2.2.1 Geometry-Conformity

In simulation scenarios, a geometry for the simulation domain is given, for which a mesh has to be generated. In this section, terms are defined to identify, if a certain mesh is the mesh of a geometry. At first, the term respect is defined for elements and sets of elements, which are compatible.

Figure 2.9: Respect property

Elements are drawn in either blue or yellow, overlapping sections are colored green. In the first picture, the yellow line respects the blue triangle and vice versa, because the intersection of the triangle and the line is the line which is a face of both. In the fourth picture, the triangle, which is overlapped by a quadrilateral, respects the blue quadrilateral but the quadrilateral does not respect the triangle because the intersection of these two elements is the triangle which in turn is a face of the triangle but not of the quadrilateral. In picture two and picture five, the yellow triangle respects the blue mesh, because it respects every mesh element. The intersection of the two quadrilaterals in picture three, visualized in green, is not a face of any quadrilateral and therefore they do not respect each other. Similarly, in picture six, the triangle and the quadrilateral do not respect each other, because the green intersection is not a face of any of these two.

In words, a set $E$ respects another set $X$, if the intersection $E \cap X$ can be represented by a union of faces of $E$. Figure 2.9 visualizes the respect property for sets and set spaces. The following holds for the respect property:

Lemma 3 (Respectness of intersection)   Let $E, A, B \in \mathfrak{L}^n$ be sets, where $E$ respects $A$ and $E$ also respects $B$. Then, $E$ respects $A \cap B$.

Proof. If either $E \cap A$, $E \cap B$, or $A \cap B$ is empty, the statement is trivially true. Let $F_1, \dots, F_k \in {\operatorname{faces}}(E)$ be the sets for representing $E \cap A = \bigcup_{i=1}^k F_i$ and $G_1, \dots, G_m$ be the sets for representing $E \cap B = \bigcup_{i=1}^m G_i$. Then,

$$E \cap (A \cap B) = (E \cap A) \cap (E \cap B) = \left(\bigcup_{i... ...\left(\bigcup_{i=1}^m G_i\right) = \bigcup_{i=1}^k \bigcup_{j=1}^m F_i \cap G_j$$ (2.4)

Because $F_i$ and $G_j$ are both faces of $E$, $F_i \cap G_j$ is also a face of $E$. Therefore, $E$ respects $A \cap B$. $\qedsymbol$

A mesh and therefore all of its elements must respect its geometry. Additionally, the mesh has to cover up the geometry.

It can be shown, that if a mesh geometry-conforms to a geometry, it also respects the geometry.

Lemma 4 (Respectness of meshes with geometry-conforming property)   Let ${\mathcal{G}}$ be a geometry, ${\mathcal{M}}$ be a mesh which geometry-conforms to ${\mathcal{G}}$, ${({\mathcal{G}}, {\widetilde{\xi}})}$ be a multi-region geometry, and ${({\mathcal{M}}, {\xi})}$ be a multi-region mesh which geometry-conforms to ${({\mathcal{G}}, {\widetilde{\xi}})}$. Then the following holds:

(i)

Every element $E$ of the mesh ${\mathcal{M}}$ respects the geometry ${\mathcal{G}}$.

(ii)

Every element $E$ of a mesh region ${\operatorname{region}}({({\mathcal{M}}, {\xi})},i)$ respects the corresponding geometry region ${\operatorname{region}}({({\mathcal{G}}, {\widetilde{\xi}})},i)$.

Proof. Every mesh element $E \in {\mathcal{M}}$ is a subset of the underlying space of the mesh. Consequently, $E$ is also a subset of the geometry ${\mathcal{G}}$ and $E \cap {\mathcal{G}}= E$. Because every element is a face of itself, $E$ respects ${\mathcal{G}}$. The second statement follows from the first. $\qedsymbol$

The term geometry-conforming of a mesh and a geometry is not related to the mesh space property conforming.

In general, a mesh ${({\mathcal{M}}, {\xi})}$, which geometry-conforms to a geometry ${\mathcal{G}}$, is not unique. It is not even guaranteed that there is a mesh in some given $n$-manifold complex, which geometry-conforms to a given geometry. For example, there is no simplex mesh which geometry-conforms to the unit three-ball. In these cases, a mesh can only approximate a given geometry. However, for each mesh, there is a unique geometry, to which the mesh is geometry-conforming to.

By definition, it is assured that the geometry of a multi-region mesh ${({\mathcal{M}}, {\xi})}$ is a geometry and ${({\mathcal{M}}, {\xi})}$ is geometry-conforming to the geometry ${\operatorname{geo}}({({\mathcal{M}}, {\xi})})$.