OperatorSplittingVarForm.cpp

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#include "OperatorSplittingVarForm.hpp"
 
#include <polyfem/assembler/AssemblerUtils.hpp>
#include <polyfem/assembler/GenericProblem.hpp>
#include <polyfem/assembler/Laplacian.hpp>
#include <polyfem/assembler/Stokes.hpp>
#include <polyfem/autogen/auto_p_bases.hpp>
#include <polyfem/autogen/auto_q_bases.hpp>
#include <polyfem/io/Evaluator.hpp>
#include <polyfem/io/MatrixIO.hpp>
#include <polyfem/problem/KernelProblem.hpp>
#include <polyfem/problem/ProblemFactory.hpp>
#include <polyfem/refinement/APriori.hpp>
#include <polyfem/solver/OperatorSplittingSolver.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/varforms/VarFormUtils.hpp>
 
#include <igl/Timer.h>
 
namespace polyfem::varform
{
    void OperatorSplittingVarForm::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_);
        write_checkpoint_ordering(writer, "pressure", pressure_space_);
        Eigen::MatrixXd velocity, pressure;
        split_solution(solution, velocity, pressure);
        writer.write_matrix("/checkpoint/state/velocity", velocity);
        writer.write_matrix("/checkpoint/state/pressure", pressure);
    }
 
    void OperatorSplittingVarForm::deserialize_checkpoint(
        const io::CheckpointReader &reader,
        Eigen::MatrixXd &solution)
    {
        VarForm::deserialize_checkpoint(reader, solution);
        validate_checkpoint_solution(solution, stacked_ndof());
        for (const std::string &path : {
                 "/checkpoint/state/velocity", "/checkpoint/state/pressure"})
            if (!reader.exists(path))
                log_and_throw_error("Operator-splitting checkpoint is missing {}.", path);
 
        Eigen::MatrixXd velocity = reader.read_matrix("/checkpoint/state/velocity");
        Eigen::MatrixXd pressure = reader.read_matrix("/checkpoint/state/pressure");
        if (velocity.rows() != primary_ndof() || velocity.cols() != 1)
            log_and_throw_error("Operator-splitting checkpoint velocity has invalid dimensions.");
        if (pressure.rows() != pressure_space_.n_bases || pressure.cols() != 1)
            log_and_throw_error("Operator-splitting checkpoint pressure has invalid dimensions.");
        if (!solution.topRows(primary_ndof()).isApprox(velocity)
            || !solution.middleRows(primary_ndof(), pressure_space_.n_bases).isApprox(pressure))
            log_and_throw_error("Operator-splitting checkpoint contains inconsistent solution blocks.");
 
        reorder_checkpoint_block(reader, "primary", space_, mesh_->dimension(), 0, velocity);
        reorder_checkpoint_block(reader, "pressure", pressure_space_, 1, 0, pressure);
        solution.topRows(primary_ndof()) = velocity;
        solution.middleRows(primary_ndof(), pressure_space_.n_bases) = pressure;
    }
 
    using namespace varform::internal;
 
    void OperatorSplittingVarForm::reset()
    {
        VarForm::reset();
        space_.reset();
        pressure_space_.reset();
        boundary_.reset();
        pressure_boundary_.reset();
        ass_vals_cache_.init_empty();
        pressure_ass_vals_cache_.init_empty();
        mass_ass_vals_cache_.init_empty(true);
        rhs_assembler_ = nullptr;
        mass_.resize(0, 0);
        rhs_.resize(0, 0);
        primary_assembler_ = nullptr;
        mass_assembler_ = nullptr;
        mixed_assembler_ = nullptr;
        pressure_assembler_ = nullptr;
        use_avg_pressure = true;
        avg_mass_ = 0;
        t0 = 0;
        time_steps = 0;
        dt = 0;
    }
 
    void OperatorSplittingVarForm::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);
        mass_assembler_ = std::make_shared<assembler::Mass>();
        mixed_assembler_ = assembler::AssemblerUtils::make_mixed_assembler(formulation);
        pressure_assembler_ = assembler::AssemblerUtils::make_assembler(assembler::AssemblerUtils::other_assembler_name(formulation));
 
