404 lines
12 KiB
Fortran
404 lines
12 KiB
Fortran
! module eos
|
|
! use constantParameters, only: dp
|
|
! implicit none
|
|
!
|
|
! private
|
|
! public:: T_to_Z
|
|
!
|
|
! contains
|
|
! pure function T_to_Z(T) result(Z)
|
|
! use constantParameters, only: eV_to_K
|
|
! use referenceValues, only: Temp_ref
|
|
! implicit none
|
|
!
|
|
! real(dp), intent(in):: T
|
|
! real(dp):: Z
|
|
!
|
|
! ! Z = (Temp_ref * T / eV_to_K)**0.6
|
|
! ! Z = max(Z, 1.0_dp)
|
|
! ! Z = min(Z, 22.0_dp)
|
|
! Z = 12.0_dp
|
|
!
|
|
! end function T_to_Z
|
|
!
|
|
! end module eos
|
|
|
|
program plasmaExpansion
|
|
use constantParameters, only: dp, kb, qe, eps_0, ev_to_K, cm3_to_m3, PI
|
|
use output
|
|
use referenceValues
|
|
! use eos, only: T_to_Z
|
|
use moduleTableBC
|
|
use omp_lib
|
|
implicit none
|
|
|
|
real(dp), parameter:: m_i = 1.9712258e-25_dp ! Tin atom mass in kg
|
|
real(dp), parameter:: gamma_i = 1.0_dp ! Adiabatic coefficient for ions
|
|
real(dp), parameter:: m_e = 9.1093837e-31_dp ! Electron mass in kg
|
|
real(dp), parameter:: gamma_e = 2.0_dp ! Adiabatic coefficient for electrons
|
|
|
|
real(dp):: r0, rf
|
|
real(dp), allocatable, dimension(:):: r
|
|
real(dp):: v0, vf
|
|
real(dp), allocatable, dimension(:):: v
|
|
real(dp):: t0, tf
|
|
real(dp):: time
|
|
real(dp):: dr, dv, dt
|
|
integer:: nr, nv, nt
|
|
integer:: i, j, t
|
|
integer:: j0 ! First integer of positive velocity
|
|
|
|
real(dp):: Temp_bc ! Temperature
|
|
real(dp):: Zave_bc ! Average charge state
|
|
real(dp):: u_bc ! Injection velocity
|
|
real(dp):: n_bc ! Injection density
|
|
real(dp):: c_s ! Ion sound speed
|
|
type(tableBC):: boundaryConditions
|
|
character(:), allocatable:: bc_file
|
|
|
|
real(dp), allocatable, dimension(:,:):: f_i, f_i_old
|
|
real(dp), allocatable, dimension(:):: f0 ! Boundary at r = x_0
|
|
real(dp), allocatable, dimension(:):: n_i
|
|
real(dp), allocatable, dimension(:):: u_i
|
|
real(dp), allocatable, dimension(:):: E_i
|
|
real(dp), allocatable, dimension(:):: T_i
|
|
real(dp), allocatable, dimension(:):: n_e
|
|
real(dp), allocatable, dimension(:):: Zave
|
|
real(dp), allocatable, dimension(:):: diag, diag_low, diag_high
|
|
real(dp), allocatable, dimension(:,:):: A
|
|
real(dp), allocatable, dimension(:):: Res
|
|
real(dp), allocatable, dimension(:):: b
|
|
integer:: info
|
|
|
|
real(dp), allocatable, dimension(:):: phi, phi_old, E, db_dphi
|
|
real(dp):: phiConv
