The boundary element method. Volume 1, Applications in thermo-fluids and acoustics PDF
Preview The boundary element method. Volume 1, Applications in thermo-fluids and acoustics
Cover Page i The Boundary Element Method Page ii To our wives: Shayla Ali and Lalitha Rajakumar and our children: Aleef & Teeasha Ali, and Vinod & Anita Rajakumar Page iii The Boundary Element Method Applications in Sound and Vibration Ashraf Ali Engineering Solution and Support (ESAS), Bellevue, Washington (Formerly with Ansys, Inc., Canonsburg, Pennsylvania) Charles Rajakumar Ansys, Inc., Canonsburg, Pennsylvania A.A.BALKEMA PUBLISHERS LISSE/ABINGDON/EXTON (PA)/TOKYO Page iv Library of Congress CataloginginPublication Data A Catalogue record for this book is available from the Library of Congress Cover design: Miranda Bourgonjen Copyright © 2004 Taylor & Francis Group plc, London, UK All rights reserved. No part of this publication or the information contained herein may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, by photocopying, recording or otherwise, without written prior permission from the publishers. Although all care is taken to ensure the integrity and quality of this publication and the information herein, no responsibility is assumed by the publishers nor the author for any damage to property or persons as a result of operation or use of this publication and/or the information contained herein. Published by: A.A.Balkema Publishers (Leiden, The Netherlands), a member of Taylor & Francis Group plc. http://balkema.tandf.co.uk and www.tandf.co.uk This edition published in the Taylor & Francis eLibrary, 2005. To purchase your own copy of this or any of Taylor & Francis or Routledge’s collection of thousands of eBooks please go to www.eBookstore.tandf.co.uk. ISBN 020302446X Master ebook ISBN ISBN 90 5809 657 2 (Print Edition) Page v Contents Preface vii Acknowledgements ix Abbreviations xi 1 Introduction 1 1.1 Why the boundary element method? 1 1.2 Typical applications of the boundary element method 2 1.3 Emergence of the boundary element method 3 1.4 History of boundary element eigenformulations 6 1.5 Organization of the book 9 2 Boundary Element Method Fundamentals 11 2.1 Introduction 11 2.2 Direct method: weighted residuals 12 2.3 Examples 19 2.4 Direct method: Green’s integral theorem 21 2.5 Indirect method 23 2.6 Body forces 26 3 Isoparametric Boundary Elements 31 3.1 Introduction 31 3.2 Twodimensional linear boundary elements 31 3.3 Higherorder elements in 2D 34 3.4 Boundary elements in 3D 36 3.5 Examples 42 4 Anisotropy, Axisymmetry and Zoning 49 4.1 Introduction 49 4.2 Anisotropic materials 49 4.3 Axisymmetric problems 51 4.4 Inhomogeneous regions and zoning 54 5 TimeHarmonic Analysis in Acoustics and Elasticity 57 5.1 Introduction 57 5.2 Acoustics 57 5.3 Elasticity 62 6 Dynamic Analysis: Acoustics and Elasticity 65 6.1 Introduction 65 6.2 Eigenvalue problem in acoustics 71 6.3 Eigenvalue problem in elasticity 72 6.4 Characteristic equation for eigenvalues 72 Page vi 7 Basics of Algebraic Eigenvalue Problem Formulation 77 7.1 Introduction 77 7.2 Development of BE algebraic eigenvalue problem 77 7.3 Formulation of Internal Cell Method 78 7.4 Example of internal cell method: rectangular plate vibration 80 8 Algebraic Eigenvalue Problem in Boundary Elements 87 8.1 Introduction 87 8.2 Eigenproblem using dual reciprocity method in acoustics 87 8.3 Eigenproblem using particular integral method in elasticity 96 9 Advanced Concepts in Boundary Element Algebraic Eigenproblem 107 9.1 Introduction 107 9.2 Algebraic eigenvalue formulation using fictitious function method 108 9.3 Example problems using fictitious function method 111 9.4 Effect of internal collocation points on eigensolutions 112 9.5 Polynomialbased particular integral method 116 9.6 Multiple reciprocity method (MRM) 121 9.7 Series expansion methods (SEM) with matrix augmentation 127 10 Acoustic FluidStructure Interaction Problems 129 10.1 Introduction 129 10.2 Boundary elementfinite element coupled eigenanalysis of fluidstructure system 130 10.3 Acoustic eigenproblem for enclosures with dissipative boundaries 137 10.4 Examples of acoustic eigenproblem with sound absorption 141 11 Solution Methods of Eigenvalue Problems 149 11.1 Introduction 149 11.2 Lanczosbased subspace approach 149 11.3 Lanczos recursion method 150 11.4 Example problems 157 11.5 Summary statements on the nonsymmetric Lanczos eigensolver 160 11.6 Damped system eigenvalue problem solution 161 11.7 Lanczos twosided recursion for the quadratic eigenvalue problem 162 11.8 Summary statements on eigenvalue computation algorithms 170 12 Discussion and Future Research 171 12.1 Discussion on boundary element eigenvalue methodologies 171 12.2 Comparison of eigenanalysis using BEM and FEM 172 12.3 Topics not covered in the book 173 12.4 Future research on BEM eigenanalysis 174 References 177 Index 187 Page vii Preface The boundary element method is a powerful discretization tool in computational mechanics. However, the eigenvalue analysis procedures within the boundary element discretization process are still in a developing stage. To our knowledge, this is the firstever book dedicated entirely to the subject of boundary element eigenvalue