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DAMASK_EICMD/processing/post/postprocessingMath.f90

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2011-11-07 23:57:33 +05:30
!$Id: postprocessingMath.f90 1054 2011-11-03 13:21:11Z MPIE\p.eisenlohr $
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!all function below are taken from math.f90
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!
module math
real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510
! *** 3x3 Identity ***
real*8, dimension(3,3), parameter :: math_I3 = &
reshape( (/ &
1.0,0.0,0.0, &
0.0,1.0,0.0, &
0.0,0.0,1.0 /),(/3,3/))
contains
!**************************************************************************
! matrix multiplication 33x33 = 3x3
!**************************************************************************
pure function math_mul33x33(A,B)
implicit none
integer i,j
real*8, dimension(3,3), intent(in) :: A,B
real*8, dimension(3,3) :: math_mul33x33
forall (i=1:3,j=1:3) math_mul33x33(i,j) = &
A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j)
return
end function math_mul33x33
!**************************************************************************
! Cramer inversion of 3x3 matrix (subroutine)
!**************************************************************************
PURE SUBROUTINE math_invert3x3(A, InvA, DetA, error)
! Bestimmung der Determinanten und Inversen einer 3x3-Matrix
! A = Matrix A
! InvA = Inverse of A
! DetA = Determinant of A
! error = logical
implicit none
logical, intent(out) :: error
real*8,dimension(3,3),intent(in) :: A
real*8,dimension(3,3),intent(out) :: InvA
real*8, intent(out) :: DetA
DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )&
- A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )&
+ A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) )
if (DetA <= tiny(DetA)) then
error = .true.
else
InvA(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA
InvA(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA
InvA(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA
InvA(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA
InvA(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA
InvA(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA
InvA(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA
InvA(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA
InvA(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA
error = .false.
endif
return
END SUBROUTINE math_invert3x3
!********************************************************************
! determinant of a 3x3 matrix
!********************************************************************
pure function math_det3x3(m)
implicit none
real*8, dimension(3,3), intent(in) :: m
real*8 math_det3x3
math_det3x3 = m(1,1)*(m(2,2)*m(3,3)-m(2,3)*m(3,2)) &
-m(1,2)*(m(2,1)*m(3,3)-m(2,3)*m(3,1)) &
+m(1,3)*(m(2,1)*m(3,2)-m(2,2)*m(3,1))
return
end function math_det3x3
!****************************************************************
pure subroutine math_pDecomposition(FE,U,R,error)
!-----FE = R.U
!****************************************************************
implicit none
real*8, intent(in) :: FE(3,3)
real*8, intent(out) :: R(3,3), U(3,3)
logical, intent(out) :: error
real*8 CE(3,3),EW1,EW2,EW3,EB1(3,3),EB2(3,3),EB3(3,3),UI(3,3),det
error = .false.
ce = math_mul33x33(transpose(FE),FE)
CALL math_spectral1(CE,EW1,EW2,EW3,EB1,EB2,EB3)
U=DSQRT(EW1)*EB1+DSQRT(EW2)*EB2+DSQRT(EW3)*EB3
call math_invert3x3(U,UI,det,error)
if (.not. error) R = math_mul33x33(FE,UI)
return
end subroutine math_pDecomposition
!**************************************************************************
! Cramer inversion of 3x3 matrix (function)
!**************************************************************************
pure function math_inv3x3(A)
! direct Cramer inversion of matrix A.
! returns all zeroes if not possible, i.e. if det close to zero
implicit none
real*8,dimension(3,3),intent(in) :: A
real*8 DetA
real*8,dimension(3,3) :: math_inv3x3
math_inv3x3 = 0.0
DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )&
- A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )&
+ A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) )
if (DetA > tiny(DetA)) then
math_inv3x3(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA
math_inv3x3(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA
math_inv3x3(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA
math_inv3x3(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA
math_inv3x3(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA
math_inv3x3(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA
math_inv3x3(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA
math_inv3x3(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA
math_inv3x3(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA
endif
return
end function math_inv3x3
!**********************************************************************
! HAUPTINVARIANTEN HI1M, HI2M, HI3M DER 3X3 MATRIX M
!**********************************************************************
PURE SUBROUTINE math_hi(M,HI1M,HI2M,HI3M)
implicit none
real*8, intent(in) :: M(3,3)
real*8, intent(out) :: HI1M, HI2M, HI3M
HI1M=M(1,1)+M(2,2)+M(3,3)
HI2M=HI1M**2/2.0-(M(1,1)**2+M(2,2)**2+M(3,3)**2)/2.0-M(1,2)*M(2,1)-M(1,3)*M(3,1)-M(2,3)*M(3,2)
HI3M=math_det3x3(M)
! QUESTION: is 3rd equiv det(M) ?? if yes, use function math_det !agreed on YES
return
END SUBROUTINE math_hi
!**********************************************************************
pure subroutine math_spectral1(M,EW1,EW2,EW3,EB1,EB2,EB3)
!**** EIGENWERTE UND EIGENWERTBASIS DER SYMMETRISCHEN 3X3 MATRIX M
implicit none
real*8, intent(in) :: M(3,3)
real*8, intent(out) :: EB1(3,3),EB2(3,3),EB3(3,3),EW1,EW2,EW3
real*8 HI1M,HI2M,HI3M,TOL,R,S,T,P,Q,RHO,PHI,Y1,Y2,Y3,D1,D2,D3
real*8 C1,C2,C3,M1(3,3),M2(3,3),M3(3,3),arg
TOL=1.e-14
CALL math_hi(M,HI1M,HI2M,HI3M)
R=-HI1M
S= HI2M
T=-HI3M
P=S-R**2.0/3.0
Q=2.0/27.0*R**3.0-R*S/3.0+T
EB1=0.0
EB2=0.0
EB3=0.0
IF((ABS(P).LT.TOL).AND.(ABS(Q).LT.TOL))THEN
! DREI GLEICHE EIGENWERTE
EW1=HI1M/3.0
EW2=EW1
EW3=EW1
! this is not really correct, but this way U is calculated
! correctly in PDECOMPOSITION (correct is EB?=I)
EB1(1,1)=1.0
EB2(2,2)=1.0
EB3(3,3)=1.0
ELSE
RHO=DSQRT(-3.0*P**3.0)/9.0
arg=-Q/RHO/2.0
if(arg.GT.1) arg=1
if(arg.LT.-1) arg=-1
PHI=DACOS(arg)
Y1=2*RHO**(1.0/3.0)*DCOS(PHI/3.0)
Y2=2*RHO**(1.0/3.0)*DCOS(PHI/3.0+2.0/3.0*PI)
Y3=2*RHO**(1.0/3.0)*DCOS(PHI/3.0+4.0/3.0*PI)
EW1=Y1-R/3.0
EW2=Y2-R/3.0
EW3=Y3-R/3.0
C1=ABS(EW1-EW2)
C2=ABS(EW2-EW3)
C3=ABS(EW3-EW1)
IF(C1.LT.TOL) THEN
! EW1 is equal to EW2
D3=1.0/(EW3-EW1)/(EW3-EW2)
M1=M-EW1*math_I3
M2=M-EW2*math_I3
EB3=math_mul33x33(M1,M2)*D3
EB1=math_I3-EB3
! both EB2 and EW2 are set to zero so that they do not
! contribute to U in PDECOMPOSITION
EW2=0.0
ELSE IF(C2.LT.TOL) THEN
! EW2 is equal to EW3
D1=1.0/(EW1-EW2)/(EW1-EW3)
M2=M-math_I3*EW2
M3=M-math_I3*EW3
EB1=math_mul33x33(M2,M3)*D1
EB2=math_I3-EB1
! both EB3 and EW3 are set to zero so that they do not
! contribute to U in PDECOMPOSITION
EW3=0.0
ELSE IF(C3.LT.TOL) THEN
! EW1 is equal to EW3
D2=1.0/(EW2-EW1)/(EW2-EW3)
M1=M-math_I3*EW1
M3=M-math_I3*EW3
EB2=math_mul33x33(M1,M3)*D2
EB1=math_I3-EB2
! both EB3 and EW3 are set to zero so that they do not
! contribute to U in PDECOMPOSITION
EW3=0.0
ELSE
! all three eigenvectors are different
D1=1.0/(EW1-EW2)/(EW1-EW3)
D2=1.0/(EW2-EW1)/(EW2-EW3)
D3=1.0/(EW3-EW1)/(EW3-EW2)
M1=M-EW1*math_I3
M2=M-EW2*math_I3
M3=M-EW3*math_I3
EB1=math_mul33x33(M2,M3)*D1
EB2=math_mul33x33(M1,M3)*D2
EB3=math_mul33x33(M1,M2)*D3
END IF
END IF
RETURN
END SUBROUTINE math_spectral1
!**************************************************************************
! volume of tetrahedron given by four vertices
!**************************************************************************
pure function math_volTetrahedron(v1,v2,v3,v4)
implicit none
real*8 math_volTetrahedron
real*8, dimension (3), intent(in) :: v1,v2,v3,v4
real*8, dimension (3,3) :: m
m(:,1) = v1-v2
m(:,2) = v2-v3
m(:,3) = v3-v4
math_volTetrahedron = math_det3x3(m)/6.0
end function math_volTetrahedron
!subroutines below are for postprocessing with python
!two small helper functions for indexing
! CAREFULL, index and location runs from 0 to N-1 (python style)
function mesh_location(idx,resolution)
integer, intent(in) :: idx
integer, intent(in) :: resolution(3)
integer :: mesh_location(3)
mesh_location = (/modulo(idx/ resolution(3) / resolution(2),resolution(1)), &
modulo(idx/ resolution(3), resolution(2)), &
modulo(idx, resolution(3))/)
end function mesh_location
function mesh_index(location,resolution)
integer, intent(in) :: location(3)
integer, intent(in) :: resolution(3)
integer :: mesh_index
mesh_index = modulo(location(3), resolution(3)) +&
(modulo(location(2), resolution(2)))*resolution(3) +&
(modulo(location(1), resolution(1)))*resolution(3)*resolution(2)
end function mesh_index
end module math
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine mesh(res_x,res_y,res_z,geomdim,defgrad_av,centroids,nodes)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to build a regular mesh of cubes for given coordinates (= center of the cubes)
!
