Program Listing for File Elasticity_Peer2018.cpp

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#include "Elasticity_Peer2018.h"
#include "SPlisHSPlasH/Simulation.h"
#include "SPlisHSPlasH/Utilities/MathFunctions.h"
#include "SPlisHSPlasH/TimeManager.h"
#include "Utilities/Timing.h"
#include "Utilities/Counting.h"

using namespace SPH;
using namespace GenParam;

std::string Elasticity_Peer2018::METHOD_NAME = "Peer et al. 2018";
int Elasticity_Peer2018::YOUNGS_MODULUS = -1;
int Elasticity_Peer2018::POISSON_RATIO = -1;
int Elasticity_Peer2018::FIXED_BOX_MIN = -1;
int Elasticity_Peer2018::FIXED_BOX_MAX = -1;
int Elasticity_Peer2018::ITERATIONS = -1;
int Elasticity_Peer2018::MAX_ITERATIONS = -1;
int Elasticity_Peer2018::MAX_ERROR = -1;
int Elasticity_Peer2018::ALPHA = -1;
int Elasticity_Peer2018::MAX_NEIGHBORS = -1;


Elasticity_Peer2018::Elasticity_Peer2018(FluidModel *model) :
    NonPressureForceBase(model)
{
    const unsigned int numParticles = model->numActiveParticles();
    m_restVolumes.resize(numParticles);
    m_current_to_initial_index.resize(numParticles);
    m_initial_to_current_index.resize(numParticles);
    m_initialNeighbors.resize(numParticles);
    m_rotations.resize(numParticles, Matrix3r::Identity());
    m_stress.resize(numParticles);
    m_L.resize(numParticles);
    m_RL.resize(numParticles);
    m_F.resize(numParticles);

    m_youngsModulus = static_cast<Real>(100000.0);
    m_poissonRatio = static_cast<Real>(0.3);
    m_iterations = 0;
    m_maxIter = 100;
    m_maxError = static_cast<Real>(1.0e-4);
    m_alpha = 0.0;
    m_maxNeighbors = -1;
    m_fixedBoxMin.setZero();
    m_fixedBoxMax.setZero();

    model->addField({ "rest volume", METHOD_NAME, FieldType::Scalar, [&](const unsigned int i) -> Real* { return &m_restVolumes[i]; }, true });
    model->addField({ "rotation", METHOD_NAME, FieldType::Matrix3, [&](const unsigned int i) -> Real* { return &m_rotations[i](0,0); } });
    model->addField({ "stress", METHOD_NAME, FieldType::Matrix3, [&](const unsigned int i) -> Real* { return &m_stress[i](0,0); } });
    model->addField({ "deformation gradient", METHOD_NAME, FieldType::Matrix3, [&](const unsigned int i) -> Real* { return &m_F[i](0,0); } });
    model->addField({ "correction matrix", METHOD_NAME, FieldType::Matrix3, [&](const unsigned int i) -> Real* { return &m_L[i](0,0); } });
}

Elasticity_Peer2018::~Elasticity_Peer2018(void)
{
    m_model->removeFieldByName("rest volume");
    m_model->removeFieldByName("rotation");
    m_model->removeFieldByName("stress");
    m_model->removeFieldByName("deformation gradient");
    m_model->removeFieldByName("correction matrix");
}

void Elasticity_Peer2018::deferredInit()
{
    initValues();
}

void Elasticity_Peer2018::initParameters()
{
    NonPressureForceBase::initParameters();

    YOUNGS_MODULUS = createNumericParameter("youngsModulus", "Young`s modulus", &m_youngsModulus);
    setGroup(YOUNGS_MODULUS, "Fluid Model|Elasticity");
    setDescription(YOUNGS_MODULUS, "Stiffness of the elastic material");
    RealParameter* rparam = static_cast<RealParameter*>(getParameter(YOUNGS_MODULUS));
    rparam->setMinValue(0.0);

    POISSON_RATIO = createNumericParameter("poissonsRatio", "Poisson`s ratio", &m_poissonRatio);
    setGroup(POISSON_RATIO, "Fluid Model|Elasticity");
    setDescription(POISSON_RATIO, "Ratio of transversal expansion and axial compression");
    rparam = static_cast<RealParameter*>(getParameter(POISSON_RATIO));
    rparam->setMinValue(static_cast<Real>(-1.0 + 1e-4));
    rparam->setMaxValue(static_cast<Real>(0.5 - 1e-4));

    ParameterBase::GetVecFunc<Real> getFct = [&]()-> Real* { return m_fixedBoxMin.data(); };
    ParameterBase::SetVecFunc<Real> setFct = [&](Real* val)
    {
        m_fixedBoxMin = Vector3r(val[0], val[1], val[2]);
        determineFixedParticles();
    };
    FIXED_BOX_MIN = createVectorParameter("fixedBoxMin", "Fixed box min", 3u, getFct, setFct);
    setGroup(FIXED_BOX_MIN, "Fluid Model|Elasticity");
    setDescription(FIXED_BOX_MIN, "Minimum point of box of which contains the fixed particles.");
    getParameter(FIXED_BOX_MIN)->setReadOnly(true);


    ParameterBase::GetVecFunc<Real> getFct2 = [&]()-> Real* { return m_fixedBoxMax.data(); };
    ParameterBase::SetVecFunc<Real> setFct2 = [&](Real* val)
    {
        m_fixedBoxMax = Vector3r(val[0], val[1], val[2]);
        determineFixedParticles();
    };
    FIXED_BOX_MAX = createVectorParameter("fixedBoxMax", "Fixed box max", 3u, getFct2, setFct2);
    setGroup(FIXED_BOX_MAX, "Fluid Model|Elasticity");
    setDescription(FIXED_BOX_MAX, "Maximum point of box of which contains the fixed particles.");
    getParameter(FIXED_BOX_MAX)->setReadOnly(true);

