Adaptive Grid Generation from Harmonic Maps on Riemannian Manifolds

摘要

The objective of Phase I of the project was to develop a new method for grid generation using harmonic maps. The coordinate transformation is obtained from the principle of least action (harmonic maps) on Riemannian manifolds. The Riemannian manifolds are formulated to reduce the rate of change of the modeled physical phenomenon when the phenomenon is considered in the new metric. As a consequence, the numerical solutions calculated on grids generated from harmonic maps on these metrics are well resolved. Existence and uniqueness conditions for harmonic maps both in 2D and 3D for general multiply connected domains have been shown. Using these findings, grid generating equations, guaranteed to have a unique one-to-one solution, were constructed. A number of examples are presented to illustrate the developed concepts.