Reference documentation for deal.II version Git 3f75665 2017-01-19 16:58:38 -0500
The step-45 tutorial program

Table of contents
  1. Introduction
  2. The commented program
  1. Results
  2. The plain program

This program was contributed by Daniel Arndt and Matthias Maier.

Introduction

In this example we present how to use periodic boundary conditions in deal.II. Periodic boundary conditions are algebraic constraints that typically occur in computations on representative regions of a larger domain that repeat in one or more directions.

An example is the simulation of the electronic structure of photonic crystals, because they have a lattice-like structure and, thus, it often suffices to do the actual computation on only one box of the lattice. To be able to proceed this way one has to assume that the computation can be periodically extended to the other boxes; this requires the solution to have a periodic structure.

Procedure

deal.II provides a number of high level entry points to impose periodic boundary conditions. The general approach to apply periodic boundary conditions consists of three steps (see also the Glossary entry on periodic boundary conditions):

  1. Create a mesh
  2. Identify those pairs of faces on different parts of the boundary across which the solution should be symmetric, using GridTools::collect_periodic_faces()
  3. Add the periodicity information to the mesh using parallel::distributed::Triangulation::add_periodicity()
  4. Add periodicity constraints using DoFTools::make_periodicity_constraints()

The second and third step are necessary for distributed meshes to ensure that cells on opposite sides of the domain but connected by periodic faces are part of the ghost layer if one of them is stored on the local processor. If the Triangulation is not a parallel::distributed::Triangulation, these steps have to be omitted.

The first step consists of collecting matching periodic faces and storing them in a std::vector of GridTools::PeriodicFacePair. This is done with the function GridTools::collect_periodic_faces() that can be invoked for example like this:

b_id1,
b_id2,
direction,
matched_pairs,
offset = <default value>,
matrix = <default value>,
first_vector_components = <default value>);

This call loops over all faces of the container dof_handler on the opposing boundaries with boundary indicator b_id1 and b_id2, respecitvely. If \(\text{vertices}_{1/2}\) are the vertices of \(\text{face}_{1/2}\), it matches pairs of faces (and dofs) such that the difference between \(\text{vertices}_2\) and \(matrix\cdot \text{vertices}_1+\text{offset}\) vanishes in every component apart from direction and stores the resulting pairs with associated data in matched_pairs. (See GridTools::orthogonal_equality() for detailed information about the matching process.)

Consider, for example, the colored unit square \(\Omega=[0,1]^2\) with boundary indicator 0 on the left, 1 on the right, 2 on the bottom and 3 on the top faces. Then,

/*b_id1*/ 0,
/*b_id2*/ 1,
/*direction*/ 0,
matched_pairs);

would yield periodicity constraints such that \(u(0,y)=u(1,y)\) for all \(y\in[0,1]\).

If we instead consider the parallelogram given by the convex hull of \((0,0)\), \((1,1)\), \((1,2)\), \((0,1)\) we can achieve the constraints \(u(0,y)=u(1,y+1)\) by specifying an offset:

/*b_id1*/ 0,
/*b_id2*/ 1,
/*direction*/ 0,
matched_pairs,
Tensor<1, 2>(0.,1.));

or

/*b_id1*/ 0,
/*b_id2*/ 1,
/*arbitrary direction*/ 0,
matched_pairs,
Tensor<1, 2>(1.,1.));

The resulting matched_pairs can be used in DoFTools::make_periodicity_constraints for populating a ConstraintMatrix with periodicity constraints:

DoFTools::make_periodicity_constraints(matched_pairs, constraints);

Apart from this high level interface there are also variants of DoFTools::make_periodicity_constraints available that combine those two steps (see the variants of DofTools::make_periodicity_constraints).

There is also a low level interface to DoFTools::make_periodicity_constraints if more flexibility is needed. The low level variant allows to directly specify two faces that shall be constrained:

using namespace DoFTools;
face_2,
constraint_matrix,
component_mask = <default value>;
face_orientation = <default value>,
face_flip = <default value>,
face_rotation = <default value>,
matrix = <default value>);

Here, we need to specify the orientation of the two faces using face_orientation, face_flip and face_orientation. For a closer description have a look at the documentation of DoFTools::make_periodicity_constraints. The remaining parameters are the same as for the high level interface apart from the self-explaining component_mask and constraint_matrix.

