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// $Id: advection.h 30942 2013-09-26 08:58:02Z kanschat $
//
// Copyright (C) 2010 - 2013 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.
//
// ---------------------------------------------------------------------
#ifndef __deal2__integrators_advection_h
#define __deal2__integrators_advection_h
#include <deal.II/base/config.h>
#include <deal.II/base/exceptions.h>
#include <deal.II/base/quadrature.h>
#include <deal.II/lac/full_matrix.h>
#include <deal.II/fe/mapping.h>
#include <deal.II/fe/fe_values.h>
#include <deal.II/meshworker/dof_info.h>
DEAL_II_NAMESPACE_OPEN
namespace LocalIntegrators
{
/**
* @brief Local integrators related to advection along a vector field and its DG formulations
*
* All advection operators depend on an advection velocity denoted by
* <b>w</b> in the formulas below. It is denoted as <tt>velocity</tt>
* in the parameter lists.
*
* The functions cell_matrix() and both upwind_value_matrix() are
* taking the equation in weak form, that is, the directional
* derivative is on the test function.
*
* @ingroup Integrators
* @author Guido Kanschat
* @date 2012
*/
namespace Advection
{
/**
* Advection along the direction <b>w</b> in weak form
* with derivative on the test function
* \f[
* m_{ij} = \int_Z u_j\,(\mathbf w \cdot \nabla) v_i \, dx.
* \f]
*
* The FiniteElement in <tt>fe</tt> may be scalar or vector valued. In
* the latter case, the advection operator is applied to each component
* separately.
*
* @param M: The advection matrix obtained as result
* @param fe: The FEValues object describing the local trial function
* space. #update_values and #update_gradients, and #update_JxW_values
* must be set.
* @param fetest: The FEValues object describing the local test
* function space. #update_values and #update_gradients must be set.
* @param velocity: The advection velocity, a vector of dimension
* <tt>dim</tt>. Each component may either contain a vector of length
* one, in which case a constant velocity is assumed, or a vector with
* as many entries as quadrature points if the velocity is not constant.
* @param factor is an optional multiplication factor for the result.
*
* @author Guido Kanschat
* @date 2012
*/
template<int dim>
void cell_matrix (
FullMatrix<double> &M,
const FEValuesBase<dim> &fe,
const FEValuesBase<dim> &fetest,
const VectorSlice<const std::vector<std::vector<double> > > &velocity,
const double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
const unsigned int t_dofs = fetest.dofs_per_cell;
const unsigned int n_components = fe.get_fe().n_components();
AssertDimension(velocity.size(), dim);
// If the size of the
// velocity vectors is one,
// then do not increment
// between quadrature points.
const unsigned int v_increment = (velocity[0].size() == 1) ? 0 : 1;
if (v_increment == 1)
{
AssertVectorVectorDimension(velocity, dim, fe.n_quadrature_points);
}
AssertDimension(M.n(), n_dofs);
AssertDimension(M.m(), t_dofs);
for (unsigned k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
const unsigned int vindex = k * v_increment;
for (unsigned j=0; j<n_dofs; ++j)
for (unsigned i=0; i<t_dofs; ++i)
for (unsigned int c=0; c<n_components; ++c)
{
double wgradv = velocity[0][vindex]
* fe.shape_grad_component(i,k,c)[0];
for (unsigned int d=1; d<dim; ++d)
wgradv += velocity[d][vindex]
* fe.shape_grad_component(i,k,c)[d];
M(i,j) -= dx * wgradv * fe.shape_value_component(j,k,c);
}
}
}
/**
* Advection residual operator in strong form
*
* \f[
* r_i = \int_Z (\mathbf w \cdot \nabla)u\, v_i \, dx.
* \f]
*/
template <int dim>
inline void
cell_residual (
Vector<double> &result,
const FEValuesBase<dim> &fe,
const std::vector<Tensor<1,dim> > &input,
const VectorSlice<const std::vector<std::vector<double> > > &velocity,
double factor = 1.)