        if (!args.contains("preset_problem"))
        {
            problem = std::make_shared<assembler::GenericTensorProblem>("GenericTensor");
            problem->clear();
 
            json tmp;
            tmp["is_time_dependent"] = is_time_dependent;
            problem->set_parameters(tmp, resources_);
 
            auto bc = args["boundary_conditions"];
            problem->set_parameters(bc, resources_);
            problem->set_parameters(args["initial_conditions"], resources_);
            problem->set_parameters(args["output"], resources_);
        }
        else
        {
            if (args["preset_problem"]["type"] == "Kernel")
            {
                problem = std::make_shared<problem::KernelProblem>("Kernel", *primary_assembler_);
                problem->clear();
            }
            else
            {
                problem = problem::ProblemFactory::factory().get_problem(args["preset_problem"]["type"]);
                problem->clear();
            }
            problem->set_parameters(args["preset_problem"], resources_);
        }
 
        problem->set_units(*primary_assembler_, units);
 
        t0 = is_time_dependent ? args["time"]["t0"].get<double>() : 0.0;
        time_steps = is_time_dependent ? args["time"]["time_steps"].get<int>() : 0;
        dt = is_time_dependent ? args["time"]["dt"].get<double>() : 0.0;
 
        assert(primary_assembler_);
        assert(primary_assembler_->name() == "OperatorSplitting");
    }
 
    void OperatorSplittingVarForm::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 = primary_ndof();
        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, pressure_space_.n_bases,
            stats_solution, *mesh_, space_.disc_orders, space_.disc_ordersq, *problem,
            timings, primary_assembler_ ? primary_assembler_->name() : name(), space_.is_iso_parametric(),
            args["output"]["advanced"]["sol_at_node"], j);
        out << j.dump(4) << std::endl;
    }
 
    io::OutputSpace OperatorSplittingVarForm::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 OperatorSplittingVarForm::compute_errors(const Eigen::MatrixXd &solution)
    {
        if (!args["output"]["advanced"]["compute_error"])
            return stats;
 
        double tend = 0;
        if (args.contains("time") && !args["time"].is_null())
        {
            if (args["time"].contains("tend"))
                tend = args["time"]["tend"].get<double>();
            else
                tend = t0 + dt * time_steps;
        }
 
        Eigen::MatrixXd velocity, pressure;
        split_solution(solution, velocity, pressure);
        stats.compute_errors(space_.n_bases, space_.basis_list(), space_.geometry_basis_list(), *mesh_, *problem, tend, velocity);
        return stats;
    }
 
    void OperatorSplittingVarForm::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 = 1.0;
        if (has_time)
        {
            if (args["time"].contains("tend"))
                tend = args["time"]["tend"].get<double>();
            else
                tend = t0 + dt * time_steps;
        }
        double local_dt = 1;
        if (has_time)
            local_dt = args["time"]["dt"];
 
        const auto opts = export_options(space);
        output_geometry_.export_data(
            space,
            output_field_function(solution, opts),
            has_time,
            tend, local_dt,
            opts,
            vis_mesh_path);
 
        Eigen::MatrixXd velocity, pressure;
        split_solution(solution, velocity, pressure);
 
        const std::string solution_path = resolve_output_path(args["output"]["data"]["solution"]);
        if (!solution_path.empty())
        {
            const int primary_rows = std::min<int>(velocity.rows(), primary_ndof());
            const Eigen::MatrixXd primary_solution = velocity.topRows(primary_rows);
            if (opts.reorder_output && space_.space_in_node_to_node.size() > 0)
            {
                const Eigen::MatrixXd nodal_solution = utils::unflatten(primary_solution, mesh_->dimension());
                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_, velocity, tend,
                stress, mises);
            if (!stress_path.empty())
                io::write_matrix(stress_path, stress);
            if (!mises_path.empty())
                io::write_matrix(mises_path, mises);
        }
    }
 
