices in Eqs. (9.68) and (9.67) can be further approximated as follows: (N ) f X  T  22 ^ mR 5 ω Re ρY3 2 VφR;1 1 jωφR ; ( Nl51 ) f X  T  ^ R;1 2 jωφR ρY3 Vφ ; dR 5 ω21 Im k5

l51

ð L=2 2L=2

PW 5 ejωt

ρgYT3 Y3 dx1 ;

Nf X

  ^ 0;1 1 φD;1 Þ 2 jωðφ0 1 φD Þ ; ρYT3 Vðφ

l51

from which the dynamic response equation of the beam can be obtained, as shown by Eq. (9.71c), that is,

(9.86.c)

Hydroelasticity theory of ship
water interactions Chapter | 9

361

FIGURE 9.5 Pendulum wave energy harvester
water interaction system.

€ W 1 ðD 1 dR ÞQ _ W 1 ðΛ 1 kÞQW 5 PW : ðI 1 mR ÞQ

(9.86d)

9.3.4.2 A wave energy harvesting device As shown in Fig. 9.5, a nonlinear pendulum consisting of a rigid massless rod of length l and mass m, connected to an electric generator D fixed at the top of a vertical rigid rod with its bottom body of total mass M floating on the water, is subjected to incident wave excitations. The vertical rod is supported by two supports, A and B, allowing its frictionless motion only along the vertical direction. The motion of mass m drives the coil of the electric generator rotating across its magnetic field, so that the mechanical motion excited by the waves is converted into electricity energy by device D. A Cartesian system o 2 x1 x2 x3 is fixed at origin o on the static-free surface, with gravitational acceleration g in the negative direction of the axis o 2 x3 . The positions of masses m and M are denoted by a rotational angle θ and a vertical displacement Z with their static positions θ 5 0 and Z 5 Z0 located at space point xiM , respectively. The effective horizontal area of the mass M is denoted by A determined by ρf gAZ0 5 2 ðm 1 MÞg. The velocities and accelerations of the _ θ, € and Z; _ Z€ , respectively. The water domain of depth H two variables θ and Z are obtained by their time derivatives θ; is represented by Ωf with its outward unit normal ν i on its boundaries: free surface Γ f , seabed Γ b , wet interface Σ, boundary Γ v on which an incident wave φ0 ðxi ; tÞ 5 φ0 ðxi ÞejΩt is applied, and Γ N denotes the far left and right undisturbed boundaries. The water is treated as a linear incompressible fluid with its motion satisfying the Laplacian equation. Here we intend to get the dynamic response of the system, from which to obtain the harvested wave energy. Investigations of nonlinear pendulum systems excited by given base motions with applications to collect wave energies were reported by Wiercigroch’s research group (Xu et al., 2005, 2007; Litaka et al., 2008, 2010; Lencia et al., 2008; Horton et al., 2011; Nandakumar et al., 2012). Obviously, this system operates on the water surface, so

362

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Solid Interaction Dynamics

that the water
structure interaction cannot be omitted in order to reveal its essential dynamic behavior. To address the related interaction mechanism, Xing et al. (2011) and Xing (2016) proposed detailed numerical formulations and numerical solution approaches, in which the electrical
mechanical interaction was also considered. Now, as an example of using the hydroelasticity theory described in the chapter, we investigate this interdisciplinary coupling system as follows. Electrical equations For an initial study, we assume that the energy convertor is a single-phase electrical generator (Nilsson and Riedel, 2011) consisting of a magnetic body, rotating with the pendulum, of intensity B, and an outside coin, fixed with the vertical rigid rod, of area S, inductance L1, and resistance R1 with negligible capacitance. At the static state, the principal N
S axis of the magnetic body is located in the vertical direction perpendicular to the horizontal coin plane, so that the induced voltage at this position is zero. When the magnetic body rotates with the pendulum, the induced voltage e(t) at the two ends of the coin supplies an energy collector of inductance L2, resistance R2, capacitance C2, and therefore a current I(t), the time change rate of electric charge Q flows in the coil, producing a torque T(t) to resist the relative rotation between the coil and the magnetic field. The Laplace electromagnetic theorem (Kittel, 1967; Nilsson and Riedel, 2011) gives the following electrical equations: e 5 2 BSθ_ sin θ;

T 5 2 BSQ_ sin θ;

Q_ 5 I:

(9.87a)

Denoting L 5 L1 1 L2 ; R 5 R1 1 R2 , and C 5 C2 , the induced voltage satisfies the following electric balance equation: ð 1 t e 5 LI_ 1 RI 1 Idt; (9.87b) C 0 from which, along with Eq. (9.87b), we obtain the following interaction equation between the electric current and pendulum rotation motion: e 5 LQ€ 1 RQ_ 1

Q 5 2 BSθ_ sin θ: C

(9.87c)

The collected power of this single-phase electrical generator can be calculated by P 5 I 2 R2 :

(9.87d)

Mechanical motion equations The mechanical system is a system with 2 degrees of freedom described by the two independent coordinates Z and θ. Applying Newton’s second law to the free-body diagram shown in Fig. 9.6, we obtain the following governing equations for the mechanical system: 2 ð 3      2 _ pdS m 1 M ml sin θ Z€ 5: (9.88) 1 ðM 1 mÞg 1 mlθ cos θ 5 4 Σ ml sin θ ml2 θ€ mgl sin θ _ BSQ sin θ Fluid domain Considering that the free surface area of the sea is very big and a finite wave height corresponding to the displacement Z of the floating mass M is very small, we adopt a linear potential wave theory to describe the fluid motion. We assume that the atmospheric pressure on the free surface is zero, so that the water pressure pðxi ; tÞ at a space point ðxi Þ can be derived from the Bernoulli Eq. (9.66h), that is, pðxi ; tÞ 5 2 ρf ðφ0;t 1 φ;t Þ 2 ρf gðZ 1 Z0 Þ;

(9.89)

where φ denotes the disturbed velocity potential. Furthermore, for similarity, we neglect the variation of the water pressure and the velocity potential on the wet interface and consider them the corresponding values at point O, x2 5 0 5 x3 . Therefore, from the Green function in Eq. (9.80b) and the Green identity in Eq. (9.80b), we obtain

Hydroelasticity theory of ship
water interactions Chapter | 9

363

FIGURE 9.6 Free-body diagrams: (A) base, (B) pendulum. M

ð

M

cðφ0 1 φ Þ 5

Σ

  M M Gðxi ; ξ i ÞZδj3 2 ðφ0 1 φ ÞG;j ðxi ; ξ i Þ ν j dΓ ;

5 A½GM Z 2 ðφ0 1 φ ÞG~ M ; GM 5 GðO; OÞ; G~ M 5 G;3 ðO; OÞ: M

M

(9.90a)

For the coordinate system shown in Fig. 9.5, the Green function in Eq. (9.77a) takes the form G5

N X n50

2

e2λn jx2 2ξ2 j cosðλn HÞ cos λn ðx3 1 HÞ : cosðλn HÞsinðλn HÞ 1 λn H

(9.90b)

From these two equations, we can obtain  M   M  M M c φ0 1 φ 5 A GM Z 2 φ0 1 φ G~ M ; 2 3   c M 1 G~ M φ0 5 A4GM Z 2 A M φ 5 ; ðc 1 AG~ M Þ

(9.90c)

which, when substituted into Eq. (9.89), gives the water pressure   M  M pðxi ; tÞ 5 2 ρf jω φ0 1 φ 1 gZ ejωt 2 ρf gZ0 :

(9.91a)

364

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Solid Interaction Dynamics

Substituting this force into Eqs. (9.90a)
(9.90c) and noting ðM 1 mÞg 5 2 ρf gz0 , we have #" #  " # " m 1 M ml sin θ Z€ dZ mlθ_ cos θ Z_ 1 ml sin θ ml2 θ_ θ€ 0 dθ 2 3

   M M 2jωρf A φ0 1 φ ejωt k 0 Z 5: 1 54 0 mgl sin θ=θ θ _ BSQ sin θ

(9.91b)

Here, k 5 ρf Ag represents a buoyance stiffness of the water, and two viscous damping coefficients dM and dθ are introduced. Combining Eqs. (9.91b) and (9.87c), we obtain the coupled equation of the system in the following matrix form: 32 3 2 32 € 3 2 m 1 M ml sin θ 0 Z_ Z dM mlθ_ cos θ 0 6 7 6 7 6 6 7 _7 €7 6 ml2 0 56 4 ml sin θ dθ 2BS sin θ 7 4 θ 514 0 54 θ 5 0 0 L Q_ Q€ 0 BS sin θ R (9.92) 3 2 32 3 2 M M k 0 0 Z 2jωρf Aðφ0 1 φ Þ 7 jωt 6 76 7 6 7e : 1 4 0 mgl sin θ=θ 0 54 θ 5 5 6 0 4 5 0 0 1=C Q 0

Solution discussion This equation is nonlinear and can be solved by using the Matlab program given by Xing (2015b); then the harvested energy can be obtained using Eq. (9.87d). Here, we do not seek the numerical results of the system but discuss its characteristics as follows. Case I: The physical mechanism of energy collector. As in practical applications, generally, the inductance L is small, and the capacitance C is quite big, so that Eq. (9.92) is approximate to Q_ 5 I 5 2 BSθ_ sin θ=R; " #" # m 1 M ml sin θ Z€ ml sin θ

ml2

θ€

" 1

0 "

1

dm

k 0

#" # Z_

mlθ_ cos θ dθ 1 B2 S2 sin2 θ=R #" # Z 0 mgl sin θ=θ

θ_ 2

3 M

M 2jωρf A φ0 1 φ ejωt 5: 54 θ 0

(9.93a)

This obviously shows that the term B2 S2 sin2 θ=R, contributed by the energy collector, plays a damping mechanism, which, when appropriately designed, can avoid some unstable resonance self-excited vibrations in designed linear or nonlinear energy harvesting systems (Xing et al., 2009b; Yang et al., 2011; Xing et al., 2011). The energy flow equation (see Xing, 2015b), corresponding to Eq. (9.93a), is given by ( " #   m 1 M ml sin θ Z€ Re Z_ θ_ Re ml sin θ ml2 θ€ " #" #   ) k 0 Z Z_ dM mlθ_ cos θ (9.93b) 1 1 2 2 2 _ 0 mgl sin θ=θ θ θ 0 dθ 1 B S sin θ=R " #     ωρf A φM 1 φM sin ωt 0 _ 5 Re Z_ θ : 0 Case II: Small rotation angle θ. In this case, we have approximated results sin θ 5 θ; can be reduced to

cos θ 5 1; so that Eq. (9.92)

Hydroelasticity theory of ship
water interactions Chapter | 9

2

m1M

6 6 mlθ 4 0

mlθ

0

32

Z€

3

2

dM

76 7 6 6€7 6 07 54 θ 5 1 4 0

ml2 0

Q€

L

2

mlθ_

0

BSθ

k

0

0

0

6 16 4 0 mgl

Z_

3

76 7 6_7 2BSθ 7 54 θ 5

dθ

0

32

365

R 32

Q_ 3 2

 M M 3 2jωρf A φ0 1 φ 76 7 6 7 jωt 6 7 6 7e : 0 7 0 54 θ 5 5 4 5 1=C Q 0 0

Z

(9.94)

In this case, the pendulum oscillates about its static equilibrium position, which is not an effective design for energy harvesting. Case III: Possible circular rotation design. We assume that θ 5 ωt, from which, when substituted into Eq. (9.93a), it follows Q_ 5 I 5 2 BSω sin ωt=R; " #" # m 1 M ml sin ωt Z€ ml sin ωt " 1

k 0

"

dM

mlω cos ωt

#" # Z_

1 0 dθ 1 B2 S2 sin2 ωt=R ω 0 3 #" # 2  M M 0 Z 2jωρf A φ0 1 φ 5ejωt ; 54 mgl sin ωt 1 0 ml2

(9.95a)

of which the first equation for the vertical motion of the system confirms its periodical characteristics with frequency ω. We can derive the energy flow equation and the collected power by considering its real part as follows (Xing, 2015b): ( " #   m 1 M ml sin ωt Z€ Z_ ω ml sin ωt ml2 0  " #   ) dM mlω cos ωt k 0 Z Z_ (9.95b) 1 1 2 2 2 0 dθ 1 B S sin ωt=R ω 0 mgl sin ωt 1 " #     ωρf A φM 1 φM sin ωt 0 5 Z_ ω ; 0 P 5 I 2 R2 5

R2 B2 S2 ω2 sin2 ωt B2 S2 ω2 sin2 ωt  : R2 R

(9.95c)

If the internal resistance R1 is much smaller than the resistance R2 for energy collection, Eq. (9.95b) can be further written in the form 8 9 < = B2 S2 ω2 sin2 ωt d 1 2 ½ðm 1 MÞZ_ 1 kZ 2  1 mlωZ_ sin ωt 1 ; dt :2 R (9.95d) M

M 2 dM Z_ 1 dθ ω2 1 mglω sin ωt 5 ωρf A φ0 1 φ Z_ sin ωt; in which the integration with respect to time t in one time period from 0 to 2π=ω and then divided by the time period 2π=ω gives a time-averaged energy flow equation: ) ( 2π=ω B2 S2 ω2 ω 1 2 2 _ _ ½ðm1MÞZ 1kZ 1mlωZ sin ωt  1 2π 2 2R 0  M M  ð 2π ρf A φ0 1 φ ω B2 S 2 ω 2 (9.95e) 1 hPd i 5 1 hP d i 5 Z_ sin θdθ; 2π 2R 0 ð ω 2π=ω _2 hP d i 5 d θ ω 2 1 dM Z dt; 2π 0

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Solid Interaction Dynamics

where we have considered the characteristics of periodic motion governed by Eq. (9.95a), and hPd i denotes the timeaveraged dissipated power by the damping of the system. The time-averaged collected power is obtained as ð 1 2π B2 S2 ω2 sin2 ωt B2 S 2 ω 2 hP i 5 dðωtÞ 5 : (9.95f) 2π 0 R 2R from which, when substituted into Eq. (9.95e), it follows M

M ρf A φ0 1 φ ω ð 2π   h Pi 1 h Pd i 5 Z_ sin θdθ 5 Pf ; 2π 0

(9.95g)

implying that the input power by the wave equals the collected and the dissipated power. To determine the condition to realize the periodical motion, we must solve the real part of Eq. (9.95a). To simplify the solution, we neglect the small damping dM and dθ , so that M

M ðm 1 MÞZ€ 1 kZ 5 2 mlω2 cos ωt 1 ωρf A φ0 1 φ sin ωt; (9.95h) mlZ€ sin ωt 1 B2 S2 ω sin2 ωt=R 1 mgl sin ωt 5 0: ~ jωt For the case with damping considered, the solution of Eq. (9.95a) can be assumed to be in a complex form Z 5 Ze with its complex amplitude, of which the real part is used to calculate the energy flow based on Eq. (9.93a). The solution of this equation is ( ) M

 2 2  1 M 2 Z5 ωρf A φ0 1 φ sin ωt 2 mlω cos ωt 1 ðm 1 MÞ B S ω sin ωt=ðmlRÞ 1 g k 5 F1 ω sin ωt 2 F2 ω2 cos ωt 1 Zg ; Z0 5 Zg 2 F2 ω2 ; Z_ 5 F1 ω2 cos ωt 1 F2 ω3 sin ωt; Z_0 5 F1 ω2 ; 2 3 2 2 ðm 1 MÞB S M M 5=k; F1 5 4ρf Aðφ0 1 φ Þ 1 mlR F2 5

ml ; k

Zg 5

(9.95i)

ðm 1 MÞg : k

The equation shows that the mass M oscillates about a point Z 5 Zg and that the initial conditions of the motion. Substituting the solution in Eq. (9.95i) into Eq. (9.95g), we have M

M 2 2 2 ρ A φ 1 φ ω ð 2π f 0 B Sω 5 Z_ sin θdθ 2R 2π 0 M

M ρf A φ0 1 φ ω ð 2π (9.95j) 5 F2 ω3 sin2 θdθ 2π 0 M

M ρf Aml φ0 1 φ ω4 5 ; 2k that is, 2 2

BS 5 R

M

M ρf Aml φ0 1 φ ω2 k

5

M

M ml φ0 1 φ ω2 g

;

(9.95k)

where the buoyance stiffness k 5 ρf Ag is introduced. The collected time-averaged power in Eq. (9.95e) is now given by

Hydroelasticity theory of ship
water interactions Chapter | 9

hP i 5

M M ml φ0 1 φ ω4 2g

:

367

(9.95l)

Eq. (9.95k) gives a necessary condition in designing the system undergoing the expected motion to harvest wave energy in Eq. (9.95l), if the damping is neglected. For the case with damping, as shown in Eq. (9.95g), the collected power will be less due to dissipation. Case IV: Resonance state. Eq. (9.95i) requires the mass M motion around the position Zg 5

ðm 1 MÞg m 1 M 5 ; k ρf A

(9.96a)

with the initial conditions, which implies that its displacement of oscillation should be large enough. Toward this aim for the given wave frequency ω, we can choose the parameters satisfying rffiffiffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffi ρf Ag k ω5 5 5 Ω; (9.96b) m1M m1M at which the system is in its resonance state: the excitation frequency equals the natural frequency Ω, and the motion Z of the mass M reaches its resonance amplitude. In this case, the inertial and stiffness forces on the left-hand side of Eq. (9.95a) cancel each other out.

Chapter 10

Variational principles for nonlinear fluid
solid interactions Chapter Outline 10.1 A short review on variational principles for nonlinear dynamical systems 10.2 Fundamental variational concepts for nonlinear systems 10.2.1 The motion of a continuum 10.2.2 Translation and transmission velocities of a curved surface 10.2.3 Time derivative of an integral over a moving volume in space 10.2.4 A local variation and a material variation 10.2.5 Local variation of an integral over a moving volume in space 10.3 Governing equations 10.3.1 Solid domain 10.3.2 Fluid domain 10.3.3 Fluid
structure interface 10.3.4 Variational conditions at initial time t1 and final time t2

370 372 372 372 373 374 376 377 377 378 381 381

10.4 Variational principles 10.4.1 Fluid motion assumed rotational 10.4.2 Fluid flow assumed irrotational 10.4.3 Discussion 10.5 Two simple examples of applications 10.5.1 A one-dimensional water
mass
spring interaction problem 10.5.2 A forced one-dimensional gas
mass
spring dynamic interaction problem 10.6 Variational principles for nonlinear elastic ship
water interactions 10.6.1 Short introduction 10.6.2 Governing equations in the moving reference frame 10.6.3 Variational formulations in the moving reference frame 10.6.4 Rigid ship dynamics 10.6.5 Offshore and hydroelastic examples

382 382 387 389 390 390 392 394 394 395 397 399 405

In Chapter 4, Variational principles of linear fluid
solid interaction systems, the variational principles for linear fluid
solid interaction (FSI) problems were investigated. These principles provide a means for the transformation of the partial differential equations governing the dynamics of a structure, fluid, or FSI system, defined by an appropriate set of physical variables (i.e., displacement, pressure, stress) into an alternative set of ordinary differential or algebraic equations amenable to numerical analysis; hence this is a numerical scheme of study, as we have used effectively in Chapter 7, Finite element models for linear fluid
solid interaction problems, to construct finite element models for linear FSI problems. However, these variational principles, based on linear assumptions, cannot be used to deal with nonlinear FSI problems. To understand this point, it is important to find the main differences in deriving variations of this functional for nonlinear FSI problems when compared with linear cases. For linear theories, we assume that the motions of the fluid and the solid are small, so that its original configuration is taken as our reference state, and there is no need to distinguish Lagrange and Euler coordinates, as well as the variations involved. For example, when we take the variation of a quantity defined in a fluid domain, we consider its boundary fixed at the original position and neglect the effect caused by boundary motion. Further, in the linear variation process, we always freely exchange the order of time and space integrations. For nonlinear cases, these are no longer valid assumptions because the large motions cause the boundary changes that have to be considered in the variation process. So that readers may learn these mathematical tools and derive the variational principles for nonlinear systems, the fundamental concepts of the variational process that are valid for nonlinear FSI systems will be discussed in detail in Section 10.2, following a short review of historical studies for the variations of nonlinear dynamic systems given in Section 10.1. Based on this knowledge and these methods, the variational principles, along with selected application examples for nonlinear FSI dynamics, are derived in

Fluid
Solid Interaction Dynamics. DOI: https://doi.org/10.1016/B978-0-12-819352-5.00010-0 © 2019 Higher Education Press. Published by Elsevier Inc. All rights reserved.

369

370

Fluid
Solid Interaction Dynamics

the subsequent subsections. The generalized theory and mathematical works adopted here were presented by Xing and Price (1997), and the applications to model the dynamics of nonlinear elastic ship
water interactions were given by Xing and Price (2000). Recently, Zhou et al. (2013) investigated an interesting phenomenon on nonlinear lowfrequency gravity waves in a water-filled cylindrical vessel subjected to high-frequency excitations using variational principles developed for nonlinear FSI systems.

10.1

A short review on variational principles for nonlinear dynamical systems

Variational principles are concerned with the stationary value of an integral. For the description of the behavior of dynamic systems, Hamilton’s principle is the most well-known, being extensively described in many textbooks (see, for example, Whittaker 1917; Gantmacher 1960). This principle states that the motion of an arbitrary dynamic Ðt system occurs in such a way that s 5 t12 Ldt becomes stationary for arbitrary possible variations of the configuration of the system between the two times t1 and t2 (t1 , t2 ), provided that the initial and final configurations are chosen to be the arguments of the Lagrangian L used to describe the configuration of the dynamic system and that the variations of these generalized coordinates must vanish at the instants t1 and t2 . Many theoretical developments and modifications to Hamilton’s principle have occurred. For example Toupin (1952) and Crandall (1957) developed complementary principles to describe dynamic systems using momentum as the variable that satisfies the dynamic equilibrium equations, whereas Green and Zerna (1954) introduced Hamilton’s principle into the theory of electrodynamics. With the development of computational techniques, variational principles are at the center of many advances in the analysis of the dynamic behavior of elastic bodies. Oden and Reddy (1983) provided an excellent description of the variational principle in theoretical mechanics, but they constrain the variations of both the displacement and the momentum to vanish at the instants t1 and t2 . This is thought to be unreasonably restrictive, and Xing (1984, 1986a,b) and Xing and Zheng (1986) have relaxed this constraint in their development of variational principles to derive solutions to electrodynamic problems and fluid
solid coupled problems subject to four types of initial and final time instant conditions. These theoretical ideas have been further extended to describe the dynamics of conservative holonomic systems (Xing and Price 1992), in which a more general variational principle framework is constructed and contains many previously quoted results. This allows the variational principles associated with many branches of applied mechanics to be discussed uniformly on the same theoretical foundation. Originally, Hamilton’s principle was created for the motions of material systems—material points, rigid bodies, and elastic bodies—so that it is based on the Lagrange reference frame adopted in solid mechanics. As previously mentioned, when this principle is used for fluid flows investigated in the Euler reference system, where the flow boundary changes are neglected in linear cases, we do not need to distinguish the Lagrange and Euler descriptions; so the principles developed in solid mechanics can be used to describe the linear dynamic problems of fluids, even though its form is different. For example, linear sloshing problems in liquid tanks and linear oscillations/acoustic analysis in air or liquid acoustic volumes can be modeled by the variational principles in the Hamilton form presented in Chapter 4, Variational principles of linear fluid
solid interaction systems. For nonlinear problems in fluid mechanics, such as nonlinear sloshing problems in liquid tanks in which the free surface is a moving boundary and subject to a nonlinear boundary condition, we have to consider the corresponding changes caused by the moving boundary defined under the spatial coordinate system. In fluid mechanics, investigations of variational principles resembling Hamilton’s principle have been undertaken by Serrin (1959), Luke (1967), Seliger and Whitham (1968), and Miles (1977). Luke incorporated a variable boundary into a variational principle developed to generate the governing equations and to analyze the behavior of gravity waves in a two-dimensional (2D) incompressible fluid. Seliger and Whitham (1968) did not examine the implications of variable boundaries but concluded that the fluid pressure variable was the Lagrangian function within the fluid variational principle. Ikegawa and Washizu (1973), utilizing the stream function, introduced a variational principle to model an incompressible gravity flow with a free surface using the finite element method, whereas Ecer et al., (1983), Ecer and Akay (1983), and Ward et al. (1988) developed variational approaches to model incompressible, viscous fluid flows. As discussed in Chapter 4, Variational principles of linear fluid
solid interaction systems, for linear FSI problems, several forms of variational principles have been developed successfully to describe the dynamic behavior of rigid or flexible structures and a fluid with or without free surface (see, for example, Xing, 1984, 1986a,b,c, 1988; Liu and Uras, 1988; Xing and Price, 1991, 1996; Bathe et al., 1995; Morand and Ohayon, 1995; Xing et al., 1996). For

Variational principles for nonlinear fluid
solid interactions Chapter | 10

371

nonlinear FSI problems, due to the complexities of the interactive mechanisms involved, only limited advances have been achieved in deriving descriptions of the nonlinear dynamical behavior of both solids and fluids. Kock and Olson (1991) presented a finite element method to analyze linear and nonlinear FSI problems by adopting a variational indicator approach based on Hamilton’s principle for inviscid, irrotational, and isentropic fluid flows throughout the fluid domain. In this analysis, entropy is assumed constant in the whole fluid domain (rather than a function of the space coordinate as assumed here), and two Lagrangian multipliers are introduced to make the global conservation of mass and the equation of continuity valid at the same time. In the construction of a variational description of the dynamic behavior of a nonlinear FSI system, two fundamental difficulties are encountered. The first concerns the concept of local (or space) variation and material variation. In a Eulerian description of the fluid field, all variables are functions of local coordinates fixed in space and time, whereas in the Lagrangian description of the structure, the motion variables are functions of the material coordinates fixed to each particle or element of the structure and time. Thus when the structure moves, the material coordinates also move from their original positions to new positions in the space. These differences are further extenuated when examining time differentials, time integrals, etc., and therefore two kinds of arguments relating to fluid and structure must be included in any proposed variational functional. The second difficulty relates to variational principles involving variable boundaries. For example, in a linear analysis with all motions assumed to be very small, the boundary of the fluid domain during motion is assumed to be the same as the original boundary in stationary equilibrium. In a nonlinear study involving large disturbances and a free fluid surface, such an assumption is invalid, and a variable-boundary fluid domain must be included in the mathematical model. Variational principles provide a means of transforming the partial differential equations associated with a set of physical variables (i.e., displacement, stress) describing the dynamics of the elastic structure, fluid, and their interaction in an alternative set of ordinary differential or algebraic equations amenable to numerical analysis and to the creation of a suitable numerical scheme of study (see, for example, Courant and Hilbert, 1962; Washizu, 1982; Oden and Reddy, 1983). Variational principles have been widely used to develop mathematical models in fluid mechanics, structural dynamics, and linear FSI problems (see, for example, Chien, 1980; Hu, 1981; Miloh, 1977; Xing and Price, 1991; Morand and Ohayon, 1995; Xing et al., 1996). Their development and application to nonlinear FSI problems are limited due to the complexity of these dynamic systems and the difficulties in satisfying the differing fundamental concepts previously mentioned. To address this, Xing and Price (1997, 2000) developed variational principles to describe the nonlinear behavior of FSI systems by constructing a unifying theory based on the studies of Gelfand and Fomin (1963) and McIver (1973) relating to the conceptual difficulties previously discussed: variational principles and their applications in fluid dynamics (Serrin, 1959; Seliger and Whitham, 1968; Miloh, 1977; Rainey, 1989; van Daalen et al., 1992; Galper and Miloh, 1995) and those for nonlinear solid mechanics (Green and Zerna, 1954; Oden and Reddy, 1983; Washizu, 1982; Xing and Price, 1996). It is assumed that the fluid is inviscid, incompressible or compressible, and with or without a free surface and that its flow can be rotational or irrotational but isentropic along the path of each fluid particle. The solid structure is treated as a nonlinear elastic body. The stationary conditions of the variational principles include the governing equations of nonlinear elastic dynamics, the consistent relationships of motions and equilibrium conditions on the FSI interface, and the governing equations for fluid dynamics and the associated boundary conditions, including those associated with nonlinear free surface disturbances. These variational principles provide the basis for developing numerical schemes of study to evaluate the nonlinear dynamic behavior of the typical fluid-structure systems illustrated in Figs. 10.2
10.4, where two simple dynamic problems were investigated to demonstrate their applicability. In nonlinear ship
water interaction dynamics, ships generally move with a mean forward velocity, and their oscillatory motions in waves are superposed on a steady flow field. It is convenient to study the motion of a ship
water interaction system relative to an equilibrium reference frame fixed at a point on the ship and moving with the ship’s forward velocity. To meet this condition for ship
water interactions, the generalized variational principles developed by Xing and Price (1997) were further derived to obtain the form for ships in the moving coordinate system (Xing and Price, 2000). The rigid ship traveling in calm water and in waves was investigated. Two hydroelastic examples were given: one involves the dynamic response of a vertical rod, fixed by a rotational spring at seabed and excited by an incident wave, and another discusses a 2D elastic beam traveling in waves. This chapter will discuss these new developments.

372

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10.2 10.2.1

Fundamental variational concepts for nonlinear systems The motion of a continuum

As discussed in Chapter 3, Fundamentals of continuum mechanics, in order to describe the motion and dynamic characteristics of a continuum (elastic solid or fluid) in a three-dimensional space, suitable systems of reference are needed (see, for example, Green and Adkins, 1960; Truesdell, 1966, 1991; Malvern, 1970; Fung, 1977). To this end, as shown in Fig. 3.1, the spatial coordinate system adopted is a fixed rectangular Cartesian frame of reference with coordinate xi ; ði 5 1; 2; 3Þ. At time t 5 t1 , a material particle located at xi 5 Xi is identified by a set of ordered real numbers Xi , referred to as the material coordinates. Since this is a symbolic coordinate used to identify a material particle, it can be chosen in different ways. For example, this role may be played by a field function αðx; tÞ representing a particular physical quantity of the continuum. As time proceeds and the material particle moves from location to location in the three-dimensional space, its history of motion can be represented by Eq. (3.1), that is, xi 5 xi ðXj ; tÞ:

(10.1)

Mathematically, this equation defines a transformation of a domain Ω1 ðX; t1 Þ into a domain Ωt ðX; tÞ, treating time t as a parameter. It is assumed that a unique inverse of this equation exists and that the Jacobian J of the transformation, defined by Eq. (3.7a), is positive, that is,

and

Xi 5 Xi ðx; tÞ;

(10.2)

   @xi  J 5   . 0: @Xj

(10.3)

If such an Eq. (10.1) or Eq. (10.2) is known for every particle in the continuum, then the history of motion of the continuum is defined. As discussed in the previous chapters of this book, this material coordinate description is always used to describe the motion of the elastic solid. The displacement, velocity, and acceleration of each particle in the elastic body are therefore a function of ðXj ; tÞ, and they take the following forms in Eqs. (3.20), (3.3), and (3.4), respectively, which, for our convenience in deriving the related equations in this chapter, are listed here again: Ui ðX; tÞ 5 xi 2 Xi ;  @xi  Dxi Vi ðX; tÞ 5 5 5 Ui;t ; @t X Dt

(10.4)

Wi ðX; tÞ 5 Vi;t 5 Ui;tt :

(10.6)

(10.5)

When describing fluid flow, it is not necessary to identify the location of every fluid particle during motion but rather the instantaneous velocity field and its evolution with time. This leads to a spatial description in which the location xi and the time t are taken as independent variables and the instantaneous velocity field of the fluid is represented by vi ðx; tÞ. By applying the material derivative definition in Eq. (3.10) to the field function ðÞ, that is, DðÞ 5 ðÞ;t 1 vi ðÞ;i ; Dt

(10.7)

the instantaneous acceleration field is given by wi ðxj ; tÞ 5

10.2.2

Dvi ðx; tÞ @vi ðX; tÞ 5 vi;t 1 vj vi; j 5 : Dt @t

(10.8)

Translation and transmission velocities of a curved surface

Let us consider a curved surface in space represented by the equation f ðxi ; tÞ 5 0;

(10.9)

Variational principles for nonlinear fluid
solid interactions Chapter | 10

373

where f ðxi ; tÞ is a continuously differentiable function. The differential of the function f ðxi ; tÞ takes the form

and therefore

df 5 f; t dt 1 f; i dxi ;

(10.10)

  f;t dt 1 gradf dr 5 0:

(10.11)

Here, dr 5 ν i dxi represents the projection of the elemental length dxi onto the normal vector ν i of the curved surface, where f; i : ν i 5  gradf 

(10.12)

From Eq. (10.11), the translation velocity of the curved surface is defined by N5

dr f; t ; 5 2  dt gradf 

(10.13)

and the projection of the velocity vi of the continuum onto the normal vector ν i of the surface takes the form vi fi : vη 5 vi ν i 5  gradf 

(10.14)

From these results, the transmission velocity of the curved surface is defined by the relative velocity Df =Dt : θ 5 N 2 vη 5 2  gradf 

(10.15a)

Physically, the translation velocity N of a curved surface is the velocity observed by an observer standing on the fixed reference coordinate system, but the transmission velocity θ represents the velocity observed by one standing on the material particle of the continuum with flow velocity vi . Therefore, if θ 5 0, this moving curved surface is a material surface, and if θ 5 2 vη , it reduces to a fixed surface in space. The concept just presented is closely linked with the ALE reference system. Actually, if we consider Eq. (10.9) representing a mesh reference curve, as shown by Eq. (3.12a) discussed in Section 3.1.3, the translation velocity N is the projection of the mesh velocity v~i in Eq. (3.15) onto the normal vector ν i of the surface represented by Eq. (10.9). In that case, we can denote the translation velocity in the form N 5 v~i ν i :

10.2.3

(10.15b)

Time derivative of an integral over a moving volume in space

Considering the relationship given by Eq. (10.15b), in Section 3.5.3.1, we presented the time derivative of an integral over a moving volume in space by Eq. (3.78a), which was directly introduced using Eq. (3.76) for the material derivative. For readers, especially students, to understand the details of the mathematics, we give here a more detailed derivation based on Fig. 10.1 in which the material boundary movement and the translation motion are clearly

FIGURE 10.1 Continuous change of the boundary of a moving region: (A) the case of θ 5 0; that is, a material region and N 5 vi ν i ; (B) the case of θ 6¼ 0 and the translation velocity N 6¼ vi ν i .

374

Fluid
Solid Interaction Dynamics

distinguished. It is assumed that Eq. (10.9) represents a convex regular region Ωðx; tÞ bounded by a surface Γðx; tÞ consisting of a finite number of parts whose outer normal forms a continuous vector field and that all regions of the solid and fluid are treated as regular. Let Fðx; tÞ represent any continuously differentiable function in Ωðx; tÞ, and ð IðtÞ 5 Fðx; tÞdΩ (10.16) Ωðx; tÞ

denotes the volume integral of this function at time t. The function IðtÞ retains dependence on t because both the integrand Fðx; tÞ and the domain Ωðx; tÞ are intrinsic functions of this parameter. As t varies, IðtÞ also varies, and therefore there exists the time derivative dI=dt. In visualizing the evaluation of this quantity (see Fig. 10.1), the boundary Γ of the region Ω at instant t translates with velocity N to the neighboring surface Γ0 of the region Ω0 at instant t 1 Δt. Thus in time Δt, the change in distance NΔt produces an elemental change in volume dΩ 5 NdtdΓ. Therefore the time derivative of IðtÞ is defined as ð  ð dI 1 5 lim Fðx; t 1 ΔtÞdΩ 2 Fðx; tÞdΩ Δt-0 Δt dt Ω0 Ω ð  ð   1 5 lim Fðx; t 1 ΔtÞ 2 Fðx; tÞ dΩ 1 Fðx; t 1 ΔtÞdΩ Δt-0 Δt Ω ΔΩ ð ð (10.17) @F 1 5 dΩ 1 lim Fðx; t 1 ΔtÞNΔtdΓ Δt-0 Δt Γ Ω @t ð ð @F dΩ 1 Fðx; tÞNdΓ: 5 Ω @t Γ From Eqs. (10.14) and (10.15a) and (10.15b), it follows that dI=dt can be rewritten as ð ð dI 5 F; t dΩ 1 Fðvi ν i 1 θÞdΓ; dt Ω Γ

(10.18)

defined at time t. If the transmission velocity θ 5 0, the domain Ω is the material domain ΩM , and the time derivative dI=dt reduces to the material derivative of the volume integral over the material domain, that is,  ð ð ð  DI DF 5 1 Fvi;i dΩ; F; t dΩ 1 Fvi ν i dΓ 5 (10.19) Dt ΩM ΓM ΩM Dt after applying Green’s theorem. From this result and subject to the continuum obeying the continuity equation, we obtain Eq. (3.73a), that is, ð ð D DF 5 dΩ: (10.20) ρFdΩ 5 ρ Dt ΩM ΩM Dt If the transmission velocity θ 5 2 vν 5 2 vi ν i , the domain Ω reduces to the fixed domain ΩF in space, and the time derivative of dI=dt reduces to the form ð dI 5 F; t dΩ: (10.21) dt ΩF

10.2.4

A local variation and a material variation

Let δx 5 δuðX; tÞ 5 δuðx; tÞ represent a virtual displacement of the particle X in the continuum from its instantaneous position x. This perturbation is produced, say, by an arbitrarily small additional internal or external force. The vector function δu is assumed to be finite valued and continuously differentiable; moreover, it conforms to any restrictions placed on the continuum position (e.g., kinematic constraints). Due to the small displacement δx, a scalar or vector field denoted by φ 5 φðx; tÞ at position x changes to φ 5 φ ðx; t; εÞ, and the original particle at x, which is now at the new

Variational principles for nonlinear fluid
solid interactions Chapter | 10

375

position x 5 x 1 εδx, acquires a field value of φ 5 φ ðx; t; εÞ. Here ε is an independent variation parameter, 21 , ε , 1. A local variation δφ in a Eulerian description and a material variation δφ in a Lagrangian description of the field function φ are defined, respectively, by Gelfand and Fomin (1963) to be  @φ ðx; t; εÞ   δφ 5 (10.22)  Bφ ðx; t; εÞ 2 φðx; t; 0Þ @ε ε50

and

  @φ ðX; t; εÞ  Dφ ðx; t; εÞ   δφ 5  5  Bφ ðx; t; εÞ 2 φðx; t; 0Þ: @ε Dε ε50 ε50

(10.23)

Further, they prove that there exists a relation between these variations of the field function φ in the form δφ 5 δφ 1 φ; i δxi 5 δφ 1 δxUrφ:

(10.24)

It is observed that δx is the initial velocity in a motion for which ε plays the role of time t. Hence the relation between the local and the material variations of a field function ðÞ is similar to the formulation denoted by Eq. (3.10) to calculate the material derivative of the velocity field vi ðx; tÞ, that is, δðÞ 5 δðÞ 1 ðÞ;i δxi :

(10.25)

From these findings, it can be shown that all local field derivatives commute but the material operators δðÞ and DðÞ=Dt both relate to a particular particle. Therefore the following exchangeable and nonexchangeable relations with respect to differential operations are valid:       DðÞ D  δðÞ;i 5 δðÞ ; i ; δðÞ; t 5 δðÞ ; t ; δ δðÞ ; (10.26a) 6¼ Dt Dt   DðÞ D ½δðÞ; (10.26b) δðÞ;i 6¼ ½δðÞ; i ; δðÞ; t 6¼ ½δðÞ; t ; δ 5 Dt Dt   DðÞ δ 5 δ½ðÞ; t 1 vi ðÞ; i  5 ½δðÞ; t 1 δvi ðÞ; i 1 vi ½δðÞ; i : (10.26c) Dt Moreover, by considering Eqs. (10.19) and (10.20) for domain ΩF ðxÞ and material domain ΩM ðx; tÞ in space, the following exchangeable relations with respect to integral operations also exist: ð t2 ð t2 ð ð δ ðÞdt 5 δðÞdt; δ ðÞdΩ 5 δðÞdΩ; (10.27a) δ

ð t2

t1

ðÞdt 5

t1

ð δ ð δ

ð t2

t1

δðÞdt;

t1

ΩM ðx; tÞ

ðÞdΩ 5

ΩM ðx; tÞ

ΩM ðx; tÞ

ðÞdΩðx; tÞ 5

ð δ

δ ð

ΩM ðx; tÞ

ΩF ðxÞ

ð

ΩM ðX; t1 Þ

ð

ρðÞdΩ 5

ΩF ðxÞ

ΩM ðx; tÞ

ρδðÞdΩ;

fδðÞ 1 ðÞ½δxi ; i gdΩ;

(10.27b) (10.27c)

ð

ΩM ðX; t1 Þ

δ½ðÞJdΩðX; t1 Þ;

(10.27d)

δðÞdΩðX; t1 Þ:

(10.27e)

ð

ðÞdΩðX; t1 Þ 5

ΩM ðX; t1 Þ

In this chapter, a local variation is adopted within the fluid domain, and a material variation is used in the solid domain. Therefore, for a function ðÞðX; tÞ defined in the material coordinate, there also exists the exchangeable relations    

DðÞðX; tÞ @ðÞðX; tÞ δ 5δ 5 δ ½ðÞðX; tÞ; t 5 ½δðÞðX; tÞ; t ; Dt @t   (10.28)

@ðÞðX; tÞ δ 5 δ ½ðÞðX; tÞ; i 5 ½δðÞðX; tÞ; i : @Xi

376

Fluid
Solid Interaction Dynamics

10.2.5

Local variation of an integral over a moving volume in space

Let the functional HðφÞ, defined over the moving region Ωðx; tÞ illustrated in Fig. 10.1, be expressible in the following form: H½φ 5

ð t2 ð t1

Ωðx; tÞ

Fðφ; φ; t ÞdΩdt;

(10.29)

where φ is a continuously differentiable function of ðx; tÞ. The local variation of this functional is defined as 1 δH 5 lim fH½φ 1 εδφ 2 H½φg; ε-0 ε

(10.30)

where ε is an arbitrary constant independent of φ, and ðx; tÞ denotes any arbitrary local variation of the function φðx; tÞ, independent of ε, satisfying the conditions δφðt1 Þ 5 0 5 δφðt2 Þ:

(10.31)

It is noted that when a local variation of the functional HðφÞ is taken, the boundary Γðx; tÞ of the region Ωðx; tÞ also experiences a variation and that the integral operations with respect to time t and to space x are not interchangeable because the boundary Γðx; tÞ moves. The substitution of Eq. (10.29) into Eq. (10.30) gives the local variation of this functional δH in the form 1 δH 5 lim ε-0 ε 1 ε-0 ε ð t 2 ð

5 lim

 ð t2 ð



ð

t1 Ωðx1εδx; tÞ ð t 2 ð t1 Ωðx; tÞ

Fðφ 1 εδφ; φ; t 1 εδφ; t ÞdΩ 2

Ωðx;tÞ

Fðφ; φ; t ÞdΩ dt

½Fðφ 1 εδφ; φ; t 1 εδφ; t Þ 2 Fðφ; φ; t ÞdΩ 1

ð ΔΩðx1εδx; tÞ

 Fðφ 1 εδφ; φ; t 1 εδφ; t ÞdΩ dt

 ð 1 5 δFdΩ 1 lim Fðφ 1 εδφ; φ; t 1 εδφ; t Þεδxi ν i dΓ dt ε-0 ε Γðx; tÞ t1 Ωðx; tÞ  ð t 2 ð ð 5 δFdΩ 1 Fðφ; φ; t Þδxi ν i dΓ dt; t1

Ωðx; tÞ

(10.32)

Γðx; tÞ

where, by comparison with Fig. 10.1, dΩ 5 εδxi ν i dΓ. From Eqs. (10.26a)
(10.26c), the first integral can be rewritten as ð t2 ð t1

ð t2 ð 

 @F @F δFdΩdt 5 δφ 1 δφ dΩdt @φ; t ; t Ωðx; tÞ t1 Ω @φ 8 9  ð t2 v v 2 ψ 1 2 gx δ > > j j j 3j > C B2 ð ρ =

f C B ~ C B ^ ^ ^ ^ dt dΩ 1 H10p p; ρf ; vi ; Φ; s; α; β; ζ; h; Ui 5 ρf B ρ v ð Φ 1 s β 1 αζ ÞdΓ ν f DΦ Dβ Dζ C > t1 > Ωf @ Γv > > A > > 2s 2α 2 (10.87) > > ; : Dt Dt Dt  ð t 2 ð ð   2 AðEij Þ 2 BðVi Þ 2 Ui F^ i dΩ 2 T^ i Ui dS dt; t1

ΩS

ST

t1

ΩS

ST

9 1 2 ðΦ;i 1 sβ 1 αζ ÞðΦ 1 sβ 1 αζ Þ=2 > ;i ð ;i ;i ;i ;i =

   B C ~ Ui 5 ^ ÞdΓ dt H9p p; ρf ; Φ; s; α; β; ζ; h; ρf @ 2 ψ 1 p 2 gx δ 2 Φ 2 sβ 2 αζ AdΩ 1 ρ^ f v^ν ðΦ 1 s^β 1 αζ j 3j ;t ;t ;t > t1 > Γv ; : Ωf ρf  ð t 2 ð ð   2 AðEij Þ 2 BðVi Þ 2 Ui F^ i dΩ 2 T^ i Ui dS dt: 8 ð t2 >

> > > vj vj 2 ψ 1 2 gxj δ3j > > > C B2 ð ð t2 > ð ρ =

DΦ Dβ Dζ C t1 > Ωf @ Γv Γφ > > A > > 2 s 2 α 2 > > ; : Dt Dt Dt  ð t 2 ð    2 AðEij Þ 2 BðVi Þ 2 Ui F^ i 1 Pi ðVi 2 Ui;t Þ 2 σij Eij 2 ðUi;j 1 Uj;i 1 Uk;i Uk;j Þ=2 dΩ t1

ð 2 ST

ΩS

) ð ^ ^ T i Ui dS 2 τ ij ηj ðUi 2 U i ÞdS dt: SU

(10.89) The variational constraint conditions of this functional at times t1 and t2 reduce to the conditions given in Eqs. (10.76)
(10.79). The stationary conditions of this functional include all the governing equations of fluid-structure dynamic interaction problems. Remark 4: The variational constraint conditions at times t1 and t2 given in Eqs. (10.76)
(10.79) can also be released, if these four conditions at times t1 and t2 are considered (see Xing and Price, 1992).

