VarForm.cpp

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#include <polyfem/varforms/VarForm.hpp>
 
#include <polyfem/assembler/AssemblerUtils.hpp>
 
#include <polyfem/io/MatrixIO.hpp>
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/varforms/VarFormUtils.hpp>
 
#include <polyfem/basis/LagrangeBasis2d.hpp>
#include <polyfem/basis/LagrangeBasis3d.hpp>
#include <polyfem/basis/SplineBasis2d.hpp>
#include <polyfem/basis/SplineBasis3d.hpp>
#include <polyfem/basis/barycentric/MVPolygonalBasis2d.hpp>
#include <polyfem/basis/barycentric/WSPolygonalBasis2d.hpp>
#include <polyfem/basis/PolygonalBasis2d.hpp>
#include <polyfem/basis/PolygonalBasis3d.hpp>
#include <polyfem/mesh/mesh2D/Mesh2D.hpp>
#include <polyfem/mesh/mesh3D/Mesh3D.hpp>
#include <polyfem/mesh/Obstacle.hpp>
#include <polyfem/refinement/APriori.hpp>
#include <polyfem/time_integrator/BDF.hpp>
#include <polyfem/time_integrator/ImplicitTimeIntegrator.hpp>
#include <polyfem/utils/Timer.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/Jacobian.hpp>
 
#include <igl/Timer.h>
#include <polyfem/utils/JSONUtils.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/StringUtils.hpp>
 
#include <fstream>
#include <limits>
 
#include <spdlog/fmt/fmt.h>
 
namespace polyfem::varform
{
    bool VarForm::read_initial_x_from_file(
        const std::string &state_path,
        const std::string &x_name,
        const bool reorder,
        const Eigen::VectorXi &in_node_to_node,
        const int dim,
        Eigen::MatrixXd &x)
    {
        if (state_path.empty())
            return false;
 
        if (!io::read_matrix(state_path, x_name, x))
        {
            logger().debug("Unable to read initial {} from file ({})", x_name, state_path);
            return false;
        }
 
        if (reorder)
        {
            const int ndof = in_node_to_node.size() * dim;
            x.topRows(ndof) = utils::reorder_matrix(x.topRows(ndof), in_node_to_node, -1, dim);
        }
 
        return true;
    }
 
    namespace
    {
 
        bool should_use_iso_parametric(const mesh::Mesh &mesh, const json &args)
        {
            if (mesh.has_poly())
                return true;
 
            if (args["space"]["basis_type"] == "Bernstein")
                return false;
 
            if (args["space"]["basis_type"] == "Spline")
                return true;
 
            if (mesh.is_rational())
                return false;
 
            if (args["space"]["use_p_ref"])
                return false;
 
            if (args["boundary_conditions"]["periodic_boundary"]["enabled"].get<bool>())
                return false;
 
            if (mesh.orders().size() <= 0)
            {
                if (args["space"]["discr_order"] == 1)
                    return true;
                return args["space"]["advanced"]["isoparametric"];
            }
 
            if (mesh.orders().minCoeff() != mesh.orders().maxCoeff())
                return false;
 
            if (args["space"]["discr_order"] == mesh.orders().minCoeff())
                return true;
 
            return args["space"]["advanced"]["isoparametric"];
        }
 
        void build_in_node_to_in_primitive(const mesh::Mesh &mesh, const mesh::MeshNodes &mesh_nodes,
                                           Eigen::VectorXi &in_node_to_in_primitive,
                                           Eigen::VectorXi &in_node_offset)
        {
            const int num_vertex_nodes = mesh_nodes.num_vertex_nodes();
            const int num_edge_nodes = mesh_nodes.num_edge_nodes();
            const int num_face_nodes = mesh_nodes.num_face_nodes();
            const int num_cell_nodes = mesh_nodes.num_cell_nodes();
 
            const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
 
            const long n_vertices = num_vertex_nodes;
            const int num_in_primitives = n_vertices + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
            const int num_primitives = mesh.n_vertices() + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
 
            in_node_to_in_primitive.resize(num_nodes);
            in_node_offset.resize(num_nodes);
 
