DAMASK_EICMD/code/DAMASK_spectral.f90

! Copyright 2011 Max-Planck-Institut fuer Eisenforschung GmbH
!
! This file is part of DAMASK,
! the Duesseldorf Advanced MAterial Simulation Kit.
!
! DAMASK is free software: you can redistribute it and/or modify
! it under the terms of the GNU General Public License as published by
! the Free Software Foundation, either version 3 of the License, or
! (at your option) any later version.
!
! DAMASK is distributed in the hope that it will be useful,
! but WITHOUT ANY WARRANTY; without even the implied warranty of
! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
! GNU General Public License for more details.
!
! You should have received a copy of the GNU General Public License
! along with DAMASK. If not, see <http://www.gnu.org/licenses/>.
!
!##############################################################
!* $Id$
!********************************************************************
! Material subroutine for BVP solution using spectral method
!
! written by P. Eisenlohr,
!            F. Roters,
!            L. Hantcherli,
!            W.A. Counts,
!            D.D. Tjahjanto,
!            C. Kords,
!            M. Diehl,
!            R. Lebensohn
!
! MPI fuer Eisenforschung, Duesseldorf
!
!********************************************************************
!     Usage:
!             - start program with DAMASK_spectral
!                       -g (--geom, --geometry) PathToGeomFile/NameOfGeom.geom
!                       -l (--load, --loadcase) PathToLoadFile/NameOfLoadFile.load
!             - PathToGeomFile will be the working directory
!             - make sure the file "material.config" exists in the working
!               directory. For further configuration use "numerics.config"
!********************************************************************
program DAMASK_spectral
!********************************************************************

 use DAMASK_interface
 use prec, only: pInt, pReal
 use IO
 use IFPORT, only: fseek, ftellI8
 use debug, only: debug_Verbosity
 use math
 use mesh, only: mesh_ipCenterOfGravity
 use CPFEM, only: CPFEM_general, CPFEM_initAll 
 use numerics, only: err_div_tol, err_stress_tol, err_stress_tolrel,&
                     relevantStrain, itmax, memory_efficient, DAMASK_NumThreadsInt,&
                     fftw_planner_flag, fftw_timelimit
 use homogenization, only: materialpoint_sizeResults, materialpoint_results
!$ use OMP_LIB                                                           ! the openMP function library

 implicit none
! variables to read from loadcase and geom file
 real(pReal), dimension(9) ::                      valueVector                        ! stores information temporarily from loadcase file
 integer(pInt), parameter ::                       maxNchunksLoadcase = &
                                                   (1_pInt + 9_pInt)*2_pInt + &       ! deformation and stress
                                                   (1_pInt + 1_pInt)*4_pInt + &       ! time, (log)incs, temp, and frequency
                                                   1_pInt + 1_pInt +2_pInt*3_pInt + & ! rotation, keyword (deg,rad), axis + value
                                                   1                                  ! dropguessing
 integer(pInt), dimension (1 + maxNchunksLoadcase*2) :: posLoadcase
 integer(pInt), parameter :: maxNchunksGeom = 7                                       ! 4 identifiers, 3 values
 integer(pInt), dimension (1 + maxNchunksGeom*2) ::  posGeom
 integer(pInt) :: myUnit, N_l, N_s, N_t, N_n, N_Fdot, headerLength                    ! numbers of identifiers
 character(len=1024) :: path, line, keyword
 logical ::  gotResolution, gotDimension, gotHomogenization

! variables storing information from loadcase file
 real(pReal), dimension (:,:,:), allocatable :: bc_deformation, &           ! applied velocity gradient or time derivative of deformation gradient
                                                bc_stress, &                ! stress BC (if applicable)
                                                bc_rotation                 ! rotation of BC (if applicable)
 real(pReal), dimension(:), allocatable ::      bc_timeIncrement, &         ! length of increment
                                                bc_temperature              ! isothermal starting conditions
 integer(pInt), dimension(:), allocatable ::    bc_steps, &                 ! number of steps
                                                bc_frequency, &             ! frequency of result writes
                                                bc_logscale                 ! linear/logaritmic time step flag
 logical, dimension(:), allocatable ::          bc_followFormerTrajectory,& ! follow trajectory of former loadcase
                                                bc_velGradApplied           ! decide wether velocity gradient or fdot is given 
 logical, dimension(:,:,:,:), allocatable ::    bc_mask                     ! mask of boundary conditions
 logical, dimension(:,:,:), allocatable ::      bc_maskvector               ! linear mask of boundary conditions                  

! variables storing information from geom file
 real(pReal) :: wgt
 real(pReal), dimension(3) ::  geomdimension    ! physical dimension of volume element in each direction
 integer(pInt) :: homog                         ! homogenization scheme used
 integer(pInt), dimension(3) :: resolution      ! resolution (number of Fourier points) in each direction
 logical :: spectralPictureMode                 ! indicating 1 to 1 mapping of FP to microstructure

! stress etc.
 real(pReal), dimension(3,3) ::                         pstress, pstress_av, defgrad_av, &
                                                        defgradAim, defgradAimOld, defgradAimCorr,&
                                                        mask_stress, mask_defgrad, fDot                          
 real(pReal), dimension(3,3,3,3) ::                     dPdF, c0_reference, c_current, s_prev, c_prev ! stiffness and compliance
 real(pReal), dimension(6) ::                           cstress                                       ! cauchy stress
 real(pReal), dimension(6,6) ::                         dsde                                          ! small strain stiffness
 real(pReal), dimension(9,9) ::                         s_prev99, c_prev99                            ! compliance and stiffness in matrix notation 
 real(pReal), dimension(:,:), allocatable ::            s_reduced, c_reduced                          ! reduced compliance and stiffness (only for stress BC)
 integer(pInt) ::                                       size_reduced                                  ! number of stress BCs

