elasticity.cc

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/*!
 *
 * Copyright (C) 2015 Technical University of Liberec.  All rights reserved.
 * 
 * This program is free software; you can redistribute it and/or modify it under
 * the terms of the GNU General Public License version 3 as published by the
 * Free Software Foundation. (http://www.gnu.org/licenses/gpl-3.0.en.html)
 * 
 * This program is distributed in the hope that it will be useful, but WITHOUT
 * ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS
 * FOR A PARTICULAR PURPOSE.  See the GNU General Public License for more details.
 *
 * 
 * @file    transport_dg.cc
 * @brief   Discontinuous Galerkin method for equation of transport with dispersion.
 * @author  Jan Stebel
 */

#include "system/sys_profiler.hh"
#include "mechanics/elasticity.hh"

#include "io/output_time.hh"
#include "quadrature/quadrature_lib.hh"
#include "fem/mapping_p1.hh"
#include "fem/fe_values.hh"
#include "fem/fe_p.hh"
#include "fem/fe_rt.hh"
#include "fem/fe_system.hh"
#include "fields/field_fe.hh"
#include "la/linsys_PETSC.hh"
#include "coupling/balance.hh"
#include "mesh/neighbours.h"

#include "fields/multi_field.hh"
#include "fields/generic_field.hh"
#include "input/factory.hh"




using namespace Input::Type;


const Record & Elasticity::get_input_type() {
    std::string equation_name = std::string(Elasticity::EqData::name()) + "_FE";
    return IT::Record(
                std::string(equation_name),
                "FEM for linear elasticity.")
           .declare_key("time", TimeGovernor::get_input_type(), Default::obligatory(),
                    "Time governor setting for the secondary equation.")
           .declare_key("balance", Balance::get_input_type(), Default("{}"),
                    "Settings for computing balance.")
           .declare_key("output_stream", OutputTime::get_input_type(), Default::obligatory(),
                    "Parameters of output stream.")
           .declare_key("solver", LinSys_PETSC::get_input_type(), Default::obligatory(),
                "Linear solver for elasticity.")
           .declare_key("input_fields", Array(
                Elasticity::EqData()
                    .make_field_descriptor_type(equation_name)),
                IT::Default::obligatory(),
                "Input fields of the equation.")
           .declare_key("output",
                EqData().output_fields.make_output_type(equation_name, ""),
                IT::Default("{ \"fields\": [ "+Elasticity::EqData::default_output_field()+" ] }"),
                "Setting of the field output.")
           .close();
}

const int Elasticity::registrar =
        Input::register_class< Elasticity, Mesh &, const Input::Record>(std::string(Elasticity::EqData::name()) + "_FE") +
        Elasticity::get_input_type().size();



namespace Mechanics {

FEObjects::FEObjects(Mesh *mesh_, unsigned int fe_order)
{
    unsigned int q_order;

    switch (fe_order)
    {
    case 1:
        q_order = 2;
        fe0_ = new FESystem<0>(std::make_shared<FE_P<0> >(1), FEVector, 3);
        fe1_ = new FESystem<1>(std::make_shared<FE_P<1> >(1), FEVector, 3);
        fe2_ = new FESystem<2>(std::make_shared<FE_P<2> >(1), FEVector, 3);
        fe3_ = new FESystem<3>(std::make_shared<FE_P<3> >(1), FEVector, 3);
        break;

    default:
        q_order=0;
        xprintf(PrgErr, "Unsupported polynomial order %d for finite elements in Elasticity", fe_order);
        break;
    }

    for (unsigned int dim = 0; dim < 4; dim++) q_[dim] = new QGauss(dim, q_order);

    map1_ = new MappingP1<1,3>;
    map2_ = new MappingP1<2,3>;
    map3_ = new MappingP1<3,3>;

    ds_ = std::make_shared<EqualOrderDiscreteSpace>(mesh_, fe0_, fe1_, fe2_, fe3_);
    dh_ = std::make_shared<DOFHandlerMultiDim>(*mesh_);

    dh_->distribute_dofs(ds_);
}


FEObjects::~FEObjects()
{
    delete fe1_;
    delete fe2_;
    delete fe3_;
    for (unsigned int dim=0; dim < 4; dim++) delete q_[dim];
    delete map1_;
    delete map2_;
    delete map3_;
}

