ReducedUnsteadyNS.C

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 * In real Time Highly Advanced Computational Applications for Finite Volumes
 * Copyright (C) 2017 by the ITHACA-FV authors
-------------------------------------------------------------------------------
 
License
    This file is part of ITHACA-FV
 
    ITHACA-FV is free software: you can redistribute it and/or modify
    it under the terms of the GNU Lesser General Public License as published by
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    but WITHOUT ANY WARRANTY; without even the implied warranty of
    MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
    GNU Lesser General Public License for more details.
 
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#include "ReducedUnsteadyNS.H"
 
 
// * * * * * * * * * * * * * * * Constructors * * * * * * * * * * * * * * * * //
 
// Constructor initialization
reducedUnsteadyNS::reducedUnsteadyNS()
{
}
 
reducedUnsteadyNS::reducedUnsteadyNS(unsteadyNS& FOMproblem)
    :
    problem(& FOMproblem)
{
    N_BC = problem->inletIndex.rows();
    Nphi_u = problem->B_matrix.rows();
    Nphi_p = problem->K_matrix.cols();
 
    // Create locally the velocity modes
    for (int k = 0; k < problem->liftfield.size(); k++)
    {
        Umodes.append((problem->liftfield[k]).clone());
    }
 
    for (int k = 0; k < problem->NUmodes; k++)
    {
        Umodes.append((problem->Umodes[k]).clone());
    }
 
    for (int k = 0; k < problem->NSUPmodes; k++)
    {
        Umodes.append((problem->supmodes[k]).clone());
    }
 
    // Create locally the pressure modes
    for (int k = 0; k < problem->NPmodes; k++)
    {
        Pmodes.append((problem->Pmodes[k]).clone());
    }
 
    newton_object_sup = newton_unsteadyNS_sup(Nphi_u + Nphi_p, Nphi_u + Nphi_p,
                        FOMproblem);
    newton_object_PPE = newton_unsteadyNS_PPE(Nphi_u + Nphi_p, Nphi_u + Nphi_p,
                        FOMproblem);
}
 
// * * * * * * * * * * * * * Operators supremizer  * * * * * * * * * * * * * //
 
// Operator to evaluate the residual for the Supremizer approach
int newton_unsteadyNS_sup::operator()(const Eigen::VectorXd& x,
                                      Eigen::VectorXd& fvec) const
{
    Eigen::VectorXd a_dot(Nphi_u);
    Eigen::VectorXd a_tmp(Nphi_u);
    Eigen::VectorXd b_tmp(Nphi_p);
    a_tmp = x.head(Nphi_u);
    b_tmp = x.tail(Nphi_p);
 
    // Choose the order of the numerical difference scheme for approximating the time derivative
    if (problem->timeDerivativeSchemeOrder == "first")
    {
        a_dot = (x.head(Nphi_u) - y_old.head(Nphi_u)) / dt;
    }
    else
    {
        a_dot = (1.5 * x.head(Nphi_u) - 2 * y_old.head(Nphi_u) + 0.5 * yOldOld.head(
                     Nphi_u)) / dt;
    }
 
    // Convective term
    Eigen::MatrixXd cc(1, 1);
    // Mom Term
    Eigen::VectorXd M1 = problem->B_matrix * a_tmp * nu;
    // Gradient of pressure
    Eigen::VectorXd M2 = problem->K_matrix * b_tmp;
    // Mass Term
    Eigen::VectorXd M5 = problem->M_matrix * a_dot;
    // Pressure Term
    Eigen::VectorXd M3 = problem->P_matrix * a_tmp;
    // Penalty term
    Eigen::MatrixXd penaltyU = Eigen::MatrixXd::Zero(Nphi_u, N_BC);
 
