ElasticVarForm.cpp

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#include "ElasticVarForm.hpp"
 
#include <polyfem/assembler/ElementAssemblyValues.hpp>
 
#include <polyfem/assembler/MultiModel.hpp>
 
#include <polyfem/autogen/auto_p_bases.hpp>
#include <polyfem/autogen/auto_q_bases.hpp>
#include <polyfem/autogen/prism_bases.hpp>
 
#include <polyfem/basis/Basis.hpp>
 
#include <polyfem/io/MshWriter.hpp>
#include <polyfem/io/MatrixIO.hpp>
#include <polyfem/io/Evaluator.hpp>
 
#include <polyfem/mesh/Obstacle.hpp>
 
#include <polyfem/problem/KernelProblem.hpp>
#include <polyfem/problem/ProblemFactory.hpp>
 
#include <polyfem/refinement/APriori.hpp>
 
#include <polyfem/solver/forms/ContactForm.hpp>
 
#include <polyfem/time_integrator/ImplicitTimeIntegrator.hpp>
#include <polyfem/utils/BoundarySampler.hpp>
#include <polyfem/utils/Jacobian.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
 
#include <algorithm>
#include <map>
#include <ostream>
 
#include <igl/Timer.h>
 
#include <spdlog/fmt/fmt.h>
 
namespace polyfem::varform
{
    void ElasticVarForm::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 ElasticVarForm::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_,
            mesh_->dimension(), 0, solution);
    }
 
    void ElasticVarForm::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;
    }
 
    void ElasticVarForm::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();
 
        primary_assembler_ = assembler::AssemblerUtils::make_assembler(formulation);
        assert(primary_assembler_->name() == formulation);
        if (args["solver"]["advanced"]["check_inversion"] == "Conservative")
        {
            if (auto elastic_assembler = std::dynamic_pointer_cast<assembler::ElasticityAssembler>(primary_assembler_))
                elastic_assembler->set_use_robust_jacobian();
        }
        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::GenericTensorProblem>("GenericTensor");
 
            problem->clear();
            json tmp;
            tmp["is_time_dependent"] = is_time_dependent;
            problem->set_parameters(tmp, resources_);
 
            // important for the BC
 
            auto bc = args["boundary_conditions"];
            problem->set_parameters(bc, resources_);
            problem->set_parameters(args["initial_conditions"], resources_);
            problem->set_parameters(args["output"], resources_);
        }
        else
        {
            if (args["preset_problem"]["type"] == "Kernel")
            {
                problem = std::make_shared<problem::KernelProblem>("Kernel", *primary_assembler_);
                problem->clear();
                problem::KernelProblem &kprob = *dynamic_cast<problem::KernelProblem *>(problem.get());
            }
            else
            {
                problem = problem::ProblemFactory::factory().get_problem(args["preset_problem"]["type"]);
                problem->clear();
            }
            // important for the BC
            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 ElasticVarForm::load_mesh(const mesh::Mesh &mesh, const json &args)
    {
        set_materials(*primary_assembler_, mesh.dimension());
        set_materials(*mass_assembler_, mesh.dimension());
        pure_mass_assembler_->set_size(mass_assembler_->size());
 
        problem->init(mesh);
 
        if (assembler::MultiModel *mm = dynamic_cast<assembler::MultiModel *>(primary_assembler_.get()))
        {
            assert(args["materials"].is_array());
 
            std::vector<std::string> materials(mesh.n_elements());
 
            std::map<int, std::string> mats;
 
            for (const auto &m : args["materials"])
                mats[m["id"].get<int>()] = m["type"];
 
            for (int i = 0; i < materials.size(); ++i)
                materials[i] = mats.at(mesh.get_body_id(i));
 
            mm->init_multimodels(materials);
        }
    }
 
    void ElasticVarForm::build_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args)
    {
        build_elastic_basis(mesh, iso_parametric, args, -1);
    }
 
    void ElasticVarForm::build_elastic_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args, const int fe_space_id)
    {
        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"], fe_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_,
            mesh.dimension(),
            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);
 
        boundary_.clear_boundary_conditions();
        problem->setup_bc(
            mesh, assembler::BoundaryKind::Dirichlet, fe_space_id,
            space_.basis_list(), boundary_.total_local_boundary,
            boundary_.local_boundary, boundary_.boundary_nodes, mesh.dimension());
        problem->setup_bc(
            mesh, assembler::BoundaryKind::Neumann, fe_space_id,
            boundary_.total_local_boundary, boundary_.local_neumann_boundary);
 
