adjoint_fluid_pnpn.f90

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!
module adjoint_fluid_pnpn
  use, intrinsic :: iso_fortran_env, only: error_unit
  use coefs, only: coef_t
  use symmetry, only: symmetry_t
  use registry, only: neko_registry
  use logger, only: neko_log, log_size, neko_log_debug
  use num_types, only: rp
  use krylov, only: ksp_monitor_t
  use pnpn_residual, only: pnpn_prs_res_t, pnpn_vel_res_t, &
       pnpn_prs_res_factory, pnpn_vel_res_factory, &
       pnpn_prs_res_stress_factory, pnpn_vel_res_stress_factory
  use rhs_maker, only: rhs_maker_sumab_t, rhs_maker_bdf_t, rhs_maker_ext_t, &
       rhs_maker_oifs_t, rhs_maker_sumab_fctry, rhs_maker_bdf_fctry, &
       rhs_maker_ext_fctry, rhs_maker_oifs_fctry
  use adjoint_fluid_scheme, only: adjoint_fluid_scheme_t
  use adjoint_fluid_scheme_incompressible, only: &
       adjoint_fluid_scheme_incompressible_t
  use device_mathops, only: device_opcolv, device_opadd2cm
  use fluid_aux, only: fluid_step_info
  use time_scheme_controller, only: time_scheme_controller_t
  use scratch_registry, only: neko_scratch_registry
  use projection, only: projection_t
  use projection_vel, only: projection_vel_t
  use device, only: device_memcpy, host_to_device, device_event_sync, &
       glb_cmd_event
  use advection_adjoint, only: advection_adjoint_t, advection_adjoint_factory
  use profiler, only: profiler_start_region, profiler_end_region
  use json_module, only: json_file, json_core, json_value
  use json_utils, only: json_get, json_get_or_default, json_extract_item
  use json_module, only: json_file
  use ax_product, only: ax_t, ax_helm_factory
  use field, only: field_t
  use dirichlet, only: dirichlet_t
  use shear_stress, only: shear_stress_t
  use wall_model_bc, only: wall_model_bc_t
  use facet_normal, only: facet_normal_t
  use non_normal, only: non_normal_t
  use checkpoint, only: chkp_t
  use mesh, only: mesh_t
  use user_intf, only: user_t
  use time_step_controller, only: time_step_controller_t
  use gs_ops, only: gs_op_add
  use neko_config, only: neko_bcknd_device
  use mathops, only: opadd2cm, opcolv
  use bc_list, only: bc_list_t
  use zero_dirichlet, only: zero_dirichlet_t
  use utils, only: neko_error, neko_type_error
  use field_math, only: field_add2, field_copy
  use bc, only: bc_t
  use file, only: file_t
  use operators, only: ortho
  use inflow, only: inflow_t
  use field_dirichlet, only: field_dirichlet_t
  use blasius, only: blasius_t
  use field_dirichlet_vector, only: field_dirichlet_vector_t
  use dong_outflow, only: dong_outflow_t
  use time_state, only: time_state_t
  use vector, only: vector_t
  use device_math, only: device_vlsc3, device_cmult
  use math, only: vlsc3, cmult
  use json_utils_ext, only: json_key_fallback
  use, intrinsic :: iso_c_binding, only: c_ptr, c_null_ptr, c_associated
  use comm, only: neko_comm, mpi_real_precision
  use mpi_f08, only: mpi_sum, mpi_max, mpi_allreduce, mpi_comm_world, &
       mpi_integer, mpi_logical, mpi_lor
 
  implicit none
  private
 
  type, public, extends(adjoint_fluid_scheme_incompressible_t) :: adjoint_fluid_pnpn_t
 
     type(field_t) :: p_res, u_res, v_res, w_res
 
     type(field_t) :: dp, du, dv, dw
 
     !
     ! Implicit operators, i.e. the left-hand-side of the Helmholz problem.
     !
 
     ! Coupled Helmholz operator for velocity
     class(ax_t), allocatable :: ax_vel
     ! Helmholz operator for pressure
     class(ax_t), allocatable :: ax_prs
 
     !
     ! Projections for solver speed-up
     !
 
     type(projection_t) :: proj_prs
     type(projection_vel_t) :: proj_vel
 
     !
     ! Special Karniadakis scheme boundary conditions in the pressure equation
     !
 
     type(facet_normal_t) :: bc_prs_surface
 
     type(facet_normal_t) :: bc_sym_surface
 
     !
     ! Boundary conditions and  lists for residuals and solution increments
     !
 
     type(zero_dirichlet_t) :: bc_vel_res
     type(zero_dirichlet_t) :: bc_du
     type(zero_dirichlet_t) :: bc_dv
     type(zero_dirichlet_t) :: bc_dw
     type(zero_dirichlet_t) :: bc_dp
 
     type(bc_list_t) :: bclst_vel_res
     type(bc_list_t) :: bclst_du
     type(bc_list_t) :: bclst_dv
     type(bc_list_t) :: bclst_dw
     type(bc_list_t) :: bclst_dp
 
 
     ! Checker for wether we have a strong pressure bc. If not, the pressure
     ! is demeaned at every time step.
     logical :: prs_dirichlet = .false.
 
 
     ! The advection operator.
     class(advection_adjoint_t), allocatable :: adv
 
     ! Time OIFS interpolation scheme for advection.
     logical :: oifs
 
     ! Time variables
     type(field_t) :: abx1, aby1, abz1
     type(field_t) :: abx2, aby2, abz2
 
     ! Advection terms for the oifs method
     type(field_t) :: advx, advy, advz
 
     class(pnpn_prs_res_t), allocatable :: prs_res
 
     class(pnpn_vel_res_t), allocatable :: vel_res
 
     class(rhs_maker_sumab_t), allocatable :: sumab
 
     class(rhs_maker_ext_t), allocatable :: makeabf
 
     class(rhs_maker_bdf_t), allocatable :: makebdf
 
     class(rhs_maker_oifs_t), allocatable :: makeoifs
 
     !  !> Adjust flow volume
     !  type(fluid_volflow_t) :: vol_flow
 
     logical :: full_stress_formulation = .false.
 