implicit none
real*8 geomdim(3)
integer res_x, res_y, res_z
real*8 wrappedCentroids(res_x+2,res_y+2,res_z+2,3)
real*8 nodes(res_x+1,res_y+1,res_z+1,3)
real*8 centroids(res_x ,res_y ,res_z ,3)
integer, dimension(3,8) :: neighbor = reshape((/ &
0, 0, 0, &
1, 0, 0, &
1, 1, 0, &
0, 1, 0, &
0, 0, 1, &
1, 0, 1, &
1, 1, 1, &
0, 1, 1 &
/), &
(/3,8/))
integer i,j,k,n
real*8, dimension(3,3) :: defgrad_av
integer, dimension(3) :: diag, shift, lookup, me, res
nodes = 0.0
diag = 1
shift = 0
lookup = 0
res = (/res_x,res_y,res_z/)
wrappedCentroids = 0.0
wrappedCentroids(2:res_x+1,2:res_y+1,2:res_z+1,:) = centroids
do k = 0,res_z+1
do j = 0,res_y+1
do i = 0,res_x+1
if (k==0 .or. k==res_z+1 .or. & ! z skin
j==0 .or. j==res_y+1 .or. & ! y skin
i==0 .or. i==res_x+1 ) then ! x skin
me = (/i,j,k/) ! me on skin
shift = sign(abs(res+diag-2*me)/(res+diag),res+diag-2*me)
lookup = me-diag+shift*res
wrappedCentroids(i+1,j+1,k+1,:) = centroids(lookup(1)+1,lookup(2)+1,lookup(3)+1,:) - &
matmul(defgrad_av, shift*geomdim)
endif
enddo; enddo; enddo
do k = 0,res_z
do j = 0,res_y
do i = 0,res_x
do n = 1,8
nodes(i+1,j+1,k+1,:) = nodes(i+1,j+1,k+1,:) + wrappedCentroids(i+1+neighbor(1,n), &
j+1+neighbor(2,n), &
k+1+neighbor(3,n), :)
enddo; enddo; enddo; enddo
nodes = nodes/8.0
end subroutine mesh
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine deformed(res_x,res_y,res_z,geomdim,defgrad,defgrad_av,coord_avgCorner)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to calculate coordinates in current configuration for given defgrad
! using linear interpolation (blurres out high frequency defomation)
!
implicit none
real*8 geomdim(3)
integer res_x, res_y, res_z
real*8 coord(8,6,res_x,res_y,res_z,3)
real*8 coord_avgOrder(8,res_x,res_y,res_z,3)
real*8 coord_avgCorner(res_x,res_y,res_z,3)
real*8 defgrad(res_x,res_y,res_z,3,3)
integer, dimension(3,8) :: corner = reshape((/ &
0, 0, 0,&
1, 0, 0,&
1, 1, 0,&
0, 1, 0,&
1, 1, 1,&
0, 1, 1,&
0, 0, 1,&
1, 0, 1 &
/), &
(/3,8/))
integer, dimension(3,8) :: step = reshape((/ &
1, 1, 1,&
-1, 1, 1,&
-1,-1, 1,&
1,-1, 1,&
-1,-1,-1,&
1,-1,-1,&
1, 1,-1,&
-1, 1,-1 &
/), &
(/3,8/))
integer, dimension(3,6) :: order = reshape((/ &
1, 2, 3,&
1, 3, 2,&
2, 1, 3,&
2, 3, 1,&
3, 1, 2,&
3, 2, 1 &
/), &
(/3,6/))
real*8 myStep(3), fones(3), parameter_coords(3)
real*8 defgrad_av(3,3)
real*8 negative(3), positive(3)
integer rear(3), init(3), ones(3), oppo(3), me(3), res(3)
integer i, j, k, s, o
print*, 'Restore geometry using linear integration'
print '(a,/,e12.5,e12.5,e12.5)', ' Dimension:', geomdim
print '(a,/,i5,i5,i5)', ' Resolution:', res_x,res_y,res_z
ones = 1
fones = 1.0
coord_avgOrder=0.0
res = (/res_x,res_y,res_z/)
do s = 0, 7 ! corners (from 0 to 7)
init = corner(:,s+1)*(res-ones) +ones
oppo = corner(:,mod((s+4),8)+1)*(res-ones) +ones
do o=1,6 ! orders ! from 1 to 6)
do k = init(order(3,o)), oppo(order(3,o)), step(order(3,o),s+1)
rear(order(2,o)) = init(order(2,o))
do j = init(order(2,o)), oppo(order(2,o)), step(order(2,o),s+1)
rear(order(1,o)) = init(order(1,o))
do i = init(order(1,o)), oppo(order(1,o)), step(order(1,o),s+1)
me(order(1,o)) = i
me(order(2,o)) = j
me(order(3,o)) = k
if ( (me(1)==init(1)).and.(me(2)==init(2)).and. (me(3)==init(3)) ) then
coord(s+1,o,me(1),me(2),me(3),:) = geomdim * (matmul(defgrad_av,corner(:,s+1)) + &
matmul(defgrad(me(1),me(2),me(3),:,:),0.5*step(:,s+1)/res))
else
myStep = (me-rear)*geomdim/res
coord(s+1,o,me(1),me(2),me(3),:) = coord(s+1,o,rear(1),rear(2),rear(3),:) + &
0.5*matmul(defgrad(me(1),me(2),me(3),:,:) + &
defgrad(rear(1),rear(2),rear(3),:,:),myStep)
endif
rear = me
enddo; enddo; enddo; enddo
do i=1,6
coord_avgOrder(s+1,:,:,:,:) = coord_avgOrder(s+1,:,:,:,:) + coord(s+1,i,:,:,:,:)/6.0
enddo
enddo
do k=0, res_z-1
do j=0, res_y-1
do i=0, res_x-1
parameter_coords = (2.0*(/i+0.0,j+0.0,k+0.0/)-real(res)+fones)/(real(res)-fones)
