ThermoElasticVarForm.cpp

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#include "ThermoElasticVarForm.hpp"
 
#include <polyfem/assembler/AssemblerUtils.hpp>
#include <polyfem/assembler/GenericProblem.hpp>
#include <polyfem/assembler/Laplacian.hpp>
#include <polyfem/assembler/MatParams.hpp>
 
#include <polyfem/io/Evaluator.hpp>
 
#include <polyfem/refinement/APriori.hpp>
 
#include <polyfem/solver/ALSolver.hpp>
#include <polyfem/solver/NLProblem.hpp>
#include <polyfem/solver/forms/BodyForm.hpp>
#include <polyfem/solver/forms/ElasticForm.hpp>
#include <polyfem/solver/forms/InertiaForm.hpp>
#include <polyfem/solver/forms/MixedAssemblerForm.hpp>
#include <polyfem/solver/forms/StackedForm.hpp>
#include <polyfem/solver/forms/lagrangian/AugmentedLagrangianForm.hpp>
#include <polyfem/solver/forms/lagrangian/BCLagrangianForm.hpp>
#include <polyfem/solver/forms/lagrangian/StackedAugmentedLagrangianForm.hpp>
 
#include <polyfem/time_integrator/ImplicitTimeIntegrator.hpp>
 
#include <polyfem/utils/Jacobian.hpp>
#include <polyfem/utils/JSONUtils.hpp>
#include <polyfem/utils/Logger.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
#include <polyfem/utils/Timer.hpp>
 
#include <igl/Timer.h>
 
#include <polysolve/linear/Solver.hpp>
#include <polysolve/nonlinear/Solver.hpp>
 
#include <algorithm>
#include <cassert>
#include <limits>
#include <map>
#include <vector>
 
namespace polyfem::varform
{
    void ThermoElasticVarForm::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, "temperature", temperature_space_);
        if (!temperature_time_integrator_)
            log_and_throw_error("Cannot checkpoint thermoelastic state without its temperature integrator.");
        temperature_time_integrator_->serialize_checkpoint(writer, "/checkpoint/state/temperature_integrator");
    }
 
    void ThermoElasticVarForm::deserialize_checkpoint(
        const io::CheckpointReader &reader,
        Eigen::MatrixXd &solution)
    {
        VarForm::deserialize_checkpoint(reader, solution);
        validate_checkpoint_solution(solution, total_ndof());
        reorder_checkpoint_block(
            reader, "primary", space_,
            mesh_->dimension(), 0, solution);
        reorder_checkpoint_block(
            reader, "temperature", temperature_space_,
            1, displacement_ndof(), solution);
    }
 
    namespace
    {
        json first_material(const json &materials)
        {
            return materials.is_array() ? materials.front() : materials;
        }
 
        void disable_newton_psd_projection(json &solver_params)
        {
            const auto disable_for_newton = [](json &params) {
                if (!params.contains("Newton") || params["Newton"].is_null())
                    params["Newton"] = json::object();
                params["Newton"]["use_psd_projection"] = false;
            };
 
            if (solver_params.contains("solver") && solver_params["solver"].is_array())
            {
                for (json &strategy : solver_params["solver"])
                {
                    const std::string type = strategy.value("type", "");
                    if (type == "Newton" || type == "SparseNewton" || type == "sparse_newton"
                        || type == "DenseNewton" || type == "dense_newton")
                        disable_for_newton(strategy);
                }
            }
            else
            {
                disable_for_newton(solver_params);
            }
        }
 
        void assert_same_space_ids(
            const json &materials,
            const int displacement_space_id,
            const int temperature_space_id)
        {
            for (const json &material : utils::json_as_array(materials))
            {
                if (material.at("displacement_space_id").get<int>() != displacement_space_id
                    || material.at("temperature_space_id").get<int>() != temperature_space_id)
                {
                    log_and_throw_error("All ThermoElasticity materials must use the same FE space ids.");
                }
            }
        }
 
        json elastic_material_from_thermo_material(const json &material)
        {
            if (!material.contains("elastic_material") || !material["elastic_material"].is_object())
                log_and_throw_error("ThermoElasticity requires elastic_material to be an elastic material object.");
 
            json elastic_material = material["elastic_material"];
            const std::string type = elastic_material.value("type", "");
            if (!assembler::AssemblerUtils::is_elastic_material(type))
                log_and_throw_error("ThermoElasticity elastic_material must be an elastic material, got '{}'.", type);
 
            if (material.contains("id"))
                elastic_material["id"] = material["id"];
            if (material.contains("rho"))
                elastic_material["rho"] = material["rho"];
 
            return elastic_material;
        }
 
        json solver_params_for_residual_mode(const json &solver_params, const bool is_residual)
        {
            json params = solver_params;
            if (is_residual)
                disable_newton_psd_projection(params);
            return params;
        }
 
        std::string elastic_formulation_from_thermo_materials(const json &materials)
        {
            std::string formulation;
            for (const json &material : utils::json_as_array(materials))
            {
                const json elastic_material = elastic_material_from_thermo_material(material);
                const std::string type = elastic_material["type"];
                if (formulation.empty())
                    formulation = type;
                else if (formulation != type)
                    formulation = "MultiModels";
            }
 
            return formulation;
        }
 
        StiffnessMatrix block_diag(const StiffnessMatrix &a, const StiffnessMatrix &b)
        {
            std::vector<Eigen::Triplet<double>> entries;
            entries.reserve(a.nonZeros() + b.nonZeros());
 
            for (int k = 0; k < a.outerSize(); ++k)
                for (StiffnessMatrix::InnerIterator it(a, k); it; ++it)
                    entries.emplace_back(it.row(), it.col(), it.value());
 
            for (int k = 0; k < b.outerSize(); ++k)
                for (StiffnessMatrix::InnerIterator it(b, k); it; ++it)
                    entries.emplace_back(a.rows() + it.row(), a.cols() + it.col(), it.value());
 
