\index{numbering}The UFC specification dictates a certain numbering of the vertices,edges etc. of the cells of a finite element mesh. First, an \emph{adhoc} numbering is picked for the vertices of each cell. Then, theremaining entities are ordered based on a simple rule, as described indetail below.\section{Basic concepts}\index{mesh entity}\index{topological dimension}The topological entities of a cell (or mesh) are referred to as\emph{mesh entities}. A mesh entity can be identified by a pair$(d, i)$, where $d$ is the topological dimension of the mesh entity and $i$is a unique index of the mesh entity. Mesh entities are numberedwithin each topological dimension from $0$ to $n_d-1$, where $n_d$ isthe number of mesh entities of topological dimension $d$.For convenience, mesh entities of topological dimension $0$ arereferred to as \emph{vertices}, entities of dimension $1$as \emph{edges}, entities of dimension $2$ as \emph{faces}, entities of\emph{codimension} $1$ as \emph{facets} and entities of codimension$0$ as \emph{cells}. These concepts are summarized inTable~\ref{tab:entities}.Thus, the vertices of a tetrahedron are identified as$v_0 = (0, 0)$, $v_1 = (0, 1)$ and $v_2 = (0, 2)$,the edges are$e_0 = (1, 0)$, $e_1 = (1, 1)$, $e_2 = (1, 2)$,$e_3 = (1, 3)$, $e_4 = (1, 4)$ and $e_5 = (1, 5)$,the faces (facets) are$f_0 = (2, 0)$, $f_1 = (2, 1)$, $f_2 = (2, 2)$ and $f_3 = (2, 3)$,and the cell itself is$c_0 = (3, 0)$.\begin{table}[H]\linespread{1.2}\selectfont  \begin{center}    \begin{tabular}{|l|c|c|}      \hline      Entity & Dimension & Codimension \\      \hline      Vertex & $0$       & -- \\      Edge   & $1$       & -- \\      Face   & $2$       & -- \\      & & \\      Facet  & --      &  $1$ \\      Cell   & --      &  $0$ \\      \hline    \end{tabular}    \caption{Named mesh entities.}    \label{tab:entities}  \end{center}\end{table}\section{Numbering of vertices}\index{vertex numbering}For simplicial cells (intervals, triangles and tetrahedra) of a finiteelement mesh, the vertices are numbered locally based on thecorresponding global vertex numbers. In particular, a tuple ofincreasing local vertex numbers corresponds to a tuple of increasingglobal vertex numbers.  This is illustrated inFigure~\ref{fig:numbering_example_triangles} for a mesh consisting oftwo triangles. \begin{figure}[htbp]  \begin{center}    \psfrag{v0}{$v_0$}    \psfrag{v1}{$v_1$}    \psfrag{v2}{$v_2$}    \psfrag{0}{$0$}    \psfrag{1}{$1$}    \psfrag{2}{$2$}    \psfrag{3}{$3$}    \includegraphics[width=8cm]{eps/numbering_example_triangles.eps}    \caption{The vertices of a simplicial mesh are numbered locally      based on the corresponding global vertex numbers.}    \label{fig:numbering_example_triangles}  \end{center}\end{figure}For non-simplicial cells (quadrilaterals and hexahedra), the numberingis arbitrary, as long as each cell is isomorphic to the correspondingreference cell by matching each vertex with the corresponding vertexin the reference cell. This is illustrated inFigure~\ref{fig:numbering_example_quadrilaterals} for a meshconsisting of two quadrilaterals.\begin{figure}[htbp]  \begin{center}    \psfrag{v0}{$v_0$}    \psfrag{v1}{$v_1$}    \psfrag{v2}{$v_2$}    \psfrag{v3}{$v_3$}    \psfrag{0}{$0$}    \psfrag{1}{$1$}    \psfrag{2}{$2$}    \psfrag{3}{$3$}    \psfrag{4}{$4$}    \psfrag{5}{$5$}    \includegraphics[width=8cm]{eps/numbering_example_quadrilaterals.eps}    \caption{The local numbering of vertices of a non-simplicial mesh      is arbitrary, as long as each cell is isomorphic to the      reference cell by matching each vertex to the corresponding      vertex of the reference cell.