Conformal marked bisection for local refinement of $n$-dimensional unstructured simplicial meshes

TL;DR

This paper introduces an $n$-dimensional marked bisection method for unstructured conformal meshes, enabling effective local refinement in adaptive applications through a novel multi-stage process.

Contribution

It develops a new multi-stage marked bisection algorithm that transitions to newest vertex bisection, ensuring conformality and reflection conditions for local mesh refinement.

Findings

The method successfully refines unstructured meshes locally.

It maintains mesh conformality and reflection properties.

The approach is suitable for high-dimensional adaptive applications.

Abstract

We present an $n$-dimensional marked bisection method for unstructured conformal meshes. We devise the method for local refinement in adaptive $n$-dimensional applications. To this end, we propose a mesh marking pre-process and three marked bisection stages. The pre-process marks the initial mesh conformingly. Then, in the first $n-1$ bisections, the method accumulates in reverse order a list of new vertices. In the second stage, the $n$-th bisection, the method uses the reversed list to cast the bisected simplices as reflected simplices, a simplex type suitable for newest vertex bisection. In the final stage, beyond the $n$-th bisection, the method switches to newest vertex bisection. To allow this switch, after the second stage, we check that under uniform bisection the mesh simplices are conformal and reflected. These conditions are sufficient to use newest vertex bisection, a bisection scheme guaranteeing key advantages for local refinement. Finally, the results show that the proposed bisection is well-suited for local refinement of unstructured conformal meshes.