     ! ======================================================================= !
     ! Addressable attributes
 
     real(kind=rp) :: norm_scaling 
     real(kind=rp) :: norm_target 
     real(kind=rp) :: norm_tolerance 
 
     ! ======================================================================= !
     ! Definition of shorthands and local variables
 
     real(kind=rp) :: norm_l2_base
 
     real(kind=rp) :: norm_l2_upper
     real(kind=rp) :: norm_l2_lower
 
     type(file_t) :: file_output
 
   contains
     procedure, pass(this) :: init => adjoint_fluid_pnpn_init
     procedure, pass(this) :: free => adjoint_fluid_pnpn_free
     procedure, pass(this) :: step => adjoint_fluid_pnpn_step
     procedure, pass(this) :: restart => adjoint_fluid_pnpn_restart
     procedure, pass(this) :: setup_bcs => adjoint_fluid_pnpn_setup_bcs
     procedure, pass(this) :: write_boundary_conditions => &
          adjoint_fluid_pnpn_write_boundary_conditions
 
     procedure, public, pass(this) :: pw_compute_ => power_iterations_compute
 
  end type adjoint_fluid_pnpn_t
 
  interface pressure_bc_factory
     module subroutine pressure_bc_factory(object, scheme, json, coef, user)
       class(bc_t), pointer, intent(inout) :: object
       type(adjoint_fluid_pnpn_t), intent(in) :: scheme
       type(json_file), intent(inout) :: json
       type(coef_t), intent(in) :: coef
       type(user_t), intent(in) :: user
     end subroutine pressure_bc_factory
  end interface pressure_bc_factory
 
  interface velocity_bc_factory
     module subroutine velocity_bc_factory(object, scheme, json, coef, user)
       class(bc_t), pointer, intent(inout) :: object
       type(adjoint_fluid_pnpn_t), intent(in) :: scheme
       type(json_file), intent(inout) :: json
       type(coef_t), intent(in) :: coef
       type(user_t), intent(in) :: user
     end subroutine velocity_bc_factory
  end interface velocity_bc_factory
 
contains
 
  subroutine adjoint_fluid_pnpn_init(this, msh, lx, params, user, chkp)
    class(adjoint_fluid_pnpn_t), target, intent(inout) :: this
    type(mesh_t), target, intent(inout) :: msh
    integer, intent(in) :: lx
    type(json_file), target, intent(inout) :: params
    type(user_t), target, intent(in) :: user
    type(chkp_t), target, intent(inout) :: chkp
    character(len=15), parameter :: scheme = 'Adjoint (Pn/Pn)'
    real(kind=rp) :: abs_tol
    character(len=LOG_SIZE) :: log_buf
    integer :: integer_val, solver_maxiter
    character(len=:), allocatable :: solver_type, precon_type
    logical :: monitor, found
    logical :: advection
    type(json_file) :: numerics_params, precon_params
 
    ! Temporary field pointers
    character(len=:), allocatable :: file_name
    character(len=256) :: header_line
 
    call this%free()
 
    ! Initialize base class
    call this%init_base(msh, lx, params, scheme, user, .true.)
 
    ! Add pressure field to the registry. For this scheme it is in the same
    ! Xh as the velocity
    call neko_registry%add_field(this%dm_Xh, 'p_adj')
    this%p_adj => neko_registry%get_field('p_adj')
 
    !
    ! Select governing equations via associated residual and Ax types
    !
 
    call json_get(params, 'case.numerics.time_order', integer_val)
    allocate(this%ext_bdf)
    call this%ext_bdf%init(integer_val)
 
    call json_get_or_default(params, "case.fluid.full_stress_formulation", &
         this%full_stress_formulation, .false.)
 
    if (this%full_stress_formulation .eqv. .true.) then
       call neko_error( &
            "Full stress formulation is not supported in the adjoint module.")
       !  ! Setup backend dependent Ax routines
       !  call ax_helm_factory(this%Ax_vel, full_formulation = .true.)
 
       !  ! Setup backend dependent prs residual routines
       !  call pnpn_prs_res_stress_factory(this%prs_res)
 
       !  ! Setup backend dependent vel residual routines
       !  call pnpn_vel_res_stress_factory(this%vel_res)
    else
       ! Setup backend dependent Ax routines
       call ax_helm_factory(this%Ax_vel, full_formulation = .false.)
 
       ! Setup backend dependent prs residual routines
       call pnpn_prs_res_factory(this%prs_res)
 
       ! Setup backend dependent vel residual routines
       call pnpn_vel_res_factory(this%vel_res)
    end if
 
    if (params%valid_path('case.fluid.nut_field')) then
       if (this%full_stress_formulation .eqv. .false.) then
          call neko_error("You need to set full_stress_formulation to " // &
               "true for the fluid to have a spatially varying " // &
               "viscocity field.")
       end if
       call json_get(params, 'case.fluid.nut_field', this%nut_field_name)
    else
       this%nut_field_name = ""
    end if
 
    ! Setup Ax for the pressure
    call ax_helm_factory(this%Ax_prs, full_formulation = .false.)
 
 
    ! Setup backend dependent summation of AB/BDF
    call rhs_maker_sumab_fctry(this%sumab)
 
    ! Setup backend dependent summation of extrapolation scheme
    call rhs_maker_ext_fctry(this%makeabf)
 
    ! Setup backend depenent contributions to F from lagged BD terms
    call rhs_maker_bdf_fctry(this%makebdf)
 
    ! Setup backend dependent summations of the OIFS method
    call rhs_maker_oifs_fctry(this%makeoifs)
 
    ! Initialize variables specific to this plan
    associate(xh_lx => this%Xh%lx, xh_ly => this%Xh%ly, xh_lz => this%Xh%lz, &
         dm_xh => this%dm_Xh, nelv => this%msh%nelv)
 
      call this%p_res%init(dm_xh, "p_res")
      call this%u_res%init(dm_xh, "u_res")
      call this%v_res%init(dm_xh, "v_res")
      call this%w_res%init(dm_xh, "w_res")
      call this%abx1%init(dm_xh, "abx1")
      call this%aby1%init(dm_xh, "aby1")
      call this%abz1%init(dm_xh, "abz1")
      call this%abx2%init(dm_xh, "abx2")
      call this%aby2%init(dm_xh, "aby2")
      call this%abz2%init(dm_xh, "abz2")
      call this%advx%init(dm_xh, "advx")
      call this%advy%init(dm_xh, "advy")
      call this%advz%init(dm_xh, "advz")
    end associate
 
    call this%du%init(this%dm_Xh, 'du')
    call this%dv%init(this%dm_Xh, 'dv')
    call this%dw%init(this%dm_Xh, 'dw')
    call this%dp%init(this%dm_Xh, 'dp')
 
    ! Set up boundary conditions
    call this%setup_bcs(user, params)
 
    ! Check if we need to output boundaries
    call json_get_or_default(params, 'case.output_boundary', found, .false.)
    if (found) call this%write_boundary_conditions()
 
    call this%proj_prs%init(this%dm_Xh%size(), this%pr_projection_dim, &
         this%pr_projection_activ_step)
 
    call this%proj_vel%init(this%dm_Xh%size(), this%vel_projection_dim, &
         this%vel_projection_activ_step)
 
