## Verma, Chaman Singh, Krishnan Suresh

Proceedings, 25th International Meshing Roundtable, Elsevier, Science Direct, September 26-30 2016

Chaman Singh Verma, University of Wisconsin, Madison, US, cverma2@wisc.edu
Krishnan Suresh, University of Wisconsin, Madison, US, ksuresh@wisc.edu

Abstract
Mesh adaptation plays a critical role in balancing computational efficiency and numerical accuracy. Three types of mesh adaptation techniques exist today, namely, mesh improvement, mesh refinement and mesh simplification, and for each of these, several strategies have been proposed. Most of these strategies yield acceptable geometric mesh quality, but provide limited control over topological quality. \par In this paper, we introduce a unified algorithm for all three types of mesh adaptation for quadrilateral meshes. The algorithm builds upon the {\em Minimum Singularity Templates} (MST) idea proposed by the authors for improving the topological quality of a quadrilateral mesh. The MST is extended here to define the concept of an $\alpha$MST where a single parameter $\alpha$ controls mesh adaptation: $\alpha = 1$ for mesh improvement, $\alpha > 1$ for mesh refinement, and $\alpha < 1$ for mesh simplification.The proposed algorithm generates mesh with high geometric and topological qualities. Further, it is non-hierarchical and stateless, and yet it provides arbitrary level of mesh adaptation. Finally, since cyclic chords can play an important role in quadrilateral mesh adaptation, we provide a simple constructive algorithm to insert such chords using the $\alpha$MST. Several examples are presented that demonstrate the robustness, efficiency, and versatility of the proposed concept and algorithm.