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Module_Thermal_Skin_GFD_Bercovici.f90
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MODULE MODULE_THERMAL_SKIN_GFD_BERCOVICI
CONTAINS
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!! Equation : Heat equation
!!!!!!!! Model : Skin theory (Balmforth2004)
!!!!!!!! Schema : Finite differenc scheme
!!!!!!!! Model couplage : Bercovici rheology (1996)
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
SUBROUTINE THERMAL_SKIN_GFD_BERCOVICI(Xi,H,P,T,Ts,BL,Dt,Dr,theta,dist,ray,M,sigma,nu,Pe,psi,delta0,el,grav,N1,F_err,z,tmps)
!*****************************************************************
! Solve for the parameter Xi, and split in Temperature and thermal layer
! from the evolution euqation using a center difference in space and a general
! theta scheme is timre
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION, DIMENSION(:,:), INTENT(IN) :: H,P
DOUBLE PRECISION , DIMENSION(:,:), INTENT(INOUT) :: Xi,T,BL,Ts
DOUBLE PRECISION , DIMENSION(:), INTENT(IN) :: dist,ray
!Parametre du model
DOUBLE PRECISION , INTENT(IN) :: Dt,Dr,theta,tmps
!Nombre sans dimensions
DOUBLE PRECISION , INTENT(IN) :: sigma,nu,Pe,psi,delta0,el,grav,N1
INTEGER, INTENT(IN) :: M, z
DOUBLE PRECISION , INTENT(INOUT) :: F_err
!Variable du sous programmes
DOUBLE PRECISION, DIMENSION(:),ALLOCATABLE :: Xi_tmps
DOUBLE PRECISION, DIMENSION(:),ALLOCATABLE :: a,b,c,d,e,f,g,k,l,S
DOUBLE PRECISION ,DIMENSION(:), ALLOCATABLE :: Xi_m
DOUBLE PRECISION :: U
INTEGER :: i,ndyke,N,Size
INTEGER :: err1,col
LOGICAL :: CHO
! Taille de la grille
ndyke=sigma/Dr
CHO=COUNT(H(:,1)>delta0)<ndyke
SELECT CASE (CHO)
CASE(.TRUE.)
N = ndyke +10 ! Cas ou on donne pas de profile initiale...
CASE(.FALSE.)
N = COUNT(H(:,1)>delta0)+10
END SELECT
! Calcule de f tmps n et n+
ALLOCATE(Xi_tmps(1:N),stat=err1)
IF (err1>1) THEN
PRINT*, 'Erreur alloc xi_tmps-xi_guess'; STOP
END IF
! col=1
! CALL TEMPERATURE_CFD_BALMFORTH(Xi_tmps,col,N,Xi,H,T,Ts,BL,P,dist,ray,Dr,nu,Pe,delta0,el,grav,N1,tmps,Dt)
!Systeme a inverser
ALLOCATE(a(1:N),b(1:N),c(1:N),d(1:N),e(1:N),S(1:N),stat=err1)
ALLOCATE(f(1:N),g(1:N),k(1:N),l(1:N),stat=err1)
IF (err1>1) THEN
PRINT*, 'Erreur allocation dans coeff Temperature'; STOP
END IF
CALL FILLING_MATRIX_2(a,b,c,d,e,f,g,k,l,S,Xi_tmps,N,H,BL,T,Ts,Xi,P,Dr,dist,ray,&
&nu,N1,Pe,delta0,el,grav,tmps,Dt,theta,psi)
a(1)=0
c(N)=0
!Inversion de la matrice
ALLOCATE(Xi_m(1:N),stat=err1)
IF (err1>1) THEN
PRINT*, 'Erreur d''allocation dans vecteur Hm'; STOP
END IF
Xi_m(:)=H(:N,1)/2.d0
CALL NONA_DIAGO(N,Xi_m,a,b,c,d,e,f,g,k,l,S)
