NAG Library Routine Document

D06DBF

INTEGER	NV1, NELT1, NEDGE1, EDGE1(3,NEDGE1), CONN1(3,NELT1), REFT1(NELT1), NV2, NELT2, NEDGE2, EDGE2(3,NEDGE2), CONN2(3,NELT2), REFT2(NELT2), NV3, NELT3, NEDGE3, EDGE3(3,), CONN3(3,), REFT3(*), ITRACE, IWORK(LIWORK), LIWORK, IFAIL
REAL (KIND=nag_wp)	EPS, COOR1(2,NV1), COOR2(2,NV2), COOR3(2,*)

3 Description

D06DBF joins together two adjacent, or overlapping, meshes. If the two meshes are adjacent then vertices belonging to the part of the boundary forming the common interface should coincide. If the two meshes overlap then vertices and triangles in the overlapping zone should coincide too.

This routine is partly derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).

4 References

None.

5 Parameters

1: $EPS$ – REAL (KIND=nag_wp)Input

On entry: the relative precision of the restitching of the two input meshes (see Section 9).

Suggested value:

0.001

Constraint:

EPS > 0.0

2: $NV1$ – INTEGERInput

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

Constraint:

NV1 \geq 3

3: $NELT1$ – INTEGERInput

On entry: the number of triangular elements in the first input mesh.

Constraint:

NELT1 \leq 2 \times NV1 - 1

4: $NEDGE1$ – INTEGERInput

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

Constraint:

NEDGE1 \geq 1

5: $COOR1 (2, NV1)$ – REAL (KIND=nag_wp) arrayInput

On entry:

COOR1 (1, i)

contains the

x

coordinate of the

i

th vertex of the first input mesh, for

i = 1, 2, \dots, NV1

; while

COOR1 (2, i)

contains the corresponding

y

coordinate.

6: $EDGE1 (3, NEDGE1)$ – INTEGER arrayInput

On entry: the specification of the boundary edges of the first input mesh.

EDGE1 (1, j)

and

EDGE1 (2, j)

contain the vertex numbers of the two end points of the

j

th boundary edge.

EDGE1 (3, j)

is a user-supplied tag for the

j

th boundary edge.

Constraint:

1 \leq EDGE1 (i, j) \leq NV1

and

EDGE1 (1, j) \neq EDGE1 (2, j)

, for

i = 1, 2

and

j = 1, 2, \dots, NEDGE1

7: $CONN1 (3, NELT1)$ – INTEGER arrayInput

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

j

CONN1 (i, j)

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

i = 1, 2, 3

and

j = 1, 2, \dots, NELT1

Constraints:

$1 \leq CONN1 (i, j) \leq NV1$ ;
$CONN1 (1, j) \neq CONN1 (2, j)$ ;
$CONN1 (1, j) \neq CONN1 (3, j)$ and $CONN1 (2, j) \neq CONN1 (3, j)$ , for $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT1$ .

8: $REFT1 (NELT1)$ – INTEGER arrayInput

On entry:

REFT1 (k)

contains the user-supplied tag of the

k

th triangle from the first input mesh, for

k = 1, 2, \dots, NELT1

9: $NV2$ – INTEGERInput

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

Constraint:

NV2 \geq 3

10: $NELT2$ – INTEGERInput

On entry: the number of triangular elements in the second input mesh.

Constraint:

NELT2 \leq 2 \times NV2 - 1

11: $NEDGE2$ – INTEGERInput

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

Constraint:

NEDGE2 \geq 1

12: $COOR2 (2, NV2)$ – REAL (KIND=nag_wp) arrayInput

On entry:

COOR2 (1, i)

contains the

x

coordinate of the

i

th vertex of the second input mesh, for

i = 1, 2, \dots, NV2

; while

COOR2 (2, i)

contains the corresponding

y

coordinate.

13: $EDGE2 (3, NEDGE2)$ – INTEGER arrayInput

On entry: the specification of the boundary edges of the second input mesh.

EDGE2 (1, j)

and

EDGE2 (2, j)

contain the vertex numbers of the two end points of the

j

th boundary edge.

EDGE2 (3, j)

is a user-supplied tag for the

j

th boundary edge.

