void
set_initial_phase_null(Calculus& calculus, Eigen::VectorXcd& initial_wave)
{
typedef typename Calculus::Cell Cell;
typedef typename Calculus::Index Index;
typedef typename Calculus::Point Point;
//! [time_phase_null]
for (int xx=9; xx<13; xx++)
for (int yy=13; yy<17; yy++)
{
const Point point(xx,yy);
const Cell cell = calculus.myKSpace.uSpel(point);
const Index index = calculus.getCellIndex(cell);
initial_wave(index) = 1;
}
//! [time_phase_null]
}
void
set_initial_phase_dir_yy(Calculus& calculus, Eigen::VectorXcd& initial_wave)
{
typedef typename Calculus::Cell Cell;
typedef typename Calculus::Index Index;
typedef typename Calculus::Point Point;
//! [time_phase_yy]
for (int xx=9; xx<13; xx++)
for (int yy=13; yy<17; yy++)
{
const Point point(xx,yy);
const Cell cell = calculus.myKSpace.uSpel(point);
const Index index = calculus.getCellIndex(cell);
initial_wave(index) = std::exp(std::complex<double>(0,2*M_PI*(yy-14.5)/8));
}
//! [time_phase_yy]
}
void propa_2d()
{
trace.beginBlock("2d propagation");
typedef DiscreteExteriorCalculus<2, 2, EigenLinearAlgebraBackend> Calculus;
typedef DiscreteExteriorCalculusFactory<EigenLinearAlgebraBackend> CalculusFactory;
{
trace.beginBlock("solving time dependent equation");
const Calculus::Scalar cc = 8; // px/s
trace.info() << "cc = " << cc << endl;
const Z2i::Domain domain(Z2i::Point(0,0), Z2i::Point(29,29));
const Calculus calculus = CalculusFactory::createFromDigitalSet(generateDiskSet(domain), false);
//! [time_laplace]
const Calculus::DualIdentity0 laplace = calculus.laplace<DUAL>() + 1e-8 * calculus.identity<0, DUAL>();
//! [time_laplace]
trace.info() << "laplace = " << laplace << endl;
trace.beginBlock("finding eigen pairs");
//! [time_eigen]
typedef Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> EigenSolverMatrix;
const EigenSolverMatrix eigen_solver(laplace.myContainer);
const Eigen::VectorXd eigen_values = eigen_solver.eigenvalues();
const Eigen::MatrixXd eigen_vectors = eigen_solver.eigenvectors();
//! [time_eigen]
trace.info() << "eigen_values_range = " << eigen_values.minCoeff() << "/" << eigen_values.maxCoeff() << endl;
trace.endBlock();
//! [time_omega]
const Eigen::VectorXd angular_frequencies = cc * eigen_values.array().sqrt();
//! [time_omega]
Eigen::VectorXcd initial_wave = Eigen::VectorXcd::Zero(calculus.kFormLength(0, DUAL));
//set_initial_phase_null(calculus, initial_wave);
set_initial_phase_dir_yy(calculus, initial_wave);
{
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, initial_wave.real());
board.saveSVG("propagation_time_wave_initial_coarse.svg");
}
//! [time_init_proj]
Eigen::VectorXcd initial_projections = eigen_vectors.transpose() * initial_wave;
//! [time_init_proj]
// low pass
//! [time_low_pass]
const Calculus::Scalar lambda_cutoff = 4.5;
const Calculus::Scalar angular_frequency_cutoff = 2*M_PI * cc / lambda_cutoff;
int cutted = 0;
for (int kk=0; kk<initial_projections.rows(); kk++)
{
const Calculus::Scalar angular_frequency = angular_frequencies(kk);
if (angular_frequency < angular_frequency_cutoff) continue;
initial_projections(kk) = 0;
cutted ++;
}
//! [time_low_pass]
trace.info() << "cutted = " << cutted << "/" << initial_projections.rows() << endl;
{
const Eigen::VectorXcd wave = eigen_vectors * initial_projections;
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, wave.real());
board.saveSVG("propagation_time_wave_initial_smoothed.svg");
}
const int kk_max = 60;
trace.progressBar(0,kk_max);
for (int kk=0; kk<kk_max; kk++)
{
//! [time_solve_time]
const Calculus::Scalar time = kk/20.;
const Eigen::VectorXcd current_projections = (angular_frequencies * std::complex<double>(0,time)).array().exp() * initial_projections.array();
