MeanField/tests/physics/gravity_monopole_accuracy.cpp

#include <catch2/catch_test_macros.hpp>
#include <catch2/matchers/catch_matchers_floating_point.hpp>

#include <algorithm>
#include <array>
#include <cmath>
#include <limits>
#include <sstream>

#include <mfem.hpp>
#include <mpi.h>

import mean_field;
import test_helpers;

namespace {
    constexpr std::array<double, 6> shell_boundaries{0.0, 0.25, 0.50, 0.75, 0.90, 1.0};
    constexpr int shell_count = static_cast<int>(shell_boundaries.size()) - 1;

    struct ShellAccumulator {
        long long points{0};
        double weight{0.0};
        double minimum_radius{std::numeric_limits<double>::infinity()};
        double maximum_radius{0.0};
        double potential_error_squared{0.0};
        double radial_error_squared{0.0};
        double tangential_squared{0.0};
    };

    struct ShellMetrics {
        long long points{0};
        double minimum_radius{0.0};
        double maximum_radius{0.0};
        double potential_rms_error{0.0};
        double radial_rms_error{0.0};
        double tangential_rms{0.0};
    };

    struct ShellMeasurement {
        std::array<ShellMetrics, shell_count> shells{};
        long long invalid_points{0};
    };

    constexpr int mapping_status_count = 8;

    struct GravitationalEnergies {
        double binding{0.0};
        double virial{0.0};
        double minimum_mapping_determinant{std::numeric_limits<double>::infinity()};
        double maximum_mapping_determinant{-std::numeric_limits<double>::infinity()};
        long long invalid_points{0};
        std::array<long long, mapping_status_count> mapping_status_counts{};
        int first_invalid_element{-1};
        int first_invalid_attribute{-1};
        int first_invalid_quadrature_point{-1};
        double first_invalid_determinant{std::numeric_limits<double>::quiet_NaN()};
    };

    constexpr int mapping_status_index(const mean_field::mapping::MappingStatus status) {
        return static_cast<int>(status);
    }

    void zero_vacuum_density(
        const mean_field::fem::FEM &f,
        mfem::GridFunction &density
    ) {
        using DomainSchema = mean_field::utils::domain::CoreEnvelopeVacuumDomainSchema;

        const mean_field::field::FieldDofMap densityMap =
            mean_field::field::make_field_dof_map<mean_field::field::Density, DomainSchema>(*f.densityFes);

        mfem::Vector densityTrue;
        density.GetTrueDofs(densityTrue);

        const mfem::Vector supportedDensity = densityMap.gather(densityTrue);
        densityMap.scatter(supportedDensity, densityTrue);
        density.SetFromTrueDofs(densityTrue);
    }

    double global_norm(
        const mfem::Vector &vector,
        MPI_Comm communicator
    ) {
        const double local_norm_squared = vector * vector;
        double global_norm_squared      = 0.0;
        MPI_Allreduce(&local_norm_squared, &global_norm_squared, 1, MPI_DOUBLE, MPI_SUM, communicator);
        return std::sqrt(global_norm_squared);
    }

    double global_dot(
        const mfem::Vector &lhs,
        const mfem::Vector &rhs,
        MPI_Comm communicator
    ) {
        const double local_dot = lhs * rhs;
        double result          = 0.0;
        MPI_Allreduce(&local_dot, &result, 1, MPI_DOUBLE, MPI_SUM, communicator);
        return result;
    }

    double global_relative_error(
        const mfem::Vector &computed,
        const mfem::Vector &reference,
        MPI_Comm communicator
    ) {
        REQUIRE(computed.Size() == reference.Size());

        mfem::Vector difference(computed);
        difference -= reference;

        return global_norm(difference, communicator) /
               std::max(global_norm(reference, communicator), std::numeric_limits<double>::epsilon());
    }

    int get_shell(const double coordinate) {
        const double clamped = std::clamp(coordinate, 0.0, std::nextafter(1.0, 0.0));

        for (int shell = 0; shell < shell_count; ++shell) {
            if (clamped < shell_boundaries[shell + 1]) {
                return shell;
            }
        }

        return shell_count - 1;
    }

    bool retryable_infinity_status(const mean_field::mapping::MappingStatus status) {
        return status == mean_field::mapping::MappingStatus::at_compactified_infinity ||
               status == mean_field::mapping::MappingStatus::outside_reference_domain ||
               status == mean_field::mapping::MappingStatus::non_finite_result ||
               status == mean_field::mapping::MappingStatus::non_positive_determinant;
    }

    class ProjectionGeometry {
    public:
        ProjectionGeometry(
            const mean_field::fem::FEM &f,
            const mean_field::mapping::DomainMapper &mapper,
            const mfem::GridFunction &displacement
        )
            : m_fem(f),
              m_mapper(mapper),
              m_displacement(displacement),
              m_workspace(f.mesh->Dimension()) {
        }

        mean_field::mapping::MappingStatus Evaluate(
            mfem::ElementTransformation &transformation,
            const mfem::IntegrationPoint &integration_point,
            mean_field::mapping::MappingPointContext &context,
            const bool permit_infinity_limit
        ) {
            m_used_infinity_limit = false;

            const int element_id  = transformation.ElementNo;
            MFEM_VERIFY(element_id >= 0, "Projection coefficient received an invalid element number.");

            const mfem::FiniteElement &displacement_element     = *m_fem.displacementFes->GetFE(element_id);
            const mfem::FiniteElement &compactification_element = *m_fem.compactificationFes->GetFE(element_id);

            mfem::Array<int> displacement_dofs;
            mfem::Array<int> compactification_dofs;

            mfem::DofTransformation *displacement_transform =
                m_fem.displacementFes->GetElementVDofs(element_id, displacement_dofs);

