BilaplacianVarForm.cpp

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#include "BilaplacianVarForm.hpp"
 
#include <polyfem/assembler/AssemblerUtils.hpp>
#include <polyfem/assembler/GenericProblem.hpp>
#include <polyfem/basis/LagrangeBasis2d.hpp>
#include <polyfem/basis/LagrangeBasis3d.hpp>
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/io/MatrixIO.hpp>
#include <polyfem/mesh/mesh2D/Mesh2D.hpp>
#include <polyfem/mesh/mesh3D/Mesh3D.hpp>
#include <polyfem/problem/ProblemFactory.hpp>
#include <polyfem/refinement/APriori.hpp>
#include <polyfem/time_integrator/BDF.hpp>
#include <polyfem/utils/Jacobian.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
#include <polyfem/utils/Timer.hpp>
#include <polyfem/varforms/VarFormUtils.hpp>
 
#include <polysolve/linear/FEMSolver.hpp>
 
namespace polyfem::varform
{
    using namespace varform::internal;
 
    void BilaplacianVarForm::reset()
    {
        VarForm::reset();
        space_.reset();
        solution_space_id_ = -1;
        auxiliary_space_id_ = -1;
        boundary_.reset();
        ass_vals_cache_.init_empty();
        mass_ass_vals_cache_.init_empty(true);
        pure_mass_ass_vals_cache_.init_empty(true);
        rhs_assembler_ = nullptr;
        mass_.resize(0, 0);
        pure_mass_.resize(0, 0);
        avg_mass_ = 0;
        rhs_.resize(0, 0);
        pressure_space_.reset();
        pressure_boundary_.reset();
        pressure_ass_vals_cache_.init_empty();
        primary_assembler_ = nullptr;
        mass_assembler_ = nullptr;
        pure_mass_assembler_ = nullptr;
        mixed_assembler_ = nullptr;
        pressure_assembler_ = nullptr;
        t0 = 0;
        time_steps = 0;
        dt = 0;
        time_integrator = nullptr;
    }
 
    void BilaplacianVarForm::init(const std::string &formulation, const Units &units, const json &args, const std::string &out_path)
    {
        VarForm::init(formulation, units, args, out_path);
        const bool is_time_dependent = args.contains("time") && !args["time"].is_null();
        const json &discr_orders = args.at("space").at("discr_order");
 
        const json &materials = args.at("materials");
        if (materials.is_array() && materials.empty())
            log_and_throw_error("Bilaplacian formulations require at least one material.");
        const json &first_material = materials.is_array() ? materials.at(0) : materials;
        solution_space_id_ = first_material.at("solution_space_id").get<int>();
        auxiliary_space_id_ = first_material.at("auxiliary_space_id").get<int>();
        if (solution_space_id_ == auxiliary_space_id_)
            log_and_throw_error("Bilaplacian solution and auxiliary fields must use different FE space IDs.");
 
        if (discr_orders.is_array())
        {
            bool has_solution_space = false;
            bool has_auxiliary_space = false;
            for (const json &entry : discr_orders)
            {
                const int fe_space_id = entry.at("fe_space").get<int>();
                has_solution_space |= fe_space_id == solution_space_id_;
                has_auxiliary_space |= fe_space_id == auxiliary_space_id_;
            }
            if (!has_solution_space || !has_auxiliary_space)
                log_and_throw_error("Bilaplacian discretization-order lists must explicitly name the solution and auxiliary FE spaces.");
        }
 
        if (materials.is_array())
        {
            for (const json &material : materials)
            {
                if (material.at("solution_space_id").get<int>() != solution_space_id_
                    || material.at("auxiliary_space_id").get<int>() != auxiliary_space_id_)
                    log_and_throw_error("All Bilaplacian materials must use the same solution and auxiliary FE space IDs.");
            }
        }
 
        primary_assembler_ = assembler::AssemblerUtils::make_assembler(formulation);
        mass_assembler_ = std::make_shared<assembler::Mass>();
        pure_mass_assembler_ = std::make_shared<assembler::HRZMass>();
        mixed_assembler_ = assembler::AssemblerUtils::make_mixed_assembler(formulation);
        pressure_assembler_ = assembler::AssemblerUtils::make_assembler(assembler::AssemblerUtils::other_assembler_name(formulation));
 
        if (!args.contains("preset_problem"))
        {
            problem = std::make_shared<assembler::GenericScalarProblem>("GenericScalar");
            problem->clear();
            json tmp;
            tmp["is_time_dependent"] = is_time_dependent;
            problem->set_parameters(tmp, root_path);
 
            auto bc = args["boundary_conditions"];
            bc["root_path"] = root_path;
            problem->set_parameters(bc, root_path);
            problem->set_parameters(args["initial_conditions"], root_path);
            problem->set_parameters(args["output"], root_path);
        }
        else
        {
            problem = problem::ProblemFactory::factory().get_problem(args["preset_problem"]["type"]);
            problem->clear();
            problem->set_parameters(args["preset_problem"], root_path);
        }
 
