condlim_test_21.f90

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MODULE boundary_test_21
  USE my_util
  USE def_type_mesh
  USE input_data
!!$ATTENTION
!!$Some subroutines have been commented to avoid warning messages when compiling executable.
!!$It can not be done in the module boundary_generic that expects all subroutines to be present.
!!$END ATTENTION
  PUBLIC :: init_velocity_pressure
!!$  PUBLIC :: init_temperature
  PUBLIC :: init_level_set
  PUBLIC :: source_in_ns_momentum
!!$  PUBLIC :: source_in_temperature
  PUBLIC :: source_in_level_set
  PUBLIC :: vv_exact
!!$  PUBLIC :: imposed_velocity_by_penalty
  PUBLIC :: pp_exact
!!$  PUBLIC :: temperature_exact
  PUBLIC :: level_set_exact
!!$  PUBLIC :: penal_in_real_space
  PUBLIC :: extension_velocity
!!$  PUBLIC :: Vexact
!!$  PUBLIC :: H_B_quasi_static
  PUBLIC :: hexact
  PUBLIC :: phiexact
  PUBLIC :: jexact_gauss
  PUBLIC :: eexact_gauss
  PUBLIC :: init_maxwell
!!$  PUBLIC :: mu_bar_in_fourier_space
!!$  PUBLIC :: grad_mu_bar_in_fourier_space
!!$  PUBLIC :: mu_in_real_space
  PUBLIC :: sigma_bar_in_fourier_space
  PRIVATE

CONTAINS
  !===============================================================================
  !                       Boundary conditions for Navier-Stokes
  !===============================================================================

  !===Initialize velocity, pressure
  SUBROUTINE init_velocity_pressure(mesh_f, mesh_c, time, dt, list_mode, &
       un_m1, un, pn_m1, pn, phin_m1, phin)
    IMPLICIT NONE
    TYPE(mesh_type)                            :: mesh_f, mesh_c
    REAL(KIND=8),                   INTENT(OUT):: time
    REAL(KIND=8),                   INTENT(IN) :: dt
    INTEGER,      DIMENSION(:),     INTENT(IN) :: list_mode
    REAL(KIND=8), DIMENSION(:,:,:), INTENT(OUT):: un_m1, un
    REAL(KIND=8), DIMENSION(:,:,:), INTENT(OUT):: pn_m1, pn, phin_m1, phin
    INTEGER                                    :: mode, i, j 
    REAL(KIND=8), DIMENSION(mesh_c%np)         :: pn_m2

    time = 0.d0
    DO i= 1, SIZE(list_mode)
       mode = list_mode(i) 
       DO j = 1, 6 
          !===velocity
          un_m1(:,j,i) = vv_exact(j,mesh_f%rr,mode,time-dt)  
          un(:,j,i) = vv_exact(j,mesh_f%rr,mode,time)
       END DO
       DO j = 1, 2
          !===pressure
          pn_m2(:)       = pp_exact(j,mesh_c%rr,mode,time-2*dt)
          pn_m1(:,j,i) = pp_exact(j,mesh_c%rr,mode,time-dt)
          pn(:,j,i) = pp_exact(j,mesh_c%rr,mode,time)
          phin_m1(:,j,i) = pn_m1(:,j,i) - pn_m2(:)
          phin(:,j,i) = pn(:,j,i) - pn_m1(:,j,i)
       ENDDO
    ENDDO
  END SUBROUTINE init_velocity_pressure

!!$  !===Initialize temperature
!!$  SUBROUTINE init_temperature(mesh, time, dt, list_mode, tempn_m1, tempn)
!!$    IMPLICIT NONE
!!$    TYPE(mesh_type)                            :: mesh
!!$    REAL(KIND=8),                   INTENT(OUT):: time
!!$    REAL(KIND=8),                   INTENT(IN) :: dt
!!$    INTEGER,      DIMENSION(:),     INTENT(IN) :: list_mode
!!$    REAL(KIND=8), DIMENSION(:,:,:), INTENT(OUT):: tempn_m1, tempn
!!$    INTEGER                                    :: mode, i, j 
!!$
!!$    time = 0.d0
!!$    DO i= 1, SIZE(list_mode)
!!$       mode = list_mode(i) 
!!$       DO j = 1, 2 
!!$          tempn_m1(:,j,i) = temperature_exact(j, mesh%rr, mode, time-dt)
!!$          tempn   (:,j,i) = temperature_exact(j, mesh%rr, mode, time)
!!$       ENDDO
!!$    ENDDO
!!$  END SUBROUTINE init_temperature