positive = fones + parameter_coords
negative = fones - parameter_coords
coord_avgCorner(i+1,j+1,k+1,:) = ( coord_avgOrder(1,i+1,j+1,k+1,:) *negative(1)*negative(2)*negative(3)&
+ coord_avgOrder(2,i+1,j+1,k+1,:) *positive(1)*negative(2)*negative(3)&
+ coord_avgOrder(3,i+1,j+1,k+1,:) *positive(1)*positive(2)*negative(3)&
+ coord_avgOrder(4,i+1,j+1,k+1,:) *negative(1)*positive(2)*negative(3)&
+ coord_avgOrder(5,i+1,j+1,k+1,:) *positive(1)*positive(2)*positive(3)&
+ coord_avgOrder(6,i+1,j+1,k+1,:) *negative(1)*positive(2)*positive(3)&
+ coord_avgOrder(7,i+1,j+1,k+1,:) *negative(1)*negative(2)*positive(3)&
+ coord_avgOrder(8,i+1,j+1,k+1,:) *positive(1)*negative(2)*positive(3))*0.125
enddo; enddo; enddo
end subroutine deformed
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine deformed_fft(res_x,res_y,res_z,geomdim,defgrad,defgrad_av,scaling,coords)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to calculate coordinates in current configuration for given defgrad
! using integration in Fourier space (more accurate than deformed(...))
!
implicit none
integer res_x, res_y, res_z
real*8 geomdim(3)
real*8 defgrad(res_x,res_y,res_z,3,3)
real*8 defgrad_av(3,3)
real*8 scaling
real*8 coords(res_x,res_y,res_z,3)
complex*16 coords_fft(res_x/2+1,res_y,res_z,3)
complex*16 defgrad_fft(res_x,res_y,res_z,3,3)
integer i, j, k
integer k_s(3)
real*8 step(3)
real*8 offset_coords(3)
real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510
integer*8 :: plan_fft(2)
print*, 'Restore geometry using FFT-based integration'
print '(a,/,e12.5,e12.5,e12.5)', ' Dimension:', geomdim
print '(a,/,i5,i5,i5)', ' Resolution:', res_x,res_y,res_z
call dfftw_plan_many_dft(plan_fft(1),3,(/res_x,res_y,res_z/),9,&
defgrad_fft,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,&
defgrad_fft,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,-1 + 32) ! -1 = FFTW_FORWARD, 32 =FFTW_PATIENT
call dfftw_plan_many_dft_c2r(plan_fft(2),3,(/res_x,res_y,res_z/),3,&
coords_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,&
coords, (/res_x, res_y,res_z/),1, res_x* res_y*res_z,32) ! 32 = FFTW_PATIENT
coords_fft = 0.0
defgrad_fft = defgrad
step(1) = geomdim(1)/real(res_x)
step(2) = geomdim(2)/real(res_y)
step(3) = geomdim(3)/real(res_z)
call dfftw_execute_dft(plan_fft(1), defgrad_fft, defgrad_fft)
do k = 1, res_z
k_s(3) = k-1
if(k > res_z/2+1) k_s(3) = k_s(3)-res_z
do j = 1, res_y
k_s(2) = j-1
if(j > res_y/2+1) k_s(2) = k_s(2)-res_y
do i = 1, res_x/2+1
k_s(1) = i-1
if(i/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)&
+ defgrad_fft(i,j,k,:,1)*geomdim(1)/(real(k_s(1))*cmplx(0.0,1.0)*pi*2.0)
if(j/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)&
+ defgrad_fft(i,j,k,:,2)*geomdim(2)/(real(k_s(2))*cmplx(0.0,1.0)*pi*2.0)
if(k/=1) coords_fft(i,j,k,:) = coords_fft(i,j,k,:)&
+ defgrad_fft(i,j,k,:,3)*geomdim(3)/(real(k_s(3))*cmplx(0.0,1.0)*pi*2.0)
enddo; enddo; enddo
call dfftw_execute_dft_c2r(plan_fft(2), coords_fft, coords)
coords = coords/real(res_x*res_y*res_z)
offset_coords = matmul(defgrad(1,1,1,:,:),step/2.0) - scaling*coords(1,1,1,:)
do k = 1, res_z; do j = 1, res_y; do i = 1, res_x
coords(i,j,k,:) = scaling*coords(i,j,k,:) + offset_coords + matmul(defgrad_av,&
(/step(1)*real(i-1),&
step(2)*real(j-1),&
step(3)*real(k-1)/))
enddo; enddo; enddo
end subroutine deformed_fft
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine volume_compare(res_x,res_y,res_z,geomdim,nodes,defgrad,volume_mismatch)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to calculate the mismatch between volume of reconstructed (compatible)
! cube and determinant of defgrad at the FP
use math
implicit none
real*8 geomdim(3)
integer res_x, res_y, res_z
real*8 nodes(res_x+1,res_y+1,res_z+1,3)
real*8 defgrad(res_x ,res_y ,res_z ,3,3)
real*8 volume_mismatch(res_x ,res_y ,res_z )
real*8 coords(8,3)
integer i,j,k
