solid_mechanics.hpp

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// Copyright (c) Lawrence Livermore National Security, LLC and
// other Serac Project Developers. See the top-level LICENSE file for
// details.
//
// SPDX-License-Identifier: (BSD-3-Clause)
 
#pragma once
 
#include <cstddef>
#include <array>
#include <functional>
#include <memory>
#include <optional>
#include <string>
#include <unordered_map>
#include <utility>
#include <vector>
 
#include "mfem.hpp"
 
#include "serac/serac_config.hpp"
#include "serac/physics/common.hpp"
#include "serac/physics/solid_mechanics_input.hpp"
#include "serac/physics/base_physics.hpp"
#include "serac/physics/boundary_conditions/components.hpp"
#include "serac/numerics/odes.hpp"
#include "serac/numerics/stdfunction_operator.hpp"
#include "serac/numerics/functional/shape_aware_functional.hpp"
#include "serac/numerics/functional/domain.hpp"
#include "serac/physics/mesh.hpp"
#include "serac/physics/state/state_manager.hpp"
#include "serac/physics/materials/solid_material.hpp"
#include "serac/infrastructure/accelerator.hpp"
#include "serac/infrastructure/profiling.hpp"
#include "serac/numerics/equation_solver.hpp"
#include "serac/numerics/functional/differentiate_wrt.hpp"
#include "serac/numerics/functional/functional.hpp"
#include "serac/numerics/functional/geometry.hpp"
#include "serac/numerics/functional/quadrature_data.hpp"
#include "serac/numerics/functional/tensor.hpp"
#include "serac/numerics/functional/tuple.hpp"
#include "serac/numerics/petsc_solvers.hpp"
#include "serac/numerics/solver_config.hpp"
#include "serac/physics/boundary_conditions/boundary_condition_manager.hpp"
#include "serac/physics/state/finite_element_dual.hpp"
#include "serac/physics/state/finite_element_state.hpp"
#include "serac/physics/state/finite_element_vector.hpp"
 
namespace serac {
namespace solid_mechanics {
 
namespace detail {
void adjoint_integrate(double dt_n, double dt_np1, mfem::HypreParMatrix* m_mat, mfem::HypreParMatrix* k_mat,
                       mfem::HypreParVector& disp_adjoint_load_vector, mfem::HypreParVector& velo_adjoint_load_vector,
                       mfem::HypreParVector& accel_adjoint_load_vector, mfem::HypreParVector& adjoint_displacement_,
                       mfem::HypreParVector& implicit_sensitivity_displacement_start_of_step_,
                       mfem::HypreParVector& implicit_sensitivity_velocity_start_of_step_,
                       mfem::HypreParVector& adjoint_essential_, BoundaryConditionManager& bcs_,
                       mfem::Solver& lin_solver);
}  // namespace detail
 
const LinearSolverOptions default_linear_options = {.linear_solver = LinearSolver::GMRES,
                                                    .preconditioner = serac::ordering == mfem::Ordering::byVDIM
                                                                          ? Preconditioner::HypreAMG
                                                                          : Preconditioner::HypreJacobi,
                                                    .relative_tol = 1.0e-6,
                                                    .absolute_tol = 1.0e-16,
                                                    .max_iterations = 500,
                                                    .print_level = 0};
 
#ifdef MFEM_USE_STRUMPACK
const LinearSolverOptions direct_linear_options = {.linear_solver = LinearSolver::Strumpack, .print_level = 0};
#else
const LinearSolverOptions direct_linear_options = {.linear_solver = LinearSolver::SuperLU, .print_level = 0};
#endif
 
const NonlinearSolverOptions default_nonlinear_options = {.nonlin_solver = NonlinearSolver::Newton,
                                                          .relative_tol = 1.0e-4,
                                                          .absolute_tol = 1.0e-8,
                                                          .max_iterations = 10,
                                                          .print_level = 1};
 
const TimesteppingOptions default_quasistatic_options = {TimestepMethod::QuasiStatic};
 
const TimesteppingOptions default_timestepping_options = {TimestepMethod::Newmark,
                                                          DirichletEnforcementMethod::RateControl};
 
}  // namespace solid_mechanics
 
template <int order, int dim, typename parameters = Parameters<>,
          typename parameter_indices = std::make_integer_sequence<int, parameters::n>>
class SolidMechanics;
 
template <int order, int dim, typename... parameter_space, int... parameter_indices>
class SolidMechanics<order, dim, Parameters<parameter_space...>, std::integer_sequence<int, parameter_indices...>>
    : public BasePhysics {
 public:
  static constexpr int VALUE = 0, DERIVATIVE = 1;
  static constexpr int SHAPE = 0;
  static constexpr auto I = Identity<dim>();
 
  static constexpr auto NUM_STATE_VARS = 2;
 
  template <typename T>
  using qdata_type = std::shared_ptr<QuadratureData<T>>;
 
  SolidMechanics(const NonlinearSolverOptions nonlinear_opts, const LinearSolverOptions lin_opts,
                 const serac::TimesteppingOptions timestepping_opts, const std::string& physics_name,
                 std::shared_ptr<serac::Mesh> serac_mesh, std::vector<std::string> parameter_names = {}, int cycle = 0,
                 double time = 0.0, bool checkpoint_to_disk = false, bool use_warm_start = true)
      : SolidMechanics(std::make_unique<EquationSolver>(nonlinear_opts, lin_opts, serac_mesh->getComm()),
                       timestepping_opts, physics_name, serac_mesh, parameter_names, cycle, time, checkpoint_to_disk,
                       use_warm_start)
  {
  }
 