DO i=1,N,1
print*,Xi(i,3)
Xi(i,3)=Xi_m(i)
END DO
! Separation variables
CALL XI_SPLIT_2(Xi,T,BL,Ts,H,N,delta0,Dt,tmps,N1,Pe,el)
Xi(N:,3) = H(N:,3)/2.D0
T(N:,3) = 0.d0
Ts(N:,3) =0.d0
BL(N:,3) = H(N:,3)/2.d0
! Calcule de l'erreur
IF (DOT_PRODUCT(Xi(:,2),Xi(:,2)) == 0D0) THEN
F_err = ABS(MAXVAL(Xi_m(:)))
ELSE
Size = COUNT(Xi(:,2)>1D-10)
F_err = ABS(MAXVAL(((Xi(:Size,3)-Xi(:Size,2))/Xi(:Size,2))))
ENDIF
DEALLOCATE(Xi_m,a,b,c,S)
DEALLOCATE(Xi_tmps)
END SUBROUTINE THERMAL_SKIN_GFD_BERCOVICI
!*****************************************************************
! Solve for T and BL from xi for balmofrth with Ts=0
!*****************************************************************
SUBROUTINE XI_SPLIT_2(Xi,T,BL,Ts,H,N,delta0,Dt,tmps,N1,Pe,el)
!*****************************************************************
! Solve for T and BL from xi for balmofrth with Ts=0
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(IN) :: H
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(INOUT) :: BL,T,Xi,Ts
!Dimensionless parameter
DOUBLE PRECISION, INTENT(IN) :: delta0,N1,Pe,el
!Parametre du model
DOUBLE PRECISION :: Dt,tmps
INTEGER, INTENT(IN) :: N
! Parametre pour le sous programme
INTEGER :: i
DOUBLE PRECISION, PARAMETER :: pi=3.14159265
DOUBLE PRECISION :: Xit,Tss,beta,d1,d2,Dr
Dr = 0.01d0
! Separation des variables
DO i=1,N
Xit = H(i,3)/6.d0
IF (Xi(i,3) <= Xit) THEN
T(i,3) = 1.d0
BL(i,3) = 3.d0*Xi(i,3)
ELSEIF (Xi(i,3)> Xit) THEN
T(i,3)= 3.d0/2.D0-(3.d0*Xi(i,3)/H(i,3))
BL(i,3)=H(i,3)/2.d0
ENDIF
END DO
END SUBROUTINE XI_SPLIT_2
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!-------------------------------------------------------------------------------------
! SUBROUTINE NONA DIAG
!-------------------------------------------------------------------------------------
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
SUBROUTINE NONA_DIAGO(N,Hm,a,b,c,d,e,f,g,k,l,S)
!*****************************************************************
! Solves for a vector Hm of length N the nano diagonal linear set
! M Hm = S, where A, B, C, D, E, F, G, K and L are the three main
! diagonals of matrix M(N,N), the other terms are 0.
! S is the right side vector.
!*****************************************************************
IMPLICIT NONE
INTEGER , INTENT(IN) :: N
INTEGER :: i
INTEGER :: err3,err4
DOUBLE PRECISION, DIMENSION(:), INTENT(IN) :: a,b,c,d,e,f,g,k,l,S
DOUBLE PRECISION, DIMENSION(:),INTENT(INOUT) :: Hm
DOUBLE PRECISION, DIMENSION(:), ALLOCATABLE :: zeta,alpha,beta,mu,xi,lambda,eta,omega,gamma
AllOCATE(zeta(1:N),alpha(1:N),beta(1:N),mu(1:N),xi(1:N),stat=err3)
