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text1r.f90
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text1r.f90
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! Data block with parameter structure
block data text1r_data_block
implicit none
include 'text1r.fh'
end block data
! Check that number of points is less then array sizes
function check_size()
include 'text1r.fh'
logical check_size
check_size = text_n.gt.MAXN
if (check_size) write (*,*) &
'Error: number of points is larger then MAXN parameter'
end
! Initialize parameter structure
subroutine text1r_init(ttc,p,nu0,r, n, itype)
implicit none
real*8 ttc,p,nu0,r
integer n, itype, i, zz,hh
logical check_size
include 'text1r.fh'
include 'he3.f90h'
text_a = he3_text_a(ttc,p)
text_b = 0
text_d = he3_text_d(ttc,p)
text_lg1 = he3_grad_lg1(ttc,p)
text_lg2 = he3_grad_lg2(ttc,p)
text_lhv = he3_text_lhv(ttc,p)
text_lsg = 3D0
text_ld = he3_ld(ttc,p)
text_lo = 0D0
text_r = r
text_h = const_2pi*nu0/he3_gyro
text_d2h = 0
text_n = n
text_err = 0
text_bnd = 0
text_abnd = acos(-0.5D0); ! -60 deg
text_bbnd = asin(sqrt(0.8D0)); ! 63.4 deg
text_energy = 0D0
if (check_size()) return
! initial conditions for alpha_n, beta_n
hh = itype/2
zz = (itype - hh*2)*2-1 ! parity of itype -1 or 1
do i=1,text_n
text_an(i) = -dble(zz)*text_abnd
text_bn(i) = 2D0*acos(0D0)*dble(hh) + &
acos(-dble(zz)*1D0/sqrt(5D0)) * &
dble(i-1)/dble(text_n-1)
if (text_bnd.ne.0) then
text_rr(i)=text_r*dble(i-1)/dble(text_n)
else
text_rr(i)=text_r*dble(i-1)/dble(text_n-1)
endif
text_bm(i)=0D0
text_vr(i)=0D0
text_vz(i)=0D0
text_vf(i)=0D0
text_lr(i)=0D0
text_lz(i)=0D0
text_lf(i)=0D0
text_w(i)=0D0
enddo
end
! Vortex cluster profile
subroutine text1r_set_vortex_cluster(omega, omega_v)
implicit none
include 'text1r.fh'
integer i
logical check_size
real*8 omega,omega_v,rr
if (check_size()) return
do i=1,text_n
if (text_bnd.ne.0) then
rr = dble(i-1)/dble(text_n)*text_r
else
rr = dble(i-1)/dble(text_n-1)*text_r
endif
! flow velocity
text_vr(i)=0D0
text_vf(i)=0D0
text_vz(i)=0D0
! vortex direction
text_lr(i)=0D0
text_lf(i)=0D0
text_lz(i)=1D0
text_w(i)=2D0*omega
! fortex-free part
if (rr > text_r*SQRT(omega_v/omega)) then
text_vf(i) = rr*(omega - omega_v*(text_r/rr)**2)
text_w(i)=0D0
endif
enddo
end
! Uniform vortex profile
subroutine text1r_set_vortex_uniform(omega, omega_v)
implicit none
include 'text1r.fh'
integer i
logical check_size
real*8 omega,omega_v,rr
if (check_size()) return
do i=1,text_n
if (text_bnd.ne.0) then
rr = dble(i-1)/dble(text_n)*text_r
else
rr = dble(i-1)/dble(text_n-1)*text_r
endif
! flow velocity
text_vr(i)=0D0
text_vf(i)=(omega-omega_v)*rr
text_vz(i)=0D0
! vortex direction
text_lr(i)=0D0
text_lf(i)=0D0
text_lz(i)=1D0
text_w(i)=2D0*omega_v
enddo
end
! Twisted-state velocity profile
subroutine text1r_set_vortex_twisted(omega, kr)
implicit none
include 'text1r.fh'
integer i
logical check_size
real*8 omega,kr,rr,r
if (check_size()) return
do i=1,text_n
if (text_bnd.ne.0) then
rr = dble(i-1)/dble(text_n)*text_r
else
rr = dble(i-1)/dble(text_n-1)*text_r
endif
r = text_r
text_vr(i)=0D0
text_vf(i)=omega*rr-(kr**2/dlog(1D0+kr**2))*omega*rr/(1D0+(kr*rr/r)**2)
text_vz(i)=-omega*r*(kr/(dlog(1D0+kr**2)*(1D0+(kr*rr/r)**2))-1D0/kr)
text_lr(i)=0D0
text_lf(i)=(kr*rr/r)/dsqrt(1D0+(kr*rr/r)**2)
text_lz(i)=1D0/dsqrt(1D0+(kr*rr/r)**2)
text_w(i)=2D0*omega*(kr**2/dlog(1D0+kr**2))*(1D0+(kr*rr/r)**2)**(-1.5D0)
enddo
end
! Print texture
subroutine text1r_print(fname)
implicit none
integer fd,i
logical check_size
real*8 r2d
character fname*(*)
include 'text1r.fh'
101 format (A,E12.4,A)
102 format (A,I4)
fd=100
open(fd, file=fname)
write (fd,101) '# Texture parameters: '
write (fd,102) '# Number of points n = ', text_n
write (fd,101) '# Cell radius r = ', text_r, ' cm'
write (fd,101) '# Mag.field H = ', text_H, ' G'
write (fd,101) '# d2H/dr2 = ', text_d2H, ' G/cm^2'
write (fd,101) '# (nu_0 = ', text_H*20.0378D0/2D0/acos(0D0), ' kHz)'
write (fd,101) '# lambda_D = ', text_ld, ' erg/cm3'
write (fd,101) '# Field par. a = ', text_a, ' erg/cm^3 1/G^2'
write (fd,101) '# lambda_G1 = ', text_lg1, ' erg/cm'
write (fd,101) '# lambda_G2 = ', text_lg2, ' erg/cm'
write (fd,101) '# (delta = ', text_lg1/text_lg2-2D0, ''
write (fd,101) '# lambda_HV = ', text_lhv, ' erg/cm3 1/G2 1/(cm/s)^2'
write (fd,101) '# lambda_SG = ', text_lsg, ' ??'
