NAG Library Function Document
nag_mesh2d_delaunay (d06abc) 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.
||nag_mesh2d_delaunay (Integer nvb,
const Integer edge,
const double weight,
const char *outfile,
nag_mesh2d_delaunay (d06abc) 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 function 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
nvb – IntegerInput
On entry: the number of vertices in the input boundary mesh.
nvint – IntegerInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
nvmax – IntegerInput
the maximum number of vertices in the mesh to be generated.
nedge – IntegerInput
the number of boundary edges in the input mesh.
edge – const IntegerInput
: the specification of the boundary edges.
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 nag_mesh2d_delaunay (d06abc). Note that the edge vertices are numbered from
and , for and .
nv – 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 .
nelt – Integer *Output
On exit: the number of triangular elements in the mesh.
coor – doubleInput/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.
conn – IntegerOutput
: the connectivity of the mesh between triangles and vertices. For each triangle
gives the indices of its three vertices (in anticlockwise order), for
. Note that the mesh vertices are numbered from
weight – const doubleInput
the dimension, dim
, 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 .
npropa – IntegerInput
: the propagation type and coefficient, the argument 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
itrace – IntegerInput
: the level of trace information required from nag_mesh2d_delaunay (d06abc).
- No output is generated.
- 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 , unless you are experienced with finite element mesh generation.
outfile – const char *Input
: the name of a file to which diagnostic output will be directed. If outfile
the diagnostic output will be directed to standard output.
fail – NagError *Input/Output
The NAG error argument (see Section 3.6
in the Essential Introduction).
6 Error Indicators and Warnings
Dynamic memory allocation failed.
On entry, argument had an illegal value.
On entry, .
On entry, .
On entry, .
On entry, .
On entry, the endpoints of the edge have the same index : and .
On entry, , and .
On entry, , , and .
Constraint: and , where denotes .
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
An error has occurred during the generation of the boundary mesh. It appears that nvmax
is not large enough:
An error has occurred during the generation of the interior mesh. Check the inputs of the boundary.
Cannot close file .
Cannot open file for writing.
On entry, and .
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 argument 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 (d06abce.c)
9.2 Program Data
Program Data (d06abce.d)
9.3 Program Results
Program Results (d06abce.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