Worst case mesh quality in the target matrix optimization paradigm

Abstract

When considering mesh quality improvement and optimization via node movement, a particularly important goal is to attempt to improve the worst quality in the mesh when that quality is unacceptable. Standard mesh optimization methods often do not address worst case quality and can make it worse. Three fundamental methods for addressing worst case quality in mesh optimization by node movement have appeared in the literature: (1) a shifted barrier approach by Barrera et. al. (Math Comput Simul 46(2):87–102, 1998), (2) a pseudo-barrier approach by Escobar et. al. (Comput Methods Appl Mech Eng 192:2775–2787, 2000), and (3) another barrier approach by Garanzha et. al. (In: IMR 2021-29th international meshing roundtable, 2021). The first two result in ‘simultaneous untangle-optimizers’ while the last addresses worst case quality in terms of the maximum value of the optimization metric. In terms of mesh optimization within the Target Matrix Optimization Paradigm (TMOP), worst case quality is defined by two quantities: the maximum value of the optimization metric ($\max \mu$) and the minimum value of the local volume ($\min \tau$), both computed over the mesh sample points. In the present paper we show that the methods by the three authors cited above can be applied to the TMOP metrics and used on both linear and high-order element meshes. Unfortunately, the first two methods increase $\max \tau$ but do not address $\max \mu$. The third method addresses $\max \mu$, but fails to address $\min \tau$. Using a composition of functions approach, the present work creates new compound metrics that simultaneously increase $\min \tau$ and decrease $\max \mu$. This goal can also be accomplished by a two-stage optimization procedure in which the first stage untangles the initial mesh and the second stage decreases $\max \mu$. Although none of these methods provide a guarantee that the worst case quality will be improved to the point that the quality becomes acceptable, it is shown by numerical examples that they can be very effective.