D06CAF uses a barycentering technique to smooth a given mesh.
| SUBROUTINE D06CAF ( | NV, NELT, NEDGE, COOR, EDGE, CONN, NVFIX, NUMFIX, ITRACE, NQINT, IWORK, LIWORK, RWORK, LRWORK, IFAIL) |
| INTEGER | NV, NELT, NEDGE, EDGE(3,NEDGE), CONN(3,NELT), NVFIX, NUMFIX(*), ITRACE, NQINT, IWORK(LIWORK), LIWORK, LRWORK, IFAIL |
| double precision | COOR(2,NV), RWORK(LRWORK) |
D06CAF uses a barycentering approach to improve the smoothness of a given mesh. The measure of quality used for a triangle
K is
where
hK is the diameter (length of the longest edge) of
K,
ρK is the radius of its inscribed circle and
α=
36
is a normalization factor chosen to give
QK=1 for an equilateral triangle.
QK ranges from
1, 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 QK for all neighbouring triangles. Some vertices may optionally be excluded from this process.
For more details about the smoothing method, especially with regard to differing quality, 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).
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.
Constraint:
NV≥3.
- 2: NELT – INTEGERInput
On entry:
the number of triangles in the input mesh.
Constraint:
NELT≤2×NV-1.
- 3: NEDGE – INTEGERInput
On entry:
the number of the boundary and interface edges in the input mesh.
Constraint:
NEDGE≥1.
- 4: COOR(2,NV) – double precision arrayInput/Output
-
On entry: COOR1i contains the x co-ordinate of the ith input mesh vertex, for i=1,2,…,NV; while COOR2i contains the corresponding y co-ordinate.
On exit:
COOR1i will contain the
x co-ordinate of the
ith smoothed mesh vertex, for
i=1,2,…,NV; while
COOR2i will contain the corresponding
y co-ordinate. Note that the co-ordinates of boundary and interface edge vertices, as well as those specified by you (see the description of
NUMFIX), are unchanged by the process.
- 5: EDGE(3,NEDGE) – INTEGER arrayInput
-
On entry: the specification of the boundary or interface edges. EDGE1j and EDGE2j contain the vertex numbers of the two end points of the jth boundary edge. EDGE3j is a user-supplied tag for the jth boundary or interface edge: EDGE3j=0 for an interior edge and has a nonzero tag otherwise.
Constraint:
1≤EDGEij≤NV and EDGE1j≠EDGE2j, for i=1,2 and j=1,2,…,NEDGE.
- 6: CONN(3,NELT) – INTEGER arrayInput
-
On entry: the connectivity of the mesh between triangles and vertices. For each triangle j, CONNij gives the indices of its three vertices (in anticlockwise order), for i=1,2,3 and j=1,2,…,NELT.
Constraint:
1≤CONNij≤NV and CONN1j≠CONN2j and CONN1j≠CONN3j and CONN2j≠CONN3j, for i=1,2,3 and j=1,2,…,NELT.
- 7: NVFIX – INTEGERInput
On entry: the number of fixed vertices in the input mesh.
Constraint:
0≤NVFIX≤NV.
- 8: NUMFIX(*) – INTEGER arrayInput
Note: the dimension of the array
NUMFIX
must be at least
max1,NVFIX.
On entry: the indices in
COOR of fixed interior vertices of the input mesh.
Constraint:
if NVFIX>0, 1≤NUMFIXi≤NV, for i=1,2,…,NVFIX.
- 9: ITRACE – INTEGERInput
On entry: the level of trace information required from D06CAF.
- ITRACE≤0
- No output is generated.
- ITRACE=1
- 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.
- ITRACE>1
- The output is similar to that produced when ITRACE=1 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 ITRACE=0, 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.
If
ITRACE=0,
NQINT is not referenced.
- 11: IWORK(LIWORK) – INTEGER arrayWorkspace
- 12: LIWORK – INTEGERInput
On entry: the dimension of the array
IWORK as declared in the (sub)program from which D06CAF is called.
Constraint:
LIWORK≥8×NELT+2×NV.
- 13: RWORK(LRWORK) – double precision arrayWorkspace
- 14: LRWORK – INTEGERInput
On entry: the dimension of the array
RWORK as declared in the (sub)program from which D06CAF is called.
Constraint:
LRWORK≥2×NV+NELT.
- 15: IFAIL – INTEGERInput/Output
-
On entry:
IFAIL must be set to
0,
-1 or 1. If you are unfamiliar with this parameter you should refer to
Section 3.3 in the Essential Introduction for details.
On exit:
IFAIL=0 unless the routine detects an error (see
Section 6).
For environments where it might be inappropriate to halt program execution when an error is detected, the value
-1 or 1 is recommended. If the output of error messages is undesirable, then the value
1 is recommended. Otherwise, if you are not familiar with this parameter, the recommended value is
0.
When the value -1 or 1 is used it is essential to test the value of IFAIL on exit.
If on entry
IFAIL=0 or
-1, explanatory error messages are output on the current error message unit (as defined by
X04AAF).
Not applicable.
Not applicable.
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.