}    \label{fig:numbering_example_quadrilaterals}  \end{center}\end{figure}\section{Numbering of other mesh entities}When the vertices have been numbered, the remaining mesh entities arenumbered within each topological dimension based on a\emph{lexicographical ordering} of the corresponding ordered tuples of\emph{non-incident vertices}.As an illustration, consider the numbering of edges (the mesh entitiesof topological dimension one) on the reference triangle inFigure~\ref{fig:orderingexample,triangle}. To number the edges of thereference triangle, we identify for each edge the correspondingnon-incident vertices. For each edge, there is only one such vertex(the vertex opposite to the edge). We thus identify the three edges inthe reference triangle with the tuples $(v_0)$, $(v_1)$ and $(v_2)$. Thefirst of these is edge $e_0$ between vertices $v_1$ and $v_2$ oppositeto vertex $v_0$, the second is edge $e_1$ between vertices $v_0$ and$v_2$ opposite to vertex $v_1$, and the third is edge $e_2$ betweenvertices $v_0$ and $v_1$ opposite to vertex $v_2$.Similarly, we identify the six edges of the reference tetrahedron withthe corresponding non-incident tuples $(v_0, v_1)$, $(v_0, v_2)$,$(v_0, v_3)$, $(v_1, v_2)$, $(v_1, v_3)$ and $(v_2, v_3)$. The first of these isedge $e_0$ between vertices $v_2$ and $v_3$ opposite to vertices $v_0$and $v_1$ as shown in Figure~\ref{fig:orderingexample,tetrahedron}.\begin{figure}[htbp]  \begin{center}    \psfrag{v0}{$v_0$}    \psfrag{v1}{$v_1$}    \psfrag{v2}{$v_2$}    \psfrag{e0}{$e_0$}    \includegraphics[width=5cm]{eps/ordering_example_triangle.eps}    \caption{Mesh entities are ordered based on a lexicographical ordering      of the corresponding ordered tuples of non-incident vertices.      The first edge $e_0$ is non-incident to vertex $v_0$.}    \label{fig:orderingexample,triangle}  \end{center}\end{figure}\begin{figure}[htbp]  \begin{center}    \psfrag{v0}{$v_0$}    \psfrag{v1}{$v_1$}    \psfrag{v2}{$v_2$}    \psfrag{v3}{$v_3$}    \psfrag{e0}{$e_0$}    \includegraphics[width=5cm]{eps/ordering_example_tetrahedron.eps}    \caption{Mesh entities are ordered based on a lexicographical ordering      of the corresponding ordered tuples of non-incident vertices.      The first edge $e_0$ is non-incident to vertices $v_0$ and $v_1$.}    \label{fig:orderingexample,tetrahedron}  \end{center}\end{figure}\subsection{Relative ordering}The relative ordering of mesh entities with respect to other incidentmesh entities follows by sorting the entities by their (global)indices. Thus, the pair of vertices incident to the first edge $e_0$of a triangular cell is $(v_1, v_2)$, not $(v_2, v_1)$. Similarly, thefirst face $f_0$ of a tetrahedral cell is incident to vertices $(v_1,v_2, v_3)$.For simplicial cells, the relative ordering in combination with theconvention of numbering the vertices locally based on global vertexindices means that two incident cells will always agree on theorientation of incident subsimplices. Thus, two incident triangleswill agree on the orientation of the common edge and two incidenttetrahedra will agree on the orientation of the common edge(s) and theorientation of the common face (if any). This is illustrated inFigure~\ref{fig:orientation_example_triangles} for two incidenttriangles sharing a common edge.\begin{figure}[htbp]  \begin{center}    \psfrag{v0}{$v_0$}    \psfrag{v1}{$v_1$}    \psfrag{v2}{$v_2$}    \psfrag{v3}{$v_3$}    \includegraphics[width=9cm]{eps/orientation_example_triangles.eps}    \caption{Two incident triangles will always agree on the      orientation of the common edge.