ScalarVarForm.cpp

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#include "ScalarVarForm.hpp"
 
#include <polyfem/assembler/AssemblerUtils.hpp>
#include <polyfem/assembler/GenericProblem.hpp>
 
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/io/MatrixIO.hpp>
 
#include <polyfem/problem/ProblemFactory.hpp>
#include <polyfem/refinement/APriori.hpp>
 
#include <polyfem/solver/NLProblem.hpp>
#include <polyfem/solver/forms/lagrangian/BCLagrangianForm.hpp>
#include <polyfem/solver/forms/lagrangian/MatrixLagrangianForm.hpp>
#include <polyfem/solver/forms/lagrangian/PeriodicBoundaryLagrangianForm.hpp>
 
#include <polyfem/time_integrator/BDF.hpp>
#include <polyfem/solver/forms/BodyForm.hpp>
#include <polyfem/solver/forms/ElasticForm.hpp>
#include <polyfem/utils/Jacobian.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
#include <polyfem/utils/Timer.hpp>
 
#include <unsupported/Eigen/SparseExtra>
 
#include <polysolve/linear/FEMSolver.hpp>
 
#include <algorithm>
 
namespace polyfem::varform
{
    namespace
    {
        void write_matrix_market(const json &args, const StiffnessMatrix &stiffness)
        {
            const std::string full_mat_path = args["output"]["data"]["full_mat"];
            if (!full_mat_path.empty())
                Eigen::saveMarket(stiffness, full_mat_path);
        }
    } // namespace
 
    void ScalarVarForm::serialize_checkpoint(
        io::CheckpointWriter &writer,
        const Eigen::MatrixXd &solution,
        const io::CheckpointMetadata &metadata) const
    {
        VarForm::serialize_checkpoint(writer, solution, metadata);
        write_checkpoint_ordering(writer, "primary", space_);
    }
 
    void ScalarVarForm::deserialize_checkpoint(
        const io::CheckpointReader &reader,
        Eigen::MatrixXd &solution)
    {
        VarForm::deserialize_checkpoint(reader, solution);
        validate_checkpoint_solution(solution, space_.ndof());
        reorder_checkpoint_block(reader, "primary", space_, 1, 0, solution);
    }
 
    void ScalarVarForm::reset()
    {
        VarForm::reset();
        space_.reset();
        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);
        primary_assembler_ = nullptr;
        mass_assembler_ = nullptr;
        pure_mass_assembler_ = nullptr;
        t0 = 0;
        time_steps = 0;
        dt = 0;
        time_integrator = nullptr;
        solve_data_ = solver::SolveData();
    }
 
    void ScalarVarForm::init(const std::string &formulation, const Units &units, const json &args, const std::string &out_path)
    {
        for (const std::string field : {"pressure_boundary", "pressure_cavity"})
        {
            const json::json_pointer path("/boundary_conditions/" + field);
            if (args.contains(path) && !args.at(path).empty())
                log_and_throw_error("{} does not support boundary_conditions.{}.", formulation, field);
        }
 
        VarForm::init(formulation, units, args, out_path);
        const bool is_time_dependent = args.contains("time") && !args["time"].is_null();
 
        primary_assembler_ = assembler::AssemblerUtils::make_assembler(formulation);
        assert(primary_assembler_->name() == formulation);
        assert(primary_assembler_->is_linear());
        assert(!primary_assembler_->is_tensor());
        mass_assembler_ = std::make_shared<assembler::Mass>();
        pure_mass_assembler_ = std::make_shared<assembler::HRZMass>();
 
        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, resources_);
 
            auto bc = args["boundary_conditions"];
            problem->set_parameters(bc, resources_);
            problem->set_parameters(args["initial_conditions"], resources_);
            problem->set_parameters(args["output"], resources_);
        }
        else
        {
            problem = problem::ProblemFactory::factory().get_problem(args["preset_problem"]["type"]);
            problem->clear();
            problem->set_parameters(args["preset_problem"], resources_);
        }
 
