The Best Approximation Method in Computational Mechanics
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 implicitly 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 functional 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 computer modelers to the already established principles and results assembled in functional analysis.
Table of Contents
Chapter 1 - Topics in Functional Analysis
1.0 Introduction
1.1 Set Theory
1.2 Functions
1.3 Matrices
1.4 Solving Matrix Systems
1.5 Metric Spaces
1.6 Linear Spaces
1.7 Normed Linear Spaces
1.8 Approximations
Chapter 2 - Integration Theory
2.0 Introduction
2.1 Reimann and Lebesgue Integrals: Step and Simple Functions
2.2 Lebesgue Measure
2.3 Measurable Functions
2.4 The Lebesgue Integral
2.4.1 Bounded Functions
2.4.2 Unbounded Functions
2.5 Key Theorems in Integration Theory
2.6 Lp Spaces
2.6.1 m-Equivalent Functions
2.6.2 The Space Lp
2.7 The Metric Space, Lp
2.8 Convergence of Sequences
2.8.1 Common Modes of Convergence
2.8.2 Convergence in Lp
2.8.3 Convergence in Measure (M)
2.8.4 Almost Uniform Convergence (AU)
2.8.5 Is the Approximation Converging?
2.8.6 Counterexamples
2.9 Capsulation
Chapter 3 - Hilbert Space and Generalized Fourier Series
3.0 Introduction
3.1 Inner Product and Hilbert Space
3.2 Best Approximations in an Inner Product Space
3.3 Approximations in L,(E)
3.3.1 Parseval's Identity
3.3.2 Bessel's Inequality
3.3 Vector Representations and Best Approximations
3.4 Computer Program
Chapter 4 – Linear Operators
4.0 Introduction
4.1 Linear Operator Theory
4.2 Operator Norms
4.3 Examples of Linear Operators in Engineering
4.4 Superposition
Chapter 5 – The Best Approximation Method
5.0 Introduction
5.1 An Inner Product for the Solution of Linear Operator Equations
5.2 Definition of Inner Product and Norm
5.3 Generalized Fourier Series
5.4 Approximation Error Evaluation
5.5 The Weighted Inner Product
5.6 Considerations in Choosing Basis Functions
5.6.1 Global Basis Elements
5.6.2 Spline Basis Functions
5.6.3 Mixed Basis Functions
Chapter 6 The Best Approximation Method: Applications
6.0 Introduction
6.1 Sensitivity of Computational Results to Variation in the Inner Product Weighting Factor
6.2 Solving Two-Dimensional Potential Problems
6.3 Application to Other Linear Operators
6.4 Computer Program: Two-Dimensional Potential Problems Using Real Variable Basis Functions
6.4.1 Introduction
6.4.2 Input Data Description
6.4.3 Computer Program Listing
6.5 Application of Computer Program
6.5.1 A Fourth Order Differential Equation
Chapter 7 – Solving Potential Problems Using the Best Approximation Method
7.0 Introduction
7.1 The Complex Variable Boundary Element Method
7.1.1 Objectives
7.1.2 Definition 7.1.1 (Working Space, WW)
7.1.3 Definition 7.1.2 (the Function w to w2)
7.1.4 Almost Everywhere (ae) Equality
7.1.5 Theorem (relationship of w to w2)
7.1.6 Theorem
7.1.7 Theorem
7.2 Mathematical Development
7.2.1 Discussion: (A Note on Hardy Spaces)
7.2.2 Theorem (Boundary Integral Representation)
7.2.3 Almost Everywhere (ae) Equivalence
7.2.4 Theorem (Uniqueness of Zero Element in WW)
7.2.5 Theorem (WW is a Vector Space)
7.2.6 Theorem (Definition of the Inner-Product)
7.2.7 Theorem (WW is an Inner-Product Space)
7.2.8 Theorem (w is a Norm on WW)
7.2.9 Theorem
7.3 The CVBEM and WW
7.3.1 Definition 7.3.1 (Angle Points)
7.3.2 Definition 7.3.2 (Boundary Element)
7.3.3 Theorem
7.3.4 Definition 7.3.3 (Linear Basis Function)
7.3.5 Theorem
7.3.6 Definition 7.3.4 (Global Trial Function)
7.3.7 Theorem
7.3.8 Discussion
7.3.9 Theorem
7.3.10 Discussion
7.3.11 Theorem (Linear Independence of Nodal Expansion Functions)
7.3.12 Discussion
7.3.13 Theorem
7.3.14 Theorem
7.3.15 Discussion
7.4 The Space WWA
7.4.1 Definition 7.4.1 (WWA)
7.4.2 Theorem
7.4.3 Theorem
7.4.4 Discussion
7.4.5 Theorem
7.4.6 Theorem
7.4.7 Discussion: Another Look at WW
7.5 Applications
7.5.1 Introduction
7.5.2 Nodal Point Placement on G
7.5.3 Potential Flow-Field (Flow-Net) Development
7.5.4 Approximate Boundary Development
7.5.5 Application Problems
7.6 Computer Program: Two-Dimensional Potential Problems using Analytic Basis Functions (CVBEM)
7.6.1 Introduction
7.6.2 CVBEM1 Program Listing
7.6.3 Input Variable Description for CVBEM1
7.6.4 CVBEM2 Program Listing
7.7 Modelling Groundwater Contaminant Transport
7.7.1 Application 1A
7.7.2 Application 1B
7.7.3 Application 2A
7.7.4 Application 2B
7.8 Three Dimensional Potential Problems
7.8.1 Approximation Error Evaluation -Approximate Boundary Method
7.8.2 Computer Implementation
7.8.3 Application
7.8.4 Trial Functions
7.8.5 Constructing the Approximate Boundary,
Chapter 8 Applications to Linear Operator Equations
8.0 Introduction
8.1 Data Fit Analysis
8.2 Ordinary Differential Equations
8.3 Best Approximation of Function
8.4 Matrix Systems
8.5 Linear Partial Differential Equations
8.6 Linear Integral Equations
8.6.1 An Inverse Problem
8.6.2 Best Approximation of the Transfer Function in a Linear Space
References
Appendix A - Derivation of CVBEM Approximation Function
Appendix B - Convergence of CVBEM Approximator
Appendix C - The Approximate Boundary for Error Analysis
Index
|
Home • Research • Resume • Publications • Expert Witness • Links • Contact