    ! Determine the time-interpolation scheme
    call json_get_or_default(params, 'case.numerics.oifs', this%oifs, .false.)
    if (params%valid_path('case.fluid.flow_rate_force')) then
       call neko_error("Flow rate forcing not available for adjoint_fluid_pnpn")
    end if
 
    ! Setup pressure solver
    call neko_log%section("Pressure solver")
 
    call json_get_or_default(params, &
         'case.fluid.pressure_solver.max_iterations', &
         solver_maxiter, 800)
    call json_get(params, 'case.fluid.pressure_solver.type', solver_type)
    call json_get(params, 'case.fluid.pressure_solver.preconditioner.type', &
         precon_type)
    call json_get(params, &
         'case.fluid.pressure_solver.preconditioner', precon_params)
    call json_get(params, 'case.fluid.pressure_solver.absolute_tolerance', &
         abs_tol)
    call json_get_or_default(params, 'case.fluid.pressure_solver.monitor', &
         monitor, .false.)
    call neko_log%message('Type       : ('// trim(solver_type) // &
         ', ' // trim(precon_type) // ')')
    write(log_buf, '(A,ES13.6)') 'Abs tol    :', abs_tol
    call neko_log%message(log_buf)
 
    call this%solver_factory(this%ksp_prs, this%dm_Xh%size(), &
         solver_type, solver_maxiter, abs_tol, monitor)
    call this%precon_factory_(this%pc_prs, this%ksp_prs, &
         this%c_Xh, this%dm_Xh, this%gs_Xh, this%bcs_prs, &
         precon_type, precon_params)
    call neko_log%end_section()
 
    ! Initialize the advection factory
    call neko_log%section("Advection factory")
    call json_get_or_default(params, 'case.fluid.advection', advection, .true.)
    call json_get(params, 'case.numerics', numerics_params)
    call advection_adjoint_factory(this%adv, numerics_params, this%c_Xh, &
         this%ulag, this%vlag, this%wlag, &
         chkp%dtlag, chkp%tlag, this%ext_bdf, &
         .not. advection)
    ! Should be in init_base maybe?
    this%chkp => chkp
    ! This is probably scheme specific
    ! Should not be init really, but more like, add fluid or something...
    call this%chkp%init(this%u_adj, this%v_adj, this%w_adj, this%p_adj)
 
    this%chkp%abx1 => this%abx1
    this%chkp%abx2 => this%abx2
    this%chkp%aby1 => this%aby1
    this%chkp%aby2 => this%aby2
    this%chkp%abz1 => this%abz1
    this%chkp%abz2 => this%abz2
    call this%chkp%add_lag(this%ulag, this%vlag, this%wlag)
 
    call neko_log%end_section()
 
    ! ------------------------------------------------------------------------ !
    ! Handling the rescaling and baseflow
 
    ! Read the norm scaling from the json file
    call json_get_or_default(params, 'norm_scaling', &
         this%norm_scaling, 0.5_rp)
 
    ! The baseflow is the solution to the forward.
    ! Userdefined baseflows can be invoked via setting initial conditions
    ! call neko_registry%add_field(this%dm_Xh, 'u')
    ! call neko_registry%add_field(this%dm_Xh, 'v')
    ! call neko_registry%add_field(this%dm_Xh, 'w')
    ! call neko_registry%add_field(this%dm_Xh, 'p')
    this%u_b => neko_registry%get_field('u')
    this%v_b => neko_registry%get_field('v')
    this%w_b => neko_registry%get_field('w')
    this%p_b => neko_registry%get_field('p')
 
    ! Read the json file
    call json_get_or_default(params, 'norm_target', &
         this%norm_target, -1.0_rp)
    call json_get_or_default(params, 'norm_tolerance', &
         this%norm_tolerance, 10.0_rp)
 
    ! Build the header
    call json_get_or_default(params, 'output_file', &
         file_name, 'power_iterations.csv')
    call this%file_output%init(trim(file_name))
    write(header_line, '(A)') 'Time, Norm, Scaling'
    call this%file_output%set_header(header_line)
 
  end subroutine adjoint_fluid_pnpn_init
 
  subroutine adjoint_fluid_pnpn_restart(this, chkp)
    class(adjoint_fluid_pnpn_t), target, intent(inout) :: this
    type(chkp_t), intent(inout) :: chkp
    real(kind=rp) :: dtlag(10), tlag(10)
    integer :: i, j, n
 
    dtlag = chkp%dtlag
    tlag = chkp%tlag
 
    n = this%u_adj%dof%size()
    if (allocated(this%chkp%previous_mesh%elements) .or. &
         chkp%previous_Xh%lx .ne. this%Xh%lx) then
       associate(u => this%u_adj, v => this%v_adj, w => this%w_adj, &
            p => this%p_adj, c_xh => this%c_Xh, ulag => this%ulag, &
            vlag => this%vlag, wlag => this%wlag)
         do concurrent(j = 1:n)
            u%x(j,1,1,1) = u%x(j,1,1,1) * c_xh%mult(j,1,1,1)
            v%x(j,1,1,1) = v%x(j,1,1,1) * c_xh%mult(j,1,1,1)
            w%x(j,1,1,1) = w%x(j,1,1,1) * c_xh%mult(j,1,1,1)
            p%x(j,1,1,1) = p%x(j,1,1,1) * c_xh%mult(j,1,1,1)
         end do
         do i = 1, this%ulag%size()
            do concurrent(j = 1:n)
               ulag%lf(i)%x(j,1,1,1) = ulag%lf(i)%x(j,1,1,1) &
                    * c_xh%mult(j,1,1,1)
               vlag%lf(i)%x(j,1,1,1) = vlag%lf(i)%x(j,1,1,1) &
                    * c_xh%mult(j,1,1,1)
               wlag%lf(i)%x(j,1,1,1) = wlag%lf(i)%x(j,1,1,1) &
                    * c_xh%mult(j,1,1,1)
            end do
         end do
       end associate
    end if
 
    if (neko_bcknd_device .eq. 1) then
       associate(u => this%u_adj, v => this%v_adj, w => this%w_adj, &
            ulag => this%ulag, vlag => this%vlag, wlag => this%wlag,&
            p => this%p_adj)
         call device_memcpy(u%x, u%x_d, u%dof%size(), &
              host_to_device, sync = .false.)
         call device_memcpy(v%x, v%x_d, v%dof%size(), &
              host_to_device, sync = .false.)
         call device_memcpy(w%x, w%x_d, w%dof%size(), &
              host_to_device, sync = .false.)
         call device_memcpy(p%x, p%x_d, p%dof%size(), &
              host_to_device, sync = .false.)
         call device_memcpy(ulag%lf(1)%x, ulag%lf(1)%x_d, &
              u%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(ulag%lf(2)%x, ulag%lf(2)%x_d, &
              u%dof%size(), host_to_device, sync = .false.)
 
