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// $Id: elasticity.h 30963 2013-09-26 18:42:28Z 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_elasticity_h
#define __deal2__integrators_elasticity_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 elasticity problems.
*
* @ingroup Integrators
* @author Guido Kanschat
* @date 2010
*/
namespace Elasticity
{
/**
* The linear elasticity operator in weak form, namely double
* contraction of symmetric gradients.
*
* \f[
* \int_Z \varepsilon(u): \varepsilon(v)\,dx
* \f]
*/
template <int dim>
inline void cell_matrix (
FullMatrix<double> &M,
const FEValuesBase<dim> &fe,
const double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
AssertDimension(fe.get_fe().n_components(), dim);
AssertDimension(M.m(), n_dofs);
AssertDimension(M.n(), n_dofs);
for (unsigned int k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int j=0; j<n_dofs; ++j)
for (unsigned int d1=0; d1<dim; ++d1)
for (unsigned int d2=0; d2<dim; ++d2)
M(i,j) += dx * .25 *
(fe.shape_grad_component(j,k,d1)[d2] + fe.shape_grad_component(j,k,d2)[d1]) *
(fe.shape_grad_component(i,k,d1)[d2] + fe.shape_grad_component(i,k,d2)[d1]);
}
}
/**
* Vector-valued residual operator for linear elasticity in weak form
*
* \f[
* - \int_Z \varepsilon(u): \varepsilon(v) \,dx
* \f]
*/
template <int dim, typename number>
inline void
cell_residual (
Vector<number> &result,
const FEValuesBase<dim> &fe,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &input,
double factor = 1.)
{
const unsigned int nq = fe.n_quadrature_points;
const unsigned int n_dofs = fe.dofs_per_cell;
AssertDimension(fe.get_fe().n_components(), dim);
AssertVectorVectorDimension(input, dim, fe.n_quadrature_points);
Assert(result.size() == n_dofs, ExcDimensionMismatch(result.size(), n_dofs));
for (unsigned int k=0; k<nq; ++k)
{
const double dx = factor * fe.JxW(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int d1=0; d1<dim; ++d1)
for (unsigned int d2=0; d2<dim; ++d2)
{
result(i) += dx * .25 *
(input[d1][k][d2] + input[d2][k][d1]) *
(fe.shape_grad_component(i,k,d1)[d2] + fe.shape_grad_component(i,k,d2)[d1]);
}
}
}
/**
* The weak boundary condition of Nitsche type for linear
* elasticity:
* @f[
* \int_F \Bigl(\gamma (u-g) \cdot v - n^T \epsilon(u) v - (u-g) \epsilon(v) n^T\Bigr)\;ds.
* @f]
*/
template <int dim>
inline void nitsche_matrix (
FullMatrix<double> &M,
const FEValuesBase<dim> &fe,
double penalty,
double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
AssertDimension(fe.get_fe().n_components(), dim);
AssertDimension(M.m(), n_dofs);
AssertDimension(M.n(), n_dofs);
for (unsigned int k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
const Point<dim> &n = fe.normal_vector(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int j=0; j<n_dofs; ++j)
for (unsigned int d1=0; d1<dim; ++d1)
{
const double u = fe.shape_value_component(j,k,d1);
const double v = fe.shape_value_component(i,k,d1);
M(i,j) += dx * 2. * penalty * u * v;
for (unsigned int d2=0; d2<dim; ++d2)
{
// v . nabla u n
M(i,j) -= .5*dx* fe.shape_grad_component(j,k,d1)[d2] *n(d2)* v;
// v (nabla u)^T n
M(i,j) -= .5*dx* fe.shape_grad_component(j,k,d2)[d1] *n(d2)* v;
// u nabla v n
M(i,j) -= .5*dx* fe.shape_grad_component(i,k,d1)[d2] *n(d2)* u;
// u (nabla v)^T n
M(i,j) -= .5*dx* fe.shape_grad_component(i,k,d2)[d1] *n(d2)* u;
}
}
}
}
/**
* Weak boundary condition for the elasticity operator by Nitsche,
* namely on the face <i>F</i> the vector
* @f[
* \int_F \Bigl(\gamma (u-g) \cdot v - n^T \epsilon(u) v - (u-g) \epsilon(v) n^T\Bigr)\;ds.
