Luca, Thank you for the help.
I now have a working Ellipsoid class:
template <int dim,int spacedim>
class Ellipsoid: public SphericalManifold<dim,spacedim>
{
public:
Ellipsoid(double,double,double);
Point<spacedim> pull_back(const Point<spacedim> &space_point) const;
Point<spacedim> push_forward(const Point<spacedim> &chart_point) const;
Point<spacedim> get_new_point(const Quadrature<spacedim> &quad) const;
Point<spacedim> grid_transform(const Point<spacedim> &X);
private:
double a,b,c;
double max_axis;
const Point<spacedim> center;
Point<dim> ellipsoid_pull_back(const Point<spacedim> &space_point) const;
Point<spacedim> ellipsoid_push_forward(const Point<dim> &chart_point)
const;
};
with member functions:
template <int dim, int spacedim>
Ellipsoid<dim,spacedim>::Ellipsoid(double a, double b, double c) :
SphericalManifold<dim,spacedim>(center), a(a), b(b),c(c), center(0,0,0)
{
max_axis = std::max(std::max(a,b),c);
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::pull_back(const Point<spacedim>
&space_point) const
{
Point<dim> chart_point = ellipsoid_pull_back(space_point);
Point<spacedim> p;
p[0] = -1; // dummy radius to match return of
SphericalManifold::pull_back()
p[1] = chart_point[0];
p[2] = chart_point[1];
return p;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::push_forward(const Point<spacedim>
&chart_point) const
{
Point<dim> p; //
p[0] = chart_point[1];
p[1] = chart_point[2];
Point<spacedim> space_point = ellipsoid_push_forward(p);
return space_point;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::get_new_point(const
Quadrature<spacedim> &quad) const
{
double u,v,w;
std::vector< Point<spacedim> > space_points;
for (unsigned int i=0; i<quad.size(); ++i)
{
u = quad.point(i)[0]/a;
v = quad.point(i)[1]/b;
w = quad.point(i)[2]/c;
space_points.push_back(Point<spacedim>(u,v,w));
}
Quadrature<spacedim> spherical_quad = Quadrature<spacedim>(space_points,
quad.get_weights());
Point<spacedim> p =
SphericalManifold<dim,spacedim>::get_new_point(spherical_quad);
double x,y,z;
x = a*p[0];
y = b*p[1];
z = c*p[2];
Point<spacedim> new_point = Point<spacedim>(x,y,z);
return new_point;
}
template <int dim,int spacedim>
Point<dim> Ellipsoid<dim,spacedim>::ellipsoid_pull_back(const
Point<spacedim> &space_point) const
{
double x,y,z, u,v,w;
// get point on ellipsoid
x = space_point[0];
y = space_point[1];
z = space_point[2];
std::cout << "using a,b,c: " << std::endl;
std::cout << a << " " << b << " " << c << std::endl;
std::cout << "from pull_back: " << std::endl;
std::cout << "space_point: " << std::endl;
std::cout << x << " " << y << " " << z << std::endl;
// map ellipsoid point onto sphere
u = x/a;
v = y/b;
w = z/c;
std::cout << "pulls back to : " << std::endl;
std::cout << u << " " << v << " " << w << std::endl;
std::cout << "on sphere." << std::endl;
Point<spacedim> p(u,v,w);
// use reference_sphere's pull_back function
Point<spacedim> q = pull_back(p);
Point<dim> chart_point;
std::cout << "sphere pull_back: " << std::endl;
std::cout << q[0] << " " << q[1] << " " << q[2] << std::endl;
std::cout << "r theta phi" << std::endl;
std::cout << "..........." << std::endl;
chart_point[0] = q[1];
chart_point[1] = q[2];
// return (theta,phi) in the chart domain
return chart_point;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::ellipsoid_push_forward(const
Point<dim> &chart_point) const
{
double theta,phi, x,y,z;
phi = chart_point[0];
theta = chart_point[1];
Point<spacedim> p(max_axis,theta,phi);
// map theta,phi in chart domain onto reference_sphere with radius
max_axis
Point<spacedim> X = push_forward(p);
// map point on sphere onto ellipsoid
x = a*X[0];
y = b*X[1];
z = c*X[2];
Point<spacedim> space_point(x,y,z);
// return point on ellipsoid
return space_point;
}
template<int dim, int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::grid_transform(const