         call device_memcpy(vlag%lf(1)%x, vlag%lf(1)%x_d, &
              v%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(vlag%lf(2)%x, vlag%lf(2)%x_d, &
              v%dof%size(), host_to_device, sync = .false.)
 
         call device_memcpy(wlag%lf(1)%x, wlag%lf(1)%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(wlag%lf(2)%x, wlag%lf(2)%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%abx1%x, this%abx1%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%abx2%x, this%abx2%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%aby1%x, this%aby1%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%aby2%x, this%aby2%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%abz1%x, this%abz1%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%abz2%x, this%abz2%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%advx%x, this%advx%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%advy%x, this%advy%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
         call device_memcpy(this%advz%x, this%advz%x_d, &
              w%dof%size(), host_to_device, sync = .false.)
       end associate
    end if
    ! Make sure that continuity is maintained (important for interpolation)
    ! Do not do this for lagged rhs
    ! (derivatives are not necessairly coninous across elements)
 
    if (allocated(this%chkp%previous_mesh%elements) &
         .or. this%chkp%previous_Xh%lx .ne. this%Xh%lx) then
       call this%gs_Xh%op(this%u_adj, gs_op_add)
       call this%gs_Xh%op(this%v_adj, gs_op_add)
       call this%gs_Xh%op(this%w_adj, gs_op_add)
       call this%gs_Xh%op(this%p_adj, gs_op_add)
 
       do i = 1, this%ulag%size()
          call this%gs_Xh%op(this%ulag%lf(i), gs_op_add)
          call this%gs_Xh%op(this%vlag%lf(i), gs_op_add)
          call this%gs_Xh%op(this%wlag%lf(i), gs_op_add)
       end do
    end if
 
  end subroutine adjoint_fluid_pnpn_restart
 
  subroutine adjoint_fluid_pnpn_free(this)
    class(adjoint_fluid_pnpn_t), intent(inout) :: this
 
    !Deallocate velocity and pressure fields
    call this%scheme_free()
 
    call this%bc_prs_surface%free()
    call this%bc_sym_surface%free()
    call this%bclst_vel_res%free()
    call this%bclst_dp%free()
    call this%proj_prs%free()
    call this%proj_vel%free()
 
    call this%p_res%free()
    call this%u_res%free()
    call this%v_res%free()
    call this%w_res%free()
 
    call this%du%free()
    call this%dv%free()
    call this%dw%free()
    call this%dp%free()
 
    call this%abx1%free()
    call this%aby1%free()
    call this%abz1%free()
 
    call this%abx2%free()
    call this%aby2%free()
    call this%abz2%free()
 
    call this%advx%free()
    call this%advy%free()
    call this%advz%free()
 
    if (allocated(this%Ax_vel)) then
       deallocate(this%Ax_vel)
    end if
 
    if (allocated(this%Ax_prs)) then
       deallocate(this%Ax_prs)
    end if
 
    if (allocated(this%prs_res)) then
       deallocate(this%prs_res)
    end if
 
    if (allocated(this%vel_res)) then
       deallocate(this%vel_res)
    end if
 
    if (allocated(this%sumab)) then
       deallocate(this%sumab)
    end if
 
    if (allocated(this%makeabf)) then
       deallocate(this%makeabf)
    end if
 
    if (allocated(this%makebdf)) then
       deallocate(this%makebdf)
    end if
 
    if (allocated(this%makeoifs)) then
       deallocate(this%makeoifs)
    end if
 
    if (allocated(this%ext_bdf)) then
       deallocate(this%ext_bdf)
    end if
 
    ! call this%vol_flow%free()
 
  end subroutine adjoint_fluid_pnpn_free
 
  subroutine adjoint_fluid_pnpn_step(this, time, dt_controller)
    class(adjoint_fluid_pnpn_t), target, intent(inout) :: this
    type(time_state_t), intent(in) :: time
    type(time_step_controller_t), intent(in) :: dt_controller
    ! number of degrees of freedom
    integer :: n
    ! Solver results monitors (pressure + 3 velocity)
    type(ksp_monitor_t) :: ksp_results(4)
 
    if (this%freeze) return
 
    n = this%dm_Xh%size()
 
    call profiler_start_region('Adjoint')
    associate(u => this%u_adj, v => this%v_adj, w => this%w_adj, &
         p => this%p_adj, &
         u_e => this%u_adj_e, v_e => this%v_adj_e, w_e => this%w_adj_e, &
         du => this%du, dv => this%dv, dw => this%dw, dp => this%dp, &
         u_b => this%u_b, v_b => this%v_b, w_b => this%w_b, &
         u_res => this%u_res, v_res => this%v_res, w_res => this%w_res, &
         p_res => this%p_res, ax_vel => this%Ax_vel, ax_prs => this%Ax_prs, &
         xh => this%Xh, &
         c_xh => this%c_Xh, dm_xh => this%dm_Xh, gs_xh => this%gs_Xh, &
         ulag => this%ulag, vlag => this%vlag, wlag => this%wlag, &
         msh => this%msh, prs_res => this%prs_res, &
         source_term => this%source_term, vel_res => this%vel_res, &
         sumab => this%sumab, &
         makeabf => this%makeabf, makebdf => this%makebdf, &
         vel_projection_dim => this%vel_projection_dim, &
         pr_projection_dim => this%pr_projection_dim, &
         oifs => this%oifs, &
         rho => this%rho, mu => this%mu, &
         f_x => this%f_adj_x, f_y => this%f_adj_y, f_z => this%f_adj_z, &
         t => time%t, tstep => time%tstep, dt => time%dt, &
         ext_bdf => this%ext_bdf, event => glb_cmd_event)
 
      ! Extrapolate the velocity if it's not done in nut_field estimation
      call sumab%compute_fluid(u_e, v_e, w_e, u, v, w, &
           ulag, vlag, wlag, ext_bdf%advection_coeffs, ext_bdf%nadv)
 
      ! Compute the source terms
      call this%source_term%compute(time)
 
      ! Add Neumann bc contributions to the RHS
      call this%bcs_vel%apply_vector(f_x%x, f_y%x, f_z%x, &
           this%dm_Xh%size(), time, strong = .false.)
 
      if (oifs) then
         call neko_error("OIFS not implemented for adjoint")
 
      else
         ! Add the advection operators to the right-hand-side.
         call this%adv%compute_adjoint(u, v, w, u_b, v_b, w_b, &
              f_x, f_y, f_z, &
              xh, this%c_Xh, dm_xh%size())
 
