NeoHookeanElasticity.cpp

Go to the documentation of this file.

#include "NeoHookeanElasticity.hpp"
 
#include <polyfem/autogen/auto_elasticity_rhs.hpp>
 
namespace polyfem::assembler
{
    namespace
    {
        bool delta(int i, int j)
        {
            return (i == j) ? true : false;
        }
    } // namespace
 
    NeoHookeanElasticity::NeoHookeanElasticity()
    {
    }
 
    void NeoHookeanElasticity::add_multimaterial(const int index, const json &params, const Units &units)
    {
        assert(size() == 2 || size() == 3);
 
        params_.add_multimaterial(index, params, size() == 3, units.stress());
    }
 
    Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 3, 1>
    NeoHookeanElasticity::compute_rhs(const AutodiffHessianPt &pt) const
    {
        assert(pt.size() == size());
        Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 3, 1> res;
 
        double lambda, mu;
        // TODO!
        params_.lambda_mu(0, 0, 0, pt(0).getValue(), pt(1).getValue(), size() == 2 ? 0. : pt(2).getValue(), 0, 0, lambda, mu);
 
        if (size() == 2)
            autogen::neo_hookean_2d_function(pt, lambda, mu, res);
        else if (size() == 3)
            autogen::neo_hookean_3d_function(pt, lambda, mu, res);
        else
            assert(false);
 
        return res;
    }
 
    Eigen::VectorXd
    NeoHookeanElasticity::assemble_gradient(const NonLinearAssemblerData &data) const
    {
        Eigen::Matrix<double, Eigen::Dynamic, 1> gradient;
 
        if (size() == 2)
        {
            switch (data.vals.basis_values.size())
            {
            case 3:
            {
                gradient.resize(6);
                compute_energy_aux_gradient_fast<3, 2>(data, gradient);
                break;
            }
            case 6:
            {
                gradient.resize(12);
                compute_energy_aux_gradient_fast<6, 2>(data, gradient);
                break;
            }
            case 10:
            {
                gradient.resize(20);
                compute_energy_aux_gradient_fast<10, 2>(data, gradient);
                break;
            }
            default:
            {
                gradient.resize(data.vals.basis_values.size() * 2);
                compute_energy_aux_gradient_fast<Eigen::Dynamic, 2>(data, gradient);
                break;
            }
            }
        }
        else // if (size() == 3)
        {
            assert(size() == 3);
            switch (data.vals.basis_values.size())
            {
            case 4:
            {
                gradient.resize(12);
                compute_energy_aux_gradient_fast<4, 3>(data, gradient);
                break;
            }
            case 10:
            {
                gradient.resize(30);
                compute_energy_aux_gradient_fast<10, 3>(data, gradient);
                break;
            }
            case 20:
            {
                gradient.resize(60);
                compute_energy_aux_gradient_fast<20, 3>(data, gradient);
                break;
            }
            default:
            {
                gradient.resize(data.vals.basis_values.size() * 3);
                compute_energy_aux_gradient_fast<Eigen::Dynamic, 3>(data, gradient);
                break;
            }
            }
        }
 
        return gradient;
    }
 
    void NeoHookeanElasticity::compute_stiffness_value(const double t,
                                                       const assembler::ElementAssemblyValues &vals,
                                                       const Eigen::MatrixXd &local_pts,
                                                       const Eigen::MatrixXd &displacement,
                                                       Eigen::MatrixXd &tensor) const
    {
        tensor.resize(local_pts.rows(), size() * size() * size() * size());
        assert(displacement.cols() == 1);
 
        Eigen::MatrixXd displacement_grad(size(), size());
 
        for (long p = 0; p < local_pts.rows(); ++p)
        {
            double lambda, mu;
            params_.lambda_mu(local_pts.row(p), vals.val.row(p), t, vals.element_id, lambda, mu);
 
            compute_diplacement_grad(size(), vals, local_pts, p, displacement, displacement_grad);
            const Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(size(), size()) + displacement_grad;
            const Eigen::MatrixXd FmT = def_grad.inverse().transpose();
            const Eigen::VectorXd FmT_vec = utils::flatten(FmT);
            const double J = def_grad.determinant();
            const double tmp1 = mu - lambda * std::log(J);
            for (int i = 0, idx = 0; i < size(); i++)
                for (int j = 0; j < size(); j++)
                    for (int k = 0; k < size(); k++)
                        for (int l = 0; l < size(); l++)
                        {
                            tensor(p, idx) = mu * delta(i, k) * delta(j, l) + tmp1 * FmT(i, l) * FmT(k, j);
                            idx++;
                        }
 
            tensor.row(p) += lambda * utils::flatten(FmT_vec * FmT_vec.transpose());
 