            // Only one node per vertex, so this is an identity map.
            in_node_to_in_primitive.head(num_vertex_nodes).setLinSpaced(num_vertex_nodes, 0, num_vertex_nodes - 1); // vertex nodes
            in_node_offset.head(num_vertex_nodes).setZero();
 
            int prim_offset = n_vertices;
            int node_offset = num_vertex_nodes;
            auto foo = [&](const int num_prims, const int num_prim_nodes) {
                if (num_prims <= 0 || num_prim_nodes <= 0)
                    return;
                const Eigen::VectorXi range = Eigen::VectorXi::LinSpaced(num_prim_nodes, 0, num_prim_nodes - 1);
                // TODO: This assumes isotropic degree of element.
                const int node_per_prim = num_prim_nodes / num_prims;
 
                in_node_to_in_primitive.segment(node_offset, num_prim_nodes) =
                    range.array() / node_per_prim + prim_offset;
 
                in_node_offset.segment(node_offset, num_prim_nodes) =
                    range.unaryExpr([&](const int x) { return x % node_per_prim; });
 
                prim_offset += num_prims;
                node_offset += num_prim_nodes;
            };
 
            foo(mesh.n_edges(), num_edge_nodes);
            foo(mesh.n_faces(), num_face_nodes);
            foo(mesh.n_cells(), num_cell_nodes);
        }
 
        bool build_in_primitive_to_primitive(
            const mesh::Mesh &mesh, const mesh::MeshNodes &mesh_nodes,
            const Eigen::VectorXi &in_ordered_vertices,
            const Eigen::MatrixXi &in_ordered_edges,
            const Eigen::MatrixXi &in_ordered_faces,
            Eigen::VectorXi &in_primitive_to_primitive)
        {
            // NOTE: Assume in_cells_to_cells is identity
            const int num_vertex_nodes = mesh_nodes.num_vertex_nodes();
            const int num_edge_nodes = mesh_nodes.num_edge_nodes();
            const int num_face_nodes = mesh_nodes.num_face_nodes();
            const int num_cell_nodes = mesh_nodes.num_cell_nodes();
            const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
 
            const long n_vertices = num_vertex_nodes;
            const int num_in_primitives = n_vertices + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
            const int num_primitives = mesh.n_vertices() + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
 
            in_primitive_to_primitive.setLinSpaced(num_in_primitives, 0, num_in_primitives - 1);
 
            igl::Timer timer;
 
            // ------------
            // Map vertices
            // ------------
 
            if (in_ordered_vertices.rows() != n_vertices)
            {
                logger().warn("Node ordering disabled, in_ordered_vertices != n_vertices, {} != {}", in_ordered_vertices.rows(), n_vertices);
                return false;
            }
 
            in_primitive_to_primitive.head(n_vertices) = in_ordered_vertices;
 
            int in_offset = n_vertices;
            int offset = mesh.n_vertices();
 
            // ---------
            // Map edges
            // ---------
 
            logger().trace("Building Mesh edges to IDs...");
            timer.start();
            const auto edges_to_ids = mesh.edges_to_ids();
            if (in_ordered_edges.rows() != edges_to_ids.size())
            {
                logger().warn("Node ordering disabled, in_ordered_edges != edges_to_ids, {} != {}", in_ordered_edges.rows(), edges_to_ids.size());
                return false;
            }
            timer.stop();
            logger().trace("Done (took {}s)", timer.getElapsedTime());
 
            logger().trace("Building in-edge to edge mapping...");
            timer.start();
            for (int in_ei = 0; in_ei < in_ordered_edges.rows(); in_ei++)
            {
                const std::pair<int, int> in_edge(
                    in_ordered_edges.row(in_ei).minCoeff(),
                    in_ordered_edges.row(in_ei).maxCoeff());
                in_primitive_to_primitive[in_offset + in_ei] =
                    offset + edges_to_ids.at(in_edge); // offset edge ids
            }
            timer.stop();
            logger().trace("Done (took {}s)", timer.getElapsedTime());
 
            in_offset += mesh.n_edges();
            offset += mesh.n_edges();
 
            // ---------
            // Map faces
            // ---------
 
            if (mesh.is_volume())
            {
                logger().trace("Building Mesh faces to IDs...");
                timer.start();
                const auto faces_to_ids = mesh.faces_to_ids();
                if (in_ordered_faces.rows() != faces_to_ids.size())
                {
                    logger().warn("Node ordering disabled, in_ordered_faces != faces_to_ids, {} != {}", in_ordered_faces.rows(), faces_to_ids.size());
                    return false;
                }
                timer.stop();
                logger().trace("Done (took {}s)", timer.getElapsedTime());
 
                logger().trace("Building in-face to face mapping...");
                timer.start();
                for (int in_fi = 0; in_fi < in_ordered_faces.rows(); in_fi++)
                {
                    std::vector<int> in_face(in_ordered_faces.cols());
                    for (int i = 0; i < in_face.size(); i++)
                        in_face[i] = in_ordered_faces(in_fi, i);
                    std::sort(in_face.begin(), in_face.end());
 
                    in_primitive_to_primitive[in_offset + in_fi] =
                        offset + faces_to_ids.at(in_face); // offset face ids
                }
                timer.stop();
                logger().trace("Done (took {}s)", timer.getElapsedTime());
 
                in_offset += mesh.n_faces();
                offset += mesh.n_faces();
            }
 
