DAMASK_EICMD/code/DAMASK_spectral.f90

! Copyright 2011 Max-Planck-Institut für Eisenforschung GmbH
!
! This file is part of DAMASK,
! the Düsseldorf 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 PathToGeomFile/NameOfGeom.geom
!               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 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, err_defgrad_tol,&
                     itmax, memory_efficient, DAMASK_NumThreadsInt       
 use homogenization, only: materialpoint_sizeResults, materialpoint_results
!$ use OMP_LIB                                                          ! the openMP function library

 implicit none
 include 'include/fftw3.f' ! header file for fftw3 (declaring variables). Library files are also needed
                           ! compile FFTW 3.2.2 with ./configure --enable-threads
! variables to read from loadcase and geom file
 real(pReal), dimension(9) ::                      valuevector           ! stores information temporarily from loadcase file
 integer(pInt), parameter ::                       maxNchunksInput = 26  ! 5 identifiers, 18 values for the matrices and 3 scalars
 integer(pInt), dimension (1+maxNchunksInput*2) :: posInput
 integer(pInt), parameter ::                       maxNchunksGeom = 7    ! 4 identifiers, 3 values
 integer(pInt), dimension (1+2*maxNchunksGeom) ::  posGeom
 integer(pInt) unit, N_l, N_s, N_t, N_n, N_freq, N_Fdot                  ! numbers of identifiers
 character(len=1024) path, line
 logical gotResolution,gotDimension,gotHomogenization
 logical, dimension(9) :: bc_maskvector

! variables storing information from loadcase file
 real(pReal)                                    time, time0, timeinc    ! elapsed time, begin of interval, time interval
 real(pReal), dimension (:,:,:), allocatable :: bc_deformation, &       ! applied velocity gradient or time derivative of deformation gradient
                                                bc_stress               ! stress BC (if applicable)
 real(pReal), dimension(:), allocatable ::      bc_timeIncrement        ! length of increment
 integer(pInt)                                  N_Loadcases, step       ! ToDo: rename?
 integer(pInt), dimension(:), allocatable ::    bc_steps, &             ! number of steps
                                                bc_frequency            ! frequency of result writes
 logical, dimension(:), allocatable ::          bc_logscale, &          ! linear/logaritmic time step flag
                                                followFormeTrajectory,& ! follow trajectory of former loadcase
                                                velGradApplied          ! decide wether velocity gradient or fdot is given 
 logical, dimension(:,:,:,:), allocatable ::    bc_mask                 ! mask of boundary conditions

! variables storing information from geom file
 real(pReal) wgt
 real(pReal), dimension(3) ::  geomdimension
 integer(pInt) homog
 integer(pInt), dimension(3) :: resolution

! stress etc.
 real(pReal), dimension(3,3) ::                         ones, zeroes, temp33_Real, damper,&
                                                        pstress, pstress_av, cstress_av, defgrad_av,&
                                                        defgradAim, defgradAimOld, defgradAimCorr, defgradAimCorrPrev,&
                                                        mask_stress, mask_defgrad, deltaF                          
 real(pReal), dimension(3,3,3,3) ::                     dPdF, c0, s0, c0_temp
! real(pReal), dimension(9,9) ::                         s099                                  ! compliance in matrix notation
 real(pReal), dimension(6) ::                           cstress                                ! cauchy stress in Mandel notation
 real(pReal), dimension(6,6) ::                         dsde, c066, s066                       ! Mandel notation of 4th order tensors
 real(pReal), dimension(:,:,:,:,:), allocatable ::      workfft, defgrad, defgradold
 real(pReal), dimension(:,:,:,:), allocatable ::        coordinates
 
! variables storing information for spectral method
 complex(pReal) ::                                      img
 complex(pReal), dimension(3,3) ::                      temp33_Complex
 real(pReal), dimension(3,3) ::                         xiDyad
 real(pReal), dimension(:,:,:,:,:,:,:), allocatable ::  gamma_hat
 real(pReal), dimension(:,:,:,:), allocatable ::        xi
 integer(pInt), dimension(3) ::                         k_s
 integer*8, dimension(2) ::                             plan_fft
 