        problem->set_units(*primary_assembler_, units);
        t0 = is_time_dependent ? args["time"]["t0"].get<double>() : 0.0;
        time_steps = is_time_dependent ? args["time"]["time_steps"].get<int>() : 0;
        dt = is_time_dependent ? args["time"]["dt"].get<double>() : 0.0;
    }
 
    void BilaplacianVarForm::save_json(const Eigen::MatrixXd &solution, std::ostream &out) const
    {
        if (!mesh_)
        {
            logger().error("Load the mesh first!");
            return;
        }
        if (solution.size() <= 0)
        {
            logger().error("Solve the problem first!");
            return;
        }
 
        logger().info("Saving json...");
        const Eigen::MatrixXd stats_solution =
            solution.rows() >= space_.n_bases
                ? solution.topRows(space_.n_bases).eval()
                : solution;
 
        nlohmann::json j;
        stats.save_json(
            args, space_.n_bases, pressure_space_.n_bases,
            stats_solution, *mesh_, space_.disc_orders, space_.disc_ordersq, *problem,
            timings, primary_assembler_ ? primary_assembler_->name() : name(), space_.is_iso_parametric(),
            args["output"]["advanced"]["sol_at_node"], j);
        out << j.dump(4) << std::endl;
    }
 
    io::OutputSpace BilaplacianVarForm::output_space() const
    {
        Eigen::VectorXi output_orders = space_.disc_orders;
        if (mesh_ && space_.disc_ordersq.size() == space_.disc_orders.size())
        {
            for (int e = 0; e < output_orders.size(); ++e)
            {
                if (mesh_->is_prism(e))
                    output_orders(e) = std::max(space_.disc_orders(e), space_.disc_ordersq(e));
            }
        }
 
        return {
            mesh_.get(),
            &space_.geometry_basis_list(),
            output_orders,
            &space_.polys,
            &space_.polys_3d,
            &boundary_.total_local_boundary,
            nullptr,
            nullptr,
            &boundary_.dirichlet_nodes,
            &boundary_.dirichlet_nodes_position};
    }
 
    io::OutStatsData BilaplacianVarForm::compute_errors(const Eigen::MatrixXd &solution)
    {
        if (!args["output"]["advanced"]["compute_error"])
            return stats;
 
        double tend = 0;
        if (!args["time"].is_null())
            tend = args["time"]["tend"];
 
        Eigen::MatrixXd value, pressure;
        split_solution(solution, value, pressure);
        stats.compute_errors(space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), *mesh_, *problem, tend, value);
        return stats;
    }
 
    void BilaplacianVarForm::export_data(const Eigen::MatrixXd &solution) const
    {
        const io::OutputSpace space = output_space();
        if (!space.mesh)
        {
            logger().error("Load the mesh first!");
            return;
        }
        if (solution.size() <= 0)
        {
            logger().error("Solve the problem first!");
            return;
        }
 
        ensure_output_sampler();
 
        const std::string vis_mesh_path = resolve_output_path(args["output"]["paraview"]["file_name"]);
        const bool has_time = args.contains("time") && !args["time"].is_null();
        double tend = has_time ? args["time"]["tend"].get<double>() : 1.0;
        double dt = 1;
        if (has_time)
            dt = args["time"]["dt"];
 
        const auto opts = export_options(space);
        output_geometry_.export_data(
            space,
            output_field_function(solution, opts),
            has_time,
            tend, dt,
            opts,
            vis_mesh_path);
 
        Eigen::MatrixXd value, pressure;
        split_solution(solution, value, pressure);
 
        const std::string solution_path = resolve_output_path(args["output"]["data"]["solution"]);
        if (!solution_path.empty())
        {
            const int primary_ndof = std::min<int>(value.rows(), space_.n_bases);
            const Eigen::MatrixXd primary_solution = value.topRows(primary_ndof);
            if (opts.reorder_output && space_.space_in_node_to_node.size() > 0)
            {
                const Eigen::MatrixXd nodal_solution = utils::unflatten(primary_solution, 1);
                Eigen::MatrixXd reordered = Eigen::MatrixXd::Zero(nodal_solution.rows(), nodal_solution.cols());
                for (int input_node = 0; input_node < space_.space_in_node_to_node.size(); ++input_node)
                {
                    const int node = space_.space_in_node_to_node(input_node);
                    if (node >= 0 && node < nodal_solution.rows() && input_node < reordered.rows())
                        reordered.row(input_node) = nodal_solution.row(node);
                }
                io::write_matrix(solution_path, reordered);
            }
            else
            {
                io::write_matrix(solution_path, primary_solution);
            }
        }
 