            return true;
        }
    } // namespace
 
    QuadratureOrders VarForm::n_boundary_samples(const int discr_order, const int gdiscr_order) const
    {
        using assembler::AssemblerUtils;
        const int n_b_samples_j = args["space"]["advanced"]["n_boundary_samples"];
        const int boundary_order = std::max(discr_order, gdiscr_order);
        const int n_b_samples = std::max(n_b_samples_j, AssemblerUtils::quadrature_order("Mass", boundary_order, AssemblerUtils::BasisType::POLY, mesh_->dimension()));
        return {{n_b_samples, n_b_samples}};
    }
 
    void VarForm::reset()
    {
        // FIXME check subclasses
        stats.reset();
        timings = io::OutRuntimeData();
        output_sampler_initialized_ = false;
        prepared_ = false;
        problem = nullptr;
        time_callback = nullptr;
        mesh_ = nullptr;
    }
 
    void VarForm::init(const std::string &formulation, const Units &units, const json &args, const std::string &out_path)
    {
        reset();
 
        this->units = units;
        this->args = args;
 
        if (utils::is_param_valid(args, "root_path"))
            root_path = args["root_path"].get<std::string>();
        else
            root_path = "";
 
        this->output_path = out_path;
        output_sampler_initialized_ = false;
    }
 
    void VarForm::set_mesh(std::unique_ptr<mesh::Mesh> mesh, const double loading_mesh_time)
    {
        mesh_ = std::move(mesh);
        timings.loading_mesh_time = loading_mesh_time;
        output_sampler_initialized_ = false;
        prepared_ = false;
        if (!mesh_)
            return;
 
        load_mesh(*mesh_, args);
    }
 
    void VarForm::prepare()
    {
        if (prepared_)
            return;
        if (!mesh_)
            log_and_throw_error("Load the mesh first!");
 
        mesh_->prepare_mesh();
        stats.compute_mesh_stats(*mesh_);
        build_basis(*mesh_, should_use_iso_parametric(*mesh_, args), args);
        assemble_rhs(*mesh_);
        assemble_mass_mat(*mesh_, args);
        prepared_ = true;
    }
 
    void VarForm::build_fe_space(
        mesh::Mesh &mesh,
        const bool iso_parametric,
        const Eigen::VectorXi &disc_orders,
        const std::string &basis_type,
        const std::string &poly_basis_type,
        const assembler::Assembler &space_assembler,
        const int value_dim,
        const int quadrature_order,
        const int mass_quadrature_order,
        const bool use_corner_quadrature,
        const int n_harmonic_samples,
        const int integral_constraints,
        FESpace &space,
        VarFormBoundaryState &boundary,
        std::shared_ptr<GeometryMapping> geometry)
    {
        using namespace mesh;
 
        const std::string space_assembler_name = space_assembler.name();
        const bool build_geom_mapping = geometry == nullptr;
 
        space.reset();
        boundary.reset();
 
        space.value_dim = value_dim;
 
        space.bases = std::make_shared<std::vector<basis::ElementBases>>();
        space.geometry = build_geom_mapping ? std::make_shared<GeometryMapping>() : std::move(geometry);
        assert(space.geometry);
 
        space.disc_orders = disc_orders;
        space.disc_ordersq = disc_orders;
 
        Eigen::MatrixXi geom_disc_orders;
        if (build_geom_mapping && !iso_parametric)
        {
            if (mesh.orders().size() <= 0)
            {
                geom_disc_orders.resizeLike(space.disc_orders);
                geom_disc_orders.setConstant(1);
            }
            else
                geom_disc_orders = mesh.orders();
 
            space.geometry->bases = std::make_shared<std::vector<basis::ElementBases>>();
            space.geometry->disc_orders = geom_disc_orders;
        }
 
        Eigen::MatrixXi geom_disc_ordersq = geom_disc_orders;
 
        logger().info("Building {} basis...", (build_geom_mapping ? (iso_parametric ? "isoparametric" : "not isoparametric") : "finite-element"));
 
        igl::Timer timer;
        timer.start();
 
        const bool has_polys = mesh.has_poly();
        std::map<int, basis::InterfaceData> poly_edge_to_data_geom;
 
        const bool use_continuous_gbasis = true;
 
        if (mesh.is_volume())
        {
            const Mesh3D &tmp_mesh = dynamic_cast<const Mesh3D &>(mesh);
 
            if (basis_type == "Spline")
            {
                space.n_bases = basis::SplineBasis3d::build_bases(
                    tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                    *space.bases, boundary.local_boundary, space.poly_edge_to_data);
            }
            else
            {
                if (build_geom_mapping && !iso_parametric)
                    space.geometry->n_bases = basis::LagrangeBasis3d::build_bases(
                        tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                        geom_disc_orders, geom_disc_ordersq, false, false, has_polys,
                        !use_continuous_gbasis, use_corner_quadrature,
                        *space.geometry->bases, boundary.local_boundary, poly_edge_to_data_geom,
                        space.geometry->mesh_nodes);
 