10.4.1.2 Compressible fluid with s;i  0 in the fluid domain Ωf If the entropy s of the fluid is treated as constant throughout the fluid domain Ωf , Eq. (10.48) is automatically satisfied, and s;i  0 in Ωf . From this condition, the vorticity ωi in Eq. (10.55) is independent of the entropy s, and the fluid field velocity v can be represented as vi 5 Φ; i 1 αζ ; i ;

(10.90)

because the term sβ ;i in Eq. (10.54) can be absorbed into Φ;i . In this case, the functionals H9 in Eq. (10.82), H8 in Eq. (10.86), H10p in Eq. (10.87), H9p in Eq. (10.88), and H14 in Eq. (10.89), respectively, reduce to the following forms: 8 9

 ð = h i ð t2

> =

2 ðΦ; i 1 αζ ; i ÞðΦ; i 1 αζ ; i Þ=2 ð B C

 B C p ^ ÞdΓ dt 5 ρf @ dΩ 1 ρ^ f v^ν ðΦ 1 αζ 2ψ1 2 gxj δ3j 2 Φ; t 2 αζ ;t A > t1 > Γv > > ; : Ωf ρf  ð t2 ð ð   2 AðEij Þ 2 BðVi Þ 2 Ui F^ i dΩ 2 T^ i Ui dS dt; ΩS

t1

(10.93)

(10.94)

ST

~ Ui ; σij ; Eij ; Pi ; Vi  H12 ½p; ρf ; vi ; Φ; α; ζ; h; 8 

ð t2 ð =

t1 > Γv ; : Ωf 2 gxj δ3j 2 Φ; t 2 αζ ; t  ð t2  ð ð   2 AðEij Þ 2 BðVi Þ 2 Ui F^ i dΩ 2 T^ i Ui dS dt t1

387

ΩS

(10.99)

ST

from H7p in Eq. (10.94). The introduction of the incompressible condition excludes the equation of fluid motion, expressed in Eq. (10.61), from the stationary conditions of the associated functionals. This is because the velocity field vi of an incompressible flow can be solved independently of the pressure p, the latter being determined from the dynamic equation after the evaluation of the velocity vi through the variation of the functionals.

10.4.2

Fluid flow assumed irrotational

In this situation, the fluid field velocity is given by vi 5 Φ;i ;

(10.100)

corresponding to the case sβ ;i 5 0 5 αζ ;i in Eq. (10.54). Further, the internal energy e of the fluid depends only on the fluid density ρf , whereas the enthalpy ψ depends only on the pressure p. Therefore, the variational principles for an irrotational fluid flow are derived as special cases of the variational principles for the rotational examples given in Section 10.4.1. These variational principles are described as follows.

10.4.2.1 Compressible fluid From functional H9 , it is concluded that, among all the admissible solid displacement Ui satisfying the straindisplacement relations in Eq. (10.36), the velocity-displacement relations in Eq. (10.37), the displacement boundary condition in Eq. (10.41), and the time instant conditions in Eq. (10.81), as well as the admissible fluid field arguments ρf , vi , and Φ satisfying Eqs. (10.70) and (10.80) and the function h~ describing the free surface disturbance, the actual motion satisfying the governing Eqs. (10.34), (10.40), (10.47), (10.62), (10.64), (10.66), (10.69), (10.72), (10.74), (10.75), and (10.90) makes the 5-argument functional 8 9

 ð = h i ð t2 < 5=ð4hÞ; 1D WðRÞ 5 A0 ð1 2 6R2 1 8R3 2 3R4 Þ; A0 5 5=ðπh2 Þ; 2D > : 105=ð16h3 Þ; 3D:

(12.61e)

(12.61f)

The performance of SPH, in terms of accuracy, stability, and computational efficiency, is influenced by the choice of kernel functions. Aluru (1999) presented a reproducing kernel particle method for meshless analysis to improve the performance of kernel functions. In the PhD thesis, Sun (2013) presented the numerical investigations of the nine commonly used kernel functions (Liu et al., 2003) in a 1D case. These nine kernel functions are as follows.

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

505

12.2.3.2 Kernel functions Quadratic (Hicks and Liebrock, 2000) 

r 2  3 12 W ðr; hÞ 5 ; 4h h

0#

r # 1: h

(12.62a)

This quadratic smoothing function was used in the grid-free finite integration method. The main advantage of this kernel function is the simplicity and ease of computation, whereas the drawback is that the first derivative is not zero on the boundary of the support domain, which means that it does not have compact support for its first derivative.

Quartic (Lucy, 1977) 

r 2

r 3

r 4  5 W ðr; hÞ 5 126 18 23 ; 4h h h h

0#

r #1 h

(12.62b)

This quartic smoothing function and its first two derivatives satisfy the compact support condition.

Johnson’s quadratic (Johnson and Beissel, 1996) W ðr; hÞ 5

  1 3 r 2 3 r 3 1 2 ; h 16 h 4h 4

0#

r #2 h

(12.62c)

The specialty of this kernel is that the first derivative increases as the particles move closer, and it decreases as they move apart. This is advantageous for adjusting the position of particles to maintain stability. However, the derivative of this kernel function is not smooth at r 5 0.

Gaussian (Gingold and Monaghan, 1977) W ðr; hÞ 5

1 ðπh2 Þn=2

e2ðr=hÞ ;

2 1 W ðr; hÞ 5 pffiffiffi e2ðr=hÞ ; πh

2

n: dimension of space; n 5 1;

0#

r #2 h

(12.62d)

Gaussian kernel was used to simulate nonspherical stars originally. It is sufficiently smooth even for the secondorder derivative. However, it is not really compact as theoretically it never goes to zero. This can result in a large support domain with an inclusion of many particles for the particle approximation.

Super Gaussian (Monaghan and Poinracic, 1985a)  3 r 2 2ðr=hÞ2 W ðr; hÞ 5 2 ; n: dimension of space; e h ðπh2 Þn=2 2   1 3 r 2 2ðr=hÞ2 r 2 W ðr; hÞ 5 pffiffiffi ; n 5 1; 0 # # 2 e πh 2 h h 1



(12.62e)

This is one of the higher-order smoothing functions that are devised from lower-order forms. Its main disadvantage is that the kernel is negative in some region of its support domain, which may lead to unphysical results for hydrodynamics problems.

506

Fluid
Solid Interaction Dynamics

Cubic spline (Monaghan and Poinracic, 1985a) 8 3 r 2 3 r 3 > > 1 2 1 > > 2 h 4 h > > C< r 3 W ðr; hÞ 5 n 1 22 h > > > 4 h > > > : 0

0,r,h h # r , 2h

;

(12.62f)

r $ 2h

where n denotes the dimension of space, and C is the normalization factor, of which the values 2/3, 10/7π, and 1/π are, respectively, for 1D, 2D (circular), and 3D (spherical) cases. The cubic spline function is the most widely used smoothing functions since it resembles a Gaussian function while having compact support. However, the second derivative of the cubic spline is a piecewise linear function, so the stability properties can be inferior to those of smoother kernels. Quartic spline 0 0 11 8   

> 4 4 4 > 1 @ r r r > > 12:5 2 @5 11:5 1 10 10:5 AA; > > 24h h h h > > > > > 0 1 > > >

4 4 < 1 r r @ 2:52 A 2 5 1:52 W ðr; hÞ 5 24h h h > > > > > > 1 r 4 > > > 2:52 > > 24h h > > > : 0

0 , r , 0:5h

0:5h # r , 1:5h 1:5h #

(12.62g)

r , 2:5h h

r $ 2:5h

Higher-order splines were introduced because they are better approximations of the Gaussian smoothing kernel and more stable. Quintic spline

0 1 8

r 5

r 5

r 5 > > 1 > @ 32 A > 2 6 22 1 15 12 > > 120h h h h > > > > > 0 1 > > > 1

r 5

r 5 < @ 32 A 2 6 22 W ðr; hÞ 5 120h h h > > > > > > 1 r 5 > > > 32 > > 120h h > > > : 0

0,r,h

h # r , 2h 2h #

(12.62h)

r , 3h h

r $ 3h

New quartic (Liu et al., 2003)   1 2 9 r 2 19 r 3 5 r 4 W ðr; hÞ 5 2 1 2 ; h 3 8 h 24 h 32 h

0#

r #2 h

(12.62i)

This quadratic smoothing function satisfies the compact support condition for the first derivative, and it has a smoother second derivative than the piecewise linear second derivative of the cubic function. Therefore the stability properties should be superior to those of the cubic function. However, the second derivative is not a monotonic function of r. This may lead to an incorrect approximation.

12.2.3.3 Numerical tested functions and error estimation The preceding kernel functions were used to approximate the following five common functions in a 1D case by Sun et al. (2011) and Sun (2013).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

F1 ðxÞ 5 x; F2 ðxÞ 5 x3 ; F3 ðxÞ 5 e2x ; F4 ðxÞ 5 sin x; F5 ðxÞ 5 tan x:

507

(12.63)

In the numerical tests, these functions first were approximated in integral form and then transformed into particle approximations. In each approximation, the effects of different smoothing lengths and different particle numbers were considered separately. Integration approximation As shown in Fig. 12.6, the 1D domain is uniformly divided into a set of points with distance δr, and the smoothing domain of each point is further divided into a set of points with spacing of δr0 . The integral approximation is formulated by ð N X   F ðxa Þ 5 F ðxb ÞW ðxa 2 xb Þdx0 5 F ðxb ÞWab δx0 ; (12.64) Ω b51 Wab 5 W ðxa 2 xb Þ: Here, δx0 is the spacing of points within the smoothing domain, that is, δx0 5 δr0 . The analytical result is calculated and compared with the integral approximation results, and then the error is obtained. Particle approximation In particle approximation, δx0 represents a particle volume, which is related to material properties. In order to stay consistent with the integral approximation, the volume of a particle should satisfy the following condition: mb δx0 5 ; (12.65a) ρb so the mass density can be expressed as ρ5

mb : δx0

(12.65b)

The mass of the system is assumed to be a unit as no specific material is considered. Since a uniform distribution of mass is preferred, the discretization in particle approximation is slightly different from integral approximation. The problem domain is just uniformly divided into a number of particles with same mass mb of spacing δr0 as shown in Fig. 12.7. From the integral approximation in Eqs. (12.64) and (12.65b), it follows that the particle approximation of the mass density N N X   X mb 0 ρ 5 W δx 5 mb Wab : ab δx0 b51 b51

Hence, the particle approximation of the function FI(x), (I 5 1, 2, 3, . . ., 9) at each particle is expressed as δr

δr′ a–1

b a

a+1

Smoothing domain of point a FIGURE 12.6 Discretization of 1D domain for an integral approximation. 1D, One-dimensional.

δr′ a–1

a

Smoothing domain of point a FIGURE 12.7 Discretization in particle approximation.

a+1

(12.65c)

508

Fluid
Solid Interaction Dynamics



N N N X  X mb X Wab F ðxa Þ 5 F ðxb ÞWab δx0 5 F ðxb ÞWab 5 F ð x b Þ PN : ρb b51 b51 b51 b51 Wab

(12.66)

Here subindex I is neglected, and we have used mb being a constant for the uniform division of the domain. The main difference between the integral approximation in Eq. (12.64) and the particle one in Eq. (12.66) is the introduction of material property in Eq. (12.66). Error parameter for accuracy estimation Smoothing length and particle numbers are the two key factors affecting the accuracy of a kernel approximation. Hence, the accuracy of each kernel function is tested with different smoothing lengths and different particle numbers. The error parameter for accuracy is defined by a mean square root (MSR): vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u N u1 X   ε5t ε2 ; εb 5 Fðxb Þ 2 Fðxb Þ ; (12.67) N b51 b where ε is the MSR error, εb is the error between the analytical results and SPH results at the bth particle, and N is the number of particles used for error analysis. In order to further analyze the relationship between the error and the number of neighboring particles N, as well as the value of smoothing length h, we respectively assume that εN 5 AN α ;

log εN 5 log A 1 α log N;

(12.68)

ln εh 5 ln B 1 β ln h;

(12.69)

and εh 5 Bhβ ;

where α and β are variables to be determined, and A and B are constant coefficients. Based on these assumptions, when logarithmic scales are used, the relationship between ε and N will appear as a straight line, as shown in Eq. (1.68), and so is the case for ε and h, as shown in Eq. (12.69).

12.2.3.4 Numerical test results Integration approximation Fig. 12.8 shows that for the five functions in Eq. (12.63), the results obtained from the integral approximation in Eq. (12.64) agree well with the analytical data except for function F 5 tan(x) due to its singularity. In this case, the problem domain should be divided into two sections to avoid incorrect computation through the singular point. The results of the integral approximations with the fixed number of neighboring particles N 5 100 for the five functions in Eq. (12.63), influenced by the smoothing length h, are shown in Fig. 12.9, where the logarithmic scales based on Eq. (12.69) are adopted. From Fig. 12.9, it can be observed that in most cases decreasing the smoothing length h reduces the error. However, the function F 5 tan(x) is a special case, for which the error is high and fluctuates. It is difficult to improve the results by decreasing the smoothing length. The reason for this is that this function is singular in the domain. Overall, the super-Gaussian kernel function provides smaller errors, and the results are not sensitive to the change of smoothing length. Using the logarithmic scales, Fig. 12.10 shows the results of the errors for five functions affected by the neighboring particles when the smoothing length is fixed at h 5 0.1. From this figure, it is noted that, generally, the five neighboring particles can produce an approximation with an error of less than 5% for all kernel functions except for function F 5 tan(x). This means that the ratio of about k 5 2 between the spacing and the smoothing domain radius (kh) can provide accurate enough results for engineering applications. Increasing the number of neighboring particles can slightly improve the accuracy of the approximation for most kernels except cubic spline. It seems that the quartic kernel provides the smallest error in most cases and that cubic spline kernel is not sensitive to the decreasing smoothing length. Particle approximations The results of particle approximations for the five functions in Eq. (12.63) are shown in Fig. 12.11 and are compared with the corresponding theoretical results, which also agree well except for function F 5 tan(x) due to its singularity

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

F=x integral approxiamtion vs analytical

F=x3 integral approxiamtion vs analytical

2.5

10

Analytical results

1.5 1 0.5 0

6 4 2

0

0.5

1 1.5 Problem domian legth

0

2

0.5

1 1.5 Problem domian legth

2

1

Analytical results Integral approximation N=100,h=0.1

1.2

0

F=sin(x) integral approxiamtion vs analytical

F=exp(–x) integral approxiamtion vs analytical

1.4

0.8

1

Function vavlue

Function vavlue

Integral approximation N=100,h=0.1

8 Function vavlue

Function vavlue

Analytical results

Integral approximation N=100,h=0.1

2

509

0.8 0.6 0.4

0.6

0.4 0.2 Analytical results

0.2 0

0 0

0.5

1

1.5

2

Integral approximation N=100,h=0.1

0

0.5

Problem domian legth

1

1.5

2

Problem domian legth F=tan(x) integral approxiamtion vs analytical

60 Analytical results

Function vavlue

40

Integral approximation N=100,h=0.1

20 0 –20 –40 –60

0

0.5

1

1.5

2

Problem domian legth FIGURE 12.8 Results of integral approximations by new quartic for the five functions in Eq. (12.63), compared with the corresponding theoretical results (Sun, 2013).

510

Fluid
Solid Interaction Dynamics

0

Error vs smoothing length in integral approximation F=x

In(average error)

–5 –10 –15 –20 –25 –30 –35 –40 –4

–3

–3.5

–3 –2.5 ln(smoothing length)

–2

–1.5

Error vs smoothing length in integral approximation F=x3

In(average error)

–4 –5 –6 –7 –8 –9 –10 –4

–5

–3.5

–3 –2.5 ln(smoothing length)

–2

–1.5

Error vs smoothing length in integral approximation F=exp(-x)

Quadratic

In(average error)

–6 –7

Quartic

–8

New-quartic

–9

Johnson-quadratic

–10 –11

Super-Gaussian

–12

Cubic-spline

–13 –4

–4

–3.5

–3 –2.5 ln(smoothing length)

–2

–1.5

Gaussian

Error vs smoothing length in integral approximation F=sin(x)

In(average error)

–5 –6 –7 –8 –9 –10 –11 –12 –4

7

–3.5

–3 –2.5 ln(smoothing length)

–2

–1.5

Error vs smoothing length in integral approximation F=tan(x)

In(average error)

6 5 4 3 2 1 0 –1 –4

–3.5

–3 –2.5 ln(smoothing length)

–2

–1.5

FIGURE 12.9 The errors ln εh in Eq. (12.69) of integral approximations of five functions affected by the smoothing length ln h, with N 5 100 (Sun, 2013).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

Error vs the number of neighbouring particles in integral approximation F=x

Error vs the number of neighbouring particles in integral approximation F=x3

0

0 –1 Log (average error)

Log (average error)

511

–10

–20

–30

–2 –3 –4 –5

–40

2

4

6

8

10

16

14

12

18

–6

20

2

4

Number of neighbouring particles

–4

–4 Log (average error)

Log (average error)

–2

–6 –8 –10 –12 6

8

10

12

14

16

10

12

14

16

18

20

Error vs the number of neighbouring particles in integral approximation F=sin(x)

–2

4

8

Number of neighbouring particles

Error vs the number of neighbouring particles in integral approximation F=exp(-x)

2

6

18

Number of neighbouring particles

20

–6 –8 –10 –12 2

4

6 8 10 12 14 16 Number of neighbouring particles

18

20

Error vs the number of neighbouring particles in integral approximation F=tan(x)

3.5

Log (average error)

3 2.5 2 1.5 1 0.5

2

4

6

8 10 12 14 16 Number of neighbouring particles

18

20

FIGURE 12.10 The errors log εN of integral approximations of five functions affected by the neighboring particle number N, with fixed h 5 0.1 and legend the same as shown in Fig. 12.9 (Sun, 2013).

point. The influence of smoothing length and the number of particles on errors in particle approximation are also studied, and the results are shown in Figs. 12.12 and 12.13, respectively. From Fig. 12.12, it is clear that a smaller smoothing length provides smaller error. In the case of linear function approximation, the error is very small, although the curves are not showing a clear trend, which implies that the particle approximation of the liner function is normally very accurate even with a large smoothing length. Therefore decreasing the smoothing length is not meaningful in this case. It seems that the super-Gaussian gives the best accuracy in this particle approximation and quartic is the second best. Comparing the integral and particle approximations, we have found that the error is reduced with smaller smoothing length. The results can also be improved by increasing the number of particles, but this is not as efficient as using smaller smoothing length. This indicates that a proper smoothing length is more important to obtain accurate approximations. For function F 5 tan(x), as the value of the function approaches infinity, its singular point within the problem

512

Fluid
Solid Interaction Dynamics

F=x particle approxiamtion vs analytical

2.5

F=x3 particle approxiamtion vs analytical

10 Analytical

Function value

Analytical

Particle approximation N=100,h=0.1

2 1.5

6

1

4

0.5

2

0

0

Particle approximation N=100,h=0.1

8

0.5 1 1.5 Problem domain length

0

2

0

F=exp(-x) particle approxiamtion vs analytical

F=sin(x) particle approxiamtion vs analytical

1.4

1 Analytical Particle approximation N=100,h=0.1

1.2

0.8

1

Function value

Function value

2

1.5 1 0.5 Problem domain length

0.8 0.6

0.6 0.4

0.4

0

Analytical Particle approximation N=100,h=0.1

0.2

0.2 0

0.5 1 1.5 Problem domain length

2

0

0

0.5 1 1.5 Problem domain length

2

F=tan(x) particle approxiamtion vs analytical 100

Function value

50 0 –50 Analytical

–100

–150

Particle approximation N=100,h=0.1

0

0.5 1 1.5 Problem domain length

2

FIGURE 12.11 Results of particle approximations by new quartic for five functions in Eq. (12.63), compared with the corresponding theoretical results (Sun, 2013).

domain, it is difficult to obtain an accurate approximation. In this case, the problem domain can be divided into two sections, and error can be assessed in each section. In both the integral and particle approximations, all the kernels produce similar results; the choice of kernel is not very important in most cases. As shown in Figs. 12.9 and 12.12, the derivatives of error with respect to smoothing length are approximately constant for the functions x3 ; e2x , and sin x in both integral and particle approximations, which implies that the second derivative of error with respect to smoothing length approximates zero and that SPH approximation has a second order

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

Error vs smoothing length in particle approximation F=x

Error vs smoothing length in particle approximation F=x3

–35

–2 –3 In(average error)

–35.5 In(average error)

513

–36 –36.5

–4 –5 –6 –7

–37 –8 –37.5 –3.4 –3.2

–3

–2.8 –2.6 –2.4 –2.2 –2 ln(smoothing length)

–9 –3.4 –3.2

–1.8 –1.6 –1.4

–6

–5

–7

–6

–9

–1.8 –1.6 –1.4

–7 –8 –9 –10

–11 –12 –3.4 –3.2

–2

Error vs smoothing length in particle approximation F=sin(x) –4

In(average error)

In(average error)

Error vs smoothing length in particle approximation F=exp(-x)

–10

–2.8 –2.6 –2.4 –2.2

ln(smoothing lenghth)

–5

–8

–3

–3

–2.8 –2.6 –2.4 –2.2

–2

ln(smoothing length)

–1.8 –1.6 –1.4

–11 –3.4 –3.2

–3

–2.8 –2.6 –2.4 –2.2 –2 ln(smoothing length)

–1.8 –1.6 –1.4

Error vs smoothing length in particle approximation F=tan(x) 2.8

In(average error)

2.7 2.6 2.5 2.4 2.3 2.2 2.1 2 –3.4 –3.2 –3 –2.8 –2.6 –2.4 –2.2 –2 –1.8 –1.6 –1.4 ln(smoothing length)

FIGURE 12.12 The errors ln εh in Eq. (12.69) of particle approximations of five functions affected by the smoothing length ln h, with N 5 100 and legend the same as shown in Fig. 12.9 (Sun, 2013).

of accuracy with smoothing length. From Figs. 12.10 and 12.13, we have found that the influence of the total number of particles on the errors of the particle approximations is less sensitive than the influence of the number of neighboring particles on the error of integral approximations. This indicates that increasing the neighboring particles is more efficient than increasing the total number of particles, and therefore the number of neighboring particles is an important factor affecting the accuracy. To summarize, integral approximation does not differ from particle approximation in most cases, which means that the particle approximation is consistent with integral approximation. Different kernel functions give similar approximations; especially for new quartic, cubic spline, quartic spline, and quintic spline functions, they provide close results in most cases. It seems that the quartic kernel function shows the best performance generally. SPH approximation has a second order of accuracy with smoothing length. Decreasing the smoothing length or increasing the number of neighboring particles is normally useful in improving accuracy.

514

Fluid
Solid Interaction Dynamics

Error vs the number of particles in particle approximation F=x3 –4

–35

–4.5 log(average error)

log(average error)

Error vs the number of particles in particle approximation F=x –34.5

–35.5 –36 –36.5 –37 –37.5 30

–6 –6.5 –7 –7.5

40

50

60

70 80 90 Number of particles

100

110

120

–7

–6.5 log(average error)

–6.5

–6

–7.5 –8 –8.5 –9

40

50

60 70 80 90 Number of particles

100

110

120

–7 –7.5 –8 –8.5 –9

–9.5 –10 30

30

Error vs the number of particles in particle approximation F=sinx

Error vs the number of particles in particle approximation F=exp(-x)

log(average error)

–5 –5.5

40

50

60 70 80 90 Numer of particles

100

110

–9.5

120

30

40

50

60 70 80 90 Number of particles

100

110

120

Error vs the number of particles in particle approximation F=tanx

log(average error)

5 4 3 2 1 30

40

50

60

70

80

90

100

110

120

Number of particles

FIGURE 12.13 The errors log εN of particle approximations of five functions affected by the particle number N, with fixed h 5 0.1 and legend the same as shown in Fig. 12.9 (Sun, 2013).

12.2.3.5 Choosing smoothing length and symmetrization of particle interaction As the numerical test results for nine kernel functions presented in Sections 12.2.3.3 and 12.2.3.4, the smoothing length h is very important in the SPH method, which directly influences the efficiency of the computation and the accuracy of the solution. Physically, in practical simulations, if the smoothing length is too small, there may not be enough particles in the support domain κh to exert forces on a given particle, which results in low accuracy. Also, if the smoothing length is too large, all the details of the particle or local properties may be smoothed out, and the accuracy suffers too. For a given accuracy, the number N of the particle inside the effective support usually is fixed. Practice shows that for 1D, 2D, and 3D problems, the number of neighboring particles and the particle itself should be about 5, 21, 57, respectively, if the particles are placed in a lattice with a smoothing length of 1.2 times the particle spacing and κ 5 2. However, in dynamic simulations, most particles are moving, and particle density varies with time, so that for a given

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

515

accuracy and computation efficiency, the smoothing length should vary with time as well. So, for problems where the fluid expands or contracts locally to maintain consistent accuracy throughout the space, a different smoothing length could be used for each particle. Therefore it is crucial to set a criterion to determine the smoothing length to be adjusted during computations in order to maintain numerical accuracy and efficiency. One way to obtain this criterion is to require that each particle has approximately the same amount of mass. For example, under the distribution of the Gaussian kernel function in Eq. (12.62d), from Eq. (12.65c), the mass density at point a is given by N N X   X ρa 5 mb Wab 5 mb b51

and the mass of particle a satisfies the equation

b51

1 ðπh2 Þn=2

e2ððxa 2xb Þ=hÞ ;

    ma 5 ρa ΔVa ~ ρa hna ;

2

(12.70a)

(12.70b)

so that, to keep particle mass ma as a constant for all particles in the supporting domain, the smoothing length ha should satisfy  1=n : (12.70c) ha ~ ρ a This can be implemented based on the following nondimensional equation by considering the averaged density to update the smoothing length, that is,  1=n h ρ 5 0 ; (12.71a) h0 ρ where the subindex 0 indicates the initial values of the smoothing length and the density, respectively, and n is the number of dimensions of the problem. Another proposed method to evolve the smoothing length is by solving the following deferential equation: dh h dρ 52 ; dt dρ dt

(12.71b)

which implies that the smoothing length should be small in the region of the density increasing with time. The updated smoothing length approach leads to its variation in time and space, which results in each particle having its own smoothing length. For two interaction particles I and J with the corresponding different smoothing lengths hI and hJ, the influencing domain of particle I may cover particle J but not necessarily vice versa. Therefore the interaction between these two particles is a violation of Newton’s third law. In order to overcome this problem, it is necessary to find a way to preserve the symmetry of particle interactions. The proposed approach is to choose a common smoothing length hIJ for the interaction of two particles by using the arithmetic or geometric mean: Arithmetic: Geometric:

hI 1 hJ ; 2 2hI hJ hIJ 5 ; hI 1 hJ hIJ 5

(12.72a)

and the choice may be hIJ 5 maxðhI ; hJ Þ or hIJ 5 minðhI ; hJ Þ: The value of the smoothing function can be obtained using the chosen smoothing length WIJ 5 WðRIJ ; hIJ Þ;

(12.72b)

where RIJ is the distance between these two particles. Another approach is to use the average smoothing function value W IJ 5 without using a symmetric smoothing length.

½WðhI Þ 1 WðhJ Þ ; 2

(12.72c)

516

Fluid
Solid Interaction Dynamics

12.2.3.6 Smoothed particle hydrodynamics approximation rules To transform the PDEs of the conservation laws in continuum mechanics into SPH formulations, the following approximation rules are used: G

The average of the product of two functions f1 and f2 can be approximated by the product of the individual ensembles, that is,       (12.73) f1 Uf2 5 f1 U f2 :

G

The gradient of a scalar field function Φ can be approximated by the gradient of its kernel representation, that is, hrΦi 5 rhΦi:

(12.74a)

The approximation in Eqs. (12.74a)
(12.74d) may be an exact relation in an infinite domain, which, by using the Green theorem, is proved as follows: ð ð 0 0 0 hrΦi 5 Φ;i ðx ÞWðx 2 x ; hÞdx 5 f½Φðx0 ÞWðx2x0 ; hÞ;i 2 Φðx0 ÞW;i ðx 2 x0 ; hÞgdx0 ðΩ ðΩ ð (12.74b) @W @W 0 0 0 5 Φðx ÞWðx 2 x ; hÞν i dx 2 Φðx0 Þ 0 ðx 2 x0 ; hÞdx0 5 Φðx0 Þ ðx 2 x0 ; hÞdx0 5 rhΦi: @xi @xi S Ω Ω Here, we have considered the first integration on the surface as vanishing. Eq. (12.74a) provides a very useful relationship, hrΦi 5 rhΦi 2 ΦhrW i;

(12.74c)

since the kernel function W is localized, and hrW i 5 0. Actually, based on Eq. (12.74a) and the utility Eq. (12.7), we have ð hrW i 5 rhW i 5 r Wð0ÞWðx 2 x0 ; hÞdx0 5 rWð0Þ 5 0: (12.74d) Ω

12.2.4

Smoothed particle hydrodynamics formulation for Navier
Stokes equations

12.2.4.1 Particle approximation of density The density approximation is very important in the SPH method since it basically determines the particle distribution and the smoothing length evolution. The following two approaches can be used to obtain the particle approximation of density. Summation density This approach directly applies the SPH approximation to the density itself. For a given particle I, the density is obtained by Eq. (12.49), that is, N X ρI 5 mJ WIJ ; (12.75) J51

where N is the number of particles in the support domain of particle I, mJ is the mass of particle J, and WIJ 5 WðxI 2 xJ ; hÞ 5 W ðjxI 2 xJ j; hÞ 5 WðRIJ ; hÞ; jxI 2 xJ j RIJ 5 ; h

(12.76)

representing the smoothing function of particle I evaluated at particle J. Here, WIJ has a unit of the inverse of volume. Eq. (12.75) states that the density of a particle can be approximated by the weighted average of the densities of the particles in the support domain of that particle. The summation density approach conserves the total mass M in the entire unchanged problem domain Ω, which may be proven as follows. Based on the unity condition of the smoothing function, we have ð ð ð 0 0 0 M 5 ρðxÞdVðxÞ 5 ρðx ÞWðx 2 x ÞdVðx Þ dVðxÞ Ω Ω Ω ð (12.77) ð ð 0 0 0 0 0 5 ρðx Þ Wðx 2 x ÞdVðxÞ dVðx Þ 5 ρðx ÞdVðx Þ 5 M; Ω

Ω

Ω

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

517

implying that the total mass is unchanged. However, when Eq. (12.75) is applied to the particles on the boundary of the problem, given the boundary particle deficiency, the result will be spurious. This edge effect could be remedied by using boundary virtual particles discussed later in the chapter. Another disadvantage for the density summation approach is that it entails more computational cost to evaluate the density before other parameters are evaluated. This method well represents the essence of the SPH approximation, so that it seems more popular in practical applications. For simulating general fluid phenomena, this approach yields better results. To improve the accuracy near both the free boundaries and the material interfaces with a density discontinuity, Eq. (12.75) is normalized with the SPH summation of the smoothing function itself over the neighboring particles as follows (Randles and Libersky, 1996): PN mJ WIJ ρI 5 PN J51 : (12.78) ðm J =ρJ ÞWIJ J51 Continuity density This approach approximates the density based on the continuity equation Dρ 5 2 ρvi;i ; Dt

(12.79)

in association with some transformation. Different transformations lead to different forms of density approximation equations. Form I: Using Eqs. (12.47) and (12.48), we approximate the velocity divergence in Eq. (12.79) so that N N X X DρI mJ J @WIJ mJ J @WIJ 5 ρI vi 5 2 ρ v : I J Dt ρ ρ i @xIi @x i J51 J J51 J

(12.80)

Form II: Consider the following expression of the derivative of the unit 0 5 1;i 5

N X mJ @WIJ J51

ρJ @xIi

;

(12.81)

from which it follows 0 5 ρI vIi

N X mJ @WIJ J51

ρJ @xIi

5 ρI

N X mJ J51

ρJ

vIi

@WIJ : @xIi

(12.82)

The summation of Eqs. (12.80) and (12.82) leads to another form of density approximation N X DρI mJ IJ @WIJ 5 ρI v ; Dt ρ i @xIi J51 J

 I  J vIJ i 5 vi 2 vi :

(12.83)

This particle approximation introduces velocity difference in the SPH formulations, which is preferred due to accounting for the relative velocities of particle pairs in the support domain. Form III: Using the continuity equation in the form Dρ 5 2 ρvi;i 5 2 ½ðρvi Þ;i 2 vi ρ;i ; Dt and particle approximation formulation in Eq. (12.47), we obtain 2 3 N DρI X m @W m @W J IJ J IJ 4 ρ J vI 5 5 2 ρ vJ i Dt ρJ ρJ J i @xIi @xIi J51 N X @WIJ 5 mJ vIJ : i @xIi J51

(12.84)

(12.85)

This equation shows clearly that the time change rate of density of a particle is closely related to the relative velocities, weighted by the gradient of the smoothing function, between this particle and all the other particles in the support domain.

518

Fluid
Solid Interaction Dynamics

The continuity density approach gives an approximation of the material derivative of the density, from which the total mass over the problem domain might be different at each time step. For simulating events with strong discontinuity, such as an explosion or a high-velocity impact, this approach is preferred.

12.2.4.2 Particle approximation of momentum The conservation law of momentum in continuum mechanics is given by Dvi 1 5 σij;j ; Dt ρ

(12.86)

in which we assume that the external body force is neglected, and σij is the stress tensor in fluids. Similar to the continuity density approach, we can obtain different forms of momentum approximation equations. Form I: Using Eqs. (12.47) and (12.48), we approximate the stress divergence in Eq. (12.86) so that N DvIi 1X mJ J @WIJ 5 σ : Dt ρI J51 ρJ ij @xIj

(12.87)

Form II: Adding the following identity derived from Eq. (12.83), 05

N N m σI X σIij X mJ @WIJ J ij @WIJ 5 ; ρI J51 ρJ @xIj ρ ρ @xIj J51 I J

(12.88)

N X σIij 1 σJij @WIJ DvIi 5 mJ : Dt ρI ρJ @xIj J51

(12.89)

we obtain

This formulation is symmetric to particle I and J, which reduces the error arising from the particle inconsistency, and it is therefore frequently used to evolve momentum. Form III: Using the identity

  Dvi 1 σij σij 5 σij;j 5 1 ρ ; Dt ρ ρ ;j ρ2 ;j

(12.90)

and particle approximation formulation in Eq. (12.47), we obtain N N σJij @WIJ σIij X DvIi X mJ @WIJ 5 mJ 1 ρ I Dt ρJ ρJ @xj ρI ρI J51 ρJ J @xIj J51 ! N X σIij σJij @WIJ 5 mJ 1 2 : @xIj ρ2I ρJ J51

(12.91)

This is another symmetric formulation evolving momentum. Introducing the constitutive equation for Stokes fluid σij 5 2 pδij 1 τ ij ;

τ ij 5 μγ ij ;

2 γ ij 5 vi;j 1 vj;i 2 vk;k δij ; 3

(12.92)

where p is the isotropic pressure, τ ij denotes viscous stress tensor, and γ ij is the tensor of shear strain rate. For linear viscous fluids, the viscous stress is proportional to the γ ij through the coefficient of dynamic viscosity μ. Eqs. (12.89) and (12.91) can be rewritten, respectively, as follows: N N X X μI γ Iij 1 μJ γ Jij @WIJ DvIi pI 1 pJ @WIJ 52 mJ 1 mJ ; I Dt ρI ρJ @xi ρI ρJ @xIj J51 J51

and  I  N N X X μI γ Iij μJ γ Jij DvIi p pJ @WIJ 52 mJ 2 1 2 1 m 1 J Dt ρI ρJ @xIi ρ2I ρ2J J51 J51

!

