HookeLinearElasticity.cpp

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#include "HookeLinearElasticity.hpp"
 
#include <polyfem/autogen/auto_elasticity_rhs.hpp>
 
namespace polyfem::assembler
{
    using namespace basis;
    namespace
    {
        template <class Matrix>
        Matrix strain_from_disp_grad(const Matrix &disp_grad)
        {
            Matrix mat = (disp_grad + disp_grad.transpose());
 
            for (int i = 0; i < mat.size(); ++i)
                mat(i) *= 0.5;
 
            return mat;
        }
 
        template <int dim>
        Eigen::Matrix<double, dim, dim> strain(
            const Eigen::MatrixXd &grad,
            const Eigen::MatrixXd &jac_it,
            int k, int coo)
        {
            Eigen::Matrix<double, dim, dim> jac;
            jac.setZero();
            jac.row(coo) = grad.row(k);
            jac = jac * jac_it;
 
            return strain_from_disp_grad(jac);
        }
 
        template <typename T, unsigned long N>
        T stress(const ElasticityTensor &elasticity_tensor, const std::array<T, N> &strain, const int j)
        {
            T res = elasticity_tensor(j, 0) * strain[0];
 
            for (unsigned long k = 1; k < N; ++k)
                res += elasticity_tensor(j, k) * strain[k];
 
            return res;
        }
    } // namespace
 
    HookeLinearElasticity::HookeLinearElasticity()
    {
    }
    HookeLinearElasticity::HookeLinearElasticity() {…}
 
    void HookeLinearElasticity::add_multimaterial(const int index, const json &params, const Units &units)
    {
        assert(size() == 2 || size() == 3);
 
        if (!params.contains("elasticity_tensor") || params["elasticity_tensor"].empty())
        {
            if (params.count("young"))
            {
                if (params["young"].is_number() && params["nu"].is_number())
                    elasticity_tensor_.set_from_young_poisson(params["young"], params["nu"], units.stress());
            }
            else if (params.count("E"))
            {
                if (params["E"].is_number() && params["nu"].is_number())
                    elasticity_tensor_.set_from_young_poisson(params["E"], params["nu"], units.stress());
            }
            else if (params.count("lambda"))
            {
                if (params["lambda"].is_number() && params["mu"].is_number())
                    elasticity_tensor_.set_from_lambda_mu(params["lambda"], params["mu"], units.stress());
            }
        }
        else
        {
            std::vector<double> entries = params["elasticity_tensor"];
            elasticity_tensor_.set_from_entries(entries, units.stress());
        }
        if (params.contains("fiber_direction"))
            fiber_direction_.add_multimaterial(index, params["fiber_direction"], units.length());
    }
    void HookeLinearElasticity::add_multimaterial(const int index, const json &params, const Units &units) {…}
 
    void HookeLinearElasticity::set_size(const int size)
    {
        Assembler::set_size(size);
        elasticity_tensor_.resize(size);
        fiber_direction_.resize(size);
    }
    void HookeLinearElasticity::set_size(const int size) {…}
 
    Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 9, 1>
    HookeLinearElasticity::assemble(const LinearAssemblerData &data) const
    {
        const Eigen::MatrixXd &gradi = data.vals.basis_values[data.i].grad;
        const Eigen::MatrixXd &gradj = data.vals.basis_values[data.j].grad;
 
        // (C : gradi) : gradj
        Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 9, 1> res(size() * size());
        res.setZero();
        assert(gradi.cols() == size());
        assert(gradj.cols() == size());
        assert(size_t(gradi.rows()) == data.vals.jac_it.size());
 
        for (long k = 0; k < gradi.rows(); ++k)
        {
            Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 9, 1> res_k(size() * size());
 
            ElasticityTensor elasticity_tensor;
            if (fiber_direction_.has_rotation())
            {
                const auto fdir_mtx_voigt = fiber_direction_.stiffness_rotation_voigt(data.vals.quadrature.points.row(k), data.vals.val.row(k), data.t, data.vals.element_id);
 
                elasticity_tensor = elasticity_tensor_;
                elasticity_tensor.rotate_stiffness(fdir_mtx_voigt);
            }
            else
            {
                elasticity_tensor = elasticity_tensor_;
            }
 
            if (size() == 2)
            {
                const Eigen::Matrix2d eps_x_i = strain<2>(gradi, data.vals.jac_it[k], k, 0);
                const Eigen::Matrix2d eps_y_i = strain<2>(gradi, data.vals.jac_it[k], k, 1);
 
                const Eigen::Matrix2d eps_x_j = strain<2>(gradj, data.vals.jac_it[k], k, 0);
                const Eigen::Matrix2d eps_y_j = strain<2>(gradj, data.vals.jac_it[k], k, 1);
 
                std::array<double, 3> e_x, e_y;
                e_x[0] = eps_x_i(0, 0);
                e_x[1] = eps_x_i(1, 1);
                e_x[2] = 2 * eps_x_i(0, 1);
 
                e_y[0] = eps_y_i(0, 0);
                e_y[1] = eps_y_i(1, 1);
                e_y[2] = 2 * eps_y_i(0, 1);
 