ALLOCATE(lambda(1:N),eta(1:N),omega(1:N),gamma(1:N),stat=err4)
IF (err3>1 .OR. err4>1) THEN
PRINT*, 'Erreur d''allocation dans vecteur P,Q'; STOP
END IF
zeta(1)=b(1)
alpha(1)=c(1)
beta(1)=d(1)
mu(1)=e(1)
xi(1)=f(1)/mu(1)
lambda(1)=g(1)/mu(1)
eta(1)=k(1)/mu(1)
omega(1)=l(1)/mu(1)
gamma(1)=S(1)/mu(1)
zeta(2)=b(2)
alpha(2)=c(2)
beta(2)=d(2)
mu(2)=e(2)-xi(1)*beta(2)
xi(2)=(f(2)-lambda(1)*beta(2))/mu(2)
lambda(2)=(g(2)-eta(1)*beta(2))/mu(2)
eta(2)=(k(2)-omega(1)*beta(2))/mu(2)
omega(2)=l(2)/mu(2)
gamma(2)=(S(2)-beta(2)*gamma(1))/mu(2)
zeta(3)=b(3)
alpha(3)=c(3)
beta(3)=d(3)-xi(1)*alpha(3)
mu(3)=e(3)-lambda(1)*alpha(3)-xi(2)*beta(3)
xi(3)=(f(3)-eta(1)*alpha(3)-lambda(2)*beta(3))/mu(3)
lambda(3)=(g(3)-omega(1)*alpha(3)-eta(2)*beta(3))/mu(3)
eta(3)=(k(3)-omega(2)*beta(3))/mu(3)
omega(3)=l(3)/mu(3)
gamma(3)=(S(3)-alpha(3)*gamma(1)-beta(3)*gamma(2))/mu(3)
zeta(4)=b(4)
alpha(4)=c(4)-xi(1)*zeta(4)
beta(4)=d(4)-lambda(1)*zeta(4)-xi(2)*alpha(4)
mu(4)=e(4)-eta(1)*zeta(4)-lambda(2)*alpha(4)-xi(3)*beta(4)
xi(4)=(f(4)-omega(1)*zeta(4)-eta(2)*alpha(4)-lambda(3)*beta(4))/mu(4)
lambda(4)=(g(4)-omega(2)*alpha(4)-eta(3)*beta(4))/mu(4)
eta(4)=(k(4)-omega(3)*beta(4))/mu(4)
omega(4)=l(4)/mu(4)
gamma(4)=(S(4)-zeta(4)*gamma(1)-alpha(4)*gamma(2)-beta(4)*gamma(3))/mu(4)
DO i=5,N
zeta(i)=b(i)-a(i)*xi(i-4)
alpha(i)=c(i)-a(i)*lambda(i-4)-xi(i-3)*zeta(i)
beta(i)=d(i)-a(i)*eta(i-4)-lambda(i-3)*zeta(i)-alpha(i)*xi(i-2)
mu(i)=e(i)-a(i)*omega(i-4)-zeta(i)*eta(i-3)-lambda(i-2)*alpha(i)-beta(i)*xi(i-1)
xi(i)=(f(i)-omega(i-3)*zeta(i)-eta(i-2)*alpha(i)-lambda(i-1)*beta(i))/mu(i)
lambda(i)=(g(i)-alpha(i)*omega(i-2)-eta(i-1)*beta(i))/mu(i)
eta(i)=(k(i)-omega(i-1)*beta(i))/mu(i)
omega(i)=l(i)/mu(i)
gamma(i)=(S(i)-a(i)*gamma(i-4)-zeta(i)*gamma(i-3)-alpha(i)*gamma(i-2)-beta(i)*gamma(i-1))/mu(i)
END DO
Hm(N)=gamma(N)
Hm(N-1)=gamma(N-1)-xi(N-1)*Hm(N)
Hm(N-2)=gamma(N-2)-lambda(N-2)*Hm(N)-xi(N-2)*Hm(N-1)
Hm(N-3)=gamma(N-3)-eta(N-3)*Hm(N)-lambda(N-3)*Hm(N-1)-xi(N-3)*Hm(N-2)
DO i=N-4,1,-1
Hm(i)=gamma(i)-xi(i)*Hm(i+1)-lambda(i)*Hm(i+2)-eta(i)*Hm(i+3)-omega(i)*Hm(i+4)
END DO
DEALLOCATE(zeta,alpha,beta,mu,xi,lambda,eta,omega,gamma)
END SUBROUTINE NONA_DIAGO
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
! SUBROUTINE TRIDIAG
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
SUBROUTINE TRIDIAG(A,B,C,S,N,U)
!*****************************************************************
! Solves for a vector U of length N the tridiagonal linear set
! M U = R, where A, B and C are the three main diagonals of matrix
! M(N,N), the other terms are 0. R is the right side vector.