  SolidMechanics(std::unique_ptr<serac::EquationSolver> solver, const serac::TimesteppingOptions timestepping_opts,
                 const std::string& physics_name, std::shared_ptr<serac::Mesh> serac_mesh,
                 std::vector<std::string> parameter_names = {}, int cycle = 0, double time = 0.0,
                 bool checkpoint_to_disk = false, bool use_warm_start = true)
      : BasePhysics(physics_name, serac_mesh, cycle, time, checkpoint_to_disk),
        displacement_(
            StateManager::newState(H1<order, dim>{}, detail::addPrefix(physics_name, "displacement"), mesh_->tag())),
        velocity_(StateManager::newState(H1<order, dim>{}, detail::addPrefix(physics_name, "velocity"), mesh_->tag())),
        acceleration_(
            StateManager::newState(H1<order, dim>{}, detail::addPrefix(physics_name, "acceleration"), mesh_->tag())),
        adjoint_displacement_(StateManager::newState(
            H1<order, dim>{}, detail::addPrefix(physics_name, "adjoint_displacement"), mesh_->tag())),
        displacement_adjoint_load_(displacement_.space(), detail::addPrefix(physics_name, "displacement_adjoint_load")),
        velocity_adjoint_load_(displacement_.space(), detail::addPrefix(physics_name, "velocity_adjoint_load")),
        acceleration_adjoint_load_(displacement_.space(), detail::addPrefix(physics_name, "acceleration_adjoint_load")),
        implicit_sensitivity_displacement_start_of_step_(displacement_.space(), "total_deriv_wrt_displacement."),
        implicit_sensitivity_velocity_start_of_step_(displacement_.space(), "total_deriv_wrt_velocity."),
        reactions_(StateManager::newDual(H1<order, dim>{}, detail::addPrefix(physics_name, "reactions"), mesh_->tag())),
        reactions_adjoint_bcs_(reactions_.space(), "reactions_shape_sensitivity"),
        nonlin_solver_(std::move(solver)),
        ode2_(displacement_.space().TrueVSize(),
              {.time = time_, .c0 = c0_, .c1 = c1_, .u = u_, .du_dt = v_, .d2u_dt2 = acceleration_}, *nonlin_solver_,
              bcs_),
        use_warm_start_(use_warm_start)
  {
    SERAC_MARK_FUNCTION;
    SLIC_ERROR_ROOT_IF(mfemParMesh().Dimension() != dim,
                       axom::fmt::format("Compile time dimension, {0}, and runtime mesh dimension, {1}, mismatch", dim,
                                         mfemParMesh().Dimension()));
 
    SLIC_ERROR_ROOT_IF(!nonlin_solver_,
                       "EquationSolver argument is nullptr in SolidMechanics constructor. It is possible that it was "
                       "previously moved.");
 
    // Check for dynamic mode
    if (timestepping_opts.timestepper != TimestepMethod::QuasiStatic) {
      ode2_.SetTimestepper(timestepping_opts.timestepper);
      ode2_.SetEnforcementMethod(timestepping_opts.enforcement_method);
      is_quasistatic_ = false;
    } else {
      is_quasistatic_ = true;
    }
 
    states_.push_back(&displacement_);
    if (!is_quasistatic_) {
      states_.push_back(&velocity_);
      states_.push_back(&acceleration_);
    }
 
    adjoints_.push_back(&adjoint_displacement_);
    duals_.push_back(&reactions_);
    dual_adjoints_.push_back(&reactions_adjoint_bcs_);
 
    // Create a pack of the primal field and parameter finite element spaces
    const mfem::ParFiniteElementSpace* test_space = &displacement_.space();
    const mfem::ParFiniteElementSpace* shape_space = &mesh_->shapeDisplacementSpace();
 
    std::array<const mfem::ParFiniteElementSpace*, NUM_STATE_VARS + sizeof...(parameter_space)> trial_spaces;
    trial_spaces[0] = &displacement_.space();
    trial_spaces[1] = &displacement_.space();
 
    SLIC_ERROR_ROOT_IF(
        sizeof...(parameter_space) != parameter_names.size(),
        axom::fmt::format("{} parameter spaces given in the template argument but {} parameter names were supplied.",
                          sizeof...(parameter_space), parameter_names.size()));
 
    if constexpr (sizeof...(parameter_space) > 0) {
      tuple<parameter_space...> types{};
      for_constexpr<sizeof...(parameter_space)>([&](auto i) {
        parameters_.emplace_back(mfemParMesh(), get<i>(types), detail::addPrefix(name_, parameter_names[i]));
        trial_spaces[i + NUM_STATE_VARS] = &(parameters_[i].state->space());
      });
    }
 
    residual_ = std::make_unique<ShapeAwareFunctional<shape_trial, test(trial, trial, parameter_space...)>>(
        shape_space, test_space, trial_spaces);
 
    // If the user wants the AMG preconditioner with a linear solver, set the pfes
    // to be the displacement
    auto* amg_prec = dynamic_cast<mfem::HypreBoomerAMG*>(&nonlin_solver_->preconditioner());
    if (amg_prec) {
      // ZRA - Iterative refinement tends to be more expensive than it is worth
      // We should add a flag allowing users to enable it
 
      // bool iterative_refinement = false;
      // amg_prec->SetElasticityOptions(&displacement_.space(), iterative_refinement);
 
      // SetElasticityOptions only works with byVDIM ordering, some evidence that it is not often optimal
      amg_prec->SetSystemsOptions(displacement_.space().GetVDim(), serac::ordering == mfem::Ordering::byNODES);
    }
 
    int true_size = velocity_.space().TrueVSize();
 
    u_.SetSize(true_size);
    v_.SetSize(true_size);
    du_.SetSize(true_size);
    predicted_displacement_.SetSize(true_size);
 
    initializeSolidMechanicsStates();
  }
 
  virtual ~SolidMechanics() {}
 
  void initializeSolidMechanicsStates()
  {
    c0_ = 0.0;
    c1_ = 0.0;
 
    time_end_step_ = 0.0;
 
    displacement_ = 0.0;
    velocity_ = 0.0;
    acceleration_ = 0.0;
 
    adjoint_displacement_ = 0.0;
    displacement_adjoint_load_ = 0.0;
    velocity_adjoint_load_ = 0.0;
    acceleration_adjoint_load_ = 0.0;
 
    implicit_sensitivity_displacement_start_of_step_ = 0.0;
    implicit_sensitivity_velocity_start_of_step_ = 0.0;
 
    reactions_ = 0.0;
    reactions_adjoint_bcs_ = 0.0;
 
