Introduction to Finite Element Analysis

Transcription

1 WILEY SERIES IN COMPUTATIONAL MECHANICS Introduction to Finite Element Analysis Formulation, Verification and Validation Barna Szabó and Ivo Babuška

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3 Introduction to Finite Element Analysis

4 WILEY SERIES IN COMPUTATIONAL MECHANICS Series Advisors: René de Borst Perumal Nithiarasu Tayfun E. Tezduyar Genki Yagawa Tarek Zohdi

5 Introduction to Finite Element Analysis Formulation, Verification and Validation Barna Szabó Washington University in St. Louis, USA Ivo Babuška The University of Texas at Austin, USA A John Wiley and Sons, Ltd., Publication

6 This edition first published 2011 c 2011 John Wiley & Sons, Ltd Registered office John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at The right of the authors to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books. Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. This publication is designed to provide accurate and authoritative information in regard to the subject matter covered. It is sold on the understanding that the publisher is not engaged in rendering professional services. If professional advice or other expert assistance is required, the services of a competent professional should be sought. MATLAB R is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not warrant the accuracy of the text or exercises in this book. This book s use or discussion of MATLAB R software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB R software. Library of Congress Cataloguing-in-Publication Data Szabó, B. A. (Barna Aladar), Introduction to finite element analysis : formulation, verification, and validation / Barna Szabó, Ivo Babuška. p. cm. Includes bibliographical references and index. ISBN (hardback) 1. Finite element method. I. Babuška, Ivo. II. Title. TA347.F5S dc A catalogue record for this book is available from the British Library. Print ISBN: epdf ISBN: obook ISBN: epub ISBN: Mobi ISBN: Set in 10/12 Times by Aptara Inc., New Delhi, India.

7 This book is dedicated to our teachers and students. If people do not believe that mathematics is simple, it is only because they do not realize how complicated life is. John von Neumann

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9 Contents About the Authors Series Preface Preface xiii xv xvii 1 Introduction Numerical simulation Conceptualization Validation Discretization Verification Decision-making Why is numerical accuracy important? Application of design rules Formulation of design rules Chapter summary 14 2 An outline of the finite element method Mathematical models in one dimension The elastic bar Conceptualization Validation The scalar elliptic boundary value problem in one dimension Approximate solution Basis functions Generalized formulation in one dimension Essential boundary conditions Neumann boundary conditions Robin boundary conditions Finite element approximations Error measures and norms The error of approximation in the energy norm 43

10 viii CONTENTS 2.5 FEM in one dimension The standard element The standard polynomial space Finite element spaces Computation of the coefficient matrices Computation of the right hand side vector Assembly Treatment of the essential boundary conditions Solution Post-solution operations Properties of the generalized formulation Uniqueness Potential energy Error in the energy norm Continuity Convergence in the energy norm Error estimation based on extrapolation The root-mean-square measure of stress Extraction methods Laboratory exercises Chapter summary 77 3 Formulation of mathematical models Notation Heat conduction The differential equation Boundary and initial conditions Symmetry, antisymmetry and periodicity Dimensional reduction The scalar elliptic boundary value problem Linear elasticity The Navier equations Boundary and initial conditions Symmetry, antisymmetry and periodicity Dimensional reduction Incompressible elastic materials Stokes flow The hierarchic view of mathematical models Chapter summary Generalized formulations The scalar elliptic problem Continuity Existence Approximation by the finite element method The principle of virtual work Elastostatic problems 117

11 CONTENTS ix Uniqueness The principle of minimum potential energy Elastodynamic models Undamped free vibration Incompressible materials The saddle point problem Poisson s ratio locking Solvability Chapter summary Finite element spaces Standard elements in two dimensions Standard polynomial spaces Trunk spaces Product spaces Shape functions Lagrange shape functions Hierarchic shape functions Mapping functions in two dimensions Isoparametric mapping Mapping by the blending function method Mapping of high-order elements Rigid body rotations Elements in three dimensions Integration and differentiation Volume and area integrals Surface and contour integrals Differentiation Stiffness matrices and load vectors Stiffness matrices Load vectors Chapter summary Regularity and rates of convergence Regularity Classification The neighborhood of singular points The Laplace equation The Navier equations Material interfaces Forcing functions acting on boundaries Strong and weak singular points Rates of convergence The choice of finite element spaces Uses of a priori information A posteriori error estimation in the energy norm Adaptive and feedback methods Chapter summary 212

