HolzapfelMaterialNonLinear.hpp

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/*!
 *  @file
 *  @brief This file contains the definition for the St. Venant Kirchhoff linear material
 *
 *  @version 1.0
 *  @date 29-07-2010
 *  @author Paolo Tricerri
 *
 *  @maintainer  Paolo Tricerri      <paolo.tricerri@epfl.ch>
 */

#ifndef _HOLZAPFELMATERIAL_H_
#define _HOLZAPFELMATERIAL_H_

#pragma GCC diagnostic ignored "-Wunused-variable"
#pragma GCC diagnostic ignored "-Wunused-parameter"


#include <lifev/structure/solver/anisotropic/StructuralAnisotropicConstitutiveLaw.hpp>

#define PI 3.14159265359

namespace LifeV
{
template <typename MeshType>
class HolzapfelMaterialNonLinear : public StructuralAnisotropicConstitutiveLaw<MeshType>
{
    //!@name Type definitions
    //@{

public:
    typedef StructuralAnisotropicConstitutiveLaw<MeshType>           super;

    typedef typename super::data_Type                data_Type;

    typedef typename super::vector_Type              vector_Type;
    typedef typename super::matrix_Type              matrix_Type;

    typedef typename super::matrixPtr_Type           matrixPtr_Type;
    typedef typename super::vectorPtr_Type           vectorPtr_Type;
    typedef typename super::dataPtr_Type             dataPtr_Type;
    typedef typename super::displayerPtr_Type        displayerPtr_Type;

    typedef typename super::mapMarkerVolumesPtr_Type mapMarkerVolumesPtr_Type;
    typedef typename super::mapMarkerVolumes_Type mapMarkerVolumes_Type;
    typedef typename mapMarkerVolumes_Type::const_iterator mapIterator_Type;

    typedef typename super::mapMarkerIndexesPtr_Type mapMarkerIndexesPtr_Type;
    typedef typename super::mapMarkerIndexes_Type    mapMarkerIndexes_Type;
    typedef typename mapMarkerIndexes_Type::const_iterator mapIteratorIndex_Type;

    typedef std::vector<typename MeshType::element_Type*> vectorVolumes_Type;
    typedef std::shared_ptr<vectorVolumes_Type>    vectorVolumesPtr_Type;

    typedef std::vector<UInt>                        vectorIndexes_Type;
    typedef std::shared_ptr<vectorIndexes_Type>    vectorIndexesPtr_Type;

    typedef typename super::FESpace_Type             FESpace_Type;
    typedef typename super::FESpacePtr_Type          FESpacePtr_Type;
    typedef typename super::ETFESpacePtr_Type        ETFESpacePtr_Type;

    //Vector for vector parameters
    typedef typename super::vectorsParameters_Type       vectorsParameters_Type;
    typedef typename super::vectorsParametersPtr_Type    vectorsParametersPtr_Type;

    typedef MatrixSmall<3, 3>                          matrixSmall_Type;

    typedef typename super::fiberFunction_Type                fiberFunction_Type;
    typedef typename super::fiberFunctionPtr_Type             fiberFunctionPtr_Type;

    typedef typename super::vectorFiberFunction_Type          vectorFiberFunction_Type;
    typedef typename super::vectorFiberFunctionPtr_Type          vectorFiberFunctionPtr_Type;

    // Typedefs for expression definitions
    typedef typename super::tensorF_Type               tensorF_Type;
    typedef typename super::determinantF_Type          determinantF_Type;
    typedef typename super::tensorC_Type               tensorC_Type;
    typedef typename super::minusT_Type                minusT_Type;
    typedef typename super::traceTensor_Type           traceTensor_Type;
    typedef typename super::powerExpression_Type       powerExpression_Type;

    // Anisotropic typedefs
    typedef typename super::interpolatedValue_Type       interpolatedValue_Type;
    typedef typename super::outerProduct_Type            outerProduct_Type;
    typedef typename super::stretch_Type                 stretch_Type;
    typedef typename super::isochoricStretch_Type        isochoricStretch_Type;
    //@}



    //! @name Constructor &  Destructor
    //@{

    HolzapfelMaterialNonLinear();

    virtual  ~HolzapfelMaterialNonLinear();

    //@}



    //!@name Methods
    //@{

    //! Setup the created object of the class StructuralIsotropicConstitutiveLaw
    /*!
      \param dFespace: the FiniteElement Space
      \param monolithicMap: the MapEpetra
      \param offset: the offset parameter used assembling the matrices
    */
    void setup ( const FESpacePtr_Type& dFESpace,
                 const ETFESpacePtr_Type& dETFESpace,
                 const std::shared_ptr<const MapEpetra>&  monolithicMap,
                 const UInt offset,const dataPtr_Type& dataMaterial);