    u_ = 0.0;
    v_ = 0.0;
    du_ = 0.0;
    predicted_displacement_ = 0.0;
  }
 
  void resetStates(int cycle = 0, double time = 0.0) override
  {
    BasePhysics::initializeBasePhysicsStates(cycle, time);
    initializeSolidMechanicsStates();
 
    if (checkpoint_to_disk_) {
      outputStateToDisk();
    } else {
      checkpoint_states_.clear();
      auto state_names = stateNames();
      for (const auto& state_name : state_names) {
        checkpoint_states_[state_name].push_back(state(state_name));
      }
    }
  }
 
  template <typename T>
  qdata_type<T> createQuadratureDataBuffer(T initial_state, const std::optional<Domain>& optional_domain = std::nullopt)
  {
    Domain domain = (optional_domain) ? *optional_domain : EntireDomain(mfemParMesh());
    return StateManager::newQuadratureDataBuffer(mesh_->tag(), domain, order, dim, initial_state);
  }
 
  template <typename AppliedDisplacementFunction>
  void setDisplacementBCs(AppliedDisplacementFunction applied_displacement, Domain& domain,
                          Components components = Component::ALL)
  {
    for (int i = 0; i < dim; ++i) {
      if (components[size_t(i)]) {
        auto mfem_coefficient_function = [applied_displacement, i](const mfem::Vector& X_mfem, double t) {
          auto X = make_tensor<dim>([&X_mfem](int k) { return X_mfem[k]; });
          return applied_displacement(X, t)[i];
        };
 
        component_disp_bdr_coef_ = std::make_shared<mfem::FunctionCoefficient>(mfem_coefficient_function);
 
        auto dof_list = domain.dof_list(&displacement_.space());
 
        // scalar ldofs -> vector ldofs
        displacement_.space().DofsToVDofs(i, dof_list);
 
        bcs_.addEssential(dof_list, component_disp_bdr_coef_, displacement_.space(), i);
      }
    }
  }
 
  void setFixedBCs(Domain& domain, Components components = Component::ALL)
  {
    auto zero_vector_function = [](tensor<double, dim>, double) { return tensor<double, dim>{}; };
    setDisplacementBCs(zero_vector_function, domain, components);
  }
 
  template <typename AppliedDisplacementFunction>
  void setDisplacementBCsByDofList(AppliedDisplacementFunction applied_displacement, const mfem::Array<int> true_dofs)
  {
    // std::function<void(const mfem::Vector&, double, mfem::Vector&)> disp
    auto mfem_vector_coefficient_function = [applied_displacement](const mfem::Vector& X_mfem, double t,
                                                                   mfem::Vector& u_mfem) {
      auto X = make_tensor<dim>([&X_mfem](int k) { return X_mfem[k]; });
      auto u = applied_displacement(X, t);
      for (int i = 0; i < dim; i++) {
        u_mfem(i) = u[i];
      }
    };
    disp_bdr_coef_ = std::make_shared<mfem::VectorFunctionCoefficient>(dim, mfem_vector_coefficient_function);
    bcs_.addEssentialByTrueDofs(true_dofs, disp_bdr_coef_, displacement_.space());
  }
 
  const FiniteElementState& state(const std::string& state_name) const override
  {
    if (state_name == "displacement") {
      return displacement_;
    } else if (state_name == "velocity") {
      return velocity_;
    } else if (state_name == "acceleration") {
      return acceleration_;
    }
 
    SLIC_ERROR_ROOT(axom::fmt::format("State '{}' requested from solid mechanics module '{}', but it doesn't exist",
                                      state_name, name_));
    return displacement_;
  }
 
  void setState(const std::string& state_name, const FiniteElementState& state) override
  {
    if (state_name == "displacement") {
      displacement_ = state;
      if (!checkpoint_to_disk_) {
        checkpoint_states_["displacement"][static_cast<size_t>(cycle_)] = displacement_;
      }
      return;
    } else if (state_name == "velocity") {
      velocity_ = state;
      if (!checkpoint_to_disk_) {
        checkpoint_states_["velocity"][static_cast<size_t>(cycle_)] = velocity_;
      }
      return;
    }
 
    SLIC_ERROR_ROOT(axom::fmt::format(
        "setState for state named '{}' requested from solid mechanics module '{}', but it doesn't exist", state_name,
        name_));
  }
 
  std::vector<std::string> stateNames() const override
  {
    if (is_quasistatic_) {
      return std::vector<std::string>{"displacement"};
    } else {
      return std::vector<std::string>{"displacement", "velocity", "acceleration"};
    }
  }
 
  template <int... active_parameters, typename callable>
  void addCustomBoundaryIntegral(DependsOn<active_parameters...>, callable qfunction,
                                 const std::optional<Domain>& optional_domain = std::nullopt)
  {
    Domain domain = (optional_domain) ? *optional_domain : EntireBoundary(mfemParMesh());
 
    residual_->AddBoundaryIntegral(Dimension<dim - 1>{}, DependsOn<0, 1, active_parameters + NUM_STATE_VARS...>{},
                                   qfunction, domain);
  }
 
  std::vector<std::string> adjointNames() const override { return std::vector<std::string>{{"displacement"}}; }
 
  const FiniteElementState& adjoint(const std::string& state_name) const override
  {
    if (state_name == "displacement") {
      return adjoint_displacement_;
    }
 
    SLIC_ERROR_ROOT(axom::fmt::format("adjoint '{}' requested from solid mechanics module '{}', but it doesn't exist",
                                      state_name, name_));
    return adjoint_displacement_;
  }
 
  std::vector<std::string> dualNames() const override { return std::vector<std::string>{{"reactions"}}; }
 
  const FiniteElementDual& dual(const std::string& dual_name) const override
  {
    if (dual_name == "reactions") {
      return reactions_;
    }
 
    SLIC_ERROR_ROOT(axom::fmt::format("dual '{}' requested from solid mechanics module '{}', but it doesn't exist",
                                      dual_name, name_));
    return reactions_;
  }
 
  const FiniteElementState& dualAdjoint(const std::string& dual_name) const override
  {
    if (dual_name == "reactions") {
      return reactions_adjoint_bcs_;
    }
 
