Curvilinear Mesh Adaptation

Abstract

This paper aims at addressing the following issue. Assume a unit square: $\varOmega = \{(x^1,x^2) \in [0,1] \times [0,1]\}$ and a Riemannian metric $g_{ij}(x^1,x^2)$ defined on U. Assume a mesh $\mathscr {T}$ of U that consist in non overlapping valid quadratic triangles that are potentially curved. Is it possible to build a unit quadratic mesh of U i.e. a mesh that has quasi-unit curvilinear edges and quasi-unit curvilinear triangles? This paper aims at providing an embryo of solution to the problem of curvilinear mesh adaptation. The method that is proposed is based on standard differential geometry concepts. At first, the concept of geodesics in Riemannian spaces is quickly presented: the geodesic between two points as well as the unit geodesic starting at a given point with a given direction are the two main tools that allow us to address our issue. Our mesh generation procedure is done in two steps. At first, points are distributed in the unit square U in a frontal fashion, ensuring that two points are never too close to each other in the geodesic sense. Then, a simple isotropic Delaunay triangulation of those points is created. Curvilinear edge swaps as then performed in order to build the unit mesh. Notions of curvilinear mesh quality is defined as well that allow to drive the edge swapping procedure. Examples of curvilinear unit meshes are finally presented.