A practical example

In the following, we show how to use the above functions in a more involved example. The task is to enforce rotated periodicity constraints for the velocity component of a Stokes flow.

On a quarter-circle defined by \(\Omega=\{{\bf x}\in(0,1)^2:\|{\bf x}\|\in (0.5,1)\}\) we are going to solve the Stokes problem

\begin{eqnarray*} -\Delta \; \textbf{u} + \nabla p &=& (\exp(-100*\|{\bf x}-(.75,0.1)^T\|^2),0)^T, \\ -\textrm{div}\; \textbf{u}&=&0,\\ \textbf{u}|_{\Gamma_1}&=&{\bf 0}, \end{eqnarray*}

where the boundary \(\Gamma_1\) is defined as \(\Gamma_1:=\{x\in \partial\Omega: \|x\|\in\{0.5,1\}\}\). For the remaining parts of the boundary we are going to use periodic boundary conditions, i.e.

\begin{align*} u_x(0,\nu)&=-u_y(\nu,0)&\nu&\in[0,1]\\ u_y(0,\nu)&=u_x(\nu,0)&\nu&\in[0,1]. \end{align*}

The commented program

This example program is a slight modification of step-22 running in parallel using Trilinos to demonstrate the usage of periodic boundary conditions in deal.II. We thus omit to discuss the majority of the source code and only comment on the parts that deal with periodicity constraints. For the rest have a look at step-22 and the full source code at the bottom.

In order to implement periodic boundary conditions only two functions have to be modified:

Setting up periodicity constraints on distributed triangulations

template <int dim>
void StokesProblem<dim>::create_mesh()
{
Point<dim> center;
const double inner_radius = .5;
const double outer_radius = 1.;
center,
inner_radius,
outer_radius,
0,
true);

Before we can prescribe periodicity constraints, we need to ensure that cells on opposite sides of the domain but connected by periodic faces are part of the ghost layer if one of them is stored on the local processor. At this point we need to think about how we want to prescribe periodicity. The vertices \(\text{vertices}_2\) of a face on the left boundary should be matched to the vertices \(\text{vertices}_1\) of a face on the lower boundary given by \(\text{vertices}_2=R\cdot \text{vertices}_1+b\) where the rotation matrix \(R\) and the offset \(b\) are given by

\begin{align*} R=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}, \quad b=\begin{pmatrix}0&0\end{pmatrix}. \end{align*}

The data structure we are saving the reuslitng information into is here based on the Triangulation.

std::vector<GridTools::PeriodicFacePair<typename parallel::distributed::Triangulation<dim>::cell_iterator> >
periodicity_vector;
FullMatrix<double> rotation_matrix(dim);
rotation_matrix[0][1]=1.;
rotation_matrix[1][0]=-1.;
GridTools::collect_periodic_faces(triangulation, 2, 3, 1,
periodicity_vector, Tensor<1, dim>(),
rotation_matrix);

Now telling the triangulation about the desired periodicity is particularly easy by just calling parallel::distributed::Triangulation::add_periodicity.

triangulation.add_periodicity(periodicity_vector);
triangulation.set_boundary(0, boundary);
triangulation.set_boundary(1, boundary);
triangulation.refine_global (4-dim);
}

Setting up periodicity constraints on distributed triangulations

template <int dim>
void StokesProblem<dim>::setup_dofs ()
{
dof_handler.distribute_dofs (fe);
std::vector<unsigned int> block_component (dim+1,0);
block_component[dim] = 1;
DoFRenumbering::component_wise (dof_handler, block_component);
std::vector<types::global_dof_index> dofs_per_block (2);
DoFTools::count_dofs_per_block (dof_handler, dofs_per_block, block_component);
const unsigned int n_u = dofs_per_block[0],
n_p = dofs_per_block[1];
{
owned_partitioning.clear();
IndexSet locally_owned_dofs = dof_handler.locally_owned_dofs();
owned_partitioning.push_back(locally_owned_dofs.get_view(0, n_u));
owned_partitioning.push_back(locally_owned_dofs.get_view(n_u, n_u+n_p));
relevant_partitioning.clear();
IndexSet locally_relevant_dofs;
DoFTools::extract_locally_relevant_dofs (dof_handler, locally_relevant_dofs);
relevant_partitioning.push_back(locally_relevant_dofs.get_view(0, n_u));
relevant_partitioning.push_back(locally_relevant_dofs.get_view(n_u, n_u+n_p));
constraints.clear ();
constraints.reinit(locally_relevant_dofs);
constraints);
dof_handler,
0,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
dof_handler,
1,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));