            mfem::DofTransformation *compactification_transform =
                m_fem.compactificationFes->GetElementDofs(element_id, compactification_dofs);

            mfem::Vector element_displacement;
            mfem::Vector element_compactification;

            m_displacement.GetSubVector(displacement_dofs, element_displacement);
            m_fem.compactificationCoordinate->GetSubVector(compactification_dofs, element_compactification);

            if (displacement_transform != nullptr) {
                displacement_transform->InvTransformPrimal(element_displacement);
            }

            if (compactification_transform != nullptr) {
                compactification_transform->InvTransformPrimal(element_compactification);
            }

            const mean_field::mapping::ElementDisplacementData displacement_data =
                mean_field::mapping::ElementDisplacementDataFromElementVDofs(
                    displacement_element, element_displacement
                );

            const mean_field::mapping::ElementCompactificationData compactification_data(
                compactification_element, element_compactification
            );

            const mean_field::mapping::ElementMappingData mapping_data{
                .displacement = displacement_data, .compactification = compactification_data
            };

            mfem::Vector compactification_shape(compactification_element.GetDof());
            compactification_element.CalcShape(integration_point, compactification_shape);

            const double coordinate = element_compactification * compactification_shape;

            mean_field::mapping::MappingStatus status =
                m_mapper.EvaluatePoint(mapping_data, transformation, integration_point, m_workspace, context);

            if (status == mean_field::mapping::MappingStatus::valid) {
                transformation.SetIntPoint(&integration_point);
                return status;
            }

            const bool infinity_request = m_mapper.IsCompactifiedElement(transformation) && coordinate >= 1.0 - 1.0e-10;

            if (!permit_infinity_limit || !infinity_request || !retryable_infinity_status(status)) {
                transformation.SetIntPoint(&integration_point);
                return status;
            }

            const mfem::IntegrationPoint &center = mfem::Geometries.GetCenter(transformation.GetGeometryType());

            constexpr std::array<double, 11> inward_fractions{1.0e-12, 1.0e-11, 1.0e-10, 1.0e-9, 1.0e-8, 1.0e-7,
                                                              1.0e-6,  1.0e-5,  1.0e-4,  1.0e-3, 1.0e-2};

            for (const double fraction : inward_fractions) {
                mfem::IntegrationPoint inward;
                inward.x      = (1.0 - fraction) * integration_point.x + fraction * center.x;
                inward.y      = (1.0 - fraction) * integration_point.y + fraction * center.y;
                inward.z      = (1.0 - fraction) * integration_point.z + fraction * center.z;
                inward.weight = integration_point.weight;

                status        = m_mapper.EvaluatePoint(mapping_data, transformation, inward, m_workspace, context);

                if (status == mean_field::mapping::MappingStatus::valid) {
                    m_used_infinity_limit = true;
                    transformation.SetIntPoint(&integration_point);
                    return status;
                }

                if (!retryable_infinity_status(status)) {
                    break;
                }
            }

            transformation.SetIntPoint(&integration_point);
            return status;
        }

        [[nodiscard]] bool UsedInfinityLimit() const noexcept {
            return m_used_infinity_limit;
        }

    private:
        const mean_field::fem::FEM &m_fem;
        const mean_field::mapping::DomainMapper &m_mapper;
        const mfem::GridFunction &m_displacement;
        mean_field::mapping::DomainMapper::Workspace m_workspace;
        bool m_used_infinity_limit{false};
    };

    class MonopolePotentialCoefficient final : public mfem::Coefficient {
    public:
        MonopolePotentialCoefficient(
            const mean_field::fem::FEM &f,
            const mean_field::mapping::DomainMapper &mapper,
            const mfem::GridFunction &displacement,
            const double mass,
            const double radius
        )
            : m_geometry(
                  f,
                  mapper,
                  displacement
              ),
              m_vacuum_attribute(field_dof_test_utils::vacuum_material_attribute),
              m_mass(mass),
              m_radius(radius) {
        }

        double Eval(
            mfem::ElementTransformation &transformation,
            const mfem::IntegrationPoint &integration_point
        ) override {
            mean_field::mapping::MappingPointContext context;

            const mean_field::mapping::MappingStatus status =
                m_geometry.Evaluate(transformation, integration_point, context, true);

            MFEM_VERIFY(
                status == mean_field::mapping::MappingStatus::valid,
                "Stateless monopole-potential projection failed with status "
                    << static_cast<int>(status) << " on element " << transformation.ElementNo << '.'
            );

            if (m_geometry.UsedInfinityLimit()) {
                return 0.0;
            }

            const double radius = context.physical_position.Norml2();

            MFEM_VERIFY(std::isfinite(radius) && radius > 0.0, "Invalid monopole projection radius.");

            if (transformation.Attribute == m_vacuum_attribute) {
                return -mean_field::utils::G * m_mass / radius;
            }

            return -mean_field::utils::G * m_mass * (3.0 * m_radius * m_radius - radius * radius) /
                   (2.0 * m_radius * m_radius * m_radius);
        }

    private:
        ProjectionGeometry m_geometry;
        int m_vacuum_attribute;
        double m_mass;
        double m_radius;
    };

    void local_to_true(
        const mfem::ParFiniteElementSpace &space,
        const mfem::Vector &local,
        mfem::Vector &true_vector
    ) {
        true_vector.SetSize(space.GetTrueVSize());
        true_vector                        = 0.0;

        const mfem::Operator *prolongation = space.GetProlongationMatrix();

        if (prolongation != nullptr) {
            prolongation->MultTranspose(local, true_vector);
        } else {
            true_vector = local;
        }
    }

