D06CAF uses a barycentering approach to improve the smoothness of a given mesh. The measure of quality used for a triangle is
where is the diameter (length of the longest edge) of , is the radius of its inscribed circle and is a normalization factor chosen to give for an equilateral triangle. ranges from , for an equilateral triangle, to , for a totally flat triangle.
D06CAF makes small perturbation to vertices (using a barycenter formula) in order to give a reasonably good value of for all neighbouring triangles. Some vertices may optionally be excluded from this process.
This routine is derived from material in the MODULEF package from INRIA (Institut National de Recherche en Informatique et Automatique).
George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris
1: NV – INTEGERInput
On entry: the total number of vertices in the input mesh.
2: NELT – INTEGERInput
On entry: the number of triangles in the input mesh.
3: NEDGE – INTEGERInput
On entry: the number of the boundary and interface edges in the input mesh.
4: COOR(,NV) – REAL (KIND=nag_wp) arrayInput/Output
On entry: contains the coordinate of the th input mesh vertex, for ; while contains the corresponding coordinate.
On exit: will contain the coordinate of the th smoothed mesh vertex, for ; while will contain the corresponding coordinate. Note that the coordinates of boundary and interface edge vertices, as well as those specified by you (see the description of NUMFIX), are unchanged by the process.
On entry: the specification of the boundary or interface 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 or interface edge: for an interior edge and has a nonzero tag otherwise.
On entry: the connectivity of the mesh between triangles and vertices. For each triangle
, gives the indices of its three vertices (in anticlockwise order), for and .
and and and , for and .
7: NVFIX – INTEGERInput
On entry: the number of fixed vertices in the input mesh.
8: NUMFIX() – INTEGER arrayInput
Note: the dimension of the array NUMFIX
must be at least
On entry: the indices in COOR of fixed interior vertices of the input mesh.
if , , for .
9: ITRACE – INTEGERInput
On entry: the level of trace information required from D06CAF.
No output is generated.
A histogram of the triangular element qualities is printed on the current advisory message unit (see X04ABF) before and after smoothing. This histogram gives the lowest and the highest triangle quality as well as the number of elements lying in each of the NQINT equal intervals between the extremes.
The output is similar to that produced when but the connectivity between vertices and triangles (for each vertex, the list of triangles in which it appears) is given.
You are advised to set , unless you are experienced with finite element meshes.
10: NQINT – INTEGERInput
On entry: the number of intervals between the extreme quality values for the input and the smoothed mesh.
On entry: the dimension of the array IWORK as declared in the (sub)program from which D06CAF is called.
13: RWORK(LRWORK) – REAL (KIND=nag_wp) arrayWorkspace
14: LRWORK – INTEGERInput
On entry: the dimension of the array RWORK as declared in the (sub)program from which D06CAF is called.
15: IFAIL – INTEGERInput/Output
On entry: IFAIL must be set to , . If you are unfamiliar with this parameter you should refer to Section 3.3 in the Essential Introduction for details.
For environments where it might be inappropriate to halt program execution when an error is detected, the value is recommended. If the output of error messages is undesirable, then the value is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is . When the value is used it is essential to test the value of IFAIL on exit.
On exit: unless the routine detects an error or a warning has been flagged (see Section 6).
6 Error Indicators and Warnings
If on entry or , explanatory error messages are output on the current error message unit (as defined by X04AAF).
Errors or warnings detected by the routine:
or for some and ,
for some ,
or for some and ,
or or for some ,
or for some if ,
A serious error has occurred in an internal call to an auxiliary routine. Check the input mesh, especially the connectivity between triangles and vertices (the parameter CONN). Setting may provide more information. If the problem persists, contact NAG.
8 Further Comments
In this example, a uniform mesh on the unit square is randomly distorted using routines from Chapter G05. D06CAF is then used to smooth the distorted mesh and recover a uniform mesh.