Dear all,
I'm am trying to reproduce with my implementation, the results in the
Photonic Crystal computations performed in [1]. Here, the author uses a
grid with an inner disk with radius R=0.475, and for FEM it is used the
software Concepts [2] that implements curvilinear elements denoted Blending
technique, or transfinite interpolation in quadrilaterals [3], [4,p. 144].
Exponential convergence was obtained for the p-FEM version.
By using the deal.II SphericalManifold, I can only get with the attached
code up to R<0.46, before I get the exception:
"The image of the mapping applied to cell with center [0.446317 -0.206317]
is distorted. The cell geometry or the mapping are invalid, giving a
non-positive volume fraction of -0.000129188 in quadrature point 54."
I am attaching a minimal test case (modified from step-10) that reproduces
the errors:
mesh_test.cc
and the code for the same mesh used in [1], adapted to deal.II standards in:
unit_cell.h
The question is then if it is possible to get this example running for
R=0.475, in order to reproduce the results in [1]. Is there something I am
doing wrong that can be improved? Can this behaviour be explained by round
off's?
Furthermore, is anyone implementing transfinite interpolation [3,4] in
deal.II?
Thanks in advance, any help is greatly appreciated!
References:
[1] Engström, C.,Spectral approximation of quadratic operator polynomials
arising in photonic band structure calculations, Numer. Math. (2014) 126:
413.
[2] Concepts web page:
http://wiki.math.ethz.ch/Concepts/WebRoot?redirectedfrom=Concepts.WebHome
[3] W. J. Gordon, C. A. Hall, Construction of curvilinear coordinate
systems and application to mesh generation Int. J. Num. Meth. Eng., Vol. 7
(1973), pp. 461-477
[4] Pavel Solin, Karel Segeth, Ivo Dolezel, Higher-Order Finite Element
Methods, Studies in Advanced Mathematics, CRC Press, 2003.
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/* ---------------------------------------------------------------------
*
* Copyright (C) 2001 - 2014 by the deal.II authors
*
* This file is part of the deal.II library.
*
* The deal.II library is free software; you can use it, 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 2.1 of the License, or (at your option) any later version.
* The full text of the license can be found in the file LICENSE at
* the top level of the deal.II distribution.
*
* ---------------------------------------------------------------------
*
* Authors: Wolfgang Bangerth, Ralf Hartmann, University of Heidelberg, 2001
*/
#include <deal.II/base/quadrature_lib.h>
#include <deal.II/base/convergence_table.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/grid/manifold_lib.h>
#include <deal.II/grid/tria.h>
#include <deal.II/grid/grid_out.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/dofs/dof_accessor.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/fe/mapping_q.h>
#include <iostream>
#include <fstream>
#include <cmath>
// added lines
#include "unit_cell.h"
#include <deal.II/numerics/data_out.h>
namespace Step10
{
using namespace dealii;
const long double pi = 3.141592653589793238462643;
template <int dim>
void gnuplot_output()
{
std::cout << "Output of grids into gnuplot files:" << std::endl
<< "===================================" << std::endl;
Triangulation<dim> triangulation;
GridGenerator::hyper_ball (triangulation);
static const SphericalManifold<dim> boundary;
triangulation.set_all_manifold_ids_on_boundary(0);
triangulation.set_manifold (0, boundary);
for (unsigned int refinement=0; refinement<2;
++refinement, triangulation.refine_global(1))
{
std::cout << "Refinement level: " << refinement << std::endl;
std::string filename_base = "ball";
filename_base += '0'+refinement;
for (unsigned int degree=1; degree<4; ++degree)
{
std::cout << "Degree = " << degree << std::endl;
const MappingQ<dim> mapping (degree);
GridOut grid_out;
GridOutFlags::Gnuplot gnuplot_flags(false, 30);
grid_out.set_flags(gnuplot_flags);
std::string filename = filename_base+"_mapping_q";
filename += ('0'+degree);
filename += ".dat";
std::ofstream gnuplot_file (filename.c_str());
grid_out.write_gnuplot (triangulation, gnuplot_file, &mapping);
}
std::cout << std::endl;
}
}
template <int dim>
void compute_pi_by_area ()
{