                space.n_bases = basis::LagrangeBasis3d::build_bases(
                    tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                    space.disc_orders, space.disc_ordersq,
                    basis_type == "Bernstein",
                    basis_type == "Serendipity",
                    has_polys, false, use_corner_quadrature,
                    *space.bases, boundary.local_boundary, space.poly_edge_to_data, space.mesh_nodes);
            }
        }
        else
        {
            const Mesh2D &tmp_mesh = dynamic_cast<const Mesh2D &>(mesh);
 
            if (basis_type == "Spline")
            {
                space.n_bases = basis::SplineBasis2d::build_bases(
                    tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                    *space.bases, boundary.local_boundary, space.poly_edge_to_data);
            }
            else
            {
                if (build_geom_mapping && !iso_parametric)
                    space.geometry->n_bases = basis::LagrangeBasis2d::build_bases(
                        tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                        geom_disc_orders, false, false, has_polys,
                        !use_continuous_gbasis, use_corner_quadrature,
                        *space.geometry->bases, boundary.local_boundary, poly_edge_to_data_geom,
                        space.geometry->mesh_nodes);
 
                space.n_bases = basis::LagrangeBasis2d::build_bases(
                    tmp_mesh, space_assembler_name, quadrature_order, mass_quadrature_order,
                    space.disc_orders,
                    basis_type == "Bernstein",
                    basis_type == "Serendipity",
                    has_polys, false, use_corner_quadrature,
                    *space.bases, boundary.local_boundary, space.poly_edge_to_data, space.mesh_nodes);
            }
        }
 
        const bool use_fe_space_as_geometry = build_geom_mapping ? iso_parametric : space.is_iso_parametric();
        build_polygonal_basis(mesh, poly_basis_type, space_assembler,
                              use_fe_space_as_geometry,
                              quadrature_order,
                              mass_quadrature_order,
                              n_harmonic_samples,
                              integral_constraints,
                              space,
                              boundary);
 
        if (build_geom_mapping)
        {
            if (iso_parametric)
                space.geometry->init_from_fe_space(space);
            else
            {
                assert(space.geometry->bases);
                assert(space.geometry->n_bases > 0);
            }
        }
 
        boundary.total_local_boundary.clear();
        for (const auto &lb : boundary.local_boundary)
            boundary.total_local_boundary.emplace_back(lb);
 
        if (build_geom_mapping)
        {
            igl::Timer timer2;
            logger().debug("Building node mapping...");
            timer2.start();
            build_node_mapping(mesh, basis_type, space, space.space_in_node_to_node, space.space_in_primitive_to_primitive);
            timer2.stop();
            logger().debug("Done (took {}s)", timer2.getElapsedTime());
        }
 
        logger().info("n_bases {}", space.n_bases);
 
        timings.building_basis_time += timer.getElapsedTime();
        logger().info(" took {}s", timings.building_basis_time);
 
        logger().info("n bases: {}", space.n_bases);
    }
 
    void VarForm::build_polygonal_basis(
        const mesh::Mesh &mesh,
        const std::string &poly_basis_type,
        const assembler::Assembler &space_assembler,
        bool iso_parametric,
        const int quadrature_order,
        const int mass_quadrature_order,
        const int n_harmonic_samples,
        const int integral_constraints,
        varform::FESpace &space,
        varform::VarFormBoundaryState &boundary)
    {
        if (space.poly_edge_to_data.empty() && space.polys.empty())
        {
            timings.computing_poly_basis_time = 0;
            return;
        }
 
        const std::string space_assembler_name = space_assembler.name();
 
        igl::Timer timer;
        timer.start();
        logger().info("Computing polygonal basis...");
 
        int new_bases = 0;
        const int dim = space_assembler.is_tensor() ? mesh.dimension() : 1;
        if (iso_parametric)
        {
            if (mesh.is_volume())
            {
                if (poly_basis_type == "MeanValue" || poly_basis_type == "Wachspress")
                    log_and_throw_error("Barycentric bases not supported in 3D");
 