!*****************************************************************
IMPLICIT NONE
DOUBLE PRECISION, DIMENSION(N), INTENT(IN) :: A,B,C,S
DOUBLE PRECISION, DIMENSION(N), INTENT(INOUT) :: U
INTEGER, INTENT(IN) :: N
INTEGER :: CODE
DOUBLE PRECISION, DIMENSION(N) :: GAM
DOUBLE PRECISION :: BET
INTEGER :: j
BET = B(1)
IF (BET == 0.D0) THEN
PRINT*,'ERROR TRIDIAG'
STOP
ENDIF
U(1) = S(1)/BET
DO J=2,N !Decomposition and forward substitution
GAM(j)=C(j-1)/BET
BET=B(j)-A(j)*GAM(j)
IF(BET.EQ.0.D0) THEN !Algorithm fails
PRINT*,'ERRORTRIDIAG2',j,N
STOP
END IF
U(j)=(S(j)-A(j)*U(j-1))/BET
END DO
DO j=N-1,1,-1 !Back substitution
U(j)=U(j)-GAM(j+1)*U(j+1)
END DO
CODE=0
RETURN
END SUBROUTINE TRIDIAG
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
! SUBROUTINE X_SPLIT
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
SUBROUTINE XI_SPLIT_BALMFORTH(Xi,T,BL,Ts,H,N,delta0,Dt,tmps,N1,Pe,el)
!*****************************************************************
! Solve for T and BL from Xi deriving Ts directly here
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(IN) :: H
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(INOUT) :: BL,T,Xi,Ts
!Dimensionless parameter
DOUBLE PRECISION, INTENT(IN) :: delta0,N1,Pe,el
!Parametre du model
DOUBLE PRECISION :: Dt,tmps
INTEGER, INTENT(IN) :: N
! Parametre pour le sous programme
INTEGER :: i
DOUBLE PRECISION, PARAMETER :: pi=3.14159265
DOUBLE PRECISION :: Xit,Tss,beta
! Separation des variables
DO i=1,N
beta = N1*Pe**(-0.5d0)/(sqrt(pi*tmps))
Xit = beta*H(i,3)**2/(6.d0*beta*H(i,3)+24.d0)
IF (Xi(i,3) <= Xit) THEN
Ts(i,3) = 3.d0*beta/4.d0*Xi(i,3)&
&-sqrt(3.d0)/4.d0*sqrt(beta*Xi(i,3)*(3.d0*Xi(i,3)*beta+8.d0))+1.d0
T(i,3) = 1.d0
BL(i,3) = 1/(Ts(i,3)*beta)*(2.d0-2.d0*Ts(i,3))
ELSEIF (Xi(i,3)> Xit) THEN
Ts(i,3) =(-12.d0*Xi(i,3)+6.d0*H(i,3))/((beta*H(i,3)+6.d0)*H(i,3))
BL(i,3) = H(i,3)/2.d0
T(i,3) = Ts(i,3)/4.d0*(beta*H(i,3)+4.d0)
ENDIF
END DO
END SUBROUTINE XI_SPLIT_BALMFORTH
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
! SUBROUTINE TEMPERATURE
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
SUBROUTINE TEMPERATURE_CFD_BALMFORTH(f,col,N,Xi,H,T,Ts,BL,P,dist,ray,Dr,nu,Pe,delta0,el,grav,N1,tmps,Dt)
!*****************************************************************
