ComponentTransportFEM.h

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#pragma once
 
#include <numeric>
#include <vector>
 
#include "ChemistryLib/ChemicalSolverInterface.h"
#include "ComponentTransportProcessData.h"
#include "MaterialLib/MPL/MaterialSpatialDistributionMap.h"
#include "MaterialLib/MPL/Medium.h"
#include "MaterialLib/MPL/Property.h"
#include "MaterialLib/MPL/Utils/FormEigenTensor.h"
#include "MathLib/LinAlg/Eigen/EigenMapTools.h"
#include "NumLib/DOF/DOFTableUtil.h"
#include "NumLib/Extrapolation/ExtrapolatableElement.h"
#include "NumLib/Fem/FiniteElement/TemplateIsoparametric.h"
#include "NumLib/Fem/InitShapeMatrices.h"
#include "NumLib/Fem/Integration/GenericIntegrationMethod.h"
#include "NumLib/Fem/ShapeMatrixPolicy.h"
#include "NumLib/Function/Interpolation.h"
#include "NumLib/NumericalStability/AdvectionMatrixAssembler.h"
#include "NumLib/NumericalStability/HydrodynamicDispersion.h"
#include "ParameterLib/Parameter.h"
#include "ProcessLib/LocalAssemblerInterface.h"
#include "ProcessLib/ProcessVariable.h"
 
namespace ProcessLib
{
namespace ComponentTransport
{
template <typename NodalRowVectorType, typename GlobalDimNodalMatrixType>
struct IntegrationPointData final
{
    IntegrationPointData(NodalRowVectorType const& N_,
                         GlobalDimNodalMatrixType const& dNdx_,
                         double const& integration_weight_)
        : N(N_), dNdx(dNdx_), integration_weight(integration_weight_)
    {
    }
 
    void pushBackState() { porosity_prev = porosity; }
    NodalRowVectorType const N;
    GlobalDimNodalMatrixType const dNdx;
    double const integration_weight;
 
    // -1 indicates that no chemical reaction takes place in the element to
    // which the integration point belongs.
    GlobalIndexType chemical_system_id = -1;
 
    double porosity = std::numeric_limits<double>::quiet_NaN();
    double porosity_prev = std::numeric_limits<double>::quiet_NaN();
    EIGEN_MAKE_ALIGNED_OPERATOR_NEW;
};
 
class ComponentTransportLocalAssemblerInterface
    : public ProcessLib::LocalAssemblerInterface,
      public NumLib::ExtrapolatableElement
{
public:
    ComponentTransportLocalAssemblerInterface() = default;
 
    virtual void setChemicalSystemID(std::size_t const /*mesh_item_id*/) = 0;
 
    void initializeChemicalSystem(
        std::size_t const mesh_item_id,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_tables,
        std::vector<GlobalVector*> const& x, double const t)
    {
        std::vector<double> local_x_vec;
 
        auto const n_processes = x.size();
        for (std::size_t process_id = 0; process_id < n_processes; ++process_id)
        {
            auto const indices =
                NumLib::getIndices(mesh_item_id, *dof_tables[process_id]);
            assert(!indices.empty());
            auto const local_solution = x[process_id]->get(indices);
            local_x_vec.insert(std::end(local_x_vec),
                               std::begin(local_solution),
                               std::end(local_solution));
        }
        auto const local_x = MathLib::toVector(local_x_vec);
 
        initializeChemicalSystemConcrete(local_x, t);
    }
 
    void setChemicalSystem(
        std::size_t const mesh_item_id,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_tables,
        std::vector<GlobalVector*> const& x, double const t, double const dt)
    {
        std::vector<double> local_x_vec;
 
        auto const n_processes = x.size();
        for (std::size_t process_id = 0; process_id < n_processes; ++process_id)
        {
            auto const indices =
                NumLib::getIndices(mesh_item_id, *dof_tables[process_id]);
            assert(!indices.empty());
            auto const local_solution = x[process_id]->get(indices);
            local_x_vec.insert(std::end(local_x_vec),
                               std::begin(local_solution),
                               std::end(local_solution));
        }
        auto const local_x = MathLib::toVector(local_x_vec);
 
        setChemicalSystemConcrete(local_x, t, dt);
    }
 
    void assembleReactionEquation(
        std::size_t const mesh_item_id,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_tables,
        std::vector<GlobalVector*> const& x, double const t, double const dt,
        GlobalMatrix& M, GlobalMatrix& K, GlobalVector& b, int const process_id)
    {
        std::vector<double> local_x_vec;
 
        auto const n_processes = x.size();
        for (std::size_t pcs_id = 0; pcs_id < n_processes; ++pcs_id)
        {
            auto const indices =
                NumLib::getIndices(mesh_item_id, *dof_tables[pcs_id]);
            assert(!indices.empty());
            auto const local_solution = x[pcs_id]->get(indices);
            local_x_vec.insert(std::end(local_x_vec),
                               std::begin(local_solution),
                               std::end(local_solution));
        }
        auto const local_x = MathLib::toVector(local_x_vec);
 
        auto const indices =
            NumLib::getIndices(mesh_item_id, *dof_tables[process_id]);
        auto const num_r_c = indices.size();
 
        std::vector<double> local_M_data;
        local_M_data.reserve(num_r_c * num_r_c);
        std::vector<double> local_K_data;
        local_K_data.reserve(num_r_c * num_r_c);
        std::vector<double> local_b_data;
        local_b_data.reserve(num_r_c);
 
        assembleReactionEquationConcrete(t, dt, local_x, local_M_data,
                                         local_K_data, local_b_data,
                                         process_id);
 
        auto const r_c_indices =
            NumLib::LocalToGlobalIndexMap::RowColumnIndices(indices, indices);
        if (!local_M_data.empty())
        {
            auto const local_M =
                MathLib::toMatrix(local_M_data, num_r_c, num_r_c);
            M.add(r_c_indices, local_M);
        }
        if (!local_K_data.empty())
        {
            auto const local_K =
                MathLib::toMatrix(local_K_data, num_r_c, num_r_c);
            K.add(r_c_indices, local_K);
        }
        if (!local_b_data.empty())
        {
            b.add(indices, local_b_data);
        }
    }
 
    virtual void postSpeciationCalculation(std::size_t const ele_id,
                                           double const t, double const dt) = 0;
 
    virtual void computeReactionRelatedSecondaryVariable(
        std::size_t const ele_id) = 0;
 
    virtual std::vector<double> const& getIntPtDarcyVelocity(
        const double t,
        std::vector<GlobalVector*> const& x,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_table,
        std::vector<double>& cache) const = 0;
 
    virtual std::vector<double> const& getIntPtMolarFlux(
        const double t,
        std::vector<GlobalVector*> const& x,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_table,
        std::vector<double>& cache) const = 0;
 
private:
    virtual void initializeChemicalSystemConcrete(
        Eigen::VectorXd const& /*local_x*/, double const /*t*/) = 0;
 
    virtual void setChemicalSystemConcrete(Eigen::VectorXd const& /*local_x*/,
                                           double const /*t*/,
                                           double const /*dt*/) = 0;
 