    mfem::Vector assemble_monopole_projection_rhs(
        mean_field::fem::FEM &f,
        const mfem::GridFunction &displacement,
        const double mass,
        const double stellar_radius
    ) {
        mfem::Vector local_rhs(f.gravityFluxFes->GetVSize());
        local_rhs = 0.0;

        mean_field::mapping::DomainMapper::Workspace workspace(f.mesh->Dimension());

        const int vacuum_attribute = field_dof_test_utils::vacuum_material_attribute;
        const int quadrature_order = 2 * f.gravityFluxFes->GetMaxElementOrder() + 8;

        for (int element_id = 0; element_id < f.mesh->GetNE(); ++element_id) {
            const mfem::FiniteElement &gravity_element          = *f.gravityFluxFes->GetFE(element_id);
            const mfem::FiniteElement &displacement_element     = *f.displacementFes->GetFE(element_id);
            const mfem::FiniteElement &compactification_element = *f.compactificationFes->GetFE(element_id);

            mfem::ElementTransformation *transformation         = f.mesh->GetElementTransformation(element_id);

            mfem::Array<int> gravity_dofs;
            mfem::Array<int> displacement_dofs;
            mfem::Array<int> compactification_dofs;

            mfem::DofTransformation *gravity_transform = f.gravityFluxFes->GetElementVDofs(element_id, gravity_dofs);
            mfem::DofTransformation *displacement_transform =
                f.displacementFes->GetElementVDofs(element_id, displacement_dofs);
            mfem::DofTransformation *compactification_transform =
                f.compactificationFes->GetElementDofs(element_id, compactification_dofs);

            mfem::Vector element_displacement;
            mfem::Vector element_compactification;

            displacement.GetSubVector(displacement_dofs, element_displacement);
            f.compactificationCoordinate->GetSubVector(compactification_dofs, element_compactification);

            if (displacement_transform != nullptr) {
                displacement_transform->InvTransformPrimal(element_displacement);
            }

            if (compactification_transform != nullptr) {
                compactification_transform->InvTransformPrimal(element_compactification);
            }

            const mean_field::mapping::ElementDisplacementData displacement_data =
                mean_field::mapping::ElementDisplacementDataFromElementVDofs(
                    displacement_element, element_displacement
                );

            const mean_field::mapping::ElementCompactificationData compactification_data(
                compactification_element, element_compactification
            );

            const mean_field::mapping::ElementMappingData mapping_data{
                .displacement = displacement_data, .compactification = compactification_data
            };

            const int dof_count = gravity_element.GetDof();
            const int dimension = transformation->GetSpaceDim();

            mfem::Vector element_rhs(dof_count);
            mfem::Vector analytic_field(dimension);
            mfem::Vector pulled_rhs_field(dimension);
            mfem::DenseMatrix vector_shape(dof_count, dimension);

            element_rhs                       = 0.0;

            const mfem::IntegrationRule &rule = mfem::IntRules.Get(transformation->GetGeometryType(), quadrature_order);

            for (int q = 0; q < rule.GetNPoints(); ++q) {
                const mfem::IntegrationPoint &point = rule.IntPoint(q);

                mean_field::mapping::VolumeMappingContext context;

                const mean_field::mapping::MappingStatus status =
                    f.domainMapperStateless->EvaluateVolume(mapping_data, *transformation, point, workspace, context);

                MFEM_VERIFY(
                    status == mean_field::mapping::MappingStatus::valid,
                    "Mapped monopole projection RHS failed with status "
                        << static_cast<int>(status) << " on element " << element_id << ", quadrature point " << q << '.'
                );

                analytic_field      = context.mapping.physical_position;

                const double radius = analytic_field.Norml2();

                MFEM_VERIFY(std::isfinite(radius) && radius > 0.0, "Invalid physical radius in projection RHS.");

                if (transformation->Attribute == vacuum_attribute) {
                    analytic_field *= mean_field::utils::G * mass / (radius * radius * radius);
                } else {
                    analytic_field *= mean_field::utils::G * mass / (stellar_radius * stellar_radius * stellar_radius);
                }

                context.mapping.mapping_jacobian.MultTranspose(analytic_field, pulled_rhs_field);

                transformation->SetIntPoint(&point);
                gravity_element.CalcVShape(*transformation, vector_shape);

                const double weight = point.weight * transformation->Weight();

                for (int i = 0; i < dof_count; ++i) {
                    for (int component = 0; component < dimension; ++component) {
                        element_rhs(i) += weight * vector_shape(i, component) * pulled_rhs_field(component);
                    }
                }
            }

            if (gravity_transform != nullptr) {
                gravity_transform->TransformDual(element_rhs);
            }

            local_rhs.AddElementVector(gravity_dofs, element_rhs);
        }

        mfem::Vector true_rhs;
        local_to_true(*f.gravityFluxFes, local_rhs, true_rhs);

        return true_rhs;
    }

    mfem::Vector project_monopole_gradient(
        mean_field::fem::FEM &f,
        const mfem::ParGridFunction &displacement,
        const double mass,
        const double stellar_radius
    ) {
        mfem::Vector displacement_true;
        displacement.GetTrueDofs(displacement_true);

        const mfem::Vector projection_rhs = assemble_monopole_projection_rhs(f, displacement, mass, stellar_radius);

        mean_field::operators::PreparedMappedHDivMassOperator mass_operator(f, *f.domainMapperStateless);
        mass_operator.Prepare(mass_operator.GetDisplacementMap().gather(displacement_true));
        const mfem::Vector reduced_projection_rhs = mass_operator.GetFluxMap().gather(projection_rhs);

        mfem::Vector projected_gradient(mass_operator.Width());
        projected_gradient = 0.0;

        mfem::CGSolver solver(f.gravityFluxFes->GetComm());
        solver.SetOperator(mass_operator);
        solver.SetRelTol(1.0e-9);
        solver.SetAbsTol(1.0e-12);
        solver.SetMaxIter(2000);
        solver.SetPrintLevel(0);
        solver.Mult(reduced_projection_rhs, projected_gradient);

        mfem::Vector projection_residual;
        mass_operator.Mult(projected_gradient, projection_residual);
        projection_residual -= reduced_projection_rhs;

        const double source_norm       = global_norm(reduced_projection_rhs, f.gravityFluxFes->GetComm());
        const double residual_norm     = global_norm(projection_residual, f.gravityFluxFes->GetComm());
        const double relative_residual = residual_norm / std::max(source_norm, std::numeric_limits<double>::epsilon());

