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Oliver Sander authored[[Imported from SVN: r8024]]
Oliver Sander authored[[Imported from SVN: r8024]]
cosseratenergystiffness.hh 13.16 KiB
#ifndef COSSERAT_ENERGY_LOCAL_STIFFNESS_HH
#define COSSERAT_ENERGY_LOCAL_STIFFNESS_HH
#include <dune/common/fmatrix.hh>
#include <dune/common/parametertree.hh>
#include <dune/grid/common/quadraturerules.hh>
#include <dune/fufem/functions/virtualgridfunction.hh>
#include <dune/fufem/boundarypatch.hh>
#include "localgeodesicfestiffness.hh"
#include "localgeodesicfefunction.hh"
#include <dune/gfe/rigidbodymotion.hh>
template<class GridView, class LocalFiniteElement, int dim>
class CosseratEnergyLocalStiffness
: public LocalGeodesicFEStiffness<GridView,LocalFiniteElement,RigidBodyMotion<dim> >
{
// grid types
typedef typename GridView::Grid::ctype DT;
typedef RigidBodyMotion<dim> TargetSpace;
typedef typename TargetSpace::ctype RT;
typedef typename GridView::template Codim<0>::Entity Entity;
// some other sizes
enum {gridDim=GridView::dimension};
/** \brief Compute the symmetric part of a matrix A, i.e. \f$ \frac 12 (A + A^T) \f$ */
static Dune::FieldMatrix<double,dim,dim> sym(const Dune::FieldMatrix<double,dim,dim>& A)
{
Dune::FieldMatrix<double,dim,dim> result;
for (int i=0; i<dim; i++)
for (int j=0; j<dim; j++)
result[i][j] = 0.5 * (A[i][j] + A[j][i]);
return result;
}
/** \brief Compute the antisymmetric part of a matrix A, i.e. \f$ \frac 12 (A - A^T) \f$ */
static Dune::FieldMatrix<double,dim,dim> skew(const Dune::FieldMatrix<double,dim,dim>& A)
{
Dune::FieldMatrix<double,dim,dim> result;
for (int i=0; i<dim; i++)
for (int j=0; j<dim; j++)
result[i][j] = 0.5 * (A[i][j] - A[j][i]);
return result;
}
/** \brief Return the square of the trace of a matrix */
static double traceSquared(const Dune::FieldMatrix<double,dim,dim>& A)
{
double trace = 0;
for (int i=0; i<dim; i++)
trace += A[i][i];
return trace*trace;
}
/** \brief Compute the (row-wise) curl of a matrix R \f$
\param DR The partial derivatives of the matrix R
*/
static Dune::FieldMatrix<double,dim,dim> curl(const Tensor3<double,dim,dim,dim>& DR)
{
Dune::FieldMatrix<double,dim,dim> result;
for (int i=0; i<dim; i++) {
result[i][0] = DR[i][2][1] - DR[i][1][2];
result[i][1] = DR[i][0][2] - DR[i][2][0];
result[i][2] = DR[i][1][0] - DR[i][0][1];
}
return result;
}
public: // for testing
/** \brief Compute the derivative of the rotation, but wrt matrix coordinates */
static void computeDR(const RigidBodyMotion<3>& value,
const Dune::FieldMatrix<double,7,gridDim>& derivative,
Tensor3<double,3,3,3>& DR)
{
// The LocalGFEFunction class gives us the derivatives of the orientation variable,
// but as a map into quaternion space. To obtain matrix coordinates we use the
// chain rule, which means that we have to multiply the given derivative with
// the derivative of the embedding of the unit quaternion into the space of 3x3 matrices.
// This second derivative is almost given by the method getFirstDerivativesOfDirectors.