            // {
            //  Eigen::MatrixXd hess = utils::unflatten(tensor.row(p), size()*size());
            //  Eigen::MatrixXd fhess;
            //  Eigen::VectorXd x0 = utils::flatten(def_grad);
            //  fd::finite_jacobian(
            //      x0, [this, lambda, mu](const Eigen::VectorXd &x1) -> Eigen::VectorXd
            //      {
            //          Eigen::MatrixXd def_grad = utils::unflatten(x1, this->size());
            //          const Eigen::MatrixXd FmT = def_grad.inverse().transpose();
            //          const double J = def_grad.determinant();
            //          Eigen::MatrixXd stress_tensor = mu * (def_grad - FmT) + lambda * std::log(J) * FmT;
            //          return utils::flatten(stress_tensor);
            //      }, fhess);
 
            //  if (!fd::compare_hessian(hess, fhess))
            //  {
            //      std::cout << "Hessian: " << hess << std::endl;
            //      std::cout << "Finite hessian: " << fhess << std::endl;
            //      log_and_throw_error("Hessian in Neohookean mismatch");
            //  }
            // }
        }
    }
 
    Eigen::MatrixXd
    NeoHookeanElasticity::assemble_hessian(const NonLinearAssemblerData &data) const
    {
        Eigen::MatrixXd hessian;
 
        if (size() == 2)
        {
            switch (data.vals.basis_values.size())
            {
            case 3:
            {
                hessian.resize(6, 6);
                hessian.setZero();
                compute_energy_hessian_aux_fast<3, 2>(data, hessian);
                break;
            }
            case 6:
            {
                hessian.resize(12, 12);
                hessian.setZero();
                compute_energy_hessian_aux_fast<6, 2>(data, hessian);
                break;
            }
            case 10:
            {
                hessian.resize(20, 20);
                hessian.setZero();
                compute_energy_hessian_aux_fast<10, 2>(data, hessian);
                break;
            }
            default:
            {
                hessian.resize(data.vals.basis_values.size() * 2, data.vals.basis_values.size() * 2);
                hessian.setZero();
                compute_energy_hessian_aux_fast<Eigen::Dynamic, 2>(data, hessian);
                break;
            }
            }
        }
        else // if (size() == 3)
        {
            assert(size() == 3);
            switch (data.vals.basis_values.size())
            {
            case 4:
            {
                hessian.resize(12, 12);
                hessian.setZero();
                compute_energy_hessian_aux_fast<4, 3>(data, hessian);
                break;
            }
            case 10:
            {
                hessian.resize(30, 30);
                hessian.setZero();
                compute_energy_hessian_aux_fast<10, 3>(data, hessian);
                break;
            }
            case 20:
            {
                hessian.resize(60, 60);
                hessian.setZero();
                compute_energy_hessian_aux_fast<20, 3>(data, hessian);
                break;
            }
            default:
            {
                hessian.resize(data.vals.basis_values.size() * 3, data.vals.basis_values.size() * 3);
                hessian.setZero();
                compute_energy_hessian_aux_fast<Eigen::Dynamic, 3>(data, hessian);
                break;
            }
            }
        }
 
        return hessian;
    }
 
    void NeoHookeanElasticity::assign_stress_tensor(const OutputData &data,
                                                    const int all_size,
                                                    const ElasticityTensorType &type,
                                                    Eigen::MatrixXd &all,
                                                    const std::function<Eigen::MatrixXd(const Eigen::MatrixXd &)> &fun) const
    {
        const auto &displacement = data.fun;
        const auto &local_pts = data.local_pts;
        const auto &bs = data.bs;
        const auto &gbs = data.gbs;
        const auto el_id = data.el_id;
        const auto t = data.t;
 
