Matching affine simplicial meshes and shape regularity

Fix an integer $d\ge 2$. The underlying coordinate index set of ${\mathbb{R}}^d$ is $I_d=\{0,\dots ,d-1\}$. In this definition the conventional labels $1\le i\le d$ are relabeled by $i\mapsto i-1$: write ${\widehat{x}}_i:=\widehat{x}(i-1)$, and let $e_i\in {\mathbb{R}}^d$ be the function on $I_d$ with $e_i(i-1)=1$ and $e_i(k)=0$ for $k\mathrm{\ne }i-1$. The vertex and barycentric labels $0,\dots ,d$ are distinct from these underlying coordinate indices. The reference open simplex is ${\widehat{S}}_d:=\{\widehat{x}\in {\mathbb{R}}^d:{\widehat{x}}_i>0(1\le i\le d),{\sum }_{i=1}^d{\widehat{x}}_i<1\},$ with closure $\overline{{\widehat{S}}_d}=\mathrm{conv}\{0,e_1,\dots ,e_d\}$.

A matching affine simplicial mesh of a bounded polyhedral open set $\mathrm{\Omega }\subset {\mathbb{R}}^d$ is a finite set $\mathcal{T}$ of closed, nondegenerate $d$-simplices such that their union is $\bar{\mathrm{\Omega }}$, their interiors are pairwise disjoint, $\mathrm{\Omega }$ is the interior of their union, and the intersection of two distinct cells is either empty or a whole face of each. Thus no lower-dimensional crack is omitted from $\mathrm{\Omega }$ while its surrounding cells are retained. Every interior facet is incident to exactly two cells and every boundary facet to one. The faces of a cell are the convex hulls of its vertex subsets. A facet means a face of dimension $d-1$.

For each cell $K\in \mathcal{T}$, an affine bijection $F_K:\overline{{\widehat{S}}_d}\to K,F_K(\widehat{x})=b_K+A_K\widehat{x},$ with invertible $A_K\in {\mathbb{R}}^{d\times d}$ is a reference map. Choose the vertex order so $\det A_K>0$; this is the orientation convention used for Piola formulas. Write $h_K:=\mathrm{diam}(K)$ and let ${\rho }_K$ be the radius of the largest Euclidean ball contained in $K$. A family $\{{\mathcal{T}}_h{\}}_{h\in H}$ is shape regular if there is a finite $\sigma$ such that $h_K\mathrm{/}{\rho }_K\le \sigma$ for every $h\in H$ and $K\in {\mathcal{T}}_h$. This universal condition is vacuous for an empty family. This definition uses radius; references using the diameter of the inscribed ball differ by a factor of two.

An affine image of the reference square is a parallelogram. A general quadrilateral requires a nonaffine map and is outside this affine definition; curved-cell approximation is not included here.