            StiffnessMatrix out(a.rows() + b.rows(), a.cols() + b.cols());
            out.setFromTriplets(entries.begin(), entries.end());
            out.makeCompressed();
            return out;
        }
 
        StiffnessMatrix identity_mass(const int size)
        {
            return utils::sparse_identity(size, size);
        }
 
    } // namespace
 
    void ThermoElasticVarForm::reset()
    {
        NonlinearElasticVarForm::reset();
        temperature_space_.reset();
        temperature_boundary_.reset();
        temperature_problem_ = nullptr;
        temperature_ass_vals_cache_.init_empty();
        temperature_mass_ass_vals_cache_.init_empty(true);
        temperature_pure_mass_ass_vals_cache_.init_empty(true);
        temperature_assembler_ = nullptr;
        thermoelastic_assembler_ = nullptr;
        temperature_mass_assembler_ = nullptr;
        temperature_pure_mass_assembler_ = nullptr;
        temperature_rhs_assembler_ = nullptr;
        temperature_rhs_density_ = std::make_shared<assembler::NoDensity>();
        temperature_mass_.resize(0, 0);
        temperature_pure_mass_.resize(0, 0);
        stacked_lumped_mass_.resize(0, 0);
        temperature_rhs_.resize(0, 0);
        temperature_time_integrator_ = nullptr;
        temperature_form_ = nullptr;
        thermoelastic_form_ = nullptr;
        temperature_body_form_ = nullptr;
        temperature_inertia_form_ = nullptr;
        stacked_form_ = nullptr;
        displacement_space_id_ = -1;
        temperature_space_id_ = -1;
        elastic_formulation_ = "NeoHookean";
    }
 
    void ThermoElasticVarForm::init(
        const std::string &formulation,
        const Units &units,
        const json &args,
        const std::string &out_path)
    {
        VarForm::init(formulation, units, args, out_path);
        read_material_space_ids(args);
 
        const bool is_time_dependent = args.contains("time") && !args["time"].is_null();
 
        primary_assembler_ = assembler::AssemblerUtils::make_assembler(elastic_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();
        }
 
        temperature_assembler_ = std::make_shared<assembler::Laplacian>("conductivity");
        thermoelastic_assembler_ = assembler::AssemblerUtils::make_mixed_nl_assembler("ThermoElasticity");
        mass_assembler_ = std::make_shared<assembler::Mass>();
        pure_mass_assembler_ = std::make_shared<assembler::HRZMass>();
        temperature_mass_assembler_ = std::make_shared<assembler::Mass>(std::make_shared<assembler::ThermalMassDensity>());
        temperature_pure_mass_assembler_ = std::make_shared<assembler::HRZMass>();
 
        problem = std::make_shared<assembler::GenericTensorProblem>("ThermoElasticDisplacement");
        problem->clear();
        temperature_problem_ = std::make_shared<assembler::GenericScalarProblem>("ThermoElasticTemperature");
        temperature_problem_->clear();
 
        json tmp;
        tmp["is_time_dependent"] = is_time_dependent;
        problem->set_parameters(tmp, resources_);
        temperature_problem_->set_parameters(tmp, resources_);
 
        auto bc = args["boundary_conditions"];
        problem->set_parameters(bc, resources_);
        temperature_problem_->set_parameters(bc, resources_);
        problem->set_parameters(args["initial_conditions"], resources_);
        temperature_problem_->set_parameters(args["initial_conditions"], resources_);
        problem->set_parameters(args["output"], resources_);
        temperature_problem_->set_parameters(args["output"], resources_);
 
        problem->set_units(*primary_assembler_, units);
        temperature_problem_->set_units(*temperature_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;
        contact_dhat_was_explicit_ = args["contact"].value("_dhat_was_explicit", false);
        this->args["contact"].erase("_dhat_was_explicit");
    }
 
    void ThermoElasticVarForm::read_material_space_ids(const json &args)
    {
        const json material = first_material(args.at("materials"));
        displacement_space_id_ = material.at("displacement_space_id").get<int>();
        temperature_space_id_ = material.at("temperature_space_id").get<int>();
        if (displacement_space_id_ == temperature_space_id_)
            log_and_throw_error("ThermoElasticity requires distinct displacement and temperature FE spaces.");
 
        elastic_formulation_ = elastic_formulation_from_thermo_materials(args.at("materials"));
        assert_same_space_ids(args.at("materials"), displacement_space_id_, temperature_space_id_);
    }
 
    json ThermoElasticVarForm::elastic_material_args() const
    {
        if (args["materials"].is_array())
        {
            json materials = json::array();
            for (const json &material : args["materials"])
                materials.push_back(elastic_material_from_thermo_material(material));
            return materials;
        }
 
