VarFormDiff.cpp

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#include <polyfem/optimization/VarFormDiff.hpp>
 
#include <polyfem/varforms/diff/DifferentiableVarForm.hpp>
 
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
#include <polyfem/utils/Types.hpp>
 
#include <polyfem/time_integrator/BDF.hpp>
#include <polyfem/time_integrator/ImplicitEuler.hpp>
 
#include <polysolve/linear/FEMSolver.hpp>
 
#include <polyfem/solver/NLProblem.hpp>
#include <polyfem/solver/NLHomoProblem.hpp>
#include <polyfem/solver/forms/BarrierContactForm.hpp>
#include <polyfem/solver/forms/SmoothContactForm.hpp>
// Below types in SolverData are forward declared, include them explicitly.
#include <polyfem/solver/forms/BodyForm.hpp>
#include <polyfem/solver/forms/FrictionForm.hpp>
#include <polyfem/solver/forms/NormalAdhesionForm.hpp>
#include <polyfem/solver/forms/TangentialAdhesionForm.hpp>
#include <polyfem/assembler/ViscousDamping.hpp>
 
#include <polyfem/optimization/DiffCache.hpp>
 
#include <ipc/ipc.hpp>
#include <ipc/potentials/friction_potential.hpp>
 
#include <Eigen/Dense>
 
#include <algorithm>
#include <vector>
#include <cassert>
#include <vector>
 
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());
        }
 
        void compute_force_jacobian_prev(const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache, const int force_step, const int sol_step, StiffnessMatrix &hessian_prev)
        {
            assert(force_step > 0);
            assert(force_step > sol_step);
 
            if (varform.primary_assembler().is_linear() && !varform.is_contact_enabled())
            {
                hessian_prev = StiffnessMatrix(varform.primary_space().ndof(), varform.primary_space().ndof());
            }
            else
            {
                const Eigen::MatrixXd u = diff_cache.u(force_step);
                const Eigen::MatrixXd u_prev = diff_cache.u(sol_step);
                const double beta = time_integrator::BDF::betas(diff_cache.bdf_order(force_step) - 1);
                const double dt = varform.solve_data()->time_integrator->dt();
 
                hessian_prev = StiffnessMatrix(u.size(), u.size());
                if (varform.get_problem().is_time_dependent())
                {
                    if (varform.solve_data()->friction_form)
                    {
                        if (sol_step == force_step - 1)
                        {
                            Eigen::MatrixXd surface_solution_prev = varform.collision_mesh().vertices(utils::unflatten(u_prev, varform.get_mesh().dimension()));
                            Eigen::MatrixXd surface_solution = varform.collision_mesh().vertices(utils::unflatten(u, varform.get_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 *>(varform.solve_data()->contact_form.get()))
                            {
                                ipc::BarrierPotential bp = barrier_contact->barrier_potential();
                                bp.set_stiffness(barrier_contact->barrier_stiffness());
                                hessian_prev =
                                    varform.solve_data()->friction_form->friction_potential().force_jacobian(
                                        diff_cache.friction_collision_set(force_step),
                                        varform.collision_mesh(),
                                        varform.collision_mesh().rest_positions(),
                                        /*lagged_displacements=*/surface_solution_prev,
                                        surface_velocities,
                                        bp,
                                        ipc::FrictionPotential::DiffWRT::LAGGED_DISPLACEMENTS)
                                    + varform.solve_data()->friction_form->friction_potential().force_jacobian(
                                          diff_cache.friction_collision_set(force_step),
                                          varform.collision_mesh(),
                                          varform.collision_mesh().rest_positions(),
                                          /*lagged_displacements=*/surface_solution_prev,
                                          surface_velocities,
                                          bp,
                                          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 = varform.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 (varform.solve_data()->tangential_adhesion_form)
                    {
 
                        if (sol_step == force_step - 1)
                        {
                            StiffnessMatrix adhesion_hessian_prev(u.size(), u.size());
 
