89,99 €
Mehr erfahren.
- Herausgeber: John Wiley & Sons
- Kategorie: Fachliteratur
- Sprache: Englisch
When using numerical simulation to make a decision, how can its reliability be determined? What are the common pitfalls and mistakes when assessing the trustworthiness of computed information, and how can they be avoided? Whenever numerical simulation is employed in connection with engineering decision-making, there is an implied expectation of reliability: one cannot base decisions on computed information without believing that information is reliable enough to support those decisions. Using mathematical models to show the reliability of computer-generated information is an essential part of any modelling effort. Giving users of finite element analysis (FEA) software an introduction to verification and validation procedures, this book thoroughly covers the fundamentals of assuring reliability in numerical simulation. The renowned authors systematically guide readers through the basic theory and algorithmic structure of the finite element method, using helpful examples and exercises throughout. * Delivers the tools needed to have a working knowledge of the finite element method * Illustrates the concepts and procedures of verification and validation * Explains the process of conceptualization supported by virtual experimentation * Describes the convergence characteristics of the h-, p- and hp-methods * Covers the hierarchic view of mathematical models and finite element spaces * Uses examples and exercises which illustrate the techniques and procedures of quality assurance * Ideal for mechanical and structural engineering students, practicing engineers and applied mathematicians * Includes parameter-controlled examples of solved problems in a companion website (href="http://www.wiley.com/go/szabo">www.wiley.com/go/szabo)
Sie lesen das E-Book in den Legimi-Apps auf:
Seitenzahl: 545
Veröffentlichungsjahr: 2011
Ähnliche
Contents
Cover Page
Series
Title Page
Copyright
Dedication
About the Authors
Series Preface
Preface
1: Introduction
1.1 Numerical simulation
1.2 Why is numerical accuracy important?
1.3 Chapter summary
2: An Outline of the Finite Element Method
2.1 Mathematical models in one dimension
2.2 Approximate solution
2.3 Generalized formulation in one dimension
2.4 Finite Element Approximations
2.5 FEM in one dimension
2.6 Properties of the generalized formulation
2.7 Error estimation based on extrapolation
2.8 Extraction methods
2.9 Laboratory exercises
2.10 Chapter summary
3: Formulation of Mathematical Models
3.1 Notation
3.2 Heat conduction
3.3 The scalar elliptic boundary value problem
3.4 Linear elasticity
3.5 Incompressible elastic materials
3.6 Stokes' flow
3.7 The hierarchic view of mathematical models
3.8 Chapter summary
4: Generalized Formulations
4.1 The scalar elliptic problem
4.2 The principle of virtual work
4.3 Elastostatic problems
4.4 Elastodynamic models
4.5 Incompressible materials
4.6 Chapter summary
5: Finite Element Spaces
5.1 Standard Elements in Two Dimensions
5.2 Standard Polynomial Spaces
5.3 Shape Functions
5.4 Mapping Functions in Two Dimensions
5.5 Elements in Three Dimensions
5.6 Integration and Differentiation
5.7 Stiffness Matrices and Load Vectors
5.8 Chapter Summary
6: Regularity and Rates of Convergence
6.1 Regularity
6.2 Classification
6.3 The Neighborhood of Singular Points
6.4 Rates of Convergence
6.5 Chapter Summary
7: Computation and Verification of Data
7.1 Computation of the Solution and Its First Derivatives
7.2 Nodal Forces
7.3 Verification of Computed Data
7.4 Flux and Stress Intensity Factors
7.5 Chapter Summary
8: What should be Computed and Why?
8.1 Basic Assumptions
8.2 Conceptualization: Drivers of Damage Accumulation
8.3 Classical Models of Metal Fatigue
8.4 Linear Elastic Fracture Mechanics
8.5 On the Existence of a Critical Distance
8.6 Driving Forces for Damage Accumulation
8.7 Cycle Counting
8.8 Validation
8.9 Chapter Summary
9: Beams, Plates and Shells
9.1 Beams
9.2 Plates
9.3 Shells
9.4 The Oak Ridge Experiments
9.5 Chapter Summary
10: Nonlinear Models
10.1 Heat Conduction
10.2 Solid Mechanics
10.3 Chapter Summary
Appendix A: Definitions
A.1 Norms and Seminorms
A.2 Normed Linear Spaces
A.3 Linear Functionals
A.4 Bilinear Forms
A.5 Convergence
A.6 Legendre Polynomials
A.7 Analytic Functions
A.8 The Schwarz Inequality for Integrals
Appendix B: Numerical Quadrature
B.1 Gaussian Quadrature
B.2 Gauss–Lobatto Quadrature
Appendix C: Properties of the Stress Tensor
C.1 The Traction Vector
C.2 Principal Stresses
C.3 Transformation of Vectors
C.4 Transformation of Stresses
Appendix D: Computation of Stress Intensity Factors
D.1 The Contour Integral Method
D.2 The Energy Release Rate
Appendix E: Saint-Venant's Principle
E.1 Green's Function for the Laplace Equation
E.2 Model Problem
Appendix F: Solutions for Selected Exercises
Bibliography
Index
Wiley Series in Computational Mechanics
Series Advisors:
René de Borst
Perumal Nithiarasu
Tayfun E. Tezduyar
Genki Yagawa
Tarek Zohdi
This edition first published 2011 © 2011 John Wiley & Sons, Ltd
Registered officeJohn Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex, PO19 8SQ, United Kingdom
For details of our global editorial offices, for customer services and for information about how to apply for permission to reuse the copyright material in this book please see our website at www.wiley.com.
