d06 Chapter Contents
d06 Chapter Introduction
NAG C Library Manual

# NAG Library Function Documentnag_mesh2d_smooth (d06cac)

## 1  Purpose

nag_mesh2d_smooth (d06cac) uses a barycentering technique to smooth a given mesh.

## 2  Specification

 #include #include
 void nag_mesh2d_smooth (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)

## 3  Description

nag_mesh2d_smooth (d06cac) 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 ${h}_{K}$ is the diameter (length of the longest edge) of $K$, ${\rho }_{K}$ is the radius of its inscribed circle and $\alpha =\frac{\sqrt{3}}{6}$ 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.
nag_mesh2d_smooth (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:     nvIntegerInput
On entry: the total number of vertices in the input mesh.
Constraint: ${\mathbf{nv}}\ge 3$.
2:     neltIntegerInput
On entry: the number of triangles in the input mesh.
Constraint: ${\mathbf{nelt}}\le 2×{\mathbf{nv}}-1$.
3:     nedgeIntegerInput
On entry: the number of the boundary and interface edges in the input mesh.
Constraint: ${\mathbf{nedge}}\ge 1$.
4:     coor[$2×{\mathbf{nv}}$]doubleInput/Output
On entry: ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)×2+0\right]$ contains the $x$ coordinate of the $\mathit{i}$th input mesh vertex, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$; while ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)×2+1\right]$ contains the corresponding $y$ coordinate.
On exit: ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)×2+0\right]$ will contain the $x$ coordinate of the $\mathit{i}$th smoothed mesh vertex, for $\mathit{i}=1,2,\dots ,{\mathbf{nv}}$; while ${\mathbf{coor}}\left[\left(\mathit{i}-1\right)×2+1\right]$ 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[$3×{\mathbf{nedge}}$]const IntegerInput
On entry: the specification of the boundary or interface edges. ${\mathbf{edge}}\left[\left(j-1\right)×3+0\right]$ and ${\mathbf{edge}}\left[\left(j-1\right)×3+1\right]$ contain the vertex numbers of the two end points of the $j$th boundary edge. ${\mathbf{edge}}\left[\left(j-1\right)×3+2\right]$ is a user-supplied tag for the $j$th boundary or interface edge: ${\mathbf{edge}}\left[\left(j-1\right)×3+2\right]=0$ for an interior edge and has a nonzero tag otherwise. Note that the edge vertices are numbered from $1$ to nv.
Constraint: $1\le {\mathbf{edge}}\left[\left(\mathit{j}-1\right)×3+\mathit{i}-1\right]\le {\mathbf{nv}}$ and ${\mathbf{edge}}\left[\left(\mathit{j}-1\right)×3+0\right]\ne {\mathbf{edge}}\left[\left(\mathit{j}-1\right)×3+1\right]$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,{\mathbf{nedge}}$.
6:     conn[$3×{\mathbf{nelt}}$]const IntegerInput
On entry: the connectivity of the mesh between triangles and vertices. For each triangle $\mathit{j}$, ${\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+\mathit{i}-1\right]$ gives the indices of its three vertices (in anticlockwise order), for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,{\mathbf{nelt}}$. Note that the mesh vertices are numbered from $1$ to nv.
Constraint: $1\le {\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+\mathit{i}-1\right]\le {\mathbf{nv}}$ and ${\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+0\right]\ne {\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+1\right]$ and ${\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+0\right]\ne {\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+2\right]$ and ${\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+1\right]\ne {\mathbf{conn}}\left[\left(\mathit{j}-1\right)×3+2\right]$, for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,{\mathbf{nelt}}$.
7:     nvfixIntegerInput
On entry: the number of fixed vertices in the input mesh.
Constraint: $0\le {\mathbf{nvfix}}\le {\mathbf{nv}}$.
8:     numfix[$\mathit{dim}$]const IntegerInput
Note: the dimension, dim, of the array numfix must be at least $\mathrm{max}\phantom{\rule{0.125em}{0ex}}\left(1,{\mathbf{nvfix}}\right)$.
On entry: the indices in coor of fixed interior vertices of the input mesh.
Constraint: if ${\mathbf{nvfix}}>0$, $1\le {\mathbf{numfix}}\left[\mathit{i}-1\right]\le {\mathbf{nv}}$, for $\mathit{i}=1,2,\dots ,{\mathbf{nvfix}}$.
9:     itraceIntegerInput
On entry: the level of trace information required from nag_mesh2d_smooth (d06cac).
${\mathbf{itrace}}\le 0$
No output is generated.
${\mathbf{itrace}}=1$
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.
${\mathbf{itrace}}>1$
The output is similar to that produced when ${\mathbf{itrace}}=1$ 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 ${\mathbf{itrace}}=0$, unless you are experienced with finite element meshes.
10:   outfileconst 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:   nqintIntegerInput
On entry: the number of intervals between the extreme quality values for the input and the smoothed mesh.
If ${\mathbf{itrace}}=0$, nqint is not referenced.
12:   failNagError *Input/Output
The NAG error argument (see Section 3.6 in the Essential Introduction).