    ITERATIONS = createNumericParameter("elasticityIterations", "Iterations", &m_iterations);
    setGroup(ITERATIONS, "Fluid Model|Elasticity");
    setDescription(ITERATIONS, "Iterations required by the elasticity solver.");
    getParameter(ITERATIONS)->setReadOnly(true);

    MAX_ITERATIONS = createNumericParameter("elasticityMaxIter", "Max. iterations (elasticity)", &m_maxIter);
    setGroup(MAX_ITERATIONS, "Fluid Model|Elasticity");
    setDescription(MAX_ITERATIONS, "Coefficient for the elasticity force computation");
    static_cast<NumericParameter<unsigned int>*>(getParameter(MAX_ITERATIONS))->setMinValue(1);

    MAX_ERROR = createNumericParameter("elasticityMaxError", "Max. elasticity error", &m_maxError);
    setGroup(MAX_ERROR, "Fluid Model|Elasticity");
    setDescription(MAX_ERROR, "Coefficient for the elasticity force computation");
    rparam = static_cast<RealParameter*>(getParameter(MAX_ERROR));
    rparam->setMinValue(1e-7);

    ALPHA = createNumericParameter("alpha", "Zero-energy modes suppression", &m_alpha);
    setGroup(ALPHA, "Fluid Model|Elasticity");
    setDescription(ALPHA, "Coefficent for zero-energy modes suppression method");
    rparam = static_cast<RealParameter*>(getParameter(ALPHA));
    rparam->setMinValue(0.0);

    MAX_NEIGHBORS = createNumericParameter("maxNeighbors", "Max. neighbors", &m_maxNeighbors);
    setGroup(MAX_NEIGHBORS, "Fluid Model|Elasticity");
    setDescription(MAX_NEIGHBORS, "Maximum number of neighbors that are considered.");
}

void Elasticity_Peer2018::determineFixedParticles()
{
    const unsigned int numParticles = m_model->numActiveParticles();

    if (!m_fixedBoxMin.isZero() || !m_fixedBoxMax.isZero())
    {
        for (int i = 0; i < (int)numParticles; i++)
        {
            const Vector3r& x = m_model->getPosition0(i);
            if ((x[0] > m_fixedBoxMin[0]) && (x[1] > m_fixedBoxMin[1]) && (x[2] > m_fixedBoxMin[2]) &&
                (x[0] < m_fixedBoxMax[0]) && (x[1] < m_fixedBoxMax[1]) && (x[2] < m_fixedBoxMax[2]))
            {
                m_model->setParticleState(i, ParticleState::Fixed);
            }
        }
    }
}

void Elasticity_Peer2018::initValues()
{
    Simulation *sim = Simulation::getCurrent();
    sim->getNeighborhoodSearch()->find_neighbors();

    FluidModel *model = m_model;
    const unsigned int numParticles = model->numActiveParticles();
    const unsigned int fluidModelIndex = model->getPointSetIndex();

    // Store the neighbors in the reference configurations and
    // compute the volume of each particle in rest state
    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            m_current_to_initial_index[i] = i;
            m_initial_to_current_index[i] = i;

            // reset particle state
            if (model->getParticleState(i) == ParticleState::Fixed)
                model->setParticleState(i, ParticleState::Active);

            // only neighbors in same phase will influence elasticity
            unsigned int numNeighbors = sim->numberOfNeighbors(fluidModelIndex, fluidModelIndex, i);
            m_initialNeighbors[i].resize(numNeighbors);
            for (unsigned int j = 0; j < numNeighbors; j++)
                m_initialNeighbors[i][j] = sim->getNeighbor(fluidModelIndex, fluidModelIndex, i, j);

            // if maxNeighbors is set, then sort all neighbors wrt. to their distance to xi
            // and only take the maxNeighbors next ones.
            if (m_maxNeighbors > 0)
            {
                struct Comparator {
                    Comparator(const Vector3r& xi, Vector3r* x) : m_xi(xi), m_x(x) {};
                    bool operator()(unsigned int a, unsigned int b)
                    {
                        return (m_x[a] - m_xi).squaredNorm() < (m_x[b] - m_xi).squaredNorm();
                    }

                    Vector3r m_xi;
                    Vector3r* m_x;
                };

                std::sort(m_initialNeighbors[i].begin(), m_initialNeighbors[i].end(), Comparator(model->getPosition0(i), &model->getPosition0(0)));
                if (m_initialNeighbors[i].size() > m_maxNeighbors)
                {
                    numNeighbors = m_maxNeighbors;
                    m_initialNeighbors[i].resize(m_maxNeighbors);
                }
            }

            // compute volume
            Real density = model->getMass(i) * sim->W_zero();
            const Vector3r &xi = model->getPosition(i);
            for (size_t j = 0; j < m_initialNeighbors[i].size(); j++)
            {
                const unsigned int neighborIndex = m_initialNeighbors[i][j];
                const Vector3r& xj = model->getPosition(neighborIndex);
                density += model->getMass(neighborIndex) * sim->W(xi - xj);
            }
            m_restVolumes[i] = model->getMass(i) / density;
            m_rotations[i].setIdentity();
        }
    }

    // mark all particles in the bounding box as fixed
    determineFixedParticles();

    computeMatrixL();
}


void Elasticity_Peer2018::step()
{
    START_TIMING("Elasticity_Peer2018")
    const unsigned int numParticles = m_model->numActiveParticles();
    if (numParticles == 0)
        return;
    const Real dt = TimeManager::getCurrent()->getTimeStepSize();

    // Init linear system solver and preconditioner
    MatrixReplacement A(3 * m_model->numActiveParticles(), matrixVecProd, (void*)this);
    //m_solver.preconditioner().init(m_model->numActiveParticles(), diagonalMatrixElement, (void*)this);

    m_solver.setTolerance(m_maxError);
    m_solver.setMaxIterations(m_maxIter);
    m_solver.compute(A);

    VectorXr b(3 * numParticles);
    VectorXr x(3 * numParticles);
    VectorXr g(3 * numParticles);

    computeRotations();
    computeRHS(b);