        return elastic_material_from_thermo_material(args["materials"]);
    }
 
    json ThermoElasticVarForm::time_integrator_args(const int fe_space_id) const
    {
        const json &integrators = args["time"]["integrator"];
        if (!integrators.is_array())
            return integrators;
 
        for (const json &integrator : integrators)
        {
            if (integrator.value("fe_space", -1) == fe_space_id)
            {
                json copy = integrator;
                copy.erase("fe_space");
                return copy;
            }
        }
 
        log_and_throw_error("Missing time integrator for FE space {}.", fe_space_id);
    }
 
    void ThermoElasticVarForm::load_mesh(const mesh::Mesh &mesh, const json &args)
    {
        assert(mesh_);
        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);
 
        const json elastic_materials = elastic_material_args();
 
        primary_assembler_->set_size(mesh.dimension());
        primary_assembler_->set_materials(body_ids, elastic_materials, units, resources_);
        thermoelastic_assembler_->set_size(mesh.dimension());
        thermoelastic_assembler_->set_materials(body_ids, args["materials"], units, resources_);
        mass_assembler_->set_size(mesh.dimension());
        mass_assembler_->set_materials(body_ids, elastic_materials, units, resources_);
        pure_mass_assembler_->set_size(mass_assembler_->size());
 
        temperature_assembler_->set_size(1);
        temperature_assembler_->set_materials(body_ids, args["materials"], units, resources_);
        temperature_mass_assembler_->set_size(1);
        temperature_mass_assembler_->set_materials(body_ids, args["materials"], units, resources_);
        temperature_pure_mass_assembler_->set_size(1);
 
        problem->init(mesh);
        temperature_problem_->init(mesh);
    }
 
    void ThermoElasticVarForm::build_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args)
    {
        assert(problem);
        assert(temperature_problem_);
        assert(primary_assembler_);
        assert(temperature_assembler_);
 
        Eigen::VectorXi displacement_orders, displacement_ordersq;
        assign_discr_orders(args["space"], displacement_space_id_, mesh, displacement_orders, displacement_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,
                displacement_orders);
 
            logger().info("min p: {} max p: {}", displacement_orders.minCoeff(), displacement_orders.maxCoeff());
        }
 
        build_fe_space(
            mesh,
            iso_parametric,
            displacement_orders,
            displacement_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);
        build_displacement_boundary(mesh);
 
        const int n_fe_bases = space_.n_bases;
        space_.n_bases += obstacle.n_vertices();
 
        logger().info("Building collision mesh...");
        build_collision_mesh(mesh, args);
        preprocess_contact_parameters();
        logger().info("Done!");
 
        for (int i = n_fe_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();
 
        build_temperature_basis(mesh, iso_parametric, args);
        build_temperature_boundary(mesh);
 
        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 (std::max(space_.n_bases, temperature_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);
            temperature_ass_vals_cache_.init(mesh.is_volume(), temperature_space_.basis_list(), temperature_space_.geometry_basis_list());
            temperature_mass_ass_vals_cache_.init(mesh.is_volume(), temperature_space_.basis_list(), temperature_space_.geometry_basis_list(), true);
            temperature_pure_mass_ass_vals_cache_.init(mesh.is_volume(), temperature_space_.basis_list(), temperature_space_.geometry_basis_list(), 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);
            temperature_ass_vals_cache_.init_empty();
            temperature_mass_ass_vals_cache_.init_empty(true);
            temperature_pure_mass_ass_vals_cache_.init_empty(true);
        }
    }
 
    void ThermoElasticVarForm::build_displacement_boundary(mesh::Mesh &mesh)
    {
        boundary_.clear_boundary_conditions();
 
        problem->setup_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            displacement_space_id_,
            space_.basis_list(),
            boundary_.total_local_boundary,
            boundary_.local_boundary,
            boundary_.boundary_nodes,
            mesh.dimension());
        std::vector<int> unused_neumann_boundary_nodes;
        problem->setup_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            displacement_space_id_,
            space_.basis_list(),
            boundary_.total_local_boundary,
            boundary_.local_neumann_boundary,
            unused_neumann_boundary_nodes,
            mesh.dimension());
 
        problem->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            displacement_space_id_,
            space_.n_bases,
            boundary_.dirichlet_nodes);
        problem->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            displacement_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, displacement_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);
    }
 
    void ThermoElasticVarForm::build_temperature_basis(mesh::Mesh &mesh, const bool iso_parametric, const json &args)
    {
        Eigen::VectorXi temperature_orders, temperature_ordersq;
        assign_discr_orders(args["space"], temperature_space_id_, mesh, temperature_orders, temperature_ordersq);
 
        build_fe_space(
            mesh,
            iso_parametric,
            temperature_orders,
            temperature_ordersq,
            args["space"]["basis_type"],
            args["space"]["poly_basis_type"],
            *temperature_assembler_,
            /*value_dim=*/1,
            args["space"]["advanced"]["quadrature_order"],
            args["space"]["advanced"]["mass_quadrature_order"],
            args["space"]["advanced"]["use_corner_quadrature"],
            args["space"]["advanced"]["n_harmonic_samples"],
            args["space"]["advanced"]["integral_constraints"],
            temperature_space_,
            temperature_boundary_,
            space_.geometry);
 