         ! At this point the RHS contains the sum of the advection operator and
         ! additional source terms, evaluated using the velocity field from the
         ! previous time-step. Now, this value is used in the explicit time
         ! scheme to advance both terms in time.
         call makeabf%compute_fluid(this%abx1, this%aby1, this%abz1,&
              this%abx2, this%aby2, this%abz2, &
              f_x%x, f_y%x, f_z%x, &
              rho%x(1,1,1,1), ext_bdf%advection_coeffs, n)
 
         ! Add the RHS contributions coming from the BDF scheme.
         call makebdf%compute_fluid(ulag, vlag, wlag, f_x%x, f_y%x, f_z%x, &
              u, v, w, c_xh%B, rho%x(1,1,1,1), dt, &
              ext_bdf%diffusion_coeffs, ext_bdf%ndiff, n)
      end if
 
      call ulag%update()
      call vlag%update()
      call wlag%update()
 
      call this%bc_apply_vel(time, strong = .true.)
      call this%bc_apply_prs(time)
 
      ! Update material properties if necessary
      call this%update_material_properties(time)
 
      ! Compute pressure residual.
      call profiler_start_region('Adjoint_pressure_residual')
 
      call prs_res%compute(p, p_res, &
           u, v, w, &
           u_e, v_e, w_e, &
           f_x, f_y, f_z, &
           c_xh, gs_xh, &
           this%bc_prs_surface, this%bc_sym_surface, &
           ax_prs, ext_bdf%diffusion_coeffs(1), dt, &
           mu, rho, event)
 
      ! De-mean the pressure residual when no strong pressure boundaries present
      if (.not. this%prs_dirichlet) call ortho(p_res%x, this%glb_n_points, n)
 
      call gs_xh%op(p_res, gs_op_add, event)
      call device_event_sync(event)
 
      ! Set the residual to zero at strong pressure boundaries.
      call this%bclst_dp%apply_scalar(p_res%x, p%dof%size(), time)
 
 
      call profiler_end_region('Adjoint_pressure_residual')
 
 
      call this%proj_prs%pre_solving(p_res%x, tstep, c_xh, n, dt_controller, &
           'Pressure')
 
      call this%pc_prs%update()
 
      call profiler_start_region('Adjoint_pressure_solve')
 
      ! Solve for the pressure increment.
      ksp_results(1) = &
           this%ksp_prs%solve(ax_prs, dp, p_res%x, n, c_xh, &
           this%bclst_dp, gs_xh)
 
 
      call profiler_end_region('Adjoint_pressure_solve')
 
      call this%proj_prs%post_solving(dp%x, ax_prs, c_xh, &
           this%bclst_dp, gs_xh, n, tstep, dt_controller)
 
      ! Update the pressure with the increment. Demean if necessary.
      call field_add2(p, dp, n)
      if (.not. this%prs_dirichlet) call ortho(p%x, this%glb_n_points, n)
 
      ! Compute velocity residual.
      call profiler_start_region('Adjoint_velocity_residual')
      call vel_res%compute(ax_vel, u, v, w, &
           u_res, v_res, w_res, &
           p, &
           f_x, f_y, f_z, &
           c_xh, msh, xh, &
           mu, rho, ext_bdf%diffusion_coeffs(1), &
           dt, dm_xh%size())
 
      call gs_xh%op(u_res, gs_op_add, event)
      call device_event_sync(event)
      call gs_xh%op(v_res, gs_op_add, event)
      call device_event_sync(event)
      call gs_xh%op(w_res, gs_op_add, event)
      call device_event_sync(event)
 
      ! Set residual to zero at strong velocity boundaries.
      call this%bclst_vel_res%apply(u_res, v_res, w_res, time)
 
 
      call profiler_end_region('Adjoint_velocity_residual')
 
      call this%proj_vel%pre_solving(u_res%x, v_res%x, w_res%x, &
           tstep, c_xh, n, dt_controller, 'Velocity')
 
      call this%pc_vel%update()
 
      call profiler_start_region("Adjoint_velocity_solve")
      ksp_results(2:4) = this%ksp_vel%solve_coupled(ax_vel, du, dv, dw, &
           u_res%x, v_res%x, w_res%x, n, c_xh, &
           this%bclst_du, this%bclst_dv, this%bclst_dw, gs_xh, &
           this%ksp_vel%max_iter)
      call profiler_end_region("Adjoint_velocity_solve")
 
      ksp_results(1)%name = 'Adjoint Pressure'
      ksp_results(2)%name = 'Adjoint Velocity U'
      ksp_results(3)%name = 'Adjoint Velocity V'
      ksp_results(4)%name = 'Adjoint Velocity W'
 
      call this%proj_vel%post_solving(du%x, dv%x, dw%x, ax_vel, c_xh, &
           this%bclst_du, this%bclst_dv, this%bclst_dw, gs_xh, n, tstep, &
           dt_controller)
 
      if (neko_bcknd_device .eq. 1) then
         call device_opadd2cm(u%x_d, v%x_d, w%x_d, &
              du%x_d, dv%x_d, dw%x_d, 1.0_rp, n, msh%gdim)
      else
         call opadd2cm(u%x, v%x, w%x, du%x, dv%x, dw%x, 1.0_rp, n, msh%gdim)
      end if
 
      if (this%forced_flow_rate) then
         call neko_error('Forced flow rate is not implemented for the adjoint')
 
      end if
 
      call fluid_step_info(time, ksp_results, &
           this%full_stress_formulation, this%strict_convergence)
 
    end associate
    call profiler_end_region('Adjoint')
 
  end subroutine adjoint_fluid_pnpn_step
 
  subroutine adjoint_fluid_pnpn_setup_bcs(this, user, params)
    use mpi_f08, only: mpi_in_place
    class(adjoint_fluid_pnpn_t), intent(inout) :: this
    type(user_t), target, intent(in) :: user
    type(json_file), intent(inout) :: params
    integer :: i, n_bcs, j, zone_size, global_zone_size, ierr
    class(bc_t), pointer :: bc_i
    type(json_core) :: core
    type(json_value), pointer :: bc_object
    type(json_file) :: bc_subdict
    logical :: found
    ! Monitor which boundary zones have been marked
    logical, allocatable :: marked_zones(:)
    integer, allocatable :: zone_indices(:)
    character(len=:), allocatable :: json_key
 
    ! Lists for the residuals and solution increments
    call this%bclst_vel_res%init()
    call this%bclst_du%init()
    call this%bclst_dv%init()
    call this%bclst_dw%init()
    call this%bclst_dp%init()
 
    call this%bc_vel_res%init_from_components(this%c_Xh)
    call this%bc_du%init_from_components(this%c_Xh)
    call this%bc_dv%init_from_components(this%c_Xh)
    call this%bc_dw%init_from_components(this%c_Xh)
    call this%bc_dp%init_from_components(this%c_Xh)
 
