Optimization-based Methods in High-Order Mesh Generation and Untangling

Abstract

High-order numerical methods for solving PDEs have the potential to deliver higher solution accuracy at a lower cost than their low-order counterparts. To fully leverage these high-order computational methods, they must be paired with a discretization of the domain that accurately captures key geometric features. In the presence of curved boundaries, this requires a high-order curvilinear mesh. Consequently, there is a lot of interest in high-order mesh generation methods. The majority of such methods warp a high-order straight-sided mesh through the following three-step process. First, they add additional nodes to a low-order mesh to create a high-order straight-sided mesh. Second, they move the newly added boundary nodes onto the curved domain (i.e., apply a boundary deformation). Finally, they compute the new locations of the interior nodes based on the boundary deformation. We have developed a mesh warping framework based on optimal weighted combinations of nodal positions. Within our framework, we develop methods for optimal affine and convex combinations of nodal positions, respectively. We demonstrate the effectiveness of the methods within our framework on a variety of high-order mesh generation examples in two and three dimensions. As with many other methods in this area, the methods within our framework do not guarantee the generation of a valid mesh. To address this issue, we have also developed two high-order mesh untangling methods. These optimization-based untangling methods formulate unconstrained optimization problems for which the objective functions are based on the unsigned and signed angles of the curvilinear elements. We demonstrate the results of our untangling methods on a variety of two-dimensional triangular meshes.