    void OperatorSplittingVarForm::load_mesh(const mesh::Mesh &mesh, const json &args)
    {
        (void)args;
        set_materials(*primary_assembler_, mesh.dimension());
        set_materials(*mass_assembler_, mesh.dimension());
        problem->init(mesh);
 
        if (mixed_assembler_)
            mixed_assembler_->set_size(mesh.dimension());
        if (pressure_assembler_)
            set_materials(*pressure_assembler_, 1);
    }
 
    int OperatorSplittingVarForm::primary_ndof() const
    {
        return mesh_ ? space_.n_bases * mesh_->dimension() : 0;
    }
 
    int OperatorSplittingVarForm::pressure_block_size() const
    {
        return pressure_space_.n_bases + (use_avg_pressure ? 1 : 0);
    }
 
    int OperatorSplittingVarForm::stacked_ndof() const
    {
        return primary_ndof() + pressure_block_size();
    }
 
    void OperatorSplittingVarForm::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 OperatorSplittingVarForm::build_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args)
    {
        assert(problem);
        assert(primary_assembler_);
        assert(mass_assembler_);
 
        Eigen::VectorXi space_disc_orders, space_disc_ordersq;
        assign_discr_orders(args["space"], mesh, space_disc_orders, space_disc_ordersq);
 
        if (args["space"]["use_p_ref"])
        {
            refinement::APriori::p_refine(
                mesh,
                args["space"]["advanced"]["B"],
                args["space"]["advanced"]["h1_formula"],
                args["space"]["discr_order"],
                args["space"]["advanced"]["discr_order_max"],
                stats,
                space_disc_orders);
 
            logger().info("min p: {} max p: {}", space_disc_orders.minCoeff(), space_disc_orders.maxCoeff());
        }
 
        build_fe_space(
            mesh,
            iso_parametric,
            space_disc_orders,
            space_disc_ordersq,
            args["space"]["basis_type"],
            args["space"]["poly_basis_type"],
            *primary_assembler_,
            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);
 
        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_.disc_orders.maxCoeff() != space_.disc_orders.minCoeff())
            log_and_throw_error("p refinement not supported in operator splitting!");
        if (!space_.poly_edge_to_data.empty())
            log_and_throw_error("Polygonal bases are not supported in operator splitting!");
 
        if (space_.n_bases <= args["solver"]["advanced"]["cache_size"])
        {
            igl::Timer cache_timer;
            cache_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);
            logger().info(" took {}s", cache_timer.getElapsedTime());
        }
        else
        {
            ass_vals_cache_.init_empty();
            mass_ass_vals_cache_.init_empty(true);
        }
 
        const auto &all_boundary = boundary_.total_local_boundary;
        const int prev_bases = space_.n_bases;
        const int prev_b_size = int(all_boundary.size());
        const bool use_corner_quadrature = args["space"]["advanced"]["use_corner_quadrature"];
        const int quadrature_order = args["space"]["advanced"]["quadrature_order"].get<int>();
        const int mass_quadrature_order = args["space"]["advanced"]["mass_quadrature_order"].get<int>();
        Eigen::VectorXi pressure_disc_orders, pressure_disc_ordersq;
        assign_discr_orders(args["space"], mesh, pressure_disc_orders, pressure_disc_ordersq);
        const std::string pressure_basis_type = args["space"]["basis_type"].get<std::string>() == "Bernstein" ? "Bernstein" : "Lagrange";
        build_fe_space(
            mesh,
            /*iso_parametric=*/true,
            pressure_disc_orders,
            pressure_disc_ordersq,
            pressure_basis_type,
            args["space"]["poly_basis_type"],
            *primary_assembler_,
            /*value_dim=*/1,
            quadrature_order,
            mass_quadrature_order,
            use_corner_quadrature,
            args["space"]["advanced"]["n_harmonic_samples"],
            args["space"]["advanced"]["integral_constraints"],
            pressure_space_,
            pressure_boundary_,
            space_.geometry);
 