! loop variables, convergence etc.
 real(pReal) guessmode, err_div, err_stress, err_defgrad, p_hat_av        
 integer(pInt)  i, j, k, l, m, n, p
 integer(pInt)  loadcase, ielem, iter, calcmode, CPFEM_mode, ierr, doesNotConverge
 logical errmatinv
 
 real(pReal) temperature                                  ! not used, but needed for call to CPFEM_general

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

 bc_maskvector = .false.
 unit = 234_pInt
 ones = 1.0_pReal; zeroes = 0.0_pReal
 img = cmplx(0.0,1.0)
 N_l = 0_pInt
 N_s = 0_pInt
 N_t = 0_pInt
 time = 0.0_pReal
 N_n = 0_pInt
 N_freq = 0_pInt
 N_Fdot = 0_pInt
 doesNotConverge = 0_pInt
 gotResolution =.false.; gotDimension =.false.; gotHomogenization = .false.
 resolution = 1_pInt
 geomdimension = 0.0_pReal
 
 temperature = 300.0_pReal

 if (IargC() /= 2) call IO_error(102)                     ! check for correct number of given arguments

! Reading the loadcase file and assign variables
 path = getLoadcaseName()
 print '(a,/,a)', 'Loadcase:      ',trim(path)
 print '(a,/,a)', 'Workingdir:    ',trim(getSolverWorkingDirectoryName())
 print '(a,/,a)', 'SolverJobName: ',trim(getSolverJobName())

 if (.not. IO_open_file(unit,path)) call IO_error(30,ext_msg = path)
 
 rewind(unit)
 do
   read(unit,'(a1024)',END = 101) line
   if (IO_isBlank(line)) cycle                            ! skip empty lines
   posInput = IO_stringPos(line,maxNchunksInput)
   do i = 1, maxNchunksInput, 1
       select case (IO_lc(IO_stringValue(line,posInput,i)))
            case('l', 'velocitygrad')
                 N_l = N_l+1
            case('fdot')
                 N_Fdot = N_Fdot+1
            case('s', 'stress', 'pk1', 'piolakirchhoff')
                 N_s = N_s+1
            case('t', 'time', 'delta')
                 N_t = N_t+1
            case('n', 'incs', 'increments', 'steps', 'logincs', 'logsteps')
                 N_n = N_n+1
            case('f', 'freq', 'frequency')
                 N_freq = N_freq+1
        end select
   enddo                                                  ! count all identifiers to allocate memory and do sanity check
 enddo

101 N_Loadcases = N_n
if ((N_l + N_Fdot /= N_n).or.(N_n /= N_t)) &              ! sanity check
  call IO_error(46,ext_msg = path)                        ! error message for incomplete inp !ToDo:change message

! allocate memory depending on lines in input file
 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 (velGradApplied(N_Loadcases));            velGradApplied = .false.
 allocate (bc_timeIncrement(N_Loadcases));          bc_timeIncrement = 0.0_pReal
 allocate (bc_steps(N_Loadcases));                  bc_steps = 0_pInt
 allocate (bc_logscale(N_Loadcases));               bc_logscale = .false.
 allocate (bc_frequency(N_Loadcases));              bc_frequency = 1_pInt
 allocate (followFormeTrajectory(N_Loadcases));     followFormeTrajectory = .true.

