4.2 Instance Interfaces and Conformity

Instances of templated structures, which are not disjoint, form interfaces. Due to Requirement 4.2, these interfaces have a dimension which is lower than the dimension of the templated structure. These interfaces between instances are especially important for templated meshes, where ${\operatorname{AT}}({\Gamma})$ has to be a valid conforming multi-region mesh: Issues might arise on interfaces between mesh instances, where the conformity can break. For example, if an algorithm performs an operation on a mesh template and ignores other mesh templates where at least two instances of each mesh template touch each other, a non-conformity will potentially be induced in the structure instance.

The main goal of this section is to identify which properties a templated structure with mesh templates has to fulfill to be a templated mesh. Identifying mesh instance interfaces and defining neighborhood terms is therefore crucial.

The motivation for the definition of ${\operatorname{interf}}_{\operatorname{geo}}$ lies in the fact that for templated structures with mesh templates, interfaces of two different instance might be empty even though the intersection of their underlying space is not. For example, the interface of the instances formed by the transformation function $T_{1,1}$ and $T_{1,2}$ of the templated structure visualized in Figure 4.4 only includes two vertices. However, the geometry instance interface of the same instances includes all vertices and edges of both instances which touch the other instance. The instance interfaces and the geometry instance interfaces are the same for templated geometries. For templated meshes, these two terms are related as indicated by Lemma 4.1.

Lemma 1 (Instance interface properties)   Let ${\mathbb{X}}$ be a templated structure with mesh templates:

(i)

Any instance interface of ${\mathbb{X}}$ respects the corresponding geometry instance interface.

(ii)

If ${\mathbb{X}}$ is a templated mesh, the underlying space of any instance interface is equal to the corresponding geometry instance interface.

(iii)

If ${\mathbb{X}}$ is a templated mesh, any instance interface is equal to the set of all boundary elements of a mesh template, which are included in the inverse geometry instance interface, transformed by the corresponding transformation function:

$${\operatorname{interf}}\left({\mathbb{X}}, T_{i,j}, \dots \right)... ...}}_{\operatorname{geo}}\left({\mathbb{X}}, T_{i,j}, \dots\right)\right)}\right)$$ (4.3)

Proof. $$

(i)

Every element $E$ in an instance interface ${\operatorname{interf}}({\mathbb{X}}, T_1, \dots, T_k)$ is in all instances $\operatorname{inst}({\mathbb{X}}, T_i)$ and therefore also in the underlying space of all instances. Consequently,
$E \cap {\operatorname{interf}}_{\operatorname{geo}}({\mathbb{X}}, T_1, \dots, T_m) = E$ and $E$ respects the geometry instance interface.

(ii)

follows from Lemma 2.1.

(iii)

From the definition of the apply-template operator follows, that
$T_{i,j}\left( {\operatorname{bnd}}\left(X_i\right) \vert_{ T_{i,j}^{-1}\left({... ...}}_{\operatorname{geo}}\left({\mathbb{X}}, T_{i,j}, \dots\right)\right)}\right)$ is a subset of ${\operatorname{interf}}\left({\mathbb{X}}, T_{i,j}, \dots \right)$. On the other hand, assume, that there is an element $E$ in ${\operatorname{interf}}\left({\mathbb{X}}, T_{i,j}, \dots \right)$, which is not in
$T_{i,j}\left( {\operatorname{bnd}}\left(X_i\right) \vert_{ T_{i,j}^{-1}\left({... ...}}_{\operatorname{geo}}\left({\mathbb{X}}, T_{i,j}, \dots\right)\right)}\right)$: Due to (ii) and the conformity of ${\operatorname{AT}}({\mathbb{X}})$, $E$ has to be a face of an element in $T_{i,j}\left( {\operatorname{bnd}}\left(X_i\right) \vert_{ T_{i,j}^{-1}\left({... ...}}_{\operatorname{geo}}\left({\mathbb{X}}, T_{i,j}, \dots\right)\right)}\right)$. However,
$T_{i,j}\left( {\operatorname{bnd}}\left(X_i\right) \vert_{ T_{i,j}^{-1}\left({... ...}}_{\operatorname{geo}}\left({\mathbb{X}}, T_{i,j}, \dots\right)\right)}\right)$ is face-complete which is a contradiction to the assumption.