    //! Compute the Stiffness matrix in StructuralSolver::buildSystem()
    /*!
      \param dataMaterial the class with Material properties data
    */
    void computeLinearStiff ( dataPtr_Type& /*dataMaterial*/,
                              const mapMarkerVolumesPtr_Type /*mapsMarkerVolumes*/,
                              const mapMarkerIndexesPtr_Type /*mapsMarkerIndexes*/);


    //! Updates the Jacobian matrix in StructualSolver::updateJacobian
    /*!
      \param disp: solution at the k-th iteration of NonLinearRichardson Method
      \param dataMaterial: a pointer to the dataType member in StructuralSolver class to get
                           the material coefficients (e.g. Young modulus, Poisson ratio..)
      \param displayer: a pointer to the Dysplaier member in the StructuralSolver class
    */
    void updateJacobianMatrix ( const vector_Type& disp,
                                const dataPtr_Type& dataMaterial,
                                const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                                const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                                const displayerPtr_Type& displayer );


    //! Updates the nonlinear terms in the Jacobian matrix in StructualSolver::updateJacobian
    /*!
      \param stiff: stiffness matrix provided from outside
      \param disp: solution at the k-th iteration of NonLinearRichardson Method
      \param dataMaterial: a pointer to the dataType member in StructuralSolver class to get
                           the material coefficients (e.g. Young modulus, Poisson ratio..)
      \param displayer: a pointer to the Dysplaier member in the StructuralSolver class
    */
    void updateNonLinearJacobianTerms ( matrixPtr_Type& jacobian,
                                        const vector_Type& disp,
                                        const dataPtr_Type& dataMaterial,
                                        const mapMarkerVolumesPtr_Type /*mapsMarkerVolumes*/,
                                        const mapMarkerIndexesPtr_Type /*mapsMarkerIndexes*/,
                                        const displayerPtr_Type& displayer );


    //! Interface method to compute the new Stiffness matrix in StructuralSolver::evalResidual and in
    //! StructuralSolver::updateSystem since the matrix is the expression of the matrix is the same.
    /*!
      \param sol:  the solution vector
      \param factor: scaling factor used in FSI
      \param dataMaterial: a pointer to the dataType member in StructuralSolver class to get the
                           material coefficients (e.g. Young modulus, Poisson ratio..)
      \param displayer: a pointer to the Dysplaier member in the StructuralSolver class
    */
    void computeStiffness ( const vector_Type& disp,
                            const UInt iter,
                            Real factor,
                            const dataPtr_Type& dataMaterial,
                            const mapMarkerVolumesPtr_Type /*mapsMarkerVolumes*/,
                            const mapMarkerIndexesPtr_Type /*mapsMarkerIndexes*/,
                            const displayerPtr_Type& displayer );


    void computeReferenceConfigurations ( const vector_Type& disp,
                      const dataPtr_Type& dataMaterial,
                      const displayerPtr_Type& displayer ){};

    //! Computes the new Stiffness vector for Neo-Hookean and Holzapfel materials in StructuralSolver
    //! given a certain displacement field.
    //! This function is used both in StructuralSolver::evalResidual and in StructuralSolver::updateSystem
    //! since the matrix is the expression of the matrix is the same.
    /*!
      \param sol:  the solution vector
      \param factor: scaling factor used in FSI
      \param dataMaterial: a pointer to the dataType member in StructuralSolver class to get
                           the material coefficients (e.g. Young modulus, Poisson ratio..)
      \param displayer: a pointer to the Dysplaier member in the StructuralSolver class
    */

    //! Computes the deformation gradient F, the cofactor matrix Cof(F),
    //! the determinant of F (J = det(F)), the trace of right Cauchy-Green tensor tr(C)
    //! This function is used in StructuralIsotropicConstitutiveLaw::computeStiffness
    /*!
      \param dk_loc: the elemental displacement
    */
    void computeKinematicsVariables ( const VectorElemental& dk_loc )
    {
    }

    //! Computes the deformation Gradient F, the cofactor of F Cof(F), the determinant of F J = det(F), the trace of C Tr(C).
    /*!
      \param dk_loc: local displacement vector
    */
    //void computeStress( const vector_Type& sol );