        problem->setup_nodal_bc(
            mesh, assembler::BoundaryKind::Dirichlet, fe_space_id, space_.n_bases, boundary_.dirichlet_nodes);
        problem->setup_nodal_bc(
            mesh, assembler::BoundaryKind::Neumann, fe_space_id, space_.n_bases, boundary_.neumann_nodes);
 
        for (const int n_id : boundary_.dirichlet_nodes)
        {
            const int tag = mesh.get_node_id(n_id);
            for (int d = 0; d < mesh.dimension(); ++d)
                if (problem->is_nodal_dimension_dirichlet(n_id, tag, d, fe_space_id))
                    boundary_.boundary_nodes.push_back(n_id * mesh.dimension() + d);
        }
        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 ElasticVarForm::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, mesh_->dimension(), 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);
    }
 
    void ElasticVarForm::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 ElasticVarForm::assemble_mass_mat(const mesh::Mesh &mesh, const json &args)
    {
        if (!problem->is_time_dependent())
        {
            avg_mass_ = 1;
            timings.assembling_mass_mat_time = 0;
            if (!primary_assembler_->is_linear())
                pure_mass_assembler_->assemble(mesh.is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), pure_mass_ass_vals_cache_, 0, pure_mass_, true);
            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);
        if (!primary_assembler_->is_linear())
            pure_mass_assembler_->assemble(mesh.is_volume(), space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), pure_mass_ass_vals_cache_, 0, pure_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 ElasticVarForm::initial_velocity(
        Eigen::MatrixXd &velocity,
        const InitialConditionOverride *override,
        const std::string &state_prefix) const
    {
        assert(rhs_assembler_ != nullptr);
        if (override && override->velocity.size() != 0)
        {
            velocity = override->velocity;
            assert(
                velocity.rows() == space_.ndof() && velocity.cols() >= 1
                && "Initial velocity override must match the simulation DOFs");
            return;
        }
 
        rhs_assembler_->initial_velocity(velocity);
    }
 
    void ElasticVarForm::initial_acceleration(
        Eigen::MatrixXd &acceleration,
        const InitialConditionOverride *override,
        const std::string &state_prefix) const
    {
        assert(rhs_assembler_ != nullptr);
        if (override && override->acceleration.size() != 0)
        {
            acceleration = override->acceleration;
            assert(
                acceleration.rows() == space_.ndof() && acceleration.cols() >= 1
                && "Initial acceleration override must match the simulation DOFs");
            return;
        }
 
        rhs_assembler_->initial_acceleration(acceleration);
    }
 
    void ElasticVarForm::initial_solution(
        Eigen::MatrixXd &solution,
        const InitialConditionOverride *override,
        const std::string &state_prefix) const
    {
        assert(rhs_assembler_ != nullptr);
        if (override && override->solution.size() != 0)
        {
            solution = override->solution;
            assert(
                solution.rows() == space_.ndof() && solution.cols() >= 1
                && "Initial solution override must match the simulation DOFs");
            return;
        }
 
        if (problem->is_time_dependent())
            rhs_assembler_->initial_solution(solution);
        else
        {
            solution.resize(rhs_.size(), 1);
            solution.setZero();
        }
    }
 
    QuadratureOrders ElasticVarForm::elastic_boundary_samples() const
    {
        const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
        return n_boundary_samples(space_.disc_orders.maxCoeff(), space_.disc_ordersq.maxCoeff(), gdiscr_order);
    }
 
    std::vector<int> ElasticVarForm::elastic_primitive_to_node() const
    {
        if (!mesh_ || !space_.geometry)
            return {};
 
        const auto &nodes = space_.is_iso_parametric() ? space_.mesh_nodes : space_.geometry->mesh_nodes;
        if (!nodes)
            return {};
 
        auto indices = nodes->primitive_to_node();
        indices.resize(mesh_->n_vertices());
        return indices;
    }
 
    std::vector<int> ElasticVarForm::elastic_node_to_primitive() const
    {
        const auto p2n = elastic_primitive_to_node();
        const int n_geometry_bases = space_.geometry ? space_.geometry->n_bases : 0;
        std::vector<int> indices(n_geometry_bases, -1);
        for (int i = 0; i < int(p2n.size()); ++i)
        {
            if (p2n[i] >= 0 && p2n[i] < int(indices.size()))
                indices[p2n[i]] = i;
        }
        return indices;
    }
 
    std::vector<io::OutputField> ElasticVarForm::elastic_output_fields(
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution,
        const io::OutputFieldOptions &options,
        const mesh::Obstacle *obstacle,
        const time_integrator::ImplicitTimeIntegrator *time_integrator,
        const std::vector<std::pair<std::string, std::shared_ptr<solver::Form>>> &named_forms,
        const solver::Form *elastic_form,
        const solver::ContactForm *contact_form) const
    {
        std::vector<io::OutputField> fields;
        if (!mesh_ || !problem || solution.size() <= 0)
            return fields;
 