 rewind(unit)
 loadcase = 0_pInt
 do
   read(unit,'(a1024)',END = 200) line
   if (IO_isBlank(line)) cycle                                                      ! skip empty lines
   loadcase = loadcase + 1
   posInput = IO_stringPos(line,maxNchunksInput)
   do j = 1,maxNchunksInput,2
     select case (IO_lc(IO_stringValue(line,posInput,j)))
       case('fdot')                                                               ! assign values for the deformation BC matrix (in case of given fdot)
         valuevector = 0.0_pReal
         forall (k = 1:9) bc_maskvector(k) = IO_stringValue(line,posInput,j+k) /= '*'
         do k = 1,9
           if (bc_maskvector(k)) valuevector(k) = IO_floatValue(line,posInput,j+k)
         enddo
         bc_mask(:,:,1,loadcase) = transpose(reshape(bc_maskvector,(/3,3/)))
         bc_deformation(:,:,loadcase) = math_transpose3x3(reshape(valuevector,(/3,3/)))
       case('l','velocitygrad')                                                     ! assign values for the deformation BC matrix (in case of given L)
         velGradApplied(loadcase) = .true.                                          ! in case of given L, set flag to true
         valuevector = 0.0_pReal
         forall (k = 1:9) bc_maskvector(k) = IO_stringValue(line,posInput,j+k) /= '*'
         do k = 1,9
           if (bc_maskvector(k)) valuevector(k) = IO_floatValue(line,posInput,j+k)
         enddo
         bc_mask(:,:,1,loadcase) = transpose(reshape(bc_maskvector,(/3,3/)))
         bc_deformation(:,:,loadcase) = math_transpose3x3(reshape(valuevector,(/3,3/)))
       case('s', 'stress', 'pk1', 'piolakirchhoff')
         valuevector = 0.0_pReal
         forall (k = 1:9) bc_maskvector(k) = IO_stringValue(line,posInput,j+k) /= '*'
         do k = 1,9
           if (bc_maskvector(k)) valuevector(k) = IO_floatValue(line,posInput,j+k)  ! assign values for the bc_stress matrix
         enddo
         bc_mask(:,:,2,loadcase) = transpose(reshape(bc_maskvector,(/3,3/)))
         bc_stress(:,:,loadcase) = math_transpose3x3(reshape(valuevector,(/3,3/)))
       case('t','time','delta')                                            ! increment time
           bc_timeIncrement(loadcase) = IO_floatValue(line,posInput,j+1)
       case('n','incs','increments','steps')                               ! bc_steps
           bc_steps(loadcase) = IO_intValue(line,posInput,j+1)
       case('logincs','logsteps')                                          ! true, if log scale
           bc_steps(loadcase) = IO_intValue(line,posInput,j+1)
           bc_logscale(loadcase) = .true.
       case('f','freq','frequency')                                        ! frequency of result writings
           bc_frequency(loadcase) = IO_intValue(line,posInput,j+1)                
       case('guessreset','dropguessing')
           followFormeTrajectory(loadcase) = .false.                       ! do not continue to predict deformation along former trajectory
     end select
 enddo; enddo

200 close(unit)

 if (followFormeTrajectory(1)) then
   call IO_warning(33,loadcase)                 ! cannot guess along trajectory for first step of first loadcase
   followFormeTrajectory(1) = .false.
 endif
 
 do loadcase = 1, N_Loadcases                                                      ! consistency checks and output
   print *, '------------------------------------------------------'
   print '(a,i5)', 'Loadcase:', loadcase
   if (.not. followFormeTrajectory(loadcase)) &
     print '(a)', 'drop guessing along trajectory'
   if (any(bc_mask(:,:,1,loadcase) == bc_mask(:,:,2,loadcase)))&
     call IO_error(31,loadcase)                                                    ! exclusive or masking only
   if (velGradApplied(loadcase)) then
     do j = 1, 3
       if (any(bc_mask(j,:,1,loadcase) == .true.) .and.&
           any(bc_mask(j,:,1,loadcase) == .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)/))','L:'        ,math_transpose3x3(bc_deformation(:,:,loadcase))
     print '(a,/,3(3(l,x)/))',    'bc_mask for L:',transpose(bc_mask(:,:,1,loadcase))
   else
     print '(a,/,3(3(f12.6,x)/))','Fdot:'       ,math_transpose3x3(bc_deformation(:,:,loadcase))
     print '(a,/,3(3(l,x)/))',    'bc_mask for Fdot:',transpose(bc_mask(:,:,1,loadcase))
   endif
   print '(a,/,3(3(f12.6,x)/))','bc_stress:',math_transpose3x3(bc_stress(:,:,loadcase))
   print '(a,/,3(3(l,x)/))',    'bc_mask for stress:'     ,transpose(bc_mask(:,:,2,loadcase))
   if (bc_timeIncrement(loadcase) < 0.0_pReal) call IO_error(34,loadcase)                 ! negative time increment
   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,i6)','incs: ',bc_steps(loadcase)
   if (bc_frequency(loadcase) < 1_pInt) call IO_error(36,loadcase)                        ! non-positive result frequency
   print '(a,i6)','freq: ',bc_frequency(loadcase)
 enddo

