General Framework for Intelligent Unstructured Mesh Generation

Delaunay mesh generators are fully automatic, fast and can handle complex geometries. However, users have very little control over the final mesh quality. The mesh size is only specified on boundary edges and faces and then graded inside. Delaunay meshes are nearly uniform far from the boundary. A better mesh would take into account the distance to the walls, the curvature of the walls, the local thickness of the domain, the proximity of some discontinuities such as corners, the trailing edge, etc. These geometric characteristics of the computational domain should drive the mesh generation. The mesh should be finer where local thickness is small. It should be stretched and aligned with the walls, the tangential size being a function of the wall curvature and the normal size being a function of the distance to the wall. What is called a good mesh also depends on the domain of application such as computational fluid dynamics, heat transfer, stress analysis, electromagnetism, etc. Good meshes always depend on the geometric characteristics of the domain, but in a different manner for different applications.;The "mesh sizing problem" is the problem of automatic unstructured mesh generation according to geometric characteristics of the computational domain and depending on the domain of application. The objective of this research proposal is to extend ongoing research in the field on anisotropic mesh adaptation to create a general framework for the mesh sizing problem.;The currently standard method for building a mesh with local size, stretching and orientation is by defining a Riemannian metric at every point of the domain and adapting the mesh according to this Riemannian metric. The Riemannian metric defines a space transformation that specifies the local size, stretching and orientation of the mesh to be built. To address the mesh sizing problem, one has to 1) compute geometric characteristics of the computational domain, 2) transform these geometric characteristics into a Riemannian metric, 3) store this Riemannian metric on a discrete support for use in an adaptive mesh generator.;These three steps are usually implemented with a regular grid or a binary tree that overlaps the computational domain. However, such a grid is not linked to the boundary of the domain and may not be fine enough where needed. The proposed approach uses an unstructured mesh of the domain. As the mesh is a discretization of the domain, it includes boundary points. Geometric characteristics will be computed on the mesh, starting from the boundary and sweeping the whole mesh with local algorithms. Problem 2) is solved by introducing parametrized templates that encapsulate the know-how of the user. These parametrized templates contain rules that transform geometric characteristics into a Riemannian metric according to the domain of application. Problem 3) is solved with anisotropic mesh adaptation. The Riemannian metric computed by steps 1) and 2) may be complex, with strong variations and directionality. A suitable mesh for a complex solution field, accurate enough but not too fine, is a mesh adapted to this field. So, the steps 1), 2) and 3) are done iteratively with a fourth, anisotropic mesh adaptation, step. This process yields two outputs for the price of one: the accurate Riemannian metric of the geometric characteristics of the domain, and the mesh itself adapted to this Riemannian metric.;Possible applications of this research include two-dimensional shape optimization. The optimizer tests different geometries that must be automatically and intelligently meshed so that a numerical simulation can be accurately performed. An example in three dimensions is the automatic meshing of complex geometries with thin parts as in the molding process. The mesh must be fine in the thickness direction and coarse in the tangential directions. The expected overall benefit of this research proposal is improved automation and more reliable numerical simulations through automatic intelligent unstructured mesh, thus reducing the engineering cost associated with the meshing process.