    // warmstart
    #pragma omp parallel for schedule(static)
    for (int i = 0; i < (int)numParticles; i++)
    {
        if (m_model->getParticleState(i) == ParticleState::Active)
            g.segment<3>(3 * i) = m_model->getVelocity(i) + dt * m_model->getAcceleration(i);
        else
            g.segment<3>(3 * i) = m_model->getVelocity(i);
    }

    // Solve linear system
    START_TIMING("Elasticity - CG solve");
    x = m_solver.solveWithGuess(b, g);
    m_iterations = (int)m_solver.iterations();
    STOP_TIMING_AVG;
    INCREASE_COUNTER("Elasticity - CG iterations", static_cast<Real>(m_iterations));

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            if (m_model->getParticleState(i) == ParticleState::Active)
            {
                Vector3r& ai = m_model->getAcceleration(i);
                ai += (1.0 / dt) * (x.segment<3>(3 * i) - m_model->getVelocity(i));
            }
        }
    }
    STOP_TIMING_AVG
}


void Elasticity_Peer2018::reset()
{
    initValues();
}

void Elasticity_Peer2018::performNeighborhoodSearchSort()
{
    const unsigned int numPart = m_model->numActiveParticles();
    if (numPart == 0)
        return;

    Simulation *sim = Simulation::getCurrent();
    auto const& d = sim->getNeighborhoodSearch()->point_set(m_model->getPointSetIndex());
    d.sort_field(&m_restVolumes[0]);
    d.sort_field(&m_rotations[0]);
    d.sort_field(&m_current_to_initial_index[0]);
    d.sort_field(&m_L[0]);

    for (unsigned int i = 0; i < numPart; i++)
        m_initial_to_current_index[m_current_to_initial_index[i]] = i;
}

void Elasticity_Peer2018::computeRotations()
{
    Simulation *sim = Simulation::getCurrent();
    const unsigned int numParticles = m_model->numActiveParticles();
    const unsigned int fluidModelIndex = m_model->getPointSetIndex();
    FluidModel *model = m_model;

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = m_current_to_initial_index[i];
            const Vector3r &xi = m_model->getPosition(i);
            const Vector3r &xi0 = m_model->getPosition0(i0);
            Matrix3r F;
            F.setZero();

            const size_t numNeighbors = m_initialNeighbors[i0].size();

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j++)
            {
                const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                // get initial neighbor index considering the current particle order
                const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                const Vector3r &xj = model->getPosition(neighborIndex);
                const Vector3r &xj0 = m_model->getPosition0(neighborIndex0);
                const Vector3r xj_xi = xj - xi;
                const Vector3r xi_xj_0 = xi0 - xj0;
                const Vector3r correctedKernel = m_L[i] * sim->gradW(xi_xj_0);
                F += m_restVolumes[neighborIndex] * xj_xi * correctedKernel.transpose();
            }

            if (sim->is2DSimulation())
                F(2, 2) = 1.0;

//              Vector3r sigma;
//              Matrix3r U, VT;
//              MathFunctions::svdWithInversionHandling(F, sigma, U, VT);
//              m_rotations[i] = U * VT;
            Quaternionr q(m_rotations[i]);
            MathFunctions::extractRotation(F, q, 10);
            m_rotations[i] = q.matrix();

            m_RL[i] = m_rotations[i] * m_L[i];
        }
    }
}

void Elasticity_Peer2018::computeMatrixL()
{
    Simulation *sim = Simulation::getCurrent();
    const unsigned int numParticles = m_model->numActiveParticles();
    const unsigned int fluidModelIndex = m_model->getPointSetIndex();

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = m_current_to_initial_index[i];
            const Vector3r &xi0 = m_model->getPosition0(i0);
            Matrix3r L;
            L.setZero();

            const size_t numNeighbors = m_initialNeighbors[i0].size();

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j++)
            {
                const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                // get initial neighbor index considering the current particle order
                const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                const Vector3r &xj0 = m_model->getPosition0(neighborIndex0);
                const Vector3r xj_xi_0 = xj0 - xi0;
                const Vector3r gradW = sim->gradW(xj_xi_0);

                // minus because gradW(xij0) == -gradW(xji0)
                L -= m_restVolumes[neighborIndex] * gradW * xj_xi_0.transpose();
            }

            // add 1 to z-component. otherwise we get a singular matrix in 2D
            if (sim->is2DSimulation())
                L(2, 2) = 1.0;

            bool invertible = false;
            L.computeInverseWithCheck(m_L[i], invertible, 1e-9);
            if (!invertible)
            {
                //MathFunctions::pseudoInverse(L, m_L[i]);
                m_L[i] = Matrix3r::Identity();
            }
        }
    }
}

#ifdef USE_AVX

void Elasticity_Peer2018::computeRHS(VectorXr & rhs)
{
    Simulation *sim = Simulation::getCurrent();
    const unsigned int numParticles = m_model->numActiveParticles();
    const unsigned int fluidModelIndex = m_model->getPointSetIndex();
    FluidModel *model = m_model;
    const Real dt = TimeManager::getCurrent()->getTimeStepSize();

    Real mu = m_youngsModulus / (static_cast<Real>(2.0) * (static_cast<Real>(1.0) + m_poissonRatio));
    Real lambda = m_youngsModulus * m_poissonRatio / ((static_cast<Real>(1.0) + m_poissonRatio) * (static_cast<Real>(1.0) - static_cast<Real>(2.0) * m_poissonRatio));

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = m_current_to_initial_index[i];
            const Vector3r &xi = m_model->getPosition(i);
            const Vector3r &xi0 = m_model->getPosition0(i0);
            const unsigned int numNeighbors = m_initialNeighbors[i0].size();

            // compute corotated deformation gradient (Eq. 18)
            Matrix3f8 F_avx;
            F_avx.setZero();
            const Vector3f8 xi_avx(xi);
            const Vector3f8 xi0_avx(xi0);
            const Matrix3f8 Ri(m_rotations[i]);
            const Matrix3f8 RLi(m_RL[i]);