write (fd,101) '# lambda/Omega = ', text_lo, ' s/rad'
write (fd,101) '# surface par d = ', text_d, ' erg/cm^2 1/G^2'
write (fd,'(A)') '#'
r2d=90D0/acos(0D0)
if (check_size()) return
103 format ('# ' A6 ' ' A9 ' ' A9 ' ' A9 &
& ' ' A9 ' ' A9 ' ' A9 &
& ' ' A9 ' ' A9 ' ' A9 &
& ' ' A9)
104 format (F7.5 ' ' F10.5 ' ' F9.5 ' ' F9.5 &
& ' ' F9.5 ' ' F9.5 ' ' F9.5 &
& ' ' F9.5 ' ' F9.5 ' ' F9.5 &
& ' ' F9.5)
write(fd, 103) 'r,cm', 'a_n,deg', 'b_n,deg', 'bm,deg', &
& 'vr,cm/s', 'vz,cm/s', 'vf,cm/s', &
& 'lr', 'lz', 'lf','w'
do i=1,text_n
write (fd,104) text_rr(i), &
& text_an(i)*r2d, text_bn(i)*r2d, &
& text_bm(i)*r2d, &
& text_vr(i),text_vz(i),text_vf(i), &
& text_lr(i),text_lz(i),text_lf(i), &
& text_w(i)
enddo
close(fd)
end
! Vary alpha_n and beta_n to get equilibrium texture
subroutine text1r_minimize(msglev)
implicit none
include 'text1r.fh'
integer maxnpar, lw, msglev
parameter (maxnpar = 2*MAXN-2)
parameter (lw = 14*maxnpar)
real*8 x(maxnpar), ex(maxnpar)
real*8 w(lw)
logical check_size
external text1r_mfunc
integer nn,i,j
! additional parameters (default values, do not change)
integer MAXIT, MAXFUN
real*8 ETA,STEPMX,ACCRCY,XTOL,MCHPR1
! update text_rr (it can depend on text_bnd)
do i=1,text_n
if (text_bnd.ne.0) then
text_rr(i)=text_r*dble(i-1)/dble(text_n)
else
text_rr(i)=text_r*dble(i-1)/dble(text_n-1)
endif
enddo
nn=2*text_n-2
MAXIT = nn/2
IF (MAXIT .GT. 50) MAXIT = 50
IF (MAXIT .LE. 0) MAXIT = 1
MAXFUN = 150*nn
ETA = .25D0
STEPMX = 1.D1
ACCRCY = 1.D2*MCHPR1()
XTOL = DSQRT(ACCRCY)
if (check_size()) return
call text1r_text2x(text_n, text_an, text_bn, x)
do i=1,20 ! we need several runs to catch small energy changes for flat textures
call lmqn(text_err, nn,x,text_energy,ex,w,lw, &
text1r_mfunc,msglev, &
MAXIT, MAXFUN, ETA, STEPMX, ACCRCY, XTOL)
if (text_err.eq.0) goto 10
if (i.eq.20) continue
do j=1,text_n
! calculation fails if initial conditions exactly
! corresponds to the energy minimum. Try to go away:
text_an(j) = text_an(j) + 0.001
call text1r_text2x(text_n, text_an, text_bn, x)
enddo
enddo
write(*,*) "ERROR CODE = ", text_err, " after ", i, " iterations"
10 call text1r_x2text(text_n, text_an, text_bn, x)
! derivatives
text_db0 = text_bn(2)/text_rr(2)
text_db1 = (text_bn(text_n)-text_bn(text_n-1))/ &
(text_rr(text_n)-text_rr(text_n-1))
text_bmax = text_bn(text_n)
! the l vector
do i=2,text_n
call n2l(text_an(i), text_bn(i), text_al(i), text_bl(i))
enddo
text_al(1)=text_al(2)
text_bl(1)=0D0
end
! calculate the l vector from n
subroutine n2l(an,bn,al,bl)
implicit none
real*8 an,bn,al,bl
real*8 nr,nf,nz,lr,lf,lz
nr=-sin(bn)*cos(an)
nf=sin(bn)*sin(an)
nz=cos(bn)
lr = 1.25D0 * nz*nr - sqrt(15D0/16D0) * nf
lf = 1.25D0 * nz*nf + sqrt(15D0/16D0) * nr
lz = -0.25D0 + 1.25D0 * nz**2
bl = acos(lz)
al = atan2(lf,-nr)
end
! subroutine text1r_minimize_btn()
! implicit none
! include 'text1r.fh'
! integer iflag
! integer maxnpar=2*MAXN-1
! integer nprocs=8
! integer lw= 3*maxnpt + 3*nprocs &
! + 4*nprocs*nprocs + 7 *(maxnpt*nprocs)
! real*8 x(maxnpar), g(maxnpar)
! real*8 w(lw)
!