        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 actual_dim = problem->is_scalar() ? 1 : mesh_->dimension();
        const auto &paraview_options = args["output"]["paraview"]["options"];
        const bool material_params = paraview_options["material"];
        const bool body_ids = paraview_options["body_ids"];
        const bool velocity = paraview_options["velocity"];
        const bool acceleration = paraview_options["acceleration"];
        const bool forces = paraview_options["forces"] && !problem->is_scalar();
        const bool tensor_values = paraview_options["tensor_values"] && !problem->is_scalar();
        const bool scalar_values = paraview_options["scalar_values"];
        const bool use_spline = args["space"]["basis_type"] == "Spline";
        const bool explicit_fields = !options.fields.empty();
 
        const auto resize_to_output_rows = [&](Eigen::MatrixXd &values) {
            if (output_rows <= values.rows())
                return;
 
            const int previous_rows = values.rows();
            values.conservativeResize(output_rows, values.cols());
            values.bottomRows(output_rows - previous_rows).setZero();
        };
 
        const auto append_obstacle_values = [&](Eigen::MatrixXd &sampled_values, const Eigen::MatrixXd &dof_values) -> bool {
            if (!obstacle || obstacle->n_vertices() <= 0)
            {
                resize_to_output_rows(sampled_values);
                return sample.points.rows() == 0 || sample.points.rows() == sampled_values.rows();
            }
 
            const bool has_obstacle_rows =
                sample.points.rows() == sampled_values.rows() + obstacle->n_vertices()
                && sample.points.cols() == obstacle->v().cols()
                && sample.points.bottomRows(obstacle->n_vertices()).isApprox(obstacle->v());
 
            if (!has_obstacle_rows)
                return sample.points.rows() == 0 || sample.points.rows() == sampled_values.rows();
 
            sampled_values.conservativeResize(sampled_values.rows() + obstacle->n_vertices(), sampled_values.cols());
            if (dof_values.rows() >= obstacle->ndof())
            {
                const Eigen::MatrixXd obstacle_values =
                    utils::unflatten(dof_values.bottomRows(obstacle->ndof()), sampled_values.cols());
                if (obstacle_values.rows() == obstacle->n_vertices()
                    && obstacle_values.cols() == sampled_values.cols())
                    sampled_values.bottomRows(obstacle->n_vertices()) = obstacle_values;
                else
                    sampled_values.bottomRows(obstacle->n_vertices()).setZero();
            }
            else
                sampled_values.bottomRows(obstacle->n_vertices()).setZero();
            return true;
        };
 
        const auto sample_dof_field = [&](const Eigen::MatrixXd &dof_values, const int field_dim, Eigen::MatrixXd &values) -> bool {
            if (dof_values.size() <= 0 || field_dim <= 0)
                return false;
 
            if (has_element_samples)
            {
                values.resize(sample.local_points.rows(), field_dim);
                for (int i = 0; i < sample.local_points.rows(); ++i)
                {
                    const int element_id = sample.element_ids(i);
                    if (element_id < 0)
                    {
                        values.row(i).setZero();
                        continue;
                    }
 
                    Eigen::MatrixXd local_sol, local_grad;
                    io::Evaluator::interpolate_at_local_vals(
                        *mesh_, field_dim, space_.basis_list(), space_.geometry_basis_list(),
                        element_id, sample.local_points.row(i), dof_values, local_sol, local_grad);
 
                    for (int d = 0; d < field_dim; ++d)
                        values(i, d) = local_sol(d);
                }
 
                return append_obstacle_values(values, dof_values);
            }
 
            if (sample.node_ids.size() > 0)
            {
                values.resize(sample.node_ids.size(), field_dim);
                for (int i = 0; i < sample.node_ids.size(); ++i)
                {
                    const int node_id = sample.node_ids(i);
                    for (int d = 0; d < field_dim; ++d)
                    {
                        const int dof = node_id * field_dim + d;
                        if (dof < 0 || dof >= dof_values.rows())
                            return false;
                        values(i, d) = dof_values(dof);
                    }
                }
 