template<> FiniteElement<0> *FEObjects::fe<0>() { return 0; }
template<> FiniteElement<1> *FEObjects::fe<1>() { return fe1_; }
template<> FiniteElement<2> *FEObjects::fe<2>() { return fe2_; }
template<> FiniteElement<3> *FEObjects::fe<3>() { return fe3_; }

template<> MappingP1<1,3> *FEObjects::mapping<1>() { return map1_; }
template<> MappingP1<2,3> *FEObjects::mapping<2>() { return map2_; }
template<> MappingP1<3,3> *FEObjects::mapping<3>() { return map3_; }

std::shared_ptr<DOFHandlerMultiDim> FEObjects::dh() { return dh_; }



} // namespace Mechanics







const Selection & Elasticity::EqData::get_bc_type_selection() {
    return Selection("Elasticity_BC_Type", "Types of boundary conditions for heat transfer model.")
            .add_value(bc_type_displacement, "displacement",
                  "Prescribed displacement.")
            .add_value(bc_type_traction, "traction",
                  "Prescribed traction.")
            .close();
}


Elasticity::EqData::EqData()
{
    *this+=bc_type
        .name("bc_type")
        .description(
        "Type of boundary condition.")
        .units( UnitSI::dimensionless() )
        .input_default("\"traction\"")
        .input_selection( get_bc_type_selection() )
        .flags_add(FieldFlag::in_rhs & FieldFlag::in_main_matrix);
        
    *this+=bc_displacement
        .name("bc_displacement")
        .description("Prescribed displacement.")
        .units( UnitSI().m() )
        .input_default("0.0")
        .flags_add(in_rhs);
        
    *this+=bc_traction
        .name("bc_traction")
        .description("Prescribed traction.")
        .units( UnitSI().Pa() )
        .input_default("0.0")
        .flags_add(in_rhs);
        
    *this+=load
        .name("load")
        .description("Prescribed load.")
        .units( UnitSI().N().m(-3) )
        .input_default("0.0")
        .flags_add(in_rhs);
    
    *this+=young_modulus
        .name("young_modulus")
        .description("Young modulus.")
        .units( UnitSI().Pa() )
         .input_default("0.0")
        .flags_add(in_main_matrix);
    
    *this+=poisson_ratio
        .name("poisson_ratio")
        .description("Poisson ratio.")
        .units( UnitSI().dimensionless() )
         .input_default("0.0")
        .flags_add(in_main_matrix);
        
    *this+=fracture_sigma
            .name("fracture_sigma")
            .description(
            "Coefficient of diffusive transfer through fractures (for each substance).")
            .units( UnitSI::dimensionless() )
            .input_default("1.0")
            .flags_add(FieldFlag::in_main_matrix);

    *this += region_id.name("region_id")
                .units( UnitSI::dimensionless())
                .flags(FieldFlag::equation_external_output);
                
    *this += subdomain.name("subdomain")
      .units( UnitSI::dimensionless() )
      .flags(FieldFlag::equation_external_output);
      
    *this+=cross_section
      .name("cross_section")
      .units( UnitSI().m(3).md() )
      .flags(input_copy & in_time_term & in_main_matrix);

    *this+=output_field
            .name("displacement")
            .units( UnitSI().m() )
            .flags(equation_result);


    // add all input fields to the output list
    output_fields += *this;

}

Elasticity::Elasticity(Mesh & init_mesh, const Input::Record in_rec)
        : EquationBase(init_mesh, in_rec),
          input_rec(in_rec),
          allocation_done(false)
{
    // Can not use name() + "constructor" here, since START_TIMER only accepts const char *
    // due to constexpr optimization.
    START_TIMER(EqData::name());

    this->eq_data_ = &data_;
    time_ = new TimeGovernor(in_rec.val<Input::Record>("time"));


    // Set up physical parameters.
    data_.set_mesh(init_mesh);
    data_.region_id = GenericField<3>::region_id(*mesh_);
    data_.subdomain = GenericField<3>::subdomain(*mesh_);
    
    // create finite element structures and distribute DOFs
    feo = new Mechanics::FEObjects(mesh_, 1);
    DebugOut().fmt("Mechanics: solution size {}\n", feo->dh()->n_global_dofs());
    