                const auto *linear_assembler = dynamic_cast<const assembler::LinearAssembler *>(&space_assembler);
                assert(linear_assembler);
                new_bases = basis::PolygonalBasis3d::build_bases(
                    *linear_assembler,
                    n_harmonic_samples,
                    dynamic_cast<const mesh::Mesh3D &>(mesh),
                    space.n_bases,
                    quadrature_order,
                    mass_quadrature_order,
                    integral_constraints,
                    *space.bases,
                    *space.bases,
                    space.poly_edge_to_data,
                    space.polys_3d);
            }
            else
            {
                const mesh::Mesh2D &mesh_2d = dynamic_cast<const mesh::Mesh2D &>(mesh);
                if (poly_basis_type == "MeanValue")
                {
                    new_bases = basis::MVPolygonalBasis2d::build_bases(
                        space_assembler.name(), dim, mesh_2d, space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        *space.bases, boundary.local_boundary, space.polys);
                }
                else if (poly_basis_type == "Wachspress")
                {
                    new_bases = basis::WSPolygonalBasis2d::build_bases(
                        space_assembler.name(), dim, mesh_2d, space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        *space.bases, boundary.local_boundary, space.polys);
                }
                else
                {
                    const auto *linear_assembler = dynamic_cast<const assembler::LinearAssembler *>(&space_assembler);
                    assert(linear_assembler);
                    new_bases = basis::PolygonalBasis2d::build_bases(
                        *linear_assembler,
                        n_harmonic_samples,
                        mesh_2d,
                        space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        integral_constraints,
                        *space.bases,
                        *space.bases,
                        space.poly_edge_to_data,
                        space.polys);
                }
            }
        }
        else
        {
            assert(space.geometry);
            assert(space.geometry->bases);
            if (mesh.is_volume())
            {
                if (poly_basis_type == "MeanValue" || poly_basis_type == "Wachspress")
                    log_and_throw_error("Barycentric bases not supported in 3D");
 
                const auto *linear_assembler = dynamic_cast<const assembler::LinearAssembler *>(&space_assembler);
                assert(linear_assembler);
                new_bases = basis::PolygonalBasis3d::build_bases(
                    *linear_assembler,
                    n_harmonic_samples,
                    dynamic_cast<const mesh::Mesh3D &>(mesh),
                    space.n_bases,
                    quadrature_order,
                    mass_quadrature_order,
                    integral_constraints,
                    *space.bases,
                    *space.geometry->bases,
                    space.poly_edge_to_data,
                    space.polys_3d);
            }
            else
            {
                const mesh::Mesh2D &mesh_2d = dynamic_cast<const mesh::Mesh2D &>(mesh);
                if (poly_basis_type == "MeanValue")
                {
                    new_bases = basis::MVPolygonalBasis2d::build_bases(
                        space_assembler.name(), dim, mesh_2d, space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        *space.bases, boundary.local_boundary, space.polys);
                }
                else if (poly_basis_type == "Wachspress")
                {
                    new_bases = basis::WSPolygonalBasis2d::build_bases(
                        space_assembler.name(), dim, mesh_2d, space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        *space.bases, boundary.local_boundary, space.polys);
                }
                else
                {
                    const auto *linear_assembler = dynamic_cast<const assembler::LinearAssembler *>(&space_assembler);
                    assert(linear_assembler);
                    new_bases = basis::PolygonalBasis2d::build_bases(
                        *linear_assembler,
                        n_harmonic_samples,
                        mesh_2d,
                        space.n_bases,
                        quadrature_order,
                        mass_quadrature_order,
                        integral_constraints,
                        *space.bases,
                        *space.geometry->bases,
                        space.poly_edge_to_data,
                        space.polys);
                }
            }
        }
 
        timer.stop();
        timings.computing_poly_basis_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.computing_poly_basis_time);
 
        space.n_bases += new_bases;
    }
 
    void VarForm::solve(Eigen::MatrixXd &sol)
    {
        prepare();
        solve_problem(sol);
    }
 
    void VarForm::build_node_mapping(
        const mesh::Mesh &mesh,
        const std::string &basis_type,
        const FESpace &space,
        Eigen::VectorXi &space_in_node_to_node,
        Eigen::VectorXi &space_in_primitive_to_primitive) const
    {
        space_in_node_to_node.resize(0);
        space_in_primitive_to_primitive.resize(0);
 
        if (basis_type == "Spline")
        {
            logger().warn("Node ordering disabled, it dosent work for splines!");
            return;
        }
 
        if (space.disc_orders.maxCoeff() >= 4 || space.disc_orders.maxCoeff() != space.disc_orders.minCoeff())
        {
            logger().warn("Node ordering disabled, it works only for p < 4 and uniform order!");
            return;
        }
 
        if (!mesh.is_conforming())
        {
            logger().warn("Node ordering disabled, not supported for non-conforming meshes!");
            return;
        }
 
        if (mesh.has_poly())
        {
            logger().warn("Node ordering disabled, not supported for polygonal meshes!");
            return;
        }
 
        if (mesh.in_ordered_vertices().size() <= 0 || mesh.in_ordered_edges().size() <= 0 || (mesh.is_volume() && mesh.in_ordered_faces().size() <= 0))
        {
            logger().warn("Node ordering disabled, input vertices/edges/faces not computed!");
            return;
        }
 