                return sample.points.rows() == 0 || sample.points.rows() == values.rows();
            }
 
            return false;
        };
 
        const auto append_sampled_dof_field = [&](const std::string &name, const Eigen::MatrixXd &dof_values, const int field_dim) {
            Eigen::MatrixXd values;
            if (sample_dof_field(dof_values, field_dim, values))
                fields.push_back({name, values, io::OutputField::Association::Point});
        };
 
        const auto append_scalar_values = [&]() {
            if (!scalar_values || problem->is_scalar() || !has_element_samples)
                return;
 
            const bool wants_scalar = options.fields.empty()
                                      || options.export_field("von_mises")
                                      || options.export_field("von_mises_avg");
            if (!wants_scalar)
                return;
 
            std::vector<assembler::Assembler::NamedMatrix> point_values;
            for (int i = 0; i < sample.local_points.rows(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id < 0)
                    continue;
 
                std::vector<assembler::Assembler::NamedMatrix> local_values;
                primary_assembler_->compute_scalar_value(
                    assembler::OutputData(sample.time, element_id, space_.basis_list()[element_id], space_.geometry_basis_list()[element_id], sample.local_points.row(i), solution),
                    local_values);
 
                if (point_values.empty())
                {
                    point_values.resize(local_values.size());
                    for (int k = 0; k < local_values.size(); ++k)
                    {
                        point_values[k].first = local_values[k].first;
                        point_values[k].second.setZero(output_rows, local_values[k].second.cols());
                    }
                }
 
                for (int k = 0; k < local_values.size(); ++k)
                    point_values[k].second.row(i) = local_values[k].second;
            }
 
            for (const auto &[name, values] : point_values)
            {
                if (options.export_field(name))
                    fields.push_back({name, values, io::OutputField::Association::Point});
            }
        };
 
        const auto append_tensor_values = [&]() {
            if (!tensor_values || problem->is_scalar() || !has_element_samples)
                return;
 
            const bool wants_tensor = options.fields.empty()
                                      || options.export_field("cauchy_stess")
                                      || options.export_field("pk1_stess")
                                      || options.export_field("pk2_stess")
                                      || options.export_field("F");
            if (!wants_tensor)
                return;
 
            std::vector<assembler::Assembler::NamedMatrix> point_values;
            for (int i = 0; i < sample.local_points.rows(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id < 0)
                    continue;
 
                std::vector<assembler::Assembler::NamedMatrix> local_values;
                primary_assembler_->compute_tensor_value(
                    assembler::OutputData(sample.time, element_id, space_.basis_list()[element_id], space_.geometry_basis_list()[element_id], sample.local_points.row(i), solution),
                    local_values);
 
                if (point_values.empty())
                {
                    point_values.resize(local_values.size());
                    for (int k = 0; k < local_values.size(); ++k)
                    {
                        point_values[k].first = local_values[k].first;
                        point_values[k].second.setZero(output_rows, local_values[k].second.cols());
                    }
                }
 
                for (int k = 0; k < local_values.size(); ++k)
                    point_values[k].second.row(i) = local_values[k].second;
            }
 
            for (const auto &[name, values] : point_values)
            {
                if (!options.export_field(name))
                    continue;
 
                const int stride = mesh_->dimension();
                assert(values.cols() % stride == 0);
                for (int i = 0; i < values.cols(); i += stride)
                {
                    const int ii = (i / stride) + 1;
                    fields.push_back({fmt::format("{:s}_{:d}", name, ii), values.middleCols(i, stride), io::OutputField::Association::Point});
                }
            }
        };
 
        const auto append_averaged_values = [&]() {
            if (use_spline || problem->is_scalar() || !has_element_samples || (!scalar_values && !tensor_values))
                return;
 
            const bool wants_avg = options.fields.empty()
                                   || options.export_field("von_mises_avg")
                                   || options.export_field("cauchy_stess_avg")
                                   || options.export_field("pk1_stess_avg")
                                   || options.export_field("pk2_stess_avg")
                                   || options.export_field("F_avg");
            if (!wants_avg)
                return;
 
