NAG Library Routine Document

d06baf (dim2_gen_boundary)

1

Purpose

2

Specification

3

Description

4

References

5

Arguments

1:     $\mathbf{nlines}$ – IntegerInput

On entry: $m$, the number of lines that define the boundary of the closed connected subdomain (this equals the number of characteristic points which separate the entire boundary $\partial \Omega$ into lines).

Constraint: $\mathbf{nlines}\ge 1$.

2:     $\mathbf{coorch}\left(2,\mathbf{nlines}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{coorch}\left(1,\mathit{i}\right)$ contains the $x$ coordinate of the $\mathit{i}$th characteristic point, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$; while $\mathbf{coorch}\left(2,i\right)$ contains the corresponding $y$ coordinate.

3:     $\mathbf{lined}\left(4,\mathbf{nlines}\right)$ – Integer arrayInput

On entry: the description of the lines that define the boundary domain. The line $\mathit{i}$, for $\mathit{i}=1,2,\dots ,m$, is defined as follows:

$\mathbf{lined}\left(1,i\right)$

The number of points on the line, including two end points.

$\mathbf{lined}\left(2,i\right)$

The first end point of the line. If $\mathbf{lined}\left(2,i\right)=j$, the coordinates of the first end point are those stored in $\mathbf{coorch}\left(:,j\right)$.

$\mathbf{lined}\left(3,i\right)$

The second end point of the line. If $\mathbf{lined}\left(3,i\right)=k$, the coordinates of the second end point are those stored in $\mathbf{coorch}\left(:,k\right)$.

$\mathbf{lined}\left(4,i\right)$

This defines the type of line segment connecting the end points. Additional information is conveyed by the numerical value of $\mathbf{lined}\left(4,i\right)$ as follows:

(i)	$\mathbf{lined}\left(4,i\right)>0$, the line is described in fbnd with $\mathbf{lined}\left(4,i\right)$ as the index. In this case, the line must be described in the trigonometric (anticlockwise) direction;
(ii)	$\mathbf{lined}\left(4,i\right)=0$, the line is a straight line;
(iii)	if $\mathbf{lined}\left(4,i\right)<0$, say ($-p$), then the line is a polygonal arc joining the end points and interior points specified in crus. In this case the line contains the points whose coordinates are stored in $\mathbf{coorch}\left(:,j\right),\mathbf{crus}\left(:,p\right),\mathbf{crus}\left(:,p+1\right),\dots ,\mathbf{crus}\left(:,p+r-3\right),\mathbf{coorch}\left(:,k\right)$ , where $z\in \left\{1,2\right\}$, $r=\mathbf{lined}\left(1,i\right)$, $j=\mathbf{lined}\left(2,i\right)$ and $k=\mathbf{lined}\left(3,i\right)$.

Constraints:

• $2\le \mathbf{lined}\left(1,i\right)$;
• $1\le \mathbf{lined}\left(2,i\right)\le \mathbf{nlines}$;
• $1\le \mathbf{lined}\left(3,i\right)\le \mathbf{nlines}$;
• $\mathbf{lined}\left(2,\mathit{i}\right)\ne \mathbf{lined}\left(3,\mathit{i}\right)$, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$.

For each line described by fbnd (lines with $\mathbf{lined}\left(4,\mathit{i}\right)>0$, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$) the two end points ($\mathbf{lined}\left(2,i\right)$ and $\mathbf{lined}\left(3,i\right)$) lie on the curve defined by index $\mathbf{lined}\left(4,i\right)$ in fbnd, i.e.,

$\mathbf{fbnd}\left(\mathbf{lined}\left(4,i\right),\mathbf{coorch}\left(1,\mathbf{lined}\left(2,i\right)\right),\mathbf{coorch}\left(2,\mathbf{lined}\left(2,i\right)\right),\mathbf{ruser},\mathbf{iuser}\right)=0;$

$\mathbf{fbnd}\left(\mathbf{lined}\left(4,\mathit{i}\right),\mathbf{coorch}\left(1,\mathbf{lined}\left(3,\mathit{i}\right)\right),\mathbf{coorch}\left(2,\mathbf{lined}\left(3,\mathit{i}\right)\right),\mathbf{ruser},\mathbf{iuser}\right)=0$, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$.

