An Alternating Mesh Quality Metric Scheme for Efficient Mesh Quality Improvement

In the numerical solution of partial differential equations (PDEs), high-quality meshes are crucial for the stability, accuracy, and convergence of the associated PDE solver. Mesh quality improvement is often performed to improve the quality of meshes before use in numerical solution of the PDE. Mesh smoothing (performed via optimization) is one popular technique for improving the mesh quality; it does so by making adjustments to the vertex locations. When an inefficient mesh quality metric is used to design the optimization problem, and hence also to measure the mesh quality within the optimization procedure, convergence of the optimization method can be much slower than desired. However, for many applications, the choice of mesh quality metric and the optimization problem should be considered fixed. In this paper, we propose a simple mesh quality metric alternation scheme for use in the mesh optimization process. The idea is to alternate the use of the original inefficient mesh quality metric with a more efficient mesh quality metric on alternate iterations of the mesh optimization procedure in order to reduce the time to convergence, while solving the original mesh quality improvement problem. Typical results of using our application scheme to solve mesh quality improvement problems yield approximately 40-55% improvement in comparison to the original mesh optimization procedure. More frequent use of the efficient metric results in greater speed-ups.