    virtual void assembleReactionEquationConcrete(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        std::vector<double>& local_M_data, std::vector<double>& local_K_data,
        std::vector<double>& local_b_data, int const transport_process_id) = 0;
};
 
template <typename ShapeFunction, int GlobalDim>
class LocalAssemblerData : public ComponentTransportLocalAssemblerInterface
{
    // When monolithic scheme is adopted, nodal pressure and nodal concentration
    // are accessed by vector index.
    static const int pressure_index = 0;
    static const int first_concentration_index = ShapeFunction::NPOINTS;
 
    static const int pressure_size = ShapeFunction::NPOINTS;
    static const int concentration_size =
        ShapeFunction::NPOINTS;  // per component
 
    using ShapeMatricesType = ShapeMatrixPolicyType<ShapeFunction, GlobalDim>;
    using ShapeMatrices = typename ShapeMatricesType::ShapeMatrices;
 
    using LocalBlockMatrixType =
        typename ShapeMatricesType::template MatrixType<pressure_size,
                                                        pressure_size>;
    using LocalSegmentVectorType =
        typename ShapeMatricesType::template VectorType<pressure_size>;
 
    using LocalMatrixType =
        Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic, Eigen::RowMajor>;
    using LocalVectorType = Eigen::Matrix<double, Eigen::Dynamic, 1>;
 
    using NodalVectorType = typename ShapeMatricesType::NodalVectorType;
    using NodalRowVectorType = typename ShapeMatricesType::NodalRowVectorType;
 
    using GlobalDimVectorType = typename ShapeMatricesType::GlobalDimVectorType;
    using GlobalDimNodalMatrixType =
        typename ShapeMatricesType::GlobalDimNodalMatrixType;
    using GlobalDimMatrixType = typename ShapeMatricesType::GlobalDimMatrixType;
 
public:
    LocalAssemblerData(
        MeshLib::Element const& element,
        std::size_t const local_matrix_size,
        NumLib::GenericIntegrationMethod const& integration_method,
        bool is_axially_symmetric,
        ComponentTransportProcessData const& process_data,
        std::vector<std::reference_wrapper<ProcessVariable>> const&
            transport_process_variables)
        : _element(element),
          _process_data(process_data),
          _integration_method(integration_method),
          _transport_process_variables(transport_process_variables)
    {
        (void)local_matrix_size;
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
        _ip_data.reserve(n_integration_points);
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        double const aperture_size = _process_data.aperture_size(0.0, pos)[0];
 
        auto const shape_matrices =
            NumLib::initShapeMatrices<ShapeFunction, ShapeMatricesType,
                                      GlobalDim>(element, is_axially_symmetric,
                                                 _integration_method);
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        for (unsigned ip = 0; ip < n_integration_points; ip++)
        {
            _ip_data.emplace_back(
                shape_matrices[ip].N, shape_matrices[ip].dNdx,
                _integration_method.getWeightedPoint(ip).getWeight() *
                    shape_matrices[ip].integralMeasure *
                    shape_matrices[ip].detJ * aperture_size);
 
            pos.setIntegrationPoint(ip);
 
            _ip_data[ip].porosity =
                medium[MaterialPropertyLib::PropertyType::porosity]
                    .template initialValue<double>(
                        pos, std::numeric_limits<double>::quiet_NaN() /*t*/);
 
            _ip_data[ip].pushBackState();
        }
    }
 
    void setChemicalSystemID(std::size_t const /*mesh_item_id*/) override
    {
        assert(_process_data.chemical_solver_interface);
        // chemical system index map
        auto& chemical_system_index_map =
            _process_data.chemical_solver_interface->chemical_system_index_map;
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
        for (unsigned ip = 0; ip < n_integration_points; ip++)
        {
            _ip_data[ip].chemical_system_id =
                chemical_system_index_map.empty()
                    ? 0
                    : chemical_system_index_map.back() + 1;
            chemical_system_index_map.push_back(
                _ip_data[ip].chemical_system_id);
        }
    }
 
    void initializeChemicalSystemConcrete(Eigen::VectorXd const& local_x,
                                          double const t) override
    {
        assert(_process_data.chemical_solver_interface);
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
        for (unsigned ip = 0; ip < n_integration_points; ip++)
        {
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& chemical_system_id = ip_data.chemical_system_id;
 
            auto const n_component = _transport_process_variables.size();
            std::vector<double> C_int_pt(n_component);
            for (unsigned component_id = 0; component_id < n_component;
                 ++component_id)
            {
                auto const concentration_index =
                    first_concentration_index +
                    component_id * concentration_size;
                auto const local_C =
                    local_x.template segment<concentration_size>(
                        concentration_index);
 
                NumLib::shapeFunctionInterpolate(local_C, N,
                                                 C_int_pt[component_id]);
            }
 
            _process_data.chemical_solver_interface
                ->initializeChemicalSystemConcrete(C_int_pt, chemical_system_id,
                                                   medium, pos, t);
        }
    }
 
    void setChemicalSystemConcrete(Eigen::VectorXd const& local_x,
                                   double const t, double dt) override
    {
        assert(_process_data.chemical_solver_interface);
 
        auto const& medium =
            _process_data.media_map.getMedium(_element.getID());
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
        for (unsigned ip = 0; ip < n_integration_points; ip++)
        {
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto& porosity = ip_data.porosity;
            auto const& porosity_prev = ip_data.porosity_prev;
            auto const& chemical_system_id = ip_data.chemical_system_id;
 
            auto const n_component = _transport_process_variables.size();
            std::vector<double> C_int_pt(n_component);
            for (unsigned component_id = 0; component_id < n_component;
                 ++component_id)
            {
                auto const concentration_index =
                    first_concentration_index +
                    component_id * concentration_size;
                auto const local_C =
                    local_x.template segment<concentration_size>(
                        concentration_index);
 
                NumLib::shapeFunctionInterpolate(local_C, N,
                                                 C_int_pt[component_id]);
            }
 
            {
                vars_prev.porosity = porosity_prev;
 
                porosity =
                    _process_data.chemically_induced_porosity_change
                        ? porosity_prev
                        : medium
                              ->property(
                                  MaterialPropertyLib::PropertyType::porosity)
                              .template value<double>(vars, vars_prev, pos, t,
                                                      dt);
 
                vars.porosity = porosity;
            }
 
            _process_data.chemical_solver_interface->setChemicalSystemConcrete(
                C_int_pt, chemical_system_id, medium, vars, pos, t, dt);
        }
    }
 
    void postSpeciationCalculation(std::size_t const ele_id, double const t,
                                   double const dt) override
    {
        if (!_process_data.chemically_induced_porosity_change)
        {
            return;
        }
 
        auto const& medium = *_process_data.media_map.getMedium(ele_id);
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(ele_id);
 
        for (auto& ip_data : _ip_data)
        {
            ip_data.porosity = ip_data.porosity_prev;
 
            _process_data.chemical_solver_interface
                ->updateVolumeFractionPostReaction(ip_data.chemical_system_id,
                                                   medium, pos,
                                                   ip_data.porosity, t, dt);
 