        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 ScalarVarForm::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);
    }
 
    void ScalarVarForm::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"], 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_);
 
        boundary_.clear_boundary_conditions();
 
        problem->update_nodes(space_.space_in_node_to_node);
        mesh.update_nodes(space_.space_in_node_to_node);
 
        problem->setup_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            /*fe_space_id=*/-1,
            space_.basis_list(),
            boundary_.total_local_boundary,
            boundary_.local_boundary,
            boundary_.boundary_nodes,
            /*value_dim=*/1);
        std::vector<int> unused_neumann_boundary_nodes;
        problem->setup_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            /*fe_space_id=*/-1,
            space_.basis_list(),
            boundary_.total_local_boundary,
            boundary_.local_neumann_boundary,
            unused_neumann_boundary_nodes,
            /*value_dim=*/1);
 
        problem->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            /*fe_space_id=*/-1,
            space_.n_bases,
            boundary_.dirichlet_nodes);
        problem->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            /*fe_space_id=*/-1,
            space_.n_bases,
            boundary_.neumann_nodes);
 
        for (const int n_id : boundary_.dirichlet_nodes)
        {
            const int tag = mesh.get_node_id(n_id);
            if (problem->is_nodal_dimension_dirichlet(n_id, tag, 0))
                boundary_.boundary_nodes.push_back(n_id);
        }
 
        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);
 
        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);
        }
    }
 
    void ScalarVarForm::build_rhs_assembler()
    {
        json rhs_solver_params = args["solver"]["linear"];
        if (!rhs_solver_params.contains("Pardiso"))
            rhs_solver_params["Pardiso"] = {};
        rhs_solver_params["Pardiso"]["mtype"] = -2;
 
        rhs_assembler_ = std::make_shared<assembler::RhsAssembler>(
            *primary_assembler_, *mesh_, nullptr,
            boundary_.dirichlet_nodes, boundary_.neumann_nodes,
            boundary_.dirichlet_nodes_position, boundary_.neumann_nodes_position,
            space_.n_bases, /*size=*/1, space_.basis_list(), space_.geometry_basis_list(), mass_ass_vals_cache_, *problem, resources_,
            args["space"]["advanced"]["bc_method"],
            rhs_solver_params,
            /*fe_space_id=*/-1);
        solve_data_.rhs_assembler = rhs_assembler_;
    }
 
    void ScalarVarForm::assemble_rhs(const mesh::Mesh &mesh)
    {
        igl::Timer timer;
        json p_params = {};
        p_params["formulation"] = primary_assembler_->name();
        {
            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, resources_);
 
        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);
    }
 
    void ScalarVarForm::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);
 
        assert(mass_.size() > 0);
 
        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_ /= mass_.rows();
        logger().info("average mass {}", avg_mass_);
 
        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();
        logger().info("sparsity: {}/{}", stats.nn_zero, stats.mat_size);
    }
 
    void ScalarVarForm::prepare_initial_solution(Eigen::MatrixXd &solution) const
    {
        if (solution.size() > 0)
            return;
        assert(rhs_assembler_ != nullptr);
        if (problem->is_time_dependent())
            rhs_assembler_->initial_solution(solution);
        else
        {
            solution.resize(rhs_.size(), 1);
            solution.setZero();
        }
    }
 
    void ScalarVarForm::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 int primary_size = space_.n_bases;
        const Eigen::MatrixXd stats_solution =
            solution.rows() >= primary_size
                ? solution.topRows(primary_size).eval()
                : solution;
 
        nlohmann::json j;
        stats.save_json(
            args, space_.n_bases, /*n_auxiliary_bases=*/0,
            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 ScalarVarForm::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 ScalarVarForm::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"];
 
        stats.compute_errors(space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), *mesh_, *problem, tend, solution);
        return stats;
    }
 
    void ScalarVarForm::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);
 