Point<spacedim> &X)
{
// transform points X from sphere onto ellipsoid
double x,y,z;
x = a*X(0);
y = b*X(1);
z = c*X(2);
return Point<spacedim>(x,y,z);
}
along with the non-member function grid_transform() (which should not be
necessary, but I cannot seem to bind the member function
Ellipsoid<dim,spacedim>::grid_transform() to an instance of Ellipsoid and a
dummy variable, as is done in step-53)
Point<3> grid_transform(const Point<3> &X)
{
// transform points X from sphere onto ellipsoid
double x,y,z;
double a = 1; double b = 3; double c = 5;
x = a*X(0);
y = b*X(1);
z = c*X(2);
return Point<3>(x,y,z);
}
At any rate, this can all be scripted using the following:
void assemble_mesh_and_manifold()
{
const int dim = 2;
const int spacedim = 3;
double a,b,c;
a = 1; b=3; c=5;
Ellipsoid<dim,spacedim> ellipsoid(a,b,c);
Triangulation<dim,spacedim> tria;
// generate coarse spherical mesh
GridGenerator::hyper_sphere (tria, Point<spacedim>(0.0,0.0,0.0), 1.0);
for (Triangulation<dim,spacedim>::active_cell_iterator
cell=tria.begin_active(); cell!=tria.end(); ++cell)
cell->set_all_manifold_ids(0);
print_mesh_info(tria, "spherical_mesh.vtk");
GridTools::transform(&grid_transform, tria);
//
//GridTools::transform(std_cxx11::bind(&Ellipsoid<dim,spacedim>::grid_transform,std_cxx11::cref(ellipsoid),std_cxx11::_1),
tria); // error when trying to bind to member function in same way as
step-53
tria.set_manifold(0,ellipsoid);
tria.refine_global(3);
print_mesh_info(tria, "ellipsoidal_mesh.vtk");
}
I have attached the entire .cc file, and the output from the above code
looks great:
<https://lh3.googleusercontent.com/-bwkb9gP-gMw/V6oMwm7nLLI/AAAAAAAAFSU/dWS9t96oEQIsqQPjWEpWfQbPmEMoQBxAwCLcB/s1600/refined_ellipsoidal_mesh0000.png>
On Tuesday, August 9, 2016 at 3:56:58 AM UTC-4, Luca Heltai wrote:
>
> There is nothing wrong with what you are doing.
>
> The problem is in the nature of your Manifold, since it contains 3
> singular points. The first is the center of the ellipsoid, while the second
> and third are the north and south poles.
>
> Try attaching a SphericalManifold to your deformed grid, and see if you
> like the result. If you do, then hack the SphericalManifold. I don’t know
> what version of deal.II you are using. Assuming you are not using the
> latest dev, if you open up the definition of spherical manifold in
> manifold_lib.cc, you’ll see that SphericalManifold::get_new_point does
> something special in the case spacedim == 3, i.e., SphericalManifold is not
> *really* a ChartManifold in 3d.
>
> In this case the ChartManifold mechanism cannot be used for the reason
> above.
>
> A quick explanation why things are not working:
>
> Consider the top cell, and let us assume for a second that the north pole
> (z=z_max) is in the center of the cell, and that the four vertices of your
> ellipsoid lie at the same z (z=h < z_max). When you transform using
> ChartManifold, the z gets mapped to the angle phi, and every point of the
> cell will have the same angle phi, and different theta. If you take the
> average of the four points in the chart manifold coordinates, you’ll see
> that the average will have an average theta (which is ok), but it will also
> have the same phi of the four surrounding points… in other words, the
> average will not be in the middle of the four points (that should be the
> north pole: a singularity of your mapping). It will lye on the same
> latitude of the other four points.
>
> ChartManifold is doing exactly what you asked it to do, but in the case of
> maps with singularities, you should not use ChartManifold…
>
> My suggestion is to derive EllipsoidManifold from SphericalManifold, and
> then overload directly get_new_point, by calling internally get_new_point
> of spherical manifold, and then projecting the result to the ellipsoid...
