StateDiff.cpp

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#include <polyfem/State.hpp>
#include <polyfem/utils/Logger.hpp>
 
#include <polyfem/time_integrator/BDF.hpp>
#include <polyfem/time_integrator/ImplicitEuler.hpp>
 
#include <polyfem/utils/BoundarySampler.hpp>
#include <polysolve/linear/FEMSolver.hpp>
#include <polyfem/utils/MaybeParallelFor.hpp>
#include <polyfem/utils/StringUtils.hpp>
#include <polyfem/io/Evaluator.hpp>
 
#include <polyfem/solver/NLProblem.hpp>
#include <polyfem/solver/NLHomoProblem.hpp>
 
#include <polyfem/solver/forms/BodyForm.hpp>
#include <polyfem/solver/forms/BarrierContactForm.hpp>
#include <polyfem/solver/forms/SmoothContactForm.hpp>
#include <polyfem/solver/forms/ElasticForm.hpp>
#include <polyfem/solver/forms/FrictionForm.hpp>
#include <polyfem/solver/forms/NormalAdhesionForm.hpp>
#include <polyfem/solver/forms/TangentialAdhesionForm.hpp>
#include <polyfem/solver/forms/InertiaForm.hpp>
#include <polyfem/solver/forms/LaggedRegForm.hpp>
#include <polyfem/assembler/ViscousDamping.hpp>
 
#include <ipc/ipc.hpp>
#include <ipc/barrier/barrier.hpp>
#include <ipc/utils/local_to_global.hpp>
#include <ipc/potentials/friction_potential.hpp>
 
#include <Eigen/Dense>
#include <unsupported/Eigen/SparseExtra>
#include <deque>
#include <map>
#include <algorithm>
 
#include <fstream>
 
using namespace polyfem::basis;
 
namespace polyfem
{
    namespace
    {
        void replace_rows_by_identity(StiffnessMatrix &reduced_mat, const StiffnessMatrix &mat, const std::vector<int> &rows)
        {
            reduced_mat.resize(mat.rows(), mat.cols());
 
            std::vector<bool> mask(mat.rows(), false);
            for (int i : rows)
                mask[i] = true;
 
            std::vector<Eigen::Triplet<double>> coeffs;
            for (int k = 0; k < mat.outerSize(); ++k)
            {
                for (StiffnessMatrix::InnerIterator it(mat, k); it; ++it)
                {
                    if (mask[it.row()])
                    {
                        if (it.row() == it.col())
                            coeffs.emplace_back(it.row(), it.col(), 1.0);
                    }
                    else
                        coeffs.emplace_back(it.row(), it.col(), it.value());
                }
            }
            reduced_mat.setFromTriplets(coeffs.begin(), coeffs.end());
        }
    } // namespace
 
    void State::get_vertices(Eigen::MatrixXd &vertices) const
    {
        vertices.setZero(mesh->n_vertices(), mesh->dimension());
 
        for (int v = 0; v < mesh->n_vertices(); v++)
            vertices.row(v) = mesh->point(v);
    }
    void State::get_vertices(Eigen::MatrixXd &vertices) const {…}
 
    void State::get_elements(Eigen::MatrixXi &elements) const
    {
        assert(mesh->is_simplicial());
 
        auto node_to_primitive_map = node_to_primitive();
 
        const auto &gbases = geom_bases();
        int dim = mesh->dimension();
        elements.setZero(gbases.size(), dim + 1);
        for (int e = 0; e < gbases.size(); e++)
        {
            int i = 0;
            for (const auto &gbs : gbases[e].bases)
                elements(e, i++) = node_to_primitive_map[gbs.global()[0].index];
        }
    }
    void State::get_elements(Eigen::MatrixXi &elements) const {…}
 
    void State::set_mesh_vertex(int v_id, const Eigen::VectorXd &vertex)
    {
        assert(vertex.size() == mesh->dimension());
        mesh->set_point(v_id, vertex);
    }
    void State::set_mesh_vertex(int v_id, const Eigen::VectorXd &vertex) {…}
 