        const std::string nodes_path = resolve_output_path(args["output"]["data"]["nodes"]);
        if (!nodes_path.empty())
        {
            Eigen::MatrixXd nodes = Eigen::MatrixXd::Zero(space_.n_bases, mesh_->dimension());
            for (const basis::ElementBases &element_bases : space_.basis_list())
                for (const basis::Basis &basis : element_bases.bases)
                    for (const auto &global : basis.global())
                        nodes.row(global.index) = global.node;
            io::write_matrix(nodes_path, nodes);
        }
 
        const std::string stress_path = resolve_output_path(args["output"]["data"]["stress_mat"]);
        const std::string mises_path = resolve_output_path(args["output"]["data"]["mises"]);
        if ((!stress_path.empty() || !mises_path.empty()) && primary_assembler_)
        {
            Eigen::MatrixXd stress;
            Eigen::VectorXd mises;
            io::Evaluator::compute_stress_at_quadrature_points(
                *mesh_, problem->is_scalar(), space_.basis_list(), space_.geometry_basis_list(),
                space_.disc_orders, space_.disc_ordersq, *primary_assembler_, value, tend,
                stress, mises);
            if (!stress_path.empty())
                io::write_matrix(stress_path, stress);
            if (!mises_path.empty())
                io::write_matrix(mises_path, mises);
        }
    }
 
    void BilaplacianVarForm::load_mesh(const mesh::Mesh &mesh, const json &args)
    {
        set_materials(*primary_assembler_, 1);
        set_materials(*mass_assembler_, 1);
        pure_mass_assembler_->set_size(mass_assembler_->size());
        problem->init(mesh);
 
        if (mixed_assembler_)
            mixed_assembler_->set_size(1);
        if (pressure_assembler_)
            set_materials(*pressure_assembler_, 1);
    }
 
    std::shared_ptr<assembler::RhsAssembler> BilaplacianVarForm::build_rhs_assembler(
        const int n_bases,
        const std::vector<basis::ElementBases> &bases,
        const assembler::AssemblyValsCache &ass_vals_cache,
        const int fe_space_id)
    {
        json rhs_solver_params = args["solver"]["linear"];
        if (!rhs_solver_params.contains("Pardiso"))
            rhs_solver_params["Pardiso"] = {};
        rhs_solver_params["Pardiso"]["mtype"] = -2;
 
        return std::make_shared<assembler::RhsAssembler>(
            *primary_assembler_, *mesh_, nullptr,
            boundary_.dirichlet_nodes, boundary_.neumann_nodes,
            boundary_.dirichlet_nodes_position, boundary_.neumann_nodes_position,
            n_bases, 1, bases, space_.geometry_basis_list(), ass_vals_cache, *problem,
            args["space"]["advanced"]["bc_method"],
            rhs_solver_params,
            fe_space_id);
    }
 
    void BilaplacianVarForm::build_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args)
    {
        assert(problem);
        assert(primary_assembler_);
        assert(mass_assembler_);
        assert(pure_mass_assembler_);
 
        Eigen::VectorXi space_disc_orders, space_disc_ordersq;
        assign_discr_orders(args["space"], solution_space_id_, mesh, space_disc_orders, space_disc_ordersq);
 
        if (args["space"]["use_p_ref"])
        {
            refinement::APriori::p_refine(
                mesh,
                args["space"]["advanced"]["B"],
                args["space"]["advanced"]["h1_formula"],
                args["space"]["discr_order"],
                args["space"]["advanced"]["discr_order_max"],
                stats,
                space_disc_orders);
 
            logger().info("min p: {} max p: {}", space_disc_orders.minCoeff(), space_disc_orders.maxCoeff());
        }
 
        build_fe_space(
            mesh,
            iso_parametric,
            space_disc_orders,
            space_disc_ordersq,
            args["space"]["basis_type"],
            args["space"]["poly_basis_type"],
            *primary_assembler_,
            /*value_dim=*/1,
            args["space"]["advanced"]["quadrature_order"],
            args["space"]["advanced"]["mass_quadrature_order"],
            args["space"]["advanced"]["use_corner_quadrature"],
            args["space"]["advanced"]["n_harmonic_samples"],
            args["space"]["advanced"]["integral_constraints"],
            space_,
            boundary_);
 
        problem->update_nodes(space_.space_in_node_to_node);
        mesh.update_nodes(space_.space_in_node_to_node);
 
        const auto &current_bases = space_.geometry_basis_list();
        if (args["space"]["advanced"]["count_flipped_els"])
            stats.count_flipped_elements(mesh, current_bases);
 
        const int n_samples = 10;
        stats.compute_mesh_size(mesh, current_bases, n_samples, args["output"]["advanced"]["curved_mesh_size"]);
 
        logger().info("flipped elements {}", stats.n_flipped);
        logger().info("h: {}", stats.mesh_size);
 