!read header of geom file to get the information needed before the complete geom file is intepretated by mesh.f90
 path = getModelName()
 print *, '------------------------------------------------------'
 print '(a,a)', 'GeomName: ',trim(path)
 if (.not. IO_open_file(unit,trim(path)//InputFileExtension)) call IO_error(101,ext_msg = trim(path)//InputFileExtension)

 rewind(unit)
 do
   read(unit,'(a1024)',END = 100) line
   if (IO_isBlank(line)) cycle                            ! skip empty lines
   posGeom = IO_stringPos(line,maxNchunksGeom)             

   select case ( IO_lc(IO_StringValue(line,posGeom,1)) )
     case ('dimension')
       gotDimension = .true.
       do i = 2,6,2
         select case (IO_lc(IO_stringValue(line,posGeom,i)))
           case('x')
              geomdimension(1) = IO_floatValue(line,posGeom,i+1)
           case('y')
              geomdimension(2) = IO_floatValue(line,posGeom,i+1)
           case('z')
              geomdimension(3) = IO_floatValue(line,posGeom,i+1)
         end select
       enddo
     case ('homogenization')
       gotHomogenization = .true.
       homog = IO_intValue(line,posGeom,2)
     case ('resolution')
       gotResolution = .true.
       do i = 2,6,2
         select case (IO_lc(IO_stringValue(line,posGeom,i)))
           case('a')
             resolution(1) = IO_intValue(line,posGeom,i+1)
           case('b')
             resolution(2) = IO_intValue(line,posGeom,i+1)
           case('c')
             resolution(3) = IO_intValue(line,posGeom,i+1)
         end select
       enddo
   end select
   if (gotDimension .and. gotHomogenization .and. gotResolution) exit
 enddo
 100 close(unit)
 
 if(mod(resolution(1),2)/=0 .or. mod(resolution(2),2)/=0 .or. mod(resolution(3),2)/=0)  call IO_error(103)
 
 print '(a,/,i4,i4,i4)','resolution a b c:', resolution
 print '(a,/,f8.4,f8.5,f8.5)','dimension x y z:', geomdimension
 print '(a,i4)','homogenization: ',homog
 
 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 (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(temperature,1_pInt,1_pInt)
 ielem = 0_pInt
 c066 = 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,0.0_pReal,ielem,1_pInt,cstress,dsde,pstress,dPdF)
   c066 = c066 + dsde
 enddo; enddo; enddo
 c066 = c066 * wgt
 c0 = math_mandel66to3333(c066)                       ! linear reference material stiffness
 call math_invert(6, c066, s066,i, errmatinv)         ! ToDo
 if(errmatinv) call IO_error(800)                     ! Matrix inversion error ToDo
 s0 = math_mandel66to3333(s066)                       ! ToDo
 
 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)*2*pi/geomdimension(3) ! 3D case  
                              xi(2,i,j,k) = real(k_s(2), pReal)*2*pi/geomdimension(2)
                              xi(1,i,j,k) = real(k_s(1), pReal)*2*pi/geomdimension(1)
  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, 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) 
 call dfftw_init_threads(ierr)
 if(ierr == 0_pInt) call IO_error(104,ierr)
 call dfftw_plan_with_nthreads(DAMASK_NumThreadsInt) 
 call dfftw_plan_many_dft_r2c(plan_fft(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_PATIENT)   
 call dfftw_plan_many_dft_c2r(plan_fft(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_PATIENT)