    SLIC_ERROR_ROOT(axom::fmt::format(
        "dualAdjoint '{}' requested from solid mechanics module '{}', but it doesn't exist", dual_name, name_));
    return reactions_adjoint_bcs_;
  }
 
  FiniteElementDual loadCheckpointedDual(const std::string& dual_name, int cycle) override
  {
    if (dual_name == "reactions") {
      auto checkpointed_sol = getCheckpointedStates(cycle);
 
      FiniteElementDual reactions(reactions_.space(), "reactions_tmp");
 
      if (is_quasistatic_) {
        reactions = (*residual_)(time_, shapeDisplacement(), checkpointed_sol.at("displacement"), acceleration_,
                                 *parameters_[parameter_indices].state...);
      } else {
        reactions = (*residual_)(time_, shapeDisplacement(), checkpointed_sol.at("displacement"),
                                 checkpointed_sol.at("acceleration"), *parameters_[parameter_indices].state...);
      }
 
      return reactions;
    }
 
    SLIC_ERROR_ROOT(
        axom::fmt::format("loadCheckpointedDual '{}' requested from solid mechanics module '{}', but it doesn't exist",
                          dual_name, name_));
    return reactions_;
  }
 
  template <int... active_parameters, typename callable, typename StateType = Nothing>
  void addCustomDomainIntegral(DependsOn<active_parameters...>, callable qfunction, Domain& domain,
                               qdata_type<StateType> qdata = NoQData)
  {
    residual_->AddDomainIntegral(Dimension<dim>{}, DependsOn<0, 1, active_parameters + NUM_STATE_VARS...>{}, qfunction,
                                 domain, qdata);
  }
 
  template <typename Material>
  struct MaterialStressFunctor {
    MaterialStressFunctor(Material material) : material_(material) {}
 
    Material material_;
 
    template <typename X, typename State, typename Displacement, typename Acceleration, typename... Params>
    auto SERAC_HOST_DEVICE operator()(double, X, State& state, Displacement displacement, Acceleration acceleration,
                                      Params... params) const
    {
      auto du_dX = get<DERIVATIVE>(displacement);
      auto d2u_dt2 = get<VALUE>(acceleration);
 
      auto stress = material_(state, du_dX, params...);
 
      return serac::tuple{material_.density * d2u_dt2, stress};
    }
  };
 
  template <int... active_parameters, typename MaterialType, typename StateType = Empty>
  void setMaterial(DependsOn<active_parameters...>, const MaterialType& material, Domain& domain,
                   qdata_type<StateType> qdata = EmptyQData)
  {
    static_assert(std::is_same_v<StateType, Empty> || std::is_same_v<StateType, typename MaterialType::State>,
                  "invalid quadrature data provided in setMaterial()");
    MaterialStressFunctor<MaterialType> material_functor(material);
    residual_->AddDomainIntegral(
        Dimension<dim>{},
        DependsOn<0, 1,
                  active_parameters + NUM_STATE_VARS...>{},  // the magic number "+ NUM_STATE_VARS" accounts for the
                                                             // fact that the displacement, acceleration, and shape
                                                             // fields are always-on and come first, so the `n`th
                                                             // parameter will actually be argument `n + NUM_STATE_VARS`
        std::move(material_functor), domain, qdata);
  }
 
  template <typename MaterialType, typename StateType = Empty>
  void setMaterial(const MaterialType& material, Domain& domain,
                   std::shared_ptr<QuadratureData<StateType>> qdata = EmptyQData)
  {
    setMaterial(DependsOn<>{}, material, domain, qdata);
  }
 
  template <typename Material>
  struct RateDependentMaterialStressFunctor {
    RateDependentMaterialStressFunctor(Material material, const double* dt) : material_(material), time_increment_(dt)
    {
    }
 
    Material material_;
 
    const double* time_increment_;
 
    template <typename X, typename State, typename Displacement, typename Acceleration, typename... Params>
    auto SERAC_HOST_DEVICE operator()(double, X, State& state, Displacement displacement, Acceleration acceleration,
                                      Params... params) const
    {
      auto du_dX = get<DERIVATIVE>(displacement);
      auto d2u_dt2 = get<VALUE>(acceleration);
 
      auto stress = material_(state, *time_increment_, du_dX, params...);
 
      return serac::tuple{material_.density * d2u_dt2, stress};
    }
  };
 
  template <int... active_parameters, typename RateDependentMaterialType, typename StateType = Empty>
  void setRateDependentMaterial(DependsOn<active_parameters...>, const RateDependentMaterialType& material,
                                Domain& domain, qdata_type<StateType> qdata = EmptyQData)
  {
    static_assert(
        std::is_same_v<StateType, Empty> || std::is_same_v<StateType, typename RateDependentMaterialType::State>,
        "invalid quadrature data provided in setMaterial()");
    RateDependentMaterialStressFunctor<RateDependentMaterialType> material_functor(material, &dt_);
    residual_->AddDomainIntegral(
        Dimension<dim>{},
        DependsOn<0, 1,
                  active_parameters + NUM_STATE_VARS...>{},  // the magic number "+ NUM_STATE_VARS" accounts for the
                                                             // fact that the displacement, acceleration, and shape
                                                             // fields are always-on and come first, so the `n`th
                                                             // parameter will actually be argument `n + NUM_STATE_VARS`
        std::move(material_functor), domain, qdata);
  }
 
  template <typename RateDependentMaterialType, typename StateType = Empty>
  void setRateDependentMaterial(const RateDependentMaterialType& material, Domain& domain,
                                std::shared_ptr<QuadratureData<StateType>> qdata = EmptyQData)
  {
    setRateDependentMaterial(DependsOn<>{}, material, domain, qdata);
  }
 
  template <typename AppliedDisplacementFunction>
  void setDisplacement(AppliedDisplacementFunction applied_displacement)
  {
    displacement_.setFromFieldFunction(applied_displacement);
  }
 
  void setDisplacement(const FiniteElementState& temp) { displacement_ = temp; }
 
  template <typename AppliedVelocityFunction>
  void setVelocity(AppliedVelocityFunction applied_velocity)
  {
    velocity_.setFromFieldFunction(applied_velocity);
  }
 
