NAG Library Function Document

nag_mesh2d_inc (d06aac)

void	nag_mesh2d_inc (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)

3 Description

nag_mesh2d_inc (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 – IntegerInput

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

Constraint:

3 \leq nvb \leq nvmax

2: nvmax – IntegerInput

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

3: nedge – IntegerInput

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

Constraint:

nedge \geq 1

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

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 nag_mesh2d_inc (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[ $2 \times nvmax$ ] – doubleInput/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[ $3 \times 2 \times (nvmax - 1)$ ] – IntegerOutput

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 doubleInput

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_BooleanInput

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 – doubleInput

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 – doubleInput

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 – IntegerInput

On entry: the level of trace information required from nag_mesh2d_inc (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 3.6 in the Essential Introduction).

6 Error Indicators and Warnings

NE_ALLOC_FAIL: Dynamic memory allocation failed.
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 endpoints 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.
NE_MESH_ERROR: An error has occurred during the generation of the interior mesh. Check the inputs of the boundary.
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

Not applicable.

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). Note that the particular mesh generated could be sensitive to the machine precision and therefore may differ from one implementation to another. Figure 2 contains the output mesh.

Figure 1: The boundary mesh of the geometry with two holes