}


void Elasticity::initialize()
{
    output_stream_ = OutputTime::create_output_stream("mechanics", input_rec.val<Input::Record>("output_stream"), time().get_unit_string());
    
    data_.set_components({"displacement"});
    data_.set_input_list( input_rec.val<Input::Array>("input_fields"), time() );
    
//     balance_ = std::make_shared<Balance>("mechanics", mesh_);
//     balance_->init_from_input(input_rec.val<Input::Record>("balance"), *time_);
    // initialization of balance object
//     data_.balance_idx_ = {
//         balance_->add_quantity("force_x"),
//         balance_->add_quantity("force_y"),
//         balance_->add_quantity("force_z")
//     };
//     balance_->units(UnitSI().kg().m().s(-2));

    // create shared pointer to a FieldFE, pass FE data and push this FieldFE to output_field on all regions
    std::shared_ptr<FieldFE<3, FieldValue<3>::VectorFixed> > output_field_ptr(new FieldFE<3, FieldValue<3>::VectorFixed>);
    output_vec = output_field_ptr->set_fe_data(feo->dh());
    data_.output_field.set_field(mesh_->region_db().get_region_set("ALL"), output_field_ptr, 0.);
    data_.output_type(OutputTime::CORNER_DATA);

    // set time marks for writing the output
    data_.output_fields.initialize(output_stream_, mesh_, input_rec.val<Input::Record>("output"), this->time());

    // equation default PETSc solver options
    std::string petsc_default_opts;
    petsc_default_opts = "-ksp_type cg -pc_type hypre -pc_hypre_type boomeramg";
    
    // allocate matrix and vector structures
    ls = new LinSys_PETSC(feo->dh()->distr().get(), petsc_default_opts);
    ( (LinSys_PETSC *)ls )->set_from_input( input_rec.val<Input::Record>("solver") );
    ls->set_solution(output_vec.petsc_vec());

    // initialization of balance object
//     balance_->allocate(feo->dh()->distr()->lsize(),
//             max(feo->fe<1>()->n_dofs(), max(feo->fe<2>()->n_dofs(), feo->fe<3>()->n_dofs())));

}


Elasticity::~Elasticity()
{
//     delete time_;

    MatDestroy(&stiffness_matrix);
    VecDestroy(&rhs);
    delete ls;
    delete feo;

}



void Elasticity::output_vector_gather()
{
    VecScatter output_scatter;
    VecScatterCreateToZero(ls->get_solution(), &output_scatter, PETSC_NULL);
    // gather solution to output_vec
    VecScatterBegin(output_scatter, ls->get_solution(), output_vec.petsc_vec(), INSERT_VALUES, SCATTER_FORWARD);
    VecScatterEnd(output_scatter, ls->get_solution(), output_vec.petsc_vec(), INSERT_VALUES, SCATTER_FORWARD);
    VecScatterDestroy(&(output_scatter));
}




void Elasticity::zero_time_step()
{
    START_TIMER(EqData::name());
    data_.mark_input_times( *time_ );
    data_.set_time(time_->step(), LimitSide::right);
    std::stringstream ss; // print warning message with table of uninitialized fields
    if ( FieldCommon::print_message_table(ss, "mechanics") ) {
        WarningOut() << ss.str();
    }

    ( (LinSys_PETSC *)ls )->set_initial_guess_nonzero();

    // check first time assembly - needs preallocation
    if (!allocation_done) preallocate();
    

    // after preallocation we assemble the matrices and vectors required for balance of forces
//     for (auto subst_idx : data_.balance_idx_)
//         balance_->calculate_instant(subst_idx, ls->get_solution());
    
//     update_solution();
    ls->start_add_assembly();
    MatSetOption(*ls->get_matrix(), MAT_KEEP_NONZERO_PATTERN, PETSC_TRUE);
    ls->mat_zero_entries();
    assemble_stiffness_matrix();
    set_sources();
    set_boundary_conditions();
    ls->finish_assembly();
    ls->apply_constrains(1.0);
    ls->solve();
    output_data();
}



void Elasticity::preallocate()
{
    // preallocate system matrix
    ls->start_allocation();
    stiffness_matrix = NULL;
    rhs = NULL;

    assemble_stiffness_matrix();
    set_sources();
    set_boundary_conditions();

    allocation_done = true;
}




void Elasticity::update_solution()
{
    START_TIMER("DG-ONE STEP");

    time_->next_time();
    time_->view("MECH");
    
