void	d06cac (Integer nv, Integer nelt, Integer nedge, double coor[], const Integer edge[], const Integer conn[], Integer nvfix, const Integer numfix[], Integer itrace, const char outfile, Integer nqint, NagError fail)

The function may be called by the names: d06cac, nag_mesh_dim2_smooth_bary or nag_mesh2d_smooth.

3 Description

d06cac uses a barycentering approach to improve the smoothness of a given mesh. The measure of quality used for a triangle

K

Q_{K} = α \frac{h_{K}}{ρ_{K}};

where

h_{K}

is the diameter (length of the longest edge) of

K

ρ_{K}

is the radius of its inscribed circle and

α = \frac{\sqrt{3}}{6}

is a normalization factor chosen to give

Q_{K} = 1

for an equilateral triangle.

Q_{K}

ranges from

1

, for an equilateral triangle, to

\infty

, for a totally flat triangle.

d06cac makes small perturbation to vertices (using a barycenter formula) in order to give a reasonably good value of

Q_{K}

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 function 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:

nv \geq 3

2: $nelt$ – Integer Input

On entry: the number of triangles in the input mesh.

Constraint:

nelt \leq 2 \times nv - 1

3: $nedge$ – Integer Input

On entry: the number of the boundary and interface edges in the input mesh.

Constraint:

nedge \geq 1

4: $coor [2 \times nv]$ – double Input/Output

Note: the

(i, j)

th element of the matrix is stored in

coor [(j - 1) \times 2 + i - 1]

On entry:

coor [(i - 1) \times 2]

contains the

x

coordinate of the

i

th input mesh vertex, for

i = 1, 2, \dots, nv

; while

coor [(i - 1) \times 2 + 1]

contains the corresponding

y

coordinate.

On exit:

coor [(i - 1) \times 2]

will contain the

x

coordinate of the

i

th smoothed mesh vertex, for

i = 1, 2, \dots, nv

; while

coor [(i - 1) \times 2 + 1]

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: $edge [3 \times nedge]$ – const Integer Input

Note: the

(i, j)

th element of the matrix is stored in

edge [(j - 1) \times 3 + i - 1]

On entry: the specification of the boundary or interface edges.

edge [(j - 1) \times 3]

and

edge [(j - 1) \times 3 + 1]

contain the vertex numbers of the two end points of the

j

th boundary edge.

edge [(j - 1) \times 3 + 2]

is a user-supplied tag for the

j

th boundary or interface edge:

edge [(j - 1) \times 3 + 2] = 0

for an interior edge and has a nonzero tag otherwise. Note that the edge vertices are numbered from

1

to nv.

Constraint:

1 \leq edge [(j - 1) \times 3 + i - 1] \leq nv

and

edge [(j - 1) \times 3] \neq edge [(j - 1) \times 3 + 1]

, for

i = 1, 2

and

j = 1, 2, \dots, nedge

6: $conn [3 \times nelt]$ – const Integer Input

Note: the

(i, j)

th element of the matrix is stored in

conn [(j - 1) \times 3 + i - 1]

On entry: the connectivity of the mesh between triangles and vertices. For each triangle

j

conn [(j - 1) \times 3 + i - 1]

gives the indices of its three vertices (in anticlockwise order), for

i = 1, 2, 3

and

j = 1, 2, \dots, nelt

. Note that the mesh vertices are numbered from

1

to nv.

Constraint:

1 \leq conn [(j - 1) \times 3 + i - 1] \leq nv

and

conn [(j - 1) \times 3] \neq conn [(j - 1) \times 3 + 1]

and

conn [(j - 1) \times 3] \neq conn [(j - 1) \times 3 + 2]

and

conn [(j - 1) \times 3 + 1] \neq conn [(j - 1) \times 3 + 2]

, for

i = 1, 2, 3

and

j = 1, 2, \dots, nelt

7: $nvfix$ – Integer Input

On entry: the number of fixed vertices in the input mesh.

Constraint:

0 \leq nvfix \leq nv

8: $numfix [\dim]$ – const Integer Input

Note: the dimension, dim, 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

1 \leq numfix [i - 1] \leq nv

, for

i = 1, 2, \dots, nvfix

9: $itrace$ – Integer Input

On entry: the level of trace information required from d06cac.

$itrace \leq 0$: No output is generated.
$itrace = 1$: A histogram of the triangular element qualities is printed 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: $outfile$ – 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.

11: $nqint$ – Integer Input

On entry: the number of intervals between the extreme quality values for the input and the smoothed mesh.

itrace = 0

, nqint is not referenced.

12: $fail$ – NagError * Input/Output

The NAG error argument (see Section 7 in the Introduction to the NAG Library CL Interface).

6 Error 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 $⟨ value ⟩$ had an illegal value.
NE_INT: On entry, $nedge = ⟨ value ⟩$ .
Constraint: $nedge \geq 1$ .

On entry, $nv = ⟨ value ⟩$ .
Constraint: $nv \geq 3$ .
NE_INT_2: On entry, $nelt = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $nelt \leq 2 \times nv - 1$ .

On entry, $nv = ⟨ value ⟩$ and $nvfix = ⟨ value ⟩$ .
Constraint: $0 \leq nvfix \leq nv$ .

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 $I = ⟨ value ⟩$ .
NE_INT_3: On entry, $numfix [I - 1] = ⟨ value ⟩$ , $I = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $numfix [I - 1] \geq 1$ and $numfix [I - 1] \leq nv$ .
NE_INT_4: On entry, $conn (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $conn (I, J) \geq 1$ and $conn (I, J) \leq nv$ , where $conn (I, J)$ denotes $conn [(J - 1) \times 3 + I - 1]$ .

On entry, $edge (I, J) = ⟨ value ⟩$ , $I = ⟨ value ⟩$ , $J = ⟨ value ⟩$ and $nv = ⟨ value ⟩$ .
Constraint: $edge (I, J) \geq 1$ and $edge (I, J) \leq nv$ , where $edge (I, J)$ denotes $edge [(J - 1) \times 3 + I - 1]$ .
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.

A serious error has occurred in an internal call to an auxiliary function. 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.
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 $⟨ value ⟩$ .
NE_NOT_WRITE_FILE: Cannot open file $⟨ value ⟩$ for writing.

7 Accuracy

Not applicable.

8 Parallelism and Performance

d06cac 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.

9 Further Comments

None.

10 Example

In this example, a uniform mesh on the unit square is randomly distorted using functions from Chapter G05. d06cac is then used to smooth the distorted mesh and recover a uniform mesh.

d06ca: FL CL CPP AD

NAG CL Interfaced06cac (dim2_​smooth_​bary)

▸▿ Contents

1 Purpose

2 Specification

3 Description

4 References

5 Arguments

6 Error Indicators and Warnings

7 Accuracy

8 Parallelism and Performance

9 Further Comments

10 Example

10.1 Program Text

10.2 Program Data

10.3 Program Results

NAG CL Interface
d06cac (dim2_smooth_bary)