  void setVelocity(const FiniteElementState& temp) { velocity_ = temp; }
 
  template <typename BodyForceType>
  struct BodyForceIntegrand {
    BodyForceType body_force_;
    BodyForceIntegrand(BodyForceType body_force) : body_force_(body_force) {}
 
    template <typename T, typename Position, typename Displacement, typename Acceleration, typename... Params>
    auto SERAC_HOST_DEVICE operator()(T t, Position position, Displacement, Acceleration, Params... params) const
    {
      return serac::tuple{-1.0 * body_force_(get<VALUE>(position), t, params...), zero{}};
    }
  };
 
  template <int... active_parameters, typename BodyForceType>
  void addBodyForce(DependsOn<active_parameters...>, BodyForceType body_force, Domain& domain)
  {
    residual_->AddDomainIntegral(Dimension<dim>{}, DependsOn<0, 1, active_parameters + NUM_STATE_VARS...>{},
                                 BodyForceIntegrand<BodyForceType>(body_force), domain);
  }
 
  template <typename BodyForceType>
  void addBodyForce(BodyForceType body_force, Domain& domain)
  {
    addBodyForce(DependsOn<>{}, body_force, domain);
  }
 
  template <int... active_parameters, typename TractionType>
  void setTraction(DependsOn<active_parameters...>, TractionType traction_function, Domain& domain)
  {
    residual_->AddBoundaryIntegral(
        Dimension<dim - 1>{}, DependsOn<0, 1, active_parameters + NUM_STATE_VARS...>{},
        [traction_function](double t, auto X, auto /* displacement */, auto /* acceleration */, auto... params) {
          auto n = cross(get<DERIVATIVE>(X));
 
          return -1.0 * traction_function(get<VALUE>(X), normalize(n), t, params...);
        },
        domain);
  }
 
  template <typename TractionType>
  void setTraction(TractionType traction_function, Domain& domain)
  {
    setTraction(DependsOn<>{}, traction_function, domain);
  }
 
  template <int... active_parameters, typename PressureType>
  void setPressure(DependsOn<active_parameters...>, PressureType pressure_function, Domain& domain)
  {
    residual_->AddBoundaryIntegral(
        Dimension<dim - 1>{}, DependsOn<0, 1, active_parameters + NUM_STATE_VARS...>{},
        [pressure_function](double t, auto X, auto displacement, auto /* acceleration */, auto... params) {
          // Calculate the position and normal in the shape perturbed deformed configuration
          auto x = X + displacement;
 
          auto n = cross(get<DERIVATIVE>(x));
 
          // serac::Functional's boundary integrals multiply the q-function output by
          // norm(cross(dX_dxi)) at that quadrature point, but if we impose a shape displacement
          // then that weight needs to be corrected. The new weight should be
          // norm(cross(dX_dxi + du_dxi + dp_dxi)) where u is displacement and p is shape displacement. This implies:
          //
          //   pressure * normalize(normal_new) * w_new
          // = pressure * normalize(normal_new) * (w_new / w_old) * w_old
          // = pressure * normalize(normal_new) * (norm(normal_new) / norm(normal_old)) * w_old
          // = pressure * (normal_new / norm(normal_new)) * (norm(normal_new) / norm(normal_old)) * w_old
          // = pressure * (normal_new / norm(normal_old)) * w_old
 
          // We always query the pressure function in the undeformed configuration
          return pressure_function(get<VALUE>(X), t, params...) * (n / norm(cross(get<DERIVATIVE>(X))));
        },
        domain);
  }
 
  template <typename PressureType>
  void setPressure(PressureType pressure_function, Domain& domain)
  {
    setPressure(DependsOn<>{}, pressure_function, domain);
  }
 
  virtual std::unique_ptr<mfem_ext::StdFunctionOperator> buildQuasistaticOperator()
  {
    // the quasistatic case is entirely described by the residual,
    // there is no ordinary differential equation
    return std::make_unique<mfem_ext::StdFunctionOperator>(
        displacement_.space().TrueVSize(),
 
        // residual function
        [this](const mfem::Vector& u, mfem::Vector& r) {
          SERAC_MARK_FUNCTION;
          const mfem::Vector res =
              (*residual_)(time_, shapeDisplacement(), u, acceleration_, *parameters_[parameter_indices].state...);
 
          // TODO this copy is required as the sundials solvers do not allow move assignments because of their memory
          // tracking strategy
          // See https://github.com/mfem/mfem/issues/3531
          r = res;
          r.SetSubVector(bcs_.allEssentialTrueDofs(), 0.0);
        },
 
        // gradient of residual function
        [this](const mfem::Vector& u) -> mfem::Operator& {
          SERAC_MARK_FUNCTION;
          auto [r, drdu] = (*residual_)(time_, shapeDisplacement(), differentiate_wrt(u), acceleration_,
                                        *parameters_[parameter_indices].state...);
          J_.reset();
          J_ = assemble(drdu);
          J_e_.reset();
          J_e_ = bcs_.eliminateAllEssentialDofsFromMatrix(*J_);
          return *J_;
        });
  }
 
  std::pair<const mfem::HypreParMatrix&, const mfem::HypreParMatrix&> stiffnessMatrix() const
  {
    SLIC_ERROR_ROOT_IF(!J_ || !J_e_, "Stiffness matrix has not yet been assembled.");
 
    return {*J_, *J_e_};
  }
 
  void completeSetup() override
  {
    if (is_quasistatic_) {
      residual_with_bcs_ = buildQuasistaticOperator();
    } else {
      // the dynamic case is described by a residual function and a second order
      // ordinary differential equation. Here, we define the residual function in
      // terms of an acceleration.
      residual_with_bcs_ = std::make_unique<mfem_ext::StdFunctionOperator>(
          displacement_.space().TrueVSize(),
 
          [this](const mfem::Vector& d2u_dt2, mfem::Vector& r) {
            add(1.0, u_, c0_, d2u_dt2, predicted_displacement_);
            const mfem::Vector res = (*residual_)(time_, shapeDisplacement(), predicted_displacement_, d2u_dt2,
                                                  *parameters_[parameter_indices].state...);
 
