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

TL;DR

This work extends the continuous mesh framework to high-order anisotropic meshing by redefining unit simplices as images of an ideal simplex under Riemannian isometries, unifying linear and curved elements. It shows that true isometric unit simplices exist only on flat manifolds, while提出ing quasi-unit concepts (QU1/QU2 and QU1p/QU2p) to enable practical high-order meshing on curved spaces. The authors demonstrate proof-of-concept 2D optimizations that produce unit or quasi-unit triangulations on both developable and non-developable surfaces, using a cost function that balances metric-weighted edge lengths and subtriangle areas. This formulation provides a principled route to metric-consistent high-order meshing and lays groundwork for error-estimation integration in high-order anisotropic adaptation. The impact lies in enabling curved, high-order meshes that respect input metrics, potentially reducing discretization error with fewer elements in complex geometries.

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.

Figures (12)

• Figure 1: Notation for the transformation between the reference, ideal and physical simplices.
• Figure 1: Uniform tilings $\mathcal{T}_{\triangle}$ (left) and $\mathcal{T}_{\widehat{K}}$ (right) of $\mathbb{R}^2$.
• Figure 1: Log-barrier function for $\epsilon = 0.2$.
• Figure 2: Developable surfaces in $\mathbb{R}^3$ and their parameter space $U \subseteq \mathbb{R}^2$. The isometric triangulations are uniform and geodesic on each surface but appear curved and distorted on $U$.
• Figure 3: Triangulations with $N = 10$ subdivisions which are isometric to $\mathcal{T}_{\triangle}$.

Theorems & Definitions (14)

• Definition 4.1
• Proposition 4.2
• Proof 1
• Definition 4.3
• Definition 5.1
• Proposition 5.2
• Proof 2
• Definition 5.3
• Proposition 5.4
• Proof 3