Program Listing for File TimeStepDFSPH.cpp

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#include "TimeStepDFSPH.h"
#include "SPlisHSPlasH/TimeManager.h"
#include "SPlisHSPlasH/SPHKernels.h"
#include "SimulationDataDFSPH.h"
#include <iostream>
#include "Utilities/Timing.h"
#include "Utilities/Counting.h"
#include "SPlisHSPlasH/Simulation.h"
#include "SPlisHSPlasH/BoundaryModel_Akinci2012.h"
#include "SPlisHSPlasH/BoundaryModel_Koschier2017.h"
#include "SPlisHSPlasH/BoundaryModel_Bender2019.h"


using namespace SPH;
using namespace std;
using namespace GenParam;


int TimeStepDFSPH::SOLVER_ITERATIONS_V = -1;
int TimeStepDFSPH::MAX_ITERATIONS_V = -1;
int TimeStepDFSPH::MAX_ERROR_V = -1;
int TimeStepDFSPH::USE_DIVERGENCE_SOLVER = -1;


TimeStepDFSPH::TimeStepDFSPH() :
    TimeStep(),
    m_simulationData()
{
    m_simulationData.init();
    m_iterationsV = 0;
    m_enableDivergenceSolver = true;
    m_maxIterationsV = 100;
    m_maxErrorV = static_cast<Real>(0.1);

    // add particle fields - then they can be used for the visualization and export
    Simulation *sim = Simulation::getCurrent();
    const unsigned int nModels = sim->numberOfFluidModels();
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nModels; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        model->addField({ "factor", FieldType::Scalar, [this, fluidModelIndex](const unsigned int i) -> Real* { return &m_simulationData.getFactor(fluidModelIndex, i); } });
        model->addField({ "advected density", FieldType::Scalar, [this, fluidModelIndex](const unsigned int i) -> Real* { return &m_simulationData.getDensityAdv(fluidModelIndex, i); } });
        model->addField({ "p / rho^2", FieldType::Scalar, [this, fluidModelIndex](const unsigned int i) -> Real* { return &m_simulationData.getPressureRho2(fluidModelIndex, i); }, true });
        model->addField({ "p_v / rho^2", FieldType::Scalar, [this, fluidModelIndex](const unsigned int i) -> Real* { return &m_simulationData.getPressureRho2_V(fluidModelIndex, i); }, true });
        model->addField({ "pressure acceleration", FieldType::Vector3, [this, fluidModelIndex](const unsigned int i) -> Real* { return &m_simulationData.getPressureAccel(fluidModelIndex, i)[0]; } });
    }
}

TimeStepDFSPH::~TimeStepDFSPH(void)
{
    // remove all particle fields
    Simulation *sim = Simulation::getCurrent();
    const unsigned int nModels = sim->numberOfFluidModels();
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nModels; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        model->removeFieldByName("factor");
        model->removeFieldByName("advected density");
        model->removeFieldByName("p / rho^2");
        model->removeFieldByName("p_v / rho^2");
        model->removeFieldByName("pressure acceleration");
    }
}

void TimeStepDFSPH::initParameters()
{
    TimeStep::initParameters();

    SOLVER_ITERATIONS_V = createNumericParameter("iterationsV", "Iterations (divergence)", &m_iterationsV);
    setGroup(SOLVER_ITERATIONS_V, "Simulation|DFSPH");
    setDescription(SOLVER_ITERATIONS_V, "Iterations required by the divergence solver.");
    getParameter(SOLVER_ITERATIONS_V)->setReadOnly(true);

    MAX_ITERATIONS_V = createNumericParameter("maxIterationsV", "Max. iterations (divergence)", &m_maxIterationsV);
    setGroup(MAX_ITERATIONS_V, "Simulation|DFSPH");
    setDescription(MAX_ITERATIONS_V, "Maximal number of iterations of the divergence solver.");
    static_cast<NumericParameter<unsigned int>*>(getParameter(MAX_ITERATIONS_V))->setMinValue(1);

    MAX_ERROR_V = createNumericParameter("maxErrorV", "Max. divergence error(%)", &m_maxErrorV);
    setGroup(MAX_ERROR_V, "Simulation|DFSPH");
    setDescription(MAX_ERROR_V, "Maximal divergence error (%).");
    static_cast<RealParameter*>(getParameter(MAX_ERROR_V))->setMinValue(static_cast<Real>(1e-6));

    USE_DIVERGENCE_SOLVER = createBoolParameter("enableDivergenceSolver", "Enable divergence solver", &m_enableDivergenceSolver);
    setGroup(USE_DIVERGENCE_SOLVER, "Simulation|DFSPH");
    setDescription(USE_DIVERGENCE_SOLVER, "Turn divergence solver on/off.");
}

void TimeStepDFSPH::step()
{
    Simulation *sim = Simulation::getCurrent();
    TimeManager *tm = TimeManager::getCurrent ();
    const Real h = tm->getTimeStepSize();
    const unsigned int nModels = sim->numberOfFluidModels();

