naginterfaces.library.mesh.dim2_​gen_​boundary

naginterfaces.library.mesh.dim2_gen_boundary(coorch, lined, crus, rate, nlcomp, lcomp, nvmax, nedmx, itrace, fbnd=None, data=None, io_manager=None)

dim2_gen_boundary generates a boundary mesh on a closed connected subdomain $\Omega$ of $R^2$.

For full information please refer to the NAG Library document for d06ba

https://support.nag.com/numeric/nl/nagdoc_30.2/flhtml/d06/d06baf.html

Parameters

coorchfloat, array-like, shape $(2,\text{nlines})$

$coorch[0,\text{i}-1]$ contains the $x$ coordinate of the $\text{i}$th characteristic point, for $\text{i}=1,2,\dots ,\text{nlines}$; while $coorch[1,i-1]$ contains the corresponding $y$ coordinate.

linedint, array-like, shape $(4,\text{nlines})$

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

$lined[0,i-1]$

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

$lined[1,i-1]$

The first end point of the line. If $lined[1,i-1]=j$, the coordinates of the first end point are those stored in $coorch[0,j-1],coorch[1,j-1]$.

$lined[2,i-1]$

The second end point of the line. If $lined[2,i-1]=k$, the coordinates of the second end point are those stored in $coorch[0,k-1],coorch[1,k-1]$.

$lined[3,i-1]$

This defines the type of line segment connecting the end points. Additional information is conveyed by the numerical value of $lined[3,i-1]$ as follows:

1. $lined[3,i-1]>0$, the line is described in $fbnd$ with $lined[3,i-1]$ as the index. In this case, the line must be described in the trigonometric (anticlockwise) direction;

2. $lined[3,i-1]=0$, the line is a straight line;

3. if $lined[3,i-1]<0$, 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 $coorch[z,j-1],crus[z,p-1],crus[z,p],\dots ,crus[z,p+r-4],coorch[z,k-1]$, where $z\in \{0,1\}$, $r=lined[0,i-1]$, $j=lined[1,i-1]$ and $k=lined[2,i-1]$.

crusfloat, array-like, shape $(2,\text{sdcrus})$

The coordinates of the intermediate points for polygonal arc lines. For a line $i$ defined as a polygonal arc (i.e., $lined[3,i-1]<0$), if $p=-lined[3,i-1]$, $crus[0,\text{k}-1]$, for $\text{k}=p,\dots ,p+lined[0,i-1]-3$, must contain the $x$ coordinate of the consecutive intermediate points for this line. Similarly $crus[1,\text{k}-1]$, for $\text{k}=p,\dots ,p+lined[0,i-1]-3$, must contain the corresponding $y$ coordinate.

ratefloat, array-like, shape $\left(\text{nlines}\right)$

$rate[\text{i}-1]$ is the geometric progression ratio between the points to be generated on the line $\text{i}$, for $\text{i}=1,2,\dots ,m$ and $lined[3,i-1]\ge 0$.

If $lined[3,i-1]<0$, $rate[i-1]$ is not referenced.

nlcompint, array-like, shape $\left(\text{ncomp}\right)$

$\left|nlcomp\right[k-1\left]\right|$ is the number of line segments in component $k$ of the contour. The line $i$ of component $k$ runs in the direction $lined[1,i-1]$ to $lined[2,i-1]$ if $nlcomp[k-1]>0$, and in the opposite direction otherwise; for $k=1,2,\dots ,n$.

lcompint, array-like, shape $\left(\text{nlines}\right)$

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

nvmaxint

The maximum number of the boundary mesh vertices to be generated.

nedmxint

The maximum number of boundary edges in the boundary mesh to be generated.

itraceint

The level of trace information required from dim2_gen_boundary.

$itrace=0$ or $itrace<-1$

No output is generated.

$itrace=1$

Output from the boundary mesh generator is printed. This output contains the input information of each line and each connected component of the boundary.

$itrace=-1$

An analysis of the output boundary mesh is printed on the file object associated with the advisory I/O unit (see FileObjManager). 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 function, calling either dim2_gen_inc(), dim2_gen_delaunay() or dim2_gen_front().

$itrace>1$

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

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

fbndNone or callable retval = fbnd(i, x, y, data=None), optional

Note: if this argument is None then a NAG-supplied facility will be used.

$fbnd$ must be supplied to calculate the value of the function which describes the curve $\{(x,y)\in R^2\text{; such that}f(x,y)=0\}$ on segments of the boundary for which $lined[3,i-1]>0$.

Parameters

iint

$lined[3,i-1]$, the reference index of the line (portion of the contour) $i$ described.

xfloat

The values of $x$ and $y$ at which $f(x,y)$ is to be evaluated.

yfloat

The values of $x$ and $y$ at which $f(x,y)$ is to be evaluated.

dataarbitrary, optional, modifiable in place

User-communication data for callback functions.