            _process_data.chemical_solver_interface->updatePorosityPostReaction(
                ip_data.chemical_system_id, medium, ip_data.porosity);
        }
    }
 
    void assemble(double const t, double const dt,
                  std::vector<double> const& local_x,
                  std::vector<double> const& /*local_x_prev*/,
                  std::vector<double>& local_M_data,
                  std::vector<double>& local_K_data,
                  std::vector<double>& local_b_data) override
    {
        auto const local_matrix_size = local_x.size();
        // Nodal DOFs include pressure
        int const num_nodal_dof = 1 + _transport_process_variables.size();
        // This assertion is valid only if all nodal d.o.f. use the same shape
        // matrices.
        assert(local_matrix_size == ShapeFunction::NPOINTS * num_nodal_dof);
 
        auto local_M = MathLib::createZeroedMatrix<LocalMatrixType>(
            local_M_data, local_matrix_size, local_matrix_size);
        auto local_K = MathLib::createZeroedMatrix<LocalMatrixType>(
            local_K_data, local_matrix_size, local_matrix_size);
        auto local_b = MathLib::createZeroedVector<LocalVectorType>(
            local_b_data, local_matrix_size);
 
        // Get block matrices
        auto Kpp = local_K.template block<pressure_size, pressure_size>(
            pressure_index, pressure_index);
        auto Mpp = local_M.template block<pressure_size, pressure_size>(
            pressure_index, pressure_index);
        auto Bp = local_b.template segment<pressure_size>(pressure_index);
 
        auto local_p = Eigen::Map<const NodalVectorType>(
            &local_x[pressure_index], pressure_size);
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        auto const number_of_components = num_nodal_dof - 1;
        for (int component_id = 0; component_id < number_of_components;
             ++component_id)
        {
            /*  Partitioned assembler matrix
             *  |  pp | pc1 | pc2 | pc3 |
             *  |-----|-----|-----|-----|
             *  | c1p | c1c1|  0  |  0  |
             *  |-----|-----|-----|-----|
             *  | c2p |  0  | c2c2|  0  |
             *  |-----|-----|-----|-----|
             *  | c3p |  0  |  0  | c3c3|
             */
            auto concentration_index =
                pressure_size + component_id * concentration_size;
 
            auto KCC =
                local_K.template block<concentration_size, concentration_size>(
                    concentration_index, concentration_index);
            auto MCC =
                local_M.template block<concentration_size, concentration_size>(
                    concentration_index, concentration_index);
            auto MCp =
                local_M.template block<concentration_size, pressure_size>(
                    concentration_index, pressure_index);
            auto MpC =
                local_M.template block<pressure_size, concentration_size>(
                    pressure_index, concentration_index);
 
            auto local_C = Eigen::Map<const NodalVectorType>(
                &local_x[concentration_index], concentration_size);
 
            assembleBlockMatrices(b, component_id, t, dt, local_C, local_p, KCC,
                                  MCC, MCp, MpC, Kpp, Mpp, Bp);
 
            if (_process_data.chemical_solver_interface)
            {
                auto const stoichiometric_matrix =
                    _process_data.chemical_solver_interface
                        ->getStoichiometricMatrix();
 
                assert(stoichiometric_matrix);
 
                for (Eigen::SparseMatrix<double>::InnerIterator it(
                         *stoichiometric_matrix, component_id);
                     it;
                     ++it)
                {
                    auto const stoichiometric_coefficient = it.value();
                    auto const coupled_component_id = it.row();
                    auto const kinetic_prefactor =
                        _process_data.chemical_solver_interface
                            ->getKineticPrefactor(coupled_component_id);
 
                    auto const concentration_index =
                        pressure_size + component_id * concentration_size;
                    auto const coupled_concentration_index =
                        pressure_size +
                        coupled_component_id * concentration_size;
                    auto KCmCn = local_K.template block<concentration_size,
                                                        concentration_size>(
                        concentration_index, coupled_concentration_index);
 
                    // account for the coupling between components
                    assembleKCmCn(component_id, t, dt, KCmCn,
                                  stoichiometric_coefficient,
                                  kinetic_prefactor);
                }
            }
        }
    }
 
    void assembleBlockMatrices(
        GlobalDimVectorType const& b, int const component_id, double const t,
        double const dt,
        Eigen::Ref<const NodalVectorType> const& C_nodal_values,
        Eigen::Ref<const NodalVectorType> const& p_nodal_values,
        Eigen::Ref<LocalBlockMatrixType> KCC,
        Eigen::Ref<LocalBlockMatrixType> MCC,
        Eigen::Ref<LocalBlockMatrixType> MCp,
        Eigen::Ref<LocalBlockMatrixType> MpC,
        Eigen::Ref<LocalBlockMatrixType> Kpp,
        Eigen::Ref<LocalBlockMatrixType> Mpp,
        Eigen::Ref<LocalSegmentVectorType> Bp)
    {
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        MaterialPropertyLib::VariableArray vars;
 
        // Get material properties
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        // Select the only valid for component transport liquid phase.
        auto const& phase = medium.phase("AqueousLiquid");
 
        // Assume that the component name is the same as the process variable
        // name. Components are shifted by one because the first one is always
        // pressure.
        auto const& component = phase.component(
            _transport_process_variables[component_id].get().getName());
 
        LocalBlockMatrixType KCC_Laplacian =
            LocalBlockMatrixType::Zero(concentration_size, concentration_size);
 
        std::vector<GlobalDimVectorType> ip_flux_vector;
        double average_velocity_norm = 0.0;
        if (!_process_data.non_advective_form)
        {
            ip_flux_vector.reserve(n_integration_points);
        }
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& w = ip_data.integration_weight;
            auto& porosity = ip_data.porosity;
 
            double C_int_pt = 0.0;
            double p_int_pt = 0.0;
 
            NumLib::shapeFunctionInterpolate(C_nodal_values, N, C_int_pt);
            NumLib::shapeFunctionInterpolate(p_nodal_values, N, p_int_pt);
 
            vars.concentration = C_int_pt;
            vars.phase_pressure = p_int_pt;
 
            // update according to a particular porosity model
            porosity = medium[MaterialPropertyLib::PropertyType::porosity]
                           .template value<double>(vars, pos, t, dt);
            vars.porosity = porosity;
 
            auto const& retardation_factor =
                component[MaterialPropertyLib::PropertyType::retardation_factor]
                    .template value<double>(vars, pos, t, dt);
 
            auto const& solute_dispersivity_transverse = medium.template value<
                double>(
                MaterialPropertyLib::PropertyType::transversal_dispersivity);
 
            auto const& solute_dispersivity_longitudinal =
                medium.template value<double>(
                    MaterialPropertyLib::PropertyType::
                        longitudinal_dispersivity);
 
            // Use the fluid density model to compute the density
            // TODO (renchao): concentration of which component as the argument
            // for calculation of fluid density
            auto const density =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template value<double>(vars, pos, t, dt);
 
            auto const decay_rate =
                component[MaterialPropertyLib::PropertyType::decay_rate]
                    .template value<double>(vars, pos, t, dt);
 
            auto const& pore_diffusion_coefficient =
                MaterialPropertyLib::formEigenTensor<GlobalDim>(
                    component[MaterialPropertyLib::PropertyType::pore_diffusion]
                        .value(vars, pos, t, dt));
 
            auto const& K = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
 
            // Use the viscosity model to compute the viscosity
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
 
