High Quality Quadrilateral Surface Meshing Without Template Restrictions: A New Approach Based on Network Flow Techniques

We investigate a purely combinatorial approach to the following mesh reenement problem: Given a coarse mesh of polygons in three-dimensional space, nd a decomposition into well-shaped quadrilaterals such that the resulting mesh is conforming and satisses prescribed local density constraints. We present a new approach based on network ow techniques. In particular, we show that this problem can eeciently be solved by a reduction to a minimum cost bidirected ow problem, if the mesh does not contain branching edges, that is, edges incident to more than two polygons. This approach handles optimization criteria such as density, angles and regularity. In our model we get rid of restrictions on the set of feasible solutions imposed by templates. On the other hand, we still use advantages of general templates with respect to mesh quality for the individual reenement of the mesh polygons. For meshes with branchings, the problem is feasible if and only if a certain system of linear equations over GF(2) has a solution. To enhance the mesh quality for meshes with branchings, we introduce a two-stage approach which rst decomposes the whole mesh into components without branchings, and then uses minimum cost bidirected ows on the components in a second phase. We report on our computational results which indicate that this approach usually leads to a very high mesh quality.