Returns

retvalfloat

The value of $f(x,y)$ at the specified point.

dataarbitrary, optional

User-communication data for callback functions.

io_managerFileObjManager, optional

Manager for I/O in this routine.

Returns

nvbint

The total number of boundary mesh vertices generated.

coorfloat, ndarray, shape $(2,nvmax)$

$coor[0,\text{i}-1]$ will contain the $x$ coordinate of the $\text{i}$th boundary mesh vertex generated, for $\text{i}=1,\dots ,nvb$; while $coor[1,i-1]$ will contain the corresponding $y$ coordinate.

nedgeint

The total number of boundary edges in the boundary mesh.

edgeint, ndarray, shape $(3,nedmx)$

The specification of the boundary edges. $edge[0,j-1]$ and $edge[1,j-1]$ will contain the vertex numbers of the two end points of the $j$th boundary edge. $edge[2,j-1]$ is a reference number for the $j$th boundary edge and

$edge[2,j-1]=lined[3,i-1]$, where $i$ and $j$ are such that the $j$th edges is part of the $i$th line of the boundary and $lined[3,i-1]\ge 0$;

$edge[2,j-1]=100+\left|lined\right[3,i-1\left]\right|$, where $i$ and $j$ are such that the $j$th edges is part of the $i$th line of the boundary and $lined[3,i-1]<0$.

Raises

NagValueError

(errno $1$)

On entry, the indices of the extremities of line $⟨value⟩$ are both equal to $⟨value⟩$.

(errno $1$)

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

(errno $1$)

On entry, $\text{ncomp}=⟨value⟩$.

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

(errno $1$)

On entry, $\text{nlines}=⟨value⟩$.

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

(errno $1$)

On entry, the geometric progression ratio between the points to be generated on line $⟨value⟩$ is $⟨value⟩$. It should be greater than $0$ unless the line segment is defined by user-supplied points.

(errno $1$)

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

(errno $1$)

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

(errno $1$)

On entry, the number of points on line $⟨value⟩$ is $⟨value⟩$. It should be greater than or equal to $2$.

(errno $1$)

On entry, $nvmax=⟨value⟩$ and $\text{nlines}=⟨value⟩$.

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

(errno $1$)

On entry, $nedmx=⟨value⟩$.

Constraint: $nedmx\ge 1$.

(errno $1$)

On entry, the line list for the separate connected component of the boundary is badly set: $lcomp[l-1]=⟨value⟩$ and $l=⟨value⟩$. It should be less than or equal to $\text{nlines}$ and greater than or equal to $1$.

(errno $1$)

On entry, end point $1$, with index $\text{K}$, does not lie on the curve representing line $\text{I}$ with index $\text{J}$: $\text{K}=⟨value⟩$, $\text{I}=⟨value⟩$, $\text{J}=⟨value⟩$, $f(x,y)=⟨value⟩$.

(errno $1$)

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

(errno $1$)

On entry, $\text{sdcrus}=⟨value⟩$ and $\text{nusmin}=⟨value⟩$.

Constraint: $\text{sdcrus}\ge \text{nusmin}$.

(errno $1$)

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

(errno $1$)

On entry, the absolute number of line segments in the $k$th component of the contour should be less than or equal to $\text{nlines}$ and greater than $0$. $k=⟨value⟩$, $nlcomp[k-1]=⟨value⟩$ and $\text{nlines}=⟨value⟩$.

(errno $2$)

An error has occurred during the generation of the boundary mesh. It appears that $nedmx$ is not large enough: $nedmx=⟨value⟩$.

(errno $3$)

An error has occurred during the generation of the boundary mesh. It appears that $nvmax$ is not large enough: $nvmax=⟨value⟩$.

(errno $4$)

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 $itrace>0$ may provide more details.

Notes

Given a closed connected subdomain $\Omega$ of $R^2$, whose boundary $\partial \Omega$ is divided by characteristic points into $m$ distinct line segments, dim2_gen_boundary generates a boundary mesh on $\partial \Omega$. Each line segment may be a straight line, a curve defined by the equation $f(x,y)=0$, or a polygonal curve defined by a set of given boundary mesh points.

This function is primarily designed for use with either dim2_gen_inc() (a simple incremental method) or dim2_gen_delaunay() (Delaunay–Voronoi method) or dim2_gen_front() (Advancing Front method) to triangulate the interior of the domain $\Omega$. For more details about the boundary and interior mesh generation, consult the D06 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).

References

George, P L and Borouchaki, H, 1998, Delaunay Triangulation and Meshing: Application to Finite Elements, Editions HERMES, Paris