std::cout << "Computation of Pi by the area:" << std::endl
<< "==============================" << std::endl;
// const QGauss<dim> quadrature(4);
for (double R=0.45; R < 0.5; R += 0.005)
for (unsigned int degree=1; degree<11; ++degree)
{
std::cout << "Radius = " << R << std::endl;
std::cout << "Degree = " << degree << std::endl;
Triangulation<dim> triangulation;
// GridGenerator::hyper_ball (triangulation);
// new lines
Geom_parameters gp;
unsigned int curved_faces_label=10;
cell_rod ( triangulation, 1.0, R, gp, false );
static const SphericalManifold<dim> boundary;
triangulation.set_manifold ( curved_faces_label, boundary ); // labels defined with the grid
// triangulation.set_all_manifold_ids_on_boundary (0);
const MappingQ<dim> mapping (degree, true);
const QGauss<dim> quadrature(degree);
// const FE_Q<dim> dummy_fe (1);
const FE_Q<dim> dummy_fe (QGaussLobatto<1>(degree + 1));
DoFHandler<dim> dof_handler (triangulation);
FEValues<dim> fe_values (mapping, dummy_fe, quadrature,
update_JxW_values);
ConvergenceTable table;
// No refinements
for (unsigned int refinement=0; refinement<1;
++refinement, triangulation.refine_global (1))
{
table.add_value("cells", triangulation.n_active_cells());
dof_handler.distribute_dofs (dummy_fe);
long double area = 0;
typename DoFHandler<dim>::active_cell_iterator
cell = dof_handler.begin_active(),
endc = dof_handler.end();
for (; cell!=endc; ++cell)
{
double r = cell->center ().distance(Point<dim>(0,0));
fe_values.reinit (cell);
for (unsigned int i=0; i<fe_values.n_quadrature_points; ++i) {
if (r < R)
area += fe_values.JxW (i);
}
};
double new_area = area/(R*R);
table.add_value("eval.pi", static_cast<double> (new_area));
table.add_value("error", static_cast<double> (std::fabs(new_area-pi)));
DataOut<dim> data_out;
data_out.attach_dof_handler (dof_handler);
Vector<double> dummy ( dof_handler.n_dofs() );
string name = "grid";
data_out.add_data_vector (dummy, name );
std::ostringstream ss;
ss <<"grid_R" << R << "_p" << degree << "_ref" << refinement << ".vtk";
std::ofstream output (ss.str().c_str());
data_out.build_patches (mapping, 8, DataOut<dim>::curved_inner_cells);
data_out.write_vtk (output);
};
table.omit_column_from_convergence_rate_evaluation("cells");
table.omit_column_from_convergence_rate_evaluation("eval.pi");
table.evaluate_all_convergence_rates(ConvergenceTable::reduction_rate_log2);
table.set_precision("eval.pi", 16);
table.set_scientific("error", true);
table.write_text(std::cout);
std::cout << std::endl;
};
}
}
int main ()
{
try
{
std::cout.precision (16);
Step10::compute_pi_by_area<2> ();
}
catch (std::exception &exc)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Exception on processing: " << std::endl
<< exc.what() << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
catch (...)
{
std::cerr << std::endl << std::endl
<< "----------------------------------------------------"
<< std::endl;
std::cerr << "Unknown exception!" << std::endl
<< "Aborting!" << std::endl
<< "----------------------------------------------------"
<< std::endl;
return 1;
}
return 0;
}
#include <deal.II/grid/tria.h>
#include <deal.II/grid/tria_accessor.h>
#include <deal.II/grid/tria_iterator.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/grid/tria_boundary_lib.h>
#include <deal.II/grid/grid_out.h>
#include <deal.II/grid/grid_in.h>
#include <deal.II/grid/grid_reordering.h>
// #include <deal.II/grid/manifold_lib.h>
#include <fstream>
#include <assert.h>
#include <map>
#define PI 3.14159265358979323846
using namespace dealii;
using namespace std;
struct Geom_parameters {
/*
* model: geometric configuration, parameters must be dependent on the model problem
*/
unsigned int n_scatterers, scatterers_coarse_cells, n_balls, coated_coarse_cells, defective_index;
double coat_radius;
bool coated=false;
bool defect=false;
bool defect_n2=false;
vector< Point<2> > ball_centers;
vector<double> radius;
vector<double> radius_PML;
vector<unsigned int> ball_labels;
// double a;
string name;
};
// In the article "Spectral approximation of quadratic operator polynomials arising in photonic band structure calculations", Christian Engström 2012,
//
// This mesh is used with R=0.475 and exponential convergence is obtained.
// Check figure 1, and figure 3 on the mentioned paper.