                Eigen::Matrix2d sigma_x;
                sigma_x << elasticity_tensor.compute_stress<3>(e_x, 0), elasticity_tensor.compute_stress<3>(e_x, 2),
                    elasticity_tensor.compute_stress<3>(e_x, 2), elasticity_tensor.compute_stress<3>(e_x, 1);
 
                Eigen::Matrix2d sigma_y;
                sigma_y << elasticity_tensor.compute_stress<3>(e_y, 0), elasticity_tensor.compute_stress<3>(e_y, 2),
                    elasticity_tensor.compute_stress<3>(e_y, 2), elasticity_tensor.compute_stress<3>(e_y, 1);
 
                res_k(0) = (sigma_x * eps_x_j).trace();
                res_k(2) = (sigma_x * eps_y_j).trace();
 
                res_k(1) = (sigma_y * eps_x_j).trace();
                res_k(3) = (sigma_y * eps_y_j).trace();
            }
            else
            {
                const Eigen::Matrix3d eps_x_i = strain<3>(gradi, data.vals.jac_it[k], k, 0);
                const Eigen::Matrix3d eps_y_i = strain<3>(gradi, data.vals.jac_it[k], k, 1);
                const Eigen::Matrix3d eps_z_i = strain<3>(gradi, data.vals.jac_it[k], k, 2);
 
                const Eigen::Matrix3d eps_x_j = strain<3>(gradj, data.vals.jac_it[k], k, 0);
                const Eigen::Matrix3d eps_y_j = strain<3>(gradj, data.vals.jac_it[k], k, 1);
                const Eigen::Matrix3d eps_z_j = strain<3>(gradj, data.vals.jac_it[k], k, 2);
 
                std::array<double, 6> e_x, e_y, e_z;
                e_x[0] = eps_x_i(0, 0);
                e_x[1] = eps_x_i(1, 1);
                e_x[2] = eps_x_i(2, 2);
                e_x[3] = 2 * eps_x_i(1, 2);
                e_x[4] = 2 * eps_x_i(0, 2);
                e_x[5] = 2 * eps_x_i(0, 1);
 
                e_y[0] = eps_y_i(0, 0);
                e_y[1] = eps_y_i(1, 1);
                e_y[2] = eps_y_i(2, 2);
                e_y[3] = 2 * eps_y_i(1, 2);
                e_y[4] = 2 * eps_y_i(0, 2);
                e_y[5] = 2 * eps_y_i(0, 1);
 
                e_z[0] = eps_z_i(0, 0);
                e_z[1] = eps_z_i(1, 1);
                e_z[2] = eps_z_i(2, 2);
                e_z[3] = 2 * eps_z_i(1, 2);
                e_z[4] = 2 * eps_z_i(0, 2);
                e_z[5] = 2 * eps_z_i(0, 1);
 
                Eigen::Matrix3d sigma_x;
                sigma_x << elasticity_tensor.compute_stress<6>(e_x, 0), elasticity_tensor.compute_stress<6>(e_x, 5), elasticity_tensor.compute_stress<6>(e_x, 4),
                    elasticity_tensor.compute_stress<6>(e_x, 5), elasticity_tensor.compute_stress<6>(e_x, 1), elasticity_tensor.compute_stress<6>(e_x, 3),
                    elasticity_tensor.compute_stress<6>(e_x, 4), elasticity_tensor.compute_stress<6>(e_x, 3), elasticity_tensor.compute_stress<6>(e_x, 2);
 
                Eigen::Matrix3d sigma_y;
                sigma_y << elasticity_tensor.compute_stress<6>(e_y, 0), elasticity_tensor.compute_stress<6>(e_y, 5), elasticity_tensor.compute_stress<6>(e_y, 4),
                    elasticity_tensor.compute_stress<6>(e_y, 5), elasticity_tensor.compute_stress<6>(e_y, 1), elasticity_tensor.compute_stress<6>(e_y, 3),
                    elasticity_tensor.compute_stress<6>(e_y, 4), elasticity_tensor.compute_stress<6>(e_y, 3), elasticity_tensor.compute_stress<6>(e_y, 2);
 