        logger().info("n temperature bases: {}", temperature_space_.n_bases);
    }
 
    void ThermoElasticVarForm::build_temperature_boundary(mesh::Mesh &mesh)
    {
        temperature_boundary_.clear_boundary_conditions();
 
        temperature_problem_->update_nodes(temperature_space_.space_in_node_to_node);
 
        temperature_problem_->setup_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            temperature_space_id_,
            temperature_space_.basis_list(),
            temperature_boundary_.total_local_boundary,
            temperature_boundary_.local_boundary,
            temperature_boundary_.boundary_nodes,
            /*value_dim=*/1);
        std::vector<int> unused_neumann_boundary_nodes;
        temperature_problem_->setup_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            temperature_space_id_,
            temperature_space_.basis_list(),
            temperature_boundary_.total_local_boundary,
            temperature_boundary_.local_neumann_boundary,
            unused_neumann_boundary_nodes,
            /*value_dim=*/1);
 
        temperature_problem_->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Dirichlet,
            temperature_space_id_,
            temperature_space_.n_bases,
            temperature_boundary_.dirichlet_nodes);
        temperature_problem_->setup_nodal_bc(
            mesh,
            assembler::BoundaryKind::Neumann,
            temperature_space_id_,
            temperature_space_.n_bases,
            temperature_boundary_.neumann_nodes);
 
        for (const int n_id : temperature_boundary_.dirichlet_nodes)
            temperature_boundary_.boundary_nodes.push_back(n_id);
 
        temperature_boundary_.normalize_boundary_nodes();
        rebuild_node_positions(temperature_space_.basis_list(), temperature_boundary_.dirichlet_nodes, temperature_boundary_.dirichlet_nodes_position);
        rebuild_node_positions(temperature_space_.basis_list(), temperature_boundary_.neumann_nodes, temperature_boundary_.neumann_nodes_position);
    }
 
    void ThermoElasticVarForm::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;
 
        solve_data_.rhs_assembler = std::make_shared<assembler::RhsAssembler>(
            *primary_assembler_, *mesh_, &obstacle,
            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,
            displacement_space_id_);
        rhs_assembler_ = solve_data_.rhs_assembler;
 
        temperature_rhs_assembler_ = std::make_shared<assembler::RhsAssembler>(
            *temperature_assembler_, *mesh_, nullptr,
            temperature_boundary_.dirichlet_nodes, temperature_boundary_.neumann_nodes,
            temperature_boundary_.dirichlet_nodes_position, temperature_boundary_.neumann_nodes_position,
            temperature_space_.n_bases, /*size=*/1,
            temperature_space_.basis_list(), temperature_space_.geometry_basis_list(),
            temperature_mass_ass_vals_cache_, *temperature_problem_, resources_,
            args["space"]["advanced"]["bc_method"],
            rhs_solver_params,
            temperature_space_id_);
    }
 
    void ThermoElasticVarForm::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_);
        temperature_problem_->set_parameters(p_params, resources_);
 
        rhs_.resize(0, 0);
        temperature_rhs_.resize(0, 0);
 
        timer.start();
        logger().info("Assigning rhs...");
 
        build_rhs_assembler();
        assert(rhs_assembler_ != nullptr);
        assert(temperature_rhs_assembler_ != nullptr);
        assert(temperature_rhs_density_ != nullptr);
        rhs_assembler_->assemble(mass_assembler_->density(), rhs_);
        rhs_ *= -1;
        temperature_rhs_assembler_->assemble(*temperature_rhs_density_, temperature_rhs_);
        temperature_rhs_ *= -1;
 
        timings.assigning_rhs_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assigning_rhs_time);
    }
 
    void ThermoElasticVarForm::assemble_mass_mat(const mesh::Mesh &mesh, const json &args)
    {
        mass_.resize(0, 0);
        pure_mass_.resize(0, 0);
        temperature_mass_.resize(0, 0);
        temperature_pure_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);
        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);
        temperature_mass_assembler_->assemble(mesh.is_volume(), temperature_space_.n_bases, temperature_space_.basis_list(), temperature_space_.geometry_basis_list(), temperature_mass_ass_vals_cache_, 0, temperature_mass_, true);
        temperature_pure_mass_assembler_->assemble(mesh.is_volume(), temperature_space_.n_bases, temperature_space_.basis_list(), temperature_space_.geometry_basis_list(), temperature_pure_mass_ass_vals_cache_, 0, temperature_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)
                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_);
            temperature_mass_ = utils::lump_matrix(temperature_mass_);
        }
 
        stacked_lumped_mass_ = block_diag(
            pure_mass_.size() > 0 ? pure_mass_ : identity_mass(displacement_ndof()),
            temperature_pure_mass_.size() > 0 ? temperature_pure_mass_ : identity_mass(temperature_ndof()));
 
        timer.stop();
        timings.assembling_mass_mat_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.assembling_mass_mat_time);
 
        stats.nn_zero = stacked_lumped_mass_.nonZeros();
        stats.num_dofs = stacked_lumped_mass_.rows();
        stats.mat_size = (long long)stacked_lumped_mass_.rows() * (long long)stacked_lumped_mass_.cols();
        logger().info("sparsity: {}/{}", stats.nn_zero, stats.mat_size);
    }
 
    void ThermoElasticVarForm::initial_temperature_solution(Eigen::MatrixXd &solution) const
    {
        assert(temperature_rhs_assembler_ != nullptr);
        temperature_rhs_assembler_->initial_solution(solution);
    }
 
