Cell

A $D$-grid can be decomposed in a group of $D$-cells called cell. Grids and cells are objects of the same topolgy. A cell can bee seen as the smallest possible grid in terms of vertices.

Let a cell be an element of the grid. It can be seen as a d-tuple of the form :

A $D$-cell is itself a hyper rectangle also known as a d-orthotope.

Cell Count

The total number of cells in a grid is given by :

$$N_C = (N_1-1) \times (N_2-1) \times \cdots \times (N_{D}-1)$$

Indexing

Each cell is indexed relatively to its origin node : $$n_{i^1i^2~\cdots~i^{D}} \Rightarrow c_{i^1i^2~\cdots~i^{D}} ~~,~~ i^j \in [1,N_j-1] ~~,~~ j \in [1,D]$$

And, like nodes, cells can be indexed regarding their tuple address.

Corner Nodes

The number of corners in a cell is :

$$2^D$$

Corner nodes ar obtained by a set of permutations about the origin :

$$\begin{array}{r|cccc} Origin & i^1 & i^1 & i^2 & \cdots & i^{D} \\ \\ & + & + & + & \cdots & + \\ \\ k & \sigma_k(i^1) & \sigma_k(i^1) & \sigma_k(i^3) & \cdots & \sigma_k(i^D) \end{array}$$

$$\\ \sigma_k(i^j) \in \{0,1\} ~~,~~ i^j \in [1,N_j-1] ~~,~~ j \in [1,D] ~~,~~ k \in [1,2^D]$$

Thus :

$$\begin{array}{r|cccc} Origin & i^1 & i^1 & i^2 & \cdots & i^{D} \\ & + & + & + & \cdots & + \\ 1 & 0 & 0 & 0 & \cdots & 0 \\ 2 & 1 & 0 & 0 & \cdots & 0 \\ 3 & 0 & 1 & 0 & \cdots & 0 \\ 4 & 1 & 1 & 0 & \cdots & 0 \\ 5 & 0 & 0 & 1 & \cdots & 0 \\ 6 & 1 & 0 & 1 & \cdots & 0 \\ 7 & 0 & 1 & 1 & \cdots & 0 \\ 8 & 1 & 1 & 1 & \cdots & 0 \\ & \vdots & \vdots & \vdots & & \vdots \\ 2^D & 1 & 1 & 1 & \cdots & 1 \end{array}$$

Note that this procedure respect the way we ordered all the nodes of the grid ; so they are also sorted in ascent order this way.