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.

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Understanding hexahedral (hex-) mesh structures is important for a number of hex-mesh generation and optimization tasks. However, due to various configurations of the singularities in a valid pure hex-mesh, the structure (or base complex) of the mesh can be arbitrarily complex. In this work, we present a first and effective method to help meshing practitioners understand the possible configurations in a valid 3D base complex for the characterization of their complexity. In particular, we propose a strategy to decompose the complex hex-mesh structure into multi-level sub-structures so that they can be studied separately, from which we identify a small set of the sub-structures that can most efficiently represent the whole mesh structure. Furthermore, from this set of sub-structures, we attempt to define the first metric for the quantification of the complexity of hex-mesh structure. To aid the exploration of the extracted multi-level structure information, we devise a visual exploration system coupled with a matrix view to help alleviate the common challenge of 3D data exploration (e.g., clutter and occlusion). We have applied our tool and metric to a large number of hex-meshes generated with different approaches to reveal different characteristics of these methods in terms of the mesh structures they can produce. We also use our metric to assess the existing structure simplification techniques in terms of their effectiveness.

Hexahedral (or Hex‐) meshes are preferred in a number of scientific and engineering simulations and analyses due to their desired numerical properties. Recent state‐of‐the‐art techniques can generate high‐quality hex‐meshes. However, they typically produce hex‐meshes with uniform element sizes and thus may fail to preserve small‐scale features on the boundary surface. In this work, we present a new framework that enables users to generate hex‐meshes with varying element sizes so that small features will be filled with smaller and denser elements, while the transition from smaller elements to larger ones is smooth, compared to the octree‐based approach. This is achieved by first detecting regions of interest (ROIs) of small‐scale features. These ROIs are then magnified using the as‐rigid‐as‐possible deformation with either an automatically determined or a user‐specified scale factor. A hex‐mesh is then generated from the deformed mesh using existing approaches that produce hex‐meshes with uniform‐sized elements. This initial hex‐mesh is then mapped back to the original volume before magnification to adjust the element sizes in those ROIs. We have applied this framework to a variety of man‐made and natural models to demonstrate its effectiveness.

Figure 1: A result of our quad mesh structure simplification framework. Left: input; middle: simplified structure; right: optimized result. Different colored blocks in the right image show different base complex components. The statistics of the meshes before and after simplification are shown by the corresponding numeric values. This result shows that our framework not only can significantly reduce the structure complexity (99%), but also preserve boundary features.

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