    Eigen::MatrixXd ThermoElasticVarForm::stacked_solution(
        const Eigen::MatrixXd &displacement,
        const Eigen::MatrixXd &temperature) const
    {
        assert(displacement.rows() == displacement_ndof());
        assert(temperature.rows() == temperature_ndof());
        assert(displacement.cols() == temperature.cols());
 
        Eigen::MatrixXd solution(displacement.rows() + temperature.rows(), displacement.cols());
        solution << displacement, temperature;
        return solution;
    }
 
    void ThermoElasticVarForm::split_solution(
        const Eigen::MatrixXd &solution,
        Eigen::MatrixXd &displacement,
        Eigen::MatrixXd &temperature) const
    {
        assert(solution.rows() == total_ndof());
        displacement = solution.topRows(displacement_ndof());
        temperature = solution.bottomRows(temperature_ndof());
    }
 
    void ThermoElasticVarForm::build_forms(Eigen::MatrixXd &solution, const double t)
    {
        assert(solution.cols() == 1);
        assert(solution.rows() == total_ndof());
 
        const bool is_time_dependent = problem->is_time_dependent();
        const double form_dt = is_time_dependent ? dt : 0.0;
 
        stacked_form_ = std::make_shared<solver::StackedForm>();
        const auto displacement_block = stacked_form_->add_block(displacement_ndof());
        const auto temperature_block = stacked_form_->add_block(temperature_ndof());
 
        Eigen::MatrixXd displacement, temperature;
        split_solution(solution, displacement, temperature);
 
        if (is_time_dependent)
        {
            solve_data_.time_integrator = time_integrator::ImplicitTimeIntegrator::construct_time_integrator(
                time_integrator_args(displacement_space_id_),
                time_integrator::ImplicitTimeIntegrator::DynamicOrder::Second);
 
            Eigen::MatrixXd displacement_solution, displacement_velocity, displacement_acceleration;
            initial_solution(displacement_solution);
            displacement_solution.col(0) = displacement;
            initial_velocity(displacement_velocity);
            initial_acceleration(displacement_acceleration);
            solve_data_.time_integrator->init(displacement_solution, displacement_velocity, displacement_acceleration, dt);
            restore_checkpoint_integrator(
                solve_data_.time_integrator, "/checkpoint/state/primary_integrator", dt,
                "primary", space_, mesh_->dimension());
        }
        else
        {
            solve_data_.time_integrator = nullptr;
        }
 
        init_forms(args, mesh_->dimension(), displacement, t);
        for (const auto &form : forms)
        {
            assert(form);
            stacked_form_->add(displacement_block, form);
        }
        solve_data_.al_form.clear();
 
        temperature_form_ = std::make_shared<solver::ElasticForm>(
            temperature_space_.n_bases, *temperature_space_.bases, temperature_space_.geometry_basis_list(),
            *temperature_assembler_, temperature_ass_vals_cache_, t, form_dt, mesh_->is_volume(),
            /*jacobian_threshold=*/0.0, solver::ElementInversionCheck::Discrete);
        stacked_form_->add(temperature_block, temperature_form_);
 
        assert(thermoelastic_assembler_);
        thermoelastic_form_ = std::make_shared<solver::MixedAssemblerForm>(
            space_.n_bases, temperature_space_.n_bases,
            space_.basis_list(), temperature_space_.basis_list(), space_.geometry_basis_list(),
            *thermoelastic_assembler_, ass_vals_cache_, temperature_ass_vals_cache_,
            t, form_dt, mesh_->is_volume());
        stacked_form_->add(displacement_block, temperature_block, thermoelastic_form_);
 
        assert(temperature_space_.disc_orders.size() > 0 && "Thermal boundary quadrature requires initialized FE orders");
        assert(temperature_space_.geometry && "Thermal boundary quadrature requires an initialized geometry mapping");
        assert(temperature_space_.geometry->disc_orders.size() > 0 && "Thermal boundary quadrature requires initialized geometry orders");
        const int gdiscr_order = mesh_->orders().size() <= 0 ? 1 : mesh_->orders().maxCoeff();
        const QuadratureOrders temperature_boundary_samples =
            n_boundary_samples(temperature_space_.disc_orders.maxCoeff(), temperature_space_.disc_ordersq.maxCoeff(), gdiscr_order);
        temperature_body_form_ = std::make_shared<solver::BodyForm>(
            temperature_ndof(), 0,
            temperature_boundary_.boundary_nodes, temperature_boundary_.local_boundary,
            temperature_boundary_.local_neumann_boundary, temperature_boundary_samples,
            temperature_rhs_, *temperature_rhs_assembler_,
            *temperature_rhs_density_,
            /*is_formulation_mixed=*/false, is_time_dependent);
        temperature_body_form_->update_quantities(t, temperature);
        stacked_form_->add(temperature_block, temperature_body_form_);
 
        if (is_time_dependent)
        {
            temperature_time_integrator_ = time_integrator::ImplicitTimeIntegrator::construct_time_integrator(
                time_integrator_args(temperature_space_id_),
                time_integrator::ImplicitTimeIntegrator::DynamicOrder::First);
 