>
> Best,
>
> Luca.
>
> > On 09 Aug 2016, at 24:13, thomas stephens <tdste...@gmail.com
> <javascript:>> wrote:
> >
> >
> >
> > I am trying to obtain the mesh for a codimension-1 ellipsoid and attach
> an ellipsoidal manifold to it in order to refine the mesh. My strategy is
> failing and I have a few questions.
> >
> > My strategy:
> > some definitions:
> > dim=2; spacedim=3; chartdim=spacedim-1; a=1; b=2;c=3;
> >
> > Set up codimension-1 Triangulation:
> >
> > Triangulation<dim,spacedim> tria;
> >
> >
> > Use GridGenerator::hyper_sphere() to obtain a codimension 1 mesh:
> >
> >
> > GridGenerator::hyper_sphere (tria, Point<spacedim>(0.0,0.0,0.0), 1.0);
> >
> > Use GridTools::transform() to map triangulation of sphere onto
> triangulation of ellipsoid (I have a question about this, see below).
> Here, grid_transform has signature Point<spacedim> grid_transform(const
> Point<spacedim> X)
> >
> > GridTools::transform(&grid_transform, tria);
> >
> > Next, subclass ChartManifold<dim,spacedim,chartdim> in order to obtain
> a push_forward() and a pull_back() function for my parametric ellipsoidal
> manifold description. Then attach that manifold to the triangulation:
> >
> > Ellipsoid<dim,spacedim,chartdim> ellipsoid(a,b,c);
> > tria.set_manifold(0,ellipsoid);
> >
> > Now refine the triangulation in order to see a refined ellipsoidal mesh:
> >
> > tria.refine_global(1);
> >
> > The Result: Complete garbage! I also don't really have an idea why this
> is not working - I've checked the dimensions on my push_forward() and
> pull_back() functions, I think that I'm setting the periodicity correctly
> in my ChartManifold superclass, and I've also checked that my simple math
> is okay. Let me know what other information would be useful. Below is
> the refined mesh for a=b=c=1. This should just be a sphere.
> >
> >
> >
> >
> > The code:
> > template <int dim,int spacedim,int chartdim=spacedim-1>
> > class Ellipsoid: public ChartManifold<dim,spacedim,chartdim>
> > {
> > public:
> >
> > Ellipsoid(double,double,double);
> >
> > Point<chartdim> pull_back(const Point<spacedim> &space_point) const;
> >
> > Point<spacedim> push_forward(const Point<chartdim> &chart_point)
> const;
> >
> > private:
> > double a,b,c;
> > double max_axis;
> > const Point<spacedim> center;
> > SphericalManifold<dim,spacedim> reference_sphere;
> >
> > };
> >
> >
> > template <int dim, int spacedim, int chartdim>
> > Ellipsoid<dim,spacedim,chartdim>::Ellipsoid(double a, double b, double
> c): ChartManifold<dim,spacedim,chartdim>(Point<chartdim>(2*numbers::PI,
> 2*numbers::PI)), a(a), b(b),c(c), center(0,0,0), reference_sphere(center)
>
> > {
> > max_axis = std::max(std::max(a,b),c);
> > }
> > template <int dim,int spacedim, int chartdim>
> > Point<chartdim> Ellipsoid<dim,spacedim,chartdim>::pull_back(const
> Point<spacedim> &space_point) const
> > {
> > double x,y,z, u,v,w;
> >
> > // get point on ellipsoid
> > x = space_point[0];
> > y = space_point[1];
> > z = space_point[2];
> >
> > std::cout << "using a,b,c: " << std::endl;
> > std::cout << a << " " << b << " " << c << std::endl;
> > std::cout << "from pull_back: " << std::endl;
> > std::cout << "space_point: " << std::endl;
> > std::cout << x << " " << y << " " << z << std::endl;
> >
> > // map ellipsoid point onto sphere
> > u = x/a;
> > v = y/b;
> > w = z/c;
> >
> > std::cout << "pulls back to : " << std::endl;
> > std::cout << u << " " << v << " " << w << std::endl;
> > std::cout << "on sphere." << std::endl;
> >
> > Point<spacedim> p(u,v,w);
> >
> > // use reference_sphere's pull_back function
> > Point<spacedim> q = reference_sphere.pull_back(p);
