Complexity and Modeling Aspects of Mesh Refinement into Quadrilaterals

Abstract.

We investigate the following mesh refinement problem: Given a mesh of polygons in three-dimensional space, find a decomposition into strictly convex quadrilaterals such that the resulting mesh is conforming and satisfies prescribed local density constraints.

The conformal mesh refinement problem is shown to be feasible if and only if a certain system of linear equations over GF(2) has a solution. To improve mesh quality with respect to optimization criteria such as density, angles, and regularity, we introduce a reduction to a minimum cost bidirected flow problem. However, this model is only applicable if the mesh does not contain branching edges, that is, edges incident to more than two polygons. The general case with branchings, however, turns out to be strongly ${\cal NP}$ -hard. To enhance the mesh quality for meshes with branchings, we introduce a two-stage approach which first decomposes the whole mesh into components without branchings, and then uses minimum cost bidirected flows on the components in a second phase.