  !===Initialize level_set
  SUBROUTINE init_level_set(vv_mesh, time, &
       dt, list_mode, level_set_m1, level_set)
    IMPLICIT NONE
    TYPE(mesh_type)                              :: vv_mesh 
    REAL(KIND=8),                     INTENT(OUT):: time
    REAL(KIND=8),                     INTENT(IN) :: dt
    INTEGER,      DIMENSION(:),       INTENT(IN) :: list_mode
    REAL(KIND=8), DIMENSION(:,:,:,:), INTENT(OUT):: level_set, level_set_m1
    INTEGER                                      :: mode, i, j, n 

    time = 0.d0
    DO i= 1, SIZE(list_mode)
       mode = list_mode(i) 
       DO j = 1, 2
          !===level_set
          DO n = 1, inputs%nb_fluid -1
             level_set_m1(n,:,j,i) = level_set_exact(n,j,vv_mesh%rr,mode,time-dt)  
             level_set(n,:,j,i) = level_set_exact(n,j,vv_mesh%rr,mode,time)
          END DO
       END DO
    END DO

  END SUBROUTINE init_level_set

  !===Source in momemtum equation. Always called.
  FUNCTION source_in_ns_momentum(TYPE, rr, mode, i, time, Re, ty, opt_density, opt_tempn) RESULT(vv)
    IMPLICIT NONE
    INTEGER     ,                             INTENT(IN) :: type
    REAL(KIND=8), DIMENSION(:,:),             INTENT(IN) :: rr
    INTEGER     ,                             INTENT(IN) :: mode, i
    REAL(KIND=8),                             INTENT(IN) :: time 
    REAL(KIND=8),                             INTENT(IN) :: re
    CHARACTER(LEN=2),                         INTENT(IN) :: ty
    REAL(KIND=8), DIMENSION(:,:,:), OPTIONAL, INTENT(IN) :: opt_density
    REAL(KIND=8), DIMENSION(:,:,:), OPTIONAL, INTENT(IN) :: opt_tempn 
    REAL(KIND=8), DIMENSION(SIZE(rr,2))                  :: vv
    REAL(KIND=8), DIMENSION(SIZE(rr,2))                  :: r, z
    INTEGER                                              :: m
    CHARACTER(LEN=2)  :: np

    IF (present(opt_tempn)) CALL error_petsc('temperature should not be present for test 21') 

    r = rr(1,:)
    z = rr(2,:)
    m = mode

    IF (m==0 .AND. type==1) THEN
       vv = 32.d0*r**3*z**2
    ELSE IF (m==0 .AND. type==5) THEN
       vv = 8.d0*r**4*z
    ELSE
       vv = 0.d0
    END IF
    RETURN

    !===Dummies variables to avoid warning
    m=i; r=time; r=re; np=ty
    IF (present(opt_density)) m=SIZE(opt_density,1)
    !===Dummies variables to avoid warning
  END FUNCTION source_in_ns_momentum

!!$  !===Extra source in temperature equation. Always called.
!!$  FUNCTION source_in_temperature(TYPE, rr, m, t)RESULT(vv)
!!$    IMPLICIT NONE
!!$    INTEGER     ,                        INTENT(IN)   :: TYPE
!!$    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
!!$    INTEGER     ,                        INTENT(IN)   :: m
!!$    REAL(KIND=8),                        INTENT(IN)   :: t   
!!$    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
!!$
!!$    vv = 0.d0
!!$    CALL error_petsc('source_in_temperature: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION source_in_temperature

  !===Extra source in level set equation. Always called.
  FUNCTION source_in_level_set(interface_nb,TYPE, rr, m, t)RESULT(vv)
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m, interface_nb
    REAL(KIND=8),                        INTENT(IN)   :: t   
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv = 0.d0
    RETURN

