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/****************************************************************************/
/* This file is part of FreeFem++. */
/* */
/* FreeFem++ is free software: you can redistribute it and/or modify */
/* it under the terms of the GNU Lesser General Public License as */
/* published by the Free Software Foundation, either version 3 of */
/* the License, or (at your option) any later version. */
/* */
/* FreeFem++ 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 Lesser General Public License for more details. */
/* */
/* You should have received a copy of the GNU Lesser General Public License */
/* along with FreeFem++. If not, see <http://www.gnu.org/licenses/>. */
/****************************************************************************/
// SUMMARY : ...
// LICENSE : LGPLv3
// ORG : LJLL Universite Pierre et Marie Curie, Paris, FRANCE
// AUTHORS : Jean-Marie Mirebeau
// E-MAIL : jean-marie.mirebeau@math.u-psud.fr
#ifndef EXAMPLE_METRICS_H
#define EXAMPLE_METRICS_H
#include "math.h"
#include "RZ.h"
// ********** functions ***********
/*
* Some riemannian metrics on the unit square or cube, for the purpose of testing algorithms.
*/
template<int whichMetric> const sym2 ExampleMetric (const R2 &P);
template<int whichMetric> const sym3 ExampleMetric3D (const R3 &P);
/********************** 2D *****************/
template<>
const sym2 ExampleMetric<0>(const R2 &P) {return sym2(1, 0, 1);}// identity
template<>
const sym2 ExampleMetric<1>(const R2 &P) { // A piecewise constant anisotropic metric
const double scal = fabs(P.x - 1 / 2.) < 1 / 6. ? 4 : 1;
return sym2(scal, -scal, 4 * scal);
}
template<>
const sym2 ExampleMetric<2>(const R2 &P) { // circle, regularity Graded.
const double delta = 0.03; // paramètre
const R2 Q = P - R2(0.5, 0.5);
const double r = Q.norm();
const double h = max(fabs(r - 1 / 2.), delta);
return sym2(1 / (h * h), 1 / h, Q);
}
template<>
const sym2 ExampleMetric<3>(const R2 &P) { // circle, regularity QuasiAcute.
const double delta = 0.4; // paramètre
const R2 Q = P - R2(0.5, 0.5);
const double r = Q.norm();
const double h = max(fabs(r - 1 / 2.), delta);
const double k = max(fabs(r - 1 / 2.), delta * delta);
return sym2(1 / (k * k), 1 / (h * h), Q);
}
template<>
const sym2 ExampleMetric<4>(const R2 &P) {return sym2(10, 0, 1);} // diagonal
template<>
const sym2 ExampleMetric<5>(const R2 &P) { // High anisotropy along the spiral r=k(theta+2 mu Pi), mu in {0,1,2}.
const double pi = 4 * atan(1);
const double width = 0.006;
const double k = 0.4 / (6 * pi);
const double mu = 100.;
const R2 Q = P - R2(0.5, 0.5);
const double r = Q.norm();
double theta = Q.x == -r ? pi : 2 * atan(Q.y / (r + Q.x)); // theta = theta >= 0 ? theta : theta+pi;
if (fabs(r - k * theta) <= width)
theta = theta + 0 * pi;
else if (fabs(r - k * (theta + 2 * pi)) <= width)
theta = theta + 2 * pi;
else if (fabs(r - k * (theta + 4 * pi)) <= width)
theta = theta + 4 * pi;
else if (fabs(r - k * (theta + 6 * pi)) <= width && theta <= 0)
theta = theta + 6 * pi;
else
return sym2(1, 0, 1); // {metric[0]=1; metric[1]=0; metric[2]=1; break;}
double c = cos(theta) - theta * sin(theta), s = sin(theta) + theta * cos(theta);// tangente à la spirale
double cOld = c;
c = -s;
s = cOld;
return sym2(1, 1 / (mu * mu), R2(c, s));
}
template<>
const sym2 ExampleMetric<6>(const R2 &P) { // high but constant anisotropy
const double mu = 30., t = 0.3;
R2 Q(cos(t), sin(t));
return sym2(1, 1 / (mu * mu), Q);
}
template<> const sym2 ExampleMetric<7>(const R2 &P) {const double s = 0.1 + (P - R2(0.1, 0.2)).norm(); return sym2() / square(s);}
template<> const sym2 ExampleMetric<8>(const R2 &P) {const double s = 0.1 + (P - R2(0.1, 0.2)).norm(); return sym2(100, 1, R2(1 / 2., sqrt(3.) / 2)) / square(s);}
template<> const sym2 ExampleMetric<9>(const R2 &P) {const double s = 0.1 + fabs(P.x); return sym2(100, 0, 1) / square(s);}
// template<int whichMetric> sym2 coExampleMetric(const R2 & P){return ExampleMetric<whichMetric>(P).comatrix();}
/************************ 3D *************************/
template<>
const sym3 ExampleMetric3D<0>(const R3 &P) {return sym3(1, 1, 1, 0, 0, 0);}
template<>
const sym3 ExampleMetric3D<1>(const R3 &P) {return sym3(1, 10, 100, 0, 0, 0);}
template<>
const sym3 ExampleMetric3D<2>(const R3 &P) {return sym3(1, 10, R3(0.1, -0.2, 0.4));}
template<>
const sym3 ExampleMetric3D<3>(const R3 &P) {// tire bouchon...
const double r0 = 0.33;
const double theta0 = 4 * M_PI;
const double delta0 = 0.06;
const double mu = 1 / 8.;
const R3 Q(P.x - 0.5, P.y - 0.5, P.z - 0.5);
const double r = sqrt(Q.x * Q.x + Q.y * Q.y);
if (fabs(r - r0) > delta0) {return sym3();}
if (square(Q.x - r * cos(theta0 * Q.z)) + square(Q.y - r * sin(theta0 * Q.z)) > square(r * delta0)) {return sym3();}
return sym3(mu * mu, 1, R3(-r0 * theta0 * sin(theta0 * Q.z), r0 * theta0 * cos(theta0 * Q.z), 1));
}
#endif
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