@WIJ : @xIj

(12.93)

(12.94)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

519

In these two formulations, the SPH approximations for the pressure and the viscous stress are separately expressed. From Eq. (12.92), the SPH approximation of shear strain rate tensor is derived as ! N N X mJ J @WIJ 2 X mJ J @WIJ I J @WIJ δ γ ij 5 vi 1 v 2 v : (12.95) ij j I I 3 J51 ρJ k @xIk ρ @xj @xi J51 J Again, from Eq. (12.81), it follows 0 5 vIj

N X mJ @WIJ J51

0 5 vIi

ρJ @xIi

N X mJ

5

J51

5

J51

ρJ

@WIJ ; @xIi

N mJ @WIJ X mJ I @WIJ 5 v ; I ρ ρ i @xIj @x j J51 J J51 J

N X mJ @WIJ

ρ @xIk J51 J

5

N X mJ J51

which, when substituted into Eq. (12.93), gives N X mJ

vIj

N X

0 5 vIk

γ Iij

ρJ

vIJ i

@WIJ @WIJ 1 vIJ j @xIj @xIi

!

ρJ

vIk

(12.96)

@WIJ ; @xIk

N 2 X mJ IJ @WIJ 2 δij v : 3 J51 ρJ k @xIk

(12.97)

This equation includes the relative velocity between two particles I and J. After the shear strain rate for two particles are calculated, the acceleration can be calculated by Eq. (12.93) or Eq. (12.94).

12.2.4.3 Particle approximation of energy The conservation law of energy states that the time change rate of energy inside an infinitesimal fluid cell equals the summation of the net heat flux into that cell, the time rate of work done by the body, and the surface forces acting on that fluid cell. If the heat flux and the body forces are neglected, the time change rate of internal energy e consists of the work done by the isotropic pressure in the volumetric strain and the energy dissipation due to the viscous shear forces and is governed by the following equation:  De p μ p μ 4 5 2 vj;j 1 γ ij vi;j 5 2 vj;j 1 γ ij ðvi;j 1 vj;i Þ 1 ðvi;j 2 vj;i Þ 2 vk;k δij Dt ρ ρ ρ 2ρ 3 (12.98) p μ γ ij γ ij ; 5 2 vj;j 1 ρ 2ρ where in the derivation process, the following identities have been used: γ ij ðvi;j 2 vj;i Þ 5 0;

γ ij δij 5 γ jj 5 0:

(12.99)

For the particle approximation of Eq. (12.98), Eq. (12.97) can be used for the second part, and the pressure work can be modeled in different ways as follows. Form I: Considering the continuity Eq. (12.84) and its particle expression in Eq. (12.85), we have p p p Dρ ; 2 vj;j 5 2 ð2 ρvj;j Þ 5 2 ρ ρ ρ Dt 2 Form II: The identity

N p @vIi pX @WIJ 5 2 mJ vIJ : i I ρ @xi ρ J51 @xIi

    p @ p @ p vj 1 vj 2 vj;j 5 2 ; ρ @xj ρ @xj ρ

(12.100)

(12.101)

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leads to another approximation, 2

N X p @vIi mJ pJ IJ @WIJ 5 v : I ρ @xi ρ2J i @xIi J51

(12.102)

Form III: Combining Eqs. (12.100) and (12.102), we obtain the most popular form of SPH approximation for the pressure work   N p @vIi 1X pI pJ IJ @WIJ 2 5 m 1 v : (12.103) J ρ @xIi 2 J51 ρ2I ρ2J i @xIi In this form, the variables appear symmetrically. Form IV: Using Eq. (12.83), we have 2 Form V: Based on the identity,

N p @vIi pI X mJ IJ @WIJ 5 v : I ρ @xi ρI J51 ρJ i @xIi

 p 1 @ @p 2 vj;j 5 2 ðpvj Þ 2 vj ; ρ ρ @xj @xj

(12.104)

(12.105)

we derive 2

N p @vIi 1X pJ @WIJ 5 mJ vIJ : i I ρ @xi ρI J51 ρJ @xIi

(12.106)

Form VI: Combining Eqs. (12.104) and (12.106), we obtain another symmetrical form of SPH approximation for the pressure work N p @vIi 1X pI 1 pJ IJ @WIJ 2 5 mJ v : (12.107) I ρ @xi 2 J51 ρI ρJ i @xIi The two most frequently used SPH approximation forms for energy equations are:   N DeI 1X pI pJ IJ @WIJ μ 5 mJ 1 vi 1 I γ Iij γ Iij ; I 2 2 2 J51 Dt @xi 2ρI ρI ρJ N DeI 1X pI 1 pJ IJ @WIJ μ 5 mJ vi 1 I γ Iij γ Iij : I Dt 2 J51 ρI ρJ @xi 2ρI

(12.108)

According to reported references, there is no obvious difference between the results obtained from these two approximation forms (Liu and Liu, 2003).

12.2.5

Numerical techniques for fluid flows

12.2.5.1 Artificial viscosity To remove numeric oscillations in fluid simulations involving shock waves, artificial viscosity was introduced. One such simulation is the von Neumann
Richtmyer artificial viscosity (Hirsch, 1988):  aN Δx2 ρðvj;j Þ2 ; vj;j , 0 DN 5 (12.109) 0; vj;j $ 0; where aN is an adjustable nondimensional constant, and another is a linear artificial viscosity  aL Δxcρvj;j ; vj;j , 0 DL 5 0; vj;j $ 0;

(12.110)

where aL is an adjustable nondimensional constant, and c denotes the speed of sound. These two artificial viscosities are widely used in fluid simulations by many numerical methods, such as FDM, FVM, FEM (Hirsch, 1988). For SPH simulations (Liu and Liu, 2003), the introduced artificial viscosity is given by

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

8 2 > < 2 αcIJ ϕIJ 1 βϕIJ ; ρIJ DIJ 5 > : 0;

IJ vIJ j xj , 0

521

(12.111)

IJ vIJ j xj $ 0;

where φIJ 5

IJ hIJ vIJ j xj ; IJ 2 xIJ j xj 1 ϕ

ϕ 5 0:1hIJ ;

cIJ 5

cI 1 cJ ; 2

I J vIJ j 5 vj 2 vj

ρIJ 5

ρI 1 ρJ ; 2

hIJ 5

hI 1 hJ ; 2

(12.112)

I J xIJ j 5 xj 2 xj :

In these equations, α and β are constants set around 1.0. The term with α produces a bulk viscosity, while the term with β suppresses particle interpenetration at a high Mach number. The factor φ prevents numerical divergences when two particles are approaching each other. The artificial viscosity given in Eq. (12.111) has the dimension of the pressure, which will be added in the pressure terms of the SPH equations. Another artificial viscosity was introduced into SPH simulations in the form qI qJ DIJ 5 2 1 2 ; ρI ρJ 8     (12.113) < 2αh ρ c vI  1 βh2 ρ vI 2 ; vI , 0 I I I j;j I I j;j j;j qI 5 : 0; vIj;j $ 0: Here, the artificial viscosity depends on the divergence of the velocity field.

12.2.5.2 Artificial heat Artificial heat is introduced into an energy equation to deal with excessive heating under severe circumstances such as the wall heating in the classic example of a stream of gas being brought to rest against a rigid wall. This artificial heat is added in the form (Liu and Liu, 2003) N X q

@WIJ IJ eI 2 eJ xIJ ; 2 j @xI IJ IJ ρ x x 1 φ j J51 IJ j j    2 qI 1 qJ     ; qI 5 2 αhI ρI cI vIj;j  1 βh2I ρI vIj;j  ; qIJ 5 2 HI 5 2

(12.114)

which will be added in the energy equation. Taking into account the artificial viscosity and heat, a very popular set of SPH formulations for the NS equations is as follows: N DρI X @WIJ 5 mJ vIJ ; i Dt @xIi J51 0 1 N σIij σJij DvIi X @WIJ 5 mJ @ 2 1 2 1 DIJ A I ; Dt @xj ρ ρ I J J51 0 1 N DeI 1X pI pJ @WIJ μ 5 mJ @ 2 1 2 1 DIJ AvIJ 1 I γ Iij γ Iij 1 HI ; i 2 J51 Dt @xIi 2ρI ρI ρJ

(12.115)

DxIj 5 vIj : Dt

12.2.5.3 Spurious zero-energy mode For some types of elements in FEM, due to an insufficient accuracy of numerical integrations, a spurious zero-energy mode with no strain or volume change in the element results in zero stress, leading to no resistance to the element

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W′

x

FIGURE 12.14 A regular particle distribution for a 1D case, which leads to zero-energy mode with W0 (0) 5 0. 1D, One-dimensional.

deformation (Bathe, 1996). For fluid analysis, the FDMs, such as a second-order central difference scheme, also suffer from this type of zero-energy mode for which the derivative of the variable at a certain grid point is zero, although the variable itself oscillates greatly. The same spurious mode problem also occurs in the SPH method when evaluating the derivatives. As shown in Fig. 12.14 for a 1D problem, the regular particle distribution leads to zero-energy mode for the first derivative calculation. To avoid the spurious zero-energy mode, the proposed method is to use two separate types of particles: one for velocity evaluation and the other for stress evaluation. Also, if the particles are not uniformly distributed, the summed contributions from these particles generally will not lead to zero derivatives.

12.2.5.4 Pressure calculation for incompressible fluids For solving compressible flows, the particle motion is driven by the pressure gradient, while particle pressure is calculated by the local particle density and internal energy by means of the state equation. However, for incompressible flows, the actual equation of the fluid state leads to an extremely small time step. Artificial compressibility To remedy this, some artificial state equation for incompressible fluids is proposed as follows (Hirsch, 1988). One of these equations is  γ ρ p5B 21 ; (12.116) ρ0 where a constant γ 5 7 is used in most circumstances, ρ0 is the reference density, and B is a problem-dependent parameter used to set a limit for the maximum change of the density, such as the initial pressure. The subtraction of 1 can remove the boundary effect for free surface flows. Another artificial compressibility equation is simpler in the form of the multiplication of the square of the speed of sound with the density, that is, p 5 c2 ρ;

(12.117)

which was used to simulate incompressible flows with a low Reynolds number. In the artificial compressibility technique, the sound speed is a key factor that deserves careful consideration, for which many discussions and investigations are available; see, for example, Hirsch (1988). Projection method Another way of computing pressure is to treat the fluid as truly incompressible and to use a Poisson equation to obtain the pressure values. This method was developed by Chorin (1967) and is called the projection method, which has proven very effective in the simulations by MPMs (Cummins and Rudman, 1999; Ataie-Ashtiani et al., 2008; Brown et al., 2001; Sun, 2013, 2016; Sun et al., 2017). The numerical process is as follows. For incompressible fluids, the mass density is a constant, and the corresponding mass conservation law requires the divergence of velocity vanishing, that is, vi;i 5 0:

(12.118)

Dvi 2 p;i 1 5 1 τ ij;j 1 gδ3i ; ρ Dt ρ

(12.119)

The momentum equation governing fluid flows is

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523

where g is the gravitation acceleration along the negative direction of x3. The proposed time integration process for Eq. (12.119) in a time step from tn to tn11 is split into two parts as follows: n11=2

vi

5 vni 1

1 n τ Δt 1 gδ3i Δt; ρ ij;j

n11=2

vn11 2 vi i

5

21 n11 p Δt; ρ ;i

(12.120)

and n11=2

vi;i

5

1 n11 p Δt; ρ ;ii

n11=2

pn11 ;ii 5

ρvi;i Δt

:

(12.121)

The first part considers the effect from the body force and the viscous stress but not the pressure gradient, so that an n11=2 unknown velocity vi is assumed, which will be determined by the second part considering the pressure gradient. Taking a divergence of the second equation in Eq. (12.120) and considering it satisfied for the time being, that is, vn11 i;i 5 0, we obtain a Poisson equation of the pressure in Eq. (12.121). From Eqs. (12.45) and (12.52), it follows that N

 1X n11=2 IJ @WIJ mJ vi ; ρ J51 @xIi N

J @W 1X IJ pn11 mJ pn11 : ;ii 5 ;i ρ J51 @xIi

(12.122)

@p xIJ @p xIJ pJ 2 pI PIJ xIJ 52 i 52 i 5 2i ; @xJi rIJ @rIJ rIJ rIJ rIJ PIJ 5 pI 2 pJ ;

(12.123)

n11=2

vi;i

52

Using Eq. (12.48), we obtain pJ;i 5

which, when substituted into Eq. (12.122), gives pn11 ;ii 5

N N

IJ @W IJ 1X mJ pn11 1 X IJ n11=2 IJ xi @WIJ 5 2 m v ; J i 2 1 ϕ2 @xI ρ J51 rIJ Δt J51 @xIi i

N X mJ pn11 xIJ @WIJ IJ

J51

i

2 1 ϕ2 @xI rIJ i

N

IJ @W ρ X IJ n11=2 52 mJ vi : Δt J51 @xIi

(12.124)

This equation can be solved efficiently. Finally, the velocity and position of each fluid particle can be renewed for the next time step using the following equation: n1ð1=2Þ

vn11 5 vi i

1 2 Pn11 Δt ρ ;i

n1ð1=2Þ 5 vi

2 Δt

N X J51

mJ



 Pn11 Pn11 @WIJ I J 1 2 1 DIJ : 2 ρ ρ @xIi

(12.125)

Here, we have used Eq. (12.52) to express the pressure gradient and introduced the artificial viscosity. Kinematic constraint method Ellero et al. (2007) presented a SPH model for incompressible fluids. As opposed to solving a pressure Poisson equation in order to get a divergence-free velocity field, the incompressibility was achieved by requiring, as a kinematic constraint, that the volume of the fluid particles was constant. They used Lagrangian multipliers to enforce this restriction. These Lagrange multipliers play the role of nonthermodynamic pressures whose actual values are fixed through kinematic restriction. The authors used the SHAKE methodology familiar in constrained molecular dynamics as an efficient method for finding the nonthermodynamic pressure satisfying the constraints. The model was tested for several flow configurations.

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12.2.5.5 Boundary treatment Free surface On the free surface, the kinematic and dynamic conditions given in Eqs. (3.131a)
(3.131c) should be stratified. The kinematical condition in Eq. (3.131a) physically implies that the free surface is a material surface that can be automatically satisfied by the moving particle method since the MPM is a Lagrangian method in which the material point motion is traced. The dynamic conditions given in Eqs. (3.131b) and (3.131c), which respectively corresponds to the cases considering and not considering the surface tension. Normally, it can be assumed that the atmospheric pressure is p0 5 0 as the initial reference state of the system; also, the dynamic pressure p derived from Eq. (3.131a) or Eq. (3.131b) should be added to the partials on the free surface identified by the following approaches. 2D case: The method presented by Koh et al. (2012), Sun et al. (2014, 2015a), and Sun (2016) can be adopted. As shown in Fig. 12.15, each particle is taken as the center of a circle discretized by 360 points (dashed line). If each of these points on a circle is completely covered by its neighbors, the particle at the center of the circle is recognized as an inner fluid particle; otherwise it is a particle on the free surface. For example, in Fig. 12.15, particle A is identified as a particle on the free surface, since it is one of the dashed points on the circle not covered by its neighbors, while particle B is an inner fluid particle. 3D case: The efficient multilevel strategy for free surface identification is widely used (Marrone, 2011; Marrone et al., 2011; Tamai and Koshizuka, 2014) and is implemented in two steps. Step 1 aims preliminarily to find the possible particles on the free surface using the neighbor particle number Np based on the condition Np , βN0 ;

(12.126)

where N0 is the neighbor particle number of a typical inner fluid particle, such as 32 for a uniform particle distribution, and β is a tuning parameter of value about 0.96. The particles satisfying Eq. (12.126) may be the ones on the free surface. Step 2 refines the searching by checking the geometry property. As shown in Fig. 12.16, a sphere of radius r 5 1.05r0 (r0 is the initial particle distance) and centered at a particle identified by Step 1 is drawn, where nz denotes a unit vector pointing from the center of the sphere to the direction of the sparsest particles. Around this vector, a circular patch on the surface of the sphere, determined by angle θ 5 π=4, is discretized by evenly distributed points. If these points are all covered by the same sphere of its neighbor particles, this particle is recognized as an inner particle; otherwise it is regarded as a free surface particle. Fixed solid boundary In most of the cases, the solid boundary in the particle method will normally be handled by the ghost (virtual) particle method that was used by Libersky and Petschek (1991) and Monaghan (1994). Liu and Gu (2001a,b) suggested using two types of virtual particles: type I virtual particles are located right on the solid boundary. Type II particles fill the boundary region outside the type I particles, which are constructed in the following way. For a real particle I located within the distance of κhI from the boundary, a vertical particle is placed symmetrically on the outside of the boundary.

FIGURE 12.15 Identification of the particles on the free surface (Sun et al., 2017).

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525

FIGURE 12.16 Identification of free surface particles for 3D cases (Sun et al., 2017). 3D, Three-dimensional.

These virtual particles have the same density and pressure as the corresponding real particles but opposite velocity. Since type II particles are not enough to fully prevent the real particles penetrating the solid boundary, the application of type I particles to produce a sufficient repulsive boundary force fIJx when a particle approaches the boundary is needed. The repulsive force is calculated using a similar approach for the molecular force of Lennar-Jones form (Liu and Liu, 2003; Sun 2013): ( h α  β i   2 D r =r 2 r =r xIJ =rIJ ; r0 =rIJ # 1 0 IJ 0 IJ x fIJ 5 (12.127) 0; r0 =rIJ . 1; where α and β are usually chosen as 12 and 4, respectively, and D is a parameter related to the problem, which should be in the same scale of the square of the largest velocity; rIJ denotes the distance between the real particle I and the virtual type I particle J, and r0 is selected approximately close to the initial particle spacing. Type I virtual particles take part in the kernel and the particle approximations for the real particles, but the position and physical variables do not evolve in the simulation process. Type II virtual particles are produced symmetrically according to the corresponding real particles, so that its parameters are determined by the related real particles in the simulation process. Enforced essential boundary conditions SPH is based on nonlocal interpolants, which makes it difficult to incorporate prescribed or developed data on an essential boundary into the discrete governing equations (Morris et al., 1997; Randles and Libersky, 1996; Takeda et al., 1994). Extra care has to be taken in order to enforce the essential boundary conditions. An approach to introduce the prescribed boundary condition into the particle equation was presented by Li and Liu (2004) as follows. Suppose particle I is a boundary particle of which the supporting domain consists of three subdomains: an interior fluid domain Ωf , a boundary domain Γ, and an outside solid domain Ωs . On the boundary domain Γ on which particle I is located, the value v^ of a variable v is prescribed. Based on the partition of unity at position I, from Eq. (12.43) with ΔVJ ρJ 5 mJ , we obtain X X X 15 ΔVJ WIJ 1 ΔVJ WIJ 1 ΔVJ WIJ ; (12.128a) JAΩf

JAΓ

JAΩs

^ it follows from which, when multiplied by the prescribed variable v, X X X ^ J WIJ 1 ^ J WIJ 1 ^ J WIJ : v^ 5 vΔV vΔV vΔV JAΩf

JAΓ

On the other hand, the kernel approximation for the value vI of the variable v gives X X X vI 5 vJ ΔVJ WIJ 1 vJ ΔVJ WIJ 1 v^J ΔVJ WIJ : JAΩf

JAΓ

(12.128b)

JAΩs

JAΩs

(12.128c)

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Since the particles on Ωs have to satisfy the prescribed boundary condition and the particles on Γ should equal the calculated value vI by kernel approximation, we have ^ J 5 v^J ; vΔV vI 5 vJ ;

JAΩs ; JAΓ;

from which, when Eq. (12.128b) is subtracted from Eq. (12.128c), we obtain P JAΩf ðvJ 2 v^ÞΔVJ WIJ P vI 5 v^ 1 ; 1 2 JAΓ ΔVJ WIJ

(12.128d)

(12.128e)

which gives the kernel approximation formulation of the boundary particles with the prescribed boundary condition satisfied. In this equation, the outside particles are not involved.

12.2.5.6 Time integration The discrete SPH equations can be integrated with standard numerical integration approaches. The explicit time integration schemes are subject to the stability condition, which requires that the computational domain of dependence in numerical simulation should include the physical domain of dependence. Therefore the maximum speed of numerical propagation must exceed the maximum speed of the physical propagation, so that the time step is taken as   hi Δt 5 min : (12.129) c Introducing viscous and external force effects into Eq. (12.129), the corresponding time step is given by 2 3 h I 5; Δtd 5 min4 cI 1 0:6ðαcI 1 β max ϕIJ Þ Δtf 5 min

h i1=2 hI f1

;

(12.130)

Δt 5 minðλ1 Δtd ; λ2 Δtf Þ; where f is the magnitude of the force per unit mass (acceleration), and λ1 and λ2 are suggested as 0.4 and 0.25, respectively. Another expression for time step is proposed as  2 h μ (12.131) Δt 5 0:125 ; ν5 ; ν ρ for considering viscous diffusion. Here the kinetic viscosity v is involved.

12.2.5.7 Particle interactions SPH method uses the smoothing function defined in a compact support domain with a finite number of particles to represent a function by particle approximations. These particles are generally referred to as nearest neighboring particles or the integration particles for the involved particle. The all-pair search approach calculates the distance rIJ from a particle I to each and every particle J in the problem domain to find the nearest particles by rIJ , κh. If the symmetric smoothing length is employed, particle I is also within the support domain of particle J. Therefore the computational time in this all-pair search approach is very long, especially for cases with a large number of particles. Monaghan and Gingold (1983) suggested using cells as a bookkeeping device to substantially save the neighboring particle search time, which is called a linked-list algorithm implementing in the following procedure (Monaghan, 1985; Dominguez et al., 2010). For a given problem, a temporary mesh with uniform spacing κh is overlain on the problem domain. For a given particle I, its nearest neighboring particles can be only in the same grid cell or in the immediately adjoining cells. Therefore the search is confined to only 3, 9, and 27 cells for 1D, 2D, and 3D space, respectively, and the search time is greatly reduced (Wro´blewski et al., 2007).

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The SPH simulations consume quite a bit of time to identify the interacting particles, after which the simulations can be carried out in the summation process over all the interacting particles. The two pairwise storage arrays can be used to store a particle ID number and its interaction particle ID number, and the two smoothing function arrays can be used to store the calculated weighted function values and its derivative values for each pair particles, based on which following summations can be effectively implemented.

12.2.5.8 Open source smoothed particle hydrodynamics solvers The SPHysics home page (http://www.wiki.machenster.ac.uk/sphsics/indes.php) provides open source SPH solvers (http://dx.doi.org/10.1016/j.cageo.2012.02.028; http://dx.doi.org/10.1016/j.cageo.2012.02.029) that can be used worldwide. SPHysics is a platform of SPH codes inspired by the formulation of Monaghan (1992) and developed jointly by researchers at the Johns Hopkins University (United States), the University of Vigo (Spain), the University of Manchester (United Kingdom), and the University of Rome La Sapienza (Italy). Developed over a number of years primarily to study free surface flow phenomena where Eulerian methods can be difficult to apply, such as waves or the impact of dam breaks on offshore structures, they have announced three available codes. Interested readers may access their webpage for more detailed information and the available codes.

12.2.6

Improved methods based on smoothed particle hydrodynamics

12.2.6.1 Corrective smoothed particle method The SPH method has many advantages in computations as just discussed, but it also has met with difficulties. One problem is the tensile instability (Swegle et al., 1995), in which the particle motion becomes unstable when particles are in a tensile stress state, which could result in particle clumping or a complete blowup in the computation. For a onedimensional von Neumann stability analysis, Swegle et al. (1995) identified a criterion for tensile instabilities in SPH. A sufficient condition for the unstable growth is σ11 W;11 . 0;

(12.132)

where σ11 5 σxx and W;11 5 d2 W=dx2 represent the stress and the second derivative of the smoothing function, respectively. As pointed out by Swegle et al. (1995), tensile instability does not happen for problems with equations of state where no tensile stress is generated. Another difficulty involves the boundaries of the problem. On the boundaries, the truncated integral in the kernel approximation and the insufficient particles in the particle approximations cause one of difficulties called the boundary deficiency problem in SPH (Monaghan, 1992; Randles and Libersky, 1996), which results in lower density near or on the boundary and finally yields spurious pressure gradients on the surface (Morris et al., 1997). To remedy these difficulties, different methods have been proposed (Liu and Liu, 2003); for example, Randles and Libersky (1996) developed a normalization formulation for the density approximation as shown in Eq. (12.78). In these modified approaches, the corrective SPM (CSPM) developed by Chen et al. (1999a,b,c, 2000) is discussed as follows. Using the Taylor series expansion at point I of function f(xi), we obtain    I I x 2 x x 2 x i j     i j f;ijI 1 ?; f ðxi Þ 5 f xIi 1 xi 2 xIi f;iI 1 (12.133a) 2 where, as used in this book, the subindex i denotes the tensor index. Multiplying Eq. (12.133a) by the smoothing function Wðxi 2 xIi Þ and integrating over the support domain Ω, as well as neglecting the derivative term, we obtain Ð f ðxÞWðx 2 xI ÞdΩ I I ; (12.133b) f ðxi Þ 5 f ðx Þ 5 ΩÐ I Ω Wðx 2 x ÞdΩ which is the corrective kernel approximation for the function f(xi). The corresponding particle approximation can be directly obtained by considering dΩJ 5 mJ =ρJ and replacing the volume integration by its summation from preceding equation, that is, PN ðmJ =ρJ ÞfJ WIJ I fI 5 f ðx Þ 5 PJ51 : (12.133c) N J51 ðmJ =ρJ ÞWIJ

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In the similar way to multiplying Eq. (12.133a) by the first derivative W;j ðxi 2 xIi Þ of the smoothing function, the three first derivatives f;iI at point I can be obtained by solving the following three coupling equations: ð ð    

  I I I f;i xi 2 xi W;j x 2 x dΩ 5 f ðxÞ 2 f ðxI Þ W;j x 2 xI dΩ: (12.133d) Ω

Ω

This equation can be rewritten in the matrix form 2 a11 a12 4 a21 a22 a31 a32 where

32 3 2 3 f;1 a13 b1 a23 54 f;2 5 5 4 b2 5; f;3 a33 b3

ð aji 5 ðxi 2 xIi ÞW;j ðx 2 xI ÞdΩ; ðΩ bj 5 ½f ðxÞ 2 f ðxI ÞW;j ðx 2 xI ÞdΩ:

(12.133e)

(12.133f)

Ω

Also, we can obtain the corresponding particle approximation in the form 2 32 3 2 3 f;1 a~11 a~12 a~13 b~1 4 a~21 a~22 a~23 54 f;2 5 5 4 b~2 5; f;3 a~31 a~32 a~33 b~3

(12.133g)

where a~ji 5 b~j 5

N X mJ 

J51 N X

ρJ

 xJi 2 xIi W;jIJ ;

mJ J f ðx Þ 2 f ðxI Þ W;jIJ : ρ J51 J

(12.133h)

Equations in Eqs. (12.133a)
(12.133h) give the fundamental formulae in the CSPM used to generate the discrete equations for PDEs of the conservation laws in continuum mechanics. It has been shown that CSPM successfully improves the boundary deficiency and reduces the tensile instability in the SPH method (Chen et al., 1999b).

12.2.6.2 Moving particle semiimplicit method The moving particle semiimplicit (MPS) method was developed by Koshizuka and Oka (1996) and has shown improvement in stability in numerical simulations (Kondo and Koshizuka, 2011). The MPS method is similar to the SPH method (Gingold and Monaghan, 1977; Lucy, 1977) in that both methods provide approximations to the strong form of the PDEs on the basis of integral interpolants. However, the MPS method applies simplified differential operator models based solely on a local weighted averaging process without taking the gradient of a kernel function. In addition, the solution process of the MPS method differs from that of the original SPH method as the solutions to PDEs are obtained through a semiimplicit prediction
correction process rather than the fully explicit process in the original SPH method. The basic formulations of MPS method are summarized as follows. Function approximation The interpolation of a field function f(x) is similar to the normalization formulation in Eq. (12.133b) and is given by f ðxI Þ 5

N X f ðxJ ÞWJI J6¼I

nI

;

where WJI is the kernel function as used in SPH, but here it takes the form  r =r; ð0 # r , re Þ; WJI 5 e 0; ðre , rÞ;

(12.134a)

(12.134b)

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529

5.0 W (r)

0.0

r/re

1.0

FIGURE 12.17 Weight or kennel function W(r).

where re is the radius of the interaction area, r 5 jxJ 2 xI j is the distance between the particles I and J, and N is the particle number in the support domain. This function is called the weight function, which does not satisfy the utility property in the MPS method. The notation nI is defined by nI 5

N X

WJI

(12.134c)

J6¼I

and is called a particle number density in MPS. Actually, the meaning of this parameter denotes the total weighted particle number in the support domain. For point I, the different particle J has a different contribution to the variable value at point I, which, measured by the weight function W as shown in Fig. 12.17, shows that the particle J near particle I has the larger contribution to the variable value at particle I. Furthermore, considering Eq. (12.49) and assuming that each particle mass mJ 5 m is a constant, we have ρI 5 mnI , implying that nI is the equivalent number of particles with the same mass m in the supporting domain around point I. The particle number in a unit volume, or by normalization, can be obtained by the following equation: nI NI 5 Ð ; (12.134d) ΩI WðrÞdΩ where the denominator is the weighed volume of the supporting domain. Therefore Eq. (12.134d) implies that the total weighted particle number is divided by the weighted volume, which gives the particle number in the unit volume around particle I. Assuming that the particles have the same mass m, the mass density of the fluid can be derived as mnI ρI 5 mNI 5 Ð : (12.134e) ΩI WðrÞdΩ Compared with Eq. (12.49), this is the normalization formulation for mass density. Modeling of incompressibility The particle number density is the equivalent number of particles, of which each carries the same mass m, in the supporting domain of the same volume; therefore a constant n0 implies an incompressible fluid, and the continuity equation is satisfied. If the particle number density n is not n0, it is implicitly corrected to n0 by adding a correction value n^ satisfying the mass conservation equation n 2 n0 2 n~ 5 5 rU~v; Δtn0 Δtn0

n 1 n~ 5 n0 ;

(12.134f)

where v~ denotes the correction value of velocity, and its divergence is the volume deformation rate, of which the positive value gives the volume increment rate. If n~ 5 n0 2 n , 0, the particle number increases corresponding to a compression case with a negative volume deformation rate at state (*) to correct it; the divergence of the correction velocity v~ should contribute a positive divergence as shown in Eq. (12.134f). Using the relationship between the pressure gradient and the velocity given in Eq. (4.9a), the velocity correction value is derived from the implicit pressure gradient term as v~ 5 2

Δt n11 rp ; ρf

(12.134g)

530

Fluid
Solid Interaction Dynamics

from which, with Eq. (12.134f), a Poisson equation of pressure is obtained: 

ρ f nI 2 n0 : Δt2 n0

ðr2 pn11 ÞI 5 2

(12.134h)

The right side is represented by the derivation of the particle number density from its constant value. Derivative approximation The derivative of a function f(x) from points I to J is calculated by   df fJ 2 fI ðfJ 2 fI ÞðrJ 2 rI Þ 5 5 ; dr I rJ 2 rI jrJ 2rI j2

(12.135a)

from which the averaged gradient of function f(x) between particle I and its neighboring particles J is weighted as follows: ðrf ÞI 5

N dX ðfJ 2 fI ÞðrJ 2 rI Þ WJI ; n0 J6¼I jrJ 2rI j2

(12.135b)

where d is the number of space dimensions. In the numerical code by Koshizuka and Oka (1996), the final gradient formulation is   N fJ 2 fImin ðrJ 2 rI Þ dX ðrf ÞI 5 0 WJI ; (12.135c) n J6¼I jrJ 2rI j2 where fImin 5 min fJ ;

(12.135d)

for all neighboring points J in the supporting domain. Approximation of Laplacian A time-dependent diffusion equation with the diffusion coefficient v is represented in Laplacian terms as @f 5 νr2 f : @t

(12.136a)

The total variance of the distribution of function by the total particles n0 is proportional to 2dνΔt during time step Δt, where d is the number of space dimensions, of which the averaged contribution by each particle is proportional to 2dνΔt=n0 . The part of quantity fI of particle I is distributed to the neighboring particle J according to the kernel function. Therefore the quantity transferred from particle I to J is assumed to be ΔfIJ 5 fJ 2 fI 5

2dνΔt fI WJI ; n0 λ

(12.136b)

where λ is a coefficient with the dimension of length square due to the second derivatives with respect to the coordinate involved on the right-hand side of Eq. (12.136c), which is taken as the averaged second moment of supporting domain of the kernel function, that is, Ð WðrÞr 2 dΩ λ 5 ΩÐ I : (12.136c) ΩI WðrÞdΩ In this model, the right-hand side in Eq. (12.136b) represents the Laplacian contribution, multiplied by νΔt, of the quantity distributed (output from I) from I to J. Therefore the quantity at point I is reduced, so that we can write Eq. (12.136b) in the following form: ðr2 f ÞI 5 2

2d fI WJI ; n0 λ

(12.136d)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

531

where a negative notation is added on the right-hand side to denote that the quantity at point I is reduced. Similarly, we can represent the quantity distributed from particle J to I (input into I) in the forms ðr2 f ÞI 5

2d 2d fJ WIJ 5 0 fJ WJI ; 0 n λ n λ

(12.136e)

where the positive notation implies the quantity input into particle I. We have used the relation WIJ 5 WJI. Considering all neighboring particles, the approximation of Laplacian contribution by all particles J to I is represented by a summation of Eqs. (12.136d) and (12.136e) overall particles, that is ðr2 f ÞI 5

N X 2d ðf 2 fI ÞWJI : 0λ J n J6¼I

(12.136f)

The approximation formulations in Eqs. (12.134a)
(12.134h), (12.135a)
(12.135d), and (12.136a)
(12.136f) in MPS for any field functions and their derivatives with Laplacian operation provide the fundamental techniques to obtain the conservation laws in the MPS model, which is similar to the SPH process, neglected here. Interested readers may refer to the original paper by Koshizuka and Oka (1996), as well as further investigations and modifications, such as Javed et al. (2013a,b, 2014a,b); Javed (2013); Sun et al. (2015a,b); and Sun (2016).

12.3

Meshfree Galerkin methods

In this subsection, we briefly present the fundamental formulation and techniques of MGMs. As an example, we summarize only the essential concept of the moving least square (MLS) method, which seems to be the basis of many other various approaches. This method uses the MLS method to generate the kernel interpolant and then, based on the Galerkin weak or variational formulations of PDEs, to seek the approximation solutions of the problems. Detailed information on these formulations with mathematical proofs may be read in Liu and Liu (2003) and Li and Liu (2004).

12.3.1

Moving least square representing kernel interpolant

The MLS method has two main technical characteristics: one is the local standard least square procedure, and the other is the moving process from local region to the whole region for a global approximation. The moving process behavior was not clearly stated until it was later elaborated by Liu et al. (1997) and again by Li and Liu (1999a,b). The detailed formulations of the MLS methods can be read in the publications by Li and Liu (1996a, 1997, 2004), which is briefly summarized as follows.