        Eigen::MatrixXd displacement_grad(size(), size());
 
        assert(displacement.cols() == 1);
 
        all.resize(local_pts.rows(), all_size);
 
        ElementAssemblyValues vals;
        vals.compute(el_id, size() == 3, local_pts, bs, gbs);
        const auto I = Eigen::MatrixXd::Identity(size(), size());
 
        for (long p = 0; p < local_pts.rows(); ++p)
        {
            compute_diplacement_grad(size(), bs, vals, local_pts, p, displacement, displacement_grad);
 
            const Eigen::MatrixXd def_grad = I + displacement_grad;
            if (type == ElasticityTensorType::F)
            {
                all.row(p) = fun(def_grad);
                continue;
            }
            const double J = def_grad.determinant();
            const Eigen::MatrixXd b = def_grad * def_grad.transpose();
 
            double lambda, mu;
            params_.lambda_mu(local_pts.row(p), vals.val.row(p), t, vals.element_id, lambda, mu);
 
            Eigen::MatrixXd stress_tensor = (lambda * std::log(J) * I + mu * (b - I)) / J;
            if (type == ElasticityTensorType::PK1)
                stress_tensor = pk1_from_cauchy(stress_tensor, def_grad);
            else if (type == ElasticityTensorType::PK2)
                stress_tensor = pk2_from_cauchy(stress_tensor, def_grad);
 
            all.row(p) = fun(stress_tensor);
        }
    }
 
    double NeoHookeanElasticity::compute_energy(const NonLinearAssemblerData &data) const
    {
        return compute_energy_aux<double>(data);
    }
 
    // Compute ∫ ½μ (tr(FᵀF) - 3 - 2ln(J)) + ½λ ln²(J) du
    template <typename T>
    T NeoHookeanElasticity::compute_energy_aux(const NonLinearAssemblerData &data) const
    {
        typedef Eigen::Matrix<T, Eigen::Dynamic, 1> AutoDiffVect;
        typedef Eigen::Matrix<T, Eigen::Dynamic, Eigen::Dynamic, 0, 3, 3> AutoDiffGradMat;
 
        AutoDiffVect local_disp;
        get_local_disp(data, size(), local_disp);
 
        AutoDiffGradMat def_grad(size(), size());
 
        T energy = T(0.0);
 
        const int n_pts = data.da.size();
        for (long p = 0; p < n_pts; ++p)
        {
            compute_disp_grad_at_quad(data, local_disp, p, size(), def_grad);
 
            // Id + grad d
            for (int d = 0; d < size(); ++d)
                def_grad(d, d) += T(1);
 
            double lambda, mu;
            params_.lambda_mu(data.vals.quadrature.points.row(p), data.vals.val.row(p), data.t, data.vals.element_id, lambda, mu);
 
            const T log_det_j = log(polyfem::utils::determinant(def_grad));
            const T val = mu / 2 * ((def_grad.transpose() * def_grad).trace() - size() - 2 * log_det_j) + lambda / 2 * log_det_j * log_det_j;
 
            energy += val * data.da(p);
        }
        return energy;
    }
 
    template <int dim>
    Eigen::Matrix<double, dim, dim> hat(const Eigen::Matrix<double, dim, 1> &x)
    {
 
        Eigen::Matrix<double, dim, dim> prod;
        prod.setZero();
 
        prod(0, 1) = -x(2);
        prod(0, 2) = x(1);
        prod(1, 0) = x(2);
        prod(1, 2) = -x(0);
        prod(2, 0) = -x(1);
        prod(2, 1) = x(0);
 
        return prod;
    }
 
    template <int dim>
    Eigen::Matrix<double, dim, 1> cross(const Eigen::Matrix<double, dim, 1> &x, const Eigen::Matrix<double, dim, 1> &y)
    {
 