            // TODO this copy is required as the sundials solvers do not allow move assignments because of their memory
            // tracking strategy
            // See https://github.com/mfem/mfem/issues/3531
            r = res;
            r.SetSubVector(bcs_.allEssentialTrueDofs(), 0.0);
          },
 
          [this](const mfem::Vector& d2u_dt2) -> mfem::Operator& {
            add(1.0, u_, c0_, d2u_dt2, predicted_displacement_);
 
            // K := dR/du
            auto K = serac::get<DERIVATIVE>((*residual_)(time_, shapeDisplacement(),
                                                         differentiate_wrt(predicted_displacement_), d2u_dt2,
                                                         *parameters_[parameter_indices].state...));
            std::unique_ptr<mfem::HypreParMatrix> k_mat(assemble(K));
 
            // M := dR/da
            auto M = serac::get<DERIVATIVE>((*residual_)(time_, shapeDisplacement(), predicted_displacement_,
                                                         differentiate_wrt(d2u_dt2),
                                                         *parameters_[parameter_indices].state...));
            std::unique_ptr<mfem::HypreParMatrix> m_mat(assemble(M));
 
            // J = M + c0 * K
            J_.reset();
            J_.reset(mfem::Add(1.0, *m_mat, c0_, *k_mat));
            J_e_.reset();
            J_e_ = bcs_.eliminateAllEssentialDofsFromMatrix(*J_);
 
            return *J_;
          });
    }
 
#ifdef SERAC_USE_PETSC
    auto* space_dep_pc =
        dynamic_cast<serac::mfem_ext::PetscPreconditionerSpaceDependent*>(&nonlin_solver_->preconditioner());
    if (space_dep_pc) {
      // This call sets the displacement ParFiniteElementSpace used to get the spatial coordinates and to
      // generate the near null space for the PCGAMG preconditioner
      space_dep_pc->SetFESpace(&displacement_.space());
    }
#endif
 
    nonlin_solver_->setOperator(*residual_with_bcs_);
 
    if (checkpoint_to_disk_) {
      outputStateToDisk();
    } else {
      checkpoint_states_.clear();
      auto state_names = stateNames();
      for (const auto& state_name : state_names) {
        checkpoint_states_[state_name].push_back(state(state_name));
      }
    }
  }
 
  void zeroEssentials(FiniteElementVector& field) const
  {
    for (const auto& essential : bcs_.essentials()) {
      field.SetSubVector(essential.getTrueDofList(), 0.0);
    }
  }
 
  void advanceTimestep(double dt) override
  {
    SERAC_MARK_FUNCTION;
    SLIC_ERROR_ROOT_IF(!residual_, "completeSetup() must be called prior to advanceTimestep(dt) in SolidMechanics.");
 
    // If this is the first call, initialize the previous parameter values as the initial values
    if (cycle_ == 0) {
      for (auto& parameter : parameters_) {
        *parameter.previous_state = *parameter.state;
      }
    }
 
    if (is_quasistatic_) {
      quasiStaticSolve(dt);
    } else {
      // The current ode interface tracks 2 times, one internally which we have a handle to via time_,
      // and one here via the step interface.
      // We are ignoring this one, and just using the internal version for now.
      // This may need to be revisited when more complex time integrators are required,
      // but at the moment, the double times creates a lot of confusion, so
      // we short circuit the extra time here by passing a dummy time and ignoring it.
      double time_tmp = time_;
      dt_ = dt;
      ode2_.Step(displacement_, velocity_, time_tmp, dt);
    }
 
    cycle_ += 1;
 
    if (checkpoint_to_disk_) {
      outputStateToDisk();
    } else {
      for (const auto& state_name : stateNames()) {
        checkpoint_states_[state_name].push_back(state(state_name));
      }
    }
 
    {
      // after finding displacements that satisfy equilibrium,
      // compute the residual one more time, this time enabling
      // the material state buffers to be updated
      residual_->updateQdata(true);
 
      reactions_ = (*residual_)(time_, shapeDisplacement(), displacement_, acceleration_,
                                *parameters_[parameter_indices].state...);
 
      residual_->updateQdata(false);
    }
 
    if (cycle_ > max_cycle_) {
      timesteps_.push_back(dt);
      max_cycle_ = cycle_;
      max_time_ = time_;
    }
  }
 
  virtual void setAdjointLoad(std::unordered_map<std::string, const serac::FiniteElementDual&> loads) override
  {
    SLIC_ERROR_ROOT_IF(loads.size() == 0, "Adjoint load container size must be greater than 0 in the solid mechanics.");
 
    auto disp_adjoint_load = loads.find("displacement");
 
    SLIC_ERROR_ROOT_IF(disp_adjoint_load == loads.end(), "Adjoint load for \"displacement\" not found.");
 
    if (disp_adjoint_load != loads.end()) {
      displacement_adjoint_load_ = disp_adjoint_load->second;
      // Add the sign correction to move the term to the RHS
      displacement_adjoint_load_ *= -1.0;
    }
 
    auto velo_adjoint_load = loads.find("velocity");
 
    if (velo_adjoint_load != loads.end()) {
      velocity_adjoint_load_ = velo_adjoint_load->second;
      // Add the sign correction to move the term to the RHS
      velocity_adjoint_load_ *= -1.0;
    }
 
    auto accel_adjoint_load = loads.find("acceleration");
 
    if (accel_adjoint_load != loads.end()) {
      acceleration_adjoint_load_ = accel_adjoint_load->second;
      // Add the sign correction to move the term to the RHS
      acceleration_adjoint_load_ *= -1.0;
    }
  }
 
  virtual void setDualAdjointBcs(std::unordered_map<std::string, const serac::FiniteElementState&> bcs) override
  {
    SLIC_ERROR_ROOT_IF(bcs.size() == 0, "Adjoint load container size must be greater than 0 in the solid mechanics.");
 
    auto reaction_adjoint_load = bcs.find("reactions");
 
