Domain Decomposition for Mean Curvature Flow of Surface Polygonal Meshes

TL;DR

This work addresses accelerating mean curvature flow (MCF) smoothing of polygonal surface meshes through domain decomposition (DDM). It develops adapted Robin and Ventcell transmission conditions within an optimized Schwarz framework, including a new discrete normal derivative, and demonstrates applicability to three polygonal Laplacians ($\Delta_A$, $\Delta_P$, $\Delta_F$). The study shows that $\Delta_P$ and $\Delta_F$ preserve texture better by minimizing tangential components, while $\Delta_A$ can introduce artifacts; the non-overlapping Robin-transmission DDM provides smooth, connected reconstructions comparable to global MCF and offers potential parallel speedups. Overall, the method provides a flexible, parallelizable approach to polygonal mesh smoothing with broad relevance to computer graphics and geometric processing, and it highlights the importance of interface treatment in domain-decomposed MCF.

Abstract

We examine the use of domain decomposition for potentially more efficient mean curvature flow of surface meshes, whose faces are arbitrary simple polygons. We first test traditional domain decomposition methods with and without overlap of deconstructed domains. And we present adapted Robin transmission conditions of optimized Schwarz method. We then analyze the resulting smoothing from the point of view of shape quality and texture deformation. By decomposing the initial mesh into two sub-meshes, we solve two smaller boundary value problems instead of one big problem, and we can process these two tasks almost entirely in parallel.

Figures (10)

• Figure 1: Mesh smoothing through mean curvature flow (MCF) of a surface polygonal mesh (far left), whose upper part has the double of the level of detail of the lower part. We decompose the mesh into two sub-meshes (separate the upper and the lower part). We then apply five iterations of MCF with our adapted Robin transmission conditions and time step $dt=0.05$. The show the resulting meshes for Laplacians of AlexaWardetzky2011 (second image), PtackovaVelho2021 (third image), and Fujiwara1995 (fourth image). The last fifth image is the resulting mesh after MCF without decomposition for the Laplacian of Fujiwara1995. We can see that even though the level of details of the sub-meshes is different and the interface is a curved line, our adapted Robin transmission conditions work well and the resulting meshes are well smoothed. In the case of the Laplacian of Fujiwara1995, our domain decomposition method actually prevents the tangential shifting of vertices caused by the different levels of details, which can be observed as the significant texture deformation in the fifth image.
• Figure 2: Smoothing out bumps on a polygonal mesh (far left) with one iteration of mean curvature flow (\ref{['eq:mcfWithBoundary']}) with $dt=0.01$, using the Laplacians of Fujiwara1995 (center left), AlexaWardetzky2011 (center right), and PtackovaVelho2021 (farright). On general polygonal meshes, the Laplacian of PtackovaVelho2021 gives better results, as the one of AlexaWardetzky2011 quickly develops artifacts, and the Laplacian of Fujiwara1995 inflicts tangential shifts resulting in texture deformation.
• Figure 3: The decomposition of a mesh $M$ into two sub-meshes $M_A$, $M_B$ without overlap. We denote the interface as $\Gamma$. The points on the interface are stored in two copies: $\Gamma_A$ belonging to $M_A$ and $\Gamma_B$ belonging to $M_B$.
• Figure 4: The decomposition of a mesh into two sub-meshes $M_A$, $M_B$ with an one-face-wide overlap. The new boundary curve $\Gamma_A$ of $M_A$ corresponds now to an interior curve in $M_B$ (dashed line), and the points on boundary $\Gamma_B$ of $M_B$ correspond to a set of interior points of $M_A$.
• Figure 5: MCF of a general polygonal mesh (upper left), which has been decomposed into two sub-meshes with an overlap (upper right). We apply two iterations of MCF with $dt = 0.05$ and employ the Laplacian of PtackovaVelho2021. The resulting mesh without decomposition is shown in the lower left and with decomposition and Schwarz alternating method in the lower right.

Theorems & Definitions (2)

• Example 2.1
• Definition 3.1