        assert(space_.basis_list().size() == pressure_space_.basis_list().size());
        for (int i = 0; i < int(pressure_space_.basis_list().size()); ++i)
        {
            quadrature::Quadrature b_quad;
            space_.basis_list()[i].compute_quadrature(b_quad);
            (*pressure_space_.bases)[i].set_quadrature([b_quad](quadrature::Quadrature &quad) { quad = b_quad; });
        }
 
        boundary_.clear_boundary_conditions();
        for (const auto &lb : all_boundary)
            boundary_.local_boundary.emplace_back(lb);
        pressure_boundary_.clear_boundary_conditions();
 
        problem->setup_bc(
            mesh, space_.n_bases,
            space_.basis_list(), space_.geometry_basis_list(), pressure_space_.basis_list(),
            boundary_.local_boundary,
            boundary_.boundary_nodes,
            boundary_.local_neumann_boundary,
            boundary_.local_pressure_boundary,
            boundary_.local_pressure_cavity,
            pressure_boundary_.boundary_nodes,
            boundary_.dirichlet_nodes, boundary_.neumann_nodes);
        pressure_boundary_.normalize_boundary_nodes();
 
        rebuild_node_positions(space_.basis_list(), boundary_.dirichlet_nodes, boundary_.dirichlet_nodes_position);
        rebuild_node_positions(space_.basis_list(), boundary_.neumann_nodes, boundary_.neumann_nodes_position);
 
        const bool has_neumann = !boundary_.local_neumann_boundary.empty() || int(boundary_.local_boundary.size()) < prev_b_size;
        use_avg_pressure = !has_neumann;
 
        for (int i = prev_bases; i < space_.n_bases; ++i)
            for (int d = 0; d < mesh.dimension(); ++d)
                boundary_.boundary_nodes.push_back(i * mesh.dimension() + d);
 
        boundary_.normalize_boundary_nodes();
 
        if (space_.n_bases <= args["solver"]["advanced"]["cache_size"])
            pressure_ass_vals_cache_.init(mesh.is_volume(), pressure_space_.basis_list(), space_.geometry_basis_list());
        else
            pressure_ass_vals_cache_.init_empty();
 
        build_rhs_assembler();
 
        logger().info("n pressure bases: {}", pressure_space_.n_bases);
    }
 
    void OperatorSplittingVarForm::assemble_rhs(const mesh::Mesh &mesh)
    {
        build_rhs_assembler();
 
        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...");
 
        assert(rhs_assembler_ != nullptr);
        rhs_assembler_->assemble(mass_assembler_->density(), rhs_);
        rhs_ *= -1;
 
        const int prev_size = rhs_.rows();
        rhs_.conservativeResize(prev_size + pressure_block_size(), rhs_.cols());
        rhs_.bottomRows(pressure_block_size()).setZero();
 
        timer.stop();
        timings.assigning_rhs_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assigning_rhs_time);
    }
 
    void OperatorSplittingVarForm::assemble_mass_mat(const mesh::Mesh &mesh, const json &args)
    {
        (void)mesh;
        (void)args;
        mass_.resize(0, 0);
        avg_mass_ = 1;
        timings.assembling_mass_mat_time = 0;
    }
 
    void OperatorSplittingVarForm::prepare_initial_solution(Eigen::MatrixXd &sol) const
    {
        if (sol.size() <= 0)
        {
            assert(rhs_assembler_ != nullptr);
            if (problem->is_time_dependent())
                rhs_assembler_->initial_solution(sol);
            else
            {
                sol.resize(rhs_.size(), 1);
                sol.setZero();
            }
        }
        if (sol.cols() > 1)
            sol.conservativeResize(Eigen::NoChange, 1);
        sol.conservativeResize(stacked_ndof(), sol.cols());
        sol.bottomRows(pressure_block_size()).setZero();
    }
 
    void OperatorSplittingVarForm::split_solution(const Eigen::MatrixXd &stacked, Eigen::MatrixXd &primary, Eigen::MatrixXd &pressure) const
    {
        const int cols = std::max(1, int(stacked.cols()));
        primary.setZero(primary_ndof(), cols);
        pressure.setZero(pressure_space_.n_bases, cols);
 
