Connectivity Oblivious Merging of Triangulations

Abstract

Fig. 1:Merging a buddha tetrahedral mesh with a background grid. Our technique is able to handle meshes with distinct levels of refinement -observe how the internal tetrahedra of the buddha have not been refined.Abstract-Simplicial meshes are extremely useful as discrete approximations of continuous spaces in numerical simulations. In some applications, however, meshes need to be modified over time. Mesh update operations are often expensive and brittle, which tends to make the numerical simulations unstable. In this paper we propose an alternative technique for updating simplicial meshes that undergo geometric and topological changes. We exploit the property that a Weighted Delaunay Triangulation (WDT) can be used to implicitly define the connectivity of a mesh. Instead of explicitly maintaining connectivity information, we simply keep a collection of weights associated with each vertex. This approach allows for a simple way to merge triangulations, which we illustrate with examples in 2D and 3D.

A Weighted Delaunay Triangulation Framework for Merging Triangulations in a Connectivity Oblivious Fashion

“…The methodology we propose can be seen as a mechanism to extend mesh generation and object modelling techniques, since most operations can be implemented within our framework. The framework we introduce in the sections that follow revise and extend the framework originally presented in [SSE*12], with an alternate way for weight computation based on a linear program formulation. We presented the mathematical derivation and provide in the Appendix details about the coefficients of the linear program formulation.…”

“…Unfortunately, this cannot be done for all meshes, as the linear program might not have a solution in the cases where the original mesh cannot be represented by a WDT. In this work we review and extend the framework presented in [SSE*12] to compute approximate vertex weights. In the original framework, we proposed one method based on a breadth‐first traversal of the mesh that locally enforces convexity constraints on the lifting construction.…”