    void State::cache_transient_adjoint_quantities(const int current_step, const Eigen::MatrixXd &sol, const Eigen::MatrixXd &disp_grad)
    {
        StiffnessMatrix gradu_h(sol.size(), sol.size());
        if (current_step == 0)
            diff_cached.init(mesh->dimension(), ndof(), problem->is_time_dependent() ? args["time"]["time_steps"].get<int>() : 0);
 
        ipc::NormalCollisions cur_collision_set;
        ipc::SmoothCollisions cur_smooth_collision_set;
        ipc::TangentialCollisions cur_friction_set;
        ipc::NormalCollisions cur_normal_adhesion_set;
        ipc::TangentialCollisions cur_tangential_adhesion_set;
 
        if (optimization_enabled == solver::CacheLevel::Derivatives)
        {
            if (!problem->is_time_dependent() || current_step > 0)
                compute_force_jacobian(sol, disp_grad, gradu_h);
 
            if (solve_data.contact_form)
            {
                if (const auto barrier_contact = dynamic_cast<const solver::BarrierContactForm*>(solve_data.contact_form.get()))
                    cur_collision_set = barrier_contact->collision_set();
                else if (const auto smooth_contact = dynamic_cast<const solver::SmoothContactForm*>(solve_data.contact_form.get()))
                    cur_smooth_collision_set = smooth_contact->collision_set();
            }
            cur_friction_set = solve_data.friction_form ? solve_data.friction_form->friction_collision_set() : ipc::TangentialCollisions();
            cur_normal_adhesion_set = solve_data.normal_adhesion_form ? solve_data.normal_adhesion_form->collision_set() : ipc::NormalCollisions();
            cur_tangential_adhesion_set = solve_data.tangential_adhesion_form ? solve_data.tangential_adhesion_form->tangential_collision_set() : ipc::TangentialCollisions();
        }
 
        if (problem->is_time_dependent())
        {
            if (args["time"]["quasistatic"].get<bool>())
            {
                diff_cached.cache_quantities_quasistatic(current_step, sol, gradu_h, cur_collision_set, cur_smooth_collision_set, cur_normal_adhesion_set, disp_grad);
            }
            else
            {
                Eigen::MatrixXd vel, acc;
                if (current_step == 0)
                {
                    if (dynamic_cast<time_integrator::BDF *>(solve_data.time_integrator.get()))
                    {
                        const auto bdf_integrator = dynamic_cast<time_integrator::BDF *>(solve_data.time_integrator.get());
                        vel = bdf_integrator->weighted_sum_v_prevs();
                    }
                    else if (dynamic_cast<time_integrator::ImplicitEuler *>(solve_data.time_integrator.get()))
                    {
                        const auto euler_integrator = dynamic_cast<time_integrator::ImplicitEuler *>(solve_data.time_integrator.get());
                        vel = euler_integrator->v_prev();
                    }
                    else
                        log_and_throw_error("Differentiable code doesn't support this time integrator!");
 
                    acc.setZero(ndof(), 1);
                }
                else
                {
                    vel = solve_data.time_integrator->compute_velocity(sol);
                    acc = solve_data.time_integrator->compute_acceleration(vel);
                }
 
                diff_cached.cache_quantities_transient(current_step, solve_data.time_integrator->steps(), sol, vel, acc, gradu_h, cur_collision_set, cur_smooth_collision_set, cur_friction_set);
            }
        }
        else
        {
            diff_cached.cache_quantities_static(sol, gradu_h, cur_collision_set, cur_smooth_collision_set, cur_friction_set, cur_normal_adhesion_set, cur_tangential_adhesion_set, disp_grad);
        }
    }
    void State::cache_transient_adjoint_quantities(const int current_step, const Eigen::MatrixXd &sol, const Eigen::MatrixXd &disp_grad) {…}
 
    void State::compute_force_jacobian(const Eigen::MatrixXd &sol, const Eigen::MatrixXd &disp_grad, StiffnessMatrix &hessian)
    {
        if (problem->is_time_dependent())
        {
            if (assembler->is_linear() && !is_contact_enabled())
                log_and_throw_adjoint_error("Differentiable transient linear solve is not supported!");
 
