d06ab:: Mesh Generation (NAG Toolbox)

The position of the internal vertices is a function position of the vertices on the given boundary. A fine mesh on the boundary results in a fine mesh in the interior. To dilute the influence of the data on the interior of the domain, the value of npropa can be changed. The propagation coefficient is calculated as:

ω = 1 + \frac{a - 1.0}{20.0}

, where

a

is the absolute value of npropa. During the process vertices are generated on edges of the mesh

T_{i}

to obtain the mesh

T_{i + 1}

in the general incremental method (consult the D06 Chapter Introduction or George and Borouchaki (1998)). This generation uses the coefficient

ω

, and it is geometric if

npropa > 0

, and arithmetic otherwise. But increasing the value of

a

may lead to failure of the process, due to precision, especially in geometries with holes. So you are advised to manipulate the argument npropa with care.

function d06ab_example


fprintf('d06ab example results\n\n');

% Boundary mesh sizes
nrae = 128;
nvc = 40;
nvb = int64(2*nrae + nvc);
edge = zeros(3, nvb, 'int64');
% Maximum number of vertices in final mesh
nvmax = 6000;
coor = zeros(2, nvmax);

% Circular boundary
centre = [1.0,0.0];
radius = 3.0;
for i = 1:nvc
  theta = 2*pi*(i-1)/nvc;
  coor(1,i) = radius*cos(theta) + centre(1);
  coor(2,i) = radius*sin(theta) + centre(2);
  edge(1,i) = i;
  edge(2,i) = i + 1;
  edge(3,i) = 1;
end
edge(2,nvc) = 1;

% data for aerofoil RAE 2822
rae = [ 0.000000  0.000000; 0.000602  0.003165; 0.002408  0.006306;
        0.005412  0.009416; 0.009607  0.012480; 0.014984  0.015489;
        0.021530  0.018441; 0.029228  0.021348; 0.038060  0.024219;
        0.048005  0.027062; 0.059039  0.029874; 0.071136  0.032644;
        0.084265  0.035360; 0.098396  0.038011; 0.113495  0.040585;
        0.129524  0.043071; 0.146447  0.045457; 0.164221  0.047729;
        0.182803  0.049874; 0.202150  0.051885; 0.222215  0.053753;
        0.242949  0.055470; 0.264302  0.057026; 0.286222  0.058414;
        0.308658  0.059629; 0.331555  0.060660; 0.354858  0.061497;
        0.378510  0.062133; 0.402455  0.062562; 0.426635  0.062779;
        0.450991  0.062774; 0.475466  0.062530; 0.500000  0.062029;
        0.524534  0.061254; 0.549009  0.060194; 0.573365  0.058845; 
        0.597545  0.057218; 0.621490  0.055344; 0.645142  0.053258;
        0.668445  0.050993; 0.691342  0.048575; 0.713778  0.046029;
        0.735698  0.043377; 0.757051  0.040641; 0.777785  0.037847;
        0.797850  0.035017; 0.817197  0.032176; 0.835779  0.029347;
        0.853553  0.026554; 0.870476  0.023817; 0.886505  0.021153;
        0.901604  0.018580; 0.915735  0.016113; 0.928864  0.013769;
        0.940961  0.011562; 0.951995  0.009508; 0.961940  0.007622;
        0.970772  0.005915; 0.978470  0.004401; 0.985016  0.003092;
        0.990393  0.002001; 0.994588  0.001137; 0.997592  0.000510;
        0.999398  0.000128; 1.000000  0.000000; 0.999398  0.000035;
        0.997592  0.000137; 0.994588  0.000296; 0.990393  0.000497;
        0.985016  0.000719; 0.978470  0.000935; 0.970772  0.001112;
        0.961940  0.001212; 0.951995  0.001197; 0.940961  0.001033;
        0.928864  0.000694; 0.915735  0.000157; 0.901604 -0.000600;
        0.886505 -0.001592; 0.870476 -0.002829; 0.853553 -0.004314;
        0.835779 -0.006048; 0.817197 -0.008027; 0.797850 -0.010244;
        0.777785 -0.012690; 0.757051 -0.015357; 0.735698 -0.018232;
        0.713778 -0.021289; 0.691342 -0.024495; 0.668445 -0.027814;
        0.645142 -0.031207; 0.621490 -0.034631; 0.597545 -0.038043;
        0.573365 -0.041397; 0.549009 -0.044642; 0.524534 -0.047719;
        0.500000 -0.050563; 0.475466 -0.053099; 0.450991 -0.055257;
        0.426635 -0.056979; 0.402455 -0.058224; 0.378510 -0.058974;
        0.354858 -0.059236; 0.331555 -0.059046; 0.308658 -0.058459;
        0.286222 -0.057547; 0.264302 -0.056376; 0.242949 -0.054994;
        0.222215 -0.053427; 0.202150 -0.051694; 0.182803 -0.049805;
        0.164221 -0.047773; 0.146447 -0.045610; 0.129524 -0.043326;
        0.113495 -0.040929; 0.098396 -0.038431; 0.084265 -0.035843;
        0.071136 -0.033170; 0.059039 -0.030416; 0.048005 -0.027586;
        0.038060 -0.024685; 0.029228 -0.021722; 0.021530 -0.018707;
        0.014984 -0.015649; 0.009607 -0.012559; 0.005412 -0.009443;
        0.002408 -0.006308; 0.000602 -0.003160];