* @f]
*
* Here, <i>u</i> is the finite element function whose values and
* gradient are given in the arguments <tt>input</tt> and
* <tt>Dinput</tt>, respectively. <i>g</i> is the inhomogeneous
* boundary value in the argument <tt>data</tt>. $n$ is the outer
* normal vector and $\gamma$ is the usual penalty parameter.
*
* @author Guido Kanschat
* @date 2013
*/
template <int dim, typename number>
void nitsche_residual (
Vector<number> &result,
const FEValuesBase<dim> &fe,
const VectorSlice<const std::vector<std::vector<double> > > &input,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &Dinput,
const VectorSlice<const std::vector<std::vector<double> > > &data,
double penalty,
double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
AssertVectorVectorDimension(input, dim, fe.n_quadrature_points);
AssertVectorVectorDimension(Dinput, dim, fe.n_quadrature_points);
AssertVectorVectorDimension(data, dim, fe.n_quadrature_points);
for (unsigned int k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
const Point<dim> &n = fe.normal_vector(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int d1=0; d1<dim; ++d1)
{
const double u= input[d1][k];
const double v= fe.shape_value_component(i,k,d1);
const double g= data[d1][k];
result(i) += dx * 2.*penalty * (u-g) * v;
for (unsigned int d2=0; d2<dim; ++d2)
{
// v . nabla u n
result(i) -= .5*dx* v * Dinput[d1][k][d2] * n(d2);
// v . (nabla u)^T n
result(i) -= .5*dx* v * Dinput[d2][k][d1] * n(d2);
// u nabla v n
result(i) -= .5*dx * (u-g) * fe.shape_grad_component(i,k,d1)[d2] * n(d2);
// u (nabla v)^T n
result(i) -= .5*dx * (u-g) * fe.shape_grad_component(i,k,d2)[d1] * n(d2);
}
}
}
}
/**
* Homogeneous weak boundary condition for the elasticity operator by Nitsche,
* namely on the face <i>F</i> the vector
* @f[
* \int_F \Bigl(\gamma u \cdot v - n^T \epsilon(u) v - u \epsilon(v) n^T\Bigr)\;ds.
* @f]
*
* Here, <i>u</i> is the finite element function whose values and
* gradient are given in the arguments <tt>input</tt> and
* <tt>Dinput</tt>, respectively. $n$ is the outer
* normal vector and $\gamma$ is the usual penalty parameter.
*
* @author Guido Kanschat
* @date 2013
*/
template <int dim, typename number>
void nitsche_residual_homogeneous (
Vector<number> &result,
const FEValuesBase<dim> &fe,
const VectorSlice<const std::vector<std::vector<double> > > &input,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &Dinput,
double penalty,
double factor = 1.)
{
const unsigned int n_dofs = fe.dofs_per_cell;
AssertVectorVectorDimension(input, dim, fe.n_quadrature_points);
AssertVectorVectorDimension(Dinput, dim, fe.n_quadrature_points);
for (unsigned int k=0; k<fe.n_quadrature_points; ++k)
{
const double dx = factor * fe.JxW(k);
const Point<dim> &n = fe.normal_vector(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int d1=0; d1<dim; ++d1)
{
const double u= input[d1][k];
const double v= fe.shape_value_component(i,k,d1);
result(i) += dx * 2.*penalty * u * v;
for (unsigned int d2=0; d2<dim; ++d2)
{
// v . nabla u n
result(i) -= .5*dx* v * Dinput[d1][k][d2] * n(d2);
// v . (nabla u)^T n
result(i) -= .5*dx* v * Dinput[d2][k][d1] * n(d2);
// u nabla v n
result(i) -= .5*dx * u * fe.shape_grad_component(i,k,d1)[d2] * n(d2);
// u (nabla v)^T n
result(i) -= .5*dx * u * fe.shape_grad_component(i,k,d2)[d1] * n(d2);
}
}
}
}
/**
* The interior penalty flux
* for symmetric gradients.