        if (space_.n_bases <= args["solver"]["advanced"]["cache_size"])
        {
            igl::Timer timer;
            timer.start();
            logger().info("Building cache...");
            ass_vals_cache_.init(mesh.is_volume(), space_.basis_list(), current_bases);
            mass_ass_vals_cache_.init(mesh.is_volume(), space_.basis_list(), current_bases, true);
            pure_mass_ass_vals_cache_.init(mesh.is_volume(), space_.basis_list(), current_bases, true);
            logger().info(" took {}s", timer.getElapsedTime());
        }
        else
        {
            ass_vals_cache_.init_empty();
            mass_ass_vals_cache_.init_empty(true);
            pure_mass_ass_vals_cache_.init_empty(true);
        }
 
        if (space_.disc_orders.maxCoeff() != space_.disc_orders.minCoeff())
            log_and_throw_error("p refinement not supported in mixed formulation!");
        if (!space_.poly_edge_to_data.empty())
            log_and_throw_error("Polygonal bases are not supported in mixed formulations!");
 
        const int prev_bases = space_.n_bases;
        const auto &all_boundary = boundary_.total_local_boundary;
        const bool use_corner_quadrature = args["space"]["advanced"]["use_corner_quadrature"];
        const int quadrature_order = args["space"]["advanced"]["quadrature_order"].get<int>();
        const int mass_quadrature_order = args["space"]["advanced"]["mass_quadrature_order"].get<int>();
        Eigen::VectorXi pressure_disc_orders, pressure_disc_ordersq;
        assign_discr_orders(args["space"], auxiliary_space_id_, mesh, pressure_disc_orders, pressure_disc_ordersq);
        // to avoid serendipity
        const std::string pressure_basis_type = args["space"]["basis_type"].get<std::string>() == "Bernstein" ? "Bernstein" : "Lagrange";
        build_fe_space(
            mesh,
            /*iso_parametric=*/true,
            pressure_disc_orders,
            pressure_disc_ordersq,
            pressure_basis_type,
            args["space"]["poly_basis_type"],
            *primary_assembler_,
            /*value_dim=*/1,
            quadrature_order,
            mass_quadrature_order,
            use_corner_quadrature,
            args["space"]["advanced"]["n_harmonic_samples"],
            args["space"]["advanced"]["integral_constraints"],
            pressure_space_,
            pressure_boundary_,
            space_.geometry);
 
        assert(space_.basis_list().size() == pressure_space_.basis_list().size());
        for (int i = 0; i < int(pressure_space_.basis_list().size()); ++i)
        {
            quadrature::Quadrature b_quad;
            space_.basis_list()[i].compute_quadrature(b_quad);
            (*pressure_space_.bases)[i].set_quadrature([b_quad](quadrature::Quadrature &quad) { quad = b_quad; });
        }
 
        boundary_.clear_boundary_conditions();
        for (const auto &lb : all_boundary)
            boundary_.local_boundary.emplace_back(lb);
        pressure_boundary_.clear_boundary_conditions();
 
        problem->setup_bc(
            mesh, space_.n_bases,
            space_.basis_list(), space_.geometry_basis_list(), pressure_space_.basis_list(),
            boundary_.local_boundary,
            boundary_.boundary_nodes,
            boundary_.local_neumann_boundary,
            boundary_.local_pressure_boundary,
            boundary_.local_pressure_cavity,
            pressure_boundary_.boundary_nodes,
            boundary_.dirichlet_nodes, boundary_.neumann_nodes);
        pressure_boundary_.normalize_boundary_nodes();
 
        rebuild_node_positions(space_.basis_list(), boundary_.dirichlet_nodes, boundary_.dirichlet_nodes_position);
        rebuild_node_positions(space_.basis_list(), boundary_.neumann_nodes, boundary_.neumann_nodes_position);
 
        for (int i = prev_bases; i < space_.n_bases; ++i)
            boundary_.boundary_nodes.push_back(i);
 
        boundary_.normalize_boundary_nodes();
 
        if (space_.n_bases <= args["solver"]["advanced"]["cache_size"])
            pressure_ass_vals_cache_.init(mesh.is_volume(), pressure_space_.basis_list(), space_.geometry_basis_list());
        else
            pressure_ass_vals_cache_.init_empty();
 
        build_rhs_assembler();
 
        logger().info("n pressure bases: {}", pressure_space_.n_bases);
    }
 
    void BilaplacianVarForm::assemble_rhs(const mesh::Mesh &mesh)
    {
        igl::Timer timer;
        json p_params = {};
        p_params["formulation"] = primary_assembler_->name();
        p_params["root_path"] = root_path;
        {
            RowVectorNd min, max, delta;
            mesh.bounding_box(min, max);
            delta = (max - min) / 2. + min;
            if (mesh.is_volume())
                p_params["bbox_center"] = {delta(0), delta(1), delta(2)};
            else
                p_params["bbox_center"] = {delta(0), delta(1)};
        }
        problem->set_parameters(p_params, root_path);
 