! Give the vector f
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION ,DIMENSION(:) , INTENT(INOUT) :: f
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(IN) :: H,Xi,T,Ts,BL,P
DOUBLE PRECISION ,DIMENSION(:), INTENT(IN) :: dist,ray
! Prametre du model
DOUBLE PRECISION ,INTENT(IN) :: Dr
INTEGER ,INTENT(IN) :: col,N
! Nombre sans dimension
DOUBLE PRECISION ,INTENT(IN) :: nu,Pe,delta0,el,grav,N1,tmps,Dt
! Parametre pour le sous programme
DOUBLE PRECISION, PARAMETER :: pi=3.14159265
DOUBLE PRECISION :: h_a,delta_a,delta_a2,eta_a,Ai,T_a
DOUBLE PRECISIOn :: omega_a,sigma_a
DOUBLE PRECISION :: h_b,delta_b,delta_b2,eta_b,Bi,T_b
DOUBLE PRECISIOn :: omega_b,sigma_b,Ts_a,Ts_b,Ds_b,Ds_a
DOUBLE PRECISION :: loss,beta
INTEGER :: i
! Remplissage de f
DO i=1,N,1
IF1:IF (i .NE. 1) THEN
eta_b=(grav*(H(i,3)-H(i-1,3))+el*(P(i,3)-P(i-1,3)))/Dr
Bi=(ray(i-1)/(dist(i)*Dr))
h_b=0.5d0*(H(i,3)+H(i-1,3))
delta_b=0.5d0*(BL(i,col)+BL(i-1,col))
delta_b2=0.5d0*(BL(i,col)**2+BL(i-1,col)**2)
T_b = 0.5d0*(T(i,col)+T(i-1,col))
Ts_b = 0.5d0*(Ts(i,col)+Ts(i-1,col))
Ds_b = T_b-Ts_b
omega_b = (eta_b*delta_b)/10.d0*(nu*(-20.d0*delta_b+30.d0*h_b)+&
&(1.d0-nu)*(6.d0*Ds_b*delta_b-15.d0*Ds_b*h_b-20.d0*T_b*delta_b+30.d0*T_b*h_b))
sigma_b = (-1.d0/210.d0)*Ds_b*delta_b2*eta_b*(nu*(-98.d0*delta_b+105.d0*h_b)+&
&(1-nu)*(22.d0*Ds_b*delta_b-35.d0*Ds_b*h_b-98.d0*T_b*delta_b+105.d0*T_b*h_b))
ENDIF IF1
IF2: IF (i .NE. N) THEN
eta_a=(grav*(H(i+1,3)-H(i,3))+el*(P(i+1,3)-P(i,3)))/Dr
Ai=(ray(i)/(dist(i)*Dr))
h_a=0.5d0*(H(i+1,3)+H(i,3))
delta_a=0.5d0*(BL(i+1,col)+BL(i,col))
delta_a2=0.5d0*(BL(i+1,col)**2+BL(i,col)**2)
T_a = 0.5d0*(T(i,col)+T(i+1,col))
Ts_a = 0.5d0*(Ts(i,col)+Ts(i+1,col))
Ds_a = T_a-Ts_a
omega_a = (eta_a*delta_a)/10.d0*(nu*(-20.d0*delta_a+30.d0*h_a)+&
&(1.d0-nu)*(6.d0*Ds_a*delta_a-15.d0*Ds_a*h_a-20.d0*T_a*delta_a+30.d0*T_a*h_a))
sigma_a = (-1.d0/210.d0)*Ds_a*delta_a2*eta_a*(nu*(-98.d0*delta_a+105.d0*h_a)+&
&(1-nu)*(22.d0*Ds_a*delta_a-35.d0*Ds_a*h_a-98.d0*T_a*delta_a+105.d0*T_a*h_a))
END IF IF2
Ai =0;Bi=0
beta = N1*Pe**(-0.5d0)/(sqrt(pi*(tmps+Dt)))
loss = Pe*beta*Ts(i,col)
loss = 2*Pe*T(i,col)/BL(i,col)
IF4: IF (i==1) THEN
f(i)=loss+Ai*Omega_a*Xi(i,col)+Ai*Sigma_a
ELSE
f(i)=Ai*Omega_a*Xi(i,col)&
&-Bi*Omega_b*Xi(i-1,col)&