            StiffnessMatrix tmp_hess;
            solve_data.nl_problem->set_project_to_psd(false);
            solve_data.nl_problem->FullNLProblem::solution_changed(sol);
            solve_data.nl_problem->FullNLProblem::hessian(sol, tmp_hess);
            hessian.setZero();
            replace_rows_by_identity(hessian, tmp_hess, boundary_nodes);
        }
        else // static formulation
        {
            if (assembler->is_linear() && !is_contact_enabled() && !is_homogenization())
            {
                hessian.setZero();
                StiffnessMatrix stiffness;
                build_stiffness_mat(stiffness);
                replace_rows_by_identity(hessian, stiffness, boundary_nodes);
            }
            else
            {
                solve_data.nl_problem->set_project_to_psd(false);
                if (is_homogenization())
                {
                    Eigen::VectorXd reduced;
                    std::shared_ptr<solver::NLHomoProblem> homo_problem = std::dynamic_pointer_cast<solver::NLHomoProblem>(solve_data.nl_problem);
                    reduced = homo_problem->full_to_reduced(sol, disp_grad);
                    solve_data.nl_problem->solution_changed(reduced);
                    solve_data.nl_problem->hessian(reduced, hessian);
                }
                else
                {
                    StiffnessMatrix tmp_hess;
                    solve_data.nl_problem->FullNLProblem::solution_changed(sol);
                    solve_data.nl_problem->FullNLProblem::hessian(sol, tmp_hess);
                    hessian.setZero();
                    replace_rows_by_identity(hessian, tmp_hess, boundary_nodes);
                }
            }
        }
    }
    void State::compute_force_jacobian(const Eigen::MatrixXd &sol, const Eigen::MatrixXd &disp_grad, StiffnessMatrix &hessian) {…}
 
    void State::compute_force_jacobian_prev(const int force_step, const int sol_step, StiffnessMatrix &hessian_prev) const
    {
        assert(force_step > 0);
        assert(force_step > sol_step);
        if (assembler->is_linear() && !is_contact_enabled())
        {
            hessian_prev = StiffnessMatrix(ndof(), ndof());
        }
        else
        {
            const Eigen::MatrixXd u = diff_cached.u(force_step);
            const Eigen::MatrixXd u_prev = diff_cached.u(sol_step);
            const double beta = time_integrator::BDF::betas(diff_cached.bdf_order(force_step) - 1);
            const double dt = solve_data.time_integrator->dt();
 
            hessian_prev = StiffnessMatrix(u.size(), u.size());
            if (problem->is_time_dependent())
            {
                if (solve_data.friction_form)
                {
                    if (sol_step == force_step - 1)
                    {
                        Eigen::MatrixXd surface_solution_prev = collision_mesh.vertices(utils::unflatten(u_prev, mesh->dimension()));
                        Eigen::MatrixXd surface_solution = collision_mesh.vertices(utils::unflatten(u, mesh->dimension()));
 
                        // TODO: use the time integration to compute the velocity
                        const Eigen::MatrixXd surface_velocities = (surface_solution - surface_solution_prev) / dt;
                        const double dv_dut = -1 / dt;
 
                        if (const auto barrier_contact = dynamic_cast<const solver::BarrierContactForm*>(solve_data.contact_form.get()))
                        {
                            hessian_prev =
                                solve_data.friction_form->friction_potential().force_jacobian(
                                    diff_cached.friction_collision_set(force_step),
                                    collision_mesh,
                                    collision_mesh.rest_positions(),
                                    /*lagged_displacements=*/surface_solution_prev,
                                    surface_velocities,
                                    barrier_contact->barrier_potential(),
                                    barrier_contact->barrier_stiffness(),
                                    ipc::FrictionPotential::DiffWRT::LAGGED_DISPLACEMENTS)
                                + solve_data.friction_form->friction_potential().force_jacobian(
                                    diff_cached.friction_collision_set(force_step),
                                    collision_mesh,
                                    collision_mesh.rest_positions(),
                                    /*lagged_displacements=*/surface_solution_prev,
                                    surface_velocities,
                                    barrier_contact->barrier_potential(),
                                    barrier_contact->barrier_stiffness(),
                                    ipc::FrictionPotential::DiffWRT::VELOCITIES)
                                    * dv_dut;
                        }
 
                        hessian_prev *= -1;
 
