Notes on visualizing high order output

Motivation

As of version 9.1, deal.II allows one to generate VTK and VTU output using high-order Lagrange cells. As their name suggests, these cells are described by a set of Lagrange points with numerical quantities attached to them. This becomes useful when you are working with high-order elements and/or high-order mappings, since these objects can be represented more accurately.

This page provides instructions on how these high-order meshes can be visualized using ParaView (note: this feature was implemented in the version 5.5, older versions will not be able to visualize these meshes).

Creating high-order output

First, we produced 2D and 3D VTU output of a spherical shell mesh. Internally, the mesh was generated using the GridGenerator::hyper_shell function, which attaches a SphericalManifold to all cells. Subsequently, the mesh was attached to a DoFHandler object with a FE_Q element of the order four. A trigonometric function was interpolated to the underlying finite-element space using the MappingQGeneric mapping of order four. Finally, we write the mesh along with the finite-element representation of the function to a VTU file using DataOut::write_vtu with high-order curved cells enabled.

Visualization

The produced VTU output was loaded into ParaView v5.7. Even though the mesh is high-order, ParaView uses (bi-/tri-)linear interpolation between the Lagrange points by default. To enable the high-order interpolation, you need to locate the property "Nonlinear Subdivision Level" in the object properties (the easiest way to do this is to search for this field by its name) and set the value to a number larger than one, according to your needs:

For our example, the 2D result looks like this:

Accordingly, here is the 3D mesh:

You can see now that cells have curved shapes and the quantity within each cell varies as a high-order (specifically, 4-th order) function.

Extra details

For more information on deal.II implementation of this feature, see the documentation of the DataOut::build_patches method. Additionally, step-53 and step-65 provide some more examples and details about high-order mappings, their internal representation and output.

Code

The images shown above were generated using the VTU files produced by the program below:

#include <deal.II/grid/tria.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/dofs/dof_handler.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/fe/mapping_q_generic.h>

#include <iostream>
#include <fstream>
#include <sstream>
#include <cmath>

using namespace dealii;

template<int dim>
class TestFunction: public Function<dim>
{
public:
  double value(const Point<dim> &p, const unsigned int component = 0) const
  {
    double v = 0;
    for(int d = 0; d < dim; ++d)
    {
      v += cos(2 * numbers::PI * p[d]);
    }

    return v;
  }
};

template<int dim>
void shell_grid(unsigned fe_order,
                unsigned mapping_order,
                unsigned output_mesh_order)
{
  FE_Q<dim> fe(fe_order);
  MappingQGeneric<dim> mapping(mapping_order);

  Triangulation<dim> triangulation;
  GridGenerator::hyper_shell(triangulation, Point<dim>(), 0.5, 1., 0, true);

  for(unsigned n = 0; n < 2; ++n)
  {
    for(auto cell: triangulation.active_cell_iterators())
    {
      for(unsigned face = 0; face < GeometryInfo<dim>::faces_per_cell; ++face)
        if(cell->face(face)->at_boundary() &&
           cell->face(face)->boundary_id() == 1)
          cell->set_refine_flag();
    }
    triangulation.execute_coarsening_and_refinement();
  }

  DoFHandler<dim> dof_handler(triangulation);
  dof_handler.distribute_dofs(fe);

  TestFunction<dim> function;
  Vector<double> vec(dof_handler.n_dofs());
  VectorTools::interpolate(mapping, dof_handler, function, vec);

  DataOutBase::VtkFlags flags;
  flags.write_higher_order_cells = true;

  std::stringstream ss;
  ss << "shell_dim=" << dim
     << "_p=" << fe_order
     << "_mapping=" << mapping_order
     << "_n=" << output_mesh_order
     << ".vtu";

  std::ofstream out(ss.str());
  DataOut<dim> data_out;
  data_out.set_flags(flags);
  data_out.attach_dof_handler(dof_handler);
  data_out.add_data_vector(vec, "vec");
  data_out.build_patches(mapping, output_mesh_order, DataOut<dim>::curved_inner_cells);
  data_out.write_vtu(out);
}

int main()
{
  const unsigned fe_order = 4;
  const unsigned mapping_order = 4;
  const unsigned output_mesh_order = 4;

  shell_grid<2>(fe_order, mapping_order, output_mesh_order);
  shell_grid<3>(fe_order, mapping_order, output_mesh_order);
}