            Eigen::MatrixXd areas(space_.n_bases, 1);
            areas.setZero();
            std::vector<assembler::Assembler::NamedMatrix> tmp_s, tmp_t;
            std::vector<Eigen::MatrixXd> avg_scalar, avg_tensor;
 
            for (int e = 0; e < int(space_.basis_list().size()); ++e)
            {
                Eigen::MatrixXd local_pts;
                if (mesh_->is_simplex(e))
                {
                    if (mesh_->dimension() == 3)
                        autogen::p_nodes_3d(space_.disc_orders(e), local_pts);
                    else
                        autogen::p_nodes_2d(space_.disc_orders(e), local_pts);
                }
                else if (mesh_->is_cube(e))
                {
                    if (mesh_->dimension() == 3)
                        autogen::q_nodes_3d(space_.disc_orders(e), local_pts);
                    else
                        autogen::q_nodes_2d(space_.disc_orders(e), local_pts);
                }
                else if (mesh_->is_prism(e))
                {
                    autogen::prism_nodes_3d(space_.disc_orders(e), space_.disc_ordersq(e), local_pts);
                }
                else
                {
                    continue;
                }
 
                const basis::ElementBases &bs = space_.basis_list()[e];
                const basis::ElementBases &gbs = space_.geometry_basis_list()[e];
 
                assembler::ElementAssemblyValues vals;
                vals.compute(e, mesh_->is_volume(), bs, gbs);
                const double area = (vals.det.array() * vals.quadrature.weights.array()).sum();
 
                if (scalar_values)
                    primary_assembler_->compute_scalar_value(assembler::OutputData(sample.time, e, bs, gbs, local_pts, solution), tmp_s);
                if (tensor_values)
                    primary_assembler_->compute_tensor_value(assembler::OutputData(sample.time, e, bs, gbs, local_pts, solution), tmp_t);
 
                if (avg_scalar.empty() && !tmp_s.empty())
                {
                    avg_scalar.resize(tmp_s.size());
                    for (auto &m : avg_scalar)
                        m.setZero(space_.n_bases, 1);
                }
                if (avg_tensor.empty() && !tmp_t.empty())
                {
                    avg_tensor.resize(tmp_t.size());
                    for (auto &m : avg_tensor)
                        m.setZero(space_.n_bases, actual_dim * actual_dim);
                }
 
                for (size_t j = 0; j < bs.bases.size(); ++j)
                {
                    const basis::Basis &b = bs.bases[j];
                    if (b.global().size() > 1)
                        continue;
 
                    const int index = b.global().front().index;
                    areas(index) += area;
                    for (int k = 0; k < tmp_s.size(); ++k)
                        avg_scalar[k](index) += tmp_s[k].second(j) * area;
                    for (int k = 0; k < tmp_t.size(); ++k)
                        avg_tensor[k].row(index) += tmp_t[k].second.row(j) * area;
                }
            }
 
            for (auto &m : avg_scalar)
                for (int i = 0; i < m.rows(); ++i)
                    if (areas(i) > 0)
                        m(i) /= areas(i);
            for (auto &m : avg_tensor)
                for (int i = 0; i < m.rows(); ++i)
                    if (areas(i) > 0)
                        m.row(i) /= areas(i);
 
            for (int k = 0; k < tmp_s.size(); ++k)
            {
                const std::string name = fmt::format("{:s}_avg", tmp_s[k].first);
                if (!options.export_field(name))
                    continue;
 
                Eigen::MatrixXd sampled;
                if (sample_dof_field(avg_scalar[k], 1, sampled))
                    fields.push_back({name, sampled, io::OutputField::Association::Point});
            }
 
            for (int k = 0; k < tmp_t.size(); ++k)
            {
                const std::string base_name = fmt::format("{:s}_avg", tmp_t[k].first);
                if (!options.export_field(base_name))
                    continue;
 
                Eigen::MatrixXd sampled;
                if (!sample_dof_field(utils::flatten(avg_tensor[k]), actual_dim * actual_dim, sampled))
                    continue;
 
                const int stride = mesh_->dimension();
                for (int i = 0; i < sampled.cols(); i += stride)
                {
                    const int ii = (i / stride) + 1;
                    fields.push_back({fmt::format("{:s}_{:d}", base_name, ii), sampled.middleCols(i, stride), io::OutputField::Association::Point});
                }
            }
        };
 
        const auto append_material_fields = [&]() {
            if (!material_params || !has_element_samples)
                return;
 
            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});
        };
 
        const auto append_body_ids = [&]() {
            if (!body_ids || !options.export_field("body_ids") || !has_element_samples)
                return;
 
            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 auto compute_traction_forces = [&]() {
            Eigen::MatrixXd traction_forces;
            traction_forces.setZero(space_.n_bases * actual_dim, 1);
 