        const int primary_rows = std::min(primary_ndof(), int(stacked.rows()));
        if (primary_rows > 0)
            primary.topRows(primary_rows) = stacked.topRows(primary_rows);
 
        if (stacked.rows() > primary_ndof())
        {
            const int pressure_rows = std::min(pressure_space_.n_bases, int(stacked.rows()) - primary_ndof());
            if (pressure_rows > 0)
                pressure.topRows(pressure_rows) = stacked.middleRows(primary_ndof(), pressure_rows);
        }
    }
 
    void OperatorSplittingVarForm::stack_solution(const Eigen::MatrixXd &primary, const Eigen::MatrixXd &pressure, Eigen::MatrixXd &stacked) const
    {
        const int cols = std::max(primary.cols(), pressure.cols());
        stacked.resize(stacked_ndof(), cols);
        stacked.setZero();
 
        const int primary_rows = std::min(primary_ndof(), int(primary.rows()));
        if (primary_rows > 0)
            stacked.topRows(primary_rows) = primary.topRows(primary_rows);
 
        const int pressure_rows = std::min(pressure_space_.n_bases, int(pressure.rows()));
        if (pressure_rows > 0)
            stacked.middleRows(primary_ndof(), pressure_rows) = pressure.topRows(pressure_rows);
    }
 
    std::vector<io::OutputField> OperatorSplittingVarForm::output_fields(
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution,
        const io::OutputFieldOptions &options) const
    {
        std::vector<io::OutputField> fields;
        if (!mesh_ || !problem || solution.size() <= 0)
            return fields;
 
        Eigen::MatrixXd velocity, pressure;
        split_solution(solution, velocity, pressure);
 
        const int field_dim = mesh_->dimension();
        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 bool export_solution_gradient =
            !options.fields.empty() && options.export_field("solution_gradient");
        const bool export_pressure_gradient =
            !options.fields.empty() && options.export_field("pressure_gradient");
 
        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 sample_vector_field = [&](const Eigen::MatrixXd &dof_values, Eigen::MatrixXd &values, Eigen::MatrixXd *gradients = nullptr) -> bool {
            if (dof_values.size() <= 0 || field_dim <= 0)
                return false;
 
            if (has_element_samples)
            {
                values.resize(sample.local_points.rows(), field_dim);
                if (gradients)
                    gradients->resize(sample.local_points.rows(), field_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)
                            gradients->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);
                    if (gradients)
                        gradients->row(i) = local_grad;
                }
 
                resize_to_output_rows(values);
                if (gradients)
                    resize_to_output_rows(*gradients);
                return true;
            }
 
            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;
        };
 
        Eigen::MatrixXd velocity_values, velocity_gradients;
        const bool sampled_velocity = sample_vector_field(
            velocity, velocity_values,
            export_solution_gradient ? &velocity_gradients : nullptr);
        if (sampled_velocity && options.export_field("velocity"))
            fields.push_back({"velocity", velocity_values, io::OutputField::Association::Point});
        if (sampled_velocity && options.export_field("solution"))
            fields.push_back({"solution", velocity_values, io::OutputField::Association::Point});
        if (sampled_velocity && export_solution_gradient)
            fields.push_back({"solution_gradient", velocity_gradients, io::OutputField::Association::Point});
 
        if (mesh_ && (options.export_field("pressure") || export_pressure_gradient))
        {
            Eigen::MatrixXd values, gradients;
            if (sample_scalar_field(
                    *mesh_, pressure_space_.basis_list(), space_.geometry_basis_list(), sample, pressure, values,
                    export_pressure_gradient ? &gradients : nullptr))
            {
                if (options.export_field("pressure"))
                    fields.push_back({"pressure", values, io::OutputField::Association::Point});
                if (export_pressure_gradient)
                    fields.push_back({"pressure_gradient", gradients, io::OutputField::Association::Point});
            }
        }
 
        const auto &paraview_options = args["output"]["paraview"]["options"];
        if (paraview_options["material"] && has_element_samples)
        {
            const auto &params = primary_assembler_->parameters();
            std::map<std::string, Eigen::MatrixXd> param_values;
            for (const auto &[p, _] : params)
                param_values[p].setZero(output_rows, 1);
 
