NAG Library Routine Document
D06ABF
1 Purpose
D06ABF generates a triangular mesh of a closed polygonal region in ${\mathbb{R}}^{2}$, given a mesh of its boundary. It uses a Delaunay–Voronoi process, based on an incremental method.
2 Specification
SUBROUTINE D06ABF ( 
NVB, NVINT, NVMAX, NEDGE, EDGE, NV, NELT, COOR, CONN, WEIGHT, NPROPA, ITRACE, RWORK, LRWORK, IWORK, LIWORK, IFAIL) 
INTEGER 
NVB, NVINT, NVMAX, NEDGE, EDGE(3,NEDGE), NV, NELT, CONN(3,2*NVMAX+5), NPROPA, ITRACE, LRWORK, IWORK(LIWORK), LIWORK, IFAIL 
REAL (KIND=nag_wp) 
COOR(2,NVMAX), WEIGHT(*), RWORK(LRWORK) 

3 Description
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).
4 References
George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris
5 Parameters
 1: NVB – INTEGERInput
On entry: the number of vertices in the input boundary mesh.
Constraint:
${\mathbf{NVB}}\ge 3$.
 2: NVINT – INTEGERInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
Constraint:
${\mathbf{NVINT}}\ge 0$.
 3: NVMAX – INTEGERInput
On entry: the maximum number of vertices in the mesh to be generated.
Constraint:
${\mathbf{NVMAX}}\ge {\mathbf{NVB}}+{\mathbf{NVINT}}$.
 4: NEDGE – INTEGERInput
On entry: the number of boundary edges in the input mesh.
Constraint:
${\mathbf{NEDGE}}\ge 1$.
 5: EDGE($3$,NEDGE) – INTEGER arrayInput
On entry: the specification of the boundary edges. ${\mathbf{EDGE}}\left(1,j\right)$ and ${\mathbf{EDGE}}\left(2,j\right)$ contain the vertex numbers of the two end points of the $j$th boundary edge. ${\mathbf{EDGE}}\left(3,j\right)$ is a usersupplied tag for the $j$th boundary edge and is not used by D06ABF.
Constraint:
$1\le {\mathbf{EDGE}}\left(\mathit{i},\mathit{j}\right)\le {\mathbf{NVB}}$ and ${\mathbf{EDGE}}\left(1,\mathit{j}\right)\ne {\mathbf{EDGE}}\left(2,\mathit{j}\right)$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,{\mathbf{NEDGE}}$.
 6: NV – INTEGEROutput
On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If ${\mathbf{NVB}}+{\mathbf{NVINT}}={\mathbf{NVMAX}}$, no interior vertices will be generated and ${\mathbf{NV}}={\mathbf{NVMAX}}$.
 7: NELT – INTEGEROutput
On exit: the number of triangular elements in the mesh.
 8: COOR($2$,NVMAX) – REAL (KIND=nag_wp) arrayInput/Output
On entry: ${\mathbf{COOR}}\left(1,\mathit{i}\right)$ contains the $x$ coordinate of the $\mathit{i}$th input boundary mesh vertex, for $\mathit{i}=1,2,\dots ,{\mathbf{NVB}}$.
${\mathbf{COOR}}\left(1,\mathit{i}\right)$ contains the $x$ coordinate of the $\left(\mathit{i}{\mathbf{NVB}}\right)$th fixed interior vertex, for $\mathit{i}={\mathbf{NVB}}+1,\dots ,{\mathbf{NVB}}+{\mathbf{NVINT}}$. For boundary and interior vertices,
${\mathbf{COOR}}\left(2,\mathit{i}\right)$ contains the corresponding $y$ coordinate, for $\mathit{i}=1,2,\dots ,{\mathbf{NVB}}+{\mathbf{NVINT}}$.
On exit: ${\mathbf{COOR}}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\left(\mathit{i}{\mathbf{NVB}}{\mathbf{NVINT}}\right)$th generated interior mesh vertex, for $\mathit{i}={\mathbf{NVB}}+{\mathbf{NVINT}}+1,\dots ,{\mathbf{NV}}$; while ${\mathbf{COOR}}\left(2,i\right)$ will contain the corresponding $y$ coordinate. The remaining elements are unchanged.
 9: CONN($3$,$2\times {\mathbf{NVMAX}}+5$) – INTEGER arrayOutput
On exit: the connectivity of the mesh between triangles and vertices. For each triangle
$\mathit{j}$, ${\mathbf{CONN}}\left(\mathit{i},\mathit{j}\right)$ gives the indices of its three vertices (in anticlockwise order), for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,{\mathbf{NELT}}$.
 10: WEIGHT($*$) – REAL (KIND=nag_wp) arrayInput