    // Term for penalty method
    if (problem->bcMethod == "penalty")
    {
        for (int l = 0; l < N_BC; l++)
        {
            penaltyU.col(l) = tauU(l,
                                   0) * (BC(l) * problem->bcVelVec[l] - problem->bcVelMat[l] *
                                         a_tmp);
        }
    }
 
    for (int i = 0; i < Nphi_u; i++)
    {
        cc = a_tmp.transpose() * Eigen::SliceFromTensor(problem->C_tensor, 0,
             i) * a_tmp;
        fvec(i) = - M5(i) + M1(i) - cc(0, 0) - M2(i);
 
        if (problem->bcMethod == "penalty")
        {
            for (int l = 0; l < N_BC; l++)
            {
                fvec(i) += penaltyU(i, l);
            }
        }
    }
 
    for (int j = 0; j < Nphi_p; j++)
    {
        int k = j + Nphi_u;
        fvec(k) = M3(j);
    }
 
    if (problem->bcMethod == "lift")
    {
        for (int j = 0; j < N_BC; j++)
        {
            fvec(j) = x(j) - BC(j);
        }
    }
 
    return 0;
}
 
// Operator to evaluate the Jacobian for the supremizer approach
int newton_unsteadyNS_sup::df(const Eigen::VectorXd& x,
                              Eigen::MatrixXd& fjac) const
{
    Eigen::NumericalDiff<newton_unsteadyNS_sup> numDiff(* this);
    numDiff.df(x, fjac);
    return 0;
}
 
// * * * * * * * * * * * * * * * Operators PPE * * * * * * * * * * * * * * * //
 
// Operator to evaluate the residual for the Pressure Poisson Equation (PPE) approach
int newton_unsteadyNS_PPE::operator()(const Eigen::VectorXd& x,
                                      Eigen::VectorXd& fvec) const
{
    Eigen::VectorXd a_dot(Nphi_u);
    Eigen::VectorXd a_tmp(Nphi_u);
    Eigen::VectorXd b_tmp(Nphi_p);
    a_tmp = x.head(Nphi_u);
    b_tmp = x.tail(Nphi_p);
 
    // Choose the order of the numerical difference scheme for approximating the time derivative
    if (problem->timeDerivativeSchemeOrder == "first")
    {
        a_dot = (x.head(Nphi_u) - y_old.head(Nphi_u)) / dt;
    }
    else
    {
        a_dot = (1.5 * x.head(Nphi_u) - 2 * y_old.head(Nphi_u) + 0.5 * yOldOld.head(
                     Nphi_u)) / dt;
    }
 
    // Convective terms
    Eigen::MatrixXd cc(1, 1);
    Eigen::MatrixXd gg(1, 1);
    // Mom Term
    Eigen::VectorXd M1 = problem->B_matrix * a_tmp * nu;
    // Gradient of pressure
    Eigen::VectorXd M2 = problem->K_matrix * b_tmp;
    // Mass Term
    Eigen::VectorXd M5 = problem->M_matrix * a_dot;
    // Pressure Term
    Eigen::VectorXd M3 = problem->D_matrix * b_tmp;
    // BC PPE
    Eigen::VectorXd M7 = problem->BC3_matrix * a_tmp * nu;
    // BC PPE time-dependents BCs
    Eigen::VectorXd M8 = problem->BC4_matrix * a_dot;
    // Penalty term
    Eigen::MatrixXd penaltyU = Eigen::MatrixXd::Zero(Nphi_u, N_BC);
 
    // Term for penalty method
    if (problem->bcMethod == "penalty")
    {
        for (int l = 0; l < N_BC; l++)
        {
            penaltyU.col(l) = tauU(l,
                                   0) * (BC(l) * problem->bcVelVec[l] - problem->bcVelMat[l] *
                                         a_tmp);
        }
    }
 
    for (int i = 0; i < Nphi_u; i++)
    {
        cc = a_tmp.transpose() * Eigen::SliceFromTensor(problem->C_tensor, 0,
             i) * a_tmp;
        fvec(i) = - M5(i) + M1(i) - cc(0, 0) - M2(i);
 
        if (problem->bcMethod == "penalty")
        {
            for (int l = 0; l < N_BC; l++)
            {
                fvec(i) += penaltyU(i, l);
            }
        }
    }
 