    START_TIMER("data reinit");
    data_.set_time(time_->step(), LimitSide::right);
    END_TIMER("data reinit");
    
    // assemble stiffness matrix
    if (stiffness_matrix == NULL
            || data_.subset(FieldFlag::in_main_matrix).changed()
            || flux_changed)
    {
        ls->start_add_assembly();
        ls->mat_zero_entries();
        assemble_stiffness_matrix();
        ls->finish_assembly();

        if (stiffness_matrix == NULL)
            MatConvert(*( ls->get_matrix() ), MATSAME, MAT_INITIAL_MATRIX, &stiffness_matrix);
        else
            MatCopy(*( ls->get_matrix() ), stiffness_matrix, DIFFERENT_NONZERO_PATTERN);
    }

    // assemble right hand side (due to sources and boundary conditions)
    if (rhs == NULL
            || data_.subset(FieldFlag::in_rhs).changed()
            || flux_changed)
    {
        ls->start_add_assembly();
        ls->rhs_zero_entries();
        set_sources();
        set_boundary_conditions();
        ls->finish_assembly();
        ls->apply_constrains(1.0);

        if (rhs == nullptr) VecDuplicate(*( ls->get_rhs() ), &rhs);
        VecCopy(*( ls->get_rhs() ), rhs);
    }

    flux_changed = false;


    START_TIMER("solve");
    ls->solve();
    END_TIMER("solve");

    calculate_cumulative_balance();
    
    output_data();

    END_TIMER("DG-ONE STEP");
}




void Elasticity::output_data()
{
    START_TIMER("MECH-OUTPUT");

    // gather the solution from all processors
    data_.output_fields.set_time( this->time().step(), LimitSide::left);
    //if (data_.output_fields.is_field_output_time(data_.output_field, this->time().step()) )
    output_vector_gather();
    data_.output_fields.output(this->time().step());
    output_stream_->write_time_frame();

//     START_TIMER("MECH-balance");
//     balance_->calculate_instant(subst_idx, ls->get_solution());
//     balance_->output();
//     END_TIMER("MECH-balance");

    END_TIMER("MECH-OUTPUT");
}



void Elasticity::calculate_cumulative_balance()
{
//     if (balance_->cumulative())
//     {
//         balance_->calculate_cumulative(subst_idx, ls->get_solution());
//     }
}








void Elasticity::assemble_stiffness_matrix()
{
  START_TIMER("assemble_stiffness");
   START_TIMER("assemble_volume_integrals");
    assemble_volume_integrals<1>();
    assemble_volume_integrals<2>();
    assemble_volume_integrals<3>();
   END_TIMER("assemble_volume_integrals");
   
  START_TIMER("assemble_fluxes_boundary");
    assemble_fluxes_boundary<1>();
    assemble_fluxes_boundary<2>();
    assemble_fluxes_boundary<3>();
   END_TIMER("assemble_fluxes_boundary");

   START_TIMER("assemble_fluxes_elem_side");
    assemble_fluxes_element_side<1>();
    assemble_fluxes_element_side<2>();
    assemble_fluxes_element_side<3>();
   END_TIMER("assemble_fluxes_elem_side");
  END_TIMER("assemble_stiffness");
}


double lame_mu(double young, double poisson)
{
    return young*0.5/(poisson+1.);
}


double lame_lambda(double young, double poisson)
{
    return young*poisson/((poisson+1.)*(1.-2.*poisson));
}


template<unsigned int dim>
void Elasticity::assemble_volume_integrals()
{
    FEValues<dim,3> fe_values(*feo->mapping<dim>(), *feo->q<dim>(), *feo->fe<dim>(),
            update_values | update_gradients | update_JxW_values | update_quadrature_points);
    const unsigned int ndofs = feo->fe<dim>()->n_dofs(), qsize = feo->q<dim>()->size();
    vector<int> dof_indices(ndofs);
    vector<arma::vec3> velocity(qsize);
    vector<double> young(qsize), poisson(qsize), csection(qsize);
    PetscScalar local_matrix[ndofs*ndofs];
    auto vec = fe_values.vector_view(0);

