NAG Library Routine Document

D06AAF

1  Purpose

2  Specification

3  Description

4  References

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($3$,$2\times \left(\mathbf{NVMAX}-1\right)$) – 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: $\mathbf{LIWORK}\ge 16\times \mathbf{NVMAX}+2\times \mathbf{NEDGE}+\max \left(4\times \mathbf{NVMAX}+2,\mathbf{NEDGE}\right)-14$.

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

$\mathbf{IFAIL}=1$

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	$\mathbf{LIWORK}<16\times \mathbf{NVMAX}+2\times \mathbf{NEDGE}+\max \left(4\times \mathbf{NVMAX}+2,\mathbf{NEDGE}\right)-14$,
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

8  Further Comments

9  Example

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