Codimensional MultiMeshing: Synchronizing the Evolution of Multiple Embedded Geometries

TL;DR

This work introduces codimensional multimeshing, a framework that encodes multiple embedded geometries of differing dimensions into a single, coherent structure via a tree of containment maps. By enabling dimension-agnostic navigation and automatic propagation of topological edits, it preserves correspondences and envelope constraints across all meshes, improving robustness for tasks like seam-preserving remeshing, periodic meshing, and multi-material simulations. The authors formalize the multimesh, implement it with a dart-based encoding, and demonstrate its utility by extending TetWild and applying it to several graphics problems, including large-scale validation on Thingi10k. The approach offers a natural alternative to ad-hoc tagging, enabling reuse of mesh algorithms across embedded geometries while maintaining topology and geometric consistency in complex workflows.

Abstract

Complex geometric tasks such as geometric modeling, physical simulation, and texture parametrization often involve the embedding of many complex sub-domains with potentially different dimensions. These tasks often require evolving the geometry and topology of the discretizations of these sub-domains, and guaranteeing a \emph{consistent} overall embedding for the multiplicity of sub-domains is required to define boundary conditions. We propose a data structure and algorithmic framework for hierarchically encoding a collection of meshes, enabling topological and geometric changes to be automatically propagated with coherent correspondences between them. We demonstrate the effectiveness of our approach in surface mesh decimation while preserving UV seams, periodic 2D/3D meshing, and extending the TetWild algorithm to ensure topology preservation of the embedded structures.

Figures (17)

• Figure 1: Containment map $\Phi_{K^k}^{L^\ell}$ from a mesh with seams to a mesh without the seam. Although all of the triangles are mapped uniquely, multiple edges on the boundary can map to the same edge on the root mesh.
• Figure 2: A multimesh of depth two, whose leaf nodes are edge meshes $E_1^1$ and $E_2^1$, has a triangle mesh node $K^2$ with a seam (red) and a root mesh $L^2$ without the seam (red).
• Figure 3: The extension $\Psi_{K^k}^{L^\ell}$, which is defined in terms of $\psi^k$ defines how the containment map $\Phi_{\overline{K}^k}^{\overline{L}^\ell}$ needs to be updated.
• Figure 4: Containment map from a $2$-dart and $1$-dart to the same $2$-dart.
• Figure 5: In a single simplex the containment map of an arbitrary dart (red square) is navigating to an anchor (green square) with switches (here switching a vertex), reading the other dart in the anchor, and then inverting the navigation.

Theorems & Definitions (7)

• Definition 1: Mesh
• Definition 2: Containment Map
• Definition 3: Multimesh
• Definition 4: Topological Operation
• Definition 5: Extension
• Definition 6: Restriction
• Definition 7: Containment Map Update