    // assemble integral over elements
    for (auto cell : feo->dh()->own_range())
    {
        if (cell.dim() != dim) continue;
        ElementAccessor<3> elm_acc = cell.elm();

        fe_values.reinit(elm_acc);
        cell.get_dof_indices(dof_indices);

        data_.cross_section.value_list(fe_values.point_list(), elm_acc, csection);
        data_.young_modulus.value_list(fe_values.point_list(), elm_acc, young);
        data_.poisson_ratio.value_list(fe_values.point_list(), elm_acc, poisson);
        
        // assemble the local stiffness matrix
        for (unsigned int i=0; i<ndofs; i++)
            for (unsigned int j=0; j<ndofs; j++)
                local_matrix[i*ndofs+j] = 0;

        for (unsigned int k=0; k<qsize; k++)
        {
            double mu = lame_mu(young[k], poisson[k]);
            double lambda = lame_lambda(young[k], poisson[k]);
            for (unsigned int i=0; i<ndofs; i++)
            {
                for (unsigned int j=0; j<ndofs; j++)
                    local_matrix[i*ndofs+j] += csection[k]*(
                                                2*mu*arma::dot(vec.sym_grad(j,k), vec.sym_grad(i,k))
                                                + lambda*vec.divergence(j,k)*vec.divergence(i,k)
                                               )*fe_values.JxW(k);
            }
        }
        ls->mat_set_values(ndofs, dof_indices.data(), ndofs, dof_indices.data(), local_matrix);
    }
}



void Elasticity::set_sources()
{
  START_TIMER("assemble_sources");
//     balance_->start_source_assembly(subst_idx);
    set_sources<1>();
    set_sources<2>();
    set_sources<3>();
//  balance_->finish_source_assembly(subst_idx);
  END_TIMER("assemble_sources");
}


template<unsigned int dim>
void Elasticity::set_sources()
{
    FEValues<dim,3> fe_values(*feo->mapping<dim>(), *feo->q<dim>(), *feo->fe<dim>(),
            update_values | update_gradients | update_JxW_values | update_quadrature_points);
    const unsigned int ndofs = feo->fe<dim>()->n_dofs(), qsize = feo->q<dim>()->size();
    vector<arma::vec3> load(qsize);
    vector<double> csection(qsize);
    vector<int> dof_indices(ndofs);
    PetscScalar local_rhs[ndofs];
    vector<PetscScalar> local_source_balance_vector(ndofs), local_source_balance_rhs(ndofs);
    auto vec = fe_values.vector_view(0);

    // assemble integral over elements
    for (auto cell : feo->dh()->own_range())
    {
        if (cell.dim() != dim) continue;
        ElementAccessor<3> elm_acc = cell.elm();

        fe_values.reinit(elm_acc);
        cell.get_dof_indices(dof_indices);

        // assemble the local stiffness matrix
        fill_n(local_rhs, ndofs, 0);
        local_source_balance_vector.assign(ndofs, 0);
        local_source_balance_rhs.assign(ndofs, 0);
        
        data_.cross_section.value_list(fe_values.point_list(), elm_acc, csection);
        data_.load.value_list(fe_values.point_list(), elm_acc, load);

        // compute sources
        for (unsigned int k=0; k<qsize; k++)
        {
            for (unsigned int i=0; i<ndofs; i++)
                local_rhs[i] += arma::dot(load[k], vec.value(i,k))*csection[k]*fe_values.JxW(k);
        }
        ls->rhs_set_values(ndofs, &(dof_indices[0]), local_rhs);

