A computational differential geometry approach to grid generation

The process of breaking up a physical domain into smaller sub-domains, known as meshing, facilitates the numerical solution of partial differential equations used to simulate physical systems. This monograph describes in detail the eminent role played by differential geometry in grid technology based on mapping. It demonstrates how the Beltrami operator helps to develop robust multidimensional grid generation codes, while supplying related numerical code. In particular, procedures for the construction of monitor metric tensors are given and their qualitative effect on the resulting mesh is analyzed. Reviewing concepts from Riemannian geometry, the book applies them to general grids with prescribed properties, and discusses the role of mean and of Gaussian curvature and other geometric characteristics for the Beltrami equations for grid generation. It addresses scientists and practitioners as well as graduate students from applied mathematics, physics, and engineering.

Part I Geometric Background to Grid Technology
1 Introductory Notions	p. 5
1.1 Representation of Physical Geometries	p. 5
1.2 General Concepts Related to Grids	p. 8
1.2.1 Grid Cells	p. 8
1.2.2 Requirements Imposed on Cells and Grids	p. 10
1.3 Grid Generation Models	p. 15
1.3.1 Mapping Approach	p. 16
1.3.2 Requirements Imposed on Mathematical Models	p. 20
1.3.3 Algebraic Methods	p. 22
1.3.4 Differential Methods	p. 23
1.3.5 Variational Methods	p. 27
1.4 Comprehensive Codes	p. 30
2 General Coordinate Systems in Domains	p. 33
2.1 Jacobi Matrix	p. 33
2.2 Coordinate Lines, Tangential Vectors, and Grid Cells	p. 34
2.3 Coordinate Surfaces and Normal Vectors	p. 36
2.4 Representation of Vectors Through the Base Vectors	p. 38
2.5 Metric Tensors	p. 40
2.5.1 Covariant Metric Tensor	p. 40
2.5.2 Line Element	p. 41
2.5.3 Contravariant Metric Tensor	p. 42
2.5.4 Relations Between Covariant and Contravariant Elements	p. 43
2.6 Cross Product	p. 44
2.6.1 Geometric Meaning	p. 44
2.6.2 Relation to Volumes	p. 45
2.6.3 Relation to Base Vectors	p. 46
2.7 Relations Concerning Second Derivatives	p. 47
2.7.1 Christoffel Symbols of Domains	p. 47
2.7.2 Differentiation of the Jacobian	p. 49
2.7.3 Basic Identity	p. 50
3 Geometry of Curves	p. 53
3.1 Curves in Multidimensional Space	p. 53
3.1.1 Definition	p. 53
3.1.2 Basic Curve Vectors	p. 54
3.2 Curves in Three-Dimensional Space	p. 55
3.2.1 Basic Vectors	p. 55
3.2.2 Curvature	p. 56
3.2.3 Torsion	p. 57
4 Multidimensional Geometry	p. 59
4.1 Tangent and Normal Vectors and Tangent Plane	p. 59
4.2 First Groundform	p. 61
4.2.1 Covariant Metric Tensor	p. 61
4.2.2 Contravariant Metric Tensor	p. 62
4.3 Generalization to Riemannian Manifolds	p. 65
4.3.1 Definition of the Manifolds	p. 65
4.3.2 Example of a Riemannian Manifold	p. 67
4.3.3 Christoffel Symbols of Manifolds	p. 68
4.4 Tensors	p. 72
4.4.1 Definition	p. 72
4.4.2 Examples of Tensors	p. 73
4.4.3 Tensor Operations	p. 76
4.5 Basic Invariants	p. 77
4.5.1 Beltrami's Differential Parameters	p. 77
4.5.2 Measure of Relative Spacing	p. 78
4.5.3 Measure of Relative Clustering	p. 80
4.5.4 Mean Curvature	p. 80
4.6 Geometry of Hypersurfaces	p. 81
4.6.1 Normal Vector to a Hypersurface	p. 81
4.6.2 Second Fundamental Form	p. 85
4.6.3 Surface Curvatures	p. 85
4.6.4 Formulas of the Mean Curvature	p. 86
4.7 Relations to the Principal Curvatures of Two-Dimensional Surfaces	p. 101
4.7.1 Second Fundamental Form	p. 101
4.7.2 Principal Curvatures	p. 101
Part II Application to Advanced Grid Technology
5 Comprehensive Grid Models	p. 111