After we provided the mesh with the necessary information for the periodicity constraints, we are now able to actual create them. For describing the matching we are using the same approach as before, i.e., the \(\text{vertices}_2\) of a face on the left boundary should be matched to the vertices \(\text{vertices}_1\) of a face on the lower boundary given by \(\text{vertices}_2=R\cdot \text{vertices}_1+b\) where the rotation matrix \(R\) and the offset \(b\) are given by

\begin{align*} R=\begin{pmatrix} 0&1\\-1&0 \end{pmatrix}, \quad b=\begin{pmatrix}0&0\end{pmatrix}. \end{align*}

These two objects not only describe how faces should be matched but also in which sense the solution should be transformed from \(\text{face}_2\) to \(\text{face}_1\).

FullMatrix<double> rotation_matrix(dim);
rotation_matrix[0][1]=1.;
rotation_matrix[1][0]=-1.;

For setting up the constraints, we first store the periodicity information in an auxiliary object of type std::vector<GridTools::PeriodicFacePair<typename DoFHandler<dim>::cell_iterator> . The periodic boundaries have the boundary indicators 2 (x=0) and 3 (y=0). All the other parameters we have set up before. In this case the direction does not matter. Due to \(\text{vertices}_2=R\cdot \text{vertices}_1+b\) this is exactly what we want.

std::vector<GridTools::PeriodicFacePair<typename DoFHandler<dim>::cell_iterator> >
periodicity_vector;
const unsigned int direction = 1;
GridTools::collect_periodic_faces(dof_handler, 2, 3, direction,
periodicity_vector, offset,
rotation_matrix);

Next we need to provide information on which vector valued components of the solution should be rotated. Since we choose here to just constraint the velocity and this starts at the first component of the solution vector we simply insert a 0:

std::vector<unsigned int> first_vector_components;
first_vector_components.push_back(0);

After setting up all the information in periodicity_vector all we have to do is to tell make_periodicity_constraints to create the desired constraints.

DoFTools::make_periodicity_constraints<DoFHandler<dim> >
(periodicity_vector, constraints, fe.component_mask(velocities),
first_vector_components);
dof_handler,
0,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
dof_handler,
1,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
}
constraints.close ();
{
(owned_partitioning, owned_partitioning,
relevant_partitioning, mpi_communicator);
(dof_handler, bsp, constraints, false,
bsp.compress();
system_matrix.reinit (bsp);
}
system_rhs.reinit (owned_partitioning,
mpi_communicator);
solution.reinit (owned_partitioning, relevant_partitioning,
mpi_communicator);
}

Results

The created output is not very surprising. We simply see that the solution is periodic with respect to the left and lower boundary:

Without the periodicity constraints we would have ended up with the following solution:

The plain program

/* ---------------------------------------------------------------------
*
* Copyright (C) 2008 - 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: Daniel Arndt, Matthias Maier, 2015
*
* Based on @ref step_22 "step-22" by Wolfgang Bangerth and Martin Kronbichler
*/
#include <deal.II/base/conditional_ostream.h>
#include <deal.II/distributed/grid_refinement.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/constraint_matrix.h>
#include <deal.II/lac/trilinos_solver.h>
#include <deal.II/lac/trilinos_precondition.h>
#include <deal.II/lac/trilinos_block_sparse_matrix.h>
#include <deal.II/lac/trilinos_parallel_block_vector.h>
#include <deal.II/lac/block_sparsity_pattern.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/tria_boundary_lib.h>
#include <deal.II/grid/grid_tools.h>
#include <deal.II/dofs/dof_renumbering.h>
#include <deal.II/dofs/dof_tools.h>
#include <deal.II/fe/fe_q.h>
#include <deal.II/fe/fe_system.h>
#include <deal.II/fe/mapping_q.h>
#include <deal.II/numerics/vector_tools.h>
#include <deal.II/numerics/data_out.h>
#include <deal.II/numerics/error_estimator.h>
namespace Step45
{
using namespace dealii;
template <int dim>
class StokesProblem
{
public:
StokesProblem (const unsigned int degree);
void run ();
private:
void create_mesh();
void setup_dofs ();
void assemble_system ();
void solve ();
void output_results (const unsigned int refinement_cycle) const;
void refine_mesh ();
const unsigned int degree;
MPI_Comm mpi_communicator;
DoFHandler<dim> dof_handler;
ConstraintMatrix constraints;
std::vector<IndexSet> owned_partitioning;
std::vector<IndexSet> relevant_partitioning;
MappingQ<dim> mapping;
};
template <int dim>
class BoundaryValues : public Function<dim>
{
public:
BoundaryValues () : Function<dim>(dim+1) {}
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &value) const;
};
template <int dim>
double
BoundaryValues<dim>::value (const Point<dim> &/*p*/,
const unsigned int component) const
{
Assert (component < this->n_components,
ExcIndexRange (component, 0, this->n_components));
return 0;
}
template <int dim>
void
BoundaryValues<dim>::vector_value (const Point<dim> &p,
Vector<double> &values) const
{
for (unsigned int c=0; c<this->n_components; ++c)
values(c) = BoundaryValues<dim>::value (p, c);
}
template <int dim>
class RightHandSide : public Function<dim>
{
public:
RightHandSide () : Function<dim>(dim+1) {}
virtual double value (const Point<dim> &p,
const unsigned int component = 0) const;
virtual void vector_value (const Point<dim> &p,
Vector<double> &value) const;
};
template <int dim>
double
RightHandSide<dim>::value (const Point<dim> &p,
const unsigned int component) const
{
const Point<dim> center(0.75, 0.1);
const double r = (p-center).norm();
if (component==0)
return std::exp(-100.*r*r);
return 0;
}
template <int dim>
void
RightHandSide<dim>::vector_value (const Point<dim> &p,
Vector<double> &values) const
{
for (unsigned int c=0; c<this->n_components; ++c)
values(c) = RightHandSide<dim>::value (p, c);
}
template <class MatrixType, class PreconditionerType>
class InverseMatrix : public Subscriptor
{
public:
InverseMatrix (const MatrixType &m,
const PreconditionerType &preconditioner,
const IndexSet &locally_owned,
const MPI_Comm &mpi_communicator);
const TrilinosWrappers::MPI::Vector &src) const;
private:
const MPI_Comm *mpi_communicator;
};
template <class MatrixType, class PreconditionerType>
InverseMatrix<MatrixType,PreconditionerType>::InverseMatrix
(const MatrixType &m,
const PreconditionerType &preconditioner,
const IndexSet &locally_owned,
const MPI_Comm &mpi_communicator)
:
matrix (&m),
preconditioner (&preconditioner),