    // search the neighbors for all particles
    sim->performNeighborhoodSearch();

#ifdef USE_PERFORMANCE_OPTIMIZATION
    // precompute the values V_j * grad W_ij for all neighbors
    START_TIMING("precomputeValues")
    precomputeValues();
    STOP_TIMING_AVG
#endif

    // compute volume/density maps boundary contribution
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
        computeVolumeAndBoundaryX();
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
        computeDensityAndGradient();

    // compute densities
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nModels; fluidModelIndex++)
        computeDensities(fluidModelIndex);

    // Compute the factor alpha_i for all particles i
    // using the equation (11) in [BK17]
    START_TIMING("computeDFSPHFactor");
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nModels; fluidModelIndex++)
        computeDFSPHFactor(fluidModelIndex);
    STOP_TIMING_AVG;

    // Perform divergence solve (see Algorithm 2 in [BK17])
    if (m_enableDivergenceSolver)
    {
        START_TIMING("divergenceSolve");
        divergenceSolve();
        STOP_TIMING_AVG
    }
    else
        m_iterationsV = 0;

    // Reset accelerations and add gravity
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nModels; fluidModelIndex++)
        clearAccelerations(fluidModelIndex);

    // Compute all nonpressure forces like viscosity, vorticity, ...
    sim->computeNonPressureForces();

    // Update the time step size, e.g. by using a CFL condition
    sim->updateTimeStepSize();

    // compute new velocities only considering non-pressure forces
    for (unsigned int m = 0; m < nModels; m++)
    {
        FluidModel *fm = sim->getFluidModel(m);
        const unsigned int numParticles = fm->numActiveParticles();
        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < (int)numParticles; i++)
            {
                if (fm->getParticleState(i) == ParticleState::Active)
                {
                    Vector3r &vel = fm->getVelocity(i);
                    vel += h * fm->getAcceleration(i);
                }
            }
        }
    }

    // Perform constant density solve (see Algorithm 3 in [BK17])
    START_TIMING("pressureSolve");
    pressureSolve();
    STOP_TIMING_AVG;

    // compute final positions
    for (unsigned int m = 0; m < nModels; m++)
    {
        FluidModel *fm = sim->getFluidModel(m);
        const unsigned int numParticles = fm->numActiveParticles();
        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < (int)numParticles; i++)
            {
                if (fm->getParticleState(i) == ParticleState::Active)
                {
                    Vector3r &xi = fm->getPosition(i);
                    const Vector3r &vi = fm->getVelocity(i);
                    xi += h * vi;
                }
            }
        }
    }

    // emit new particles and perform an animation field step
    sim->emitParticles();
    sim->animateParticles();

    // Compute new time
    tm->setTime (tm->getTime () + h);
}


void TimeStepDFSPH::pressureSolve()
{
    const Real h = TimeManager::getCurrent()->getTimeStepSize();
    const Real h2 = h*h;
    const Real invH = static_cast<Real>(1.0) / h;
    const Real invH2 = static_cast<Real>(1.0) / h2;
    Simulation *sim = Simulation::getCurrent();
    const unsigned int nFluids = sim->numberOfFluidModels();

    // Compute rho_adv
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        const Real density0 = model->getDensity0();
        const int numParticles = (int)model->numActiveParticles();
        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Compute rho_adv,i^(0) (see equation in Section 3.3 in [BK17])
                // using the velocities after the non-pressure forces were applied.
                computeDensityAdv(fluidModelIndex, i, h, density0);

                // In the end of Section 3.3 [BK17] we have to multiply the density
                // error with the factor alpha_i divided by h^2. Hence, we multiply
                // the factor directly by 1/h^2 here.
                m_simulationData.getFactor(fluidModelIndex, i) *= invH2;

                // For the warm start we use 0.5 times the old pressure value.
                // Note: We divide the value by h^2 since we multiplied it by h^2 at the end of
                // the last time step to make it independent of the time step size.
#ifdef USE_WARMSTART
                if (m_simulationData.getDensityAdv(fluidModelIndex, i) > 1.0)
                    m_simulationData.getPressureRho2(fluidModelIndex, i) = static_cast<Real>(0.5) * min(m_simulationData.getPressureRho2(fluidModelIndex, i), static_cast<Real>(0.00025)) * invH2;
                else
                    m_simulationData.getPressureRho2(fluidModelIndex, i) = 0.0;
#else
                // If we don't use a warm start, we directly compute a pressure value
                // by multiplying the density error with the factor.
                //m_simulationData.getPressureRho2(fluidModelIndex, i) = 0.0;
                const Real s_i = static_cast<Real>(1.0) - m_simulationData.getDensityAdv(fluidModelIndex, i);
                const Real residuum = min(s_i, static_cast<Real>(0.0));     // r = b - A*p
                m_simulationData.getPressureRho2(fluidModelIndex, i) = -residuum * m_simulationData.getFactor(fluidModelIndex, i);
#endif
            }
        }
    }

    m_iterations = 0;

    // Start solver

    Real avg_density_err = 0.0;
    bool chk = false;


