d06ccc 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

#include <nag.h>

void	d06ccc (Integer nv, Integer nelt, Integer nedge, Integer nnzmax, Integer nnz, double coor[], Integer edge[], Integer conn[], Integer irow[], Integer icol[], Integer itrace, const char outfile, NagError *fail)

The function may be called by the names: d06ccc, nag_mesh_dim2_renumber or nag_mesh2d_renum.

3 Description

d06ccc 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 function 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). d06ccc also returns the sparsity structure of the matrix associated with the renumbered mesh.

This function 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 function from which d06ccc 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: $coor [2 \times nv]$ – double Input/Output

Note: the

(i, j)

th element of the matrix is stored in

coor [(j - 1) \times 2 + i - 1]

On entry:

coor [(i - 1) \times 2]

contains the

x

coordinate of the

i

th input mesh vertex, for

i = 1, 2, \dots, nv

; while

coor [(i - 1) \times 2 + 1]

contains the corresponding

y

coordinate.

On exit:

coor [(i - 1) \times 2]

will contain the

x

coordinate of the

i

th renumbered mesh vertex, for

i = 1, 2, \dots, nv

; while

coor [(i - 1) \times 2 + 1]

will contain the corresponding

y

coordinate.

7: $edge [3 \times nedge]$ – Integer Input/Output

Note: the

(i, j)

th element of the matrix is stored in

edge [(j - 1) \times 3 + i - 1]

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

edge [(j - 1) \times 3]

and

edge [(j - 1) \times 3 + 1]

contain the vertex numbers of the two end points of the

j

th boundary edge.

edge [(j - 1) \times 3 + 2]

is a user-supplied tag for the

j

th boundary or interface edge:

edge [(j - 1) \times 3 + 2] = 0

for an interior edge and has a nonzero tag otherwise. Note that the edge vertices are numbered from

1

to nv.

Constraint:

1 \leq edge [(j - 1) \times 3 + i - 1] \leq nv

and

edge [(j - 1) \times 3] \neq edge [(j - 1) \times 3 + 1]

, for

i = 1, 2

and

j = 1, 2, \dots, nedge

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

8: $conn [3 \times nelt]$ – Integer Input/Output

Note: the

(i, j)

th element of the matrix is stored in

conn [(j - 1) \times 3 + i - 1]

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

j

conn [(j - 1) \times 3 + i - 1]

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

i = 1, 2, 3

and

j = 1, 2, \dots, nelt

. Note that the mesh vertices are numbered from

1

to nv.

Constraint:

1 \leq conn [(j - 1) \times 3 + i - 1] \leq nv

and

conn [(j - 1) \times 3] \neq conn [(j - 1) \times 3 + 1]

and

conn [(j - 1) \times 3] \neq conn [(j - 1) \times 3 + 2]

and

conn [(j - 1) \times 3 + 1] \neq conn [(j - 1) \times 3 + 2]

, 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 Output

10: $icol [nnzmax]$ – Integer 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 d06ccc.

$itrace \leq 0$: 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: $outfile$ – const char * Input

On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.

13: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM: On entry, argument $⟨ value ⟩$ had an illegal value.
NE_FAIL_SPARSITY: An error has occurred during the computation of the compact sparsity of the finite element matrix. Check the Triangle/Vertices connectivity.
NE_INT: On entry, $nedge = ⟨ value ⟩$ .
Constraint: $nedge \geq 1$ .

On entry, $nv = ⟨ value ⟩$ .
Constraint: $nv \geq 3$ .
NE_INT_2: On entry, $nelt = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $nelt \leq 2 \times nv - 1$ .

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 $I = ⟨ value ⟩$ .
NE_INT_3: On entry, $nnzmax = ⟨ value ⟩$ , $nelt = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $nnzmax \geq (4 \times nelt + nv)$ and $nnzmax \leq {nv}^{2}$ .
NE_INT_4: On entry, $conn (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $conn (I, J) \geq 1$ and $conn (I, J) \leq nv$ , where $conn (I, J)$ denotes $conn [(J - 1) \times 3 + I - 1]$ .

On entry, $edge (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $edge (I, J) \geq 1$ and $edge (I, J) \leq nv$ , where $edge (I, J)$ denotes $edge [(J - 1) \times 3 + I - 1]$ .
NE_INTERNAL_ERROR: An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.

A serious error has occurred in an internal call to the renumbering function. Check the input mesh especially the connectivity. Seek expert help.
NE_NO_LICENCE: Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_CLOSE_FILE: Cannot close file $⟨ value ⟩$ .
NE_NOT_WRITE_FILE: Cannot open file $⟨ value ⟩$ for writing.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d06ccc 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 function. 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 (d06aac) and it has

250

vertices and

402

triangles (see Figure 1 in Section 10.3). The function d06bac 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).