A Study On Adaptive Surface Mesh Generation

Mesh generation is the key pre-process of numerical simulations based on partial differential equations in engineering and scientific computing. One of its important research areas is surface mesh generation. Not only could surface mesh be applied directly to surface modeling, but also it is the input of volume mesh generation. Besides numerical simulations, surface mesh generation also is of many other applications in various fields, such as computer graphics and geographic information systems.As an input of surface mesh generation, complex geometry models usually contain many features. To generate good meshes for numerical analyses, the mesh size should be small near the features to achieve high geometry accuracy and element quality, and large elsewhere to avoid increasing the number of mesh elements unnecessarily. However, manual size control on complex models is time consuming and prone to errors to achieve such goals. Instead, adaptive mesh generation is capable of overcoming this bottleneck problem, effectively.Based on the Delaunay method and Riemannian metric, in this thesis, adaptive mesh generation for all kinds of surfaces, including planes, continuous curved surfaces and discrete curved surfaces, is systematically studied. Additionally, a parallel algorithm for planar mesh generation is also presented.Firstly, some fundamental algorithms for adaptive surface mesh generation are discussed, mainly including planar anisotropic mesh generation algorithms and adaptive mesh size control algorithms. Riemannian metric is introduced to define mesh sizes and to calculate point-to-point distances, thus a traditional isotropic Bowyer-Watson kernel for Delaunay mesh generation is revised for Riemannian context of planar anisotropic mesh generation. To recognize shape proximity over plane or curved surfaces, constrained Delaunay triangulations of sample points on boundary curves are used to produce the discrete medial axes, with help of Euclid metric for planes or Riemannian metric for curved surfaces, respectively. Coupled with curvature calculations, adaptive surface meshes of planar or curved surfaces can be produced. Moreover, by combining adaptive and traditional mesh size control strategies, a general framework for adaptive mesh generation over curved surfaces is presented.Taking composite Coons surfaces as an example, the above fundamental algorithms are applied for continuous curved surface mesh generation. The approaches to calculate Riemannian metric and curvature are studied, and two kinds of Riemannian metrics are discussed: (1) The intrinsic metric is used to recognize proximity, for adaptive mesh size control together with curvatures. (2) The Riemannian metric coupled with mesh sizes is used to generate anisotropic meshes on parametric planes. Combining these two kinds of metrics and a surface mesh generation algorithm based on mapping transformation, an adaptive mesh generator for continuous curved surfaces is implemented.Taking STL surfaces as an example, the above fundamental algorithms are applied for discrete curved surface mesh generation. Firstly, a fast surface mesh generation algorithm is proposed. The initial model is divided into many sub-domains with features preserved at sub-domain boundaries, and then the parametric plane for every sub-domain is constructed to dedimension the sub-domain mesh generation, where a planar isotropic algorithm is applied. To enhance this algorithm, G~1 continuous triangular B-B surfaces are reconstructed for sub-domains to calculate the Riemannian metrics for points on parametric planes. In the Riemannian context, high quality isotropic surface meshes are generated with anisotropic meshes generated in parametric planes, and constrained Delaunay triangulations for sample points on surface boundaries are computed to recognize proximity. Coupled with curvature calculation, adaptive meshes for STL models can be generated.Parallel mesh generation is used to generate large-scale meshes for engineering and scientific computing. With a geometry domain decomposition as the pre-process, a stable, effective and scalable parallel planar Delaunay mesh generator is built on the serial isotropic mesh generation algorithm. It defines Sub-Domain Graph (SDG) to represent the sub-domain connections, then partitions the graph dynamically to achieve dual goals of loading balance and communication minimization, and finally generates distributed meshes with high partitioning quality simultaneously with parallel mesh generation. Compared with traditional algorithms based on mesh graph partitioning, this algorithm can reduce or eliminate the cost of mesh repartitioning, which can accelerate the whole simulation process significantly for large-scale meshes.