    // Perform solver iterations
    while ((!chk || (m_iterations < m_minIterations)) && (m_iterations < m_maxIterations))
    {
        chk = true;
        for (unsigned int i = 0; i < nFluids; i++)
        {
            FluidModel *model = sim->getFluidModel(i);
            const Real density0 = model->getDensity0();

            avg_density_err = 0.0;
            pressureSolveIteration(i, avg_density_err);

            // Maximal allowed density fluctuation
            const Real eta = m_maxError * static_cast<Real>(0.01) * density0;  // maxError is given in percent
            chk = chk && (avg_density_err <= eta);
        }

        m_iterations++;
    }

    INCREASE_COUNTER("DFSPH - iterations", static_cast<Real>(m_iterations));

    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        const int numParticles = (int)model->numActiveParticles();
        const Real density0 = model->getDensity0();

        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Time integration of the pressure accelerations to get new velocities
                computePressureAccel(fluidModelIndex, i, density0, m_simulationData.getPressureRho2Data(), true);
                model->getVelocity(i) += h * m_simulationData.getPressureAccel(fluidModelIndex, i);
            }
        }
    }
#ifdef USE_WARMSTART
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel* model = sim->getFluidModel(fluidModelIndex);
        const int numParticles = (int)model->numActiveParticles();
        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Multiply by h^2, the time step size has to be removed
                // to make the pressure value independent
                // of the time step size
                m_simulationData.getPressureRho2(fluidModelIndex, i) *= h2;
            }
        }
    }
#endif
}

void TimeStepDFSPH::divergenceSolve()
{
    // Init parameters

    const Real h = TimeManager::getCurrent()->getTimeStepSize();
    const Real invH = static_cast<Real>(1.0) / h;
    Simulation *sim = Simulation::getCurrent();
    const unsigned int maxIter = m_maxIterationsV;
    const Real maxError = m_maxErrorV;
    const unsigned int nFluids = sim->numberOfFluidModels();

    // Compute divergence of velocity field
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        const int numParticles = (int)model->numActiveParticles();

        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Compute rho_adv,i^(0) (see equation (9) in Section 3.2 [BK17])
                // using the velocities after the non-pressure forces were applied.
                computeDensityChange(fluidModelIndex, i, h);

                Real densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
                densityAdv = max(densityAdv, static_cast<Real>(0.0));

                unsigned int numNeighbors = 0;
                for (unsigned int pid = 0; pid < sim->numberOfPointSets(); pid++)
                    numNeighbors += sim->numberOfNeighbors(fluidModelIndex, pid, i);

                // in case of particle deficiency do not perform a divergence solve
                if (!sim->is2DSimulation())
                {
                    if (numNeighbors < 20)
                        densityAdv = 0.0;
                }
                else
                {
                    if (numNeighbors < 7)
                        densityAdv = 0.0;
                }

                // In equation (11) [BK17] we have to multiply the divergence
                // error with the factor divided by h. Hence, we multiply the factor
                // directly by 1/h here.
                m_simulationData.getFactor(fluidModelIndex, i) *= invH;

                // For the warm start we use 0.5 times the old pressure value.
                // Divide the value by h. We multiplied it by h at the end of
                // the last time step to make it independent of the time step size.
#ifdef USE_WARMSTART_V
                if (densityAdv > 0.0)
                    m_simulationData.getPressureRho2_V(fluidModelIndex, i) = static_cast<Real>(0.5) * min(m_simulationData.getPressureRho2_V(fluidModelIndex, i), static_cast<Real>(0.5)) * invH;
                else
                    m_simulationData.getPressureRho2_V(fluidModelIndex, i) = 0.0;
#else
                // If we don't use a warm start, directly compute a pressure value
                // by multiplying the divergence error with the factor.
                m_simulationData.getPressureRho2_V(fluidModelIndex, i) = densityAdv * m_simulationData.getFactor(fluidModelIndex, i);
#endif
            }
        }
    }

    m_iterationsV = 0;

    // Start solver

    Real avg_density_err = 0.0;
    bool chk = false;

    // Perform solver iterations
    while ((!chk || (m_iterationsV < 1)) && (m_iterationsV < maxIter))
    {
        chk = true;
        for (unsigned int i = 0; i < nFluids; i++)
        {
            FluidModel *model = sim->getFluidModel(i);
            const Real density0 = model->getDensity0();

            avg_density_err = 0.0;
            divergenceSolveIteration(i, avg_density_err);

            // Maximal allowed density fluctuation
            // use maximal density error divided by time step size
            const Real eta = (static_cast<Real>(1.0) / h) * maxError * static_cast<Real>(0.01) * density0;  // maxError is given in percent
            chk = chk && (avg_density_err <= eta);
        }

        m_iterationsV++;
    }

    INCREASE_COUNTER("DFSPH - iterationsV", static_cast<Real>(m_iterationsV));


    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel *model = sim->getFluidModel(fluidModelIndex);
        const int numParticles = (int)model->numActiveParticles();
        const Real density0 = model->getDensity0();