                        // {
                        //  Eigen::MatrixXd X = collision_mesh.rest_positions();
                        //  Eigen::VectorXd x = utils::flatten(surface_solution_prev);
                        //  const double barrier_stiffness = solve_data.contact_form->barrier_stiffness();
                        //  const double dhat = solve_data.contact_form->dhat();
                        //  const double mu = solve_data.friction_form->mu();
                        //  const double epsv = solve_data.friction_form->epsv();
 
                        //  Eigen::MatrixXd fgrad;
                        //  fd::finite_jacobian(
                        //      x, [&](const Eigen::VectorXd &y) -> Eigen::VectorXd
                        //      {
                        //          Eigen::MatrixXd fd_Ut = utils::unflatten(y, surface_solution_prev.cols());
 
                        //          ipc::TangentialCollisions fd_friction_constraints;
                        //          ipc::NormalCollisions fd_constraints;
                        //          fd_constraints.set_use_convergent_formulation(solve_data.contact_form->use_convergent_formulation());
                        //          fd_constraints.set_enable_shape_derivatives(true);
                        //          fd_constraints.build(collision_mesh, X + fd_Ut, dhat);
 
                        //          fd_friction_constraints.build(
                        //              collision_mesh, X + fd_Ut, fd_constraints, dhat, barrier_stiffness,
                        //              mu);
 
                        //          return fd_friction_constraints.compute_potential_gradient(collision_mesh, (surface_solution - fd_Ut) / dt, epsv);
 
                        //      }, fgrad, fd::AccuracyOrder::SECOND, 1e-8);
 
                        //  logger().trace("force Ut derivative error {} {}", (fgrad - hessian_prev).norm(), hessian_prev.norm());
                        // }
 
                        hessian_prev = collision_mesh.to_full_dof(hessian_prev); // / (beta * dt) / (beta * dt);
                    }
                    else
                    {
                        // const double alpha = time_integrator::BDF::alphas(std::min(diff_cached.bdf_order(force_step), force_step) - 1)[force_step - sol_step - 1];
                        // Eigen::MatrixXd velocity = collision_mesh.map_displacements(utils::unflatten(diff_cached.v(force_step), collision_mesh.dim()));
                        // hessian_prev = diff_cached.friction_collision_set(force_step).compute_potential_hessian( //
                        //          collision_mesh, velocity, solve_data.friction_form->epsv(), false) * (-alpha / beta / dt);
 
                        // hessian_prev = collision_mesh.to_full_dof(hessian_prev);
                    }
                }
 
                if (solve_data.tangential_adhesion_form)
                {
 
                    if (sol_step == force_step - 1)
                    {
                        StiffnessMatrix adhesion_hessian_prev(u.size(), u.size());
 
                        Eigen::MatrixXd surface_solution_prev = collision_mesh.vertices(utils::unflatten(u_prev, mesh->dimension()));
                        Eigen::MatrixXd surface_solution = collision_mesh.vertices(utils::unflatten(u, mesh->dimension()));
 
                        // TODO: use the time integration to compute the velocity
                        const Eigen::MatrixXd surface_velocities = (surface_solution - surface_solution_prev) / dt;
                        const double dv_dut = -1 / dt;
 
                        adhesion_hessian_prev =
                            solve_data.tangential_adhesion_form->tangential_adhesion_potential().force_jacobian(
                                diff_cached.tangential_adhesion_collision_set(force_step),
                                collision_mesh,
                                collision_mesh.rest_positions(),
                                /*lagged_displacements=*/surface_solution_prev,
                                surface_velocities,
                                solve_data.normal_adhesion_form->normal_adhesion_potential(),
                                1,
                                ipc::TangentialPotential::DiffWRT::LAGGED_DISPLACEMENTS)
                            + solve_data.tangential_adhesion_form->tangential_adhesion_potential().force_jacobian(
                                  diff_cached.tangential_adhesion_collision_set(force_step),
                                  collision_mesh,
                                  collision_mesh.rest_positions(),
                                  /*lagged_displacements=*/surface_solution_prev,
                                  surface_velocities,
                                  solve_data.normal_adhesion_form->normal_adhesion_potential(),
                                  1,
                                  ipc::TangentialPotential::DiffWRT::VELOCITIES)
                                  * dv_dut;
 