12.3.1.1 Local standard least square approximation Let uðx; x~ Þ be a sufficiently smooth function defined on a simply connected open domain in the space, that is, ΩARn . Considering a fixed point x~ AΩ, we can define a local function  uðx; x~ Þ; xABlρ ð~xÞ l u ðx; x~ Þ 5 (12.137a) 0; otherwise; where Blρ ð~xÞ denotes an effective spherical ball with its dilation parameter ρ 5 ð1B3Þ 3 particle space that is similar to the smoothing length of SPH. According to the well-known Stone-Weierstrass theorem (Rudin, 1976), we are able to approximate u(x) by a local polynomial series:     l X ~ x 2 x~ l T x2x u ðx; x~ Þ 5 Pα Dα ð~x; ρÞ 5 P Dð~x; ρÞ; (12.137b) ρ ρ α51 where

T Dð~x; ρÞ 5 D1 D2 ? Dl ;

T PðxÞ 5 P1 ðxÞ P2 ðxÞ ? P3 ðxÞ :

(12.137c)

D is an unknown vector, and Pα ðxÞ is an independent polynomial basis with P1(x) 5 1. Since the polynomial series can be sufficiently smoothed by choosing a suitable term number l, the local approximation of the function in

532

Fluid
Solid Interaction Dynamics

Eq. (12.137b) is sufficiently smooth. For vectors: PðxÞ 5 1 PðxÞ 5 1 PðxÞ 5 1

examples, in 1D, 2D, and 3D cases, we have the following polynomial basis

T x x2 ? xl21 ;

T x1 x2 x21 x1 x2 x22 ; x1 x2 x3 x21 x22 x23 x1 x2

x2 x3

x3 x1

T

(12.137d) :

The error of the approximation in Eq. (12.137b) for the function u(x) at point I is given by   ~ T xI 2 x rðxI ; x~ Þ 5 P Dð~x; ρÞ 2 uðxI Þ; ρ

(12.138a)

of which the weighted square residual is defined as RðDÞ 5

N X

r 2 ðxI ; x~ ÞWρ ð~x 2 xI Þ;

(12.138b)

I51

where Wρ ð~x 2 xI Þ denotes the kernel function as a weight at point I. To minimum the residual in Eq. (12.138b), the standard least square method can be used to obtain the optimal coefficient vector D, satisfying     N X ~ 2 xI x~ 2 xI T x P (12.138c) uðxI Þ 2 P Dð~x; ρÞ Wρ ð~x 2 xI Þ 5 0; ρ ρ I51 from which we obtain

  N X x~ 2 xI P uðxI ÞWρ ð~x 2 xI Þ; ρ I51

(12.138d)

    N X x 2 xI T x 2 xI P P Wρ ðx 2 xI Þ; ρ ρ I51

(12.138e)

Dð~xÞ 5 M21 ð~xÞ and the moment matrix defined by MðxÞ 5 or in an integration form



ð MðxÞ 5

P Ω

   x2y T x2y P Wρ ðx 2 yÞdΩy : ρ ρ

(12.138f)

The moment matrix is always positive since Pα ðxÞ ðα 5 1; 2; . . .; lÞ are linearly independent, so that it is invertible. Substituting Eq. (12.138d) into Eq. (12.137b), we obtain the local standard weighted least square approximation of the function in the form     N X x~ 2 x x~ 2 xI ul ðx; x~ Þ 5 PT M21 ð~xÞ P uðxI ÞWρ ð~x 2 xI Þ: (12.138g) ρ ρ I51

12.3.1.2 Moving process for global approximation Eq. (12.138f) is optical in a local region Blρ ð~xÞ. To extend this approximation to the whole region, a moving process is ensured, which requires the fixed point x~ -x, that is,   N X uðxÞ 5 lim ul ðx; x~ Þ 5 PT ð0ÞM21 ðxÞ P x 2 xI uðxI ÞWρ ðx 2 xI Þ: (12.139a) x~ -x ρ I51 We define the correction function



 x 2 xI Cρ ðx; xI Þ 5 P ð0ÞM ðxÞP ρ     x 2 xI x 2 xI 5 BT ðx; ρÞ P 5 PT Bðx; ρÞ; ρ ρ T

21

Bðx; ρÞ 5 M21 ðxÞPð0Þ;

(12.139b)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

533

so that we can obtain the kernel function produced by the MLS process κρ ðx 2 xI ; xÞ 5 Cρ ðx; xI ÞWρ ðx 2 xI Þ;

(12.139c)

and the corresponding function approximation uρ ðxÞ 5

N X

uðxI Þκρ ðx 2 xI ; xÞ;

(12.139d)

uðyÞκρ ðx 2 y; xÞdΩ:

(12.139e)

I51

or its integration form

ð uρ ðxÞ 5

Ω

12.3.1.3 An example It would be helpful to examine Eqs. (12.139a)
(12.139e) through a pedagogic example. Now we consider the case of the linear polynomial basis

T

T Pð0Þ 5 1 0 0 0 ; (12.140a) PðxÞ 5 1 x1 x2 x3 ; of which the moment matrix can be derived using Eq. (12.138f), that is, 2 3 1 ð 6 7 x 2 y1 x1 2 y1 x3 2 y3 6 ðx1 2 y1 Þ=ρ1 7 1 1 MðxÞ 5 6 Wρ ðx 2 yÞdΩy ; 7 ρ1 ρ1 ρ3 Ω 4 ðx1 2 y1 Þ=ρ1 5

(12.140b)

ðx3 2 y3 Þ=ρ3 which may be rewritten in compact form:

where

ð  mαβ 5

Ω

m00 6m 6 10 MðxÞ 5 6 4 m20

m01 m11

m02 m12

m21

m22

3 m03 m13 7 7 7 5 MT ðxÞ; m23 5

m30

m31

m32

m33

xα 2 yα ρα

2



 xβ 2 yβ Wρ ðx 2 yÞdΩy ; ρβ

x0 2 y0 5 1; ρ0

(12.140c)

ð0 # α; β # 3Þ:

(12.140d)

The corresponding correction function in Eq. (12.139b) is expressed as 2 3 1 7

A 6 6 ðx1 2 y1 Þ=ρ1 7 Cρ ðx; xI Þ 5 1 0 0 0 6 7 jMj 4 ðx1 2 y1 Þ=ρ1 5 5

A11 

5 PT

A12 A13 jMj

ðx3 2 y3 Þ=ρ3 2 3 1

6 A14 6 ðx1 2 y1 Þ=ρ1 7 7 6 7 4 ðx1 2 y1 Þ=ρ1 5

 x 2 xI Bðx; ρÞ; ρ

ðx3 2 y3 Þ=ρ3

where the matrix A denotes the adjoint matrix of the moment matrix M(x) of which an element     ; α~ 6¼ α; β~ 6¼ β; Aαβ 5 ð21Þα1β mα~ β~  333

(12.140e)

(12.140f)

534

Fluid
Solid Interaction Dynamics

and

Bðx; ρÞ 5

12.3.2

A11

A12 A13 jMj

A14

:

(12.140g)

Shepard interpolant

Eq. (12.139d) is also considered as an interpolation of a function, which may be rewritten in the form uðxÞ 5

N X

uI NI ðx; ρÞ;

NI ðx; ρÞ 5 κρ ðx 2 xI ; xÞ;

(12.141a)

I51

which, when the polynomial series in Eq. (12.137d) is used, satisfies uðxI Þ 5 uI ;

(12.141b)

NI ðxJ Þ 5 δIJ ;

(12.141c)

but the interpolation shape function does not satisfy

which is exclusively reserved for the term interpolation. A special case is l 5 1 in which, when considering Eqs. (12.138a)
(12.138g) and (12.139a)
(12.139e), we have PðxÞ 5 1; N X Mð0Þ 5 Wðx 2 xI Þ;

1

M21 ð0Þ 5 PN

I51

I51

Wðx 2 xI Þ

; (12.141d)

Wðx 2 xI Þ κðx 2 xI ; xÞ 5 PN : I51 Wðx 2 xI Þ It is clear that N X

0 , κðx 2 xI ; xÞ , 1;

κðx 2 xI ; xÞ 5 1:

(12.141e)

I51

This particular interpolant was first studied by Shepard (1968), from which the interpolation in Eq. (12.141a) becomes PN uI Wðx 2 xI Þ uðxÞ 5 PI51 : (12.141f) N I51 Wðx 2 xI Þ As suggested by Shepard, a remedy for the interpolation property is to choose the singular weight function 1 ; jx2xI jα

Wðx 2 xI Þ 5

(12.142a)

where α is a positive even integer. Therefore the Shepard interpolant, based on Eq. (12.142a), is κS ðx 2 xI ; xÞ 5

1

P jx2xI jα NJ51

α

1=jx2xJ j

5

11

PN

J6¼I

1 jx 2 xI jα =jx 2 xJ jα

;

(12.142b)

which satisfies the following equations: κS ðxJ 2 xI ; xJ Þ 5 δIJ ; N X κS ðx 2 xI ; xÞ 5 1; I51

rκS ðx2xI Þx5J 5 0:

0 # κS ðx 2 xI ; xÞ # 1; lim κS ðx 2 xI ; xÞ 5

x-N

1 ; N

(12.142c)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

12.3.3

535

Orthogonal basis for local approximations

To reduce the time consumed in simulations, orthogonal basis would be a good choice. Based on the linear independent basis vector in Eq. (12.137c), the following orthogonal basis can be constructed: q1 ðx; x~ Þ 5 P1 ðx; x~ Þ 5 1;   p2 ; q1 x~  q1 ðx; x~ Þ; q2 ðx; x~ Þ 5 P2 ðx; x~ Þ 2  q1 ; q1 x~     p3 ; q1 x~ p3 ; q2 x~  q1 ðx; x~ Þ 2   q2 ðx; x~ Þ; q3 ðx; x~ Þ 5 P3 ðx; x~ Þ 2  q1 ; q1 x~ q2 ; q2 x~ ^ qα ðx; x~ Þ 5 Pα ðx; x~ Þ 2

  α21 X pα ; qβ x~   qβ ðx; x~ Þ; qβ ; qβ x~ β51

(12.143a)

α 5 1; 2; . . .; l:

Here, the following inner product definition is used ð   f ; g x~ 5 f ð~x 2 xÞgð~x 2 xÞWρ ð~x 2 xÞdΩ:

(12.143b)

Ω

Because of orthogonalization, the local moment matrix is diagonal, and its inversion is 1  : qα ; qα x~

(12.143c)



T Dð~xÞ 5 D1 D2 ? DN ; PN qα ðxI ; x~ ÞuI Wρ ð~x 2 xI Þ  Dα ð~xÞ 5 I51  : qα ; qα x~

(12.143d)

M21 ð~xÞ 5 diagðaα Þ;

aα 5 

Thus the vector in Eq. (12.138d) becomes

The global approximation correction function in Eq. (12.139b) is now given by Cρ ðx; xI Þ 5

l X qα ðx; xÞqα ðxI ; xÞ   ; qα ; qα x~ α51

(12.143e)

which can be used in Eqs. (12.139c) and (12.139d) to give the global approximation formulations based on the orthogonal basis in Eq. (12.143a). The Lagrangian polynomial (Waring, 1779) N x 2 xI Lðx; xI Þ 5 Π (12.144a) I50 xJ 2 xI I 6¼ J

is a typical example of an orthogonal polynomial that satisfies the following interpolation and orthogonal conditions: LðxJ ; xI Þ 5 δIJ ; N N X X ½Lðxα ; xJ ÞLðxα ; xI Þ 5 ½δαJ δαI  5 NδIJ : α51

12.3.4

(12.144b)

α51

Applications of the moving least square reproducing kernels

The kernel function given in Eq. (12.139c) can be used in numerical simulations in the following aspects: 1. By choosing different polynomial series and weight functions, this general equation produces the different kernel functions with sufficiently smooth characteristics to be used in MPMs, such as SPH and MPS discussed in Section 12.2. 2. This general function provides a shape function defined at point I, that is,

536

Fluid
Solid Interaction Dynamics

NI ðx; ρÞ 5 κρ ðx 2 xI ; xÞ;

(12.145a)

which can be used like the Rayleigh
Ritz functions in Section 1.4.3.3 or interpolation functions in FEA to obtain the approximation solutions of PDEs modeled by the variational principles, weighted residual method discussed in Chapter 1, Introduction. The references by Li and Liu (1996a, 1997, 2004) present various problems in continuum mechanics in the weak form of the functionals or Galerkin formulations and then solve them by the assumed solution uðxÞ 5

N X

NI ðx; ρÞQI ;

(12.145b)

I51

where QI denotes the node variables in analysis, which may be a time function for the dynamic problems. For examples, Belytschko et al. (1994a,b) adopted the MLS interpolant and the weak Galerkin formulation to simulate a crack growth problem in a linear elastic solid.

12.4

Mixed finite element—smoothed particle method for fluid
solid interaction problems

Now we discuss the proposed MFE
SPMs for nonlinear FSI problems. Compared with the FE
CFD method discussed in Chapter 11, Mixed finite element
computational fluid dynamics method for nonlinear fluid
solid interactions, the MFE
SPM method is more convenient in simulating problems with violent fluid motions, such as breaking waves and solid
fluid separations, since SPM models the fluids as many individual material particles that are easily traced even in violent flow cases. Here, SPM is used as a generalized term to include various MPMs, such as the SPH, MPS, and MLS methods.

12.4.1

Generalized solution procedure

As discussed in Section 11.3 for solving nonlinear FSI problems using the MFE
CFD approach, we also have two general options—the simultaneous integration and the partitioned iteration—to be chosen for the MFE
SPM dealing with nonlinear FSI problems. But the partitioned iterative procedure is suggested for combining the available powerful FEM codes for solid structures and well developed SPMs codes for fluids. Obviously, as reported in some references, it is possible to use SPM techniques to both fluids and solids modeling. However, the numerical practices show that for a solid continuum system defined in a 2D or 3D domain, it is not difficult to model it by the SPM, but for many complex structures often used in engineering, SPM modeling might not be effective. The combination of FEM for structures and SPM for fluids may be the most appropriate approach to solve nonlinear fluid
structure interaction problems involving violent fluid motions. In this combined method, the structures can be exactly modeled by the nonlinear FEM described in Section 11.1, and the fluids can be exactly simulated by the SPM discussed in Sections 12.2 and 12.3, as well as other by similar methods reported in the references. In this simulation, since both the fluid and the solid undergo their nonlinear mechanical process, the solutions for the fluid and the solid are also based on their iterative calculations. The flowchart of the numerical solution is given in Fig. 12.18, where the parameter α is a relaxing factor to modify the approximate intermediate solutions for numerical convergence purposes, which can be determined for different methods by experience or numerical tests in calculations. Based on theoretical analysis (Aitken, 1937; see also Gianola and Schaffer, 1987), this relaxation factor can be updated to α in the next iteration calculated by the increment, denoted by ΔðÞ, of the variables in the form α 5

2αðΔU ÞT ΔðU 2 UÞ   : ΔðU 2 UÞ

(12.146a)

The simulations consist of the following main steps: 1. Input the initial conditions: geometrical sizes, material constants, initial values of variables such as solid displacement and velocity, fluid pressure and velocity, boundary conditions, etc. 2. Do fluid SPM iterative calculations based on the prescribed initial solid information to obtain the fluid pressure P ðt 1 ΔtÞ, velocity, etc. 3. Do solid FE iterative calculations based on the fluid information obtained in step 2 to obtain the solid displacement U  ðt 1 ΔtÞ, velocity, and accelerations, etc.

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

Start t=0

537

Yes Stop No Time end

Initial P(t),U(t)

Fluid SPM calculation P* (t + Δt)

U = U + α(U*–U)

t = t + Δt Yes

⏐⏐U*–U⏐⏐< ε

Solid FEM calculation U* (t + Δt)

No

FIGURE 12.18 Flowchart of the partitioned iteration of mixed FEM
SPM for nonlinear fluid
structure interaction dynamics. FEM, FE method. FE, finite element; SPM, smoothed particle method.

4. Make a convergence check to determine whether the error is less than the allowed error ε, that is, :U  2 U: , ε. Time goes to next time step; otherwise, modify the solid displacement to U 5 U 1 αðU  2 UÞ to repeat the calculations in steps 2 and 3 until convergence reached. 5. Check the time period of interest in the FSI problem. If it has not been reached, go to next time step to repeat steps 2
4 until it is reached. The detailed numerical implementation process depends on the models adopted for solids and fluids. For nonlinear fluid
structure interaction problems in which the structure undergoes motions with no large rigid motions but only large elastic deformations, the updated FE formulation could be effective in simulating its motions. If the structure undergoes large rigid motions with small, relatively elastic deformation, FEM may be not effective. For example, for a structure undergoing motions with no elastic deformation, there is no elastic deformation energy anywhere in the rigid body, so that there is no need to use elements describing the local deformation at every point. Therefore for nonlinear rigid body fluid interactions, it is more effective to use the methods in rigid body dynamics to model the motions in the body. In Section 11.3.1.2, we gave the mathematical formulations governing rigid body motions coupled with fluid flows. In maritime structures
water interactions, such as nonlinear ship
water interactions or missile into or out of the water problems, the structure normally undergoes very large rigid motions with relatively small elastic deformation. To model this type of nonlinear FSI problem, it is convenient to separate the rigid motion and small elastic motion to establish the governing equations describing the solid motion, discussed in the next subsection.

12.4.2 Modeling of fluid
solid interaction involving large rigid motions with small elastic deformation The generalized mathematical modeling of the MFE
SPM for violent water
structure interactions, with freak waves and with the structure undergoing a large rigid motion plus a small elastic deformation, is presented by Xing et al. (2016). In this model, the total displacement of the structure is expressed by a rigid motion of 6 degrees of freedom (DOF) plus a small displacement relative to the rigid motion. Since the elastic deformation is small, its deformation is governed by a linear elastic theory, and a mode summation approach is valid in reducing the final size in the numerical equations. The elastic mode shapes of elastic structure can be obtained by FEA or by the available theoretical mode functions for some structural members, such as rods, beams, shafts, plates, etc. The water is assumed to be inviscid and incompressible, and its motion is governed by a nonlinear NS equation. On the coupling interface where no FS

538

Fluid
Solid Interaction Dynamics

separation occurs, the equilibrium and consistency conditions are required. The governing equations of the nonlinear FSI systems are as follows.

12.4.2.1 Governing equations As shown in Fig. 12.19, a solid occupies a material domain Ωs of boundary S 5 ST , SU , Σ with its outside normal vector denoted by ηi . A traction force T^ i and a displacement U^ i are, respectively, given on part of the boundary ST and SU. The solid body is floating on the surface of the water, and no displacement boundary SU is given, of which the dry structure is considered a free
free body in the space. The solid body interacts on its wet interface Σ with the water in the domain Ωf of boundary Γ 5 Γf , Γb , Σ , Γv1 , Γv2 with its outside normal vector ν i . The dynamic pressure on the free surface Γf is assumed to be the atmosphere denoted by p 5 0, and the sea bottom Γb is considered a fixed boundary with the water velocity vi 5 0. The possible velocities v^i may be given on the boundaries Γv1 or Γv2 ; for example, an inflow velocity is given on Γv1 , while Γv2 is treated as an infinite boundary with no disturbance of v^i 5 0. Three coordinate systems are defined. A Cartesian system o 2 x1 x2 x3 with positive x3 in the vertical direction is fixed at a point o on the free surface in the static equilibrium state of the system. A moving system O 2 y1 y2 y3 , with its axis O 2 yi parallel to the axis o 2 xi , (i 5 1, 2, 3), respectively, is fixed at the mass center O of the solid, which moves with the mass center. Beside these two coordinate systems, a principal inertial material system O 2 X1 X2 X3 is defined, of which the origin and three axes are located at the same positions of the system o 2 x1 x2 x3 at time t 5 0. The position coordinates of a material point at time t 5 0 are denoted by Xj to identify this material particle, which moves to a new position xj at time t. The motion of this material point Xj is denoted by a summation of a translation Uio of the mass center, a rigid rotation θi about three axes, and small elastic displacement Uj of the structure. Using the notations of Cartesian tensors and summation convention, such as the Kronecker delta δij and permutation symbol eijk , and denoting the rigid rotation by a tensor Rij ðθk Þ with its time derivative and partial derivative with respect to θi given in Eqs. (10.156) and (10.157), that is, @Rij 5 eimk Rkj ; @θm

R_ij 5 eimk θ_ m Rkj ;

(12.146b)

from which, along with the preceding assumptions, we can derive the dynamic equations of the system as follows. Since the material coordinate system is fixed at the mass center of the body and it is chosen as the principal inertial system, its first-order inertial moment and the second-order cross-inertial products vanish, and the inertial matrix J is a diagonal matrix, that is,  ð ð 0; j 6¼ i ρs Xi dΩ 5 0; ρs Xi Xj dΩ 5 Jij 5 (12.146c) JII ; j 5 I 5 i: Ωs Ωs The total mass M of the body is calculated by the following integration over the body volume: ð ρs dΩ 5 M:

(12.146d)

Ωs

x3y3 X3

y3

θ3

X2

X3

P

p

x

x2y2 X2

O

FIGURE 12.19 Scheme of nonlinear FSI system involving large structural rigid motions with small elastic deformation (Sun, 2016). FSI, Fluid
solid interaction.

t=0

ρgg

x1y1 X1

Ωs

θ1 y1

Ωf Γv1

y2

O Σ

Γb

θ2

t=t1 X1

Γv2

ρfg Fluid field modeled by particle method

Γf

u

x

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

539

Solid domain Displacement and velocity fields The position and velocity of particle Xj at time t, respectively, are xi 5 Xi 1 Uio 1 Rij Xj 1 Rij Uj ; o x_i 5 U_ i 1 R_ij Xj 1 R_ij Uj 1 Rij U_ j o 5 U_ i

(12.147a)

1 eimk θ_ m Rkj ðXj 1 Uj Þ 1 Rij U_ j :

Here the elastic displacement Ui of the material point Xj is measured in the material coordinate system, so that a rigid rotation is required to transfer it into the fixed coordinate system. Introducing a mode transformation matrix Φi with its related generalized coordinate q(t), the small elastic displacement can be expressed in the following summation form: 2 3 q1 6

6 q2 7 7 (12.147b) Uj 5 Φj q 5 φj1 φj2 ? φjN 6 7; 4 ^ 5 qN where φjα ðXÞ and qα(t) represent, respectively, the αth elastic mode shape of the dry structure and the corresponding generalized coordinate, and N denotes the retained mode number. These elastic modes satisfy the orthogonal relationships ( ð 0; α 6¼ β T ρs φiα φiβ 5 mαβ 5 ; m 5 diagðmα Þ; mα ; α 5 β Ωs ( ð 0; α 6¼ β (12.147c) ϕTiα;j Cijkl φkβ;l dΩ 5 kαβ 5 ; k 5 diagðkα Þ; k α5β α; Ωs kα m21 k 5 diagðω~ 2α Þ; ω~ 2α 5 : mα Here the diagonal matrices m and k, respectively, are the generalized mass and stiffness matrices of the elastic structure, and ω~ 2α is the square of the αth natural frequency of the structure. The elastic mode shape and the corresponding generalized coordinate are the functions of material coordinate X and time t, respectively. These mode shape functions can be obtained by traditional FEA or theoretical mode functions of the dry structure as previously mentioned. Based on Eq. (12.147b), the elastic structure with an infinite number of DOF is now approximated by a discrete system of DOF N, and the integrated body is now modeled by a system of DOF N 1 6 consisting of 6 rigid DOF and N elastic DOF for numerical simulation. For a free
free elastic structure with no any external forces applied on it, when it vibrates in an elastic natural mode φjα , its inertial force must be in dynamic equilibrium at each time, which implies ð ð 2 ρs φiα q€α dΩ 5 2 q€α ρs φiα dΩ 5 0; Ωs ð Ωs (12.147d) ρs φiα dΩ 5 0: Ωs

Substituting Eq. (12.147b) into Eq. (12.147a), we obtain the displacement and velocity field of the body in the following mode summation form ui 5 xi 2 Xi 5 Uio 1 Rij Xj 1 Rij Φj q; o Vi 5 U_ i 1 R_ij Xj 1 R_ij Φj q 1 Rij Φj q_ o 5 U_ i

(12.147e)

_ 1 eimk θ_ m Rkj ðXj 1 Φj qÞ 1 Rij Φj q:

Eq. (12.147e) can be expressed in the matrix form 2 03 2 03 _

U

U V 5 I A RΦ 4 θ_ 5; u 5 I R RΦ 4 X 5; q q_

U 5 U1

U2

U3

T

5 Φq;

(12.147f)

540

Fluid
Solid Interaction Dynamics

where u 5 u1

u2

X 5 X1

X2

Φ 5 ΦT1 2 6 6 A56 4

T

u3

X3

ΦT2

U0 5 U10

;

T

ΦT3

U20

θ 5 θ1

;

T

θ2

I 5 IT1

;

0

I3 RðX 1 ΦqÞ

0

0

I2 RðX 1 ΦqÞ 0

I1 5 1 0 0 ; I2 5 0 1

U30 θ3

T

T

;

;

IT3 ;

IT2

0

3

(12.147g)

7 7 I1 RðX 1 ΦqÞ 7; 5 0



0 ; I3 5 0 0 1 :

Strain field and strain energy density The linear strain tensor of the small deformation of the elastic body can be calculated by Eq. (4.50), that is, Eij 5

1 1 ðUi;j 1 Uj;i Þ 5 ðΦi;j 1 Φj;i Þq; 2 2

(12.148a)

from which the elastic strain energy density of the body is given by e5

1 1 1 Eij Cijkl Ekl 5 Ui;j Cijkl Uk;l 5 qT ΦTi;j Cijkl Φk;l q; 2 2 2

(12.148b)

where Cijkl is the elastic tensor as given in Section 3.4, and its symmetry has been used to derive Eq. (12.148b). The total strain energy of the elastic body is given by ð 1 1 ~ ΠðqÞ 5 edΩ 5 qT kq 5 q~ T Kq; 2 2 Ωs 2 3 0 0 0

T 6 7 (12.148c) q~ 5 U0T θT qT ; K 5 4 0 0 0 5; 0 0 k ð k 5 ΦTi;j Cijkl Φk;l dΩ 5 kT : Ωs

Here k is the generalized elastic stiffness matrix of the body as defined by Eq. (12.147c). Kinetic energy density Based on Eq. (12.147f), the kinetic energy density of the body per unit volume is calculated by k5

5

1 ρ VT V 2 s 1h 2

T θ_

_ 0T U 2

I

1 T6 5 q_~ 4 AT 2 ΦT RT

2 3 I i 6 T 7 q_ T 4 A 5 I ΦT R T A AT A ΦT R T A

RΦ

A 3

7 AT RΦ 5q_~ ΦT Φ

2 03 _ U

6 7 RΦ 4 θ_ 5 q_

;

(12.149a)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

541

where the rotation matrix satisfying RT R 5 I has been used. The total kinetic energy of the body is given by ð 1 T _ 0 _ _ _ 5 kdΩ 5 q_~ Mq; ~ TðU ; θ; θ; q; qÞ 2 Ωs 2 3 (12.149b) ð I A RΦ T T T 4 5 Mðθ; qÞ 5 ρs A A A A RΦ dΩ; Ωs ΦT RT ΦT RT A ΦT Φ where the matrix Mðθ;qÞ is the mass matrix. Based on Eqs. (12.146c), (12.147c), and (12.147d), we obtain ð ð ρs AdΩ 5 0; ρs AT dΩ 5 0; Ωs Ωs ð ð ρs RΦdΩ 5 0; ρs ΦT RT dΩ 5 0; Ωs Ωs ð ð ~ I Þ; ~ ρs AT AdΩ 5 diagðA ρs AT RΦdΩ 5 B; Ωs Ωs ð ~ 1 5 ρ ðqT ΦT 1 XT ÞRT IT I2 RðX 1 ΦqÞdΩ; A s 2 Ω ð s ~ 2 5 ρ ðqT ΦT 1 XT ÞRT IT I3 RðX 1 ΦqÞdΩ; A s 3 Ω ð s ~ 3 5 ρ ðqT ΦT 1 XT ÞRT IT I1 RðX 1 ΦqÞdΩ; A s 1 Ωs ð ρs ΦT ΦdΩ 5 diagðmα Þ 5 m;

(12.149c)

Ωs

so that the mass matrix Mðθ;qÞ is given by

2

MI

6 Mðθ; qÞ 5 4 0 0

0

3 0 ~ 7 B 5 5 MT ðθ; qÞ:

~ IÞ diagðA T ~ B m

(12.149d)

Generalized force We assume that the solid body is subjected a traction force T^ i on its surface ST and that a traction force caused by the water pressure p, that is, fip 5 2 pηi , on the wet surface Σ. The components of these two traction forces are defined in the material coordinate system, so that their components relative to the Eulerian system are obtained by the corresponding rotation, that is,

^ ^ 5 T^ 1 T^ 2 T^ 3 T ; T i 5 Rij T^ j ; T 5 RT; T (12.150a) p p f i 5 2 pRij ηj ; f 5 2 pRη: The body force of the solid is the gravitational force defined in the Eulerian system, which is represented by

(12.150b) f g 5 0 0 2ρs g : To obtain the generalized force vector Q, we use the virtual work δW produced by all forces when the system is subjected to a virtual displacement 2 03 δU

6 7 δu 5 vδt 5 I A RΦ 4 δθ 5; (12.150c) δq

542

Fluid
Solid Interaction Dynamics

to obtain the virtual work

ð δW 5

ð δu f dΩ 1 T g

Ωs

5 δU0T 5 δU0T

δθT

ð

T p

δu f dS 1 δuT TdS ST ð  ð ð

T g T p T T H f dΩ 1 H f dS 1 H TdS δq Σ

Ωs

Σ

δqT Q 5 δq~ T Q;

δθT

HT 5 I

and the generalized force vector

ST

A

RΦ

(12.150d)

T

ð

Qðp; θ; qÞ 5 F 1 F 1 F ; F 5 HT TdS; ST ð ð ð g p T g T p F 5 H f dΩ; F 5 H f dS 5 2 pHT RηdS: g

p

T

Ωs

T

Σ

(12.150e)

Σ

This generalized force vector is a function of the fluid pressure, rotation, and elastic deformation of the structure. Dynamic equation for numerical simulation When the potential energy in Eq. (12.148c), the kinetic energy in Eq. (12.149c), and the generalized force in Eq. (12.150e) are substituted into the Lagrangian
Hamilton equation (Meirovitch, 1997),       d @T @T @Π 2 1 5 Q; (12.151a) dt @q_~ @q~ @q~ ~ as given in Section 2.2, to obtain the following we can use the derivative rule of a scalar with respect to vector q, dynamic equation in the numerical simulation form: _~ q_~ 1 Kq~ 5 Qðp; θ; qÞ; Mðθ; qÞq€~ 1 Cðθ; q; qÞ  dMðθ; qÞ  T @Mðθ; qÞ C θ; q; q_~ 5 2 q_~ dt 2@q~ 2 3 @Mðθ; qÞ T @Mðθ; qÞ 5 5 q_~ T @Mðθ; qÞ 5 q~_ 4 2 @q~ 2@q~ 2@q~ 5

(12.151b)

N 16 X @M q_~I : @q~I I51

Obviously, Eq. (12.151a) is a nonlinear matrix equation with the DOF N 1 6 consisting of three rigid translations of the mass center, three rigid rotations about the mass center, as well as N elastic degrees describing the elastic deformation of the structure. The mass matrix involving the rigid rotation and the elastic deformation and the damping matrix are affected not only by the geometrical position of the structure but also by its velocity. The generalized force is a function of the geometrical position and the fluid dynamic pressure, which is caused by the FSI. Fluid domain Navier
Stokes equation The water is considered an incompressible perfect fluid, so that its divergence of velocity vanishes and its motion is governed by Eq. (3.135) with the body gravity force fi 5 2 gδi3 in the coordinate system defined in Fig. 12.19: p;i in Ωf ; v_i 5 2 gδi3 2 ; vi;i 5 0; (12.153a) ρf or in the matrix form

Dv rp 5g2 ; Dt ρf

rUv 5 0;

where v and g denote the velocity vector and the body force vector as follows:

T

T g 5 0 0 2g : v 5 v1 v2 v3 ;

(12.153b)

(12.153c)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

543

Based on the MPS method given in Section 12.2.6.2, the fluid velocity, the gradient, and the Laplacian of the pressure with their values at position rn of the time step n can be determined by Eqs. (12.134a), (12.135c), and (12.136f), respectively, that is, vn ðxI Þ 5

N X vn ðxJ ÞWJI

nI

J6¼I

;

 n  min N pJ 2 pn; ðrJ 2 rI Þ dX I ðrp ÞI 5 0 WJI ; n J6¼I jrJ 2rI j2 n

ðr2 pn ÞI 5

N X  2d  n pJ 2 pnI WJI : 0 n λ J6¼I

(12.153d)

(12.153e)

(12.153f)

Now, using the projection method discussed in Section 12.2.5.4, the intermediate velocity marked by *, without considering the pressure gradient contribution, and the corresponding intermediate position of particle can be calculated as v 5 vn 1 Δtg;

r 5 rn 1 Δtv :

(12.154a)

To include the pressure contribution, the velocity at time step n 1 1 is calculated by vn11 5 v 2 Δt

rpn11 ; ρf

(12.154b)

from which, when a diligence operation is taken and considering the incompressibility condition in Eq. (12.153b) valid at time n 1 1, we obtain a pressure Poisson equation: r2 pn11 5

ρf rUv ; Δt

(12.154c)

which has its particle form in Eq. (12.134h), that is, 

ðr p

2 n11

ρ f n 2 n0 ÞI 5 2 2 I 0 : Δt n

(12.154d)

For pressure calculation accuracy, this equation is further modified to include the numerically accumulated error in the mass density at time step n (Sun et al., 2015a,b; Sun, 2016), so that the final Poisson equation is given by 0 1  0 n 0 ρ n 2 n n 2 n f ðr2 pn11 ÞI 5 2 2 @ I 0 1 α I 0 A Δt n n 2 3 (12.154e)  ρf ð1 1 αÞn0 2 nI 2 αnnI 5; 5 24 Δt n0 where

 8   0 > n  > n 2 n >  1 ΔtjrUvn j; ðn0 2 nn ÞrUvn $ 0;  > > > <  n0   α 5   > 0 n >  n 2 n > > ;  > ðn0 2 nn ÞrUvn , 0: > :  n0 

(12.154f)

Physically, ðn0 2 nn ÞrUvn . 0 implies that both (n0
nn) and rUvn are negative or positive. The negative case implies the fluid is compressed at time n and is further compressed due to the motion with rUvn , 0, while the positive case means the reverse: the fluid expands at time n, and further expands by the flow of rUvn . 0; therefore an additional term ΔtjrUvn j is added in the Poisson equation. However, when ðn0 2 nn ÞrUvn , 0, the particle density (n0
nn) and the velocity divergence rUvn have different developing directions that amend each other, and no term ΔtjrUvn j is added in the modified parameter α.

544

Fluid
Solid Interaction Dynamics

Combining Eqs. (12.153f) and (12.154e), we obtain   N X  ρf nI 1 αnnI 2 ð1 1 αÞn0 2d  n11 n11 p 2 p W 5 ; JI J n0 λ I Δt2 n0 J6¼I which is the equation to solve the pressure expressed in the following matrix form: 2 3n 2 3n11 2 3n H11 ? H1N p1 N1 HP 5 4 ^ & ^ 5 4 ^ 5 5 4 ^ 5 5 N; HN1 ? HNN pN NN

(X

where

N

L6¼I

HIJ 5

2d WLI ; ðn0 λÞI

(12.155a)

(12.155b)

J 5 I;

2 2d WJI ; J 6¼ I; ðn0 λÞI "  # ρf nI 1 αnnI 2 ð1 1 αÞn0 NI 5 2 : Δt n0

(12.155c)

This equation is a simultaneous equation expressed by a linear symmetric coefficient matrix H, of which the solution gives the pressure at time n 1 1. When the pressure is obtained, the velocity and the location of the fluid can be updated by vn11 5 r

n11

vn 2 Δtrpn11 ; ρf

5 r 1 Δtv n

n11

(12.156)

:

Fluid boundary conditions For the fluid boundaries without FSI, such as free surface and fixed solid boundaries, the treatment approaches described in Section 12.2.5.5 can be used. For the FSI boundary, the motion of the solid body is also not known, so a special consideration is required. Fluid
solid interaction boundary condition Pressure Neumann condition On the FSI boundary, both a force equilibrium and a geometrical consistency must be satisfied. The force equilibrium condition has been considered by the solid dynamic equation in Eq. (12.151b), where the generalized force Qðp; θ; qÞ is a function of the fluid pressure p. Therefore we only need to consider the geometrical consistency condition given in Eq. (4.16a), which is rewritten in the reference system defined in Fig. 12.19, where the outside normal vector of fluid ν 5 2 η (for the solid): _ νn11 Urpn11 5 ρf ðνn11 Ug 2 νn11 UV

n11

Þ:

(12.157a)

_ n11 and the unit normal vector νn11 of the FSI interface are also In this equation, both the solid acceleration V unknown at time step n 1 1, so that an iteration process is needed. As an approximation, their values at time n may be _ n11 can be derived by using Eq. (12.147f) while the used in the corresponding explicit form. The solid acceleration V gradient of the pressure can be expressed in particle form as given by Eq. (12.135.6) in numerical calculations. Laplacian operator compensation near fluid
solid interaction interface For the fluid particles near the FSI interface, the Laplacian operator is modified to be consistent with the Newman condition in Eq. (12.157a). As shown in Fig. 12.20, a virtual particle in the solid is positioned along the normal direction ν and away from the FSI interface by a distance dr0. This virtual particle is also included in the discretization equation of the Laplacian operator. Based on Eq. (12.157a), the pressure of the fluid pF is derived as _ 0; pF 5 pS 1 ρf ðνUg 2 νUVÞdr where pF denotes the pressure of the solid particle.

(12.157b)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

re

545

Virtual particle

ν dr0

r

Fluid particle under concern dr0

Boundary particle

FIGURE 12.20 Pressure modification on FSI interface (Sun, 2016). FSI, Fluid
solid interaction.

Intermediate velocity of fluid
solid interaction boundary The intermediate velocity v obtained from Eq. (12.154b) should guarantee the velocity consistency condition on the FSI interface, as suggested by Brown et al. (2001). Eq. (12.157a) is just the geometrical condition in the pressure gradient form, from which, when it combines with Eq. (12.154b), it follows _ νUv 5 νU½vn11 1 Δtðg 2 V

n11

Þ;

(12.158a)

from which, when the fluid velocity equals the solid velocity, that is, vn11 5 Vn11 , it follows _ νUv 5 νU½Vn11 1 Δtðg 2 V

n11

Þ;

(12.158b)

_ which may be used to modify the intermediate velocity v . Actually, if we approximate ΔtV Eq. (12.158b) further reduces to

n11

νUv 5 νUðΔtg 1 Vn Þ:

5 Vn11 2 Vn , (12.158c)

This equation is just the projection of Eq. (12.154a) on the unit normal direction. Therefore if the fluid velocity exactly satisfies vn11 5 Vn11 on the FSI boundary, the modification by Eq. (12.158b) seems unnecessary. However, in numerical simulations, the condition vn11 5 Vn11 can never be exactly satisfied due to the numerical error at each time step, so that the modification in Eq. (12.158b) is needed. For a perfect fluid, along the tangent direction of an FSI interface, the fluid velocity and the solid velocity can be different, and there is no geometrical condition in the tangent direction. For the viscous fluid, the nonslip condition requires the fluid and solid to have the same velocity on the FSI interface; as a result, Eq. (12.158b) should be modified to _ n11 Þ: v 5 Vn11 1 Δtðg 2 V

12.4.2.2 Integrated coupling equations Here we list the integrated coupling equations used in numerical simulations.

(12.158d)

546

Fluid
Solid Interaction Dynamics

Solid domain Eqs. (12.147a), (12.147f), and (12.151b) should be satisfied at time step n 1 1; they are rewritten as Mðθ; qÞn11 q€~

_~ n11 q_~ n11 1 Kq~ n11 5 Qðp; θ; qÞn11 ; 1 Cðθ; q; qÞ 2 0 3n11 U

6 7 n11 n11 n11 u 5 I R R Φ 4X5 ; n11

Vn11 5 I An11

q 2 0 3n11 _ U

6 7 n11 R Φ 4 θ_ 5 ; q_

(12.159a)

(12.159b)

Un11 5 Φqn11 ; xn11 5 X 1 U0n11 1 Rn11 X 1 Rn11 Un11 ; in which, the elastic mode matrix Φ, the elastic stiffness matrix K, and the material coordinate vector X depend on only the geometrical and physical properties of the body and do not change with time. Fluid domain Eqs. (12.154a), (12.155b), (12.55c), (12.153e), and (12.156) are satisfied at time n 1 1 as follows: v 5 vn 1 Δtg; r 5 rn 1 Δtv ; 2 3n 2 3n11 2 3n H11 ? H1N p1 N1 Hn Pn11 5 4 ^ & ^ 5 4 ^ 5 5 4 ^ 5 5 Nn ; HN1 ? HNN pN NN

(X N

HIJn

L6¼I

5

2d WLI ; ðn0 λÞI

ðrp

 n11  min N pJ 2 pn11; ðrJ 2 rI Þ dX I ÞI 5 0 WJI ; 2 n J6¼I jrJ 2rI j vn11 5 r

n11

vn 2 Δtrpn11 ; ρf

5 r 1 Δtv n

n11

(12.160b)

J 5 I;

2 2d WJI ; J 6¼ I; ðn0 λÞI 2 3  n 0 ρ n 1 αnI 2 ð1 1 αÞn 5 f NIn 5 2 4 I ; Δt n0

n11

(12.160a)

(12.160c)

(12.161a)

(12.161b)

:

Fluid
solid interaction interface The coupling conditions in Eqs. (12.157a) and (12.158b) are rewritten as

 _ n11 ; νn11 Urpn11 5 ρf νn11 Ug 2 νn11 UV h

i _ n11 : νn11 Uv 5 νn11 U Vn11 1 Δt g 2 V Initial conditions At the initial time t 5 0, the initial position and the velocity of the solid and fluid are given by

(12.162a) (12.162b)

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

547

Solid body Initial position: Mass center: Material point:

x0o 5 0; x0 5 X:

(12.163a)

Initial displacement and velocity: Mass center: Rigidrotation: Elastic deformation:

_ ; U0 5 0 5 U _ θ 5 0 5 θ; _ U 5 0 5 U: 0

(12.163b)

Fluid domain Initial position, velocity and pressure: Position vector: r 5 r0 ; Velocity: v 5 v0 ; Pressure: p 5 2 xi δ3i g:

(12.164)

12.4.2.3 Numerical simulation process As discussed in Chapter 11, Mixed finite element
computational fluid dynamics method for nonlinear fluid
solid interactions, for nonlinear FSI problems, the partitioned iteration solution is suggested. The numerical process follows the flowchart given in Fig. 12.18 and entails three main steps. 

1. Starting from the initial position of the system (n 5 0), in which the position u~ n11 5 un with the corresponding velocity and acceleration on the FSI interface prescribed, use the fluid equations in Eqs. (12.160a)
(12.160c), (12.161a), (12.161b), (12.162a), and (12.162b) to solve a fluid problem with the prescribed initial fluid pressure and  velocity fields, as well as its boundary conditions in order to obtain an intermediate pressure P ðt 1 ΔtÞ 5 p n11 ,   velocity v n11 , and position vector r n11 .  2. The resultant pressure p n11 is substituted into the solid Eqs. (12.159a) and (12.159b), and the solid equations are    solved to obtain the solid displacement u n11 , velocity u_ n11 , and acceleration u~ n11 using the time integration approaches.   3. Compare the displacement u n11 and the testing displacement u~ n11 by checking whether the convergence  obtained   condition u n11 2 u~ n11  , ε is reached. If it is reached, go to next time step; otherwise modify     u~ n11 5 u~ n11 1 αðu n11 2 u~ n11 Þ to repeat the preceding process. The relaxation factor α can be updated by Eq. (12.146a).

12.4.3

Application examples

Here, based on the numerical methods formulated in this chapter, some application examples of nonlinear FSI problems completed by our PhD students, colleagues, and the author (see Sun et al., 2011, 2012, 2013, 2014, 2015a,b; 2016, 2017; Sun, 2013, 2016; Javid et al., 2013a,b, 2014a,b, 2016; Javed, 2015) are presented to demonstrate the proposed numerical methods.

12.4.3.1 Rigid wedge dropping on the water Water impact is an important problem in marine and offshore engineering (Khabakhpasheva and Korobkin, 2003, 2013). Wedge dropping tests have been used to study the loads of a ship slamming into water to determine safety considerations in the design of marine structures. As the velocity of the dropping wedge depends on FSI interactions, simulations could be difficult if grid-based methods are employed to treat breaking waves on the free surface and moving FSI boundaries. Oger et al. (2006) applied the WCSPH method to wedge dropping simulation using denser particles in the impact area, and the smoothing length was changed depending on the requirement of accuracy to ensure an acceptable level of density fluctuation in the fluid. Gong et al. (2009) proposed an alternative method by using a sponge layer on the bottom of the tank to adjust the density calculation of the fluid particles. When Shao’s ISPH method was applied to water entry of a free-falling

548

Fluid
Solid Interaction Dynamics

wedge, mirror particles were used on the moving solid (Shao, 2009). With Lee’s ISPH method, the proposed two boundary treatments can be applied, which simplifies the model generation and reduces computation time. In this example, simulated by Sun et al. (2011) and Sun (2013), as shown in Fig. 12.21, a symmetric wedge with a dead-rise angle θ 5 π=6 (an angle measured upward from a horizontal plane at keel level) dropping into water is simulated using ISPH with the denser wall particle boundary treatment. In the simulation, the particle spacing is 0.01 m, but near the boundary it is 0.005 m, and the time step is 0.001 second. The weight of the dropping wedge is 241 kg, the width of the wedge is 0.5 m, and the length is 1 m. The tank size is 2 m 3 1 m. In the simulation, the wedge is placed just above the free surface of the calm water with a dropping velocity of 6.15 m/s given from the 2D experiment done by Zhao et al. (1997). The wedge is allowed to move only in the vertical direction, and its motion follows the equation of motion for rigid body. The resultant water pattern generated in the simulation is shown in Fig. 12.22, which is compared with the photo from the experiment by Tveitnes et al. (2008). When the falling wedge hits the water surface, the surface breaks because of the strong impact, and the water is pushed up around the wedge. The breaking wave pattern obtained from the SPH simulation is marked with a black line for easy comparison with the experiment. The velocity of the wedge

1m 0.5 m

θ

1m

2m

FIGURE 12.21 Sketch of a rigid wedge dropping into the water (Sun, 2013).