        const std::string solution_path = resolve_output_path(args["output"]["data"]["solution"]);
        if (!solution_path.empty())
        {
            const int primary_ndof = std::min<int>(solution.rows(), space_.n_bases);
            const Eigen::MatrixXd primary_solution = solution.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_, solution, tend,
                stress, mises);
            if (!stress_path.empty())
                io::write_matrix(stress_path, stress);
            if (!mises_path.empty())
                io::write_matrix(mises_path, mises);
        }
    }
 
    std::vector<io::OutputField> ScalarVarForm::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;
 
        assert(problem->is_scalar());
        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>(solution.rows(), space_.n_bases);
        const Eigen::MatrixXd primary_solution = solution.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(
                    solution, 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});
        }
 
        return fields;
    }
 
    void ScalarVarForm::build_stiffness_mat(StiffnessMatrix &stiffness)
    {
        igl::Timer timer;
        timer.start();
        logger().info("Assembling stiffness mat...");
        assert(primary_assembler_->is_linear());
        assert(problem->is_scalar());
 
        primary_assembler_->assemble(mesh_->is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), ass_vals_cache_, 0, 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();
        logger().info("sparsity: {}/{}", stats.nn_zero, stats.mat_size);
 
        write_matrix_market(args, stiffness);
    }
 
    void ScalarVarForm::solve_linear_system(
        const std::unique_ptr<polysolve::linear::Solver> &solver,
        StiffnessMatrix &A,
        Eigen::VectorXd &b,
        const bool compute_spectrum,
        Eigen::MatrixXd &sol)
    {
        assert(primary_assembler_->is_linear());
        assert(problem->is_scalar());
        assert(rhs_assembler_ != nullptr);
 
        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_problem_mixed=*/false,
            /*use_avg_pressure=*/false);
 
        sol = x;
        solver->get_info(stats.solver_info);
 
        const auto error = (A * x - b).norm();
        if (error > 1e-4)
            logger().error("Solver error: {}", error);
        else
            logger().debug("Solver error: {}", error);
    }
 
    void ScalarVarForm::solve_linear_system_with_constraints(
        const std::unique_ptr<polysolve::linear::Solver> &solver,
        StiffnessMatrix &A,
        Eigen::VectorXd &b,
        const bool compute_spectrum,
        const QuadratureOrders &boundary_samples,
        const double time,
        Eigen::MatrixXd &sol)
    {
        const json &periodic_conditions = args["boundary_conditions"]["periodic"];
        const json &zero_mean = args["constraints"]["zero_mean"];
        const bool add_zero_mean =
            zero_mean.is_boolean()
                ? zero_mean.get<bool>()
                : (zero_mean.is_array()
                   && std::find(zero_mean.begin(), zero_mean.end(), 0) != zero_mean.end());
        const bool has_global_constraints = !periodic_conditions.empty() || add_zero_mean;
 
        if (!has_global_constraints)
        {
            solve_linear_system(solver, A, b, compute_spectrum, sol);
            return;
        }
 
        if (!zero_mean.is_boolean() && !zero_mean.is_array())
            log_and_throw_error("constraints.zero_mean must be a boolean or a list of FE-space IDs");
 
        StiffnessMatrix constraint_mass = mass_;
        if (constraint_mass.rows() != A.rows() || constraint_mass.cols() != A.cols())
        {
            mass_assembler_->assemble(
                mesh_->is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(),
                mass_ass_vals_cache_, 0, constraint_mass, true);
        }
        if (constraint_mass.rows() != A.rows() || constraint_mass.cols() != A.cols())
            log_and_throw_error("Unable to assemble scalar constraint mass matrix for {} DoFs", A.rows());
 
        std::vector<std::shared_ptr<solver::AugmentedLagrangianForm>> constraint_forms;
        if (!boundary_.boundary_nodes.empty())
        {
            constraint_forms.push_back(std::make_shared<solver::BCLagrangianForm>(
                A.rows(), boundary_.boundary_nodes, boundary_.local_boundary,
                boundary_.local_neumann_boundary, boundary_samples, constraint_mass,
                *rhs_assembler_, /*obstacle_ndof=*/0, problem->is_time_dependent(), time));
        }
 