    !===Dummies variables to avoid warning
    n=type; n=SIZE(rr,1); n=m; n=interface_nb; r=t
    !===Dummies variables to avoid warning
  END FUNCTION source_in_level_set

  !===Velocity for boundary conditions in Navier-Stokes.
  !===Can be used also to initialize velocity in: init_velocity_pressure_temperature 
  FUNCTION vv_exact(TYPE,rr,m,t) RESULT(vv)
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER,                             INTENT(IN)   :: m
    REAL(KIND=8),                        INTENT(IN)   :: t
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv = 0.d0
    RETURN

    !===Dummies variables to avoid warning
    n=type; n=SIZE(rr,1); n=m; r=t
    !===Dummies variables to avoid warning
  END FUNCTION vv_exact

!!$ !===Solid velocity imposed when using penalty technique
!!$ !===Defined in Fourier space on mode 0 only.
!!$ FUNCTION imposed_velocity_by_penalty(rr,t) RESULT(vv)
!!$    IMPLICIT NONE 
!!$    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
!!$    REAL(KIND=8),                        INTENT(IN)   :: t
!!$    REAL(KIND=8), DIMENSION(SIZE(rr,2),6)             :: vv
!!$
!!$    vv=0.d0
!!$    RETURN
!!$  END FUNCTION imposed_velocity_by_penalty

  !===Pressure for boundary conditions in Navier-Stokes.
  !===Can be used also to initialize pressure in the subroutine init_velocity_pressure.
  !===Use this routine for outflow BCs only.
  !===CAUTION: Do not enfore BCs on pressure where normal component 
  !            of velocity is prescribed.
  FUNCTION pp_exact(TYPE,rr,m,t) RESULT (vv)
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m
    REAL(KIND=8),                        INTENT(IN)   :: t
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv 
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv = 0.d0
    RETURN

    !===Dummies variables to avoid warning
    n=type; n=SIZE(rr,1); n=m; r=t
    !===Dummies variables to avoid warning
  END FUNCTION pp_exact

!!$  !===Temperature for boundary conditions in temperature equation.
!!$  FUNCTION temperature_exact(TYPE,rr,m,t) RESULT (vv)
!!$    IMPLICIT NONE
!!$    INTEGER     ,                        INTENT(IN)   :: TYPE
!!$    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
!!$    INTEGER     ,                        INTENT(IN)   :: m
!!$    REAL(KIND=8),                        INTENT(IN)   :: t
!!$    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
!!$
!!$    vv = 0.d0
!!$    CALL error_petsc('temperature_exact: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION temperature_exact

  !===Can be used to initialize level set in the subroutine init_level_set.
  FUNCTION level_set_exact(interface_nb,TYPE,rr,m,t)  RESULT (vv)
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m, interface_nb
    REAL(KIND=8),                        INTENT(IN)   :: t
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
    REAL(KIND=8) :: r

    IF (interface_nb==1) THEN
       IF (m==0 .AND. TYPE ==1) then
          vv = 1.d0 + rr(2,:)
       ELSE IF (m==1 .AND. type==1) THEN
          vv = rr(1,:)
       ELSE
          vv = 0.d0
       END IF
    ELSE 
       CALL error_petsc(' BUG in level_set_exact, we should compute only 1 level set')
    END IF
    RETURN

    !===Dummies variables to avoid warning
    r=t
    !===Dummies variables to avoid warning
  END FUNCTION level_set_exact

!!$  !===Penalty coefficient (if needed)
!!$  !===This coefficient is equal to zero in subdomain
!!$  !===where penalty is applied (penalty is zero in solid)
!!$  FUNCTION penal_in_real_space(mesh,rr_gauss,angles,nb_angles,nb,ne,time) RESULT(vv)
!!$    IMPLICIT NONE
!!$    TYPE(mesh_type)                            :: mesh
!!$    REAL(KIND=8), DIMENSION(:,:), INTENT(IN)   :: rr_gauss
!!$    REAL(KIND=8), DIMENSION(:), INTENT(IN)     :: angles
!!$    INTEGER,                    INTENT(IN)     :: nb_angles
!!$    INTEGER,                    INTENT(IN)     :: nb, ne
!!$    REAL(KIND=8),               INTENT(IN)     :: time
!!$    REAL(KIND=8), DIMENSION(nb_angles,ne-nb+1) :: vv
!!$
!!$    vv = 1.d0
!!$    CALL error_petsc('penal_in_real_space: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION penal_in_real_space

