DAMASK_EICMD/src/homogenization_RGC.f90

!--------------------------------------------------------------------------------------------------
!> @author Denny Tjahjanto, Max-Planck-Institut für Eisenforschung GmbH
!> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH
!> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH
!> @brief Relaxed grain cluster (RGC) homogenization scheme
!> Ngrains is defined as p x q x r (cluster)
!--------------------------------------------------------------------------------------------------
module homogenization_RGC
 use prec, only: &
   pReal, &
   pInt

 implicit none
 private
 integer(pInt),              dimension(:),       allocatable,        public :: &
   homogenization_RGC_sizeState, &
   homogenization_RGC_sizePostResults
 integer(pInt),              dimension(:,:),     allocatable,target, public :: &
   homogenization_RGC_sizePostResult
 character(len=64),          dimension(:,:),     allocatable,target, public :: &
   homogenization_RGC_output                                                                        ! name of each post result output
 integer(pInt),              dimension(:),       allocatable,target, public :: &
   homogenization_RGC_Noutput                                                                 !< number of outputs per homog instance
 integer(pInt),              dimension(:,:),     allocatable,        private :: &
   homogenization_RGC_Ngrains
 real(pReal),                dimension(:,:),     allocatable,        private :: &
   homogenization_RGC_dAlpha, &
   homogenization_RGC_angles
 real(pReal),                dimension(:,:,:,:), allocatable,        private :: &
   homogenization_RGC_orientation
 real(pReal),                dimension(:),       allocatable,        private :: &
   homogenization_RGC_xiAlpha, &
   homogenization_RGC_ciAlpha
 enum, bind(c) 
   enumerator :: undefined_ID, &
                 constitutivework_ID, &
                 penaltyenergy_ID, &
                 volumediscrepancy_ID, &
                 averagerelaxrate_ID,&
                 maximumrelaxrate_ID,&
                 ipcoords_ID,&
                 magnitudemismatch_ID,&
                 avgdefgrad_ID,&
                 avgfirstpiola_ID
 end enum
 integer(kind(undefined_ID)), dimension(:,:),     allocatable,       private :: &
   homogenization_RGC_outputID                                                                 !< ID of each post result output

 public :: &
   homogenization_RGC_init, &
   homogenization_RGC_partitionDeformation, &
   homogenization_RGC_averageStressAndItsTangent, &
   homogenization_RGC_updateState, &
   homogenization_RGC_postResults
 private :: &
   homogenization_RGC_stressPenalty, &
   homogenization_RGC_volumePenalty, &
   homogenization_RGC_grainDeformation, &
   homogenization_RGC_surfaceCorrection, &
   homogenization_RGC_equivalentModuli, &
   homogenization_RGC_relaxationVector, &
   homogenization_RGC_interfaceNormal, &
   homogenization_RGC_getInterface, &
   homogenization_RGC_grain1to3, &
   homogenization_RGC_grain3to1, &
   homogenization_RGC_interface4to1, &
   homogenization_RGC_interface1to4

contains

!--------------------------------------------------------------------------------------------------
!> @brief allocates all necessary fields, reads information from material configuration file
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_init(fileUnit)
#if defined(__GFORTRAN__) || __INTEL_COMPILER >= 1800
 use, intrinsic :: iso_fortran_env, only: &
   compiler_version, &
   compiler_options
#endif
 use prec, only: &
   pReal, &
   pInt 
 use debug, only: &
  debug_level, &
  debug_homogenization, &
  debug_levelBasic, &
  debug_levelExtensive
 use math, only: &
   math_Mandel3333to66,&
   math_Voigt66to3333, &
   math_I3, &
   math_sampleRandomOri,&
   math_EulerToR,&
   INRAD
 use mesh, only: &
   mesh_maxNips, &
   mesh_NcpElems,& 
   mesh_element, &
   FE_Nips, &
   FE_geomtype
 use IO
 use material
 use config_material

 implicit none
 integer(pInt), intent(in) :: fileUnit                                                              !< file pointer to material configuration
 integer(pInt), allocatable, dimension(:) :: chunkPos
 integer :: &
   homog, &
   NofMyHomog, &
   o, &
   instance, &
   sizeHState
 integer(pInt) :: section=0_pInt, maxNinstance, i,j,e, mySize, myInstance
 character(len=65536) :: &
   tag = '', &
   line = ''
 
