Hexahedral mesh generation for configurations sharing the same topology with solid torus

Abstract The generation of hexahedral meshes for complex configurations, which is related to many frontier problems in mathematics, has long been a challenge for scientists and engineers. To date, no sufficient mathematical theories have been developed to guarantee the existence of a hexahedral mesh for every arbitrarily complex configuration, necessitating a combination of mathematics and engineering to achieve this generation process. Considering that the curvature and other geometric features vary considerably across complex configurations, we approach the topic from new perspectives. Namely, we incorporate computational conformal geometry into the generation process: using this technique, geometry information is more logically classified, and geometries with large deformations are hence more easily accommodated. In practice, several major branches of computational conformal geometry have been applied in our method, ranging from the generation of closed isoparametric curves on a 2-D manifold and the generation of discrete minimal surfaces to surface registrations among these discrete minimal surfaces. Moreover, some traditional methods, such as butterfly topology, have also been applied in our methods. Consequently, a balance between high automation and good mesh quality is achievable within our framework. To test this approach, illustrative examples are presented in this paper. The experimental results show that our theory and algorithm are robust. The generated hexahedral meshes can be used as input for subsequent finite element analysis calculation.