        if (!space.mesh_nodes)
        {
            logger().warn("Node ordering disabled, FE space does not expose mesh nodes!");
            return;
        }
 
        const int num_vertex_nodes = space.mesh_nodes->num_vertex_nodes();
        const int num_edge_nodes = space.mesh_nodes->num_edge_nodes();
        const int num_face_nodes = space.mesh_nodes->num_face_nodes();
        const int num_cell_nodes = space.mesh_nodes->num_cell_nodes();
 
        const int num_nodes = num_vertex_nodes + num_edge_nodes + num_face_nodes + num_cell_nodes;
        const long n_vertices = num_vertex_nodes;
        const int num_in_primitives = n_vertices + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
        const int num_primitives = mesh.n_vertices() + mesh.n_edges() + mesh.n_faces() + mesh.n_cells();
 
        igl::Timer timer;
 
        logger().trace("Building in-node to in-primitive mapping...");
        timer.start();
        Eigen::VectorXi in_node_to_in_primitive;
        Eigen::VectorXi in_node_offset;
        build_in_node_to_in_primitive(mesh, *space.mesh_nodes, in_node_to_in_primitive, in_node_offset);
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
 
        logger().trace("Building in-primitive to primitive mapping...");
        timer.start();
        bool ok = build_in_primitive_to_primitive(
            mesh, *space.mesh_nodes,
            mesh.in_ordered_vertices(),
            mesh.in_ordered_edges(),
            mesh.in_ordered_faces(),
            space_in_primitive_to_primitive);
        timer.stop();
        logger().trace("Done (took {}s)", timer.getElapsedTime());
 
        if (!ok)
        {
            space_in_node_to_node.resize(0);
            space_in_primitive_to_primitive.resize(0);
            return;
        }
 
        const auto &tmp = space.mesh_nodes->in_ordered_vertices();
        int max_tmp = -1;
        for (auto v : tmp)
            max_tmp = std::max(max_tmp, v);
 
        space_in_node_to_node.resize(max_tmp + 1);
        for (int i = 0; i < tmp.size(); ++i)
        {
            if (tmp[i] >= 0)
                space_in_node_to_node[tmp[i]] = i;
        }
    }
 
    void VarForm::assign_discr_orders(const json &discr_order, const mesh::Mesh &mesh, Eigen::VectorXi &disc_orders)
    {
        disc_orders.resize(mesh.n_elements());
 
        if (discr_order.is_number_integer())
        {
            disc_orders.setConstant(discr_order);
        }
        else if (discr_order.is_string())
        {
            const std::string discr_orders_path = utils::resolve_path(discr_order, root_path);
            Eigen::MatrixXi tmp;
            io::read_matrix(discr_orders_path, tmp);
            assert(tmp.size() == disc_orders.size());
            assert(tmp.cols() == 1);
            disc_orders = tmp;
        }
        else if (discr_order.is_array())
        {
            const auto b_discr_orders = discr_order;
 
            std::map<int, int> b_orders;
            for (size_t i = 0; i < b_discr_orders.size(); ++i)
            {
                assert(b_discr_orders[i]["id"].is_array() || b_discr_orders[i]["id"].is_number_integer());
 
                const int order = b_discr_orders[i]["order"];
                for (const int id : utils::json_as_array<int>(b_discr_orders[i]["id"]))
                {
                    b_orders[id] = order;
                    logger().trace("bid {}, discr {}", id, order);
                }
            }
 
            for (int e = 0; e < mesh.n_elements(); ++e)
            {
                const int bid = mesh.get_body_id(e);
                const auto order = b_orders.find(bid);
                if (order == b_orders.end())
                {
                    logger().debug("Missing discretization order for body {}; using 1", bid);
                    b_orders[bid] = 1;
                    disc_orders[e] = 1;
                }
                else
                {
                    disc_orders[e] = order->second;
                }
            }
        }
        else
        {
            logger().error("space/discr_order must be either a number a path or an array");
            throw std::runtime_error("invalid json");
        }
    }
 
    void VarForm::save_json(const Eigen::MatrixXd &solution) const
    {
        const std::string out_path = resolve_output_path(args["output"]["json"]);
        if (out_path.empty())
            return;
 
        std::ofstream file(out_path);
        if (!file.is_open())
        {
            logger().error("Unable to save simulation JSON to {}", out_path);
            return;
        }
        save_json(solution, file);
    }
 
    void VarForm::set_materials(assembler::Assembler &assembler, const int size) const
    {
        assert(mesh_ != nullptr);
        assembler.set_size(size);
 
        if (!utils::is_param_valid(args, "materials"))
            return;
 