|
|
real(dp):: phi0
|
|
real(dp):: T_e
|
|
! real(dp):: phiF
|
|
integer:: k
|
|
|
|
real(dp), allocatable, dimension(:):: fCum_i
|
|
real(dp):: rCum
|
|
integer:: rCum_index
|
|
|
|
! Set number of threads
|
|
call omp_set_num_threads(8)
|
|
|
|
! Set reference numbers (in SI units)
|
|
Temp_ref = 30.0_dp * eV_to_K
|
|
n_ref = 1.0e20_dp * cm3_to_m3
|
|
t_ref = sqrt(eps_0 * m_i / (n_ref * 1.0_dp * qe**2)) ! 1.0_dp represents Z = 1 for reference values
|
|
u_ref = sqrt(kb * Temp_ref / m_i)
|
|
L_ref = u_ref * t_ref
|
|
phi_ref = kb * Temp_ref / qe
|
|
|
|
! Set input parameters (remember these have to be in non-dimensional units)
|
|
c_s = sqrt(9.0_dp * gamma_i * 1.0_dp)
|
|
bc_file = 'bc.csv'
|
|
call boundaryConditions%init(bc_file)
|
|
|
|
! Set domain boundaries (non-dimensional units)
|
|
r0 = 10.0e-6_dp / L_ref
|
|
rf = 100.0e-6_dp / L_ref
|
|
dr = 2.0e1_dp
|
|
nr = nint((rf - r0) / dr) + 1
|
|
dr = (rf - r0) / float(nr-1)
|
|
allocate(r(1:nr))
|
|
do i = 1, nr
|
|
r(i) = dr * float(i-1) + r0
|
|
end do
|
|
|
|
! Set position to calculate cumulative sum of f (non-dimensional units)
|
|
rCum = 80.0e-6 / L_ref
|
|
|
|
! Index for cumulative sum
|
|
rCum_index = minloc(abs(r - rCum), 1)
|
|
|
|
v0 =-1.0e1_dp*c_s
|
|
vf = 2.0e1_dp*c_s
|
|
dv = 1.0e-1_dp
|
|
nv = nint((vf - v0) / dv) + 1
|
|
dv = (vf - v0) / float(nv-1)
|
|
allocate(v(1:nv))
|
|
do j = 1, nv
|
|
v(j) = dv * float(j-1) + v0
|
|
end do
|
|
! Shift v mesh so it passes by 0
|
|
v = v - (minval(abs(v)))
|
|
j0 = minloc(abs(v), 1)
|
|
if (v(j0) < 0.0_dp) then
|
|
j0 = j0 + 1
|
|
end if
|
|
|
|
t0 = 0.0_dp
|
|
tf = 1.0e-7_dp / t_ref
|
|
! tf = 1.0e1_dp * (rf - r0) / c_s
|
|
dt = 1.0e-2_dp*dr/c_s
|
|
nt = nint((tf - t0) / dt)
|
|
dt = (tf - t0) / float(nt)
|
|
|
|
everyOutput = nint(1.0e-9_dp/t_ref/dt)
|
|
if (everyOutput == 0) then
|
|
everyOutput = 1
|
|
|
|
end if
|
|
|
|
everyWrite = everyOutput/10
|
|
if (everyWrite == 0) then
|
|
everyWrite = 1
|
|
|
|
end if
|
|
|
|
write(*, '(A,ES0.4e3)') 'CFL: ', dt*vf/dr
|
|
|
|
! Allocate vectors
|
|
allocate(f_i(1:nr,1:nv), f_i_old(1:nr,1:nv))
|
|
allocate(n_i(1:nr))
|
|
allocate(u_i(1:nr), E_i(1:nr), T_i(1:nr))
|
|
allocate(Zave(1:nr))
|
|
allocate(n_e(1:nr))
|
|
allocate(phi(1:nr), phi_old(1:nr), E(1:nr))
|
|
allocate(fCum_i(1:nv))
|
|
f_i = 0.0_dp
|
|
f_i_old = 0.0_dp
|
|
n_i = 0.0_dp
|
|
u_i = 0.0_dp
|
|
E_i = 0.0_dp
|
|
T_i = 0.0_dp
|
|
n_e = 0.0_dp
|
|
T_e = 0.0_dp