! if (text_n.gt.MAXN) then
! write (*,*) 'Error: number of points is larger then MAXN parameter'
! return
! endif
!
! call ab2x(text_n, text_an, text_bn, x)
! call btnez(n,x,f,g, w, lw, text1r_mfunc, iflag)
! call x2ab(text_n, text_an, text_bn, x)
! end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Various checks
subroutine text1r_selfcheck(msglev)
implicit none
include 'text1r.fh'
integer msglev, i
logical check_size
if (check_size()) return
call text1r_conv_selfcheck( &
text_n,text_an,text_bn, 1D-10)
do i=2,text_n
call text1r_ebulk_selfcheck( &
text_rr(i), text_an(i), text_bn(i), text_bm(i), &
text_vz(i), text_vr(i), text_vf(i), &
text_lz(i), text_lr(i), text_lf(i), &
text_w(i), 1D-3, 1D-4)
call text1r_esurf_selfcheck( &
text_an(i),text_bn(i), 1D-3, 1D-4)
call text1r_egrad_selfcheck( &
text_rr(i),text_an(i),text_bn(i), 1D-3, 1D-4)
enddo
call text1r_eint_selfcheck( &
text_n,text_an,text_bn, 1D-3, 1D-2)
call text1r_mfunc_selfcheck( &
text_n,text_an,text_bn, 1D-3, 1D-2)
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Convert alpha and beta vectors to x vector
! x(1:2n-1) = [alpha(1:n) beta(2:n)]
! beta(1) is thrown away (it is 0)
subroutine text1r_text2x(n,a,b, x)
implicit none
integer i,n
real*8 a(n),b(n),x(2*n-2)
do i=2,n
x(i-1) = -sin(b(i))*cos(a(i))/(1D0+cos(b(i)))
x(n-1 + i-1) = sin(b(i))*sin(a(i))/(1D0+cos(b(i)))
enddo
end
! Convert dE/dAi, dE/dBi to dE/dXi
subroutine text1r_text2ex(n,a,b, ea,eb,ex)
implicit none
integer i,n
real*8 a(n),b(n),ea(n),eb(n),ex(2*n-2)
real*8 dau,dav,dbu,dbv,cb
do i=2,n
cb=cos(b(i))
if (1D0-cb.gt.1D-30) then
dau = sin(b(i))*sin(a(i))/(1D0-cb)
dav = sin(b(i))*cos(a(i))/(1D0-cb)
dbu = -sin(b(i))*cos(a(i))*sqrt((1D0+cb)/(1D0-cb))
dbv = sin(b(i))*sin(a(i))*sqrt((1D0+cb)/(1D0-cb))
else
dau = sin(a(i)) * 2D0/1D-30
dav = cos(a(i)) * 2D0/1D-30
dbu = -2D0*cos(a(i))
dbv = 2D0*sin(a(i))
endif
ex(i-1) = ea(i)*dau + eb(i)*dbu
ex(n-1 + i-1) = ea(i)*dav + eb(i)*dbv
enddo
end
! Convert x to alpha, beta vectors
! beta(0) is set to 0
subroutine text1r_x2text(n,a,b, x)
implicit none
integer i,n
real*8 a(n),b(n),x(2*n-2),u,v
do i=n,2,-1
u = x(i-1)
v = x(i-1+n-1)
b(i) = acos((1D0-u**2-v**2)/(1D0+u**2+v**2))
if (abs(b(i)).gt.1D-10) then
a(i) = 4D0*atan(1D0)-atan2(v, u)
else if (i.lt.n) then
a(i)=a(i+1)
else
a(i)=0D0
endif
! write(*,*) a(i),b(i),u,v
enddo
a(1)=a(2)
b(1)=0D0
end
! Self test for convert functions
subroutine text1r_conv_selfcheck(n, a,b, e)
implicit none
include 'text1r.fh'
integer n,i
real*8 a(n),b(n),x(2*MAXN-2),e
real*8 an(MAXN),bn(MAXN)
real*8 E1,Ea1(MAXN),Eb1(MAXN)
real*8 E2,Ea2(MAXN),Eb2(MAXN)
real*8 der1,der2
! check conversions a,b -> x - a,b
call text1r_text2x(n, a, b, x)
call text1r_x2text(n, an, bn, x)
do i=1,n
if (dabs(a(i) - an(i)) > e) then
write (*,*) 'text1r_conv_selfcheck failed for A->X->A conversion:'
write (*,*) 'i: ', i, ' A1: ', an(i), ' A2: ', a(i)
endif
if (dabs(b(i) - bn(i)) > e) then
write (*,*) 'text1r_conv_selfcheck failed for B->X->B conversion:'
write (*,*) 'i: ', i, ' B1: ', bn(i), ' B2: ', b(i)
endif
enddo
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Wrapper for egrad function for using in the TN.