        INFO("Mapped H(div) projection converged = " << solver.GetConverged());
        INFO("Mapped H(div) projection iterations = " << solver.GetNumIterations());
        INFO("Mapped H(div) projection reported final norm = " << solver.GetFinalNorm());
        INFO("Mapped H(div) projection direct residual norm = " << residual_norm);
        INFO("Mapped H(div) projection direct relative residual = " << relative_residual);

        REQUIRE(std::isfinite(relative_residual));
        REQUIRE(relative_residual < 1.0e-8);
        return mass_operator.GetFluxMap().scatter(projected_gradient);
    }

    double mapped_hdiv_relative_error(
        mean_field::fem::FEM &f,
        const mfem::ParGridFunction &displacement,
        const mfem::Vector &computed,
        const mfem::Vector &reference
    ) {
        REQUIRE(computed.Size() == reference.Size());

        mfem::Vector displacement_true;
        displacement.GetTrueDofs(displacement_true);

        mean_field::operators::PreparedMappedHDivMassOperator mass_operator(f, *f.domainMapperStateless);
        mass_operator.Prepare(mass_operator.GetDisplacementMap().gather(displacement_true));

        mfem::Vector difference(computed);
        difference -= reference;

        mfem::Vector difference_action;
        mfem::Vector reference_action;

        const mfem::Vector reduced_difference = mass_operator.GetFluxMap().gather(difference);
        const mfem::Vector reduced_reference  = mass_operator.GetFluxMap().gather(reference);
        mass_operator.Mult(reduced_difference, difference_action);
        mass_operator.Mult(reduced_reference, reference_action);

        MPI_Comm communicator          = f.gravityFluxFes->GetComm();

        const double difference_energy = global_dot(reduced_difference, difference_action, communicator);
        const double reference_energy  = global_dot(reduced_reference, reference_action, communicator);

        REQUIRE(difference_energy >= -1.0e-12 * std::abs(reference_energy));
        REQUIRE(reference_energy > 0.0);

        return std::sqrt(std::max(0.0, difference_energy) / reference_energy);
    }

    ShellMeasurement measure_exterior_shells(
        mean_field::fem::FEM &f,
        const mean_field::physics::GravitySolution &solution,
        const mfem::GridFunction &displacement,
        const double mass
    ) {
        std::array<ShellAccumulator, shell_count> local{};
        long long local_invalid_points = 0;

        mean_field::mapping::DomainMapper::Workspace workspace(f.mesh->Dimension());

        const int vacuum_attribute = field_dof_test_utils::vacuum_material_attribute;
        const int quadrature_order =
            2 * std::max(f.gravityPotentialFes->GetMaxElementOrder(), f.gravityFluxFes->GetMaxElementOrder()) + 8;

        for (int element_id = 0; element_id < f.mesh->GetNE(); ++element_id) {
            mfem::ElementTransformation *transformation = f.mesh->GetElementTransformation(element_id);

            if (transformation->Attribute != vacuum_attribute) {
                continue;
            }

            const mfem::FiniteElement &displacement_element     = *f.displacementFes->GetFE(element_id);
            const mfem::FiniteElement &compactification_element = *f.compactificationFes->GetFE(element_id);

            mfem::Array<int> displacement_dofs;
            mfem::Array<int> compactification_dofs;

            mfem::DofTransformation *displacement_transform =
                f.displacementFes->GetElementVDofs(element_id, displacement_dofs);
            mfem::DofTransformation *compactification_transform =
                f.compactificationFes->GetElementDofs(element_id, compactification_dofs);

            mfem::Vector element_displacement;
            mfem::Vector element_compactification;

            displacement.GetSubVector(displacement_dofs, element_displacement);
            f.compactificationCoordinate->GetSubVector(compactification_dofs, element_compactification);

            if (displacement_transform != nullptr) {
                displacement_transform->InvTransformPrimal(element_displacement);
            }

            if (compactification_transform != nullptr) {
                compactification_transform->InvTransformPrimal(element_compactification);
            }

            const mean_field::mapping::ElementDisplacementData displacement_data =
                mean_field::mapping::ElementDisplacementDataFromElementVDofs(
                    displacement_element, element_displacement
                );

            const mean_field::mapping::ElementCompactificationData compactification_data(
                compactification_element, element_compactification
            );

            const mean_field::mapping::ElementMappingData mapping_data{
                .displacement = displacement_data, .compactification = compactification_data
            };

            mfem::Vector compactification_shape(compactification_element.GetDof());

            const mfem::IntegrationRule &rule = mfem::IntRules.Get(transformation->GetGeometryType(), quadrature_order);

            for (int q = 0; q < rule.GetNPoints(); ++q) {
                const mfem::IntegrationPoint &point = rule.IntPoint(q);

                transformation->SetIntPoint(&point);
                compactification_element.CalcShape(point, compactification_shape);

                const double coordinate = element_compactification * compactification_shape;
                const int shell         = get_shell(coordinate);

                mfem::Vector reference_field(3);
                mfem::Vector physical_field(3);
                mfem::Vector physical_position(3);

