Isometric simplices for high-order anisotropic mesh adaptation. Part I: Definition and existence of isometric triangulations

TL;DR

This paper extends the concept of isometric simplices to high-order curved elements within anisotropic mesh adaptation, providing foundational definitions and proofs of concept for curved high-order mesh generation.

Contribution

It introduces a new framework for defining and analyzing isometric high-order simplices using Riemannian isometries, advancing anisotropic high-order mesh adaptation methods.

Findings

Defined high-order isometric simplices using Riemannian isometries.

Extended the notion of quasi-unitness to curved high-order elements.

Presented proofs of concept for 2D high-order isometric meshes.

Abstract

Anisotropic mesh adaptation with Riemannian metrics has proven effective for generating straight-sided meshes with anisotropy induced by the geometry of interest and/or the resolved physics. Within the continuous mesh framework, anisotropic meshes are thought of as discrete counterparts to Riemannian metrics. Ideal, or unit, simplicial meshes consist only of simplices whose edges exhibit unit or quasi-unit length with respect to a given Riemannian metric. Recently, mesh adaptation with high-order (i.e., curved) elements has grown in popularity in the meshing community, as the additional flexibility of high-order elements can further reduce the approximation error. However, a complete and competitive methodology for anisotropic and high-order mesh adaptation is not yet available. The goal of this paper is to address a key aspect of metric-based high-order mesh adaptation, namely, the adequacy between a Riemannian metric and high-order simplices. This is done by extending the notions of unit simplices and unit meshes, central to the continuous mesh framework, to high-order elements. The existing definitions of a unit simplex are reviewed, then a broader definition involving Riemannian isometries is introduced to handle curved and high-order simplices. Similarly, the notion of quasi-unitness is extended to curved simplices to tackle the practical generation of high-order meshes. Proofs of concept for unit and (quasi-)isometric meshes are presented in two dimensions.