        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Time integration of the pressure accelerations
                computePressureAccel(fluidModelIndex, i, density0, m_simulationData.getPressureRho2VData(), true);
                model->getVelocity(i) += h * m_simulationData.getPressureAccel(fluidModelIndex, i);

                m_simulationData.getFactor(fluidModelIndex, i) *= h;
            }
        }
    }
#ifdef USE_WARMSTART_V
    for (unsigned int fluidModelIndex = 0; fluidModelIndex < nFluids; fluidModelIndex++)
    {
        FluidModel* model = sim->getFluidModel(fluidModelIndex);
        const int numParticles = (int)model->numActiveParticles();
        #pragma omp parallel default(shared)
        {
            #pragma omp for schedule(static)
            for (int i = 0; i < numParticles; i++)
            {
                // Multiply by h, the time step size has to be removed
                // to make the pressure value independent
                // of the time step size
                m_simulationData.getPressureRho2_V(fluidModelIndex, i) *= h;
            }
        }
    }
#endif
}


void TimeStepDFSPH::pressureSolveIteration(const unsigned int fluidModelIndex, Real &avg_density_err)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const Real density0 = model->getDensity0();
    const int numParticles = (int)model->numActiveParticles();
    if (numParticles == 0)
        return;

    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();
    const Real h = TimeManager::getCurrent()->getTimeStepSize();
    const Real invH = static_cast<Real>(1.0) / h;

    Real density_error = 0.0;

    #pragma omp parallel default(shared)
    {
        // Compute pressure accelerations using the current pressure values.
        // (see Algorithm 3, line 7 in [BK17])
        #pragma omp for schedule(static)
        for (int i = 0; i < numParticles; i++)
        {
            computePressureAccel(fluidModelIndex, i, density0, m_simulationData.getPressureRho2Data());
        }

        // Update pressure values
        #pragma omp for reduction(+:density_error) schedule(static)
        for (int i = 0; i < numParticles; i++)
        {
            if (model->getParticleState(i) != ParticleState::Active)
                continue;

            Real aij_pj = compute_aij_pj(fluidModelIndex, i);
            aij_pj *= h * h;

            // Compute source term: s_i = 1 - rho_adv
            // Note: that due to our multiphase handling, the multiplier rho0
            // is missing here
            const Real& densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
            const Real s_i = static_cast<Real>(1.0) - densityAdv;


            // Update the value p/rho^2 (in [BK17] this is kappa/rho):
            //
            // alpha_i = -1 / (a_ii * rho_i^2)
            // p_rho2_i = (p_i / rho_i^2)
            //
            // Therefore, the following lines compute the Jacobi iteration:
            // p_i := p_i + 1/a_ii (source_term_i - a_ij * p_j)
            Real& p_rho2_i = m_simulationData.getPressureRho2(fluidModelIndex, i);
            const Real residuum = min(s_i - aij_pj, static_cast<Real>(0.0));     // r = b - A*p
            //p_rho2_i -= residuum * m_simulationData.getFactor(fluidModelIndex, i);

            p_rho2_i = max(p_rho2_i - 0.5 * (s_i - aij_pj) * m_simulationData.getFactor(fluidModelIndex, i), 0.0);

            // Compute the sum of the density errors
            density_error -= density0 * residuum;
        }
    }

    // Compute the average density error
    avg_density_err = density_error / numParticles;
}

void TimeStepDFSPH::divergenceSolveIteration(const unsigned int fluidModelIndex, Real &avg_density_err)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const Real density0 = model->getDensity0();
    const int numParticles = (int)model->numActiveParticles();
    if (numParticles == 0)
        return;

    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();
    const Real h = TimeManager::getCurrent()->getTimeStepSize();
    const Real invH = static_cast<Real>(1.0) / h;

    Real density_error = 0.0;

    #pragma omp parallel default(shared)
    {
        // Compute pressure accelerations using the current pressure values.
        // (see Algorithm 2, line 7 in [BK17])
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            computePressureAccel(fluidModelIndex, i, density0, m_simulationData.getPressureRho2VData());
        }

        // Update pressure
        #pragma omp for reduction(+:density_error) schedule(static)
        for (int i = 0; i < numParticles; i++)
        {
            Real aij_pj = compute_aij_pj(fluidModelIndex, i);
            aij_pj *= h;

            // Compute source term: s_i = -d rho / dt
            const Real& densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
            const Real s_i = -densityAdv;

            // Update the value p/rho^2:
            //
            // alpha_i = -1 / (a_ii * rho_i^2)
            // pv_rho2_i = (pv_i / rho_i^2)
            //
            // Therefore, the following line computes the Jacobi iteration:
            // pv_i := pv_i + 1/a_ii (source_term_i - a_ij * pv_j)
            Real& pv_rho2_i = m_simulationData.getPressureRho2_V(fluidModelIndex, i);
            Real residuum = min(s_i - aij_pj, static_cast<Real>(0.0));     // r = b - A*p

            unsigned int numNeighbors = 0;
            for (unsigned int pid = 0; pid < sim->numberOfPointSets(); pid++)
                numNeighbors += sim->numberOfNeighbors(fluidModelIndex, pid, i);