FIGURE 12.22 Breaking wave pattern compared with the experiment photo taken by Tveitnes et al. (2008) and Sun (2013).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

549

6.50 Experimental (Zhao et.al., 1997)

6.00

Vy(m/s)

5.50 5.00 4.50 4.00 3.50 3.00 0.00

0.01

0.02

0.03

Time (s)

FIGURE 12.23 Wedge dropping velocity compared with the experiment result given by Zhao et al. (1997) and Sun (2013).

Experimental (Zhao et.al., 1997) Analytical

7000.0 6000.0

Fy/N

5000.0 4000.0 3000.0 2000.0 1000.0 0.0 0.00

0.01

0.02

0.03

Time (s)

FIGURE 12.24 Impact force on the wedge compared with the experiment/analytical results given by Zhao et al. (1997) and Sun (2013).

after impact and the impact forces on the wedge from the water are compared with the experimental data and analytical results given by Zhao et al. (1997) in Figs. 12.23 and 12.24, respectively. Fig. 12.23 shows good agreement between the velocity obtained by the current SPH method and the experiment result, and it is slightly lower than the experimental values in the later stage with a maximum error of 2%. Also, it is seen that the wedge dropping velocity decreases more rapidly after 0.007 second since the water buoyancy produces an upward acceleration after the wedge enters the water. As observed in Fig. 12.24, the fluid force initially increases steadily and then slows down before reaching the peak at around 0.015 second, after which it starts to decrease. The computed force values are slightly overpredicated at first, and then it is underpredicated for a short period of time, but it is higher than experimental values at the last stage. Overall, both the dropping velocity and the vertical fluid force obtained from the proposed SPH method agree well with experimental values. To investigate the effect of different parameters on the water entry process and to provide some guidelines for engineering applications, the following three cases are studied by Sun (2013): 1. Case 1: Different wedge masses with a dead-rise angle of 30 degrees and initial dropping velocity of 6.15 m/s 2. Case 2: Different initial dropping velocities with a wedge mass of 241 kg and dead-rise angle of 30 degrees 3. Case 3: Different dead-rise angles with wedge mass of 241 kg and initial dropping velocity of 6.15 m/s Fig. 12.25 shows the results affected by different wedge masses, from which it is clear that, as the wedge mass increases, both the wedge velocity and the water force on the wedge are increased. This may be explained by the momentum theorem. Assume that during the impact time period Δt, the wedge of mass M is subject to the fluid impact force F, and its velocity reduces from the initial value V0 to the ending value Vt. The momentum theorem for this impact process is given by MðVt 2 V0 Þ 5 ðMg 2 FÞΔt; from which it follows

 F Vt 5 V 0 1 g 2 Δt; M

(12.165a)



(12.165b)

Vertical velocity (m/s)

550

Fluid
Solid Interaction Dynamics

FIGURE 12.25 Time history of wedge velocities (top) and fluid forces (bottom) affected by different wedge masses (Sun, 2013).

M=141 kg M=241 kg M=341 kg

6.5 6 5.5 5 4.5 4 3.5 3 0

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

Fluid force (N)

7000

M=141 kg M=241 kg M=341 kg M=441 kg

6000 5000 4000 3000 2000 1000 0 0

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

Vertical velocity (m/s)

6.5

FIGURE 12.26 Time history of dropping velocities (top) and fluid forces (bottom) affected by different initial wedge velocities (Sun, 2013).

V0=6.15 m/s V0=5.15 m/s V0=4.15 m/s

6 5.5 5 4.5 4 3.5 3 0

0.005

0.01

0.015 Time (s)

0.02

Fluid force(N)

6000

0.025

0.03

V0=6.15 m/s V0=5.15 m/s V0=4.15 m/s

5000 4000 3000 2000 1000 0 0

0.005

0.01

0.015

0.02

0.025

0.03

Time (s)

  V 0 2 Vt F5M g1 5 Mðg 1 at Þ: Δt

(12.165c)

Eq. (12.165b) indicates that for the same impulse FΔt, the wedge velocity Vt increases as the mass increases, while Eq. (12.165c) shows that for the same velocity change rate at, the impact force also increases with a larger mass. Fig. 12.26 shows that the decreasing rate of the wedge velocity increases with increasing initial dropping velocity before time 0.0125 but that, after this time, the three curves tend to be parallel with almost the same rate. For the fluid force, larger initial velocity generates higher fluid force at the early stage with a larger peak value. By contrast, the force values become almost the same at the later stage, which is tending to the static buoyance with the wedge velocity further reduced.

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

FIGURE 12.27 Time history of dropping velocities (top) and fluid forces (bottom) affected by different dead-rise angles (Sun, 2013).

6.5 Vertical velocity (m/s)

551

6 5.5 5 4.5

Dead-rise angle=30º

4

Dead-rise angle=45º

3.5

Dead-rise angle=60º

3 0

0.005

0.01

0.015 Time (s)

0.02

Fluid force (N)

6000

0.025

0.03

Dead-rise angle=30º Dead-rise angle=45º

5000

Dead-rise angle=60º 4000 3000 2000 1000 0 0

0.005

0.01

0.015 Time (s)

0.02

0.025

0.03

Fig. 12.27 shows that fluid force acting on a wedge is smaller for the wedge with a larger dead-rise angle and that consequently the velocity decreases much more slowly. This is because the vertical force on the wedge is a projection of fluid pressure on the vertical line, which is proportional to the cosine of the dead-rise angle, so that a larger dead-rise angle leads to a smaller force.

12.4.3.2 Flexible wedge dropping on the water Further simulation of a wedge dropping problem including wedge elastic deformation was completed by Sun et al. (2015a) and Sun (2016). Fig. 12.28 shows the initial configuration of the problem, of which the tree different flexible cases with the corresponding parameters listed in Table 12.1 are considered. In the simulations, the symmetrical wedge is assumed to be constructed by a rigid frame marked by shading lines and the two inclined elastic beams on the bottom surface. The integrated system is considered a 2D symmetrical one with its symmetrical motions. The mass center of the wedge undergoes a large rigid motion, and its bottom elastic beams move with the mass center and undergo their small elastic deformation modeled by the mode summation method. For each beam, three theoretical modes are taken into account, and the corresponding first three circular frequencies are: 96.210, 602.943, and 1688.238. The MPS method described in Section 12.2.6.2 is adopted in modeling the fluid motion. The fluid field is discretized by particles with the initial space of 0.005 m, which results in 38,400 fluid particles and a total 40,122 particles including solid particles on the FSI interface. The time interval is determined by the Courant
Friedrichs
Lewy condition with a maximum limit of 0.0002 second. It is worth mentioning that the majority of the computational time is used for the fluid solver, that is, MPS part. The time used for the structure solver is negligible since the number of DOF of the solid is only 4, consisting of a rigid and three elastic ones. The convergence error is set to 1025 using the averaged coupling root mean square (RMS) (CRMS) as the error criterion. vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u NFSI u 1 X   CRMS 5 t ðu n11 2 u~ n11 ÞI Uðu n11 2 u~ n11 ÞI ; (12.166) NFSI I51 



where NFSI denotes the particle number on the FSI interface, and ðu n11 2 u~ n11 ÞI denotes the difference of the coupling var  iable values of particle I at time step n 1 1 between the starting value u~ n11 and the ending value u n11 in the iterations. The accelerations of the flexible wedge, calculated by the modified MPS, are compared with the results from experimental, rigid body simulation and Wagner’s theory (Panciroli et al., 2012, 2013; Panciroli, 2013, respectively) in

552

Fluid
Solid Interaction Dynamics

FIGURE 12.28 Initial configuration of flexible wedge dropping on the water (Sun, 2016).

Y 0.8 m

Equivalent dropping height H*

0.8 m 0.6 m

X

0 1.6 m

TABLE 12.1 The parameters of three cases of flexible wedge. Case 1 Material

Case 2

E-glass (woven)/epoxy

Case 3 Aluminum

Young modulus, E (GPa)

30.3

68

Density, ρ (kg/m )

2015

2700

3

Mass of the rig (kg/m)

22

Length of each bottom, L (m)

0.3

Thickness (mm)

2.2

Dead-rise angle, β (degree)

2

30

Entry speed (m/s)

4.29

5.57

4

Equivalent height, H* (m)

0.938

1.5813

0.8155

Fig. 12.29. As shown in Fig. 12.29A and B for elastic wedges, the numerical results coincide with the experimental data for the main trend; the first impact pressure peak as well as the oscillating tend to a constant value with time until the end of the simulation. It can be seen that for the two elastic cases there is a trough at about 0.025 second on both the experimental and the numerical curves, which is not shown on the curves for the rigid cases given in Fig. 12.29C and D. The numerical curves show an overshot of the second acceleration peak at the time earlier than the experimental curve. This is probably caused by a 2D beam approximation in the simulations for the 3D experiment; the real 3D bottom is a plate with different natural frequencies from those of the beam. The pressure and velocity contours are shown in Figs. 12.30 and 12.31, respectively. Due to the flexibility of the wedge bottom, the cavitation starts to develop from about t 5 0.02 second and vanishes until about t 5 0.04 second. Because the current model involves only the water phase, the dynamics caused by the entrapped air between the wedge bottom and water could not be captured correctly. Fig. 12.32 shows the deformation of the flexible bottoms at some typical time instants. During the initial stage of the impact, the bottoms are bended down by the coupling effect of the inertia and the concentrated impact force near the wedge tip. After about t 5 0.02 second, the deformation of the beam starts to bounce back toward the symmetry line of the wedge.

(A) 250

(C) 250 Rigid - Modified MPS+rigid-body dynamics Rigid - Wagner theory

Flexible - Modified MPS+CRMS Flexible - Experiment

200

150

acc (m/s2)

acc (m/s2)

200

100

150 100

50 50 0 0

0.01

0.02

0.03

0.04

0.05

Time (s)

(B)

0

0.06

0.02

0.03

0.04

0.05

0.06

Time (s) 400

Flexible - Modified MPS+CRMS Flexible - Experiment

350

Rigid - Modified MPS+rigid-body dynamics Rigid - Wagner theory

350

300

300

250

acc (m/s2)

acc (m/s2)

0.01

(D)

400

200 150 100

250 200 150

50

100

0

50

–50

0

0

0.01

0.02

0.03

Time (s)

0.04

0.05

0.06

0

0

0.01

0.02

0.03

0.04

0.05

0.06

Time (s)

FIGURE 12.29 Acceleration of the flexible/rigid wedge obtained by MPS and compared with the experiment or Wagner’s theory (Panciroli et al., 2012, 2013; Panciroli, 2013): (A) elastic case 1, (B) elastic case 2, (C) rigid with case 1 entry speed 4.29 m/s, (D) rigid with case 2 entry speed 5.57 m/s (Sun, 2016). MPS, Moving particle semiimplicit.

0.6

0.6

0.6

0.5

0.5

0.5

0.4

y (m)

0.7

0.4 0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0 0.3

0.4

0.5

0.6

0.7

0.8 x (m)

0.9

1

1.1

1.2

1.3

0 0.3

0.4

0.5

0.6

0.7

0.8

t=0.02 s

0.8 x (m)

0.9

1

1.1

1.2

1.3

0.6

0.6

0.5

0.5

0.5

y (m)

0.6

y (m)

0.7

0.4 0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0

0

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0 0.3

0.4

0.5

0.6

0.7

0.9

1

1.1

1.2

1.3

0.8

t=0.035 s

0.6

0.5

0.5

y (m)

0.6

0.5

y (m)

0.6

0.4

0.4 0.3

0.3

0.2

0.2

0.2

0.1

0.1

0.1

0.5

0.6

0.7

0.8 x (m)

0.9

1

1.1

1.2

1.3

0

0.3

0.4

0.5

0.6

0.7

0.8 x (m)

FIGURE 12.30 Water pressure contours at different times obtained by case 2 simulation (Sun, 2016).

0.9

1

1.1

1.2

1.3

t=0.03 s

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

1.3

0.9

1

1.1

1.2

1.3

1

1.1

1.2

1.3

t=0.06 s

0.4

0.3

0.4

0.8

0.8

t=0.04 s

0.7

0.3

0.7

x (m)

0.7

0

0.6

x (m)

0.8 0.7

0.8

0.5

0.4

0.3

0.4

0.4

0.8

t=0.025 s

0.7

0.3

0.3

x (m)

0.7

0.4

t=0.015 s

0.4

0.3

x (m)

y (m)

0.8

t=0.01 s

0.7

0.8

y (m)

0.8

t=0.005 s

0.7

y (m)

y (m)

0.8

0

0.3

0.4

0.5

0.6

0.7

0.8 x (m)

0.9

0.8

0.8

y (m)

y (m)

0.6

0.5

0.4

0.3

0.7

0.6

0.6

0.5

0.4

0.5

0.6

0.7

0.8 x (m)

0.9

1

1.1

1.2

0.4

0.3

0.3

1.3

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

0.2

1.3

0.8

0.6

0.5

y (m)

0.6

y (m)

0.6

0.5

0.4

0.4

0.3

0.3

0.3

0.6

0.7

0.8 x (m)

0.7

0.9

1

1.1

1.2

0.2

1.3

0.8

0.2 0.3

0.4

0.5

0.6

0.7

0.8

t = 0.035 s 0.7

0.7

0.6

0.6

0.8 x (m)

0.9

1

1.1

1.2

1.3

1

1.1

1.2

1.3

1

1.1

1.2

1.3

0.5

0.4

0.5

0.6

t = 0.03 s 0.7

0.4

0.5

t = 0.025 s 0.7

0.3

0.4

0.8

t = 0.02 s 0.7

0.2

0.3

x (m)

0.8

y (m)

0.5

0.4

0.2 0.3

t = 0.015 s

0.7

y (m)

0.7

0.2

0.8

t = 0.01 s

t = 0.005 s

0.8 x (m)

0.9

1

1.1

1.2

1.3

t = 0.04 s

0.3

0.4

0.5

0.6

0.7

0.8

0.8 x (m)

0.9

t = 0.06 s

0.5

y (m)

y (m)

y (m)

0.7

0.5

0.6

0.5 0.4

0.4

0.3

0.3

0.4

0.3 0.2

0.2

0.3

0.4

0.5

0.6

0.7

0.8 x (m)

0.9

1

1.1

1.2

1.3

0.3

0.4

0.5

0.6

0.7

FIGURE 12.31 Velocity contours at different times obtained by case 2 simulation (Sun, 2016).

0.8

x (m)

0.9

1

1.1

1.2

1.3

0.3

0.4

0.5

0.6

0.7

0.8 x (m)

0.9

0.75 0.8

0.75 0.7

0.75

t = 0.01 s

0.7

0.65

0.6

0.6

0.55

0.5

0.7

t = 0.015 s

0.65

y(m)

y(m)

y(m)

t = 0.005 s

0.65

0.6

0.6

0.7

0.8

0.9

1

0.55

0.5

1.1

0.6

0.7

0.8

0.9

1

1.1

0.5

0.6

0.7

x(m)

x(m)

0.8

0.9

1

1.1

0.9

1

1.1

1

1.1

x(m) 0.75

0.75 0.75 0.7 0.7

t = 0.025 s

0.7

0.65

t = 0.03 s

0.65 y(m)

y(m)

y(m)

t = 0.02 s 0.65

0.6

0.6 0.6 0.55 0.55 0.55 0.5 0.5 0.5 0.5

0.6

0.7

0.8 x(m)

0.9

1

1.1

0.5

0.6

0.7

0.8 x(m)

0.9

1

1.1

0.5

0.7

0.65

y(m)

y(m)

y(m)

t = 0.06 s

0.6

0.5

0.5

0.45

0.5

0.5

0.6

0.7

0.8 x(m)

0.9

1

1.1

0.55

0.55 0.55

0.45 0.5

0.8 x(m)

0.6

t = 0.04 s

0.65

0.6

0.7

0.65

0.7

t = 0.035 s

0.6

0.6

0.7

0.8 x(m)

FIGURE 12.32 Beam elastic deformation at different times obtained by case 2 simulation (Sun, 2016).

0.9

1

1.1

0.5

0.6

0.7

0.8 x(m)

0.9

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

(A)

557

200 SPH Modified MPS+CRMS Experiment

0 –200

Strain (x10–6)

–400 –600 –800 –1000 –1200 –1400 –1600 –1800

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t (s) (B)

800 SPH Modified MPS+CRMS Experiment

600

Strain (x10–6)

400 200 0 –200 –400 –600 –800 –1000

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1

t (s) FIGURE 12.33 The beam strains (case 3) at points (A) 30 mm and (B) 120 mm from beam tip by MPS: (A) at 30 mm from beam tip and compared with the ones by SPH and experiment (Panciroli, 2013; Sun, 2016). MPS, Moving particle semiimplicit; SPH, smoothed particle hydrodynamics.

This process is also reflected in the time history of strain that is monitored at two different locations on the upper surface of each bottom beam, that is, 30 and 120 mm away from the wedge tip respectively, as shown in Fig. 12.33. The positive part of the strain in Fig. 12.33B, monitored at 120 mm (about the middle of the bottom), represents the initial downbending stage of the beam shown in Fig. 12.32. After that, the strain remains negative because of the pressure of the surrounding water. The numerical strain record matches well with the experimental data.

12.4.3.3 Two-dimensional water dam breaking impact on rigid/flexible beams Spring-supported rigid beam Fig. 12.34 shows the sketch of a water dam break impact to a rigid uniform beam of length L 5 2 m, total mass M 5 1 kg, and supported by a rotation spring of stiffness K 5 1500 N m at the base. The rotation angle θ of this rigid beam about it base is governed by its dynamic equation derived by the Newton’s second law, that is, 1 I θ€ 1 Kθ 2 MLg sin θ 5 T; 2 T5

NFSI X J51

(12.167) pJ YJ ;

558

Fluid
Solid Interaction Dynamics

FIGURE 12.34 Sketch of a 2D water dam break impact to a spring-supported rigid beam (Sun, 2016). 2D, Twodimensional.

2m

θb

0.6 m

1.2 m

3.22 m

where I 5 ML2/3 is the rotation moment of inertia of the beam about the beam base point. The torque T of fluid pressure is obtained by a summation of the product of each particle pressure pJ and distance YJ of its impact point along the beam to the base point, since the fluid pressure is always perpendicular to the beam line. The geometrical parameters of the initial water dam and the flow domain are indicated in the figure. The initial particle distance is chosen as 0.01 m, which results in 7200 fluid particles and a total 8652 particles, including the solid ones on the FSI interface. The MPS method was used to simulate this problem by Sun (2016). The pressure contour and free surface profiles at selected time instants are shown in Fig. 12.35, from which it can be seen that the distribution of the pressure is quite smooth in the space domain. Two major impacts are found in two durations, t 5 0.75
2 seconds and t 5 5.1
5.9 seconds, during which the large fluid pressure generated from the falling of the water column pushes the beam to relatively large angles. Fig. 12.36A shows the time history of the rotational angle of the beam, on which the two significant angle impulses are consistent with the time of the two major impacts observed in Fig. 12.35. Except for these violent interactions, the water pressure applied on the beam is relatively much smaller. In the video recording of the simulation results, it was seen that the beam oscillates in a period 0.2 second, which is larger than the period 0.1879 second of its natural frequency due to the FSI effect with additional water mass on the beam. In order to investigate the effect of the beam rotation on the pressure field, the time history of the pressure monitored at a beam point 0.16 m above the right corner of the beam is compared in Fig. 12.36B: case 1 for the springsupported rotational beam and case 2 for the nonrotational fixed one. From this figure it can be seen that the time history during the first major impact in case 1 is basically same as the one in case 2 for the rigid fixed beam case. However, the peak pressure value in case 1 during the second major impact is much larger than in case 2. This is because at the beginning of the second impact, the beam rotates back, so that the water front and the beam move toward each other and have a more vigorous impact. After t 5 6 seconds for the rotational beam case, both the rotational angle and the water pressure show an oscillation of the frequency near the natural frequency of the rotational beam, which is not observed for case 2, fixed rigid beam shown in Fig. 12.36B. Fixed elastic beam Fig. 12.37 shows a 2D elastic beam
water dam FSI system with the related geometrical parameters given. The elastic beam is fixed at its base, and its parameters are chosen as: Young’s modulus E 5 0.2 GPa, thickness D 5 0.006 m, mass density m 5 47.16 kg/m, and the moment of inertia of the cross section I 5 1:8 3 1028 m4 . Based on the FE
MPS method proposed for a nonlinear FSI, for the water domain, the initial particle distance is 0.004 m, which results in 1250 fluid particles and a total 1736 particles for the integrated system; the beam’s motion is modeled by the linear FEM. The pressure contours and the beam deformation at some selected time instants are shown in Fig. 12.38. The deflection X of the beam top is shown in Fig. 12.39A, from which it can be seen that the period of its oscillation is about 0.6 second, very close to its first natural period 0.5971 second, with the modal shape shown in Fig. 12.37 by the dashed line The time histories of water pressure monitored at point Y 5 0.02 m of both the rigid and the elastic wall cases are given in Fig. 12.39B, which shows the very small fluctuation of pressure for both cases with the negligible difference between them. This implies that the small elastic deformation of the beam will not greatly change the fluid motion and its pressure field.

2.5

2.5

2.5

2.5

2

2

2

2

t=0.75 s

t=1.25 s

t=1.56 s 1.5

y (m )

y (m )

y (m )

t=2 s

1.5

1.5

y (m )

1.5

1

1

1

1

0.5

0.5

0.5

0.5

0

0.5

1

1.5

2

2.5

3

3.5

0

0

0.5

1

1.5

2 x (m)

x (m) 2.5

2.5

3

3.5

0

4

2.5

2

2

t=2.2 s

0

0.5

1

2.5

3

3.5

0

4

0

2.5

2

2

t=5.1 s

1.5

1

1

1

1

0.5

0.5

0.5

0.5

0

0

0.5

1

1.5

2 x (m)

2.5

3

3.5

0

0 0

0.5

1

1.5

2

2.5

3

x (m)

FIGURE 12.35 Pressure contour and free surface profiles at selected time instants (Sun, 2016).

3.5

0.5

1

1.5

2 x (m)

2.5

3

3.5

4

3

3.5

t=5.9 s

1.5

y (m )

y (m )

y (m )

1.5

2 x (m)

2.5

t=3 s

1.5

1.5

y (m )

0

0

0.5

1

1.5

2 x (m)

2.5

3

3.5

4

0

0

0.5

1

1.5

2 x (m)

2.5

560

Fluid
Solid Interaction Dynamics

(A) 0.4

Arc degree

0.3 0.2 0.1 0 –0.1

0

1

2

3

4

5

6

7

8

9

10

Time (s) (B) 8000 Elastic beam Rigid beam

p (Pa)

6000 4000 2000 0

0

1

2

3

4

5

6

7

8

9

10

Time (s) FIGURE 12.36 Time history of the rotation angle (A) and the pressure at monitor point (B): case 1, elastic beam for spring-supported rotational one; case 2, rigid beam for fixed rigid one (Sun, 2016).

y

η (s)

0.304 m

0.2 m

x 0.1 m

0 0.352 m

FIGURE 12.37 Sketch of water dam break impact to an elastic beam wall (Sun, 2016).

12.4.3.4 Flow-induced vibration of two-dimensional cylinder As reported by Javed (2015) and Javed et al. (2016), a coupled meshfree
mesh-based fluid solver was employed for flow-induced vibration problems. As shown in Fig. 12.40, fluid domain is modeled by a hybrid grid consisting of a body conformal meshfree nodal cloud around the solid object and a static Cartesian grid surrounding the meshfree cloud in the far-field. The meshfree nodal cloud (Fig. 12.41) provides flexibility in dealing with solid motion by moving and

t=0.24 s 0.3

0.3

0.3

0.25

0.25

0.25

0.25

0.2

0.2

0.2

0.2

0.15

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0.15

0.1

0.1

0.1

0.1

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0.2 x (m)

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0 0.34

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0

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t=0.9705 s

0.15

0.1

0

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0

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0.1

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x (m)

0.3

0

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0 0.34

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0.3

0.3

0.25

0.25

0.2

0.2

y (m)

0.05

y (m)

0

y (m)

y (m)

y (m)

y (m)

y (m) 0.15

0

t=0.65 s

y (m)

0.3

0.15

0.1

0.05

0.05

0

0

0.05

0.1

FIGURE 12.38 Pressure contour, free surface profiles, and beam deformations at selected time instants (Sun, 2016).

0.15

0.2 x (m)

0.25

0.3

0.35

0.4

0.36 x (m)

0.37

t=1.32 s

0.15

0.1

x (m)

0.35

0 0.34

0.35

0.36 x (m)

0.37

562

Fluid
Solid Interaction Dynamics

X coordinate

(A) 0.38 0.37 0.36 0.35 0.34

0

1

2

3

4

5 Time (s)

6

7

8

9

10

(B) 5000 Elastic beam Rigid beam

p (Pa)

4000 3000 2000 1000 0

0

1

2

3

4

5 Time (s)

6

7

8

9

FIGURE 12.39 Time histories of: (A) the beam deflection at its top and (B) the water pressure at beam point Y 5 0.02 m (Sun, 2016).

FIGURE 12.40 Hybrid grid configuration in fluid domain (Javed, 2015).

FIGURE 12.41 Nodes in a support domain of point x1 in the meshfree zone (Javed, 2015).

10

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

563

FIGURE 12.42 A 2D cylinder vibration induced by an incoming flow (Javed, 2015).2D, Two-dimensional.

morphing along with the solid boundary without necessitating remeshing. The Cartesian grid, on the other hand, provides improved performance by allowing the use of the computationally efficient mesh-based method. Flow equations, in ALE formulation, are solved by local radial basis function (RBF) in FD mode on moving meshfree nodes. Conventional FD is used over static Cartesian grid for flow equations in the Eulerian formulation. The equations for solid motion are solved using the classical Runge
Kutta method. Closed coupling is introduced between fluid and structural solvers by using a subiterative prediction
correction algorithm. In order to reduce computational overhead due to subiterations, only near-field flow in the meshfree zone is solved during inner iterations. The full fluid domain is solved during outer iterations only when the convergence at the solid
fluid interface has already been reached. In the meshfree zone, adaptive sizing of the influence domain is introduced to maintain a suitable number of neighboring particles. A hybrid grid has been found to be useful in improving computational performance by faster computing over a Cartesian grid, as well as by reducing the number of computations in the fluid domain with FSI. The solution scheme was tested for problems relating to flow-induced cylindrical and aerofoil vibrations. The results are found to be in good agreement with previous work and experimental results. Here, as an example, we discuss the problem on flow-induced vibration of a 2D cylinder as follows. Motion equation of two-dimensional cylinder Fig. 12.42 shows the sketch of a 2D cylinder in the flow, of which the motion equations can be derived using Newton’s second law in the following forms: mx€ 1 cx x_ 1 kx x 5 DðtÞ; my€ 1 cyðy_ 1 ky y 5 LðtÞ; DðtÞ 5

Σ

σ1j ηj dS;

ð LðtÞ 5

Σ

σ2j ηj dS;

(12.168)

where m, c, and k denote the mass, viscous damping coefficient, and the stiffness with subindex x and y to distinguish two directions of the cylinder. D(t) and L(t) represent the resultant fluid drag and lift forces on the cylinder, obtained by the integration with respect to the Cauchy stress σij on the cylinder surface Σ of outside unit normal vector ηj . Fluid domain In the active meshfree zone, the fluid velocity is represented by an interpolation based on nodes shown in Fig. 12.41 and using a set of interpolation functions called RBFs (Kansa, 1990). The spatial derivatives of the velocity can be directly obtained by conducting the derivative operation of the RBFs, from which the original PDE of flows can be transformed into an ODE of time. Actually, this method belongs in the category discussed in Section 12.3. In the nonactive meshfree zone, the meshfree notes are overlapped by a Cartesian grid in order to match the stationary Cartesian grid in the far-field, so the ALE method is adopted to account for nodal movement.

564

Fluid
Solid Interaction Dynamics

For numerical simulations, the following nondimensional parameters are defined: Reynolds number: Reduced velocity: Lift coefficient: Drag coefficient:

ρUD ; μ U vr 5 ; fN D L CL 5 ; ρU 2 D D CD 5 ; ρU 2 D

Re 5

Mass ratio:

m 5

Frequency ratio:

fNx ; fNy

(12.169)

m ; displace fluid mass

where ρ, U, and μ denote the mass density, free stream velocity, and dynamic viscosity of the fluid, respectively; D, fNx, and fNy represent the diameter and the horizontal and vertical natural frequencies of the cylinder. In the numerical simulations, the Reynolds number 150, the frequency ratio 2, the mass ratio 2, and reduced velocity range 1
12 are chosen, but the damping coefficients for cylinder are set to zero. As shown in Fig. 12.43, the chosen parameters with the full fluid domain size are the same as the reported investigation and experiment by Dahl et al. (2010) and Zhou and Tu (2012), from which the dual resonance was found. The size of the active meshfree zone around the cylinder is set at 3D 3 3D, outside of which an overlapped meshfree zone extends by a length of 1.5D in all four directions. Solutions are obtained for both ordered and randomized meshfree nodal arrangements. Resultant amplitudes of crossflow and inflow vibrations, RMS values of the lift coefficient, and mean values of the drag coefficient are shown in Fig. 12.44 along with numerical solutions obtained by Zhou and Tu (2012) and experimental results by Dahl et al. (2010). It can be observed that the results do not change significantly with randomization of meshfree nodes. Vibration amplitudes and lift and drag coefficients tend to increase dramatically as the resonance conditions are approached near to vr 5 6, at which the crossflow amplitude achieves its maximum value 0.908. However, inflow and crossflow vibration amplitudes are very low outside the resonance range from vr 5 6. These observations are in agreement with the results by Dahl et al. (2010) and Zhou and Tu (2012). Fig. 12.45 shows the vortex structures produced by the vibrations of a 2-DOF cylinder with a frequency ratio fNx =fNy 5 2:0 at different reduced velocities.

FIGURE 12.43 The geometrical configuration of the FSI system (top), and the active and inactive meshfree grid around 2D cylinder and Cartesian grid far from the cylinder (bottom) in the numerical simulation (Javed, 2015). FSI, Fluid
solid interaction; 2D, two-dimensional.

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

0.35

1.4

0.3

1

0.25

0.8

0.2

0.6

0.15

0.4

0.1

0.2

0.05 2

8 4 6 →Reduced velocity (vr)

10

0

12

4.5

1.2

4

1

3.5

0.8 0.6

0.2

1.5 1 4 6 8 →Reduced velocity (vr)

8

10

12

10

12

2.5 2

2

6

3

0.4

0

4

→Reduced velocity (vr)

1.4

0

2

0

mean CD

rms(CL)

Xmax/D

Ymax/D

1.2

0 0

565

10

12

0

2

8 4 6 →Reduced velocity (vr)

FIGURE 12.44 Nondimensional vibration amplitudes of cylinder in crossflow Ymax/D and inflow Xmax/D, RMS lift coefficient CL, and mean drag coefficient CD obtained in the simulation with parameters (m 5 2 5 fNx =fNy , Re 5 150) and meshfree nodes: ordered 2 O 2 and random 2Δ 2 , which is compared with the results
*
by Zhou and Tu (2012) based on the same parameters, and the experimental result (----) based on parameters (m 5 5:7; fNx =fNy 5 1:9, Re 5 15,000
60,000) by Dahl et al. (2010) and (Javed, 2015). RMS, Root mean square.

FIGURE 12.45 Vortex structure produced by 2-DOF cylinder vibrations in the frequency ratio fNx =fNy 5 2:0 and at different reduced velocities (Javed, 2015). DOF, Degrees of freedom.

566

Fluid
Solid Interaction Dynamics

12.4.3.5 Wave/wind energy harvesting systems Fundamental principle for wave/wind energy harvesting device design With increasing requirements for green energy to reduce environment pollution, scientists and engineers have put more effort into extracting energy from sea waves and wind. Investigations on wave energy harvesting devices have attracted wide interest around the world (for example, see Thorpe, 1999; Falnes, 2002; Office of Naval Research, 2003; Bedard et al., 2005; Rhinefrank, 2005; Wave Dragon, 2005; U.S. Department of the Interior, 2006; JAMSTEC, 2006; Ocean Power Technologies, 2006; Ocean Power Delivery Ltd., 2006; Wave Plane Production, 2006). As discussed in a review paper by Xing (2016), among these designs, the fundamental principle is to use waves/winds to excite mechanical motions of energy harvesting devices and then to convert mechanical energy to storable energies. Therefore the motions of energy harvesting devices excited by fluids are required to be as large as possible. To reach this aim, the following physical mechanisms may be adopted. Interdisciplinary research on fluid
solid interaction and electric
mechanical interaction As it is well-known that sea wave/wind energy harvesting devices are operated on or in fluids, the total system is typically an FSI. The dynamic behaviors of devices designed in dry cases are affected by fluids. A minor change of the characteristic frequency of a device can cause a large difference of dynamic response, as demonstrated by Xing et al. (2009b). Therefore FSIs have to be considered to design effective energy harvesting devices. Further as studied in the papers by Xing et al. (2009b, 2011), electric units in energy harvesting devices cause mechanical behaviors affected by electromagnetic dynamics involving electric
mechanical interactions (EMIs). As a result, for efficient energy harvesting designs, interdisciplinary research concerning linear/nonlinear dynamics, fluids, solids, and electric systems, as well as their interactions, is necessary. Resonance Resonance is a vibration to be avoided in engineering designs of structures, since a very small excitation force from the environment causes a very large motion of the system, so that the designed structure can fail in its dynamic strength or cannot work normally in the unwanted dynamic environment. However, in a reverse consideration, the resonance phenomenon can be adopted in order to design a linear device with its natural frequency close to the wave frequency, whose resonance is expected to create large mechanical motions by excited waves for energy harvesting devices. In considering FSI and using the developed numerical method (Xing et al., 1991, 1996) with computer code fluid
structure interaction analysis program (Xing, 1992a,b, 1995a,b), a wave energy harvesting device
water interaction system was subjected to the wave maker excitation in a towing tank (Xing et al., 2009b). The results demonstrated that FSI changes the natural frequency of the device designed in dry cases and therefore obviously affects the system efficiency. More importantly, this investigation revealed that the collected energy plays a role like an active damping added to the system, so that it is useful to keep a stable oscillation at its resonance state of the system. In Section 7.4.9.7, we have discussed this linear oscillator
water interaction system based on the resonance of FSI system to harvest large amounts of wave energy. Periodical orbit of nonlinear system Another idea is to design a nonlinear energy harvesting system and to use its inherent large stable orbit motion to extract energy. A number of researchers (Xu et al., 2005, 2007; Litaka et al., 2008, 2010; Lencia et al., 2008; Horton et al., 2011; Nandakumar et al., 2012; Pavlovskaia et al., 2012) proposed and investigated the possible rotational motions of a nonlinear pendulum subject to different base motion excitations aiming to harvest wave energy, but unfortunately they did not consider FSI effect. To amend this, Xing et al. (2011) proposed a mathematical model to study the dynamics of nonlinear oscillators, pendulums, and smooth and discontinuous (SD) oscillators (Cao et al., 2006, 2008a,b) coupling with water, as well as an electric-magnetic energy converter aiming to harvest wave energy. In the model, the pressure in the finite water domain is chosen to describe the water motions, and the paper qualitatively discusses the integrated FSI/EMI characteristics and proposed numerical solution approaches. Moreover, using the potential of velocity as a variable and based on the Green function satisfying the free surface wave condition and the Green identity, in Section 9.3.4.2 we discussed a nonlinear pendulum
water interaction system (Xing et al., 2011) to use its periodical solution for harvesting energy. By this investigation, we have further demonstrated that the energy converter in integrated systems acts as a damping mechanism to keep the excited large periodical motions in a stable state of motion. The nonlinear energy flow theory (Xing, 2015a,b) can be used to analyze more complex system based on energy conservation laws. Aerofoil flutter system Flutter is a very harmful vibration for airplanes that has to be avoided (Bisplinghoff et al., 1955; Bisplinghoff, 1958; Bisplinghoff and Ashley, 1962; Fung, 1955, 1969). However, harmful flutter vibration could be used to harvest wave/wind energy. The idea is to design an aerofoil with the energy converter excited by flows. Since the energy converter has a damping effect (Xing et al., 2009b; 2011; Xing 2016), the flutter generated is a stable oscillation to be practically used to harvest wave/wind energy.

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

567

FIGURE 12.46 Schematic model of oscillating aerofoil power extraction device (Yang, 2013).