                        adhesion_hessian_prev *= -1;
 
                        adhesion_hessian_prev = collision_mesh.to_full_dof(adhesion_hessian_prev); // / (beta * dt) / (beta * dt);
 
                        hessian_prev += adhesion_hessian_prev;
                    }
                }
 
                if (damping_assembler->is_valid() && sol_step == force_step - 1) // velocity in damping uses BDF1
                {
                    utils::SparseMatrixCache mat_cache;
                    StiffnessMatrix damping_hessian_prev(u.size(), u.size());
                    damping_prev_assembler->assemble_hessian(mesh->is_volume(), n_bases, false, bases, geom_bases(), ass_vals_cache, force_step * args["time"]["dt"].get<double>() + args["time"]["t0"].get<double>(), dt, u, u_prev, mat_cache, damping_hessian_prev);
 
                    hessian_prev += damping_hessian_prev;
                }
 
                if (sol_step == force_step - 1)
                {
                    StiffnessMatrix body_force_hessian(u.size(), u.size());
                    solve_data.body_form->hessian_wrt_u_prev(u_prev, force_step * dt, body_force_hessian);
                    hessian_prev += body_force_hessian;
                }
            }
        }
    }
    void State::compute_force_jacobian_prev(const int force_step, const int sol_step, StiffnessMatrix &hessian_prev) const {…}
 
    void State::solve_adjoint_cached(const Eigen::MatrixXd &rhs)
    {
        diff_cached.cache_adjoints(solve_adjoint(rhs));
    }
    void State::solve_adjoint_cached(const Eigen::MatrixXd &rhs) {…}
 
    Eigen::MatrixXd State::solve_adjoint(const Eigen::MatrixXd &rhs) const
    {
        if (problem->is_time_dependent())
            return solve_transient_adjoint(rhs);
        else
            return solve_static_adjoint(rhs);
    }
    Eigen::MatrixXd State::solve_adjoint(const Eigen::MatrixXd &rhs) const {…}
 
    Eigen::MatrixXd State::solve_static_adjoint(const Eigen::MatrixXd &adjoint_rhs) const
    {
        Eigen::MatrixXd b = adjoint_rhs;
 
        Eigen::MatrixXd adjoint;
        if (lin_solver_cached)
        {
            b(boundary_nodes, Eigen::all).setZero();
 
            StiffnessMatrix A = diff_cached.gradu_h(0);
            const int full_size = A.rows();
            const int problem_dim = problem->is_scalar() ? 1 : mesh->dimension();
            int precond_num = problem_dim * n_bases;
 
            b.conservativeResizeLike(Eigen::MatrixXd::Zero(A.rows(), b.cols()));
 
            std::vector<int> boundary_nodes_tmp;
            if (has_periodic_bc())
            {
                boundary_nodes_tmp = periodic_bc->full_to_periodic(boundary_nodes);
                precond_num = periodic_bc->full_to_periodic(A);
                b = periodic_bc->full_to_periodic(b, true);
            }
            else
                boundary_nodes_tmp = boundary_nodes;
 
            adjoint.setZero(ndof(), adjoint_rhs.cols());
            for (int i = 0; i < b.cols(); i++)
            {
                Eigen::VectorXd x, tmp;
                tmp = b.col(i);
                dirichlet_solve_prefactorized(*lin_solver_cached, A, tmp, boundary_nodes_tmp, x);
 
                if (has_periodic_bc())
                    adjoint.col(i) = periodic_bc->periodic_to_full(full_size, x);
                else
                    adjoint.col(i) = x;
            }
        }
        else
        {
            auto solver = polysolve::linear::Solver::create(args["solver"]["adjoint_linear"], adjoint_logger());
 
            StiffnessMatrix A = diff_cached.gradu_h(0); // This should be transposed, but A is symmetric in hyper-elastic and diffusion problems
 
            /*
            For non-periodic problems, the adjoint solution p's size is the full size in NLProblem
            For periodic problems, the adjoint solution p's size is the reduced size in NLProblem
            */
            if (!is_homogenization())
            {
                adjoint.setZero(ndof(), adjoint_rhs.cols());
                for (int i = 0; i < b.cols(); i++)
                {
                    Eigen::VectorXd tmp = b.col(i);
                    tmp(boundary_nodes).setZero();
 