real*8 vol_initial
print*, 'Calculating volume mismatch'
vol_initial = geomdim(1)*geomdim(2)*geomdim(3)/real(res_x)/real(res_y)/real(res_z)
do k = 1,res_z
do j = 1,res_y
do i = 1,res_x
coords(1,:) = nodes(i ,j ,k ,:)
coords(2,:) = nodes(i+1,j ,k ,:)
coords(3,:) = nodes(i+1,j+1,k ,:)
coords(4,:) = nodes(i ,j+1,k ,:)
coords(5,:) = nodes(i ,j, k+1,:)
coords(6,:) = nodes(i+1,j ,k+1,:)
coords(7,:) = nodes(i+1,j+1,k+1,:)
coords(8,:) = nodes(i ,j+1,k+1,:)
volume_mismatch(i,j,k) = abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(8,:),coords(4,:))) &
+ abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(8,:),coords(5,:))) &
+ abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(3,:),coords(4,:))) &
+ abs(math_volTetrahedron(coords(7,:),coords(1,:),coords(3,:),coords(2,:))) &
+ abs(math_volTetrahedron(coords(7,:),coords(5,:),coords(2,:),coords(6,:))) &
+ abs(math_volTetrahedron(coords(7,:),coords(5,:),coords(2,:),coords(1,:)))
volume_mismatch(i,j,k) = volume_mismatch(i,j,k)/math_det3x3(defgrad(i,j,k,:,:))
enddo; enddo; enddo
volume_mismatch = volume_mismatch/vol_initial
end subroutine volume_compare
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine shape_compare(res_x,res_y,res_z,geomdim,nodes,centroids,defgrad,shape_mismatch)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to calculate the mismatch between the vectors from the central point to
! the corners of reconstructed (combatible) volume element and the vectors calculated by deforming
! the initial volume element with the current deformation gradient
implicit none
real*8 geomdim(3)
integer res_x, res_y, res_z
real*8 nodes(res_x+1,res_y+1,res_z+1,3)
real*8 centroids(res_x ,res_y ,res_z ,3)
real*8 defgrad(res_x ,res_y ,res_z ,3,3)
real*8 shape_mismatch(res_x ,res_y ,res_z)
real*8 coords_initial(8,3)
integer i,j,k
print*, 'Calculating shape mismatch'
coords_initial(1,:) = (/-geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/)
coords_initial(2,:) = (/+geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/)
coords_initial(3,:) = (/+geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/)
coords_initial(4,:) = (/-geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),-geomdim(3)/2.0/real(res_z)/)
coords_initial(5,:) = (/-geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/)
coords_initial(6,:) = (/+geomdim(1)/2.0/real(res_x),-geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/)
coords_initial(7,:) = (/+geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/)
coords_initial(8,:) = (/-geomdim(1)/2.0/real(res_x),+geomdim(2)/2.0/real(res_y),+geomdim(3)/2.0/real(res_z)/)
do i=1,8
enddo
do k = 1,res_z
do j = 1,res_y
do i = 1,res_x
shape_mismatch(i,j,k) = &
sqrt(sum((nodes(i ,j ,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(1,:)))**2.0))&
+ sqrt(sum((nodes(i+1,j ,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(2,:)))**2.0))&
+ sqrt(sum((nodes(i+1,j+1,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(3,:)))**2.0))&
+ sqrt(sum((nodes(i ,j+1,k ,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(4,:)))**2.0))&
+ sqrt(sum((nodes(i ,j, k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(5,:)))**2.0))&
+ sqrt(sum((nodes(i+1,j ,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(6,:)))**2.0))&
+ sqrt(sum((nodes(i+1,j+1,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(7,:)))**2.0))&
+ sqrt(sum((nodes(i ,j+1,k+1,:) - centroids(i,j,k,:) - matmul(defgrad(i,j,k,:,:), coords_initial(8,:)))**2.0))
enddo; enddo; enddo
end subroutine shape_compare
!++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine inverse_reconstruction(res_x,res_y,res_z,reference_configuration,current_configuration,defgrad)
!++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! Routine to calculate deformation gradient from reference and current configuration
! NOT WORKING BY NOW!!!!!!!!!!!!!