    //! ShowMe method of the class (saved on a file the stiffness vector and the jacobian)
    void showMe ( std::string const& fileNameVectStiff,
                  std::string const& fileNameJacobain );

    //! Compute the First Piola Kirchhoff Tensor
    /*!
       \param firstPiola Epetra_SerialDenseMatrix that has to be filled
       \param tensorF Epetra_SerialDenseMatrix the deformation gradient
       \param cofactorF Epetra_SerialDenseMatrix cofactor of F
       \param invariants std::vector with the invariants of C and the detF
       \param material UInt number to get the material parameteres form the VenantElasticData class
    */
    void computeLocalFirstPiolaKirchhoffTensor ( Epetra_SerialDenseMatrix& firstPiola,
                                                 const Epetra_SerialDenseMatrix& tensorF,
                                                 const Epetra_SerialDenseMatrix& cofactorF,
                                                 const std::vector<Real>& invariants,
                                                 const UInt marker);

    //! Compute the First Piola Kirchhoff Tensor
    /*!
       \param disp the displacement field from which we compute the fisrt piola-Kirchhoff tensor
       \param sigma_1 the first column of the Cauchy stress tensor
       \param sigma_2 the second column of the Cauchy stress tensor
       \param sigma_3 the third column of the Cauchy stress tensor
    */
    void computeCauchyStressTensor ( const vectorPtr_Type disp,
                     const QuadratureRule& evalQuad,
                     vectorPtr_Type sigma_1,
                     vectorPtr_Type sigma_2,
                     vectorPtr_Type sigma_3);


    void setupFiberDirections( vectorFiberFunctionPtr_Type vectorOfFibers );


    void evaluateActivationFibers( const vector_Type&  displacement,
                                   vector_Type&  fourthInvariant){}
    //@}

    //! @name Get Methods
    //@{

    //! Get the Stiffness matrix
    matrixPtr_Type  const stiffMatrix() const
    {
        return super::M_jacobian;
    }


    //! Get the stiffness vector
    vectorPtr_Type  const stiffVector() const
    {
        return M_stiff;
    }

    //! Specific for multimechanism
    vectorPtr_Type  const selectionCriterion( const UInt /*i*/ ) const
    {
      vectorPtr_Type empty( new vector_Type( this->M_dispFESpace->map() ) );
      return empty;
    }


    vectorPtr_Type  const activationDisplacement( const UInt /*i*/ ) const
    {
      vectorPtr_Type empty( new vector_Type( this->M_dispFESpace->map() ) );
      return empty;
    }

    vectorPtr_Type  const activatedUnitFiber( const UInt /*i*/ ) const
    {
      vectorPtr_Type empty( new vector_Type( this->M_dispFESpace->map() ) );
      return empty;
    }

    vectorPtr_Type  const activatedDeterminant( const UInt /*i*/ ) const
    {
      vectorPtr_Type empty( new vector_Type( this->M_dispFESpace->map() ) );
      return empty;
    }



    void apply ( const vector_Type& sol, vector_Type& res,
                 const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                 const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                 const displayerPtr_Type displayer) ;

    //@}



protected:

    //! construct the vectors for the parameters
    /*!
      \param VOID
      \return VOID
    */
    // void setupVectorsParameters ( void );


    //! Vector: stiffness non-linear
    vectorPtr_Type                         M_stiff;

    //Create the indentity for F
    matrixSmall_Type                      M_identity;

};





template <typename MeshType>
HolzapfelMaterialNonLinear<MeshType>::HolzapfelMaterialNonLinear() :
    super            ( ),
    M_stiff          ( ),
    M_identity       ( )
{
}





template <typename MeshType>
HolzapfelMaterialNonLinear<MeshType>::~HolzapfelMaterialNonLinear()
{}





template <typename MeshType>
void
HolzapfelMaterialNonLinear<MeshType>::setup ( const FESpacePtr_Type&                       dFESpace,
                          const ETFESpacePtr_Type&                     dETFESpace,
                          const std::shared_ptr<const MapEpetra>&   monolithicMap,
                          const UInt                                  offset,
                          const dataPtr_Type& dataMaterial)
{

    this->M_dataMaterial                = dataMaterial;
    this->M_dispFESpace                 = dFESpace;
    this->M_dispETFESpace               = dETFESpace;
    this->M_localMap                    = monolithicMap;
    this->M_offset                      = offset;

    M_stiff.reset                       (new vector_Type (*this->M_localMap) );

