Understanding FEDOF in easifem (Part 1)
FEDOF
FEDOF denotes the finite element degree of freedom. The concept of degree of freedom depends upon the type of finite element approximation (basis functions) used for the problem.
These series of notes will present the FEDOF concept for for H1 conforming Hierarchical and Lagrange basis functions.
- For Lagrange polynomials the degree of freedoms are associatd with the nodes of the mesh. In this case, a node implies a point in the mesh. This point can be a vertex, somewhere on the edge, face, or interior of the element.
- For Hierarchical polynmials the degree of freedoms are associated with the modes. They can be associated with nodes, edges, faces and interior of the elements. In this case the node has an abstract meaning. But we will associcate them with the vertex, edge, face, and interior basis functions.
- In the case of H1 conforming basis functions with Hierarchical polynomials, the orientation of edge and faces with respect to the master element is very important. However, for Lagrange polynomials the orientation is not so much needed if we generate the nodes correctly for higher order mesh.
Mesh related stuff
In this note we will focus on 2D mesh with quadrilateral.
Information about this domain is given below:
engine: GMSH 4.1 0 8
version: 4.1
nsd: 2
minNptrs: 1
maxNptrs: 9
minElemNum: 1
maxElemNum: 16
tNodes: 9
tEntitiesForNodes: 9
tEntitiesForElements: 9
tElements: 4, 8, 4, 0
total mesh of volume: 0
total mesh of surface: 1
total mesh of curve: 4
total mesh of point: 4
The information about the cell mesh is given below.
total nodes: 9
total elements: 4
tEdges: 0
tFaces: 12
For a 2D-mesh, we do not define the edges. For 2D mesh we only have Cell, faces, and vertices.
From the domain we can get the pointer to mesh of cells. In this case a cell is made of quadrilateral elements.
cellMesh => dom%GetMeshPointer()
We can get the maximum number of nodes in an element by using GetMaxNNE() method.
maxNNE = cellMesh%GetMaxNNE()
For this mesh the value of maxNNE will be 4, as the maximum number of nodes in a quadrilateral element is 4.
You can think maxNNE as the maximum number of vertex connectivity for elements of mesh.
We can get the global element number of a local element number by using following.
iel = cellMesh%GetGlobalElemNumber(localElement=1)
Global element number is what you see in the mesh file, it the element number assigned by the mesh generator. Whereas, local element number is the location of this element inside the cellMesh object. We do not hope you to know the local element number, therefore, you can convert global element number to local element by using the method GetLocalElemNumber().
In this case iel will be 13.
Vertex connectivity of mesh
The vertex connectivity of this element can be obtained by using the following code.
con = cellMesh%GetConnectivity(globalElement=1, islocal=yes, opt="V")
To get vertex connectivity we use opt="V", for face connectivity we will use opt="F", for edge connectivity we will use opt="E", and for cell connectivity we will use opt="C", and for all connectivity we will use opt="A".
The vertex connectivity of global element 13 (local element 1) is given below.
1 5 9 8
Note that the vertex connectivity is returned in terms of global vertex numbers.
Face connectivity of mesh
Now lets get the face connectivity for this element.
We can enquire about the total number of faces in the mesh by using the method GetTotalFaces() as shown below.
CALL Display(cellMesh%GetTotalFaces(), "Total faces in mesh:")
The face connectivity of global Element 13 (local element 1) can be obtained by using the following.
con = cellMesh%GetConnectivity(globalElement=1, islocal=yes, opt="F")
CALL Display(con, "Connectivity of faces for globalElement=1: ")
The result is given below.
1 2 3 4
These are local face numbers. Note that there are no global face numbers because the mesh file does not provide face number. Therefore, the concept of global or local face number is not valid here.
We can get the orientation of these faces by using GetFaceOrientation(), the face orientation is a two dimensional array, where first dimension contains the face orientation flag in two dimensions and the second dimension contains the local face number. The following code is related to getting the face orientation.
The number of rows in faceOrientation should be 3.
CALL Reallocate(orient, 3, SIZE(con))
CALL cellMesh%GetFaceOrientation(globalElement=1, islocal=yes, ans=orient, &
nrow=nrow, ncol=ncol)
CALL Display(orient(1:nrow, 1:ncol), &
"Face orientations for globalElement=1: ")
The result is given below.
Face orientations for globalElement=1:
---------------------------------------
1 1 -1 -1
Let's understand the concept of face orientation.
The vertex connectivity of global element 13 is given by [1 5 9 8]. The faces are oriented in the counter clockwise direction.
- Local face 1 is form vertex 1 to vertex 5
- Local face 2 is form vertex 5 to vertex 9
- Local face 3 is form vertex 9 to vertex 8
- Local face 4 is form vertex 8 to vertex 1
In 2D, positive orientation of a face means from small to large vertex number. Therefore,
- Local face 1, form vertex 1 to vertex 5, has positive orientation.
- Local face 2, form vertex 5 to vertex 9, has positive orientation.
- Local face 3, form vertex 9 to vertex 8, has negative orientation.
- Local face 4, form vertex 8 to vertex 1, has negative orientation.
If you try to get the edge data for 2D mesh you will get nothing because edge data is note defined for 2D mesh.
Getting vertex connectivity of the a face. To get the node numbers of a face we need to describe the globalElement and the local face number as shown below.
con = cellMesh%GetFacetConnectivity(globalElement=1, iface=1, islocal=yes)
CALL Display(con, "Node numbers of face 1 of element 1: ")
con = cellMesh%GetFacetConnectivity(globalElement=1, iface=2, islocal=yes)
CALL Display(con, "Node numbers of face 2 of element 1: ")
con = cellMesh%GetFacetConnectivity(globalElement=1, iface=3, islocal=yes)
CALL Display(con, "Node numbers of face 3 of element 1: ")
con = cellMesh%GetFacetConnectivity(globalElement=1, iface=4, islocal=yes)
CALL Display(con, "Node numbers of face 4 of element 1: ")
The result is given below.
Node numbers of face 1 of element 1: [1 5]
Node numbers of face 2 of element 1: [5 9]
Node numbers of face 3 of element 1: [9 8]
Node numbers of face 4 of element 1: [8 1]
Note these face connectivities are for local face of element.
Getting all connectivity of mesh
we can get all connectivity of an element by using the following code.
con = cellMesh%GetConnectivity(globalElement=1, islocal=yes, opt='V')
CALL Display(con, "Connectivity of vertex for localElement=1: ")
Scalar FEDOF
Getting the degree of freedom for scalar field.
- Vertex DOF connectivity for localElement=1:
1 5 9 8
- Face DOF connectivity for localElement=1:
10 11 12 13
- Cell DOF connectivity for localElement=1:
22
- All DOF connectivity for localElement=1:
1 5 9 8 10 11 12 13 22
There is another way to get FEDOF connectivity.
From the cellMesh we get the connectivity with opt "V", "F", "C", and "A" for vertex, face, cell, and all DOF connectivity respectively. Then for each of these abstract node number we can call following methods on FEDOF object.
GetVertextDOF()GetFaceDOF()GetCellDOF()