!
use math
implicit none
integer res_x, res_y, res_z
real*8 reference_configuration(res_x+1,res_y+1,res_z+1,3)
real*8 current_configuration(res_x+1,res_y+1,res_z+1,3)
real*8 defgrad(res_x,res_y,res_z,3,3)
real*8 delta, tolerance, res, res_center
real*8 reference(8,3)
real*8 current(8,3)
real*8 defgrad_temp(3,3)
real*8 dres_dF(3,3)
real*8 identity(3,3)
real*8 ref_bar(3)
real*8 current_bar(3)
real*8 r(8)
real*8 differentiate(9,3,3)
integer i, j, k, m, l, x, y, o
identity = 0.0
identity(1,1) = 1.0
identity(2,2) = 1.0
identity(3,3) = 1.0
differentiate = 0.0
tolerance = 1e-10
delta = 1e-9
k = 0
do j = 1, 3; do i = 1, 3
k = k+1
differentiate(k,i,j) = 1.0
enddo; enddo
do k = 1, res_z
do j = 1, res_y
do i = 1, res_x
reference(1,:) = reference_configuration(i ,j ,k ,:)
reference(2,:) = reference_configuration(i+1,j ,k ,:)
reference(3,:) = reference_configuration(i+1,j+1,k ,:)
reference(4,:) = reference_configuration(i ,j+1,k ,:)
reference(5,:) = reference_configuration(i ,j ,k+1,:)
reference(6,:) = reference_configuration(i+1,j ,k+1,:)
reference(7,:) = reference_configuration(i+1,j+1,k+1,:)
reference(8,:) = reference_configuration(i ,j+1,k+1,:)
current(1,:) = current_configuration(i ,j ,k ,:)
current(2,:) = current_configuration(i+1,j ,k ,:)
current(3,:) = current_configuration(i+1,j+1,k ,:)
current(4,:) = current_configuration(i ,j+1,k ,:)
current(5,:) = current_configuration(i ,j ,k+1,:)
current(6,:) = current_configuration(i+1,j ,k+1,:)
current(7,:) = current_configuration(i+1,j+1,k+1,:)
current(8,:) = current_configuration(i ,j+1,k+1,:)
do o=1,3
ref_bar(o) = sum(reference(:,o))/8.0
current_bar(o) = sum(current(:,o))/8.0
enddo
do o=1,8
reference(o,:) = reference(o,:) -ref_bar
current(o,:) = current(o,:) -current_bar
enddo
defgrad_temp = identity
res_center = 2.0*tolerance
o=0
do while(res_center >= tolerance)
o = o + 1
do l = 1,8 ! loop over corners
r(l) = sqrt(sum((current(l,:)-matmul(defgrad_temp,reference(l,:)))**2)) ! corner distance
enddo
res_center = sum(r*r) ! current residuum
print*, 'res_center', res_center
m=0
do y=1,3; do x=1,3 ! numerical differentiation
m = m+1
do l = 1,8
r(l) = sqrt(sum((current(l,:)-matmul((defgrad_temp+differentiate(m,:,:)*delta),reference(l,:)))**2)) ! corner distance
enddo
res = sum(r*r)
print*,'res step', m, res
dres_dF(x,y) = (res-res_center)/delta
enddo; enddo
print*, 'dres_dF', dres_dF
print*, 'deltadef', math_inv3x3(dres_dF)*res_center
defgrad_temp = defgrad_temp - math_inv3x3(dres_dF)*res_center ! Newton--Raphson
print*, o, res_center
! pause
enddo
defgrad(i,j,k,:,:) = defgrad_temp
enddo; enddo; enddo
end subroutine inverse_reconstruction
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine tensor_avg(res_x,res_y,res_z,tensor,avg)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!calculate average of tensor field
!
implicit none
integer res_x, res_y, res_z
real*8 tensor(res_x,res_y,res_z,3,3)
real*8 avg(3,3)
real*8 wgt
integer m,n
wgt = 1/real(res_x*res_y*res_z)
do m = 1,3; do n = 1,3
avg(m,n) = sum(tensor(:,:,:,m,n)) * wgt
enddo; enddo
end subroutine tensor_avg
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine logstrain_spat(res_x,res_y,res_z,defgrad,logstrain_field)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!calculate logarithmic strain in spatial configuration for given defgrad field
!
use math
implicit none
integer res_x, res_y, res_z
integer i, j, k
real*8 defgrad(res_x,res_y,res_z,3,3)
real*8 logstrain_field(res_x,res_y,res_z,3,3)
real*8 temp33_Real(3,3), temp33_Real2(3,3)
real*8 eigenvectorbasis(3,3,3)
real*8 eigenvalue(3)
logical errmatinv
do k = 1, res_z; do j = 1, res_y; do i = 1, res_x
call math_pDecomposition(defgrad(i,j,k,:,:),temp33_Real2,temp33_Real,errmatinv) !store R in temp33_Real
temp33_Real2 = math_inv3x3(temp33_Real)
temp33_Real = math_mul33x33(defgrad(i,j,k,:,:),temp33_Real2) ! v = F o inv(R), store in temp33_Real2
call math_spectral1(temp33_Real, eigenvalue(1), eigenvalue(2), eigenvalue(3),&
eigenvectorbasis(1,:,:), eigenvectorbasis(2,:,:), eigenvectorbasis(3,:,:))
eigenvalue = log(sqrt(eigenvalue))
logstrain_field(i,j,k,:,:) = eigenvalue(1)*eigenvectorbasis(1,:,:)+&
eigenvalue(2)*eigenvectorbasis(2,:,:)+&
eigenvalue(3)*eigenvectorbasis(3,:,:)
enddo; enddo; enddo
end subroutine logstrain_spat
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine logstrain_mat(res_x,res_y,res_z,defgrad,logstrain_field)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!calculate logarithmic strain in material configuration for given defgrad field
!