            Eigen::MatrixXd rhos = Eigen::MatrixXd::Zero(output_rows, 1);
            const auto &density = mass_assembler_->density();
            for (int i = 0; i < sample.local_points.rows(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id < 0)
                    continue;
 
                for (const auto &[p, func] : params)
                    param_values.at(p)(i) = func(sample.local_points.row(i), sample.points.row(i), sample.time, element_id);
                rhos(i) = density(sample.local_points.row(i), sample.points.row(i), sample.time, element_id);
            }
 
            for (const auto &[field_name, values] : param_values)
                if (options.export_field(field_name))
                    fields.push_back({field_name, values, io::OutputField::Association::Point});
            if (options.export_field("rho"))
                fields.push_back({"rho", rhos, io::OutputField::Association::Point});
        }
 
        if (paraview_options["body_ids"] && options.export_field("body_ids") && has_element_samples)
        {
            Eigen::MatrixXd ids = Eigen::MatrixXd::Zero(output_rows, 1);
            for (int i = 0; i < sample.element_ids.size(); ++i)
            {
                const int element_id = sample.element_ids(i);
                if (element_id >= 0)
                    ids(i) = mesh_->get_body_id(element_id);
            }
            fields.push_back({"body_ids", ids, io::OutputField::Association::Point});
        }
 
        return fields;
    }
 
    void OperatorSplittingVarForm::solve_problem(
        Eigen::MatrixXd &sol,
        const InitialConditionOverride *initial_condition_override,
        const ForwardStepCallback &post_step)
    {
        assert(!initial_condition_override && "Operator splitting does not support initial-condition overrides");
        assert(!post_step && "Operator splitting does not support post-step callbacks");
 
        stats.spectrum.setZero();
 
        if (!problem->is_time_dependent())
            log_and_throw_error("Operator splitting requires a transient problem.");
 
        igl::Timer timer;
        timer.start();
        logger().info("Solving {}", primary_assembler_->name());
 
        prepare_initial_solution(sol);
 
        Eigen::MatrixXd velocity, pressure;
        split_solution(sol, velocity, pressure);
        stack_solution(velocity, pressure, sol);
        save_timestep(t0, 0, t0, dt, sol);
 
        Eigen::MatrixXd local_pts;
        const auto &gbases = space_.geometry_basis_list();
        assert(space_.disc_orders.size() > 0 && "Operator splitting requires initialized FE orders");
        const int discr_order = space_.disc_orders.maxCoeff();
        if (mesh_->dimension() == 2)
        {
            if (gbases[0].bases.size() == 3)
                autogen::p_nodes_2d(discr_order, local_pts);
            else
                autogen::q_nodes_2d(discr_order, local_pts);
        }
        else
        {
            if (gbases[0].bases.size() == 4)
                autogen::p_nodes_3d(discr_order, local_pts);
            else
                autogen::q_nodes_3d(discr_order, local_pts);
        }
 
        std::vector<int> bnd_nodes;
        bnd_nodes.reserve(boundary_.boundary_nodes.size() / std::max(1, mesh_->dimension()));
        for (auto it = boundary_.boundary_nodes.begin(); it != boundary_.boundary_nodes.end(); ++it)
        {
            if (!(*it % mesh_->dimension()))
                continue;
            bnd_nodes.push_back(*it / mesh_->dimension());
        }
 
        const int n_el = int(space_.basis_list().size());
        const int shape = gbases[0].bases.size();
 
        auto fluid_assembler = std::dynamic_pointer_cast<assembler::OperatorSplitting>(primary_assembler_);
        if (!fluid_assembler)
            log_and_throw_error("Invalid assembler {}!", primary_assembler_ ? primary_assembler_->name() : name());
        const double viscosity = fluid_assembler->viscosity()(0, 0, 0, 0, 0);
        assert(viscosity >= 0);
 