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j += 8)
            {
                const unsigned int count = std::min(numNeighbors - j, 8u);
                std::array<unsigned int, 8> indices;
                generateIndices(m_initial_to_current_index.data(), &m_initialNeighbors[i0][j], indices, count);

                const Vector3f8 xj0_avx = convertVec_zero(&m_initialNeighbors[i0][j], &model->getPosition0(0), count);
                const Vector3f8 xj_avx = convertVec_zero(&indices[0], &model->getPosition(0), count);
                const Scalarf8 Vj0_avx = convert_zero(&indices[0], &m_restVolumes[0], count);

                const Vector3f8 xj_xi = xj_avx - xi_avx;
                const Vector3f8 xi_xj_0 = xi0_avx - xj0_avx;
                const Vector3f8 correctedRotatedKernel = RLi * CubicKernel_AVX::gradW(xi_xj_0);

                Matrix3f8 dyad;
                dyadicProduct((xj_xi - Ri * (xj0_avx - xi0_avx)), correctedRotatedKernel, dyad);
                F_avx += dyad * Vj0_avx;
            }

            m_F[i] = F_avx.reduce();
            m_F[i] += Matrix3r::Identity();

            if (sim->is2DSimulation())
                m_F[i](2, 2) = 1.0;

            // compute Cauchy strain: epsilon = 0.5 (F + F^T) - I
            Vector6r strain;
            strain[0] = m_F[i](0, 0) - static_cast<Real>(1.0);                      // \epsilon_{00}
            strain[1] = m_F[i](1, 1) - static_cast<Real>(1.0);                      // \epsilon_{11}
            strain[2] = m_F[i](2, 2) - static_cast<Real>(1.0);                      // \epsilon_{22}
            strain[3] = static_cast<Real>(0.5) * (m_F[i](0, 1) + m_F[i](1, 0));         // \epsilon_{01}
            strain[4] = static_cast<Real>(0.5) * (m_F[i](0, 2) + m_F[i](2, 0));         // \epsilon_{02}
            strain[5] = static_cast<Real>(0.5) * (m_F[i](1, 2) + m_F[i](2, 1));         // \epsilon_{12}

            // First Piola Kirchhoff stress = 2 mu epsilon + lambda trace(epsilon) I
            const Real trace = strain[0] + strain[1] + strain[2];
            const Real ltrace = lambda*trace;
            Matrix3r& stress = m_stress[i];
            stress(0, 0) = static_cast<Real>(2.0) * mu * strain[0] + ltrace;
            stress(1, 1) = static_cast<Real>(2.0) * mu * strain[1] + ltrace;
            stress(2, 2) = static_cast<Real>(2.0) * mu * strain[2] + ltrace;

            stress(0, 1) = static_cast<Real>(2.0) * mu * strain[3];
            stress(1, 0) = static_cast<Real>(2.0) * mu * strain[3];
            stress(0, 2) = static_cast<Real>(2.0) * mu * strain[4];
            stress(2, 0) = static_cast<Real>(2.0) * mu * strain[4];
            stress(1, 2) = static_cast<Real>(2.0) * mu * strain[5];
            stress(2, 1) = static_cast<Real>(2.0) * mu * strain[5];
        }
    }

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            if (model->getParticleState(i) == ParticleState::Active)
            {
                const unsigned int i0 = m_current_to_initial_index[i];
                const Vector3r& xi0 = m_model->getPosition0(i0);
                const unsigned int numNeighbors = m_initialNeighbors[i0].size();

                // Compute elastic force
                Vector3f8 force_avx;
                force_avx.setZero();
                const Scalarf8 Vi0_avx(m_restVolumes[i]);
                const Vector3f8 xi0_avx(xi0);
                const Matrix3f8 RLi(m_RL[i]);
                const Matrix3f8 stress_i(m_stress[i]);

                for (unsigned int j = 0; j < numNeighbors; j += 8)
                {
                    const unsigned int count = std::min(numNeighbors - j, 8u);
                    std::array<unsigned int, 8> indices;
                    generateIndices(m_initial_to_current_index.data(), &m_initialNeighbors[i0][j], indices, count);

                    const Matrix3f8& RLj = convertMat_zero(&indices[0], &m_RL[0], count);
                    const Vector3f8 xj0_avx = convertVec_zero(&m_initialNeighbors[i0][j], &model->getPosition0(0), count);
                    const Scalarf8 Vj0_avx = convert_zero(&indices[0], &m_restVolumes[0], count);
                    const Vector3f8 xi_xj_0 = xi0_avx - xj0_avx;
                    const Vector3f8 gradW = CubicKernel_AVX::gradW(xi_xj_0);
                    const Vector3f8 correctedRotatedKernel_i = RLi * gradW;
                    const Vector3f8 correctedRotatedKernel_j = RLj * gradW;

                    const Matrix3f8& stress_j = convertMat_zero(&indices[0], &m_stress[0], count);
                    Vector3f8 PWi = stress_i * correctedRotatedKernel_i;
                    Vector3f8 PWj = stress_j * correctedRotatedKernel_j;
                    force_avx += (PWi + PWj) * Vi0_avx * Vj0_avx;
                }
                Vector3r force = force_avx.reduce();

                if (m_alpha != 0.0)
                {
                    // Ganzenmüller, G.C. 2015. An hourglass control algorithm for Lagrangian
                    // Smooth Particle Hydrodynamics. Computer Methods in Applied Mechanics and
                    // Engineering 286, 87.106.
                    Vector3r fi_hg;
                    fi_hg.setZero();
                    const Vector3r& xi = m_model->getPosition(i);
                    for (unsigned int j = 0; j < numNeighbors; j++)
                    {
                        const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                        // get initial neighbor index considering the current particle order
                        const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                        const Vector3r& xj = model->getPosition(neighborIndex);
                        const Vector3r& xj0 = m_model->getPosition0(neighborIndex0);