The right of the authors to be identified as the authors of this work has been asserted in accordance with the Copyright, Designs and Patents Act 1988.
All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, except as permitted by the UK Copyright, Designs and Patents Act 1988, without the prior permission of the publisher.
Wiley also publishes its books in a variety of electronic formats. Some content that appears in print may not be available in electronic books.
Designations used by companies to distinguish their products are often claimed as trademarks. All brand names and product names used in this book are trade names, service marks, trademarks or registered trademarks of their respective owners. The publisher is not associated with any product or vendor mentioned in this book. 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® 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 MATLABa® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular pedagogical approach or particular use of the MATLAB® software.
Library of Congress Cataloguing-in-Publication Data
Szabó, B. A. (Barna Aladar), 1935- Introduction to finite element analysis : formulation, verification, and validation / Barna Szabó, Ivo Babuška. p. cm. Includes bibliographical references and index. ISBN 978-0-470-97728-6 (hardback) 1. Finite element method. I. Babuška, Ivo. II. Title. TA347.F5S979 2011 620.001′51825–dc22 2010051233
A catalogue record for this book is available from the British Library.
Print ISBN: 9780470977286 ePDF ISBN: 9781119993827 oBook ISBN: 9781119993834 ePub ISBN: 9781119993483 Mobi ISBN: 9781119993490
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
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®. 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 on the p-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.
Series Preface
The series on Computational Mechanics will be a conveniently identifiable set of books covering interrelated subjects that have been receiving much attention in recent years and need to have a place in senior undergraduate and graduate school curricula, and in engineering practice. The subjects will cover applications and methods categories. They will range from biomechanics to fluid-structure interactions to multiscale mechanics and from computational geometry to meshfree techniques to parallel and iterative computing methods. Application areas will be across the board in a wide range of industries, including civil, mechanical, aerospace, automotive, environmental and biomedical engineering. Practicing engineers, researchers and software developers at universities, industry and government laboratories, and graduate students will find this book series to be an indispensible source for new engineering approaches, interdisciplinary research, and a comprehensive learning experience in computational mechanics.
This book, written by two well-recognized, leading experts on finite element analysis, gives an introduction to finite element analysis with an emphasis on validation – the process to ascertain that the mathematical/numerical model meets acceptance criteria – and verification – the process for acceptability of the approximate solution and computed data. The systematic treatment of formulation, verification and validation procedures is a distinguishing feature of this book and sets it apart from other texts on finite elements. It encapsulates contemporary research on proper model selection and control of modelling errors. Another unique feature of the book is that with a minimum of mathematical requisites it bridges the gap between engineering and mathematically-oriented introductory text books into finite elements.
Preface
Increasingly, engineering decisions are based on computed information with the expectation that the computed information will provide a reliable quantitative estimate of some attributes of a physical system or process. The question of how much reliance on computed information can be justified is being asked with increasing frequency and urgency. Assurance of the reliability of computed information has two key aspects: (a) selection of a suitable mathematical model and (b) approximation of the solution of the corresponding mathematical problem. The process by which it is ascertained that a mathematical model meets necessary criteria for acceptance (i.e., it is not unsuitable for purposes of analysis) is called validation. The process by which it is ascertained that the approximate solution, as well as the data computed from the approximate solution, meet necessary conditions for acceptance, given the goals of computation, is called verification. This book addresses the problems of verification and validation.
Obtaining approximate solutions for mathematical models with guaranteed accuracy is one of the principal goals of research in finite element analysis. An important result obtained in the mid-1980s was that exponential rates of convergence can be achieved through proper design of the finite element mesh and proper assignment of polynomial degrees for a large and important class of problems that includes elasticity, heat conduction and similar problems. This made it feasible to estimate and control the errors of discretization for many practical problems.
At present the problems of proper model selection and control of modeling errors are at the forefront of research. The concepts of hierarchic models and modeling strategies have been developed. Progress in this area makes many important practical applications possible.
The distinguishing feature of this book is that it presents a systematic treatment of formulation, verification and validation procedures, illustrated by examples. We believe that users of finite element analysis (FEA) software products must have a basic understanding of how mathematical models are constructed; what are the essential assumptions incorporated in a mathematical model; what is the algorithmic structure of the finite element method; how the discretization parameters affect the accuracy of the finite element solution; how the accuracy of the computed data can be assessed; and how to avoid common pitfalls and mistakes. Our primary objective in assembling the material presented in this book is to provide a basic working knowledge of the finite element method. A link to the student edition of a professional FEA software product called StressCheck® is provided in the companion website (www.wiley.com/go/szabo) to enable readers to perform computational experiments.1 Another important objective of this book is to prepare readers to follow and understand new developments in the field of FEA through continued self-study.
Engineering students typically take only one course in FEA, consisting of approximately 15 weeks of instruction (45 lecture hours). We have organized the material in this book so as to make efficient use of the available time. The book is written in such a way that the prerequisites are minimal. Junior standing in engineering with some background in potential flow and strength of materials are sufficient. For this reason the mathematical content is focused on the introduction of the essential concepts and terminology necessary for understanding applications of FEA in elasticity and heat conduction. Some key theorems are proven in a simple setting.
We would like to thank Dr. Norman F. Knight, Jr. and Dr. Sebastian Nervi for reviewing and commenting on the manuscript.
Barna Szabó Washington University in St. Louis, USA
Ivo Babuška The University of Texas at Austin, USA
1 StressCheck® is a trademark of Engineering Software Research and Development, Inc., St. Louis, Missouri, USA.