    M_identity (0, 0) = 1.0;
    M_identity (0, 1) = 0.0;
    M_identity (0, 2) = 0.0;
    M_identity (1, 0) = 0.0;
    M_identity (1, 1) = 1.0;
    M_identity (1, 2) = 0.0;
    M_identity (2, 0) = 0.0;
    M_identity (2, 1) = 0.0;
    M_identity (2, 2) = 1.0;

    this->M_epsilon = this->M_dataMaterial->smoothness();
    //this->setupVectorsParameters( );
}


// template <typename MeshType>
// void
// HolzapfelMaterialNonLinear<MeshType>::setupVectorsParameters ( void )
// {
//     // Paolo Tricerri: February, 20th
//     // In each class, the name of the parameters has to inserted in the law
//     // TODO: move the saving of the material parameters to more abstract objects
//     //       such that in the class of the material we do not need to call explicitly
//     //       the name of the parameter.

//     // Number of volume on the local part of the mesh
//     UInt nbElements = this->M_dispFESpace->mesh()->numVolumes();

//     // Parameter alpha
//     // 1. resize the vector in the first element of the vector.
//     (* (this->M_vectorsParameters) ) [0].resize ( nbElements );

//     // Parameter gamma
//     (* (this->M_vectorsParameters) ) [1].resize ( nbElements );

//     // Parameter bulk
//     (* (this->M_vectorsParameters) ) [2].resize ( nbElements );

//     for (UInt i (0); i < nbElements; i++ )
//     {
//         // Extracting the marker
//         UInt markerID = this->M_dispFESpace->mesh()->element ( i ).markerID();

//         Real alpha = this->M_dataMaterial->alpha ( markerID );
//         Real gamma = this->M_dataMaterial->gamma ( markerID );
//         Real bulk  = this->M_dataMaterial->bulk ( markerID );

//         ( (* (this->M_vectorsParameters) ) [0]) [ i ] = alpha;
//         ( (* (this->M_vectorsParameters) ) [1]) [ i ] = gamma;
//         ( (* (this->M_vectorsParameters) ) [2]) [ i ] = bulk;
//     }
// }


template <typename MeshType>
void
HolzapfelMaterialNonLinear<MeshType>::setupFiberDirections ( vectorFiberFunctionPtr_Type vectorOfFibers  )
{
    // Allocating the right space for the vector of fiber function
    this->M_vectorOfFibers.reset( new vectorFiberFunction_Type( ) );

    // In this method, the vector of fiber functions has to be properly set  in the main
    // of the test. The functions are then projected on a FESpace for the integration

    // Number of volume on the local part of the mesh
    UInt nbFamilies = (*vectorOfFibers).size();

    ASSERT( nbFamilies == this->M_dataMaterial->numberFibersFamilies(),
        " The number of families set in the test is different from the one in the data" );

    this->M_vectorOfFibers->resize( nbFamilies );

    for( UInt k(0); k < nbFamilies; k++ )
    {
        ( *(this->M_vectorOfFibers) )[ k ] = ( *vectorOfFibers )[ k ];
    }

    // Setting the vectors that will be used
    this->M_vectorInterpolated.resize( nbFamilies );

    for( UInt k(0); k < nbFamilies; k++ )
    {
        this->M_vectorInterpolated[ k ].reset( new vector_Type(*this->M_localMap) );
        this->M_dispFESpace->interpolate ( *( ( *(this->M_vectorOfFibers) )[ k ] ) ,
                                           * ( ( this->M_vectorInterpolated )[ k ] ),
                                           0.0 );
    }

}


template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::computeLinearStiff (dataPtr_Type& /*dataMaterial*/,
                                                                 const mapMarkerVolumesPtr_Type /*mapsMarkerVolumes*/,
                                                                 const mapMarkerIndexesPtr_Type /*mapsMarkerIndexes*/)
{
    //! Empty method for exponential material
}





template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::updateJacobianMatrix ( const vector_Type&       disp,
                                                                    const dataPtr_Type&      dataMaterial,
                                                                    const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                                                                    const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                                                                    const displayerPtr_Type& displayer )
{

    this->M_jacobian.reset (new matrix_Type (*this->M_localMap) );

    displayer->leaderPrint (" \n*********************************\n  ");
    updateNonLinearJacobianTerms (this->M_jacobian, disp, this->M_dataMaterial, mapsMarkerVolumes, mapsMarkerIndexes, displayer);
    displayer->leaderPrint (" \n*********************************\n  ");

}

template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::updateNonLinearJacobianTerms ( matrixPtr_Type&         jacobian,
                                                                          const vector_Type&     disp,
                                                                          const dataPtr_Type&     dataMaterial,
                                                                          const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                                                                          const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                                                                          const displayerPtr_Type& displayer )
{
    using namespace ExpressionAssembly;

    // In the following definitions, the critical template argument is MapEpetra
    // When other maps will be available in LifeV, the Holzapfel class and its mother
    // should have as template the MeshType and the MapType.

