Lubby2.cpp

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#include "Lubby2.h"
 
#include <Eigen/LU>
 
#include "MaterialLib/MPL/Utils/GetSymmetricTensor.h"
 
namespace MPL = MaterialPropertyLib;
 
namespace MaterialLib
{
namespace Solids
{
namespace Lubby2
{
template <int DisplacementDim>
Eigen::Matrix<double, Lubby2<DisplacementDim>::JacobianResidualSize,
              Lubby2<DisplacementDim>::KelvinVectorSize>
calculatedGdEBurgers()
{
    Eigen::Matrix<double, Lubby2<DisplacementDim>::JacobianResidualSize,
                  Lubby2<DisplacementDim>::KelvinVectorSize>
        dGdE =
            Eigen::Matrix<double, Lubby2<DisplacementDim>::JacobianResidualSize,
                          Lubby2<DisplacementDim>::KelvinVectorSize>::Zero();
    dGdE.template topLeftCorner<Lubby2<DisplacementDim>::KelvinVectorSize,
                                Lubby2<DisplacementDim>::KelvinVectorSize>()
        .diagonal()
        .setConstant(-2.);
    return dGdE;
}
 
template <int DisplacementDim, typename LinearSolver>
MathLib::KelvinVector::KelvinMatrixType<DisplacementDim> tangentStiffnessA(
    double const GM0, double const KM0, LinearSolver const& linear_solver)
{
    // Calculate dGdE for time step
    auto const dGdE = calculatedGdEBurgers<DisplacementDim>();
 
    // Consistent tangent from local Newton iteration of material
    // functionals.
    // Only the upper left block is relevant for the global tangent.
    static int const KelvinVectorSize =
        MathLib::KelvinVector::kelvin_vector_dimensions(DisplacementDim);
    using KelvinMatrix =
        MathLib::KelvinVector::KelvinMatrixType<DisplacementDim>;
 
    KelvinMatrix const dzdE =
        linear_solver.solve(-dGdE)
            .template topLeftCorner<KelvinVectorSize, KelvinVectorSize>();
 
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
    auto const& P_sph = Invariants::spherical_projection;
    auto const& P_dev = Invariants::deviatoric_projection;
 
    KelvinMatrix C = GM0 * dzdE * P_dev + 3. * KM0 * P_sph;
    return C;
};
 
template <int DisplacementDim>
std::optional<std::tuple<typename Lubby2<DisplacementDim>::KelvinVector,
                         std::unique_ptr<typename MechanicsBase<
                             DisplacementDim>::MaterialStateVariables>,
                         typename Lubby2<DisplacementDim>::KelvinMatrix>>
Lubby2<DisplacementDim>::integrateStress(
    MaterialPropertyLib::VariableArray const& variable_array_prev,
    MaterialPropertyLib::VariableArray const& variable_array, double const t,
    ParameterLib::SpatialPosition const& x, double const dt,
    typename MechanicsBase<DisplacementDim>::MaterialStateVariables const&
        material_state_variables) const
{
    auto const& eps_m = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array.mechanical_strain);
    auto const& eps_m_prev = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array_prev.mechanical_strain);
    auto const& sigma_prev = std::get<MPL::SymmetricTensor<DisplacementDim>>(
        variable_array_prev.stress);
 
    using Invariants = MathLib::KelvinVector::Invariants<KelvinVectorSize>;
 
    assert(dynamic_cast<MaterialStateVariables const*>(
               &material_state_variables) != nullptr);
    MaterialStateVariables state(
        static_cast<MaterialStateVariables const&>(material_state_variables));
    state.setInitialConditions();
 
    auto local_lubby2_properties =
        detail::LocalLubby2Properties<DisplacementDim>{t, x, _mp};
 
    // calculation of deviatoric parts
    auto const& P_dev = Invariants::deviatoric_projection;
    KelvinVector const eps_m_d_i = P_dev * eps_m;
    KelvinVector const eps_m_d_t = P_dev * eps_m_prev;
 
    // initial guess as elastic predictor.
    KelvinVector sigd_j = 2.0 * (eps_m_d_i - state.eps_M_t - state.eps_K_t);
    // Note: sigd_t contains dimensionless stresses!
    KelvinVector sigd_t = P_dev * sigma_prev / local_lubby2_properties.GM0;
 