! write header of output file
 open(538,file=trim(getSolverWorkingDirectoryName())//trim(getSolverJobName())&
                                                    //'.spectralOut',form='UNFORMATTED')       
 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 (false: linear, true: log)
 write(538), 'frequencies', bc_frequency                                   ! one entry per loadcase
 write(538), 'times', bc_timeIncrement                                     ! one entry per loadcase
 bc_steps(1) = bc_steps(1)+1                                               ! +1 to store initial situation
 write(538), 'increments', bc_steps                                        ! one entry per loadcase
 bc_steps(1) = bc_steps(1)-1                                               ! re-adjust for correct looping
 write(538), 'eoh'                                                         ! end of header

 write(538)  materialpoint_results(:,1,:)                                  ! initial (non-deformed) results
! Initialization done

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

   if (followFormeTrajectory(loadcase)) then
     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))
   damper = ones/10
   deltaF = 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) == .true.) 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+bc_timeIncrement(loadcase)/time0)**( step   *1.0/(bc_steps(loadcase))))  &
                               - ((1.0+bc_timeIncrement(loadcase)/time0)**((step-1)*1.0/(bc_steps(loadcase)))) )
        endif
     else                                                                                         ! linear scale
        timeinc = bc_timeIncrement(loadcase)/bc_steps(loadcase)
     endif
     
     time = time + timeinc

! update macroscopic deformation gradient (defgrad BC)
     
     if (velGradApplied(loadcase)) &                                                 ! calculate deltaF from given L and current F
       deltaF = math_mul33x33(bc_deformation(:,:,loadcase), defgradAim)

     temp33_Real = defgradAim                                            
     defgradAim = defgradAim &                                                                         
                + guessmode * mask_stress * (defgradAim - defgradAimOld) &      
                + mask_defgrad * deltaF * 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 (velGradApplied(loadcase)) &       ! using velocity gradient to calculate new deformation gradient (if not guessing)
         deltaF = math_mul33x33(bc_deformation(:,:,loadcase),defgradold(i,j,k,:,:))
       defgrad(i,j,k,:,:) = defgrad(i,j,k,:,:) &        ! decide if guessing along former trajectory or apply homogeneous addon (addon only for applied deformation)
                            + guessmode * (defgrad(i,j,k,:,:) - defgradold(i,j,k,:,:))&
                            + (1.0_pReal-guessmode) * mask_defgrad * deltaF *timeinc
       defgradold(i,j,k,:,:) = temp33_Real   
     enddo; enddo; enddo

     guessmode = 1.0_pReal                              ! keep guessing along former trajectory during same loadcase

     if(all(bc_mask(:,:,1,loadcase))) then
       calcmode = 1_pInt                                ! if no stress BC is given (calmode 0 is not needed)
     else
       calcmode = 0_pInt                                ! start calculation of BC fulfillment
     endif

     CPFEM_mode = 1_pInt                                ! winding forward
     iter = 0_pInt
     err_div= 2.0_pReal * err_div_tol                   ! go into loop 
     defgradAimCorr = 0.0_pReal                         ! reset damping calculation
     damper = damper * 0.9_pReal
    
!*************************************************************
! convergence loop
     do while(iter < itmax .and. &
             (err_div     > err_div_tol    .or. &
              err_stress  > err_stress_tol .or. &
              err_defgrad > err_defgrad_tol))
       iter = iter + 1_pInt
       if (iter == itmax) doesNotConverge = doesNotConverge + 1
       print*, ' '
       print '(3(A,I5.5,tr2))', ' Loadcase = ',loadcase, ' Step = ',step, ' Iteration = ',iter 
       cstress_av = 0.0_pReal
       workfft = 0.0_pReal                               ! needed because of the padding for FFTW
!*************************************************************
       
! adjust defgrad to fulfill BCs 
       select case (calcmode)
       case (0)
         print *, '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,timeinc,ielem,1_pInt,&
                              cstress,dsde, pstress, dPdF)
         enddo; enddo; enddo
         