            GlobalDimMatrixType const K_over_mu = K / mu;
            GlobalDimVectorType const velocity =
                _process_data.has_gravity
                    ? GlobalDimVectorType(-K_over_mu *
                                          (dNdx * p_nodal_values - density * b))
                    : GlobalDimVectorType(-K_over_mu * dNdx * p_nodal_values);
 
            const double drho_dp =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::phase_pressure,
                        pos, t, dt);
 
            const double drho_dC =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::concentration, pos,
                        t, dt);
 
            GlobalDimMatrixType const hydrodynamic_dispersion =
                NumLib::computeHydrodynamicDispersion(
                    _process_data.stabilizer, _element.getID(),
                    pore_diffusion_coefficient, velocity, porosity,
                    solute_dispersivity_transverse,
                    solute_dispersivity_longitudinal);
 
            const double R_times_phi(retardation_factor * porosity);
            GlobalDimVectorType const mass_density_flow = velocity * density;
            auto const N_t_N = (N.transpose() * N).eval();
 
            if (_process_data.non_advective_form)
            {
                MCp.noalias() += N_t_N * (C_int_pt * R_times_phi * drho_dp * w);
                MCC.noalias() += N_t_N * (C_int_pt * R_times_phi * drho_dC * w);
                KCC.noalias() -= dNdx.transpose() * mass_density_flow * N * w;
            }
            else
            {
                ip_flux_vector.emplace_back(mass_density_flow);
                average_velocity_norm += velocity.norm();
            }
            MCC.noalias() += N_t_N * (R_times_phi * density * w);
            KCC.noalias() += N_t_N * (decay_rate * R_times_phi * density * w);
            KCC_Laplacian.noalias() +=
                dNdx.transpose() * hydrodynamic_dispersion * dNdx * density * w;
 
            MpC.noalias() += N_t_N * (porosity * drho_dC * w);
 
            // Calculate Mpp, Kpp, and bp in the first loop over components
            if (component_id == 0)
            {
                Mpp.noalias() += N_t_N * (porosity * drho_dp * w);
                Kpp.noalias() +=
                    dNdx.transpose() * K_over_mu * dNdx * (density * w);
 
                if (_process_data.has_gravity)
                {
                    Bp.noalias() += dNdx.transpose() * K_over_mu * b *
                                    (density * density * w);
                }
            }
        }
 
        if (!_process_data.non_advective_form)
        {
            NumLib::assembleAdvectionMatrix(
                _process_data.stabilizer,
                _ip_data,
                ip_flux_vector,
                average_velocity_norm /
                    static_cast<double>(n_integration_points),
                KCC_Laplacian);
        }
 
        KCC.noalias() += KCC_Laplacian;
    }
 
    void assembleKCmCn(int const component_id, double const t, double const dt,
                       Eigen::Ref<LocalBlockMatrixType> KCmCn,
                       double const stoichiometric_coefficient,
                       double const kinetic_prefactor)
    {
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        MaterialPropertyLib::VariableArray vars;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
        auto const& component = phase.component(
            _transport_process_variables[component_id].get().getName());
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& w = ip_data.integration_weight;
            auto& porosity = ip_data.porosity;
 
            auto const retardation_factor =
                component[MaterialPropertyLib::PropertyType::retardation_factor]
                    .template value<double>(vars, pos, t, dt);
 
            porosity = medium[MaterialPropertyLib::PropertyType::porosity]
                           .template value<double>(vars, pos, t, dt);
 
            auto const density =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template value<double>(vars, pos, t, dt);
 
            KCmCn.noalias() -= w * N.transpose() * stoichiometric_coefficient *
                               kinetic_prefactor * retardation_factor *
                               porosity * density * N;
        }
    }
 
    void assembleForStaggeredScheme(double const t, double const dt,
                                    Eigen::VectorXd const& local_x,
                                    Eigen::VectorXd const& local_x_prev,
                                    int const process_id,
                                    std::vector<double>& local_M_data,
                                    std::vector<double>& local_K_data,
                                    std::vector<double>& local_b_data) override
    {
        if (process_id == _process_data.hydraulic_process_id)
        {
            assembleHydraulicEquation(t, dt, local_x, local_x_prev,
                                      local_M_data, local_K_data, local_b_data);
        }
        else
        {
            // Go for assembling in an order of transport process id.
            assembleComponentTransportEquation(t, dt, local_x, local_x_prev,
                                               local_M_data, local_K_data,
                                               local_b_data, process_id);
        }
    }
 
    void assembleHydraulicEquation(double const t,
                                   double const dt,
                                   Eigen::VectorXd const& local_x,
                                   Eigen::VectorXd const& local_x_prev,
                                   std::vector<double>& local_M_data,
                                   std::vector<double>& local_K_data,
                                   std::vector<double>& local_b_data)
    {
        auto const local_p =
            local_x.template segment<pressure_size>(pressure_index);
        auto const local_C = local_x.template segment<concentration_size>(
            first_concentration_index);
        auto const local_C_prev =
            local_x_prev.segment<concentration_size>(first_concentration_index);
 
        auto local_M = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_M_data, pressure_size, pressure_size);
        auto local_K = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_K_data, pressure_size, pressure_size);
        auto local_b = MathLib::createZeroedVector<LocalSegmentVectorType>(
            local_b_data, pressure_size);
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& w = ip_data.integration_weight;
            auto& porosity = ip_data.porosity;
            auto const& porosity_prev = ip_data.porosity_prev;
 
            double C_int_pt = 0.0;
            double p_int_pt = 0.0;
 
            NumLib::shapeFunctionInterpolate(local_C, N, C_int_pt);
            NumLib::shapeFunctionInterpolate(local_p, N, p_int_pt);
 
            vars.concentration = C_int_pt;
            vars.phase_pressure = p_int_pt;
 
            //  porosity
            {
                vars_prev.porosity = porosity_prev;
 
                porosity =
                    _process_data.chemically_induced_porosity_change
                        ? porosity_prev
                        : medium[MaterialPropertyLib::PropertyType::porosity]
                              .template value<double>(vars, vars_prev, pos, t,
                                                      dt);
 
                vars.porosity = porosity;
            }
 
            // Use the fluid density model to compute the density
            // TODO: Concentration of which component as one of arguments for
            // calculation of fluid density
            auto const density =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template value<double>(vars, pos, t, dt);
 
            auto const& K = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
 
            // Use the viscosity model to compute the viscosity
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
 
            GlobalDimMatrixType const K_over_mu = K / mu;
 
            const double drho_dp =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::phase_pressure,
                        pos, t, dt);
            const double drho_dC =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::concentration, pos,
                        t, dt);
 
            // matrix assembly
            local_M.noalias() += w * N.transpose() * porosity * drho_dp * N;
            local_K.noalias() +=
                w * dNdx.transpose() * density * K_over_mu * dNdx;
 
            if (_process_data.has_gravity)
            {
                local_b.noalias() +=
                    w * density * density * dNdx.transpose() * K_over_mu * b;
            }
 