    // Definition of F
    tensorF_Type F = ExpressionDefinitions::deformationGradient( this->M_dispETFESpace,  disp, this->M_offset, this->M_identity );

    // Definition of J
    determinantF_Type J = ExpressionDefinitions::determinantF( F );

    //Definition of C
    tensorC_Type C = ExpressionDefinitions::tensorC( transpose(F), F );

    // Definition of J^-(2/3) = det( C ) using the isochoric/volumetric decomposition
    powerExpression_Type  Jel = ExpressionDefinitions::powerExpression( J , (-2.0/3.0) );

    // Definition of F^-T
    minusT_Type  F_T = ExpressionDefinitions::minusT( F );

    // Update the heaviside function for the stretch of the fibers
    * (jacobian) *= 0.0;

    displayer->leaderPrint ("   Non-Linear S - updating non linear terms in the Jacobian Matrix (N fibers Holzapfel): \n");

    for( UInt i(0); i < this->M_vectorInterpolated.size(); i++ )
    {

        displayer->leaderPrint ("                ", i + 1,"-th fiber family \n" );
        // Defining the expression for the i-th fiber
        // Definitions of the quantities which depend on the fiber directions e.g. I_4^i
        interpolatedValue_Type fiberIth = ExpressionDefinitions::interpolateFiber( this->M_dispETFESpace, *(this->M_vectorInterpolated[ i ] ) );

        // Definition of the tensor M = ithFiber \otimes ithFiber
        // At the moment, it's automatic that the method constructs the expression M = ithFiber \otimes ithFiber
        // For a more general case, the file ExpressionDefinitions.hpp should be changed
        outerProduct_Type Mith = ExpressionDefinitions::fiberTensor( fiberIth );

        // Definition of the fourth invariant : I_4^i = C:Mith
        stretch_Type IVith = ExpressionDefinitions::fiberStretch( C, Mith );

        // Definition of the fouth isochoric invariant : J^(-2.0/3.0) * I_4^i
        isochoricStretch_Type IVithBar = ExpressionDefinitions::isochoricFourthInvariant( Jel, IVith );

        // first term:
        // (-4.0/3.0) * aplha_i * J^(-2.0/3.0) * \bar{I_4} * ( \bar{I_4} - 1 ) * exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // (epsilon / PI) * ( 1/ (1 + epsilon^2 * ( \bar{I_4} - 1 )^2 ) ) * ( F^-T : dF ) ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( -4.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel *
                   IVithBar * ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   derAtan( IVithBar - value(1.0), this->M_epsilon, ( 1.0 / PI ) ) *
                   dot( F_T, grad(phi_j) ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;


        // second term
        // 2.0 * aplha_i * J^(-4.0/3.0) * ( \bar{I_4} - 1 ) * exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // (epsilon / PI) * ( 1/ (1 + epsilon^2 * ( \bar{I_4} - 1 )^2 ) ) * ( dF^T*F : M + F^T*dF:M ) ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) *
                   ( IVithBar - value(1.0) ) * Jel * Jel *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   derAtan( IVithBar - value(1.0), this->M_epsilon, ( 1.0 / PI ) ) *
                   dot( transpose( grad(phi_j) ) * F + transpose(F) * grad(phi_j) , Mith ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;


        // third term
        // ( -4.0/3.0 ) * alpha_i * J^(-2.0/3.0) * ( \bar{I_4} - 1 ) * exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( F^-T : dF) * ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( -4.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) *
                   Jel * ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( F_T , grad(phi_j) ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;

        // fourth term
        // ( -4.0/3.0 ) * alpha_i * J^(-2.0/3.0) * \bar{I_4} * exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( F^-T : dF) * ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( -4.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) *
                   Jel * IVithBar *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( F_T , grad(phi_j) ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;


        // fifth term
        // 2.0 * aplha_i * J^(-4.0/3.0) * exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( dF^T*F : M + F^T*dF:M ) ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * Jel *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( transpose( grad(phi_j) ) * F + transpose(F) * grad(phi_j) , Mith ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;