       ! c0_temp = 0.0_pReal    !for calculation of s0 ToDo
         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,timeinc,ielem,1_pInt,&
                              cstress,dsde, pstress, dPdF)
           CPFEM_mode = 2_pInt
         ! c0_temp = c0_temp + dPdF                                         ToDo
           workfft(i,j,k,:,:) = pstress                                     ! build up average P-K stress 
           cstress_av = cstress_av + math_mandel6to33(cstress)              ! build up average Cauchy stress
         enddo; enddo; enddo

       ! call math_invert(9, math_plain3333to99(c0_temp),s099,i,errmatinv)        ToDo       
       ! if(errmatinv) call IO_error(800,ext_msg = "problem in c0 inversion") ToDo
       ! s0 = math_plain99to3333(s099) *real(resolution(1)*resolution(2)*resolution(3), pReal) ! average s0 for calculation of BC ToDo
         
         cstress_av = cstress_av * wgt
         do n = 1,3; do m = 1,3
           pstress_av(m,n) = sum(workfft(1:resolution(1),1:resolution(2),1:resolution(3),m,n)) * wgt
           defgrad_av(m,n) = sum(defgrad(1:resolution(1),1:resolution(2),1:resolution(3),m,n)) * wgt
         enddo; enddo

         err_stress = maxval(abs(mask_stress * (pstress_av - bc_stress(:,:,loadcase))))
         err_stress_tol = maxval(abs(pstress_av))*err_stress_tolrel
         
         print*, 'Correcting deformation gradient to fullfill BCs'
         defgradAimCorrPrev = defgradAimCorr
         defgradAimCorr     = -mask_stress * math_mul3333xx33(s0, (mask_stress*(pstress_av - bc_stress(:,:,loadcase))))

         do m=1,3; do n =1,3                                        ! calculate damper (correction is far too strong) !ToDo: Check for better values
           if ( sign(1.0_pReal,defgradAimCorr(m,n))/=sign(1.0_pReal,defgradAimCorrPrev(m,n))) then
             damper(m,n) = max(0.01_pReal,damper(m,n)*0.8)
           else
             damper(m,n) = min(1.0_pReal,damper(m,n) *1.2)
           endif
         enddo; enddo
         defgradAimCorr = mask_Stress*(damper * defgradAimCorr)
         defgradAim = defgradAim + defgradAimCorr

         do m = 1,3; do n = 1,3
           defgrad(:,:,:,m,n) = defgrad(:,:,:,m,n) + (defgradAim(m,n) - defgrad_av(m,n)) ! anticipated target minus current state
         enddo; enddo
         err_div = 2.0_pReal * err_div_tol
         err_defgrad = maxval(abs(mask_defgrad * (defgrad_av - defgradAim)))
         print '(a,/,3(3(f12.7,x)/))', ' Deformation Gradient:',math_transpose3x3(defgrad_av)
         print '(a,/,3(3(f10.4,x)/))', ' Piola-Kirchhoff Stress / MPa: ',math_transpose3x3(pstress_av)/1.e6
         print '(2(a,E8.2))', ' error stress:               ',err_stress, '  Tol. = ', err_stress_tol*0.8
         print '(2(a,E8.2))', ' error deformation gradient: ',err_defgrad,'  Tol. = ', err_defgrad_tol
         if(err_stress < err_stress_tol*0.8) then 
           calcmode = 1_pInt
         endif
             
! Using the spectral method to calculate the change of deformation gradient, check divergence of stress field in fourier space
       case (1)
         print *, '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_pInt
           call CPFEM_general(3, coordinates(1:3,i,j,k), defgradold(i,j,k,:,:), defgrad(i,j,k,:,:),&
                              temperature,timeinc,ielem,1_pInt,&
                              cstress,dsde, pstress, dPdF)
         enddo; enddo; enddo
         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,:,:),&
                              temperature,timeinc,ielem,1_pInt,&
                              cstress,dsde, pstress, dPdF)
           CPFEM_mode = 2_pInt
           workfft(i,j,k,:,:) = pstress 
           cstress_av = cstress_av + math_mandel6to33(cstress)          
         enddo; enddo; enddo       
         cstress_av = cstress_av * wgt
         do n = 1,3; do m = 1,3
           pstress_av(m,n) = sum(workfft(1:resolution(1),1:resolution(2),1:resolution(3),m,n)) * wgt
         enddo; enddo
         
