NAG Toolbox: nag_mesh_2d_renumber (d06cc)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

1:     $\mathrm{nnzmax}$ – int64int32nag_int scalar

The maximum number of nonzero entries in the matrix based on the input mesh. It is the dimension of the arrays irow and icol as declared in the function from which nag_mesh_2d_renumber (d06cc) is called.

Constraint: $4\times \mathbf{nelt}+\mathbf{nv}\le \mathbf{nnzmax}\le {\mathbf{nv}}^2$.

2:     $\mathrm{coor}\left(2,\mathbf{nv}\right)$ – double array

$\mathbf{coor}\left(1,\mathit{i}\right)$ contains the $x$ coordinate of the $\mathit{i}$th input mesh vertex, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$; while $\mathbf{coor}\left(2,\mathit{i}\right)$ contains the corresponding $y$ coordinate.

3:     $\mathrm{edge}\left(3,\mathbf{nedge}\right)$ – int64int32nag_int array

The specification of the boundary or interface edges. $\mathbf{edge}\left(1,j\right)$ and $\mathbf{edge}\left(2,j\right)$ contain the vertex numbers of the two end points of the $j$th boundary edge. $\mathbf{edge}\left(3,j\right)$ is a user-supplied tag for the $j$th boundary or interface edge: $\mathbf{edge}\left(3,j\right)=0$ for an interior edge and has a nonzero tag otherwise.

Constraint: $1\le \mathbf{edge}\left(\mathit{i},\mathit{j}\right)\le \mathbf{nv}$ and $\mathbf{edge}\left(1,\mathit{j}\right)\ne \mathbf{edge}\left(2,\mathit{j}\right)$, for $\mathit{i}=1,2$ and $\mathit{j}=1,2,\dots ,\mathbf{nedge}$.

4:     $\mathrm{conn}\left(3,\mathbf{nelt}\right)$ – int64int32nag_int array

The connectivity of the mesh between triangles and vertices. For each triangle $\mathit{j}$, $\mathbf{conn}\left(\mathit{i},\mathit{j}\right)$ gives the indices of its three vertices (in anticlockwise order), for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

Constraint: $1\le \mathbf{conn}\left(\mathit{i},\mathit{j}\right)\le \mathbf{nv}$ and $\mathbf{conn}\left(1,\mathit{j}\right)\ne \mathbf{conn}\left(2,\mathit{j}\right)$ and $\mathbf{conn}\left(1,\mathit{j}\right)\ne \mathbf{conn}\left(3,\mathit{j}\right)$ and $\mathbf{conn}\left(2,\mathit{j}\right)\ne \mathbf{conn}\left(3,\mathit{j}\right)$, for $\mathit{i}=1,2,3$ and $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

5:     $\mathrm{itrace}$ – int64int32nag_int scalar

The level of trace information required from nag_mesh_2d_renumber (d06cc).

$\mathbf{itrace}\le 0$

No output is generated.

$\mathbf{itrace}=1$

Information about the effect of the renumbering on the finite element matrix are output. This information includes the half bandwidth and the sparsity structure of this matrix before and after renumbering.

$\mathbf{itrace}>1$

The output is similar to that produced when $\mathbf{itrace}=1$ but the sparsities (for each row of the matrix, indices of nonzero entries) of the matrix before and after renumbering are also output.

Optional Input Parameters

1:     $\mathrm{nv}$ – int64int32nag_int scalar

Default: the dimension of the array coor.

The total number of vertices in the input mesh.

Constraint: $\mathbf{nv}\ge 3$.

2:     $\mathrm{nelt}$ – int64int32nag_int scalar

Default: the dimension of the array conn.

The number of triangles in the input mesh.

Constraint: $\mathbf{nelt}\le 2\times \mathbf{nv}-1$.

3:     $\mathrm{nedge}$ – int64int32nag_int scalar

Default: the dimension of the array edge.

The number of boundary edges in the input mesh.

Constraint: $\mathbf{nedge}\ge 1$.

Output Parameters

1:     $\mathrm{nz}$ – int64int32nag_int scalar

The number of nonzero entries in the matrix based on the input mesh.

2:     $\mathrm{coor}\left(2,\mathbf{nv}\right)$ – double array

$\mathbf{coor}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\mathit{i}$th renumbered mesh vertex, for $\mathit{i}=1,2,\dots ,\mathbf{nv}$; while $\mathbf{coor}\left(2,\mathit{i}\right)$ will contain the corresponding $y$ coordinate.

3:     $\mathrm{edge}\left(3,\mathbf{nedge}\right)$ – int64int32nag_int array

The renumbered specification of the boundary or interface edges.

4:     $\mathrm{conn}\left(3,\mathbf{nelt}\right)$ – int64int32nag_int array

The renumbered connectivity of the mesh between triangles and vertices.

