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quasisymmetry_grad_B_tensor.f90
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subroutine quasisymmetry_grad_B_tensor
use quasisymmetry_variables
implicit none
real(dp), dimension(:,:,:), allocatable :: grad_B_tensor
real(dp), dimension(:), allocatable :: div_B, should_be_curvature
real(dp), dimension(:,:), allocatable :: should_be_curvature_times_normal, should_be_normal_Cartesian
integer :: j, k
real(dp) :: iota_N
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
allocate(grad_B_tensor(N_phi,3,3))
if (allocated(modBinv_sqrt_half_grad_B_colon_grad_B)) deallocate(modBinv_sqrt_half_grad_B_colon_grad_B)
allocate(modBinv_sqrt_half_grad_B_colon_grad_B(N_phi))
iota_N = iota + axis_helicity*nfp
! The formula below is derived in "20181018-02 Gradient of B from near-axis expansion- v2.docx"
modBinv_sqrt_half_grad_B_colon_grad_B = 0
do j = 1, 3
do k = 1, 3
grad_B_tensor(:,k,j) = sign_psi * B0 * B0_over_abs_G0 * ( &
(sign_G*sign_psi*abs_G0_over_B0*curvature*tangent_Cartesian(:,j) &
+ (d_X1c_d_zeta*Y1s + iota_N*X1c*Y1c) * normal_Cartesian(:,j) &
+ (d_Y1c_d_zeta*Y1s - d_Y1s_d_zeta*Y1c + sign_G*sign_psi*abs_G0_over_B0*torsion + iota_N*(Y1s*Y1s + Y1c*Y1c)) * binormal_Cartesian(:,j) &
) * normal_Cartesian(:,k) &
+ ((-sign_G*sign_psi*abs_G0_over_B0*torsion - iota_N*X1c*X1c) * normal_Cartesian(:,j) &
+ (X1c*d_Y1s_d_zeta - iota_N*X1c*Y1c) * binormal_Cartesian(:,j) &
) * binormal_Cartesian(:,k)) &
+ sign_G*curvature*B0*normal_Cartesian(:,j)*tangent_Cartesian(:,k)
modBinv_sqrt_half_grad_B_colon_grad_B = modBinv_sqrt_half_grad_B_colon_grad_B + grad_B_tensor(:,k,j) ** 2
end do
end do
modBinv_sqrt_half_grad_B_colon_grad_B = sqrt((0.5d+0) * modBinv_sqrt_half_grad_B_colon_grad_B) / B0
!if (.true.) then
if (.false.) then
! Sanity tests
print *,"modBinv_sqrt_half_grad_B_colon_grad_B:",modBinv_sqrt_half_grad_B_colon_grad_B
allocate(div_B(N_phi))
allocate(should_be_curvature(N_phi))
allocate(should_be_curvature_times_normal(N_phi,3))
allocate(should_be_normal_Cartesian(N_phi,3))
div_B = grad_B_tensor(:,1,1) + grad_B_tensor(:,2,2) + grad_B_tensor(:,3,3)
print *,"div_B:",div_B
print *,"max(abs(div_B)):",maxval(abs(div_B))
print *," "
print *,"grad_B_tensor(:,1,2):",grad_B_tensor(:,1,2)
print *,"grad_B_tensor(:,2,1):",grad_B_tensor(:,2,1)
print *,"max(abs(grad_B_tensor(:,1,2) - grad_B_tensor(:,2,1))):",maxval(abs(grad_B_tensor(:,1,2)-grad_B_tensor(:,2,1)))
print *," "
print *,"grad_B_tensor(:,1,3):",grad_B_tensor(:,1,3)
print *,"grad_B_tensor(:,3,1):",grad_B_tensor(:,3,1)
print *,"max(abs(grad_B_tensor(:,1,3) - grad_B_tensor(:,3,1))):",maxval(abs(grad_B_tensor(:,1,3)-grad_B_tensor(:,3,1)))
print *," "
print *,"grad_B_tensor(:,2,3):",grad_B_tensor(:,2,3)
