NAG Toolbox: nag_mesh_2d_gen_inc (d06aa)

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

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

The specification of the boundary 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 edge and is not used by nag_mesh_2d_gen_inc (d06aa).

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

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

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

3:     $\mathrm{bspace}\left(\mathbf{nvb}\right)$ – double array

The desired mesh spacing (triangle diameter, which is the length of the longer edge of the triangle) near the boundary vertices.

Constraint: $\mathbf{bspace}\left(\mathit{i}\right)>0.0$, for $\mathit{i}=1,2,\dots ,\mathbf{nvb}$.

4:     $\mathrm{smooth}$ – logical scalar

Indicates whether or not mesh smoothing should be performed.

If $\mathbf{smooth}=\mathit{true}$, the smoothing is performed; otherwise no smoothing is performed.

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

The level of trace information required from nag_mesh_2d_gen_inc (d06aa).

$\mathbf{itrace}\le 0$

No output is generated.

$\mathbf{itrace}\ge 1$

Output from the meshing solver is printed on the current advisory message unit (see nag_file_set_unit_advisory (x04ab)). This output contains details of the vertices and triangles generated by the process.

You are advised to set $\mathbf{itrace}=0$, unless you are experienced with finite element mesh generation.

Optional Input Parameters

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

Default: the dimension of the array bspace.

The number of vertices in the input boundary mesh.

Constraint: $3\le \mathbf{nvb}\le \mathbf{nvmax}$.

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

Default: the dimension of the array coor.

The maximum number of vertices in the mesh to be generated.

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

Default: the dimension of the array edge.

The number of boundary edges in the input mesh.

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

4:     $\mathrm{coef}$ – double scalar

Default: $0.75$.

The coefficient in the stopping criteria for the generation of interior vertices. This argument controls the triangle density and the number of triangles generated is in $\mathit{O}\left({\mathbf{coef}}^2\right)$. The mesh will be finer if coef is greater than $0.7165$ and $0.75$ is a good value.

5:     $\mathrm{power}$ – double scalar

Default: $0.25$.

Controls the rate of change of the mesh size during the generation of interior vertices. The smaller the value of power, the faster the decrease in element size away from the boundary.

Constraint: $0.1\le \mathbf{power}\le 10.0$.

Output Parameters

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

The total number of vertices in the output mesh (including both boundary and interior vertices). If $\mathbf{nvb}=\mathbf{nvmax}$, no interior vertices will be generated and $\mathbf{nv}=\mathbf{nvb}$.

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

The number of triangular elements in the mesh.

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

$\mathbf{coor}\left(1,\mathit{i}\right)$ will contain the $x$ coordinate of the $\left(\mathit{i}-\mathbf{nvb}\right)$th generated interior mesh vertex; while $\mathbf{coor}\left(2,\mathit{i}\right)$ will contain the corresponding $y$ coordinate, for $\mathit{i}=\mathbf{nvb}+1,\dots ,\mathbf{nv}$. The remaining elements are unchanged.

4:     $\mathrm{conn}\left(3,2\times \left(\mathbf{nvmax}-1\right)\right)$ – 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 $\mathit{j}=1,2,\dots ,\mathbf{nelt}$.

5:     $\mathrm{ifail}$ – 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,	$\mathbf{nvb}<3$ or $\mathbf{nvb}>\mathbf{nvmax}$,
or	$\mathbf{nedge}<1$,
or	$\mathbf{edge}\left(i,j\right)<1$ or $\mathbf{edge}\left(i,j\right)>\mathbf{nvb}$, 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	$\mathbf{bspace}\left(i\right)\le 0.0$, for some $i=1,2,\dots ,\mathbf{nvb}$,
or	$\mathbf{power}<0.1$ or $\mathbf{power}>10.0$,
or	$\mathit{liwork}<16\times \mathbf{nvmax}+2\times \mathbf{nedge}+\max \left(4\times \mathbf{nvmax}+2,\mathbf{nedge}\right)-14$,
or	$\mathit{lrwork}<\mathbf{nvmax}$.

   $\mathbf{ifail}=2$

An error has occurred during the generation of the interior mesh. Check the definition of the boundary (arguments coor and edge) as well as the orientation of the boundary (especially in the case of a multiple connected component boundary). Setting $\mathbf{itrace}>0$ may provide more details.

   $\mathbf{ifail}=-99$

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: d06aa_example

function d06aa_example


fprintf('d06aa 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);

[nv, nelt, coor, conn, ifail] = d06aa( ...
				       edge, coor, bspace, smooth, itrace);
fprintf('\nnv   = %d\n', nv);
fprintf('nelt = %d\n', nelt);
% Plot mesh
fig1 = figure;
triplot(transpose(double(conn(:,1:nelt))), coor(1,:), coor(2,:));
axis equal;

d06aa example results


nv   = 250
nelt = 402