! pointwise data 
 real(pReal), dimension(:,:,:,:,:), allocatable ::      workfft, defgrad, defgradold
 real(pReal), dimension(:,:,:,:), allocatable ::        coordinates
 real(pReal), dimension(:,:,:), allocatable ::          temperature

 
! variables storing information for spectral method
 real(pReal), dimension(3,3) ::                         xiDyad                                   ! product of wave vectors
 real(pReal), dimension(:,:,:,:,:,:,:), allocatable ::  gamma_hat                                ! gamma operator (field) for spectral method
 real(pReal), dimension(:,:,:,:), allocatable ::        xi                                       ! wave vector field 
 integer(pInt), dimension(3) ::                         k_s                                
 integer*8, dimension(2) ::                             fftw_plan                                ! plans for fftw (forward and backward)
 integer*8 ::                                           fftw_flag                                ! planner flag for fftw
 
! loop variables, convergence etc.
 real(pReal) :: time, time0, timeinc                                                             ! elapsed time, begin of interval, time interval 
 real(pReal) :: guessmode, err_div, err_stress, p_hat_avg
 complex(pReal), parameter ::                           img = cmplx(0.0,1.0)
 real(pReal), dimension(3,3), parameter ::              ones = 1.0_pReal, zeroes = 0.0_pReal
 complex(pReal), dimension(3,3) ::                      temp33_Complex
 real(pReal), dimension(3,3) ::                         temp33_Real
 integer(pInt) :: i, j, k, l, m, n, p
 integer(pInt) :: N_Loadcases, loadcase, step, iter, ielem, CPFEM_mode, ierr, notConvergedCounter, writtenOutCounter
 logical errmatinv
 integer*8 :: incPosition

!Initializing
!$ call omp_set_num_threads(DAMASK_NumThreadsInt)         ! set number of threads for parallel execution set by DAMASK_NUM_THREADS

 print*, ''
 print*, '<<<+-  DAMASK_spectral init  -+>>>'
 print*, '$Id$'
 print*, ''
 
 myUnit = 234_pInt

 N_l = 0_pInt
 N_Fdot = 0_pInt
 N_t = 0_pInt
 N_n = 0_pInt
 
 time = 0.0_pReal

 notConvergedCounter = 0_pInt
 resolution = 1_pInt
 geomdimension = 0.0_pReal
 
 if (command_argument_count() /= 4) call IO_error(102)         ! check for correct number of given arguments

! Reading the loadcase file and allocate variables
 path = getLoadcaseName()
 !$OMP CRITICAL (write2out)
 print '(a)', '******************************************************'
 print '(a,a)', 'Working Directory:    ',trim(getSolverWorkingDirectoryName())
 print '(a,a)', 'Solver Job Name:      ',trim(getSolverJobName())
 !$OMP END CRITICAL (write2out)
 if (.not. IO_open_file(myUnit,path)) call IO_error(30,ext_msg = trim(path))

 rewind(myUnit)
 do
   read(myUnit,'(a1024)',END = 100) line
   if (IO_isBlank(line)) cycle                                ! skip empty lines
   posLoadcase = IO_stringPos(line,maxNchunksLoadcase)
   do i = 1, maxNchunksLoadcase, 1                            ! reading compulsory parameters for loadcase
       select case (IO_lc(IO_stringValue(line,posLoadcase,i)))
            case('l', 'velocitygrad', 'velgrad')
                 N_l = N_l+1
            case('fdot')
                 N_Fdot = N_Fdot+1
            case('t', 'time', 'delta')
                 N_t = N_t+1
            case('n', 'incs', 'increments', 'steps', 'logincs', 'logsteps')
                 N_n = N_n+1
        end select
   enddo                                                      ! count all identifiers to allocate memory and do sanity check
 enddo

100 N_Loadcases = N_n
 if ((N_l + N_Fdot /= N_n) .or. (N_n /= N_t)) &               ! sanity check
   call IO_error(37,ext_msg = trim(path))                     ! error message for incomplete loadcase

 allocate (bc_deformation(3,3,N_Loadcases));        bc_deformation = 0.0_pReal
 allocate (bc_stress(3,3,N_Loadcases));             bc_stress = 0.0_pReal
 allocate (bc_mask(3,3,2,N_Loadcases));             bc_mask = .false.
 allocate (bc_maskvector(9,2,N_Loadcases));         bc_maskvector = .false.
 allocate (bc_velGradApplied(N_Loadcases));         bc_velGradApplied = .false.
 allocate (bc_timeIncrement(N_Loadcases));          bc_timeIncrement = 0.0_pReal
 allocate (bc_temperature(N_Loadcases));            bc_temperature = 300.0_pReal
 allocate (bc_steps(N_Loadcases));                  bc_steps = 0_pInt
 allocate (bc_logscale(N_Loadcases));               bc_logscale = 0_pInt
 allocate (bc_frequency(N_Loadcases));              bc_frequency = 1_pInt
 allocate (bc_followFormerTrajectory(N_Loadcases)); bc_followFormerTrajectory = .true.
 allocate (bc_rotation(3,3,N_Loadcases));           bc_rotation = 0.0_pReal

