5.2.4 Vertex Insertion and Refinement

Figure 5.14: Vertex insertion in the interior of a template

When inserting a vertex into a mesh template, it has to be distinguished, if the vertex is to be inserted in the interior or on the boundary of the mesh template. Adding a vertex in the interior of the mesh template does not conflict with Lemma 4.2 and is a safe operation as visualized in Figure 5.14. On the other hand, inserting a vertex on the boundary potentially affects other mesh templates, if the newly inserted vertex is part of any geometry instance interface.

Figure 5.15: Conformity issues with insertion of multiple vertices

If two vertices (highlighted in red) are inserted in the template $X_1$, two different edge configurations are possible for the front face. The same edge configuration has to be applied to the back side to maintain the conformity of the structure instance.

A vertex inserted on a boundary which is part of any geometry instance interface induces the insertion of vertices in all direct and indirect neighboring mesh templates. This is always possible for 2D meshes, even for templated meshes with irregular instance graphs. However, for 3D meshes, not only the inserted vertex has an effect on the neighboring templates, but also all additionally inserted edges have to match at all instance interfaces. Especially, if multiple vertices are inserted at once, the edges have to be taken into account (cf. Figure 5.15). However, the edge configuration is unique when inserting a single vertex into the boundary of a template of a 3D templated simplex mesh.

As mention in Section 3.2.3, edge bi- or trisection is often used in element refinement algorithms, where one or two new vertices are inserted on every edge marked for refinement. The refinement algorithm remains unchanged for edges which are not part of the boundary. For all other edges, the refinement algorithm can be adapted using the vertex insertion described above.