5:     $\mathrm{irow}\left(\mathbf{nnzmax}\right)$ – int64int32nag_int array

6:     $\mathrm{icol}\left(\mathbf{nnzmax}\right)$ – int64int32nag_int array

The first nz elements contain the row and column indices of the nonzero elements supplied in the finite element matrix $A$.

7:     $\mathrm{ifail}$ – int64int32nag_int scalar

$\mathbf{ifail}=\mathbf{0}$ unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

   $\mathbf{ifail}=1$

On entry,	$\mathbf{nv}<3$,
or	$\mathbf{nelt}>2\times \mathbf{nv}-1$,
or	$\mathbf{nedge}<1$,
or	$\mathbf{nnzmax}<4\times \mathbf{nelt}+\mathbf{nv}$ or $\mathbf{nnzmax}>{\mathbf{nv}}^2$
or	$\mathbf{conn}\left(i,j\right)<1$ or $\mathbf{conn}\left(i,j\right)>\mathbf{nv}$ for some $i=1,2,3$ and $j=1,2,\dots ,\mathbf{nelt}$,
or	$\mathbf{conn}\left(1,j\right)=\mathbf{conn}\left(2,j\right)$ or $\mathbf{conn}\left(1,j\right)=\mathbf{conn}\left(3,j\right)$ or $\mathbf{conn}\left(2,j\right)=\mathbf{conn}\left(3,j\right)$ for some $j=1,2,\dots ,\mathbf{nelt}$,
or	$\mathbf{edge}\left(i,j\right)<1$ or $\mathbf{edge}\left(i,j\right)>\mathbf{nv}$ for some $i=1,2$ and $j=1,2,\dots ,\mathbf{nedge}$,
or	$\mathbf{edge}\left(1,j\right)=\mathbf{edge}\left(2,j\right)$ for some $j=1,2,\dots ,\mathbf{nedge}$,
or	$\mathit{liwork}<\max \left(\mathbf{nnzmax},20\times \mathbf{nv}\right)$,
or	$\mathit{lrwork}<\mathbf{nv}$.

   $\mathbf{ifail}=2$

A serious error has occurred during the computation of the compact sparsity of the finite element matrix or in an internal call to the renumbering function. Check the input mesh, especially the connectivity between triangles and vertices (the argument conn). If the problem persists, contact NAG.

   $\mathbf{ifail}=-99$

An unexpected error has been triggered by this routine. Please contact NAG.

   $\mathbf{ifail}=-399$

Your licence key may have expired or may not have been installed correctly.

   $\mathbf{ifail}=-999$

Dynamic memory allocation failed.

Accuracy

Further Comments

Example

Open in the MATLAB editor: d06cc_example

function d06cc_example


fprintf('d06cc example results\n\n');


edge = zeros(3, 100, 'int64');
coor = zeros(2, 250);

% Define boundaries
ncirc     = 3; % 3 circles
nvertices = [40, 30, 30];
radii     = [1, 0.49, 0.15];
centres   = [0, 0; -0.5, 0; -0.5, 0.65];

% First circle is outer circle
csign = 1;
i1 = 0;
nvb = 0;
for icirc = 1:ncirc
   for i = 0:nvertices(icirc)-1
      i1 = i1+1;
      theta = 2*pi*i/nvertices(icirc);
      coor(1,i1) = radii(icirc)*cos(theta) + centres(icirc, 1);
      coor(2,i1) = csign*radii(icirc)*sin(theta) +  centres(icirc, 2);
      edge(1,i1) = i1;
      edge(2,i1) = i1 + 1;
      edge(3,i1) = 1;
   end
   edge(2,i1) = nvb + 1;
   nvb = nvb + nvertices(icirc);
   % Subsequent circles are inner circles
   csign = -1;
end
nedge = nvb;

% Initialise mesh control parameters
bspace = zeros(1, 100);
bspace(1:nvb) = 0.05;
smooth = true;
itrace = int64(0);

nnzmax = int64(3000);

% Mesh geometry
[nv, nelt, coor, conn, ifail] = d06aa(edge, coor, bspace, smooth, itrace);

% Compute the sparsity of the FE matrix from the input geometry
[nz, irow, icol, ifail] = d06cb(nv, nnzmax, conn, 'nelt', nelt);

fprintf('\nNumber of non-zero entries in input mesh:  %d\n', nz);

% Plot sparsity of input mesh
fig1 = figure;
plot(irow(1:double(nz)), icol(1:double(nz)), '.');
title ('Input Mesh');
set(gca,'YDir','reverse');


% Call the renumbering routine and get the new sparsity
[nz, coor, edge, conn, irow, icol, ifail] = ...
    d06cc(nnzmax, coor, edge, conn, itrace, 'nelt', nelt);

fprintf('Number of non-zero entries in output mesh: %d\n', nz);
% Plot smoothed mesh
fig2 = figure;
plot(irow(1:double(nz)), icol(1:double(nz)), '.');
title ('Output Mesh');
set(gca,'YDir','reverse');

d06cc example results


Number of non-zero entries in input mesh:  1556
Number of non-zero entries in output mesh: 1556