12 x CONTENTS 7 Computation and verification of data Computation of the solution and its first derivatives Nodal forces Nodal forces in the h-version Nodal forces in the p-version Nodal forces and stress resultants Verification of computed data Flux and stress intensity factors The Laplace equation Planar elasticity Chapter summary What should be computed and why? Basic assumptions Conceptualization: drivers of damage accumulation Classical models of metal fatigue Models of damage accumulation Notch sensitivity The theory of critical distances Linear elastic fracture mechanics On the existence of a critical distance Driving forces for damage accumulation Cycle counting Validation Chapter summary Beams, plates and shells Beams The Timoshenko beam The Bernoulli Euler beam Plates The Reissner Mindlin plate The Kirchhoff plate Enforcement of continuity: the HCT element Shells Hierarchic thin-solid models The Oak Ridge experiments Description Conceptualization Verification Validation: comparison of predicted and observed data Discussion Chapter summary Nonlinear models Heat conduction Radiation Nonlinear material properties 298

13 CONTENTS xi 10.2 Solid mechanics Large strain and rotation Structural stability and stress stiffening Plasticity Mechanical contact Chapter summary 313 A Definitions 315 A.1 Norms and seminorms 315 A.2 Normed linear spaces 316 A.3 Linear functionals 316 A.4 Bilinear forms 316 A.5 Convergence 317 A.6 Legendre polynomials 317 A.7 Analytic functions 318 A.7.1 Analytic functions in R A.7.2 Analytic curves in R A.8 The Schwarz inequality for integrals 319 B Numerical quadrature 321 B.1 Gaussian quadrature 322 B.2 Gauss Lobatto quadrature 323 C Properties of the stress tensor 325 C.1 The traction vector 325 C.2 Principal stresses 326 C.3 Transformation of vectors 327 C.4 Transformation of stresses 328 D Computation of stress intensity factors 331 D.1 The contour integral method 331 D.2 The energy release rate 333 D.2.1 Symmetric (Mode I) loading 333 D.2.2 Antisymmetric (Mode II) loading 334 D.2.3 Combined (Mode I and Mode II) loading 335 D.2.4 Computation by the stiffness derivative method 335 E Saint-Venant s principle 337 E.1 Green s function for the Laplace equation 337 E.2 Model problem 338 F Solutions for selected exercises 345 Bibliography 353 Index 359

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15 About the Authors Barna Szabó is co-founder and president of Engineering Software Research and Development, Inc. (ESRD), the company that produces the professional finite element analysis software StressCheck R. Prior to his retirement from the School of Engineering and Applied Science of Washington University in 2006 he served as the Albert P. and Blanche Y. Greensfelder Professor of Mechanics. His primary research interest is assurance of quality and reliability in the numerical simulation of structural and mechanical systems by the finite element method. He has published over 150 papers in refereed technical journals, several of them in collaboration with Professor Ivo Babuška, with whom he also published a book on finite element analysis (John Wiley & Sons, Inc., 1991). He is a founding member and Fellow of the US Association for Computational Mechanics. Among his honors are election to the Hungarian Academy of Sciences as External Member and an honorary doctorate. Ivo Babuška s research has been concerned mainly with the reliability of computational analysis of mathematical problems and their applications, especially by the finite element method. He was the first to address a posteriori error estimation and adaptivity in finite element analysis. His research papers on these subjects published in the 1970s have been widely cited. His joint work with Barna Szabó onthep-version of the finite element method established the theoretical foundations and the algorithmic structure for this method. His recent work has been concerned with the mathematical formulation and treatment of uncertainties which are present in every mathematical model. In recognition of his numerous important contributions, Professor Babuška received may honors, which include honorary doctorates, medals and prizes and election to prestigious academies.