A Computational Differential Geometry Approach to Grid Generation

2. Auflage 2006
ISBN: 978-3-540-34236-6
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Adobe DRM (»Systemvoraussetzungen)

E-Book, Englisch, 294 Seiten

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 gives a detailed treatment of applications of geometric methods to advanced grid technology. It focuses on and describes a comprehensive approach based on the numerical solution of inverted Beltramian and diffusion equations with respect to monitor metrics for generating both structured and unstructured grids in domains and on surfaces. In this second edition the author takes a more detailed and practice-oriented approach towards explaining how to implement the method by:

• Employing geometric and numerical analyses of monitor metrics as the basis for developing efficient tools for controlling grid properties.
• Describing new grid generation codes based on finite differences for generating both structured and unstructured surface and domain grids.
• Providing examples of applications of the codes to the generation of adaptive, field-aligned, and balanced grids, to the solutions of CFD and magnetized plasmas problems.

The book addresses both scientists and practitioners in applied mathematics and numerical solution of field problems.

Autoren/Hrsg.

Inhaltsverzeichnis

1;Preface to the Second Edition;5
2;Contents;11
3;Geometric Background to Grid Technology;16
3.1;1 Introductory Notions;19
3.1.1;1.1 Representation of Physical Geometries;19
3.1.2;1.2 General Concepts Related to Grids;22
3.1.3;1.3 Grid Generation Models;30
3.1.4;1.4 Comprehensive Codes;46
3.2;2 General Coordinate Systems in Domains;49
3.2.1;2.1 Jacobi Matrix;49
3.2.2;2.2 Coordinate Lines, Tangential Vectors, and Grid Cells;50
3.2.3;2.3 Coordinate Surfaces and Normal Vectors;52
3.2.4;2.4 Representation of Vectors Through the Base Vectors;54
3.2.5;2.5 Metric Tensors;56
3.2.6;2.6 Cross Product;60
3.2.7;2.7 Relations Concerning Second Derivatives;63
3.3;3 Geometry of Curves;69
3.3.1;3.1 Curves in Multidimensional Space;69
3.3.2;3.2 Curves in Three-Dimensional Space;71
3.4;4 Multidimensional Geometry;75
3.4.1;4.1 Tangent and Normal Vectors and Tangent Plane;75
3.4.2;4.2 First Groundform;77
3.4.3;4.3 Generalization to Riemannian Manifolds;81
3.4.4;4.4 Tensors;88
3.4.5;4.5 Basic Invariants;95
3.4.6;4.6 Geometry of Hypersurfaces;99
3.4.7;4.7 Relations to the Principal Curvatures of Two- Dimensional Surfaces;120
4;Algorithms and Applications of Advanced Grid Technology;128
4.1;5 Comprehensive Grid Models;131
4.1.1;5.1 Formulation of Differential Grid Generators;133
4.1.2;5.2 Variational Formulations;145
4.1.3;5.3 Formulation of Monitor Metrics;154
4.2;6 Inverted Equations;175
4.2.1;6.1 General Forms of Equations;175
4.2.2;6.2 Equations for Classical Monitor Metrics;182
4.2.3;6.3 Role of the Mean Curvature;196
4.2.4;6.4 Practical Grid Equations;221
4.3;7 Numerical Implementation of Grid Generators;233
4.3.1;7.1 Method of Fractional Steps;233
4.3.2;7.2 Method of Minimization of Energy Functional;250
4.3.3;7.3 Generation of Multi-Block Grids;269
4.3.4;7.4 Application of Layer-Type Functions to Grid Codes;281
5;References;293
6;Index;303