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subroutine NSSLR2(keff, ng, bndtl, xxx, dely, diff, sigr, scat, chi, nusigf, L, R)
!
!-----------------------------------------------------------------------
!
!Purpose:
! Compute the 2D ANM coupling matrices for a single node.
!
!Copyright:
! Copyright (C) 2023 Ecole Polytechnique de Montreal
! This library is free software; you can redistribute it and/or
! modify it under the terms of the GNU Lesser General Public
! License as published by the Free Software Foundation; either
! version 2.1 of the License, or (at your option) any later version
!
!Author(s): A. Hebert
!
!Parameters: input
! keff effective multiplication factor.
! ng number of energy groups.
! bndtl set to 'flat' or 'quadratic'.
! xxx node support along X-axis.
! dely node width along Y-axis.
! diff diffusion coefficient array (cm).
! sigr removal cross section array (cm-1).
! scat P0 scattering cross section matrix (cm^-1).
! chi fission spectrum array.
! nusigf nu*fission cross section array (cm^-1).
!
!Parameters: output
! L left nodal coupling matrix.
! R right nodal coupling matrix.
!
!-----------------------------------------------------------------------
!
!----
! subroutine arguments
!----
integer, intent(in) :: ng
real, intent(in) :: keff, xxx(4), dely
real, dimension(ng), intent(in) :: diff, sigr, chi, nusigf
real, dimension(ng,ng), intent(in) :: scat
character(len=12), intent(in) :: bndtl
real(kind=8), dimension(ng,8*ng), intent(out) :: L, R
!----
! local variables
!----
real(kind=8) :: m0(3,3),m2(3,3),m3(2,3),m4(1,3),Lambda_r,sqla,mmax2
!----
! allocatable arrays
!----
complex(kind=8), allocatable, dimension(:,:) :: T,Lambda
real(kind=8), allocatable, dimension(:,:) :: F,DI,T_r,TI,S,Mm,Mp,Nm,Np, &
& GAR1,GAR2,GAR3,GAR4,Vm,Vp,Um,Up,MAT1,MAT2,S7
!----
! scratch storage allocation
!----
allocate(F(ng,ng),T_r(ng,ng),T(ng,ng),TI(ng,ng),DI(ng,ng), &
& Lambda(ng,ng),S(ng,ng),Mm(2*ng,2*ng),Mp(2*ng,2*ng),Nm(ng,2*ng), &
& Np(ng,2*ng),GAR1(ng,2*ng),GAR2(ng,2*ng),GAR3(ng,8*ng), &
& GAR4(ng,8*ng),Vm(2*ng,3*ng),Vp(2*ng,3*ng),Um(ng,3*ng), &
& Up(ng,3*ng),MAT1(ng,8*ng),MAT2(ng,8*ng))
!