                solution.gradPhi.GetVectorValue(element_id, point, reference_field);

                mean_field::mapping::VolumeMappingContext context;

                const mean_field::mapping::MappingStatus status =
                    f.domainMapperStateless->EvaluateVolume(mapping_data, *transformation, point, workspace, context);

                if (status != mean_field::mapping::MappingStatus::valid) {
                    ++local_invalid_points;
                    continue;
                }

                physical_position = context.mapping.physical_position;
                mean_field::mapping::MapHDivFluxToPhysical(context.mapping, reference_field, physical_field);

                const double radius = physical_position.Norml2();

                if (!std::isfinite(radius) || radius <= 0.0) {
                    ++local_invalid_points;
                    continue;
                }

                mfem::Vector radial_unit(physical_position);
                radial_unit /= radius;

                const double radial_field = physical_field * radial_unit;

                mfem::Vector tangential_field(physical_field);
                tangential_field.Add(-radial_field, radial_unit);

                const double potential           = solution.phi.GetValue(element_id, point);

                const double scaled_potential    = -radius * potential / (mean_field::utils::G * mass);
                const double scaled_radial_field = radius * radius * radial_field / (mean_field::utils::G * mass);
                const double scaled_tangential_field =
                    radius * radius * tangential_field.Norml2() / (mean_field::utils::G * mass);

                if (!std::isfinite(scaled_potential) || !std::isfinite(scaled_radial_field) ||
                    !std::isfinite(scaled_tangential_field)) {
                    ++local_invalid_points;
                    continue;
                }

                const double weight           = point.weight * transformation->Weight();

                ShellAccumulator &accumulator = local[shell];

                ++accumulator.points;
                accumulator.weight += weight;

                accumulator.minimum_radius = std::min(accumulator.minimum_radius, radius);
                accumulator.maximum_radius = std::max(accumulator.maximum_radius, radius);

                accumulator.potential_error_squared += weight * (scaled_potential - 1.0) * (scaled_potential - 1.0);
                accumulator.radial_error_squared += weight * (scaled_radial_field - 1.0) * (scaled_radial_field - 1.0);
                accumulator.tangential_squared += weight * scaled_tangential_field * scaled_tangential_field;
            }
        }

        MPI_Comm communicator = f.gravityFluxFes->GetComm();

        ShellMeasurement measurement;

        MPI_Allreduce(&local_invalid_points, &measurement.invalid_points, 1, MPI_LONG_LONG, MPI_SUM, communicator);

        for (int shell = 0; shell < shell_count; ++shell) {
            long long points = 0;

            MPI_Allreduce(&local[shell].points, &points, 1, MPI_LONG_LONG, MPI_SUM, communicator);

            const double local_sums[4]{
                local[shell].weight, local[shell].potential_error_squared, local[shell].radial_error_squared,
                local[shell].tangential_squared
            };

            double sums[4]{};

            MPI_Allreduce(local_sums, sums, 4, MPI_DOUBLE, MPI_SUM, communicator);

            double minimum_radius = 0.0;
            double maximum_radius = 0.0;

            MPI_Allreduce(&local[shell].minimum_radius, &minimum_radius, 1, MPI_DOUBLE, MPI_MIN, communicator);
            MPI_Allreduce(&local[shell].maximum_radius, &maximum_radius, 1, MPI_DOUBLE, MPI_MAX, communicator);

            measurement.shells[shell] = {
                .points         = points,
                .minimum_radius = minimum_radius,
                .maximum_radius = maximum_radius,
                .potential_rms_error =
                    sums[0] > 0.0 ? std::sqrt(sums[1] / sums[0]) : std::numeric_limits<double>::infinity(),
                .radial_rms_error =
                    sums[0] > 0.0 ? std::sqrt(sums[2] / sums[0]) : std::numeric_limits<double>::infinity(),
                .tangential_rms = sums[0] > 0.0 ? std::sqrt(sums[3] / sums[0]) : std::numeric_limits<double>::infinity()
            };
        }

        return measurement;
    }

    GravitationalEnergies compute_stellar_energies(
        mean_field::fem::FEM &f,
        const mfem::GridFunction &density,
        const mean_field::physics::GravitySolution &solution,
        const mfem::GridFunction &displacement
    ) {
        mean_field::mapping::DomainMapper::Workspace workspace(f.mesh->Dimension());

        double local_binding             = 0.0;
        double local_virial              = 0.0;
        long long local_invalid_points   = 0;
        double local_minimum_determinant = std::numeric_limits<double>::infinity();
        double local_maximum_determinant = -std::numeric_limits<double>::infinity();

        const int vacuum_attribute       = field_dof_test_utils::vacuum_material_attribute;

        const int order =
            2 * std::max(f.gravityPotentialFes->GetMaxElementOrder(), f.gravityFluxFes->GetMaxElementOrder()) + 8;
        std::array<long long, mapping_status_count> local_status_counts{};

        for (int element_id = 0; element_id < f.mesh->GetNE(); ++element_id) {
            mfem::ElementTransformation *transformation = f.mesh->GetElementTransformation(element_id);
            if (transformation->Attribute == vacuum_attribute) {
                continue;
            }

            const mfem::FiniteElement &displacement_element     = *f.displacementFes->GetFE(element_id);
            const mfem::FiniteElement &compactification_element = *f.compactificationFes->GetFE(element_id);

            mfem::Array<int> displacement_dofs;
            mfem::Array<int> compactification_dofs;

            mfem::DofTransformation *displacement_transform =
                f.displacementFes->GetElementVDofs(element_id, displacement_dofs);
            mfem::DofTransformation *compactification_transform =
                f.compactificationFes->GetElementDofs(element_id, compactification_dofs);