For all lines described as polygonal arcs (lines with $\mathbf{lined}\left(4,\mathit{i}\right)<0$, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$) the sets of intermediate points (i.e.,$\left[-\mathbf{lined}\left(4,i\right):-\mathbf{lined}\left(4,i\right)+\mathbf{lined}\left(1,i\right)-3\right]$ for all $i$ such that $\mathbf{lined}\left(4,i\right)<0$) are not overlapping. This can be expressed as:

$$-\mathbf{lined}\left(4,i\right)+\mathbf{lined}\left(1,i\right)-3=\sum _{\left\{i,\mathbf{lined}\left(4,i\right)<0\right\}}\left\{\mathbf{lined}\left(1,i\right)-2\right\}$$

or

$$-\mathbf{lined}\left(4,i\right)+\mathbf{lined}\left(1,i\right)-2=-\mathbf{lined}\left(4,j\right)\text{,}$$

for a $j$ such that $j=1,2,\dots ,\mathbf{nlines}$, $j\ne i$ and $\mathbf{lined}\left(4,j\right)<0$.

4:     $\mathbf{fbnd}$ – real (Kind=nag_wp) Function, supplied by the user.External Procedure

fbnd must be supplied to calculate the value of the function which describes the curve $\left\{\left(x,y\right)\in {ℝ}^2\text{; such that }f\left(x,y\right)=0\right\}$ on segments of the boundary for which $\mathbf{lined}\left(4,i\right)>0$. If there are no boundaries for which $\mathbf{lined}\left(4,i\right)>0$ fbnd will never be referenced by d06baf and fbnd may be the dummy function d06bad. (d06bad is included in the NAG Library.)

The specification of fbnd is:

Fortran Interface

Function fbnd ( i, x, y, ruser, iuser) Real (Kind=nag_wp) :: fbnd Integer, Intent (In) :: i Integer, Intent (Inout) :: iuser(*) Real (Kind=nag_wp), Intent (In) :: x, y Real (Kind=nag_wp), Intent (Inout) :: ruser(*)

C Header Interface

#include <nagmk26.h> double  fbnd (const Integer *i, const double *x, const double *y, double ruser[], Integer iuser[])

1:     $\mathbf{i}$ – IntegerInput

On entry: $\mathbf{lined}\left(4,i\right)$, the reference index of the line (portion of the contour) $i$ described.

2:     $\mathbf{x}$ – Real (Kind=nag_wp)Input

3:     $\mathbf{y}$ – Real (Kind=nag_wp)Input

On entry: the values of $x$ and $y$ at which $f\left(x,y\right)$ is to be evaluated.

4:     $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

5:     $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

fbnd is called with the arguments ruser and iuser as supplied to d06baf. You should use the arrays ruser and iuser to supply information to fbnd.

fbnd must either be a module subprogram USEd by, or declared as EXTERNAL in, the (sub)program from which d06baf is called. Arguments denoted as Input must not be changed by this procedure.

Note: fbnd should not return floating-point NaN (Not a Number) or infinity values, since these are not handled by d06baf. If your code inadvertently does return any NaNs or infinities, d06baf is likely to produce unexpected results.

5:     $\mathbf{crus}\left(2,\mathbf{sdcrus}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: the coordinates of the intermediate points for polygonal arc lines. For a line $i$ defined as a polygonal arc (i.e., $\mathbf{lined}\left(4,i\right)<0$), if $p=-\mathbf{lined}\left(4,i\right)$, $\mathbf{crus}\left(1,\mathit{k}\right)$, for $\mathit{k}=p,\dots ,p+\mathbf{lined}\left(1,i\right)-3$, must contain the $x$ coordinate of the consecutive intermediate points for this line. Similarly $\mathbf{crus}\left(2,\mathit{k}\right)$, for $\mathit{k}=p,\dots ,p+\mathbf{lined}\left(1,i\right)-3$, must contain the corresponding $y$ coordinate.

6:     $\mathbf{sdcrus}$ – IntegerInput

On entry: the second dimension of the array crus as declared in the (sub)program from which d06baf is called.