    ! Special PnPn boundary conditions for pressure
    call this%bc_prs_surface%init_from_components(this%c_Xh)
    call this%bc_sym_surface%init_from_components(this%c_Xh)
 
    json_key = 'case.adjoint_fluid.boundary_conditions'
 
    ! Populate bcs_vel and bcs_prs based on the case file
    if (params%valid_path(json_key)) then
       call params%info(json_key, n_children = n_bcs)
       call params%get_core(core)
       call params%get(json_key, bc_object, found)
 
       !
       ! Velocity bcs
       !
       call this%bcs_vel%init(n_bcs)
 
       allocate(marked_zones(size(this%msh%labeled_zones)))
       marked_zones = .false.
 
       do i = 1, n_bcs
          ! Create a new json containing just the subdict for this bc
          call json_extract_item(core, bc_object, i, bc_subdict)
 
          call json_get(bc_subdict, "zone_indices", zone_indices)
 
          ! Check that we are not trying to assing a bc to zone, for which one
          ! has already been assigned and that the zone has more than 0 size
          ! in the mesh.
          do j = 1, size(zone_indices)
             zone_size = this%msh%labeled_zones(zone_indices(j))%size
             call mpi_allreduce(zone_size, global_zone_size, 1, &
                  mpi_integer, mpi_max, neko_comm, ierr)
 
             if (global_zone_size .eq. 0) then
                write(error_unit, '(A, A, I0, A, A, I0, A)') "*** ERROR ***: ",&
                     "Zone index ", zone_indices(j), &
                     " is invalid as this zone has 0 size, meaning it does ", &
                     "not in the mesh. Check adjoint boundary condition ", &
                     i, "."
                error stop
             end if
 
             if (marked_zones(zone_indices(j)) .eqv. .true.) then
                write(error_unit, '(A, A, I0, A, A, A, A)') "*** ERROR ***: ", &
                     "Zone with index ", zone_indices(j), &
                     " has already been assigned a boundary condition. ", &
                     "Please check your boundary_conditions entry for the ", &
                     "adjoint and make sure that each zone index appears ", &
                     "only in a single boundary condition."
                error stop
             else
                marked_zones(zone_indices(j)) = .true.
             end if
          end do
 
          bc_i => null()
          call velocity_bc_factory(bc_i, this, bc_subdict, this%c_Xh, user)
 
          ! Not all bcs require an allocation for velocity in particular,
          ! so we check.
          if (associated(bc_i)) then
 
             ! We need to treat mixed bcs separately because they are by
             ! convention marked weak and currently contain nested
             ! bcs, some of which are strong.
             select type (bc_i)
             type is (symmetry_t)
                ! Symmetry has 3 internal bcs, but only one actually contains
                ! markings.
                ! Symmetry's apply_scalar doesn't do anything, so we need to
                ! mark individual nested bcs to the du,dv,dw, whereas the
                ! vel_res can just get symmetry as a whole, because on this
                ! list we call apply_vector.
                ! Additionally we have to mark the special surface bc for p.
                call this%bclst_vel_res%append(bc_i)
                call this%bc_du%mark_facets(bc_i%bc_x%marked_facet)
                call this%bc_dv%mark_facets(bc_i%bc_y%marked_facet)
                call this%bc_dw%mark_facets(bc_i%bc_z%marked_facet)
 
                call this%bcs_vel%append(bc_i)
 
                call this%bc_sym_surface%mark_facets(bc_i%marked_facet)
             type is (non_normal_t)
                ! This is a bc for the residuals and increments, not the
                ! velocity itself. So, don't append to bcs_vel
                call this%bclst_vel_res%append(bc_i)
                call this%bc_du%mark_facets(bc_i%bc_x%marked_facet)
                call this%bc_dv%mark_facets(bc_i%bc_y%marked_facet)
                call this%bc_dw%mark_facets(bc_i%bc_z%marked_facet)
             type is (shear_stress_t)
                ! Same as symmetry
                call this%bclst_vel_res%append(bc_i%symmetry)
                call this%bclst_du%append(bc_i%symmetry%bc_x)
                call this%bclst_dv%append(bc_i%symmetry%bc_y)
                call this%bclst_dw%append(bc_i%symmetry%bc_z)
 
                call this%bcs_vel%append(bc_i)
             type is (wall_model_bc_t)
                ! Same as symmetry
                call this%bclst_vel_res%append(bc_i%symmetry)
                call this%bclst_du%append(bc_i%symmetry%bc_x)
                call this%bclst_dv%append(bc_i%symmetry%bc_y)
                call this%bclst_dw%append(bc_i%symmetry%bc_z)
 
                call this%bcs_vel%append(bc_i)
             class default
 
                ! For the default case we use our dummy zero_dirichlet bcs to
                ! mark the same faces as in ordinary velocity dirichlet
                ! conditions.
                ! Additionally we mark the special PnPn pressure  bc.
                if (bc_i%strong .eqv. .true.) then
                   call this%bc_vel_res%mark_facets(bc_i%marked_facet)
                   call this%bc_du%mark_facets(bc_i%marked_facet)
                   call this%bc_dv%mark_facets(bc_i%marked_facet)
                   call this%bc_dw%mark_facets(bc_i%marked_facet)
 
                   call this%bc_prs_surface%mark_facets(bc_i%marked_facet)
                end if
 
                call this%bcs_vel%append(bc_i)
             end select
          end if
       end do
 
       ! Make sure all labeled zones with non-zero size have been marked
       do i = 1, size(this%msh%labeled_zones)
          if ((this%msh%labeled_zones(i)%size .gt. 0) .and. &
               (marked_zones(i) .eqv. .false.)) then
             write(error_unit, '(A, A, I0)') "*** ERROR ***: ", &
                  "No adjoint boundary condition assigned to zone ", i
             error stop
          end if
       end do
 
       !
       ! Pressure bcs
       !
       call this%bcs_prs%init(n_bcs)
 
       do i = 1, n_bcs
          ! Create a new json containing just the subdict for this bc
          call json_extract_item(core, bc_object, i, bc_subdict)
          bc_i => null()
          call pressure_bc_factory(bc_i, this, bc_subdict, this%c_Xh, user)
 
          ! Not all bcs require an allocation for pressure in particular,
          ! so we check.
          if (associated(bc_i)) then
             call this%bcs_prs%append(bc_i)
 