 rewind(myUnit)
 loadcase = 0_pInt
 do
   read(myUnit,'(a1024)',END = 101) line
   if (IO_isBlank(line)) cycle                                                    ! skip empty lines
   loadcase = loadcase + 1
   bc_rotation(:,:,loadcase) = math_I3                                            ! assume no rotation, overwrite later in case rotation of loadcase is given
   posLoadcase = IO_stringPos(line,maxNchunksLoadcase)
   do j = 1,maxNchunksLoadcase
     select case (IO_lc(IO_stringValue(line,posLoadcase,j)))
       case('fdot','l','velocitygrad')                                            ! assign values for the deformation BC matrix
         bc_velGradApplied(loadcase) = (IO_lc(IO_stringValue(line,posLoadcase,j)) == 'l' .or. &
                                     IO_lc(IO_stringValue(line,posLoadcase,j)) == 'velocitygrad')         ! in case of given L, set flag to true
         valueVector = 0.0_pReal
         forall (k = 1:9) bc_maskvector(k,1,loadcase) = IO_stringValue(line,posLoadcase,j+k) /= '*'
         do k = 1,9
           if (bc_maskvector(k,1,loadcase)) valueVector(k) = IO_floatValue(line,posLoadcase,j+k)
         enddo
         bc_mask(:,:,1,loadcase) = transpose(reshape(bc_maskvector(1:9,1,loadcase),(/3,3/)))
         bc_deformation(:,:,loadcase) = math_plain9to33(valueVector)
       case('s', 'stress', 'pk1', 'piolakirchhoff')
         valueVector = 0.0_pReal
         forall (k = 1:9) bc_maskvector(k,2,loadcase) = IO_stringValue(line,posLoadcase,j+k) /= '*'
         do k = 1,9
           if (bc_maskvector(k,2,loadcase)) valueVector(k) = IO_floatValue(line,posLoadcase,j+k)  ! assign values for the bc_stress matrix
         enddo
         bc_mask(:,:,2,loadcase) = transpose(reshape(bc_maskvector(1:9,2,loadcase),(/3,3/)))
         bc_stress(:,:,loadcase) = math_plain9to33(valueVector)
       case('t','time','delta')                                            ! increment time
         bc_timeIncrement(loadcase) = IO_floatValue(line,posLoadcase,j+1)
       case('temp','temperature')                                          ! starting temperature
         bc_temperature(loadcase) = IO_floatValue(line,posLoadcase,j+1)
       case('n','incs','increments','steps')                               ! bc_steps
         bc_steps(loadcase) = IO_intValue(line,posLoadcase,j+1)
       case('logincs','logsteps')                                          ! true, if log scale
         bc_steps(loadcase) = IO_intValue(line,posLoadcase,j+1)
         bc_logscale(loadcase) = 1_pInt
       case('f','freq','frequency')                                        ! frequency of result writings
         bc_frequency(loadcase) = IO_intValue(line,posLoadcase,j+1)                
       case('guessreset','dropguessing')
         bc_followFormerTrajectory(loadcase) = .false.                     ! do not continue to predict deformation along former trajectory
       case('rotation','rot','euler')
         p = 0_pInt                                                        ! assuming values given in radians
         k = 1_pInt                                                        ! assuming keyword indicating degree/radians
         select case (IO_lc(IO_stringValue(line,posLoadcase,j+1)))
           case('deg','degree')
             p = 1_pInt                                                    ! for conversion from degree to radian           
           case('rad','radian') 
           case default               
             k = 0_pInt                                                    ! imediately reading in angles, assuming radians
         end select
         do l = 0,4,2                                                      ! looping to find keywords
           select case (IO_lc(IO_stringValue(line,posLoadcase,j+1 +k +l)))
             case('x')
               temp33_Real(1,1) = IO_floatValue(line,posLoadcase,j+1 +k +l +1) * real(p,pReal) * inRad
             case('y')
               temp33_Real(2,1) = IO_floatValue(line,posLoadcase,j+1 +k +l +1) * real(p,pReal) * inRad 
             case('z')
               temp33_Real(3,1) = IO_floatValue(line,posLoadcase,j+1 +k +l +1) * real(p,pReal) * inRad
           end select
         enddo
         bc_rotation(:,:,loadcase) = math_EulerToR(temp33_Real(:,1))
     end select
 enddo; enddo

101 close(myUnit)

!read header of geom file to get the information needed before the complete geom file is intepretated by mesh.f90
 gotResolution =.false.
 gotDimension =.false.
 gotHomogenization = .false.
 spectralPictureMode = .false.
 
 path = getModelName()
 !$OMP CRITICAL (write2out)
 print '(a)', '******************************************************'
 print '(a,a)', 'Geom File Name:       ',trim(path)//'.geom'
 print '(a)', '------------------------------------------------------'
 !$OMP END CRITICAL (write2out)
 if (.not. IO_open_file(myUnit,trim(path)//InputFileExtension))&
        call IO_error(101,ext_msg = trim(path)//InputFileExtension)

 rewind(myUnit)
 read(myUnit,'(a1024)') line
 posGeom = IO_stringPos(line,2)
 keyword = IO_lc(IO_StringValue(line,posGeom,2))
 if (keyword(1:4) == 'head') then
   headerLength = IO_intValue(line,posGeom,1) + 1_pInt
 else
   call IO_error(42)
 endif
 