                        // Note: Ganzenmueller defines xij = xj-xi
                        const Vector3r xi_xj = -(xi - xj);
                        const Real xixj_l = xi_xj.norm();
                        if (xixj_l > 1.0e-6)
                        {
                            // Note: Ganzenmueller defines xij = xj-xi
                            const Vector3r xi_xj_0 = -(xi0 - xj0);
                            const Real xixj0_l2 = xi_xj_0.squaredNorm();
                            const Real W0 = sim->W(xi_xj_0);

                            const Vector3r xij_i = m_F[i] * m_rotations[i] * xi_xj_0;
                            const Vector3r xji_j = -m_F[neighborIndex] * m_rotations[neighborIndex] * xi_xj_0;
                            const Vector3r epsilon_ij_i = xij_i - xi_xj;
                            const Vector3r epsilon_ji_j = xji_j + xi_xj;

                            const Real delta_ij_i = epsilon_ij_i.dot(xi_xj) / xixj_l;
                            const Real delta_ji_j = -epsilon_ji_j.dot(xi_xj) / xixj_l;

                            fi_hg -= m_restVolumes[neighborIndex] * W0 / xixj0_l2 * (delta_ij_i + delta_ji_j) * xi_xj / xixj_l;
                        }
                    }
                    fi_hg *= m_alpha * m_youngsModulus * m_restVolumes[i];
                    force += fi_hg;
                    //model->getAcceleration(i) += fi_hg / model->getMass(i);
                }
                rhs.segment<3>(3 * i) = model->getVelocity(i) + dt / model->getMass(i) * force;
            }
            else
                rhs.segment<3>(3 * i) = model->getVelocity(i);
        }
    }
}

void Elasticity_Peer2018::matrixVecProd(const Real* vec, Real *result, void *userData)
{
    Simulation *sim = Simulation::getCurrent();
    Elasticity_Peer2018 * elasticity = static_cast<Elasticity_Peer2018*>(userData);
    FluidModel *model = elasticity->getModel();
    const unsigned int numParticles = model->numActiveParticles();
    const Real dt = TimeManager::getCurrent()->getTimeStepSize();

    const auto &current_to_initial_index = elasticity->m_current_to_initial_index;
    const auto &initial_to_current_index = elasticity->m_initial_to_current_index;
    const auto &initialNeighbors = elasticity->m_initialNeighbors;
    const auto &restVolumes = elasticity->m_restVolumes;
    const auto &rotations = elasticity->m_rotations;
    const auto &L = elasticity->m_L;
    const auto &RL = elasticity->m_RL;
    auto &stress = elasticity->m_stress;
    const Real youngsModulus = elasticity->m_youngsModulus;
    const Real poissonRatio = elasticity->m_poissonRatio;

    Real mu = youngsModulus / (static_cast<Real>(2.0) * (static_cast<Real>(1.0) + poissonRatio));
    Real lambda = youngsModulus * poissonRatio / ((static_cast<Real>(1.0) + poissonRatio) * (static_cast<Real>(1.0) - static_cast<Real>(2.0) * poissonRatio));

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = current_to_initial_index[i];
            const Vector3r &pi = Eigen::Map<const Vector3r>(&vec[3 * i], 3);
            const Vector3r &xi0 = model->getPosition0(i0);
            const unsigned int numNeighbors = initialNeighbors[i0].size();

            // compute corotated deformation gradient (Eq. 18)
            Matrix3f8 nablaU_avx;
            nablaU_avx.setZero();
            const Vector3f8 pi_avx(pi);
            const Vector3f8 xi0_avx(xi0);
            const Matrix3f8 RLi(RL[i]);

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j += 8)
            {
                const unsigned int count = std::min(numNeighbors - j, 8u);
                std::array<unsigned int, 8> indices;
                elasticity->generateIndices(initial_to_current_index.data(), &initialNeighbors[i0][j], indices, count);

                const Vector3f8 xj0_avx = convertVec_zero(&initialNeighbors[i0][j], &model->getPosition0(0), count);
                const Vector3f8 pj_avx = convertVec_zero(&indices[0], &vec[0], count);
                const Scalarf8 Vj0_avx = convert_zero(&indices[0], &restVolumes[0], count);
                const Vector3f8 pj_pi = pj_avx - pi_avx;
                const Vector3f8 xi_xj_0 = xi0_avx - xj0_avx;
                const Vector3f8 correctedRotatedKernel = RLi * CubicKernel_AVX::gradW(xi_xj_0);

                Matrix3f8 dyad;
                dyadicProduct(pj_pi, correctedRotatedKernel, dyad);
                nablaU_avx += dyad * Vj0_avx;
            }
            Matrix3r nablaU = nablaU_avx.reduce();
            nablaU *= dt;
            // compute Cauchy strain: epsilon = 0.5 (nablaU + nablaU^T)
            Vector6r strain;
            strain[0] = nablaU(0, 0);                                   // \epsilon_{00}
            strain[1] = nablaU(1, 1);                                   // \epsilon_{11}
            strain[2] = nablaU(2, 2);                                   // \epsilon_{22}
            strain[3] = static_cast<Real>(0.5) * (nablaU(0, 1) + nablaU(1, 0));         // \epsilon_{01}
            strain[4] = static_cast<Real>(0.5) * (nablaU(0, 2) + nablaU(2, 0));         // \epsilon_{02}
            strain[5] = static_cast<Real>(0.5) * (nablaU(1, 2) + nablaU(2, 1));         // \epsilon_{12}