            // in case of particle deficiency do not perform a divergence solve
            if (!sim->is2DSimulation())
            {
                if (numNeighbors < 20)
                    residuum = 0.0;
            }
            else
            {
                if (numNeighbors < 7)
                    residuum = 0.0;
            }
            //pv_rho2_i -= residuum * m_simulationData.getFactor(fluidModelIndex, i);
            pv_rho2_i = max(pv_rho2_i - 0.5*(s_i - aij_pj) * m_simulationData.getFactor(fluidModelIndex, i), 0.0);


            // Compute the sum of the divergence errors
            density_error -= density0 * residuum;
        }
    }

    // Compute the average divergence error
    avg_density_err = density_error / numParticles;
}



void TimeStepDFSPH::reset()
{
    TimeStep::reset();
    m_simulationData.reset();
    m_iterations = 0;
    m_iterationsV = 0;
}

void TimeStepDFSPH::performNeighborhoodSearchSort()
{
    m_simulationData.performNeighborhoodSearchSort();
}

void TimeStepDFSPH::emittedParticles(FluidModel *model, const unsigned int startIndex)
{
    m_simulationData.emittedParticles(model, startIndex);
}

void TimeStepDFSPH::resize()
{
    m_simulationData.init();
}

#ifdef USE_AVX

void TimeStepDFSPH::computeDFSPHFactor(const unsigned int fluidModelIndex)
{
    // Init parameters

    Simulation* sim = Simulation::getCurrent();
    const unsigned int nFluids = sim->numberOfFluidModels();
    FluidModel* model = sim->getFluidModel(fluidModelIndex);
    const int numParticles = (int)model->numActiveParticles();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    #pragma omp parallel default(shared)
    {
        // Compute pressure stiffness denominator

        #pragma omp for schedule(static)
        for (int i = 0; i < numParticles; i++)
        {
            // Compute gradient dp_i/dx_j * (1/kappa)  and dp_j/dx_j * (1/kappa)
            // (see Equation (8) and the previous one [BK17])
            // Note: That in all quantities rho0 is missing due to our
            // implementation of multiphase simulations.
            const Vector3r& xi = model->getPosition(i);

            Real sum_grad_p_k;
            Vector3r grad_p_i;
            Vector3f8 xi_avx(xi);
            Scalarf8 sum_grad_p_k_avx(0.0f);
            Vector3f8 grad_p_i_avx;
            grad_p_i_avx.setZero();

            // Fluid
            forall_fluid_neighbors_avx_nox(
                compute_xj(fm_neighbor, pid);
                compute_Vj(fm_neighbor);
                compute_Vj_gradW();
                sum_grad_p_k_avx += V_gradW.squaredNorm();
                grad_p_i_avx += V_gradW;
            );

            // Boundary
            if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
            {
                forall_boundary_neighbors_avx(
                    const Scalarf8 V_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVolume(0), count);
                    const Vector3f8 grad_p_j = CubicKernel_AVX::gradW(xj_avx - xi_avx) * V_avx;
                    grad_p_i_avx -= grad_p_j;
                );
            }

            sum_grad_p_k = sum_grad_p_k_avx.reduce();
            grad_p_i[0] = grad_p_i_avx.x().reduce();
            grad_p_i[1] = grad_p_i_avx.y().reduce();
            grad_p_i[2] = grad_p_i_avx.z().reduce();

            if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
            {
                forall_density_maps(
                    grad_p_i -= gradRho;
                );
            }
            else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
            {
                forall_volume_maps(
                    const Vector3r grad_p_j = -Vj * sim->gradW(xi - xj);
                    grad_p_i -= grad_p_j;
                );
            }

            sum_grad_p_k += grad_p_i.squaredNorm();

            // Compute factor alpha_i / rho_i (see Equation (11) in [BK17])
            Real& factor = m_simulationData.getFactor(fluidModelIndex, i);
            if (sum_grad_p_k > m_eps)
                factor = static_cast<Real>(1.0) / (sum_grad_p_k);
            else
                factor = 0.0;
        }
    }
}

void TimeStepDFSPH::computeDensityAdv(const unsigned int fluidModelIndex, const unsigned int i, const Real h, const Real density0)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const Real &density = model->getDensity(i);
    Real &densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
    const Vector3r &xi = model->getPosition(i);
    const Vector3r &vi = model->getVelocity(i);
    Real delta = 0.0;
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    Scalarf8 delta_avx(0.0f);
    const Vector3f8 xi_avx(xi);
    Vector3f8 vi_avx(vi);

    // Fluid
    forall_fluid_neighbors_avx_nox(
        compute_xj(fm_neighbor, pid);
        compute_Vj(fm_neighbor);
        compute_Vj_gradW();
        const Vector3f8 vj_avx = convertVec_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &fm_neighbor->getVelocity(0), count);
        delta_avx += (vi_avx - vj_avx).dot(V_gradW);
    );