        std::vector<int> body_ids(mesh_->n_elements());
        for (int i = 0; i < mesh_->n_elements(); ++i)
            body_ids[i] = mesh_->get_body_id(i);
 
        assembler.set_materials(body_ids, args["materials"], units, root_path);
    }
 
    void VarForm::ensure_output_sampler() const
    {
        if (output_sampler_initialized_)
            return;
 
        const io::OutputSpace space = output_space();
        if (space.mesh)
        {
            output_geometry_.init_sampler(*space.mesh, args["output"]["paraview"]["vismesh_rel_area"]);
            output_geometry_.build_grid(*space.mesh, args["output"]["advanced"]["sol_on_grid"]);
        }
        output_sampler_initialized_ = true;
    }
 
    io::OutGeometryData::ExportOptions VarForm::export_options(const io::OutputSpace &space) const
    {
        return io::OutGeometryData::ExportOptions(
            args,
            space.mesh->is_linear(),
            space.mesh->has_prism(),
            problem_dimension() == 1);
    }
 
    io::OutputFieldFunction VarForm::output_field_function(const Eigen::MatrixXd &solution, const io::OutGeometryData::ExportOptions &opts) const
    {
        return [this, &solution, fields = opts.fields](const io::OutputSample &sample) {
            return output_fields(
                sample, solution,
                io::OutputFieldOptions{
                    sample.requested_fields.empty() ? fields : sample.requested_fields});
        };
    }
 
    int VarForm::problem_dimension() const
    {
        if (!problem)
            return 0;
        if (problem->is_scalar())
            return 1;
        return mesh_ ? mesh_->dimension() : 0;
    }
 
    std::shared_ptr<time_integrator::BDF> VarForm::make_bdf_time_integrator() const
    {
        const json &config = args["time"]["integrator"];
        const std::string type = config.is_object() ? config.at("type").get<std::string>() : config.get<std::string>();
 
        if (type == "ImplicitEuler" || type == "implict_euler")
            return std::make_shared<time_integrator::BDF>();
 
        if (!utils::StringUtils::startswith(type, "BDF"))
            log_and_throw_error("First-order transient formulations require ImplicitEuler or BDF, got {}.", type);
 
        auto integrator = std::make_shared<time_integrator::BDF>(
            type == "BDF" ? 1 : std::stoi(type.substr(3)));
        if (config.is_object() && config.contains("steps"))
            integrator->set_parameters(config);
        return integrator;
    }
 
    void VarForm::save_step_state(
        const double t0,
        const double dt,
        const int t,
        const time_integrator::ImplicitTimeIntegrator *time_integrator,
        const bool rest_mesh_written) const
    {
        const int global_t = output_file_index(t);
        const std::string state_path = resolve_output_path(fmt::format(args["output"]["data"]["state"], global_t));
        if (!state_path.empty() && time_integrator)
            time_integrator->save_state(state_path);
 
        save_restart_json(t0, dt, t, rest_mesh_written);
    }
 
    void VarForm::save_timestep(const double time, const int t, const double t0, const double dt, const Eigen::MatrixXd &solution) const
    {
        const io::OutputSpace space = output_space();
        if (!space.mesh || !args["output"]["advanced"]["save_time_sequence"])
            return;
        const int global_t = output_file_index(t);
        if (global_t % args["output"]["paraview"]["skip_frame"].get<int>())
            return;
 
        ensure_output_sampler();
 
        logger().trace("Saving VTU...");
        const std::string step_name = args["output"]["advanced"]["timestep_prefix"];
        const auto opts = export_options(space);
        output_geometry_.save_vtu(
            resolve_output_path(fmt::format(step_name + "{:d}.vtu", global_t)),
            space, output_field_function(solution, opts), time, dt,
            opts);
 
        output_geometry_.save_pvd(
            resolve_output_path(args["output"]["paraview"]["file_name"]),
            [step_name](int i) { return fmt::format(step_name + "{:d}.vtm", i); },
            global_t, t0, dt, args["output"]["paraview"]["skip_frame"].get<int>());
    }
 
    void VarForm::save_subsolve(const int i, const int t, const Eigen::MatrixXd &solution) const
    {
        const io::OutputSpace space = output_space();
        if (!space.mesh || !args["output"]["advanced"]["save_solve_sequence_debug"].get<bool>())
            return;
 
        const bool has_time = args.contains("time") && !args["time"].is_null();
        double dt = 1;
        if (has_time)
            dt = args["time"]["dt"];
 
        ensure_output_sampler();
        const auto opts = export_options(space);
        output_geometry_.save_vtu(
            resolve_output_path(fmt::format("solve_{:d}.vtu", i)),
            space, output_field_function(solution, opts), t, dt,
            opts);
    }
 