// However, since the directors of a given unit quaternion are the _columns_ of the
// corresponding orthogonal matrix, we need to invert the i and j indices
//
// So, if I am not mistaken, DR[i][j][k] contains \partial R_ij / \partial k
Tensor3<double,3 , 3, 4> dd_dq;
value.q.getFirstDerivativesOfDirectors(dd_dq);
DR = 0;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++)
for (int k=0; k<gridDim; k++)
for (int l=0; l<4; l++)
DR[i][j][k] += dd_dq[j][i][l] * derivative[l+3][k];
}
public:
/** \brief Constructor with a set of material parameters
* \param parameters The material parameters
*/
CosseratEnergyLocalStiffness(const Dune::ParameterTree& parameters,
const BoundaryPatch<GridView>* neumannBoundary,
const Dune::VirtualFunction<Dune::FieldVector<double,gridDim>, Dune::FieldVector<double,3> >* neumannFunction)
: neumannBoundary_(neumannBoundary),
neumannFunction_(neumannFunction)
{
// The shell thickness
thickness_ = parameters.template get<double>("thickness");
// Lame constants
mu_ = parameters.template get<double>("mu");
lambda_ = parameters.template get<double>("lambda");
// Cosserat couple modulus, preferably 0
mu_c_ = parameters.template get<double>("mu_c");
// Length scale parameter
L_c_ = parameters.template get<double>("L_c");
// Curvature exponent
q_ = parameters.template get<double>("q");
}
/** \brief Assemble the energy for a single element */
RT energy (const Entity& e,
const LocalFiniteElement& localFiniteElement,
const std::vector<TargetSpace>& localSolution) const;
RT quadraticMembraneEnergy(const Dune::FieldMatrix<double,3,3>& U) const
{
Dune::FieldMatrix<double,3,3> UMinus1 = U;
for (int i=0; i<dim; i++)
UMinus1[i][i] -= 1;
return mu_ * sym(UMinus1).frobenius_norm2()
+ mu_c_ * skew(UMinus1).frobenius_norm2()
+ (mu_*lambda_)/(2*mu_ + lambda_) * traceSquared(sym(UMinus1));
}
RT curvatureEnergy(const Tensor3<double,3,3,3>& DR) const
{
return mu_ * std::pow(L_c_ * curl(DR).frobenius_norm(),q_);
}
RT bendingEnergy(const Dune::FieldMatrix<double,dim,dim>& R, const Tensor3<double,3,3,3>& DR) const
{
// The derivative of the third director
/** \brief There is no real need to have DR3 as a separate object */
Dune::FieldMatrix<double,3,3> DR3;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++)
DR3[i][j] = DR[i][2][j];
// left-multiply with R^T
Dune::FieldMatrix<double,3,3> RT_DR3;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++) {
RT_DR3[i][j] = 0;
for (int k=0; k<3; k++)
RT_DR3[i][j] += R[k][i] * DR3[k][j];
}
return mu_ * sym(RT_DR3).frobenius_norm2()
+ mu_c_ * skew(RT_DR3).frobenius_norm2()
/** \todo Is this sym correct? It is in the paper, but not in the notes */
+ mu_*lambda_/(2*mu_+lambda_) * traceSquared(sym(RT_DR3));
}
/** \brief The shell thickness */
double thickness_;
/** \brief Lame constants */
double mu_, lambda_;
/** \brief Cosserat couple modulus, preferably 0 */
double mu_c_;
/** \brief Length scale parameter */
double L_c_;
/** \brief Curvature exponent */
double q_;
/** \brief The Neumann boundary */
const BoundaryPatch<GridView>* neumannBoundary_;
/** \brief The function implementing the Neumann data */
const Dune::VirtualFunction<Dune::FieldVector<double,gridDim>, Dune::FieldVector<double,3> >* neumannFunction_;
};
template <class GridView, class LocalFiniteElement, int dim>
typename CosseratEnergyLocalStiffness<GridView,LocalFiniteElement,dim>::RT
CosseratEnergyLocalStiffness<GridView,LocalFiniteElement,dim>::
energy(const Entity& element,
const LocalFiniteElement& localFiniteElement,
const std::vector<RigidBodyMotion<dim> >& localSolution) const
{
RT energy = 0;
LocalGeodesicFEFunction<gridDim, double, LocalFiniteElement, TargetSpace> localGeodesicFEFunction(localFiniteElement,
localSolution);
int quadOrder = (element.type().isSimplex()) ? localFiniteElement.localBasis().order()
: localFiniteElement.localBasis().order() * gridDim;
const Dune::QuadratureRule<double, gridDim>& quad
= Dune::QuadratureRules<double, gridDim>::rule(element.type(), quadOrder);
for (size_t pt=0; pt<quad.size(); pt++) {
// Local position of the quadrature point
const Dune::FieldVector<double,gridDim>& quadPos = quad[pt].position();
const double integrationElement = element.geometry().integrationElement(quadPos);
const Dune::FieldMatrix<double,gridDim,gridDim>& jacobianInverseTransposed = element.geometry().jacobianInverseTransposed(quadPos);
double weight = quad[pt].weight() * integrationElement;
// The value of the local function
RigidBodyMotion<dim> value = localGeodesicFEFunction.evaluate(quadPos);
// The derivative of the local function defined on the reference element
Dune::FieldMatrix<double, TargetSpace::EmbeddedTangentVector::dimension, gridDim> referenceDerivative = localGeodesicFEFunction.evaluateDerivative(quadPos);
// The derivative of the function defined on the actual element
Dune::FieldMatrix<double, TargetSpace::EmbeddedTangentVector::dimension, gridDim> derivative(0);
for (size_t comp=0; comp<referenceDerivative.N(); comp++)
jacobianInverseTransposed.umv(referenceDerivative[comp], derivative[comp]);
/////////////////////////////////////////////////////////
// compute U, the Cosserat strain
/////////////////////////////////////////////////////////
dune_static_assert(dim>=gridDim, "Codim of the grid must be nonnegative");
//
Dune::FieldMatrix<double,dim,dim> R;
value.q.matrix(R);
/* Compute F, the deformation gradient.