> > Point<chartdim> chart_point;
> >
> >
> > std::cout << "sphere pull_back: " << std::endl;
> > std::cout << q[0] << " " << q[1] << " " << q[2] << std::endl;
> > std::cout << "r theta phi" << std::endl;
> > std::cout << "..........." << std::endl;
> >
> > chart_point[0] = q[1];
> > chart_point[1] = q[2];
> >
> > // return (theta,phi) in the chart domain
> > return chart_point;
> >
> > }
> > template <int dim,int spacedim, int chartdim>
> > Point<spacedim> Ellipsoid<dim,spacedim,chartdim>::push_forward(const
> Point<chartdim> &chart_point) const
> > {
> > double theta,phi, x,y,z;
> >
> > phi = chart_point[0];
> > theta = chart_point[1];
> >
> >
> > Point<spacedim> p(max_axis,theta,phi);
> > // map theta,phi in chart domain onto reference_sphere with radius
> max_axis
> > Point<spacedim> X = reference_sphere.push_forward(p);
> >
> > // map point on sphere onto ellipsoid
> >
> > x = a*X[0];
> > y = b*X[1];
> > z = c*X[2];
> >
> > Point<spacedim> space_point(x,y,z);
> >
> > // return point on ellipsoid
> > return space_point;
> > }
> >
> >
> > Point<3> grid_transform (const Point<3> &X)
> > {
> > // transform points on sphere onto ellipsoid
> > double a,b,c;
> > a = 1.0; b = 1.0; c = 1.0;
> >
> > double x,y,z;
> > x = a*X(0);
> > y = b*X(1);
> > z = c*X(2);
> >
> > return Point<3>(x,y,z);
> > }
> >
> > void assemble_mesh_and_manifold()
> > {
> >
> > const int dim = 2;
> > const int spacedim = 3;
> > const int chartdim = 2;
> >
> > double a,b,c;
> > a = 1; b=1; c=1;
> >
> > Ellipsoid<dim,spacedim,chartdim> ellipsoid(a,b,c);
> >
> > Triangulation<dim,spacedim> tria;
> >
> > // generate coarse spherical mesh
> > GridGenerator::hyper_sphere (tria, Point<spacedim>(0.0,0.0,0.0), 1.0);
> > for (Triangulation<dim,spacedim>::active_cell_iterator
> cell=tria.begin_active(); cell!=tria.end(); ++cell)
> > cell->set_all_manifold_ids(0);
> >
> > print_mesh_info(tria, "spherical_mesh.vtk");
> >
> > GridTools::transform(&grid_transform, tria);
> >
> > tria.set_manifold(0,ellipsoid);
> >
> > tria.refine_global(1);
> >
> > print_mesh_info(tria, "ellipsoidal_mesh.vtk");
> >
> > }
> >
> > int main ()
> > {
> > assemble_mesh_and_manifold();
> > }
> >
> > Attached is the .vtk of my refined ellipsoidal mesh and the .cc file
> that generates this output (mostly reproduced above.
> >
> > Thank you,
> > Tom
> >
> > --
> > The deal.II project is located at http://www.dealii.org/
> > For mailing list/forum options, see
> https://groups.google.com/d/forum/dealii?hl=en
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> > For more options, visit https://groups.google.com/d/optout.
> > <sphere_to_ellipsoid.cc><ellipsoidal_mesh.vtk>
>
>
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/* ---------------------------------------------------------------------
*
* sphere_to_ellipsoid.cc_
modified from step-49 of the tutorial July 25, 2016
Purpose: to generate a grid using dealii tools, modify it according to a
smooth function, and then attach a manifold to it.
* Copyright (C) 2013 - 2015 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.
*
* ---------------------------------------------------------------------
*
* Author: Timo Heister, Texas A&M University, 2013
*/
// This tutorial program is odd in the sense that, unlike for most other
// steps, the introduction already provides most of the information on how to
// use the various strategies to generate meshes. Consequently, there is
// little that remains to be commented on here, and we intersperse the code
// with relatively little text. In essence, the code here simply provides a
// reference implementation of what has already been described in the
// introduction.