            mfem::Vector element_displacement;
            mfem::Vector element_compactification;

            displacement.GetSubVector(displacement_dofs, element_displacement);
            f.compactificationCoordinate->GetSubVector(compactification_dofs, element_compactification);

            if (displacement_transform != nullptr) {
                displacement_transform->InvTransformPrimal(element_displacement);
            }

            if (compactification_transform != nullptr) {
                compactification_transform->InvTransformPrimal(element_compactification);
            }

            const mean_field::mapping::ElementDisplacementData displacement_data =
                mean_field::mapping::ElementDisplacementDataFromElementVDofs(
                    displacement_element, element_displacement
                );

            const mean_field::mapping::ElementCompactificationData compactification_data(
                compactification_element, element_compactification
            );

            const mean_field::mapping::ElementMappingData mapping_data{
                .displacement = displacement_data, .compactification = compactification_data
            };

            const mfem::IntegrationRule &rule = mfem::IntRules.Get(transformation->GetGeometryType(), order);

            for (int q = 0; q < rule.GetNPoints(); ++q) {
                const mfem::IntegrationPoint &point = rule.IntPoint(q);

                mean_field::mapping::VolumeMappingContext context;

                const mean_field::mapping::MappingStatus status =
                    f.domainMapperStateless->EvaluateVolume(mapping_data, *transformation, point, workspace, context);

                const double mapping_determinant = context.mapping.mapping_determinant;
                if (std::isfinite(mapping_determinant)) {
                    local_minimum_determinant = std::min(local_minimum_determinant, mapping_determinant);
                    local_maximum_determinant = std::max(local_maximum_determinant, mapping_determinant);
                }

                const int status_index = mapping_status_index(status);
                MFEM_VERIFY(status_index >= 0 && status_index < mapping_status_count, "Unexpected mapping status.");
                ++local_status_counts[status_index];

                if (status != mean_field::mapping::MappingStatus::valid) {
                    ++local_invalid_points;
                    continue;
                }

                if (status != mean_field::mapping::MappingStatus::valid) {
                    ++local_invalid_points;
                    continue;
                }

                mfem::Vector reference_field(3);
                mfem::Vector physical_field(3);

                solution.gradPhi.GetVectorValue(element_id, point, reference_field);

                mean_field::mapping::MapHDivFluxToPhysical(context.mapping, reference_field, physical_field);

                const double rho = density.GetValue(element_id, point);
                const double phi = solution.phi.GetValue(element_id, point);

                local_binding += 0.5 * rho * phi * context.quadrature.weight;
                local_virial -= rho * (context.mapping.physical_position * physical_field) * context.quadrature.weight;
            }
        }

        GravitationalEnergies energies;

        MPI_Comm communicator = f.densityFes->GetComm();

        MPI_Allreduce(&local_binding, &energies.binding, 1, MPI_DOUBLE, MPI_SUM, communicator);
        MPI_Allreduce(&local_virial, &energies.virial, 1, MPI_DOUBLE, MPI_SUM, communicator);
        MPI_Allreduce(&local_invalid_points, &energies.invalid_points, 1, MPI_LONG_LONG, MPI_SUM, communicator);
        MPI_Allreduce(
            local_status_counts.data(), energies.mapping_status_counts.data(), mapping_status_count, MPI_LONG_LONG,
            MPI_SUM, communicator
        );
        MPI_Allreduce(
            &local_minimum_determinant, &energies.minimum_mapping_determinant, 1, MPI_DOUBLE, MPI_MIN, communicator
        );
        MPI_Allreduce(
            &local_maximum_determinant, &energies.maximum_mapping_determinant, 1, MPI_DOUBLE, MPI_MAX, communicator
        );

        return energies;
    }
} // namespace

TEST_CASE(
    "Gravity Field Monopole Accuracy And Projection Floor",
    tags::gravity_analytic_accuracy
) {
    auto args              = test_utils::setup_args();
    args.p.rtol            = 1.0e-13;
    args.p.max_iters       = std::max(args.p.max_iters, 1000);

    mean_field::fem::FEM f = mean_field::fem::setup_fem(args.mesh_file, args, 0);

    REQUIRE(f.domainMapperStateless != nullptr);
    REQUIRE(f.domainMapperStateless != nullptr);
    REQUIRE(f.compactificationCoordinate != nullptr);

    const double radius        = mean_field::utils::RADIUS;
    const double mass          = mean_field::utils::MASS;

    const double density_value = mass / ((4.0 / 3.0) * M_PI * radius * radius * radius);

    mfem::ParGridFunction displacement(f.displacementFes.get());
    displacement    = 0.0;

    *f.displacement = 0.0;

    mfem::GridFunction density(f.densityFes.get());
    density = density_value;

    zero_vacuum_density(f, density);

    mean_field::analysis::conserve_mass(f, density, mass);

    f.com = mean_field::analysis::get_com(f, density);
    f.Q   = mean_field::physics::compute_quadrupole_moment_tensor(f, density, f.com);

    const mean_field::physics::GravitySolution numerical_solution =
        mean_field::physics::solve_gravity_field(f, args, density, displacement);