 rewind(myUnit)
 do i = 1, headerLength
   read(myUnit,'(a1024)') line
   posGeom = IO_stringPos(line,maxNchunksGeom)             
   select case ( IO_lc(IO_StringValue(line,posGeom,1)) )
     case ('dimension')
       gotDimension = .true.
       do j = 2,6,2
         select case (IO_lc(IO_stringValue(line,posGeom,j)))
           case('x')
              geomdimension(1) = IO_floatValue(line,posGeom,j+1)
           case('y')
              geomdimension(2) = IO_floatValue(line,posGeom,j+1)
           case('z')
              geomdimension(3) = IO_floatValue(line,posGeom,j+1)
         end select
       enddo
     case ('homogenization')
       gotHomogenization = .true.
       homog = IO_intValue(line,posGeom,2)
     case ('resolution')
       gotResolution = .true.
       do j = 2,6,2
         select case (IO_lc(IO_stringValue(line,posGeom,j)))
           case('a')
             resolution(1) = IO_intValue(line,posGeom,j+1)
           case('b')
             resolution(2) = IO_intValue(line,posGeom,j+1)
           case('c')
             resolution(3) = IO_intValue(line,posGeom,j+1)
         end select
       enddo
     case ('picture')
       spectralPictureMode = .true.
   end select
 enddo
 close(myUnit)
 if (.not.(gotDimension .and. gotHomogenization .and. gotResolution)) call IO_error(45)
 
 if(mod(resolution(1),2_pInt)/=0_pInt .or.&
    mod(resolution(2),2_pInt)/=0_pInt .or.&
   (mod(resolution(3),2_pInt)/=0_pInt .and. resolution(3)/= 1_pInt))  call IO_error(103)

 !$OMP CRITICAL (write2out)
 print '(a,/,i12,i12,i12)','resolution a b c:', resolution
 print '(a,/,f12.5,f12.5,f12.5)','dimension x y z:', geomdimension
 print '(a,i5)','homogenization:      ',homog
 print '(a,L)','spectralPictureMode: ',spectralPictureMode
 print '(a)', '******************************************************'
 print '(a,a)','Loadcase File Name:   ',trim(getLoadcaseName())
 if (bc_followFormerTrajectory(1)) then
   call IO_warning(33)                 ! cannot guess along trajectory for first step of first loadcase
   bc_followFormerTrajectory(1) = .false.
 endif
! consistency checks and output of loadcase
 do loadcase = 1, N_Loadcases
   print '(a)', '------------------------------------------------------'
   print '(a,i5)', 'Loadcase:            ', loadcase
   if (.not. bc_followFormerTrajectory(loadcase)) &
     print '(a)', 'Drop Guessing Along Trajectory'
   if (any(bc_mask(:,:,1,loadcase) .eqv. bc_mask(:,:,2,loadcase)))&                ! exclusive or masking only
     call IO_error(31,loadcase)
     if (any(bc_mask(1:3,1:3,2,loadcase).and.transpose(bc_mask(1:3,1:3,2,loadcase)).and.& !checking if no rotation is allowed by stress BC
           reshape((/.false.,.true.,.true.,.true.,.false.,.true.,.true.,.true.,.false./),(/3,3/))))&
     call IO_error(38,loadcase)
   if (bc_velGradApplied(loadcase)) then
     do j = 1, 3
       if (any(bc_mask(j,:,1,loadcase) .eqv. .true.) .and.&
           any(bc_mask(j,:,1,loadcase) .eqv. .false.)) call IO_error(32,loadcase)     ! each line should be either fully or not at all defined
     enddo
     print '(a,/,3(3(f12.6,x)/))','Velocity Gradient:',    merge(math_transpose3x3(bc_deformation(:,:,loadcase)),&
                                              reshape(spread(DAMASK_NaN,1,9),(/3,3/)),&
                                              transpose(bc_mask(:,:,1,loadcase)))
   else
     print '(a,/,3(3(f12.6,x)/))','Change of Deformation Gradient:', merge(math_transpose3x3(bc_deformation(:,:,loadcase)),&
                                             reshape(spread(DAMASK_NaN,1,9),(/3,3/)),&
                                             transpose(bc_mask(:,:,1,loadcase)))
   endif
   print '(a,/,3(3(f12.6,x)/))','Stress Boundary Condition/MPa:',merge(math_transpose3x3(bc_stress(:,:,loadcase)),&
                                                 reshape(spread(DAMASK_NaN,1,9),(/3,3/)),&
                                                 transpose(bc_mask(:,:,2,loadcase)))*1e-6
   if (any(bc_rotation(:,:,loadcase)/=math_I3)) &
     print '(a,/,3(3(f12.6,x)/))','Rotation of BCs:',math_transpose3x3(bc_rotation(:,:,loadcase))
   if (bc_timeIncrement(loadcase) < 0.0_pReal) call IO_error(34,loadcase)                 ! negative time increment
   print '(a,f12.6)','Temperature: ',bc_temperature(loadcase)
   print '(a,f12.6)','Time:        ',bc_timeIncrement(loadcase)
   if (bc_steps(loadcase) < 1_pInt) call IO_error(35,loadcase)                            ! non-positive increment count
   print '(a,i5)','Increments:         ',bc_steps(loadcase)
   if (bc_frequency(loadcase) < 1_pInt) call IO_error(36,loadcase)                        ! non-positive result frequency
   print '(a,i5)','Freq. of Output:    ',bc_frequency(loadcase)
 enddo
 print '(a)', '******************************************************'
 !$OMP END CRITICAL (write2out)

 allocate (defgrad    (  resolution(1),resolution(2),resolution(3),3,3));  defgrad     = 0.0_pReal
 allocate (defgradold (  resolution(1),resolution(2),resolution(3),3,3));  defgradold  = 0.0_pReal
 allocate (coordinates(3,resolution(1),resolution(2),resolution(3)));      coordinates = 0.0_pReal
 allocate (temperature(  resolution(1),resolution(2),resolution(3)));      temperature = bc_temperature(1)  ! start out isothermally
 allocate (xi         (3,resolution(1)/2+1,resolution(2),resolution(3)));  xi          =0.0_pReal 

 wgt = 1.0_pReal/real(resolution(1)*resolution(2)*resolution(3), pReal)
 defgradAim    = math_I3
 defgradAimOld = math_I3
 defgrad_av    = math_I3
 