Note: the dimension of the array
WEIGHT
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{NVINT}}\right)$.
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 ${\mathbf{NVINT}}>0$, ${\mathbf{WEIGHT}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{NVINT}}$.
 11: NPROPA – INTEGERInput
On entry: 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.
Constraint:
${\mathbf{NPROPA}}\ne 0$.
 12: ITRACE – INTEGERInput
On entry: the level of trace information required from D06ABF.
 ${\mathbf{ITRACE}}\le 0$
 No output is generated.
 ${\mathbf{ITRACE}}\ge 1$
 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 ${\mathbf{ITRACE}}=0$, unless you are experienced with finite element mesh generation.
 13: RWORK(LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
 14: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which D06ABF is called.
Constraint:
${\mathbf{LRWORK}}\ge 12\times {\mathbf{NVMAX}}+15$.
 15: IWORK(LIWORK) – INTEGER arrayWorkspace
 16: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which D06ABF is called.
Constraint:
${\mathbf{LIWORK}}\ge 6\times {\mathbf{NEDGE}}+32\times {\mathbf{NVMAX}}+2\times {\mathbf{NVB}}+78$.
 17: IFAIL – INTEGERInput/Output

On entry:
IFAIL must be set to
$0$,
$1\text{ or}1$. 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
$1\text{ or}1$ is recommended. If the output of error messages is undesirable, then the value
$1$ is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
$0$.
When the value $\mathbf{1}\text{ or}\mathbf{1}$ is used it is essential to test the value of IFAIL on exit.
On exit:
${\mathbf{IFAIL}}={\mathbf{0}}$ unless the routine detects an error or a warning has been flagged (see
Section 6).
6 Error Indicators and Warnings
If on entry
${\mathbf{IFAIL}}={\mathbf{0}}$ or
${{\mathbf{1}}}$, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Errors or warnings detected by the routine:
 ${\mathbf{IFAIL}}=1$
On entry,  ${\mathbf{NVB}}<3$, 
or  ${\mathbf{NVINT}}<0$, 
or  ${\mathbf{NVB}}+{\mathbf{NVINT}}>{\mathbf{NVMAX}}$, 
or  ${\mathbf{NEDGE}}<1$, 
or  ${\mathbf{EDGE}}\left(i,j\right)<1$ or ${\mathbf{EDGE}}\left(i,j\right)>{\mathbf{NVB}}$, for some $i=1,2$ and $j=1,2,\dots ,{\mathbf{NEDGE}}$, 
or  ${\mathbf{EDGE}}\left(1,j\right)={\mathbf{EDGE}}\left(2,j\right)$, for some $j=1,2,\dots ,{\mathbf{NEDGE}}$, 
or  ${\mathbf{NPROPA}}=0$; 
or  if ${\mathbf{NVINT}}>0$, ${\mathbf{WEIGHT}}\left(i\right)\le 0.0$, for some $i=1,2,\dots ,{\mathbf{NVINT}}$; 
or  ${\mathbf{LRWORK}}<12\times {\mathbf{NVMAX}}+15$, 
or  ${\mathbf{LIWORK}}<6\times {\mathbf{NEDGE}}+32\times {\mathbf{NVMAX}}+2\times {\mathbf{NVB}}+78$. 
 ${\mathbf{IFAIL}}=2$
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
${\mathbf{ITRACE}}>0$ may provide more details.
 ${\mathbf{IFAIL}}=3$
An error has occurred during the generation of the boundary mesh. It appears that
NVMAX is not large enough.
7 Accuracy
Not applicable.
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:
$\omega =1+\frac{a1.0}{20.0}$, where
$a$ is the absolute value of
NPROPA. During the process vertices are generated on edges of the mesh
${\mathcal{T}}_{i}$ to obtain the mesh
${\mathcal{T}}_{i+1}$ in the general incremental method (consult the
D06 Chapter Introduction or
George and Borouchaki (1998)). This generation uses the coefficient
$\omega $, and it is geometric if
${\mathbf{NPROPA}}>0$, and arithmetic otherwise. But increasing the value of
$a$ may lead to failure of the process, due to precision, especially in geometries with holes. So you are advised to manipulate the parameter
NPROPA 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.
9 Example
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 $\left(1.0,0.0\right)$ with a radius $3$, 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
$296$ vertices and
$296$ 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 ${\mathbf{NPROPA}}=5$ (middle left), $1$ (middle right),
$1$ (bottom left) and $5$ (bottom right) of a double RAE wings inside a circle geometry