Adapting Meshes by Deformation Numerical Examples and Applications

Summary

The goal of this paper is to give a strictly derived definition for an optimal deformation of a given simplicial grid and to discuss in detail some properties and useful applications. Many grid handling approaches generate or adapt a mesh using combinatorial techniques such as refining or coarsening of simplices or continuously adding new points and reconnecting the vertices. Different to those methods we keep the connectivity of the vertices fixed solely changing their location and thereby deforming the mesh. It can rigorously be demonstrated that under a few reasonable assumptions the functional describing optimality of a mesh is of a type wellknown from elasticity theory. This approach is also suitable for numerical usage. It is possible to improve and adapt 3D grids by a stable minimization algorithm with useful properties. Interesting applications are the reconstruction of a mesh for a domain with a free boundary, the concentration of a mesh at singularities, or its compression near an arbitrary given boundary to resolve a layer structure.