        Eigen::Matrix<double, dim, 1> z;
        z.setZero();
 
        z(0) = x(1) * y(2) - x(2) * y(1);
        z(1) = x(2) * y(0) - x(0) * y(2);
        z(2) = x(0) * y(1) - x(1) * y(0);
 
        return z;
    }
 
    template <int n_basis, int dim>
    void NeoHookeanElasticity::compute_energy_aux_gradient_fast(const NonLinearAssemblerData &data, Eigen::Matrix<double, Eigen::Dynamic, 1> &G_flattened) const
    {
        assert(data.x.cols() == 1);
 
        const int n_pts = data.da.size();
 
        Eigen::Matrix<double, n_basis, dim> local_disp(data.vals.basis_values.size(), size());
        local_disp.setZero();
        for (size_t i = 0; i < data.vals.basis_values.size(); ++i)
        {
            const auto &bs = data.vals.basis_values[i];
            for (size_t ii = 0; ii < bs.global.size(); ++ii)
            {
                for (int d = 0; d < size(); ++d)
                {
                    local_disp(i, d) += bs.global[ii].val * data.x(bs.global[ii].index * size() + d);
                }
            }
        }
 
        Eigen::Matrix<double, dim, dim> def_grad(size(), size());
 
        Eigen::Matrix<double, n_basis, dim> G(data.vals.basis_values.size(), size());
        G.setZero();
 
        for (long p = 0; p < n_pts; ++p)
        {
            Eigen::Matrix<double, n_basis, dim> grad(data.vals.basis_values.size(), size());
 
            for (size_t i = 0; i < data.vals.basis_values.size(); ++i)
            {
                grad.row(i) = data.vals.basis_values[i].grad.row(p);
            }
 
            Eigen::Matrix<double, dim, dim> jac_it = data.vals.jac_it[p];
 
            // Id + grad d
            def_grad = local_disp.transpose() * grad * jac_it + Eigen::Matrix<double, dim, dim>::Identity(size(), size());
 
            const double J = def_grad.determinant();
            const double log_det_j = log(J);
 
            Eigen::Matrix<double, dim, dim> delJ_delF(size(), size());
            delJ_delF.setZero();
 
            if (dim == 2)
            {
 
                delJ_delF(0, 0) = def_grad(1, 1);
                delJ_delF(0, 1) = -def_grad(1, 0);
                delJ_delF(1, 0) = -def_grad(0, 1);
                delJ_delF(1, 1) = def_grad(0, 0);
            }
 
            else if (dim == 3)
            {
 
                Eigen::Matrix<double, dim, 1> u(def_grad.rows());
                Eigen::Matrix<double, dim, 1> v(def_grad.rows());
                Eigen::Matrix<double, dim, 1> w(def_grad.rows());
 
                u = def_grad.col(0);
                v = def_grad.col(1);
                w = def_grad.col(2);
 
                delJ_delF.col(0) = cross<dim>(v, w);
                delJ_delF.col(1) = cross<dim>(w, u);
                delJ_delF.col(2) = cross<dim>(u, v);
            }
 
            double lambda, mu;
            params_.lambda_mu(data.vals.quadrature.points.row(p), data.vals.val.row(p), data.t, data.vals.element_id, lambda, mu);
 
            Eigen::Matrix<double, n_basis, dim> delF_delU = grad * jac_it;
 
            Eigen::Matrix<double, dim, dim> gradient_temp = mu * def_grad - mu * (1 / J) * delJ_delF + lambda * log_det_j * (1 / J) * delJ_delF;
            Eigen::Matrix<double, n_basis, dim> gradient = delF_delU * gradient_temp.transpose();
 
            double val = mu / 2 * ((def_grad.transpose() * def_grad).trace() - size() - 2 * log_det_j) + lambda / 2 * log_det_j * log_det_j;
 
            G.noalias() += gradient * data.da(p);
        }
 
        Eigen::Matrix<double, dim, n_basis> G_T = G.transpose();
 
        constexpr int N = (n_basis == Eigen::Dynamic) ? Eigen::Dynamic : n_basis * dim;
        Eigen::Matrix<double, N, 1> temp(Eigen::Map<Eigen::Matrix<double, N, 1>>(G_T.data(), G_T.size()));
        G_flattened = temp;
    }
 
    template <int n_basis, int dim>
    void NeoHookeanElasticity::compute_energy_hessian_aux_fast(const NonLinearAssemblerData &data, Eigen::MatrixXd &H) const
    {
        assert(data.x.cols() == 1);
 
        constexpr int N = (n_basis == Eigen::Dynamic) ? Eigen::Dynamic : n_basis * dim;
        const int n_pts = data.da.size();
 