5.1 Formulation of a Differential Grid Generator	p. 112
5.1.1 Beltramian Operator	p. 112
5.1.2 Boundary Value Problem for Grid Equations	p. 113
5.1.3 Interpretation as a Multidimensional Equidistribution Principle	p. 116
5.1.4 Examples of Familiar Grid Equations	p. 116
5.1.5 Realization of Specified Grids	p. 118
5.1.6 Formulation of Monitor Metrics	p. 122
5.2 Variational Formulation	p. 125
5.2.1 Functional of Grid Smoothness	p. 125
5.2.2 Geometric Interpretation	p. 127
5.2.3 Relation to Harmonic Functions	p. 130
5.2.4 Application to Adaptive Grid Generation	p. 132
5.3 Conclusion	p. 134
6 Relations to Monitor Manifolds	p. 137
6.1 Computation of Geometric Characteristics	p. 138
6.1.1 Recursive Representation of the Monitor Metric	p. 138
6.1.2 Jacobian of the Covariant Metric Tensor	p. 139
6.1.3 Contravariant Metric Tensor	p. 140
6.1.4 Beltrami's Mixed and First Differential Parameters	p. 142
6.1.5 Christoffel Symbols	p. 142
6.1.6 Mixed Derivatives	p. 143
6.1.7 Beltrami's Second Differential Parameter	p. 144
6.1.8 Geometric Characteristics for a Spherical Monitor Metric	p. 144
6.1.9 Computation of Geometric Characteristics of a Monitor Manifold	p. 145
6.2 Geometric Characteristics of Monitor Surfaces	p. 147
6.2.1 Computation of the Elements of the Contravariant Metric Tensor	p. 148
6.2.2 Beltrami's Mixed and First Differential Parameters	p. 153
6.2.3 Christoffel Symbols	p. 154
6.2.4 Mixed Derivatives	p. 155
6.2.5 Beltrami's Second Differential Parameter	p. 156
6.3 Particular Two-Dimensional Case	p. 158
6.3.1 Preliminary Results	p. 159
6.3.2 Contravariant Metric Tensor	p. 162
6.3.3 Beltrami's Mixed and First Differential Parameters	p. 164
7 Grid Equations with Respect to Intermediate Transformations	p. 169
7.1 Relations to Comprehensive Equations	p. 169
7.2 Resolved Grid Equations	p. 171
7.2.1 Basic Elliptic Operator	p. 171
7.2.2 General Grid Equations	p. 171
7.2.3 Equations for a Spherical Monitor Metric	p. 172
7.2.4 Equations for a Spherical Monitor Metric Over a Surface	p. 173
7.2.5 Domain Grid Equations with Respect to a Monitor Surface	p. 174
7.2.6 Domain Grid Equations with Respect to a Monitor Metric	p. 176
7.2.7 Surface Grid Equations with Respect to a Monitor Surface	p. 179
7.2.8 Surface Grid Equations with Respect to a Monitor Metric	p. 182
7.3 Role of the Mean Curvature in the Grid Equations	p. 183
7.4 Practical Grid Equations	p. 185
7.4.1 Equations for Generating Grids on Curves	p. 186
7.4.2 Equations for Generating Grids on Two-Dimensional Surfaces	p. 188
7.4.3 Equations for Generating Grids in Domains	p. 191
8 Control of Grid Clustering	p. 195
8.1 Fundamental Formula for Grid Clustering	p. 196
8.1.1 Relative Spacing Between Coordinate Surfaces	p. 197
8.1.2 Rate of Change of the Relative Spacing	p. 197
8.1.3 Relations to Geometry Characteristics	p. 199
8.1.4 Basic Relation to Grid Coordinates	p. 205
8.1.5 Remarks	p. 205
8.1.6 Grid Behavior near Boundary Segments of a Monitor Surface	p. 208
8.2 Application of Theorem to Popular Elliptic Models	p. 210
8.2.1 Control of Grids near Boundaries of Domains	p. 210
8.2.2 Diffusion Equations	p. 227
8.2.3 Control of Grid Spacing near Boundaries of Physical Surfaces	p. 229
9 Numerical Implementation of Grid Generator	p. 241
9.1 One-Dimensional Equation	p. 241
9.1.1 Numerical Algorithm	p. 241
9.2 Two-Dimensional Equations	p. 243
9.2.1 Algorithms for Generating Grids in Two-Dimensional Domains	p. 244
9.2.2 Algorithm for Generating Grids on Two-Dimensional Surfaces	p. 250
9.3 Three-Dimensional Equations	p. 253
References	p. 255
Index	p. 263

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