                    Eigen::VectorXd x;
                    x.setZero(tmp.size());
                    dirichlet_solve(*solver, A, tmp, boundary_nodes, x, A.rows(), "", false, false, false);
 
                    adjoint.col(i) = x;
                    adjoint(boundary_nodes, i) = -b(boundary_nodes, i);
                }
            }
            else
            {
                solver->analyze_pattern(A, A.rows());
                solver->factorize(A);
 
                adjoint.setZero(adjoint_rhs.rows(), adjoint_rhs.cols());
                for (int i = 0; i < b.cols(); i++)
                {
                    Eigen::MatrixXd tmp = b.col(i);
 
                    Eigen::VectorXd x;
                    x.setZero(tmp.size());
                    solver->solve(tmp, x);
                    x.conservativeResize(adjoint.rows());
 
                    adjoint.col(i) = x;
                }
            }
        }
 
        return adjoint;
    }
    Eigen::MatrixXd State::solve_static_adjoint(const Eigen::MatrixXd &adjoint_rhs) const {…}
 
    Eigen::MatrixXd State::solve_transient_adjoint(const Eigen::MatrixXd &adjoint_rhs) const
    {
        const double dt = args["time"]["dt"];
        const int time_steps = args["time"]["time_steps"];
 
        int bdf_order = 1;
        if (args["time"]["integrator"].is_string())
            bdf_order = 1;
        else if (args["time"]["integrator"]["type"] == "ImplicitEuler")
            bdf_order = 1;
        else if (args["time"]["integrator"]["type"] == "BDF")
            bdf_order = args["time"]["integrator"]["steps"].get<int>();
        else
            log_and_throw_adjoint_error("Integrator type not supported for differentiability.");
 
        assert(adjoint_rhs.cols() == time_steps + 1);
 
        const int cols_per_adjoint = time_steps + 1;
        Eigen::MatrixXd adjoints;
        adjoints.setZero(ndof(), cols_per_adjoint * 2);
 
        // set dirichlet rows of mass to identity
        StiffnessMatrix reduced_mass;
        replace_rows_by_identity(reduced_mass, mass, boundary_nodes);
 
        Eigen::MatrixXd sum_alpha_p, sum_alpha_nu;
        for (int i = time_steps; i >= 0; --i)
        {
            {
                sum_alpha_p.setZero(ndof(), 1);
                sum_alpha_nu.setZero(ndof(), 1);
 
                const int num = std::min(bdf_order, time_steps - i);
 
                Eigen::VectorXd bdf_coeffs = Eigen::VectorXd::Zero(num);
                for (int j = 0; j < bdf_order && i + j < time_steps; ++j)
                    bdf_coeffs(j) = -time_integrator::BDF::alphas(std::min(bdf_order - 1, i + j))[j];
 
                sum_alpha_p = adjoints.middleCols(i + 1, num) * bdf_coeffs;
                sum_alpha_nu = adjoints.middleCols(cols_per_adjoint + i + 1, num) * bdf_coeffs;
            }
 
            Eigen::VectorXd rhs_ = -reduced_mass.transpose() * sum_alpha_nu - adjoint_rhs.col(i);
            for (int j = 1; j <= bdf_order; j++)
            {
                if (i + j > time_steps)
                    break;
 
                StiffnessMatrix gradu_h_prev;
                compute_force_jacobian_prev(i + j, i, gradu_h_prev);
                Eigen::VectorXd tmp = adjoints.col(i + j) * (time_integrator::BDF::betas(diff_cached.bdf_order(i + j) - 1) * dt);
                tmp(boundary_nodes).setZero();
                rhs_ += -gradu_h_prev.transpose() * tmp;
            }
 
            if (i > 0)
            {
                double beta_dt = time_integrator::BDF::betas(diff_cached.bdf_order(i) - 1) * dt;
 
                rhs_ += (1. / beta_dt) * (diff_cached.gradu_h(i) - reduced_mass).transpose() * sum_alpha_p;
 
