NAG Library Routine Document
D06ABF generates a triangular mesh of a closed polygonal region in , given a mesh of its boundary. It uses a Delaunay–Voronoi process, based on an incremental method.
|SUBROUTINE D06ABF (
||NVB, NVINT, NVMAX, NEDGE, EDGE, NV, NELT, COOR, CONN, WEIGHT, NPROPA, ITRACE, RWORK, LRWORK, IWORK, LIWORK, IFAIL)
||NVB, NVINT, NVMAX, NEDGE, EDGE(3,NEDGE), NV, NELT, CONN(3,2*NVMAX+5), NPROPA, ITRACE, LRWORK, IWORK(LIWORK), LIWORK, IFAIL
||COOR(2,NVMAX), WEIGHT(*), RWORK(LRWORK)
D06ABF generates the set of interior vertices using a Delaunay–Voronoi process, based on an incremental method. It allows you to specify a number of fixed interior mesh vertices together with weights which allow concentration of the mesh in their neighbourhood. 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).
George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris
- 1: NVB – INTEGERInput
On entry: the number of vertices in the input boundary mesh.
- 2: NVINT – INTEGERInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
- 3: NVMAX – INTEGERInput
On entry: the maximum number of vertices in the mesh to be generated.
- 4: NEDGE – INTEGERInput
On entry: the number of boundary edges in the input mesh.
- 5: EDGE(,NEDGE) – INTEGER arrayInput
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 D06ABF.
and , for and .
- 6: NV – INTEGEROutput
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 .
- 7: NELT – INTEGEROutput
On exit: the number of triangular elements in the mesh.
- 8: COOR(,NVMAX) – REAL (KIND=nag_wp) arrayInput/Output
On entry: contains the coordinate of the th input boundary mesh vertex, for .
contains the coordinate of the th fixed interior vertex, for . For boundary and interior vertices,
contains the corresponding coordinate, for .
On exit: will contain the coordinate of the th generated interior mesh vertex, for ; while will contain the corresponding coordinate. The remaining elements are unchanged.
- 9: CONN(,) – INTEGER arrayOutput
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 .
- 10: WEIGHT() – REAL (KIND=nag_wp) arrayInput
the dimension of the array WEIGHT
must be at least
On entry: the weight of fixed interior vertices. It is the diameter of triangles (length of the longer edge) created around each of the given interior vertices.
if , , for .
- 11: NPROPA – INTEGERInput
: the propagation type and coefficient, the parameter NPROPA
is used when the internal points are created. They are distributed in a geometric manner if NPROPA
is positive and in an arithmetic manner if it is negative. For more details see Section 8
- 12: ITRACE – INTEGERInput
: the level of trace information required from D06ABF.
- 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.
- 13: RWORK(LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
- 14: LRWORK – INTEGERInput
: the dimension of the array RWORK
as declared in the (sub)program from which D06ABF is called.
- 15: IWORK(LIWORK) – INTEGER arrayWorkspace
- 16: LIWORK – INTEGERInput
: the dimension of the array IWORK
as declared in the (sub)program from which D06ABF is called.
- 17: IFAIL – INTEGERInput/Output
must be set to
. 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
is recommended. If the output of error messages is undesirable, then the value
is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
. When the value is used it is essential to test the value of IFAIL on exit.
unless the routine detects an error or a warning has been flagged (see Section 6
6 Error Indicators and Warnings
If on entry
, explanatory error messages are output on the current error message unit (as defined by X04AAF
Errors or warnings detected by the routine:
|or|| or , for some and ,|
|or||, for some ,|
|or||if , , for some ;|
An error has occurred during the generation of the interior mesh. Check the definition of the boundary (arguments COOR
) as well as the orientation of the boundary (especially in the case of a multiple connected component boundary). Setting
may provide more details.
An error has occurred during the generation of the boundary mesh. It appears that NVMAX
is not large enough.
The position of the internal vertices is a function position of the vertices on the given boundary. A fine mesh on the boundary results in a fine mesh in the interior. To dilute the influence of the data on the interior of the domain, the value of NPROPA
can be changed. The propagation coefficient is calculated as:
is the absolute value of NPROPA
. During the process vertices are generated on edges of the mesh
to obtain the mesh
in the general incremental method (consult the D06 Chapter Introduction
or George and Borouchaki (1998)
). This generation uses the coefficient
, and it is geometric if
, and arithmetic otherwise. But increasing the value of
may lead to failure of the process, due to precision, especially in geometries with holes. So you are advised to manipulate the parameter NPROPA
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 wings inside an exterior circle) is meshed using a Delaunay–Voronoi method. The exterior circle is centred at the point with a radius , the first RAE wing begins at the origin and it is normalized, and the last wing is a result from the first one after a translation, a scale reduction and a rotation. To be able to carry out some realistic computation on that geometry, some interior points have been introduced to have a finer mesh in the wake of those airfoils.
The boundary mesh has
edges (see Figure 1
top). Note that the particular mesh generated could be sensitive to the machine precision
and therefore may differ from one implementation to another. The interior meshes for different values of NPROPA
are given in Figure 1
9.1 Program Text
Program Text (d06abfe.f90)
9.2 Program Data
Program Data (d06abfe.d)
9.3 Program Results
Program Results (d06abfe.r)
Figure 1: The boundary mesh (top), the interior mesh with (middle left), (middle right),
(bottom left) and (bottom right) of a double RAE wings inside a circle geometry