SpatialIntegralForms.cpp

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#include "SpatialIntegralForms.hpp"
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/utils/MaybeParallelFor.hpp>
 
#include <polyfem/State.hpp>
#include <polyfem/assembler/Mass.hpp>
 
#include <polyfem/solver/NLProblem.hpp>
#include <polyfem/solver/NLHomoProblem.hpp>
 
#include <polyfem/utils/IntegrableFunctional.hpp>
 
using namespace polyfem::utils;
 
namespace polyfem::solver
{
    namespace
    {
        bool delta(int i, int j)
        {
            return (i == j) ? true : false;
        }
 
        double dot(const Eigen::MatrixXd &A, const Eigen::MatrixXd &B) { return (A.array() * B.array()).sum(); }
 
        class LocalThreadScalarStorage
        {
        public:
            double val;
            assembler::ElementAssemblyValues vals;
            QuadratureVector da;
 
            LocalThreadScalarStorage()
            {
                val = 0;
            }
        };
 
        class LocalThreadVecStorage
        {
        public:
            Eigen::MatrixXd vec;
            assembler::ElementAssemblyValues vals;
            QuadratureVector da;
 
            LocalThreadVecStorage(const int size)
            {
                vec.resize(size, 1);
                vec.setZero();
            }
        };
    } // namespace
 
    double SpatialIntegralForm::value_unweighted_step(const int time_step, const Eigen::VectorXd &x) const
    {
        assert(time_step < state_.diff_cached.size());
        return AdjointTools::integrate_objective(state_, get_integral_functional(), state_.diff_cached.u(time_step), ids_, spatial_integral_type_, time_step);
    }
 
    void SpatialIntegralForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        assert(time_step < state_.diff_cached.size());
        gradv = weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::Shape, &state_, x, [this, time_step, &x]() {
            Eigen::VectorXd term;
            AdjointTools::compute_shape_derivative_functional_term(this->state_, this->state_.diff_cached.u(time_step), this->get_integral_functional(), this->ids_, this->spatial_integral_type_, term, time_step);
            return term;
        });
        gradv += variable_to_simulations_.apply_parametrization_jacobian(ParameterType::PeriodicShape, &state_, x, [this, time_step, &x]() {
            Eigen::VectorXd term;
            AdjointTools::compute_shape_derivative_functional_term(this->state_, this->state_.diff_cached.u(time_step), this->get_integral_functional(), this->ids_, this->spatial_integral_type_, term, time_step);
            term *= this->weight();
 
            const Eigen::VectorXd adjoint_rhs = this->compute_adjoint_rhs_step(time_step, x, state_);
            const NLHomoProblem &homo_problem = *std::dynamic_pointer_cast<NLHomoProblem>(state_.solve_data.nl_problem);
            const Eigen::VectorXd full_shape_deriv = homo_problem.reduced_to_full_shape_derivative(state_.diff_cached.disp_grad(), adjoint_rhs);
            term += utils::flatten(utils::unflatten(full_shape_deriv, state_.mesh->dimension())(state_.primitive_to_node(), Eigen::all));
 
            return term;
        });
    }
 
    Eigen::VectorXd SpatialIntegralForm::compute_adjoint_rhs_step(const int time_step, const Eigen::VectorXd &x, const State &state) const
    {
        if (&state != &state_)
            return Eigen::VectorXd::Zero(state.ndof());
 
        assert(time_step < state_.diff_cached.size());
 
        Eigen::VectorXd rhs;
        AdjointTools::dJ_du_step(state, get_integral_functional(), state.diff_cached.u(time_step), ids_, spatial_integral_type_, time_step, rhs);
 
        return rhs * weight();
    }
 
    IntegrableFunctional ElasticEnergyForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        const std::string formulation = state_.formulation();
 
        j.set_j([formulation, this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), 1);
            const int dim = u.cols();
            Eigen::MatrixXd grad_u_q;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                grad_u_q = utils::unflatten(grad_u.row(q), u.cols());
                if (formulation == "LinearElasticity")
                {
                    grad_u_q = (grad_u_q + grad_u_q.transpose()).eval() / 2.;
                    val(q) = mu(q) * (grad_u_q.transpose() * grad_u_q).trace() + lambda(q) / 2. * grad_u_q.trace() * grad_u_q.trace();
                }
                else if (formulation == "NeoHookean")
                {
                    Eigen::MatrixXd def_grad = grad_u_q + Eigen::MatrixXd::Identity(dim, dim);
                    double log_det_j = log(def_grad.determinant());
                    val(q) = mu(q) / 2 * ((def_grad.transpose() * def_grad).trace() - dim - 2 * log_det_j) + lambda(q) / 2 * log_det_j * log_det_j;
                }
                else
                    log_and_throw_adjoint_error("[{}] Unknown formulation {}!", name(), formulation);
            }
        });
 