print *,"grad_B_tensor(:,3,2):",grad_B_tensor(:,3,2)
print *,"max(abs(grad_B_tensor(:,2,3) - grad_B_tensor(:,3,2))):",maxval(abs(grad_B_tensor(:,2,3)-grad_B_tensor(:,3,2)))
! kappa \vec{N} = (1/|B|) grad |B| = (1/B^2) (grad \vec{B}) dot \vec{B} = (sign_G/|B|) (grad \vec{B}) dot \vec{t}
should_be_curvature_times_normal = 0
do j = 1, 3
do k = 1, 3
should_be_curvature_times_normal(:,k) = should_be_curvature_times_normal(:,k) + (1/B0) * grad_B_tensor(:,k,j) * tangent_Cartesian(:,j)
end do
end do
should_be_curvature = sqrt(should_be_curvature_times_normal(:,1) ** 2 + should_be_curvature_times_normal(:,2) ** 2 + should_be_curvature_times_normal(:,3) ** 2)
do k = 1, 3
should_be_normal_Cartesian(:,k) = should_be_curvature_times_normal(:,k) / should_be_curvature
end do
print *," "
print *,"should_be_curvature:",should_be_curvature
print *," curvature:",curvature
print *,"max(abs(curvature - should_be_curvature)):",maxval(abs(curvature - should_be_curvature))
print *," "
print *,"max(abs(normal_Cartesian - should_be_normal_Cartesian)):",maxval(abs(normal_Cartesian - should_be_normal_Cartesian))
print *," "
deallocate(div_B, should_be_curvature, should_be_curvature_times_normal, should_be_normal_Cartesian)
end if
call quasisymmetry_max_modBinv_sqrt_half_grad_B_colon_grad_B()
deallocate(grad_B_tensor)
end subroutine quasisymmetry_grad_B_tensor
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!1
subroutine quasisymmetry_max_modBinv_sqrt_half_grad_B_colon_grad_B
use quasisymmetry_variables
implicit none
real(dp), dimension(:), allocatable :: modBinv_sqrt_half_grad_B_colon_grad_B_sin, modBinv_sqrt_half_grad_B_colon_grad_B_cos, modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed
integer :: n, index_of_max
real(dp) :: phi_of_max_modBinv_sqrt_half_grad_B_colon_grad_B, fmin_tolerance, quasisymmetry_fmin, maxval_modBinv_sqrt_half_grad_B_colon_grad_B
! Search for maximum using Fourier interpolation...
index_of_max = maxloc(modBinv_sqrt_half_grad_B_colon_grad_B,1)
maxval_modBinv_sqrt_half_grad_B_colon_grad_B = modBinv_sqrt_half_grad_B_colon_grad_B(index_of_max)
!!$ if (maxval_modBinv_sqrt_half_grad_B_colon_grad_B > max_precise_modBinv_sqrt_half_grad_B_colon_grad_B) then
!!$ max_modBinv_sqrt_half_grad_B_colon_grad_B = maxval_modBinv_sqrt_half_grad_B_colon_grad_B
!!$ if (verbose) then
!!$ print *,"maxval(modBinv_sqrt_half_grad_B_colon_grad_B): ",maxval_modBinv_sqrt_half_grad_B_colon_grad_B
!!$ print "(a)"," modBinv_sqrt_half_grad_B_colon_grad_B > max_precise_modBinv_sqrt_half_grad_B_colon_grad_B so skipping precise solve."
!!$ end if
!!$ return
!!$ end if
! In preparation for searching for max modBinv_sqrt_half_grad_B_colon_grad_B, Fourier transform the modBinv_sqrt_half_grad_B_colon_grad_B.