//
// http://link.springer.com/article/10.1007/s00211-013-0568-y
//
// Software Concepts:
// http://wiki.math.ethz.ch/Concepts/WebRoot?redirectedfrom=Concepts.WebHome
void cell_rod ( Triangulation<2> &tria, double L, double r, // 2*r/L < 1.0
Geom_parameters &gp, bool mesh_out)
{
double l = 0.5*L, d = 0.5*r, q=1.0/sqrt(2.0); // q: corner points factor
const unsigned int n = 2, n_vert = 24+1, n_cell = 19+1;
bool info = false, plot_eps = true; // for printing cells, faces and vertex values
int fv = 4, fc = 1; // f: fixed, displacement of index on the vertexes and cells ... inner objects
// geometric information-----------------------
gp.n_scatterers = 1;
gp.scatterers_coarse_cells = 12;
gp.n_balls = 1; // just one Manifold ... everything centered at origin
gp.ball_centers.resize( gp.n_balls );
gp.radius.resize(gp.n_balls);
gp.radius[0]=r;
gp.ball_centers[0] = Point<2>( 0.0, 0.0 );
gp.name = "cell_rod";
if(mesh_out) printf("%% defined by using %d balls\n", gp.n_balls);
std::vector<Point<2> > vertices(n_vert);
vertices[0] = Point<2>( 0.0, 0.0 );
// inner 4 squares
vertices[1] = Point<2>( d, 0.0 );
vertices[2] = Point<2>( .5*d, .5*d );
vertices[3] = Point<2>( 0.0, d );
vertices[4] = Point<2>( -.5*d, .5*d );
vertices[5] = Point<2>( -d, 0.0 );
vertices[6] = Point<2>( -.5*d, -.5*d );
vertices[7] = Point<2>( 0.0, -d );
vertices[8] = Point<2>( .5*d, -.5*d );
// touching circle
vertices[9] = Point<2>( r, 0.0 );
vertices[10] = Point<2>( r*q, r*q );
vertices[11] = Point<2>( 0.0, r );
vertices[12] = Point<2>( -r*q, r*q );
vertices[13] = Point<2>( -r, 0.0 );
vertices[14] = Point<2>( -r*q, -r*q );
vertices[15] = Point<2>( 0.0, -r );
vertices[16] = Point<2>( r*q, -r*q );
vertices[17] = Point<2>( l, 0.0 );
vertices[18] = Point<2>( l, l );
vertices[19] = Point<2>( 0.0, l );
vertices[20] = Point<2>( -l, l );
vertices[21] = Point<2>( -l, 0.0 );
vertices[22] = Point<2>( -l, -l );
vertices[23] = Point<2>( 0.0, -l );
vertices[24] = Point<2>( l, -l );
//vertices[] = Point<2>( , );
std::vector< std::vector<int> > cell_v (n_cell, std::vector<int>(4));
cell_v[0][0] = 0; // cell, i = 0
cell_v[0][1] = 8;
cell_v[0][2] = 2;
cell_v[0][3] = 1;
cell_v[1][0] = 0; // cell, i = 1
cell_v[1][1] = 2;
cell_v[1][2] = 4;
cell_v[1][3] = 3;
cell_v[2][0] = 0; // cell, i = 2
cell_v[2][1] = 4;
cell_v[2][2] = 6;
cell_v[2][3] = 5;
cell_v[3][0] = 0; // cell, i = 3
cell_v[3][1] = 6;
cell_v[3][2] = 8;
cell_v[3][3] = 7;
cell_v[4][0] = 8; // cell, i = 4
cell_v[4][1] = 16;
cell_v[4][2] = 1;
cell_v[4][3] = 9;
cell_v[5][0] = 2; // cell, i = 5
cell_v[5][1] = 1;
cell_v[5][2] = 10;
cell_v[5][3] = 9;
cell_v[6][0] = 2; // cell, i = 6
cell_v[6][1] = 10;
cell_v[6][2] = 3;
cell_v[6][3] = 11;
cell_v[7][0] = 4; // cell, i = 7
cell_v[7][1] = 3;
cell_v[7][2] = 12;
cell_v[7][3] = 11;
cell_v[8][0] = 4; // cell, i = 8
cell_v[8][1] = 12;
cell_v[8][2] = 5;
cell_v[8][3] = 13;
cell_v[9][0] = 6; // cell, i = 9
cell_v[9][1] = 5;
cell_v[9][2] = 14;
cell_v[9][3] = 13;
cell_v[10][0] = 6; // cell, i = 10
cell_v[10][1] = 14;
cell_v[10][2] = 7;
cell_v[10][3] = 15;
cell_v[11][0] = 8; // cell, i = 11
cell_v[11][1] = 7;
cell_v[11][2] = 16;
cell_v[11][3] = 15;
// cell 12
cell_v[12][0] = 16; // cell, i = 12
cell_v[12][1] = 24;
cell_v[12][2] = 9;
cell_v[12][3] = 17;
// cell 13
cell_v[13][0] = 10; // cell, i = 13
cell_v[13][1] = 9;
cell_v[13][2] = 18;
cell_v[13][3] = 17;
// cell 14
cell_v[14][0] = 10; // cell, i = 14
cell_v[14][1] = 18;
cell_v[14][2] = 11;
cell_v[14][3] = 19;
// cell 15