mpi_communicator (&mpi_communicator),
tmp(locally_owned, mpi_communicator)
{}
template <class MatrixType, class PreconditionerType>
void InverseMatrix<MatrixType,PreconditionerType>::vmult
{
SolverControl solver_control (src.size(), 1e-6*src.l2_norm());
TrilinosWrappers::SolverCG cg (solver_control,
tmp = 0.;
cg.solve (*matrix, tmp, src, *preconditioner);
dst = tmp;
}
template <class PreconditionerType>
class SchurComplement : public TrilinosWrappers::SparseMatrix
{
public:
SchurComplement (const TrilinosWrappers::BlockSparseMatrix &system_matrix,
const InverseMatrix<TrilinosWrappers::SparseMatrix,
PreconditionerType> &A_inverse,
const IndexSet &owned_pres,
const MPI_Comm &mpi_communicator);
const TrilinosWrappers::MPI::Vector &src) const;
private:
const SmartPointer<const InverseMatrix<TrilinosWrappers::SparseMatrix,
PreconditionerType> > A_inverse;
mutable TrilinosWrappers::MPI::Vector tmp1, tmp2;
};
template <class PreconditionerType>
SchurComplement<PreconditionerType>::
SchurComplement (const TrilinosWrappers::BlockSparseMatrix &system_matrix,
const InverseMatrix<TrilinosWrappers::SparseMatrix,
PreconditionerType> &A_inverse,
const IndexSet &owned_vel,
const MPI_Comm &mpi_communicator)
:
system_matrix (&system_matrix),
A_inverse (&A_inverse),
tmp1 (owned_vel, mpi_communicator),
tmp2 (tmp1)
{}
template <class PreconditionerType>
void SchurComplement<PreconditionerType>::vmult
{
system_matrix->block(0,1).vmult (tmp1, src);
A_inverse->vmult (tmp2, tmp1);
system_matrix->block(1,0).vmult (dst, tmp2);
}
template <int dim>
StokesProblem<dim>::StokesProblem (const unsigned int degree)
:
degree (degree),
mpi_communicator (MPI_COMM_WORLD),
triangulation (mpi_communicator),
fe (FE_Q<dim>(degree+1), dim,
FE_Q<dim>(degree) , 1),
dof_handler (triangulation),
pcout (std::cout,
Utilities::MPI::this_mpi_process(mpi_communicator) == 0),
mapping(degree+1)
{}
template <int dim>
void StokesProblem<dim>::create_mesh()
{
Point<dim> center;
const double inner_radius = .5;
const double outer_radius = 1.;
center,
inner_radius,
outer_radius,
0,
true);
std::vector<GridTools::PeriodicFacePair<typename parallel::distributed::Triangulation<dim>::cell_iterator> >
periodicity_vector;
FullMatrix<double> rotation_matrix(dim);
rotation_matrix[0][1]=1.;
rotation_matrix[1][0]=-1.;
GridTools::collect_periodic_faces(triangulation, 2, 3, 1,
periodicity_vector, Tensor<1, dim>(),
rotation_matrix);
triangulation.add_periodicity(periodicity_vector);
triangulation.set_boundary(0, boundary);
triangulation.set_boundary(1, boundary);
triangulation.refine_global (4-dim);
}
template <int dim>
void StokesProblem<dim>::setup_dofs ()
{
dof_handler.distribute_dofs (fe);
std::vector<unsigned int> block_component (dim+1,0);
block_component[dim] = 1;
DoFRenumbering::component_wise (dof_handler, block_component);
std::vector<types::global_dof_index> dofs_per_block (2);
DoFTools::count_dofs_per_block (dof_handler, dofs_per_block, block_component);
const unsigned int n_u = dofs_per_block[0],
n_p = dofs_per_block[1];
{
owned_partitioning.clear();
IndexSet locally_owned_dofs = dof_handler.locally_owned_dofs();
owned_partitioning.push_back(locally_owned_dofs.get_view(0, n_u));
owned_partitioning.push_back(locally_owned_dofs.get_view(n_u, n_u+n_p));
relevant_partitioning.clear();
IndexSet locally_relevant_dofs;
DoFTools::extract_locally_relevant_dofs (dof_handler, locally_relevant_dofs);
relevant_partitioning.push_back(locally_relevant_dofs.get_view(0, n_u));
relevant_partitioning.push_back(locally_relevant_dofs.get_view(n_u, n_u+n_p));
constraints.clear ();
constraints.reinit(locally_relevant_dofs);
constraints);
dof_handler,
0,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