        rhs_.resize(0, 0);
 
        timer.start();
        logger().info("Assigning rhs...");
 
        build_rhs_assembler();
        assert(rhs_assembler_ != nullptr);
        rhs_assembler_->assemble(mass_assembler_->density(), rhs_);
        rhs_ *= -1;
 
        timings.assigning_rhs_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assigning_rhs_time);
 
        const int prev_size = rhs_.rows();
        rhs_.conservativeResize(prev_size + pressure_space_.n_bases, rhs_.cols());
        if (boundary_.local_neumann_boundary.empty())
        {
            rhs_.bottomRows(pressure_space_.n_bases).setZero();
        }
        else
        {
            Eigen::MatrixXd tmp = Eigen::MatrixXd::Zero(pressure_space_.n_bases, 1);
            auto tmp_rhs_assembler = build_rhs_assembler(pressure_space_.n_bases, pressure_space_.basis_list(), pressure_ass_vals_cache_, auxiliary_space_id_);
            const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
            const QuadratureOrders boundary_samples = n_boundary_samples(space_.disc_orders.maxCoeff(), space_.disc_ordersq.maxCoeff(), gdiscr_order);
            tmp_rhs_assembler->set_bc(
                std::vector<mesh::LocalBoundary>(), std::vector<int>(), boundary_samples, boundary_.local_neumann_boundary, tmp);
            rhs_.bottomRows(pressure_space_.n_bases) = tmp;
        }
    }
 
    void BilaplacianVarForm::assemble_mass_mat(const mesh::Mesh &mesh, const json &args)
    {
        if (!problem->is_time_dependent())
        {
            avg_mass_ = 1;
            timings.assembling_mass_mat_time = 0;
            return;
        }
 
        mass_.resize(0, 0);
        igl::Timer timer;
        timer.start();
        logger().info("Assembling mass mat...");
        mass_assembler_->assemble(mesh.is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), mass_ass_vals_cache_, 0, mass_, true);
        avg_mass_ = 0;
        for (int k = 0; k < mass_.outerSize(); ++k)
            for (StiffnessMatrix::InnerIterator it(mass_, k); it; ++it)
            {
                assert(it.col() == k);
                avg_mass_ += it.value();
            }
        avg_mass_ /= std::max(1, int(mass_.rows()));
        if (args["solver"]["advanced"]["lump_mass_matrix"])
            mass_ = utils::lump_matrix(mass_);
        timer.stop();
        timings.assembling_mass_mat_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assembling_mass_mat_time);
        stats.nn_zero = mass_.nonZeros();
        stats.num_dofs = mass_.rows();
        stats.mat_size = (long long)mass_.rows() * (long long)mass_.cols();
    }
 
    int BilaplacianVarForm::stacked_ndof() const
    {
        return space_.n_bases + pressure_space_.n_bases;
    }
 
    void BilaplacianVarForm::prepare_initial_solution(Eigen::MatrixXd &sol) const
    {
        if (sol.size() <= 0)
        {
            assert(rhs_assembler_ != nullptr);
            const bool was_solution_loaded = read_initial_x_from_file(
                resolve_input_path(args["input"]["data"]["state"]), "u",
                args["input"]["data"]["reorder"], space_.space_in_node_to_node,
                /*dim=*/1, sol);
 
            if (!was_solution_loaded)
            {
                if (problem->is_time_dependent())
                    rhs_assembler_->initial_solution(sol);
                else
                {
                    sol.resize(rhs_.size(), 1);
                    sol.setZero();
                }
            }
        }
        if (sol.cols() > 1)
            sol.conservativeResize(Eigen::NoChange, 1);
        sol.conservativeResize(stacked_ndof(), sol.cols());
        sol.bottomRows(pressure_space_.n_bases).setZero();
    }
 
    void BilaplacianVarForm::split_solution(const Eigen::MatrixXd &stacked, Eigen::MatrixXd &primary, Eigen::MatrixXd &pressure) const
    {
        const int cols = std::max(1, int(stacked.cols()));
        primary.setZero(space_.n_bases, cols);
        pressure.setZero(pressure_space_.n_bases, cols);
        const int primary_rows = std::min(space_.n_bases, int(stacked.rows()));
        if (primary_rows > 0)
            primary.topRows(primary_rows) = stacked.topRows(primary_rows);
        if (stacked.rows() > space_.n_bases)
        {
            const int pressure_rows = std::min(pressure_space_.n_bases, int(stacked.rows()) - space_.n_bases);
            if (pressure_rows > 0)
                pressure.topRows(pressure_rows) = stacked.middleRows(space_.n_bases, pressure_rows);
        }
    }
 
    void BilaplacianVarForm::build_stiffness_mat(StiffnessMatrix &stiffness)
    {
        igl::Timer timer;
        timer.start();
        logger().info("Assembling stiffness mat...");
 