// @sect3{Include files}
#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/manifold_lib.h>
#include <deal.II/grid/grid_out.h>
#include <deal.II/grid/grid_in.h>
#include <iostream>
#include <fstream>
#include <map>
using namespace dealii;
// @sect3{Generating output for a given mesh}
// The following function generates some output for any of the meshes we will
// be generating in the remainder of this program. In particular, it generates
// the following information:
//
// - Some general information about the number of space dimensions in which
// this mesh lives and its number of cells.
// - The number of boundary faces that use each boundary indicator, so that
// it can be compared with what we expect.
//
// Finally, the function outputs the mesh in encapsulated postscript (EPS)
// format that can easily be visualized in the same way as was done in step-1.
template <int dim,int spacedim>
void print_mesh_info(const Triangulation<dim,spacedim> &tria,
const std::string &filename)
{
std::cout << "Mesh info:" << std::endl
<< " dimension: " << dim << std::endl
<< " no. of cells: " << tria.n_active_cells() << std::endl;
// Next loop over all faces of all cells and find how often each
// boundary indicator is used (recall that if you access an element
// of a std::map object that doesn't exist, it is implicitly created
// and default initialized -- to zero, in the current case -- before
// we then increment it):
{
std::map<unsigned int, unsigned int> boundary_count;
typename Triangulation<dim,spacedim>::active_cell_iterator
cell = tria.begin_active(),
endc = tria.end();
for (; cell!=endc; ++cell)
{
for (unsigned int face=0; face<GeometryInfo<dim>::faces_per_cell; ++face)
{
if (cell->face(face)->at_boundary())
boundary_count[cell->face(face)->boundary_id()]++;
}
}
std::cout << " boundary indicators: ";
for (std::map<unsigned int, unsigned int>::iterator it=boundary_count.begin();
it!=boundary_count.end();
++it)
{
std::cout << it->first << "(" << it->second << " times) ";
}
std::cout << std::endl;
}
// Finally, produce a graphical representation of the mesh to an output
// file:
std::ofstream out (filename.c_str());
GridOut grid_out;
grid_out.write_vtk (tria, out);
std::cout << " written to " << filename
<< std::endl
<< std::endl;
}
template <int dim,int spacedim>
class Ellipsoid: public SphericalManifold<dim,spacedim>
{
public:
Ellipsoid(double,double,double);
Point<spacedim> pull_back(const Point<spacedim> &space_point) const;
Point<spacedim> push_forward(const Point<spacedim> &chart_point) const;
Point<spacedim> get_new_point(const Quadrature<spacedim> &quad) const;
Point<spacedim> grid_transform(const Point<spacedim> &X);
private:
double a,b,c;
double max_axis;
const Point<spacedim> center;
Point<dim> ellipsoid_pull_back(const Point<spacedim> &space_point) const;
Point<spacedim> ellipsoid_push_forward(const Point<dim> &chart_point) const;
};
template <int dim, int spacedim>
Ellipsoid<dim,spacedim>::Ellipsoid(double a, double b, double c) : SphericalManifold<dim,spacedim>(center), a(a), b(b),c(c), center(0,0,0)
{
max_axis = std::max(std::max(a,b),c);
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::pull_back(const Point<spacedim> &space_point) const
{
Point<dim> chart_point = ellipsoid_pull_back(space_point);
Point<spacedim> p;
p[0] = -1; // dummy radius to match return of SphericalManifold::pull_back()
p[1] = chart_point[0];
p[2] = chart_point[1];
return p;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::push_forward(const Point<spacedim> &chart_point) const
{
Point<dim> p; //
p[0] = chart_point[1];
p[1] = chart_point[2];
Point<spacedim> space_point = ellipsoid_push_forward(p);
