Preface -- Acknowledgment: -- Contents -- Chapter 1 Basic Concepts -- 1.1 The Physics Simulation Process -- 1.2 Mesh Enhancement using Elliptic Methods -- 1.3 A Brief Primer on Meshes -- 1.3.1 Mesh Topology -- 1.3.2 Structured Meshes -- 1.3.3 Unstructured Meshes -- 1.3.4 Basic Grid Generation Algorithms -- 1.3.5 Mesh Quality Issues -- Chapter 2 Computational Geometry and Geometric Data Structures -- 2.1 Geometric Modeling -- 2.1.1 Curves -- 2.1.2 Properties of Parametric Curves -- 2.1.3 Constructing Parametric Curves -- 2.1.4 Parametric Surfaces -- 2.1.5 CSG Geometry -- 2.2 Mesh Operations On Geometry -- 2.2.1 Boundary Discretization -- 2.2.2 Boundary Node Redistribution -- 2.3 Geometric Data Structures -- 2.3.1 IGES Data -- 2.3.2 Stereo Lithography -- Chapter 3 Discretization Methods for Differential Equations -- 3.1 Finite Difference Methods -- 3.1.1 Finite Difference Operators -- 3.1.2 Non-uniform Difference Approximation -- 3.1.3 Finite Differences Applied to Differential Equations -- 3.1.4 Consistency of Finite Difference Approximations -- 3.1.5 Stability of Finite Difference Approximations -- 3.1.6 Implicit Finite Difference Approximations -- 3.2 Weak Solutions -- 3.2.1 Hilbert Spaces -- 3.2.2 Sobolev Spaces -- 3.2.3 Boundary Value Problem for Elliptic Operators -- 3.2.4 Weak Solution Methods -- 3.3 Applications of Weighted Residual Methods -- 3.3.1 The Subdomain Method -- 3.3.2 Finite Volume Methods -- 3.3.3 The Galerkin Method -- 3.4 The Finite Element Method -- 3.4.1 One-Dimensional Finite Element Methods -- 3.4.2 Two-Dimensional Finite Element Methods. -- 3.4.3 Three-Dimensional Elements -- 3.4.4 Computation of Element Matrices Using Gaussian Quadrature -- 3.4.5 A Brief Introduction to Errors in the Finite Element Method -- Chapter 4 Solving the Mesh Enhancement Algebraic Equation System -- 4.1 Introduction.

4.2 Selected Matrix Solution Methods -- 4.2.1 Jacobi Iterative Relaxation -- 4.2.2 Gauss-Seidel Relaxation -- 4.2.3 Other Iterative Approaches -- 4.3 Solution of Nonlinear Problems -- 4.3.1 Picard Linearization -- 4.3.2 Newton Linearization -- 4.4 Nonlinear Solution Methods for Finite Element Applications -- 4.4.1 Picard Linearization -- 4.4.2 Newton Linearization -- Chapter 5 The Geometry of Surfaces in Euclidean Space -- 5.1 The First Fundamental Form -- 5.2 The Second Fundamental Form -- 5.3 Intrinsic Equations of a Surface -- 5.3.1 The Gauss-Weingarten Equations -- 5.3.2 The Theorem of Gauss and the Codazzi Equations -- 5.4 Covariant Differentiation -- 5.5 Geodesics in a Riemann Space -- 5.6 Curvature of Space -- Chapter 6 Special Coordinate Systems -- 6.1 Isothermal Coordinates -- 6.1.1 Beltrami Equation -- 6.1.2 Complex Form -- 6.1.3 Solution of the Beltrami Equation -- 6.1.4 Teichmuller Spaces -- 6.2 Harmonic Coordinates -- 6.2.1 Sign Convention and Basic Formulas -- 6.2.2 The Ricci Tensor in Harmonic Coordinates -- 6.3 Conclusions -- Chapter 7 Elliptic Mesh Enhancement Equation Systems -- 7.1 Khamayseh-Mastin Mesh Enhancement -- 7.2 Special Cases -- 7.2.1 Thompson-Thames-Mastin Generator -- 7.2.2 Winslow Grid Generator -- 7.3 Connection with Harmonic Coordinates -- 7.4 Variational Methods -- 7.5 Surface Meshes and Adaptive Methods -- 7.5.1 Numerical Examples -- 7.5.2 Adaptive Grid Methods -- Chapter 8 Structured Mesh Smoothing and Enhancement -- 8.1 Introduction -- 8.1.1 Boundary Curve Discretization, Boundary Conditions, and Structured Terminology -- 8.1.2 Development of a Mesh Quality Metric -- 8.1.3 Introduction to Two-Dimensional Structured Mesh Methods -- 8.2 Prescriptive Methods -- 8.2.1 Thompson-Thames-Mastin Generator -- 8.2.2 Specification of Control Functions via an Algebraic Transformation to a Parameter Space.

8.3 Laplace-Beltrami Mesh Enhancement Using a Target Metric -- 8.3.1 The Discrete Laplace-Beltrami System -- 8.3.2 Geometric Interpretation of the Metric Terms -- 8.3.3 The Target Metric Tensor -- 8.4 Using a Target Metric Surface to Implement Solution Adaptivity -- 8.5 Rudimentary Methods for Unstructured Meshes -- 8.5.1 Laplace Smoothing -- 8.5.2 Ad hoc Laplace-Beltrami Methods -- 8.5.3 Winslow-based Unstructured Methods -- 8.6 Three-Dimensional Structured Laplace-Beltrami Methods -- 8.7 Conclusions -- Chapter 9 Mesh Enhancement Methods for Unstructured Meshes -- 9.1 Laplace-Beltrami Enhancement Equations in Two and Three Dimensions -- 9.2 Approximation of the Laplace-Beltrami Equation System -- 9.2.1 Two-Dimensional Finite Element Implementation -- 9.2.2 Three-Dimensional Finite Element Implementation -- 9.2.3 Target Metric Tensor Estimation: Metric Equidistribution -- 9.2.4 Target Metric Tensor Estimation: Coarse Graining -- 9.2.5 Implementation Details and Summary -- 9.2.6 Two-Dimensional Examples -- 9.2.7 Three-Dimensional Examples -- 9.3 Quantitative Mesh Comparisons -- 9.4 Conclusions -- Appendix A Differential Manifolds and Fiber Bundles -- A.1 Scalar Field -- A.2 Vectors and Tensors -- A.3 Infinitesimal Transformations -- A.4 Other Transformation Laws -- A.5 Fiber Bundles -- A.5.1 Differentiable Manifolds -- A.5.2 Principal Fiber Bundles -- A.5.3 Associated Fiber Bundles -- A.6 Connections in Vector Bundles -- A.7 Riemannian Manifolds -- Bibliography -- Index.