            // First Piola Kirchhoff stress = 2 mu epsilon + lambda trace(epsilon) I
            const Real trace = strain[0] + strain[1] + strain[2];
            const Real ltrace = lambda*trace;
            stress[i](0, 0) = static_cast<Real>(2.0) * mu * strain[0] + ltrace;
            stress[i](1, 1) = static_cast<Real>(2.0) * mu * strain[1] + ltrace;
            stress[i](2, 2) = static_cast<Real>(2.0) * mu * strain[2] + ltrace;

            stress[i](0, 1) = static_cast<Real>(2.0) * mu * strain[3];
            stress[i](1, 0) = static_cast<Real>(2.0) * mu * strain[3];
            stress[i](0, 2) = static_cast<Real>(2.0) * mu * strain[4];
            stress[i](2, 0) = static_cast<Real>(2.0) * mu * strain[4];
            stress[i](1, 2) = static_cast<Real>(2.0) * mu * strain[5];
            stress[i](2, 1) = static_cast<Real>(2.0) * mu * strain[5];
        }
    }

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            if (model->getParticleState(i) == ParticleState::Active)
            {
                const unsigned int i0 = current_to_initial_index[i];
                const Vector3r& xi0 = model->getPosition0(i0);
                const unsigned int numNeighbors = initialNeighbors[i0].size();

                // Compute elastic force
                Vector3f8 force_avx;
                force_avx.setZero();
                const Scalarf8 Vi0_avx(restVolumes[i]);
                const Vector3f8 xi0_avx(xi0);
                const Matrix3f8 RLi(RL[i]);
                const Matrix3f8 stress_i(stress[i]);
                for (unsigned int j = 0; j < numNeighbors; j += 8)
                {
                    const unsigned int count = std::min(numNeighbors - j, 8u);
                    std::array<unsigned int, 8> indices;
                    elasticity->generateIndices(initial_to_current_index.data(), &initialNeighbors[i0][j], indices, count);

                    const Matrix3f8& RLj = convertMat_zero(&indices[0], &RL[0], count);
                    const Vector3f8 xj0_avx = convertVec_zero(&initialNeighbors[i0][j], &model->getPosition0(0), count);
                    const Scalarf8 Vj0_avx = convert_zero(&indices[0], &restVolumes[0], count);
                    const Vector3f8 xi_xj_0 = xi0_avx - xj0_avx;
                    const Vector3f8 gradW = CubicKernel_AVX::gradW(xi_xj_0);
                    const Vector3f8 correctedRotatedKernel_i = RLi * gradW;
                    const Vector3f8 correctedRotatedKernel_j = RLj * gradW;

                    const Matrix3f8& stress_j = convertMat_zero(&indices[0], &stress[0], count);
                    Vector3f8 PWi = stress_i * correctedRotatedKernel_i;
                    Vector3f8 PWj = stress_j * correctedRotatedKernel_j;
                    force_avx += (PWi + PWj) * Vi0_avx * Vj0_avx;
                }

                const Vector3r force = force_avx.reduce();
                const Real factor = dt / model->getMass(i);
                result[3 * i] = vec[3 * i] - factor * force[0];
                result[3 * i + 1] = vec[3 * i + 1] - factor * force[1];
                result[3 * i + 2] = vec[3 * i + 2] - factor * force[2];
            }
            else
            {
                result[3 * i] = vec[3 * i];
                result[3 * i + 1] = vec[3 * i + 1];
                result[3 * i + 2] = vec[3 * i + 2];
            }
        }
    }
}


#else


void Elasticity_Peer2018::computeRHS(VectorXr & rhs)
{
    Simulation *sim = Simulation::getCurrent();
    const unsigned int numParticles = m_model->numActiveParticles();
    const unsigned int fluidModelIndex = m_model->getPointSetIndex();
    FluidModel *model = m_model;
    const Real dt = TimeManager::getCurrent()->getTimeStepSize();

    Real mu = m_youngsModulus / (static_cast<Real>(2.0) * (static_cast<Real>(1.0) + m_poissonRatio));
    Real lambda = m_youngsModulus * m_poissonRatio / ((static_cast<Real>(1.0) + m_poissonRatio) * (static_cast<Real>(1.0) - static_cast<Real>(2.0) * m_poissonRatio));

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = m_current_to_initial_index[i];
            const Vector3r &xi = m_model->getPosition(i);
            const Vector3r &xi0 = m_model->getPosition0(i0);
            const size_t numNeighbors = m_initialNeighbors[i0].size();

            // compute corotated deformation gradient (Eq. 18)
            m_F[i].setZero();

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j++)
            {
                const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                // get initial neighbor index considering the current particle order
                const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                const Vector3r &xj = model->getPosition(neighborIndex);
                const Vector3r &xj0 = m_model->getPosition0(neighborIndex0);
                const Vector3r xj_xi = xj - xi;
                const Vector3r xi_xj_0 = xi0 - xj0;
                const Vector3r correctedRotatedKernel = m_RL[i] * sim->gradW(xi_xj_0);
                m_F[i] += m_restVolumes[neighborIndex] * (xj_xi - m_rotations[i]*(xj0-xi0)) * correctedRotatedKernel.transpose();
            }

            m_F[i] += Matrix3r::Identity();

            if (sim->is2DSimulation())
                m_F[i](2, 2) = 1.0;

            // compute Cauchy strain: epsilon = 0.5 (F + F^T) - I
            Vector6r strain;
            strain[0] = m_F[i](0, 0) - static_cast<Real>(1.0);                      // \epsilon_{00}
            strain[1] = m_F[i](1, 1) - static_cast<Real>(1.0);                      // \epsilon_{11}
            strain[2] = m_F[i](2, 2) - static_cast<Real>(1.0);                      // \epsilon_{22}
            strain[3] = static_cast<Real>(0.5) * (m_F[i](0, 1) + m_F[i](1, 0));         // \epsilon_{01}
            strain[4] = static_cast<Real>(0.5) * (m_F[i](0, 2) + m_F[i](2, 0));         // \epsilon_{02}
            strain[5] = static_cast<Real>(0.5) * (m_F[i](1, 2) + m_F[i](2, 1));         // \epsilon_{12}