  !===Extension of the velocity field in the solid.
  !===Used when temperature or Maxwell equations are solved.
  !===It extends the velocity field on the Navier-Stokes domain to a
  !===velocity field on the temperature and the Maxwell domain.
  !===It is also used if problem type=mxw and restart velocity
  !===is set to true in data (type problem denoted mxx in the code).
  FUNCTION extension_velocity(TYPE, H_mesh, mode, t, n_start) RESULT(vv)
    IMPLICIT NONE
    TYPE(mesh_type),                     INTENT(IN)   :: h_mesh     
    INTEGER     ,                        INTENT(IN)   :: type, n_start
    INTEGER,                             INTENT(IN)   :: mode 
    REAL(KIND=8),                        INTENT(IN)   :: t
    REAL(KIND=8), DIMENSION(H_Mesh%np)                :: vv
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv = 0.d0
    RETURN

    !===Dummies variables to avoid warning
    n=h_mesh%np; r=t; n=type; n=mode; n=n_start
    !===Dummies variables to avoid warning
  END FUNCTION extension_velocity

  !===============================================================================
  !                       Boundary conditions for Maxwell
  !===============================================================================
!!$  !===Velocity used in the induction equation.
!!$  !===Used only if problem type is mxw and restart velocity is false
!!$  FUNCTION Vexact(m, H_mesh) RESULT(vv)  !Set uniquement a l'induction
!!$    IMPLICIT NONE
!!$    TYPE(mesh_type),                       INTENT(IN) :: H_mesh 
!!$    INTEGER,                               INTENT(IN) :: m
!!$    REAL(KIND=8), DIMENSION(H_mesh%np,6)              :: vv
!!$
!!$    vv = 0.d0
!!$  END FUNCTION Vexact

!!$  !===Magnetic field and magnetic induction for quasi-static approximation
!!$  !===if needed
!!$  FUNCTION H_B_quasi_static(char_h_b, rr, m) RESULT(vv) 
!!$    IMPLICIT NONE
!!$    CHARACTER(LEN=1),                    INTENT(IN)   :: char_h_b
!!$    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
!!$    INTEGER,                             INTENT(IN)   :: m
!!$    REAL(KIND=8), DIMENSION(SIZE(rr,2),6)             :: vv
!!$
!!$    vv = 0.d0
!!$    RETURN
!!$  END FUNCTION H_B_quasi_static

  !===Magnetic field for boundary conditions in the Maxwell equations.
  FUNCTION hexact(H_mesh,TYPE, rr, m, mu_H_field, t) RESULT(vv) 
    IMPLICIT NONE
    TYPE(mesh_type),                     INTENT(IN)   :: h_mesh
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m
    REAL(KIND=8),                        INTENT(IN)   :: t 
    REAL(KIND=8), DIMENSION(:),          INTENT(IN)   :: mu_h_field
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv
    REAL(KIND=8) :: r
    INTEGER      :: n

    IF (m==0.AND. type==1) THEN
       vv = rr(1,:)**3
    ELSE IF (m==0.AND.type==5) THEN
       vv = -4.d0*rr(1,:)**2*rr(2,:)
    ELSE
       vv = 0.d0
    END IF
    RETURN

    !===Dummies variables to avoid warning
    n=h_mesh%np; r=mu_h_field(1); r=t
    !===Dummies variables to avoid warning
  END FUNCTION hexact

  !===Scalar potential for boundary conditions in the Maxwell equations.
  FUNCTION phiexact(TYPE, rr, m, mu_phi,t) RESULT(vv) 
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:,:),        INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m
    REAL(KIND=8),                        INTENT(IN)   :: mu_phi, t
    REAL(KIND=8), DIMENSION(SIZE(rr,2))               :: vv   
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv=0.d0
    CALL error_petsc('Phiexact: should not be called for this test')
    RETURN