    for (int j = 0; j < Nphi_p; j++)
    {
        int k = j + Nphi_u;
        gg = a_tmp.transpose() * Eigen::SliceFromTensor(problem->gTensor, 0,
             j) * a_tmp;
        fvec(k) = M3(j, 0) + gg(0, 0) - M7(j, 0);
 
        if (problem->timedepbcMethod == "yes")
        {
            fvec(k) += M8(j, 0);
        }
    }
 
    if (problem->bcMethod == "lift")
    {
        for (int j = 0; j < N_BC; j++)
        {
            fvec(j) = x(j) - BC(j);
        }
    }
 
    return 0;
}
 
// Operator to evaluate the Jacobian for the supremizer approach
int newton_unsteadyNS_PPE::df(const Eigen::VectorXd& x,
                              Eigen::MatrixXd& fjac) const
{
    Eigen::NumericalDiff<newton_unsteadyNS_PPE> numDiff(* this);
    numDiff.df(x, fjac);
    return 0;
}
 
 
// * * * * * * * * * * * * * Solve Functions supremizer * * * * * * * * * * * //
 
void reducedUnsteadyNS::solveOnline_sup(Eigen::MatrixXd vel,
                                        int startSnap)
{
    M_Assert(exportEvery >= dt,
             "The time step dt must be smaller than exportEvery.");
    M_Assert(storeEvery >= dt,
             "The time step dt must be smaller than storeEvery.");
    M_Assert(ITHACAutilities::isInteger(storeEvery / dt) == true,
             "The variable storeEvery must be an integer multiple of the time step dt.");
    M_Assert(ITHACAutilities::isInteger(exportEvery / dt) == true,
             "The variable exportEvery must be an integer multiple of the time step dt.");
    M_Assert(ITHACAutilities::isInteger(exportEvery / storeEvery) == true,
             "The variable exportEvery must be an integer multiple of the variable storeEvery.");
    int numberOfStores = round(storeEvery / dt);
 
    if (problem->bcMethod == "lift")
    {
        vel_now = setOnlineVelocity(vel);
    }
    else if (problem->bcMethod == "penalty")
    {
        vel_now = vel;
    }
 
    // Create and resize the solution vector
    y.resize(Nphi_u + Nphi_p, 1);
    y.setZero();
    y.head(Nphi_u) = ITHACAutilities::getCoeffs(problem->Ufield[startSnap],
                     Umodes);
    y.tail(Nphi_p) = ITHACAutilities::getCoeffs(problem->Pfield[startSnap],
                     Pmodes);
    int nextStore = 0;
    int counter2 = 0;
 
    // Change initial condition for the lifting function
    if (problem->bcMethod == "lift")
    {
        for (int j = 0; j < N_BC; j++)
        {
            y(j) = vel_now(j, 0);
        }
    }
 
    // Set some properties of the newton object
    newton_object_sup.nu = nu;
    newton_object_sup.y_old = y;
    newton_object_sup.yOldOld = newton_object_sup.y_old;
    newton_object_sup.dt = dt;
    newton_object_sup.BC.resize(N_BC);
    newton_object_sup.tauU = tauU;
 
    for (int j = 0; j < N_BC; j++)
    {
        newton_object_sup.BC(j) = vel_now(j, 0);
    }
 
    // Set number of online solutions
    int Ntsteps = static_cast<int>((finalTime - tstart) / dt);
    int onlineSize = static_cast<int>(Ntsteps / numberOfStores);
    online_solution.resize(onlineSize);
    // Set the initial time
    time = tstart;
    // Counting variable
    int counter = 0;
    // Create vector to store temporal solution and save initial condition as first solution
    Eigen::MatrixXd tmp_sol(Nphi_u + Nphi_p + 1, 1);
    tmp_sol(0) = time;
    tmp_sol.col(0).tail(y.rows()) = y;
    online_solution[counter] = tmp_sol;
    counter ++;
    counter2++;
    nextStore += numberOfStores;
    // Create nonlinear solver object
    Eigen::HybridNonLinearSolver<newton_unsteadyNS_sup> hnls(newton_object_sup);
    // Set output colors for fancy output
    Color::Modifier red(Color::FG_RED);
    Color::Modifier green(Color::FG_GREEN);
    Color::Modifier def(Color::FG_DEFAULT);
 
    while (time < finalTime)
    {
        time = time + dt;
 
