NAG C Library Function Document

nag_mesh2d_front (d06acc)


nag_mesh2d_front (d06acc) generates a triangular mesh of a closed polygonal region in 2, given a mesh of its boundary. It uses an Advancing Front process, based on an incremental method.


#include <nag.h>
#include <nagd06.h>
void  nag_mesh2d_front (Integer nvb, Integer nvint, Integer nvmax, Integer nedge, const Integer edge[], Integer *nv, Integer *nelt, double coor[], Integer conn[], const double weight[], Integer itrace, const char *outfile, NagError *fail)


nag_mesh2d_front (d06acc) generates the set of interior vertices using an Advancing Front 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


1:     nvb IntegerInput
On entry: the number of vertices in the input boundary mesh.
Constraint: nvb3.
2:     nvint IntegerInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
Constraint: nvint0.
3:     nvmax IntegerInput
On entry: the maximum number of vertices in the mesh to be generated.
Constraint: nvmaxnvb+nvint.
4:     nedge IntegerInput
On entry: the number of boundary edges in the input mesh.
Constraint: nedge1.
5:     edge[3×nedge] const IntegerInput
Note: the i,jth element of the matrix is stored in edge[j-1×3+i-1].
On entry: the specification of the boundary edges. edge[j-1×3] and edge[j-1×3+1] contain the vertex numbers of the two end points of the jth boundary edge. edge[j-1×3+2] is a user-supplied tag for the jth boundary edge and is not used by nag_mesh2d_front (d06acc). Note that the edge vertices are numbered from 1 to nvb.
Constraint: 1edge[j-1×3+i-1]nvb and edge[j-1×3]edge[j-1×3+1], for i=1,2 and j=1,2,,nedge.
6:     nv Integer *Output
On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If nvb+nvint=nvmax, no interior vertices will be generated and nv=nvmax.
7:     nelt Integer *Output
On exit: the number of triangular elements in the mesh.
8:     coor[2×nvmax] doubleInput/Output
Note: the i,jth element of the matrix is stored in coor[j-1×2+i-1].
On entry: coor[i-1×2] contains the x coordinate of the ith input boundary mesh vertex, for i=1,2,,nvb. coor[i-1×2] contains the x coordinate of the i-nvbth fixed interior vertex, for i=nvb+1,,nvb+nvint. For boundary and interior vertices, coor[i-1×2+1] contains the corresponding y coordinate, for i=1,2,,nvb+nvint.
On exit: coor[i-1×2] will contain the x coordinate of the i-nvb-nvintth generated interior mesh vertex, for i=nvb+nvint+1,,nv; while coor[i-1×2+1] will contain the corresponding y coordinate. The remaining elements are unchanged.
9:     conn[3×2×nvmax+5] IntegerOutput
Note: the i,jth element of the matrix is stored in conn[j-1×3+i-1].
On exit: the connectivity of the mesh between triangles and vertices. For each triangle j, conn[j-1×3+i-1] gives the indices of its three vertices (in anticlockwise order), for i=1,2,3 and j=1,2,,nelt. Note that the mesh vertices are numbered from 1 to nv.
10:   weight[dim] const doubleInput
Note: the dimension, dim, of the array weight must be at least max1,nvint.
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.
Constraint: if nvint>0, weight[i-1]>0.0, for i=1,2,,nvint.
11:   itrace IntegerInput
On entry: the level of trace information required from nag_mesh2d_front (d06acc).
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 itrace=0, unless you are experienced with finite element mesh generation.
12:   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.
13:   fail NagError *Input/Output
The NAG error argument (see Section 3.7 in How to Use the NAG Library and its Documentation).

Error Indicators and Warnings

Dynamic memory allocation failed.
See Section in How to Use the NAG Library and its Documentation for further information.
On entry, argument value had an illegal value.
On entry, nedge=value.
Constraint: nedge1.
On entry, nvb=value.
Constraint: nvb3.
On entry, nvint=value.
Constraint: nvint0.
On entry, the end points of the edge J have the same index I: J=value and I=value.
On entry, nv=value, nvint=value and nvmax=value.
Constraint: nvb+nvintnvmax.
On entry, nvb=value, nvint=value and nvmax=value.
Constraint: nvmaxnvb+nvint.
On entry, edgeI,J=value, I=value, J=value and nvb=value.
Constraint: edgeI,J1 and edgeI,Jnvb, where edgeI,J denotes edge[J-1×3+I-1].
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.
See Section 2.7.6 in How to Use the NAG Library and its Documentation for further information.
An error has occurred during the generation of the interior mesh. Check the definition of the boundary (arguments coor and edge) as well as the orientation of the boundary (especially in the case of a multiple connected component boundary). Setting itrace>0 may provide more details.
Your licence key may have expired or may not have been installed correctly.
See Section 2.7.5 in How to Use the NAG Library and its Documentation for further information.
Cannot close file value.
Cannot open file value for writing.
On entry, weight[I-1]=value and I=value.
Constraint: weight[I-1]>0.0.


Not applicable.

Parallelism and Performance

nag_mesh2d_front (d06acc) makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the x06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this function. Please also consult the Users' Note for your implementation for any additional implementation-specific information.

Further Comments

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. During the process vertices are generated on edges of the mesh Ti to obtain the mesh Ti+1 in the general incremental method (consult the d06 Chapter Introduction or George and Borouchaki (1998)).
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 1.5,0.0 with a radius 4.5, the first wing begins at the origin and it is normalized, finally the last wing is also normalized and begins at the point 0.8,-0.3. 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 120 vertices and 120 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.

Program Text

Program Text (d06acce.c)

Program Data

Program Data (d06acce.d)

Program Results

Program Results (d06acce.r)

The boundary mesh (top), the interior mesh (bottom) of a double wing inside a circle geometry
Figure 1: The boundary mesh (top), the interior mesh (bottom) of a
double wing inside a circle geometry