            // coupling term
            {
                double const C_dot = (C_int_pt - N.dot(local_C_prev)) / dt;
 
                local_b.noalias() -=
                    w * N.transpose() * porosity * drho_dC * C_dot;
            }
        }
    }
 
    void assembleComponentTransportEquation(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        Eigen::VectorXd const& local_x_prev, std::vector<double>& local_M_data,
        std::vector<double>& local_K_data,
        std::vector<double>& /*local_b_data*/, int const transport_process_id)
    {
        auto const local_p =
            local_x.template segment<pressure_size>(pressure_index);
        auto const local_C = local_x.template segment<concentration_size>(
            first_concentration_index +
            (transport_process_id - 1) * concentration_size);
        auto const local_p_prev =
            local_x_prev.segment<pressure_size>(pressure_index);
 
        NodalVectorType local_T;
        if (_process_data.temperature)
        {
            local_T =
                _process_data.temperature->getNodalValuesOnElement(_element, t);
        }
 
        auto local_M = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_M_data, concentration_size, concentration_size);
        auto local_K = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_K_data, concentration_size, concentration_size);
 
        LocalBlockMatrixType KCC_Laplacian =
            LocalBlockMatrixType::Zero(concentration_size, concentration_size);
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        std::vector<GlobalDimVectorType> ip_flux_vector;
        double average_velocity_norm = 0.0;
        if (!_process_data.non_advective_form)
        {
            ip_flux_vector.reserve(n_integration_points);
        }
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
        // Hydraulic process id is 0 and thus transport process id starts
        // from 1.
        auto const component_id = transport_process_id - 1;
        auto const& component = phase.component(
            _transport_process_variables[component_id].get().getName());
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& w = ip_data.integration_weight;
            auto& porosity = ip_data.porosity;
            auto const& porosity_prev = ip_data.porosity_prev;
 
            double C_int_pt = 0.0;
            double p_int_pt = 0.0;
 
            NumLib::shapeFunctionInterpolate(local_C, N, C_int_pt);
            NumLib::shapeFunctionInterpolate(local_p, N, p_int_pt);
 
            vars.concentration = C_int_pt;
            vars.phase_pressure = p_int_pt;
 
            if (_process_data.temperature)
            {
                vars.temperature = N.dot(local_T);
            }
 
            // porosity
            {
                vars_prev.porosity = porosity_prev;
 
                porosity =
                    _process_data.chemically_induced_porosity_change
                        ? porosity_prev
                        : medium[MaterialPropertyLib::PropertyType::porosity]
                              .template value<double>(vars, vars_prev, pos, t,
                                                      dt);
 
                vars.porosity = porosity;
            }
 
            auto const& retardation_factor =
                component[MaterialPropertyLib::PropertyType::retardation_factor]
                    .template value<double>(vars, pos, t, dt);
 
            auto const& solute_dispersivity_transverse = medium.template value<
                double>(
                MaterialPropertyLib::PropertyType::transversal_dispersivity);
            auto const& solute_dispersivity_longitudinal =
                medium.template value<double>(
                    MaterialPropertyLib::PropertyType::
                        longitudinal_dispersivity);
 
            // Use the fluid density model to compute the density
            auto const density =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template value<double>(vars, pos, t, dt);
            auto const decay_rate =
                component[MaterialPropertyLib::PropertyType::decay_rate]
                    .template value<double>(vars, pos, t, dt);
 
            auto const& pore_diffusion_coefficient =
                MaterialPropertyLib::formEigenTensor<GlobalDim>(
                    component[MaterialPropertyLib::PropertyType::pore_diffusion]
                        .value(vars, pos, t, dt));
 
            auto const& K = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
            // Use the viscosity model to compute the viscosity
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
 
            GlobalDimMatrixType const K_over_mu = K / mu;
            GlobalDimVectorType const velocity =
                _process_data.has_gravity
                    ? GlobalDimVectorType(-K_over_mu *
                                          (dNdx * local_p - density * b))
                    : GlobalDimVectorType(-K_over_mu * dNdx * local_p);
 
            GlobalDimMatrixType const hydrodynamic_dispersion =
                NumLib::computeHydrodynamicDispersion(
                    _process_data.stabilizer, _element.getID(),
                    pore_diffusion_coefficient, velocity, porosity,
                    solute_dispersivity_transverse,
                    solute_dispersivity_longitudinal);
 
            double const R_times_phi = retardation_factor * porosity;
            auto const N_t_N = (N.transpose() * N).eval();
 
            if (_process_data.non_advective_form)
            {
                const double drho_dC =
                    phase[MaterialPropertyLib::PropertyType::density]
                        .template dValue<double>(
                            vars, MaterialPropertyLib::Variable::concentration,
                            pos, t, dt);
                local_M.noalias() +=
                    N_t_N * (R_times_phi * C_int_pt * drho_dC * w);
            }
 
            local_M.noalias() += N_t_N * (R_times_phi * density * w);
 
            // coupling term
            if (_process_data.non_advective_form)
            {
                double const p_dot = (p_int_pt - N.dot(local_p_prev)) / dt;
 
                const double drho_dp =
                    phase[MaterialPropertyLib::PropertyType::density]
                        .template dValue<double>(
                            vars, MaterialPropertyLib::Variable::phase_pressure,
                            pos, t, dt);
 
                local_K.noalias() +=
                    N_t_N * ((R_times_phi * drho_dp * p_dot) * w) -
                    dNdx.transpose() * velocity * N * (density * w);
            }
            else
            {
                ip_flux_vector.emplace_back(velocity * density);
                average_velocity_norm += velocity.norm();
            }
            local_K.noalias() +=
                N_t_N * (decay_rate * R_times_phi * density * w);
 
            KCC_Laplacian.noalias() += dNdx.transpose() *
                                       hydrodynamic_dispersion * dNdx *
                                       (density * w);
        }
 
        if (!_process_data.non_advective_form)
        {
            NumLib::assembleAdvectionMatrix(
                _process_data.stabilizer,
                _ip_data,
                ip_flux_vector,
                average_velocity_norm /
                    static_cast<double>(n_integration_points),
                KCC_Laplacian);
        }
        local_K.noalias() += KCC_Laplacian;
    }
 
    void assembleWithJacobianForStaggeredScheme(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        Eigen::VectorXd const& local_x_prev, int const process_id,
        std::vector<double>& /*local_M_data*/,
        std::vector<double>& /*local_K_data*/,
        std::vector<double>& local_b_data,
        std::vector<double>& local_Jac_data) override
    {
        if (process_id == _process_data.hydraulic_process_id)
        {
            assembleWithJacobianHydraulicEquation(t, dt, local_x, local_x_prev,
                                                  local_b_data, local_Jac_data);
        }
        else
        {
            int const component_id = process_id - 1;
            assembleWithJacobianComponentTransportEquation(
                t, dt, local_x, local_x_prev, local_b_data, local_Jac_data,
                component_id);
        }
    }
 
    void assembleWithJacobianHydraulicEquation(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        Eigen::VectorXd const& local_x_prev, std::vector<double>& local_b_data,
        std::vector<double>& local_Jac_data)
    {
        auto const p = local_x.template segment<pressure_size>(pressure_index);
        auto const c = local_x.template segment<concentration_size>(
            first_concentration_index);
 
        auto const p_prev = local_x_prev.segment<pressure_size>(pressure_index);
        auto const c_prev =
            local_x_prev.segment<concentration_size>(first_concentration_index);
 
        auto local_Jac = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_Jac_data, pressure_size, pressure_size);
        auto local_rhs = MathLib::createZeroedVector<LocalSegmentVectorType>(
            local_b_data, pressure_size);
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& w = ip_data.integration_weight;
            auto& phi = ip_data.porosity;
            auto const& phi_prev = ip_data.porosity_prev;
 
            double const p_ip = N.dot(p);
            double const c_ip = N.dot(c);
 
            double const cdot_ip = (c_ip - N.dot(c_prev)) / dt;
 
            vars.phase_pressure = p_ip;
            vars.concentration = c_ip;
 