        // sixth term
        // (-8.0/3.0) * aplha_i * gamma_i * J^(-2.0/3.0) * \bar{I_4} * ( \bar{I_4} - 1.0 )^2 *  exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( F^-T:dF ) ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( -8.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) *
                   Jel * IVithBar * ( IVithBar - value(1.0) ) * ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( F_T , grad(phi_j) ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;


        // seventh term
        // 4.0 * aplha_i * gamma_i * J^(-4.0/3.0) * ( \bar{I_4} - 1.0 )^2 *  exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( dF^T*F : M + F^T*dF:M ) ( ( F * M - (1.0/3.0) * I_4 * F^-T ) : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( 4.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) *
                   Jel * Jel *  ( IVithBar - value(1.0) ) * ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( transpose( grad(phi_j) ) * F + transpose(F) * grad(phi_j) , Mith ) *
                   dot( F * Mith - value(1.0/3.0) * IVith * F_T, grad(phi_i) )
                   ) >> jacobian;

        // tenth term
        // 2.0 * aplha_i * J^(-2.0/3.0) * ( \bar{I_4} - 1.0 ) *  exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( dF*M : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel *  ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( grad(phi_j) * Mith, grad(phi_i) )
                   ) >> jacobian;

        // eleventh term
        // (2.0/3.0) * aplha_i * \bar{I_4} * ( \bar{I_4} - 1.0 ) *  exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( F^-T * dF^T * F^-T  : d\phi )
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( 2.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * IVithBar *  ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( F_T * transpose( grad(phi_j) ) * F_T, grad(phi_i) )
                   ) >> jacobian;


        // twelveth term
        // (-2.0/3.0) * aplha_i * J^(-2.0/3.0)  * ( \bar{I_4} - 1.0 ) *  exp( gamma_i * ( \bar{I_4} - 1 )^2 ) *
        // ( ( 1 / PI ) * atan(\epsilon(\bar{I_4} - 1)) + 1/2  ) * ( dF^T * F + F^T * dF  : Mith ) ( F^-T : d \phi)
        integrate( elements ( this->M_dispETFESpace->mesh() ),
                   this->M_dispFESpace->qr(),
                   this->M_dispETFESpace,
                   this->M_dispETFESpace,
                   value( -2.0/3.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel *  ( IVithBar - value(1.0) ) *
                   exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) ) ) *
                   atan( IVithBar - value(1.0), this->M_epsilon, ( 1 / PI ), (1.0/2.0) ) *
                   dot( transpose( grad(phi_j) ) * F + transpose(F) * grad(phi_j) , Mith ) *
                   dot( F_T , grad( phi_i ) )
                   ) >> jacobian;



    } // closing loop on fibers

    jacobian->globalAssemble();
}





template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::computeStiffness ( const vector_Type& disp,
                                  const UInt /*iter*/,
                                  Real /*factor*/,
                                  const dataPtr_Type& dataMaterial,
                                  const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                                  const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                                  const displayerPtr_Type& displayer )
{

    using namespace ExpressionAssembly;

    M_stiff.reset (new vector_Type (*this->M_localMap) );
    * (M_stiff) *= 0.0;

    displayer->leaderPrint (" \n*********************************\n  ");
    displayer->leaderPrint (" Non-Linear S-  Computing the Holzapfel nonlinear stiffness vector (N fibers Holzapfel)");
    displayer->leaderPrint (" \n*********************************\n  ");

    // For anisotropic part of the Piola-Kirchhoff is assemble summing up the parts of the
    // Piola-Kirchhoff using the fiber index

    // Here the fiber vector at the quadrature node is deduced using the method
    // Interpolate value of the expression template.

    // Defining quantities which depend of the displacement field and not on the fiber direction

    // Definition of F
    tensorF_Type F = ExpressionDefinitions::deformationGradient( this->M_dispETFESpace,  disp, this->M_offset, this->M_identity );

    // Definition of J
    determinantF_Type J = ExpressionDefinitions::determinantF( F );

    //Definition of C
    tensorC_Type C = ExpressionDefinitions::tensorC( transpose(F), F );

    // Definition of J^-(2/3) = det( C ) using the isochoric/volumetric decomposition
    powerExpression_Type  Jel = ExpressionDefinitions::powerExpression( J , (-2.0/3.0) );

    // Definition of F^-T
    minusT_Type  F_T = ExpressionDefinitions::minusT( F );

    for( UInt i(0); i < this->M_vectorInterpolated.size() ; i++ )
      {
          // As in other classes, the specialization of the MapType = MapEpetra makes this expression
          // not always usable. When other maps will be available in LifeV, the class should be re-templated.

