D06 Chapter Contents
D06 Chapter Introduction
NAG Library Manual

# NAG Library Routine DocumentD06AAF

Note:  before using this routine, please read the Users' Note for your implementation to check the interpretation of bold italicised terms and other implementation-dependent details.

## 1  Purpose

D06AAF generates a triangular mesh of a closed polygonal region in ${ℝ}^{2}$, given a mesh of its boundary. It uses a simple incremental method.

## 2  Specification

 SUBROUTINE D06AAF ( NVB, NVMAX, NEDGE, EDGE, NV, NELT, COOR, CONN, BSPACE, SMOOTH, COEF, POWER, ITRACE, RWORK, LRWORK, IWORK, LIWORK, IFAIL)
 INTEGER NVB, NVMAX, NEDGE, EDGE(3,NEDGE), NV, NELT, CONN(3,2*(NVMAX-1)), ITRACE, LRWORK, IWORK(LIWORK), LIWORK, IFAIL REAL (KIND=nag_wp) COOR(2,NVMAX), BSPACE(NVB), COEF, POWER, RWORK(LRWORK) LOGICAL SMOOTH

## 3  Description

D06AAF 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 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: $3\le {\mathbf{NVB}}\le {\mathbf{NVMAX}}$.
2:     NVMAX – INTEGERInput
On entry: the maximum number of vertices in the mesh to be generated.
3:     NEDGE – INTEGERInput
On entry: the number of boundary edges in the input mesh.
Constraint: ${\mathbf{NEDGE}}\ge 1$.
4:     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 user-supplied tag for the $j$th boundary edge and is not used by D06AAF.
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 $\mathit{j}=1,2,\dots ,{\mathbf{NEDGE}}$.
5:     NV – INTEGEROutput
On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If ${\mathbf{NVB}}={\mathbf{NVMAX}}$, no interior vertices will be generated and ${\mathbf{NV}}={\mathbf{NVB}}$.
6:     NELT – INTEGEROutput
On exit: the number of triangular elements in the mesh.
7:     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; while ${\mathbf{COOR}}\left(2,\mathit{i}\right)$ contains the corresponding $y$ coordinate, for $\mathit{i}=1,2,\dots ,{\mathbf{NVB}}$.
On exit: ${\mathbf{COOR}}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\left(\mathit{i}-{\mathbf{NVB}}\right)$th generated interior mesh vertex; while ${\mathbf{COOR}}\left(2,\mathit{i}\right)$ will contain the corresponding $y$ coordinate, for $\mathit{i}={\mathbf{NVB}}+1,\dots ,{\mathbf{NV}}$. The remaining elements are unchanged.
8:     CONN($3$,$2×\left({\mathbf{NVMAX}}-1\right)$) – 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 $\mathit{j}=1,2,\dots ,{\mathbf{NELT}}$.
9:     BSPACE(NVB) – REAL (KIND=nag_wp) arrayInput
On entry: the desired mesh spacing (triangle diameter, which is the length of the longer edge of the triangle) near the boundary vertices.
Constraint: ${\mathbf{BSPACE}}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,{\mathbf{NVB}}$.
10:   SMOOTH – LOGICALInput
On entry: indicates whether or not mesh smoothing should be performed.
If ${\mathbf{SMOOTH}}=\mathrm{.TRUE.}$, the smoothing is performed; otherwise no smoothing is performed.
11:   COEF – REAL (KIND=nag_wp)Input
On entry: the coefficient in the stopping criteria for the generation of interior vertices. This parameter controls the triangle density and the number of triangles generated is in $\mathit{O}\left({{\mathbf{COEF}}}^{2}\right)$. The mesh will be finer if COEF is greater than $0.7165$ and $0.75$ is a good value.
Suggested value: $0.75$.
12:   POWER – REAL (KIND=nag_wp)Input
On entry: 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.
Suggested value: $0.25$.
Constraint: $0.1\le {\mathbf{POWER}}\le 10.0$.
13:   ITRACE – INTEGERInput
On entry: the level of trace information required from D06AAF.
${\mathbf{ITRACE}}\le 0$
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.
14:   RWORK(LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
15:   LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which D06AAF is called.
Constraint: ${\mathbf{LRWORK}}\ge {\mathbf{NVMAX}}$.
16:   IWORK(LIWORK) – INTEGER arrayWorkspace
17:   LIWORK – INTEGERInput
On entry: the dimension of the array IWORK as declared in the (sub)program from which D06AAF is called.
Constraint: ${\mathbf{LIWORK}}\ge 16×{\mathbf{NVMAX}}+2×{\mathbf{NEDGE}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(4×{\mathbf{NVMAX}}+2,{\mathbf{NEDGE}}\right)-14$.
18:   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 ${\mathbf{NVB}}>{\mathbf{NVMAX}}$, 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 ${\mathbf{BSPACE}}\left(i\right)\le 0.0$, for some $i=1,2,\dots ,{\mathbf{NVB}}$, or ${\mathbf{POWER}}<0.1$ or ${\mathbf{POWER}}>10.0$, or ${\mathbf{LIWORK}}<16×{\mathbf{NVMAX}}+2×{\mathbf{NEDGE}}+\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(4×{\mathbf{NVMAX}}+2,{\mathbf{NEDGE}}\right)-14$, or ${\mathbf{LRWORK}}<{\mathbf{NVMAX}}$.
${\mathbf{IFAIL}}=2$
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.

## 7  Accuracy

Not applicable.

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, BSPACE, COEF and POWER. But you are advised to manipulate the last two parameters 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 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 $1.0$, the first interior circle is centred at the point $\left(-0.5,0.0\right)$ with a radius $0.49$, and the second one is centred at the point $\left(-0.5,0.65\right)$ with a radius $0.15$. Note that the points $\left(-1.0,0.0\right)$ and $\left(-0.5,0.5\right)$) are points of ‘near tangency’ between the exterior circle and the first and second circles.
The boundary mesh has $100$ vertices and $100$ 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 (d06aafe.f90)

### 9.2  Program Data

Program Data (d06aafe.d)

### 9.3  Program Results

Program Results (d06aafe.r)

Figure 1: The boundary mesh of the geometry with two holes
Figure 2: Interior mesh of the geometry with two holes