! quadratic leakage and boundary conditions
xmm=xxx(1) ; xm=xxx(2) ; xp=xxx(3) ; xpp=xxx(4) ; delx=xp-xm ;
if(xmm == -99999.) then
! Vacuum or zero flux node at left boundary
xmm=2.0*xm-xp
m0(:3,1)=1.0d0 ; m0(1,2)=(xmm+xm)/2.0d0 ; m0(1,3)=(xmm**2+xmm*xm+xm**2)/3.0d0
m0(2,2)=(xm+xp)/2.0d0 ; m0(2,3)=(xm**2+xm*xp+xp**2)/3.0d0
m0(3,2)=(xp+xpp)/2.0d0 ; m0(3,3)=(xp**2+xp*xpp+xpp**2)/3.0d0
call ALINVD(3,m0,3,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(1)')
m0(:3,1)=0.0d0
elseif(xpp == -99999.) then
! Vacuum or zero flux node at right boundary
xpp=2.0*xp-xm
m0(:3,1)=1.0d0 ; m0(1,2)=(xmm+xm)/2.0d0 ; m0(1,3)=(xmm**2+xmm*xm+xm**2)/3.0d0
m0(2,2)=(xm+xp)/2.0d0 ; m0(2,3)=(xm**2+xm*xp+xp**2)/3.0d0
m0(3,2)=(xp+xpp)/2.0d0 ; m0(3,3)=(xp**2+xp*xpp+xpp**2)/3.0d0
call ALINVD(3,m0,3,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(2)')
m0(:3,3)=0.0d0
else
! Internal node
m0(:3,1)=1.0d0 ; m0(1,2)=(xmm+xm)/2.0d0 ; m0(1,3)=(xmm**2+xmm*xm+xm**2)/3.0d0
m0(2,2)=(xm+xp)/2.0d0 ; m0(2,3)=(xm**2+xm*xp+xp**2)/3.0d0
m0(3,2)=(xp+xpp)/2.0d0 ; m0(3,3)=(xp**2+xp*xpp+xpp**2)/3.0d0
call ALINVD(3,m0,3,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(3)')
endif
if(bndtl == 'flat') then
! flat leakage approximation
m0(:3,:3)=0.0d0 ; m0(1,2)=1.0d0
endif
!----
! compute matrices L and R
!----
Mm(:,:)=0.0d0
Mp(:,:)=0.0d0
Nm(:,:)=0.0d0
Np(:,:)=0.0d0
DI(:,:)=0.0d0
Vm(:,:)=0.0d0
Vp(:,:)=0.0d0
Um(:,:)=0.0d0
Up(:,:)=0.0d0
do ig=1,ng
do jg=1,ng
if(ig == jg) then
F(ig,ig)=(chi(ig)*nusigf(ig)/keff-sigr(ig))/diff(ig)
else
F(ig,jg)=(chi(ig)*nusigf(jg)/keff+scat(ig,jg))/diff(ig)
endif
enddo
DI(ig,ig)=1./diff(ig)
enddo
maxiter=40
call ALHQR(ng,ng,F,maxiter,iter,T,Lambda)
mmax2=0.0d0
do ig=1,ng
do jg=1,ng
mmax2=max(mmax2,abs(aimag(T(ig,jg))))
enddo
enddo
if(mmax2 > 1.0e-6) then
write(6,'(3h T=)')
do ig=1,ng
write(6,'(1p,12e12.4)') T(ig,:)
enddo
call XABORT('NSSLR2: complex eigenvalues.')
endif
T_r(:,:)=real(T(:,:),8)
do ig=1,ng
Lambda_r=real(Lambda(ig,ig),8)
sqla=sqrt(abs(Lambda_r))
m2(:3,:3)=0.0d0
m2(1,1)=1.0d0/Lambda_r ; m2(1,3)=-2.0d0/Lambda_r**2