! Calculate f and g from x values.
! x as array of both alpha and beta values
! g is array of both ga, gb
subroutine text1r_mfunc(nx,x,e,ex)
!! Wrapper for egrad function for using in the TN.
implicit none
include 'text1r.fh'
integer i,nx,n
real*8 e, x(nx),ex(nx)
real*8 a(MAXN), b(MAXN)
real*8 ea(MAXN), eb(MAXN)
n=nx/2+1
call text1r_x2text(n, a,b, x)
call text1r_eint(n,a,b,e,ea,eb)
call text1r_text2ex(n, a,b,ea,eb, ex)
end
! Self test for mfunc derivatives
subroutine text1r_mfunc_selfcheck(n, a,b, d, e)
implicit none
include 'text1r.fh'
integer n,i
real*8 a(n),b(n),d,e
real*8 x(2*MAXN-2)
real*8 E1,Ex1(2*MAXN-2)
real*8 E2,Ex2(2*MAXN-2)
real*8 der1,der2
call text1r_text2x(n, a, b, x)
call text1r_mfunc(2*n-2, x, e1, ex1)
do i=1,2*n-2
x(i) = x(i) + d
call text1r_mfunc(2*n-2, x, e2, ex2)
x(i) = x(i) - d
der1=(e2-e1)/d
der2=(ex2(i)+ex1(i))/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write (*,*) 'text1r_mfunc_selfcheck failed for derivative dE/dXi:'
write (*,*) 'i: ', i, ' (e2-e1)/dx: ', der1, ' ex*dx: ', der2
endif
enddo
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!! Calculate total free energy E and derivatives dE/da(i), dE/db(i)
!
! E = Int e(r, a(r),b(r), da/dr, db/dr,...) r dr
! Gaussian quadrature is used for calculating integral. For each
! mesh point i = 1:N-1 energies em,ep is calculated in points
! i+sm, i+sp (sm/sp = (3 +/- 3/sqrt(3))/6), and
! (em*rm + ep*rp)*dr/2 is added to the sum
!
! Derivatives dE/da(i), dE/db(i) are also needed.
! Change DA of a(i) (or b(i)) affects 4 terms in E intergral:
! at i-sp, i-sm (for i!=0), i+sp, i+sm (for i!=n)
! Changes of a in these points are DA*sm, DA*sp, DA*sp, DA*sm
! Changes of a' in these points are DA/dr, DA/dr, -DA/dr, -DA/dr
! We need to calculate dE/DA = Sum(dE/da * da + dE/da' * da')/DA
! We also need r*dr/2 eactor as in energy calculation
!
! Strightforward approach is to calculate sum for these 4 points
! (Ea*sm*dr - Eda)*r/2 for i+sp, i!=n
! (Ea*sp*dr - Eda)*r/2 for i+sm, i!=n
! (Ea*sp*dr + Eda)*r/2 for i-sm, i!=0
! (Ea*sm*dr + Eda)*r/2 for i-sp, i!=0
! but we can calculate E* only in two points instead of 4
! and add some terms to both dE/da(i) and dE/da(i+1)
subroutine text1r_eint(n,a,b,ei,eai,ebi)
implicit none
include 'text1r.fh'
integer i,n
real*8 a(n), b(n), eai(n), ebi(n)
real*8 rp,rm,bp,bm,ap,am,ei
real*8 da,db
real*8 E0,Ea,Eb,Eda,Edb
real*8 E1, E2,E1a,E1b,E2a,E2b
real*8 dx,sp,sm
! points for Gaussian quadrature
sp = (3D0 + sqrt(3D0))/6D0
sm = (3D0 - sqrt(3D0))/6D0
! grid step
dx = 1D0/dble(n-1)
do i=1,n
eai(i)=0D0
ebi(i)=0D0
enddo
ei=0D0
do i=1,n-1
! gaussian quadrature points
rp = (dble(i-1)+sp)*dx
rm = (dble(i-1)+sm)*dx
! bulk energy at the i point
if (i.eq.1) then
call text1r_ebulk(i*dx, a(i),b(i), text_bm(i), &
text_vz(i), text_vr(i), text_vf(i), &
text_lz(i), text_lr(i), text_lf(i), &
text_w(i), E1, E1a, E1b)
else
E1=E2
E1a=E2a
E1b=E2b
endif
! bulk energy at the i+1 point