Constraint: $\mathbf{sdcrus}\ge {\displaystyle \sum _{\left\{i,\mathbf{lined}\left(4,i\right)<0\right\}}}\left\{\mathbf{lined}\left(1,i\right)-2\right\}$.

7:     $\mathbf{rate}\left(\mathbf{nlines}\right)$ – Real (Kind=nag_wp) arrayInput

On entry: $\mathbf{rate}\left(\mathit{i}\right)$ is the geometric progression ratio between the points to be generated on the line $\mathit{i}$, for $\mathit{i}=1,2,\dots ,m$ and $\mathbf{lined}\left(4,i\right)\ge 0$.

If $\mathbf{lined}\left(4,i\right)<0$, $\mathbf{rate}\left(i\right)$ is not referenced.

Constraint: if $\mathbf{lined}\left(4,i\right)\ge 0$, $\mathbf{rate}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,\mathbf{nlines}$.

8:     $\mathbf{ncomp}$ – IntegerInput

On entry: $n$, the number of separately connected components of the boundary.

Constraint: $\mathbf{ncomp}\ge 1$.

9:     $\mathbf{nlcomp}\left(\mathbf{ncomp}\right)$ – Integer arrayInput

On entry: $\left|\mathbf{nlcomp}\left(k\right)\right|$ is the number of line segments in component $k$ of the contour. The line $i$ of component $k$ runs in the direction $\mathbf{lined}\left(2,i\right)$ to $\mathbf{lined}\left(3,i\right)$ if $\mathbf{nlcomp}\left(k\right)>0$, and in the opposite direction otherwise; for $k=1,2,\dots ,n$.

Constraints:

• $1\le \left|\mathbf{nlcomp}\left(\mathit{k}\right)\right|\le \mathbf{nlines}$, for $\mathit{k}=1,2,\dots ,\mathbf{ncomp}$;
• ${\displaystyle \sum _{k=1}^n}\left|\mathbf{nlcomp}\left(k\right)\right|=\mathbf{nlines}$.

10:   $\mathbf{lcomp}\left(\mathbf{nlines}\right)$ – Integer arrayInput

On entry: lcomp must contain the list of line numbers for the each component of the boundary. Specifically, the line numbers for the $\mathit{k}$th component of the boundary, for $\mathit{k}=1,2,\dots ,\mathbf{ncomp}$, must be in elements $l1-1$ to $l2-1$ of lcomp, where $l2={\displaystyle \sum _{i=1}^k}\left|\mathbf{nlcomp}\left(i\right)\right|$ and $l1=l2+1-\left|\mathbf{nlcomp}\left(\mathit{k}\right)\right|$.

Constraint: $\mathbf{lcomp}$ must hold a valid permutation of the integers $\left[1,\mathbf{nlines}\right]$.

11:   $\mathbf{nvmax}$ – IntegerInput

On entry: the maximum number of the boundary mesh vertices to be generated.

Constraint: $\mathbf{nvmax}\ge \mathbf{nlines}$.

12:   $\mathbf{nedmx}$ – IntegerInput

On entry: the maximum number of boundary edges in the boundary mesh to be generated.

Constraint: $\mathbf{nedmx}\ge 1$.

13:   $\mathbf{nvb}$ – IntegerOutput

On exit: the total number of boundary mesh vertices generated.

14:   $\mathbf{coor}\left(2,\mathbf{nvmax}\right)$ – Real (Kind=nag_wp) arrayOutput

On exit: $\mathbf{coor}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\mathit{i}$th boundary mesh vertex generated, for $\mathit{i}=1,2,\dots ,\mathbf{nvb}$; while $\mathbf{coor}\left(2,i\right)$ will contain the corresponding $y$ coordinate.

15:   $\mathbf{nedge}$ – IntegerOutput

On exit: the total number of boundary edges in the boundary mesh.