            Eigen::MatrixXd uv, points, normals;
            Eigen::VectorXd weights;
            Eigen::VectorXi global_primitive_ids;
            assembler::ElementAssemblyValues vals;
 
            for (const auto &lb : boundary_.total_local_boundary)
            {
                const int e = lb.element_id();
                const bool has_samples = utils::BoundarySampler::boundary_quadrature(
                    lb, elastic_boundary_samples(), *mesh_, false, uv, points, normals, weights, global_primitive_ids);
                if (!has_samples)
                    continue;
 
                const basis::ElementBases &bs = space_.basis_list()[e];
                const basis::ElementBases &gbs = space_.geometry_basis_list()[e];
                vals.compute(e, mesh_->is_volume(), points, bs, gbs);
 
                for (int n = 0; n < normals.rows(); ++n)
                {
                    Eigen::MatrixXd deform_mat = Eigen::MatrixXd::Zero(actual_dim, actual_dim);
                    for (const auto &b : vals.basis_values)
                    {
                        for (const auto &g : b.global)
                        {
                            for (int d = 0; d < actual_dim; ++d)
                                deform_mat.row(d) += solution(g.index * actual_dim + d) * b.grad.row(n);
                        }
                    }
 
                    Eigen::MatrixXd trafo = vals.jac_it[n].inverse() + deform_mat;
                    normals.row(n) = normals.row(n) * trafo.inverse();
                    normals.row(n).normalize();
                }
 
                std::vector<assembler::Assembler::NamedMatrix> tensor_flat;
                primary_assembler_->compute_tensor_value(assembler::OutputData(sample.time, e, bs, gbs, points, solution), tensor_flat);
 
                for (long n = 0; n < vals.basis_values.size(); ++n)
                {
                    const assembler::AssemblyValues &v = vals.basis_values[n];
                    const int g_index = v.global[0].index * actual_dim;
                    for (int q = 0; q < points.rows(); ++q)
                    {
                        assert(tensor_flat[0].first == "cauchy_stess");
                        Eigen::MatrixXd stress_tensor = utils::unflatten(tensor_flat[0].second.row(q), actual_dim);
                        traction_forces.block(g_index, 0, actual_dim, 1) += stress_tensor * normals.row(q).transpose() * v.val(q) * weights(q);
                    }
                }
            }
 
            return traction_forces;
        };
 
        const auto append_traction_force = [&]() {
            if (problem->is_scalar() || !explicit_fields || !options.export_field("traction_force"))
                return;
 
            if (has_element_samples && sample.normals.rows() == sample.local_points.rows() && sample.primitive_ids.size() == sample.local_points.rows())
            {
                const Eigen::MatrixXd displaced_normals = displaced_output_normals(sample, solution);
                const Eigen::MatrixXd &normals = displaced_normals.rows() == sample.normals.rows() ? displaced_normals : sample.normals;
                Eigen::MatrixXd values = Eigen::MatrixXd::Zero(output_rows, actual_dim);
                for (int i = 0; i < sample.local_points.rows(); ++i)
                {
                    const int element_id = sample.element_ids(i);
                    if (element_id < 0)
                        continue;
 
                    std::vector<assembler::Assembler::NamedMatrix> tensor_flat;
                    primary_assembler_->compute_tensor_value(
                        assembler::OutputData(sample.time, element_id, space_.basis_list()[element_id], space_.geometry_basis_list()[element_id], sample.local_points.row(i), solution),
                        tensor_flat);
 
                    assert(tensor_flat[0].first == "cauchy_stess");
                    Eigen::Map<Eigen::MatrixXd> tensor(tensor_flat[0].second.data(), actual_dim, actual_dim);
                    values.row(i) = normals.row(i) * tensor;
 
                    double area = 0;
                    const int primitive_id = sample.primitive_ids(i);
                    if (mesh_->is_volume())
                    {
                        if (mesh_->is_simplex(element_id))
                            area = mesh_->tri_area(primitive_id);
                        else if (mesh_->is_cube(element_id))
                            area = mesh_->quad_area(primitive_id);
                        else if (mesh_->is_prism(element_id))
                            area = mesh_->n_face_vertices(primitive_id) == 4 ? mesh_->quad_area(primitive_id) : mesh_->tri_area(primitive_id);
                    }
                    else
                    {
                        area = mesh_->edge_length(primitive_id);
                    }
                    values.row(i) *= area;
                }
                fields.push_back({"traction_force", values, io::OutputField::Association::Point});
                return;
            }
 
