Abstract
We propose a novel deviation-based vertex reordering method for 2D mesh quality improvement. We reorder free vertices based on how likely this is to improve the quality of adjacent elements, based on the gradient of the element quality with respect to the vertex location. Specifically, we prioritize the free vertex with large differences between the best and the worst-quality element around the free vertex. Our method performs better than existing vertex reordering methods since it is based on the theory of non-smooth optimization. The downhill simplex method is employed to solve the mesh optimization problem for improving the worst element quality. Numerical results show that the proposed vertex reordering techniques improve both the worst and average element, compared to those with existing vertex reordering techniques.
Highlights
When the finite element method (FEM) is used to solve certain partial differential equations (PDEs), the quality of a mesh element is determined geometrically
We have proposed a novel vertex reordering technique for 2D mesh quality improvement such that the propagation of inequality in the quality of mesh elements is accelerated
Our idea is to prioritize the free vertex with large differences between the best and the worst-quality element, in the pseudo-active set
Summary
When the finite element method (FEM) is used to solve certain partial differential equations (PDEs), the quality of a mesh element is determined geometrically. In the local mesh optimization technique, a natural question arises about the order in which the vertices are moved: is it possible to further improve the mesh quality or the efficiency (fewer iterations/less time) of the optimization by changing the vertex ordering? Sastry et al developed a global formulation for the objective function based on the log-barrier technique, which improved the quality of all poor elements in a mesh, not just the worst one [7]. We observed that there were both high- and poor-quality elements, and the quality of the poor elements were being improved at the cost of the quality of good elements This observation led us to develop our inequality-based vertex reordering technique. Our heuristic algorithm performs better than prior vertex reordering techniques because it is based on the theory of non-smooth optimization
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