dof_handler,
1,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
FullMatrix<double> rotation_matrix(dim);
rotation_matrix[0][1]=1.;
rotation_matrix[1][0]=-1.;
Tensor<1,dim> offset;
std::vector<GridTools::PeriodicFacePair<typename DoFHandler<dim>::cell_iterator> >
periodicity_vector;
const unsigned int direction = 1;
GridTools::collect_periodic_faces(dof_handler, 2, 3, direction,
periodicity_vector, offset,
rotation_matrix);
std::vector<unsigned int> first_vector_components;
first_vector_components.push_back(0);
DoFTools::make_periodicity_constraints<DoFHandler<dim> >
(periodicity_vector, constraints, fe.component_mask(velocities),
first_vector_components);
dof_handler,
0,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
dof_handler,
1,
BoundaryValues<dim>(),
constraints,
fe.component_mask(velocities));
}
constraints.close ();
{
(owned_partitioning, owned_partitioning,
relevant_partitioning, mpi_communicator);
(dof_handler, bsp, constraints, false,
bsp.compress();
system_matrix.reinit (bsp);
}
system_rhs.reinit (owned_partitioning,
mpi_communicator);
solution.reinit (owned_partitioning, relevant_partitioning,
mpi_communicator);
}
template <int dim>
void StokesProblem<dim>::assemble_system ()
{
system_matrix=0.;
system_rhs=0.;
QGauss<dim> quadrature_formula(degree+2);
FEValues<dim> fe_values (mapping, fe, quadrature_formula,
const unsigned int dofs_per_cell = fe.dofs_per_cell;
const unsigned int n_q_points = quadrature_formula.size();
FullMatrix<double> local_matrix (dofs_per_cell, dofs_per_cell);
Vector<double> local_rhs (dofs_per_cell);
std::vector<types::global_dof_index> local_dof_indices (dofs_per_cell);
const RightHandSide<dim> right_hand_side;
std::vector<Vector<double> > rhs_values (n_q_points,
Vector<double>(dim+1));
const FEValuesExtractors::Vector velocities (0);
const FEValuesExtractors::Scalar pressure (dim);
std::vector<SymmetricTensor<2,dim> > symgrad_phi_u (dofs_per_cell);
std::vector<double> div_phi_u (dofs_per_cell);
std::vector<double> phi_p (dofs_per_cell);
cell = dof_handler.begin_active(),
endc = dof_handler.end();
for (; cell!=endc; ++cell)
if (cell->is_locally_owned())
{
fe_values.reinit (cell);
local_matrix = 0;
local_rhs = 0;
right_hand_side.vector_value_list(fe_values.get_quadrature_points(),
rhs_values);
for (unsigned int q=0; q<n_q_points; ++q)
{
for (unsigned int k=0; k<dofs_per_cell; ++k)
{
symgrad_phi_u[k] = fe_values[velocities].symmetric_gradient (k, q);
div_phi_u[k] = fe_values[velocities].divergence (k, q);
phi_p[k] = fe_values[pressure].value (k, q);
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
{
for (unsigned int j=0; j<=i; ++j)
{
local_matrix(i,j) += (symgrad_phi_u[i] * symgrad_phi_u[j]
- div_phi_u[i] * phi_p[j]
- phi_p[i] * div_phi_u[j]
+ phi_p[i] * phi_p[j])
* fe_values.JxW(q);
}
const unsigned int component_i =
local_rhs(i) += fe_values.shape_value(i,q) *
rhs_values[q](component_i) *
fe_values.JxW(q);
}
}
for (unsigned int i=0; i<dofs_per_cell; ++i)
for (unsigned int j=i+1; j<dofs_per_cell; ++j)
local_matrix(i,j) = local_matrix(j,i);
cell->get_dof_indices (local_dof_indices);
constraints.distribute_local_to_global (local_matrix, local_rhs,
local_dof_indices,
system_matrix, system_rhs);
}
system_matrix.compress (VectorOperation::add);
pcout << " Computing preconditioner..." << std::endl << std::flush;
}
template <int dim>
void StokesProblem<dim>::solve ()
{
A_preconditioner.initialize(system_matrix.block(0,0));
const InverseMatrix<TrilinosWrappers::SparseMatrix,
A_inverse (system_matrix.block(0,0),
A_preconditioner,
owned_partitioning[0],
mpi_communicator);
TrilinosWrappers::MPI::BlockVector tmp (owned_partitioning,
mpi_communicator);
{
