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.

2 Specification

Fortran Interface

Subroutine d06ccf (

nv, nelt, nedge, nnzmax, nnz, coor, edge, conn, irow, icol, itrace, iwork, liwork, rwork, lrwork, ifail)

Integer, Intent (In)	::	nv, nelt, nedge, nnzmax, itrace, liwork, lrwork
Integer, Intent (Inout)	::	edge(3,nedge), conn(3,nelt), ifail
Integer, Intent (Out)	::	nnz, irow(nnzmax), icol(nnzmax), iwork(liwork)
Real (Kind=nag_wp), Intent (Inout)	::	coor(2,nv)
Real (Kind=nag_wp), Intent (Out)	::	rwork(lrwork)

C Header Interface

#include <nag.h>

void

d06ccf_ (const Integer *nv, const Integer *nelt, const Integer *nedge, const Integer *nnzmax, Integer *nnz, double coor[], Integer edge[], Integer conn[], Integer irow[], Integer icol[], const Integer *itrace, Integer iwork[], const Integer *liwork, double rwork[], const Integer *lrwork, Integer *ifail)

The routine may be called by the names d06ccf or nagf_mesh_dim2_renumber.

3 Description

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

a_{i j}

such that:

a_{i j} \neq 0 \Rightarrow i ​ and ​ j ​ are vertices belonging to the same triangle.

This routine reduces the bandwidth

m

, which is the smallest integer such that

a_{i j} \neq 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).

4 References

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

5 Arguments

1: $nv$ – Integer Input

On entry: the total number of vertices in the input mesh.

Constraint:

nv \geq 3

2: $nelt$ – Integer Input

On entry: the number of triangles in the input mesh.

Constraint:

nelt \leq 2 \times nv - 1

3: $nedge$ – Integer Input

On entry: the number of boundary edges in the input mesh.

Constraint:

nedge \geq 1

4: $nnzmax$ – Integer Input

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 \times nelt + nv \leq nnzmax \leq {nv}^{2}

5: $nnz$ – Integer Output

On exit: the number of nonzero entries in the matrix based on the input mesh.

6: $coor (2, nv)$ – Real (Kind=nag_wp) array Input/Output

On entry:

coor (1, i)

contains the

x

coordinate of the

i

th input mesh vertex, for

i = 1, 2, \dots, nv

; while

coor (2, i)

contains the corresponding

y

coordinate.

On exit:

coor (1, i)

will contain the

x

coordinate of the

i

th renumbered mesh vertex, for

i = 1, 2, \dots, nv

; while

coor (2, i)

will contain the corresponding

y

coordinate.

7: $edge (3, nedge)$ – Integer array Input/Output

On entry: the specification of the boundary or interface edges.

edge (1, j)

and

edge (2, j)

contain the vertex numbers of the two end points of the

j

th boundary edge.

edge (3, j)

is a user-supplied tag for the

j

th boundary or interface edge:

edge (3, j) = 0

for an interior edge and has a nonzero tag otherwise.

Constraint:

1 \leq edge (i, j) \leq nv

and

edge (1, j) \neq edge (2, j)

, for

i = 1, 2

and

j = 1, 2, \dots, nedge

On exit: the renumbered specification of the boundary or interface edges.

8: $conn (3, nelt)$ – Integer array Input/Output

On entry: the connectivity of the mesh between triangles and vertices. For each triangle

j

conn (i, j)

gives the indices of its three vertices (in anticlockwise order), for

i = 1, 2, 3

and

j = 1, 2, \dots, nelt

Constraint:

1 \leq conn (i, j) \leq nv

and

conn (1, j) \neq conn (2, j)

and

conn (1, j) \neq conn (3, j)

and

conn (2, j) \neq conn (3, j)

, for

i = 1, 2, 3

and

j = 1, 2, \dots, nelt

On exit: the renumbered connectivity of the mesh between triangles and vertices.

9: $irow (nnzmax)$ – Integer array Output

10: $icol (nnzmax)$ – Integer array Output

On exit: the first nnz elements contain the row and column indices of the nonzero elements supplied in the finite element matrix

A

11: $itrace$ – Integer Input

On entry: the level of trace information required from d06ccf.

$itrace \leq 0$: 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 array Workspace

13: $liwork$ – Integer Input

On entry: the dimension of the array iwork as declared in the (sub)program from which d06ccf is called.

Constraint:

liwork \geq \max (nnzmax, 20 \times nv)

14: $rwork (lrwork)$ – Real (Kind=nag_wp) array Workspace

15: $lrwork$ – Integer Input

On entry: the dimension of the array rwork as declared in the (sub)program from which d06ccf is called.

Constraint:

lrwork \geq nv

16: $ifail$ – Integer Input/Output

On entry: ifail must be set to

0

−1

1

to set behaviour on detection of an error; these values have no effect when no error is detected.

A value of

0

causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of

−1

means that an error message is printed while a value of

1

means that it is not.

If halting is not appropriate, the value

−1

1

is recommended. If message printing is undesirable, then the value

1

is recommended. Otherwise, the value

0

is recommended. When the value $- 1$ or $1$ is used it is essential to test the value of ifail on exit.

On exit:

ifail = 0

unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry

ifail = 0

−1

, explanatory error messages are output on the current error message unit (as defined by x04aaf).

Errors or warnings detected by the routine:

$ifail = 1$: On entry, $conn (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $conn (I, J) \geq 1$ and $conn (I, J) \leq nv$ .

On entry, $edge (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $edge (I, J) \geq 1$ and $edge (I, J) \leq nv$ .

On entry, $liwork = ⟨ value ⟩$ and $LIWKMN = ⟨ value ⟩$ .
Constraint: $liwork \geq LIWKMN$ .

On entry, $lrwork = ⟨ value ⟩$ and $LRWKMN = ⟨ value ⟩$ .
Constraint: $lrwork \geq LRWKMN$ .

On entry, $nedge = ⟨ value ⟩$ .
Constraint: $nedge \geq 1$ .

On entry, $nelt = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $nelt \leq 2 \times nv - 1$ .

On entry, $nnzmax = ⟨ value ⟩$ , $nelt = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $nnzmax \geq (4 \times nelt + nv)$ and $nnzmax \leq {nv}^{2}$ .

On entry, $nv = ⟨ value ⟩$ .
Constraint: $nv \geq 3$ .

On entry, the end points of the edge $J$ have the same index $I$ : $J = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .

On entry, vertices $1$ and $2$ of the triangle $K$ have the same index $I$ : $K = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .

On entry, vertices $1$ and $3$ of the triangle $K$ have the same index $I$ : $K = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .

On entry, vertices $2$ and $3$ of the triangle $K$ have the same index $I$ : $K = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .

On the computation of the compact sparsity of the finite element matrix, an error has occurred. liwork has at least to be greater than $⟨ value ⟩$ .

$ifail = 2$: An error has occurred during the computation of the compact sparsity of the finite element matrix. Check the Triangle/Vertices connectivity.

$ifail = 3$: A serious error has occurred in an internal call to the renumbering routine. Check the input mesh especially the connectivity. Seek expert help.

$ifail = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.

$ifail = - 999$: Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d06ccf is threaded by NAG for parallel execution in multithreaded implementations of the NAG Library.

Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

9 Further Comments

None.

10 Example

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 in Section 10.3). 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 and 3 in Section 10.3).