NAG FL Interface
d06caf (dim2_​smooth_​bary)

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1 Purpose

d06caf uses a barycentering technique to smooth a given mesh.

2 Specification

Fortran Interface
Subroutine d06caf ( nv, nelt, nedge, coor, edge, conn, nvfix, numfix, itrace, nqint, iwork, liwork, rwork, lrwork, ifail)
Integer, Intent (In) :: nv, nelt, nedge, edge(3,nedge), conn(3,nelt), nvfix, numfix(*), itrace, nqint, liwork, lrwork
Integer, Intent (Inout) :: ifail
Integer, Intent (Out) :: iwork(liwork)
Real (Kind=nag_wp), Intent (Inout) :: coor(2,nv)
Real (Kind=nag_wp), Intent (Out) :: rwork(lrwork)
C Header Interface
#include <nag.h>
void  d06caf_ (const Integer *nv, const Integer *nelt, const Integer *nedge, double coor[], const Integer edge[], const Integer conn[], const Integer *nvfix, const Integer numfix[], const Integer *itrace, const Integer *nqint, Integer iwork[], const Integer *liwork, double rwork[], const Integer *lrwork, Integer *ifail)
The routine may be called by the names d06caf or nagf_mesh_dim2_smooth_bary.

3 Description

d06caf uses a barycentering approach to improve the smoothness of a given mesh. The measure of quality used for a triangle K is
QK=αhKρK;  
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).

4 References

George P L and Borouchaki H (1998) Delaunay Triangulation and Meshing: Application to Finite Elements Editions HERMES, Paris

5 Arguments

1: nv Integer Input
On entry: the total number of vertices in the input mesh.
Constraint: nv3.
2: nelt Integer Input
On entry: the number of triangles in the input mesh.
Constraint: nelt2×nv-1.
3: nedge Integer Input
On entry: the number of the boundary and interface edges in the input mesh.
Constraint: nedge1.
4: coor(2,nv) Real (Kind=nag_wp) array Input/Output
On entry: coor(1,i) contains the x coordinate of the ith input mesh vertex, for i=1,2,,nv; while coor(2,i) contains the corresponding y coordinate.
On exit: coor(1,i) will contain the x coordinate of the ith smoothed mesh vertex, for i=1,2,,nv; while coor(2,i) will contain the corresponding y 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.
5: edge(3,nedge) Integer array Input
On entry: the specification of the boundary or interface edges. edge(1,j) and edge(2,j) contain the vertex numbers of the two end points of the jth boundary edge. edge(3,j) is a user-supplied tag for the jth boundary or interface edge: edge(3,j)=0 for an interior edge and has a nonzero tag otherwise.
Constraint: 1edge(i,j)nv and edge(1,j)edge(2,j), for i=1,2 and j=1,2,,nedge.
6: conn(3,nelt) Integer array Input
On entry: the connectivity of the mesh between triangles and vertices. For each triangle j, conn(i,j) gives the indices of its three vertices (in anticlockwise order), for i=1,2,3 and j=1,2,,nelt.
Constraint: 1conn(i,j)nv and conn(1,j)conn(2,j) and conn(1,j)conn(3,j) and conn(2,j)conn(3,j), for i=1,2,3 and j=1,2,,nelt.
7: nvfix Integer Input
On entry: the number of fixed vertices in the input mesh.
Constraint: 0nvfixnv.
8: numfix(*) Integer array Input
Note: the dimension of the array numfix must be at least max(1,nvfix).
On entry: the indices in coor of fixed interior vertices of the input mesh.
Constraint: if nvfix>0, 1numfix(i)nv, for i=1,2,,nvfix.
9: itrace Integer Input
On entry: the level of trace information required from d06caf.
itrace0
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 Integer Input
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 array Workspace
12: liwork Integer Input
On entry: the dimension of the array iwork as declared in the (sub)program from which d06caf is called.
Constraint: liwork8×nelt+2×nv.
13: rwork(lrwork) Real (Kind=nag_wp) array Workspace
14: lrwork Integer Input
On entry: the dimension of the array rwork as declared in the (sub)program from which d06caf is called.
Constraint: lrwork2×nv+nelt.
15: ifail Integer Input/Output
On entry: ifail must be set to 0, −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 −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 −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 -1 or 1 is used it is essential to test the value of ifail on exit.
On exit: ifail=0 unless the routine detects an error or a warning has been flagged (see Section 6).

6 Error Indicators and Warnings

If on entry ifail=0 or −1, explanatory error messages are output on the current error message unit (as defined by x04aaf).
Errors or warnings detected by the routine:
ifail=1
On entry, conn(I,J)=value, I=value, J=value and nv=value.
Constraint: conn(I,J)1 and conn(I,J)nv.
On entry, edge(I,J)=value, I=value, J=value and nv=value.
Constraint: edge(I,J)1 and edge(I,J)nv.
On entry, liwork=value and LIWKMN=value.
Constraint: liworkLIWKMN.
On entry, lrwork=value and LRWKMN=value.
Constraint: lrworkLRWKMN.
On entry, nedge=value.
Constraint: nedge1.
On entry, nelt=value and nv=value.
Constraint: nelt2×nv-1.
On entry, numfix(I)=value, I=value and nv=value.
Constraint: numfix(I)1 and numfix(I)nv.
On entry, nv=value.
Constraint: nv3.
On entry, nv=value and nvfix=value.
Constraint: 0nvfixnv.
On entry, the end points of the edge J have the same index I: J=value and I=value.
On entry, vertices 1 and 2 of the triangle K have the same index I: K=value and I=value.
On entry, vertices 1 and 3 of the triangle K have the same index I: K=value and I=value.
On entry, vertices 2 and 3 of the triangle K have the same index I: K=value and I=value.
ifail=2
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 argument conn). Setting itrace>1 may provide more information. If the problem persists, contact NAG.
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.
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.
ifail=-999
Dynamic memory allocation failed.
See Section 9 in the Introduction to the NAG Library FL Interface for further information.

7 Accuracy

Not applicable.

8 Parallelism and Performance

Background information to multithreading can be found in the Multithreading documentation.
d06caf 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.

9 Further Comments

None.

10 Example

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.

10.1 Program Text

Program Text (d06cafe.f90)

10.2 Program Data

Program Data (d06cafe.d)

10.3 Program Results

Program Results (d06cafe.r)
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 "d06cafe_distorted.r" Example Program Randomly distorted uniform mesh
GnuplotProduced by GNUPLOT 5.4 patchlevel 6 "d06cafe_smoothed.r" Distorted mesh smoothed and a uniform mesh recovered