    // Calculate effective stress and update material properties
    double sig_eff = Invariants::equivalentStress(sigd_j);
    local_lubby2_properties.update(sig_eff);
 
    using LocalJacobianMatrix =
        Eigen::Matrix<double, KelvinVectorSize * 3, KelvinVectorSize * 3,
                      Eigen::RowMajor>;
 
    // Linear solver for the newton loop is required after the loop with the
    // same matrix. This saves one decomposition.
    Eigen::FullPivLU<LocalJacobianMatrix> linear_solver;
 
    // Different solvers are available for the solution of the local system.
    // TODO Make the following choice of linear solvers available from the
    // input file configuration:
    //      K_loc.partialPivLu().solve(-res_loc);
    //      K_loc.fullPivLu().solve(-res_loc);
    //      K_loc.householderQr().solve(-res_loc);
    //      K_loc.colPivHouseholderQr().solve(res_loc);
    //      K_loc.fullPivHouseholderQr().solve(-res_loc);
    //      K_loc.llt().solve(-res_loc);
    //      K_loc.ldlt().solve(-res_loc);
 
    LocalJacobianMatrix K_loc;
    {  // Local Newton solver
        using LocalResidualVector =
            Eigen::Matrix<double, KelvinVectorSize * 3, 1>;
 
        auto const update_residual = [&](LocalResidualVector& residual)
        {
            calculateResidualBurgers(dt, eps_m_d_i, eps_m_d_t, sigd_j, sigd_t,
                                     state.eps_K_j, state.eps_K_t,
                                     state.eps_M_j, state.eps_M_t, residual,
                                     local_lubby2_properties);
        };
 
        auto const update_jacobian = [&](LocalJacobianMatrix& jacobian)
        {
            calculateJacobianBurgers(
                t, x, dt, jacobian, sig_eff, sigd_j, state.eps_K_j,
                local_lubby2_properties);  // for solution dependent Jacobians
        };
 
        auto const update_solution = [&](LocalResidualVector const& increment)
        {
            // increment solution vectors
            sigd_j.noalias() += increment.template segment<KelvinVectorSize>(
                KelvinVectorSize * 0);
            state.eps_K_j.noalias() +=
                increment.template segment<KelvinVectorSize>(KelvinVectorSize *
                                                             1);
            state.eps_M_j.noalias() +=
                increment.template segment<KelvinVectorSize>(KelvinVectorSize *
                                                             2);
 
            // Calculate effective stress and update material properties
            sig_eff = MathLib::KelvinVector::Invariants<
                KelvinVectorSize>::equivalentStress(sigd_j);
            local_lubby2_properties.update(sig_eff);
        };
 
        auto newton_solver = NumLib::NewtonRaphson<
            decltype(linear_solver), LocalJacobianMatrix,
            decltype(update_jacobian), LocalResidualVector,
            decltype(update_residual), decltype(update_solution)>(
            linear_solver, update_jacobian, update_residual, update_solution,
            _nonlinear_solver_parameters);
 
        auto const success_iterations = newton_solver.solve(K_loc);
 
        if (!success_iterations)
        {
            return {};
        }
 
        // If the Newton loop didn't run, the linear solver will not be
        // initialized.
        // This happens usually for the first iteration of the first timestep.
        if (*success_iterations == 0)
        {
            linear_solver.compute(K_loc);
        }
    }
 
    KelvinMatrix C =
        tangentStiffnessA<DisplacementDim>(local_lubby2_properties.GM0,
                                           local_lubby2_properties.KM0,
                                           linear_solver);
 
    // Hydrostatic part for the stress and the tangent.
    double const delta_eps_m_trace = Invariants::trace(eps_m - eps_m_prev);
    double const sigma_trace_prev = Invariants::trace(sigma_prev);
    KelvinVector const sigma =
        local_lubby2_properties.GM0 * sigd_j +
        (local_lubby2_properties.KM0 * delta_eps_m_trace +
         sigma_trace_prev / 3.) *
            Invariants::identity2;
    return {std::make_tuple(
        sigma,
        std::unique_ptr<
            typename MechanicsBase<DisplacementDim>::MaterialStateVariables>{
            new MaterialStateVariables{state}},
        C)};
}
 
template <int DisplacementDim>
void Lubby2<DisplacementDim>::calculateResidualBurgers(
    const double dt,
    const KelvinVector& strain_curr,
    const KelvinVector& strain_t,
    const KelvinVector& stress_curr,
    const KelvinVector& stress_t,
    const KelvinVector& strain_Kel_curr,
    const KelvinVector& strain_Kel_t,
    const KelvinVector& strain_Max_curr,
    const KelvinVector& strain_Max_t,
    ResidualVector& res,
    detail::LocalLubby2Properties<DisplacementDim> const& properties) const
{
    // calculate stress residual
    res.template segment<KelvinVectorSize>(0).noalias() =
        (stress_curr - stress_t) -
        2. * ((strain_curr - strain_t) - (strain_Kel_curr - strain_Kel_t) -
              (strain_Max_curr - strain_Max_t));
 
