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Chapter Introduction
NAG Toolbox

NAG Toolbox: nag_mesh_2d_gen_inc (d06aa)


    1  Purpose
    2  Syntax
    7  Accuracy
    9  Example


nag_mesh_2d_gen_inc (d06aa) generates a triangular mesh of a closed polygonal region in 2, given a mesh of its boundary. It uses a simple incremental method.


[nv, nelt, coor, conn, ifail] = d06aa(edge, coor, bspace, smooth, itrace, 'nvb', nvb, 'nvmax', nvmax, 'nedge', nedge, 'coef', coef, 'power', power)
[nv, nelt, coor, conn, ifail] = nag_mesh_2d_gen_inc(edge, coor, bspace, smooth, itrace, 'nvb', nvb, 'nvmax', nvmax, 'nedge', nedge, 'coef', coef, 'power', power)


nag_mesh_2d_gen_inc (d06aa) generates the set of interior vertices using a process based on a simple incremental method. A smoothing of the mesh is optionally available. For more details about the triangulation method, 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).


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


Compulsory Input Parameters

1:     edge3nedge int64int32nag_int array
The specification of the boundary edges. edge1j and edge2j contain the vertex numbers of the two end points of the jth boundary edge. edge3j is a user-supplied tag for the jth boundary edge and is not used by nag_mesh_2d_gen_inc (d06aa).
Constraint: 1edgeijnvb and edge1jedge2j, for i=1,2 and j=1,2,,nedge.
2:     coor2nvmax – double array
coor1i contains the x coordinate of the ith input boundary mesh vertex; while coor2i contains the corresponding y coordinate, for i=1,2,,nvb.
3:     bspacenvb – double array
The desired mesh spacing (triangle diameter, which is the length of the longer edge of the triangle) near the boundary vertices.
Constraint: bspacei>0.0, for i=1,2,,nvb.
4:     smooth – logical scalar
Indicates whether or not mesh smoothing should be performed.
If smooth=true, the smoothing is performed; otherwise no smoothing is performed.
5:     itrace int64int32nag_int scalar
The level of trace information required from nag_mesh_2d_gen_inc (d06aa).
No output is generated.
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 itrace=0, unless you are experienced with finite element mesh generation.

Optional Input Parameters

1:     nvb int64int32nag_int scalar
Default: the dimension of the array bspace.
The number of vertices in the input boundary mesh.
Constraint: 3nvbnvmax.
2:     nvmax int64int32nag_int scalar
Default: the dimension of the array coor.
The maximum number of vertices in the mesh to be generated.
3:     nedge int64int32nag_int scalar
Default: the dimension of the array edge.
The number of boundary edges in the input mesh.
Constraint: nedge1.
4:     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 Ocoef2. The mesh will be finer if coef is greater than 0.7165 and 0.75 is a good value.
5:     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: 0.1power10.0.

Output Parameters

1:     nv int64int32nag_int scalar
The total number of vertices in the output mesh (including both boundary and interior vertices). If nvb=nvmax, no interior vertices will be generated and nv=nvb.
2:     nelt int64int32nag_int scalar
The number of triangular elements in the mesh.
3:     coor2nvmax – double array
coor1i will contain the x coordinate of the i-nvbth generated interior mesh vertex; while coor2i will contain the corresponding y coordinate, for i=nvb+1,,nv. The remaining elements are unchanged.
4:     conn32×nvmax-1 int64int32nag_int array
The connectivity of the mesh between triangles and vertices. For each triangle j, connij gives the indices of its three vertices (in anticlockwise order), for i=1,2,3 and j=1,2,,nelt.
5:     ifail int64int32nag_int scalar
ifail=0 unless the function detects an error (see Error Indicators and Warnings).

Error Indicators and Warnings

Errors or warnings detected by the function:
On entry,nvb<3 or nvb>nvmax,
oredgeij<1 or edgeij>nvb, for some i=1,2 and j=1,2,,nedge,
oredge1j=edge2j, for some j=1,2,,nedge,
orbspacei0.0, for some i=1,2,,nvb,
orpower<0.1 or power>10.0,
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 itrace>0 may provide more details.
An unexpected error has been triggered by this routine. Please contact NAG.
Your licence key may have expired or may not have been installed correctly.
Dynamic memory allocation failed.


Not applicable.

Further Comments

The position of the internal vertices is a function of the positions of the vertices on the given boundary. A fine mesh on the boundary results in a fine mesh in the interior. The algorithm allows you to obtain a denser interior mesh by varying nvmax, bspace, coef and power. But you are advised to manipulate the last two arguments with care.
You are advised to take care to set the boundary inputs properly, especially for a boundary with multiply connected components. The orientation of the interior boundaries should be in clockwise order and opposite to that of the exterior boundary. If the boundary has only one connected component, its orientation should be anticlockwise.


In this example, a geometry with two holes (two interior circles inside an exterior one) is meshed using the simple incremental method (see the D06 Chapter Introduction). The exterior circle is centred at the origin with a radius 1.0, the first interior circle is centred at the point -0.5,0.0 with a radius 0.49, and the second one is centred at the point -0.5,0.65 with a radius 0.15. Note that the points -1.0,0.0 and -0.5,0.5) are points of ‘near tangency’ between the exterior circle and the first and second circles.
The boundary mesh has 100 vertices and 100 edges. Note that the particular mesh generated could be sensitive to the machine precision and therefore may differ from one implementation to another.
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;
  edge(2,i1) = nvb + 1;
  nvb = nvb + nvertices(icirc);
  % Subsequent circles are inner circles
  csign = -1;
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

PDF version (NAG web site, 64-bit version, 64-bit version)
Chapter Contents
Chapter Introduction
NAG Toolbox

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