    SLIC_ERROR_ROOT_IF(reaction_adjoint_load == bcs.end(), "Adjoint load for \"reaction\" not found.");
 
    if (reaction_adjoint_load != bcs.end()) {
      reactions_adjoint_bcs_ = reaction_adjoint_load->second;
    }
  }
 
  void reverseAdjointTimestep() override
  {
    SERAC_MARK_FUNCTION;
    SLIC_ERROR_ROOT_IF(cycle_ <= min_cycle_,
                       "Maximum number of adjoint timesteps exceeded! The number of adjoint timesteps must equal the "
                       "number of forward timesteps");
 
    cycle_--;  // cycle is now at n \in [0,N-1]
 
    double dt_np1_to_np2 = getCheckpointedTimestep(cycle_ + 1);
    const double dt_n_to_np1 = getCheckpointedTimestep(cycle_);
 
    auto end_step_solution = getCheckpointedStates(cycle_ + 1);
 
    displacement_ = end_step_solution.at("displacement");
 
    if (is_quasistatic_) {
      quasiStaticAdjointSolve(dt_n_to_np1);
    } else {
      SLIC_ERROR_ROOT_IF(ode2_.GetTimestepper() != TimestepMethod::Newmark,
                         "Only Newmark implemented for transient adjoint solid mechanics.");
 
      // Load the end of step velo, accel from the previous cycle
 
      velocity_ = end_step_solution.at("velocity");
      acceleration_ = end_step_solution.at("acceleration");
 
      // K := dR/du
      auto K = serac::get<DERIVATIVE>((*residual_)(time_, shapeDisplacement(), differentiate_wrt(displacement_),
                                                   acceleration_, *parameters_[parameter_indices].state...));
      std::unique_ptr<mfem::HypreParMatrix> k_mat(assemble(K));
 
      // M := dR/da
      auto M = serac::get<DERIVATIVE>((*residual_)(time_, shapeDisplacement(), displacement_,
                                                   differentiate_wrt(acceleration_),
                                                   *parameters_[parameter_indices].state...));
      std::unique_ptr<mfem::HypreParMatrix> m_mat(assemble(M));
 
      auto& lin_solver = nonlin_solver_->linearSolver();
      solid_mechanics::detail::adjoint_integrate(
          dt_n_to_np1, dt_np1_to_np2, m_mat.get(), k_mat.get(), displacement_adjoint_load_, velocity_adjoint_load_,
          acceleration_adjoint_load_, adjoint_displacement_, implicit_sensitivity_displacement_start_of_step_,
          implicit_sensitivity_velocity_start_of_step_, reactions_adjoint_bcs_, bcs_, lin_solver);
    }
 
    time_end_step_ = time_;
    time_ -= dt_n_to_np1;
  }
 
  FiniteElementDual computeTimestepSensitivity(size_t parameter_field) override
  {
    SLIC_ASSERT_MSG(parameter_field < sizeof...(parameter_indices),
                    axom::fmt::format("Invalid parameter index '{}' requested for sensitivity."));
 
    auto drdparam = serac::get<DERIVATIVE>(d_residual_d_[parameter_field](time_end_step_));
    auto drdparam_mat = assemble(drdparam);
 
    drdparam_mat->MultTranspose(adjoint_displacement_, *parameters_[parameter_field].sensitivity);
 
    return *parameters_[parameter_field].sensitivity;
  }
 
  const FiniteElementDual& computeTimestepShapeSensitivity() override
  {
    auto drdshape =
        serac::get<DERIVATIVE>((*residual_)(time_end_step_, differentiate_wrt(shapeDisplacement()), displacement_,
                                            acceleration_, *parameters_[parameter_indices].state...));
 
    auto drdshape_mat = assemble(drdshape);
 
    drdshape_mat->MultTranspose(adjoint_displacement_, shape_displacement_dual_);
 
    return shapeDisplacementSensitivity();
  }
 
  const std::unordered_map<std::string, const serac::FiniteElementDual&> computeInitialConditionSensitivity()
      const override
  {
    return {{"displacement", implicit_sensitivity_displacement_start_of_step_},
            {"velocity", implicit_sensitivity_velocity_start_of_step_}};
  }
 
  const serac::FiniteElementState& displacement() const { return displacement_; };
 
  const serac::FiniteElementState& velocity() const { return velocity_; };
 
  const serac::FiniteElementState& acceleration() const { return acceleration_; };
 
  const serac::FiniteElementDual& reactions() const { return reactions_; };
 
 protected:
  using trial = H1<order, dim>;
 
  using test = H1<order, dim>;
 
  using shape_trial = H1<SHAPE_ORDER, dim>;
 
  FiniteElementState displacement_;
 
  FiniteElementState velocity_;
 
  FiniteElementState acceleration_;
 
  // In the case of transient dynamics, this is more like an adjoint_acceleration
  FiniteElementState adjoint_displacement_;
 
  FiniteElementDual displacement_adjoint_load_;
 
  FiniteElementDual velocity_adjoint_load_;
 
  FiniteElementDual acceleration_adjoint_load_;
 
  FiniteElementDual implicit_sensitivity_displacement_start_of_step_;
 
  FiniteElementDual implicit_sensitivity_velocity_start_of_step_;
 
  FiniteElementDual reactions_;
 
  FiniteElementState reactions_adjoint_bcs_;
 
  std::unique_ptr<ShapeAwareFunctional<shape_trial, test(trial, trial, parameter_space...)>> residual_;
 
  std::unique_ptr<mfem_ext::StdFunctionOperator> residual_with_bcs_;
 
  std::unique_ptr<EquationSolver> nonlin_solver_;
 
  mfem_ext::SecondOrderODE ode2_;
 
  std::unique_ptr<mfem::HypreParMatrix> J_;
 
  std::unique_ptr<mfem::HypreParMatrix> J_e_;
 
  mfem::Vector predicted_displacement_;
 
  mfem::Vector du_;
 
  mfem::Vector u_;
 
  mfem::Vector v_;
 
  double c0_;
 
  double c1_;
 
  double time_end_step_;
 
  bool use_warm_start_;
 
  std::shared_ptr<mfem::VectorCoefficient> disp_bdr_coef_;
 
  std::shared_ptr<mfem::Coefficient> component_disp_bdr_coef_;
 
  std::array<std::function<decltype((*residual_)(DifferentiateWRT<1>{}, 0.0, shape_displacement_, displacement_,
                                                 acceleration_, *parameters_[parameter_indices].state...))(double)>,
             sizeof...(parameter_indices)>
      d_residual_d_ = {[&](double _t) {
        return (*residual_)(DifferentiateWRT<NUM_STATE_VARS + 1 + parameter_indices>{}, _t, shape_displacement_,
                            displacement_, acceleration_, *parameters_[parameter_indices].state...);
      }...};
 
  virtual void quasiStaticSolve(double dt)
  {
    // warm start must be called prior to the time update so that the previous Jacobians can be used consistently
    // throughout.
    warmStartDisplacement(dt);
    time_ += dt;
 