allocate(modBinv_sqrt_half_grad_B_colon_grad_B_sin((N_phi+1)/2))
allocate(modBinv_sqrt_half_grad_B_colon_grad_B_cos((N_phi+1)/2))
allocate(modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed(N_phi))
modBinv_sqrt_half_grad_B_colon_grad_B_sin(1) = 0
modBinv_sqrt_half_grad_B_colon_grad_B_cos(1) = sum(modBinv_sqrt_half_grad_B_colon_grad_B) / N_phi
do n=1,((N_phi-1)/2)
modBinv_sqrt_half_grad_B_colon_grad_B_sin(n+1) = sum(modBinv_sqrt_half_grad_B_colon_grad_B * sin_n_phi(:,n+1)) * 2 / (N_phi)
modBinv_sqrt_half_grad_B_colon_grad_B_cos(n+1) = sum(modBinv_sqrt_half_grad_B_colon_grad_B * cos_n_phi(:,n+1)) * 2 / (N_phi)
end do
!!$ modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed = modBinv_sqrt_half_grad_B_colon_grad_B_cos(1)
!!$ do n = 1,((N_phi-1)/2)
!!$ modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed = modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed + modBinv_sqrt_half_grad_B_colon_grad_B_sin(n+1) * sin_n_phi(:,n+1) + modBinv_sqrt_half_grad_B_colon_grad_B_cos(n+1) * cos_n_phi(:,n+1)
!!$ end do
!!$
!!$ print *,"modBinv_sqrt_half_grad_B_colon_grad_B:"
!!$ print *,modBinv_sqrt_half_grad_B_colon_grad_B
!!$ print *,"modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed:"
!!$ print *,modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed
!!$
!!$ print *,"modBinv_sqrt_half_grad_B_colon_grad_B_sin:",modBinv_sqrt_half_grad_B_colon_grad_B_sin
!!$ print *,"modBinv_sqrt_half_grad_B_colon_grad_B_cos:",modBinv_sqrt_half_grad_B_colon_grad_B_cos
!!$ print *,"index_of_max:",index_of_max
!!$ print *,"d_phi:",d_phi,"(index_of_max-1)*d_phi:",(index_of_max-1)*d_phi
!!$ print *,"Interpolated modBinv_sqrt_half_grad_B_colon_grad_B at index_of_max:",-minus_modBinv_sqrt_half_grad_B_colon_grad_B((index_of_max-1)*d_phi)
fmin_tolerance = 0
phi_of_max_modBinv_sqrt_half_grad_B_colon_grad_B = quasisymmetry_fmin((index_of_max-2)*d_phi, (index_of_max)*d_phi, &
minus_modBinv_sqrt_half_grad_B_colon_grad_B, fmin_tolerance)
max_modBinv_sqrt_half_grad_B_colon_grad_B = -minus_modBinv_sqrt_half_grad_B_colon_grad_B(phi_of_max_modBinv_sqrt_half_grad_B_colon_grad_B)
if (verbose) then
print *,"maxval(modBinv_sqrt_half_grad_B_colon_grad_B): ",maxval_modBinv_sqrt_half_grad_B_colon_grad_B
print *,"max modBinv_sqrt_half_grad_B_colon_grad_B from fmin:",max_modBinv_sqrt_half_grad_B_colon_grad_B
end if
if (maxval_modBinv_sqrt_half_grad_B_colon_grad_B > max_modBinv_sqrt_half_grad_B_colon_grad_B * (1 + 1.0d-10)) then
print *,"Error! Something went wrong with the max_modBinv_sqrt_half_grad_B_colon_grad_B search on proc",mpi_rank,". maxval_modBinv_sqrt_half_grad_B_colon_grad_B=",maxval_modBinv_sqrt_half_grad_B_colon_grad_B,", max_modBinv_sqrt_half_grad_B_colon_grad_B=",max_modBinv_sqrt_half_grad_B_colon_grad_B,", modBinv_sqrt_half_grad_B_colon_grad_B=",modBinv_sqrt_half_grad_B_colon_grad_B
!if (maxval_modBinv_sqrt_half_grad_B_colon_grad_B < max_precise_modBinv_sqrt_half_grad_B_colon_grad_B) stop ! If the modBinv_sqrt_half_grad_B_colon_grad_B is larger than this, we don't care about the solution much, so don't bother aborting.
end if
deallocate(modBinv_sqrt_half_grad_B_colon_grad_B_sin, modBinv_sqrt_half_grad_B_colon_grad_B_cos, modBinv_sqrt_half_grad_B_colon_grad_B_reconstructed)
contains
real(dp) function minus_modBinv_sqrt_half_grad_B_colon_grad_B(this_phi)
implicit none
real(dp) :: f, this_phi
integer :: nn
f = modBinv_sqrt_half_grad_B_colon_grad_B_cos(1)
do nn = 1,((N_phi-1)/2)
f = f + modBinv_sqrt_half_grad_B_colon_grad_B_sin(nn+1) * sin(nn*nfp*this_phi) + modBinv_sqrt_half_grad_B_colon_grad_B_cos(nn+1) * cos(nn*nfp*this_phi)
end do
minus_modBinv_sqrt_half_grad_B_colon_grad_B = -f
return
end function minus_modBinv_sqrt_half_grad_B_colon_grad_B
end subroutine quasisymmetry_max_modBinv_sqrt_half_grad_B_colon_grad_B