          // Defining the expression for the i-th fiber
          // Definitions of the quantities which depend on the fiber directions e.g. I_4^i
          interpolatedValue_Type fiberIth = ExpressionDefinitions::interpolateFiber( this->M_dispETFESpace, *(this->M_vectorInterpolated[ i ] ) );

      // Definition of the tensor M = ithFiber \otimes ithFiber
      // At the moment, it's automatic that the method constructs the expression M = ithFiber \otimes ithFiber
      // For a more general case, the file ExpressionDefinitions.hpp should be changed
      outerProduct_Type Mith = ExpressionDefinitions::fiberTensor( fiberIth );

          // Definition of the fourth invariant : I_4^i = C:Mith
      stretch_Type IVith = ExpressionDefinitions::fiberStretch( C, Mith );

          // Definition of the fouth isochoric invariant : J^(-2.0/3.0) * I_4^i
      isochoricStretch_Type IVithBar = ExpressionDefinitions::isochoricFourthInvariant( Jel, IVith );

          // First term:
          // 2 alpha_i J^(-2.0/3.0) ( \bar{I_4} - 1 ) exp( gamma_i * (\bar{I_4} - 1)^2 ) * F M : \grad phi_i
          // where alpha_i and gamma_i are the fiber parameters and M is the 2nd order tensor of type f_i \otimes \ f_i
          integrate ( elements ( this->M_dispETFESpace->mesh() ),
                      this->M_dispFESpace->qr(),
                      this->M_dispETFESpace,
                      atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                      value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                      exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                      dot( F * Mith, grad( phi_i ) )
                      ) >> this->M_stiff;

          // Second term:
          // 2 alpha_i J^(-2.0/3.0) ( \bar{I_4} - 1 ) exp( gamma_i * (\bar{I_4} - 1)^2 ) * ( 1.0/3.0 * I_4 ) F^-T : \grad phi_i
          // where alpha_i and gamma_i are the fiber parameters and M is the 2nd order tensor of type f_i \otimes \ f_i
          integrate ( elements ( this->M_dispETFESpace->mesh() ),
                      this->M_dispFESpace->qr(),
                      this->M_dispETFESpace,
                      atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                      value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                      exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                      value( -1.0/3.0 ) * IVith *
                      dot( F_T , grad( phi_i ) )
            ) >> this->M_stiff;


      }

    this->M_stiff->globalAssemble();
}


template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::showMe ( std::string const& fileNameStiff,
                                                      std::string const& fileNameJacobian )
{
    this->M_stiff->spy (fileNameStiff);
    this->M_jacobian->spy (fileNameJacobian);
}


template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::apply ( const vector_Type& sol,
                                                   vector_Type& res, const mapMarkerVolumesPtr_Type mapsMarkerVolumes,
                                                   const mapMarkerIndexesPtr_Type mapsMarkerIndexes,
                                                   const displayerPtr_Type displayer)
{
    computeStiffness (sol, 0, 0, this->M_dataMaterial, mapsMarkerVolumes, mapsMarkerIndexes, displayer);
    res += *M_stiff;
}


template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::computeLocalFirstPiolaKirchhoffTensor ( Epetra_SerialDenseMatrix& firstPiola,
                                                                                     const Epetra_SerialDenseMatrix& tensorF,
                                                                                     const Epetra_SerialDenseMatrix& cofactorF,
                                                                                     const std::vector<Real>& invariants,
                                                                                     const UInt marker)
{

  // Still Need to Define P

}

template <typename MeshType>
void HolzapfelMaterialNonLinear<MeshType>::computeCauchyStressTensor ( const vectorPtr_Type disp,
                                                                       const QuadratureRule& evalQuad,
                                                                       vectorPtr_Type sigma_1,
                                                                       vectorPtr_Type sigma_2,
                                                                       vectorPtr_Type sigma_3)
{

    using namespace ExpressionAssembly;

    // Definition of F
    tensorF_Type F = ExpressionDefinitions::deformationGradient( this->M_dispETFESpace,  *disp, this->M_offset, this->M_identity );

    // Definition of J
    determinantF_Type J = ExpressionDefinitions::determinantF( F );

    //Definition of C
    tensorC_Type C = ExpressionDefinitions::tensorC( transpose(F), F );

    // Definition of J^-(2/3) = det( C ) using the isochoric/volumetric decomposition
    powerExpression_Type  Jel = ExpressionDefinitions::powerExpression( J , (-2.0/3.0) );

    // Definition of F^-T
    minusT_Type  F_T = ExpressionDefinitions::minusT( F );

    for( UInt i(0); i < this->M_vectorInterpolated.size() ; i++ )
    {
          // As in other classes, the specialization of the MapType = MapEpetra makes this expression
          // not always usable. When other maps will be available in LifeV, the class should be re-templated.