             ! Mark strong bcs in the dummy dp bc to force zero change.
             if (bc_i%strong .eqv. .true.) then
                call this%bc_dp%mark_facets(bc_i%marked_facet)
             end if
 
          end if
 
       end do
    else
       ! Check that there are no labeled zones, i.e. all are periodic.
       do i = 1, size(this%msh%labeled_zones)
          if (this%msh%labeled_zones(i)%size .gt. 0) then
             call neko_error("No boundary_conditions entry in the case file!")
          end if
       end do
 
    end if
 
    call this%bc_prs_surface%finalize()
    call this%bc_sym_surface%finalize()
 
    call this%bc_vel_res%finalize()
    call this%bc_du%finalize()
    call this%bc_dv%finalize()
    call this%bc_dw%finalize()
    call this%bc_dp%finalize()
 
    call this%bclst_vel_res%append(this%bc_vel_res)
    call this%bclst_du%append(this%bc_du)
    call this%bclst_dv%append(this%bc_dv)
    call this%bclst_dw%append(this%bc_dw)
    call this%bclst_dp%append(this%bc_dp)
 
    ! If we have no strong pressure bcs, we will demean the pressure
    this%prs_dirichlet = .not. this%bclst_dp%is_empty()
    call mpi_allreduce(mpi_in_place, this%prs_dirichlet, 1, &
         mpi_logical, mpi_lor, neko_comm)
 
  end subroutine adjoint_fluid_pnpn_setup_bcs
 
  subroutine adjoint_fluid_pnpn_write_boundary_conditions(this)
    class(adjoint_fluid_pnpn_t), target, intent(inout) :: this
    type(dirichlet_t) :: bdry_mask
    type(field_t), pointer :: bdry_field
    type(file_t) :: bdry_file
    integer :: temp_index, i
    class(bc_t), pointer :: bci
    character(len=LOG_SIZE) :: log_buf
 
    call neko_log%section("Adjoint boundary conditions")
    write(log_buf, '(A)') &
         'Marking using integer keys in boundary_adjoint0.f00000'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') 'Condition-value pairs: '
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  no_slip                         = 1'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  velocity_value                  = 2'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  outflow, normal_outflow (+dong) = 3'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  symmetry                        = 4'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  user_velocity_pointwise         = 5'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  periodic                        = 6'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  user_velocity                   = 7'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  user_pressure                   = 8'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  shear_stress                    = 9'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  wall_modelling                  = 10'
    call neko_log%message(log_buf)
    write(log_buf, '(A)') '  blasius_profile                 = 11'
    call neko_log%message(log_buf)
    call neko_log%end_section()
 
    call neko_scratch_registry%request_field(bdry_field, temp_index, .true.)
 
 
 
    call bdry_mask%init_from_components(this%c_Xh, 6.0_rp)
    call bdry_mask%mark_zone(this%msh%periodic)
    call bdry_mask%finalize()
    call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
    call bdry_mask%free()
 
    do i = 1, this%bcs_prs%size()
       bci => this%bcs_prs%get(i)
       select type (bc => bci)
       type is (zero_dirichlet_t)
          call bdry_mask%init_from_components(this%c_Xh, 3.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (dong_outflow_t)
          call bdry_mask%init_from_components(this%c_Xh, 3.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (field_dirichlet_t)
          call bdry_mask%init_from_components(this%c_Xh, 8.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       end select
    end do
 
    do i = 1, this%bcs_vel%size()
       bci => this%bcs_vel%get(i)
       select type (bc => bci)
       type is (zero_dirichlet_t)
          call bdry_mask%init_from_components(this%c_Xh, 1.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (inflow_t)
          call bdry_mask%init_from_components(this%c_Xh, 2.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (symmetry_t)
          call bdry_mask%init_from_components(this%c_Xh, 4.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (field_dirichlet_vector_t)
          call bdry_mask%init_from_components(this%c_Xh, 7.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (shear_stress_t)
          call bdry_mask%init_from_components(this%c_Xh, 9.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (wall_model_bc_t)
          call bdry_mask%init_from_components(this%c_Xh, 10.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       type is (blasius_t)
          call bdry_mask%init_from_components(this%c_Xh, 11.0_rp)
          call bdry_mask%mark_facets(bci%marked_facet)
          call bdry_mask%finalize()
          call bdry_mask%apply_scalar(bdry_field%x, this%dm_Xh%size())
          call bdry_mask%free()
       end select
    end do
 
 
    call bdry_file%init('boundary_adjoint.fld')
    call bdry_file%write(bdry_field)
 
    call neko_scratch_registry%relinquish_field(temp_index)
  end subroutine adjoint_fluid_pnpn_write_boundary_conditions
 
  ! End of section to verify
  ! ========================================================================== !
 
  subroutine rescale_fluid(fluid_data, scale)
 
    class(adjoint_fluid_pnpn_t), intent(inout) :: fluid_data
    real(kind=rp), intent(in) :: scale
 
    ! Local variables
    integer :: i
 
    ! Scale the velocity fields
    if (neko_bcknd_device .eq. 1) then
       call device_cmult(fluid_data%u_adj%x_d, scale, fluid_data%u_adj%size())
       call device_cmult(fluid_data%v_adj%x_d, scale, fluid_data%v_adj%size())
       call device_cmult(fluid_data%w_adj%x_d, scale, fluid_data%w_adj%size())
    else
       call cmult(fluid_data%u_adj%x, scale, fluid_data%u_adj%size())
       call cmult(fluid_data%v_adj%x, scale, fluid_data%v_adj%size())
       call cmult(fluid_data%w_adj%x, scale, fluid_data%w_adj%size())
    end if
 
    ! Scale the right hand sides
    if (neko_bcknd_device .eq. 1) then
       call device_cmult(fluid_data%f_adj_x%x_d, scale, &
            fluid_data%f_adj_x%size())
       call device_cmult(fluid_data%f_adj_y%x_d, scale, &
            fluid_data%f_adj_y%size())
       call device_cmult(fluid_data%f_adj_z%x_d, scale, &
            fluid_data%f_adj_z%size())
       ! HARRY
       ! maybe the abx's too
       call device_cmult(fluid_data%abx1%x_d, scale, fluid_data%abx1%size())
       call device_cmult(fluid_data%aby1%x_d, scale, fluid_data%aby1%size())
       call device_cmult(fluid_data%abz1%x_d, scale, fluid_data%abz1%size())
       call device_cmult(fluid_data%abx2%x_d, scale, fluid_data%abx2%size())
       call device_cmult(fluid_data%aby2%x_d, scale, fluid_data%aby2%size())
       call device_cmult(fluid_data%abz2%x_d, scale, fluid_data%abz2%size())
 
    else
       call cmult(fluid_data%f_adj_x%x, scale, fluid_data%f_adj_x%size())
       call cmult(fluid_data%f_adj_y%x, scale, fluid_data%f_adj_y%size())
       call cmult(fluid_data%f_adj_z%x, scale, fluid_data%f_adj_z%size())
 
       call cmult(fluid_data%abx1%x, scale, fluid_data%abx1%size())
       call cmult(fluid_data%aby1%x, scale, fluid_data%aby1%size())
       call cmult(fluid_data%abz1%x, scale, fluid_data%abz1%size())
 
       call cmult(fluid_data%abx2%x, scale, fluid_data%abx2%size())
       call cmult(fluid_data%aby2%x, scale, fluid_data%aby2%size())
       call cmult(fluid_data%abz2%x, scale, fluid_data%abz2%size())
    end if
 
