updated appendix for meshes

\end{cppcode}

\chapter{Ordering of Mesh Subentities}

\chapter{Mesh Entity Topologies}

\label{appendix:topologies}

\inline{Update for polygonal and polyhedral topologies}

The cells of a mesh are specified by their subvertices and every other subentity is determined implicitly by the cell topology and its subvertices.

Hence, the subvertices of a cell must follow a specific order which is shown in \cref{fig:entity topologies} for the entity topologies currently built into the TNL library.

The ordering of the subvertices is inspired by the VTK library \cite{VTK} and extended with a specific numbering of other subentities which can be utilized by numerical schemes.

In non-polyhedral meshes, cells are specified by their subvertices and every other subentity is determined implicitly by the cell topology and its subvertices.

Hence, the subvertices of a cell must follow a specific order which is shown in \cref{fig:entity topologies} for the entity topologies currently implemented in TNL (except polyhedron).

The ordering of the subvertices is inspired by VTK \cite{VTK} and extended with a specific numbering of other subentities which can be utilized by numerical schemes.

For example, the faces of a simplex are numbered such that each face has the same local index as the opposite vertex.

\begin{figure}[h]

    \end{subfigure}

    \\

    \vspace{1em}

    \begin{subfigure}{.45\textwidth}

    \begin{subfigure}{.3\textwidth}

        \centering

        \includegraphics[scale=0.8]{./figures/meshes/topologies/triangle.pdf}

        \caption{triangle}

    \end{subfigure}

    \begin{subfigure}{.45\textwidth}

    \begin{subfigure}{.3\textwidth}

        \centering

        \includegraphics[scale=0.8]{./figures/meshes/topologies/quadrilateral.pdf}

        \caption{quadrilateral}

    \end{subfigure}

    \begin{subfigure}{.3\textwidth}

        \centering

        \includegraphics[scale=0.8]{./figures/meshes/topologies/polygon.pdf}

        \caption{polygon}

    \end{subfigure}

    \\

    \vspace{1em}

    \begin{subfigure}{.45\textwidth}

        \includegraphics[scale=0.8]{./figures/meshes/topologies/hexahedron.pdf}

        \caption{hexahedron}

    \end{subfigure}

    \caption{Built-in entity shapes and the ordering of subvertices and faces.}

    \\

    \vspace{1em}

    \begin{subfigure}{.45\textwidth}

        \centering

        \includegraphics[scale=0.8]{./figures/meshes/topologies/wedge.pdf}

        \caption{wedge}

    \end{subfigure}

    \begin{subfigure}{.45\textwidth}

        \centering

        \includegraphics[scale=0.8]{./figures/meshes/topologies/pyramid.pdf}

        \caption{pyramid}

    \end{subfigure}

    \caption{Mesh entity topologies in the TNL library and the ordering of subvertices and faces.}

    \label{fig:entity topologies}

\end{figure}

label("$v_7$", V7, NW+(0.0,0.5));

pair x = (V0+V1+V2+V3)/4;

pair y = x+(0.2, -0.5);

pair y = x+(0.8, -0.2);

draw(x--y, PEN+dotted, ARROW);

label("$f_0$ (bottom)", y, SE);

x = (V0+V1+V5+V4)/4 + (0.05, 0.1);

y = x + (-0.25, -0.9);

x = (V0+V1+V5+V4)/4 + (0.1, 0.1);

y = x + (-0.2, -0.75);

draw(x--y, PEN+dotted, ARROW);

label("$f_1$ (front)", y, S);

label("$f_2$ (right)", y, E);

x = (V2+V3+V6+V7)/4 + (-0.05, -0.1);

y = x + (0.4, 0.9);

y = x + (0.4, 0.75);

draw(x--y, PEN+dotted, ARROW);

label("$f_3$ (back)", y, NE);

label("$f_4$ (left)", y, S);

x = (V4+V5+V6+V7)/4;

y = x + (-0.2, 0.5);

y = x + (-0.8, 0.2);

draw(x--y, PEN+dotted, ARROW);

label("$f_5$ (top)", y, N);

label("$f_5$ (top)", y, W);

pen PEN = fontsize(10) + linewidth(1.0);

unitsize(2cm);

pair v0 = (0.0, 0.0);

pair v1 = (1.0, 0.1);

pair v2 = (0.5, 0.3);

pair v3 = (0.6, 0.75);

pair v4 = (1.1, 1.0);

pair vn2 = (0.5, 1.1);

pair vn1 = (-0.1, 0.8);

dot(v0, black+5.0);

dot(v1, black+5.0);

dot(v2, black+5.0);

dot(v3, black+5.0);

dot(v4, black+5.0);

dot(vn2, black+5.0);

dot(vn1, black+5.0);

draw(v0--v1--v2--v3--v4--vn2--vn1--cycle, PEN);

label("$v_0$", v0, SW+(0.0,-0.5));

label("$v_1$", v1, SE+(0.0,-0.5));

label("$v_2$", v2, W+(0.0,-0.5));

label("$v_{n-2}$", vn2, N+(0.0,0.5));

label("$v_{n-1}$", vn1, W+(0.0,0.5));

label("$f_0$", (0.5, 0.05), S);

label("$f_1$", (0.7, 0.35), E);

label("$f_2$", (0.43, 0.4), N);

label("$f_{n-2}$", (0.35, 1.0), NW);

label("$f_{n-1}$", (-0.05, 0.4), W);