*/
template <int dim>
inline void ip_matrix (
FullMatrix<double> &M11,
FullMatrix<double> &M12,
FullMatrix<double> &M21,
FullMatrix<double> &M22,
const FEValuesBase<dim> &fe1,
const FEValuesBase<dim> &fe2,
const double pen,
const double int_factor = 1.,
const double ext_factor = -1.)
{
const unsigned int n_dofs = fe1.dofs_per_cell;
AssertDimension(fe1.get_fe().n_components(), dim);
AssertDimension(fe2.get_fe().n_components(), dim);
AssertDimension(M11.m(), n_dofs);
AssertDimension(M11.n(), n_dofs);
AssertDimension(M12.m(), n_dofs);
AssertDimension(M12.n(), n_dofs);
AssertDimension(M21.m(), n_dofs);
AssertDimension(M21.n(), n_dofs);
AssertDimension(M22.m(), n_dofs);
AssertDimension(M22.n(), n_dofs);
const double nu1 = int_factor;
const double nu2 = (ext_factor < 0) ? int_factor : ext_factor;
const double penalty = .5 * pen * (nu1 + nu2);
for (unsigned int k=0; k<fe1.n_quadrature_points; ++k)
{
const double dx = fe1.JxW(k);
const Point<dim> &n = fe1.normal_vector(k);
for (unsigned int i=0; i<n_dofs; ++i)
for (unsigned int j=0; j<n_dofs; ++j)
for (unsigned int d1=0; d1<dim; ++d1)
{
const double u1 = fe1.shape_value_component(j,k,d1);
const double u2 = fe2.shape_value_component(j,k,d1);
const double v1 = fe1.shape_value_component(i,k,d1);
const double v2 = fe2.shape_value_component(i,k,d1);
M11(i,j) += dx * penalty * u1*v1;
M12(i,j) -= dx * penalty * u2*v1;
M21(i,j) -= dx * penalty * u1*v2;
M22(i,j) += dx * penalty * u2*v2;
for (unsigned int d2=0; d2<dim; ++d2)
{
// v . nabla u n
M11(i,j) -= .25 * dx * nu1 * fe1.shape_grad_component(j,k,d1)[d2] * n(d2) * v1;
M12(i,j) -= .25 * dx * nu2 * fe2.shape_grad_component(j,k,d1)[d2] * n(d2) * v1;
M21(i,j) += .25 * dx * nu1 * fe1.shape_grad_component(j,k,d1)[d2] * n(d2) * v2;
M22(i,j) += .25 * dx * nu2 * fe2.shape_grad_component(j,k,d1)[d2] * n(d2) * v2;
// v (nabla u)^T n
M11(i,j) -= .25 * dx * nu1 * fe1.shape_grad_component(j,k,d2)[d1] * n(d2) * v1;
M12(i,j) -= .25 * dx * nu2 * fe2.shape_grad_component(j,k,d2)[d1] * n(d2) * v1;
M21(i,j) += .25 * dx * nu1 * fe1.shape_grad_component(j,k,d2)[d1] * n(d2) * v2;
M22(i,j) += .25 * dx * nu2 * fe2.shape_grad_component(j,k,d2)[d1] * n(d2) * v2;
// u nabla v n
M11(i,j) -= .25 * dx * nu1 * fe1.shape_grad_component(i,k,d1)[d2] * n(d2) * u1;
M12(i,j) += .25 * dx * nu1 * fe1.shape_grad_component(i,k,d1)[d2] * n(d2) * u2;
M21(i,j) -= .25 * dx * nu2 * fe2.shape_grad_component(i,k,d1)[d2] * n(d2) * u1;
M22(i,j) += .25 * dx * nu2 * fe2.shape_grad_component(i,k,d1)[d2] * n(d2) * u2;
// u (nabla v)^T n
M11(i,j) -= .25 * dx * nu1 * fe1.shape_grad_component(i,k,d2)[d1] * n(d2) * u1;
M12(i,j) += .25 * dx * nu1 * fe1.shape_grad_component(i,k,d2)[d1] * n(d2) * u2;
M21(i,j) -= .25 * dx * nu2 * fe2.shape_grad_component(i,k,d2)[d1] * n(d2) * u1;
M22(i,j) += .25 * dx * nu2 * fe2.shape_grad_component(i,k,d2)[d1] * n(d2) * u2;
}
}
}
}
/**
* Elasticity residual term for the symmetric interior penalty method.