                            Eigen::MatrixXd surface_solution_prev = varform.collision_mesh().vertices(utils::unflatten(u_prev, varform.get_mesh().dimension()));
                            Eigen::MatrixXd surface_solution = varform.collision_mesh().vertices(utils::unflatten(u, varform.get_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 =
                                varform.solve_data()->tangential_adhesion_form->tangential_adhesion_potential().force_jacobian(
                                    diff_cache.tangential_adhesion_collision_set(force_step),
                                    varform.collision_mesh(),
                                    varform.collision_mesh().rest_positions(),
                                    /*lagged_displacements=*/surface_solution_prev,
                                    surface_velocities,
                                    varform.solve_data()->normal_adhesion_form->normal_adhesion_potential(),
                                    ipc::TangentialPotential::DiffWRT::LAGGED_DISPLACEMENTS)
                                + varform.solve_data()->tangential_adhesion_form->tangential_adhesion_potential().force_jacobian(
                                      diff_cache.tangential_adhesion_collision_set(force_step),
                                      varform.collision_mesh(),
                                      varform.collision_mesh().rest_positions(),
                                      /*lagged_displacements=*/surface_solution_prev,
                                      surface_velocities,
                                      varform.solve_data()->normal_adhesion_form->normal_adhesion_potential(),
                                      ipc::TangentialPotential::DiffWRT::VELOCITIES)
                                      * dv_dut;
 
                            adhesion_hessian_prev *= -1;
 
                            adhesion_hessian_prev = varform.collision_mesh().to_full_dof(adhesion_hessian_prev); // / (beta * dt) / (beta * dt);
 
                            hessian_prev += adhesion_hessian_prev;
                        }
                    }
 
                    if (varform.damping_assembler() && varform.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());
                        varform.damping_prev_assembler()->assemble_hessian(varform.get_mesh().is_volume(), varform.primary_space().n_bases, false, varform.primary_space().basis_list(), varform.primary_space().geometry_basis_list(), varform.assembly_cache(), force_step * varform.get_args()["time"]["dt"].get<double>() + varform.get_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());
                        varform.solve_data()->body_form->hessian_wrt_u_prev(u_prev, force_step * dt, body_force_hessian);
                        hessian_prev += body_force_hessian;
                    }
                }
            }
        }
 
        Eigen::MatrixXd solve_static_adjoint(const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache, const Eigen::MatrixXd &adjoint_rhs)
        {
 
            Eigen::MatrixXd b = adjoint_rhs;
 
            Eigen::MatrixXd adjoint;
            {
                auto solver = polysolve::linear::Solver::create(varform.get_args()["solver"]["adjoint_linear"], adjoint_logger());
 
                StiffnessMatrix A = diff_cache.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 (!varform.is_homogenization())
                {
                    adjoint.setZero(varform.primary_space().ndof(), adjoint_rhs.cols());
                    for (int i = 0; i < b.cols(); i++)
                    {
                        Eigen::VectorXd tmp = b.col(i);
                        tmp(varform.boundary_state().boundary_nodes).setZero();
 
                        Eigen::VectorXd x;
                        x.setZero(tmp.size());
                        dirichlet_solve(*solver, A, tmp, varform.boundary_state().boundary_nodes, x, A.rows(), "", false, false, false);
 
                        adjoint.col(i) = x;
                        adjoint(varform.boundary_state().boundary_nodes, i) = -b(varform.boundary_state().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 solve_transient_adjoint(const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache, const Eigen::MatrixXd &adjoint_rhs)
        {
 
            const double dt = varform.get_args()["time"]["dt"];
            const int time_steps = varform.get_args()["time"]["time_steps"];
 
            int bdf_order = 1;
            if (varform.get_args()["time"]["integrator"].is_string())
                bdf_order = 1;
            else if (varform.get_args()["time"]["integrator"]["type"] == "ImplicitEuler")
                bdf_order = 1;
            else if (varform.get_args()["time"]["integrator"]["type"] == "BDF")
                bdf_order = varform.get_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(varform.primary_space().ndof(), cols_per_adjoint * 2);
 
            // set dirichlet rows of mass to identity
            StiffnessMatrix reduced_mass;
            replace_rows_by_identity(reduced_mass, varform.mass_matrix(), varform.boundary_state().boundary_nodes);
 