&+Ai*Sigma_a-Bi*Sigma_b &
&+loss
END IF IF4
END DO
END SUBROUTINE TEMPERATURE_CFD_BALMFORTH
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
! SUBROUTINE JACOBIENNE TEMPERATURE
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
SUBROUTINE FILLING_MATRIX(a,b,c,S,Xi_tmps,N,H,BL,T,Ts,Xi,P,Dr,dist,ray,nu,N1,Pe,delta0,el,grav,tmps,Dt,theta,psi)
!*****************************************************************
! Give the jacobian coeficient a1,b1,c1
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION ,DIMENSION(:) , INTENT(INOUT) :: a,b,c,S,Xi_tmps
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(IN) :: H,Xi,T,Ts,BL,P
DOUBLE PRECISION ,DIMENSION(:), INTENT(IN) :: dist,ray
! Prametre du model
DOUBLE PRECISION ,INTENT(IN) :: Dr
INTEGER ,INTENT(IN) :: N
! Nombre sans dimension
DOUBLE PRECISION ,INTENT(IN) :: nu,Pe,delta0,el,grav,N1,tmps,Dt,theta,psi
! Parametre pour le sous programme
DOUBLE PRECISION, PARAMETER :: pi=3.14159265
DOUBLE PRECISION :: h_a,delta_a,delta_a2,eta_a,Ai,T_a
DOUBLE PRECISIOn :: omega_a,sigma_a
DOUBLE PRECISION :: h_b,delta_b,delta_b2,eta_b,Bi,T_b
DOUBLE PRECISIOn :: omega_b,sigma_b,Ts_a,Ts_b,Ds_b,Ds_a
DOUBLE PRECISION :: loss,beta
INTEGER :: i,col
! Remplissage de f
col = 2
DO i=1,N,1
IF1:IF (i .NE. 1) THEN
eta_b=(grav*(H(i,3)-H(i-1,3))+el*(P(i,3)-P(i-1,3)))/Dr
Bi=(ray(i-1)/(dist(i)*Dr))
h_b=0.5d0*(H(i,3)+H(i-1,3))
delta_b=0.5d0*(BL(i,col)+BL(i-1,col))
delta_b2=0.5d0*(BL(i,col)**2+BL(i-1,col)**2)
T_b = 0.5d0*(T(i,col)+T(i-1,col))
Ts_b = 0.5d0*(Ts(i,col)+Ts(i-1,col))
Ds_b = T_b-Ts_b
omega_b = (eta_b*delta_b)/10.d0*(nu*(-20.d0*delta_b+30.d0*h_b)+&
&(1.d0-nu)*(6.d0*Ds_b*delta_b-15.d0*Ds_b*h_b-20.d0*T_b*delta_b+30.d0*T_b*h_b))
sigma_b = (-1.d0/210.d0)*Ds_b*delta_b2*eta_b*(nu*(-98.d0*delta_b+105.d0*h_b)+&
&(1-nu)*(22.d0*Ds_b*delta_b-35.d0*Ds_b*h_b-98.d0*T_b*delta_b+105.d0*T_b*h_b))
ENDIF IF1
IF2: IF (i .NE. N) THEN
eta_a=(grav*(H(i+1,3)-H(i,3))+el*(P(i+1,3)-P(i,3)))/Dr
Ai=(ray(i)/(dist(i)*Dr))
h_a=0.5d0*(H(i+1,3)+H(i,3))
delta_a=0.5d0*(BL(i+1,col)+BL(i,col))
delta_a2=0.5d0*(BL(i+1,col)**2+BL(i,col)**2)
T_a = 0.5d0*(T(i,col)+T(i+1,col))
Ts_a = 0.5d0*(Ts(i,col)+Ts(i+1,col))
Ds_a = T_a-Ts_a
omega_a = (eta_a*delta_a)/10.d0*(nu*(-20.d0*delta_a+30.d0*h_a)+&