formulations. All the techniques of boundary element eigenvalue analysis currently available in the literature are reviewed and presented in the book. For each technique, a detailed theoretical formulation is presented, followed by numerical illustrations. The advantages and disadvantages of each method in terms of computational efficiencies, generalities, and formulation difficulties are also presented. The book includes detailed formulations on linear and quadratic eigensolvers for unsymmetric matrices since boundary element matrices are naturally unsymmetric. The book also sheds light on the ongoing debate on the choice of technique, the relative merits of eigenanalyses based on the boundary element and finite element methods, the unresolved issues that require immediate attention and the future direction of research in this area. The modefrequency analyses of vibrating structures and the computation of resonant frequencies of acoustical cavities are now routinely performed in the industry. The eigenanalysis based on the boundary element method holds promise of becoming a userfriendly and popular procedure with practicing engineers simply because here it can avoid the tedious and timeconsuming process of creating an adequate mesh for their models. Some applications include: Elasticity Area: 1. Machines (automobiles, aircraft, etc.); 2. Machine parts such as connectors, shafts, gears, fasteners such as screws, pins, etc; 3. Other equipment that is subjected to vibrations during normal operation; 4. Structures (bridges, buildings, etc.). Acoustics Area: 1. Acoustic enclosures such as auditoriums, theaters, passengercar and train cabins, etc.; 2. HiFi sound equipment such as loudspeakers; 3. Fluidfilled structures such as oil tankers; 4. Noise control of structures such as automobile mufflers; aircraft fuselages, rooms housing vibrating machines, etc. Furthermore, eigenvalue analysis forms the basis for subsequent modebased dynamic analyses, such as mode superposition transient analysis, spectrum analysis, and random vibration analysis. Page viii Before the advent of practical numerical methods like the finite element method, engineers conducted experiments on prototypes for determining natural frequencies. Starting in the 1970s, computer programs based on the finite element method were available. Although the finite element method is a versatile computational technique, it requires a much longer datapreparation time than the boundary element method. Specifically, engineers are forced to spend a significant amount of time in generating an adequate mesh for the model problem. Despite the introduction of a few automatic meshgeneration algorithms in the commercial finite element programs, engineers still continue to struggle in creating meshes of acceptable quality. The boundary element method, on the other hand, is a boundary technique where only the boundary of the domain is required to be meshed, thereby simplifying data preparation efforts significantly. Among other benefits, the overall physical time spent by engineers to perform the analysis is reduced significantly, and the analysis process becomes more userfriendly. The book presents the eigenvalue analysis techniques that use the boundary element method. The boundary element method does not easily lend itself to eigenvalue formulations, especially algebraic eigenvalue formulations. Consequently, publications on boundary element algebraic eigenvalue formulations did not come out until the mid1980s, although the boundary element method has been around since the late 1960s. However, nonalgebraic boundary element eigenvalue analysis, which is not a practical analysis technique, appeared in the literature of the mid1970s. For historical reasons, the book presents some materials related to the nonalgebraic boundary element eigenvalue analysis techniques. Three general purpose boundary element computer programs (GPBEST, BEASY, and SYSNOISE) offer boundary element eigenvalue analysis capabilities. This book will hopefully satisfy the needs of engineers to acquire a detailed knowledge on the subject. The capabilities of the commercial programs, such as those mentioned, may be enhanced through the implementation of some of the different methods of performing boundary element eigenvalue analysis presented in the book. This book should also encourage the development of new and more powerful computer programs on boundary element eigenvalue analysis. The book can be used in the graduate classes on “computational mechanics” and “boundary element methods”. The researchers in universities and industries, practicing engineers, mathematicians, computer scientists, physicists, chemists and chemical engineers, and researchers in biomedical fields can also use it as a reference. Ashraf Ali Seattle, Washington Charles Rajakumar Pittsburgh, Pennsylvania Page ix Acknowledgements We are indebted to Ansys, Inc., Canonsburg, Pennsylvania for getting us interested in the subject of boundary element method. We are also grateful to Sidney Solomon of The Solomon Press of New York in giving us encouragement and valuable suggestions in the preparation and completion of the book.
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