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors_avx(
            const Scalarf8 Vj_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVolume(0), count);
            const Vector3f8 vj_avx = convertVec_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVelocity(0), count);
            delta_avx += Vj_avx * (vi_avx - vj_avx).dot(CubicKernel_AVX::gradW(xi_avx - xj_avx));
        );
    }

    delta = delta_avx.reduce();

    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xi, vj);
            delta -= (vi - vj).dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xj, vj);
            delta += Vj * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }

    densityAdv = density / density0 + h*delta;
}

void TimeStepDFSPH::computeDensityChange(const unsigned int fluidModelIndex, const unsigned int i, const Real h)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const Vector3r &xi = model->getPosition(i);
    const Vector3r &vi = model->getVelocity(i);
    unsigned int numNeighbors = 0;
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    Scalarf8 densityAdv_avx(0.0f);
    const Vector3f8 xi_avx(xi);
    Vector3f8 vi_avx(vi);

    // Fluid
    forall_fluid_neighbors_avx_nox(
        compute_xj(fm_neighbor, pid);
        compute_Vj(fm_neighbor);
        compute_Vj_gradW();
        const Vector3f8 vj_avx = convertVec_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &fm_neighbor->getVelocity(0), count);
        densityAdv_avx += (vi_avx - vj_avx).dot(V_gradW);
    );

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors_avx(
            const Scalarf8 Vj_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVolume(0), count);
            const Vector3f8 vj_avx = convertVec_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVelocity(0), count);
            densityAdv_avx += Vj_avx * (vi_avx - vj_avx).dot(CubicKernel_AVX::gradW(xi_avx - xj_avx));
        );
    }

    Real &densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
    densityAdv = densityAdv_avx.reduce();

    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xi, vj);
            densityAdv -= (vi - vj).dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xj, vj);
            densityAdv += Vj * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }
}

void TimeStepDFSPH::computePressureAccel(const unsigned int fluidModelIndex, const unsigned int i, const Real density0, std::vector<std::vector<Real>>& pressure_rho2, const bool applyBoundaryForces)
{
    Simulation* sim = Simulation::getCurrent();
    FluidModel* model = sim->getFluidModel(fluidModelIndex);
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    Vector3r& ai = m_simulationData.getPressureAccel(fluidModelIndex, i);

    if (model->getParticleState(i) != ParticleState::Active)
        return;

    // p_rho2_i = (p_i / rho_i^2)
    const Real p_rho2_i = pressure_rho2[fluidModelIndex][i];
    const Vector3r &xi = model->getPosition(i);

    Scalarf8 p_rho2_i_avx(p_rho2_i);
    const Vector3f8 xi_avx(xi);
    Vector3f8 delta_ai_avx;
    delta_ai_avx.setZero();

    // Fluid
    forall_fluid_neighbors_avx_nox(
        compute_xj(fm_neighbor, pid);
        compute_Vj(fm_neighbor);
        compute_Vj_gradW();
        const Scalarf8 densityFrac_avx(fm_neighbor->getDensity0() / density0);

        // p_rho2_j = (p_j / rho_j^2)
        const Scalarf8 p_rho2_j_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &pressure_rho2[pid][0], count);
        const Scalarf8 pSum = p_rho2_i_avx + densityFrac_avx * p_rho2_j_avx;
        delta_ai_avx -= V_gradW * pSum;
    )

    // Boundary
    if (fabs(p_rho2_i) > m_eps)
    {
        if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
        {
            const Scalarf8 mi_avx(model->getMass(i));
            forall_boundary_neighbors_avx(
                const Scalarf8 Vj_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVolume(0), count);

                // Directly update velocities instead of storing pressure accelerations
                const Vector3f8 a = -CubicKernel_AVX::gradW(xi_avx - xj_avx) * (Vj_avx * p_rho2_i_avx);
                delta_ai_avx += a;

                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj_avx, -a * mi_avx, count);
            );
        }
    }

    ai[0] = delta_ai_avx.x().reduce();
    ai[1] = delta_ai_avx.y().reduce();
    ai[2] = delta_ai_avx.z().reduce();

    if (fabs(p_rho2_i) > m_eps)
    {
        if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
        {
            forall_density_maps(
                const Vector3r a = (Real) 1.0 * p_rho2_i * gradRho;
                ai += a;

                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj, -model->getMass(i) * a);
            );
        }
        else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
        {
            forall_volume_maps(
                const Vector3r grad_p_j = -Vj * sim->gradW(xi - xj);
                const Vector3r a = (Real) 1.0 * p_rho2_i * grad_p_j;
                ai += a;

                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj, -model->getMass(i) * a);
            );
        }
    }
}