        // Set time-dependent BCs
        if (problem->timedepbcMethod == "yes" )
        {
            for (int j = 0; j < N_BC; j++)
            {
                newton_object_sup.BC(j) = vel_now(j, counter);
            }
        }
 
        Eigen::VectorXd res(y);
        res.setZero();
        hnls.solve(y);
 
        if (problem->bcMethod == "lift")
        {
            for (int j = 0; j < N_BC; j++)
            {
                if (problem->timedepbcMethod == "no" )
                {
                    y(j) = vel_now(j, 0);
                }
                else if (problem->timedepbcMethod == "yes" )
                {
                    y(j) = vel_now(j, counter);
                }
            }
        }
 
        newton_object_sup.operator()(y, res);
        newton_object_sup.yOldOld = newton_object_sup.y_old;
        newton_object_sup.y_old = y;
        std::cout << "################## Online solve N° " << counter <<
                  " ##################" << std::endl;
        Info << "Time = " << time << endl;
 
        if (res.norm() < 1e-5)
        {
            std::cout << green << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                      hnls.iter << " iterations " << def << std::endl << std::endl;
        }
        else
        {
            std::cout << red << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                      hnls.iter << " iterations " << def << std::endl << std::endl;
        }
 
        tmp_sol(0) = time;
        tmp_sol.col(0).tail(y.rows()) = y;
 
        if (counter == nextStore)
        {
            if (counter2 >= online_solution.size())
            {
                online_solution.append(tmp_sol);
            }
            else
            {
                online_solution[counter2] = tmp_sol;
            }
 
            nextStore += numberOfStores;
            counter2 ++;
        }
 
        counter ++;
    }
 
    // Export the solution
    ITHACAstream::exportMatrix(online_solution, "red_coeff", "python",
                               "./ITHACAoutput/red_coeff");
    ITHACAstream::exportMatrix(online_solution, "red_coeff", "matlab",
                               "./ITHACAoutput/red_coeff");
}
 
// * * * * * * * * * * * * * * * Solve Functions PPE * * * * * * * * * * * * * //
 
void reducedUnsteadyNS::solveOnline_PPE(Eigen::MatrixXd vel,
                                        int startSnap)
{
    M_Assert(exportEvery >= dt,
             "The time step dt must be smaller than exportEvery.");
    M_Assert(storeEvery >= dt,
             "The time step dt must be smaller than storeEvery.");
    M_Assert(ITHACAutilities::isInteger(storeEvery / dt) == true,
             "The variable storeEvery must be an integer multiple of the time step dt.");
    M_Assert(ITHACAutilities::isInteger(exportEvery / dt) == true,
             "The variable exportEvery must be an integer multiple of the time step dt.");
    M_Assert(ITHACAutilities::isInteger(exportEvery / storeEvery) == true,
             "The variable exportEvery must be an integer multiple of the variable storeEvery.");
    int numberOfStores = round(storeEvery / dt);
 
    if (problem->bcMethod == "lift")
    {
        vel_now = setOnlineVelocity(vel);
    }
    else if (problem->bcMethod == "penalty")
    {
        vel_now = vel;
    }
 
    // Create and resize the solution vector
    y.resize(Nphi_u + Nphi_p, 1);
    y.setZero();
    // Set Initial Conditions
    y.head(Nphi_u) = ITHACAutilities::getCoeffs(problem->Ufield[startSnap],
                     Umodes);
    y.tail(Nphi_p) = ITHACAutilities::getCoeffs(problem->Pfield[startSnap],
                     Pmodes);
    int nextStore = 0;
    int counter2 = 0;
 