            //  porosity
            {
                vars_prev.porosity = phi_prev;
 
                phi = _process_data.chemically_induced_porosity_change
                          ? phi_prev
                          : medium[MaterialPropertyLib::PropertyType::porosity]
                                .template value<double>(vars, vars_prev, pos, t,
                                                        dt);
 
                vars.porosity = phi;
            }
 
            auto const rho = phase[MaterialPropertyLib::PropertyType::density]
                                 .template value<double>(vars, pos, t, dt);
 
            auto const k = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
 
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
 
            auto const drho_dp =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::phase_pressure,
                        pos, t, dt);
            auto const drho_dc =
                phase[MaterialPropertyLib::PropertyType::density]
                    .template dValue<double>(
                        vars, MaterialPropertyLib::Variable::concentration, pos,
                        t, dt);
 
            // matrix assembly
            local_Jac.noalias() += w * N.transpose() * phi * drho_dp / dt * N +
                                   w * dNdx.transpose() * rho * k / mu * dNdx;
 
            local_rhs.noalias() -=
                w * N.transpose() * phi *
                    (drho_dp * N * p_prev + drho_dc * cdot_ip) +
                w * rho * dNdx.transpose() * k / mu * dNdx * p;
 
            if (_process_data.has_gravity)
            {
                local_rhs.noalias() +=
                    w * rho * dNdx.transpose() * k / mu * rho * b;
            }
        }
    }
 
    void assembleWithJacobianComponentTransportEquation(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        Eigen::VectorXd const& local_x_prev, std::vector<double>& local_b_data,
        std::vector<double>& local_Jac_data, int const component_id)
    {
        auto const concentration_index =
            first_concentration_index + component_id * concentration_size;
 
        auto const p = local_x.template segment<pressure_size>(pressure_index);
        auto const c =
            local_x.template segment<concentration_size>(concentration_index);
        auto const c_prev =
            local_x_prev.segment<concentration_size>(concentration_index);
 
        NodalVectorType T;
        if (_process_data.temperature)
        {
            T = _process_data.temperature->getNodalValuesOnElement(_element, t);
        }
 
        auto local_Jac = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_Jac_data, concentration_size, concentration_size);
        auto local_rhs = MathLib::createZeroedVector<LocalSegmentVectorType>(
            local_b_data, concentration_size);
 
        LocalBlockMatrixType KCC_Laplacian =
            LocalBlockMatrixType::Zero(concentration_size, concentration_size);
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        std::vector<GlobalDimVectorType> ip_flux_vector;
        double average_velocity_norm = 0.0;
        ip_flux_vector.reserve(n_integration_points);
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
        auto const& component = phase.component(
            _transport_process_variables[component_id].get().getName());
 
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& w = ip_data.integration_weight;
            auto& phi = ip_data.porosity;
            auto const& phi_prev = ip_data.porosity_prev;
 
            double const p_ip = N.dot(p);
            double const c_ip = N.dot(c);
 
            vars.phase_pressure = p_ip;
            vars.concentration = c_ip;
 
            if (_process_data.temperature)
            {
                vars.temperature = N.dot(T);
            }
 
            // porosity
            {
                vars_prev.porosity = phi_prev;
 
                phi = _process_data.chemically_induced_porosity_change
                          ? phi_prev
                          : medium[MaterialPropertyLib::PropertyType::porosity]
                                .template value<double>(vars, vars_prev, pos, t,
                                                        dt);
 
                vars.porosity = phi;
            }
 
            auto const R =
                component[MaterialPropertyLib::PropertyType::retardation_factor]
                    .template value<double>(vars, pos, t, dt);
 
            auto const alpha_T = medium.template value<double>(
                MaterialPropertyLib::PropertyType::transversal_dispersivity);
            auto const alpha_L = medium.template value<double>(
                MaterialPropertyLib::PropertyType::longitudinal_dispersivity);
 
            auto const rho = phase[MaterialPropertyLib::PropertyType::density]
                                 .template value<double>(vars, pos, t, dt);
            // first-order decay constant
            auto const alpha =
                component[MaterialPropertyLib::PropertyType::decay_rate]
                    .template value<double>(vars, pos, t, dt);
 
            auto const Dp = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                component[MaterialPropertyLib::PropertyType::pore_diffusion]
                    .value(vars, pos, t, dt));
 
            auto const k = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
            // Darcy flux
            GlobalDimVectorType const q =
                _process_data.has_gravity
                    ? GlobalDimVectorType(-k / mu * (dNdx * p - rho * b))
                    : GlobalDimVectorType(-k / mu * dNdx * p);
 
            GlobalDimMatrixType const D = NumLib::computeHydrodynamicDispersion(
                _process_data.stabilizer, _element.getID(), Dp, q, phi, alpha_T,
                alpha_L);
 
            // matrix assembly
            local_Jac.noalias() +=
                w * rho * N.transpose() * phi * R * (alpha + 1 / dt) * N;
 
            KCC_Laplacian.noalias() += w * rho * dNdx.transpose() * D * dNdx;
 
            auto const cdot = (c - c_prev) / dt;
            local_rhs.noalias() -=
                w * rho * N.transpose() * phi * R * N * (cdot + alpha * c);
 
            ip_flux_vector.emplace_back(q * rho);
            average_velocity_norm += q.norm();
        }
 
        NumLib::assembleAdvectionMatrix(
            _process_data.stabilizer, _ip_data, ip_flux_vector,
            average_velocity_norm / static_cast<double>(n_integration_points),
            KCC_Laplacian);
 
        local_rhs.noalias() -= KCC_Laplacian * c;
 
        local_Jac.noalias() += KCC_Laplacian;
    }
 
    void assembleReactionEquationConcrete(
        double const t, double const dt, Eigen::VectorXd const& local_x,
        std::vector<double>& local_M_data, std::vector<double>& local_K_data,
        std::vector<double>& local_b_data,
        int const transport_process_id) override
    {
        auto const local_C = local_x.template segment<concentration_size>(
            first_concentration_index +
            (transport_process_id - 1) * concentration_size);
 
        auto local_M = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_M_data, concentration_size, concentration_size);
        auto local_K = MathLib::createZeroedMatrix<LocalBlockMatrixType>(
            local_K_data, concentration_size, concentration_size);
        auto local_b = MathLib::createZeroedVector<LocalVectorType>(
            local_b_data, concentration_size);
 
