Unstructured Background Mesh Based Element Sizing Field For Adaptive Parametric Surface Mesh Generation

Mesh generation is the pre-process of various numerical methods such as finite element methods. The ability to generate good-quality meshes is very important for the success of numerical methods. Here, a good quality mesh refers to a mesh that can model a physical problem at a highest accuracy with a least number of freedoms. Therefore, element sizes of a region must adapt to the variations of geometry and physical features defined in that region. In practice, such an element-sizing field is represented on a background mesh. The previous studies mainly focus on the element-sizing algorithms that employ Cartesian meshes as background meshes. For complex geometry models, the magnitude of the employed Cartesian mesh used to define the element-sizing field may be comparable with the final mesh used for numerical simulations. Therefore, the Cartesian mesh based element-sizing schemes are expensive in terms of computational time and storage requirement.As contrast, the unstructured mesh has a more flexible topological structure and its local update does not need to be propagated; hence, it can represent a size field with reduced memory and timing requirements. In this study, a new element-sizing scheme based on unstructured meshes is presented and applied in geometry-based adaptive mesh generation of planar and parametric surfaces. In general, three main issues are resolved in this study.Firstly, the basic concepts of element sizes and element-sizing fields and mathematical meanings behind these concepts are discussed. In addition, well gradation is the fundamental requirement for a size field. In this study, this requirement is formularised and automatic algorithms are suggested to smooth a size field to be a well-graded one.Secondly, an element-sizing framework that considers the requirement of geometry-based adaptive mesh generation is presented for planar surfaces. It involves the modules and algorithms for the creation, combination and smoothing of element-sizing fields. Compared with previous studies, the proposed framework are enhanced in the following aspects.(1) An Delaunay triangulation based scheme for geometry proximity calculation is proposed to help automate the setup process of the geometry-adaptive size field.(2) The boundary recovery procedure of the Delaunay meshing algorithm is enhanced so that the proposed framework can manage the problems with geometric or physical features inside the problem domain. Moreover, these feature curves can intersect with each other.(3) Without the help of some auxiliary structures, a brute-force procedure that queries the size value of a point in an unstructured background mesh is very slow and its practical efficiency is unacceptable. An improved walk-through algorithm that does not depend on any auxiliary structure is suggested in this study.(4) In the initial background mesh, most of the mesh nodes lie on the geometry boundary. If small sizes are defined on these nodes, the interior sizes interpolated with a linear scheme are very small as well, and tiny elements will be generated in both the domain boundary and interior. Solutions are integrated in our framework to help generate a coarse mesh in the interior of the domain.Finally, the above framework for plane surfaces is extended for parametric surfaces by using Riemannian metric tools. Many issues that are specific for parametric surfaces are resolved. By integrating the extended element-sizing framework, an advancing front surface mesher is enhanced with the fully automatic capability of generating geometry-based adaptive meshes.