return space_point;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::get_new_point(const Quadrature<spacedim> &quad) const
{
double u,v,w;
std::vector< Point<spacedim> > space_points;
for (unsigned int i=0; i<quad.size(); ++i)
{
u = quad.point(i)[0]/a;
v = quad.point(i)[1]/b;
w = quad.point(i)[2]/c;
space_points.push_back(Point<spacedim>(u,v,w));
}
Quadrature<spacedim> spherical_quad = Quadrature<spacedim>(space_points, quad.get_weights());
Point<spacedim> p = SphericalManifold<dim,spacedim>::get_new_point(spherical_quad);
double x,y,z;
x = a*p[0];
y = b*p[1];
z = c*p[2];
Point<spacedim> new_point = Point<spacedim>(x,y,z);
return new_point;
}
template <int dim,int spacedim>
Point<dim> Ellipsoid<dim,spacedim>::ellipsoid_pull_back(const Point<spacedim> &space_point) const
{
double x,y,z, u,v,w;
// get point on ellipsoid
x = space_point[0];
y = space_point[1];
z = space_point[2];
std::cout << "using a,b,c: " << std::endl;
std::cout << a << " " << b << " " << c << std::endl;
std::cout << "from pull_back: " << std::endl;
std::cout << "space_point: " << std::endl;
std::cout << x << " " << y << " " << z << std::endl;
// map ellipsoid point onto sphere
u = x/a;
v = y/b;
w = z/c;
std::cout << "pulls back to : " << std::endl;
std::cout << u << " " << v << " " << w << std::endl;
std::cout << "on sphere." << std::endl;
Point<spacedim> p(u,v,w);
// use reference_sphere's pull_back function
Point<spacedim> q = pull_back(p);
Point<dim> chart_point;
std::cout << "sphere pull_back: " << std::endl;
std::cout << q[0] << " " << q[1] << " " << q[2] << std::endl;
std::cout << "r theta phi" << std::endl;
std::cout << "..........." << std::endl;
chart_point[0] = q[1];
chart_point[1] = q[2];
// return (theta,phi) in the chart domain
return chart_point;
}
template <int dim,int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::ellipsoid_push_forward(const Point<dim> &chart_point) const
{
double theta,phi, x,y,z;
phi = chart_point[0];
theta = chart_point[1];
Point<spacedim> p(max_axis,theta,phi);
// map theta,phi in chart domain onto reference_sphere with radius max_axis
Point<spacedim> X = push_forward(p);
// map point on sphere onto ellipsoid
x = a*X[0];
y = b*X[1];
z = c*X[2];
Point<spacedim> space_point(x,y,z);
// return point on ellipsoid
return space_point;
}
template<int dim, int spacedim>
Point<spacedim> Ellipsoid<dim,spacedim>::grid_transform(const Point<spacedim> &X)
{
// transform points X from sphere onto ellipsoid
double x,y,z;
x = a*X(0);
y = b*X(1);
z = c*X(2);
return Point<spacedim>(x,y,z);
}
Point<3> grid_transform(const Point<3> &X)
{
// transform points X from sphere onto ellipsoid
double x,y,z;
double a = 1; double b = 3; double c = 5;
x = a*X(0);
y = b*X(1);
z = c*X(2);
return Point<3>(x,y,z);
}
void assemble_mesh_and_manifold()
{
const int dim = 2;
const int spacedim = 3;
double a,b,c;
a = 1; b=3; c=5;
Ellipsoid<dim,spacedim> ellipsoid(a,b,c);
Triangulation<dim,spacedim> tria;
// generate coarse spherical mesh
GridGenerator::hyper_sphere (tria, Point<spacedim>(0.0,0.0,0.0), 1.0);
for (Triangulation<dim,spacedim>::active_cell_iterator cell=tria.begin_active(); cell!=tria.end(); ++cell)
cell->set_all_manifold_ids(0);
print_mesh_info(tria, "spherical_mesh.vtk");
GridTools::transform(&grid_transform, tria);
//
//GridTools::transform(std_cxx11::bind(&Ellipsoid<dim,spacedim>::grid_transform,std_cxx11::cref(ellipsoid),std_cxx11::_1), tria); // error when trying to bind to member function in same way as step-53
tria.set_manifold(0,ellipsoid);
tria.refine_global(3);
print_mesh_info(tria, "ellipsoidal_mesh.vtk");
}
// @sect3{The main function}
// Finally, the main function. There isn't much to do here, only to call the
// subfunctions.
int main ()
{
assemble_mesh_and_manifold();
}