Here, we present the two types of flutter aerofoil energy harvesting systems: the active and the semipassive designs. As discussed in the review papers (Young et al., 2014a,b; Xiao and Zhu, 2014), the full- and semipassive designs require an excitation by an input energy from another source to keep the device in the expected motion, while the full active one does not need this extraexcitation energy so that is more conducive to harvesting natural energies. Active aerofoil The energy harvesting device studied by Yang et al. (2011) is shown in Fig. 12.46. A 2-DOF aerofoil is placed in a uniform airflow of velocity U, with h denoting heaving displacement and α being positive nose up, representing pitching angle. The moving coil is rigidly connected to aerofoil at its elastic axis by a massless bar. The static equilibrium position where h 5 0 5 α is taken as a reference position. The aerofoil is considered a unit span in the direction perpendicular to the paper plane. Based on quasisteady aerodynamic theory, its governing equations of motion are written as follows (Fung, 1969; Dovel and Ligamov, 1988; Lee and Jiang, 1999). Equations of aerofoil motion mh€ 1 Sα€ 1 ch h_ 1 kh h 5 F 2 FM ;

Sh€ 1 Iα α€ 1 cα α_ 1 kα α 5 M;

kh 5 kh1 1 kh2 h2 ;

kα 5 kα1 1 kα2 α2 ; (12.170)

where m, S, and I are the total mass of the aerofoil and electric coil, the static and second moments of the aerofoil mass, respectively; c and k, respectively, represent the damping and stiffness coefficients identified by subindex h for vertical motion and α for pitching rotation; F and M denote the resultant lift and its moment acting on the aerofoil by the flow, and FM is the magnetic force. The stiffness coefficients of springs are assumed nonlinear, denoted by Eq. (12.170). Eq. (12.170) are written in a nondimensional form  2  ω ω  1 Fm b ξv 1 xα αv 1 2ζ ξ  ξ 0 1 Cl ðτ Þ 1 ξ 1 β ξ ξ3 5 2 ; (12.171a)  U U πμ mU 2  xa ζ 1  2 ξv 1 αv 1 2 α α0 1  2 α 1 β α α3 5 Cm ðτ Þ; U πμra2 ra2 U where h U Ut ωξ ; ω5 ξ 5 ; U 5 ; τ5 ; b bωα b ωα sffiffiffiffiffiffiffi kα1 ch cα ωα 5 ; ζξ 5 ; ζα 5 ; Iα 2mωξ 2Iα ωα

sffiffiffiffiffiffi kh1 ; ωξ 5 m μ5

m ; πρb2

xα 5

S ; mb

rα 5

sffiffiffiffiffiffiffiffi Iα ; mb2

(12.171b)

βξ 5

kh2 ; kh1

βα 5

kα2 : kα1 (12.171c)

568

Fluid
Solid Interaction Dynamics

Here, a prime ðÞ0 denotes derivative of ðÞ with respect to nondimensional time τ, and μ, xα, and rα are the mass ratio, the distance of mass center after the elastic axis, and the radius of gyration about the elastic axis, respectively; the values of lift force and moment were obtained by adopting a quasisteady aerodynamic theory (Fung, 1969), that is, F 5 2 ρbU 2 Cl ðτ Þ;

(12.172a)

M 5 2ρb U Cm ðτ Þ;

(12.172b)

2

2

where Cl ðτ Þ and Cm ðτ Þ are the nondimensional lift and pitching moment coefficients, respectively. Given incompressible flow, they are given by the expressions as in Fung (1969) and Lee and Jiang (1999), that is,       ðτ 1 1 ðσÞ 0 0 ðσÞ 0 2 ah α ð0Þ φðτ Þ 1 2π φðτ 2 σÞ α ðσÞ 1 ξvðσÞ 1 2 ah α αv dσπ Cl ðτ Þ 5 2π αð0Þ 1 ξ ð0Þ 1 2 2 (12.172c) 0 1 ðξv 2 ah αv 1 α0 Þ;

      π 1 π 0 π 1 1 ah ðξv 2 ah αvÞ 2 2 ah α 2 αv 1 π 1 ah αð0Þ 1 ξ0 ð0Þ 1 2 ah α0ð0Þ φðτ Þ 2 2 2 16 2 2    ð τ  1 1 1 ah 2 ah αvðσÞ dσ; 1π φðτ 2 σÞ α0 ðσÞ 1 ξvðσÞ 1 2 2 0

C m ðτ Þ 5

(12.172d)

where the Wagner function φðτ Þ is expressed as (Fung, 1969) φðτ Þ 5 1 2 Ψ1 e2E1 τ 2 Ψ2 e2E2 τ :

(12.172e)

Here, the constants’ values Ψ1 5 0:165, Ψ2 5 0:0335; E1 5 0:0455, and E2 5 0:3 are given by Jones (1940). Electromagnetic power generation Electric power generation obeys the Laplace theorem describing electromagnetic phenomena (Kittel, 1967; Xing et al., 2009b): a voltage will be induced by a moving coil of effective length l in the magnetic field of intensity B, that is, _ eðtÞ 5 Blh:

(12.173a)

When the electric coil is in an electric circuit with resistance R and inductance L, a dynamic current i is introduced. Assuming the electrical conductance is zero, the equation for the electric circuit is obtained by applying a Kirchhoff’s voltage law: diðtÞ 1 RiðtÞ 2 e 5 0: dt Combining Eqs. (12.173a) and (12.173b), we obtain a differential equation with respect to τ: L

i0 1

Rb Blb 0 i2 ξ 5 0: LU L

(12.173b)

(12.173c)

Since the coil moves in the magnetic field, an electromagnetic force will be introduced FM 5 iBl:

(12.173d)

pg ðτ Þ 5 Ri2 :

(12.173e)

The power harvested by resistor R is

Solution approach The integral terms in Eqs. (12.172c) and (12.172d) make it difficult to study the system’s dynamical behaviors analytically. To assist in investigation, as used by Lee and Jiang (1999), four new variables are introduced: ðτ ðτ 2E1 ðτ2σÞ w1 5 e αðσÞdσ; w2 5 e2E2 ðτ2σÞ αðσÞdσ; 0 ðτ ð 0τ (12.174) 2E1 ðτ2σÞ w3 5 e ξðσÞdσ; w4 5 e2E2 ðτ2σÞ ξ ðσÞdσ: 0

0

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

569

Based on this definition, Eqs. (12.171a), (12.171b), and (12.173b) can be rewritten as c0 ξv 1 c1 αv 1 c2 ξ 0 1 c3 α0 1 c4 ξ 1 c5 ξ3 1c6 α 1 c7 w1 1 c8 w2 1 c9 w3 1 c10 w4 1 c11 i 5 f ðτ Þ; 0

0

d0 ξv 1 d1 αv 1 d2 α 1 d3 α 1 d4 α 1 d5 ξ 1d6 ξ 1 d7 w1 1 d8 w2 1 d9 w3 1 d10 w4 5 gðτ Þ; 3

0

0

n1 ξ 1 n2 i 1 n3 i 5 0; where 2 f ðτ Þ 5 μ

(12.175a) (12.175b) (12.175c)

     1 2 ah αð0Þ 1 ξ ð0Þ ψ1 E1 e2E1 τ 1 ψ2 E2 e2E2 τ ; 2

(12.175d)

ð1 1 2ah Þf ðτ Þ ; 2ra2

(12.175e)

gð τ Þ 5 2 with the following coefficients:

1 ah ω 2 ; c1 5 xα 2 ; c2 5 2ζ ξ  1 ð1 2 Ψ1 2 Ψ2 Þ; μ U μ μ

2 2 1 c3 5 ½1 1 ð1 2 2ah Þð1 2 Ψ1 2 Ψ2 Þ; c4 5 Uω 1 ðE1 Ψ1 1 E2 Ψ2 Þ; μ μ 8 9   < =

2 2 1 c5 5 Uω β ξ ; c6 5 2 a h ð E 1 Ψ 1 1 E 2 Ψ2 Þ ; ð 1 2 Ψ1 2 Ψ 2 Þ 1 ; μ: 2 2 3 2 3     2 1 2 1 c7 5 E1 Ψ1 41 2 E1 2 ah 5; c8 5 E2 Ψ2 41 2 E2 2 ah 5 ; μ 2 μ 2 c0 5 1 1

c9 5 2

2E21 Ψ1 2E2 Ψ2 ; c10 5 2 2 ; μ μ

xa ah 8a2 1 1 ζ 1 2 2ah 2 2 ; d1 5 1 1 h 2 ; d2 5 2 α 2 2 8μra ra μra U 2μra2 8 9   = 1 1 1 2ah < 1 2 a d3 5  2 ð E Ψ 1 E Ψ Þ 2 2 ð 1 2 Ψ 2 Ψ Þ ; h 1 1 2 2 1 2 ; U 2μra2 : 2 d0 5

d4 5

βα ð1 2 Ψ1 2 Ψ2 Þð1 1 2ah Þ ;  2 ; d5 5 2 μra2 U

d6 5 2

d8 5 2 d10 5

(12.175f)

   1 2 ah E1 Ψ1 ð1 1 2ah Þ 1 2 E1 2

E 1 Ψ 1 1 E 2 Ψ2 ð1 1 2ah Þ; d7 5 2 μra2    1 E2 Ψ2 ð1 1 2ah Þ 1 2 E2 2 ah 2 μra2

μra2

; d9 5

E21 Ψ1 ð1 1 2ah Þ μra2

E22 Ψ2 ð1 1 2ah Þ ; μra2

n1 5 2

Blb Rb ; n2 5 1; n3 5 : L LU

Introducing nine new parameters, x1 5 α; x2 5 α0 ; x3 5 ξ; x4 5 ξ 0 ; x5 5 w1 ; x6 5 w2 ; x7 5 w3 ; x8 5 w4 ; x9 5 i;

(12.176a)

570

Fluid
Solid Interaction Dynamics

we transform Eqs. (12.175a)
(12.175c) into a set of first-order differential equations: 0

x1 5 x2 ; c 0 H 2 d0 P 0 x2 5 ; d0 c1 2 c0 d1 0

x3 5 x4 ; 0

x4 5 2

c 1 H 1 d1 P ; d0 c 1 2 c 0 d1

0

x 5 5 x 1 2 E1 x 5 ;

(12.176b)

0

x 6 5 x 1 2 E2 x 6 ; 0

x 7 5 x 3 2 E1 x 7 ; 0

x 8 5 x 3 2 E2 x 8 ; n1 x 4 1 n3 x 9 0 x9 5 2 ; n2 where P 5 c2 x4 1 c3 x2 1 c4 x3 1 c5 x33 1 c6 x1 1c7 x5 1 c8 x6 1 c9 x7 1 c10 x8 1 c11 x9 2 f ðτ Þ;

(12.176c)

H 5 d2 x2 1 d3 x1 1 d4 x31 1 d5 x4 1 d6 x3 1d7 x5 1 d8 x6 1 d9 x7 1 d10 x8 2 gðτ Þ:

(12.176d)

By using a fourth-order Runge
Kutta method, those first-order differential equations can be numerically integrated. Solution results Fig. 12.47 shows the phase portrait obtained, which clearly indicates the limit cycle oscillation of the aerofoil. It is interesting to notice that incorporated electric generator results in decreases of amplitudes of both pitching and heaving motions, implying a damping effect on the integrated system. While the pitching amplitude reduces slightly, that of the heave decreases sharply. The reason is that the electromagnetic force created by the coil acts as resistance for the heaving motion. The differences in displacement and velocity between the two cases, with and without considering the electrical
magnetic system, can be used to estimate the energy harvested. Fig. 12.48 shows the time history of the dynamic current of the system, which oscillates at a constant amplitude of about 8.63 A. This finding demonstrates that, by adding a magnetic energy converter, the unstable flutter aerofoil system behaves as a stable oscillation system due to the energy converter’s damping, which is very important to guarantee its practical applications. The maximum instantaneous power generation, which is approximately 75.58 W, can thus be obtained by using Eq. (12.173e). Assuming that the current variation is sinusoidal, we obtain the average power of 37.79 W. Noticing that in all formulations, only a unit span value is considered, if the span of the aerofoil is 5 m, the average power would be about 188.95 W. Semiactive aerofoil For the preceding active aerofoil calculations, the quasisteady aerodynamic theory in terms of the prescribed fluid lift and moment functions in Eqs. (12.172a)
(12.172d) has not accurately given the dynamic forces in the process.

FIGURE 12.47 Phase portrait showing the limit cycle oscillations of the system: (A) pitch motion and (B) heave motion. Solid line: without electro magnetic generator; dashed line: with electromagnetic generator ðU 5 6:5; ω 5 0:2; ζ ξ 5 0:01; ζ α 5 0:01; μ 5 100, xα 5 0:25, rα 5 0:5, ah 5 2 0:5, β ξ 5 1, β α 5 1:5; B 5 0:5 T, l 5 20 m, R 5 1 Ω, L 5 0:05 H, b 5 1 m; ρ 5 1:293 kg=m3 Þ (Yang, 2013).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

571

FIGURE 12.48 Dynamic current at steady state considering the electromagnetic generator. Parameter values given in the caption for Fig. 12.46 (Yang, 2013).

FIGURE 12.49 Aerofoil flutter system studied by Javed (2015) and Javed et al. (2016).

Javed (2015) and Javed et al. (2016) adopted a coupled meshfree
mesh-based fluid solver described in Section 12.4.3.4 to simulate a semiactive aerofoil system, as shown in Fig. 12.49. In this case, the aerofoil is subjected to a prescribed pitching motion about its elastic axis and is allowed to move freely along the heave axis due to fluid forces. The aerofoil is mounted on a translational spring-damper system. When the aerofoil is subjected to periodic pitch oscillation, it causes a corresponding variation in the fluid forces over time. These time-varying fluid forces induce heaving motion. Such mechanisms have recently gained attention for their potential application in tidal and wind energy extraction systems (Wu et al., 2014; Deng et al., 2015), in which the pitching motion is prescribed by other excitation and both the vertical and the pitching motions are used to harvest the flow energy. Equations of aerofoil motions Aerofoil NACA0015 is chosen of chord length c with its electric axis located at a distance c/3 from the leading edge to complete the numerical simulation. The vertical h(t) and pitching θðtÞ motions of the aerofoil in the fluid are governed by the nonlinear PDEs similar to the ones given by Dubcova et al. (2009), that is, 2 ^ mh€ 1 Sθ€ cos θ 2 Sθ_ sin θ 1 ch h_ 1 kh h 5 LðtÞ 1 L; ^ Sh€ cos θ 1 Iθ θ€ 1 cθ θ_ 1 kθ θ 5 MðtÞ 1 M;

(12.177a)

which, when the pitching angle is small, reduces to ^ mh€ 1 Sθ€ 1 ch h_ 1 kh h 5 LðtÞ 1 L; € _ € ^ Sh 1 Iθ θ 1 cθ θ 1 kθ θ 5 MðtÞ 1 M;

(12.177b)

572

Fluid
Solid Interaction Dynamics

which is similar to Eq. (12.170), except that no magnetic force is included. Here, L^ and M^ denote external excitations to produce the prescribed pitch motion, and other notations are the same as the ones in Eq. (12.170), except α is replaced by the prescribed rotation angle θ. The produced lift force L(t) is calculated by Eq. (12.168), and the moment M(t) of the lift force is given by ð MðtÞ 5 e3ki r k σij ηj dS; (12.177c) Σ

where rk denotes the position vector from the elastic axis to a lift point on the surface of aerofoil. The prescribed pitching motion is defined by θðtÞ 5 θ0 cos ωt;

(12.178)

where θ0 and ω denote the amplitude and frequency of this harmonic motion, respectively. Fluid motion The fluid of mass density ρ and the free stream velocity U are assumed to be viscous with their chosen Reynolds number Re 5 ρUc=μ 5 1100; 1000, of which the flow is predominantly laminar (Kinsey and Dumas, 2008). The grid arrangement of the fluid domain is given by Fig. 12.40, for which Fig. 12.50 gives more details. As sketched in Fig. 12.50B, the aerofoil is placed at a distance of 4c from the inlet and 12c from outlet, and the width of fluid domain is set to 10c. The dimensions of active meshfree zone around the aerofoil are set to 1.35c 3 1.6c. The velocity of the flow particle on the aerofoil surface is required to equal the velocity of aerofoil motion, and the pressure value at the boundary is obtained by solving the pressure Poisson equation based on the Neumann boundary condition, as discussed in Eqs. (12.162a) and (12.162b). A total of 300 nodes are placed on the aerofoil surface, and the computational domain comprises a total of 25,880 meshfree and 65,354 Cartesian nodes. The movement of mesh is accomplished by displacing the meshfree zone according to the prescribed pitching motion. The time step is chosen as 0.001 second. Solution The solution process is the same as used for the flow-induced vibrations given in Section 12.4.3.4, of which the details can be found in the thesis by Javed (2015) and the paper by Javed et al. (2016). The aerofoil equation is solved in a nondimensional form by using the following parameters: m 5

m ; 0:5ρc2

c 5

c ; 0:5ρUc

k 5

k ; 0:5ρU 2

f 5

ω : 2π

(12.179)

FIGURE 12.50 Hybrid grid around aerofoil: (A) arrangement of Cartesian grid and meshfree particles, (B) full domain, (C) leading edge, (D) trailing edge (Javed, 2015).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

573

TABLE 12.2 Comparison of the simulation result by Javed et al. (2016) (J) with that by Wu et al. (2014) (W) and that by Deng et al. (2015) (D). Mechanical parameters θ0 15

(degree)

m*

c*

k*

f*

1

10

10

0.2

π

30

75

75

0.1022

0

Re

Source

CLmax

CD

1100

W

0.704

0.179

J

0.690

0.170

0.1

0.12

0.22

1000

max CM

W

0.905

0.345

J

0.885

0.334

D

2.0

0.33

J

2.017

0.31

D

2.8

0.6

J

2.550

0.56

The simulations are run at two chosen Reynolds Re 5 1100, 1000 with four different mechanical parameters. The simulation results are compared with the ones reported by Wu et al. (2014) and Deng et al. (2015) in Table 12.2, from which a good agreement is found. The time histories of heave/pitch displacements and lift/drag coefficients of the aerofoil subjected to the prescribed pitch harmonic motion in Eq. (12.178) with its amplitude θ0 5 76:33 degrees and frequency ω 5 0:28π are shown in Fig. 12.51. The obtained vorticity profiles at selected time instants are given in Fig. 12.52. From these two figures, it is observed that with the pitch angle increasing in the initial phase t , T/8 of oscillation, the lift achieves its maximum value, and the flow remains largely attached with the aerofoil’s top surface as shown in Fig. 12.52A. The first peak in lift profile appears at around t 5 T/8. Increasing lift also causes an increase in drag, and therefore the drag coefficient also increases. This initial rise in lift is followed by flow separation close to the leading edge, as shown in Fig. 12.52B, causing a reduction in lift. Subsequently, the detached leading edge vortex reattaches with the aerofoil close to its trailing edge (Fig. 12.52C) at about t 5 3T/8, causing a second peak in the lift profile. However, as the leading edge vortex leaves the aerofoil from the trailing edge and moves farther downstream, a sharp decline in lift is observed between 3T/ 8 , t , 5T/8. The lift coefficient reduces to zero and then shows a similar profile in the negative direction. These results are in good agreement with the reference values by Kinsey and Dumas (2008). Fig. 12.53 shows the two time histories of lift coefficient affected by the excitation frequency. It can be observed that the peak value of the lift coefficient increases at a higher frequency f  . Similar behavior was observed by Wu et al. (2014). Harvested energy As mentioned, this semiactive energy harvesting system requires the external force M^ to keep the aerofoil in the given pitch motion, so that external input power is needed, which reduces the net harvesting power of the system. In this simulation, the main aim is to investigate the dynamic behavior of the aerofoil energy harvesting system based on the NS PDEs solved by th meshfree particle approach, and the energy converter is not included. We have learned that the energy converter in the system acts as a damping mechanism; therefore we can consider the damping coefficients ch and cθ in Eqs. (12.177a)
(12.177c) to reflect the energy collection damping for the heave and pitch motions, respectively, of the aerofoil. From this assumption, we can estimate the harvested power of the system as follows. Instantaneous energy flow equilibrium equations The dynamic equations in Eqs. (12.177a)
(12.177c) are now rewritten in matrix form:         h LðtÞ m S cos θ h€ kh 0 L^ ch 2Sθ_ sin θ h_ 5 1 ^ : (12.180) 1 1 _ € MðtÞ 0 kθ θ S cos θ Iθ θ M 0 cθ θ

Premultiplying Eq. (12.180) by the row matrix h_ θ_ , we obtain its energy flow equilibrium equation (Xing and Price, 1999; Xing, 2015) in the form

1

100

h(t) θ(t)

50

0

0

–0.5

–1

θ(t)

h(t)

0.5

–50

0

T/4

T/2 Time (t)

3T/4

T

2

–100

4

CL

1

3

0

2

–1

1

–2

0

T/4

T/2 Time (t)

3T/4

T

CD

CL

CD

0

FIGURE 12.51 Time histories of the heave displacement h(t), pitch angle θðtÞ, lift coefficient CL, and drag coefficient CD of the NACA0015 aerofoil subjected to the prescribed pitch motion in Eq. (12.178) at Re 5 1100, θ0 5 76:33 degrees, and ω 5 0:28π (Javed, 2015).

FIGURE 12.52 Instantaneous vorticity profiles around NACA0015 at Re 5 1100, Re 5 1100, θ0 5 76:33 degrees, and ω 5 0:28π (Javed, 2015).

FIGURE 12.53 The time history of lift coefficients of NACA0015 aerofoil subject to the pitch oscillation with its amplitude θ0 5 75 degrees and two frequencies, f  5 0:12; 0:22 (Re 5 1000, m 5 0:1022, c 5 π, k 5 0) (Javed, 2015).

Mixed finite element—smoothed particle methods for nonlinear fluid
solid interactions Chapter | 12

  



^ dT dΠ 1 m S cos θ h_ ^I 5 h_ θ_ L ; 1 PH 1 5 P^I ðtÞ 1 PF ðtÞ; T 5 ; P h_ θ_ S cos θ Iθ θ_ M^ dt dt 2     

kh 0

ch 0

LðtÞ 1 h h_ h θ PH 5 h_ θ_ ; PF 5 h_ θ_ ; Π5 ; 0 cθ θ_ 0 kθ θ MðtÞ 2 where we have used the following equation in deriving this result: " #" #  " #



m S cos θ h€ dT 1 0 2Sθ_ sin θ h_ _ _ _ _ 5 h θ 1 h θ dt 2 S cos θ Iθ θ_ 2Sθ_ sin θ 0 θ€ ( " # " # " # )

m S cos θ h€ 0 2Sθ_ sin θ h_ _ _ 1 : 5 h θ S cos θ Iθ θ_ θ€ 0 0

575

(12.181a)

(12.181b)

Here, T and Π, respectively, denote the kinetic energy and potential energy of the system, while PH is the harvested power, and P^I and PF represent the input power of the external excitation force and the power done by the aerodynamic forces, respectively. Time-averaged energy flow equilibrium equation Taking a time average of Eqs. (12.181a) and (12.181b) by the operation for a time function P, ð hPi 5 1T^ Pdt 0T^

over a vibration period T^ 5 2π=ω, we obtain the time-averaged energy flow equilibrium equation as         dT dΠ 1 hP H i 1 5 P^I ðtÞ 1 PF ðtÞ : dt dt Net harvested averaged power

  hPN i 5 hPH i 2 P^I

(12.182a)

(12.182b)

(12.182c)

Here it is necessary to remember the following important points for designing nonlinear energy harvest systems: 1. The dynamic equation shown in Eq. (12.180) is a nonlinear equation, in which the related matrices, especially the fluid dynamic forces, depend on the motion of the system, so that the excited motion and the fluid forces of the nonlinear system produced by the harmonic pitch oscillation are not necessarily harmonic variables with the same frequency as the pitch oscillation. The energy flow variables in Eqs. (12.181a), (12.181b), and (12.182a)
(12.182c) generally must be calculated by numerical integrations. 2. For a nonlinear system, the time-averaged kinetic/potential energies do not necessarily vanish (Xing, 2015), imply^ and ΠðtÞ 6¼ Πðt 1 TÞ; ^ therefore the time change rates of the kinetic and ing that, in general cases, TðtÞ 6¼ Tðt 1 TÞ potential energies in Eq. (12.182b) remain. Physically, this represents that some energy exchange occurs between the solid system and the flow and/or excitation forces during a given time period. 3. For the semiactive aerofoil system, the arrangement of the vertical/rotational stiffens, providing the possibility of adjusting the natural frequencies of the solid system in order to obtain the required large motions and the harvested energy. To confirm this, analysis of the net harvested averaged power affected by the design parameters is needed.

Appendix

Numerical methods solving finite element dynamic equations This appendix gives some numerical methods, developed by Xing (2005), in FORTRAN designed to solve dynamic finite element (FE) equations. The principles, solution processes, examples, and comparison of five time integration methods—Newmark, Wilson-θ, Hilber-α, Hilber collocation, and central difference—which are often used to solve dynamic FE equations, are presented. A computer code Program of Time Element Methods (PTEM), developed by the author, is introduced and presented. This code provides a generalized functional module using these five algorithms to solve the FE equation t M x€ 1 t C x_ 1 t Kx 5 Rðt; x; x_ Þ describing linear and nonlinear structural dynamics. This functional program module is conveniently incorporated into any computer program used to investigate complex dynamics problems in maritime engineering.

A.1

Introduction

Direct integration methods, or time element methods, are important and necessary approaches in analyzing dynamic response problems of linear and nonlinear dynamic systems (Bathe and Wilson, 1976; Bathe, 1982, 1986, 1996; Belytschko and Hughes 1983; Zheng and Tan, 1985; Zheng 1986). Essentially, direct numerical integration methods are based on two ideas. First, instead of satisfying the FE dynamic equation at any time t, these methods are aimed to satisfy it only at discrete time intervals Δt apart. The second is that a variation of displacements, velocities, and accelerations within each time interval is assumed. In depending on the different discrete points and variation assumptions used, there are different time integration schemes with different performances of accuracy, stability, and cost of solution. The central difference method is an often used explicit method with three-point interpolation. Houbolt (1950) developed a four-point implicit scheme. Newmark (1959) introduced two parameters γ and β to represent displacement and velocity, respectively, and created a very useful algorithm called the Newmark method. Hilber and Taylor (1977) introduced another parameter, α, in order to control numerical damping and to improve accuracy, into the Newmark scheme and created the Hilber-α method. Wilson et al. (1973), assuming a linear acceleration distribution over a time interval (t,t 1 θΔt), constructed the Wilson-θ method. Combining the ideas of the Wilson and Newmark methods, Hilber and Hughes (1978) developed another method called the Hilber collocation method. Zienkiewiez (1977) investigated the constructions of the Newmark and other time integration methods using a weighted residual approach. Based on variational principles and interpolations in the time domain, Xing and Zheng (1985, 1987) studied existing methods and created a new approach. In an integrated variational frame for the interpolations in space and time domains, the concept of a four-dimensional FE method was proposed. In the current FE software, the five methods—the central difference, Newmark, Wilson-θ, Hilber-α, Hilber collocation schemes—are widely used. This appendix gives a description and comparison of these five methods. A computer code PTEM, developed by the author, is introduced and listed here for application by readers.

A.2 A.2.1

Fundamental iteration formulations Generalized dynamic equation

In a linear FE analysis (FEA), a generalized dynamic equation is expressed as M€x 1 C_x 1 Kx 5 R;

(A.1)

577

578

Appendix: Numerical methods solving finite element dynamic equations

where M, C, and K are constant mass, damping, and stiffness matrices of the system, x, x_ , and x€ , represent the displacement, velocity, and acceleration vectors, and R is a time-dependent load vector. For nonlinear systems, Eq. (A.1) can be replaced by the following increment form: t

MΔ€x 1 t CΔ_x 1 t KΔx 5 Δt R;

(A.2)

where tM, tC, and tK denote the tangent mass, damping, and stiffness matrices at time t, and Δt R is the load increment. Since at each time step, the tangent mass, damping, and stiffness matrices are recalculated, the cost of solving Eq. (A.2) is quite high. For practical problems, nonlinearity is mainly caused by stiffness and damping but not involving the mass of the system. To reduce the computational cost, Eq. (A.2) is represented as t 1 Δt

M x€ ðkÞ 1 0 CΔ_xðkÞ 1 0 KΔxðkÞ 5 t 1 Δt R 2 t 1 Δt Fd ð_xðk21Þ Þ 2 t 1 Δt Fs ðxðk21Þ Þ;

t 1 Δt ðkÞ

5 t 1 Δt xðk 2 1Þ 1 ΔxðkÞ ;

t 1 Δt ðkÞ

5 t 1 Δt _x ðk 2 1Þ 1 Δ_xðkÞ ;

x

x_

(A.3)


k 5 1; 2; 3; :::



where 0C and 0K are the initial damping and stiffness matrices at time t 5 0, and Fd and Fs represent the damping and stiffness forces that can be expressed in a summation of linear and nonlinear components as follows: d Fd 5 0 C x_ 1 F~ ð_xÞ;

s Fs 5 0 Kx 1 F~ ðxÞ:

(A.4)

Eq. (A.3) is now rewritten in the form t 1 Δt

t 1 Δt ~ R 2 t 1 Δt F~ d ð_xðk21Þ Þ 2 t 1 Δt F~ s ðxðk21Þ Þ 5 t 1 Δt R; ðM€xðkÞ 1 C_xðkÞ 1 KxðkÞ Þ 5

(A.5)

where for convenience the superscripts of the damping and stiffness matrices are neglected. Eq. (A.5) provides an iteration formulation to obtain a solution x(k) at iteration k from previous result x(k21). The following three convergence criteria may be chosen to check whether the convergence is reached at iteration k.

A.2.2

Iteration criterions

A.2.2.1 Energy criterion ΔxðkÞ  t 1 Δt ½R 2 Fd ð_xðk21Þ 2 Fs ðxðk21Þ 2 M€xðk21Þ  Δxð1Þ U½t 1 Δt R 2 t Fd 2 t Fs 2 Mt1Δt x€ ð0Þ 

(A.6)

# ETOL:

A.2.2.2 Force criterion :t1Δt ½R2Fd ð_xðk21Þ 2Fs ðxðk21Þ 2M€xðk21Þ :2 :½t1Δt R2 t Fd 2 t Fs 2Mt1Δt x€ ð0Þ :2

(A.7)

# RTOL:

A.2.2.3 Displacement criterion :ΔxðkÞ :2 # DTOL: :xðkÞ :2

Here :a:2 5

pffiffiffiffiffiffiffiffi aUa, ETOL, RTOL, and DTOL represent accuracy errors defined by users.

(A.8)

Appendix: Numerical methods solving finite element dynamic equations

A.3 A.3.1

579

Five numerical integration schemes Newmark method

The Newmark method assumes that the velocity and displacement vectors at time step n 1 1 take the forms x_ n11 5 x_ n 1 ½ð1 2 γÞ€xn 1 γ x€ n11 ; xn11 5 xn 1 x_ n Δt 1 ½ð0:5 2 βÞ€xn 1 β x€ n11 Δt2 ;

(A.9)

~ n11 ; M€xn11 1 C_xn11 1 Kxn11 5 R from which the displacement, velocity, and acceleration vectors xn11, x_ n11 , and x€ n11 at time step n 1 1 are derived from their values at the previous time step n. The unconditional stable condition of the Newmark method requires that γ $ 0:5;

A.3.2

β $ 0:25ð0:51γÞ2 :

(A.10)

Wilson-θ method

This method assumes that the acceleration in time interval τAðt; t 1 θΔtÞ varies in a linear form: τ ð€xn1θΔt 2 x€ n Þ: x€ n1τ 5 x€ n 1 θΔt

(A.11)

The integration of this equation gives that x_ n1τ 5 x_ n 1 x€ n τ 1

τ2 ð€xn1θΔt 2 x€ n Þ; 2θΔt

τ2 τ3 xn1τ 5 xn 1 x_ n τ 1 x€ n 1 ð€xn1θΔt 2 x€ n Þ; 2 6θΔt

(A.12)

which, at τ 5 θΔt, x_ n1θΔt , and x€ n1θΔt , can be solved in terms of xn1θΔt . The substitution of the results of x_ n1θΔt and x€ n1θΔt into the dynamic equation M€xn1θΔt 1 C_xn1θΔt 1 Kxn1θΔt ~ n1θΔt 5 R ~ n 1 θðR ~ n1Δt 2 R ~ nÞ 5R

(A.13)

provides a solution for xn1θΔt . The solution of xn1Δt , x_ n1Δt , and x€ n1Δt can be obtained by   3 6 6 x€ n11 5 1 2 x€ n 2 2 x_ n 1 3 2 ðxn1θΔt 2 xn Þ; θ θ Δt θ Δt x_ n11 5 x_ n 1

(A.14a)

Δt ð€xn11 1 x€ n Þ; 2

Δt2 xn11 5 xn 1 Δtx_ n 1 ð€xn11 1 2€xn Þ: 6

(A.14b)

The unconditional stable condition of the method requires θ $ 1:37.

A.3.3

Hilber-α method

Based on the Newmark method, Hilber introduced a parameter α into Eq. (A.9) and obtained that ~ n11 ; M€xn11 1 C_xn11 1 K½xn11 ð1 1 αÞ 2 αxn  5 R

(A.15)

580

Appendix: Numerical methods solving finite element dynamic equations

developing a modification algorithm. The parameter α controls human-made damping and increases the accuracy. The unconditional stable condition of this method requires that γ 5 0:5 2 α; β 5 0:25ð1 2 αÞ;

A.3.4

20:5 # α # 0 :

(A.16)

Hilber collocation method

Combining the Wilson-θ and Newmark methods, Hilber constructed a collocation scheme using the following form: x€ n1θ 5 x€ n 1 θð€xn11 2 x€ n Þ; x_ n1θ 5 x_ n 1 θΔt½ð1 2 γÞ€xn 1 γ x€ n1θ ; xn1θ 5 xn 1 θΔtx_ n 1 ðθΔtÞ2 ½ð0:5 2 βÞ€xn 1 β x€ n1θ ; ~ n1θ ; M€xn1θ 5 C€xn1θ 1 Kxn1θ 5 R

(A.17)

in which x_ n1θ and x€ n1θ are represented in terms of xn1θ , and then xn1θ is obtained by solving ^ ^ n1θ 5 R; Kx 1 γ ^5 C 1 K; M1 K βθΔt βðθΔtÞ2 ^ 5 ð1 2 θÞR ~ n 1 θR ~ n11 ; R 0 1 3 2 1 1 1 x_ n 1 @ 2 1Ax€ n 5; xn 1 1 M4 βðθΔtÞ 2β βðθΔtÞ2 0 1 0 1 2 3 γ γ γ xn 1 @ 2 1Ax_ n 1 @ 2 1AθΔtx€ n 5: 1 C4 βðθΔtÞ β 2β

(A.18)

Hilber demonstrated that this method is unconditionally stable provided that γ 5 0;

θ 2θ2 2 1 $β$ ; 2ðθ 1 1Þ 4ð2θ3 2 1Þ

θ$1

(A.19)

This method reduces to a Newmark method of γ 5 0:5 and β 5 0:25 if θ 5 1 and to a Wilson-θ method if γ 5 1=2 and β 5 1=6.

A.3.5

Central difference method

For the central difference method, it is assumed that

x_ n 5

1 ðxn11 2 xn21 Þ; 2Δt (A.20a)

1 x€ n 5 2 ðxn11 2 2xn 1 xn21 Þ; Δt ~ n; M€xn 1 C_xn 1 Kxn 5 R

(A.20b)

Appendix: Numerical methods solving finite element dynamic equations

581

from which it follows that ^ ^ n11 5 R; Kx ^ 5 1 M 1 1 C; K Δt2 2Δt 0 1 0 1 2 1 1 ^ 5R ~ n 2 @K 2 R CAxn21 ; M A xn 2 @ 2 M 2 Δt2 Δt 2Δt x21 5 x0 2 Δtx_ 0 1

(A.21)

Δt2 x€ 0 : 2

The central difference method is an explicit method for which iteration is not needed, and therefore it is widely used in nonlinear analysis. This method is conditionally stable, and the stable condition requires Δt #

Tm ; π

(A.22)

where Tm is the minimum vibration period of the problem.

A.4

Stability and accuracy

As previously described, the four methods, except for the central difference method, are unconditionally stable, implying that for any time step Δt, an approximate solution can be definitely obtained. However, for a large time step Δt, the accuracy of the result is lower but at less of a cost of calculation. Here, the stability and accuracy analysis of the methods are given for a reference in applications.

A.4.1

Stability and spectral radius

The stability of an integration method is determined by examining the behavior of the numerical solution for arbitrary initial conditions. Assume that the iteration formulation of a method to determine the solution t 1 Δt x^ from the solution t x^ is defined as t 1 Δt

x^ 5 At x^ ;

(A.23)

where A represents the matrix operator of a method. Let ρ(A) be the spectral radius of A defined as ρðAÞ 5 max jλj j;

j 5 1; 2; 3; . . . :

(A.24)

The necessary and sufficient condition of a stable time integration method is ρðAÞ # 1:

A.4.2

(A.25)

Period elongation and amplitude decay

Various studies of the accuracies of time integration methods have been reported. Here, we only summarize some important solution characteristics. For this purpose, we consider the solution of the initial value problem defined by x€ 1 ω2 x 5 0;

0

x 5 1:0; 0 x_ 5 0:0;

(A.26)

for which the exact solution is x 5 cos ωt. However, there is a period elongation PE and an amplitude decay AD in the approximate solution obtained by time integration methods. It has been defined that Percentage amplitude decay 5 AD 3 100%; Percentage period elongation 5

PE 3 100%; T

(A.27)

582

Appendix: Numerical methods solving finite element dynamic equations

to describe the accuracy of a method. Also, a damping factor η has been introduced as AD 5 1 2 e22πη ;

η5

2lnð1 2 ADÞ : 2π

(A.28)

For the often used time integration methods, the references by Hilber et al. (1977) and Hilber and Hughes (1978) present the spectral radii ρ(A), the percentage amplitude decay, the damping factor, and the percentage of period elongation as functions of Δt/T. These curves provide a good comparison of the discussed time integration methods. From the point of view of engineering applications, all of the five methods can provide sufficient accuracy for engineering analysis, provided a suitable time step chosen.

A.5

Implementation

The implementation process of these five time integration methods is described as follows.

A.5.1

Initial calculations

1. Form mass, stiffness, and damping matrices M, K, and C. 2. Initialize 0 x, 0 x_ , and 0 x€ 5 M21 ð0 R 2 C0 x_ 2 K0 xÞ. For the central difference method, you need to calculate 21 x 5 0 x 5 0 x_ Δt 1 0:50 x€ Δt2 . 3. Select time step Δt , Δtcr , and calculate the integration constants.

A.5.1.1 Newmark method γ $ 0:5; Ak 5 1;

β $ 0:25ð0:51γÞ2 ; A0 5 1ðβΔt2 Þ; A1 5 γ=ðβΔtÞ;

A2 5 1=ðβΔtÞ;

A3 5 1ð2βÞ 2 1;

A5 5 ðγ=β 2 2ÞΔt=2;

A4 5 γ=β 2 1;

A6 5 ð1 2 γÞΔt;

A7 5 γΔt:

A.5.1.2 Hilber-α method 0:5 # α # 0;

γ 5 0:5 2 α; β 5 0:25ð12αÞ2 ;

Ak 5 1 1 α;

A0 5 1=ðβΔt2 Þ;

A2 5 1=ðβΔtÞ;

A1 5 γ=ðβΔtÞ;

A3 5 1=ð2βÞ 2 1;

A5 5 ðγ=β 2 2ÞΔt=2;

A4 5 γ=β 2 1;

A6 5 ð1 2 γÞΔt;

A7 5 γΔt:

A.5.1.3 Wilson-θ method θ $ 1:37;

Ak 5 1;

A0 5 6=ðθΔt2 Þ; A3 5 θΔt=2; A6 5 1 2 3=θ;

A1 5 3=ðθΔtÞ; A2 5 2A1 ;

A4 5 A0 =θ;

A5 5 2 A2 =θ;

A7 5 Δt=2; A8 5 Δt2 =6:

Appendix: Numerical methods solving finite element dynamic equations

583

A.5.1.4 Hilber collocation method θ $ 1;

γ 5 0:5;

Ak 5 1;

θ 2θ2 2 1 $β$ ; 2ðθ 1 1Þ 4ð2θ3 2 1Þ

A0 5 1=ðβθ2 Δt2 Þ;

A1 5 γ=ðβθΔtÞ;

A2 5 1=ðβθΔtÞ; A3 5 1=ð2βÞ 2 1; A4 5 γ=β 2 1;   γ A0 21 A5 5 2 1 θΔt; A6 5 ; A7 5 2 ; 2β θ βθ Δt A8 5 2

1 ; 2βθ

B1 5 Δtγ;

A9 5 ð1 2 γÞΔt;

B2 5 ð0:5 2 βÞθ2 Δt2 ;

B3 5 βθ2 Δt2 :

A.5.1.5 Central difference method Δt # Tn =π 5 Δtcr ; Ak 5 0; A0 5 1=Δt2 ;

A.5.2

A1 5 1ð2ΔtÞ;

A2 5 2A0 :

Form effective stiffness matrix ^ 5 A0 M 1 A1 C 1 Ak K: K

A.5.3

Triangular decomposition of matrix K^ ^ 5 LDLT : K

A.5.4

(A.29a)

(A.29b)

Calculations at each time step

1. Form effective loads at time t 1 Δt. For a convenience in equation expression, we define that t

R 5 MðA0 t x 1 A2 t x_ 1 A3 t x€ Þ 1 CðA1 t x 1 A4 t x_ 1 A5 t x€ Þ:

(A.30)

Newmark method t 1 Δt

^ 5 t 1 Δt R 1 t R: R

(A.31)

t 1 Δt

^ 5 t 1 Δt R 1 t R: R

(A.32)

Hilber-α method

Wilson-θ method t 1 Δt

^ 5 A2 t R 1 θt 1 Δt R 2 A2 Mt x€ R 2 Kðt x 1 A3 t x_ 1 A5 t x€ Þ 2 Cðt x 1 A4 t x€ Þ:

(A.33)

Hilber collocation method t 1 Δt

^ 5 ð1 2 θÞt R 1 θt 1 Δt R 1 t R: R

(A.34)

584

Appendix: Numerical methods solving finite element dynamic equations

Central difference method t 1 Δt

^ 5 t R 1 MðA2 t x 2 A0 t 2 Δt xÞ 2 Kt x 1 CA1 t 1 Δt x: R

(A.35)

2. Solve for x^ at time t 1 Δt: ^ x 5 LDLT x^ 5 t 1 Δt R: ^ K^

(A.36)

3. Evaluate displacements, velocities, and accelerations at time t 1 Δt. Newmark and Hilber-α methods t 1 Δt

x 5 x^ ;

t 1 Δt

x€ 5 A0 ð^x 2 t x^ Þ 2 A2 t x_ 2 A3 t x€ ;

(A.37)

x 5 t x_ 1 A6 t x€ 1 A7 t 1 Δt x€ :

t 1 Δt _

Wilson-θ method t 1 Δt

x€ 5 A4 ð^x 2 t x^ Þ 1 A5 t x_ 1 A6 t x€ ; x 5 t x_ 1 A7 ðt x€ 1 t 1 Δt x€ Þ;

t 1 Δt _ t 1 Δt

(A.38)

x 5 t x 1 Δtt x_ 1 A8 ð2t x€ 1 t 1 Δt x€ Þ:

Hilber collocation method t 1 Δt

x€ 5 A6 ð^x 2 t x^ Þ 1 A7 t x_ 1 A8 t x€ ; x 5 t x_ 1 A9 t x€ 1 B1 t 1 Δt x€ ;

t 1 Δt _ t 1 Δt

x 5 x 1 Δt x 1 B2 x€ 1 B3 t

t_

t

(A.39) t 1 Δt

x€ :

Central difference method t 1 Δt

x 5 x^ ;

x 5 A1 ðt 1 Δt x 2 t 2 Δt xÞ;

t_ t

A.6

(A.40)

x€ 5 A0 ðt 1 Δt x 2 2t x 1 t 2 Δt xÞ:

Time element program

The time element program PTEM, developed by Xing (1989), is a functional module that includes the five time integration methods just discussed. This program is used to solve linear and nonlinear dynamic FEA equations to obtain displacements, velocities, and accelerations of the system at time t 1 Δt from their values at time t. It can be incorporated into an FEA program as a functional module or directly used to solve dynamic equations obtained using other software.

A.6.1

Main characteristics

1. The program consists of about 1200 FORTRAN sentences. All functions of the program are completed in the module. It provides a convenient interface for users to connect the module into other FEA programs. 2. There are three forms to solve Eq. (A.1), which is controlled by a control variable JC as follows: JC 5 1: The matrices M, C, and K are assumed unchanged at each time step, and load R is dependent on time only. This function is mainly used to solve linear problems with no iteration required at each time step. JC 5 2: The matrices M, C, and K are still assumed unchanged at each time step, but the load R is dependent on unknown displacement and velocity. For the four time integration methods except the central difference method, iteration is required at each time step. Users can choose the convergence criteria. This process is mainly used to solve problems with weak nonlinearity. For problems with strong nonlinearity, iteration convergence may not reach, and the central difference method can be used.