    void VarForm::notify_time_step(const int t, const int time_steps, const double t0, const double dt) const
    {
        if (time_callback)
            time_callback(t, time_steps, t0 + dt * t, t0 + dt * time_steps);
    }
 
    void VarForm::save_restart_json(const double t0, const double dt, const int t, const bool rest_mesh_written) const
    {
        const std::string restart_json_path = args["output"]["restart_json"];
        if (restart_json_path.empty())
            return;
 
        const int global_t = output_file_index(t);
 
        json restart_json;
        restart_json["root_path"] = root_path;
        restart_json["common"] = root_path;
        restart_json["time"] = {{"t0", t0 + dt * t}};
        restart_json["output"] = {{"data", {{"file_index_offset", global_t}}}};
 
        restart_json["space"] = R"({
            "remesh": {
                "collapse": {
                    "abs_max_edge_length": -1,
                    "rel_max_edge_length": -1
                }
            }
        })"_json;
 
        const double starting_min_edge_length = stats.min_edge_length;
        restart_json["space"]["remesh"]["collapse"]["abs_max_edge_length"] = std::min(
            args["space"]["remesh"]["collapse"]["abs_max_edge_length"].get<double>(),
            starting_min_edge_length * args["space"]["remesh"]["collapse"]["rel_max_edge_length"].get<double>());
        restart_json["space"]["remesh"]["collapse"]["rel_max_edge_length"] = std::numeric_limits<float>::max();
 
        std::string rest_mesh_path = args["output"]["data"]["rest_mesh"].get<std::string>();
        if (!rest_mesh_path.empty())
        {
            if (!rest_mesh_written)
                logger().warn("Restart JSON for {} references a rest mesh that this formulation does not write.", name());
 
            rest_mesh_path = resolve_output_path(fmt::format(args["output"]["data"]["rest_mesh"], global_t));
 
            std::vector<json> patch;
            if (args["geometry"].is_array())
            {
                const std::vector<json> in_geometry = args["geometry"];
                for (int i = 0; i < in_geometry.size(); ++i)
                {
                    if (!in_geometry[i]["is_obstacle"].get<bool>())
                    {
                        patch.push_back({
                            {"op", "remove"},
                            {"path", fmt::format("/geometry/{}", i)},
                        });
                    }
                }
 
                const int remaining_geometry = in_geometry.size() - patch.size();
                assert(remaining_geometry >= 0);
 
                patch.push_back({
                    {"op", "add"},
                    {"path", fmt::format("/geometry/{}", remaining_geometry > 0 ? "0" : "-")},
                    {"value",
                     {
                         {"mesh", rest_mesh_path},
                     }},
                });
            }
            else
            {
                assert(args["geometry"].is_object());
                patch.push_back({
                    {"op", "remove"},
                    {"path", "/geometry"},
                });
                patch.push_back({
                    {"op", "replace"},
                    {"path", "/geometry"},
                    {"value",
                     {
                         {"mesh", rest_mesh_path},
                     }},
                });
            }
 
            restart_json["patch"] = patch;
        }
 
        restart_json["input"] = {{
            "data",
            {
                {"state", resolve_output_path(fmt::format(args["output"]["data"]["state"], global_t))},
            },
        }};
 
        std::ofstream file(resolve_output_path(fmt::format(restart_json_path, global_t)));
        file << restart_json;
    }
 
    int VarForm::output_file_index(const int t) const
    {
        return t + args["output"]["data"]["file_index_offset"].get<int>();
    }
 
    std::string VarForm::resolve_input_path(const std::string &path, const bool only_if_exists) const
    {
        return utils::resolve_path(path, root_path, only_if_exists);
    }
 
    std::string VarForm::resolve_output_path(const std::string &path) const
    {
        if (output_path.empty() || path.empty() || std::filesystem::path(path).is_absolute())
        {
            return path;
        }
        return std::filesystem::weakly_canonical(std::filesystem::path(output_path) / path).string();
    }
 
    void VarForm::rebuild_node_positions(
        const std::vector<basis::ElementBases> &bases,
        const std::vector<int> &node_ids,
        std::vector<RowVectorNd> &positions)
    {
        positions.resize(node_ids.size());
        for (int n = 0; n < int(node_ids.size()); ++n)
        {
            const int node_id = node_ids[n];
            bool found = false;
            for (const auto &bs : bases)
            {
                for (const auto &b : bs.bases)
                {
                    for (const auto &lg : b.global())
                    {
                        if (lg.index == node_id)
                        {
                            positions[n] = lg.node;
                            found = true;
                            break;
                        }
                    }
 
                    if (found)
                        break;
                }
 
                if (found)
                    break;
            }
            assert(found);
        }
    }
} // namespace polyfem::varform