    // calculate Kelvin strain residual
    res.template segment<KelvinVectorSize>(KelvinVectorSize).noalias() =
        (strain_Kel_curr - strain_Kel_t) -
        dt / (2. * properties.etaK) *
            (properties.GM0 * stress_curr -
             2. * properties.GK * strain_Kel_curr);
 
    // calculate Maxwell strain residual
    res.template segment<KelvinVectorSize>(2 * KelvinVectorSize).noalias() =
        (strain_Max_curr - strain_Max_t) -
        dt * 0.5 * properties.GM0 / properties.etaM * stress_curr;
}
 
template <int DisplacementDim>
void Lubby2<DisplacementDim>::calculateJacobianBurgers(
    double const t,
    ParameterLib::SpatialPosition const& x,
    const double dt,
    JacobianMatrix& Jac,
    double s_eff,
    const KelvinVector& sig_i,
    const KelvinVector& eps_K_i,
    detail::LocalLubby2Properties<DisplacementDim> const& properties) const
{
    Jac.setZero();
 
    // build G_11
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(0, 0)
        .diagonal()
        .setConstant(1.);
 
    // build G_12
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(0, KelvinVectorSize)
        .diagonal()
        .setConstant(2.);
 
    // build G_13
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(0,
                                                           2 * KelvinVectorSize)
        .diagonal()
        .setConstant(2.);
 
    // build G_21
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(KelvinVectorSize, 0)
        .noalias() =
        -0.5 * dt * properties.GM0 / properties.etaK * KelvinMatrix::Identity();
    if (s_eff > 0.)
    {
        KelvinVector const eps_K_aid =
            1. / (properties.etaK * properties.etaK) *
            (properties.GM0 * sig_i - 2. * properties.GK * eps_K_i);
 
        KelvinVector const dG_K = 1.5 * _mp.mK(t, x)[0] * properties.GK *
                                  properties.GM0 / s_eff * sig_i;
        KelvinVector const dmu_vK = 1.5 * _mp.mvK(t, x)[0] * properties.GM0 *
                                    properties.etaK / s_eff * sig_i;
        Jac.template block<KelvinVectorSize, KelvinVectorSize>(KelvinVectorSize,
                                                               0)
            .noalias() += 0.5 * dt * eps_K_aid * dmu_vK.transpose() +
                          dt / properties.etaK * eps_K_i * dG_K.transpose();
    }
 
    // build G_22
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(KelvinVectorSize,
                                                           KelvinVectorSize)
        .diagonal()
        .setConstant(1. + dt * properties.GK / properties.etaK);
 
    // nothing to do for G_23
 
    // build G_31
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(2 * KelvinVectorSize,
                                                           0)
        .noalias() =
        -0.5 * dt * properties.GM0 / properties.etaM * KelvinMatrix::Identity();
    if (s_eff > 0.)
    {
        KelvinVector const dmu_vM = 1.5 * _mp.mvM(t, x)[0] * properties.GM0 *
                                    properties.etaM / s_eff * sig_i;
        Jac.template block<KelvinVectorSize, KelvinVectorSize>(
               2 * KelvinVectorSize, 0)
            .noalias() += 0.5 * dt * properties.GM0 /
                          (properties.etaM * properties.etaM) * sig_i *
                          dmu_vM.transpose();
    }
 
    // nothing to do for G_32
 
    // build G_33
    Jac.template block<KelvinVectorSize, KelvinVectorSize>(2 * KelvinVectorSize,
                                                           2 * KelvinVectorSize)
        .diagonal()
        .setConstant(1.);
}
 
template class Lubby2<2>;
template class Lubby2<3>;
 
}  // namespace Lubby2
}  // namespace Solids
}  // namespace MaterialLib