          // Defining the expression for the i-th fiber
          // Definitions of the quantities which depend on the fiber directions e.g. I_4^i
          interpolatedValue_Type fiberIth = ExpressionDefinitions::interpolateFiber( this->M_dispETFESpace, *(this->M_vectorInterpolated[ i ] ) );

          // Definition of the tensor M = ithFiber \otimes ithFiber
          // At the moment, it's automatic that the method constructs the expression M = ithFiber \otimes ithFiber
          // For a more general case, the file ExpressionDefinitions.hpp should be changed
          outerProduct_Type Mith = ExpressionDefinitions::fiberTensor( fiberIth );

          // Definition of the fourth invariant : I_4^i = C:Mith
          stretch_Type IVith = ExpressionDefinitions::fiberStretch( C, Mith );

          // Definition of the fouth isochoric invariant : J^(-2.0/3.0) * I_4^i
          isochoricStretch_Type IVithBar = ExpressionDefinitions::isochoricFourthInvariant( Jel, IVith );

          // First term:
          // 2 alpha_i J^(-2.0/3.0) ( \bar{I_4} - 1 ) exp( gamma_i * (\bar{I_4} - 1)^2 ) * F M : \grad phi_i
          // where alpha_i and gamma_i are the fiber parameters and M is the 2nd order tensor of type f_i \otimes \ f_i

          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          F * Mith ) *
                                                        transpose(F) , 0 ), phi_i )
                        ) >> sigma_1;

          // Second term:
          // 2 alpha_i J^(-2.0/3.0) ( \bar{I_4} - 1 ) exp( gamma_i * (\bar{I_4} - 1)^2 ) * ( 1.0/3.0 * I_4 ) F^-T : \grad phi_i
          // where alpha_i and gamma_i are the fiber parameters and M is the 2nd order tensor of type f_i \otimes \ f_i
          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          value( -1.0/3.0 ) * IVith *
                                                          F_T ) *
                                                        transpose(F) , 0 ), phi_i )
                        ) >> sigma_1;
          sigma_1->globalAssemble();

          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          F * Mith ) *
                                                        transpose(F) , 1 ), phi_i )
                        ) >> sigma_2;

          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          value( -1.0/3.0 ) * IVith *
                                                          F_T ) *
                                                        transpose(F) , 1 ), phi_i )
                        ) >> sigma_2;
          sigma_2->globalAssemble();

          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          F * Mith ) *
                                                        transpose(F) ,  2 ), phi_i )
                        ) >> sigma_3;

          evaluateNode( elements( this->M_dispETFESpace->mesh() ),
                        evalQuad,
                        this->M_dispETFESpace,
                        meas_K * dot( vectorFromMatrix( ( 1 / J ) *
                                                        ( atan( IVithBar - value(1.0) , this->M_epsilon, ( 1 / PI ), ( 1.0/2.0 )  ) *
                                                          value( 2.0 ) * value( this->M_dataMaterial->ithStiffnessFibers( i ) ) * Jel * ( IVithBar - value(1.0) ) *
                                                          exp( value( this->M_dataMaterial->ithNonlinearityFibers( i ) ) * ( IVithBar- value(1.0) ) * ( IVithBar- value(1.0) )  ) *
                                                          value( -1.0/3.0 ) * IVith *
                                                          F_T ) *
                                                        transpose(F) , 2 ), phi_i )
                        ) >> sigma_3;
          sigma_3->globalAssemble();
    }

}


template <typename MeshType>
inline StructuralAnisotropicConstitutiveLaw<MeshType>* createHolzapfelMaterialNonLinear()
{
    return new HolzapfelMaterialNonLinear<MeshType >();
}

namespace
{
static bool registerHL = StructuralAnisotropicConstitutiveLaw<LifeV::RegionMesh<LinearTetra> >::StructureAnisotropicMaterialFactory::instance().registerProduct ( "holzapfel", &createHolzapfelMaterialNonLinear<LifeV::RegionMesh<LinearTetra> > );
}

} //Namespace LifeV

#endif /* __EXPONENTIALMATERIAL_H */