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03/22/05 Colloquium
Dr. Andrew Zardecki
University of California,
Los Alamos National Laboratory
University of California,
Los Alamos National Laboratory
Harmonic Morphisms for Grid
Smoothing and Adaptation
Abstract:
Numerical grid generation enables one to compute solutions to partial differential equations of fluid dynamics in physical regions with complex geometry. The optimal mesh, obtained by modifying the location of nodes, involves grid smoothing. In the adaptation process the grid is adjusted to correspond to variations of physical quantities or to surface gradients.
The purpose of this talk is twofold. First, the basic elliptic grid smoothing techniques will be established. Second, the interplay of the smoothing techniques and different branches of mathematics (complex functions, differential geometry, functional analysis) will be emphasized. A broad class of elliptic smoothing equations can be formulated within the framework of harmonic mappings. These arise from a variational problem dealing with the energy of maps between two Riemannian manifolds and its solution, the harmonic maps. A harmonic morphism is a harmonic map that is semiconformal (i.e. the map's differential, at the points where it does not vanish, is conformal and surjective). Based on the notion of a harmonic morphism, adaptive surface grid equations, applicable to both structured and unstructured grids, are formulated. The weak form of the grid equations is solved using the finite element method, which reduces the grid equations to a nonlinear, algebraic set. Computational examples illustrate the applicability of this approach.