Constraint:

1 \leq EDGE2 (i, j) \leq NV2

and

EDGE2 (1, j) \neq EDGE2 (2, j)

, for

i = 1, 2

and

j = 1, 2, \dots, NEDGE2

14: $CONN2 (3, NELT2)$ – INTEGER arrayInput

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

j

CONN2 (i, j)

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

i = 1, 2, 3

and

j = 1, 2, \dots, NELT2

Constraints:

$1 \leq CONN2 (i, j) \leq NV2$ ;
$CONN2 (1, j) \neq CONN2 (2, j)$ ;
$CONN2 (1, j) \neq CONN2 (3, j)$ and $CONN2 (2, j) \neq CONN2 (3, j)$ , for $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT2$ .

15: $REFT2 (NELT2)$ – INTEGER arrayInput

On entry:

REFT2 (k)

contains the user-supplied tag of the

k

th triangle from the second input mesh, for

k = 1, 2, \dots, NELT2

16: $NV3$ – INTEGEROutput

On exit: the total number of vertices in the resulting mesh.

17: $NELT3$ – INTEGEROutput

On exit: the number of triangular elements in the resulting mesh.

18: $NEDGE3$ – INTEGEROutput

On exit: the number of boundary edges in the resulting mesh.

19: $COOR3 (2 *)$ – REAL (KIND=nag_wp) arrayOutput

Note: the second dimension of the array COOR3 must be at least

NV1 + NV2

On exit:

COOR3 (1, i)

will contain the

x

coordinate of the

i

th vertex of the resulting mesh, for

i = 1, 2, \dots, NV3

; while

COOR3 (2, i)

will contain the corresponding

y

coordinate.

20: $EDGE3 (3 *)$ – INTEGER arrayOutput

Note: the second dimension of the array EDGE3 must be at least

NEDGE1 + NEDGE2

. This may be reduced to

NEDGE3

once that value is known.

On exit: the specification of the boundary edges of the resulting mesh.

EDGE3 (i, j)

will contain the vertex number of the

i

th end point (

i = 1, 2

) of the

j

th boundary or interface edge.

If the two meshes overlap,

EDGE3 (3, j)

will contain the same tag as the corresponding edge belonging to the first and/or the second input mesh.

If the two meshes are adjacent,

(i)	if the $j$ th edge is part of the partition interface, then $EDGE3 (3, j)$ will contain the value $1000 \times k_{1} + k_{2}$ where $k_{1}$ and $k_{2}$ are the tags for the same edge of the first and the second mesh respectively;
(ii)	otherwise, $EDGE3 (3, j)$ will contain the same tag as the corresponding edge belonging to the first and/or the second input mesh.

21: $CONN3 (3 *)$ – INTEGER arrayOutput

Note: the second dimension of the array CONN3 must be at least

NELT1 + NELT2

. This may be reduced to

NELT3

once that value is known.

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

CONN3 (i, j)

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

i = 1, 2, 3

and

j = 1, 2, \dots, NELT3

22: $REFT3 (*)$ – INTEGER arrayOutput

Note: the dimension of the array REFT3 must be at least

NELT1 + NELT2

. This may be reduced to

NELT3

once that value is known.

On exit: if the two meshes form a partition,

REFT3 (k)

will contain the same tag as the corresponding triangle belonging to the first or the second input mesh, for

k = 1, 2, \dots, NELT3

. If the two meshes overlap, then

REFT3 (k)

will contain the value

1000 \times k_{1} + k_{2}

where

k_{1}

and

k_{2}

are the user-supplied tags for the same triangle of the first and the second mesh respectively, for

k = 1, 2, \dots, NELT3

23: $ITRACE$ – INTEGERInput

On entry: the level of trace information required from D06DBF.

$ITRACE \leq 0$: No output is generated.
$ITRACE \geq 1$: Details about the common vertices, edges and triangles to both meshes are printed on the current advisory message unit (see X04ABF).

24: $IWORK (LIWORK)$ – INTEGER arrayWorkspace

25: $LIWORK$ – INTEGERInput

On entry: the dimension of the array IWORK as declared in the (sub)program from which D06DBF is called.