         print *, 'Calculating equilibrium using spectral method'
         err_div = 0.0_pReal; p_hat_av = 0.0_pReal
         call dfftw_execute_dft_r2c(plan_fft(1),workfft,workfft)           ! FFT of pstress
         do m = 1,3                                                        ! L infinity Norm of stress tensor 
           p_hat_av = max(p_hat_av, sum(abs(workfft(1,1,1,m,:) + workfft(2,1,1,m,:)*img)))     
         enddo
         do k = 1, resolution(3); do j = 1, resolution(2); do i = 1, resolution(1)/2+1 
           err_div = max(err_div, maxval(abs(math_mul33x3_complex(workfft(i*2-1,j,k,:,:)+& ! L infinity norm of div(stress)
                                                                  workfft(i*2,  j,k,:,:)*img,xi(:,i,j,k)))))
         enddo; enddo; enddo         
         err_div = err_div/p_hat_av                                        ! Criterion as supposed in Suquet 2001

         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, 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)                                 ! change of average strain
           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)                                 ! change of average strain
           workfft(i*2  ,j,k,:,:) = aimag(temp33_Complex) 
          enddo; enddo; enddo
         endif
         
         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(plan_fft(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) + mask_defgrad(m,n)*(defgradAim(m,n) - defgrad_av(m,n))  ! anticipated target minus current state on components with prescribed deformation
         enddo; enddo
         
         err_stress = maxval(abs(mask_stress * (pstress_av - bc_stress(:,:,loadcase))))
         err_stress_tol = maxval(abs(pstress_av))*err_stress_tolrel                       ! accecpt relative error specified
         err_defgrad = maxval(abs(mask_defgrad * (defgrad_av - defgradAim)))  
         
         print '(2(a,E8.2))', ' error divergence:           ',err_div,    '  Tol. = ', err_div_tol
         print '(2(a,E8.2))', ' error stress:               ',err_stress, '  Tol. = ', err_stress_tol 
         print '(2(a,E8.2))', ' error deformation gradient: ',err_defgrad,'  Tol. = ', err_defgrad_tol 

         if((err_stress > err_stress_tol .or. err_defgrad > err_defgrad_tol) .and. err_div < err_div_tol) then  ! change to calculation of BCs, reset damper etc.
           calcmode = 0_pInt
           defgradAimCorr = 0.0_pReal
           damper = damper * 0.9_pReal
         endif     
       end select 
     enddo    ! end looping when convergency is achieved 
     
     if (mod(step,bc_frequency(loadcase)) == 0_pInt) &                          ! at output frequency
       write(538)  materialpoint_results(:,1,:)                                 ! write result to file
     
     print '(a,x,f12.7)'         , ' Determinant of Deformation Aim: ', math_det3x3(defgradAim)
     print '(a,/,3(3(f12.7,x)/))', ' Deformation Aim:     ',math_transpose3x3(defgradAim)
     print '(a,/,3(3(f12.7,x)/))', ' Deformation Gradient:',math_transpose3x3(defgrad_av) 
     print '(a,/,3(3(f10.4,x)/))', ' Cauchy Stress / MPa: ',math_transpose3x3(cstress_av)/1.e6
     print '(a,/,3(3(f10.4,x)/))', ' Piola-Kirchhoff Stress / MPa: ',math_transpose3x3(pstress_av)/1.e6
     print '(A)', '************************************************************'
   enddo  ! end looping over steps in current loadcase
 enddo    ! end looping over loadcases
 print '(a,i10,a)', 'A Total of ', doesNotConverge, ' Steps did not converge!'
 close(538)
 call dfftw_destroy_plan(plan_fft(1)); call dfftw_destroy_plan(plan_fft(2))
    
end program DAMASK_spectral

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

 integer(pInt) id

 stop
end subroutine