NAG Library Routine Document

D06CCF

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Parameters

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

▸▿ 10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

1 Purpose

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

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
REAL (KIND=nag_wp)	COOR(2,NV), RWORK(LRWORK)

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 Parameters

1: $NV$ – INTEGERInput

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

Constraint:

NV \geq 3

2: $NELT$ – INTEGERInput

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

Constraint:

NELT \leq 2 \times NV - 1

3: $NEDGE$ – INTEGERInput

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

Constraint:

NEDGE \geq 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 \times NELT + NV \leq NNZMAX \leq {NV}^{2}

5: $NNZ$ – INTEGEROutput

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

6: $COOR (2, NV)$ – REAL (KIND=nag_wp) arrayInput/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 arrayInput/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 arrayInput/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 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.

$ITRACE \leq 0$: 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 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 \geq \max (NNZMAX, 20 \times NV)

14: $RWORK (LRWORK)$ – REAL (KIND=nag_wp) 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 \geq 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.

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.

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,	$NV < 3$ ,
or	$NELT > 2 \times NV - 1$ ,
or	$NEDGE < 1$ ,
or	$NNZMAX < 4 \times NELT + NV$ or $NNZMAX > {NV}^{2}$
or	$CONN (i, j) < 1$ or $CONN (i, j) > NV$ for some $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT$ ,
or	$CONN (1, j) = CONN (2, j)$ or $CONN (1, j) = CONN (3, j)$ or $CONN (2, j) = CONN (3, j)$ for some $j = 1, 2, \dots, NELT$ ,
or	$EDGE (i, j) < 1$ or $EDGE (i, j) > NV$ for some $i = 1, 2$ and $j = 1, 2, \dots, NEDGE$ ,
or	$EDGE (1, j) = EDGE (2, j)$ for some $j = 1, 2, \dots, NEDGE$ ,
or	$LIWORK < \max (NNZMAX, 20 \times NV)$ ,
or	$LRWORK < NV$ .

$IFAIL = 2$: A serious error has occurred during the computation of the compact sparsity of the finite element matrix or in an internal call to the renumbering routine. Check the input mesh, especially the connectivity between triangles and vertices (the parameter CONN). If the problem persists, contact NAG.

$IFAIL = - 99$: An unexpected error has been triggered by this routine. Please contact NAG.
See Section 3.8 in the Essential Introduction for further information.

$IFAIL = - 399$: Your licence key may have expired or may not have been installed correctly.
See Section 3.7 in the Essential Introduction for further information.

$IFAIL = - 999$: Dynamic memory allocation failed.
See Section 3.6 in the Essential Introduction 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).