! Initialization of CPFEM_general (= constitutive law) and of deformation gradient field
 call CPFEM_initAll(bc_temperature(1),1_pInt,1_pInt)

 ielem = 0_pInt
 c_current = 0.0_pReal 
 do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)
   defgradold(i,j,k,:,:) = math_I3                       ! no deformation at the beginning
   defgrad(i,j,k,:,:) = math_I3 
   ielem = ielem +1 
   coordinates(1:3,i,j,k) = mesh_ipCenterOfGravity(1:3,1,ielem)  ! set to initial coordinates ToDo: SHOULD BE UPDATED TO CURRENT POSITION IN FUTURE REVISIONS!!!
   call CPFEM_general(2,coordinates(1:3,i,j,k),math_I3,math_I3,temperature(i,j,k),0.0_pReal,ielem,1_pInt,cstress,dsde,pstress,dPdF)
   c_current = c_current + dPdF
 enddo; enddo; enddo
 c0_reference = c_current * wgt  ! linear reference material stiffness
 c_prev = c0_reference
 
 if (debug_verbosity > 1) then
  !$OMP CRITICAL (write2out)
    write (6,*) 'First Call to CPFEM_general finished'
  !$OMP END CRITICAL (write2out)
 endif
 
 do k = 1, resolution(3)                              ! calculation of discrete angular frequencies, ordered as in FFTW (wrap around)
    k_s(3) = k-1
    if(k > resolution(3)/2+1) k_s(3) = k_s(3)-resolution(3)
    do j = 1, resolution(2)
      k_s(2) = j-1
      if(j > resolution(2)/2+1) k_s(2) = k_s(2)-resolution(2) 
      do i = 1, resolution(1)/2+1
        k_s(1) = i-1
        xi(3,i,j,k) = 0.0_pReal                                                  ! 2D case
        if(resolution(3) > 1) xi(3,i,j,k) = real(k_s(3), pReal)/geomdimension(3) ! 3D case  
                              xi(2,i,j,k) = real(k_s(2), pReal)/geomdimension(2)
                              xi(1,i,j,k) = real(k_s(1), pReal)/geomdimension(1)
 enddo; enddo; enddo
! remove highest frequencies for calculation of divergence (CAREFULL, they will be used for pre calculatet gamma operator!) 
 do k = 1,resolution(3); do j = 1,resolution(2); do i = 1,resolution(1)/2+1
   if(k==resolution(3)/2+1) xi(3,i,j,k)= 0.0_pReal
   if(j==resolution(2)/2+1) xi(2,i,j,k)= 0.0_pReal
   if(i==resolution(1)/2+1) xi(1,i,j,k)= 0.0_pReal
 enddo; enddo; enddo
 
 if(memory_efficient) then                            ! allocate just single fourth order tensor
   allocate (gamma_hat(1,1,1,3,3,3,3)); gamma_hat = 0.0_pReal
 else                                                 ! precalculation of gamma_hat field
   allocate (gamma_hat(resolution(1)/2+1,resolution(2),resolution(3),3,3,3,3)); gamma_hat = 0.0_pReal
   do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)/2+1
     if (any(xi(:,i,j,k) /= 0.0_pReal)) then     
       do l = 1,3; do m = 1,3
          xiDyad(l,m) = xi(l,i,j,k)*xi(m,i,j,k)
       enddo; enddo
       temp33_Real = math_inv3x3(math_mul3333xx33(c0_reference, xiDyad)) 
     else
        xiDyad  = 0.0_pReal
        temp33_Real = 0.0_pReal
     endif 
     do l=1,3; do m=1,3; do n=1,3; do p=1,3
       gamma_hat(i,j,k, l,m,n,p) = - 0.25*(temp33_Real(l,n)+temp33_Real(n,l)) *&
                                          (xiDyad(m,p)+xiDyad(p,m))
     enddo; enddo; enddo; enddo         
   enddo; enddo; enddo
 endif

 allocate (workfft(resolution(1)+2,resolution(2),resolution(3),3,3)); workfft = 0.0_pReal

! Initialization of fftw (see manual on fftw.org for more details)

 if(DAMASK_NumThreadsInt>0_pInt) then
   call dfftw_init_threads(ierr)
   if(ierr == 0_pInt) call IO_error(104,ierr)
   call dfftw_plan_with_nthreads(DAMASK_NumThreadsInt) 
 endif
 
 !call dfftw_timelimit(fftw_timelimit)
 select case(IO_lc(fftw_planner_flag))
   case('estimate','fftw_estimate')
     fftw_flag = 64
   case('measure','fftw_measure')
     fftw_flag = 0
   case('patient','fftw_patient')
     fftw_flag= 32
   case('exhaustive','fftw_exhaustive')
     fftw_flag = 8 
 end select
   
 call dfftw_plan_many_dft_r2c(fftw_plan(1),3,(/resolution(1),resolution(2),resolution(3)/),9,&
   workfft,(/resolution(1)  +2,resolution(2),resolution(3)/),1,(resolution(1)  +2)*resolution(2)*resolution(3),&
   workfft,(/resolution(1)/2+1,resolution(2),resolution(3)/),1,(resolution(1)/2+1)*resolution(2)*resolution(3),fftw_flag)   
 call dfftw_plan_many_dft_c2r(fftw_plan(2),3,(/resolution(1),resolution(2),resolution(3)/),9,&
   workfft,(/resolution(1)/2+1,resolution(2),resolution(3)/),1,(resolution(1)/2+1)*resolution(2)*resolution(3),&
   workfft,(/resolution(1)  +2,resolution(2),resolution(3)/),1,(resolution(1)  +2)*resolution(2)*resolution(3),fftw_flag)
 !$OMP CRITICAL (write2out)
 if (debug_verbosity > 1) then
    write (6,*) 'FFTW initialized'
 endif