Appendix: Numerical methods solving finite element dynamic equations

585

JC 5 3: The matrices M, C, and K are changed at each time step, but the load R is independent of only the time function unknown variables. During the calculation at each time step the matrices M, C, and K are required to be updated using FEA software. There are three flags JCM, JCC, and JCK to control the changes of matrices M, C, and K, respectively. 3. A control flag IJK is designed for users to choose one of the five time integration methods. 4. The program can automatically check the parameters α, β, γ, and θ to satisfy the stability conditions. 5. The load functions designed in the program include sinusoidal functions, discrete time functions given by input data, and the nonlinear spring function FUNC1 5 Ax2 1 Bx3

(A.41)

FUNC1 5 A_x2 1 B_x3 ;

(A.42)

and nonlinear damping function

where A and B are constants defined by users. Users can input their load functions, defined by themselves, and define the starting time of each load. 6. Output results are the displacements, velocities, and accelerations at the selected degrees of freedom (DOFs) by users. Users can choose the time intervals of the output data.

A.6.2

Flowchart of the program

A flowchart of the program is shown in Fig. A.1. All the input data, the values of control variables, and the method parameters are supplied by an INPUT.DAT file prepared by users. A file OUTPUT.DAT produced automatically by the program gives the output results.

A.7 A.7.1

Examples Example 1

A nondamping free vibration of a system with 1 DOF is defined by the following equation: x€ 1 x 5 0;

xð0Þ 5 1:0;

_ 5 1:0 ; xð0Þ

(A.43)

which has a theoretical solution x5

 pffiffiffi π 2 cos t 5 : 4

(A.44)

Table A.1 lists the displacements calculated using the program in association with the parameters Δt 5 π=8, γ 5 0:5, β 5 0:25, α 5 20:01, and θ 5 1:4 for the Wilson-θ method and θ 5 1:001 for the Hilber collocation method.

A.7.2

Example 2

Let us consider a system with 2 DOFs described by the following equation: 2 4

2 0 0 1

2

54

x€1 x€2

3

2

514

2 3 0 4 554 5 0 x2 x1

3

32

2

6

22

22

4

54

2 3 0 4 5 5 4 5: 0 x_2 x_1

3

32

x1 x2

3

2

554

0

3 5;

10 (A.45)

586

Appendix: Numerical methods solving finite element dynamic equations

FIGURE A.1 The flowchart of the PTEM. PTEM, Program of Time Element Methods.

Appendix: Numerical methods solving finite element dynamic equations

587

TABLE A.1 Displacements calculated in Section A.7.1. t Δt

Newmark

Wilson-θ

Hilber-α

Hilber collocation

Central difference

Theoretical

0

1.0000

1.0000

1.0000

1.0000

1.0000

1.0000

2

1.4141

1.4256

1.4139

1.4142

1.4283

1.4142

4

1.0195

1.0532

1.0196

1.0196

1.0095

1.0008

6

0.0419

0.0974

0.0426

0.0420

20.0080

0.0017

8

20.9598

20.9063

20.9586

20.9596

201.0207

20.9984

10

21.4125

21.3996

21.4119

21.4125

21.4281

21.4142

12

21.0574

21.1115

21.0582

21.0574

20.9886

21.0024

TABLE A.2 Displacement x1 calculated in Section A.7.2. t Δt

Newmark

Wilson-θ

Hilber-α

Hilber collocation

Central difference

Theoretical

0

0.0000

0.0000

0.0000

0.0000

0.0000

0.0000

2

0.0504

0.0525

0.0507

0.0505

0.0307

0.0381

4

0.4846

0.4896

0.4847

0.4845

0.4871

0.4860

6

1.5805

1.5425

1.5790

1.5803

1.7009

1.6570

8

2.7607

2.6702

2.7578

2.7605

2.9134

2.8068

10

2.8505

2.8182

2.8502

2.8506

2.7711

2.8057

11

2.2840

2.3340

2.2865

2.2844

2.0368

2.1306

The theoretical solution of this problem is given by



1 5 x1 5 x2 3 5

22 4

"


pffiffiffi  # 1 2 cos 2t
pffiffiffi  ; 1 2 cos 5t

(A.46)

pffiffiffi pffiffiffi which is a summation of two vibrations of periods T1 5 2π and T2 5 2π= 5. The time step Δt 5 0.1T2 is chosen in the calculations. The results obtained using PTEM are listed in Table A.2. The solutions for the Newmark and Wilson-θ methods are exactly same as the ones reported by Bathe and Wilson (1976) and by Bathe (1982, 1986).

A.7.3

Example 3

Here, a weak nonlinear problem of a system with 100 DOFs, as shown in Fig. A.2, is considered. For the ith degree of the system, the mass mi and the spring force fi are given as mi 5 1; fi 5 δi 1 0:2δ3i ;

(A.47)

where δi represents the elongation of the ith spring. Therefore the spring force of each spring is nonlinear. The Rayleigh damping of the system is assumed, given by C 5 0:01M 1 0:005K:

(A.48)

588

Appendix: Numerical methods solving finite element dynamic equations

C1,K1,m1

C100,K100,m100

•••

•

FIGURE A.2 A nonlinear system of 100 degrees of freedom.

TABLE A.3 Displacement x100 calculated in Section A.7.3. t Δt 0 48

Newmark

Wilson-θ

Hilber-α

20.000

20.000

20.000

8.7725

8.7537

Hilber collocation 20.000

8.7732

8.7727

Central difference 20.000 8.7838

Zheng and Tan 20.000 8.7210

108

23.130

23.245

23.132

23.131

23.028

23.123

149

4.008

3.879

4.006

4.007

4.113

4.080

206

23.597

23.588

23.596

23.596

23.646

23.622

247

2 6.765

26.740

26.765

26.766

26.857

26.862

287

26.985

26.928

26.984

26.985

27.088

26.988

The dynamic response of the system excited by five forces   πt Ri 5 0:005i sin ; i 5 20; 40; 60; 80; 100 50

(A.49)

is calculated using PTEM. The initial conditions are xi ð0Þ 5 0:2i;

x_i ð0Þ 5 0;

ði 5 1; 2; 3; . . .; 100Þ :

(A.50)

If the nonlinear part of total stiffness is relatively smaller than its linear part, we consider a constant linear stiffness 2 3 2 21 6 21 & & 7 7 K56 (A.51) 4 & 2 215 21 1 s and a nonlinear force term F~ , in which an element is given by h i s F~ i 5 0:2 ðxi 2xi21 Þ3 1 ðxi 2xi11 Þ3 ;

x0 5 x101 5 0:

(A.52)

The equation describing the system is now written as s M€x 1 C_x 1 Kx 5 R 1 F~ :

(A.53)

The time steps Δt 5 0.25 for the central difference method and Δt 5 2.5 for the other four methods are used. The parameters for each time integration method are the same as in Sections A.7.1 and A.7.2. The results obtained are shown and compared in Table A.3, in which the results by Zheng and Tan (1985) and Zheng (1986) are listed.

Appendix: Numerical methods solving finite element dynamic equations

A.8

Program of Time Element Methods

589

590

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

591

592

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

593

594

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

595

596

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

597

598

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

599

600

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

601

602

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

603

604

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

605

606

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

607

608

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

609

610

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

611

612

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

613

614

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

615

616

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

A.9

617

User manual

The variables used in the input file INPUT.DAT follow the rules of FORTAN. The users need to generate this INPUT. DAT as their input file to use the program. In Section A.10, the input files for the three examples are listed. The structure of the input file is defined as follows: Line 1: Maximum 72 letters allowed for users to define the title of problem Line 2: Main control data of 20 integer variables IJK, JC, NN, NWM, NWC, NWK, NITER, NPRT, NINT, NC, NO, NLD, NLDT, NLDU, NLDV, NAD, NAT, JCM, JCC, JCK The physical meanings of these variables are: IJK: Time element method control 5 1 Newmark method 5 2 Wilson-θ method 5 3 Hilber-α modification method 5 4 Hilber allocation method 5 5 Central difference method

618

Appendix: Numerical methods solving finite element dynamic equations

JC: Solution method control with the definition 51 M, C, K matrices are not changed at each time step, and loadR(t) is time function only, no iteration for the solution 52 M, C, K matrices are not changed at each time step, but loadR depends on the unknown solution and iteration needed except using the central difference method 53 M, C, K matrices are changed, which is required to be input at each time step, the load R(t) is time function only and no iteration needed NN: NWM: NWC: NWK: NITER: NPRT:

The order of matrices M, C, and K The length of matrix M in a one-dimensional array form The length of matrix C in a one-dimensional array form The length of matrix K in a one-dimensional array form Maximum iteration time number, automatically set to 30, if it is 0 Print control 5 1 Print the results during the process of calculation 5 2 Not print the results during the process of calculation NNIT: Load function control 5 0 Load not involving acceleration 5 2 Load involving acceleration (option not for this version) NC: Time step number for output results NO: Number of DOF that the results require NLD: Number of load functions, NLD 5 NLDT 1 NLDU 1 NLDV NLDT: Number of load functions as time function only NLDU: Number of nonlinear spring forces in the form fI 5 AI ðxI 2xE Þ2 1 BI ðxI 2xE Þ3 ; AI ; BI : constants to be given by users; xI : displacement at end I of spring; where force acted; xE : displacement at another end of sprint:

NLDV: Number of nonlinear damping forces in the form fI 5 AI ðx_I 2 x_E Þ2 1 BI ðx_I 2 x_E Þ3 ; AI ; BI : constants to be given by users; x_I : velocity at end I of damper; where damping force acted; x_E : velocity at another end of damper: NAD: Number of DOF where loads acted. If a DOF has several different forces, the total number of different forces minus 1 should be added into NAD NAT: Number of loads acting times JCM: M matrix change control at each step 5 1 Change 5 0 Not change JCC: C matrix change control at each step 5 1 Change 5 0 Not change JCK: K matrix change control at each step 5 1 Change 5 0 Not change

Appendix: Numerical methods solving finite element dynamic equations

619

Line 3: Eight parameters of time elements TS, TE, DT, GAMMA, BETA, THETA, ALPHA, TOL, defined as: TS: The starting time of solution TE: The ending time of solution DT: Time step (second) GAMMA: Parameter γ BETA: Parameter β THETA: Parameter θ ALPHA: Parameter α TOL: Convergence accuracy, which is set to 1025, if given as 0 Line 4: Loads arriving time data, total NAT real numbers for array AT(NAT) Time load function cards: Total NLDT sets of data. If NLDT 5 0, it is not needed. For each time function, the following data lines are required. Line 1:

Line 2: Line 3:

Line 4:

NLP: Load function control 5 0 Sinusoidal function .0 Discrete time function, NLP equals the data number SFTR: A coefficient factor Title of function: Total maximum 60 letters Sinusoidal function (only for NLP 5 0 case) in the form f ðtÞ 5 SFTR 3 SINð2π 3 FREQ 3 t 1 2π 3 PHASEÞ FREQ: Frequency (Hz) PHASE: Phase angle 5 phase degree/360 Discrete time function data (only for NLP . 0); total NLP sets data of time T and function value for the array (T(J), P(J), J 5 1, NLP), of which the ending time T(NLP) . TE, the ending time of solution

Nonlinear spring force data cards (only used when NLDU . 0): Total NLDU sets data into array AU(NLDU, 2) in the form ðAUðI; 1Þ; AUðI; 2Þ; I 5 1; NLDUÞ; AUðI; 1Þ 5 AI ; AUðI; 2Þ 5 BI : Nonlinear damping force data cards (only used when NLDV . 0): Total NLDV sets data into array AV(NLDV, 2) in the form ðAVðI; 1Þ; AVðI; 2Þ; I 5 1; NLDVÞ; AVðI; 1Þ 5 AI ; AVðI; 2Þ 5 BI : Original matrices data cards: 1.

Mass matrix RM(NWM) and its address array IM(NNN), NNN 5 NN 1 1, NWM 5 IM (NNN) 2 1 RM(NWM): One-dimensional stored mass matrix IM(NNN): The address of diagonal elements of mass matrix M in RM array

2.

Damping matrix RC(NWC) and its address array IC(NNN) (only used for NWC . 0 case) NNN 5 NN 1 1, NWC 5 IC(NNN) 2 1 RC(NWC): One-dimensional stored damping matrix IC(NNN): The address of diagonal elements of damping matrix C in RC array

620

Appendix: Numerical methods solving finite element dynamic equations

3.

Stiffness matrix RK(NWK) and its address array IK(NNN), NNN 5 NN 1 1, NWK 5 IK(NNN) 2 1 RK(NWK): One-dimensional stored stiffness matrix IK(NNN): The address of diagonal elements of stiffness matrix K in RK array

To illustrate this data structure, we give a symmetric matrix M and its corresponding arrays RM and IM as follows: 2 3 a e 0 0 6 b 0 07 7; RM 5 ða; b; c; d; e; f Þ; IM 5 ð1; 2; 4; 5; 7Þ; NNN 5 5: M56 4 d r5 e Here, for example, the third diagonal element d of M matrix locates at the fourth position in array RM, so that IM (3) 5 4. IM(5) 5 Length of RM 1 1 5 7. Initial conditions cards: U0(NN): Initial displacement vector, NN number of real numbers U1(NN): Initial velocity vector, NN umber of real numbers Load information data matrix NADF (NAD,4) and factor PFCTOR (NAD): Total NAD lines of which each line inputs the following five data elements: NADF(I,1): NADF(I,2):

NADF(I,3): NADF(I,4): PFCTOR(I):

Degree number at which Load I acts. This can be repeated to deal with the case if the same degree number is subject to different loads. Load I’s function ID number, which is in the order: NLDT time functions,NLDU nonlinear spring forces, NLDV nonlinear damping forces. Starting time number of load I, which is an integer of 1BNAT For the time function, set to 0; for nonlinear force function I at degree NADF(I,1), here input another end degree number. Load I’s coefficient factor

Elements change data of matrices RM, RC, RK: From the second time step to the NT step, calculated by NT 5 IDINT ((TE 2 TS)/DT) 2 1 NOUT 5 NT/NC 1 1 NT 5 (NOUT 2 1) 3 NC IF (IJK.EQ.5) NT 5 NT 1 1 to input the following data: JCM 5 1 input RM (NWM) JCC 5 1 input RC (NWC) JCK 5 1 input RK (NWK)

A.10

Input and output files for Examples 1 and 2 in Section A.7

As examples, we give the input and output files for Examples 1 and 2 discussed in Section A.7, which should help readers use the program manual for their problems of interest. The data files for Example 3 are quite a bit longer, so they are left out here.

Appendix: Numerical methods solving finite element dynamic equations

A.10.1

Example 1

INPUT.DAT Input file for using the Newmark method

621

622

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

623

624

Appendix: Numerical methods solving finite element dynamic equations

Appendix: Numerical methods solving finite element dynamic equations

625

626

Appendix: Numerical methods solving finite element dynamic equations

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Index Note: Page numbers followed by “f” and “t” refer to figures and tables, respectively.

A Absorption condition, 101, 136 Acceleration, 372 Acceleration velocity potential forms, 124 127 Accuracy, 581 582 Acoustic radiation in infinite fluid domain, 322 326 governing equations of problem, 322 323 solution approach mode functions of fluid corresponding Ith solid mode, 324 mode summation for solid structure, 323 324 mode summation of pressure and FSI equation, 324 325 of underwater small ball, 325 326 fluid solid interaction radiation equation, 326 pressure mode, 326 vibration of elastic ball, 325 Acoustics, 34 36 Acoustoelastic model, 104 105 Active aerofoil, 567 570 electromagnetic power generation, 568 equations of aerofoil motion, 567 568 solution approach, 568 570 solution results, 570 Admissible displacement field, 15 Aeroacoustics of turbomachines, 33 Aeroelasticity, 28, 37, 333 334 of air structure interactions, 32 33 of turbomachines, 33 Aerofoil flutter system, 566 567, 571f motion equations, 567 568, 571 572 AIAA. See American Institute of Aeronautics and Astronautics (AIAA) Air liquid interface, interactions on, 111 112 boundary conditions on infinity and moving structures, 115 116 conditions of surface tension interactions, 112 115 pressure acceleration model, 112 synthesized formulations, 111 velocity potential displacement model, 111 112 Air liquid shell interaction system, 106, 107f Airplane VLFS water interaction dynamic problems, 293

ALE system. See Arbitrary Lagrange Euler system (ALE system) All-pair search approach, 526 Almansi’s strain, 65 66 American Institute of Aeronautics and Astronautics (AIAA), 32 33 Amplification factor, 436 437, 439 Amplitude decay, 581 582 AMR. See Applied Mechanics Reviews (AMR) Angular momentum conservation, 84 85 ANSYS software, 307 Antinodes, 199 Applied Mechanics Reviews (AMR), 26, 35 Approximate equation by Rayleigh Ritz method, 155 157, 159 160 Approximate solution, 393 394 Approximate theory for long waves, 206 208, 206f Arbitrary integrands material derivatives in arbitrary Lagrange Euler system, 75 76 referential coordinates, 77 spatial coordinates, 76 77 material derivatives in spatial system, 75 76 Arbitrary Lagrange Euler system (ALE system), 57, 75, 82 83, 86, 90 91 arbitrary integrand material derivatives in, 75 76 coordinate, 61 62 description, 423 424, 488 of water motion, 462 frame of reference, 60 62 reference system, 60 updated ALE formulations in CFD, 423 449 consistency, stability, and convergence of numerical schemes, 434 443 discretization techniques, 424 433 in finite difference method and FE method, 443 449 Arbitrary Lagrangian Eulerian and Fluid Structure Interaction: Numerical Simulation, 41 Arbitrary mixed Euler Lagrange description, 38 Arbitrary vectors, proving identity of, 53 Area element transformation, 67 68 Area ratio, 67 68 Artificial compressibility, 452, 463, 522 Artificial heat, 521 Artificial viscosity, 520 521

ASME conferences, 29 30 Atmosphere pressure, 349 Average smoothing function value, 515

B Barotropic fluid, 73 BE. See Boundary element (BE) Beam water interaction, 172 178 beam equations, 172 dry mode functions of beam, 177 178 radiation case, 178 solution, 172 176 system, 178 two-sided water case, 178 water depth less than beam height, 176 BEM. See Boundary element method (BEM) Bernoulli beam, 172 Bernoulli equation, 97 99, 337, 347 Bessel equation, 180 181 Bessel functions, 180 181, 181f Body force, 70 Boundary conditions, 10, 117, 121, 140, 179, 347, 378, 380 381, 459 460 on infinity and moving structures, 115 116 Boundary element (BE), 285 equation for water, 291 293 model of fluid domain, 298 299 Boundary element method (BEM), 38 39, 285 formulation, 285 293 fundamental solution of Laplace equation, 285 286 Boundary integral equation method, 285 Green’s second identity, 287 288 Green’s third identity, 288 mass spring unit landing on mass floating, 290 293 numerical solution, 289 290 Boundary treatment, 524 526 enforced essential boundary conditions, 525 526 fixed solid boundary, 524 525 free surface, 524 Boussinesq equation, 195 Burgers’ equation, 192

C Cartesian system, 538 coordinate system, 294, 322

651

652

Index

Cartesian tensor. See also Stress tensor differentiation of function, 45 e δ identity, 45 index forms of important variables, 47 Kronecker delta, 44 notation, 294 295 permutation symbol, 45 primary identities, 48 quotient rule, 47 rotation of coordinates, 46f summation convention, 44 transformation of coordinates, 46 vector, 43 Cauchy deformation tensor, 65 Cauchy stress, 70 tensor, 411 CEAS. See Council of European Aerospace Society (CEAS) Cell face areas, 433, 433f Cell-centered FVM, 432 Cell-vertex FVM, 432 Central difference approximation, 25 method, 577, 580 581, 583 CFD. See Computational fluid dynamics (CFD) Chinese Society of Theoretical and Applied Mechanics (CSTAM), 39 40 Classic linear beam bending theory, 139 Classical Runge Kutta method, 560 563 Coefficient matrix, 177 178 Coefficient of viscosity, 72 Collocation method, 24 Complementary energy model, 116 120 Complex instant power, 185 186 Complex modulus, 135 Complex variational formulation, 137 138 Compressible fluid, 382 388 Compressible water, 398 Computational fluid dynamics (CFD), 25, 409, 487 updated ALE formulations in, 423 449 Computational Fluid Structure Interaction: Methods and Applications, 41 Computational methods, 33, 38 40, 370 Computer code design, 261 Conditional stability example, 437 Conservation of energy, 379 of energy and equation of energy, 87 92 arbitrary Lagrange Euler system, 90 91 Lagrange system, 89 90 spatial system, 88 89 updated arbitrary Lagrange Euler system, 91 updated Lagrange system, 92 equation of mass, 458 459 equation of momentum, 459 of identity of particles, 379 of mass and equation of continuity, 82 83 arbitrary Lagrange Euler system, 82 83 Lagrange system, 82 spatial system, 82 updated arbitrary Lagrange Euler system, 83 updated Lagrange system, 83

of momentum and equations of motion, 83 87 arbitrary Lagrange Euler system, 86 Lagrange system, 85 spatial system, 84 85 updated arbitrary Lagrange Euler system, 86 updated Lagrange system, 86 87 Conservative discretization, 430 431, 431f Consistence condition, 42 Consistency, 434 of kernel approximation, 497 499 of particle approximation, 501 502 Constitutive equation, 72 73, 117, 121, 378, 459 fluids, 72 73 solids, 72 in tensor form, 53 Continuity density, 517 518 Continuity equation, 379 Continuous derivative function, 410 411 Continuously differentiable function, 373 374 Continuum mechanics, 423 analysis of deformation, 64 70 constitutive equation, 72 73 descriptions of motion of continuum, 57 64 laws of conservation, 73 92 Navier Stokes equations and boundary conditions, 92 101 stress tensor, 70 71 Control cell volumes, 433 Convective velocity. See Relative velocity Convergence, 435 condition, 42 Coordinate interpolations, 418 Coordinate systems, 336, 355 356 Corrective smoothed particle method (CSPM), 527 528 Council of European Aerospace Society (CEAS), 32 33 Coupling equation, 300 Coupling root mean square (CRMS), 551 Courant Friedrichs Lewy condition, 437 Cross-multiplication of vector, 52 53 CSPM. See Corrective smoothed particle method (CSPM) CSTAM. See Chinese Society of Theoretical and Applied Mechanics (CSTAM) Cubic spline function, 506 Cylinder and numerical simulation domain, 477f Cylinder rotating about mass center, 481 482 Cylinder with 2 degrees of freedom of motion, 482 484, 483f, 484f

D d’Alembert principle, 410 411 d’Alembert’s solution, 189 190, 190f Damping, 134 136, 261 complex modulus, 135 factor, 168 radiation and absorption conditions, 136 Rayleigh damping, 135 136

structural, 134 135 viscous, 134 Dam water interactions, 28, 106, 106f Dam water pond excited by earthquake, 168 172 problem and governing equation, 168 169 solution, 169 171 water domain and boundaries, 168 169 Deep water waves, 198 199 Deformation analysis of, 64 70 displacement and strain, 64 69 gradient, 64 65 velocity field and rate of deformation of fluids, 69 70 Degrees of freedom (DOF), 537 538, 585 Delta function property, 493 Denominator layout convention. See Hessian formulation Differential matrix equation for FDM, 448 formulations, 452 Differentiation of function, 45 Diffracted potential of beam-like ship, 360 Diffraction, 333 334 Diffusion error, 440 Dilatational wave in solids, 93 94 Dirac delta function, 492 Direct integration method, 424 425, 577 Discrete equation and constrains, 431 Discretization discretized matrix form in FVM, 443 of kinematic condition on free surface, 465 spatial derivatives, 465 time derivatives, 465 methods and principles, 42 of momentum equation, 463 464 spatial derivatives, 464 time derivatives, 463 464 techniques of CFD finite difference method, 424 430 FVM, 430 433 Dispersive wave, 190 191 Displacement, 10, 64 69, 372, 539 540 boundary condition, 22 23, 395, 420 422 criterion, 578 gradient, 65 66 interpolations, 315 316, 418 of mass, 164 solution of solid mechanics, 92 94 governing equations, 93 linear waves in solids, 93 94 solution equation, 93 Displacement consistency model for solid substructure, 241 246 rule and method for mode reduction, 244 245 reduction depending on vibration shape factor, 245 sets of mode vectors consistent displacement modes between interfaces, 242 243 normalized modes of fixed interface substructure, 243 synthesis of substructure equations, 245 246

Index

Displacement-based element, 413 414 Displacement acceleration pressure form, 129 130 Displacement potential form, 130 132 Displacement pressure displacement pressure velocity potential form, 127 128 model, 233 239 potential forms, 124 127 Displacement strain relationship, 10 Displacement velocity potential FE model, 209, 211 233. See also Mixed finite element method (MFE method) examples, 221 233 FE solution of one-dimensional fluid structure problem, 224 233 mode solution of spring mass water interaction system, 221 224 finite element equations, 213 215 governing equations, 212 natural vibration dynamic response, 219 220 free vibration, 218 219 quadratic eigenvalue problem, 215 216 solution of natural frequencies and modes, 217 218 problem description, 211 variational formulation, 212 Displacement velocity potential form, 120 124. See also Pressure acceleration form governing equations, 120 122 conditions on interaction interfaces, 121 122 fluid subdomains, 121 solid structure, 121 variational formulation, 122 124 Dissipation error, 440 wave, 191 variational formulations for, 133 138 complex variational formulation, 137 138 damping types, 134 136 virtual variational formulations, 136 137 DOF. See Degrees of freedom (DOF) Dry harvest device, 277 278 affected by water buoyancy, 278 Dry mode functions of beam, 177 178 Dynamic equation, 116 117, 121, 377, 379 380, 412, 414, 420 422 derived from functional, 392 393 with implicit time integration, 413 with incremental decompositions, 413, 415 linearization, 413, 415 for numerical simulation, 542 of solid motion, 466 Dynamic response analysis, 261 and energy harvest system, 280 283 of fixed rigid rod to incident wave, 405 406, 405f of FSI system, 219 220 mode summation approach in state space, 219 220 noncoupling mode method, 220

of interaction system, 360 361 of 1D FSI system to pressure wave, 146 150 force vibration, 149 150 governing equations, 146 natural vibration, 147 149 solution approach, 146 147 of Sommerfeld system, 153 154 of structures to explosions in water, 29 of 2D beam-like ship to incident wave, 358 361 diffracted potential of beam-like ship, 360 radiated potential of beam-like ship, 359 360 of 2D beam-water system excited by pressure wave, 265 266 of 2D dam-water system, 264 265 of 2D fluid container, 263 264 Dynamic(s), 34 35 absorber, 232 displacement of center of cylinder, 473f equilibrium equation, 10, 410 411 force coefficient, 473f pressure, 171 172, 349

E Earthquake, dam water pond excited by, 168 172 Earthquake loads, liquid storage tank excited by, 266 267 Eigenvalue equations, 174 176 Elastic mode shape, 539 natural mode, 539 Elastodynamics, 211 Electrical equations, 274 275, 362 Electrical equilibrium principle, 274 275 Electrical mechanical interaction, 361 362 Electric mechanical interaction, 566 Electromagnetic force, 70 Electromagnetic power generation, 568 Endo and Yago’s test, 311 312 Energy conservation, 379 and equation, 87 92 convertor, 362 criterion, 578 flow theory, 214 215 harvest device, 274 275 design, 277 electrical equations, 274 275 mechanical motion equations, 275 harvest system, 280 283 harvested, 573 575 integration of surface tension, 114 115 pressure form, 115 velocity potential form, 114 115 kinetic, 204 mechanical, 204 205 particle approximation, 519 520 potential, 204 ENOC. See European Nonlinear Dynamics Conference (ENOC) Equilibrium of infinitesimal element, 95, 96f

653

Error equation, 435 436 estimation, 506 508 parameter for accuracy estimation, 508 Euler reference system, 370 Eulerian approach, 210 211 Eulerian coordinate system, 430 431 Eulerian description, 409 410 Eulerian system, 541 European Nonlinear Dynamics Conference (ENOC), 35 Explicit nonlinear dynamics equation, 417 External load vector, 419 421

F Far-field boundary, 338 Far-field water pressure, 101 FD. See Finite difference (FD) FDM. See Finite difference method (FDM) FE. See Finite element (FE) FEA. See Finite element analysis (FEA) FE BE method. See Finite element boundary element method (FE BE method) FEM. See Finite element method (FEM) Fifth International Congress on Sound and Vibration (ICSV5), 34 Finite difference (FD), 423 approximation, 3, 9 for first-order derivative, 425 for second-order derivative, 426 FD-type mesh, 432 Finite difference method (FDM), 25 26, 424 430, 443 449, 488 difference formulas with multiple points, 426 generalized numerical conservative laws, 448 449 higher-order derivatives, 428 429 implicit FD formulas and general generation, 429 multidimensional FD formulas, 430 nonuniform meshes, 430 operator methods for FD formulas, 426 427 Finite element (FE), 285, 409, 487 equations, 213 215, 257 258, 577 FE-type mesh, 432 matrices, 413 416 Finite element analysis (FEA), 9, 488, 577 578 Finite element method (FEM), 3, 18 23, 209, 409, 488. See also Mixed finite element boundary element method (Mixed FE BE method); Time integration methods displacement boundary conditions, 22 23 displacement velocity potential, 211 233 element analysis, 18 20 element assembling, 20 22 for linear FSI, 210 211 mesh, 18 mixed, 233 239 of solid substructures, 296 298 finite element equation, 296 297 mode equation, 297 298 SS approaches, 239 283

654

Index

Finite element method (FEM) (Continued) UL formulation in, 410 422 Finite element boundary element method (FE BE method), 334, 358 Finite volume method (FVM), 39 40, 430 433, 443 449, 488 conservative discretization, 430 431 evaluation formulas for, 432 433 fundamental concepts, 431 432 generalized numerical conservative laws in, 443 448 First-order derivative, 425 First-order inertial moment, 538 in material system, 402 Fixed coordinate system, 475 Fixed cylinder, 480 481 Fixed elastic beam, 558 559 Fixed solid boundary, 524 525 Flexible structure subject waves equation, 351 Flexible wedge dropping on water, 551 557, 552f, 552t, 553f beam elastic deformation, 556f beam strains, 557f Floating beam landing beam lands on, 303 311 mass damper spring unit landing on, 302 303 Floating fluid solid interaction interface (Floating FSI interface) interaction on, 109 110 pressure acceleration model, 110 velocity potential displacement model, 110 natural vibration of vertical system with, 158 161 approximate equation by Rayleigh Ritz method, 159 160 approximate solutions, 160 161 governing equations and corresponding functional equation, 158 159 Floating rigid mass vibrations on water, 178 182 governing equations, 179 solution, 180 182 velocity potential and pressure, 182 Flow equations with quasi-linear form, 440 443 stationary shallow-water equations, 442 443 two first-order equations in 2D space, 442 Flow-induced vibrations, 35 36 in pipes, 35 36 of two-dimensional cylinder, 560 565 fluid domain, 563 565 hybrid grid configuration in fluid domain, 562f motion equation of two-dimensional cylinder, 563 nodes in support domain of point, 562f 2D cylinder vibration induced by incoming flow, 563f Flows around bluff body cases to calculating in simulation, 478t comparison of numerical result, 478t

cylinder rotating about mass center, 481 482 cylinder with 2 degrees of freedom of motion, 482 484 fixed cylinder, 480 481 overset grids, 475 477 problem, 477 479 time histories of drag and lift coefficients, 479f Fluid domain, 275 276, 295, 313 314, 362, 369 370, 378 388, 458 459, 546 547, 563 565, 564f. See also Solid domain boundary conditions, 380 381, 459 460, 544 boundary element model, 298 299 conservation of energy, 379 of identity of particles, 379 dynamic equation, 379 380 equation of continuity, 379 Navier Stokes equation, 542 544 nondimensional vibration amplitudes, 565f state equation, 378 vortex structure, 565f Fluid flows assumed irrotational, 387 389 compressible fluid, 387 388 incompressible fluid, 388 389 numerical techniques for, 520 527 artificial heat, 521 artificial viscosity, 520 521 boundary treatment, 524 526 open source SPH solvers, 527 particle interactions, 526 527 pressure calculation for incompressible fluids, 522 523 spurious zero-energy mode, 521 522 time integration, 526 Fluid motion, 572 assumed rotational, 382 387 compressible fluid, 382 386 incompressible fluid, 386 387 Fluid Structure Interaction, 40 41 Fluid(s), 72 73 boundary conditions, 544 deformation rate, 69 70 density, 386 dynamic analysis, 422 functional variation, 141 interface reconstruction methods, 39 pressure on wet interface, 349 sloshing pressure, 210 subdomains, 117, 121 velocity potential interpolation, 316 Fluid body interaction, 477 Fluid cylinder interaction, 484 Fluid fluid model, 39 Fluid mass spring interaction system, 467 469, 467f, 468f, 469f Fluid solid analyses, 39 Fluid solid interaction (FSI), 3, 39, 145, 148 149, 333 334, 369 370, 377f, 389, 409, 566. See also Nonlinear fluid solid interactions (Nonlinear FSIs)

aim and characteristics, 42 43 approaches to deriving numerical equations, 9 26 books on, 40 41 boundary condition intermediate velocity, 545 Laplacian operator compensation near FSI interface, 544 pressure Neumann condition, 544 dynamics, 57 and characteristics, 3 5 feedback loop in flutter, 6f force relationships, 5f historical remarkable events and progress, 28 29 influential review papers, 35 40 interface, 117 118, 121, 546 natural vibration of FSI system, 154 158 problem, 290, 536 575 in engineering, 5 7, 6f, 7f radiation equation, 326 terms and subdisciplines in literatures, 26 28 world-recognized conferences, 29 35 Fluid structure interaction (FSI), 209, 240f conferences by WIT, 30 31 interface, 381, 461 462 conditions, 296 problems, 145 solid/fluid natural frequencies effect on, 238 239 Fluid Structure Interaction Analysis Program (FSIAP), 211, 233 computer code, 261, 262f, 273 274, 279 Flutter, 566 Fluxes through cell faces, 433 Force criterion, 578 Forced one-dimensional gas mass spring dynamic interaction problem, 392 394 approximate solution, 393 394 dynamic equations derived from functional, 392 393 Forced vibration, 149 150, 471, 473 474. See also Natural vibration dynamic moment, 472f dynamic rotation angle, 472f general solution of, 13 14 wave height, 472f Form effective stiffness matrix, 583 Formulated hydrodynamic forces, 333 334 4 X 4 square matrix, 175 Fourier series, 435 436 Fourier transformation method, 101 Fourth International Congress on Sound and Vibration (ICSV4), 34 Frame of reference, 334 336, 458. See also Moving reference frame arbitrary Lagrange Euler, 60 62 fixed rectangular Cartesian, 372 material, 57 59, 458 spatial, 59 Free index, 43 Free surface, 95, 370, 380 381, 451 452, 524

Index

conditions for incompressible potential flows, 99 fluid, 471 474 Green functions, 351 equations for, 351 352 three-dimensional case, 354 two-dimensional case, 352 354 of water, 112 wave consideration, 173 174 Free vibration, 218 219, 470 473, 473f dynamic moment, 471f dynamic rotation angle, 471f general solution of, 13 wave height, 471f Free-body diagrams, 363f Free-space Green’s function, 285 286 Free free beam theoretical mode functions of, 303 306 theoretical natural modes, 360 361 Frequency of encounter of ship, 343 frequency-domain second-order wave force approach, 394 shift, 252 256 algorithm, 261 constant pressure mode with zero frequency, 253 FSI case, 255 256 global eigenvalue problem, 253 SS method with no FSI, 253 254 of wave encounter, 343 FSI. See Fluid solid interaction (FSI); Fluid structure interaction (FSI) FSIAP. See Fluid Structure Interaction Analysis Program (FSIAP) Fundamental solution of Laplace equation, 285 286, 286f Fundamental variational concepts for nonlinear systems, 372 377 Fundamentals of Fluid Solid Interactions: Analytical and Computational Approaches, 41 FVM. See Finite volume method (FVM)

G Galerkin method, 24 Gas liquid interaction interfaces, 118, 122 Gauss’s theorem. See Green theorem Gaussian kernel, 505 Generalized 3D Laplace equation, 195 196 Generalized acoustic radiation problems, 327 332 governing equations, 327 image method, 327 331 quarter radiation, 331 332 Generalized coordinates, 370 equation of structure, 357 Generalized dynamic equation, 577 578 Generalized equation of motion, 350 351 equation of flexible structure subject waves, 351 equation of static equilibrium, 350 equation of steady motion, 350 351

Generalized fluid forces, 349 350 Generalized force, 541 542 Generalized FSIAP, 261 Generalized mass and stiffness, 13 Generalized nonlinear water wave equation, 192 193 Generalized nonlinear wave equation, 193 Generalized numerical conservative laws in FDM differential matrix equation, 448 numerical matrix equation, 448 449 in FVM, 443 448 discretized matrix form, 443 matrix integration form, 443 numerical process, 443 448 Generalized solution procedure, 536 537 Governing equations, 336 339, 377 382, 458 462, 538 545, 538f dynamic equation for numerical simulation, 542 far-field boundary, 338 fluid domain, 378 381, 458 459, 542 544 fluid solid interaction boundary condition, 544 545 fluid structure interaction interface, 461 462 interface, 381 generalized force, 541 542 interaction interface, 338 339 kinetic energy density, 540 541 in moving reference frame, 395 397 ship structure, 336 solid domain, 377 378, 460 461, 539 540 strain field and strain energy density, 540 variational conditions at initial time and final time, 381 382 water, 337 339 of water domain, 451 452 Gravitation force, 70 Gravitational acceleration, 334 335 Gravitational force, 339, 541 Green deformation tensor, 65 Green functions, 298 299 free surface, 351 354 of Helmholtz equation, 320 321 Green identity, 321 322 Green third identity, 288, 355 357 Green’s second identity, 287 288 and solution of ship excited by waves, 354 357 Green theorem, 73 74, 74f, 222 223, 287, 315 316, 340, 393, 430 431, 497 Green’s strain tensor, 65 66 Grid-based method limitations for violent flows, 488 Ground motion, 256 dynamic response to, 231 Ground vertical motion, 259 260 Group velocity, 202 203, 202f

H Half-infinite water domain, 334 Hamilton principle in analytical dynamics, 104

655

Hamilton’s principle, 140 141 Hankel function, 101 Harmonic force, 178 Harmonic wave solution, 190 Harvested energy, 573 575 “Hat” Ritz function, 26, 26f Helmholtz equation, Green functions of, 320 321 Hessian formulation, 49 Higher-order derivatives, 428 429 Higher-order splines, 506 Hilber collocation method, 577, 580, 583 Hilber-α method, 579 580, 582 Horizontal motion of beam section center due to bending, 139 140 Hull slamming, 38 Hybrid displacement model of solid substructure, 246 249 rule and method of mode reduction, 247 248 sets of mode vector, 246 247 synthesis of substructure equations, 248 249 Hybrid grid around aerofoil, 572f Hydroelasticity, 37 38 theory of ship water interactions fundamentals for ship water interactions, 334 341 incident waves, 341 343 linear hydroelasticity theory, 343 367 of water structure interactions, 31 32

I IACM. See International Association of Computational Mechanics (IACM) ICSV. See International Congress on Sound and Vibration (ICSV) ICSV4. See Fourth International Congress on Sound and Vibration (ICSV4) ICSV5. See Fifth International Congress on Sound and Vibration (ICSV5) ICTAM. See International Congress of Theoretical and Applied Mechanics (ICTAM) ICVE 2015. See International Conference of Vibration Engineering 2015 (ICVE 2015) IFASD. See International Forum on Aeroelasticity and Structural Dynamics (IFASD) IIAV. See International Institute of Acoustics and Vibration (IIAV) Image method, 327 331 radiation pressure for quarter water with free surface wave, 330 331 with no free surface wave, 330 relationships valid for symmetrical image system, 329 Implicit finite difference formulas and general generation, 429 Implicit Lagrangian method, 447 Implicit nonlinear dynamics equation, 417 Implicit time integration, dynamic equation with, 413, 415