            Eigen::MatrixXd temperature_solution;
            initial_temperature_solution(temperature_solution);
            temperature_solution.col(0) = temperature;
            Eigen::MatrixXd temperature_velocity = Eigen::MatrixXd::Zero(temperature_solution.rows(), temperature_solution.cols());
            Eigen::MatrixXd temperature_acceleration = Eigen::MatrixXd::Zero(temperature_solution.rows(), temperature_solution.cols());
            temperature_time_integrator_->init(temperature_solution, temperature_velocity, temperature_acceleration, dt);
            restore_checkpoint_integrator(
                temperature_time_integrator_, "/checkpoint/state/temperature_integrator", dt,
                "temperature", temperature_space_, 1);
 
            temperature_inertia_form_ = std::make_shared<solver::InertiaForm>(temperature_mass_, *temperature_time_integrator_);
            if (!temperature_boundary_.boundary_nodes.empty())
            {
                temperature_inertia_form_->set_x_tilde_updater(
                    [this, temperature_boundary_samples](
                        const double t,
                        const Eigen::VectorXd &,
                        Eigen::VectorXd &target) {
                        Eigen::MatrixXd projected_target = target;
                        const std::vector<mesh::LocalBoundary> empty_neumann_boundary;
                        temperature_rhs_assembler_->set_bc(
                            temperature_boundary_.local_boundary,
                            temperature_boundary_.boundary_nodes,
                            temperature_boundary_samples,
                            empty_neumann_boundary,
                            projected_target,
                            Eigen::MatrixXd(),
                            t);
                        assert(projected_target.cols() == 1);
                        target = projected_target.col(0);
                    });
            }
            stacked_form_->add(temperature_block, temperature_inertia_form_);
 
            update_transient_form_weights();
        }
        else
        {
            temperature_time_integrator_ = nullptr;
            temperature_inertia_form_ = nullptr;
        }
 
        forms.clear();
        forms.push_back(stacked_form_);
        for (const auto &form : forms)
            form->set_output_dir(output_path);
 
        solve_data_.al_form.clear();
        if (!boundary_.boundary_nodes.empty() || !temperature_boundary_.boundary_nodes.empty())
        {
            auto stacked_al = std::make_shared<solver::StackedAugmentedLagrangianForm>();
            const auto displacement_al_block = stacked_al->add_block(displacement_block.size());
            const auto temperature_al_block = stacked_al->add_block(temperature_block.size());
 
            if (!boundary_.boundary_nodes.empty())
            {
                stacked_al->add(
                    displacement_al_block,
                    std::make_shared<solver::BCLagrangianForm>(
                        displacement_block.size(),
                        boundary_.boundary_nodes, boundary_.local_boundary, boundary_.local_neumann_boundary,
                        elastic_boundary_samples(), mass_, *rhs_assembler_,
                        obstacle.n_vertices() * mesh_->dimension(), is_time_dependent, t));
            }
 
            if (!temperature_boundary_.boundary_nodes.empty())
            {
                stacked_al->add(
                    temperature_al_block,
                    std::make_shared<solver::BCLagrangianForm>(
                        temperature_block.size(),
                        temperature_boundary_.boundary_nodes, temperature_boundary_.local_boundary,
                        temperature_boundary_.local_neumann_boundary, temperature_boundary_samples,
                        temperature_mass_, *temperature_rhs_assembler_,
                        /*obstacle_ndof=*/0, is_time_dependent, t));
            }
 
            solve_data_.al_form.push_back(stacked_al);
        }
    }
 
    void ThermoElasticVarForm::update_transient_form_weights()
    {
        assert(problem->is_time_dependent());
        assert(solve_data_.time_integrator && "Transient thermoelasticity requires a displacement time integrator");
        assert(temperature_time_integrator_);
 
        solve_data_.update_dt();
 
        const double displacement_scaling = solve_data_.time_integrator->acceleration_scaling();
        const double temperature_scaling = temperature_time_integrator_->acceleration_scaling();
 
        if (temperature_form_)
            temperature_form_->set_weight(temperature_scaling);
        if (temperature_body_form_)
            temperature_body_form_->set_weight(temperature_scaling);
        if (thermoelastic_form_)
            thermoelastic_form_->set_row_weights(displacement_scaling, temperature_scaling);
    }
 
    void ThermoElasticVarForm::solve_nonlinear_step(const int step, Eigen::MatrixXd &solution)
    {
        assert(solve_data_.nl_problem != nullptr && "Thermoelastic forms must initialize the nonlinear problem before solving");
        solver::NLProblem &nl_problem = *solve_data_.nl_problem;
 
        const json nonlinear_params = solver_params_for_residual_mode(args["solver"]["nonlinear"], nl_problem.is_residual());
        const json al_nonlinear_params = solver_params_for_residual_mode(args["solver"]["augmented_lagrangian"]["nonlinear"], nl_problem.is_residual());
 
        std::shared_ptr<polysolve::nonlinear::Solver> nl_solver =
            polysolve::nonlinear::Solver::create(
                nonlinear_params, args["solver"]["linear"],
                units.characteristic_length(), logger());
 