$\qedsymbol$

Figure 4.5: Non-trivial interface of a template with itself

The dimension of instance interface elements is always less than the dimension of the original structure. In particular, an element in the instance interface of $k$ different instances has a dimension of at most ${\operatorname{DIM}}({\Lambda})-k+1$. Every template has a trivial instance interface with itself: ${\operatorname{interf}}({\mathbb{X}}, T_{i, g},T_{i, g}) = \operatorname{inst}({\mathbb{X}},i,g)$. A template might also have additional interfaces with itself using different transformation functions. For example, in a templated structure based on a reflection, the template has an instance interface with itself using different transformation functions (cf. Figure 4.5).

Two instances $\operatorname{inst}({\mathbb{X}},i,g)$ and $\operatorname{inst}({\mathbb{X}},j,h)$ are said to be neighbors, if their geometry instance interface is not empty: ${\operatorname{interf}}_{\operatorname{geo}}({\mathbb{X}}, T_{i,g}, T_{j,h}) \neq \emptyset$. Two templates are said to be neighbors, if there are instances $\operatorname{inst}({\mathbb{X}},i,g)$ and $\operatorname{inst}({\mathbb{X}},j,h)$ which are neighbors.

Figure 4.6: Instance graph

Every edge of the graph represents a non-empty instance interface and is indicated using an arrow pointing from and to the boundary sections of the instance interface.

Using this neighborhood definition, a graph can be created, where each node is a template and two nodes are connected, if they are neighbors. Because two templates can have multiple different neighboring instances, this graph is in fact a multi-graph, where each edge represents the combination of the transformation functions which form the neighboring instance interface. This graph, called the instance graph of the templated structure, is reflexive (every template is neighbor of itself) and undirected (if template ${\mathbb{X}}_i$ is a neighbor of ${\mathbb{X}}_j$, then ${\mathbb{X}}_j$ is also a neighbor of ${\mathbb{X}}_i$). Figure 4.6 shows an example of an instance graph of a templated structure.

The requirement for a templated mesh given in Definition 4.4, being ${\operatorname{AT}}({\mathbb{X}})$ to be a valid conforming multi-region mesh, is hard to evaluate in practice. Therefore, more practicable lemmas and instruments are given in this section to verify, if a templated structure with mesh templates is a templated mesh.

Lemma 2 (Interface conformity Lemma)   Let ${\Gamma}$ be a templated structure with mesh templates. ${\Gamma}$ is a templated mesh, i.e., ${\operatorname{AT}}({\Gamma})$ is conforming, if for all neighboring instances $\operatorname{inst}({\Gamma},i,g)$ and $\operatorname{inst}({\Gamma},j,h)$, the following holds:

(i)

The underlying space of the instance interface is equal to their geometry instance interface: ${\operatorname{us}}({\operatorname{interf}}({\Gamma}, T_{i,g}, T_{j,h})) = {\operatorname{interf}}_{\operatorname{geo}}({\Gamma}, T_{i,g}, T_{j,h})$.

(ii)

The boundary elements of both templates transformed to the instance interface are the same: $T_{i,g}( {\mathbb{IP}}_{T_{i,g}, T_{j,h}} ) = T_{j,h}({\mathbb{IP}}_{T_{j,h}, T_{i,g}})$, where

$${\mathbb{IP}}_{T_{i,g}, T_{j,h}} := {\operatorname{bnd}}(\operato... ...1}( {\operatorname{interf}}_{\operatorname{geo}}({\Gamma}, T_{i,g}, T_{j,h}))}.$$ (4.4)

Requirement (i) ensures that the transformed boundary mesh of a template respects the corresponding geometry instance interface; requirement (ii) states that for every two neighboring mesh templates, the mesh instance interface is conforming.

Proof. Every mesh template ${\Gamma}_i$ is already conforming, so non-conformities can only occur between two elements $A,B \in {\operatorname{AT}}({\Gamma})$ which are in different mesh instances. Let $A$ and $B$ be any two elements in the structure instance ${\operatorname{AT}}({\Gamma})$ where $A$ is in the instance of template $X_i$ with transformation function $T_{i,g}$ and $B$ is in the instance of template $X_j$ with a transformation function $T_{j,h}$. Due to Requirement 4.2 of Definition 4.3, the intersection of $A$ and $B$ is either empty (if $X_i$ and $X_j$ are no neighbors) or included in the geometry instance interface of the instances. If their intersection is not empty, Requirement (i) ensures that both $A$ and $B$ are also included in the geometry instance interface. Due to (ii), $A$ and $B$ are in both interfaces and their intersection is a face of both. Therefore, ${\operatorname{AT}}({\Gamma})$ is conforming. $\qedsymbol$