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        MaterialPropertyLib::VariableArray vars;
        MaterialPropertyLib::VariableArray vars_prev;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const component_id = transport_process_id - 1;
        for (unsigned ip(0); ip < n_integration_points; ++ip)
        {
            pos.setIntegrationPoint(ip);
 
            auto& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const w = ip_data.integration_weight;
            auto& porosity = ip_data.porosity;
            auto const& porosity_prev = ip_data.porosity_prev;
            auto const chemical_system_id = ip_data.chemical_system_id;
 
            double C_int_pt = 0.0;
            NumLib::shapeFunctionInterpolate(local_C, N, C_int_pt);
 
            vars.concentration = C_int_pt;
 
            auto const porosity_dot = (porosity - porosity_prev) / dt;
 
            // porosity
            {
                vars_prev.porosity = porosity_prev;
 
                porosity =
                    _process_data.chemically_induced_porosity_change
                        ? porosity_prev
                        : medium[MaterialPropertyLib::PropertyType::porosity]
                              .template value<double>(vars, vars_prev, pos, t,
                                                      dt);
            }
 
            local_M.noalias() += w * N.transpose() * porosity * N;
 
            local_K.noalias() += w * N.transpose() * porosity_dot * N;
 
            if (chemical_system_id == -1)
            {
                continue;
            }
 
            auto const C_post_int_pt =
                _process_data.chemical_solver_interface->getConcentration(
                    component_id, chemical_system_id);
 
            local_b.noalias() +=
                w * N.transpose() * porosity * (C_post_int_pt - C_int_pt) / dt;
        }
    }
 
    std::vector<double> const& getIntPtDarcyVelocity(
        const double t,
        std::vector<GlobalVector*> const& x,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_table,
        std::vector<double>& cache) const override
    {
        assert(x.size() == dof_table.size());
 
        auto const n_processes = x.size();
        std::vector<std::vector<double>> local_x;
        local_x.reserve(n_processes);
 
        for (std::size_t process_id = 0; process_id < n_processes; ++process_id)
        {
            auto const indices =
                NumLib::getIndices(_element.getID(), *dof_table[process_id]);
            assert(!indices.empty());
            local_x.push_back(x[process_id]->get(indices));
        }
 
        // only one process, must be monolithic.
        if (n_processes == 1)
        {
            auto const local_p = Eigen::Map<const NodalVectorType>(
                &local_x[0][pressure_index], pressure_size);
            auto const local_C = Eigen::Map<const NodalVectorType>(
                &local_x[0][first_concentration_index], concentration_size);
            return calculateIntPtDarcyVelocity(t, local_p, local_C, cache);
        }
 
        // multiple processes, must be staggered.
        {
            constexpr int pressure_process_id = 0;
            constexpr int concentration_process_id = 1;
            auto const local_p = Eigen::Map<const NodalVectorType>(
                &local_x[pressure_process_id][0], pressure_size);
            auto const local_C = Eigen::Map<const NodalVectorType>(
                &local_x[concentration_process_id][0], concentration_size);
            return calculateIntPtDarcyVelocity(t, local_p, local_C, cache);
        }
    }
 
    std::vector<double> const& calculateIntPtDarcyVelocity(
        const double t,
        Eigen::Ref<const NodalVectorType> const& p_nodal_values,
        Eigen::Ref<const NodalVectorType> const& C_nodal_values,
        std::vector<double>& cache) const
    {
        auto const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        cache.clear();
        auto cache_mat = MathLib::createZeroedMatrix<
            Eigen::Matrix<double, GlobalDim, Eigen::Dynamic, Eigen::RowMajor>>(
            cache, GlobalDim, n_integration_points);
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        MaterialPropertyLib::VariableArray vars;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
 
        for (unsigned ip = 0; ip < n_integration_points; ++ip)
        {
            auto const& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& porosity = ip_data.porosity;
 
            pos.setIntegrationPoint(ip);
 
            double C_int_pt = 0.0;
            double p_int_pt = 0.0;
 
            NumLib::shapeFunctionInterpolate(C_nodal_values, N, C_int_pt);
            NumLib::shapeFunctionInterpolate(p_nodal_values, N, p_int_pt);
 
            vars.concentration = C_int_pt;
            vars.phase_pressure = p_int_pt;
            vars.porosity = porosity;
 
            // TODO (naumov) Temporary value not used by current material
            // models. Need extension of secondary variables interface.
            double const dt = std::numeric_limits<double>::quiet_NaN();
            auto const& K = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
            GlobalDimMatrixType const K_over_mu = K / mu;
 
            cache_mat.col(ip).noalias() = -K_over_mu * dNdx * p_nodal_values;
            if (_process_data.has_gravity)
            {
                auto const rho_w =
                    phase[MaterialPropertyLib::PropertyType::density]
                        .template value<double>(vars, pos, t, dt);
                // here it is assumed that the vector b is directed 'downwards'
                cache_mat.col(ip).noalias() += K_over_mu * rho_w * b;
            }
        }
 
        return cache;
    }
 
    Eigen::Map<const Eigen::RowVectorXd> getShapeMatrix(
        const unsigned integration_point) const override
    {
        auto const& N = _ip_data[integration_point].N;
 
        // assumes N is stored contiguously in memory
        return Eigen::Map<const Eigen::RowVectorXd>(N.data(), N.size());
    }
 
    Eigen::Vector3d getFlux(MathLib::Point3d const& pnt_local_coords,
                            double const t,
                            std::vector<double> const& local_x) const override
    {
        auto const local_p = Eigen::Map<const NodalVectorType>(
            &local_x[pressure_index], pressure_size);
        auto const local_C = Eigen::Map<const NodalVectorType>(
            &local_x[first_concentration_index], concentration_size);
 
        // Eval shape matrices at given point
        // Note: Axial symmetry is set to false here, because we only need dNdx
        // here, which is not affected by axial symmetry.
        auto const shape_matrices =
            NumLib::computeShapeMatrices<ShapeFunction, ShapeMatricesType,
                                         GlobalDim>(
                _element, false /*is_axially_symmetric*/,
                std::array{pnt_local_coords})[0];
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        MaterialPropertyLib::VariableArray vars;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
 
        // local_x contains the local concentration and pressure values
        double c_int_pt;
        NumLib::shapeFunctionInterpolate(local_C, shape_matrices.N, c_int_pt);
        vars.concentration = c_int_pt;
 
        double p_int_pt;
        NumLib::shapeFunctionInterpolate(local_p, shape_matrices.N, p_int_pt);
        vars.phase_pressure = p_int_pt;
 
        // TODO (naumov) Temporary value not used by current material models.
        // Need extension of secondary variables interface.
        double const dt = std::numeric_limits<double>::quiet_NaN();
        auto const K = MaterialPropertyLib::formEigenTensor<GlobalDim>(
            medium[MaterialPropertyLib::PropertyType::permeability].value(
                vars, pos, t, dt));
 
        auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                            .template value<double>(vars, pos, t, dt);
        GlobalDimMatrixType const K_over_mu = K / mu;
 