Constraint:

LIWORK \geq 2 \times NV1 + 3 \times NV2 + NELT1 + NELT2 + NEDGE1 + NEDGE2 + 1024

26: $IFAIL$ – 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,	$EPS \leq 0.0$ ,
or	$NV1 < 3$ ,
or	$NELT1 > 2 \times NV1 - 1$ ,
or	$NEDGE1 < 1$ ,
or	$EDGE1 (i, j) < 1$ or $EDGE1 (i, j) > NV1$ for some $i = 1, 2$ and $j = 1, 2, \dots, NEDGE1$ ,
or	$EDGE1 (1, j) = EDGE1 (2, j)$ for some $j = 1, 2, \dots, NEDGE1$ ,
or	$CONN1 (i, j) < 1$ or $CONN1 (i, j) > NV1$ for some $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT1$ ,
or	$CONN1 (1, j) = CONN1 (2, j)$ or $CONN1 (1, j) = CONN1 (3, j)$ or $CONN1 (2, j) = CONN1 (3, j)$ for some $j = 1, 2, \dots, NELT1$ ,
or	$NV2 < 3$ ,
or	$NELT2 > 2 \times NV2 - 1$ ,
or	$NEDGE2 < 1$ ,
or	$EDGE2 (i, j) < 1$ or $EDGE2 (i, j) > NV2$ for some $i = 1, 2$ and $j = 1, 2, \dots, NEDGE2$ ,
or	$EDGE2 (1, j) = EDGE2 (2, j)$ for some $j = 1, 2, \dots, NEDGE2$ ,
or	$CONN2 (i, j) < 1$ or $CONN2 (i, j) > NV2$ for some $i = 1, 2, 3$ and $j = 1, 2, \dots, NELT2$ ,
or	$CONN2 (1, j) = CONN2 (2, j)$ or $CONN2 (1, j) = CONN2 (3, j)$ or $CONN2 (2, j) = CONN2 (3, j)$ for some $j = 1, 2, \dots, NELT2$ ,
or	$LIWORK < 2 \times NV1 + 3 \times NV2 + NELT1 + NELT2 + NEDGE1 + NEDGE2 + 1024$ .

$IFAIL = 2$: Using the input precision EPS, the routine has detected fewer than two coincident vertices between the two input meshes. You are advised to try another value of EPS; if this error still occurs the two meshes are probably not stitchable.

$IFAIL = 3$: A serious error has occurred in an internal call to the restitching routine. You should check the input of the two meshes, especially the edge/vertex and/or the triangle/vertex connectivities. If the problem persists, contact NAG.

$IFAIL = 4$: The routine has detected a different number of coincident triangles from the two input meshes in the overlapping zone. You should check the input of the two meshes, especially the triangle/vertex connectivities.

$IFAIL = 5$: The routine has detected a different number of coincident edges from the two meshes on the partition interface. You should check the input of the two meshes, especially the edge/vertex connectivities.

$IFAIL = - 99$: 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

Not applicable.

9 Further Comments

D06DBF finds all the common vertices between the two input meshes using the relative precision of the restitching parameter EPS. You are advised to vary the value of EPS in the neighbourhood of

0.001

with

ITRACE \geq 1

to get the optimal value for the meshes under consideration.

10 Example

For this routine two examples are presented. There is a single example program for D06DBF, with a main program and the code to solve the two example problems given in Example 1 (EX1) and Example 2 (EX2).

Example 1 (EX1)

This example involves the unit square

{[0, 1]}^{2}

meshed uniformly, and then translated by a vector

\vec{u} = (\begin{matrix} u_{1} \\ u_{2} \end{matrix})

(using D06DAF). This translated mesh is then restitched with the original mesh. Two cases are considered:

(a)	overlapping meshes ( $u_{1} = 15.0$ , $u_{2} = 17.0$ ),
(b)	partitioned meshes ( $u_{1} = 19.0$ , $u_{2} = 0.0$ ).

The mesh on the unit square has

400

vertices,

722

triangles and its boundary has

76

edges. In the partitioned case the resulting geometry is shown in Figure 1 in Section 10.3 while the restitched mesh is shown in Figure 2 in Section 10.3. In the overlapping case the geometry and mesh are shown in Figure 3 and Figure 4 in Section 10.3.

Example 2 (EX2)

This example restitches three geometries by calling the routine D06DBF twice. The result is a mesh with three partitions. The first geometry is meshed by the Delaunay–Voronoi process (using D06ABF), the second one meshed by an Advancing Front algorithm (using D06ACF), while the third one is the result of a rotation (by

- π / 2

) of the second one (using D06DAF). The resulting geometry is shown in Figure 5 in Section 10.3 and restitched mesh in Figure 6 in Section 10.3.

NAG Library Routine DocumentD06DBF

▸▿ 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

NAG Library Routine Document

D06DBF