                {
                    StiffnessMatrix A = diff_cached.gradu_h(i).transpose();
                    Eigen::VectorXd b_ = rhs_;
                    b_(boundary_nodes).setZero();
 
                    auto solver = polysolve::linear::Solver::create(args["solver"]["adjoint_linear"], adjoint_logger());
 
                    Eigen::VectorXd x;
                    dirichlet_solve(*solver, A, b_, boundary_nodes, x, A.rows(), "", false, false, false);
                    adjoints.col(i + cols_per_adjoint) = x;
                }
 
                // TODO: generalize to BDFn
                Eigen::VectorXd tmp = rhs_(boundary_nodes);
                if (i + 1 < cols_per_adjoint)
                    tmp += (-2. / beta_dt) * adjoints(boundary_nodes, i + 1);
                if (i + 2 < cols_per_adjoint)
                    tmp += (1. / beta_dt) * adjoints(boundary_nodes, i + 2);
 
                tmp -= (diff_cached.gradu_h(i).transpose() * adjoints.col(i + cols_per_adjoint))(boundary_nodes);
                adjoints(boundary_nodes, i + cols_per_adjoint) = tmp;
                adjoints.col(i) = beta_dt * adjoints.col(i + cols_per_adjoint) - sum_alpha_p;
            }
            else
            {
                adjoints.col(i) = -reduced_mass.transpose() * sum_alpha_p;
                adjoints.col(i + cols_per_adjoint) = rhs_; // adjoint_nu[0] actually stores adjoint_mu[0]
            }
        }
        return adjoints;
    }
    Eigen::MatrixXd State::solve_transient_adjoint(const Eigen::MatrixXd &adjoint_rhs) const {…}
 
    void State::compute_surface_node_ids(const int surface_selection, std::vector<int> &node_ids) const
    {
        node_ids = {};
 
        const auto &gbases = geom_bases();
        for (const auto &lb : total_local_boundary)
        {
            const int e = lb.element_id();
            for (int i = 0; i < lb.size(); ++i)
            {
                const int primitive_global_id = lb.global_primitive_id(i);
                const int boundary_id = mesh->get_boundary_id(primitive_global_id);
                const auto nodes = gbases[e].local_nodes_for_primitive(primitive_global_id, *mesh);
 
                if (boundary_id == surface_selection)
                {
                    for (long n = 0; n < nodes.size(); ++n)
                    {
                        const int g_id = gbases[e].bases[nodes(n)].global()[0].index;
 
                        if (std::count(node_ids.begin(), node_ids.end(), g_id) == 0)
                            node_ids.push_back(g_id);
                    }
                }
            }
        }
    }
    void State::compute_surface_node_ids(const int surface_selection, std::vector<int> &node_ids) const {…}
 
    void State::compute_total_surface_node_ids(std::vector<int> &node_ids) const
    {
        node_ids = {};
 
        const auto &gbases = geom_bases();
        for (const auto &lb : total_local_boundary)
        {
            const int e = lb.element_id();
            for (int i = 0; i < lb.size(); ++i)
            {
                const int primitive_global_id = lb.global_primitive_id(i);
                const auto nodes = gbases[e].local_nodes_for_primitive(primitive_global_id, *mesh);
 
                for (long n = 0; n < nodes.size(); ++n)
                {
                    const int g_id = gbases[e].bases[nodes(n)].global()[0].index;
 
                    if (std::count(node_ids.begin(), node_ids.end(), g_id) == 0)
                        node_ids.push_back(g_id);
                }
            }
        }
    }
    void State::compute_total_surface_node_ids(std::vector<int> &node_ids) const {…}
 
    void State::compute_volume_node_ids(const int volume_selection, std::vector<int> &node_ids) const
    {
        node_ids = {};
 
        const auto &gbases = geom_bases();
        for (int e = 0; e < gbases.size(); e++)
        {
            const int body_id = mesh->get_body_id(e);
            if (body_id == volume_selection)
                for (const auto &gbs : gbases[e].bases)
                    for (const auto &g : gbs.global())
                        node_ids.push_back(g.index);
        }
    }
    void State::compute_volume_node_ids(const int volume_selection, std::vector<int> &node_ids) const {…}
 
} // namespace polyfem