{
const unsigned int nq = fe.n_quadrature_points;
const unsigned int n_dofs = fe.dofs_per_cell;
Assert(input.size() == nq, ExcDimensionMismatch(input.size(), nq));
Assert(result.size() == n_dofs, ExcDimensionMismatch(result.size(), n_dofs));
AssertDimension(velocity.size(), dim);
const unsigned int v_increment = (velocity[0].size() == 1) ? 0 : 1;
if (v_increment == 1)
{
AssertVectorVectorDimension(velocity, dim, fe.n_quadrature_points);
}
for (unsigned k=0; k<nq; ++k)
{
const double dx = factor * fe.JxW(k);
for (unsigned i=0; i<n_dofs; ++i)
for (unsigned int d=0; d<dim; ++d)
result(i) += dx * input[k][d]
* fe.shape_value(i,k) * velocity[d][k * v_increment];
}
}
/**
* Vector-valued advection residual operator in strong form
*
*
* \f[
* r_i = \int_Z \bigl((\mathbf w \cdot \nabla) \mathbf u\bigr) \cdot\mathbf v_i \, dx.
* \f]
*/
template <int dim>
inline void
cell_residual (
Vector<double> &result,
const FEValuesBase<dim> &fe,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &input,
const VectorSlice<const std::vector<std::vector<double> > > &velocity,
double factor = 1.)
{
const unsigned int nq = fe.n_quadrature_points;
const unsigned int n_dofs = fe.dofs_per_cell;
const unsigned int n_comp = fe.get_fe().n_components();
AssertVectorVectorDimension(input, n_comp, fe.n_quadrature_points);
Assert(result.size() == n_dofs, ExcDimensionMismatch(result.size(), n_dofs));
AssertDimension(velocity.size(), dim);
const unsigned int v_increment = (velocity[0].size() == 1) ? 0 : 1;
if (v_increment == 1)
{
AssertVectorVectorDimension(velocity, dim, fe.n_quadrature_points);
}
for (unsigned k=0; k<nq; ++k)
{
const double dx = factor * fe.JxW(k);
for (unsigned i=0; i<n_dofs; ++i)
for (unsigned int c=0; c<n_comp; ++c)
for (unsigned int d=0; d<dim; ++d)
result(i) += dx * input[c][k][d]
* fe.shape_value_component(i,k,c) * velocity[d][k * v_increment];
}
}
/**
* Upwind flux at the boundary
* for weak advection
* operator. This is the value of
* the trial function at the
* outflow boundary and zero
* else:
* @f[
* a_{ij} = \int_{\partial\Omega}
* [\mathbf w\cdot\mathbf n]_+
* u_i v_j \, ds
* @f]
*
* The <tt>velocity</tt> is
* provided as a VectorSlice,
* having <tt>dim</tt> vectors,
* one for each velocity
* component. Each of the
* vectors must either have only
* a single entry, if t he
* advection velocity is
* constant, or have an entry
* for each quadrature point.
*
* The finite element can have
* several components, in which
* case each component is
* advected by the same velocity.