 write(6,'(/,a)')   ' <<<+-  homogenization_'//HOMOGENIZATION_RGC_label//' init  -+>>>'
 write(6,'(/,a)')   ' Tjahjanto et al., International Journal of Material Forming, 2(1):939–942, 2009'
 write(6,'(/,a)')   ' https://doi.org/10.1007/s12289-009-0619-1'
 write(6,'(/,a)')   ' Tjahjanto et al., Modelling and Simulation in Materials Science and Engineering, 18:015006, 2010'
 write(6,'(/,a)')   ' https://doi.org/10.1088/0965-0393/18/1/015006'
 write(6,'(a15,a)') ' Current time: ',IO_timeStamp()
#include "compilation_info.f90"

 maxNinstance = int(count(homogenization_type == HOMOGENIZATION_RGC_ID),pInt)
 if (maxNinstance == 0_pInt) return
  if (iand(debug_level(debug_HOMOGENIZATION),debug_levelBasic) /= 0_pInt) &
   write(6,'(a16,1x,i5,/)') '# instances:',maxNinstance
 allocate(homogenization_RGC_sizeState(maxNinstance),                            source=0_pInt)
 allocate(homogenization_RGC_sizePostResults(maxNinstance),                      source=0_pInt)
 allocate(homogenization_RGC_Noutput(maxNinstance),                              source=0_pInt)
 allocate(homogenization_RGC_Ngrains(3,maxNinstance),                            source=0_pInt)
 allocate(homogenization_RGC_ciAlpha(maxNinstance),                              source=0.0_pReal)
 allocate(homogenization_RGC_xiAlpha(maxNinstance),                              source=0.0_pReal)
 allocate(homogenization_RGC_dAlpha(3,maxNinstance),                             source=0.0_pReal)
 allocate(homogenization_RGC_angles(3,maxNinstance),                             source=400.0_pReal)
 allocate(homogenization_RGC_output(maxval(homogenization_Noutput),maxNinstance))
                                                              homogenization_RGC_output=''
 allocate(homogenization_RGC_outputID(maxval(homogenization_Noutput),maxNinstance),source=undefined_ID)
 allocate(homogenization_RGC_sizePostResult(maxval(homogenization_Noutput),maxNinstance),&
                                                                                 source=0_pInt)
 allocate(homogenization_RGC_orientation(3,3,mesh_maxNips,mesh_NcpElems),        source=0.0_pReal)
   homogenization_RGC_orientation = spread(spread(math_I3,3,mesh_maxNips),4,mesh_NcpElems)          ! initialize to identity
 
 rewind(fileUnit)
 do while (trim(line) /= IO_EOF .and. IO_lc(IO_getTag(line,'<','>'))/=material_partHomogenization) ! wind forward to <homogenization>
   line = IO_read(fileUnit)
 enddo

 parsingFile: do while (trim(line) /= IO_EOF)                                                                    ! read through sections of homogenization part
   line = IO_read(fileUnit)
   if (IO_isBlank(line)) cycle                                                                      ! skip empty lines
   if (IO_getTag(line,'<','>') /= '') then                                                          ! stop at next part
     line = IO_read(fileUnit, .true.)                                                               ! reset IO_read
     exit                                                                                           
   endif
   if (IO_getTag(line,'[',']') /= '') then                                                          ! next section
     section = section + 1_pInt
     cycle
   endif
   if (section > 0_pInt ) then                                                                      ! do not short-circuit here (.and. with next if-statement). It's not safe in Fortran
     if (homogenization_type(section) == HOMOGENIZATION_RGC_ID) then                                ! one of my sections
       i = homogenization_typeInstance(section)                                                     ! which instance of my type is present homogenization
       chunkPos = IO_stringPos(line)
       tag = IO_lc(IO_stringValue(line,chunkPos,1_pInt))                                           ! extract key
       select case(tag)
         case ('(output)')
             homogenization_RGC_Noutput(i) = homogenization_RGC_Noutput(i) + 1_pInt
             homogenization_RGC_output(homogenization_RGC_Noutput(i),i) = &
                                                           IO_lc(IO_stringValue(line,chunkPos,2_pInt))         
           select case(IO_lc(IO_stringValue(line,chunkPos,2_pInt)))
             case('constitutivework')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = constitutivework_ID
             case('penaltyenergy')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = penaltyenergy_ID
             case('volumediscrepancy')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = volumediscrepancy_ID
             case('averagerelaxrate')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = averagerelaxrate_ID
             case('maximumrelaxrate')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = maximumrelaxrate_ID
             case('magnitudemismatch')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = magnitudemismatch_ID
             case('ipcoords')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = ipcoords_ID
             case('avgdefgrad','avgf')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = avgdefgrad_ID
             case('avgp','avgfirstpiola','avg1stpiola')
               homogenization_RGC_outputID(homogenization_RGC_Noutput(i),i) = avgfirstpiola_ID
             case default
               homogenization_RGC_Noutput(i) = homogenization_RGC_Noutput(i) -1_pInt  ! correct for invalid

