Weighted Squared Volume Minimization (WSVM) for Generating Uniform Tetrahedral Meshes

TL;DR

This paper presents a new algorithm, Weighted Squared Volume Minimization (WSVM), for generating high-quality tetrahedral meshes from closed triangle meshes that employs a new energy function integrating weighted squared volumes for tetrahedral elements.

Abstract

This paper presents a new algorithm, Weighted Squared Volume Minimization (WSVM), for generating high-quality tetrahedral meshes from closed triangle meshes. Drawing inspiration from the principle of minimal surfaces that minimize squared surface area, WSVM employs a new energy function integrating weighted squared volumes for tetrahedral elements. When minimized with constant weights, this energy promotes uniform volumes among the tetrahedra. Adjusting the weights to account for local geometry further achieves uniform dihedral angles within the mesh. The algorithm begins with an initial tetrahedral mesh generated via Delaunay tetrahedralization and proceeds by sequentially minimizing volume-oriented and then dihedral angle-oriented energies. At each stage, it alternates between optimizing vertex positions and refining mesh connectivity through the iterative process. The algorithm operates fully automatically and requires no parameter tuning. Evaluations on a variety of 3D models demonstrate that WSVM consistently produces tetrahedral meshes of higher quality, with fewer slivers and enhanced uniformity compared to existing methods. Check out further details at the project webpage: https://kaixinyu-hub.github.io/WSVM.github.io.

Figures (11)

• Figure 1: Minimizing volume-oriented energy promotes uniformity in the volumes of tetrahedral elements. As the resolution of the boundary mesh increases, the generated tetrahedral meshes also exhibit a greater number of degrees of freedom. Consequently, this leads to improved uniformity in tetrahedral volumes, as demonstrated by a reduced standard deviation in the volume distribution.
• Figure 2: Algorithmic pipeline. (a) Our method takes a set of watertight, high-quality manifold triangular meshes as the input boundaries. (b) The initial tetrahedral mesh, constructed via Delaunay tetrahedralization, contains numerous slivers. Slivers with dihedral angles less than $30^\circ$ are rendered in red. Minimizing the volume-oriented energy improves the uniformity of the volume of tetrahedra. (c) Subsequently, minimizing the diheral angle-oriented energy further reduces the number of slivers. (d) The final tetrahedral mesh exhibits significantly improved mesh quality. The red and blue histograms represent the volume distribution and dihedral angle distribution of tetrahedral elements, respectively. In each angle histogram, we indicate the range of dihedral angles $[\theta_{\min}, \theta_{\max}]$ via short solid lines and the average of the minimal and maximal angles $\theta_{\min}^{\mathrm{avg}}$ and $\theta_{\max}^{\mathrm{avg}}$ via long dashed lines.
• Figure 3: The flip operations for removing edges and faces. (a) The 2-to-3 flip, used for removing the red face. (b) A specific instance of an $n$-to-$m$ flip, where $n = m = 4$, is illustrated. This is achieved by applying a 2-to-3 flip followed by a 3-to-2 flip to remove the red edge.
• Figure 4: Comparison of WSVM with various methods across different metrics. In the scatter plots, data points in the top-right corner represent better performance for the minimum dihedral angle. Data points in the bottom-left corner indicate better performance for the maximum dihedral angle, condition number, edge ratio, and equiangle skewness. Lower data points in the proportion of bad elements plot suggest a smaller percentage of poor-quality elements.
• Figure 5: Comparison of WSVM with various methods, for the first two models, elements with dihedral angles less than 30 degrees are rendered in red. For the latter two models, we show the mesh visualization colored by equiangle skewness values, along with corresponding histograms indicating the distribution of skewness values. The histograms highlight the minimum, average, and maximum skewness values, represented by green bars on the plot.

Theorems & Definitions (9)

• Definition 1
• Theorem 1
• Theorem 2
• proof
• Lemma 1
• proof : Proof of Lemma \ref{['equ:lemma2_subj']}
• Lemma 2
• proof : Proof of Lemma \ref{['equ:lemma1_obj']}
• proof