            append_sampled_dof_field("traction_force", compute_traction_forces(), actual_dim);
        };
 
        append_scalar_values();
        append_tensor_values();
        append_averaged_values();
        append_material_fields();
        append_body_ids();
 
        if ((paraview_options["jacobian_validity"] || (explicit_fields && options.export_field("validity")))
            && has_element_samples
            && sample.primitive_ids.size() == 0)
        {
            const auto invalid_elements = utils::count_invalid(
                mesh_->dimension(), space_.basis_list(), space_.geometry_basis_list(), 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});
        }
 
        if (problem->is_time_dependent())
        {
            if (velocity && options.export_field("velocity"))
                append_sampled_dof_field(
                    "velocity",
                    time_integrator ? time_integrator->v_prev() : Eigen::VectorXd::Zero(solution.size()),
                    actual_dim);
            if (acceleration && options.export_field("acceleration"))
                append_sampled_dof_field(
                    "acceleration",
                    time_integrator ? time_integrator->a_prev() : Eigen::VectorXd::Zero(solution.size()),
                    actual_dim);
        }
 
        if (forces)
        {
            const double s = time_integrator ? time_integrator->acceleration_scaling() : 1;
            for (const auto &[name, form] : named_forms)
            {
                const std::string field_name = name + "_forces";
                if (!options.export_field(field_name))
                    continue;
 
                Eigen::VectorXd force;
                if (form && form->enabled())
                {
                    form->first_derivative(solution, force);
                    force *= -1.0 / s;
                }
                else
                {
                    force.setZero(solution.size());
                }
                append_sampled_dof_field(field_name, force, actual_dim);
            }
        }
 
        append_traction_force();
 
        if (explicit_fields && options.export_field("gradient_of_elastic_potential") && elastic_form)
        {
            Eigen::VectorXd potential_grad;
            elastic_form->first_derivative(solution, potential_grad);
            append_sampled_dof_field("gradient_of_elastic_potential", potential_grad, actual_dim);
        }
 
        if (explicit_fields && options.export_field("gradient_of_contact_potential") && contact_form && contact_form->weight() > 0)
        {
            Eigen::VectorXd potential_grad;
            contact_form->first_derivative(solution, potential_grad);
            potential_grad *= -contact_form->barrier_stiffness() / contact_form->weight();
            append_sampled_dof_field("gradient_of_contact_potential", potential_grad, actual_dim);
        }
 
        append_primary_output_fields(fields, sample, solution, options, obstacle);
        return fields;
    }
 
    void ElasticVarForm::append_primary_output_fields(
        std::vector<io::OutputField> &fields,
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution,
        const io::OutputFieldOptions &options,
        const mesh::Obstacle *obstacle) const
    {
        if (!mesh_ || solution.size() <= 0)
            return;
 
        const int dim = mesh_->dimension();
        const bool has_element_samples =
            sample.local_points.rows() > 0
            && sample.local_points.rows() == sample.element_ids.size();
        const bool export_solution_gradient =
            !options.fields.empty() && options.export_field("solution_gradient");
 
        Eigen::MatrixXd values, gradients;
        if (has_element_samples)
        {
            values.resize(sample.local_points.rows(), dim);
            if (export_solution_gradient)
                gradients.resize(sample.local_points.rows(), dim * 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.row(i).setZero();
                    if (gradients.rows() > 0)
                        gradients.row(i).setZero();
                    continue;
                }
 
                Eigen::MatrixXd local_value, local_gradient;
                io::Evaluator::interpolate_at_local_vals(
                    *mesh_, dim, space_.basis_list(), space_.geometry_basis_list(),
                    element_id, sample.local_points.row(i), solution,
                    local_value, local_gradient);
                values.row(i) = local_value;
                if (gradients.rows() > 0)
                    gradients.row(i) = local_gradient;
            }
 