cell_v[15][0] = 12; // cell, i = 15
cell_v[15][1] = 11;
cell_v[15][2] = 20;
cell_v[15][3] = 19;
// cell 16
cell_v[16][0] = 12; // cell, i = 16
cell_v[16][1] = 20;
cell_v[16][2] = 13;
cell_v[16][3] = 21;
// cell 17
cell_v[17][0] = 14; // cell, i = 17
cell_v[17][1] = 13;
cell_v[17][2] = 22;
cell_v[17][3] = 21;
// cell 18
cell_v[18][0] = 14; // cell, i = 18
cell_v[18][1] = 22;
cell_v[18][2] = 15;
cell_v[18][3] = 23;
// cell 19
cell_v[19][0] = 16; // cell, i =
cell_v[19][1] = 15;
cell_v[19][2] = 24;
cell_v[19][3] = 23;
if (info) {
for (unsigned int i = 0; i < n_cell; ++i)
if(info) printf("cell[%d] = (%d,%d,%d,%d)\n", i, cell_v[i][0], cell_v[i][1],cell_v[i][2],cell_v[i][3]);
}
std::vector<CellData<2> > cells (n_cell, CellData<2>());
unsigned int cell_idx = 0;
for (unsigned int i=0; i < n_cell; ++i) {
// printf("%d: ", i);
for (unsigned int j=0; j < 4; ++j) {
cells[i].vertices[j] = cell_v[i][j];
if(info) printf("%d ",cell_v[i][j]);
}
if(info) printf(";\n");
cell_idx++;
}
for (unsigned int i=0; i < n_vert; ++i) {
if(info) printf(" %f, %f;\n", vertices[i][0],vertices[i][1] );
}
if(info) printf("\n\n");
//printf("after grid ordering\n");
tria.create_triangulation (
std::vector<Point<2> >(&vertices[0], &vertices[n_vert]),
cells,
SubCellData()); // no boundary information
if(info) printf("after creating triangulation\n");
// colorizing ------------------------------------------------------------------------------------
double eps = 1e-5, mid;
unsigned int label=10;
// cell way
for (Triangulation<2>::active_cell_iterator
cell = tria.begin_active(); cell!=tria.end(); ++cell)
{
// cell->set_all_manifold_ids (label);
for (unsigned int f=0;
f < GeometryInfo<2>::faces_per_cell;++f) {
const Point<2> face_center = cell->face(f)->center(),
p0 = cell->face(f)->vertex(0) , p1 = cell->face(f)->vertex(1);
double d0 = p0.distance( Point<2>( 0.0,0.0 ) ), d1 = p1.distance( Point<2>( 0.0,0.0 ) );
if(info) printf(" f=%d, dist=%f\n", f, cell->face(f)->center().distance( Point<2>( 0.0,0.0 ) ) );
if( (cell->face(f)->center().distance( Point<2>( 0.0,0.0 )) - mid) < eps ) {
if(info) printf(" ... set one! ... dist=%f\n",
cell->face(f)->center().distance( Point<2>( 0.0,0.0 ) ) );
cell->face(f)->set_all_manifold_ids (label+1);
} // if face of circle
if ( abs((d0+d1)-2*r) < eps ) { // disk faces
cell->face(f)->set_all_manifold_ids (label);
} // outer faces
} // face
if ( cell->center().distance( Point<2>( 0.0,0.0 ) ) < eps ) { // eliminating origin
cell->set_all_manifold_ids (label+1);
}
} // cell
// end-colorizing --------------------------------------------------------------------------------
if(mesh_out) { // printing the resulting geometry
//printf("printing original mesh to eps ... \n");
std::ofstream gridfile ("coarse.eps");
GridOut grid_out;
grid_out.write_eps (tria, gridfile);
// printing vertices
printf("vert=[\n");
for (unsigned int j = 0; j < n_vert; j++) {
printf("\t%.3f, %.3f;\n", vertices[j](0), vertices[j](1));
}
printf("];\n\n");
// printing cell indexes
printf("cells=[\n");
for (unsigned int j = 0; j < n_cell; j++) {
printf("\t%d, %d,%d, %d;\n", cell_v[j][0], cell_v[j][1], cell_v[j][2], cell_v[j][3] );
}
printf("];\n");
}
}
void cell_rod ( Triangulation<2> &tria, double L, double diameter_ratio) {
Geom_parameters gp;
cell_rod ( tria, L, diameter_ratio, gp, false);
}
void cell_rod ( Triangulation<2> &tria, double L, double diameter_ratio,
Geom_parameters &gp) {
cell_rod ( tria, L, diameter_ratio, gp, false);
}