    // Change initial condition for the lifting function
    if (problem->bcMethod == "lift")
    {
        for (int j = 0; j < N_BC; j++)
        {
            y(j) = vel_now(j, 0);
        }
    }
 
    // Set some properties of the newton object
    newton_object_PPE.nu = nu;
    newton_object_PPE.y_old = y;
    newton_object_PPE.yOldOld = newton_object_PPE.y_old;
    newton_object_PPE.dt = dt;
    newton_object_PPE.BC.resize(N_BC);
    newton_object_PPE.tauU = tauU;
 
    for (int j = 0; j < N_BC; j++)
    {
        newton_object_PPE.BC(j) = vel_now(j, 0);
    }
 
    // Set number of online solutions
    int Ntsteps = static_cast<int>((finalTime - tstart) / dt);
    int onlineSize = static_cast<int>(Ntsteps / numberOfStores);
    online_solution.resize(onlineSize);
    // Set the initial time
    time = tstart;
    // Counting variable
    int counter = 0;
    // Create vector to store temporal solution and save initial condition as first solution
    Eigen::MatrixXd tmp_sol(Nphi_u + Nphi_p + 1, 1);
    tmp_sol(0) = time;
    tmp_sol.col(0).tail(y.rows()) = y;
    online_solution[counter] = tmp_sol;
    counter ++;
    counter2++;
    nextStore += numberOfStores;
    // Create nonlinear solver object
    Eigen::HybridNonLinearSolver<newton_unsteadyNS_PPE> hnls(newton_object_PPE);
    // Set output colors for fancy output
    Color::Modifier red(Color::FG_RED);
    Color::Modifier green(Color::FG_GREEN);
    Color::Modifier def(Color::FG_DEFAULT);
 
    // Start the time loop
    while (time < finalTime)
    {
        time = time + dt;
 
        // Set time-dependent BCs
        if (problem->timedepbcMethod == "yes" )
        {
            for (int j = 0; j < N_BC; j++)
            {
                newton_object_PPE.BC(j) = vel_now(j, counter);
            }
        }
 
        Eigen::VectorXd res(y);
        res.setZero();
        hnls.solve(y);
 
        if (problem->bcMethod == "lift")
        {
            for (int j = 0; j < N_BC; j++)
            {
                if (problem->timedepbcMethod == "no" )
                {
                    y(j) = vel_now(j, 0);
                }
                else if (problem->timedepbcMethod == "yes" )
                {
                    y(j) = vel_now(j, counter);
                }
            }
        }
 
        newton_object_PPE.operator()(y, res);
        newton_object_PPE.yOldOld = newton_object_PPE.y_old;
        newton_object_PPE.y_old = y;
        std::cout << "################## Online solve N° " << counter <<
                  " ##################" << std::endl;
        Info << "Time = " << time << endl;
 
        if (res.norm() < 1e-5)
        {
            std::cout << green << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                      hnls.iter << " iterations " << def << std::endl << std::endl;
        }
        else
        {
            std::cout << red << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                      hnls.iter << " iterations " << def << std::endl << std::endl;
        }
 
        tmp_sol(0) = time;
        tmp_sol.col(0).tail(y.rows()) = y;
 
        if (counter == nextStore)
        {
            if (counter2 >= online_solution.size())
            {
                online_solution.append(tmp_sol);
            }
            else
            {
                online_solution[counter2] = tmp_sol;
            }
 
            nextStore += numberOfStores;
            counter2 ++;
        }
 
        counter ++;
    }
 
    // Export the solution
    ITHACAstream::exportMatrix(online_solution, "red_coeff", "python",
                               "./ITHACAoutput/red_coeff");
    ITHACAstream::exportMatrix(online_solution, "red_coeff", "matlab",
                               "./ITHACAoutput/red_coeff");
}
 