TrilinosWrappers::MPI::Vector schur_rhs (owned_partitioning[1],
mpi_communicator);
A_inverse.vmult (tmp.block(0), system_rhs.block(0));
system_matrix.block(1,0).vmult (schur_rhs, tmp.block(0));
schur_rhs -= system_rhs.block(1);
SchurComplement<TrilinosWrappers::PreconditionJacobi>
schur_complement (system_matrix, A_inverse,
owned_partitioning[0],
mpi_communicator);
SolverControl solver_control (solution.block(1).size(),
1e-6*schur_rhs.l2_norm());
TrilinosWrappers::PreconditionAMG preconditioner;
preconditioner.initialize (system_matrix.block(1,1));
InverseMatrix<TrilinosWrappers::SparseMatrix,
TrilinosWrappers::PreconditionAMG>
m_inverse (system_matrix.block(1,1), preconditioner,
owned_partitioning[1], mpi_communicator);
cg.solve (schur_complement,
tmp.block(1),
schur_rhs,
preconditioner);
constraints.distribute (tmp);
solution.block(1)=tmp.block(1);
}
{
system_matrix.block(0,1).vmult (tmp.block(0), tmp.block(1));
tmp.block(0) *= -1;
tmp.block(0) += system_rhs.block(0);
A_inverse.vmult (tmp.block(0), tmp.block(0));
constraints.distribute (tmp);
solution.block(0)=tmp.block(0);
}
}
template <int dim>
void
StokesProblem<dim>::output_results (const unsigned int refinement_cycle) const
{
std::vector<std::string> solution_names (dim, "velocity");
solution_names.push_back ("pressure");
std::vector<DataComponentInterpretation::DataComponentInterpretation>
data_component_interpretation
data_component_interpretation
DataOut<dim> data_out;
data_out.attach_dof_handler (dof_handler);
data_out.add_data_vector (solution, solution_names,
data_component_interpretation);
Vector<float> subdomain (triangulation.n_active_cells());
for (unsigned int i=0; i<subdomain.size(); ++i)
subdomain(i) = triangulation.locally_owned_subdomain();
data_out.add_data_vector (subdomain, "subdomain");
data_out.build_patches (mapping, degree+1);
std::ostringstream filename;
filename << "solution-"
<< Utilities::int_to_string (refinement_cycle, 2)
<< "."
<< Utilities::int_to_string (triangulation.locally_owned_subdomain(),2)
<< ".vtu";
std::ofstream output (filename.str().c_str());
data_out.write_vtu (output);
if (Utilities::MPI::this_mpi_process(MPI_COMM_WORLD) == 0)
{
std::vector<std::string> filenames;
for (unsigned int i=0; i<Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD); ++i)
filenames.push_back (std::string("solution-") +
Utilities::int_to_string (refinement_cycle, 2) +
"." +
".vtu");
const std::string
pvtu_master_filename = ("solution-" +
Utilities::int_to_string (refinement_cycle, 2) +
".pvtu");
std::ofstream pvtu_master (pvtu_master_filename.c_str());
data_out.write_pvtu_record (pvtu_master, filenames);
}
}
template <int dim>
void
StokesProblem<dim>::refine_mesh ()
{
Vector<float> estimated_error_per_cell (triangulation.n_active_cells());
QGauss<dim-1>(degree+1),
solution,
estimated_error_per_cell,
fe.component_mask(pressure));
estimated_error_per_cell,
0.3, 0.0);
triangulation.execute_coarsening_and_refinement ();
}
template <int dim>
void StokesProblem<dim>::run ()
{
create_mesh();
for (unsigned int refinement_cycle = 0; refinement_cycle<9;
++refinement_cycle)
{
pcout << "Refinement cycle " << refinement_cycle << std::endl;
if (refinement_cycle > 0)
refine_mesh ();
setup_dofs ();
pcout << " Assembling..." << std::endl << std::flush;
assemble_system ();
pcout << " Solving..." << std::flush;
solve ();
output_results (refinement_cycle);
pcout << std::endl;
}
}
}
int main (int argc, char *argv[])
{
try
{
using namespace dealii;
using namespace Step45;
Utilities::MPI::MPI_InitFinalize mpi_initialization (argc, argv, 1);
StokesProblem<2> flow_problem(1);
flow_problem.run ();
}
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;
}