        logger().info("Matrices assembly...");
        StiffnessMatrix stiffness_viscosity, mixed_stiffness, velocity_mass, pressure_stiffness;
 
        assembler::Laplacian lapl_assembler;
        lapl_assembler.set_size(1);
        lapl_assembler.assemble(mesh_->is_volume(), space_.n_bases, space_.basis_list(), gbases, ass_vals_cache_, 0, stiffness_viscosity);
        mass_assembler_->set_size(1);
        mass_assembler_->assemble(mesh_->is_volume(), space_.n_bases, space_.basis_list(), gbases, mass_ass_vals_cache_, 0, mass_, true);
 
        lapl_assembler.assemble(mesh_->is_volume(), pressure_space_.n_bases, pressure_space_.basis_list(), gbases, pressure_ass_vals_cache_, 0, pressure_stiffness);
 
        mixed_assembler_->assemble(mesh_->is_volume(), pressure_space_.n_bases, space_.n_bases, pressure_space_.basis_list(), space_.basis_list(), gbases,
                                   pressure_ass_vals_cache_, ass_vals_cache_, 0, mixed_stiffness);
        mass_assembler_->set_size(mesh_->dimension());
        mass_assembler_->assemble(mesh_->is_volume(), space_.n_bases, space_.basis_list(), gbases, mass_ass_vals_cache_, 0, velocity_mass, true);
        mixed_stiffness = mixed_stiffness.transpose();
        logger().info("Matrices assembly ends!");
 
        solver::OperatorSplittingSolver solver(
            *mesh_, shape, n_el, boundary_.local_boundary, boundary_.boundary_nodes, pressure_boundary_.boundary_nodes, bnd_nodes, mass_,
            stiffness_viscosity, pressure_stiffness, velocity_mass, dt, viscosity, args["solver"]["linear"]);
 
        const QuadratureOrders boundary_samples = n_boundary_samples(space_.disc_orders.maxCoeff(), space_.disc_ordersq.maxCoeff(), discr_order);
        for (int t = 1; t <= time_steps; ++t)
        {
            const double time = t0 + t * dt;
            logger().info("{}/{} steps, t={}s", t, time_steps, time);
 
            logger().info("Advection...");
            if (args["space"]["advanced"]["use_particle_advection"])
                solver.advection_FLIP(*mesh_, gbases, space_.basis_list(), velocity, dt, local_pts);
            else
                solver.advection(*mesh_, gbases, space_.basis_list(), velocity, dt, local_pts);
            logger().info("Advection finished!");
 
            rhs_assembler_->set_bc(
                boundary_.local_boundary, boundary_.boundary_nodes, boundary_samples, boundary_.local_neumann_boundary, velocity, Eigen::MatrixXd(), time);
 
            logger().info("Solving diffusion...");
            if (viscosity > 0)
                solver.solve_diffusion_1st(mass_, bnd_nodes, velocity);
            logger().info("Diffusion solved!");
 
            solver.external_force(*mesh_, *primary_assembler_, gbases, space_.basis_list(), dt, velocity, local_pts, problem, time);
 
            logger().info("Pressure projection...");
            solver.solve_pressure(mixed_stiffness, pressure_boundary_.boundary_nodes, velocity, pressure);
            solver.projection(space_.n_bases, gbases, space_.basis_list(), pressure_space_.basis_list(), local_pts, pressure, velocity);
            logger().info("Pressure projection finished!");
 
            pressure = pressure / dt;
 
            rhs_assembler_->set_bc(
                boundary_.local_boundary, boundary_.boundary_nodes, boundary_samples, boundary_.local_neumann_boundary, velocity, Eigen::MatrixXd(), time);
 
            stack_solution(velocity, pressure, sol);
            save_timestep(time, t, t0, dt, sol);
            save_step_state(t0, dt, t, sol, nullptr);
            notify_time_step(t, time_steps, t0, dt);
        }
 
        timer.stop();
        timings.solving_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.solving_time);
    }
} // namespace polyfem::varform