        StiffnessMatrix main_stiffness, mixed_stiffness, aux_stiffness;
        primary_assembler_->assemble(mesh_->is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), ass_vals_cache_, 0, main_stiffness);
        mixed_assembler_->assemble(mesh_->is_volume(), pressure_space_.n_bases, space_.n_bases, pressure_space_.basis_list(), space_.basis_list(), space_.geometry_basis_list(), pressure_ass_vals_cache_, ass_vals_cache_, 0, mixed_stiffness);
        pressure_assembler_->assemble(mesh_->is_volume(), pressure_space_.n_bases, pressure_space_.basis_list(), space_.geometry_basis_list(), pressure_ass_vals_cache_, 0, aux_stiffness);
 
        assembler::AssemblerUtils::merge_mixed_matrices(
            space_.n_bases, pressure_space_.n_bases, 1, /*add_average=*/false,
            main_stiffness, mixed_stiffness, aux_stiffness, stiffness);
 
        timer.stop();
        timings.assembling_stiffness_mat_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assembling_stiffness_mat_time);
        stats.nn_zero = stiffness.nonZeros();
        stats.num_dofs = stiffness.rows();
        stats.mat_size = (long long)stiffness.rows() * (long long)stiffness.cols();
        write_matrix_market(args, stiffness);
    }
 
    void BilaplacianVarForm::solve_linear_system(
        const std::unique_ptr<polysolve::linear::Solver> &solver,
        StiffnessMatrix &A,
        Eigen::VectorXd &b,
        const bool compute_spectrum,
        Eigen::MatrixXd &sol)
    {
        Eigen::VectorXd x;
        stats.spectrum = dirichlet_solve(
            *solver,
            A,
            b,
            boundary_.boundary_nodes,
            x,
            space_.n_bases,
            args["output"]["data"]["stiffness_mat"],
            compute_spectrum,
            /*is_fluid=*/false,
            /*use_avg_pressure=*/false);
        sol = x;
        solver->get_info(stats.solver_info);
    }
 
    void BilaplacianVarForm::solve_static_linear(Eigen::MatrixXd &sol)
    {
        auto solver = polysolve::linear::Solver::create(args["solver"]["linear"], logger());
        logger().info("{}...", solver->name());
        const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
        const QuadratureOrders boundary_samples = n_boundary_samples(space_.disc_orders.maxCoeff(), space_.disc_ordersq.maxCoeff(), gdiscr_order);
        rhs_assembler_->set_bc(
            boundary_.local_boundary, boundary_.boundary_nodes, boundary_samples,
            (primary_assembler_->name() != "Bilaplacian") ? boundary_.local_neumann_boundary : std::vector<mesh::LocalBoundary>(), rhs_);
        StiffnessMatrix A;
        build_stiffness_mat(A);
        Eigen::VectorXd b = rhs_;
        solve_linear_system(solver, A, b, args["output"]["advanced"]["spectrum"], sol);
    }
 
    void BilaplacianVarForm::solve_transient_linear(Eigen::MatrixXd &sol)
    {
        auto solver = polysolve::linear::Solver::create(args["solver"]["linear"], logger());
        logger().info("{}...", solver->name());
 
        Eigen::MatrixXd value, pressure;
        split_solution(sol, value, pressure);
        auto bdf = time_integrator::ImplicitTimeIntegrator::construct_bdf_integrator(
            args["time"]["integrator"]);
        bdf->init(
            value,
            Eigen::MatrixXd::Zero(value.rows(), value.cols()),
            Eigen::MatrixXd::Zero(value.rows(), value.cols()),
            dt);
        time_integrator = bdf;
 
        save_timestep(t0, 0, t0, dt, sol);
 
        Eigen::MatrixXd current_rhs = rhs_;
        StiffnessMatrix stiffness, expanded_mass;
        build_stiffness_mat(stiffness);
        expand_primary_matrix(stacked_ndof(), mass_, expanded_mass);
        const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
        const QuadratureOrders boundary_samples = n_boundary_samples(space_.disc_orders.maxCoeff(), space_.disc_ordersq.maxCoeff(), gdiscr_order);
 
        for (int t = 1; t <= time_steps; ++t)
        {
            const double time = t0 + t * dt;
            rhs_assembler_->compute_energy_grad(
                boundary_.local_boundary, boundary_.boundary_nodes, mass_assembler_->density(), boundary_samples, boundary_.local_neumann_boundary, rhs_, time,
                current_rhs);
            rhs_assembler_->set_bc(
                boundary_.local_boundary, boundary_.boundary_nodes, boundary_samples, boundary_.local_neumann_boundary, current_rhs, value, time);
 
            if (current_rhs.rows() != stacked_ndof())
            {
                const int old_rows = current_rhs.rows();
                current_rhs.conservativeResize(stacked_ndof(), current_rhs.cols());
                if (stacked_ndof() > old_rows)
                    current_rhs.bottomRows(stacked_ndof() - old_rows).setZero();
            }
            current_rhs.bottomRows(pressure_space_.n_bases).setZero();
 
