A Computational Differential Geometry Approach to Grid Generation
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Beschreibung
Grid technology whose achievements have significant impact on the efficiency of numerical codes still remains a rapidly advancing field of computational and applied mathematics. New achievements are being added by the creation of more sophisticated techniques, modification of the available methods, and implementation of more subtle tools as well as the results of the theories of differential equations, calculas of variations, and Riemannian geometry being applied to the formulation of grid models and analysis of grid properties. The development of comprehensive differential and variational grid gen eration techniques reviewed in the monographs of J. F. Thompson, Z. U. A. Warsi, C. W. Mastin, P. Knupp, S. Steinberg, V. D. Liseikin has been largely based on a popular concept in accordance with which a grid model realizing the required grid properties should be formulated through a linear combina tion of basic and control grid operators with weights. A typical basic grid operator is the operator responsible for the well-posedness of the grid model and construction of unfolding grids, e. g. the Laplace equations (generalized Laplace equations for surfaces) or the functional of grid smoothness which produces fixed nonfolding grids while grid clustering is controlled by source terms in differential grid formulations or by an adaptation functional in vari ational models. However, such a formulation does not obey the fundamental invariance laws with respect to parameterizations of physical geometries. It frequently results in cumbersome governing grid equations whose choice of weight and control functions provide conflicting grid requirements.
Geometric methods in grid generation is a fairly recent subject with many applications in scientific computing So far the literature on the subject is sparse Includes supplementary material: sn.pub/extras
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. The implementation of the Beltrami operator helps to develop robust multidimensional grid generation codes. The book reviews concepts from Riemannian geometry, 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. The book also includes numerical codes based on the Beltrami equations. It addresses scientists and practitioners as well as graduate students from applied mathematics, physics, and engineering.
Autor*in
Vladimir D. Liseikin
Themen in »A Computational Differential Geometry Approach to Grid Generation«
Beltramian Equations Gaussian curvature Grid Generation Quasiconformal Grids Riemannian geometry Scientific Computing curvature differential geometry manifold