call text1r_ebulk((i+1)*dx, a(i+1),b(i+1), text_bm(i+1), &
text_vz(i+1), text_vr(i+1), text_vf(i+1), &
text_lz(i+1), text_lr(i+1), text_lf(i+1), &
text_w(i+1), E2, E2a, E2b)
! interpolate bulk energy and derivatives to rp,rm points
ei = ei + rp*(sp*E2 + sm*E1)*0.5D0*dx
eai(i) = eai(i) + E1a*sm*dx*rp*0.5D0
ebi(i) = ebi(i) + E1b*sm*dx*rp*0.5D0
eai(i+1) = eai(i+1) + E2a*sp*dx*rp*0.5D0
ebi(i+1) = ebi(i+1) + E2b*sp*dx*rp*0.5D0
ei = ei + rm*(sm*E2 + sp*E1)*0.5D0*dx
eai(i) = eai(i) + E1a*sp*dx*rm*0.5D0
ebi(i) = ebi(i) + E1b*sp*dx*rm*0.5D0
eai(i+1) = eai(i+1) + E2a*sm*dx*rm*0.5D0
ebi(i+1) = ebi(i+1) + E2b*sm*dx*rm*0.5D0
! calculate gradient energy in rp, rm points
ap = sp*a(i+1)+sm*a(i)
am = sm*a(i+1)+sp*a(i)
bp = sp*b(i+1)+sm*b(i)
bm = sm*b(i+1)+sp*b(i)
da = (a(i+1)-a(i))/dx
db = (b(i+1)-b(i))/dx
call text1r_egrad(rp, ap,bp,da,db, E0,Ea,Eb,Eda,Edb)
ei = ei + rp*E0*0.5D0*dx
eai(i) = eai(i) + (Ea*sm*dx - Eda)*rp*0.5D0
ebi(i) = ebi(i) + (Eb*sm*dx - Edb)*rp*0.5D0
eai(i+1) = eai(i+1) + (Ea*sp*dx + Eda)*rp*0.5D0
ebi(i+1) = ebi(i+1) + (Eb*sp*dx + Edb)*rp*0.5D0
call text1r_egrad(rm, am,bm,da,db, E0,Ea,Eb,Eda,Edb)
ei = ei + rm*E0*0.5D0*dx
eai(i) = eai(i) + (Ea*sp*dx - Eda)*rm*0.5D0
ebi(i) = ebi(i) + (Eb*sp*dx - Edb)*rm*0.5D0
eai(i+1) = eai(i+1) + (Ea*sm*dx + Eda)*rm*0.5D0
ebi(i+1) = ebi(i+1) + (Eb*sm*dx + Edb)*rm*0.5D0
enddo
!! strict boundary conditions: add one more point (n+1)
!! with fixed angles text_abnd, text_bbnd
if (text_bnd.ne.0) then
! gaussian quadrature points
rp = (dble(n-1)+sp)*dx
rm = (dble(n-1)+sm)*dx
! use already calculated values an n point
E1=E2
E1a=E2a
E1b=E2b
! bulk energy at the n+1 point; use all parameters from the
! n point
call text1r_ebulk(text_r,text_abnd, text_bbnd, &
text_bm(n), &
text_vz(n), text_vr(n), text_vf(n), &
text_lz(n), text_lr(n), text_lf(n), &
text_w(n), E2, E2a, E2b)
! interpolate bulk energy and derivatives to rp,rm points
ei = ei + rp*(sp*E2 + sm*E1)*0.5D0*dx
eai(n) = eai(n) + E1a*sm*dx*rp*0.5D0
ebi(n) = ebi(n) + E1b*sm*dx*rp*0.5D0
ei = ei + rm*(sm*E2 + sp*E1)*0.5D0*dx
eai(n) = eai(n) + E1a*sp*dx*rm*0.5D0
ebi(n) = ebi(n) + E1b*sp*dx*rm*0.5D0
! calculate gradient energy in rp, rm points
ap = sp*text_abnd+sm*a(n)
am = sm*text_abnd+sp*a(n)
bp = sp*text_bbnd+sm*b(n)
bm = sm*text_bbnd+sp*b(n)
da = (text_abnd-a(n))/dx
db = (text_bbnd-b(n))/dx
call text1r_egrad(rp, ap,bp,da,db, E0,Ea,Eb,Eda,Edb)
ei = ei + rp*E0*0.5D0*dx
eai(n) = eai(n) + (Ea*sm*dx - Eda)*rp*0.5D0
ebi(n) = ebi(n) + (Eb*sm*dx - Edb)*rp*0.5D0
call text1r_egrad(rm, am,bm,da,db, E0,Ea,Eb,Eda,Edb)
ei = ei + rm*E0*0.5D0*dx
eai(n) = eai(n) + (Ea*sp*dx - Eda)*rm*0.5D0
ebi(n) = ebi(n) + (Eb*sp*dx - Edb)*rm*0.5D0
else
! surface energy
call text1r_esurf(a(n),b(n),E0,Ea,Eb)
ei = ei + E0
eai(n) = eai(n) + Ea
ebi(n) = ebi(n) + Eb
endif
end
! Self test for eint derivatives
subroutine text1r_eint_selfcheck(n, a,b, d, e)
implicit none
integer n,i
real*8 a(n),b(n),d,e
real*8 E1,Ea1(n),Eb1(n)
real*8 E2,Ea2(n),Eb2(n)
real*8 der1,der2