            StiffnessMatrix A = expanded_mass / bdf->beta_dt() + stiffness;
            Eigen::VectorXd b = Eigen::VectorXd::Zero(stacked_ndof());
            b.head(space_.n_bases) = (mass_ * bdf->weighted_sum_x_prevs()) / bdf->beta_dt();
            for (int i : boundary_.boundary_nodes)
                b[i] = 0;
            b += current_rhs;
 
            solve_linear_system(solver, A, b, args["output"]["advanced"]["spectrum"].get<bool>() && t == time_steps, sol);
            split_solution(sol, value, pressure);
            bdf->update_quantities(value.col(0));
 
            save_timestep(time, t, t0, dt, sol);
            save_step_state(t0, dt, t, time_integrator.get());
            logger().info("{}/{}  t={}", t, time_steps, time);
            notify_time_step(t, time_steps, t0, dt);
        }
    }
 
    void BilaplacianVarForm::solve_problem(Eigen::MatrixXd &sol)
    {
        stats.spectrum.setZero();
        igl::Timer timer;
        timer.start();
        logger().info("Solving {}", primary_assembler_->name());
        prepare_initial_solution(sol);
        if (problem->is_time_dependent())
            solve_transient_linear(sol);
        else
        {
            time_integrator = nullptr;
            solve_static_linear(sol);
        }
        timer.stop();
        timings.solving_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.solving_time);
    }
 
    std::vector<io::OutputField> BilaplacianVarForm::output_fields(
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution,
        const io::OutputFieldOptions &options) const
    {
        std::vector<io::OutputField> fields;
        if (!mesh_ || !problem || solution.size() <= 0)
            return fields;
 
        Eigen::MatrixXd value, pressure;
        split_solution(solution, value, pressure);
 
        const bool has_element_samples = sample.local_points.rows() > 0 && sample.local_points.rows() == sample.element_ids.size();
        const int output_rows = sample.points.rows() > 0 ? sample.points.rows() : std::max<int>(sample.local_points.rows(), sample.node_ids.size());
        const int primary_ndof = std::min<int>(value.rows(), space_.n_bases);
        const Eigen::MatrixXd primary_solution = value.topRows(primary_ndof);
 
        const auto sample_dof_field = [&](const Eigen::MatrixXd &dof_values, Eigen::MatrixXd &values, Eigen::MatrixXd *gradients = nullptr) -> bool {
            if (dof_values.size() <= 0)
                return false;
 
            if (has_element_samples)
            {
                values.resize(sample.local_points.rows(), 1);
                if (gradients)
                    gradients->resize(sample.local_points.rows(), mesh_->dimension());
                for (int i = 0; i < sample.local_points.rows(); ++i)
                {
                    const int element_id = sample.element_ids(i);
                    if (element_id < 0)
                    {
                        values(i) = 0;
                        if (gradients)
                            gradients->row(i).setZero();
                        continue;
                    }
 
                    Eigen::MatrixXd local_sol, local_grad;
                    io::Evaluator::interpolate_at_local_vals(
                        *mesh_, 1, space_.basis_list(), space_.geometry_basis_list(),
                        element_id, sample.local_points.row(i), dof_values, local_sol, local_grad);
                    values(i) = local_sol(0);
                    if (gradients)
                        gradients->row(i) = local_grad;
                }
 
                if (output_rows > values.rows())
                {
                    const int previous_rows = values.rows();
                    values.conservativeResize(output_rows, Eigen::NoChange);
                    values.bottomRows(output_rows - previous_rows).setZero();
                    if (gradients)
                    {
                        gradients->conservativeResize(output_rows, Eigen::NoChange);
                        gradients->bottomRows(output_rows - previous_rows).setZero();
                    }
                }
                return true;
            }
 
            if (sample.node_ids.size() > 0)
            {
                values.resize(sample.node_ids.size(), 1);
                for (int i = 0; i < sample.node_ids.size(); ++i)
                {
                    const int node_id = sample.node_ids(i);
                    if (node_id < 0 || node_id >= dof_values.rows())
                        return false;
                    values(i) = dof_values(node_id);
                }
                return sample.points.rows() == 0 || sample.points.rows() == values.rows();
            }
 
            return false;
        };
 
