The Best Approximation Method in Computational Mechanics

258 Pages·1993·8.31 MB·English

by  Theodore V. Hromadka II PhD

The Best Approximation Method in
Computational Mechanics
Theodore V. Hromadka IT
The Best Approximation
Method in
Computational Mechanics
With 35 Figures
Springer-Verlag
London Berlin Heidelberg New York
Paris Tokyo Hong Kong
Barcelona Budapest
Theodore V. Hromadka II, PhD, PhD, PH, RCE
California State University, Fullerton and Computational Hydrology Institute,
Irvine, USA
ISBN-13: 978-1-4471-2022-3 e-ISBN-13: 978-1-4471-2020-9
DOl: 10.1007/978-1-4471-2020-9
British Library Cataloguing in Publication Data
Hromadka, T.V.
Best Approximation Method in
Computational Mechanics
I. Title
620.100285
ISBN-13: 978-1-4471-2022-3
Library of Congress Cataloging-in-Publication Data
Hromadka, Theodore V.
The best approximation method in computational mechanics 1
Theodore V. Hromadka II.
p. cm.
Includes bibliographical references and index.
ISBN-13: 978-1-4471-2022-3
1. Functional analysis. I. Title.
QA320.H76 1992 92-33754
515'.7--dc20 CIP
Apart from any fair dealing for the purposes of research or private study, or criticism or
review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may
only be reproduced, stored or transmitted, in any form or by any means, with the prior
permission in writing of the publishers, or in the case of repro graphic reproduction in
accordance with the terms of licences issued by the Copyright Licensing Agency. Enquiries
concerning reproduction outside those terms should be sent to the publishers.
C Springer-Verlag London Limited 1993
Softcover reprint ofthe hardcover 1st edition 1993
The publisher makes no representation, express or implied, with regard to the accuracy of the
information contained in this book and cannot accept any legal responsibility or liability for
any errors or omissions that may be made.
Typesetting: Camera ready by author
69/3830-543210 Printed on acid-free paper
To Laura
Acknowledgements
The author pays acknowledgements to Dr. C.C. Yen of Williamson &
Schmid, Tustin, California, who carefully reviewed the manuscript many
times, and also helped prepared several of the computer applications.
Acknowledgements are also paid to Ms. Phyllis Williams, who typed and
modified the various versions of the manuscript. Thanks are given to Mr.
Bill Burchard, who prepared the several figures. And finally, thanks are
given to my wife, Laura, who supported me throughout this project.
Contents
Chapter 1 Topics in Functional Analysis 1
1.0 Introduction 1
1.1 Set Theory 2
1.2 Functions 5
1.3 Matrices 7
1.4 Solving Matrix Systems 9
1.5 Metric Spaces 18
1.6 Linear Spaces 22
1.7 Normed Linear Spaces 25
1.8 Approximations 31
Chapter 2 Integration Theory 37
2.0 Introduction 37
2.1 Reimann and Lebesgue Integrals: Step and
Simple Functions 37
2.2 Lebesgue Measure 38
2.3 Measurable Functions 40
2.4 The Lebesgue Integral 41
2.4.1 Bounded Functions 42
2.4.2 Unbounded Functions 44
2.5 Key Theorems in Integration Theory 47
2.6 Lp Spaces 49
2.6.1 m-Equivalent Functions 49
2.6.2 The Space Lp 50
2.7 The Metric Space, Lp 51
2.8 Convergence of Sequences 51
2.8.1 Common Modes of Convergence 51
2.8.2 Convergence in Lp 52
2.8.3 Convergence in Measure (M) 52
2.8.4 Almost Uniform Convergence·(AU) 52
2.8.5 Is the Approximation Converging? 52
2.8.6 Counterexamples 53
2.9 Capsulation 55
Chapter 3 Hilbert Space and Generalized Fourier Series 57
3.0 Introduction 57
3.1 Inner Product and Hilbert Space 58
3.2 Best Approximations in an Inner Product Space 62
3.3 Approximations in ~(E) 70
3.3.1 Parseval's Identity 71
3.3.2 Bessel's Inequality 71
x Contents
3.4 Vector Representations and Best Approximations 71
3.5 Computer Program 82
Chapter 4 Linear Operators 89
4.0 Introduction 89
4.1 Linear Operator Theory 89
4.2 Operator Norms 93
4.3 Examples of Linear Operators in Engineering 97
4.4 Superposition 101
Chapter 5 The Best Approximation Method 104
5.0 Introduction 104
5.1 An Inner Product for the Solution of Linear Operator
Equations 104
5.2 Definition of Inner Product and Norm 106
5.3 Generalized Fourier Series 108
5.4 Approximation Error Evaluation 117
5.5 The Weighted Inner Product 124
5.6 Considerations in Choosing Basis Functions 128
5.6.1 Global Basis Elements 128
5.6.2 Spline Basis Functions 129
5.6.3 Mixed Basis Functions 133
Chapter 6 The Best Approximation Method: Applications 134
6.0 Introduction 134
6.1 Sensitivity of Computational Results to Variation in
the Inner Product Weighting Factor 134
6.2 Solving Two-Dimensional Potential Problems 137
6.3 Application to Other Linear Operators 146
6.4 Computer Program: Two-Dimensional Potential
Problems Using Real Variable Basis Functions 150
6.4.1 Introduction 150
6.4.2 Input Data Description 152
6.4.3 Computer Program Listing 154
6.5 Application of Computer Program 166
6.5.1 A Fourth Order Differential Equation 167
Chapter 7 Solving Potential Problems using the Best