        for (const json &condition : periodic_conditions)
        {
            const int fe_space = condition.value("fe_space", -1);
            if (fe_space >= 0 && fe_space != 0)
                continue;
 
            const std::array<int, 2> boundary_ids = {{condition["boundary_ids"][0].get<int>(),
                                                      condition["boundary_ids"][1].get<int>()}};
            constraint_forms.push_back(std::make_shared<solver::PeriodicBoundaryLagrangianForm>(
                A.rows(), /*value_dim=*/1, *mesh_, space_.basis_list(),
                boundary_.total_local_boundary, boundary_ids,
                condition.value("tolerance", 1e-5)));
        }
 
        if (add_zero_mean)
        {
            const Eigen::VectorXd weights = constraint_mass * Eigen::VectorXd::Ones(A.rows());
            const double weight_sum = weights.cwiseAbs().sum();
            if (weight_sum <= 0)
                log_and_throw_error("Unable to assemble a scalar zero-mean constraint");
 
            std::vector<Eigen::Triplet<double>> entries;
            entries.reserve(weights.size());
            for (int dof = 0; dof < weights.size(); ++dof)
                if (weights(dof) != 0)
                    entries.emplace_back(0, dof, weights(dof) / weight_sum);
 
            StiffnessMatrix C(1, A.rows());
            C.setFromTriplets(entries.begin(), entries.end());
            constraint_forms.push_back(std::make_shared<solver::MatrixLagrangianForm>(
                C, Eigen::MatrixXd::Zero(1, 1)));
        }
 
        if (constraint_forms.empty())
        {
            solve_linear_system(solver, A, b, compute_spectrum, sol);
            return;
        }
 
        auto constraint_solver = polysolve::linear::Solver::create(args["solver"]["linear"], logger());
        std::shared_ptr<polysolve::linear::Solver> shared_constraint_solver(std::move(constraint_solver));
        solver::NLProblem constrained_problem(
            A.rows(), time, {}, constraint_forms, shared_constraint_solver,
            /*char_length=*/1, /*char_force=*/1, constraint_mass, /*dimension=*/1);
 
        const StiffnessMatrix full_A = A;
        const Eigen::VectorXd affine_offset =
            constrained_problem.reduced_to_full(Eigen::VectorXd::Zero(constrained_problem.reduced_size()));
        b = constrained_problem.full_to_reduced_grad(b - full_A * affine_offset);
        constrained_problem.full_hessian_to_reduced_hessian(A);
 
        Eigen::VectorXd reduced_solution;
        stats.spectrum = dirichlet_solve(
            *solver, A, b, {}, reduced_solution, A.rows(),
            args["output"]["data"]["stiffness_mat"],
            compute_spectrum,
            /*is_problem_mixed=*/false, /*use_avg_pressure=*/false);
        sol = constrained_problem.reduced_to_full(reduced_solution);
        solver->get_info(stats.solver_info);
 
        const double error = (A * reduced_solution - b).norm();
        if (error > 1e-4)
            logger().error("Solver error: {}", error);
        else
            logger().debug("Solver error: {}", error);
    }
 
    void ScalarVarForm::solve_static(Eigen::MatrixXd &sol, const ForwardStepCallback &post_step)
    {
        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_with_constraints(
            solver, A, b, args["output"]["advanced"]["spectrum"],
            boundary_samples, /*time=*/1, sol);
 
        // Static scalar solves do not use solver forms, but the current adjoint
        // implementation requires them through SolveData to compute force derivatives.
        // TODO: Remove this hack
        solve_data_.elastic_form = std::make_shared<solver::ElasticForm>(
            space_.n_bases, *space_.bases, space_.geometry_basis_list(),
            *primary_assembler_, ass_vals_cache_, 0, 0, mesh_->is_volume(),
            args["solver"]["advanced"]["jacobian_threshold"],
            args["solver"]["advanced"]["check_inversion"],
            args["solver"]["advanced"]["conservative_max_iter"]);
        solve_data_.body_form = std::make_shared<solver::BodyForm>(
            space_.ndof(), 0, boundary_.boundary_nodes, boundary_.local_boundary,
            boundary_.local_neumann_boundary, boundary_samples, rhs_, *rhs_assembler_,
            mass_assembler_->density(), false, false);
        solve_data_.body_form->update_quantities(0, sol);
        if (post_step)
            post_step(0, sol);
    }
 
