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///
/// This file is part of Rheolef.
///
/// Copyright (C) 2000-2009 Pierre Saramito <Pierre.Saramito@imag.fr>
///
/// Rheolef is free software; you can redistribute it and/or modify
/// it under the terms of the GNU General Public License as published by
/// the Free Software Foundation; either version 2 of the License, or
/// (at your option) any later version.
///
/// Rheolef is distributed in the hope that it will be useful,
/// but WITHOUT ANY WARRANTY; without even the implied warranty of
/// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
/// GNU General Public License for more details.
///
/// You should have received a copy of the GNU General Public License
/// along with Rheolef; if not, write to the Free Software
/// Foundation, Inc., 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA
///
/// =========================================================================
#include "rheolef/quadrature.h"
#include "rheolef/gauss_jacobi.h"
using namespace rheolef;
using namespace std;
/* ------------------------------------------
* Gauss quadrature formulae on the triangle
* order r : exact for all polynoms of P_r
*
* References for optimal formulae :
* 1) G. Dhatt and G. Touzot,
* Une presentation de la methode des elements finis,
* Maloine editeur, Paris
* Deuxieme edition,
* 1984,
* page 297
* 2) M. Crouzeix et A. L. Mignot
* Exercices d'analyse des equations differentielles,
* Masson,
* 1986,
* p. 61
* PROBLEM: direct optimized formulae have hard-coded
* coefficients in double precision.
* TODO: compute numeric value at the fly why full
* precision.
* ------------------------------------------
*/
void
quadrature_on_geo::init_triangle (quadrature_option_type opt)
{
// -------------------------------------------------------------------------
// special case : superconvergent patch recovery nodes & weights
// -------------------------------------------------------------------------
quadrature_option_type::family_type f = opt.get_family();
if (f == quadrature_option_type::superconvergent) {
switch (opt.get_order()) {
case 0:
case 1:
f = quadrature_option_type::gauss;
break;
case 2:
f = quadrature_option_type::middle_edge;
break;
default:
error_macro ("unsupported superconvergent("<<opt.get_order()<<")");
}
}
// -------------------------------------------------------------------------
// special case : Middle Edge formulae
// e.g. when using special option in riesz_representer
// -------------------------------------------------------------------------
if (f == quadrature_option_type::middle_edge) {
switch (opt.get_order()) {
case 0 :
case 1 :
case 2 :
// middle edge formulae:
wx(x(Float(0.5), Float(0.0)), 1./6);
wx(x(Float(0.0), Float(0.5)), 1./6);
wx(x(Float(0.5), Float(0.5)), 1./6);
break;
default:
error_macro ("unsupported Middle-Edge("<<opt.get_order()<<")");
}
return;
}
// -------------------------------------------------------------------------
// special case : Gauss-Lobatto points
// e.g. when using special option in riesz() function
// -------------------------------------------------------------------------
if (f == quadrature_option_type::gauss_lobatto) {
switch (opt.get_order()) {
case 0 :
case 1 :
// trapezes:
wx(x(Float(0), Float(0)), 1./6);
wx(x(Float(1), Float(0)), 1./6);
wx(x(Float(0), Float(1)), 1./6);
break;
case 2 :
// simpson:
wx(x(Float(0), Float(0)), 3/Float(120));
wx(x(Float(1), Float(0)), 3/Float(120));
wx(x(Float(0), Float(1)), 3/Float(120));
wx(x(Float(0.5), Float(0.0)), 8/Float(120));
wx(x(Float(0.0), Float(0.5)), 8/Float(120));
wx(x(Float(0.5), Float(0.5)), 8/Float(120));
wx(x(1/Float(3), 1/Float(3)), 27/Float(120));
break;
default:
error_macro ("unsupported Gauss-Lobatto("<<opt.get_order()<<")");
}
return;
}
// -------------------------------------------------------------------------
// default & general case : Gauss points
// -------------------------------------------------------------------------
assert_macro (f == quadrature_option_type::gauss,
"unsupported quadrature family \"" << opt.get_family_name() << "\"");
switch (opt.get_order()) {
case 0 :
case 1 :
// better than the general case:
// r=0 => 2 nodes
// r=1 => 4 nodes
// here : 1 node
wx(x(1/Float(3), 1/Float(3)), 1./2);
break;
case 2 :
// better than the general case:
// r=2 => 6 nodes
// here : 3 node
wx(x(1/Float(6), 1/Float(6)), 1/Float(6));
wx(x(4/Float(6), 1/Float(6)), 1/Float(6));
wx(x(1/Float(6), 4/Float(6)), 1/Float(6));
break;
#if !(defined(_RHEOLEF_HAVE_CLN) || defined(_RHEOLEF_HAVE_DOUBLEDOUBLE) || defined(_RHEOLEF_HAVE_LONG_DOUBLE))
// numerical values are inlined in double precision : 15 digits !
case 3 :
case 4 : {
// better than the general case:
// r=3 => 9 nodes
// r=4 => 12 nodes
// here : 6 node
Float a = 0.445948490915965;
Float b = 0.091576213509771;
Float wa = 0.111690794839005;
Float wb = 0.054975871827661;
wx(x(a, a), wa);
wx(x(1-2*a, a), wa);
wx(x(a, 1-2*a), wa);
wx(x(b, b), wb);
wx(x(1-2*b, b), wb);
wx(x(b, 1-2*b), wb);
break;
}
case 5 :
case 6 : {
// better than the general case:
// r=5 => 16 nodes
// r=6 => 20 nodes
// here : 12 node
Float a = 0.063089014491502;
Float b = 0.249286745170910;
Float c = 0.310352451033785;
Float d = 0.053145049844816;
Float wa = 0.025422453185103;
Float wb = 0.058393137863189;
Float wc = 0.041425537809187;
wx(x(a, a), wa);
wx(x(1-2*a, a), wa);
wx(x(a, 1-2*a), wa);
wx(x(b, b), wb);
wx(x(1-2*b, b), wb);
wx(x(b, 1-2*b), wb);
wx(x(c, d), wc);
wx(x(d, c), wc);
wx(x(1-(c+d), c), wc);
wx(x(1-(c+d), d), wc);
wx(x(c, 1-(c+d)), wc);
wx(x(d, 1-(c+d)), wc);
break;
}
#endif // BIGFLOAT
default: {
// Gauss-Legendre quadrature formulae
// where Legendre = Jacobi(alpha=0,beta=0) polynoms
size_t r = opt.get_order();
size_type n0 = n_node_gauss(r);
size_type n1 = n_node_gauss(r+1);
vector<Float> zeta0(n0), omega0(n0);
vector<Float> zeta1(n1), omega1(n1);
gauss_jacobi (n0, 0, 0, zeta0.begin(), omega0.begin());
gauss_jacobi (n1, 0, 0, zeta1.begin(), omega1.begin());
for (size_type i = 0; i < n0; i++) {
for (size_type j = 0; j < n1; j++) {
// we transform the square into the triangle
Float eta_0 = (1+zeta0[i])*(1-zeta1[j])/4;
Float eta_1 = (1+zeta1[j])/2;
Float J = (1-zeta1[j])/8;
wx (x(eta_0,eta_1), J*omega0[i]*omega1[j]);
}
}
}
}
}
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