5.2 Mesh Adaptation

In this section, selected algorithms presented in Section 3.2 are modified for templated meshes. These modifications are discussed and investigated. Algorithms presented in this section take a templated mesh $\Gamma$ as the main input to generate an adapted templated mesh $\Omega$ where ${\operatorname{AT}}(\Gamma )$ and ${\operatorname{AT}}(\Omega )$ geometry-conform to the same geometry. The input and the output of a mesh adaptation algorithm are even geometry-conforming to the same templated geometry in ideal situations.

Figure 5.6: Non-locality of operations in the structure instance

Because the operation is actually performed on the corresponding template vertex, the operation is not local in the structure instance.

Operations on elements in the structure instance of a templated mesh might affect other parts of the structure instance. Even if an algorithm performs an operation which would be local in ${\operatorname{AT}}({\Gamma})$, the effect is potentially non-local as due to the instance-template-relation (cf. Figure 5.6). For almost every operation performed on a templated mesh, all consequences for ${\operatorname{AT}}({\Gamma})$ have to be considered. In some cases, an operation which is beneficial in one area of the mesh potentially has negative impacts in other areas. For example, a refinement operation of one template element automatically refines all instances of that element potentially yielding non-desired element sizes in some ares of the structure instance. However, if such an operation is necessary or highly beneficial, like mesh refinement in an adaptive process, template splitting and cloning (cf. Section 5.2.1) is an option.

Figure 5.7: Conformity issues with operations performed on template boundaries

Moving the vertex (encircled in red) breaks the conformity of the structure instance.

For operations on or near template boundaries, which are part of any instance interface, Lemma 4.2 has to be fulfilled. This results in less freedom for operations, like vertex movement-based quality improvement algorithms, and is especially problematic for templated meshes with irregular instance graphs. However, valid operations in the interior of a template have no negative effects on the conformity of the structure instance. An example of algorithmic issues due to Lemma 4.2 is shown in Figure 5.7.

In the following, a selection of mesh adaptation algorithms for templated meshes are presented and discussed in detail.