//         for (unsigned int i=0; i<ndofs; i++)
//         {
//             for (unsigned int k=0; k<qsize; k++)
//                 local_source_balance_vector[i] -= 0;//sources_sigma[k]*fe_values[vec].value(i,k)*fe_values.JxW(k);
// 
//             local_source_balance_rhs[i] += local_rhs[i];
//         }
//         balance_->add_source_matrix_values(subst_idx, elm_acc.region().bulk_idx(), dof_indices, local_source_balance_vector);
//         balance_->add_source_vec_values(subst_idx, elm_acc.region().bulk_idx(), dof_indices, local_source_balance_rhs);
    }
}







template<unsigned int dim>
void Elasticity::assemble_fluxes_boundary()
{
    FESideValues<dim,3> fe_values_side(*feo->mapping<dim>(), *feo->q<dim-1>(), *feo->fe<dim>(),
            update_values | update_gradients | update_side_JxW_values | update_normal_vectors | update_quadrature_points);
    const unsigned int ndofs = feo->fe<dim>()->n_dofs();
    vector<int> side_dof_indices(ndofs);
    PetscScalar local_matrix[ndofs*ndofs];

    // assemble boundary integral
    for (unsigned int iedg=0; iedg<feo->dh()->n_loc_edges(); iedg++)
    {
        Edge *edg = &mesh_->edges[feo->dh()->edge_index(iedg)];
        if (edg->n_sides > 1) continue;
        // check spatial dimension
        if (edg->side(0)->dim() != dim-1) continue;
        // skip edges lying not on the boundary
        if (edg->side(0)->cond() == NULL) continue;

        SideIter side = edg->side(0);
        ElementAccessor<3> cell = side->element();
        feo->dh()->cell_accessor_from_element(cell.idx()).get_dof_indices(side_dof_indices);
        fe_values_side.reinit(cell, side->side_idx());
//         unsigned int bc_type = data_.bc_type.value(side->centre(), side->cond()->element_accessor());
 
//         for (unsigned int i=0; i<ndofs; i++)
//             for (unsigned int j=0; j<ndofs; j++)
//                 local_matrix[i*ndofs+j] = 0;
//         
//         ls->mat_set_values(ndofs, side_dof_indices.data(), ndofs, side_dof_indices.data(), local_matrix);
    }
}


arma::mat33 mat_t(const arma::mat33 &m, const arma::vec3 &n)
{
  arma::mat33 mt = m - m*arma::kron(n,n.t());
  return mt;
}


template<unsigned int dim>
void Elasticity::assemble_fluxes_element_side()
{
    if (dim == 1) return;
    FEValues<dim-1,3> fe_values_sub(*feo->mapping<dim-1>(), *feo->q<dim-1>(), *feo->fe<dim-1>(),
            update_values | update_gradients | update_JxW_values | update_quadrature_points);
    FESideValues<dim,3> fe_values_side(*feo->mapping<dim>(), *feo->q<dim-1>(), *feo->fe<dim>(),
            update_values | update_gradients | update_side_JxW_values | update_normal_vectors | update_quadrature_points);
 
    vector<FEValuesSpaceBase<3>*> fv_sb(2);
    const unsigned int ndofs_side = feo->fe<dim>()->n_dofs();    // number of local dofs
    const unsigned int ndofs_sub  = feo->fe<dim-1>()->n_dofs();
    const unsigned int qsize = feo->q<dim-1>()->size();     // number of quadrature points
    vector<vector<int> > side_dof_indices(2);
    vector<unsigned int> n_dofs = { ndofs_sub, ndofs_side };
    vector<double> frac_sigma(qsize);
    vector<double> csection_lower(qsize), csection_higher(qsize), young(qsize), poisson(qsize), alpha(qsize);
    PetscScalar local_matrix[2][2][(ndofs_side)*(ndofs_side)];
    PetscScalar local_rhs[2][ndofs_side];
    auto vec_side = fe_values_side.vector_view(0);
    auto vec_sub = fe_values_sub.vector_view(0);

    // index 0 = element with lower dimension,
    // index 1 = side of element with higher dimension
    side_dof_indices[0].resize(ndofs_sub);
    side_dof_indices[1].resize(ndofs_side);
    fv_sb[0] = &fe_values_sub;
    fv_sb[1] = &fe_values_side;

    // assemble integral over sides
    for (unsigned int inb=0; inb<feo->dh()->n_loc_nb(); inb++)
    {
        Neighbour *nb = &mesh_->vb_neighbours_[feo->dh()->nb_index(inb)];
        // skip neighbours of different dimension
        if (nb->element()->dim() != dim-1) continue;

        ElementAccessor<3> cell_sub = nb->element();
        feo->dh()->cell_accessor_from_element(cell_sub.idx()).get_dof_indices(side_dof_indices[0]);
        fe_values_sub.reinit(cell_sub);