        j.set_dj_dgradu([formulation, this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), grad_u.cols());
            Eigen::MatrixXd grad_u_q, def_grad, FmT, stress;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                grad_u_q = utils::unflatten(grad_u.row(q), u.cols());
                if (formulation == "LinearElasticity")
                {
                    stress = mu(q) * (grad_u_q + grad_u_q.transpose()) + lambda(q) * grad_u_q.trace() * Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols());
                }
                else if (formulation == "NeoHookean")
                {
                    def_grad = Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols()) + grad_u_q;
                    FmT = def_grad.inverse().transpose();
                    stress = mu(q) * (def_grad - FmT) + lambda(q) * std::log(def_grad.determinant()) * FmT;
                }
                else
                    log_and_throw_adjoint_error("[{}] Unknown formulation {}!", name(), formulation);
                val.row(q) = utils::flatten(stress);
            }
        });
 
        return j;
    }
 
    void ElasticEnergyForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        SpatialIntegralForm::compute_partial_gradient_step(time_step, x, gradv);
        gradv += weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::LameParameter, &state_, x, [this]() {
            log_and_throw_adjoint_error("[{}] Doesn't support derivatives wrt. material!", name());
            return Eigen::VectorXd::Zero(0).eval();
        });
    }
 
    IntegrableFunctional StressNormForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        const std::string formulation = state_.formulation();
        const int power = in_power_;
 
        j.set_j([formulation, power, &state = std::as_const(state_)](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), 1);
            const double dt = state.problem->is_time_dependent() ? state.args["time"]["dt"].get<double>() : 0;
 
            Eigen::MatrixXd grad_u_q, stress, grad_unused;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                if (formulation == "Laplacian")
                    stress = grad_u.row(q);
                else
                {
                    vector2matrix(grad_u.row(q), grad_u_q);
                    state.assembler->compute_stress_grad_multiply_mat(OptAssemblerData(params.t, dt, params.elem, local_pts.row(q), pts.row(q), grad_u_q), Eigen::MatrixXd::Zero(grad_u_q.rows(), grad_u_q.cols()), stress, grad_unused);
                }
                val(q) = pow(stress.squaredNorm(), power / 2.);
            }
        });
 
        j.set_dj_dgradu([formulation, power, &state = std::as_const(state_)](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), grad_u.cols());
            const double dt = state.problem->is_time_dependent() ? state.args["time"]["dt"].get<double>() : 0;
            const int dim = state.mesh->dimension();
 
            if (formulation == "Laplacian")
            {
                for (int q = 0; q < grad_u.rows(); q++)
                {
                    const double coef = power * pow(grad_u.row(q).squaredNorm(), power / 2. - 1.);
                    val.row(q) = coef * grad_u.row(q);
                }
            }
            else
            {
                Eigen::MatrixXd grad_u_q, stress, stress_dstress;
                for (int q = 0; q < grad_u.rows(); q++)
                {
                    vector2matrix(grad_u.row(q), grad_u_q);
                    state.assembler->compute_stress_grad_multiply_stress(OptAssemblerData(params.t, dt, params.elem, local_pts.row(q), pts.row(q), grad_u_q), stress, stress_dstress);
 
                    const double coef = power * pow(stress.squaredNorm(), power / 2. - 1.);
                    val.row(q) = coef * utils::flatten(stress_dstress);
                }
            }
        });
 
        return j;
    }
 
    void StressNormForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        SpatialIntegralForm::compute_partial_gradient_step(time_step, x, gradv);
        gradv += weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::LameParameter, &state_, x, [this]() {
            log_and_throw_adjoint_error("[{}] Doesn't support derivatives wrt. material!", name());
            return Eigen::VectorXd::Zero(0).eval();
        });
    }
 
    IntegrableFunctional ComplianceForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        if (state_.formulation() != "LinearElasticity")
            log_and_throw_adjoint_error("[{}] Only Linear Elasticity formulation is supported!", name());
 
        j.set_j([this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), 1);
 
            Eigen::MatrixXd grad_u_q, stress;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                vector2matrix(grad_u.row(q), grad_u_q);
                stress = mu(q) * (grad_u_q + grad_u_q.transpose()) + lambda(q) * grad_u_q.trace() * Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols());
                val(q) = (stress.array() * grad_u_q.array()).sum();
            }
        });
 
        j.set_dj_dgradu([](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), grad_u.cols());
            const int dim = u.cols();
 