Approximation Method 170
7.0 Introduction 170
7.1 The Complex Variable Boundary Element Method 171
7.1.1 Objectives 171
7.1.2 Definition 7.1.1 (Working Space, Wo) 171
II wi I
7.1.3 Definition 7.1.2 (the Function to
\I wll 2) 172
Contents xi
7.1.4 Almost Everywhere (ae) Equality 172
7.1.5 Theorem (relationship of II wll to IIwl10 172
7.1.6 Theorem 173
7.1.7 Theorem 173
7.2 Mathematical Development 174
7.2.1 Discussion: (A Note on Hardy Spaces) 174
7.2.2 Theorem (Boundary Integral Representation) 174
7.2.3 Almost Everywhere (ae) Equivalence 174
7.2.4 Theorem (Uniqueness of Zero Element in W0 ) 175
7.2.5 Theorem (W 0 is a Vector Space) 175
7.2.6 Theorem (Definition of the Inner-Product) 176
7.2.7 Theorem (W 0 is an Inner-Product Space) 176
(11wll
7.2.8 Theorem is a Norm on Wo) 176
7.2.9 Theorem 176
7.3 The CVBEM and W0 176
7.3.1 Definition 7.3.1 (Angle Points) 176
7.3.2 Definition 7.3.2 (Boundary Element) 177
7.3.3 Theorem 177
7.3.4 Definition 7.3.3 (Linear Basis Function) 177
7.3.5 Theorem 177
7.3.6 Definition 7.3.4 (Global Trial Function) 177
7.3.7 Theorem 178
7.3.8 Discussion 178
7.3.9 Theorem 178
7.3.10 Discussion 178
7.3.11 Theorem (Linear Independence of Nodal
Expansion Functions) 180
7.3.12 Discussion 181
7.3.13 Theorem 181
7.3.14 Theorem 182
7.3.15 Discussion 182
7.4 The Space Wl 183
7.4.1 Definition 7.4.1 (WOA) 183
7.4.2 Theorem 183
7.4.3 Theorem 183
7.4.4 Discussion 184
7.4.5 Theorem 184
7.4.6 Theorem 185
7.4.7 Discussion: Another Look at W0 185
7.5 Applications 185
7.5.1 Introduction 185
7.5.2 Nodal Point Placement on r 186
7.5.3 Potential Flow-Field (Flow-Net) Development 186
7.5.4 Approximate Boundary Development 186
7.5.5 Application Problems 187
xii Contents
7.6 Computer Program: Two-Dimensional Potential
Problems using Analytic Basis Functions (CVBEM) 187
7.6.1 Introduction 187
7.6.2 CVBEM1 Program Listing 191
7.6.3 Input Variable Description for CVBEM1 203
7.6.4 CVBEM2 Program Listing 204
7.7 ModeIling Groundwater Contaminant Transport 213
7.7.1 Application lA 214
7.7.2 Application lB 214
7.7.3 Application 2A 214
7.7.4 Application 2B 214
7.8 Three Dimensional Potential Problems 217
7.8.1 Approximation Error Evaluation -
Approximate Boundary Method 217
7.8.2 Computer Implementation 218
7.8.3 Application 219
7.8.4 Trial Functions 219
f
7.8.5 Constructing the Approximate Boundary, 221
Chapter 8 Applications to Linear Operator Equations 222
8.0 Introduction 222
8.1 Data Fit Analysis 222
8.2 Ordinary Differential Equations 223
8.3 Best Approximation of Function 226
8.4 Matrix Systems 228
8.5 Linear Partial Differential Equations 230
8.6 Linear Integral Equations 233
8.6.1 An Inverse Problem 234
8.6.2 Best Approximation of the Transfer Function
in a Linear Space 236
References 238
Appendix A Derivation of CVBEM Approximation
Function 239
Appendix B Convergence of CVBEM Approximator 243
Appendix C The Approximate Boundary for Error
Analysis 245
Index 249
CHAPTER 1
TOPICS IN FUNCTIONAL ANALYSIS
1.0. Introduction
With the overwhelming use of computers in engineering, science,
and physics, the ability to approximately solve complex mathematical
systems of equations is almost commonplace. And yet, despite the vast
quantities of synthetic data one sometimes isn't quite sure whether
these approximations are valid. A nagging question haunts the analyst
as to whether the extrapolation of true data just achieved by the
computer program really represents reality, or merely represents some
impossible result that was generated by a collection of small but
accumulative errors in both analysis and computation.
In order to investigate the validity of the computational results, a
return to mathematical analysis of the computational scheme may be
necessary. Consequently computer modelers, both experienced and
novices, need to become familiar with the bodies of mathematical
literature generally classified as functional analysis and the more
recent numerical analysis. Many questions regarding the validity and
competence of computer program algorithms can be answered with
usage of theorems in functional analysis. Issues regarding algorithm
convergence and stability oftentimes can be addressed in terms of
concepts in functional analysis.
Fortunately many of the important concepts of functional and
numerical analysis can be communicated in a readable setting without
use of elaborate proofs and derivations. Oftentimes, the fine details
accounted for in the detailed proof are not at issue in the underlying
space of functions that the analyst is implicity using to develop the
approximation.
A goal of this book is to present possibly the more important and
useful functional analysis concepts that may serve the computer
modeler in his/her search for "truth". The book may serve as an
introduction to a fanctional analysis course, or may also serve as an
introduction to mathematical analysis of computer modeling
algorithms. In any event, the book may direct the attention of