call text1r_eint(n,a,b,e1,ea1,eb1)
do i=2,n
b(i) = b(i) + d
call text1r_eint(n,a,b,e2,ea2,eb2)
b(i) = b(i) - d
der1=(e2-e1)/d
der2=(eb2(i)+eb1(i))/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write (*,*) 'text1r_eint_selfcheck failed for derivative dE/dBi:'
write (*,*) 'i: ', i, ' (e2-e1)/db: ', der1, ' ea1*db: ', der2
endif
a(i) = a(i) + d
call text1r_eint(n,a,b,e2,ea2,eb2)
a(i) = a(i) - d
der1=(e2-e1)/d
der2=(ea2(i)+ea1(i))/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write (*,*) 'text1r_eint_selfcheck failed for derivative dE/dAi:'
write (*,*) 'i: ', i, ' (e2-e1)/da: ', der1, ' ea1: ', der2
endif
enddo
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!! Calculate He3 bulk energy E and derivatives dE/da, dE/db
!! in 1d radial coordinated as a function of r, n-vector angles a and b,
!! Energy is divided by (a r H^2).
!! parameters used:
!! chia*(nub/nu0)^2, for non-zero apsi
!! lo for non-zero rotation
!! vd for non-zero flow
!! de and xi
subroutine text1r_ebulk(r, a,b, bm, vz,vr,vf, lz,lr,lf, w,&
E,Ea,Eb)
implicit none
include 'text1r.fh'
real*8 r,a,b,E,Ea,Eb
real*8 bm, vz,vr,vf, lz,lr,lf, w
real*8 nz,nr,nf, rzz,rzr,rzf
real*8 sin_a, sin_b, cos_a, cos_b, cos2b, sin2b
real*8 con1, con2, help, c,s, H
real*8 s3,s5,vd
vd = dsqrt(0.4D0 * text_a / text_lhv)
s3 = sqrt(3D0)
s5 = sqrt(5D0)
cos_a = cos(a)
sin_a = sin(a)
cos_b = cos(b)
sin_b = sin(b)
cos2b = cos(2D0*b)
sin2b = sin(2D0*b)
nr=-sin_b*cos_a
nf=sin_b*sin_a
nz=cos_b
c=-0.25D0 !\cos\theta
s=SQRT(15D0)/4D0 !\sin\theta
! H*Rij = cos(t) H + (1-cos(t))(H n) n - sin(t)(H x n)
rzr=(1D0-c)*nz*nr-s*nf ! H*Rij/abs(H)
rzf=(1D0-c)*nz*nf+s*nr
rzz=c+(1D0-c)*nz**2
E = 0D0
Ea = 0D0
Eb = 0D0
! magnetic free energy F_DH = -a * (n H)^2
E = E + sin_b**2
Eb = Eb + sin2b
! spin-orbit free energy
H = text_h + text_d2h*r**2
help = 15D0 * text_ld / text_a / H**2 * sin(bm/2D0)**2
E = E + help * sin_b**2
Eb = Eb + help * sin2b
! flow free energy F_HV
! v_d = sqrt(2/5 a/lhv)
E = E - 2D0/5D0 / (vd**2) * (rzr*vr+rzf*vf+rzz*vz)**2
help = vr*(-(1D0-c)*cos2b*cos_a - s*cos_b*sin_a) &
+ vf*( (1D0-c)*cos2b*sin_a - s*cos_b*cos_a) &
+ vz*(-(1D0-c)*sin2b)
Eb = Eb - 4D0/5D0 / (vd**2) * help * (rzr*vr+rzf*vf+rzz*vz)
help = vr*((1D0-c)*sin_b*cos_b*sin_a - s*sin_b*cos_a) &
+ vf*((1D0-c)*sin_b*cos_b*cos_a + s*sin_b*sin_a)
Ea = Ea - 4D0/5D0 / (vd**2) * help *(rzr*vr+rzf*vf+rzz*vz)
! vortex free energy F_LH
E = E + text_lo * w/5D0 * (rzr*lr+rzf*lf+rzz*lz)**2
help = lr*(-(1D0-c)*cos2b*cos_a - s*cos_b*sin_a) &
+ lf*( (1D0-c)*cos2b*sin_a - s*cos_b*cos_a) &
+ lz*(-(1D0-c)*sin2b)
Eb = Eb + text_lo*w*2D0/5D0 * help * (rzr*lr+rzf*lf+rzz*lz)
help = lr*((1D0-c)*sin_b*cos_b*sin_a - s*sin_b*cos_a) &
+ lf*((1D0-c)*sin_b*cos_b*cos_a + s*sin_b*sin_a)
Ea = Ea + text_lo*w*2D0/5D0 * help * (rzr*lr+rzf*lf+rzz*lz)
end
! Self test for ebulk derivatives
subroutine text1r_ebulk_selfcheck(r,a,b, bm, vz,vr,vf, lz,lr,lf, w, d, e)
implicit none
real*8 r,a,b,d,e
real*8 E1,Ea1,Eb1