Eigen::MatrixXd reducedUnsteadyNS::penalty_sup(Eigen::MatrixXd& vel_now,
        Eigen::MatrixXd& tauIter,
        int startSnap)
{
    // Initialize new value on boundaries
    Eigen::MatrixXd valBC = Eigen::MatrixXd::Zero(N_BC, timeStepPenalty);
    // Initialize old values on boundaries
    Eigen::MatrixXd valBC0 = Eigen::MatrixXd::Zero(N_BC, timeStepPenalty);
    int Iter = 0;
    Eigen::VectorXd diffvel =  (vel_now.col(timeStepPenalty - 1) - valBC.col(
                                    timeStepPenalty - 1));
    diffvel = diffvel.cwiseAbs();
 
    while (diffvel.maxCoeff() > tolerancePenalty && Iter < maxIterPenalty)
    {
        if ((valBC.col(timeStepPenalty - 1) - valBC0.col(timeStepPenalty - 1)).sum() !=
                0)
        {
            for (int j = 0; j < N_BC; j++)
            {
                tauIter(j, 0) = tauIter(j, 0) * diffvel(j) / tolerancePenalty;
            }
        }
 
        std::cout << "Solving for penalty factor(s): " << tauIter << std::endl;
        std::cout << "number of iterations: " << Iter << std::endl;
        //  Set the old boundary value to the current value
        valBC0  = valBC;
        y.resize(Nphi_u + Nphi_p, 1);
        y.setZero();
        y.head(Nphi_u) = ITHACAutilities::getCoeffs(problem->Ufield[startSnap],
                         Umodes);
        y.tail(Nphi_p) = ITHACAutilities::getCoeffs(problem->Pfield[startSnap],
                         Pmodes);
        // Set some properties of the newton object
        newton_object_sup.nu = nu;
        newton_object_sup.y_old = y;
        newton_object_sup.dt = dt;
        newton_object_sup.BC.resize(N_BC);
        newton_object_sup.tauU = tauIter;
 
        // Set boundary conditions
        for (int j = 0; j < N_BC; j++)
        {
            newton_object_sup.BC(j) = vel_now(j, 0);
        }
 
        // Create nonlinear solver object
        Eigen::HybridNonLinearSolver<newton_unsteadyNS_sup> hnls(newton_object_sup);
        // Set output colors for fancy output
        Color::Modifier red(Color::FG_RED);
        Color::Modifier green(Color::FG_GREEN);
        Color::Modifier def(Color::FG_DEFAULT);
        // Set initially for convergence check
        Eigen::VectorXd res(y);
        res.setZero();
 
        // Start the time loop
        for (int i = 1; i < timeStepPenalty; i++)
        {
            // Set boundary conditions
            for (int j = 0; j < N_BC; j++)
            {
                if (problem->timedepbcMethod == "yes" )
                {
                    newton_object_sup.BC(j) = vel_now(j, i);
                }
                else
                {
                    newton_object_sup.BC(j) = vel_now(j, 0);
                }
            }
 
            Eigen::VectorXd res(y);
            res.setZero();
            hnls.solve(y);
            newton_object_sup.operator()(y, res);
            newton_object_sup.y_old = y;
 
            if (res.norm() < 1e-5)
            {
                std::cout << green << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                          hnls.iter << " iterations " << def << std::endl << std::endl;
            }
            else
            {
                std::cout << red << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                          hnls.iter << " iterations " << def << std::endl << std::endl;
            }
 
            volVectorField U_rec("U_rec", Umodes[0] * 0);
 
            for (int j = 0; j < Nphi_u; j++)
            {
                U_rec += Umodes[j] * y(j);
            }
 
            for (int k = 0; k < problem->inletIndex.rows(); k++)
            {
                int BCind = problem->inletIndex(k, 0);
                int BCcomp = problem->inletIndex(k, 1);
                valBC(k, i) = U_rec.boundaryFieldRef()[BCind][0].component(BCcomp);
            }
        }
 
        for (int j = 0; j < N_BC; j++)
        {
            diffvel(j) = abs(abs(vel_now(j, timeStepPenalty - 1)) - abs(valBC(j,
                             timeStepPenalty - 1)));
        }
 
        std::cout << "max error: " << diffvel.maxCoeff() << std::endl;
        // Count the number of iterations
        Iter ++;
    }
 
    std::cout << "Final penalty factor(s): " << tauIter << std::endl;
    std::cout << "Iterations: " << Iter - 1 << std::endl;
    return tauIter;
}
 