% Transform RAE 2822 for secondary foil by rotating (theta),
% contracting (0.4) and translating (Ttrans).
theta = pi/12.0;
c = cos(theta);
s = sin(theta);
Trot = [c s;-s,c];
Ttrans = [0.75+0.25*c; -0.25*s];
coor(1:2,nvc+1:nvc+nrae) = transpose(rae);
coor(1:2,nvc+nrae+1:nvb) = 0.4*Trot*transpose(rae);
for i = nvc+nrae+1:nvb
  coor(1:2,i) = coor(1:2,i) + Ttrans;
end

% Edges
for i = 1:nvb
  edge(1,i) = i;
  edge(2,i) = i + 1;
  edge(3,i) = 0;
end
edge(2,nvc) = 1;
edge(2,nvc+nrae) = nvc + 1;
edge(2,nvb) = nvc + nrae + 1;

% Interior vertices on the wake of the aerofiols
d_interior = 2.5/(nvc+1);
for i = 1:nvc
  i1 = nvc + i;
  coor(1,nvb+i) = 1.38 + i*d_interior;
  coor(2,nvb+i) = -tan(theta)*(coor(1,nvb+i)-0.75);
end

% Weights for interior vertices
weight(1:nvc) = 0.01;
      
% Generate mesh using arithmetic propgation coefficient 1.2
npropa = int64(-5);
itrace = int64(0);
[nv, nelt, coor, conn, ifail] = ...
d06ab( ...
       nvb, edge, coor, weight, npropa, 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; % To ensure that circles look like circles
title('Triangulation for Aerofoils RAE 2822 and wake vertices');

On entry,	$nvb < 3$ ,
or	$nvint < 0$ ,
or	$nvb + nvint > nvmax$ ,
or	$nedge < 1$ ,
or	$edge (i, j) < 1$ or $edge (i, j) > nvb$ , for some $i = 1, 2$ and $j = 1, 2, \dots, nedge$ ,
or	$edge (1, j) = edge (2, j)$ , for some $j = 1, 2, \dots, nedge$ ,
or	$npropa = 0$ ;
or	if $nvint > 0$ , $weight (i) \leq 0.0$ , for some $i = 1, 2, \dots, nvint$ ;
or	$lrwork < 12 \times nvmax + 15$ ,
or	$liwork < 6 \times nedge + 32 \times nvmax + 2 \times nvb + 78$ .

NAG Toolbox: nag_mesh_2d_gen_delaunay (d06ab)

▸▿ Contents

Purpose

Syntax

Description

References

Parameters

Compulsory Input Parameters

Optional Input Parameters

Output Parameters

Error Indicators and Warnings

Accuracy

Further Comments

Example