Real TimeStepDFSPH::compute_aij_pj(const unsigned int fluidModelIndex, const unsigned int i)
{
    Simulation* sim = Simulation::getCurrent();
    FluidModel* model = sim->getFluidModel(fluidModelIndex);
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    // Compute A*p which is the change of the density when applying the
    // pressure forces.
    // \sum_j a_ij * p_j = h^2 \sum_j V_j (a_i - a_j) * gradW_ij
    // This is the RHS of Equation (12) in [BK17]
    const Vector3r& xi = model->getPosition(i);
    const Vector3r& ai = m_simulationData.getPressureAccel(fluidModelIndex, i);
    const Vector3f8 xi_avx(xi);
    const Vector3f8 ai_avx(ai);
    Scalarf8 aij_pj_avx;
    aij_pj_avx.setZero();

    // Fluid
    forall_fluid_neighbors_avx_nox(
        compute_xj(fm_neighbor, pid);
        compute_Vj(fm_neighbor);
        compute_Vj_gradW();

        const Vector3f8 aj_avx = convertVec_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &m_simulationData.getPressureAccel(pid, 0), count);
        aij_pj_avx += (ai_avx - aj_avx).dot(V_gradW);
    );

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors_avx(
            const Scalarf8 Vj_avx = convert_zero(&sim->getNeighborList(fluidModelIndex, pid, i)[j], &bm_neighbor->getVolume(0), count);
            aij_pj_avx += Vj_avx * ai_avx.dot(CubicKernel_AVX::gradW(xi_avx - xj_avx));
        );
    }

    Real aij_pj = aij_pj_avx.reduce();

    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            aij_pj -= ai.dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            aij_pj += Vj * ai.dot(sim->gradW(xi - xj));
        );
    }
    return aij_pj;
}



#else

void TimeStepDFSPH::computeDFSPHFactor(const unsigned int fluidModelIndex)
{
    // Init parameters

    Simulation *sim = Simulation::getCurrent();
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const int numParticles = (int) model->numActiveParticles();

    #pragma omp parallel default(shared)
    {
        // Compute pressure stiffness denominator

        #pragma omp for schedule(static)
        for (int i = 0; i < numParticles; i++)
        {
            // Compute gradient dp_i/dx_j * (1/kappa)  and dp_j/dx_j * (1/kappa)
            // (see Equation (8) and the previous one [BK17])
            // Note: That in all quantities rho0 is missing due to our
            // implementation of multiphase simulations.
            const Vector3r &xi = model->getPosition(i);
            Real sum_grad_p_k = 0.0;
            Vector3r grad_p_i;
            grad_p_i.setZero();

            // Fluid
            forall_fluid_neighbors(
                const Vector3r grad_p_j = -fm_neighbor->getVolume(neighborIndex) * sim->gradW(xi - xj);
                sum_grad_p_k += grad_p_j.squaredNorm();
                grad_p_i -= grad_p_j;
            );

            // Boundary
            if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
            {
                forall_boundary_neighbors(
                    const Vector3r grad_p_j = -bm_neighbor->getVolume(neighborIndex) * sim->gradW(xi - xj);
                    grad_p_i -= grad_p_j;
                );
            }

            else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
            {
                forall_density_maps(
                    grad_p_i -= gradRho;
                );
            }
            else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
            {
                forall_volume_maps(
                    const Vector3r grad_p_j = -Vj * sim->gradW(xi - xj);
                    grad_p_i -= grad_p_j;
                );
            }

            sum_grad_p_k += grad_p_i.squaredNorm();

            // Compute factor as: factor_i = -1 / (a_ii * rho_i^2)
            // where a_ii is the diagonal entry of the linear system
            // for the pressure A * p = source term
            Real &factor = m_simulationData.getFactor(fluidModelIndex, i);
            if (sum_grad_p_k > m_eps)
                factor = static_cast<Real>(1.0) / (sum_grad_p_k);
            else
                factor = 0.0;
        }
    }
}

void TimeStepDFSPH::computeDensityAdv(const unsigned int fluidModelIndex, const unsigned int i, const Real h, const Real density0)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    const Real &density = model->getDensity(i);
    Real &densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
    const Vector3r &xi = model->getPosition(i);
    const Vector3r &vi = model->getVelocity(i);
    Real delta = 0.0;
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    // Fluid
    forall_fluid_neighbors(
        const Vector3r & vj = fm_neighbor->getVelocity(neighborIndex);
        delta += (vi - vj).dot(sim->gradW(xi - xj));
        //delta += fm_neighbor->getVolume(neighborIndex) * (vi - vj).dot(sim->gradW(xi - xj));
    );
    // assumes that all fluid particles have the same volume
    delta *= model->getVolume(i);

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors(
            const Vector3r &vj = bm_neighbor->getVelocity(neighborIndex);
            delta += bm_neighbor->getVolume(neighborIndex) * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xi, vj);
            delta -= (vi - vj).dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xj, vj);
            delta += Vj * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }

    densityAdv = density / density0 + h*delta;
}

void TimeStepDFSPH::computeDensityChange(const unsigned int fluidModelIndex, const unsigned int i, const Real h)
{
    Simulation *sim = Simulation::getCurrent();
    FluidModel *model = sim->getFluidModel(fluidModelIndex);
    Real &densityAdv = m_simulationData.getDensityAdv(fluidModelIndex, i);
    const Vector3r &xi = model->getPosition(i);
    const Vector3r& vi = model->getVelocity(i);
    densityAdv = 0.0;
    unsigned int numNeighbors = 0;
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    // Fluid
    forall_fluid_neighbors(
        const Vector3r & vj = fm_neighbor->getVelocity(neighborIndex);
        densityAdv += (vi - vj).dot(sim->gradW(xi - xj));
    );
    // assumes that all fluid particles have the same volume
    densityAdv *= model->getVolume(i);