        if (nl_problem.uses_lagging())
            nl_problem.init_lagging(solution);
 
        const auto update_displacement_barrier_stiffness = [&](const Eigen::VectorXd &x) {
            const Eigen::VectorXd displacement = x.head(displacement_ndof());
            solve_data_.update_barrier_stiffness(displacement);
        };
 
        if (!solve_data_.al_form.empty())
        {
            solver::ALSolver al_solver(
                solve_data_.al_form,
                args["solver"]["augmented_lagrangian"]["initial_weight"],
                args["solver"]["augmented_lagrangian"]["scaling"],
                args["solver"]["augmented_lagrangian"]["max_weight"],
                args["solver"]["augmented_lagrangian"]["eta"],
                update_displacement_barrier_stiffness);
 
            al_solver.post_subsolve = [&](const double al_weight) {
                stats.solver_info.push_back(
                    {{"type", al_weight > 0 ? "al" : "rc"},
                     {"t", step},
                     {"info", nl_solver->info()}});
                if (al_weight > 0)
                    stats.solver_info.back()["weight"] = al_weight;
                save_subsolve(stats.solver_info.size(), step, solution);
            };
 
            al_solver.solve_al(
                nl_problem, solution, al_nonlinear_params,
                args["solver"]["linear"], units.characteristic_length(), nl_solver);
            al_solver.solve_reduced(
                nl_problem, solution, nonlinear_params,
                args["solver"]["linear"], units.characteristic_length(), nl_solver);
            return;
        }
 
        Eigen::VectorXd x = solution;
        nl_problem.init(x);
        update_displacement_barrier_stiffness(x);
        nl_problem.normalize_forms();
        nl_solver->minimize(nl_problem, x);
        nl_problem.finish();
        solution = x;
        stats.solver_info.push_back({{"type", "rc"}, {"t", step}, {"info", nl_solver->info()}});
        save_subsolve(stats.solver_info.size(), step, solution);
    }
 
    void ThermoElasticVarForm::solve_problem(
        Eigen::MatrixXd &sol,
        const InitialConditionOverride *initial_condition_override,
        const ForwardStepCallback &post_step)
    {
        assert(!initial_condition_override && "Thermoelasticity does not support initial-condition overrides");
        assert(!post_step && "Thermoelasticity does not support post-step callbacks");
 
        stats.spectrum.setZero();
 
        igl::Timer timer;
        timer.start();
        logger().info("Solving ThermoElasticity");
 
        {
            POLYFEM_SCOPED_TIMER("Setup RHS");
 
            if (sol.size() <= 0)
            {
                Eigen::MatrixXd displacement, temperature;
                initial_solution(displacement);
                initial_temperature_solution(temperature);
                const int cols = std::max(displacement.cols(), temperature.cols());
                if (displacement.cols() != cols)
                    displacement.conservativeResize(Eigen::NoChange, cols);
                if (temperature.cols() != cols)
                    temperature.conservativeResize(Eigen::NoChange, cols);
                sol = stacked_solution(displacement, temperature);
            }
 
            if (sol.cols() > 1)
                sol.conservativeResize(Eigen::NoChange, 1);
        }
 
        build_forms(sol, problem->is_time_dependent() ? t0 + dt : 1.0);
 
        double characteristic_length = 0;
        if (args["solver"]["advanced"]["characteristic_length"] > 0)
            characteristic_length = args["solver"]["advanced"]["characteristic_length"];
        else
        {
            RowVectorNd min, max;
            mesh_->bounding_box(min, max);
            characteristic_length = (max - min).norm();
        }
 
        double characteristic_force_density = 0;
        if (args["solver"]["advanced"]["characteristic_force_density"] <= 0)
        {
            logger().warn("No user-specified force density was provided, defaulting to 10000.");
            characteristic_force_density = 10000;
        }
        else
            characteristic_force_density = args["solver"]["advanced"]["characteristic_force_density"];
 
        solve_data_.nl_problem = std::make_shared<solver::NLProblem>(
            total_ndof(), problem->is_time_dependent() ? t0 + dt : 1.0,
            forms, solve_data_.al_form,
            polysolve::linear::Solver::create(args["solver"]["linear"], logger()),
            characteristic_length, characteristic_force_density,
            stacked_lumped_mass_.size() > 0 ? stacked_lumped_mass_ : identity_mass(total_ndof()),
            mesh_->dimension(),
            problem->is_time_dependent());
        solve_data_.nl_problem->init(sol);
        solve_data_.nl_problem->update_quantities(problem->is_time_dependent() ? t0 + dt : 1.0, sol);
        stats.solver_info = json::array();
 
        if (!problem->is_time_dependent())
        {
            solve_nonlinear_step(0, sol);
        }
        else
        {
            save_timestep(t0, 0, t0, dt, sol);
            for (int t = 1; t <= time_steps; ++t)
            {
                const double time = t0 + dt * t;
                solve_nonlinear_step(t, sol);
 
                save_timestep(time, t, t0, dt, sol);
 
