Simplicial Complex in 3D

We describe general idea of the data structure generated for three dimensional triangular grids, suppose $N,NE,NF,NT$ represents the number of vertices, edges, faces and elements resprectively.

Contents

Construct Data Structure

elem is $NT\times 4$ matrix with $NT$ be the number of the tetrahedrons. elem(i,:) contain the global index of the vertex of the i-th tetrahedron. we sort the vertices of elem such that:elem(i,1)<elem(i,2)<elem(i,3)<elem(i,4). In this way, the volume is not always positive, because we sacrifice the righthand orientation of tet. We also sort vertices of face such that:face(j,1)<face(j,2)<face(j,3),and the vertices of edge such that: edge(k,1)<edge(k,2).

The local ordering of face in a tetrahedron is locFace = [2 3 4; 1 3 4; 1 2 4; 1 2 3]. The local ordering of edge in a face is locEdge = [2 3;1 3;1 2]. Now the ordering of local indices is always consistent with that of global indices.

In the following example, elem2face, a $NT\times 4$ matrix, is the elementwise pointer from elem to face indicies. elem(i,:) stores the global indicies of faces for tetrahedron i. elem2edge, a $NT\times 6$ matrix, is the elementwise pointer from elem to edge indicies. elem(j,:) stores the global indicies of edges for tetrahedron j.

Example

Generate

elem = [1 4 5 8;1 4 5 7];
node = [1,0,0;1,1,1;1,-1,-1; 0,1,0; 0,0,0;1,1,-1; 0,0,1;-1,-1,-1];
[elem2edge,~,edge] = dof3edge(elem);
[elem2face,~,face] = dof3RT0(elem);
figure(1);clf;
showmesh3(node,elem);
view(-20,20);
findnode3(node,[1,4,5,7,8]);
findelem3(node,elem);
display(edge); display(elem2edge);
display(face); display(elem2face);

edge =

           1           4
           1           5
           1           7
           1           8
           4           5
           4           7
           4           8
           5           7
           5           8


elem2edge =

           1           2           4           5           7           9
           1           2           3           5           6           8


face =

           1           4           5
           1           4           7
           1           4           8
           1           5           7
           1           5           8
           4           5           7
           4           5           8


elem2face =

           7           5           3           1
           6           4           2           1

The directions.

Direction of face.

While we sort the elem, face and edge in the increasing order of vertices. We determine the direction of a face [i j k] by the righthand rule. The direction of face is not always outword normal direction of a certain tetrahedron. We denote the direction as +1 if the direction of a face is the same with the normal direction in a certain elem, and -1 otherwise. Then for locface of a tetrahedron, the directions should be sign(volume)*[+1 -1 +1 -1].

Direction of edge.

Similarly, we determine the direction of a edge [i j] is from the node with smaller global index to bigger one(that is from i to j). The direction of edge is not always consistent with the counter clockwise direction in a face. When the edge direction is the same with the local counter clockwise direction, we set the direction as +1, otherwise -1. Then for locedge of a face, the directions should be [+1 -1 +1].

Direction of the basis.

According to our sorted elem and face, the RT0 basis on face [i j k]. Defined as $\phi_{i,j,k} = 2(\lambda_i \bigtriangledown \lambda_j \times \bigtriangledown \lambda_k+ \lambda_j \bigtriangledown \lambda_k \times \bigtriangledown \lambda_i+\lambda_k \bigtriangledown \lambda_i \times \bigtriangledown \lambda_j)$ The direction is always consistent with the face direction.

Inward normal direction.

Dlambda(t,:,1) is the gradient of lambda of the 1st index of the t-th tet. The direction of Dlambda is always inward normal direction.

[Dlambda, ~] = gradbasis3(node,elem(2,:));
display(Dlambda);

Dlambda(:,:,1) =

     1     0     0


Dlambda(:,:,2) =

     0     1     0


Dlambda(:,:,3) =

    -1    -1    -1


Dlambda(:,:,4) =

     0     0     1


Published with MATLAB® 7.11