        ElementAccessor<3> cell = nb->side()->element();
        feo->dh()->cell_accessor_from_element(cell.idx()).get_dof_indices(side_dof_indices[1]);
        fe_values_side.reinit(cell, nb->side()->side_idx());

        // Element id's for testing if they belong to local partition.
        bool own_element_id[2];
        own_element_id[0] = feo->dh()->cell_accessor_from_element(cell_sub.idx()).is_own();
        own_element_id[1] = feo->dh()->cell_accessor_from_element(cell.idx()).is_own();
        
        data_.cross_section.value_list(fe_values_sub.point_list(), cell_sub, csection_lower);
        data_.cross_section.value_list(fe_values_sub.point_list(), cell, csection_higher);
        data_.fracture_sigma.value_list(fe_values_sub.point_list(), cell_sub, frac_sigma);
        data_.young_modulus.value_list(fe_values_sub.point_list(), cell_sub, young);
        data_.poisson_ratio.value_list(fe_values_sub.point_list(), cell_sub, poisson);
        
        for (unsigned int n=0; n<2; ++n)
        {
            for (unsigned int i=0; i<ndofs_side; i++)
            {
                for (unsigned int m=0; m<2; ++m)
                {
                    for (unsigned int j=0; j<ndofs_side; j++)
                        local_matrix[n][m][i*(ndofs_side)+j] = 0;
                }
                local_rhs[n][i] = 0;
            }
        }

        // set transmission conditions
        for (unsigned int k=0; k<qsize; k++)
        {
            arma::vec3 nv = fe_values_side.normal_vector(k);
            double mu = lame_mu(young[k], poisson[k]);
            double lambda = lame_lambda(young[k], poisson[k]);
            
            for (int n=0; n<2; n++)
            {
                if (!own_element_id[n]) continue;

                for (unsigned int i=0; i<n_dofs[n]; i++)
                {
                    arma::vec3 vi = (n==0)?arma::zeros(3):vec_side.value(i,k);
                    arma::vec3 vf = (n==1)?arma::zeros(3):vec_sub.value(i,k);
                    arma::mat33 gvft = (n==0)?vec_sub.grad(i,k):arma::zeros(3,3);
                    
                    for (int m=0; m<2; m++)
                        for (unsigned int j=0; j<n_dofs[m]; j++) {
                            arma::vec3 ui = (m==0)?arma::zeros(3):vec_side.value(j,k);
                            arma::vec3 uf = (m==1)?arma::zeros(3):vec_sub.value(j,k);
                            arma::mat33 guit = (m==1)?mat_t(vec_side.grad(j,k),nv):arma::zeros(3,3);
                            double divuit = (m==1)?arma::trace(guit):0;
                            
                            local_matrix[n][m][i*n_dofs[m] + j] +=
                                    (
                                     frac_sigma[k]*arma::dot(vf-vi,
                                      2/csection_lower[k]*(mu*(uf-ui)+(mu+lambda)*(arma::dot(uf-ui,nv)*nv))
                                      + mu*arma::trans(guit)*nv
                                      + lambda*divuit*nv
                                     )
                                     - arma::dot(gvft, mu*arma::kron(nv,ui.t()) + lambda*arma::dot(ui,nv)*arma::eye(3,3))
                                    )*fv_sb[0]->JxW(k);
                        }
                }
            }
        }
            
        for (unsigned int n=0; n<2; ++n)
        {
            for (unsigned int m=0; m<2; ++m)
                ls->mat_set_values(n_dofs[n], side_dof_indices[n].data(), n_dofs[m], side_dof_indices[m].data(), local_matrix[n][m]);
            
            ls->rhs_set_values(n_dofs[n], side_dof_indices[n].data(), local_rhs[n]);
        }
    }
}







void Elasticity::set_boundary_conditions()
{
  START_TIMER("assemble_bc");
//     balance_->start_flux_assembly(subst_idx);
    set_boundary_conditions<1>();
    set_boundary_conditions<2>();
    set_boundary_conditions<3>();
//  balance_->finish_flux_assembly(subst_idx);
  END_TIMER("assemble_bc");
}



