Evaluating Hex-mesh Quality Metrics via Correlation Analysis

Abstract

Hexahedral (hex-) meshes are important for solving partial differential equations (PDEs) in applications of scientific computing and mechanical engineering. Many methods have been proposed aiming to generate hex-meshes with high scaled Jacobians. While it is well established that a hex-mesh should be inversion-free (i.e. have a positive Jacobian measured at every corner of its hexahedron), it is not well-studied that whether the scaled Jacobian is the most effective indicator of the quality of simulations performed on inversion-free hex-meshes given the existing dozens of quality metrics for hex-meshes. Due to the challenge of precisely defining the relations among metrics, studying the correlations among different quality metrics and their correlations with the stability and accuracy of the simulations is a first and effective approach to address the above question. In this work, we propose a correlation analysis framework to systematically study these correlations. Specifically, given a large hex-mesh dataset, we classify the existing quality metrics into groups based on their correlations, which characterizes their similarity in measuring the quality of hex-elements. In addition, we rank the individual metrics based on their correlations with the accuracy and stability metrics for simulations that solve a number of elliptic PDE problems. Our preliminary experiments suggest that metrics that assess the conditioning of the elements are more correlated to the quality of solving elliptic PDEs than the others. Furthermore, an inversion-free hex-mesh with higher average quality (measured by any quality metrics) usually leads to a more accurate and stable computation of elliptic PDEs. To support our correlation study and address the lack of a publicly available large hex-mesh dataset with sufficiently varying quality metric values, we also propose a two-level perturbation strategy to generate the desired dataset from a small number of meshes to exclude the influences of element numbers, vertex connectivity, and volume sizes to our study.