Polyhedral Discretizations for Elliptic PDEs

TL;DR

This paper benchmarks BFEM and VEM against traditional FEM on polyhedral and polygonal meshes for elliptic PDEs relevant to computer graphics, using a large-scale dataset spanning multiple domains, problems, and meshing strategies. It systematically compares solve-time versus accuracy across direct and iterative solvers in 2D and 3D, highlighting that high-quality triangulations often outperform polygonal meshes under direct solves, while polygonal methods can be competitive with iterative solvers and under higher-order or barycentric formulations. Key findings show higher-order and barycentric approaches can close the gap with VEM in some cases, but robustness to mesh quality and non-conforming meshes remains a challenge, especially for polygonal discretizations. The work provides a replicable benchmark and practical guidance for developers of polyhedral meshing techniques and related solvers, while outlining future avenues such as higher-order barycentric bases and distributed 3D computations to realize potential advantages of polygonal discretizations.

Abstract

We study the use of polyhedral discretizations for the solution of heat diffusion and elastodynamic problems in computer graphics. Polyhedral meshes are more natural for certain applications than pure triangular or quadrilateral meshes, which thus received significant interest as an alternative representation. We consider finite element methods using barycentric coordinates as basis functions and the modern virtual finite element approach. We evaluate them on a suite of classical graphics problems to understand their benefits and limitations compared to standard techniques on simplicial discretizations. Our analysis provides recommendations and a benchmark for developing polyhedral meshing techniques and corresponding analysis techniques.

Figures (26)

• Figure 1: Several frames of a soft stuck puzzle piece falling under gravity using a Voronoi (top), tiled (middle), and triangular (bottom) mesh with a similar number of vertices. The three simulations require approximately the same time to solve the linear system, while the tiled simulation has fewer Newton iterations.
• Figure 2: Domains used in our case study.
• Figure 3: Visualization of the solution for different heat sources (first five figures) and the biaxial load (last figure).
• Figure 4: Visualization of the solution for elastic problems.
• Figure 5: Displacement polyhedral meshes by displacing a grid.