Mesh Element Quality and Mesh Quality

For many boundary values problems, the shape of the mesh elements directly affects the numerical convergence and accuracy of the solution [54][105]. A mesh element is said to have a good quality, if these effects are beneficial. However, it mainly depends on the application, the boundary value problem, and the element type, which shapes of an element are considered to be good. There are many different methods on how to measure the quality of mesh elements [66][131]. The following geometric properties are needed for most of these measures.

$\begin{defn} % latex2html id marker 2522 [Shortest edge, longest edge] Let $E$\ ... ... $l_\textnormal{max}(E)$\ is defined as the largest face edge of $E$. \end{defn}$

$\begin{defn}[Volume, relative volume] Let $\mu$\ be the Lebesgue measure and $\m... ...torname{Vol}}^\star}(E) := \mathcal{H}^{{\operatorname{DIM}}(E)}(E)$. \end{defn}$

$\begin{defn}[$n$-circumball, circumradius] Let $S \subseteq {\mathbb{R}}^n$\ be ... ...(S)$\ is defined as the radius of the smallest $n$-circumball of $S$. \end{defn}$

$\begin{defn} % latex2html id marker 2534 [Min-containment ball, inradius, inball... ...\overline{\mathcal{B}}_r^n(\bm{x}) \subseteq E$\ is called an inball. \end{defn}$

**Figure 2.10:** Inballs for a rectangle, for a triangle, and for a tetrahedron
$\begin{subfigure} % latex2html id marker 2542 [b]{0.3\textwidth} \centering \i... ...h=2.8cm]{figures/inball_rect} \caption{Inballs of a rectangle} \end{subfigure}$ $\begin{subfigure} % latex2html id marker 2549 [b]{0.3\textwidth} \centering \i... ...idth=2.1cm]{figures/inball_tri} \caption{Inball of a triangle} \end{subfigure}$ $\begin{subfigure} % latex2html id marker 2556 [b]{0.3\textwidth} \centering \i... ...=2.24cm]{figures/inball_tet} \caption{Inball of a tetrahedron} \end{subfigure}$ Note, that the inball for the rectangle is not unique.

While there is one unique min-containment ball for every mesh element, there might be more than one inball for certain elements. For example, a non-quadratic rectangle has infinitely many inballs. However, the inball of a simplex is unique. Inballs for a quadrilateral, a triangle, and a tetrahedral are visualized in Figure 2.10.

Element measures are mappings which assign a quality value to each cell of a mesh. When using the FEM, angles near $180\degree$ cause interpolation errors which further lead to discretization errors. On the other hand, angles near 0 lead to ill-conditioned stiffness-matrices [66].

$\begin{defn} % latex2html id marker 2569 [Smallest/largest angle, smallest/large... ...orname{max}}(E)$\ is similarly defined as the largest interior angle. \end{defn}$

The smallest and largest angles of a tetrahedron are equal to the smallest and largest dihedral angles, respectively. For lines with non-planar co-facets (cf. Definition A.11), e.g., the lines of a quadrilateral, all interior angles along the line have to be taken into account when finding the smallest angle.

$\begin{defn}[Radius-edge ratio] Let $S \subseteq {\mathbb{R}}^n$\ be a simplex, ... ...-edge}}}(S) := {R_{\operatorname{circ}}}(E) / l_\textnormal{min}(E)$. \end{defn}$

If the spatial dimension is two, the minimal angle and the radius-edge ratio are related: ${\phi_{\operatorname{radius-edge}}}(S) = 1/(2\sin(\angle_{\operatorname{min}}(S)))$ . However, the radius-edge ratio is a flawed measure for tetrahedrons in ${\mathbb{R}}^3$ , because slivers, i.e., tetrahedrons which are not considered to be good elements, can have radius-edge ratios as small as $1 / \sqrt{2}$ . Other measures like the radius ratio or the numerically more robust volume-length measure, offer improved quality measurements for mesh elements in ${\mathbb{R}}^3$ .

$\begin{defn} % latex2html id marker 2579 [Radius ratio] Let $E$\ be a simplex me... ...E) := {R_{\operatorname{in}}}(E) / {R_{\operatorname{min-cont}}}(E)$. \end{defn}$

$\begin{defn} % latex2html id marker 2581 [Volume-length measure] Let $E$\ be a m... ...ame{faces}}_2(E)}{{{\operatorname{Vol}}^\star}(F)^2}} } \end{equation}\end{defn}$

Based on an element quality measure, a quality score can be calculated. The only difference between a quality measure and a quality score is, that elements with good quality have a high score. The fundamental element-level quality score enables to evaluate an entire multi-region mesh quality.

$\begin{defn} % latex2html id marker 2591 [Quality vector] Let ${({\mathcal{M}}, ... ...ty scores of all cells $C \in {\operatorname{cells}}({\mathcal{M}})$. \end{defn}$

The quality of two meshes can now be compared by using a lexicographically ordering on the quality vectors. A mesh ${\mathcal{M}}_1$ has a better quality than a mesh ${\mathcal{M}}_2$ , if $\Psi _\Phi ({\mathcal{M}}_1) > \Psi _\Phi ({\mathcal{M}}_2)$ . Creating the entire quality vector of a mesh is too time-consuming. However, identifying the worst elements is usually sufficient [75]. Nevertheless, for theoretical analysis, the quality vector with the lexicographical ordering still plays an important role.

2.2.2 Mesh Element Quality and Mesh Quality