## 6  Error Indicators and Warnings

NE_ALLOC_FAIL
Dynamic memory allocation failed.
On entry, argument $〈\mathit{\text{value}}〉$ had an illegal value.
NE_INT
On entry, ${\mathbf{nedge}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nedge}}\ge 1$.
On entry, ${\mathbf{nv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nv}}\ge 3$.
NE_INT_2
On entry, ${\mathbf{nelt}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{nelt}}\le 2×{\mathbf{nv}}-1$.
On entry, ${\mathbf{nv}}=〈\mathit{\text{value}}〉$ and ${\mathbf{nvfix}}=〈\mathit{\text{value}}〉$.
Constraint: $0\le {\mathbf{nvfix}}\le {\mathbf{nv}}$.
On entry, the endpoints of the edge $j$ have the same index $i$: $j=〈\mathit{\text{value}}〉$ and $i=〈\mathit{\text{value}}〉$.
On entry, vertices $1$ and $2$ of the triangle $k$ have the same index $i$: $k=〈\mathit{\text{value}}〉$ and $i=〈\mathit{\text{value}}〉$.
On entry, vertices $1$ and $3$ of the triangle $k$ have the same index $i$: $k=〈\mathit{\text{value}}〉$ and $i=〈\mathit{\text{value}}〉$.
On entry, vertices $2$ and $3$ of the triangle $k$ have the same index $i$: $k=〈\mathit{\text{value}}〉$ and $i=〈\mathit{\text{value}}〉$.
NE_INT_3
On entry, ${\mathbf{numfix}}\left[i-1\right]=〈\mathit{\text{value}}〉$, $i=〈\mathit{\text{value}}〉$ and ${\mathbf{nv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{numfix}}\left[i-1\right]\ge 1$ and ${\mathbf{numfix}}\left[i-1\right]\le {\mathbf{nv}}$.
NE_INT_4
On entry, ${\mathbf{CONN}}\left(i,j\right)=〈\mathit{\text{value}}〉$, $i=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{nv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{CONN}}\left(i,j\right)\ge 1$ and ${\mathbf{CONN}}\left(i,j\right)\le {\mathbf{nv}}$, where ${\mathbf{CONN}}\left(i,j\right)$ denotes ${\mathbf{conn}}\left[\left(j-1\right)×3+i-1\right]$.
On entry, ${\mathbf{EDGE}}\left(i,j\right)=〈\mathit{\text{value}}〉$, $i=〈\mathit{\text{value}}〉$, $j=〈\mathit{\text{value}}〉$ and ${\mathbf{nv}}=〈\mathit{\text{value}}〉$.
Constraint: ${\mathbf{EDGE}}\left(i,j\right)\ge 1$ and ${\mathbf{EDGE}}\left(i,j\right)\le {\mathbf{nv}}$, where ${\mathbf{EDGE}}\left(i,j\right)$ denotes ${\mathbf{edge}}\left[\left(j-1\right)×3+i-1\right]$.
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.
A serious error has occurred in an internal call to an auxiliary function. Check the input mesh especially the connectivity. Seek expert help.
NE_NOT_CLOSE_FILE
Cannot close file $〈\mathit{\text{value}}〉$.
NE_NOT_WRITE_FILE
Cannot open file $〈\mathit{\text{value}}〉$ for writing.

Not applicable.

Not applicable.

## 9  Example

In this example, a uniform mesh on the unit square is randomly distorted using functions from Chapter g05. nag_mesh2d_smooth (d06cac) is then used to smooth the distorted mesh and recover a uniform mesh.

### 9.1  Program Text

Program Text (d06cace.c)

### 9.2  Program Data

Program Data (d06cace.d)

### 9.3  Program Results

Program Results (d06cace.r)