m2(2,2)=1.0d0/Lambda_r ; m2(3,3)=1.0d0/Lambda_r
m2(:3,:3)=matmul(m2(:3,:3),m0(:3,:3))
m3(1,1)=1.0d0 ; m3(1,2)=(xm+xp)/2. ; m3(1,3)=(xm**2+xm*xp+xp**2)/3.0d0
m3(2,1)=0.0d0 ; m3(2,2)=-1.0d0 ; m3(2,3)=-2.0d0*xm
m3(:2,:3)=matmul(m3(:2,:3),m2(:3,:3))
Vm(ig,ig)=m3(1,1) ; Vm(ig,ng+ig)=m3(1,2) ; Vm(ig,2*ng+ig)=m3(1,3) ;
Vm(ng+ig,ig)=m3(2,1) ; Vm(ng+ig,ng+ig)=m3(2,2) ; Vm(ng+ig,2*ng+ig)=m3(2,3) ;
m3(1,1)=1.0d0 ; m3(1,2)=(xm+xp)/2.0d0 ; m3(1,3)=(xm**2+xm*xp+xp**2)/3.0d0
m3(2,1)=0.0d0 ; m3(2,2)=-1.0d0 ; m3(2,3)=-2.0d0*xp
m3(:2,:3)=matmul(m3(:2,:3),m2(:3,:3))
Vp(ig,ig)=m3(1,1) ; Vp(ig,ng+ig)=m3(1,2) ; Vp(ig,2*ng+ig)=m3(1,3) ;
Vp(ng+ig,ig)=m3(2,1) ; Vp(ng+ig,ng+ig)=m3(2,2) ; Vp(ng+ig,2*ng+ig)=m3(2,3) ;
m4(1,1)=1.0d0 ; m4(1,2)=xm ; m4(1,3)=xm**2
m4(:1,:3)=matmul(m4(:1,:3),m2(:3,:3))
Um(ig,ig)=m4(1,1) ; Um(ig,ng+ig)=m4(1,2) ; Um(ig,2*ng+ig)=m4(1,3) ;
m4(1,1)=1.0d0 ; m4(1,2)=xp ; m4(1,3)=xp**2
m4(:1,:3)=matmul(m4(:1,:3),m2(:3,:3))
Up(ig,ig)=m4(1,1) ; Up(ig,ng+ig)=m4(1,2) ; Up(ig,2*ng+ig)=m4(1,3) ;
if(delx*sqla < 1.e-6) then
if(Lambda_r >= 0) then
Mm(ig,ig)=-(delx*sqla)**6/5040.+(delx*sqla)**4/120.-(delx*sqla)**2/6.+1.
Mm(ig,ng+ig)=(delx*sqla)**5/720.-(delx*sqla)**3/24.+(delx*sqla)/2.
Mm(ng+ig,ng+ig)=-sqla
Mp(ng+ig,ig)=((delx*sqla)**6/120.-(delx*sqla)**4/6.+(delx*sqla)**2)/delx
Mp(ng+ig,ng+ig)=(-(delx*sqla)**5/24.+(delx*sqla)**3/2.-(delx*sqla))/delx
Nm(ig,ig)=1.
Np(ig,ig)=-(delx*sqla)**6/720.+(delx*sqla)**4/24.-(delx*sqla)**2/2.+1.
Np(ig,ng+ig)=(delx*sqla)**5/120.-(delx*sqla)**3/6.+(delx*sqla)
else
Mm(ig,ig)=(delx*sqla)**4/120.+(delx*sqla)**3/24.+(delx*sqla)**2/6.+(delx*sqla)/2. + 1.
Mm(ig,ng+ig)=-(delx*sqla)**3/24.+(delx*sqla)**2/6.-(delx*sqla)/2. + 1.
Mm(ng+ig,ig)=-sqla ; Mm(ng+ig,ng+ig)=sqla ;
Mp(ng+ig,ig)=(-(delx*sqla)**4/6.-(delx*sqla)**3/2.-(delx*sqla)**2-(delx*sqla))/delx
Mp(ng+ig,ng+ig)=(-(delx*sqla)**4/6+(delx*sqla)**3/2.-(delx*sqla)**2+(delx*sqla))/delx
Nm(ig,ig)=1. ; Nm(ig,ng+ig)=1. ;
Np(ig,ig)=(delx*sqla)**4/24.+(delx*sqla)**3/6.+(delx*sqla)**2/2.+(delx*sqla)+1.
Np(ig,ng+ig)=(delx*sqla)**4/24.-(delx*sqla)**3/6.+(delx*sqla)**2/2.-(delx*sqla)+1.