        Eigen::Matrix<double, n_basis, dim> local_disp(data.vals.basis_values.size(), size());
        local_disp.setZero();
        for (size_t i = 0; i < data.vals.basis_values.size(); ++i)
        {
            const auto &bs = data.vals.basis_values[i];
            for (size_t ii = 0; ii < bs.global.size(); ++ii)
            {
                for (int d = 0; d < size(); ++d)
                {
                    local_disp(i, d) += bs.global[ii].val * data.x(bs.global[ii].index * size() + d);
                }
            }
        }
 
        Eigen::Matrix<double, dim, dim> def_grad(size(), size());
 
        for (long p = 0; p < n_pts; ++p)
        {
            Eigen::Matrix<double, n_basis, dim> grad(data.vals.basis_values.size(), size());
 
            for (size_t i = 0; i < data.vals.basis_values.size(); ++i)
            {
                grad.row(i) = data.vals.basis_values[i].grad.row(p);
            }
 
            Eigen::Matrix<double, dim, dim> jac_it = data.vals.jac_it[p];
 
            // Id + grad d
            def_grad = local_disp.transpose() * grad * jac_it + Eigen::Matrix<double, dim, dim>::Identity(size(), size());
 
            const double J = def_grad.determinant();
            double log_det_j = log(J);
 
            Eigen::Matrix<double, dim, dim> delJ_delF(size(), size());
            delJ_delF.setZero();
            Eigen::Matrix<double, dim * dim, dim * dim> del2J_delF2(size() * size(), size() * size());
            del2J_delF2.setZero();
 
            if (dim == 2)
            {
                delJ_delF(0, 0) = def_grad(1, 1);
                delJ_delF(0, 1) = -def_grad(1, 0);
                delJ_delF(1, 0) = -def_grad(0, 1);
                delJ_delF(1, 1) = def_grad(0, 0);
 
                del2J_delF2(0, 3) = 1;
                del2J_delF2(1, 2) = -1;
                del2J_delF2(2, 1) = -1;
                del2J_delF2(3, 0) = 1;
            }
            else if (size() == 3)
            {
                Eigen::Matrix<double, dim, 1> u(def_grad.rows());
                Eigen::Matrix<double, dim, 1> v(def_grad.rows());
                Eigen::Matrix<double, dim, 1> w(def_grad.rows());
 
                u = def_grad.col(0);
                v = def_grad.col(1);
                w = def_grad.col(2);
 
                delJ_delF.col(0) = cross<dim>(v, w);
                delJ_delF.col(1) = cross<dim>(w, u);
                delJ_delF.col(2) = cross<dim>(u, v);
 
                del2J_delF2.template block<dim, dim>(0, 6) = hat<dim>(v);
                del2J_delF2.template block<dim, dim>(6, 0) = -hat<dim>(v);
                del2J_delF2.template block<dim, dim>(0, 3) = -hat<dim>(w);
                del2J_delF2.template block<dim, dim>(3, 0) = hat<dim>(w);
                del2J_delF2.template block<dim, dim>(3, 6) = -hat<dim>(u);
                del2J_delF2.template block<dim, dim>(6, 3) = hat<dim>(u);
            }
 
            double lambda, mu;
            params_.lambda_mu(data.vals.quadrature.points.row(p), data.vals.val.row(p), data.t, data.vals.element_id, lambda, mu);
 
            Eigen::Matrix<double, dim * dim, dim * dim> id = Eigen::Matrix<double, dim * dim, dim * dim>::Identity(size() * size(), size() * size());
 
            Eigen::Matrix<double, dim * dim, 1> g_j = Eigen::Map<const Eigen::Matrix<double, dim * dim, 1>>(delJ_delF.data(), delJ_delF.size());
 
            Eigen::Matrix<double, dim * dim, dim * dim> hessian_temp = (mu * id) + (((mu + lambda * (1 - log_det_j)) / (J * J)) * (g_j * g_j.transpose())) + (((lambda * log_det_j - mu) / (J)) * del2J_delF2);
 
