d06aaf 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 routine is derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).
-
1:
– Integer
Input
-
On entry: the number of vertices in the input boundary mesh.
Constraint:
.
-
2:
– Integer
Input
-
On entry: the maximum number of vertices in the mesh to be generated.
-
3:
– Integer
Input
-
On entry: the number of boundary edges in the input mesh.
Constraint:
.
-
4:
– Integer array
Input
-
On entry: the specification of the boundary edges. and contain the vertex numbers of the two end points of the th boundary edge. is a user-supplied tag for the th boundary edge and is not used by d06aaf.
Constraint:
and , for and .
-
5:
– Integer
Output
-
On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If , no interior vertices will be generated and .
-
6:
– Integer
Output
-
On exit: the number of triangular elements in the mesh.
-
7:
– Real (Kind=nag_wp) array
Input/Output
-
On entry: contains the coordinate of the th input boundary mesh vertex; while contains the corresponding coordinate, for .
On exit: will contain the coordinate of the th generated interior mesh vertex; while will contain the corresponding coordinate, for . The remaining elements are unchanged.
-
8:
– Integer array
Output
-
On exit: the connectivity of the mesh between triangles and vertices. For each triangle
, gives the indices of its three vertices (in anticlockwise order), for and .
-
9:
– Real (Kind=nag_wp) array
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:
, for .
-
10:
– Logical
Input
-
On entry: indicates whether or not mesh smoothing should be performed.
If , the smoothing is performed; otherwise no smoothing is performed.
-
11:
– Real (Kind=nag_wp)
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
. The mesh will be finer if
coef is greater than
and
is a good value.
Suggested value:
.
-
12:
– Real (Kind=nag_wp)
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:
.
Constraint:
.
-
13:
– Integer
Input
-
On entry: the level of trace information required from
d06aaf.
- No output is generated.
- Output from the meshing solver is printed on the current advisory message unit (see x04abf). This output contains details of the vertices and triangles generated by the process.
You are advised to set , unless you are experienced with finite element mesh generation.
-
14:
– Real (Kind=nag_wp) array
Workspace
-
15:
– Integer
Input
-
On entry: the dimension of the array
rwork as declared in the (sub)program from which
d06aaf is called.
Constraint:
.
-
16:
– Integer array
Workspace
-
17:
– Integer
Input
-
On entry: the dimension of the array
iwork as declared in the (sub)program from which
d06aaf is called.
Constraint:
.
-
18:
– Integer
Input/Output
-
On entry:
ifail must be set to
,
or
to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of means that an error message is printed while a value of means that it is not.
If halting is not appropriate, the value
or
is recommended. If message printing is undesirable, then the value
is recommended. Otherwise, the value
is recommended.
When the value or is used it is essential to test the value of ifail on exit.
On exit:
unless the routine detects an error or a warning has been flagged (see
Section 6).
If on entry
or
, explanatory error messages are output on the current error message unit (as defined by
x04aaf).
-
On entry, and .
Constraint: .
On entry, , , and .
Constraint: and .
On entry, and .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, and .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, the end points of the edge have the same index : and .
Not applicable.
Background information to multithreading can be found in the
Multithreading documentation.
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.
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
, the first interior circle is centred at the point
with a radius
, and the second one is centred at the point
with a radius
. Note that the points
and
) are points of ‘near tangency’ between the exterior circle and the first and second circles.
The boundary mesh has
vertices and
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.