real*8 E2,Ea2,Eb2
real*8 der1,der2
real*8 bm, vz,vr,vf, lz,lr,lf, w
call text1r_ebulk(r,a,b, bm, vz,vr,vf, lz,lr,lf, w, E1,Ea1,Eb1)
call text1r_ebulk(r,a+d,b, bm, vz,vr,vf, lz,lr,lf, w, E2,Ea2,Eb2)
der1=(E2-E1)/d
der2=(Ea2+Ea1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_ebulk_selfcheck failed for dE/da:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/da: ', der1, ' Ea: ', der2
endif
call text1r_ebulk(r,a,b+d, bm, vz,vr,vf, lz,lr,lf, w, E2,Ea2,Eb2)
der1=(E2-E1)/d
der2=(Eb2+Eb1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_ebulk_selfcheck failed for dE/db:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/db: ', der1, ' Eb: ', der2
endif
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!! Calculate He3 gradient energy E and derivatives dE/da, dE/db, dE/da', dE/db'
!! in 1d radial coordinated as a function of r, n-vector angles a and b,
!! and derivatives a'=da/dr, b'=db/dr.
!! Energy is divided by (a H^2).
subroutine text1r_egrad(r,a,b,da,db, E,Ea,Eb,Eda,Edb)
implicit none
include 'text1r.fh'
real*8 r,a,b,da,db,E,Ea,Eb,Eda,Edb
real*8 H
real*8 sin_a, sin_b, cos_a, cos_b
real*8 con1, con2, con3, help
real*8 s3,s5
s3 = sqrt(3D0)
s5 = sqrt(5D0)
cos_a = cos(a)
sin_a = sin(a)
cos_b = cos(b)
sin_b = sin(b)
E = 0D0
Ea = 0D0
Eb = 0D0
Eda = 0D0
Edb = 0D0
H = text_h + text_d2h*r**2
con1 = 5D0*(text_lg2 + text_lg1/2D0) &
/text_a/H**2/text_r**2
E = E + con1*(db**2 + (sin_b**2)*da**2 + (sin_b**2)/r**2) ! (\nabla n)^2 (?)
Eda = Eda + con1*2D0*da*sin_b**2
Edb = Edb + con1*2D0*db
Eb = Eb + con1 * 2D0*sin_b*cos_b*(da**2 + 1D0/r**2)
con2 = - 5D0/16D0*text_lg1 &
/text_a/H**2/text_r**2
help=(s5*sin_a-s3*cos_b*cos_a)*db + &
(s5*cos_b*cos_a+s3*sin_a)*sin_b*da + &
(s5*cos_b*sin_a-s3*cos_a)*sin_b/r ! s3 \div n + s5 n \rot n (?)
E = E + con2 * help**2
Eda = Eda + 2D0*con2*help * (s5*cos_b*cos_a + s3*sin_a)*sin_b
Edb = Edb + 2D0*con2*help * (s5*sin_a - s3*cos_b*cos_a)
Ea = Ea + 2D0*con2*help* &
( (s5*cos_a + s3*cos_b*sin_a)*db &
- (s5*cos_b*sin_a - s3*cos_a)*sin_b*da &
+ (s5*cos_b*cos_a + s3*sin_a)*sin_b/r)
Eb = Eb + 2D0*con2*help* &
( (s3*db - s5*sin_b*da)*sin_b*cos_a &
+ (s5*cos_b*cos_a + s3*sin_a)*cos_b*da &
+ (s5*cos_b*sin_a - s3*cos_a)*cos_b/r &
- s5*sin_b*sin_a*sin_b/r)
con3 = 5D0*text_lg1 &
/text_a/H**2/text_r**2
E = E + con3 * cos_b*sin_b*db/r
Edb = Edb + con3 * cos_b*sin_b/r
Eb = Eb + con3 * (cos_b**2 - sin_b**2)*db/r
end
! Self test for egrad derivatives
subroutine text1r_egrad_selfcheck(r,a,b, d, e)
implicit none
real*8 r,a,b,d,e,g0
real*8 E1,Ea1,Eb1,Eda1,Edb1
real*8 E2,Ea2,Eb2,Eda2,Edb2
real*8 der1,der2
g0=1D-3
call text1r_egrad(r, a, b,g0,g0,E1,Ea1,Eb1,Eda1,Edb1)
call text1r_egrad(r, a+d,b,g0,g0,E2,Ea2,Eb2,Eda2,Edb2)
der1=(E2-E1)/d
der2=(Ea2+Ea1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_egrad_selfcheck failed for dE/da:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/da: ', der1, ' Ea: ', der2
endif
call text1r_egrad(r, a,b+d,g0,g0,E2,Ea2,Eb2,Eda2,Edb2)