! write header of output file
 open(538,file=trim(getSolverWorkingDirectoryName())//trim(getSolverJobName())&
                                                    //'.spectralOut',form='UNFORMATTED',status='REPLACE')!,access='DIRECT')
 write(538), 'load', trim(getLoadcaseName())
 write(538), 'workingdir', trim(getSolverWorkingDirectoryName())
 write(538), 'geometry', trim(getSolverJobName())//InputFileExtension
 write(538), 'resolution', resolution
 write(538), 'dimension', geomdimension
 write(538), 'materialpoint_sizeResults', materialpoint_sizeResults
 write(538), 'loadcases', N_Loadcases
 write(538), 'logscale', bc_logscale                                       ! one entry per loadcase (0: linear, 1: log)
 write(538), 'frequencies', bc_frequency                                   ! one entry per loadcase
 write(538), 'times', bc_timeIncrement                                     ! one entry per loadcase
 write(538), 'increments'                                                  ! one entry per loadcase
 incPosition = ftellI8(538)
 write(538), 0_pInt                                                        ! end of header
 write(538), 'eoh'                                                         ! end of header
 write(538),  materialpoint_results(:,1,:)                                  ! initial (non-deformed) results
 ierr = fseek(538,incPosition,0)
 if (ierr/=0_pInt ) call IO_error(107,ext_msg = 'inc 1')
 writtenOutCounter = 1_pInt
 write(538), writtenOutCounter                                                        ! 0 inc is written out
!$OMP END CRITICAL (write2out)
! Initialization done

!*************************************************************
! Loop over loadcases defined in the loadcase file
 do loadcase = 1, N_Loadcases
!*************************************************************
   time0 = time                                                            ! loadcase start time                

   if (bc_followFormerTrajectory(loadcase)) then                              ! continue to guess along former trajectory where applicable
     guessmode = 1.0_pReal
   else
     guessmode = 0.0_pReal                                                 ! change of load case, homogeneous guess for the first step
   endif
   
   mask_defgrad =  merge(ones,zeroes,bc_mask(:,:,1,loadcase))
   mask_stress =  merge(ones,zeroes,bc_mask(:,:,2,loadcase))
   size_reduced = count(bc_maskvector(1:9,2,loadcase))
   allocate (c_reduced(size_reduced,size_reduced));          c_reduced = 0.0_pReal
   allocate (s_reduced(size_reduced,size_reduced));          s_reduced = 0.0_pReal

   timeinc = bc_timeIncrement(loadcase)/bc_steps(loadcase)                ! only valid for given linear time scale. will be overwritten later in case loglinear scale is used
   fDot = bc_deformation(:,:,loadcase)                                    ! only valid for given fDot. will be overwritten later in case L is given
!*************************************************************
! loop oper steps defined in input file for current loadcase
   do step = 1, bc_steps(loadcase)
!*************************************************************
     if (bc_logscale(loadcase) == 1_pInt) then                                                    ! loglinear scale
        if (loadcase == 1_pInt) then                                                              ! 1st loadcase of loglinear scale            
            if (step == 1_pInt) then                                                              ! 1st step of 1st loadcase of loglinear scale
                timeinc = bc_timeIncrement(1)*(2.0**(1 - bc_steps(1)))                            ! assume 1st step is equal to 2nd 
            else                                                                                  ! not-1st step of 1st loadcase of loglinear scale
                timeinc = bc_timeIncrement(1)*(2.0**(step - (1 + bc_steps(1))))
            endif
        else                                                                                      ! not-1st loadcase of loglinear scale
            timeinc = time0 * (  ((1.0_pReal+bc_timeIncrement(loadcase)/time0)**(float( step   )/(bc_steps(loadcase))))  &
                               - ((1.0_pReal+bc_timeIncrement(loadcase)/time0)**(float((step-1))/(bc_steps(loadcase)))) )
        endif
     endif
     time = time + timeinc
     
     if (bc_velGradApplied(loadcase)) &                                                 ! calculate fDot from given L and current F
       fDot = math_mul33x33(bc_deformation(1:3,1:3,loadcase), defgradAim)

!winding forward of deformation aim
     temp33_Real = defgradAim                                            
     defgradAim = defgradAim &                                                                         
                + guessmode * mask_stress * (defgradAim - defgradAimOld) &      
                + mask_defgrad * fDot * timeinc 
     defgradAimOld = temp33_Real
 
! update local deformation gradient
     do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)
       temp33_Real = defgrad(i,j,k,:,:)
       if (bc_velGradApplied(loadcase)) &                                                  ! use velocity gradient to calculate new deformation gradient (if not guessing)
         fDot = math_mul33x33(bc_deformation(1:3,1:3,loadcase),defgradold(i,j,k,1:3,1:3))
         defgrad(i,j,k,1:3,1:3) = defgrad(i,j,k,1:3,1:3) &                                      ! decide if guessing along former trajectory or apply homogeneous addon
                            + guessmode * (defgrad(i,j,k,1:3,1:3) - defgradold(i,j,k,1:3,1:3))& ! guessing... 
                            + (1.0_pReal-guessmode) * mask_defgrad * fDot *timeinc    ! apply the prescribed value where deformation is given if not guessing
         defgradold(i,j,k,1:3,1:3) = temp33_Real   
     enddo; enddo; enddo
     guessmode = 1.0_pReal                                                              ! keep guessing along former trajectory during same loadcase