16:   $\mathbf{edge}\left(3,\mathbf{nedmx}\right)$ – Integer arrayOutput

On exit: the specification of the boundary edges. $\mathbf{edge}\left(1,j\right)$ and $\mathbf{edge}\left(2,j\right)$ will contain the vertex numbers of the two end points of the $j$th boundary edge. $\mathbf{edge}\left(3,j\right)$ is a reference number for the $j$th boundary edge and

• $\mathbf{edge}\left(3,j\right)=\mathbf{lined}\left(4,i\right)$, where $i$ and $j$ are such that the $j$th edges is part of the $i$th line of the boundary and $\mathbf{lined}\left(4,i\right)\ge 0$;
• $\mathbf{edge}\left(3,j\right)=100+\left|\mathbf{lined}\left(4,i\right)\right|$, where $i$ and $j$ are such that the $j$th edges is part of the $i$th line of the boundary and $\mathbf{lined}\left(4,i\right)<0$.

17:   $\mathbf{itrace}$ – IntegerInput

On entry: the level of trace information required from d06baf.

$\mathbf{itrace}=0$ or $\mathbf{itrace}<-1$

No output is generated.

$\mathbf{itrace}=1$

Output from the boundary mesh generator is printed on the current advisory message unit (see x04abf). This output contains the input information of each line and each connected component of the boundary.

$\mathbf{itrace}=-1$

An analysis of the output boundary mesh is printed on the current advisory message unit. This analysis includes the orientation (clockwise or anticlockwise) of each connected component of the boundary. This information could be of interest to you, especially if an interior meshing is carried out using the output of this routine, calling either d06aaf, d06abf or d06acf.

$\mathbf{itrace}>1$

The output is similar to that produced when $\mathbf{itrace}=1$, but the coordinates of the generated vertices on the boundary are also output.

You are advised to set $\mathbf{itrace}=0$, unless you are experienced with finite element mesh generation.

18:   $\mathbf{iuser}\left(*\right)$ – Integer arrayUser Workspace

19:   $\mathbf{ruser}\left(*\right)$ – Real (Kind=nag_wp) arrayUser Workspace

iuser and ruser are not used by d06baf, but are passed directly to fbnd and may be used to pass information to this routine.

20:   $\mathbf{rwork}\left(\mathbf{lrwork}\right)$ – Real (Kind=nag_wp) arrayWorkspace

21:   $\mathbf{lrwork}$ – IntegerInput

On entry: the dimension of the array rwork as declared in the (sub)program from which d06baf is called.

Constraint: $\mathbf{lrwork}\ge 2\times \left(\mathbf{nlines}+\mathbf{sdcrus}\right)+2\times {\max }_{i=1,2,\dots ,m}\left\{\mathbf{lined}\left(1,i\right)\right\}\times \mathbf{nlines}$.

22:   $\mathbf{iwork}\left(\mathbf{liwork}\right)$ – Integer arrayWorkspace

23:   $\mathbf{liwork}$ – IntegerInput

On entry: the dimension of the array iwork as declared in the (sub)program from which d06baf is called.

Constraint: $\mathbf{liwork}\ge {\displaystyle \sum _{\left\{i,\mathbf{lined}\left(4,i\right)<0\right\}}}\left\{\mathbf{lined}\left(1,i\right)-2\right\}+8\times \mathbf{nlines}+\mathbf{nvmax}+3\times \mathbf{nedmx}+2\times \mathbf{sdcrus}$.

24:   $\mathbf{ifail}$ – IntegerInput/Output

On entry: ifail must be set to $0$, $-1\text{ or }1$. If you are unfamiliar with this argument you should refer to Section 3.4 in How to Use the NAG Library and its Documentation 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 argument, 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, end point $1$, with index $\mathit{K}$, does not lie on the curve representing line $\mathit{I}$ with index $\mathit{J}$: $\mathit{K}=〈\mathit{\text{value}}〉$, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathit{J}=〈\mathit{\text{value}}〉$, $f\left(x,y\right)=〈\mathit{\text{value}}〉$.

On entry, end point $2$, with index $\mathit{K}$, does not lie on the curve representing line $\mathit{I}$ with index $\mathit{J}$: $\mathit{K}=〈\mathit{\text{value}}〉$, $\mathit{I}=〈\mathit{\text{value}}〉$, $\mathit{J}=〈\mathit{\text{value}}〉$, $f\left(x,y\right)=〈\mathit{\text{value}}〉$.