template<unsigned int dim>
void Elasticity::set_boundary_conditions()
{
    FESideValues<dim,3> fe_values_side(*feo->mapping<dim>(), *feo->q<dim-1>(), *feo->fe<dim>(),
            update_values | update_gradients | update_normal_vectors | update_side_JxW_values | update_quadrature_points);
    const unsigned int ndofs = feo->fe<dim>()->n_dofs(), qsize = feo->q<dim-1>()->size();
    vector<int> side_dof_indices(ndofs);
    unsigned int loc_b=0;
    double local_rhs[ndofs];
    vector<PetscScalar> local_flux_balance_vector(ndofs);
    PetscScalar local_flux_balance_rhs;
    vector<arma::vec3> bc_values(qsize), bc_traction(qsize);
    vector<double> csection(qsize);
    auto vec = fe_values_side.vector_view(0);

    for (auto cell : feo->dh()->own_range())
    {
        ElementAccessor<3> elm = cell.elm();
        if (elm->boundary_idx_ == nullptr) continue;

        for (unsigned int si=0; si<elm->n_sides(); ++si)
        {
            const Edge *edg = elm.side(si)->edge();
            if (edg->n_sides > 1) continue;
            // skip edges lying not on the boundary
            if (edg->side(0)->cond() == NULL) continue;

            if (edg->side(0)->dim() != dim-1)
            {
                if (edg->side(0)->cond() != nullptr) ++loc_b;
                continue;
            }

            SideIter side = edg->side(0);
            ElementAccessor<3> cell = side->element();
            ElementAccessor<3> bc_cell = side->cond()->element_accessor();

            unsigned int bc_type = data_.bc_type.value(side->centre(), bc_cell);

            fe_values_side.reinit(cell, side->side_idx());

            data_.cross_section.value_list(fe_values_side.point_list(), cell, csection);
            // The b.c. data are fetched for all possible b.c. types since we allow
            // different bc_type for each substance.
            data_.bc_displacement.value_list(fe_values_side.point_list(), bc_cell, bc_values);
            data_.bc_traction.value_list(fe_values_side.point_list(), bc_cell, bc_traction);

            feo->dh()->cell_accessor_from_element(cell.idx()).get_dof_indices(side_dof_indices);

            fill_n(local_rhs, ndofs, 0);
            local_flux_balance_vector.assign(ndofs, 0);
            local_flux_balance_rhs = 0;

            if (bc_type == EqData::bc_type_displacement)
            {
                // We solve a local problem to determine which local dof 
                // to assign the boundary value to.
                // TODO: Should be possibly optimized by inverting the matrix only once within reference element.
                arma::mat d_mat(ndofs,ndofs);
                arma::vec d_vec(ndofs);
                for (unsigned int i=0; i<ndofs; i++)
                {
                    d_vec(i) = 0;
                    for (unsigned int k=0; k<qsize; k++)
                        d_vec(i) += arma::dot(vec.value(i,k),bc_values[k])*fe_values_side.JxW(k);
                    for (unsigned int j=i; j<ndofs; j++)
                    {
                        d_mat(i,j) = 0;
                        for (unsigned int k=0; k<qsize; k++)
                            d_mat(i,j) += arma::dot(vec.value(i,k),vec.value(j,k))*fe_values_side.JxW(k);
                        d_mat(j,i) = d_mat(i,j);
                    }
                }
                arma::vec d_val = pinv(d_mat)*d_vec;
                
                for (unsigned int i=0; i<ndofs; i++)
                    if (norm(d_mat.row(i)) > 0)
                        ls->add_constraint(side_dof_indices[i], d_val(i));
            }
            else if (bc_type == EqData::bc_type_traction)
            {
              for (unsigned int k=0; k<qsize; k++)
              {
                for (unsigned int i=0; i<ndofs; i++)
                  local_rhs[i] += csection[k]*arma::dot(vec.value(i,k),bc_traction[k])*fe_values_side.JxW(k);
              }
            }
            ls->rhs_set_values(ndofs, &(side_dof_indices[0]), local_rhs);


            
//             balance_->add_flux_matrix_values(subst_idx, loc_b, side_dof_indices, local_flux_balance_vector);
//             balance_->add_flux_vec_value(subst_idx, loc_b, local_flux_balance_rhs);
            ++loc_b;
        }
    }
}