D06CCF renumbers the vertices of a given mesh using a Gibbs method, in order the reduce the bandwidth of Finite Element matrices associated with that mesh.
| SUBROUTINE D06CCF ( | NV, NELT, NEDGE, NNZMAX, NNZ, COOR, EDGE, CONN, IROW, ICOL, ITRACE, IWORK, LIWORK, RWORK, LRWORK, IFAIL) |
| INTEGER | NV, NELT, NEDGE, NNZMAX, NNZ, EDGE(3,NEDGE), CONN(3,NELT), IROW(NNZMAX), ICOL(NNZMAX), ITRACE, IWORK(LIWORK), LIWORK, LRWORK, IFAIL |
| double precision | COOR(2,NV), RWORK(LRWORK) |
D06CCF uses a Gibbs method to renumber the vertices of a given mesh in order to reduce the bandwidth of the associated finite element matrix
A. This matrix has elements
aij such that:
This routine reduces the bandwidth
m, which is the smallest integer such that
aij≠0 whenever
i-j>m (see
Gibbs et al. (1976) for details about that method). D06CCF also returns the sparsity structure of the matrix associated with the renumbered mesh.
This routine is derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).
Gibbs N E, Poole W G Jr and Stockmeyer P K (1976) An algorithm for reducing the bandwidth and profile of a sparse matrix
SIAM J. Numer. Anal. 13 236–250
- 1: NV – INTEGERInput
On entry:
the total number of vertices in the input mesh.
Constraint:
NV≥3.
- 2: NELT – INTEGERInput
On entry:
the number of triangles in the input mesh.
Constraint:
NELT≤2×NV-1.
- 3: NEDGE – INTEGERInput
On entry:
the number of the boundary edges in the input mesh.
Constraint:
NEDGE≥1.
- 4: NNZMAX – INTEGERInput
On entry:
the maximum number of nonzero entries in the matrix based on the input mesh. It is the dimension of the arrays
IROW and
ICOL as declared in the subroutine from which D06CCF is called.
Constraint:
4×NELT+NV≤NNZMAX≤NV2.
- 5: NNZ – INTEGEROutput
On exit: the number of nonzero entries in the matrix based on the input mesh.
- 6: COOR(2,NV) – double precision arrayInput/Output
-
On entry: COOR1i contains the x co-ordinate of the ith input mesh vertex, for i=1,…,NV; while COOR2i contains the corresponding y co-ordinate.
On exit: COOR1i will contain the x co-ordinate of the ith renumbered mesh vertex, for i=1,…,NV; while COOR2i will contain the corresponding y co-ordinate.
- 7: EDGE(3,NEDGE) – INTEGER arrayInput/Output
-
On entry: the specification of the boundary or interface edges. EDGE1j and EDGE2j contain the vertex numbers of the two end points of the jth boundary edge. EDGE3j is a user-supplied tag for the jth boundary or interface edge: EDGE3j=0 for an interior edge and has a nonzero tag otherwise.
On exit: the renumbered specification of the boundary or interface edges.
Constraint:
1≤EDGEij≤NV and EDGE1j≠EDGE2j, for i=1,2 and j=1,2,…,NEDGE.
- 8: CONN(3,NELT) – INTEGER arrayInput/Output
-
On entry: the connectivity of the mesh between triangles and vertices. For each triangle j, CONNij gives the indices of its three vertices (in anticlockwise order), for i=1,2,3 and j=1,2,…,NELT.
On exit: the renumbered connectivity of the mesh between triangles and vertices.
Constraint:
1≤CONNij≤NV and CONN1j≠CONN2j and CONN1j≠CONN3j and CONN2j≠CONN3j, for i=1,2,3 and j=1,2,…,NELT.
- 9: IROW(NNZMAX) – INTEGER arrayOutput
- 10: ICOL(NNZMAX) – INTEGER arrayOutput
On exit: the first
NNZ elements contain the row and column indices of the nonzero elements supplied in the finite element matrix
A.
- 11: ITRACE – INTEGERInput
On entry: the level of trace information required from D06CCF.
- ITRACE≤0
- No output is generated.
- ITRACE=1
- Information about the effect of the renumbering on the finite element matrix are output. This information includes the half bandwidth and the sparsity structure of this matrix before and after renumbering.
- ITRACE>1
- The output is similar to that produced when ITRACE=1 but the sparsities (for each row of the matrix, indices of nonzero entries) of the matrix before and after renumbering are also output.
- 12: IWORK(LIWORK) – INTEGER arrayWorkspace
- 13: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which D06CCF is called.
Constraint:
LIWORK≥maxNNZMAX,20×NV.
- 14: RWORK(LRWORK) – double precision arrayWorkspace
- 15: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which D06CCF is called.
Constraint:
LRWORK≥NV.
- 16: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Not applicable.
Not applicable.
In this example, a geometry with two holes (two interior circles inside an exterior one) is considered. The geometry has been meshed using the simple incremental method (
D06AAF) and it has
250 vertices and
402 triangles (see
Figure 1). The routine
D06BAF is used to renumber the vertices, and one can see the benefit in terms of the sparsity of the finite element matrix based on the renumbered mesh (see
Figure 2).
Figure 1: Mesh of the geometry
Figure 2: Sparsity of the matrix before (top) and after (bottom) the renumbering