*
* @author Guido Kanschat
* @date 2013
*/
template<int dim, typename number>
void
ip_residual(
Vector<number> &result1,
Vector<number> &result2,
const FEValuesBase<dim> &fe1,
const FEValuesBase<dim> &fe2,
const VectorSlice<const std::vector<std::vector<double> > > &input1,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &Dinput1,
const VectorSlice<const std::vector<std::vector<double> > > &input2,
const VectorSlice<const std::vector<std::vector<Tensor<1,dim> > > > &Dinput2,
double pen,
double int_factor = 1.,
double ext_factor = -1.)
{
const unsigned int n1 = fe1.dofs_per_cell;
AssertDimension(fe1.get_fe().n_components(), dim);
AssertDimension(fe2.get_fe().n_components(), dim);
AssertVectorVectorDimension(input1, dim, fe1.n_quadrature_points);
AssertVectorVectorDimension(Dinput1, dim, fe1.n_quadrature_points);
AssertVectorVectorDimension(input2, dim, fe2.n_quadrature_points);
AssertVectorVectorDimension(Dinput2, dim, fe2.n_quadrature_points);
const double nu1 = int_factor;
const double nu2 = (ext_factor < 0) ? int_factor : ext_factor;
const double penalty = .5 * pen * (nu1 + nu2);
for (unsigned int k=0; k<fe1.n_quadrature_points; ++k)
{
const double dx = fe1.JxW(k);
const Point<dim> &n = fe1.normal_vector(k);
for (unsigned int i=0; i<n1; ++i)
for (unsigned int d1=0; d1<dim; ++d1)
{
const double v1 = fe1.shape_value_component(i,k,d1);
const double v2 = fe2.shape_value_component(i,k,d1);
const double u1 = input1[d1][k];
const double u2 = input2[d1][k];
result1(i) += dx * penalty * u1*v1;
result1(i) -= dx * penalty * u2*v1;
result2(i) -= dx * penalty * u1*v2;
result2(i) += dx * penalty * u2*v2;
for (unsigned int d2=0; d2<dim; ++d2)
{
// v . nabla u n
result1(i) -= .25*dx* (nu1*Dinput1[d1][k][d2]+nu2*Dinput2[d1][k][d2]) * n(d2) * v1;
result2(i) += .25*dx* (nu1*Dinput1[d1][k][d2]+nu2*Dinput2[d1][k][d2]) * n(d2) * v2;
// v . (nabla u)^T n
result1(i) -= .25*dx* (nu1*Dinput1[d2][k][d1]+nu2*Dinput2[d2][k][d1]) * n(d2) * v1;
result2(i) += .25*dx* (nu1*Dinput1[d2][k][d1]+nu2*Dinput2[d2][k][d1]) * n(d2) * v2;
// u nabla v n
result1(i) -= .25*dx* nu1*fe1.shape_grad_component(i,k,d1)[d2] * n(d2) * (u1-u2);
result2(i) -= .25*dx* nu2*fe2.shape_grad_component(i,k,d1)[d2] * n(d2) * (u1-u2);
// u (nabla v)^T n
result1(i) -= .25*dx* nu1*fe1.shape_grad_component(i,k,d2)[d1] * n(d2) * (u1-u2);
result2(i) -= .25*dx* nu2*fe2.shape_grad_component(i,k,d2)[d1] * n(d2) * (u1-u2);
}
}
}
}
}
}
DEAL_II_NAMESPACE_CLOSE
#endif
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