const Eigen::VectorXcd current_wave = eigen_vectors * current_projections;
//! [time_solve_time]
std::stringstream ss;
ss << "propagation_time_wave_solution_" << std::setfill('0') << std::setw(3) << kk << ".svg";
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-1, 1));
board << Calculus::DualForm0(calculus, current_wave.real());
board.saveSVG(ss.str().c_str());
trace.progressBar(kk+1,kk_max);
}
trace.info() << endl;
trace.endBlock();
}
{
trace.beginBlock("forced oscillations");
const Z2i::Domain domain(Z2i::Point(0,0), Z2i::Point(50,50));
const Calculus calculus = CalculusFactory::createFromDigitalSet(generateDiskSet(domain), false);
const Calculus::DualIdentity0 laplace = calculus.laplace<DUAL>();
trace.info() << "laplace = " << laplace << endl;
trace.beginBlock("finding eigen pairs");
typedef Eigen::SelfAdjointEigenSolver<Eigen::MatrixXd> EigenSolverMatrix;
const EigenSolverMatrix eigen_solver(laplace.myContainer);
const Eigen::VectorXd laplace_eigen_values = eigen_solver.eigenvalues();
const Eigen::MatrixXd laplace_eigen_vectors = eigen_solver.eigenvectors();
trace.info() << "eigen_values_range = " << laplace_eigen_values.minCoeff() << "/" << laplace_eigen_values.maxCoeff() << endl;
trace.endBlock();
Calculus::DualForm0 concentration(calculus);
{
const Z2i::Point point(25,25);
const Calculus::Cell cell = calculus.myKSpace.uSpel(point);
const Calculus::Index index = calculus.getCellIndex(cell);
concentration.myContainer(index) = 1;
}
{
Board2D board;
board << domain;
board << concentration;
board.saveSVG("propagation_forced_concentration.svg");
}
for (int ll=0; ll<6; ll++)
{
//! [forced_lambda]
const Calculus::Scalar lambda = 4*20.75/(1+2*ll);
//! [forced_lambda]
trace.info() << "lambda = " << lambda << endl;
//! [forced_dalembert_eigen]
const Eigen::VectorXd dalembert_eigen_values = (laplace_eigen_values.array() - (2*M_PI/lambda)*(2*M_PI/lambda)).array().inverse();
const Eigen::MatrixXd concentration_to_wave = laplace_eigen_vectors * dalembert_eigen_values.asDiagonal() * laplace_eigen_vectors.transpose();
//! [forced_dalembert_eigen]
//! [forced_wave]
Calculus::DualForm0 wave(calculus, concentration_to_wave * concentration.myContainer);
//! [forced_wave]
wave.myContainer /= wave.myContainer(calculus.getCellIndex(calculus.myKSpace.uSpel(Z2i::Point(25,25))));
{
trace.info() << "saving samples" << endl;
const Calculus::Properties properties = calculus.getProperties();
{
std::stringstream ss;
ss << "propagation_forced_samples_hv_" << ll << ".dat";
std::ofstream handle(ss.str().c_str());
for (int kk=0; kk<=50; kk++)
{
const Z2i::Point point_horizontal(kk,25);
const Z2i::Point point_vertical(25,kk);
const Calculus::Scalar xx = 2 * (kk/50. - .5);
handle << xx << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_horizontal) << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_vertical) << endl;
}
}
{
std::stringstream ss;
ss << "propagation_forced_samples_diag_" << ll << ".dat";
std::ofstream handle(ss.str().c_str());
for (int kk=0; kk<=50; kk++)
{
const Z2i::Point point_diag_pos(kk,kk);
const Z2i::Point point_diag_neg(kk,50-kk);
const Calculus::Scalar xx = 2 * sqrt(2) * (kk/50. - .5);
handle << xx << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_diag_pos) << " ";
handle << sample_dual_0_form<Calculus>(properties, wave, point_diag_neg) << endl;
}
}
}
{
std::stringstream ss;
ss << "propagation_forced_wave_" << ll << ".svg";
Board2D board;
board << domain;
board << CustomStyle("KForm", new KFormStyle2D(-.5, 1));
board << wave;
board.saveSVG(ss.str().c_str());
}
}
trace.endBlock();
}
trace.endBlock();
}