endif
else if(Lambda_r >= 0) then
Mm(ig,ig)=(sin(sqla*xp)-sin(sqla*xm))/(delx*sqla)
Mm(ig,ng+ig)=-(cos(sqla*xp)-cos(sqla*xm))/(delx*sqla)
Mm(ng+ig,ig)=sqla*sin(sqla*xm)
Mm(ng+ig,ng+ig)=-sqla*cos(sqla*xm)
Mp(ng+ig,ig)=sqla*sin(sqla*xp)
Mp(ng+ig,ng+ig)=-sqla*cos(sqla*xp)
Nm(ig,ig)=cos(sqla*xm)
Nm(ig,ng+ig)=sin(sqla*xm)
Np(ig,ig)=cos(sqla*xp)
Np(ig,ng+ig)=sin(sqla*xp)
else
Mm(ig,ig)=exp(sqla*xm)*(exp(sqla*(xp-xm))-1.0d0)/(delx*sqla)
Mm(ig,ng+ig)=-exp(-sqla*xm)*(exp(-sqla*(xp-xm))-1.0d0)/(delx*sqla)
Mm(ng+ig,ig)=-sqla*exp(sqla*xm)
Mm(ng+ig,ng+ig)=sqla*exp(-sqla*xm)
Mp(ng+ig,ig)=-sqla*exp(sqla*xp)
Mp(ng+ig,ng+ig)=sqla*exp(-sqla*xp)
Nm(ig,ig)=exp(sqla*xm)
Nm(ig,ng+ig)=exp(-sqla*xm)
Np(ig,ig)=exp(sqla*xp)
Np(ig,ng+ig)=exp(-sqla*xp)
endif
Mp(ig,ig)=Mm(ig,ig)
Mp(ig,ng+ig)=Mm(ig,ng+ig)
enddo
!
TI(:,:)=T_r(:,:)
call ALINVD(2*ng,Mm,2*ng,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(4)')
call ALINVD(2*ng,Mp,2*ng,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(5)')
call ALINVD(ng,TI,ng,ier)
if(ier /= 0) call XABORT('NSSLR2: singular matrix.(6)')
!
GAR1=matmul(Nm,Mm) ! ng,2*ng
GAR2=matmul(Np,Mp) ! ng,2*ng
S=matmul(TI,DI) ! ng,ng
!
MAT1(:ng,:2*ng)=GAR1(:ng,:2*ng)
MAT1(:ng,2*ng+1:5*ng)=-Um(:ng,:3*ng)/dely+matmul(GAR1(:ng,:2*ng),Vm(:2*ng,:3*ng))/dely
MAT1(:ng,5*ng+1:8*ng)=Um(:ng,:3*ng)/dely-matmul(GAR1(:ng,:2*ng),Vm(:2*ng,:3*ng))/dely
MAT2(:ng,:2*ng)=GAR2(:ng,:2*ng)
MAT2(:ng,2*ng+1:5*ng)=-Up(:ng,:3*ng)/dely+matmul(GAR2(:ng,:2*ng),Vp(:2*ng,:3*ng))/dely
MAT2(:ng,5*ng+1:8*ng)=Up(:ng,:3*ng)/dely-matmul(GAR2(:ng,:2*ng),Vp(:2*ng,:3*ng))/dely
!
GAR3=matmul(T_r,MAT1) ! ng,8*ng
GAR4=matmul(T_r,MAT2) ! ng,8*ng
L(:ng,:ng)=matmul(GAR3(:ng,:ng),TI(:ng,:ng))
R(:ng,:ng)=matmul(GAR4(:ng,:ng),TI(:ng,:ng))
allocate(S7(7*ng,7*ng))
S7(:,:)=0.0d0 ! 7*ng,7*ng
do i=1,7
S7((i-1)*ng+1:i*ng,(i-1)*ng+1:i*ng)=S(:ng,:ng)
enddo
L(:ng,ng+1:8*ng)=matmul(GAR3(:ng,ng+1:8*ng),S7(:7*ng,:7*ng))
R(:ng,ng+1:8*ng)=matmul(GAR4(:ng,ng+1:8*ng),S7(:7*ng,:7*ng))
!----
! scratch storage deallocation
!----
deallocate(S7,MAT2,MAT1,Up,Um,Vp,Vm,GAR4,GAR3,GAR2,GAR1,Np,Nm,Mp,Mm,S, &
& Lambda,DI,TI,T,T_r,F)
end subroutine NSSLR2
|