|
|
Zave = 0.0_dp
|
|
phi = 0.0_dp
|
|
phi_old = 0.0_dp
|
|
E = 0.0_dp
|
|
fCum_i = 0.0_dp
|
|
|
|
! Allocate matrix for Poisson equation
|
|
allocate(diag(1:nr), diag_low(1:nr-1), diag_high(1:nr-1))
|
|
allocate(b(1:nr))
|
|
allocate(db_dphi(1:nr))
|
|
diag = 0.0_dp
|
|
diag_low = 0.0_dp
|
|
diag_high = 0.0_dp
|
|
b = 0.0_dp
|
|
db_dphi = 0.0_dp
|
|
diag = -2.0_dp / dr**2
|
|
diag_low = 1.0_dp / dr**2 - 1.0_dp / (r(2:nr) * dr)
|
|
diag_high = 1.0_dp / dr**2 + 1.0_dp / (r(1:nr-1) * dr)
|
|
diag(1) = 1.0_dp ! Dirichlet
|
|
diag_high(1) = 0.0_dp ! Dirichlet
|
|
! diag_high(1) = 2.0_dp / dr**2 ! Neumann
|
|
! diag(nr) = 1.0_dp ! Dirichlet
|
|
! diag_low(nr-1) = 0.0_dp ! Dirichlet
|
|
diag_low(nr-1) = 2.0_dp / dr**2 ! Neumann
|
|
|
|
allocate(A(1:nr,1:nr))
|
|
A = 0.0_dp
|
|
A(1,1) = diag(1)
|
|
A(1,2) = diag_high(1)
|
|
do i = 2, nr - 1
|
|
A(i, i-1) = diag_low(i-1)
|
|
A(i, i) = diag(i)
|
|
A(i, i+1) = diag_high(i)
|
|
end do
|
|
A(nr,nr-1) = diag_low(nr-1)
|
|
A(nr,nr) = diag(nr)
|
|
|
|
allocate(Res(1:nr))
|
|
Res = 0.0_dp
|
|
|
|
! Set boundary values
|
|
phi0 = 1.0e2_dp / phi_ref ! Dirichlet
|
|
phi(1) = phi0 ! Dirichlet
|
|
! phi0 = phi(1) ! Neumann
|
|
allocate(f0(j0:nv))
|
|
f0 = 0.0_dp
|
|
|
|
! Output initial values
|
|
call createPath()
|
|
call setTimeFormat(nt)
|
|
t = 0
|
|
call writeOutputRef()
|
|
call writeOutputF(t, dt, nr, r, nv, v, f_i_old)
|
|
call writeOutputPhi(t, dt, nr, r, phi, E, n_e)
|
|
call writeOutputMom(t, dt, nr, r, n_i, u_i, T_i, Zave)
|
|
|
|
! Main loop
|
|
do t = 1, nt
|
|
time = t * dt + t0
|
|
call boundaryConditions%get(time, n_bc, u_bc, Temp_bc, Zave_bc)
|
|
call writeOutputBoundary(t, dt, n_bc, u_bc, Temp_bc, Zave_bc)
|
|
u_bc = sqrt(Zave_bc * Temp_bc)
|
|
! f0(j0:nv) = v(j0:nv)**2 / sqrt(PI*Temp_bc**3) * exp(-(v(j0:nv) - u_bc)**2 / Temp_bc)
|
|
f0(j0:nv) = 1.0_dp / sqrt(PI*Temp_bc) * exp(-(v(j0:nv) - u_bc)**2 / Temp_bc)
|
|
f0 = f0 * n_bc / (sum(f0)*dv)
|
|
T_e = Temp_bc
|
|
|
|
! Boundary conditions
|
|
! r = r0, v>0
|
|
f_i_old(1,j0:nv) = f0
|
|
f_i(1,j0:nv) = f_i_old(1,j0:nv)
|
|
T_i(1) = Temp_bc
|
|
Zave(1) = Zave_bc
|
|
! r = rf, v<0
|
|
f_i_old(nr,1:j0-1) = 0.0_dp
|
|
f_i(nr,1:j0-1) = f_i_old(nr,1:j0-1)
|
|
! set edge velocities to 0
|
|
f_i_old(:,1) = 0.0_dp
|
|
f_i_old(:,nv) = 0.0_dp
|
|
! Advect in the r direction
|
|
!$omp parallel do
|
|
do i = 1, nr
|
|
! Advect negative velocity
|
|
if (i < nr) then
|
|