&(1.d0-nu)*(6.d0*Ds_a*delta_a-15.d0*Ds_a*h_a-20.d0*T_a*delta_a+30.d0*T_a*h_a))
sigma_a = (-1.d0/210.d0)*Ds_a*delta_a2*eta_a*(nu*(-98.d0*delta_a+105.d0*h_a)+&
&(1-nu)*(22.d0*Ds_a*delta_a-35.d0*Ds_a*h_a-98.d0*T_a*delta_a+105.d0*T_a*h_a))
END IF IF2
beta = N1*Pe**(-0.5d0)/(sqrt(pi*(tmps+Dt)))
loss = Pe*beta*Ts(i,col)
loss = 2*Pe*T(i,col)/BL(i,col)
Ai =0;Bi=0
IF3:IF (i==1) THEN
a(i) = 0.d0
b(i) = (1.d0-Psi)-Dt*theta*Ai*Omega_a
c(i) = 0.d0
S(i) = (1.d0-theta)*Xi_tmps(i)+(1.d0-Psi)*Xi(i,1)+Dt*theta*(Ai*Sigma_a-Bi*Sigma_b+loss)
ELSEIF (i == N) THEN
a(i) =0.d0
b(i) = (1.d0-Psi)
S(i) = Dt*theta*loss
ELSE
a(i) = Dt*theta*Bi*Omega_b
b(i) = (1.d0-Psi)-Dt*theta*Ai*Omega_a
c(i) = 0.d0
S(i) = (1.d0-theta)*Xi_tmps(i)+(1.d0-Psi)*Xi(i,1)+Dt*theta*(Ai*Sigma_a-Bi*Sigma_b+loss)
END IF IF3
ENDDO
END SUBROUTINE FILLING_MATRIX
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
! SUBROUTINE JACOBIENNE TEMPERATURE
!-------------------------------------------------------------------------------------
!-------------------------------------------------------------------------------------
SUBROUTINE FILLING_MATRIX_2(a,b,c,d,e,f,g,k,l,S,Xi_tmps,N,H,BL,T,Ts,Xi,P,Dr,dist,ray,&
&nu,N1,Pe,delta0,el,grav,tmps,Dt,theta,psi)
!*****************************************************************
! On essay avec un third upwaind order scheme
!*****************************************************************
IMPLICIT NONE
! Tableaux
DOUBLE PRECISION ,DIMENSION(:) , INTENT(INOUT) :: a,b,c,d,e,f,g,k,l,S,Xi_tmps
DOUBLE PRECISION ,DIMENSION(:,:), INTENT(IN) :: H,Xi,T,Ts,BL,P
DOUBLE PRECISION ,DIMENSION(:), INTENT(IN) :: dist,ray
! Prametre du model
DOUBLE PRECISION ,INTENT(IN) :: Dr
INTEGER ,INTENT(IN) :: N
! Nombre sans dimension
DOUBLE PRECISION ,INTENT(IN) :: nu,Pe,delta0,el,grav,N1,tmps,Dt,theta,psi
! Parametre pour le sous programme
DOUBLE PRECISION, PARAMETER :: pi=3.14159265
DOUBLE PRECISION :: h_a,delta_a,delta_a2,eta_a,Ai,T_a
DOUBLE PRECISIOn :: omega_a,sigma_a
DOUBLE PRECISION :: h_b,delta_b,delta_b2,eta_b,Bi,T_b
DOUBLE PRECISION :: omega_b,sigma_b,Ts_a,Ts_b,Ds_b,Ds_a
DOUBLE PRECISION :: loss,beta
INTEGER :: i,col
! Remplissage de f
col = 2
DO i=1,N,1
IF1:IF (i .NE. 1) THEN
eta_b=(grav*(H(i,3)-H(i-1,3))+el*(P(i,3)-P(i-1,3)))/Dr
Bi=(ray(i-1)/(dist(i)*Dr))*Dt