            Eigen::Matrix<double, dim * dim, N> delF_delU_tensor(jac_it.size(), grad.size());
 
            for (size_t i = 0; i < local_disp.rows(); ++i)
            {
                for (size_t j = 0; j < local_disp.cols(); ++j)
                {
                    Eigen::Matrix<double, dim, dim> temp(size(), size());
                    temp.setZero();
                    temp.row(j) = grad.row(i);
                    temp = temp * jac_it;
                    Eigen::Matrix<double, dim * dim, 1> temp_flattened(Eigen::Map<Eigen::Matrix<double, dim * dim, 1>>(temp.data(), temp.size()));
                    delF_delU_tensor.col(i * size() + j) = temp_flattened;
                }
            }
 
            Eigen::Matrix<double, N, N> hessian = delF_delU_tensor.transpose() * hessian_temp * delF_delU_tensor;
 
            double val = mu / 2 * ((def_grad.transpose() * def_grad).trace() - size() - 2 * log_det_j) + lambda / 2 * log_det_j * log_det_j;
 
            H += hessian * data.da(p);
        }
    }
 
    void NeoHookeanElasticity::compute_stress_grad_multiply_mat(
        const OptAssemblerData &data,
        const Eigen::MatrixXd &mat,
        Eigen::MatrixXd &stress,
        Eigen::MatrixXd &result) const
    {
        const double t = data.t;
        const int el_id = data.el_id;
        const Eigen::MatrixXd &local_pts = data.local_pts;
        const Eigen::MatrixXd &global_pts = data.global_pts;
        const Eigen::MatrixXd &grad_u_i = data.grad_u_i;
 
        double lambda, mu;
        params_.lambda_mu(local_pts, global_pts, t, el_id, lambda, mu);
 
        Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_i.rows(), grad_u_i.cols()) + grad_u_i;
        Eigen::MatrixXd FmT = def_grad.inverse().transpose();
 
        stress = mu * (def_grad - FmT) + lambda * std::log(def_grad.determinant()) * FmT;
        result = mu * mat + FmT * mat.transpose() * FmT * (mu - lambda * std::log(def_grad.determinant())) + lambda * (FmT.array() * mat.array()).sum() * FmT;
    }
 
    void NeoHookeanElasticity::compute_stress_grad_multiply_stress(
        const OptAssemblerData &data,
        Eigen::MatrixXd &stress,
        Eigen::MatrixXd &result) const
    {
        const double t = data.t;
        const int el_id = data.el_id;
        const Eigen::MatrixXd &local_pts = data.local_pts;
        const Eigen::MatrixXd &global_pts = data.global_pts;
        const Eigen::MatrixXd &grad_u_i = data.grad_u_i;
 
        double lambda, mu;
        params_.lambda_mu(local_pts, global_pts, t, el_id, lambda, mu);
 
        Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_i.rows(), grad_u_i.cols()) + grad_u_i;
        Eigen::MatrixXd FmT = def_grad.inverse().transpose();
 
        stress = mu * (def_grad - FmT) + lambda * std::log(def_grad.determinant()) * FmT;
        result = mu * stress + FmT * stress.transpose() * FmT * (mu - lambda * std::log(def_grad.determinant())) + lambda * (FmT.array() * stress.array()).sum() * FmT;
    }
 
    void NeoHookeanElasticity::compute_stress_grad_multiply_vect(
        const OptAssemblerData &data,
        const Eigen::MatrixXd &vect,
        Eigen::MatrixXd &stress,
        Eigen::MatrixXd &result) const
    {
        const double t = data.t;
        const int el_id = data.el_id;
        const Eigen::MatrixXd &local_pts = data.local_pts;
        const Eigen::MatrixXd &global_pts = data.global_pts;
        const Eigen::MatrixXd &grad_u_i = data.grad_u_i;
 
        double lambda, mu;
        params_.lambda_mu(local_pts, global_pts, t, el_id, lambda, mu);
 
        Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_i.rows(), grad_u_i.cols()) + grad_u_i;
        Eigen::MatrixXd FmT = def_grad.inverse().transpose();
 
        stress = mu * (def_grad - FmT) + lambda * std::log(def_grad.determinant()) * FmT;
        result.setZero(size() * size(), size());
        if (vect.rows() == 1)
            for (int i = 0; i < size(); ++i)
                for (int j = 0; j < size(); ++j)
                    for (int l = 0; l < size(); ++l)
                    {
                        result(i * size() + j, l) += mu * vect(i) * ((j == l) ? 1 : 0);
                        // For some reason, the second gives lower error. From the formula, though, it should be the following.
                        // result(i * size() + j, l) += (mu - lambda * std::log(def_grad.determinant())) * FmT(j, l) * (FmT.col(i).array() * vect.transpose().array()).sum();
                        result(i * size() + j, l) += (mu - lambda * std::log(def_grad.determinant())) * FmT(i, l) * (FmT.col(j).array() * vect.transpose().array()).sum();
                        result(i * size() + j, l) += lambda * FmT(i, j) * (FmT.col(l).array() * vect.transpose().array()).sum();
                    }
        else
            for (int i = 0; i < size(); ++i)
                for (int j = 0; j < size(); ++j)
                    for (int k = 0; k < size(); ++k)
                    {
                        result(i * size() + j, k) += mu * vect(j) * ((i == k) ? 1 : 0);
                        // For some reason, the second gives lower error. From the formula, though, it should be the following.
                        // result(i * size() + j, k) += (mu - lambda * std::log(def_grad.determinant())) * FmT(k, j) * (FmT.row(i).array() * vect.transpose().array()).sum();
                        result(i * size() + j, k) += (mu - lambda * std::log(def_grad.determinant())) * FmT(k, i) * (FmT.row(j).array() * vect.transpose().array()).sum();
                        result(i * size() + j, k) += lambda * FmT(i, j) * (FmT.row(k).array() * vect.transpose().array()).sum();
                    }
    }
 
    void NeoHookeanElasticity::compute_dstress_dmu_dlambda(
        const OptAssemblerData &data,
        Eigen::MatrixXd &dstress_dmu,
        Eigen::MatrixXd &dstress_dlambda) const
    {
        const double t = data.t;
        const int el_id = data.el_id;
        const Eigen::MatrixXd &local_pts = data.local_pts;
        const Eigen::MatrixXd &global_pts = data.global_pts;
        const Eigen::MatrixXd &grad_u_i = data.grad_u_i;
 
        Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_i.rows(), grad_u_i.cols()) + grad_u_i;
        Eigen::MatrixXd FmT = def_grad.inverse().transpose();
        dstress_dmu = def_grad - FmT;
        dstress_dlambda = std::log(def_grad.determinant()) * FmT;
    }
 
    std::map<std::string, Assembler::ParamFunc> NeoHookeanElasticity::parameters() const
    {
        std::map<std::string, ParamFunc> res;
        const auto &params = params_;
        const int size = this->size();
 
        res["lambda"] = [&params](const RowVectorNd &uv, const RowVectorNd &p, double t, int e) {
            double lambda, mu;
 
            params.lambda_mu(uv, p, t, e, lambda, mu);
            return lambda;
        };
 
        res["mu"] = [&params](const RowVectorNd &uv, const RowVectorNd &p, double t, int e) {
            double lambda, mu;
 
            params.lambda_mu(uv, p, t, e, lambda, mu);
            return mu;
        };
 
        res["E"] = [&params, size](const RowVectorNd &uv, const RowVectorNd &p, double t, int e) {
            double lambda, mu;
            params.lambda_mu(uv, p, t, e, lambda, mu);
 
            if (size == 3)
                return mu * (3.0 * lambda + 2.0 * mu) / (lambda + mu);
            else
                return 2 * mu * (2.0 * lambda + 2.0 * mu) / (lambda + 2.0 * mu);
        };
 
        res["nu"] = [&params, size](const RowVectorNd &uv, const RowVectorNd &p, double t, int e) {
            double lambda, mu;
 
            params.lambda_mu(uv, p, t, e, lambda, mu);
 
            if (size == 3)
                return lambda / (2.0 * (lambda + mu));
            else
                return lambda / (lambda + 2.0 * mu);
        };
 
        return res;
    }
 
} // namespace polyfem::assembler