                Eigen::Matrix3d sigma_z;
                sigma_z << elasticity_tensor.compute_stress<6>(e_z, 0), elasticity_tensor.compute_stress<6>(e_z, 5), elasticity_tensor.compute_stress<6>(e_z, 4),
                    elasticity_tensor.compute_stress<6>(e_z, 5), elasticity_tensor.compute_stress<6>(e_z, 1), elasticity_tensor.compute_stress<6>(e_z, 3),
                    elasticity_tensor.compute_stress<6>(e_z, 4), elasticity_tensor.compute_stress<6>(e_z, 3), elasticity_tensor.compute_stress<6>(e_z, 2);
 
                res_k(0) = (sigma_x * eps_x_j).trace();
                res_k(3) = (sigma_x * eps_y_j).trace();
                res_k(6) = (sigma_x * eps_z_j).trace();
 
                res_k(1) = (sigma_y * eps_x_j).trace();
                res_k(4) = (sigma_y * eps_y_j).trace();
                res_k(7) = (sigma_y * eps_z_j).trace();
 
                res_k(2) = (sigma_z * eps_x_j).trace();
                res_k(5) = (sigma_z * eps_y_j).trace();
                res_k(8) = (sigma_z * eps_z_j).trace();
            }
 
            res += res_k * data.da(k);
        }
 
        return res;
    }
    HookeLinearElasticity::assemble(const LinearAssemblerData &data) const {…}
 
    void HookeLinearElasticity::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;
 
        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);
 
        for (long p = 0; p < local_pts.rows(); ++p)
        {
            compute_diplacement_grad(size(), bs, vals, local_pts, p, displacement, displacement_grad);
 
            ElasticityTensor elasticity_tensor;
            if (fiber_direction_.has_rotation())
            {
                const auto fdir_mtx_voigt = fiber_direction_.stiffness_rotation_voigt(local_pts.row(p), vals.val.row(p), data.t, el_id);
 
                elasticity_tensor = elasticity_tensor_;
                elasticity_tensor.rotate_stiffness(fdir_mtx_voigt);
            }
            else
            {
                elasticity_tensor = elasticity_tensor_;
            }
 
            if (type == ElasticityTensorType::F)
            {
                all.row(p) = fun(displacement_grad + Eigen::MatrixXd::Identity(size(), size()));
                continue;
            }
 
            Eigen::MatrixXd strain = (displacement_grad + displacement_grad.transpose()) / 2;
            Eigen::MatrixXd sigma(size(), size());
 
            if (size() == 2)
            {
                std::array<double, 3> eps;
                eps[0] = strain(0, 0);
                eps[1] = strain(1, 1);
                eps[2] = 2 * strain(0, 1);
 
                sigma << elasticity_tensor.compute_stress<3>(eps, 0), elasticity_tensor.compute_stress<3>(eps, 2),
                    elasticity_tensor.compute_stress<3>(eps, 2), elasticity_tensor.compute_stress<3>(eps, 1);
            }
            else
            {
                std::array<double, 6> eps;
                eps[0] = strain(0, 0);
                eps[1] = strain(1, 1);
                eps[2] = strain(2, 2);
                eps[3] = 2 * strain(1, 2);
                eps[4] = 2 * strain(0, 2);
                eps[5] = 2 * strain(0, 1);
 
                sigma << elasticity_tensor.compute_stress<6>(eps, 0), elasticity_tensor.compute_stress<6>(eps, 5), elasticity_tensor.compute_stress<6>(eps, 4),
                    elasticity_tensor.compute_stress<6>(eps, 5), elasticity_tensor.compute_stress<6>(eps, 1), elasticity_tensor.compute_stress<6>(eps, 3),
                    elasticity_tensor.compute_stress<6>(eps, 4), elasticity_tensor.compute_stress<6>(eps, 3), elasticity_tensor.compute_stress<6>(eps, 2);
            }
 
            if (type == ElasticityTensorType::PK1)
                sigma = pk1_from_cauchy(sigma, displacement_grad + Eigen::MatrixXd::Identity(size(), size()));
            else if (type == ElasticityTensorType::PK2)
                sigma = pk2_from_cauchy(sigma, displacement_grad + Eigen::MatrixXd::Identity(size(), size()));
 
            all.row(p) = fun(sigma);
        }
    }
    void HookeLinearElasticity::assign_stress_tensor( {…}
 
    Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 3, 1>
    HookeLinearElasticity::compute_rhs(const AutodiffHessianPt &pt) const
    {
        assert(pt.size() == size());
        Eigen::Matrix<double, Eigen::Dynamic, 1, 0, 3, 1> res;
 
        if (size() == 2)
            autogen::hooke_2d_function(pt, elasticity_tensor_, res);
        else if (size() == 3)
            autogen::hooke_3d_function(pt, elasticity_tensor_, res);
        else
            assert(false);
 
        return res;
    }
    HookeLinearElasticity::compute_rhs(const AutodiffHessianPt &pt) const {…}
 
    std::map<std::string, Assembler::ParamFunc> HookeLinearElasticity::parameters() const
    {
        std::map<std::string, ParamFunc> res;
        const auto &elast_tensor = elasticity_tensor();
        const int size = this->size() == 2 ? 3 : 6;
 
        for (int i = 0; i < size; ++i)
        {
            for (int j = i; j < size; ++j)
            {
                res[fmt::format("C_{}{}", i, j)] = [&elast_tensor, i, j](const RowVectorNd &, const RowVectorNd &, double, int) {
                    return elast_tensor(i, j);
                };
            }
        }
        return res;
    }
    std::map<std::string, Assembler::ParamFunc> HookeLinearElasticity::parameters() const {…}
 
    double HookeLinearElasticity::compute_energy(const NonLinearAssemblerData &data) const
    {
        return compute_energy_aux<double>(data);
    }
    double HookeLinearElasticity::compute_energy(const NonLinearAssemblerData &data) const {…}
 
    Eigen::VectorXd HookeLinearElasticity::assemble_gradient(const NonLinearAssemblerData &data) const
    {
        const int n_bases = data.vals.basis_values.size();
        return polyfem::gradient_from_energy(
            size(), n_bases, data,
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 6, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 8, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 12, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 18, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 24, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 30, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 60, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, 81, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, Eigen::Dynamic, 1, 0, SMALL_N, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::Matrix<double, Eigen::Dynamic, 1, 0, BIG_N, 1>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar1<double, Eigen::VectorXd>>(data); });
    }
    Eigen::VectorXd HookeLinearElasticity::assemble_gradient(const NonLinearAssemblerData &data) const {…}
 
    Eigen::MatrixXd HookeLinearElasticity::assemble_hessian(const NonLinearAssemblerData &data) const
    {
        const int n_bases = data.vals.basis_values.size();
        return polyfem::hessian_from_energy(
            size(), n_bases, data,
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 6, 1>, Eigen::Matrix<double, 6, 6>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 8, 1>, Eigen::Matrix<double, 8, 8>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 12, 1>, Eigen::Matrix<double, 12, 12>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 18, 1>, Eigen::Matrix<double, 18, 18>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 24, 1>, Eigen::Matrix<double, 24, 24>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 30, 1>, Eigen::Matrix<double, 30, 30>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 60, 1>, Eigen::Matrix<double, 60, 60>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, 81, 1>, Eigen::Matrix<double, 81, 81>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::Matrix<double, Eigen::Dynamic, 1, 0, SMALL_N, 1>, Eigen::Matrix<double, Eigen::Dynamic, Eigen::Dynamic, 0, SMALL_N, SMALL_N>>>(data); },
            [&](const NonLinearAssemblerData &data) { return compute_energy_aux<DScalar2<double, Eigen::VectorXd, Eigen::MatrixXd>>(data); });
    }
    Eigen::MatrixXd HookeLinearElasticity::assemble_hessian(const NonLinearAssemblerData &data) const {…}
 
    template <typename T>
    T HookeLinearElasticity::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 disp_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(), disp_grad);
 
            AutoDiffGradMat strain = strain_from_disp_grad(disp_grad);
            AutoDiffGradMat stress_tensor(size(), size());
 
            if (size() == 2)
            {
                std::array<T, 3> eps;
                eps[0] = strain(0, 0);
                eps[1] = strain(1, 1);
                eps[2] = 2 * strain(0, 1);
 
                stress_tensor << stress(elasticity_tensor_, eps, 0), stress(elasticity_tensor_, eps, 2),
                    stress(elasticity_tensor_, eps, 2), stress(elasticity_tensor_, eps, 1);
            }
            else
            {
                std::array<T, 6> eps;
                eps[0] = strain(0, 0);
                eps[1] = strain(1, 1);
                eps[2] = strain(2, 2);
                eps[3] = 2 * strain(1, 2);
                eps[4] = 2 * strain(0, 2);
                eps[5] = 2 * strain(0, 1);
 
                stress_tensor << stress(elasticity_tensor_, eps, 0), stress(elasticity_tensor_, eps, 5), stress(elasticity_tensor_, eps, 4),
                    stress(elasticity_tensor_, eps, 5), stress(elasticity_tensor_, eps, 1), stress(elasticity_tensor_, eps, 3),
                    stress(elasticity_tensor_, eps, 4), stress(elasticity_tensor_, eps, 3), stress(elasticity_tensor_, eps, 2);
            }
 
            energy += (stress_tensor * strain).trace() * data.da(p);
        }
 
        return energy * 0.5;
    }
    T HookeLinearElasticity::compute_energy_aux(const NonLinearAssemblerData &data) const {…}
} // namespace polyfem::assembler