h_b=0.5d0*(H(i,3)+H(i-1,3))
delta_b=0.5d0*(BL(i,col)+BL(i-1,col))
delta_b2=0.5d0*(BL(i,col)**2+BL(i-1,col)**2)
T_b = 0.5d0*(T(i,col)+T(i-1,col))
Ts_b = 0.5d0*(Ts(i,col)+Ts(i-1,col))
Ds_b = T_b-Ts_b
omega_b = (eta_b*delta_b)/10.d0*(nu*(-20.d0*delta_b+30.d0*h_b)+&
&(1.d0-nu)*(6.d0*Ds_b*delta_b-15.d0*Ds_b*h_b-20.d0*T_b*delta_b+30.d0*T_b*h_b))
sigma_b = (-1.d0/210.d0)*Ds_b*delta_b2*eta_b*(nu*(-98.d0*delta_b+105.d0*h_b)+&
&(1-nu)*(22.d0*Ds_b*delta_b-35.d0*Ds_b*h_b-98.d0*T_b*delta_b+105.d0*T_b*h_b))
ENDIF IF1
IF2: IF (i .NE. N) THEN
eta_a=(grav*(H(i+1,3)-H(i,3))+el*(P(i+1,3)-P(i,3)))/Dr
Ai=(ray(i)/(dist(i)*Dr))*Dt
h_a=0.5d0*(H(i+1,3)+H(i,3))
delta_a=0.5d0*(BL(i+1,col)+BL(i,col))
delta_a2=0.5d0*(BL(i+1,col)**2+BL(i,col)**2)
T_a = 0.5d0*(T(i,col)+T(i+1,col))
Ts_a = 0.5d0*(Ts(i,col)+Ts(i+1,col))
Ds_a = T_a-Ts_a
omega_a = (eta_a*delta_a)/10.d0*(nu*(-20.d0*delta_a+30.d0*h_a)+&
&(1.d0-nu)*(6.d0*Ds_a*delta_a-15.d0*Ds_a*h_a-20.d0*T_a*delta_a+30.d0*T_a*h_a))
sigma_a = (-1.d0/210.d0)*Ds_a*delta_a2*eta_a*(nu*(-98.d0*delta_a+105.d0*h_a)+&
&(1-nu)*(22.d0*Ds_a*delta_a-35.d0*Ds_a*h_a-98.d0*T_a*delta_a+105.d0*T_a*h_a))
END IF IF2
! beta = N1*Pe**(-0.5d0)/(sqrt(pi*(tmps+Dt)))
! loss = Pe*beta*Ts(i,col)
loss = 2*Pe*T(i,col)/BL(i,col)*Dt
IF3:IF (i==1) THEN
a(i) =0.d0
c(i) = 0.d0
d(i) = 0.d0
e(i) = (1.d0-Psi)+(2.d0*Bi-5.d0*Ai)/6.d0+(5.d0*Bi+Ai)/6.d0
f(i) = -2*Ai/6.d0
f(i) = 0.d0; g(i) =0.d0; k(i) =0.d0; l(i)=0.d0
S(i) = (1.d0-Psi)*Xi(i,1)+Ai*Sigma_a-Bi*Sigma_b+loss
ELSEIF (i==2) THEN
a(i) =0.d0
c(i) = 0.d0
d(i) = (5.d0*Bi+Ai)/6.d0-Bi/6.d0
e(i) = (1.d0-Psi)+(2.d0*Bi-5.d0*Ai)/6.d0
f(i) = -2*Ai/6.d0
f(i) = 0.d0; g(i) =0.d0; k(i) =0.d0; l(i)=0.d0
S(i) = (1.d0-Psi)*Xi(i,1)+Ai*Sigma_a-Bi*Sigma_b+loss
ELSEIF (i == N) THEN
a(i) =0.d0
c(i) = -Bi/6.d0
d(i) = (5.d0*Bi)/6.d0
e(i) = (1.d0-Psi)+(2.d0*Bi)/6.d0
f(i) = 0.d0
f(i) = 0.d0; g(i) =0.d0; k(i) =0.d0; l(i)=0.d0
S(i) = (1.d0-Psi)*Xi(i,1)+Ai*Sigma_a-Bi*Sigma_b+loss
ELSE
a(i) =0.d0
c(i) = -Bi/6.d0
d(i) = (5.d0*Bi+Ai)/6.d0
e(i) = (1.d0-Psi)+(2.d0*Bi-5.d0*Ai)/6.d0
f(i) = -2*Ai/6.d0
f(i) = 0.d0; g(i) =0.d0; k(i) =0.d0; l(i)=0.d0
S(i) = (1.d0-Psi)*Xi(i,1)+Ai*Sigma_a-Bi*Sigma_b+loss
END IF IF3
ENDDO
END SUBROUTINE FILLING_MATRIX_2
END MODULE MODULE_THERMAL_SKIN_GFD_BERCOVICI