Elliptic smoothing of high-order meshes using the Winslow equations

With the ever increasing desire to improve accuracy of numerical simulations, obtaining high quality meshes is of paramount importance. In light of this, the generation of valid meshes that take into account the curvature of the edges is a challenging problem that cannot be avoided to correctly solve partial differential equations. In this thesis, a method to generate high-order curvilinear finite element (FE) meshes is presented, initially proposed by Persson and Fortunato. The method is based on a smoothing technique using the Winslow non-linear elliptic equations, which translate a harmonic transformation from an initial straight-sided mesh, towards the final physical surface, taking into account its curved boundaries. These partial differential equations are solved using a continuous Galerkin FE discretisation and a splitting of the equations, allowing for an efficient resolution procedure using Picard iterations. A systematic approach is followed in this work, beginning with the smoothing of linear meshes using the Winslow equations. The description of a linear elasticity FE solver then follows, and finally the high-order mesh generation technique is implemented. Examples on two dimensional isotropic as well as boundary layer meshes show high quality results, with limited computational costs.