3.1 Geometrical Mesh Quality

Well shaped elements are especially important for finite element methods. Certain geometrical quality measures and ratios can be defined to evaluate the shape of an element. Such purely geometrical criteria lead by nature to an isotropic mesh density distribution. They are implemented without the knowledge of the physical problem at hand. Various types of not so well shaped elements are depicted in Fig. 3.1.

Figure 3.1: Various types of not so well shaped elements and some parameters.

Some simple parameters of triangles in two dimensions and tetrahedra in three dimensions are the edge lengths $l_{i}$, (dihedral) angles $\alpha_{i}$, volume $V$, circumsphere radius $R$, insphere radius $r$, and normal distance $d_{i}$ of a vertex from its opposite edge or triangle. The height of an element is defined as $d_{\min}$. It can be desirable to avoid elements with a too small height or too obtuse (dihedral) angles. While such large angles are related to the error of a finite element approximation, small angles can have a negative effect on the condition number of the stiffness matrix [5,80]. The following are some examples for the definition of an element measure where greater values of $Q$ denote a higher quality shape.

$$Q_{1} = \frac{l_{\min}}{R}$$ (3.1)

$$Q_{2} = \frac{r}{l_{\max}}$$ (3.2)

$$Q_{3} = \frac{r}{R}$$ (3.3)

$$Q_{4} = \frac{l_{\min}}{l_{\max}}$$ (3.4)

$$Q_{5} = \frac{V}{l_{\max}^{3}}$$ (3.5)

To be more precise one has to note that not all of these definitions provide a well behaved quality measure in all dimensions. For example only $Q_{2,3,5}$ satisfy

$$\lim_{area, volume \rightarrow 0} Q = 0$$ (3.6)

$Q_{3}$ is actually the inverted aspect ratio as it is commonly defined for three-dimensional elements. $Q_{5}$ has been used by [8,87]. $Q_{1}$ behaves well in two dimensions, but it is inapt to capture the shape of three-dimensional sliver elements as depicted in Fig. 3.1. The volume of such a sliver element can be made arbitrarily small while at the same time $Q_{1}$ remains a positive constant. The dihedral angles of the sliver element can be changed to the better or to the worse while the measure $Q_{1}$ can be kept constant. One can gradually transform a well shaped element into a sliver element by moving one vertex without changing $l_{\min}, R$ and hence without changing $Q_{1}$. This interesting fact follows from the important relation between the angles of an element and its $l_{i}, R$ parameters. Only in two dimensions it is possible to derive a formula for triangles which describes the relation between the edge length and its opposite angle (Fig. 3.2 and Fig. 3.3).

$$\sin \alpha_{i} = \frac{l_{i}}{2R}$$ (3.7)

Figure 3.2: The relation between the edge length and its opposite angle in a triangle follows from $2\xi = 180-2\psi -2\zeta = 180-2\alpha _{i}$ and therefore $l_{i}=2R\sin(90-\xi)=2R\sin(\alpha_{i})$.

Figure 3.3: With constant edge length and circumsphere radius the opposite dihedral angle in a tetrahedron can have arbitrary values.

Assuming that $R$ is constant, the smallest angle will correspond to the shortest edge. Hence, a minimum bound for $Q_{1}$ also is a bound to the smallest angle in the triangle. Lacking this relationship in three dimensions many quality criteria fail to guarantee bounded dihedral angles. $Q_{4}$ has more the nature of a one-dimensional measure. It fails even in two dimensions to avoid badly shaped triangles. With a fixed $Q_{4}$ a triangle may have any obtuse angle. It will depend on the application whether or not angles are important and which criteria prove to be useful.

An important conclusion can be drawn. Whatever means are pursued to improve the element quality, the employed technique must fit to the applied measure. Otherwise termination is not ensured. If the number of needles and caps should be reduced by means of refinement but the number of slivers by local transformations, the former must be distinguished from the latter. The refinement should then be controlled by e.g. $Q_{1}$ which will not detect the slivers. And the local transformations could be applied in a following step for elements which do not comply with e.g. $Q_{3}$.