NAG Library Function Document
nag_mesh2d_inc (d06aac) generates a triangular mesh of a closed polygonal region in , given a mesh of its boundary. It uses a simple incremental method.
||nag_mesh2d_inc (Integer nvb,
const Integer edge,
const double bspace,
const char *outfile,
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).
George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris
nvb – IntegerInput
the number of vertices in the input boundary mesh.
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_inc (d06aac). 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; 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.
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
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.
, for .
smooth – Nag_BooleanInput
: indicates whether or not mesh smoothing should be performed.
If , the smoothing is performed; otherwise no smoothing is performed.
coef – doubleInput
: 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
is a good value.
power – doubleInput
: 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.
itrace – IntegerInput
: the level of trace information required from nag_mesh2d_inc (d06aac).
- 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, and .
On entry, the endpoints of the edge have the same index : 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 interior mesh. Check the inputs of the boundary.
Cannot close file .
Cannot open file for writing.
On entry, .
On entry, .
On entry, and .
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
. 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
) are points of ‘near tangency’ between the exterior circle and the first and second circles.
The boundary mesh has
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.
9.1 Program Text
Program Text (d06aace.c)
9.2 Program Data
Program Data (d06aace.d)
9.3 Program Results
Program Results (d06aace.r)
Figure 1: The boundary mesh of the geometry with two holes
Figure 2: Interior mesh of the geometry with two holes