The Best Approximation Method in Computational Mechanics

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.- 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.- 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 L2(E).- 3.3.1 Parseval’s Identity.- 3.3.2 Bessel’s Inequality.- 3.4 Vector Representations and Best Approximations.- 3.5 Computer Program.- 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.- 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.- 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.- 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, W?).- 7.1.3 Definition 7.1.2 (the Function ??? to ???2).- 7.1.4 Almost Everywhere (ae) Equality.- 7.1.5 Theorem (relationship of ??? to ???2).- 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 W?).- 7.2.5 Theorem (W? is a Vector Space).- 7.2.6 Theorem (Definition of the Inner-Product).- 7.2.7 Theorem (W? is an Inner-Product Space).- 7.2.8 Theorem (??? is a Norm on W?).- 7.2.9 Theorem.- 7.3 The CVBEM and W?.- 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 W?A.- 7.4.1 Definition 7.4.1 (W?A).- 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 W?.- 7.5 Applications.- 7.5.1 Introduction.- 7.5.2 Nodal Point Placement on ?.- 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, ?.- 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.