            // First Piola Kirchhoff stress = 2 mu epsilon + lambda trace(epsilon) I
            const Real trace = strain[0] + strain[1] + strain[2];
            const Real ltrace = lambda*trace;
            Matrix3r& stress = m_stress[i];
            stress(0, 0) = static_cast<Real>(2.0) * mu * strain[0] + ltrace;
            stress(1, 1) = static_cast<Real>(2.0) * mu * strain[1] + ltrace;
            stress(2, 2) = static_cast<Real>(2.0) * mu * strain[2] + ltrace;

            stress(0, 1) = static_cast<Real>(2.0) * mu * strain[3];
            stress(1, 0) = static_cast<Real>(2.0) * mu * strain[3];
            stress(0, 2) = static_cast<Real>(2.0) * mu * strain[4];
            stress(2, 0) = static_cast<Real>(2.0) * mu * strain[4];
            stress(1, 2) = static_cast<Real>(2.0) * mu * strain[5];
            stress(2, 1) = static_cast<Real>(2.0) * mu * strain[5];
        }
    }

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = m_current_to_initial_index[i];
            const Vector3r &xi0 = m_model->getPosition0(i0);
            const size_t numNeighbors = m_initialNeighbors[i0].size();

            // Compute elastic force
            Vector3r force;
            force.setZero();
            for (unsigned int j = 0; j < numNeighbors; j++)
            {
                const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                // get initial neighbor index considering the current particle order
                const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                const Vector3r &xj0 = m_model->getPosition0(neighborIndex0);
                const Vector3r xi_xj_0 = xi0 - xj0;
                const Vector3r correctedRotatedKernel_i = m_RL[i] * sim->gradW(xi_xj_0);
                const Vector3r correctedRotatedKernel_j = -m_RL[neighborIndex] * sim->gradW(xi_xj_0);
                const Vector3r PWi = m_stress[i] * correctedRotatedKernel_i;
                const Vector3r PWj = m_stress[neighborIndex] * correctedRotatedKernel_j;
                force += m_restVolumes[i] * m_restVolumes[neighborIndex] * (PWi - PWj);
            }

            if (m_alpha != 0.0)
            {
                // Ganzenmüller, G.C. 2015. An hourglass control algorithm for Lagrangian
                // Smooth Particle Hydrodynamics. Computer Methods in Applied Mechanics and
                // Engineering 286, 87.106.
                Vector3r fi_hg;
                fi_hg.setZero();
                const Vector3r &xi = m_model->getPosition(i);
                for (unsigned int j = 0; j < numNeighbors; j++)
                {
                    const unsigned int neighborIndex = m_initial_to_current_index[m_initialNeighbors[i0][j]];
                    // get initial neighbor index considering the current particle order
                    const unsigned int neighborIndex0 = m_initialNeighbors[i0][j];

                    const Vector3r &xj = model->getPosition(neighborIndex);
                    const Vector3r &xj0 = m_model->getPosition0(neighborIndex0);

                    // Note: Ganzenmueller defines xij = xj-xi
                    const Vector3r xi_xj = -(xi - xj);
                    const Real xixj_l = xi_xj.norm();
                    if (xixj_l > 1.0e-6)
                    {
                        // Note: Ganzenmueller defines xij = xj-xi
                        const Vector3r xi_xj_0 = -(xi0 - xj0);
                        const Real xixj0_l2 = xi_xj_0.squaredNorm();
                        const Real W0 = sim->W(xi_xj_0);

                        const Vector3r xij_i = m_F[i] * m_rotations[i] * xi_xj_0;
                        const Vector3r xji_j = -m_F[neighborIndex] * m_rotations[neighborIndex] * xi_xj_0;
                        const Vector3r epsilon_ij_i = xij_i - xi_xj;
                        const Vector3r epsilon_ji_j = xji_j + xi_xj;

                        const Real delta_ij_i = epsilon_ij_i.dot(xi_xj) / xixj_l;
                        const Real delta_ji_j = -epsilon_ji_j.dot(xi_xj) / xixj_l;

                        fi_hg -= m_restVolumes[neighborIndex] * W0 / xixj0_l2 * (delta_ij_i + delta_ji_j) * xi_xj / xixj_l;
                    }
                }
                fi_hg *= m_alpha * m_youngsModulus * m_restVolumes[i];
                model->getAcceleration(i) += fi_hg / model->getMass(i);
            }
            if (model->getParticleState(i) == ParticleState::Active)
                rhs.segment<3>(3 * i) = model->getVelocity(i) + dt * (model->getAcceleration(i) + 1.0 / model->getMass(i) * force);
            else
                rhs.segment<3>(3 * i) = model->getVelocity(i);
        }
    }
}

void Elasticity_Peer2018::matrixVecProd(const Real* vec, Real *result, void *userData)
{
    Simulation *sim = Simulation::getCurrent();
    Elasticity_Peer2018 * elasticity = static_cast<Elasticity_Peer2018*>(userData);
    FluidModel *model = elasticity->getModel();
    const unsigned int numParticles = model->numActiveParticles();
    const Real dt = TimeManager::getCurrent()->getTimeStepSize();

    const auto &current_to_initial_index = elasticity->m_current_to_initial_index;
    const auto &initial_to_current_index = elasticity->m_initial_to_current_index;
    const auto &initialNeighbors = elasticity->m_initialNeighbors;
    const auto &restVolumes = elasticity->m_restVolumes;
    const auto &rotations = elasticity->m_rotations;
    const auto &L = elasticity->m_L;
    const auto &RL = elasticity->m_RL;
    auto &stress = elasticity->m_stress;
    const Real youngsModulus = elasticity->m_youngsModulus;
    const Real poissonRatio = elasticity->m_poissonRatio;

    Real mu = youngsModulus / (static_cast<Real>(2.0) * (static_cast<Real>(1.0) + poissonRatio));
    Real lambda = youngsModulus * poissonRatio / ((static_cast<Real>(1.0) + poissonRatio) * (static_cast<Real>(1.0) - static_cast<Real>(2.0) * poissonRatio));

