Beijing Institute of Mathematical Sciences and Applications

Zheng Wei Liu , Shing-Tung Yau , Hui Zhao

Xianfeng David Gu

Friday, April 1, 2022 9:30 AM - 10:30 AM

Zoom 388 528 9728 (BIMSA)

esh generation plays a fundamental role in CAD, CAE and many engineering fields. After tens of years of intensive research, meshing on surfaces and volumes with complicated topological and geometric features remains challenging. The major difficulties include 1. complex topology and geometry of the input shapes; 2. anisotropy; 3. global structured meshes with high conformality and so on. In this talk, we propose a systematic method to tackle these challenges based on modern topology and geometry theories, especially conformal geometry and algebraic geometry. The method is rigorous with solid foundation, practical with broad applications in manufacturing industry and medical imaging field. 1. For shapes with complex topology and geometry, we can conformally map them to canonical planar domains and use 2D algorithms to generate high quality planar meshes, then pull the meshes back to the surfaces. The key point is that the mapping is angle-preserving (conformal) , therefore it preserves the quality of the meshes from 2D to 3D. The mapping is produced by Ricci flow method, which is the tool used by Perelman to prove Poincare's conjecture. 2.For anisotropic requirements, we can use quasi-conformal geometric methods to handle them. The anisotropic conditions are encoded to the so-called Beltrami coefficient, which can be used to construct an auxiliary Riemannian metric. Under the auxiliary metric, anisotropic meshes are converted to isotropic meshes. 3. For globally structured meshes, e.g. quad or hex meshes, the configuration of the singularities are crucial. For a long time, the placements of singularities are mainly heuristic. We have found the PDEs to govern the singularities based on Abel-Jacobi theory in algebraic geometry, and developed an automatic algorithm to generate quad-meshes. Then we extend the algorithm for hex-mesh generation.

David Gu is a New York Empire Innovation Professor at the Department of Computer Science, Stony Brook University. He received his Ph.D degree from the Department of Computer Science, Harvard University in 2003， supervised by the Fields medalist, Prof. Shing-Tung Yau and B.S. degree from the Tsinghua University, Beijing, China in 1995. His research focuses on applying modern geometry in engineering and medical fields. With Prof. Yau and his collaborators, David systematically develops discrete theories and computational algorithms in the interdisciplinary field: Computational Conformal Geometry, Computational Optimal Transportation, and applied them in engineering and medical imaging fields.