f_i(i,1:j0-1) = f_i_old(i,1:j0-1) - v(1:j0-1)*dt/dr/r(i)**2*(r(i+1)**2*f_i_old(i+1,1:j0-1) - &
|
|
r(i )**2*f_i_old(i ,1:j0-1))
|
|
end if
|
|
! Advect positive velocity
|
|
if (i > 1) then
|
|
f_i(i,j0:nv) = f_i_old(i, j0:nv) - v( j0:nv)*dt/dr/r(i)**2*(r(i )**2*f_i_old(i , j0:nv) - &
|
|
r(i-1)**2*f_i_old(i-1, j0:nv))
|
|
end if
|
|
|
|
n_i(i) = sum(f_i(i,:))*dv
|
|
if (n_i(i) > 1.0e-10_dp) then
|
|
u_i(i) = sum(v(:) *f_i(i,:))*dv / n_i(i)
|
|
E_i(i) = sum(v(:)**2*f_i(i,:))*dv / n_i(i)
|
|
T_i(i) = 2.0_dp*E_i(i) - 2.0_dp*u_i(i)**2
|
|
Zave(i) = Zave_bc
|
|
|
|
else
|
|
u_i(i) = 0.0_dp
|
|
T_i(i) = 0.0_dp
|
|
Zave(i) = 0.0_dp
|
|
end if
|
|
|
|
end do
|
|
!$omp end parallel do
|
|
|
|
! Assume quasi-neutrality to start iterating
|
|
n_e = Zave * n_i
|
|
|
|
! Solve Poission (maximum number of iterations, break if convergence is reached before)
|
|
do k = 1, 1000
|
|
! Store previous value
|
|
phi_old = phi
|
|
|
|
! Diagonal matrix for Newton integration scheme
|
|
db_dphi = n_e / T_e ! Isotropic
|
|
diag = -2.0_dp / dr**2 - db_dphi
|
|
diag_low = 1.0_dp / dr**2 - 1.0_dp / (r(2:nr) * dr)
|
|
diag_high = 1.0_dp / dr**2 + 1.0_dp / (r(1:nr-1) * dr)
|
|
diag(1) = 1.0_dp ! Dirichlet
|
|
diag_high(1) = 0.0_dp ! Dirichlet
|
|
! diag(nr) = 1.0_dp ! Dirichlet
|
|
! diag_low(nr-1) = 0.0_dp ! Dirichlet
|
|
diag_low(nr-1) = 2.0_dp / dr**2 - db_dphi(nr) ! Neumann
|
|
|
|
! Calculate charge density
|
|
b = -(Zave*n_i - n_e)
|
|
! Apply boundary conditions
|
|
b(1) = phi0 ! Dirichlet
|
|
! b(nr) = 0.0_dp ! Dirichlet
|
|
|
|
! Calculate residual
|
|
!$omp parallel workshare
|
|
Res = -(MATMUL(A, phi_old) - b)
|
|
!$omp end parallel workshare
|
|
|
|
! Iterate system
|
|
call dgtsv(nr, 1, diag_low, diag, diag_high, Res, nr, info)
|
|
phi = phi_old + 1.0e-1_dp*Res
|
|
! phi0=phi(1) ! Neumann
|
|
|
|
! Calculate distribution of electrons
|
|
n_e = Zave(1) * n_i(1) * exp((phi- phi0) / T_e) ! Isothermal (Boltzmann)
|
|
! n_e = Zave(1) * n_i(1) * (1.0_dp + ((gamma_e - 1.0_dp)/gamma_e*(phi-phi0)/T_e)**(1.0_dp/(gamma_e - 1.0_dp))) ! Not working
|
|
|
|
! Check if the solution has converged
|
|
phiConv = maxval(abs(Res),1)
|
|
if (phiConv < 1.0e-3_dp) then
|
|
exit
|
|
|
|
end if
|
|
|
|
! ! Calculate new potential to ensure 0 current at the edge
|
|
! if (n_i(nr) > 1.0e-10_dp) then
|
|
! phiF = phi0 + T_i(1) * log((2.0_dp*sqrt(pi)*Zave(nr)*n_i(nr)*u_i(nr)) / (Zave(1)*n_i(1)*sqrt(m_i*T_i(1)/m_e)))
|
|
!