        const auto &paraview_options = args["output"]["paraview"]["options"];
        if (has_element_samples && problem->has_exact_sol() && sample.points.rows() == output_rows)
        {
            Eigen::MatrixXd exact;
            problem->exact(sample.points, sample.time, exact);
            if (exact.rows() == output_rows)
            {
                if (options.export_field("exact"))
                    fields.push_back({"exact", exact, io::OutputField::Association::Point});
                if (options.export_field("error"))
                {
                    Eigen::MatrixXd values;
                    if (sample_dof_field(primary_solution, values))
                        fields.push_back({"error", (values - exact).rowwise().norm(), io::OutputField::Association::Point});
                }
            }
        }
 
        if ((paraview_options["nodes"] || (!options.fields.empty() && options.export_field("nodes")))
            && has_element_samples
            && sample.primitive_ids.size() == 0)
        {
            Eigen::MatrixXd dof_ids(primary_ndof, 1);
            dof_ids.col(0).setLinSpaced(primary_ndof, 0, primary_ndof - 1);
            Eigen::MatrixXd values;
            if (sample_dof_field(dof_ids, values))
                fields.push_back({"nodes", values, io::OutputField::Association::Point});
        }
 
        if ((paraview_options["jacobian_validity"] || (!options.fields.empty() && options.export_field("validity")))
            && has_element_samples
            && mesh_->dimension() == 1
            && sample.primitive_ids.size() == 0)
        {
            const auto invalid_elements = utils::count_invalid(mesh_->dimension(), space_.basis_list(), space_.geometry_basis_list(), primary_solution);
            Eigen::MatrixXd validity = Eigen::MatrixXd::Zero(output_rows, 1);
            for (int i = 0; i < sample.element_ids.size(); ++i)
                validity(i) = std::find(invalid_elements.begin(), invalid_elements.end(), sample.element_ids(i)) != invalid_elements.end();
            fields.push_back({"validity", validity, io::OutputField::Association::Point});
        }
 
        const bool export_solution_gradient =
            !options.fields.empty() && options.export_field("solution_gradient");
        if (options.export_field("solution") || export_solution_gradient)
        {
            Eigen::MatrixXd values, gradients;
            if (sample_dof_field(
                    value, values,
                    export_solution_gradient ? &gradients : nullptr))
            {
                if (options.export_field("solution"))
                    fields.push_back({"solution", values, io::OutputField::Association::Point});
                if (export_solution_gradient)
                    fields.push_back({"solution_gradient", gradients, io::OutputField::Association::Point});
            }
        }
 
        if (paraview_options["material"] && has_element_samples)
        {
            const auto &params = primary_assembler_->parameters();
            std::map<std::string, Eigen::MatrixXd> param_values;
            for (const auto &[p, _] : params)
                param_values[p].setZero(output_rows, 1);
 
            Eigen::MatrixXd rhos = Eigen::MatrixXd::Zero(output_rows, 1);
            const auto &density = mass_assembler_->density();
            for (int i = 0; i < sample.local_points.rows(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id < 0)
                    continue;
 
                for (const auto &[p, func] : params)
                    param_values.at(p)(i) = func(sample.local_points.row(i), sample.points.row(i), sample.time, element_id);
                rhos(i) = density(sample.local_points.row(i), sample.points.row(i), sample.time, element_id);
            }
 
            for (const auto &[name, values] : param_values)
                if (options.export_field(name))
                    fields.push_back({name, values, io::OutputField::Association::Point});
            if (options.export_field("rho"))
                fields.push_back({"rho", rhos, io::OutputField::Association::Point});
        }
 
        if (paraview_options["body_ids"] && options.export_field("body_ids") && has_element_samples)
        {
            Eigen::MatrixXd ids = Eigen::MatrixXd::Zero(output_rows, 1);
            for (int i = 0; i < sample.element_ids.size(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id >= 0)
                    ids(i) = mesh_->get_body_id(element_id);
            }
            fields.push_back({"body_ids", ids, io::OutputField::Association::Point});
        }
 
        const bool export_pressure_gradient =
            !options.fields.empty() && options.export_field("pressure_gradient");
        if (mesh_ && (options.export_field("pressure") || export_pressure_gradient || (!options.fields.empty() && options.export_field("auxiliary"))))
        {
            Eigen::MatrixXd values, gradients;
            if (sample_scalar_field(
                    *mesh_, pressure_space_.basis_list(), space_.geometry_basis_list(), sample, pressure, values,
                    export_pressure_gradient ? &gradients : nullptr))
            {
                if (options.export_field("pressure"))
                    fields.push_back({"pressure", values, io::OutputField::Association::Point});
                if (export_pressure_gradient)
                    fields.push_back({"pressure_gradient", gradients, io::OutputField::Association::Point});
                if (!options.fields.empty() && options.export_field("auxiliary"))
                    fields.push_back({"auxiliary", values, io::OutputField::Association::Point});
            }
        }
        return fields;
    }
} // namespace polyfem::varform