Dynamic Mesh Refinement with Quad Trees and Off-Centers

Many algorithms exist for producing quality meshes when the input point cloud is known a priori. However, modern finite element simulations and graphics applications need to change the input set during the simulation dynamically. In this paper, we show a dynamic algorithm for building and maintaining a quadtree under insertions into and deletions from an input point set in any fixed dimension. This algorithm runs in O(lg L/s) time per update, where L/s is the spread of the input. The result of the dynamic quadtree can be combined with a postprocessing step to generate and maintain a simplicial mesh under dynamic changes in the same asymptotic runtime. The mesh output by the dynamic algorithm is of good quality (it has no small dihedral angle), and is optimal in size. This gives the first time-optimal dynamic algorithm that outputs good quality meshes in any dimension. As a second result, we dynamize the quadtree postprocessing technique of Har-Peled and Üngör for generating meshes in two dimensions. When composed with the dynamic quadtree algorithm, the resulting algorithm yields quality meshes that are the smallest known in practice, while guaranteeing the same asymptotic optimality guarantees. † Toyota Technological Institute at Chicago This work was supported in part by the National Science Foundation under grants ACI 0086093, CCR-0085982 and CCR0122581.