d06acc generates a triangular mesh of a closed polygonal region in , given a mesh of its boundary. It uses an Advancing Front process, based on an incremental method.
The function may be called by the names: d06acc, nag_mesh_dim2_gen_front or nag_mesh2d_front.
3Description
d06acc generates the set of interior vertices using an Advancing Front 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: – IntegerInput
On entry: the number of vertices in the input boundary mesh.
Constraint:
.
2: – IntegerInput
On entry: the number of fixed interior mesh vertices to which a weight will be applied.
Constraint:
.
3: – IntegerInput
On entry: the maximum number of vertices in the mesh to be generated.
Constraint:
.
4: – IntegerInput
On entry: the number of boundary edges in the input mesh.
Constraint:
.
5: – const IntegerInput
Note: the th element of the matrix is stored in .
On entry: the specification of the boundary edges. and contain the vertex numbers of the two end points of the th boundary edge. is a user-supplied tag for the th boundary edge and is not used by d06acc. Note that the edge vertices are numbered from to nvb.
Constraint:
and , for and .
6: – Integer *Output
On exit: the total number of vertices in the output mesh (including both boundary and interior vertices). If , no interior vertices will be generated and .
7: – Integer *Output
On exit: the number of triangular elements in the mesh.
8: – doubleInput/Output
Note: the th element of the matrix is stored in .
On entry: contains the coordinate of the th input boundary mesh vertex, for .
contains the coordinate of the th fixed interior vertex, for . For boundary and interior vertices,
contains the corresponding coordinate, for .
On exit: will contain the coordinate of the th generated interior mesh vertex, for ; while will contain the corresponding coordinate. The remaining elements are unchanged.
9: – IntegerOutput
Note: the th element of the matrix is stored in .
On exit: the connectivity of the mesh between triangles and vertices. For each triangle
, gives the indices of its three vertices (in anticlockwise order), for and . Note that the mesh vertices are numbered from to nv.
10: – const doubleInput
Note: the dimension, dim, of the array weight
must be at least
.
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 , , for .
11: – IntegerInput
On entry: the level of trace information required from d06acc.
No output is generated.
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 , unless you are experienced with finite element mesh generation.
12: – 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.
13: – 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 had an illegal value.
NE_INT
On entry, .
Constraint: .
On entry, .
Constraint: .
On entry, .
Constraint: .
NE_INT_2
On entry, the end points of the edge have the same index : and .
NE_INT_3
On entry, , and .
Constraint: .
On entry, , and .
Constraint: .
NE_INT_4
On entry, , , and .
Constraint: and , where denotes .
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. 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 may provide more details.
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 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 .
NE_NOT_WRITE_FILE
Cannot open file for writing.
NE_REAL_ARRAY_INPUT
On entry, and .
Constraint: .
7Accuracy
Not applicable.
8Parallelism and Performance
Background information to multithreading can be found in the Multithreading documentation.
d06acc 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. During the process vertices are generated on edges of the mesh to obtain the mesh in the general incremental method (consult the D06 Chapter Introduction or George and Borouchaki (1998)).
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 with a radius , the first wing begins at the origin and it is normalized, finally the last wing is also normalized and begins at the point . 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 airfoils.
The boundary mesh has vertices and edges (see Figure 1 top). Note that the particular mesh generated could be sensitive to the machine precision and, therefore, may differ from one implementation to another.