Papers »

Universal meshes for domains with corners

A universal mesh is a background mesh which can be used to achieve conforming meshes for a family of domains. Different domains are immersed into a universal mesh, and only a small number of vertices near the boundary are perturbed to achieve a conforming mesh for each domain. Previous study suggested an algorithm which identifies a set of edges called positive edges, and maps them onto the boundary by the closest point projection. The algorithm guarantees to achieve conforming meshes for domains with C2-regular boundaries, but it does not handle domains with less regular boundaries such as with corners. We extend the algorithm so that it is applicable on any domain with a boundary that is a piece-wise C2-regular curve with finite number of non C2 continuous points.

The key idea is that the mapping of the positive edges must be guaranteed to be a homeomorphism, and the mapping should not flip any triangle. We use the closest point projection and focus on how to choose an appropriate set of positive edges. For a simple approach, C2-regular curve segments near a corner are approximated as line segments. Instead of selecting positive edges at once, positive sides are chosen separately for each segment forming the corner. By choosing appropriate sides, we can guarantee the projection to be a homeomorphism. To claim this to be true, we need the restriction that all triangles in the universal mesh are acute triangles. We then extend the case to corners with curved segments. Given the local maximum curvature of the curve segments and the maximum angle of the mesh, we suggest an upper bound for the mesh size such that the algorithm is applicable for corners with curve segments.

Author(s):

Kyuwon Kim    
Stanford University
United States

Adrian Lew    
Stanford University
United States