An algorithm for triangulating smooth three‐dimensional domains immersed in universal meshes

Abstract

We describe an algorithm to recover a boundary-fitting triangulation for a bounded C 2 -regular domain Ω ⊂ R 3 immersed in a nonconforming background mesh of tetrahedra. The algorithm consists in identifying a polyhedral domain h bounded by facets in the background mesh and morphing h into a boundary-fitting polyhedral approximation Ω h of Ω. We discuss assumptions on the regularity of the domain, on element sizes and on specific angles in the background mesh that appear to render the algorithm robust. With the distinctive feature of involving just small perturbations of a few elements of the background mesh that are in the vicinity of the immersed boundary, the algorithm is designed to benefit numerical schemes for simulating free and moving boundary problems. In such problems, it is now possible to immerse an evolving geometry in the same background mesh, called a universal mesh, and recover conforming discretizations for it. In particular, the algorithm entirely avoids remeshing-type operations and its complexity scales approximately linearly with the number of elements in the vicinity of the immersed boundary. We include detailed examples examining its performance. KEYWORDSbackground meshes, evolving boundary, immersed boundary, mesh relaxation, moving mesh, 3D meshing 84

DVRlib: A C++ library for geometric mesh improvement using Directional Vertex Relaxation

“…In this context, we mention that DVRlib is an integral component of on-going research on universal meshes for discretizing evolving domains (Rangarajan, Kabaria, & Lew, 2019). We refer to Sharma & Rangarajan (2019) for an application that uses DVRlib to adapt triangle meshes to conform to moving contact interfaces that arise when simulating elastic contact.…”

A shape optimization approach for simulating contact of elastic membranes with rigid obstacles

“…We mention that Algorithm 3 differs from the meshing algorithm described in Rangarajan and Lew only in the choice of the vertex relaxation algorithm. A three‐dimensional analog of Algorithm 3 for triangulating domains immersed in tetrahedral meshes is described and tested in Rangarajan et al Cognizant of the fact that details and analyses supporting Algorithm 3 are well documented in the literature, in the following, we only provide a terse description of the meshing algorithm with sufficient detail to facilitate a correct implementation. We mention conditions on the mesh size and on angles in the universal mesh, and on the regularity of the immersed domain assumed in the algorithm.…”

A Front Advancing Adaptive Triangular Mesh Dynamic Generation Algorithm and Its Application in 3D Geological Modeling

“…Research based on the triangulated network generation algorithm can be divided into two streams based on their objectives. The first stream focuses on the construction method of a background mesh [29] that provides size information for the AFT algorithm. The second stream focuses on the calculation method and unit construction strategy of candidate mesh points under the influence of the frontier edge of complex morphology.…”