            Eigen::MatrixXd grad_u_q, stress;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                vector2matrix(grad_u.row(q), grad_u_q);
                stress = mu(q) * (grad_u_q + grad_u_q.transpose()) + lambda(q) * grad_u_q.trace() * Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols());
                val.row(q) = 2 * utils::flatten(stress);
            }
        });
 
        return j;
    }
 
    void ComplianceForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        const double dt = state_.problem->is_time_dependent() ? state_.args["time"]["dt"].get<double>() : 0;
        const double t = state_.problem->is_time_dependent() ? dt * time_step + state_.args["time"]["t0"].get<double>() : 0;
 
        SpatialIntegralForm::compute_partial_gradient_step(time_step, x, gradv);
        gradv = weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::LameParameter, &state_, x, [this, t, dt, time_step, &x]() {
            const auto &bases = state_.bases;
            Eigen::VectorXd term = Eigen::VectorXd::Zero(bases.size() * 2);
            const int dim = state_.mesh->dimension();
 
            for (int e = 0; e < bases.size(); e++)
            {
                assembler::ElementAssemblyValues vals;
                state_.ass_vals_cache.compute(e, state_.mesh->is_volume(), bases[e], state_.geom_bases()[e], vals);
 
                const quadrature::Quadrature &quadrature = vals.quadrature;
                Eigen::VectorXd da = vals.det.array() * quadrature.weights.array();
 
                Eigen::MatrixXd u, grad_u;
                io::Evaluator::interpolate_at_local_vals(e, dim, dim, vals, state_.diff_cached.u(time_step), u, grad_u);
 
                Eigen::MatrixXd grad_u_q;
                for (int q = 0; q < quadrature.weights.size(); q++)
                {
                    double lambda, mu;
                    lambda = state_.assembler->parameters().at("lambda")(quadrature.points.row(q), vals.val.row(q), t, e);
                    mu = state_.assembler->parameters().at("mu")(quadrature.points.row(q), vals.val.row(q), t, e);
 
                    vector2matrix(grad_u.row(q), grad_u_q);
 
                    Eigen::MatrixXd f_prime_dmu, f_prime_dlambda;
                    state_.assembler->compute_dstress_dmu_dlambda(OptAssemblerData(t, dt, e, quadrature.points.row(q), vals.val.row(q), grad_u_q), f_prime_dmu, f_prime_dlambda);
 
                    term(e + bases.size()) += dot(f_prime_dmu, grad_u_q) * da(q);
                    term(e) += dot(f_prime_dlambda, grad_u_q) * da(q);
                }
            }
            return term;
        });
    }
 
    IntegrableFunctional PositionForm::get_integral_functional() const
    {
        IntegrableFunctional j;
        const int dim = dim_;
 
        j.set_j([dim](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val = u.col(dim) + pts.col(dim);
        });
 
        j.set_dj_du([dim](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), u.cols());
            val.col(dim).setOnes();
        });
 
        j.set_dj_dx([dim](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(pts.rows(), pts.cols());
            val.col(dim).setOnes();
        });
 
        return j;
    }
 
    IntegrableFunctional AccelerationForm::get_integral_functional() const
    {
        IntegrableFunctional j;
        const int dim = this->dim_;
 
        j.set_j([dim, &state = std::as_const(this->state_)](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            Eigen::MatrixXd acc, grad_acc;
            io::Evaluator::interpolate_at_local_vals(*(state.mesh), state.problem->is_scalar(), state.bases, state.geom_bases(), params.elem, local_pts, state.diff_cached.acc(params.step), acc, grad_acc);
 
            val = acc.col(dim);
        });
 
        j.set_dj_du([dim, this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), u.cols());
            log_and_throw_adjoint_error("[{}] Gradient not implemented!", name());
        });
 
        return j;
    }
 
    IntegrableFunctional KineticForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        j.set_j([this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            Eigen::MatrixXd v, grad_v;
            io::Evaluator::interpolate_at_local_vals(*(state_.mesh), state_.problem->is_scalar(), state_.bases, state_.geom_bases(), params.elem, local_pts, state_.diff_cached.v(params.step), v, grad_v);
 
            val.setZero(u.rows(), 1);
            for (int q = 0; q < v.rows(); q++)
            {
                const double rho = state_.mass_matrix_assembler->density()(local_pts.row(q), pts.row(q), params.t, params.elem);
                val(q) = 0.5 * rho * v.row(q).squaredNorm();
            }
        });
 
        j.set_dj_du([this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(u.rows(), u.cols());
 
            log_and_throw_adjoint_error("[{}] Gradient not implemented!", name());
        });
 
        return j;
    }
 
    void StressForm::compute_partial_gradient_step(const int time_step, const Eigen::VectorXd &x, Eigen::VectorXd &gradv) const
    {
        SpatialIntegralForm::compute_partial_gradient_step(time_step, x, gradv);
        gradv += weight() * variable_to_simulations_.apply_parametrization_jacobian(ParameterType::LameParameter, &state_, x, [this]() {
            log_and_throw_adjoint_error("[{}] Doesn't support derivatives wrt. material!", name());
            return Eigen::VectorXd::Zero(0).eval();
        });
    }
 