}    \label{fig:orientation_example_triangles}  \end{center}\end{figure}\subsection{Limitations} The UFC specification is only concerned with the ordering of meshentities with respect to entities of larger topological dimension. Inother words, the UFC specification is only concerned with the orderingof incidence relations of the class $d - d'$ where $d > d'$. Forexample, the UFC specification is not concerned with the ordering ofincidence relations of the class $0 - 1$, that is, the ordering ofedges incident to vertices.\newpage\section{Numbering schemes for reference cells}The numbering scheme is demonstrated below for cellsisomorphic to each of the five reference cells.\subsection{Numbering for intervals}\begin{table}[H]\linespread{1.2}\selectfont  \begin{center}    \begin{tabular}{|c|c|c|}      \hline      Entity & Incident vertices & Non-incident vertices \\      \hline      \hline      $v_0 = (0, 0)$ & $(v_0)$ & $(v_1)$ \\      \hline      $v_1 = (0, 1)$ & $(v_1)$ & $(v_0)$ \\      \hline      $c_0 = (1, 0)$ & $(v_0, v_1)$ & $\emptyset$ \\      \hline    \end{tabular}    \caption{Numbering of mesh entities on intervals.}    \label{tab:interval,entities}  \end{center}\end{table}\subsection{Numbering for triangular cells}\begin{table}[H]\linespread{1.2}\selectfont  \begin{center}    \begin{tabular}{|c|c|c|}      \hline      Entity & Incident vertices & Non-incident vertices \\      \hline      \hline      $v_0 = (0, 0)$ & $(v_0)$ & $(v_1, v_2)$ \\      \hline      $v_1 = (0, 1)$ & $(v_1)$ & $(v_0, v_2)$ \\      \hline      $v_2 = (0, 2)$ & $(v_2)$ & $(v_0, v_1)$ \\      \hline      $e_0 = (1, 0)$ & $(v_1, v_2)$ & $(v_0)$ \\      \hline      $e_1 = (1, 1)$ & $(v_0, v_2)$ & $(v_1)$ \\      \hline      $e_2 = (1, 2)$ & $(v_0, v_1)$ & $(v_2)$ \\      \hline      $c_0 = (2, 0)$ & $(v_0, v_1, v_2)$ & $\emptyset$ \\      \hline    \end{tabular}    \caption{Numbering of mesh entities on triangular cells.}    \label{tab:triangle,entities}  \end{center}\end{table}\subsection{Numbering for quadrilateral cells}\begin{table}[H]\linespread{1.1}\selectfont  \begin{center}    \begin{tabular}{|c|c|c|}      \hline      Entity & Incident vertices & Non-incident vertices \\      \hline      \hline      $v_0 = (0, 0)$ & $(v_0)$ & $(v_1, v_2, v_3)$ \\      \hline      $v_1 = (0, 1)$ & $(v_1)$ & $(v_0, v_2, v_3)$ \\      \hline      $v_2 = (0, 2)$ & $(v_2)$ & $(v_0, v_1, v_3)$ \\      \hline      $v_3 = (0, 3)$ & $(v_3)$ & $(v_0, v_1, v_2)$ \\      \hline      $e_0 = (1, 0)$ & $(v_2, v_3)$ & $(v_0, v_1)$ \\      \hline      $e_1 = (1, 1)$ & $(v_1, v_2)$ & $(v_0, v_3)$ \\      \hline      $e_2 = (1, 2)$ & $(v_0, v_3)$ & $(v_1, v_2)$ \\      \hline      $e_3 = (1, 3)$ & $(v_0, v_1)$ & $(v_2, v_3)$ \\      \hline      $c_0 = (2, 0)$ & $(v_0, v_1, v_2, v_3)$ & $\emptyset$ \\      \hline    \end{tabular}    \caption{Numbering of mesh entities on quadrilateral cells.}    \label{tab:quadrilateral,entities}  \end{center}\end{table}\newpage\subsection{Numbering for tetrahedral cells}\begin{table}[H]\linespread{1.1}\selectfont  \begin{center}    \begin{tabular}{|c|c|c|}      \hline      Entity & Incident vertices & Non-incident vertices \\      \hline      \hline      $v_0 = (0, 0)$ & $(v_0)$ & $(v_1, v_2, v_3)$ \\      \hline      $v_1 = (0, 1)$ & $(v_1)$ & $(v_0, v_2, v_3)$ \\      \hline      $v_2 = (0, 2)$ & $(v_2)$ & $(v_0, v_1, v_3)$ \\      \hline      $v_3 = (0, 3)$ & $(v_3)$ & $(v_0, v_1, v_2)$ \\      \hline      $e_0 = (1, 0)$ & $(v_2, v_3)$ & $(v_0, v_1)$ \\      \hline      $e_1 = (1, 1)$ & $(v_1, v_3)$ & $(v_0, v_2)$ \\      \hline      $e_2 = (1, 2)$ & $(v_1, v_2)$ & $(v_0, v_3)$ \\      \hline      $e_3 = (1, 3)$ & $(v_0, v_3)$ & $(v_1, v_2)$ \\      \hline      $e_4 = (1, 4)$ & $(v_0, v_2)$ & $(v_1, v_3)$ \\      \hline      $e_5 = (1, 5)$ & $(v_0, v_1)$ & $(v_2, v_3)$ \\      \hline      $f_0 = (2, 0)$ & $(v_1, v_2, v_3)$ & $(v_0)$ \\      \hline      $f_1 = (2, 1)$ & $(v_0, v_2, v_3)$ & $(v_1)$ \\      \hline      $f_2 = (2, 2)$ & $(v_0, v_1, v_3)$ & $(v_2)$ \\      \hline      $f_3 = (2, 3)$ & $(v_0, v_1, v_2)$ & $(v_3)$ \\      \hline      $c_0 = (3, 0)$ & $(v_0, v_1, v_2, v_3)$ & $\emptyset$ \\      \hline    \end{tabular}    \caption{Numbering of mesh entities on tetrahedral cells.}        \label{tab:tetrahedron,entities}   \end{center}\end{table}\vfill\newpage\subsection{Numbering for hexahedral cells}\vspace{-0.5cm}\begin{table}[H]\small\linespread{1.2}\selectfont  \begin{center}    \begin{tabular}{|c|c|c|}      \hline      Entity & Incident vertices & Non-incident vertices \\      \hline      \hline      $v_0 = (0, 0)$ & $(v_0)$ & $(v_1, v_2, v_3, v_4, v_5, v_6, v_7)$ \\      \hline      $v_1 = (0, 1)$ & $(v_1)$ & $(v_0, v_2, v_3, v_4, v_5, v_6, v_7)$ \\      \hline      $v_2 = (0, 2)$ & $(v_2)$ & $(v_0, v_1, v_3, v_4, v_5, v_6, v_7)$ \\      \hline      $v_3 = (0, 3)$ & $(v_3)$ & $(v_0, v_1, v_2, v_4, v_5, v_6, v_7)$ \\      \hline      $v_4 = (0, 4)$ & $(v_4)$ & $(v_0, v_1, v_2, v_3, v_5, v_6, v_7)$ \\      \hline      $v_5 = (0, 5)$ & $(v_5)$ & $(v_0, v_1, v_2, v_3, v_4, v_6, v_7)$ \\      \hline      $v_6 = (0, 6)$ & $(v_6)$ & $(v_0, v_1, v_2, v_3, v_4, v_5, v_7)$ \\      \hline      $v_7 = (0, 7)$ & $(v_7)$ & $(v_0, v_1, v_2, v_3, v_4, v_5, v_6)$ \\      \hline      $e_0 = (1, 0)$ & $(v_6, v_7)$ & $(v_0, v_1, v_2, v_3, v_4, v_5)$ \\      \hline      $e_1 = (1, 1)$ & $(v_5, v_6)$ & $(v_0, v_1, v_2, v_3, v_4, v_7)$ \\      \hline      $e_2 = (1, 2)$ & $(v_4, v_7)$ & $(v_0, v_1, v_2, v_3, v_5, v_6)$ \\      \hline      $e_3 = (1, 3)$ & $(v_4, v_5)$ & $(v_0, v_1, v_2, v_3, v_6, v_7)$ \\      \hline      $e_4 = (1, 4)$ & $(v_3, v_7)$ & $(v_0, v_1, v_2, v_4, v_5, v_6)$ \\      \hline      $e_5 = (1, 5)$ & $(v_2, v_6)$ & $(v_0, v_1, v_3, v_4, v_5, v_7)$ \\      \hline      $e_6 = (1, 6)$ & $(v_2, v_3)$ & $(v_0, v_1, v_4, v_5, v_6, v_7)$ \\      \hline      $e_7 = (1, 7)$ & $(v_1, v_5)$ & $(v_0, v_2, v_3, v_4, v_6, v_7)$ \\      \hline      $e_8 = (1, 8)$ & $(v_1, v_2)$ & $(v_0, v_3, v_4, v_5, v_6, v_7)$ \\      \hline      $e_9 = (1, 9)$ & $(v_0, v_4)$ & $(v_1, v_2, v_3, v_5, v_6, v_7)$ \\      \hline      $e_{10} = (1, 10)$ & $(v_0, v_3)$ & $(v_1, v_2, v_4, v_5, v_6, v_7)$ \\      \hline      $e_{11} = (1, 11)$ & $(v_0, v_1)$ & $(v_2, v_3, v_4, v_5, v_6, v_7)$ \\      \hline      $f_0 = (2, 0)$ & $(v_4, v_5, v_6, v_7)$ & $(v_0, v_1, v_2, v_3)$ \\      \hline      $f_1 = (2, 1)$ & $(v_2, v_3, v_6, v_7)$ & $(v_0, v_1, v_4, v_5)$ \\      \hline      $f_2 = (2, 2)$ & $(v_1, v_2, v_5, v_6)$ & $(v_0, v_3, v_4, v_7)$ \\      \hline      $f_3 = (2, 3)$ & $(v_0, v_3, v_4, v_7)$ & $(v_1, v_2, v_5, v_6)$ \\      \hline      $f_4 = (2, 4)$ & $(v_0, v_1, v_4, v_5)$ & $(v_2, v_3, v_6, v_7)$ \\      \hline      $f_5 = (2, 5)$ & $(v_0, v_1, v_2, v_3)$ & $(v_4, v_5, v_6, v_7)$ \\      \hline      $c_0 = (3, 0)$ & $(v_0, v_1, v_2, v_3, v_4, v_5, v_6, v_7)$ & $\emptyset$ \\      \hline    \end{tabular}    \caption{Numbering of mesh entities on hexahedral cells.}    \label{tab:hexahedron,entities}  \end{center}\end{table}