Solid Geometry Processing on Deconstructed Domains

TL;DR

This paper introduces a novel approach for solid geometry processing by decomposing complex shapes into overlapping subdomains, simplifying tetrahedral meshing and coupling solutions along boundary surfaces, improving robustness and accuracy.

Contribution

It proposes a new coupling method for solid domains along boundary surfaces, addressing challenges in meshing and solution coupling in complex shapes.

Findings

The method outperforms previous coupling techniques in convergence tests.

It effectively handles complex boundary surfaces in solid geometry processing.

Qualitative applications demonstrate improved results across various PDEs.

Abstract

Many tasks in geometry processing are modeled as variational problems solved numerically using the finite element method. For solid shapes, this requires a volumetric discretization, such as a boundary conforming tetrahedral mesh. Unfortunately, tetrahedral meshing remains an open challenge and existing methods either struggle to conform to complex boundary surfaces or require manual intervention to prevent failure. Rather than create a single volumetric mesh for the entire shape, we advocate for solid geometry processing on deconstructed domains, where a large and complex shape is composed of overlapping solid subdomains. As each smaller and simpler part is now easier to tetrahedralize, the question becomes how to account for overlaps during problem modeling and how to couple solutions on each subdomain together algebraically. We explore how and why previous coupling methods fail, and propose a method that couples solid domains only along their boundary surfaces. We demonstrate the superiority of this method through empirical convergence tests and qualitative applications to solid geometry processing on a variety of popular second-order and fourth-order partial differential equations.