            Eigen::MatrixXd sum_alpha_p, sum_alpha_nu;
            for (int i = time_steps; i >= 0; --i)
            {
                {
                    sum_alpha_p.setZero(varform.primary_space().ndof(), 1);
                    sum_alpha_nu.setZero(varform.primary_space().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(varform, diff_cache, i + j, i, gradu_h_prev);
                    Eigen::VectorXd tmp = adjoints.col(i + j) * (time_integrator::BDF::betas(diff_cache.bdf_order(i + j) - 1) * dt);
                    tmp(varform.boundary_state().boundary_nodes).setZero();
                    rhs_ += -gradu_h_prev.transpose() * tmp;
                }
 
                if (i > 0)
                {
                    double beta_dt = time_integrator::BDF::betas(diff_cache.bdf_order(i) - 1) * dt;
 
                    rhs_ += (1. / beta_dt) * (diff_cache.gradu_h(i) - reduced_mass).transpose() * sum_alpha_p;
 
                    {
                        StiffnessMatrix A = diff_cache.gradu_h(i).transpose();
                        Eigen::VectorXd b_ = rhs_;
                        b_(varform.boundary_state().boundary_nodes).setZero();
 
                        auto solver = polysolve::linear::Solver::create(varform.get_args()["solver"]["adjoint_linear"], adjoint_logger());
 
                        Eigen::VectorXd x;
                        dirichlet_solve(*solver, A, b_, varform.boundary_state().boundary_nodes, x, A.rows(), "", false, false, false);
                        adjoints.col(i + cols_per_adjoint) = x;
                    }
 
                    // TODO: generalize to BDFn
                    Eigen::VectorXd tmp = rhs_(varform.boundary_state().boundary_nodes);
                    if (i + 1 < cols_per_adjoint)
                        tmp += (-2. / beta_dt) * adjoints(varform.boundary_state().boundary_nodes, i + 1);
                    if (i + 2 < cols_per_adjoint)
                        tmp += (1. / beta_dt) * adjoints(varform.boundary_state().boundary_nodes, i + 2);
 
                    tmp -= (diff_cache.gradu_h(i).transpose() * adjoints.col(i + cols_per_adjoint))(varform.boundary_state().boundary_nodes);
                    adjoints(varform.boundary_state().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 solve_adjoint(const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache, const Eigen::MatrixXd &rhs)
        {
            if (varform.get_problem().is_time_dependent())
                return solve_transient_adjoint(varform, diff_cache, rhs);
            else
                return solve_static_adjoint(varform, diff_cache, rhs);
        }
    } // namespace
 
    void solve_adjoint_cached(const varform::DifferentiableVarForm &varform, DiffCache &diff_cache, const Eigen::MatrixXd &rhs)
    {
        diff_cache.cache_adjoints(solve_adjoint(varform, diff_cache, rhs));
    }
 
    Eigen::MatrixXd get_adjoint_mat(const varform::DifferentiableVarForm &varform, const DiffCache &diff_cache, int type)
    {
        assert(diff_cache.adjoint_mat().size() > 0);
 
        if (varform.get_problem().is_time_dependent())
        {
            if (type == 0)
                return diff_cache.adjoint_mat().leftCols(diff_cache.adjoint_mat().cols() / 2);
            else if (type == 1)
                return diff_cache.adjoint_mat().middleCols(diff_cache.adjoint_mat().cols() / 2, diff_cache.adjoint_mat().cols() / 2);
            else
                log_and_throw_adjoint_error("Invalid adjoint type!");
        }
 
        return diff_cache.adjoint_mat();
    }
 
    void compute_surface_node_ids(const varform::DifferentiableVarForm &varform, const int surface_selection, std::vector<int> &node_ids)
    {
 
        node_ids = {};
 
        const auto &gbases = varform.primary_space().geometry_basis_list();
        for (const auto &lb : varform.boundary_state().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 = varform.get_mesh().get_boundary_id(primitive_global_id);
                const auto nodes = gbases[e].local_nodes_for_primitive(primitive_global_id, varform.get_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 compute_total_surface_node_ids(const varform::DifferentiableVarForm &varform, std::vector<int> &node_ids)
    {
 
        node_ids = {};
 
        const auto &gbases = varform.primary_space().geometry_basis_list();
        for (const auto &lb : varform.boundary_state().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, varform.get_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 compute_volume_node_ids(const varform::DifferentiableVarForm &varform, const int volume_selection, std::vector<int> &node_ids)
    {
 
        node_ids = {};
 
        const auto &gbases = varform.primary_space().geometry_basis_list();
        for (int e = 0; e < gbases.size(); e++)
        {
            const int body_id = varform.get_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);
        }
    }
 
} // namespace polyfem