High-order mesh untangling and smoothing using the hierarchical smoother

Abstract

In the last years, high-order methods are emerging to simulate physical phenomena in appliedsciences and engineering. However, these kind of methods rely on a high-order discretization. One ofthe most used methods to generate a high-order mesh is the a posteriori approach. First, a linearmesh that contains the required features for the simulation is generated. Then, the interpolationdegree of the mesh is increased, and the additional nodes are inserted. However, when the high-ordernodes are inserted, low-quality or inverted elements may appear if high-curved surfaces are presentin the geometry. Thus, mesh untangling and smoothing is a key step of the meshing process to obtainvalid high-order discretizations. Usually, the smoothing process is performed in two steps, as shownin [1,2]. First, the boundary nodes are moved by taking into account the quality of the boundaryelements. Then, the quality of the maximal dimension elements is optimized by moving the positionof the inner nodes. However, the boundary nodes position may constrain the quality of the wholemesh. Thus, we propose a novel framework in which we only consider the quality of the maximaldimension elements [3]. To this end, we minimize an objective function defined as the distortion ofthe maximal dimension elements, by moving the position of the inner nodes, as well as the positionin the parametric space of the face and edge nodes. Although the objective function is definedglobally, for implementation purposes, we propose to perform a node-by-node relocation process byusing a non-linear Gauss-Seidel approach. That is, we first move the edge nodes, then the face nodes,and finally the inner nodes, until convergence is achieved.