void	d06aac (Integer nvb, Integer nvmax, Integer nedge, const Integer edge[], Integer nv, Integer nelt, double coor[], Integer conn[], const double bspace[], Nag_Boolean smooth, double coef, double power, Integer itrace, const char outfile, NagError fail)

The function may be called by the names: d06aac, nag_mesh_dim2_gen_inc or nag_mesh2d_inc.

3 Description

d06aac generates the set of interior vertices using a process based on a simple incremental method. A smoothing of the mesh is optionally available. For more details about the triangulation method, consult the D06 Chapter Introduction as well as George and Borouchaki (1998).

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

4 References

George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris

5 Arguments

1: $nvb$ – Integer Input

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

Constraint:

3 \leq nvb \leq nvmax

2: $nvmax$ – Integer Input

On entry: the maximum number of vertices in the mesh to be generated.

3: $nedge$ – Integer Input

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

Constraint:

nedge \geq 1

4: $edge [3 \times nedge]$ – const Integer Input

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 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 edge and is not used by d06aac. Note that the edge vertices are numbered from

1

to nvb.

Constraint:

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

and

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

, for

i = 1, 2

and

j = 1, 2, \dots, nedge

5: $nv$ – Integer * Output

On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If

nvb = nvmax

, no interior vertices will be generated and

nv = nvb

6: $nelt$ – Integer * Output

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

7: $coor [2 \times nvmax]$ – 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 boundary mesh vertex; while

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

contains the corresponding

y

coordinate, for

i = 1, 2, \dots, nvb

On exit:

coor [(i - 1) \times 2]

will contain the

x

coordinate of the

(i - nvb)

th generated interior mesh vertex; while

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

will contain the corresponding

y

coordinate, for

i = nvb + 1, \dots, nv

. The remaining elements are unchanged.

8: $conn [3 \times 2 \times (nvmax - 1)]$ – Integer Output

Note: the

(i, j)

th element of the matrix is stored in

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

On exit: 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.

9: $bspace [nvb]$ – const double Input

On entry: the desired mesh spacing (triangle diameter, which is the length of the longer edge of the triangle) near the boundary vertices.

Constraint:

bspace [i - 1] > 0.0

, for

i = 1, 2, \dots, nvb

10: $smooth$ – Nag_Boolean Input

On entry: indicates whether or not mesh smoothing should be performed.

smooth = Nag_TRUE

, the smoothing is performed; otherwise no smoothing is performed.

11: $coef$ – double Input

On entry: the coefficient in the stopping criteria for the generation of interior vertices. This argument controls the triangle density and the number of triangles generated is in

O ({coef}^{2})

. The mesh will be finer if coef is greater than

0.7165

and

0.75

is a good value.

Suggested value:

0.75

12: $power$ – double Input

On entry: controls the rate of change of the mesh size during the generation of interior vertices. The smaller the value of power, the faster the decrease in element size away from the boundary.

Suggested value:

0.25

Constraint:

0.1 \leq power \leq 10.0

13: $itrace$ – Integer Input

On entry: the level of trace information required from d06aac.

$itrace \leq 0$: No output is generated.
$itrace \geq 1$: Output from the meshing solver is printed. This output contains details of the vertices and triangles generated by the process.

You are advised to set

itrace = 0

, unless you are experienced with finite element mesh generation.

14: $outfile$ – 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.

15: $fail$ – 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_INT: On entry, $nedge = ⟨ value ⟩$ .
Constraint: $nedge \geq 1$ .
NE_INT_2: On entry, $nvb = ⟨ value ⟩$ and $nvmax = ⟨ value ⟩$ .
Constraint: $3 \leq nvb \leq nvmax$ .

On entry, the end points of the edge $J$ have the same index $I$ : $J = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .
NE_INT_4: On entry, $edge (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nvb = ⟨ value ⟩$ .
Constraint: $edge (I, J) \geq 1$ and $edge (I, J) \leq nvb$ , 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.
NE_MESH_ERROR: An error has occurred during the generation of the interior mesh. Check the definition of the boundary (arguments coor and edge) as well as the orientation of the boundary (especially in the case of a multiple connected component boundary). Setting $itrace > 0$ may provide more details.
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.
NE_REAL: On entry, $power = ⟨ value ⟩$ .
Constraint: $power \leq 10.0$ .

On entry, $power = ⟨ value ⟩$ .
Constraint: $power \geq 0.1$ .
NE_REAL_ARRAY_INPUT: On entry, $bspace [I - 1] = ⟨ value ⟩$ and $I = ⟨ value ⟩$ .
Constraint: $bspace [I - 1] > 0.0$ .

7 Accuracy

Not applicable.

8 Parallelism and Performance

d06aac is not threaded in any implementation.

9 Further Comments

The position of the internal vertices is a function of the positions of the vertices on the given boundary. A fine mesh on the boundary results in a fine mesh in the interior. The algorithm allows you to obtain a denser interior mesh by varying nvmax, bspace, coef and power. But you are advised to manipulate the last two arguments with care.

You are advised to take care to set the boundary inputs properly, especially for a boundary with multiply connected components. The orientation of the interior boundaries should be in clockwise order and opposite to that of the exterior boundary. If the boundary has only one connected component, its orientation should be anticlockwise.

10 Example

In this example, a geometry with two holes (two interior circles inside an exterior one) is meshed using the simple incremental method (see the D06 Chapter Introduction). The exterior circle is centred at the origin with a radius

1.0

, the first interior circle is centred at the point

(- 0.5, 0.0)

with a radius

0.49

, and the second one is centred at the point

(- 0.5, 0.65)

with a radius

0.15

. Note that the points

(- 1.0, 0.0)

and

(- 0.5, 0.5)

) are points of ‘near tangency’ between the exterior circle and the first and second circles.

The boundary mesh has

100

vertices and

100

edges (see Figure 1 in Section 10.3). Note that the particular mesh generated could be sensitive to the machine precision and, therefore, may differ from one implementation to another. Figure 2 in Section 10.3 contains the output mesh.

d06aa: FL CL CPP AD

NAG CL Interfaced06aac (dim2_​gen_​inc)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

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 CL Interface
d06aac (dim2_gen_inc)