     CPFEM_mode = 1_pInt                                                                ! winding forward
     iter = 0_pInt
     err_div = 2.0_pReal * err_div_tol                                                  ! go into loop 

     if(size_reduced > 0_pInt) then                                                     ! calculate compliance in case stress BC is applied
       c_prev99 = math_Plain3333to99(c_prev)
       k = 0_pInt                                                                       ! build reduced stiffness
       do n = 1,9
         if(bc_maskvector(n,2,loadcase)) then
           k = k + 1_pInt
           j = 0_pInt
           do m = 1,9
             if(bc_maskvector(m,2,loadcase)) then
               j = j + 1_pInt
               c_reduced(k,j) = c_prev99(n,m)
       endif; enddo; endif; enddo
       call math_invert(size_reduced, c_reduced, s_reduced, i, errmatinv)               ! invert reduced stiffness
       if(errmatinv) call IO_error(800)
       s_prev99 = 0.0_pReal                                                             ! build full compliance
       k = 0_pInt
       do n = 1,9
         if(bc_maskvector(n,2,loadcase)) then
           k = k + 1_pInt
           j = 0_pInt
           do m = 1,9
           if(bc_maskvector(m,2,loadcase)) then
                 j = j + 1_pInt
                 s_prev99(n,m) = s_reduced(k,j)
       endif; enddo; endif; enddo
       s_prev = (math_Plain99to3333(s_prev99))
     endif
!*************************************************************
! convergence loop
     do while(iter < itmax .and. &
             (err_div     > err_div_tol    .or. &
              err_stress  > err_stress_tol)) 
       iter = iter + 1_pInt
       print '(A)', '************************************************************'
       print '(3(A,I5.5,tr2)A)', '**** Loadcase = ',loadcase, 'Step = ',step, 'Iteration = ',iter,'****'
       print '(A)', '************************************************************'
       workfft = 0.0_pReal                                                            ! needed because of the padding for FFTW
!*************************************************************
       do n = 1,3; do m = 1,3
         defgrad_av(m,n) = sum(defgrad(:,:,:,m,n)) * wgt
       enddo; enddo
       print '(a,/,3(3(f12.7,x)/))', 'Deformation Gradient:',math_transpose3x3(defgrad_av)
       
       print '(A,/)', '== Update Stress Field (Constitutive Evaluation P(F)) ======'
       ielem = 0_pInt
       do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)
         ielem = ielem + 1
         call CPFEM_general(3,&                                                       ! collect cycle
                            coordinates(1:3,i,j,k), defgradold(i,j,k,:,:), defgrad(i,j,k,:,:),&
                            temperature(i,j,k),timeinc,ielem,1_pInt,&
                            cstress,dsde, pstress, dPdF)
       enddo; enddo; enddo
       
       c_current = 0.0_pReal
       ielem = 0_pInt       
       do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)
         ielem = ielem + 1_pInt
         call CPFEM_general(CPFEM_mode,&                                              ! first element in first iteration retains CPFEM_mode 1, 
                            coordinates(1:3,i,j,k),&
                            defgradold(i,j,k,:,:), defgrad(i,j,k,:,:),&               ! others get 2 (saves winding forward effort)
                            temperature(i,j,k),timeinc,ielem,1_pInt,&
                            cstress,dsde, pstress,dPdF)
         CPFEM_mode = 2_pInt
         workfft(i,j,k,:,:) = pstress                                     ! build up average P-K stress 
         c_current = c_current + dPdF
       enddo; enddo; enddo
       
       do n = 1,3; do m = 1,3
         pstress_av(m,n) = sum(workfft(1:resolution(1),:,:,m,n)) * wgt
       enddo; enddo
       print '(a,/,3(3(f12.7,x)/))', 'Piola-Kirchhoff Stress / MPa: ',math_transpose3x3(pstress_av)/1.e6

       err_stress_tol = 0.0_pReal
       if(size_reduced > 0_pInt) then                                                                            ! calculate stress BC if applied
         err_stress = maxval(abs(mask_stress * (pstress_av - bc_stress(1:3,1:3,loadcase))))                      ! maximum deviaton (tensor norm not applicable)
         err_stress_tol = maxval(abs(mask_defgrad * pstress_av)) * err_stress_tolrel                             ! don't use any tensor norm because the comparison should be coherent
         print '(A,/)', '== Correcting Deformation Gradient to Fullfill BCs ========='
         print '(2(a,E10.5)/)', 'Error Stress = ',err_stress, ', Tol. = ', err_stress_tol 
         defgradAimCorr = - math_mul3333xx33(s_prev, ((pstress_av - bc_stress(1:3,1:3,loadcase))))               ! residual on given stress components
         defgradAim = defgradAim + defgradAimCorr
         print '(a,/,3(3(f12.7,x)/))', 'Deformation Aim:     ',math_transpose3x3(defgradAim)
         print '(a,x,f12.7,/)'       , 'Determinant of Deformation Aim: ', math_det3x3(defgradAim)
       endif
       print '(A,/)', '== Calculating Equilibrium Using Spectral Method ===========' 
       call dfftw_execute_dft_r2c(fftw_plan(1),workfft,workfft)                                                   ! FFT of pstress
 
