Contents
Dof on Edges in Three Dimensions
We describe general idea of the data structures generated in subroutine dof3edge for three dimensional tetrahedron grids.
help dof3edge
DOF3EDGE dof structure for the lowest order edge elements.
[elem2edge,edge] = DOF3EDGE(elem) constructs data structure
for the lowest order edge element. elem is the connectivity matrix for a
3-D triangulation. elem2edge is the elementwise pointer from elem to dof
indices. edge is the edge matrix. The orientation of edge is
from the node with smaller index to bigger one. The orientation of local
edges of a tetrahedron formed by [1 2 3 4] is lexicographic order:
[1 2], [1 3], [1 4], [2 3], [2 4], [3 4]
[elem2edge,edge,elem2edgeSign] = DOF3EDGE(elem) also output elem2edgeSign
which records the consistency of the local and global edge orientation.
If elem is ascend ordered, then elem2edgeSign is 1 and not needed.
See also dofedge.
Doc: <a href="matlab:ifem dof3edgedoc">dof3edgedoc</a>
Copyright (C) Long Chen. See COPYRIGHT.txt for details.
Local Labeling of DOFs
node = [1,0,0; 0,1,0; 0,0,0; 0,0,1]; elem = [1 2 3 4]; localEdge = [1 2; 1 3; 1 4; 2 3; 2 4; 3 4]; edge = zeros(20,2); edge([1 12 5 20 11 4],:) = localEdge; elem2edge = [1 12 5 20 11 4]; figure(1); clf; set(gcf,'Units','normal'); set(gcf,'Position',[0,0,0.6,0.4]); subplot(1,2,1) showmesh3(node,elem); view(-14,12); findnode3(node); findedge(node,localEdge,'all'); subplot(1,2,2) showmesh3(node,elem); view(-14,12); % findnode3(node); findedge(node,edge,elem2edge');
The six dofs associated to edges in a tetrahedron is sorted in the ordering [1 2; 1 3; 1 4; 2 3; 2 4; 3 4]. Here [1 2 3 4] are local indices of vertices.
For a triangulation consisting of several tetrahedrons, all local edges are collected and the duplication is eliminated to form edges of the triangulation. Therefore the global indices of edges is different with the local one. elem2edge(:,1:6) records the mapping from local to global indices.
display(elem2edge)
elem2edge =
1 12 5 20 11 4
Orientation of Edges
node = [1,0,0; 0,1,0; 0,0,0; 0,0,1]; elem = [2 4 3 1]; localEdge = [elem(:,1) elem(:,2); elem(:,1) elem(:,3); elem(:,1) elem(:,4); ... elem(:,2) elem(:,3); elem(:,2) elem(:,4); elem(:,3) elem(:,4)]; edge = sort(localEdge,2); elem2edge = [1 2 3 4 5 6]; elem2edgeSign = [1 1 -1 -1 -1 -1]; figure(1); clf; set(gcf,'Units','normal'); set(gcf,'Position',[0,0,0.6,0.4]); subplot(1,2,1) showmesh3(node,elem); view(-14,12); tempnode = node(elem(:),:); findnode3(tempnode); findedge(node,localEdge,'all','vec'); subplot(1,2,2) showmesh3(node,elem); view(-14,12); findnode3(node); findedge(node,edge,'all','vec');
The edge [i,j] is orientated in the direction such that i<j, where i,j are global indices. The edges formed by local indices may not be consistent with this orientation. Therefore elem2edgeSign is used to indicate the consistency.
The nodal indices in the left figure is local while that in the right is the global one. The local direction and global direction of edges is also indicated by different edge vectors.
display(elem2edge) display(elem2edgeSign)
elem2edge =
1 2 3 4 5 6
elem2edgeSign =
1 1 -1 -1 -1 -1
An Example
[node,elem] = cubemesh([0,1,0,1,0,1],1); [elem2edge,edge,elem2edgeSign] = dof3edge(elem); figure(1); clf; showmesh3(node,elem,[],'FaceAlpha',0.25); view(-30,10); findedge(node,edge,'all','vec'); findelem3(node,elem,[1 3]'); display(elem2edge); display(elem2edgeSign);
elem2edge =
1 2 6 8 10 12
3 2 6 11 13 12
4 5 6 15 16 18
4 7 6 17 16 19
1 5 6 9 10 18
3 7 6 14 13 19
elem2edgeSign =
1 1 1 1 1 1
1 1 1 -1 1 1
1 1 1 1 1 1
1 1 1 1 1 -1
1 1 1 1 1 1
1 1 1 1 1 -1