           end select
         case ('clustersize')
           homogenization_RGC_Ngrains(1,i) = IO_intValue(line,chunkPos,2_pInt)
           homogenization_RGC_Ngrains(2,i) = IO_intValue(line,chunkPos,3_pInt)
           homogenization_RGC_Ngrains(3,i) = IO_intValue(line,chunkPos,4_pInt)
           if (homogenization_Ngrains(section) /= product(homogenization_RGC_Ngrains(1:3,i))) &
             call IO_error(211_pInt,ext_msg=trim(tag)//' ('//HOMOGENIZATION_RGC_label//')')
         case ('scalingparameter')
           homogenization_RGC_xiAlpha(i) = IO_floatValue(line,chunkPos,2_pInt)
         case ('overproportionality')
           homogenization_RGC_ciAlpha(i) = IO_floatValue(line,chunkPos,2_pInt)
         case ('grainsize')
           homogenization_RGC_dAlpha(1,i) = IO_floatValue(line,chunkPos,2_pInt)
           homogenization_RGC_dAlpha(2,i) = IO_floatValue(line,chunkPos,3_pInt)
           homogenization_RGC_dAlpha(3,i) = IO_floatValue(line,chunkPos,4_pInt)
         case ('clusterorientation')
           homogenization_RGC_angles(1,i) = IO_floatValue(line,chunkPos,2_pInt)
           homogenization_RGC_angles(2,i) = IO_floatValue(line,chunkPos,3_pInt)
           homogenization_RGC_angles(3,i) = IO_floatValue(line,chunkPos,4_pInt)

       end select
     endif  
   endif
 enddo parsingFile

!--------------------------------------------------------------------------------------------------
! * assigning cluster orientations
 elementLooping: do e = 1_pInt,mesh_NcpElems
   if (homogenization_type(mesh_element(3,e)) == HOMOGENIZATION_RGC_ID) then
     myInstance = homogenization_typeInstance(mesh_element(3,e))
     if (all (homogenization_RGC_angles(1:3,myInstance) >= 399.9_pReal)) then
       homogenization_RGC_orientation(1:3,1:3,1,e) = math_EulerToR(math_sampleRandomOri())
       do i = 1_pInt,FE_Nips(FE_geomtype(mesh_element(2,e)))
         if (microstructure_elemhomo(mesh_element(4,e))) then
           homogenization_RGC_orientation(1:3,1:3,i,e) = homogenization_RGC_orientation(1:3,1:3,1,e)
         else
           homogenization_RGC_orientation(1:3,1:3,i,e) = math_EulerToR(math_sampleRandomOri())
         endif
       enddo
     else
       do i = 1_pInt,FE_Nips(FE_geomtype(mesh_element(2,e)))
         homogenization_RGC_orientation(1:3,1:3,i,e) = &
                                   math_EulerToR(homogenization_RGC_angles(1:3,myInstance)*inRad)
       enddo
     endif
   endif
 enddo elementLooping

 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
   do i = 1_pInt,maxNinstance
     write(6,'(a15,1x,i4,/)')     'instance:  ', i
     write(6,'(a25,3(1x,i8))')    'cluster size:         ',(homogenization_RGC_Ngrains(j,i),j=1_pInt,3_pInt)
     write(6,'(a25,1x,e10.3)')    'scaling parameter:    ', homogenization_RGC_xiAlpha(i)
     write(6,'(a25,1x,e10.3)')    'over-proportionality: ', homogenization_RGC_ciAlpha(i)
     write(6,'(a25,3(1x,e10.3))') 'grain size:           ',(homogenization_RGC_dAlpha(j,i),j=1_pInt,3_pInt)
     write(6,'(a25,3(1x,e10.3))') 'cluster orientation:  ',(homogenization_RGC_angles(j,i),j=1_pInt,3_pInt)
   enddo
 endif
!--------------------------------------------------------------------------------------------------
 initializeInstances: do homog = 1_pInt, material_Nhomogenization
   myHomog: if (homogenization_type(homog) == HOMOGENIZATION_RGC_ID) then
     NofMyHomog = count(material_homog == homog)
     instance = homogenization_typeInstance(homog)