der1=(E2-E1)/d
der2=(Eb2+Eb1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_egrad_selfcheck failed for dE/db:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/db: ', der1, ' Eb: ', der2
endif
call text1r_egrad(r, a,b,g0+d,g0,E2,Ea2,Eb2,Eda2,Edb2)
der1=(E2-E1)/d
der2=(Eda2+Eda1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_egrad_selfcheck failed for dE/d(da):'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/d(da): ', der1, ' Eda: ', der2
endif
call text1r_egrad(r, a,b,g0,g0+d,E2,Ea2,Eb2,Eda2,Edb2)
der1=(E2-E1)/d
der2=(Edb2+Edb1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_egrad_selfcheck failed for dE/d(db):'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/d(db): ', der1, ' Edb: ', der2
endif
end
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
! Calculate E, dE/da, dE/db, dE/da', dE/db' at surface
! parameters used: dar,xir,lsg
subroutine text1r_esurf(a,b,E,Ea,Eb)
implicit none
include 'text1r.fh'
integer i
real*8 a,b,E,Ea,Eb
real*8 nr,nf,nz, H
real*8 sin_a, sin_b, cos_a, cos_b, sin2b,cos2b
real*8 s3,s5,de, xir,dar,bar
H = text_h + text_d2h*text_r**2
dar = text_d / (text_a*text_r)
bar = text_b / (text_a*H**2*text_r)
de = text_lg1/text_lg2 - 2D0
xir = sqrt(65D0/8D0 * text_lg2/text_a)/ H / text_r
s3 = sqrt(3D0)
s5 = sqrt(5D0)
E=0D0
Ea=0D0
Eb=0D0
cos_a = cos(a)
sin_a = sin(a)
cos_b = cos(b)
sin_b = sin(b)
sin2b = sin(2D0*b)
cos2b = cos(2D0*b)
nr=-sin_b*cos_a
nf=sin_b*sin_a
nz=cos_b
! dar is d/a/r
! E = E - 5D0*dar*(s5*nz*nr-s3*nf)**2/16D0
! Eb = Eb + 5D0*dar*(s5*nz*nr-s3*nf)*(s5*cos2b*cos_a+s3*cos_b*sin_a)/8D0
! Ea = Ea - 5D0*dar*(s5*nz*nr-s3*nf)*(s5*nz*nf + s3*nr)/8D0
E = E - 5D0/16D0*dar*(s5*nz*nr+s3*nf)**2
Ea = Ea - 5D0/8D0*dar*(s5*nz*nr+s3*nf)* &
sin_b*(s3*cos_a + s5*cos_b*sin_a)
Eb = Eb - 5D0/8D0*dar*(s5*nz*nr+s3*nf)* &
(-s5*cos_a*cos_b**2 + s3*cos_b*sin_a + s5*cos_a*sin_b**2)
! from bending free energy
! xir = xi_H = sqrt(65 lg2 /(8 a)) / H/R
E = E + 4D0*(2D0+de)*xir**2*sin_b**2/13D0
Eb = Eb + 4D0*(2D0+de)*xir**2*sin2b/13D0
! lsg ~ 3
E = E - 2D0*text_lsg*xir**2*sin_b**2/13D0
Eb = Eb - 2D0*text_lsg*xir**2*sin2b/13D0
! bar = b/a/r
E = E - bar* (nr**2 - 5D0/18D0*nr**4)
Ea = Ea - bar* (2D0*nr - 10D0/9D0*nr**3)*sin_a*sin_b
Eb = Eb + bar* (2D0*nr - 10D0/9D0*nr**3)*cos_a*cos_b
end
! Self test for esurf derivatives
subroutine text1r_esurf_selfcheck(a,b, d, e)
implicit none
real*8 a,b,d,e
real*8 E1,Ea1,Eb1
real*8 E2,Ea2,Eb2
real*8 der1,der2
call text1r_esurf(a,b,E1,Ea1,Eb1)
call text1r_esurf(a+d,b,E2,Ea2,Eb2)
der1=(E2-E1)/d
der2=(Ea2+Ea1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_esurf_selfcheck failed for dE/da:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/da: ', der1, ' Ea: ', der2
endif
call text1r_esurf(a,b+d,E2,Ea2,Eb2)
der1=(E2-E1)/d
der2=(Eb2+Eb1)/2D0
if ( dabs( der1/der2 - 1D0 ) > e ) then
write(*,*) 'text1r_esurf_selfcheck failed for dE/db:'
write(*,*) ' a: ', a, ' b: ', b, ' (E2-E1)/db: ', der1, ' Eb: ', der2
endif
end