d06abc 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 function may be called by the names: d06abc, nag_mesh_dim2_gen_delaunay or nag_mesh2d_delaunay.
3Description
d06abc 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 function 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.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{edge}}\left[(j-1)\times 3+i-1\right]$.
On entry: the specification of the boundary edges. ${\mathbf{edge}}\left[\left(j-1\right)\times 3\right]$ and ${\mathbf{edge}}\left[\left(j-1\right)\times 3+1\right]$ contain the vertex numbers of the two end points of the $j$th boundary edge. ${\mathbf{edge}}\left[\left(j-1\right)\times 3+2\right]$ is a user-supplied tag for the $j$th boundary edge and is not used by d06abc. Note that the edge vertices are numbered from $1$ to nvb.
Constraint:
$1\le {\mathbf{edge}}\left[\left(\mathit{j}-1\right)\times 3+\mathit{i}-1\right]\le {\mathbf{nvb}}$ and ${\mathbf{edge}}\left[\left(\mathit{j}-1\right)\times 3\right]\ne {\mathbf{edge}}\left[\left(\mathit{j}-1\right)\times 3+1\right]$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,{\mathbf{nedge}}$.
6: $\mathbf{nv}$ – Integer *Output
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: $\mathbf{nelt}$ – Integer *Output
On exit: the number of triangular elements in the mesh.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{coor}}\left[(j-1)\times 2+i-1\right]$.
On entry: ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)\times 2\right]$ contains the $x$ coordinate of the $\mathit{i}$th input boundary mesh vertex, for $\mathit{i}=1,2,\dots ,{\mathbf{nvb}}$.
${\mathbf{coor}}\left[\left(\mathit{i}-1\right)\times 2\right]$ 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,
${\mathbf{coor}}\left[\left(\mathit{i}-1\right)\times 2+1\right]$ contains the corresponding $y$ coordinate, for $\mathit{i}=1,2,\dots ,{\mathbf{nvb}}+{\mathbf{nvint}}$.
On exit: ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)\times 2\right]$ 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 ${\mathbf{coor}}\left[\left(i-1\right)\times 2+1\right]$ will contain the corresponding $y$ coordinate. The remaining elements are unchanged.
Note: the $(i,j)$th element of the matrix is stored in ${\mathbf{conn}}\left[(j-1)\times 3+i-1\right]$.
On exit: the connectivity of the mesh between triangles and vertices. For each triangle
$\mathit{j}$, ${\mathbf{conn}}\left[\left(\mathit{j}-1\right)\times 3+\mathit{i}-1\right]$ gives the indices of its three vertices (in anticlockwise order), for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,{\mathbf{nelt}}$. Note that the mesh vertices are numbered from $1$ to nv.
Note: the dimension, dim, 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 ${\mathbf{nvint}}>0$, ${\mathbf{weight}}\left[\mathit{i}-1\right]>0.0$, 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 d06abc.
${\mathbf{itrace}}\le 0$
No output is generated.
${\mathbf{itrace}}\ge 1$
Output from the meshing solver is printed. 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: $\mathbf{outfile}$ – const char *Input
On entry: the name of a file to which diagnostic output will be directed. If outfile is NULL the diagnostic output will be directed to standard output.
14: $\mathbf{fail}$ – NagError *Input/Output
The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).
6Error Indicators and Warnings
NE_ALLOC_FAIL
Dynamic memory allocation failed.
See Section 3.1.2 in the Introduction to the NAG Library CL Interface for further information.
NE_BAD_PARAM
On entry, argument $\u27e8\mathit{\text{value}}\u27e9$ had an illegal value.
NE_INT
On entry, ${\mathbf{nedge}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nedge}}\ge 1$.
On entry, ${\mathbf{npropa}}=0$.
On entry, ${\mathbf{nvb}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvb}}\ge 3$.
On entry, ${\mathbf{nvint}}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{nvint}}\ge 0$.
NE_INT_2
On entry, the end points of the edge $\mathit{J}$ have the same index $\mathit{I}$: $\mathit{J}=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$.
NE_INT_3
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: ${\mathbf{nvmax}}\ge {\mathbf{nvb}}+{\mathbf{nvint}}$.
NE_INT_4
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 ${\mathbf{edge}}(\mathit{I},\mathit{J})\le {\mathbf{nvb}}$, where ${\mathbf{edge}}(\mathit{I},\mathit{J})$ denotes ${\mathbf{edge}}\left[\left(\mathit{J}-1\right)\times 3+\mathit{I}-1\right]$.
NE_INTERNAL_ERROR
An internal error has occurred in this function. Check the function call and any array sizes. If the call is correct then please contact NAG for assistance.
See Section 7.5 in the Introduction to the NAG Library CL Interface for further information.
NE_MESH_ERROR
An error has occurred during the generation of the boundary mesh. It appears that nvmax is not large enough: ${\mathbf{nvmax}}=\u27e8\mathit{\text{value}}\u27e9$.
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.
NE_NO_LICENCE
Your licence key may have expired or may not have been installed correctly.
See Section 8 in the Introduction to the NAG Library CL Interface for further information.
NE_NOT_CLOSE_FILE
Cannot close file $\u27e8\mathit{\text{value}}\u27e9$.
NE_NOT_WRITE_FILE
Cannot open file $\u27e8\mathit{\text{value}}\u27e9$ for writing.
NE_REAL_ARRAY_INPUT
On entry, ${\mathbf{weight}}\left[\mathit{I}-1\right]=\u27e8\mathit{\text{value}}\u27e9$ and $\mathit{I}=\u27e8\mathit{\text{value}}\u27e9$.
Constraint: ${\mathbf{weight}}\left[\mathit{I}-1\right]>0.0$.
7Accuracy
Not applicable.
8Parallelism and Performance
d06abc 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 function. 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.