    ! Scale the lag terms
    if (neko_bcknd_device .eq. 1) then
       do i = 1, fluid_data%ulag%size()
          call device_cmult(fluid_data%ulag%lf(i)%x_d, &
               scale, fluid_data%ulag%lf(i)%size())
       end do
 
       do i = 1, fluid_data%vlag%size()
          call device_cmult(fluid_data%vlag%lf(i)%x_d, &
               scale, fluid_data%vlag%lf(i)%size())
       end do
 
       do i = 1, fluid_data%wlag%size()
          call device_cmult(fluid_data%wlag%lf(i)%x_d, &
               scale, fluid_data%wlag%lf(i)%size())
       end do
    else
       do i = 1, fluid_data%ulag%size()
          call cmult(fluid_data%ulag%lf(i)%x, &
               scale, fluid_data%ulag%lf(i)%size())
       end do
 
       do i = 1, fluid_data%vlag%size()
          call cmult(fluid_data%vlag%lf(i)%x, &
               scale, fluid_data%vlag%lf(i)%size())
       end do
 
       do i = 1, fluid_data%wlag%size()
          call cmult(fluid_data%wlag%lf(i)%x, &
               scale, fluid_data%wlag%lf(i)%size())
       end do
    end if
 
  end subroutine rescale_fluid
 
  function norm(x, y, z, B, volume, n)
    use mpi_f08, only: mpi_in_place
 
    integer, intent(in) :: n
    real(kind=rp), dimension(n), intent(in) :: x, y, z
    real(kind=rp), dimension(n), intent(in) :: b
    real(kind=rp), intent(in) :: volume
 
    real(kind=rp) :: norm
 
    norm = vlsc3(x, x, b, n) + vlsc3(y, y, b, n) + vlsc3(z, z, b, n)
 
    call mpi_allreduce(mpi_in_place, norm, 1, &
         mpi_real_precision, mpi_sum, mpi_comm_world)
 
    norm = sqrt(norm / volume)
  end function norm
 
  function device_norm(x_d, y_d, z_d, B_d, volume, n)
    use mpi_f08, only: mpi_in_place
 
    type(c_ptr), intent(in) :: x_d, y_d, z_d
    type(c_ptr), intent(in) :: B_d
    real(kind=rp), intent(in) :: volume
    integer, intent(in) :: n
 
    real(kind=rp) :: device_norm
 
    device_norm = device_vlsc3(x_d, x_d, b_d, n) + &
         device_vlsc3(y_d, y_d, b_d, n) + &
         device_vlsc3(z_d, z_d, b_d, n)
 
    call mpi_allreduce(mpi_in_place, device_norm, 1, &
         mpi_real_precision, mpi_sum, mpi_comm_world)
 
    device_norm = sqrt(device_norm / volume)
 
  end function device_norm
 
  subroutine power_iterations_compute(this, t, tstep)
    class(adjoint_fluid_pnpn_t), target, intent(inout) :: this
    real(kind=rp), intent(in) :: t
    integer, intent(in) :: tstep
 
    ! Local variables
    real(kind=rp) :: scaling_factor
    real(kind=rp) :: norm_l2, norm_l2_base
    character(len=256) :: log_message
    type(vector_t) :: data_line
    integer :: n
 
    n = this%c_Xh%dof%size()
    if (tstep .eq. 1) then
       if (neko_bcknd_device .eq. 1) then
          norm_l2_base = device_norm(this%u_adj%x_d, this%v_adj%x_d, &
               this%w_adj%x_d, &
               this%c_Xh%B_d, this%c_Xh%volume, n)
       else
          norm_l2_base = this%norm_scaling * norm(this%u_adj%x, this%v_adj%x, &
               this%w_adj%x, &
               this%c_Xh%B, this%c_Xh%volume, n)
       end if
       if (this%norm_target .lt. 0.0_rp) then
          this%norm_target = norm_l2_base
       end if
 
       this%norm_l2_upper = this%norm_tolerance * this%norm_target
       this%norm_l2_lower = this%norm_target / this%norm_tolerance
 
    end if
 
    ! Compute the norm of the velocity field and eigenvalue estimate
    if (neko_bcknd_device .eq. 1) then
       norm_l2 = device_norm(this%u_adj%x_d, this%v_adj%x_d, this%w_adj%x_d, &
            this%c_Xh%B_d, this%c_Xh%volume, n)
    else
       norm_l2 = norm(this%u_adj%x, this%v_adj%x, this%w_adj%x, &
            this%c_Xh%B, this%c_Xh%volume, n)
    end if
    norm_l2 = sqrt(this%norm_scaling) * norm_l2
    scaling_factor = 1.0_rp
 
    ! Rescale the flow if necessary
    if (norm_l2 .gt. this%norm_l2_upper &
         .or. norm_l2 .lt. this%norm_l2_lower) then
       scaling_factor = this%norm_target / norm_l2
       call rescale_fluid(this, scaling_factor)
       norm_l2 = this%norm_target
 
       if (tstep .eq. 1) then
          scaling_factor = 1.0_rp
       end if
    end if
 
    ! Log the results
    !call neko_log%section('Power Iterations', lvl = NEKO_LOG_DEBUG)
    call neko_log%section('Power Iterations')
 
    write (log_message, '(A7,E20.14)') 'Norm: ', norm_l2
    call neko_log%message(log_message, lvl = neko_log_debug)
    write (log_message, '(A7,E20.14)') 'Scaling: ', scaling_factor
    call neko_log%message(log_message, lvl = neko_log_debug)
 
    ! Save to file
    call data_line%init(2)
    data_line%x = [norm_l2, scaling_factor]
    call this%file_output%write(data_line, t)
 
    !call neko_log%end_section('Power Iterations', lvl = NEKO_LOG_DEBUG)
    call neko_log%end_section('Power Iterations')
  end subroutine power_iterations_compute
 
 
end module adjoint_fluid_pnpn