On entry, $\mathbf{liwork}=〈\mathit{\text{value}}〉$ and $\mathrm{LIWKMN}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{liwork}\ge \mathrm{LIWKMN}$.

On entry, $\mathbf{lrwork}=〈\mathit{\text{value}}〉$ and $\mathrm{LRWKMN}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{lrwork}\ge \mathrm{LRWKMN}$.

On entry, $\mathbf{ncomp}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{ncomp}\ge 1$.

On entry, $\mathbf{nedmx}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nedmx}\ge 1$.

On entry, $\mathbf{nlines}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nlines}\ge 1$.

On entry, $\mathbf{nvmax}=〈\mathit{\text{value}}〉$ and $\mathbf{nlines}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{nvmax}\ge \mathbf{nlines}$.

On entry, $\mathbf{sdcrus}=〈\mathit{\text{value}}〉$ and $\mathit{nusmin}=〈\mathit{\text{value}}〉$.
Constraint: $\mathbf{sdcrus}\ge \mathit{nusmin}$.

On entry, the absolute number of line segments in the $k$th component of the contour should be less than or equal to nlines and greater than $0$. $k=〈\mathit{\text{value}}〉$, $\mathbf{nlcomp}\left(k\right)=〈\mathit{\text{value}}〉$ and $\mathbf{nlines}=〈\mathit{\text{value}}〉$.

On entry, the geometric progression ratio between the points to be generated on line $〈\mathit{\text{value}}〉$ is $〈\mathit{\text{value}}〉$. It should be greater than $0$ unless the line segment is defined by user-supplied points.

On entry, the index of the first end point of line $〈\mathit{\text{value}}〉$ is $〈\mathit{\text{value}}〉$. It should be greater than or equal to $1$ and less than or equal to $\mathbf{nlines}=〈\mathit{\text{value}}〉$.

On entry, the index of the second end point of line $〈\mathit{\text{value}}〉$ is $〈\mathit{\text{value}}〉$. It should be greater than or equal to $1$ and less than or equal to $\mathbf{nlines}=〈\mathit{\text{value}}〉$.

On entry, the indices of the extremities of line $〈\mathit{\text{value}}〉$ are both equal to $〈\mathit{\text{value}}〉$.

On entry, the line list for the separate connected component of the boundary is badly set: $\mathbf{lcomp}\left(l\right)=〈\mathit{\text{value}}〉$ and $l=〈\mathit{\text{value}}〉$. It should be less than or equal to nlines and greater than or equal to $1$.

On entry, the number of points on line $〈\mathit{\text{value}}〉$ is $〈\mathit{\text{value}}〉$. It should be greater than or equal to $2$.

On entry, there is a correlation problem between the user-supplied coordinates and the specification of the polygonal arc representing line $\mathit{I}=〈\mathit{\text{value}}〉$ with the index in $\mathbf{crus}=〈\mathit{\text{value}}〉$.

On entry, the sum of absolute values of all numbers of line segments is different from nlines. The sum of all the elements of $\mathbf{nlcomp}=〈\mathit{\text{value}}〉$. $\mathbf{nlines}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=2$

An error has occurred during the generation of the boundary mesh. It appears that nedmx is not large enough: $\mathbf{nedmx}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=3$

An error has occurred during the generation of the boundary mesh. It appears that nvmax is not large enough: $\mathbf{nvmax}=〈\mathit{\text{value}}〉$.

$\mathbf{ifail}=4$

An error has occurred during the generation of the boundary mesh. Check the definition of each line (the argument lined) and each connected component of the boundary (the arguments nlcomp, and lcomp, as well as the coordinates of the characteristic points. Setting $\mathbf{itrace}>0$ may provide more details.

$\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

See Section 3.9 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

See Section 3.8 in How to Use the NAG Library and its Documentation for further information.

$\mathbf{ifail}=-999$

Dynamic memory allocation failed.

See Section 3.7 in How to Use the NAG Library and its Documentation for further information.

7

Accuracy

8

Parallelism and Performance

9

Further Comments

10

Example

10.1

Program Text

Program Text (d06bafe.f90)

10.2

Program Data

Program Data (d06bafe.d)

10.3

Program Results

Program Results (d06bafe.r)