                Eigen::MatrixXd displacement, temperature;
                split_solution(sol, displacement, temperature);
                solve_data_.time_integrator->update_quantities(displacement);
                temperature_time_integrator_->update_quantities(temperature);
                update_transient_form_weights();
                solve_data_.update_barrier_stiffness(displacement);
                solve_data_.nl_problem->update_quantities(t0 + (t + 1) * dt, sol);
 
                logger().info("{}/{}  t={}", t, time_steps, time);
                notify_time_step(t, time_steps, t0, dt);
                save_step_state(t0, dt, t, sol, solve_data_.time_integrator.get());
            }
        }
 
        timer.stop();
        timings.solving_time = timer.getElapsedTime();
        logger().info(" took {}s", timings.solving_time);
    }
 
    io::OutStatsData ThermoElasticVarForm::compute_errors(const Eigen::MatrixXd &solution)
    {
        if (!args["output"]["advanced"]["compute_error"])
            return stats;
 
        Eigen::MatrixXd displacement, temperature;
        split_solution(solution, displacement, temperature);
 
        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, displacement);
        return stats;
    }
 
    std::vector<io::OutputField> ThermoElasticVarForm::output_fields(
        const io::OutputSample &sample,
        const Eigen::MatrixXd &solution,
        const io::OutputFieldOptions &options) const
    {
        Eigen::MatrixXd displacement, temperature;
        split_solution(solution, displacement, temperature);
 
        std::vector<io::OutputField> fields =
            NonlinearElasticVarForm::output_fields(sample, displacement, options);
        fields.erase(
            std::remove_if(
                fields.begin(), fields.end(),
                [](const io::OutputField &field) {
                    return field.name == "solution" || field.name == "solution_gradient";
                }),
            fields.end());
 
        if (!mesh_ || temperature.size() <= 0)
            return fields;
        if (sample.domain == io::OutputSample::Domain::Contact)
            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 auto has_field = [&](const std::string &name) {
            return std::any_of(fields.begin(), fields.end(), [&](const io::OutputField &field) {
                return field.name == name;
            });
        };
 
        const auto append_material_fields = [&](const assembler::Assembler &assembler) {
            const auto &paraview_options = args["output"]["paraview"]["options"];
            if (!paraview_options["material"] || !has_element_samples)
                return;
 
            const auto params = assembler.parameters();
            std::map<std::string, Eigen::MatrixXd> param_values;
            for (const auto &[p, _] : params)
                param_values[p].setZero(output_rows, 1);
 
            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);
            }
 
            for (const auto &[name, values] : param_values)
                if (options.export_field(name) && !has_field(name))
                    fields.push_back({name, values, io::OutputField::Association::Point});
        };
 
        const auto sample_temperature = [&](Eigen::MatrixXd &values, Eigen::MatrixXd *gradients = nullptr) -> bool {
            if (has_element_samples)
            {
                values.resize(sample.local_points.rows(), 1);
                if (gradients)
                    gradients->resize(sample.local_points.rows(), mesh_->dimension());
                for (int i = 0; i < sample.local_points.rows(); ++i)
                {
                    const int element_id = sample.element_ids(i);
                    if (element_id < 0)
                    {
                        values(i) = 0;
                        if (gradients)
                            gradients->row(i).setZero();
                        continue;
                    }
 
                    Eigen::MatrixXd local_sol, local_grad;
                    io::Evaluator::interpolate_at_local_vals(
                        *mesh_, 1, temperature_space_.basis_list(), temperature_space_.geometry_basis_list(),
                        element_id, sample.local_points.row(i), temperature, local_sol, local_grad);
                    values(i) = local_sol(0);
                    if (gradients)
                        gradients->row(i) = local_grad;
                }
 
                if (output_rows > values.rows())
                {
                    const int previous_rows = values.rows();
                    values.conservativeResize(output_rows, Eigen::NoChange);
                    values.bottomRows(output_rows - previous_rows).setZero();
                    if (gradients)
                    {
                        gradients->conservativeResize(output_rows, Eigen::NoChange);
                        gradients->bottomRows(output_rows - previous_rows).setZero();
                    }
                }
                return true;
            }
 
            if (sample.node_ids.size() > 0)
            {
                values.resize(sample.node_ids.size(), 1);
                for (int i = 0; i < sample.node_ids.size(); ++i)
                {
                    const int node_id = sample.node_ids(i);
                    if (node_id < 0 || node_id >= temperature.rows())
                        return false;
                    values(i) = temperature(node_id);
                }
                return sample.points.rows() == 0 || sample.points.rows() == values.rows();
            }
 
            return false;
        };
 
        const bool export_temperature_gradient =
            !options.fields.empty() && options.export_field("temperature_gradient");
        if (options.export_field("temperature") || export_temperature_gradient)
        {
            Eigen::MatrixXd values, gradients;
            if (sample_temperature(values, export_temperature_gradient ? &gradients : nullptr))
            {
                if (options.export_field("temperature"))
                    fields.push_back({"temperature", values, io::OutputField::Association::Point});
                if (export_temperature_gradient)
                    fields.push_back({"temperature_gradient", gradients, io::OutputField::Association::Point});
            }
        }
 
        if (thermoelastic_assembler_)
            append_material_fields(*thermoelastic_assembler_);
        if (temperature_assembler_)
            append_material_fields(*temperature_assembler_);
        if (temperature_mass_assembler_)
            append_material_fields(*temperature_mass_assembler_);
 
        return fields;
    }
} // namespace polyfem::varform