Past Event: Oden Institute Seminar

Quadrilateral and Hexahedral Mesh Generation Based on Surface Foliation Theory

David Xianfeng Gu, Associate Professor, Department of Computer Science, State University of New York at Stony Brook

Abstract

For the purpose of isogeometric analysis, tetrahedral meshes need to be converted to structured hexahedral meshes, which have regular tensor product structure for T-Spline fitting. This theoretic work proposes a novel surface quadrilateral meshing method, colorable quad-mesh, which leads to the structured hexahedral mesh of the enclosed volume for high genus surfaces. The work proves the equivalence relations among colorable quad-meshes, finite measured foliations and Strebel differentials on the surface. This trinity theorem lays down the theoretic foundation for quadrilateral/ hexahedral mesh generation, and leads to practical, automatic algorithms. The work proposes the following algorithm: the user input a set of disjoint, simple loops on a high genus surface, and specify a height parameter for each loop; a unique Strebel differential is computed with the combinatorial type and the heights prescribed by the user input; the Strebel differential assigns a flat metric on the surface and decompose the surface into cylinders; a colorable quad-mesh is generated by splitting each cylinder into two quadrilaterals, followed by subdivision; the surface cylindrical decomposition is extended inward to induce a solid cylindrical decomposition of the volume; the hexadhedral meshing is generated for each volumetric cylinder and then glued together to form a global consistent hex-mesh. The method is rigorous, geometric, automatic and conformal to the geometry. This work focuses on the theoretic aspects of the framework, the algorithmic details will be given in the future expositions. Bio: David Gu got his PhD from Harvard university, supervised by a Fields medalist Prof. Shing-Tung Yau. Dr. Gu's research focuses on applying modern geometry for engineering and medical applications. Prof. Yau and Dr. Gu together founded an interdisciplinary field - Computational Conformal Geometry, and applied the method in graphics, vision, geometric modeling, networking and medical imaging. Dr. Gu won NSF Career Award in 2005, Morningside Gold Medal in applied mathematics in 2013.