A Computational Differential Geometry Approach to Grid Generation (eBook)

Preface to the Second Edition 5
Contents 11
Geometric Background to Grid Technology 16
1 Introductory Notions 19
1.1 Representation of Physical Geometries 19
1.2 General Concepts Related to Grids 22
1.3 Grid Generation Models 30
1.4 Comprehensive Codes 46
2 General Coordinate Systems in Domains 49
2.1 Jacobi Matrix 49
2.2 Coordinate Lines, Tangential Vectors, and Grid Cells 50
2.3 Coordinate Surfaces and Normal Vectors 52
2.4 Representation of Vectors Through the Base Vectors 54
2.5 Metric Tensors 56
2.6 Cross Product 60
2.7 Relations Concerning Second Derivatives 63
3 Geometry of Curves 69
3.1 Curves in Multidimensional Space 69
3.2 Curves in Three-Dimensional Space 71
4 Multidimensional Geometry 75
4.1 Tangent and Normal Vectors and Tangent Plane 75
4.2 First Groundform 77
4.3 Generalization to Riemannian Manifolds 81
4.4 Tensors 88
4.5 Basic Invariants 95
4.6 Geometry of Hypersurfaces 99
4.7 Relations to the Principal Curvatures of Two- Dimensional Surfaces 120
Algorithms and Applications of Advanced Grid Technology 128
5 Comprehensive Grid Models 131
5.1 Formulation of Differential Grid Generators 133
5.2 Variational Formulations 145
5.3 Formulation of Monitor Metrics 154
6 Inverted Equations 175
6.1 General Forms of Equations 175
6.2 Equations for Classical Monitor Metrics 182
6.3 Role of the Mean Curvature 196
6.4 Practical Grid Equations 221
7 Numerical Implementation of Grid Generators 233
7.1 Method of Fractional Steps 233
7.2 Method of Minimization of Energy Functional 250
7.3 Generation of Multi-Block Grids 269
7.4 Application of Layer-Type Functions to Grid Codes 281
References 293
Index 303