Eigen::MatrixXd reducedUnsteadyNS::penalty_PPE(Eigen::MatrixXd& vel_now,
        Eigen::MatrixXd& tauIter,
        int startSnap)
{
    // Initialize new value on boundaries
    Eigen::MatrixXd valBC = Eigen::MatrixXd::Zero(N_BC, timeStepPenalty);
    // Initialize old values on boundaries
    Eigen::MatrixXd valBC0 = Eigen::MatrixXd::Zero(N_BC, timeStepPenalty);
    int Iter = 0;
    Eigen::VectorXd diffvel =  (vel_now.col(timeStepPenalty - 1) - valBC.col(
                                    timeStepPenalty - 1));
    diffvel = diffvel.cwiseAbs();
 
    while (diffvel.maxCoeff() > tolerancePenalty && Iter < maxIterPenalty)
    {
        if ((valBC.col(timeStepPenalty - 1) - valBC0.col(timeStepPenalty - 1)).sum() !=
                0)
        {
            for (int j = 0; j < N_BC; j++)
            {
                tauIter(j, 0) = tauIter(j, 0) * diffvel(j) / tolerancePenalty;
            }
        }
 
        std::cout << "Solving for penalty factor(s): " << tauIter << std::endl;
        std::cout << "number of iterations: " << Iter << std::endl;
        //  Set the old boundary value to the current value
        valBC0  = valBC;
        y.resize(Nphi_u + Nphi_p, 1);
        y.setZero();
        y.head(Nphi_u) = ITHACAutilities::getCoeffs(problem->Ufield[startSnap],
                         Umodes);
        y.tail(Nphi_p) = ITHACAutilities::getCoeffs(problem->Pfield[startSnap],
                         Pmodes);
        // Set some properties of the newton object
        newton_object_PPE.nu = nu;
        newton_object_PPE.y_old = y;
        newton_object_PPE.yOldOld = newton_object_PPE.y_old;
        newton_object_PPE.dt = dt;
        newton_object_PPE.BC.resize(N_BC);
        newton_object_PPE.tauU = tauIter;
 
        // Set boundary conditions
        for (int j = 0; j < N_BC; j++)
        {
            newton_object_PPE.BC(j) = vel_now(j, 0);
        }
 
        // Create nonlinear solver object
        Eigen::HybridNonLinearSolver<newton_unsteadyNS_PPE> hnls(newton_object_PPE);
        // Set output colors for fancy output
        Color::Modifier red(Color::FG_RED);
        Color::Modifier green(Color::FG_GREEN);
        Color::Modifier def(Color::FG_DEFAULT);
        // Set initially for convergence check
        Eigen::VectorXd res(y);
        res.setZero();
 
        // Start the time loop
        for (int i = 1; i < timeStepPenalty; i++)
        {
            // Set boundary conditions
            for (int j = 0; j < N_BC; j++)
            {
                if (problem->timedepbcMethod == "yes" )
                {
                    newton_object_PPE.BC(j) = vel_now(j, i);
                }
                else
                {
                    newton_object_PPE.BC(j) = vel_now(j, 0);
                }
            }
 