    // this method is essentially equivalent to the 1-liner
    // u += dot(inv(J), dot(J_elim[:, dofs], (U(t + dt) - u)[dofs]));
    nonlin_solver_->solve(displacement_);
  }
 
  virtual void quasiStaticAdjointSolve(double /*dt*/)
  {
    auto [_, drdu] = (*residual_)(time_, shapeDisplacement(), differentiate_wrt(displacement_), acceleration_,
                                  *parameters_[parameter_indices].state...);
    J_.reset();
    J_ = assemble(drdu);
 
    auto J_T = std::unique_ptr<mfem::HypreParMatrix>(J_->Transpose());
 
    J_e_.reset();
    J_e_ = bcs_.eliminateAllEssentialDofsFromMatrix(*J_T);
 
    auto& constrained_dofs = bcs_.allEssentialTrueDofs();
 
    mfem::EliminateBC(*J_T, *J_e_, constrained_dofs, reactions_adjoint_bcs_, displacement_adjoint_load_);
    for (int i = 0; i < constrained_dofs.Size(); i++) {
      int j = constrained_dofs[i];
      displacement_adjoint_load_[j] = reactions_adjoint_bcs_[j];
    }
 
    auto& lin_solver = nonlin_solver_->linearSolver();
    lin_solver.SetOperator(*J_T);
    lin_solver.Mult(displacement_adjoint_load_, adjoint_displacement_);
 
    // Reset the equation solver to use the full nonlinear residual operator.  MRT, is this needed?
    nonlin_solver_->setOperator(*residual_with_bcs_);
  }
 
  mfem::Array<int> calculateConstrainedDofs(std::function<bool(const mfem::Vector&)> is_node_constrained,
                                            std::optional<int> component = {}) const
  {
    // Get the nodal positions for the displacement vector in grid function form
    mfem::ParGridFunction nodal_positions(
        const_cast<mfem::ParFiniteElementSpace*>(&displacement_.space()));  // MRT mfem const correctness issue
    mfemParMesh().GetNodes(nodal_positions);
 
    const int num_nodes = nodal_positions.Size() / dim;
    mfem::Array<int> constrained_dofs;
 
    for (int i = 0; i < num_nodes; i++) {
      // Determine if this "local" node (L-vector node) is in the local true vector. I.e. ensure this node is not a
      // shared node owned by another processor
      int idof = mfem::Ordering::Map<serac::ordering>(nodal_positions.FESpace()->GetNDofs(),
                                                      nodal_positions.FESpace()->GetVDim(), i, 0);
      if (nodal_positions.ParFESpace()->GetLocalTDofNumber(idof) >= 0) {
        mfem::Vector node_coords(dim);
        mfem::Array<int> node_dofs;
        for (int d = 0; d < dim; d++) {
          // Get the local dof number for the prescribed component
          int local_vector_dof = mfem::Ordering::Map<serac::ordering>(nodal_positions.FESpace()->GetNDofs(),
                                                                      nodal_positions.FESpace()->GetVDim(), i, d);
 
          // Save the spatial position for this coordinate dof
          node_coords(d) = nodal_positions(local_vector_dof);
 
          // Check if this component of the displacement vector is constrained
          bool is_active_component = true;
          if (component) {
            if (*component != d) {
              is_active_component = false;
            }
          }
 
          if (is_active_component) {
            // Add the true dof for this component to the related dof list
            node_dofs.Append(nodal_positions.ParFESpace()->GetLocalTDofNumber(local_vector_dof));
          }
        }
 
        // Do the user-defined spatial query to determine if these dofs should be constrained
        if (is_node_constrained(node_coords)) {
          constrained_dofs.Append(node_dofs);
        }
 
        // Reset the temporary container for the dofs associated with a particular node
        node_dofs.DeleteAll();
      }
    }
    return constrained_dofs;
  }
 
  void warmStartDisplacement(double dt)
  {
    SERAC_MARK_FUNCTION;
 
    du_ = 0.0;
    for (auto& bc : bcs_.essentials()) {
      // apply the future boundary conditions, but use the most recent Jacobians stiffness.
      bc.setDofs(du_, time_ + dt);
    }
 
    auto& constrained_dofs = bcs_.allEssentialTrueDofs();
    for (int i = 0; i < constrained_dofs.Size(); i++) {
      int j = constrained_dofs[i];
      du_[j] -= displacement_(j);
    }
 
    if (use_warm_start_) {
      // Update external forcing
      auto r = (*residual_)(time_ + dt, shapeDisplacement(), displacement_, acceleration_,
                            *parameters_[parameter_indices].state...);
 
      // use the most recently evaluated Jacobian
      auto [_, drdu] = (*residual_)(time_, shapeDisplacement(), differentiate_wrt(displacement_), acceleration_,
                                    *parameters_[parameter_indices].previous_state...);
      J_.reset();
      J_ = assemble(drdu);
      J_e_.reset();
      J_e_ = bcs_.eliminateAllEssentialDofsFromMatrix(*J_);
 
      r *= -1.0;
 
      mfem::EliminateBC(*J_, *J_e_, constrained_dofs, du_, r);
      for (int i = 0; i < constrained_dofs.Size(); i++) {
        int j = constrained_dofs[i];
        r[j] = du_[j];
      }
 
      auto& lin_solver = nonlin_solver_->linearSolver();
 
      lin_solver.SetOperator(*J_);
 
      lin_solver.Mult(r, du_);
    }
 
    displacement_ += du_;
    dt_ = dt;
  }
};
 
}  // namespace serac