HIGH FIDELITY INTERVAL ASSIGNMENT

Abstract

Quadrilateral meshing algorithms impose certain constraints on the number of intervals or mesh edges of the curves bounding a surface. When constructing a conformal mesh of a collection of adjoining surfaces, the constraints for all of the surfaces must be simultaneously satisfied. These constraints can be formulated as an integer linear program. Not all solutions to this problem are equally desirable, however. The user typically indicates a goal (soft-set) or required (hard-set) number of intervals for each curve. The hard-sets constrain the problem further, while the soft-sets influence the objective function. This paper describes an algorithm for solving this interval assignment problem. In a good solution, for each curve the positive or negative difference between its goal and assigned number of intervals is small relative to its number of goal intervals. The algorithm solves a series of linear programs, and comes close to minimizing the maximum lexicographic vector of these weighted differences. Then the algorithm solves a nearby mixed-integer linear program to satisfy certain "sum-even" constraints. The algorithm reliably produces numbers of intervals that are very close to the user's desires and is easily extendible to new constraints. Earlier versions of the algorithm1 were slower than alternative algorithms, but this is no longer a significant issue; in practice, the running time is a minor fraction of the time to mesh, even in models composed of thousands of curves.