Partial Differential Equations in Applied Mathematics (Jun 2022)
Geometric parameters in the target matrix mesh optimization paradigm
Abstract
The Target Matrix Optimization Paradigm for improving mesh quality has always had, as its agenda, to create a direct path between application mesh quality requirements and a set of target matrices that can result in an optimal mesh that meets the requirements. A major step forward in this agenda is made through the introduction of a standard set of geometric parameters which can be used to analyze Jacobian-based quality metrics and to establish a fully geometric method of mesh optimization. While Jacobian-based metrics have been widely used in the past, it has not been explicitly recognized that these kinds of metrics are expressible in terms of geometric parameters related to the local volume, orientation, skew, and aspect ratio of the Jacobian and the target matrices. When this is done, one can express the global minimum of a metric in terms of relationships between the geometric parameters. This provides a better understanding of Jacobian-based metrics that have been used in the past, as well as the opportunity to devise more effective metrics. Ideally, optimization metrics should have global minimizers at which the geometric parameters of the Jacobian matrix are equal to the geometric parameters of the target. This observation allows a classification of metrics into those having a metric type and those which do not. It is shown that there are eight theoretical metric types, including the shape and shape+size types used informally in the past. Concrete metrics exist for each of the eight theoretical types. However, not all concrete typed metrics are equally well-posed. In particular, it is important that there exist concrete metrics of each metric type that are convex (or polyconvex, or invex). A well-posed typed metric should also simplify target construction. These additional requirements mean that only six of the eight theoretical metric types can be populated with well-posed, typed metrics. Nevertheless, the most important metric types are covered with well-posed metrics. With this framework, the burden of defining mesh quality is properly placed on the application because it is their requirements that determine which of the eight metric types should be used in the optimization. When this has been determined, the path toward appropriate target construction is clear.