656

Index

Incident water waves equation, 341 342 Incident wave dynamic response of fixed rigid rod to, 405 406 equation of incident water waves, 341 342 frequency of wave encounter, 343 linear plane gravity waves, 342 two-dimensional beam-like ship to, 358 361 Incompressible fluids, 97, 386 389 Incompressible potential flows, 98 Incompressible SPH (ISPH), 490 491 Incompressible water, 399 Increment strain, 419, 421 Incremental decompositions, 413 414 Index forms of important variables, 47, 48t Index notations, proving identities using, 54 55 Infinite water domain, 334 Infinity boundaries, conditions on, 118, 122 Influential review papers, 35 40 Inhomogeneous Helmholtz equation, 321 Instantaneous energy flow equilibrium equations, 573 Institute of Sound and Vibration Research, 34 Integral representation of function, 492 496 consistency of kernel approximation, 497 499 first derivative, 494 495 rules for nth-order accuracy, 496 497 second derivative, 495 496 Integrated coupling equations, 545 547 fluid domain, 546 547 fluid solid interaction interface, 546 initial conditions, 546 547 solid domain, 546 Integrated coupling fields, solution of, 8 9 Integrated dynamic equations, 452 453 Integration approximation, 507 508, 509f, 510f, 511f Integration particles for involved particle, 526 Interaction interface, 338 339, 344 fluid pressure on wet interface, 349 kinematic condition, 348 Interdisciplinary research, 566 Intermediate velocity of FSI boundary, 545 Internal energy of incompressible fluid, 386 387 per unit mass of fluid, 378 Internal force vector, 21 International Association of Computational Mechanics (IACM), 33 International Conference of Vibration Engineering 2015 (ICVE 2015), 39 40 International Congress of Theoretical and Applied Mechanics (ICTAM), 31 International Congress on Sound and Vibration (ICSV), 34 International Forum on Aeroelasticity and Structural Dynamics (IFASD), 32 33 International Institute of Acoustics and Vibration (IIAV), 34 International Society of Offshore and Polar Engineers (ISOPE), 32

International Union for Theoretical and Applied Mechanics conferences (IUTAM conferences), 31 Interpolation functions, 18 19 Irrotational water flow, 396 ISOPE. See International Society of Offshore and Polar Engineers (ISOPE) ISPH. See Incompressible SPH (ISPH) Iteration criterions, 578 equation, 422

J Jacobi elliptic functions, 193 expansion method, 193 194 Jacobi elliptic sine function, 193 Jacobian formulation, 49 Jacobian matrix, 50 Johnson’s quadratic function, 505 Jump condition and equation, 81 82

K KdV equation. See Korteweg de Vries equation (KdV equation) Kernel approximation, 492 consistency of, 497 499 Kernel functions, 492, 505 506, 530 Kernel interpolant, MLS representing, 531 534 example, 533 534 local standard least square approximation, 531 532 moving process for global approximation, 532 533 Kinematic condition, 348 on free surface, 465 constraint method, 523 variable, 412 Kinetic energy, 411 density, 140, 540 541 of wave, 204 Korteweg de Vries equation (KdV equation), 189, 192 solution of, 194 195 Kronecker delta, 44, 52

L Lagrange system, 82, 85, 89 90 Lagrangian description, 409 410 Lagrangian displacement element model, 210 Lagrangian multiplier approach, 127, 129, 246 Lagrangian polynomial, 535 Lame constants, 72 Landing beam lands on floating beam, 303 311, 305f geometrical and physical parameters, 306 nondimensional form of results, 306 311 theoretical mode functions of free free beam, 303 306 Laplace electromagnetic theorem, 274 275

Laplace equation, fundamental solution of, 285 286, 286f Laplace law, 95 Laplace transformation method, 232 233 Laplacian approximation, 530 531 Laplacian equation, 179 Laplacian operator compensation near FSI interface, 544, 545f Laws of conservation, 73 92 arbitrary Lagrange Euler system, 79 integration form, 79 differential form, 79 of energy and equation of energy, 87 92 general forms, 77 80 Green theorem, 73 74 jump condition and equation, 81 82 Lagrange system, 78 79 differential form, 79 integration form, 78 of mass and equation of continuity, 82 83 material derivatives of arbitrary integrands in arbitrary Lagrange Euler system, 76 77 of arbitrary integrands in spatial system, 75 76 of volume integral with mass density, 74 75 of momentum and equations of motion, 83 87 updated arbitrary Lagrange Euler system, 79 80 differential form, 80 integration form, 79 80 updated Lagrange system, 80 differential form, 80 integration form, 80 Lax Wendroff theorem, 431 Least squares method, 24 Legendre transformation, 127 Line element transformation, 66 Linear elasticity theory, 209 Linear equation, 190 191 Linear fluid solid interaction, 369 370 problems 1D problems, 146 161 2D problems, 161 178 3D problems, 178 188 systems, 145 Linear FSI problems, 209 FE models for, 210 211 Linear gravity wave condition, 179 Linear hydroelasticity theory, 343 367 equations in modal space, 344 351 linearized governing equations, 343 344 numerical solutions, 351 358 Linear incompressible fluid, 361 Linear models, 389 Linear plane gravity waves, 342 Linear water waves approximate theory for long waves, 206 208 plane water wave, 197 206 three-dimensional water waves, 195 197 Linear water structure interactions, 334

Index

Linear wave in fluids, 99 101 radiation conditions, 100 101 wave equations, 99 100 in solids, 93 94 dilatational wave in solids, 93 94 shear wave in solids, 94 theory, 199 Linearization of dynamic equation, 413, 415 Linearized Bernoulli equation, 195 196 Linearized governing equations, 343. See also Governing equations interaction interface, 344 ship structure, 343 344 water, 344 Linearized method, 217 Liquefied natural gas (LNG), 37, 211 Liquid Sloshing Dynamics: Theory and Applications, 41 Liquid shell interaction system, 6, 7f LNG. See Liquefied natural gas (LNG) Local standard least square approximation, 531 532 Local variation, 374 375 of integral over moving volume in space, 376 377 Longitudinal strain, 419, 421

M Manmade elements, 18 Mass conservation, 455 Mass matrix, 419 421 Mass motion equation, 179 Mass-spring system coupled to 1D infinite fluid domain, 150 153, 153f complex eigenvalues of problem, 151 153 governing equations, 150 151 unit landing on mass floating, 290 293 boundary element equation for water, 291 293 equations of motion, 290 291 initial conditions, 290 Mass damper spring unit landing on floating beam, 302 303 Material coordinate, 61 62 Material derivatives, 59 of arbitrary integrands in arbitrary Lagrange Euler system, 76 77 in spatial system, 75 76 of volume integral with mass density, 74 75 Material frame of reference, 57 59 Material variation, 374 375 Matrix by scalar, 50 Matrix calculus, 48 51 derivatives with matrices, 50 51 matrix by matrix, 51 matrix by scalar, 50 scalar by matrix, 51 derivatives with vectors, 50 scalar by vector, 50

vector by scalar, 50 vector by vector, 50 identities, 51 scalar by vector, 51, 52t vector by vector, 49t, 51 matrix derivatives, 49, 49t Matrix equation of displacement-based element, 413 414 dynamic equation with implicit time integration, 413, 415 finite element matrices, 413 416 for radiated velocity potential, 357 Matrix integration form, 443 Matrix scaling process, 210 211 Mean square root (MSR), 508 Mechanical energy conservation, 443 of wave, 204 205 Mechanical motion equations, 275, 362 Mechanics, 3, 4f Mesh and control volume, 432 distortion, 487 points, 60 velocity, 60 Meshfree Galerkin methods (MGMs), 489 490, 531 536 MLS representing kernel interpolant, 531 534 MLS reproducing kernels applications, 535 536 orthogonal basis for local approximations, 535 Shepard interpolant, 534 Meshfree particle methods (MPMs), 487 key ideas, 488 489 MFE method. See Mixed finite element method (MFE method) MFE SPM. See Mixed finite element smoothed particle method (MFE SPM) MGMs. See Meshfree Galerkin methods (MGMs) Mirror image method for acoustic radiations of underwater structures, 320 332 acoustic radiation in infinite fluid domain, 322 326 generalized acoustic radiation problems, 327 332 Green functions of Helmholtz equation, 320 321 Green identity, 321 322 Mixed coordinate, 60 Mixed energy models, 124 127 Mixed FE BE method. See Mixed finite element boundary element method (Mixed FE BE method) Mixed FE computational fluid dynamics method differential matrix formulations, 452 equations, 456 457 governing equations of water domain, 451 452 integrated dynamic equations, 452 453

657

nonlinear rod water interaction system, 451f numerical equations, 453 456 solutions for nonlinear FSI problems, 449 457 direct or simultaneous integration, 449 450 partitioned iteration, 450 UL formulation in FEM, 410 422 updated ALE formulations in CFD, 423 449 Mixed FE FD numerical method, 424 Mixed FE FDM, numerical examples by, 457 485 ALE description of water motion, 462 flows around bluff body, 475 485 fluid mass spring interaction system, 467 469 nonlinear rigid body water fluid solid interaction systems, 458 462 numerical formulations, 463 466 prescribed motion of rigid cylinder floating in water, 474 475 spring-supported fluid rigid-dam interaction system, 469 471 two-dimensional rigid body floating in freesurface fluid, 471 474 Mixed finite element method (MFE method), 209, 211, 233 239, 487. See also Displacement velocity potential FE model general description of problem, 233 solid/fluid natural frequencies effect on FSI process, 238 239 symmetric matrix equations and approximations, 236 238 variational formulation, 233 234 Mixed finite element boundary element method (Mixed FE BE method), 293, 296 312, 351. See also Finite element method (FEM) BEM of fluid domain, 298 299 for dynamic response of structures excited by incident water waves, 312 320 general description of problem and governing equations, 313 314 mixed FE BE equation, 315 320 solution approach, 314 variational formulation, 314 315 finite element model of solid substructures, 296 298 general description of problem, 294 295 governing equations fluid domain, 295 fluid structure interaction interface conditions, 296 initial conditions, 296 solid substructures, 295 296 mixed FE BE equation, 299 300 numerical solution procedure characteristics of coupling equation, 300 landing beam lands on floating beam, 303 311 mass damper spring unit landing on floating beam, 302 303

658

Index

Mixed finite element boundary element method (Mixed FE BE method) (Continued) nonsymmetrical behavior, 300 numerical integration, 301 simulation for car running test, 311 312 solution strategies, 301 time integration scheme, 301 302 for VLFS subjected to airplane landing impacts, 293 Mixed finite element smoothed particle method (MFE SPM), 536 575, 537f for FSI problems, 536 575 application examples, 547 575 flexible wedge dropping on water, 551 557 flow-induced vibration of two-dimensional cylinder, 560 565 generalized solution procedure, 536 537 modeling of FSI involving large rigid motions, 537 547 rigid wedge dropping on water, 547 551 2D water dam breaking impact on rigid/ flexible beams, 557 559 wave/wind energy harvesting systems, 566 575 Mixed model of substructure subdomain, 251 252 set of mode vectors, 251 252 synthesis of SS equations, 252 MLS method. See Moving least square method (MLS method) Modal damping parameters, 135 Modal space, equations in generalized equation of motion, 350 351 generalized fluid forces, 349 350 interaction interface, 348 349 mode summation method for ship motions, 344 346 water domain, 346 347 Mode equation of ship motions, 346 Mode summation method, 177 for ship motions, 344 346 mode equation of ship motions, 346 natural frequencies and modes with orthogonality, 344 346 in state space, 219 220 in state space, 232 233 Modified first-kind Bessel function, 181 Moment matrix, 532 533 Momentum conservation, 455 equation discretization, 463 464 governing fluid flows, 522 523 particle approximation, 518 519 Mooring systems, 38 Morison’s approximate equation, 153 Motion equations, 290 291, 336 for rigid ship, 399 403 of two-dimensional cylinder, 563 Motion of continuum, 57 64, 372 arbitrary Lagrange Euler frame of reference, 60 62

material frame of reference, 57 59 spatial frame of reference, 59 updated arbitrary Lagrange Euler system, 63 64 updated Lagrangian system, 63 Motion of cylinder, 477 Moving coordinate system, 399 Moving least square method (MLS method), 531 representing kernel interpolant, 531 534 reproducing kernels applications, 535 536 Moving particle semiimplicit method (MPS method), 528 531 approximation of Laplacian, 530 531 derivative approximation, 530 function approximation, 528 529 modeling of incompressibility, 529 530 Moving reference frame. See also Frame of reference governing equations in, 395 397 ship structure, 395 water, 396 397 water ship interface, 397 variational formulations in, 397 399 compressible water, 398 incompressible water, 399 MPMs. See Meshfree particle methods (MPMs) MPS method. See Moving particle semiimplicit method (MPS method) MSR. See Mean square root (MSR) Multidimensional finite difference formulas, 430 Multiple points, difference formulas with, 426

N Natural characteristics of system, 11 Natural elements, 18 Natural frequencies and modes of FSI system, 217 218 linearized method, 217 orthogonality with matrix, 218 symmetric formulation, 218 with orthogonality, 344 346 Natural vibration, 10 11, 147 149. See also Forced vibration of dynamic system, 101 of fluid solid interaction system, 154 158 approximate equation by Rayleigh Ritz method, 155 157 approximate solutions, 157 158 governing equations and corresponding functional, 155 of FSI system, 279 280 of two-dimensional fluid container, 263 264 of vertical system with floating FSI interface, 158 161 Navier’s equation, 183 Navier Stokes equations (NS equations), 191, 542 544 and boundary conditions, 92 101 Bernoulli equation and potential flows, 97 99

displacement solution of solid mechanics, 92 94 linear waves in fluids, 99 101 velocity pressure solution equations of fluid mechanics, 94 97 solvers, 487 SPH formulation for, 516 520 particle approximation of density, 516 518 particle approximation of energy, 519 520 particle approximation of momentum, 518 519 system, 430 431 Nearest neighboring particles, 526 Net harvested averaged power, 575 New quartic function, 506 Newmark method, 303, 577, 579, 582 Newton Raphson iteration, 416 No free surface waves, 173 Nodal point stress vector, 419, 421 Nodes, 199 Noise reductions, damping material layer effect on, 186 188 power reduction factor, 187 188 pressure reduction factor, 186 187 relative power amplitude, 186 relative pressure amplitude, 186 Noncoupling free vibration of system, 585 mode method, 220 Nondimensional forms, 459 Nonfloating fluid solid interaction interface, interactions on, 109 pressure acceleration model, 109 velocity potential displacement model, 109 Nonlinear coupled systems, 377 Nonlinear dynamical behavior, 370 371 Nonlinear dynamical FSI systems, 389 Nonlinear dynamical systems, 369 370 variational principles for, 370 371 Nonlinear elastic dynamics, 371 Nonlinear elastic ship water interactions, 369 370 variational principles for, 394 395 governing equations in moving reference frame, 395 397 offshore and hydroelastic examples, 405 407 rigid ship dynamics, 399 405 variational formulations in moving reference frame, 397 399 Nonlinear elastic structure, 377 Nonlinear equations solution, 416 417 Newton Raphson iteration, 416 solution of explicit nonlinear dynamics equation, 417 of implicit nonlinear dynamics equation, 417 Nonlinear fluid solid interactions (Nonlinear FSIs), 487. See also Fluid solid interaction (FSI)

Index

fundamental of SPH, 492 531 variational concepts for nonlinear systems, 372 377 governing equations, 377 382 history and developments with applications, 489 492 key ideas of meshfree particle methods, 488 489 limitations of grid-based methods for violent flows, 488 MFE SPM for FSI problems, 536 575 problems, 424 430 simple examples of applications, 390 394 1D water mass spring interaction problem, 390 392 forced 1D gas mass spring dynamic interaction problem, 392 394 variational principles, 382 389 for nonlinear dynamical systems, 370 371 for nonlinear elastic ship water interactions, 394 407 Nonlinear FSIs. See Nonlinear fluid solid interactions (Nonlinear FSIs) Nonlinear mathematical model, 466 Nonlinear pendulum, 361 Nonlinear rigid body water FSI systems governing equations, 458 462 reference frames, 458 Nonlinear rigid ship water interaction system, 402 Nonlinear ship water interactions, 537 Nonlinear solid mechanics, 424 Nonlinear system concepts local variation of integral over moving volume in space, 376 377 and material variation, 374 375 motion of continuum, 372 time derivative of integral over moving volume in space, 373 374 translation and transmission velocities of curved surface, 372 373 Nonlinear wave equations, 189, 191 195 analytical solutions, 193 195 Jacobi elliptic function expansion method, 193 194 Jacobi elliptic functions, 193 solution of Boussinesq equation, 195 solution of KdV equation, 194 195 Burgers’ equation, 192 and characteristic solution, 191 192 generalized nonlinear water wave equation, 192 193 KdV equation, 192 Nonsingular fluid matrix, 236 237 Nonsingular solid mass matrix, 237 Nonsingular solid stiffness matrix, 237 238 Nonsymmetrical behavior, 300 Nonuniform meshes, 430, 430f Nonzero vectors, 55 Normalized modes of fixed interface substructure, 243

NS equations. See Navier Stokes equations (NS equations) nth-order accuracy, rules for, 496 497 Numerator layout convention. See Jacobian formulation Numerical equations, 48, 453 456, 454f approaches to deriving, 9 26 analytical solution, 10 14 FEM, 18 23 finite difference method, 25 26 problem and governing equations, 10 uniform rod exciting by dynamic forces, 10f variational formulations and Rayleigh Ritz method, 15 17 weighted residual methods, 23 24 Numerical errors, spectral analysis of, 439 440 Numerical formulations, 463 466 artificial compressibility, 463 discretization of kinematic condition on free surface, 465 discretization of momentum equation, 463 464 dynamic equation of solid motion, 466 numerical solution, 466 Numerical implementation process, 536 Numerical integration schemes, 579 581 central difference method, 580 581 Hilber collocation method, 580 Hilber-α method, 579 580 Newmark method, 579 Wilson-θ method, 579 Numerical matrix equation for FDM, 448 449 Numerical methods solving finite element dynamic equations, 577 direct integration methods, 577 fundamental iteration formulations, 577 578 implementation process of time integration methods, 582 584 input and output files, 620 626 numerical integration schemes, 579 581 program of time element methods, 589 617 stability and accuracy, 581 582 time element program, 584 585 user manual, 617 620 Numerical process, 443 448 for fluid flows, 520 527 Numerical simulation, 547 dynamic equation for, 542 Numerical solutions, 351 358, 466 examples, 358 367 dynamic response of 2D beam-like ship to incident wave, 358 361 wave energy harvesting device, 361 367 free surface Green functions, 351 354 Green identity and solution of ship excited by waves, 354 357 solution of ship water interaction in hydroelasticity, 357 358 Numerical tested functions, 506 508

659

O ODEs. See Ordinary differential equations (ODEs) Office of Naval Research (ONR), 34 Offshore and hydroelastic examples, 405 407 dynamic response of fixed rigid rod to incident wave, 405 406 two-dimensional elastic beam traveling in waves, 406 407 Offshore engineering, 172 One dimensional FSI problem (1D FSI problem), 145 dynamic response to pressure wave, 146 150 of Sommerfeld system, 153 154 mass-spring system coupled to 1D infinite fluid domain, 150 153 natural vibration with free surface wave, 154 158 of vertical system with floating FSI interface, 158 161 in SS methods, 256 261 finite element equation, 257 258 governing equations, 256 257 ground vertical motion, 259 260 natural vibration, 258 259 pressure excitation on free surface, 260 261 variational formulation, 257 One-dimension (1D) compressible gas structure dynamic interaction system, 392, 392f example, 418 422 coordinate interpolations, 418 displacement interpolations, 418 iteration equation, 422 total Lagrangian formulations, 419 421 UL formulations, 421 422 FE solution of 1D fluid structure problem, 224 233 complex mode method, 228 229 complex response function, 231 232 dynamic response to ground motion, 231 finite element equation, 226 228 governing equations, 224 225 linearized solution, 229 231 mode summation method in state space, 232 233 natural vibration, 228 variational formulation, 225 226 motion, 189 pressure wave in water, 261 263 Sommerfeld systems, 145 water mass spring interaction problem, 390 392, 390f ONR. See Office of Naval Research (ONR) Open source SPH solvers, 527 Operator methods for finite difference formulas, 426 427 Ordinary differential equations (ODEs), 489 Original statically equilibrium configuration, 409 410 Orthogonal basis for local approximations, 535

660

Index

Orthogonal relationships of natural modes, 11 13 Orthogonality with matrix, 218 natural frequencies and modes with, 344 346 relationship, 165 166 Oscillating aerofoil power extraction device, 567f Overset grids, 475 477

P Partial differential equations, 371 Particle approximation, 499 502, 507 513, 512f, 513f, 514f consistency, 501 502 of density, 516 518 continuity density, 517 518 summation density, 516 517 of energy, 519 520 of function, 499 501 of momentum, 518 519 techniques in deriving SPH formulations, 501 Particle inconsistency, 501 502 Particle interactions, 526 527 Particle motions, orbit of for stationary wave, 200 for traveling waves, 201 Particle numbers, 508 density, 528 529 Particle volume, 507 Partitioned iteration, 39 40, 450, 450f Pendulum wave energy harvester water interaction system, 361f Perfect fluids, 96 Period elongation, 581 582 Periodical orbit of nonlinear system, 566 Permutation symbol, 45, 53 Phase velocity, 190, 202 203, 202f Physical mechanism of energy collector, 364 Physical vibration, 143 Piola stress tensor, 377 378 Piola Kirchhoff stress, 71 Pipe vibrations inducing by flowing fluids, 28 29 Pipes conveying fluid description and assumptions of problem, 139 140 geometrical relationships, 139 140 natural vibrations and approximate frequency, 142 143 variational formulations for, 138 143 variational stationary conditions for governing equations, 141 142 governing equation of system, 141 142 variation of fluid functional, 141 variation of solid functional, 141 Plane water wave. See also Three-dimensional water waves deep and shallow water waves, 198 199 governing equations, 197 group velocity and phase velocity, 202 203

solution by variable separation method, 197 198 stationary wave or standing wave, 199 200 traveling waves, 200 202 wave energies, 203 205 transmission, 205 wave resistance, 206 Poisson equation of pressure, 529 530 Poisson’s ratio, 72, 264 265 of wood, 267 268 Polynomial form, 503 504 Potential energy density, 140 of elements, 20 21 of wave, 204 Potential flows, 97 99 Power flow theory, 185 Prescribed motion of rigid cylinder floating in water, 474 475 cylinder heaving in viscous fluid, 475f velocity-vector plots and free-surface disturbance, 476f Pressure calculation for incompressible fluids, 522 523 artificial compressibility, 522 kinematic constraint method, 523 projection method, 522 523 contour, 552, 554f, 559f, 561f equilibrium model of fluid domain, 249 251 sets of mode vector, 250 synthesis of subdomain equations, 250 251 form, 111 impulse form, 132 133 Neumann condition, 544 reduction factor, 186 187 in traveling waves, 202 of water relative, 164 Pressure acceleration form, 116 120. See also Displacement velocity potential form governing equations, 116 118 conditions on interaction interfaces, 117 118 fluid subdomains, 117 solid structure, 116 117 variational formulation, 118 120 Pressure acceleration model, 109 110, 112 Program of time element methods (PTEM), 577, 586f Projection method, 522 523

Q Quadratic eigenvalue problem, 215 216 pure imaginary eigenvalues, 216 Quadratic smoothing function, 503 505 Quarter radiation, 331 332 case for free surface wave, 332 case for no free surface wave, 331 Quartic spline function, 506 Quasi-linear form, 440 443 Quasicoupling approximation method, 8

Quintic spline function, 506 Quotient rule, 47

R Radial basis function (RBF), 560 563 Radiated potential of beam-like ship, 359 360 Radiated velocity potential, 357 Radiation conditions, 100 101, 136, 338 involving free surface and pressure waves, 100 101 Sommerfeld condition, 100 effect, 418 of mixed compressive and gravity waves, 164 168 functions Xn, 166 167 governing equations, 164 165 orthogonality relationship, 165 166 solution, 165 167 potentials, 333 334 pressure for quarter water with free surface wave, 330 331 with no free surface wave, 330 RAeS. See Royal Aeronautical Society (RAeS) RASD. See Recent advances in structural dynamics (RASD) Rayleigh damping, 135 136 matrix, 300 Rayleigh Ritz method, 3, 15 17, 142, 145 approximate equation by, 155 157, 159 160 functions, 535 RBF. See Radial basis function (RBF) Recent advances in structural dynamics (RASD), 34 Reference frames. See Frame of reference Referential coordinate system, 60 62 Relative power amplitude, 186 Relative pressure amplitude, 186 Relative velocity, 61 Resonance, 231, 566 state, 367 Resonance scattering technique (RST), 36 Resultant external force vector of ship, 402 Resultant external moment tensor about mass center, 402 Resultant water force moment tensor about mass center, 402 vector on ship, 402 Reynolds number, 97 Right-half water domain, 453, 453f Rigid ship dynamics, 399 405 motion equations for rigid ship, 399 403 sea keeping problem, 404 405 steady-state problem, 403 404 Rigid ship travelling in calm water, 403 404 in waves, 404 405 Rigid wedge dropping on water, 547 551, 548f, 549f time history of dropping velocities, 550f, 551f time history of wedge velocities, 550f

Index

Rigid/flexible beams, 557 559 Ritz functions, 17 18 Root mean square (RMS), 551 Rotation angle of beam cross section, 139 tensor, 65 66 and time derivatives, 460 Royal Aeronautical Society (RAeS), 32 33 RST. See Resonance scattering technique (RST)

S Same fluid, 111 112 Scalar by matrix, 51 Scalar by vector, 50 identities, 51, 52t Sea keeping problem, 404 405 Second Piola Kirchhoff stress, 412 Second-order derivative, 426 Second-order inertial moment in material system, 402 Semiactive aerofoil, 570 575 equations of aerofoil motions, 571 572 fluid motion, 572 harvested energy, 573 575 simulation result, 573t, 574f solution, 572 573 Semicoupling approach, 8 SFI. See Solid fluid interactions (SFI) Shallow water, 182 waves, 198 199 Shape function, 497 498 Shear wave in solids, 94 Shepard interpolant, 534 Ship excited by waves, 354 357 Ship motion mode summation method for, 344 346 theory, 333 334 Ship structure, 336, 343 344, 395 Ship water interactions, 29, 334 341, 335f, 336f, 371 dynamics, 334 equations for static equilibrium state, 339 340 for steady motion, 340 341 frames of reference, 334 336 governing equations, 336 339 solution in hydroelasticity FE BE method, 358 generalized coordinate equation of structure, 357 matrix equation for radiated velocity potential, 357 solution process of ship water interaction, 357 358 system, 106, 106f Simultaneous integration, 424 425 Singularity distribution methods, 334 Slender Structure and Axial/Cross Flows, 41 Slender structure fluid interactions, 138 Sloshing, 29 dynamics, 36 37 modes of 2D rectangular water container, 161 164

governing equations, 162 surface tension effect, 163 164 variable separation method for solution, 162 163 sloshing-induced slamming, 37 Small elastic deformation, 537 547 governing equations, 538 545 integrated coupling equations, 545 547 numerical simulation process, 547 Small rotation angle, 364 365 Smoothed particle hydrodynamics (SPH), 490 531 application on fluid solid interaction problems, 491 492 on incompressible fluid, 490 491 on multiphase flow, 491 on solids, 491 approximation rules, 516 construction of smoothing functions, 503 516 formulation for Navier Stokes equations, 516 520 improving methods based on CSPM, 527 528 MPS method, 528 531 interpolations, 492 499 numerical techniques for fluid flows, 520 527 particle approximation, 499 502 Smoothed particle method (SPM), 39 40, 487 Smoothing function, 492 493 construction of, 503 516 Kernel functions, 505 506 numerical test results, 508 513 numerical tested functions and error estimation, 506 508 polynomial form, 503 504 symmetrization of particle interaction, 514 515 kernel function, 492 Smoothing length, 508, 514 515 Solid domain, 377 378, 389, 460 461, 539 540, 546. See also Fluid domain boundary conditions, 378 constitutive equations, 378 displacement and velocity fields, 539 540 dynamic equation, 377 strain-displacement and velocitydisplacement relations, 378 Solid dynamics total Lagrangian formulation in, 412 414 UL formulation dynamic equation, 414 with incremental decompositions, 414 415 linearization of dynamic equation, 415 matrix equation of displacement-based element, 415 416 Solid(s), 72, 183 body, 547 boundary, 94 95 continuum system, 536 dam water interaction system, 3 4, 4f

661

dynamic equation of solid motion, 466 functional variation, 141 substructures, 295 296 finite element model of, 296 298 Solid fluid interactions (SFI), 27 Solid solid model, 39 Solution approaches to FSI problems, 8 9. See also Fluid solid interaction (FSI) approximate solution with no FSI, 8 quasicoupling approximation method, 8 solution of integrated coupling fields, 8 9 Sommerfeld condition, 100 Sommerfeld radiation condition, 321, 351 Sommerfeld system, dynamic response of, 153 154 Sound speed, 522 Spatial derivatives, 464 465 Spatial frame of reference, 59 Spatial system, 74 75, 78, 82, 84 85, 88 89 arbitrary integrand material derivatives in, 75 76 differential form, 78 integration form, 78 Spectral analysis of numerical errors, 439 440 Spectral radius, 581 SPH. See Smoothed particle hydrodynamics (SPH) SPM. See Smoothed particle method (SPM) Spring-supported fluid rigid-dam interaction system, 469 471, 470f forced vibration, 471 free vibration, 470 Spring-supported rigid beam, 557 558, 558f, 560f Spring mass water interaction system, mode solution of, 221 224 approximate solution by mode reduction of water domain, 223 224 governing equations, 221 222 variational formulation, 222 223 Spurious zero-energy mode, 521 522 SS methods. See Substructure subdomain methods (SS methods) Stability, 435, 581 582 condition, 42 von Neumann method for stability analysis, 435 440 Standing wave. See Stationary wave Static equilibrium equation, 350 state, 470 equations for, 339 340 Static variable, 412 Static water pressure, 349 Stationary shallow-water equations, 442 443 Stationary wave, 199 200. See also Traveling waves orbit of particle motions, 200 velocity field of water waves, 199 200 wave profile on free surface, 199 Steady motion equation, 350 351 equations for, 340 341 pressure, 349

662

Index

Steady-state case, 333 334 problem, 403 404 Stiffness matrices, 419, 421 Stokes fluid, 96 Stone-Weierstrass theorem, 531 Strain, 64 69 energy density, 540 field, 540 tensors, 64 66 Strain-displacement, 116, 121 matrices, 419, 421 relation, 295, 378, 382 Strain-rate velocity relation, 459 Stress principle, 70f virtual work, 411 412 Stress tensor, 70 71. See also Cartesian tensor Cauchy’s stress, 70 Piola Kirchhoff stress, 71 Stress strain relationship, 10, 295 Stretch ratio, 66 67 Structural damping, 134 135 Structure acoustic volume interaction system, 267 271 Subdomain method, 24 Subspace iteration method, 261 Substructure subdomain methods (SS methods), 209, 239 283 application examples, 261 283 dynamic response of 2D beam-water system, 265 266 dynamic response of 2D dam-water system, 264 265 liquid storage tank excited by earthquake loads, 266 267 natural vibration and dynamic response of 2D fluid container, 263 264 one-dimensional pressure wave in water, 261 263 structure acoustic volume interaction system, 267 271 wave energy harvest device water interaction system, 271 283 computer code design, 261 displacement consistency model for solid substructure, 241 246 hybrid displacement model of solid substructure, 246 249 mixed model, 251 252 one-dimensional fluid structure interaction problem, 256 261 pressure equilibrium model of fluid domain, 249 251 techniques for FSI systems frequency shift, 252 256 ground motion, 256 variational formulations, 241 Summation convention, 44, 52 Summation density approach, 501, 516 517 Super Gaussian, 505 Surface force, 70 Surface tension effect on 2D rectangular water container, 163 164

Surface tension interactions on air liquid interface, 113 114 conditions of, 112 115 energy integration of surface tension, 114 115 synthesized equations on surface tension interface, 112 113 pressure form, 113 surface membrane acceleration form, 113 velocity potential form, 113 Symmetric FE method, 210 211 Symmetric matrix approach, 210 211 equations and approximations, 236 238 nonsingular fluid matrix, 236 237 nonsingular solid mass matrix, 237 nonsingular solid stiffness matrix, 237 238 Symmetrization methods, 210 211 of particle interaction, 514 515

T Tacoma suspension bridge, 28 Tangent vector of vector, 50 Taylor series expansion, 497 498, 527 528 Tensile instability, 491 Tensor, 46 47 equation in coordinate form, 54 operation, 53 Texas Institute for Computational Mechanics (TICOM), 36 Thermodynamic quantities, 378 Third mapping, 60 Three field variational formulations, 127 130 displacement acceleration pressure form, 129 130 displacement pressure velocity potential form, 127 128 Three-dimension (3D), 333 334 case, 354 FSI problem, 145 floating rigid mass vibrations on water, 178 182 water spherical shell damping layer interaction system, 182 188 hydroelasticity of ships, 38 method, 32 open space, 490 space, 61 62 Three-dimensional water waves. See also Plane water wave governing equations, 195 196 initial conditions, 196 197 TICOM. See Texas Institute for Computational Mechanics (TICOM) Time averaged power, 185 Time derivatives, 463 465 of integral over moving volume in space, 373 374 Time element methods. See Direct integration method

Time element program, 584 585 characteristics, 584 585 displacements, 587t, 588t examples, 585 588 flowchart of program, 585 nonlinear system of 100 degrees of freedom, 588f program, 589 617 Time function, 143 Time integration methods, 301 302, 526 calculations, 583 584 form effective stiffness matrix, 583 implementation process of, 582 584 initial calculations, 582 583 triangular decomposition of matrix, 583 Time-averaged collected power, 365 366 Total Lagrangian formulations, 419 421 displacement boundary condition, 420 421 dynamic equation, 420 increment strain, 419 longitudinal strain, 419 mass matrix and external load vector, 419 420 in solid dynamics dynamic equation, 412 incremental decompositions, 413 with incremental decompositions, 413 linearization of dynamic equation, 413 matrix equation of displacement-based element, 413 414 stiffness matrices and nodal point stress vector, 419 strain displacement matrices, 419 Total mass of ship, 402 Traction, 10, 71 boundary condition, 336 Transformation of coordinates, 46 Translation motion equation of mass center, 460 velocity of curved surface, 372 373 Transmission velocity of curved surface, 372 373 Traveling waves, 200 202. See also Stationary wave orbit of particle motion, 201 pressure, 202 velocity field of water waves, 201 wave profile on free surface, 201 Triangular decomposition of matrix, 583 Truncation error, 424 425 Turbomachines, aeroacoustics and aeroelasticity of, 33 Turbulent flows in curved pipes, 35 Two first-order equations in two-dimensional space, 442 Two-dimension (2D), 333 334 beam-like ship to incident wave, 358 361, 358f case, 352 354 Cartesian coordinate system, 303 cylinder, 560 565 dynamic response of 2D beam-water system, 265 266 of 2D dam-water system, 264 265

Index

elastic beam traveling in waves, 406 407, 406f first-order equations in2D space, 442 flows, 38 fluid container, 263 264 FSI problem, 145 beam water interaction, 172 178 dam water pond excited by earthquake, 168 172 radiations of mixed compressive and gravity waves, 164 168 sloshing modes of 2D rectangular water container, 161 164 linear hydroelasticity theory, 333 334 rigid body floating in free-surface fluid, 471 474 forced vibration, 473 474 free vibration, 471 473 water dam breaking impact on rigid/flexible beams, 557 559 fixed elastic beam, 558 559 spring-supported rigid beam, 557 558 Two-point tensor, 64 65 Two-step approximate method, 3 4 Type I virtual particles, 524 525 Type II virtual particles, 524 525

U Unconditionally unstable example, 437 Undisturbed condition, 338 Unit normal vector of deformed area element, 68 Unity function, 492 Unsteady-state case, 333 334 Updated arbitrary Lagrange Euler system, 63 64, 83, 86, 91 Updated Lagrange (UL) system, 63, 80, 83, 86 87, 92, 410 formulation in FEM, 410 422 expression of stress virtual work, 411 412 formulation in solid dynamics, 412 416 one-dimensional example, 418 422 principle of virtual work and dynamic equilibrium equations, 410 411 solution of nonlinear equations, 416 417

V Variable separation method for solution, 162 163 Variational formulations, 15 17, 257 for dissipative systems, 133 138 for pipes conveying fluid, 138 143 Variational function of system, 140 141 Variational principles, 145, 160, 382 389. See also Fluid solid interaction (FSI) complementary energy model, 116 120 fluid domain, 387 388 fluid flow assumed irrotational, 387 389 fluid motion assumed rotational, 382 387 fluid solid interaction, 389

formulations with displacement potential or pressure impulse, 130 133 historic background, 104 105 of linear fluid solid interaction systems, 103 linear models, 389 mixed energy models, 124 127 for nonlinear dynamical systems, 370 371 potential energy model, 120 124 problems and interaction conditions, 105 116 geometric and dynamic conditions on material interfaces, 107 109 interactions on air liquid interface, 111 112 interactions on floating fluid solid interaction interface, 109 110 interactions on nonfloating fluid solid interaction interface, 109 solid domain, 389 three field variational formulations, 127 130 Vector, 43 by scalar, 50 by vector, 50 identities, 49t, 51 Vector identity (e δ identity), 45, 47 Velocity, 372 contour, 552, 555f field of water waves in stationary wave, 199 200 in traveling waves, 201 fields, 69 70, 539 540 Velocity potential, 346 347 velocity potential displacement model, 109 112 Velocity-displacement relations, 378, 382 Velocity potential form, 111 Velocity pressure solution equations of fluid mechanics, 94 97 cases of fluids, 96 97 governing equations, 94 95 Navier Stokes equation, 96 Vertical component of fluid speed and pressure, 140 Very large floating structure (VLFS), 6, 7f, 30 31, 285 mixed FE BEM for, 293 312 Vibrations, 34 35 Virtual particles, 524 525 Virtual variational formulations, 136 137 Virtual work principle, 410 411, 411f Viscous damping, 134 VIV. See Vortex-induced vibration (VIV) VLFS. See Very large floating structure (VLFS) Volume element transformation, 68 69, 69f Volume integral material derivatives with mass density, 74 75 Arbitrary Lagrange Euler system, 75 spatial system, 74 75 Volume ratio, 68 69, 69f Von Karman’s method, 29 von Neumann method for stability analysis, 435 440

663

amplification factor, 436 437 conditional stability example, 437 error equation and Fourier series, 435 436 formation, 438 439 spectral analysis of numerical errors, 439 440 unconditionally unstable example, 437 Vortex-induced vibration (VIV), 36

W Water, 184, 337 339, 344, 396 397 domain Bernoulli equation, 347 and boundaries, 168 169 boundary conditions, 347 governing equations of, 451 452 potential of velocity, 346 347 floating rigid mass vibrations on, 178 182 motion ALE description of, 462 equation, 179 pressure, 454 Water ship interface, 397 Water solid interactions, 37 38 Water spherical shell damping layer interaction system, 182 188 effect on noise reduction, 186 188 general solution, 184 185 power flow, 185 mathematical model, 183 184 solids, 183 water, 184 Water structure interactions, 450 451 hydroelasticity of, 31 32 Wave energy, 203 205 harvesting device, 361 367 electrical equations, 362 fluid domain, 362 mechanical motion equations, 362 solution discussion, 362 364 kinetic energy, 204 mechanical energy, 204 205 potential energy, 204 transmission, 205 Wave energy harvest device water interaction system, 271 283, 274f dry harvest device, 277 278 affected by water buoyancy, 278 dynamic response and energy harvest system, 280 283 energy harvest device, 274 275 design, 277 finite element equation, 276 277 fluid domain, 275 276 governing equations, 274 natural vibration of FSI system, 279 280 variational formulation, 276 wave energy harvest device, 274f Wave profile on free surface in group velocity and phase velocity, 202 in stationary wave, 199 in traveling waves, 201

664

Index

Wave Propagation in Infinite Domains: With Applications to Structure Interaction, 41 Wave/wind energy harvesting systems active aerofoil, 567 570 fundamental principle for, 566 567 aerofoil flutter system, 566 567 interdisciplinary research, 566 periodical orbit of nonlinear system, 566 resonance, 566 semiactive aerofoil, 570 575 Wavefront surface and unit normal vector, 441f Wave(s) encounter frequency, 343 equations, 99 100, 184

number, 321 preliminaries of d’Alembert’s solution, 189 190, 190f dispersive wave, 190 191 dissipation wave, 191 linear water waves, 195 208 nonlinear waves, 191 195 propagation, 189 resistance, 206 wave-induced seafloor dynamics, 37 WCCM. See World Congress on Computational Mechanics (WCCM) Weakly compressible SPH (WCSPH), 490 491 Weierstrass elliptic function method, 193

Weighted residual methods, 23 24 Weighted square residual, 532 Wessex Institute of Technology (WIT), 30 31 Wet interface, fluid pressure on, 349 Wilson-θ method, 261, 579, 582 WIT. See Wessex Institute of Technology (WIT) World Congress on Computational Mechanics (WCCM), 33 World-recognized conferences, 29 35

Y Young’s modulus, 234, 264 265, 267 268, 279, 558