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            const unsigned int i0 = current_to_initial_index[i];
            const Vector3r &pi = Eigen::Map<const Vector3r>(&vec[3 * i], 3);
            const Vector3r &xi0 = model->getPosition0(i0);
            const size_t numNeighbors = initialNeighbors[i0].size();

            // compute corotated deformation gradient (Eq. 18)
            Matrix3r nablaU;
            nablaU.setZero();

            // Fluid
            for (unsigned int j = 0; j < numNeighbors; j++)
            {
                const unsigned int neighborIndex = initial_to_current_index[initialNeighbors[i0][j]];
                // get initial neighbor index considering the current particle order
                const unsigned int neighborIndex0 = initialNeighbors[i0][j];

                const Vector3r &pj = Eigen::Map<const Vector3r>(&vec[3 * neighborIndex], 3);
                const Vector3r &xj0 = model->getPosition0(neighborIndex0);
                const Vector3r pj_pi = pj - pi;
                const Vector3r xi_xj_0 = xi0 - xj0;
                const Vector3r correctedRotatedKernel = RL[i] * sim->gradW(xi_xj_0);
                nablaU += restVolumes[neighborIndex] * pj_pi * correctedRotatedKernel.transpose();
            }
            nablaU *= dt;
            // compute Cauchy strain: epsilon = 0.5 (nablaU + nablaU^T)
            Vector6r strain;
            strain[0] = nablaU(0, 0);                                   // \epsilon_{00}
            strain[1] = nablaU(1, 1);                                   // \epsilon_{11}
            strain[2] = nablaU(2, 2);                                   // \epsilon_{22}
            strain[3] = static_cast<Real>(0.5) * (nablaU(0, 1) + nablaU(1, 0));         // \epsilon_{01}
            strain[4] = static_cast<Real>(0.5) * (nablaU(0, 2) + nablaU(2, 0));         // \epsilon_{02}
            strain[5] = static_cast<Real>(0.5) * (nablaU(1, 2) + nablaU(2, 1));         // \epsilon_{12}

            // First Piola Kirchhoff stress = 2 mu epsilon + lambda trace(epsilon) I
            const Real trace = strain[0] + strain[1] + strain[2];
            const Real ltrace = lambda*trace;
            stress[i](0, 0) = static_cast<Real>(2.0) * mu * strain[0] + ltrace;
            stress[i](1, 1) = static_cast<Real>(2.0) * mu * strain[1] + ltrace;
            stress[i](2, 2) = static_cast<Real>(2.0) * mu * strain[2] + ltrace;

            stress[i](0, 1) = static_cast<Real>(2.0) * mu * strain[3];
            stress[i](1, 0) = static_cast<Real>(2.0) * mu * strain[3];
            stress[i](0, 2) = static_cast<Real>(2.0) * mu * strain[4];
            stress[i](2, 0) = static_cast<Real>(2.0) * mu * strain[4];
            stress[i](1, 2) = static_cast<Real>(2.0) * mu * strain[5];
            stress[i](2, 1) = static_cast<Real>(2.0) * mu * strain[5];
        }
    }

    #pragma omp parallel default(shared)
    {
        #pragma omp for schedule(static)
        for (int i = 0; i < (int)numParticles; i++)
        {
            if (model->getParticleState(i) == ParticleState::Active)
            {
                const unsigned int i0 = current_to_initial_index[i];
                const Vector3r& xi0 = model->getPosition0(i0);
                const size_t numNeighbors = initialNeighbors[i0].size();

                // Compute elastic force
                Vector3r force;
                force.setZero();
                for (unsigned int j = 0; j < numNeighbors; j++)
                {
                    const unsigned int neighborIndex = initial_to_current_index[initialNeighbors[i0][j]];
                    // get initial neighbor index considering the current particle order
                    const unsigned int neighborIndex0 = initialNeighbors[i0][j];

                    const Vector3r& xj0 = model->getPosition0(neighborIndex0);
                    const Vector3r xi_xj_0 = xi0 - xj0;
                    const Vector3r gradW = sim->gradW(xi_xj_0);
                    const Vector3r correctedRotatedKernel_i = RL[i] * gradW;
                    const Vector3r correctedRotatedKernel_j = -RL[neighborIndex] * gradW;
                    const Vector3r PWi = stress[i] * correctedRotatedKernel_i;
                    const Vector3r PWj = stress[neighborIndex] * correctedRotatedKernel_j;
                    force += restVolumes[i] * restVolumes[neighborIndex] * (PWi - PWj);
                }

                const Real factor = dt / model->getMass(i);
                result[3 * i] = vec[3 * i] - factor * force[0];
                result[3 * i + 1] = vec[3 * i + 1] - factor * force[1];
                result[3 * i + 2] = vec[3 * i + 2] - factor * force[2];
            }
            else
            {
                result[3 * i] = vec[3 * i];
                result[3 * i + 1] = vec[3 * i + 1];
                result[3 * i + 2] = vec[3 * i + 2];
            }
        }
    }
}

#endif





void SPH::Elasticity_Peer2018::saveState(BinaryFileWriter &binWriter)
{
    binWriter.writeBuffer((char*)m_current_to_initial_index.data(), m_current_to_initial_index.size() * sizeof(unsigned int));
    binWriter.writeBuffer((char*)m_initial_to_current_index.data(), m_initial_to_current_index.size() * sizeof(unsigned int));
    binWriter.writeBuffer((char*)m_L.data(), m_L.size() * sizeof(Matrix3r));
}

void SPH::Elasticity_Peer2018::loadState(BinaryFileReader &binReader)
{
    binReader.readBuffer((char*)m_current_to_initial_index.data(), m_current_to_initial_index.size() * sizeof(unsigned int));
    binReader.readBuffer((char*)m_initial_to_current_index.data(), m_initial_to_current_index.size() * sizeof(unsigned int));
    binReader.readBuffer((char*)m_L.data(), m_L.size() * sizeof(Matrix3r));
}