    IntegrableFunctional StressForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        std::string formulation = state_.formulation();
        auto dimensions = dimensions_;
 
        j.set_j([formulation, dimensions, this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), 1);
 
            Eigen::MatrixXd grad_u_q, stress;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                vector2matrix(grad_u.row(q), grad_u_q);
                if (formulation == "LinearElasticity")
                {
                    stress = mu(q) * (grad_u_q + grad_u_q.transpose()) + lambda(q) * grad_u_q.trace() * Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols());
                }
                else if (formulation == "NeoHookean")
                {
                    Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols()) + grad_u_q;
                    Eigen::MatrixXd FmT = def_grad.inverse().transpose();
                    stress = mu(q) * (def_grad - FmT) + lambda(q) * std::log(def_grad.determinant()) * FmT;
                }
                else
                    log_and_throw_adjoint_error("[{}] Unknown formulation {}!", name(), formulation);
                val(q) = stress(dimensions[0], dimensions[1]);
            }
        });
 
        j.set_dj_dgradu([formulation, dimensions, this](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setZero(grad_u.rows(), grad_u.cols());
 
            const int dim = state_.mesh->dimension();
            Eigen::MatrixXd grad_u_q, stiffness, stress;
            for (int q = 0; q < grad_u.rows(); q++)
            {
                stiffness.setZero(1, dim * dim * dim * dim);
                vector2matrix(grad_u.row(q), grad_u_q);
 
                if (formulation == "LinearElasticity")
                {
                    stress = mu(q) * (grad_u_q + grad_u_q.transpose()) + lambda(q) * grad_u_q.trace() * Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols());
                    for (int i = 0, idx = 0; i < dim; i++)
                        for (int j = 0; j < dim; j++)
                            for (int k = 0; k < dim; k++)
                                for (int l = 0; l < dim; l++)
                                {
                                    stiffness(idx++) = mu(q) * delta(i, k) * delta(j, l) + mu(q) * delta(i, l) * delta(j, k) + lambda(q) * delta(i, j) * delta(k, l);
                                }
                }
                else if (formulation == "NeoHookean")
                {
                    Eigen::MatrixXd def_grad = Eigen::MatrixXd::Identity(grad_u_q.rows(), grad_u_q.cols()) + grad_u_q;
                    Eigen::MatrixXd FmT = def_grad.inverse().transpose();
                    stress = mu(q) * (def_grad - FmT) + lambda(q) * std::log(def_grad.determinant()) * FmT;
                    Eigen::VectorXd FmT_vec = utils::flatten(FmT);
                    double J = def_grad.determinant();
                    double tmp1 = mu(q) - lambda(q) * std::log(J);
                    for (int i = 0, idx = 0; i < dim; i++)
                        for (int j = 0; j < dim; j++)
                            for (int k = 0; k < dim; k++)
                                for (int l = 0; l < dim; l++)
                                {
                                    stiffness(idx++) = mu(q) * delta(i, k) * delta(j, l) + tmp1 * FmT(i, l) * FmT(k, j);
                                }
                    stiffness += lambda(q) * utils::flatten(FmT_vec * FmT_vec.transpose()).transpose();
                }
                else
                    log_and_throw_adjoint_error("[{}] Unknown formulation {}!", name(), formulation);
 
                val.row(q) = stiffness.block(0, (dimensions[0] * dim + dimensions[1]) * dim * dim, 1, dim * dim);
            }
        });
 
        return j;
    }
 
    IntegrableFunctional VolumeForm::get_integral_functional() const
    {
        IntegrableFunctional j;
 
        j.set_j([](const Eigen::MatrixXd &local_pts, const Eigen::MatrixXd &pts, const Eigen::MatrixXd &u, const Eigen::MatrixXd &grad_u, const Eigen::VectorXd &lambda, const Eigen::VectorXd &mu, const Eigen::MatrixXd &reference_normals, const assembler::ElementAssemblyValues &vals, const IntegrableFunctional::ParameterType &params, Eigen::MatrixXd &val) {
            val.setOnes(grad_u.rows(), 1);
        });
 
        return j;
    }
} // namespace polyfem::solver