    !===Dummies variables to avoid warning
    n=type; n=SIZE(rr,1); n=m; r=mu_phi; r=t
    !===Dummies variables to avoid warning
  END FUNCTION phiexact

  !===Current in Ohm's law. Curl(H) = sigma(E + uxB) + current
  FUNCTION jexact_gauss(TYPE, rr, m, mu_phi, sigma, mu_H, t, mesh_id, opt_B_ext) RESULT(vv) 
    IMPLICIT NONE
    INTEGER     ,                        INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:),          INTENT(IN)   :: rr
    INTEGER     ,                        INTENT(IN)   :: m 
    REAL(KIND=8),                        INTENT(IN)   :: mu_phi, sigma, mu_h, t 
    INTEGER     ,                        INTENT(IN)   :: mesh_id
    REAL(KIND=8), DIMENSION(6), OPTIONAL,INTENT(IN)   :: opt_b_ext 
    REAL(KIND=8)                                      :: vv
    REAL(KIND=8) :: r
    INTEGER      :: n

    IF(m==0.AND.type==3) THEN
       vv = 8.d0*rr(1)*rr(2)
    ELSE
       vv = 0.d0
    END IF
    RETURN

    !===Dummies variables to avoid warning
    r=rr(1); r=mu_phi; r=sigma; r=mu_h; r=t; n=type; n=m; n=mesh_id
    IF (present(opt_b_ext)) r=opt_b_ext(1)
    !===Dummies variables to avoid warning
  END FUNCTION jexact_gauss

  !===Electric field for Neumann BC (cf. doc)
  FUNCTION eexact_gauss(TYPE, rr, m, mu_phi, sigma, mu_H, t) RESULT(vv)
    IMPLICIT NONE
    INTEGER,                             INTENT(IN)   :: type
    REAL(KIND=8), DIMENSION(:),          INTENT(IN)   :: rr
    INTEGER,                             INTENT(IN)   :: m
    REAL(KIND=8),                        INTENT(IN)   :: mu_phi, sigma, mu_h, t
    REAL(KIND=8)                                      :: vv 
    REAL(KIND=8) :: r
    INTEGER      :: n

    vv = 0.d0
    RETURN

    !===Dummies variables to avoid warning
    r=rr(1); r=mu_phi; r=sigma; r=mu_h; r=t; n=type; n=m
    !===Dummies variables to avoid warning
  END FUNCTION eexact_gauss

  !===Initialization of magnetic field and scalar potential (if present)
  SUBROUTINE init_maxwell(H_mesh, phi_mesh, time, dt, mu_H_field, mu_phi, &
       list_mode, hn1, hn, phin1, phin)
    IMPLICIT NONE
    TYPE(mesh_type)                            :: h_mesh, phi_mesh     
    REAL(KIND=8),                   INTENT(OUT):: time
    REAL(KIND=8),                   INTENT(IN) :: dt
    REAL(KIND=8), DIMENSION(:),     INTENT(IN) :: mu_h_field
    REAL(KIND=8),                   INTENT(IN) :: mu_phi
    INTEGER,      DIMENSION(:),     INTENT(IN) :: list_mode    
    REAL(KIND=8), DIMENSION(:,:,:), INTENT(OUT):: hn, hn1
    REAL(KIND=8), DIMENSION(:,:,:), INTENT(OUT):: phin, phin1
    INTEGER                                    :: i, k

    time = -dt
    DO k=1,6
       DO i=1, SIZE(list_mode)
          hn1(:,k,i) = hexact(h_mesh,k, h_mesh%rr, list_mode(i), mu_h_field, time)
          IF (inputs%nb_dom_phi>0) THEN
             IF (k<3) THEN
                phin1(:,k,i) = phiexact(k, phi_mesh%rr, list_mode(i) , mu_phi, time)
             ENDIF
          END  IF
       ENDDO
    ENDDO

    time = time + dt 
    DO k=1,6
       DO i=1, SIZE(list_mode)
          hn(:,k,i) = hexact(h_mesh,k, h_mesh%rr, list_mode(i), mu_h_field, time)
          IF (inputs%nb_dom_phi>0) THEN
             IF (k<3) THEN
                phin(:,k,i) = phiexact(k, phi_mesh%rr, list_mode(i), mu_phi, time)
             ENDIF
          END  IF
       ENDDO
    ENDDO
  END SUBROUTINE init_maxwell