*/
template <int dim>
void upwind_value_matrix(
FullMatrix<double> &M,
const FEValuesBase<dim> &fe,
const FEValuesBase<dim> &fetest,
const VectorSlice<const std::vector<std::vector<double> > > &velocity,
double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
const unsigned int t_dofs = fetest.dofs_per_cell;
unsigned int n_components = fe.get_fe().n_components();
AssertDimension (M.m(), n_dofs);
AssertDimension (M.n(), n_dofs);
AssertDimension(velocity.size(), dim);
const unsigned int v_increment = (velocity[0].size() == 1) ? 0 : 1;
if (v_increment == 1)
{
AssertVectorVectorDimension(velocity, dim, fe.n_quadrature_points);
}
for (unsigned k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
double nv = 0.;
for (unsigned int d=0; d<dim; ++d)
nv += fe.normal_vector(k)[d] * velocity[d][k * v_increment];
if (nv > 0)
{
for (unsigned i=0; i<t_dofs; ++i)
for (unsigned j=0; j<n_dofs; ++j)
{
if (fe.get_fe().is_primitive())
M(i,j) += dx * nv * fe.shape_value(i,k) * fe.shape_value(j,k);
else
for (unsigned int c=0; c<n_components; ++c)
M(i,j) += dx * nv * fetest.shape_value_component(i,k,c)
* fe.shape_value_component(j,k,c);
}
}
}
}
/**
* Upwind flux in the interior
* for weak advection
* operator. Matrix entries
* correspond to the upwind value
* of the trial function, multiplied
* by the jump of the test
* functions
* @f[
* a_{ij} = \int_F \left|\mathbf w
* \cdot \mathbf n\right|
* u^\uparrow
* (v^\uparrow-v^\downarrow)
* \,ds
* @f]
*
* The <tt>velocity</tt> is
* provided as a VectorSlice,
* having <tt>dim</tt> vectors,
* one for each velocity
* component. Each of the
* vectors must either have only
* a single entry, if t he
* advection velocity is
* constant, or have an entry
* for each quadrature point.
*
* The finite element can have
* several components, in which
* case each component is
* advected the same way.
*/
template <int dim>
void upwind_value_matrix (
FullMatrix<double> &M11,
FullMatrix<double> &M12,
FullMatrix<double> &M21,
FullMatrix<double> &M22,
const FEValuesBase<dim> &fe1,
const FEValuesBase<dim> &fe2,
const FEValuesBase<dim> &fetest1,
const FEValuesBase<dim> &fetest2,
const VectorSlice<const std::vector<std::vector<double> > > &velocity,
const double factor = 1.)
{
const unsigned int n1 = fe1.dofs_per_cell;
// Multiply the quadrature point
// index below with this factor to
// have simpler data for constant
// velocities.
AssertDimension(velocity.size(), dim);
const unsigned int v_increment = (velocity[0].size() == 1) ? 0 : 1;
if (v_increment == 1)
{
AssertVectorVectorDimension(velocity, dim, fe1.n_quadrature_points);
}
for (unsigned k=0; k<fe1.n_quadrature_points; ++k)
{
double nbeta = fe1.normal_vector(k)[0] * velocity[0][k * v_increment];
for (unsigned int d=1; d<dim; ++d)
nbeta += fe1.normal_vector(k)[d] * velocity[d][k * v_increment];
const double dx_nbeta = factor * nbeta * fe1.JxW(k);
for (unsigned i=0; i<n1; ++i)
for (unsigned j=0; j<n1; ++j)
if (fe1.get_fe().is_primitive())
{
if (nbeta > 0)
{
M11(i,j) += dx_nbeta
* fe1.shape_value(j,k)
* fetest1.shape_value(i,k);
M21(i,j) -= dx_nbeta
* fe1.shape_value(j,k)
* fetest2.shape_value(i,k);
}
else
{
M22(i,j) -= dx_nbeta
* fe2.shape_value(j,k)
* fetest2.shape_value(i,k);
M12(i,j) += dx_nbeta
* fe2.shape_value(j,k)
* fetest1.shape_value(i,k);
}
}
else
{
for (unsigned int d=0; d<fe1.get_fe().n_components(); ++d)
if (nbeta > 0)
{
M11(i,j) += dx_nbeta
* fe1.shape_value_component(j,k,d)
* fetest1.shape_value_component(i,k,d);
M21(i,j) -= dx_nbeta
* fe1.shape_value_component(j,k,d)
* fetest2.shape_value_component(i,k,d);
}
else
{
M22(i,j) -= dx_nbeta
* fe2.shape_value_component(j,k,d)
* fetest2.shape_value_component(i,k,d);
M12(i,j) += dx_nbeta
* fe2.shape_value_component(j,k,d)
* fetest1.shape_value_component(i,k,d);
}
}
}
}
}
}
DEAL_II_NAMESPACE_CLOSE
#endif
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