|
|
! else
|
|
! phiF = phi(nr-5)
|
|
!
|
|
! end if
|
|
|
|
end do
|
|
|
|
! Calculate electric field
|
|
E(1) = - (phi(2) - phi(1)) / dr ! Dirichlet
|
|
! E(1) = 0.0_dp ! Neumann
|
|
!$omp parallel do
|
|
do i = 2, nr-1
|
|
E(i) = - 0.5_dp*(phi(i+1) - phi(i-1)) / dr
|
|
|
|
end do
|
|
!$omp end parallel do
|
|
! E(nr) = - (phi(nr) - phi(nr-1)) / dr ! Dirichlet
|
|
E(nr) = 0.0_dp ! Neumann
|
|
|
|
! Update intermediate f
|
|
f_i_old = f_i
|
|
|
|
! Advect in the v direction
|
|
! i = 1, v<0
|
|
i = 1
|
|
if (E(i) >= 0.0_dp) then
|
|
f_i(i,2:j0-2) = f_i_old(i,2:j0-2) - Zave(i)*E(i)*dt/dv*(f_i_old(i,2:j0-2) - f_i_old(i,1:j0-3))
|
|
else
|
|
f_i(i,2:j0-2) = f_i_old(i,2:j0-2) - Zave(i)*E(i)*dt/dv*(f_i_old(i,3:j0-1) - f_i_old(i,2:j0-2))
|
|
|
|
end if
|
|
! i = 2, nr-1; all v
|
|
!$omp parallel do
|
|
do i = 2, nr-1
|
|
if (E(i) >= 0.0_dp) then
|
|
f_i(i,2:nv-1) = f_i_old(i,2:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,2:nv-1) - f_i_old(i,1:nv-2))
|
|
else
|
|
f_i(i,2:nv-1) = f_i_old(i,2:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,3:nv) - f_i_old(i,2:nv-1))
|
|
|
|
end if
|
|
|
|
end do
|
|
!$omp end parallel do
|
|
! i = nr, v>=0
|
|
i = nr
|
|
if (E(i) >= 0.0_dp) then
|
|
f_i(i,j0+1:nv-1) = f_i_old(i,j0+1:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,j0+1:nv-1) - f_i_old(i,j0:nv-2))
|
|
else
|
|
f_i(i,j0+1:nv-1) = f_i_old(i,j0+1:nv-1) - Zave(i)*E(i)*dt/dv*(f_i_old(i,j0+2:nv) - f_i_old(i,j0+1:nv-1))
|
|
|
|
end if
|
|
|
|
! Reset values for next iteration
|
|
f_i_old = f_i
|
|
fCum_i = fCum_i + f_i_old(rCum_index,:)
|
|
|
|
! Write output
|
|
if (mod(t,everyOutput) == 0 .or. t == nt) then
|
|
call writeOutputF(t, dt, nr, r, nv, v, f_i_old)
|
|
call writeOutputPhi(t, dt, nr, r, phi, E, n_e)
|
|
call writeOutputMom(t, dt, nr, r, n_i, u_i, T_i, Zave)
|
|
call writeOutputFCum(t, dt, r(rCum_index), nv, v, fCum_i)
|
|
|
|
end if
|
|
|
|
! Write progress
|
|
if (mod(t,everyWrite) == 0) then
|
|
write (*, '(I10, A, I10)' ) t, '/', nt
|
|
write (*, '(A, ES0.4e3,","ES0.4e3)') 'phi max,min: ', maxval(phi)*phi_ref, minval(phi)*phi_ref
|
|
|
|
end if
|
|
|
|
end do
|
|
|
|
end program plasmaExpansion
|
|
|