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors(
            const Vector3r &vj = bm_neighbor->getVelocity(neighborIndex);
            densityAdv += bm_neighbor->getVolume(neighborIndex) * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xi, vj);
            densityAdv -= (vi - vj).dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            Vector3r vj;
            bm_neighbor->getPointVelocity(xj, vj);
            densityAdv += Vj * (vi - vj).dot(sim->gradW(xi - xj));
        );
    }
}

void TimeStepDFSPH::computePressureAccel(const unsigned int fluidModelIndex, const unsigned int i, const Real density0, std::vector<std::vector<Real>>& pressure_rho2, const bool applyBoundaryForces)
{
    Simulation* sim = Simulation::getCurrent();
    FluidModel* model = sim->getFluidModel(fluidModelIndex);
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    Vector3r& ai = m_simulationData.getPressureAccel(fluidModelIndex, i);
    ai.setZero();

    if (model->getParticleState(i) != ParticleState::Active)
        return;

    // p_rho2_i = (p_i / rho_i^2)
    const Real p_rho2_i = pressure_rho2[fluidModelIndex][i];
    const Vector3r &xi = model->getPosition(i);

    // Fluid
    forall_fluid_neighbors(
        // p_rho2_j = (p_j / rho_j^2)
        const Real p_rho2_j = pressure_rho2[pid][neighborIndex];
        const Real pSum = p_rho2_i + fm_neighbor->getDensity0()/density0 * p_rho2_j;
        if (fabs(pSum) > m_eps)
        {
            const Vector3r grad_p_j = -fm_neighbor->getVolume(neighborIndex) * sim->gradW(xi - xj);
            ai += pSum * grad_p_j;
        }
    )

    // Boundary
    if (fabs(p_rho2_i) > m_eps)
    {
        if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
        {
            forall_boundary_neighbors(
                const Vector3r grad_p_j = -bm_neighbor->getVolume(neighborIndex) * sim->gradW(xi - xj);

                const Vector3r a = (Real) 1.0 * p_rho2_i * grad_p_j;
                ai += a;
                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj, -model->getMass(i) * a);
            );
        }
        else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
        {
            forall_density_maps(
                const Vector3r a = (Real) 1.0 * p_rho2_i * gradRho;
                ai += a;
                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj, -model->getMass(i) * a);
            );
        }
        else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
        {
            forall_volume_maps(
                const Vector3r grad_p_j = -Vj * sim->gradW(xi - xj);
                const Vector3r a = (Real) 1.0 * p_rho2_i * grad_p_j;
                ai += a;

                if (applyBoundaryForces)
                    bm_neighbor->addForce(xj, -model->getMass(i) * a);
            );
        }
    }
}


Real TimeStepDFSPH::compute_aij_pj(const unsigned int fluidModelIndex, const unsigned int i)
{
    Simulation* sim = Simulation::getCurrent();
    FluidModel* model = sim->getFluidModel(fluidModelIndex);
    const unsigned int nFluids = sim->numberOfFluidModels();
    const unsigned int nBoundaries = sim->numberOfBoundaryModels();

    // Compute A*p which is the change of the density when applying the
    // pressure forces.
    // \sum_j a_ij * p_j = h^2 \sum_j V_j (a_i - a_j) * gradW_ij
    // This is the RHS of Equation (12) in [BK17]
    const Vector3r& xi = model->getPosition(i);
    const Vector3r& ai = m_simulationData.getPressureAccel(fluidModelIndex, i);
    Real aij_pj = 0.0;

    // Fluid
    forall_fluid_neighbors(
        const Vector3r & aj = m_simulationData.getPressureAccel(pid, neighborIndex);
        //aij_pj += fm_neighbor->getVolume(neighborIndex) * (ai - aj).dot(sim->gradW(xi - xj));
        aij_pj += (ai - aj).dot(sim->gradW(xi - xj));
    );
    // assumes that all fluid particles have the same volume
    aij_pj *= model->getVolume(i);

    // Boundary
    if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Akinci2012)
    {
        forall_boundary_neighbors(
            aij_pj += bm_neighbor->getVolume(neighborIndex) * ai.dot(sim->gradW(xi - xj));
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Koschier2017)
    {
        forall_density_maps(
            aij_pj -= ai.dot(gradRho);
        );
    }
    else if (sim->getBoundaryHandlingMethod() == BoundaryHandlingMethods::Bender2019)
    {
        forall_volume_maps(
            aij_pj += Vj * ai.dot(sim->gradW(xi - xj));
        );
    }
    return aij_pj;
}


#endif