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.
The routine may be called by the names d06abf or nagf_mesh_dim2_gen_delaunay.
3Description
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).
4References
George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris
5Arguments
1: $\mathbf{nvb}$ – IntegerInput
On entry: the number of vertices in the input boundary mesh.
Constraint:
${\mathbf{nvb}}\ge 3$.
2: $\mathbf{nvint}$ – IntegerInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
Constraint:
${\mathbf{nvint}}\ge 0$.
3: $\mathbf{nvmax}$ – IntegerInput
On entry: the maximum number of vertices in the mesh to be generated.
On entry: the specification of the boundary edges. ${\mathbf{edge}}(1,j)$ and ${\mathbf{edge}}(2,j)$ contain the vertex numbers of the two end points of the $j$th boundary edge. ${\mathbf{edge}}(3,j)$ is a user-supplied tag for the $j$th boundary edge and is not used by d06abf.
Constraint:
$1\le {\mathbf{edge}}(\mathit{i},\mathit{j})\le {\mathbf{nvb}}$ and ${\mathbf{edge}}(1,\mathit{j})\ne {\mathbf{edge}}(2,\mathit{j})$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,{\mathbf{nedge}}$.
6: $\mathbf{nv}$ – 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: $\mathbf{nelt}$ – IntegerOutput
On exit: the number of triangular elements in the mesh.
8: $\mathbf{coor}(2,{\mathbf{nvmax}})$ – Real (Kind=nag_wp) arrayInput/Output
On entry: ${\mathbf{coor}}(1,\mathit{i})$ contains the $x$ coordinate of the $\mathit{i}$th input boundary mesh vertex, for $\mathit{i}=1,2,\dots ,{\mathbf{nvb}}$.
${\mathbf{coor}}(1,\mathit{i})$ contains the $x$ coordinate of the $(\mathit{i}-{\mathbf{nvb}})$th fixed interior vertex, for $\mathit{i}={\mathbf{nvb}}+1,\dots ,{\mathbf{nvb}}+{\mathbf{nvint}}$. For boundary and interior vertices,
${\mathbf{coor}}(2,\mathit{i})$ contains the corresponding $y$ coordinate, for $\mathit{i}=1,2,\dots ,{\mathbf{nvb}}+{\mathbf{nvint}}$.
On exit: ${\mathbf{coor}}(1,\mathit{i})$ will contain the $x$ coordinate of the $(\mathit{i}-{\mathbf{nvb}}-{\mathbf{nvint}})$th generated interior mesh vertex, for $\mathit{i}={\mathbf{nvb}}+{\mathbf{nvint}}+1,\dots ,{\mathbf{nv}}$; while ${\mathbf{coor}}(2,i)$ will contain the corresponding $y$ coordinate. The remaining elements are unchanged.
On exit: the connectivity of the mesh between triangles and vertices. For each triangle
$\mathit{j}$, ${\mathbf{conn}}(\mathit{i},\mathit{j})$ 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: $\mathbf{weight}\left(*\right)$ – Real (Kind=nag_wp) arrayInput
Note: the dimension of the array weight
must be at least
$\mathrm{max}\phantom{\rule{0.125em}{0ex}}(1,{\mathbf{nvint}})$.
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: $\mathbf{npropa}$ – IntegerInput
On entry: the propagation type and coefficient, the argument 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 9.
Constraint:
${\mathbf{npropa}}\ne 0$.
12: $\mathbf{itrace}$ – 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: $\mathbf{rwork}\left({\mathbf{lrwork}}\right)$ – Real (Kind=nag_wp) arrayWorkspace
14: $\mathbf{lrwork}$ – IntegerInput
On entry: the dimension of the array rwork as declared in the (sub)program from which d06abf is called.
On entry: ifail must be set to $0$, $\mathrm{-1}$ or $1$ to set behaviour on detection of an error; these values have no effect when no error is detected.
A value of $0$ causes the printing of an error message and program execution will be halted; otherwise program execution continues. A value of $\mathrm{-1}$ means that an error message is printed while a value of $1$ means that it is not.
If halting is not appropriate, the value $\mathrm{-1}$ or $1$ is recommended. If message printing is undesirable, then the value $1$ is recommended. Otherwise, the value $0$ is recommended. When the value $-\mathbf{1}$ 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).
6Error Indicators and Warnings
If on entry ${\mathbf{ifail}}=0$ or $\mathrm{-1}$, 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, ${\mathbf{edge}}(\mathit{I},\mathit{J})=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$, $\mathit{J}=\u27e8\mathit{\text{value}}\u27e9$ and ${\mathbf{nvb}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{edge}}(\mathit{I},\mathit{J})\ge 1$ and ${\mathbf{edge}}(\mathit{I},\mathit{J})\le {\mathbf{nvb}}$.
On entry, ${\mathbf{liwork}}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{LIWKMN}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{liwork}}\ge \mathrm{LIWKMN}$.
On entry, ${\mathbf{lrwork}}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathrm{LRWKMN}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{lrwork}}\ge \mathrm{LRWKMN}$.
On entry, ${\mathbf{nedge}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nedge}}\ge 1$.
On entry, ${\mathbf{npropa}}=0$.
On entry, ${\mathbf{nvb}}=\u27e8\mathit{\text{value}}\u27e9$, ${\mathbf{nvint}}=\u27e8\mathit{\text{value}}\u27e9$
and ${\mathbf{nvmax}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvmax}}\ge {\mathbf{nvb}}+{\mathbf{nvint}}$.
On entry, ${\mathbf{nvb}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvb}}\ge 3$.
On entry, ${\mathbf{nvint}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvint}}\ge 0$.
On entry, the end points of the edge $\mathit{J}$ have the same index $\mathit{I}$: $\mathit{J}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$.
On entry, ${\mathbf{weight}}\left(\mathit{I}\right)=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{weight}}\left(\mathit{I}\right)>0.0$.
${\mathbf{ifail}}=2$
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. 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.
An error has occurred during the generation of the boundary mesh. It appears that nvmax is not large enough: ${\mathbf{nvmax}}=\u27e8\mathit{\text{value}}\u27e9$.
${\mathbf{ifail}}=-99$
An unexpected error has been triggered by this routine. Please
contact NAG.
See Section 7 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-399$
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library FL Interface for further information.
${\mathbf{ifail}}=-999$
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d06abf makes calls to BLAS and/or LAPACK routines, which may be threaded within the vendor library used by this implementation. Consult the documentation for the vendor library for further information.
Please consult the X06 Chapter Introduction for information on how to control and interrogate the OpenMP environment used within this routine. Please also consult the Users' Note for your implementation for any additional implementation-specific information.
9Further Comments
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: $\omega =1+\frac{a-1.0}{20.0}$, 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 argument 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.
10Example
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 $(1.0,0.0)$ with a radius $3$. The main wing, using aerofoil RAE 2822 data, lies between the origin and the centre of the circle, while the secondary aerofoil is produced from the first by performing 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 aerofoils.
The boundary mesh has $296$ vertices and $296$ edges (see Section 10.3 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 Section 10.3.