        GlobalDimVectorType q = -K_over_mu * shape_matrices.dNdx * local_p;
        auto const rho_w = phase[MaterialPropertyLib::PropertyType::density]
                               .template value<double>(vars, pos, t, dt);
        if (_process_data.has_gravity)
        {
            q += K_over_mu * rho_w * b;
        }
        Eigen::Vector3d flux(0.0, 0.0, 0.0);
        flux.head<GlobalDim>() = rho_w * q;
        return flux;
    }
 
    void computeSecondaryVariableConcrete(
        double const t,
        double const /*dt*/,
        Eigen::VectorXd const& local_x,
        Eigen::VectorXd const& /*local_x_prev*/) override
    {
        auto const local_p =
            local_x.template segment<pressure_size>(pressure_index);
        auto const local_C = local_x.template segment<concentration_size>(
            first_concentration_index);
 
        std::vector<double> ele_velocity;
        calculateIntPtDarcyVelocity(t, local_p, local_C, ele_velocity);
 
        auto const n_integration_points =
            _integration_method.getNumberOfPoints();
        auto const ele_velocity_mat =
            MathLib::toMatrix(ele_velocity, GlobalDim, n_integration_points);
 
        auto const ele_id = _element.getID();
        Eigen::Map<LocalVectorType>(
            &(*_process_data.mesh_prop_velocity)[ele_id * GlobalDim],
            GlobalDim) =
            ele_velocity_mat.rowwise().sum() / n_integration_points;
    }
 
    void computeReactionRelatedSecondaryVariable(
        std::size_t const ele_id) override
    {
        auto const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        if (_process_data.chemically_induced_porosity_change)
        {
            auto const& medium = *_process_data.media_map.getMedium(ele_id);
 
            for (auto& ip_data : _ip_data)
            {
                ip_data.porosity = ip_data.porosity_prev;
 
                _process_data.chemical_solver_interface
                    ->updatePorosityPostReaction(ip_data.chemical_system_id,
                                                 medium, ip_data.porosity);
            }
 
            (*_process_data.mesh_prop_porosity)[ele_id] =
                std::accumulate(_ip_data.begin(), _ip_data.end(), 0.,
                                [](double const s, auto const& ip)
                                { return s + ip.porosity; }) /
                n_integration_points;
        }
 
        std::vector<GlobalIndexType> chemical_system_indices;
        chemical_system_indices.reserve(n_integration_points);
        std::transform(_ip_data.begin(), _ip_data.end(),
                       std::back_inserter(chemical_system_indices),
                       [](auto const& ip_data)
                       { return ip_data.chemical_system_id; });
 
        _process_data.chemical_solver_interface->computeSecondaryVariable(
            ele_id, chemical_system_indices);
    }
 
    std::vector<double> const& getIntPtMolarFlux(
        const double t,
        std::vector<GlobalVector*> const& x,
        std::vector<NumLib::LocalToGlobalIndexMap const*> const& dof_tables,
        std::vector<double>& cache) const override
    {
        std::vector<double> local_x_vec;
 
        auto const n_processes = x.size();
        for (std::size_t process_id = 0; process_id < n_processes; ++process_id)
        {
            auto const indices =
                NumLib::getIndices(_element.getID(), *dof_tables[process_id]);
            assert(!indices.empty());
            auto const local_solution = x[process_id]->get(indices);
            local_x_vec.insert(std::end(local_x_vec),
                               std::begin(local_solution),
                               std::end(local_solution));
        }
        auto const local_x = MathLib::toVector(local_x_vec);
 
        auto const p = local_x.template segment<pressure_size>(pressure_index);
        auto const c = local_x.template segment<concentration_size>(
            first_concentration_index);
 
        auto const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        cache.clear();
        auto cache_mat = MathLib::createZeroedMatrix<
            Eigen::Matrix<double, GlobalDim, Eigen::Dynamic, Eigen::RowMajor>>(
            cache, GlobalDim, n_integration_points);
 
        ParameterLib::SpatialPosition pos;
        pos.setElementID(_element.getID());
 
        auto const& b =
            _process_data
                .projected_specific_body_force_vectors[_element.getID()];
 
        MaterialPropertyLib::VariableArray vars;
 
        auto const& medium =
            *_process_data.media_map.getMedium(_element.getID());
        auto const& phase = medium.phase("AqueousLiquid");
 
        int const component_id = 0;
        auto const& component = phase.component(
            _transport_process_variables[component_id].get().getName());
 
        for (unsigned ip = 0; ip < n_integration_points; ++ip)
        {
            auto const& ip_data = _ip_data[ip];
            auto const& N = ip_data.N;
            auto const& dNdx = ip_data.dNdx;
            auto const& phi = ip_data.porosity;
 
            pos.setIntegrationPoint(ip);
 
            double const p_ip = N.dot(p);
            double const c_ip = N.dot(c);
 
            vars.concentration = c_ip;
            vars.phase_pressure = p_ip;
            vars.porosity = phi;
 
            double const dt = std::numeric_limits<double>::quiet_NaN();
 
            auto const& k = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                medium[MaterialPropertyLib::PropertyType::permeability].value(
                    vars, pos, t, dt));
            auto const mu = phase[MaterialPropertyLib::PropertyType::viscosity]
                                .template value<double>(vars, pos, t, dt);
            auto const rho = phase[MaterialPropertyLib::PropertyType::density]
                                 .template value<double>(vars, pos, t, dt);
 
            // Darcy flux
            GlobalDimVectorType const q =
                _process_data.has_gravity
                    ? GlobalDimVectorType(-k / mu * (dNdx * p - rho * b))
                    : GlobalDimVectorType(-k / mu * dNdx * p);
 
            auto const alpha_T = medium.template value<double>(
                MaterialPropertyLib::PropertyType::transversal_dispersivity);
            auto const alpha_L = medium.template value<double>(
                MaterialPropertyLib::PropertyType::longitudinal_dispersivity);
            auto const Dp = MaterialPropertyLib::formEigenTensor<GlobalDim>(
                component[MaterialPropertyLib::PropertyType::pore_diffusion]
                    .value(vars, pos, t, dt));
 
            // Hydrodynamic dispersion
            GlobalDimMatrixType const D = NumLib::computeHydrodynamicDispersion(
                _process_data.stabilizer, _element.getID(), Dp, q, phi, alpha_T,
                alpha_L);
 
            cache_mat.col(ip).noalias() = q * c_ip - phi * D * dNdx * c;
        }
 
        return cache;
    }
 
    void postTimestepConcrete(Eigen::VectorXd const& /*local_x*/,
                              Eigen::VectorXd const& /*local_x_prev*/,
                              double const /*t*/, double const /*dt*/,
                              bool const /*use_monolithic_scheme*/,
                              int const /*process_id*/) override
    {
        unsigned const n_integration_points =
            _integration_method.getNumberOfPoints();
 
        for (unsigned ip = 0; ip < n_integration_points; ip++)
        {
            _ip_data[ip].pushBackState();
        }
    }
 
private:
    MeshLib::Element const& _element;
    ComponentTransportProcessData const& _process_data;
 
    NumLib::GenericIntegrationMethod const& _integration_method;
    std::vector<std::reference_wrapper<ProcessVariable>> const
        _transport_process_variables;
 
    std::vector<
        IntegrationPointData<NodalRowVectorType, GlobalDimNodalMatrixType>,
        Eigen::aligned_allocator<
            IntegrationPointData<NodalRowVectorType, GlobalDimNodalMatrixType>>>
        _ip_data;
};
 
}  // namespace ComponentTransport
}  // namespace ProcessLib