In the continuum case this is
\f$ \hat{F} = \nabla m \f$
In the case of a shell it is
\f$ \hat{F} = (\nabla m | \overline{R}_3) \f$
*/
Dune::FieldMatrix<double,dim,dim> F;
for (int i=0; i<dim; i++)
for (int j=0; j<gridDim; j++)
F[i][j] = derivative[i][j];
for (int i=0; i<dim; i++)
for (int j=gridDim; j<dim; j++)
F[i][j] = R[i][j];
// U = R^T F
Dune::FieldMatrix<double,dim,dim> U;
for (int i=0; i<dim; i++)
for (int j=0; j<dim; j++) {
U[i][j] = 0;
for (int k=0; k<dim; k++)
U[i][j] += R[k][i] * F[k][j];
}
//////////////////////////////////////////////////////////
// Compute the derivative of the rotation
// Note: we need it in matrix coordinates
//////////////////////////////////////////////////////////
Tensor3<double,3,3,3> DR;
computeDR(value, derivative, DR);
// Add the local energy density
if (gridDim==2) {
energy += weight * thickness_ * quadraticMembraneEnergy(U);
energy += weight * thickness_ * curvatureEnergy(DR);
energy += weight * std::pow(thickness_,3) / 12.0 * bendingEnergy(R,DR);
} else if (gridDim==3) {
energy += weight * quadraticMembraneEnergy(U);
energy += weight * curvatureEnergy(DR);
} else
DUNE_THROW(Dune::NotImplemented, "CosseratEnergyStiffness for 1d grids");
}
//////////////////////////////////////////////////////////////////////////////
// Assemble boundary contributions
//////////////////////////////////////////////////////////////////////////////
assert(neumannFunction_);
for (typename Entity::LeafIntersectionIterator it = element.ileafbegin(); it != element.ileafend(); ++it) {
if (not neumannBoundary_ or not neumannBoundary_->contains(*it))
continue;
const Dune::QuadratureRule<double, gridDim-1>& quad
= Dune::QuadratureRules<double, gridDim-1>::rule(it->type(), quadOrder);
for (size_t pt=0; pt<quad.size(); pt++) {
// Local position of the quadrature point
const Dune::FieldVector<double,gridDim>& quadPos = it->geometryInInside().global(quad[pt].position());
const double integrationElement = it->geometry().integrationElement(quad[pt].position());
// The value of the local function
RigidBodyMotion<dim> value = localGeodesicFEFunction.evaluate(quadPos);
// Value of the Neumann data at the current position
Dune::FieldVector<double,3> neumannValue;
if (dynamic_cast<const VirtualGridViewFunction<GridView,Dune::FieldVector<double,3> >*>(neumannFunction_))
dynamic_cast<const VirtualGridViewFunction<GridView,Dune::FieldVector<double,3> >*>(neumannFunction_)->evaluateLocal(element, quadPos, neumannValue);
else
neumannFunction_->evaluate(it->geometry().global(quad[pt].position()), neumannValue);
// Constant Neumann force in z-direction
energy += thickness_ * (neumannValue * value.r) * quad[pt].weight() * integrationElement;
}
}
return energy;
}
#endif