use math
implicit none
integer res_x, res_y, res_z
integer i, j, k
real*8 defgrad(res_x,res_y,res_z,3,3)
real*8 logstrain_field(res_x,res_y,res_z,3,3)
real*8 temp33_Real(3,3), temp33_Real2(3,3)
real*8 eigenvectorbasis(3,3,3)
real*8 eigenvalue(3)
logical errmatinv
do k = 1, res_z; do j = 1, res_y; do i = 1, res_x
call math_pDecomposition(defgrad(i,j,k,:,:),temp33_Real,temp33_Real2,errmatinv) !store U in temp33_Real
call math_spectral1(temp33_Real, eigenvalue(1), eigenvalue(2), eigenvalue(3),&
eigenvectorbasis(1,:,:), eigenvectorbasis(2,:,:), eigenvectorbasis(3,:,:))
eigenvalue = log(sqrt(eigenvalue))
logstrain_field(i,j,k,:,:) = eigenvalue(1)*eigenvectorbasis(1,:,:)+&
eigenvalue(2)*eigenvectorbasis(2,:,:)+&
eigenvalue(3)*eigenvectorbasis(3,:,:)
enddo; enddo; enddo
end subroutine logstrain_mat
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine calculate_cauchy(res_x,res_y,res_z,defgrad,p_stress,c_stress)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!calculate cauchy stress for given PK1 stress and defgrad field
!
use math
implicit none
integer res_x, res_y, res_z
integer i, j, k
real*8 defgrad(res_x,res_y,res_z,3,3)
real*8 p_stress(res_x,res_y,res_z,3,3)
real*8 c_stress(res_x,res_y,res_z,3,3)
real*8 jacobi
c_stress = 0.0
do k = 1, res_z; do j = 1, res_y; do i = 1, res_x
jacobi = math_det3x3(defgrad(i,j,k,:,:))
c_stress(i,j,k,:,:) = matmul(p_stress(i,j,k,:,:),transpose(defgrad(i,j,k,:,:)))/jacobi
enddo; enddo; enddo
end subroutine calculate_cauchy
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine calculate_mises(res_x,res_y,res_z,tensor,vm)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
!calculate von Mises equivalent of tensor field
!
implicit none
integer res_x, res_y, res_z
integer i, j, k
real*8 tensor(res_x,res_y,res_z,3,3)
real*8 vm(res_x,res_y,res_z,1)
real*8 deviator(3,3)
real*8 delta(3,3)
real*8 J_2
delta =0.0
delta(1,1) = 1.0
delta(2,2) = 1.0
delta(3,3) = 1.0
do k = 1, res_z; do j = 1, res_y; do i = 1, res_x
deviator = tensor(i,j,k,:,:) - 1.0/3.0*tensor(i,j,k,1,1)*tensor(i,j,k,2,2)*tensor(i,j,k,3,3)*delta
J_2 = deviator(1,1)*deviator(2,2)&
+ deviator(2,2)*deviator(3,3)&
+ deviator(1,1)*deviator(3,3)&
- (deviator(1,2))**2&
- (deviator(2,3))**2&
- (deviator(1,3))**2
vm(i,j,k,:) = sqrt(3*J_2)
enddo; enddo; enddo
end subroutine calculate_mises
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine divergence_fft(res_x,res_y,res_z,vec_tens,geomdim,field,divergence_field)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! calculates divergence field using integration in Fourier space
!use vec_tens to decide if tensor (3) or vector (1)
implicit none
integer res_x, res_y, res_z, vec_tens
real*8 geomdim(3)
real*8 field(res_x,res_y,res_z,vec_tens,3)
real*8 field_copy(res_x,res_y,res_z,vec_tens,3)
real*8 xi(res_x,res_y,res_z,3)
real*8 divergence_field(res_x,res_y,res_z,vec_tens)
complex*16 divergence_field_fft(res_x/2+1,res_y,res_z,vec_tens)
complex*16 field_fft(res_x,res_y,res_z,vec_tens,3)
complex*16 img
integer i, j, k
real*8, parameter :: pi = 3.14159265358979323846264338327950288419716939937510
integer*8 :: plan_fft(2)
img = cmplx(0.0,1.0)
call dfftw_plan_many_dft_r2c(plan_fft(1),3,(/res_x,res_y,res_z/),vec_tens*3,&
field_copy,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,&
field_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,32) ! 32 =FFTW_PATIENT
call dfftw_plan_many_dft_c2r(plan_fft(2),3,(/res_x,res_y,res_z/),vec_tens,&
divergence_field_fft,(/res_x/2+1,res_y,res_z/),1,(res_x/2+1)*res_y*res_z,&
divergence_field,(/res_x,res_y,res_z/),1,res_x*res_y*res_z,32) ! 32 = FFTW_PATIENT
! field_copy is destroyed during plan creation
field_copy = field
call dfftw_execute_dft_r2c(plan_fft(1), field_copy, field_fft)
xi = 0.0
! Alternative calculation of discrete frequencies k_s, ordered as in FFTW (wrap around)
! do k = 0,res_z/2 -1
! do j = 0,res_y/2 -1
! do i = 0,res_x/2 -1
! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/-i,-j,-k/)/geomdim