       p_hat_avg = sqrt(maxval (math_eigenvalues3x3(math_mul33x33(workfft(1,1,1,1:3,1:3),&                       ! L_2 norm of average stress in fourier space,  
                                                math_transpose3x3(workfft(1,1,1,1:3,1:3))))))                    ! ignore imaginary part as it is always zero for real only input))
       err_div = 0.0_pReal
       do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)/2+1
         err_div = err_div + sqrt(sum((math_mul33x3_complex(workfft(i*2-1,j,k,1:3,1:3)+&                         ! avg of L_2 norm of div(stress) in fourier space (Suquet small strain)
                                                            workfft(i*2,  j,k,1:3,1:3)*img,xi(1:3,i,j,k)))**2.0))                                                    
       enddo; enddo; enddo

       err_div = err_div*wgt/p_hat_avg*(minval(geomdimension)*wgt**(-1/4))                                       ! weigthting,  multiplying by minimum dimension to get rid of dimension dependency and phenomenologigal factor wgt**(-1/4) to get rid of resolution dependency

       if(memory_efficient) then                                                                                 ! memory saving version, on-the-fly calculation of gamma_hat
         do k = 1, resolution(3); do j = 1, resolution(2) ;do i = 1, resolution(1)/2+1                         
           if (any(xi(:,i,j,k) /= 0.0_pReal)) then     
             do l = 1,3; do m = 1,3
               xiDyad(l,m) = xi(l,i,j,k)*xi(m,i,j,k)
             enddo; enddo
             temp33_Real = math_inv3x3(math_mul3333xx33(c0_reference, xiDyad)) 
           else
             xiDyad = 0.0_pReal
             temp33_Real = 0.0_pReal
           endif 
           do l=1,3; do m=1,3; do n=1,3; do p=1,3
             gamma_hat(1,1,1, l,m,n,p) = - 0.25_pReal*(temp33_Real(l,n)+temp33_Real(n,l))*&
                                                       (xiDyad(m,p) +xiDyad(p,m))
           enddo; enddo; enddo; enddo   
           do m = 1,3; do n = 1,3
             temp33_Complex(m,n) = sum(gamma_hat(1,1,1,m,n,:,:) *(workfft(i*2-1,j,k,:,:)&                 
                                                                 +workfft(i*2  ,j,k,:,:)*img))
           enddo; enddo
           workfft(i*2-1,j,k,:,:) = real (temp33_Complex) 
           workfft(i*2  ,j,k,:,:) = aimag(temp33_Complex)
         enddo; enddo; enddo
       else                                                                                                       ! use precalculated gamma-operator
         do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)/2+1
           do m = 1,3; do n = 1,3
             temp33_Complex(m,n) = sum(gamma_hat(i,j,k, m,n,:,:) *(workfft(i*2-1,j,k,:,:)&
                                                                 + workfft(i*2  ,j,k,:,:)*img))
           enddo; enddo
           workfft(i*2-1,j,k,:,:) = real (temp33_Complex)
           workfft(i*2  ,j,k,:,:) = aimag(temp33_Complex) 
         enddo; enddo; enddo
       endif

! average strain 
       workfft(1,1,1,:,:) = defgrad_av - math_I3                                        ! zero frequency (real part)
       workfft(2,1,1,:,:) = 0.0_pReal                                                   ! zero frequency (imaginary part)
       
       call dfftw_execute_dft_c2r(fftw_plan(2),workfft,workfft)
       defgrad = defgrad + workfft(1:resolution(1),:,:,:,:)*wgt
       do m = 1,3; do n = 1,3
         defgrad_av(m,n) = sum(defgrad(:,:,:,m,n))*wgt
         defgrad(:,:,:,m,n) = defgrad(:,:,:,m,n) + (defgradAim(m,n) - defgrad_av(m,n))  ! anticipated target minus current state
       enddo; enddo

       print '(2(a,E10.5)/)', 'Error Divergence = ',err_div,    ', Tol. = ', err_div_tol
       
     enddo    ! end looping when convergency is achieved 
     
     c_prev = c_current*wgt        ! calculate stiffness for next step
     
    ! if (mod(step,bc_frequency(loadcase)) == 0_pInt) then                          ! at output frequency
  !     ierr = fseek(538,0,2)
   !    if (ierr/=0_pInt ) call IO_error(107,ext_msg = 'searching backwards')
    !   write(538),  materialpoint_results(:,1,:)                                 ! write result to file
    !   writtenOutCounter = writtenOutCounter + 1_pInt 
     !  ierr = fseek(538,incPosition,0)
      ! if (ierr/=0_pInt ) call IO_error(107,ext_msg = 'searching Position of Inc')
     !  write(538), writtenOutCounter                                 ! write result to file
    ! endif
     if(err_div<=err_div_tol .and. err_stress<=err_stress_tol) then
       print '(2(A,I5.5),A,/)', '== Step = ',step, ' of Loadcase = ',loadcase, ' Converged =============='
     else
       print '(2(A,I5.5),A,/)', '== Step = ',step, ' of Loadcase = ',loadcase, ' NOT Converged =========='
       notConvergedCounter = notConvergedCounter + 1
     endif
   enddo  ! end looping over steps in current loadcase
   deallocate(c_reduced)
   deallocate(s_reduced)
   enddo    ! end looping over loadcases
   print '(A,/)', '############################################################'
   print '(a,i5.5,a)', 'A Total of ', notConvergedCounter, ' Steps did not Converge!'
   print '(a,i5.5,a)', 'A Total of ', writtenOutCounter, ' Steps are written to File!'
 close(538)
 call dfftw_destroy_plan(fftw_plan(1)); call dfftw_destroy_plan(fftw_plan(2))
    
end program DAMASK_spectral

!********************************************************************
! quit subroutine to satisfy IO_error
!
!********************************************************************
subroutine quit(id)
 use prec
 implicit none

 integer(pInt) id

 stop
end subroutine