    void ScalarVarForm::solve_transient(Eigen::MatrixXd &sol, const ForwardStepCallback &post_step)
    {
        assert(problem->is_time_dependent());
        assert(rhs_assembler_ != nullptr);
 
        auto solver = polysolve::linear::Solver::create(args["solver"]["linear"], logger());
        logger().info("{}...", solver->name());
 
        auto bdf = time_integrator::ImplicitTimeIntegrator::construct_bdf_integrator(
            args["time"]["integrator"]);
        bdf->init(sol, Eigen::VectorXd::Zero(sol.size()), Eigen::VectorXd::Zero(sol.size()), dt);
        time_integrator = bdf;
        restore_checkpoint_integrator(
            time_integrator, "/checkpoint/state/primary_integrator", dt,
            "primary", space_, 1);
 
        save_timestep(t0, 0, t0, dt, sol);
        if (post_step)
            post_step(0, sol);
 
        Eigen::MatrixXd current_rhs = rhs_;
 
        StiffnessMatrix stiffness;
        build_stiffness_mat(stiffness);
 
        const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
        const QuadratureOrders n_b_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(), n_b_samples,
                boundary_.local_neumann_boundary, rhs_, time, current_rhs);
 
            rhs_assembler_->set_bc(
                boundary_.local_boundary, boundary_.boundary_nodes, n_b_samples, boundary_.local_neumann_boundary, current_rhs, sol, time);
 
            StiffnessMatrix A = mass_ / bdf->beta_dt() + stiffness;
            Eigen::VectorXd b = (mass_ * bdf->weighted_sum_x_prevs()) / bdf->beta_dt();
            for (int i : boundary_.boundary_nodes)
                b[i] = 0;
            b += current_rhs;
 
            solve_linear_system_with_constraints(
                solver, A, b,
                args["output"]["advanced"]["spectrum"].get<bool>() && t == time_steps,
                n_b_samples, time, sol);
            if (post_step)
                post_step(t, sol);
 
            bdf->update_quantities(sol);
            save_timestep(time, t, t0, dt, sol);
            save_step_state(t0, dt, t, sol, time_integrator.get());
 
            logger().info("{}/{}  t={}", t, time_steps, time);
            notify_time_step(t, time_steps, t0, dt);
        }
    }
 
    void ScalarVarForm::solve_problem(
        Eigen::MatrixXd &sol,
        const InitialConditionOverride *initial_condition_override,
        const ForwardStepCallback &post_step)
    {
        assert(
            (!initial_condition_override
             || (initial_condition_override->velocity.size() == 0
                 && initial_condition_override->acceleration.size() == 0))
            && "Scalar formulations do not accept initial velocity or acceleration overrides");
 
        stats.spectrum.setZero();
 
        igl::Timer timer;
        timer.start();
        logger().info("Solving {}", primary_assembler_->name());
 
        {
            POLYFEM_SCOPED_TIMER("Setup RHS");
 
            if (initial_condition_override && initial_condition_override->solution.size() != 0)
            {
                sol = initial_condition_override->solution;
                assert(
                    sol.rows() == space_.ndof() && sol.cols() == 1
                    && "Scalar initial solution override must match the simulation DOFs and have one column");
            }
            else if (sol.size() <= 0)
                prepare_initial_solution(sol);
 
            if (sol.cols() > 1)
                sol.conservativeResize(Eigen::NoChange, 1);
        }
 
        time_integrator = nullptr;
        if (problem->is_time_dependent())
            solve_transient(sol, post_step);
        else
            solve_static(sol, post_step);
 
        timer.stop();
        timings.solving_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.solving_time);
    }
} // namespace polyfem::varform