    MonopolePotentialCoefficient potential_coefficient(f, *f.domainMapperStateless, displacement, mass, radius);

    mean_field::physics::GravitySolution projected_solution(f);
    projected_solution.phi     = 0.0;
    projected_solution.gradPhi = 0.0;

    projected_solution.phi.ProjectCoefficient(potential_coefficient);

    const mfem::Vector projected_gradient_true = project_monopole_gradient(f, displacement, mass, radius);

    projected_solution.gradPhi.SetFromTrueDofs(projected_gradient_true);

    const ShellMeasurement numerical     = measure_exterior_shells(f, numerical_solution, displacement, mass);

    const ShellMeasurement projected     = measure_exterior_shells(f, projected_solution, displacement, mass);

    const GravitationalEnergies energies = compute_stellar_energies(f, density, numerical_solution, displacement);

    mfem::Vector numerical_gradient;
    mfem::Vector numerical_potential;
    mfem::Vector projected_gradient;
    mfem::Vector projected_potential;

    numerical_solution.gradPhi.GetTrueDofs(numerical_gradient);
    numerical_solution.phi.GetTrueDofs(numerical_potential);

    projected_solution.gradPhi.GetTrueDofs(projected_gradient);
    projected_solution.phi.GetTrueDofs(projected_potential);

    MPI_Comm communicator = f.gravityFluxFes->GetComm();

    const double gradient_projection_gap =
        mapped_hdiv_relative_error(f, displacement, numerical_gradient, projected_gradient);

    const double potential_projection_gap =
        global_relative_error(numerical_potential, projected_potential, communicator);

    double maximum_numerical_potential_error = 0.0;
    double maximum_projected_potential_error = 0.0;
    double maximum_numerical_radial_error    = 0.0;
    double maximum_projected_radial_error    = 0.0;
    double maximum_numerical_tangential      = 0.0;
    double maximum_projected_tangential      = 0.0;

    std::ostringstream report;

    for (int shell = 0; shell < shell_count; ++shell) {
        const ShellMetrics &numerical_shell = numerical.shells[shell];
        const ShellMetrics &projected_shell = projected.shells[shell];

        maximum_numerical_potential_error =
            std::max(maximum_numerical_potential_error, numerical_shell.potential_rms_error);

        maximum_projected_potential_error =
            std::max(maximum_projected_potential_error, projected_shell.potential_rms_error);

        maximum_numerical_radial_error = std::max(maximum_numerical_radial_error, numerical_shell.radial_rms_error);

        maximum_projected_radial_error = std::max(maximum_projected_radial_error, projected_shell.radial_rms_error);

        maximum_numerical_tangential   = std::max(maximum_numerical_tangential, numerical_shell.tangential_rms);

        maximum_projected_tangential   = std::max(maximum_projected_tangential, projected_shell.tangential_rms);

        report << "shell " << shell << " xi=[" << shell_boundaries[shell] << ", " << shell_boundaries[shell + 1]
               << ")\n"
               << "  radius=[" << numerical_shell.minimum_radius << ", " << numerical_shell.maximum_radius << "]\n"
               << "  potential error: solved=" << numerical_shell.potential_rms_error
               << ", projection=" << projected_shell.potential_rms_error << '\n'
               << "  radial error: solved=" << numerical_shell.radial_rms_error
               << ", projection=" << projected_shell.radial_rms_error << '\n'
               << "  tangential amplitude: solved=" << numerical_shell.tangential_rms
               << ", projection=" << projected_shell.tangential_rms << '\n';
    }

    const double analytic_energy   = -3.0 * mean_field::utils::G * mass * mass / (5.0 * radius);

    const double binding_error     = std::abs(energies.binding - analytic_energy) / std::abs(analytic_energy);
    const double virial_error      = std::abs(energies.virial - analytic_energy) / std::abs(analytic_energy);
    const double consistency_error = std::abs(energies.binding - energies.virial) / std::abs(energies.binding);

    INFO(report.str());

    INFO("Gradient solution/projection mapped H(div) gap = " << gradient_projection_gap);
    INFO("Potential solution/projection DOF gap = " << potential_projection_gap);
    INFO("Analytic energy = " << analytic_energy);
    INFO("Computed binding energy = " << energies.binding);
    INFO("Computed virial energy = " << energies.virial);
    INFO("Relative binding error = " << binding_error);
    INFO("Relative virial error = " << virial_error);
    INFO("Relative virial consistency error = " << consistency_error);

    REQUIRE(numerical.invalid_points == 0);
    REQUIRE(projected.invalid_points == 0);
    REQUIRE(energies.invalid_points == 0);

    for (int shell = 0; shell < shell_count; ++shell) {
        REQUIRE(numerical.shells[shell].points > 0);
        REQUIRE(projected.shells[shell].points > 0);
    }

    CHECK(maximum_numerical_potential_error < 5.0e-2);
    CHECK(maximum_projected_potential_error < 5.0e-2);
    CHECK(maximum_numerical_radial_error < 5.0e-3);
    CHECK(maximum_projected_radial_error < 5.0e-3);
    CHECK(maximum_numerical_tangential < 5.0e-3);
    CHECK(maximum_projected_tangential < 5.0e-3);
    CHECK(gradient_projection_gap < 5.0e-3);
    CHECK(potential_projection_gap < maximum_numerical_potential_error);

    constexpr double virial_target = 1.0e-6;

    CHECK(binding_error < virial_target);
    CHECK(virial_error < virial_target);
    CHECK(consistency_error < virial_target);
}

TEST_CASE(
    "Gravity Field Virial Consistency Across Volume Preserving Deformation",
    tags::gravity_consistency_accuracy
) {
    auto args              = test_utils::setup_args();
    args.p.rtol            = 1.0e-13;
    args.p.max_iters       = std::max(args.p.max_iters, 1000);

    mean_field::fem::FEM f = mean_field::fem::setup_fem(args.mesh_file, args, 0);