!!$  !===Analytical permeability (if needed)
!!$  !===This function is not needed unless the flag
!!$  !===     ===Use FEM Interpolation for magnetic permeability  (true/false)
!!$  !===is activated and set to .FALSE. in the data data file. Default is .TRUE.
!!$  FUNCTION mu_bar_in_fourier_space(H_mesh,nb,ne,pts,pts_ids) RESULT(vv)
!!$    IMPLICIT NONE
!!$    TYPE(mesh_type), INTENT(IN)                :: H_mesh
!!$    REAL(KIND=8), DIMENSION(ne-nb+1)           :: vv
!!$    INTEGER,     INTENT(IN)                    :: nb, ne
!!$    REAL(KIND=8),DIMENSION(2,ne-nb+1),OPTIONAL :: pts
!!$    INTEGER,     DIMENSION(ne-nb+1), OPTIONAL  :: pts_ids
!!$
!!$    vv = 1.d0
!!$    CALL error_petsc('mu_bar_in_fourier_space: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION mu_bar_in_fourier_space

!!$  !===Analytical mu_in_fourier_space (if needed)
!!$  !===This function is not needed unless the flag
!!$  !===     ===Use FEM Interpolation for magnetic permeability  (true/false)
!!$  !===is activated and set to .FALSE. in the data data file. Default is .TRUE.
!!$  FUNCTION grad_mu_bar_in_fourier_space(pt,pt_id) RESULT(vv)
!!$    IMPLICIT NONE
!!$    REAL(KIND=8),DIMENSION(2), INTENT(in):: pt
!!$    INTEGER,DIMENSION(1), INTENT(in)     :: pt_id
!!$    REAL(KIND=8),DIMENSION(2)            :: vv
!!$
!!$    vv=0.d0
!!$    CALL error_petsc('grad_mu_bar_in_fourier_space: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION grad_mu_bar_in_fourier_space

!!$  !===Analytical permeability, mu in real space (if needed)
!!$  FUNCTION mu_in_real_space(H_mesh,angles,nb_angles,nb,ne,time) RESULT(vv)
!!$    IMPLICIT NONE
!!$    TYPE(mesh_type), INTENT(IN)                :: H_mesh
!!$    REAL(KIND=8), DIMENSION(:), INTENT(IN)     :: angles
!!$    INTEGER, INTENT(IN)                        :: nb_angles
!!$    INTEGER, INTENT(IN)                        :: nb, ne
!!$    REAL(KIND=8), INTENT(IN)                   :: time
!!$    REAL(KIND=8), DIMENSION(nb_angles,ne-nb+1) :: vv
!!$
!!$    vv = 1.d0
!!$    CALL error_petsc('mu_in_real_space: should not be called for this test')
!!$    RETURN
!!$  END FUNCTION mu_in_real_space

  !=== Only used for multiphase flow with variable electrical conductivity
  !===This function is not needed unless the following line is in the data
  !=== ===Conductivity of fluid 0, fluid 1, ...
  !=== The above line should be followed by the value of the conductivity in the different fluids.
  !=== The result vv has to smaller than the effective conductivity on every nodes of the mesh.
  FUNCTION sigma_bar_in_fourier_space(H_mesh) RESULT(vv)
    USE input_data
    IMPLICIT NONE
    TYPE(mesh_type), INTENT(IN)                :: h_mesh
    REAL(KIND=8), DIMENSION(SIZE(H_mesh%rr,2)) :: vv

    vv = 0.5d0*minval(inputs%sigma_fluid)
    RETURN
  END FUNCTION sigma_bar_in_fourier_space
  
END MODULE boundary_test_21