! xi(1+i, 1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/ i,-j,-k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+j, 1+mod(res_z-k,res_z),:) = (/-i, j,-k/)/geomdim
! xi(1+i, 1+j, 1+mod(res_z-k,res_z),:) = (/ i, j,-k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+k, :) = (/-i,-j, k/)/geomdim
! xi(1+i, 1+mod(res_y-j,res_y),1+k, :) = (/ i,-j, k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+j, 1+k, :) = (/-i, j, k/)/geomdim
! xi(1+i, 1+j, 1+k, :) = (/ i, j, k/)/geomdim
! xi(1+i, 1+j, 1+k, :) = (/ i, j, k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+j, 1+k, :) = (/-i, j, k/)/geomdim
! xi(1+i, 1+mod(res_y-j,res_y),1+k, :) = (/ i,-j, k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+k, :) = (/-i,-j, k/)/geomdim
! xi(1+i, 1+j, 1+mod(res_z-k,res_z),:) = (/ i, j,-k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+j, 1+mod(res_z-k,res_z),:) = (/-i, j,-k/)/geomdim
! xi(1+i, 1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/ i,-j,-k/)/geomdim
! xi(1+mod(res_x-i,res_x),1+mod(res_y-j,res_y),1+mod(res_z-k,res_z),:) = (/-i,-j,-k/)/geomdim
! enddo; enddo; enddo
do k = 0, res_z-1
do j = 0, res_y-1
do i = 0, res_x/2
xi(i+1,j+1,k+1,:) = (/real(i),real(j),real(k)/)/geomdim
if(k==res_z/2) xi(i+1,j+1,k+1,3)= 0.0 ! set highest frequencies to zero
if(j==res_y/2) xi(i+1,j+1,k+1,2)= 0.0
if(i==res_x/2) xi(i+1,j+1,k+1,1)= 0.0
enddo; enddo; enddo
do k = 1, res_z
do j = 1, res_y
do i = 1, res_x/2+1
divergence_field_fft(i,j,k,1) = sum(field_fft(i,j,k,1,:)*xi(i,j,k,:))
if(vec_tens == 3) then
divergence_field_fft(i,j,k,2) = sum(field_fft(i,j,k,2,:)*xi(i,j,k,:))
divergence_field_fft(i,j,k,3) = sum(field_fft(i,j,k,3,:)*xi(i,j,k,:))
endif
enddo; enddo; enddo
divergence_field_fft = divergence_field_fft*img*2.0*pi
call dfftw_execute_dft_c2r(plan_fft(2), divergence_field_fft, divergence_field)
end subroutine divergence_fft
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
subroutine divergence(res_x,res_y,res_z,vec_tens,order,geomdim,field,divergence_field)
!+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
! calculates divergence field using FDM with variable accuracy
!use vec_tes to decide if tensor (3) or vector (1)
use math
implicit none
integer res_x, res_y, res_z, vec_tens, order
integer coordinates(6,3)
real*8 geomdim(3)
real*8 field(res_x,res_y,res_z,vec_tens,3)
real*8 divergence_field(res_x,res_y,res_z,vec_tens)
integer i, j, k, m, l
real*8, dimension(4,4) :: FDcoefficient = reshape((/ & !from http://en.wikipedia.org/wiki/Finite_difference_coefficients
1.0/2.0, 0.0, 0.0, 0.0,&
2.0/3.0,-1.0/12.0, 0.0, 0.0,&
3.0/4.0,-3.0/20.0,1.0/ 60.0, 0.0,&
4.0/5.0,-1.0/ 5.0,4.0/105.0,-1.0/280.0/),&
(/4,4/))
divergence_field = 0.0
order = order + 1
do k = 0, res_z-1; do j = 0, res_y-1; do i = 0, res_x-1
do m = 1, order
coordinates(1,:) = mesh_location(mesh_index((/i+m,j,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
coordinates(2,:) = mesh_location(mesh_index((/i-m,j,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
coordinates(3,:) = mesh_location(mesh_index((/i,j+m,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
coordinates(4,:) = mesh_location(mesh_index((/i,j-m,k/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
coordinates(5,:) = mesh_location(mesh_index((/i,j,k+m/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
coordinates(6,:) = mesh_location(mesh_index((/i,j,k-m/),(/res_x,res_y,res_z/)),(/res_x,res_y,res_z/)) + (/1,1,1/)
do l = 1, vec_tens
divergence_field(i+1,j+1,k+1,l) = divergence_field(i+1,j+1,k+1,l) + FDcoefficient(m,order) * &
((field(coordinates(1,1),coordinates(1,2),coordinates(1,3),l,1)- &
field(coordinates(2,1),coordinates(2,2),coordinates(2,3),l,1))*real(res_x)/geomdim(1) +&
(field(coordinates(3,1),coordinates(3,2),coordinates(3,3),l,2)- &
field(coordinates(4,1),coordinates(4,2),coordinates(4,3),l,2))*real(res_y)/geomdim(2) +&
(field(coordinates(5,1),coordinates(5,2),coordinates(5,3),l,3)- &
field(coordinates(6,1),coordinates(6,2),coordinates(6,3),l,3))*real(res_z)/geomdim(3))
enddo
enddo
enddo; enddo; enddo
end subroutine divergence