            if (obstacle && obstacle->n_vertices() > 0
                && sample.points.rows() == values.rows() + obstacle->n_vertices()
                && sample.points.cols() == obstacle->v().cols()
                && sample.points.bottomRows(obstacle->n_vertices()).isApprox(obstacle->v()))
            {
                values.conservativeResize(values.rows() + obstacle->n_vertices(), Eigen::NoChange);
                if (solution.rows() >= obstacle->ndof())
                    values.bottomRows(obstacle->n_vertices()) =
                        utils::unflatten(solution.bottomRows(obstacle->ndof()), dim);
                else
                    values.bottomRows(obstacle->n_vertices()).setZero();
                if (gradients.rows() > 0)
                {
                    gradients.conservativeResize(values.rows(), Eigen::NoChange);
                    gradients.bottomRows(obstacle->n_vertices()).setZero();
                }
            }
        }
        else if (sample.node_ids.size() > 0)
        {
            values.resize(sample.node_ids.size(), dim);
            for (int i = 0; i < sample.node_ids.size(); ++i)
            {
                for (int d = 0; d < dim; ++d)
                {
                    const int dof = sample.node_ids(i) * dim + d;
                    if (dof < 0 || dof >= solution.rows())
                        return;
                    values(i, d) = solution(dof);
                }
            }
        }
        else
        {
            return;
        }
 
        if (sample.points.rows() > 0 && values.rows() != sample.points.rows())
            return;
        if (options.export_field("displacement"))
            fields.push_back({"displacement", values, io::OutputField::Association::Point});
        if (options.export_field("solution"))
            fields.push_back({"solution", values, io::OutputField::Association::Point});
        if (export_solution_gradient && gradients.rows() == values.rows())
            fields.push_back({"solution_gradient", gradients, io::OutputField::Association::Point});
 
        if (options.export_field("displaced_normals"))
        {
            Eigen::MatrixXd normals = displaced_output_normals(sample, solution);
            if (normals.rows() == values.rows())
                fields.push_back({"displaced_normals", normals, io::OutputField::Association::Point});
        }
    }
 
    Eigen::MatrixXd ElasticVarForm::displaced_output_normals(
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution) const
    {
        if (!mesh_
            || sample.normals.rows() == 0
            || sample.normals.rows() != sample.local_points.rows()
            || sample.local_points.rows() != sample.element_ids.size())
            return {};
 
        const int dim = mesh_->dimension();
        Eigen::MatrixXd displaced_normals = sample.normals;
        for (int i = 0; i < sample.local_points.rows(); ++i)
        {
            const int element_id = sample.element_ids(i);
            if (element_id < 0)
                continue;
 
            Eigen::MatrixXd local_value, local_gradient;
            io::Evaluator::interpolate_at_local_vals(
                *mesh_, dim, space_.basis_list(), space_.geometry_basis_list(),
                element_id, sample.local_points.row(i), solution,
                local_value, local_gradient);
 
            Eigen::MatrixXd deformation = Eigen::MatrixXd::Identity(dim, dim);
            for (int d = 0; d < dim; ++d)
                deformation.row(d) += local_gradient.block(0, d * dim, 1, dim);
            displaced_normals.row(i) = sample.normals.row(i) * deformation.inverse();
            displaced_normals.row(i).normalize();
        }
        return displaced_normals;
    }
 
    io::OutputSpace ElasticVarForm::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 ElasticVarForm::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 ElasticVarForm::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 dim = mesh_->dimension();
            const int primary_ndof = std::min<int>(solution.rows(), space_.n_bases * dim);
            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, dim);
                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);
        }
    }
 
    void ElasticVarForm::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 * mesh_->dimension();
        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, 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;
    }
 
    void ElasticVarForm::save_elastic_step_state(
        const double t0,
        const double dt,
        const int t,
        const Eigen::MatrixXd &solution,
        const time_integrator::ImplicitTimeIntegrator *time_integrator) const
    {
        if (!mesh_)
            return;
        save_step_state(t0, dt, t, solution, time_integrator);
    }
 
    void ElasticVarForm::build_mesh_matrices(Eigen::MatrixXd &V, Eigen::MatrixXi &F) const
    {
        assert(mesh_);
        assert(space_.basis_list().size() == mesh_->n_elements());
        const size_t n_vertices = space_.n_bases - n_obstacle_vertices();
        const int dim = mesh_->dimension();
 
        V.resize(n_vertices, dim);
        F.resize(space_.basis_list().size(), dim + 1);
 
        for (int i = 0; i < space_.basis_list().size(); i++)
        {
            const basis::ElementBases &element = space_.basis_list()[i];
            for (int j = 0; j < element.bases.size(); j++)
            {
                const basis::Basis &basis = element.bases[j];
                assert(basis.global().size() == 1);
                V.row(basis.global()[0].index) = basis.global()[0].node;
                if (j < F.cols())
                    F(i, j) = basis.global()[0].index;
            }
        }
    }
 
} // namespace polyfem::varform