    REQUIRE(f.domainMapperStateless != nullptr);
    REQUIRE(f.domainMapperStateless != nullptr);

    const double radius          = mean_field::utils::RADIUS;
    const double mass            = mean_field::utils::MASS;

    const double central_density = 15.0 * mass / (8.0 * M_PI * radius * radius * radius);

    auto density_function        = [central_density, radius](const mfem::Vector &position) {
        const double normalized_radius_squared = (position * position) / (radius * radius);
        return central_density * std::max(0.0, 1.0 - normalized_radius_squared);
    };

    mfem::FunctionCoefficient density_coefficient(density_function);
    mfem::GridFunction density(f.densityFes.get());
    density.ProjectCoefficient(density_coefficient);

    zero_vacuum_density(f, density);

    mean_field::analysis::conserve_mass(f, density, mass);

    constexpr std::array<double, 7> amplitudes{0.0, 0.02, 0.05, 0.1, 0.2, 0.5, 1.0};

    std::array<double, amplitudes.size()> consistency_errors{};
    std::array<double, amplitudes.size()> normalized_quadrupoles{};
    std::array<double, amplitudes.size()> binding_energies{};
    std::array<double, amplitudes.size()> virial_energies{};

    std::ostringstream report;

    for (std::size_t index = 0; index < amplitudes.size(); ++index) {
        const double amplitude = amplitudes[index];

        const double x_scale   = 1.0 + 0.15 * amplitude;
        const double y_scale   = 1.0 - 0.05 * amplitude;
        const double z_scale   = 1.0 / (x_scale * y_scale);

        REQUIRE_THAT(x_scale * y_scale * z_scale, Catch::Matchers::WithinAbs(1.0, 1.0e-14));

        auto displacement_function = [x_scale, y_scale, z_scale](const mfem::Vector &position, mfem::Vector &value) {
            value.SetSize(3);

            value(0) = (x_scale - 1.0) * position(0);
            value(1) = (y_scale - 1.0) * position(1);
            value(2) = (z_scale - 1.0) * position(2);
        };

        mfem::VectorFunctionCoefficient displacement_coefficient(3, displacement_function);
        mfem::ParGridFunction displacement(f.displacementFes.get());
        displacement.ProjectCoefficient(displacement_coefficient);

        *f.displacement = displacement;

        f.com           = mean_field::analysis::get_com(f, density);
        f.Q             = mean_field::physics::compute_quadrupole_moment_tensor(f, density, f.com);
        const mfem::FiniteElementSpace *nodal_space = f.mesh->GetNodalFESpace();

        const mean_field::physics::GravitySolution solution =
            mean_field::physics::solve_gravity_field(f, args, density, displacement);
        const GravitationalEnergies energies = compute_stellar_energies(f, density, solution, displacement);

        CAPTURE(amplitude, x_scale, y_scale, z_scale);
        INFO(
            "Mapping status valid = "
            << energies.mapping_status_counts[mapping_status_index(mean_field::mapping::MappingStatus::valid)]
        );
        INFO(
            "Mapping status invalid_dimension = "
            << energies
                   .mapping_status_counts[mapping_status_index(mean_field::mapping::MappingStatus::invalid_dimension)]
        );
        INFO(
            "Mapping status non_finite_input = "
            << energies
                   .mapping_status_counts[mapping_status_index(mean_field::mapping::MappingStatus::non_finite_input)]
        );
        INFO(
            "Mapping status invalid_reference_radius = " << energies.mapping_status_counts[mapping_status_index(
                mean_field::mapping::MappingStatus::invalid_reference_radius
            )]
        );
        INFO(
            "Mapping status at_compactified_infinity = " << energies.mapping_status_counts[mapping_status_index(
                mean_field::mapping::MappingStatus::at_compactified_infinity
            )]
        );
        INFO(
            "Mapping status outside_reference_domain = " << energies.mapping_status_counts[mapping_status_index(
                mean_field::mapping::MappingStatus::outside_reference_domain
            )]
        );
        INFO(
            "Mapping status non_finite_result = "
            << energies
                   .mapping_status_counts[mapping_status_index(mean_field::mapping::MappingStatus::non_finite_result)]
        );
        INFO(
            "Mapping status non_positive_determinant = " << energies.mapping_status_counts[mapping_status_index(
                mean_field::mapping::MappingStatus::non_positive_determinant
            )]
        );
        INFO("Total invalid mapping points = " << energies.invalid_points);
        INFO("Mesh nodal order = " << (nodal_space != nullptr ? nodal_space->GetMaxElementOrder() : -1));
        INFO("Displacement order = " << f.displacementFes->GetMaxElementOrder());
        INFO("Minimum discrete mapping determinant = " << energies.minimum_mapping_determinant);
        INFO("Maximum discrete mapping determinant = " << energies.maximum_mapping_determinant);

        REQUIRE(energies.invalid_points == 0);
        REQUIRE(std::isfinite(energies.binding));
        REQUIRE(std::isfinite(energies.virial));
        REQUIRE(energies.binding < 0.0);
        REQUIRE(energies.virial < 0.0);

        binding_energies[index]       = energies.binding;
        virial_energies[index]        = energies.virial;

        consistency_errors[index]     = std::abs(energies.binding - energies.virial) / std::abs(energies.binding);

        normalized_quadrupoles[index] = f.Q.FNorm() / (mass * radius * radius);

        report << "amplitude=" << amplitude << ", scales=(" << x_scale << ", " << y_scale << ", " << z_scale
               << "), normalized quadrupole=" << normalized_quadrupoles[index]
               << ", binding=" << binding_energies[index] << ", virial=" << virial_energies[index]
               << ", consistency error=" << consistency_errors[index] << '\n';
    }

    INFO(report.str());

    for (std::size_t index = 1; index < amplitudes.size(); ++index) {
        CHECK(normalized_quadrupoles[index] > normalized_quadrupoles[index - 1]);
    }

    CHECK(consistency_errors[0] < 1.0e-5);
    for (std::size_t index = 1; index < amplitudes.size(); ++index) {
        CHECK(consistency_errors[index] < 5.0e-5);
    }
}