            Eigen::VectorXd res(y);
            res.setZero();
            hnls.solve(y);
            newton_object_PPE.operator()(y, res);
            newton_object_PPE.yOldOld = newton_object_PPE.y_old;
            newton_object_PPE.y_old = y;
 
            if (res.norm() < 1e-5)
            {
                std::cout << green << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                          hnls.iter << " iterations " << def << std::endl << std::endl;
            }
            else
            {
                std::cout << red << "|F(x)| = " << res.norm() << " - Minimun reached in " <<
                          hnls.iter << " iterations " << def << std::endl << std::endl;
            }
 
            volVectorField U_rec("U_rec", Umodes[0] * 0);
 
            for (int j = 0; j < Nphi_u; j++)
            {
                U_rec += Umodes[j] * y(j);
            }
 
            for (int k = 0; k < problem->inletIndex.rows(); k++)
            {
                int BCind = problem->inletIndex(k, 0);
                int BCcomp = problem->inletIndex(k, 1);
                valBC(k, i) = U_rec.boundaryFieldRef()[BCind][0].component(BCcomp);
            }
        }
 
        for (int j = 0; j < N_BC; j++)
        {
            diffvel(j) = abs(abs(vel_now(j, timeStepPenalty - 1)) - abs(valBC(j,
                             timeStepPenalty - 1)));
        }
 
        std::cout << "max error: " << diffvel.maxCoeff() << std::endl;
        // Count the number of iterations
        Iter ++;
    }
 
    std::cout << "Final penalty factor(s): " << tauIter << std::endl;
    std::cout << "Iterations: " << Iter - 1 << std::endl;
    return tauIter;
}
 
void reducedUnsteadyNS::reconstruct(bool exportFields, fileName folder)
{
    if (exportFields)
    {
        mkDir(folder);
        ITHACAutilities::createSymLink(folder);
    }
 
    int counter = 0;
    int nextwrite = 0;
    List < Eigen::MatrixXd> CoeffU;
    List < Eigen::MatrixXd> CoeffP;
    List <double> tValues;
    CoeffU.resize(0);
    CoeffP.resize(0);
    tValues.resize(0);
    int exportEveryIndex = round(exportEvery / storeEvery);
 
    for (int i = 0; i < online_solution.size(); i++)
    {
        if (counter == nextwrite)
        {
            Eigen::MatrixXd currentUCoeff;
            Eigen::MatrixXd currentPCoeff;
            currentUCoeff = online_solution[i].block(1, 0, Nphi_u, 1);
            currentPCoeff = online_solution[i].bottomRows(Nphi_p);
            CoeffU.append(currentUCoeff);
            CoeffP.append(currentPCoeff);
            nextwrite += exportEveryIndex;
            double timeNow = online_solution[i](0, 0);
            tValues.append(timeNow);
        }
 
        counter++;
    }
 
    volVectorField uRec("uRec", Umodes[0] * 0);
    volScalarField pRec("pRec", problem->Pmodes[0] * 0);
    uRecFields = problem->L_U_SUPmodes.reconstruct(uRec, CoeffU, "uRec");
    pRecFields = problem->Pmodes.reconstruct(pRec, CoeffP, "pRec");
 
    if (exportFields)
    {
        ITHACAstream::exportFields(uRecFields, folder,
                                   "uRec");
        ITHACAstream::exportFields(pRecFields, folder,
                                   "pRec");
    }
}
 
Eigen::MatrixXd reducedUnsteadyNS::setOnlineVelocity(Eigen::MatrixXd vel)
{
    assert(problem->inletIndex.rows() == vel.rows()
           && "Imposed boundary conditions dimensions do not match given values matrix dimensions");
    Eigen::MatrixXd vel_scal;
    vel_scal.resize(vel.rows(), vel.cols());
 
    for (int k = 0; k < problem->inletIndex.rows(); k++)
    {
        int p = problem->inletIndex(k, 0);
        int l = problem->inletIndex(k, 1);
        scalar area = gSum(problem->liftfield[0].mesh().magSf().boundaryField()[p]);
        scalar u_lf = gSum(problem->liftfield[k].mesh().magSf().boundaryField()[p] *
                           problem->liftfield[k].boundaryField()[p]).component(l) / area;
 
        for (int i = 0; i < vel.cols(); i++)
        {
            vel_scal(k, i) = vel(k, i) / u_lf;
        }
    }
 
    return vel_scal;
}
 
//************************************************************************* //