! *  Determine size of postResults array
     outputsLoop: do o = 1_pInt, homogenization_RGC_Noutput(instance)
       select case(homogenization_RGC_outputID(o,instance))
        case(constitutivework_ID,penaltyenergy_ID,volumediscrepancy_ID, &
            averagerelaxrate_ID,maximumrelaxrate_ID)
         mySize = 1_pInt
        case(ipcoords_ID,magnitudemismatch_ID)
         mySize = 3_pInt
        case(avgdefgrad_ID,avgfirstpiola_ID)
         mySize = 9_pInt
        case default
         mySize = 0_pInt
       end select

       outputFound: if (mySize > 0_pInt) then
        homogenization_RGC_sizePostResult(o,instance) = mySize
        homogenization_RGC_sizePostResults(instance) = &
          homogenization_RGC_sizePostResults(instance) + mySize
       endif outputFound
     enddo outputsLoop
     
     sizeHState = &
         3_pInt*(homogenization_RGC_Ngrains(1,instance)-1_pInt)* &
               homogenization_RGC_Ngrains(2,instance)*homogenization_RGC_Ngrains(3,instance) &
         + 3_pInt*homogenization_RGC_Ngrains(1,instance)*(homogenization_RGC_Ngrains(2,instance)-1_pInt)* &
               homogenization_RGC_Ngrains(3,instance) &
         + 3_pInt*homogenization_RGC_Ngrains(1,instance)*homogenization_RGC_Ngrains(2,instance)* &
              (homogenization_RGC_Ngrains(3,instance)-1_pInt) &
         + 8_pInt   ! (1) Average constitutive work, (2-4) Overall mismatch, (5) Average penalty energy, 
                    ! (6) Volume discrepancy, (7) Avg relaxation rate component, (8) Max relaxation rate component
                    
! allocate state arrays
     homogState(homog)%sizeState = sizeHState
     homogState(homog)%sizePostResults = homogenization_RGC_sizePostResults(instance)
     allocate(homogState(homog)%state0   (sizeHState,NofMyHomog), source=0.0_pReal)
     allocate(homogState(homog)%subState0(sizeHState,NofMyHomog), source=0.0_pReal)
     allocate(homogState(homog)%state    (sizeHState,NofMyHomog), source=0.0_pReal)

   endif myHomog
 enddo initializeInstances
 


end subroutine homogenization_RGC_init


!--------------------------------------------------------------------------------------------------
!> @brief partitions the deformation gradient onto the constituents
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_partitionDeformation(F,avgF,ip,el)
 use debug, only: &
   debug_level, &
   debug_homogenization, &
   debug_levelExtensive
 use mesh, only: &
   mesh_element
 use material, only: &
   homogenization_maxNgrains, &
   homogenization_Ngrains,&
   homogenization_typeInstance

 implicit none
 real(pReal),   dimension (3,3,homogenization_maxNgrains), intent(out) :: F                         !< partioned F  per grain
 real(pReal),   dimension (3,3),                           intent(in)  :: avgF                      !< averaged F
 integer(pInt),                                            intent(in)  :: &
   ip, &                                                                                            !< integration point number
   el                                                                                               !< element number
 real(pReal),   dimension (3) :: aVect,nVect
 integer(pInt), dimension (4) :: intFace
 integer(pInt), dimension (3) :: iGrain3
 integer(pInt) :: homID, iGrain,iFace,i,j
 integer(pInt),                                            parameter  :: nFace = 6_pInt

!--------------------------------------------------------------------------------------------------
! compute the deformation gradient of individual grains due to relaxations
 homID = homogenization_typeInstance(mesh_element(3,el))
 F = 0.0_pReal
 do iGrain = 1_pInt,homogenization_Ngrains(mesh_element(3,el))
   iGrain3 = homogenization_RGC_grain1to3(iGrain,homID)
   do iFace = 1_pInt,nFace
     intFace = homogenization_RGC_getInterface(iFace,iGrain3)                                       ! identifying 6 interfaces of each grain

     aVect = homogenization_RGC_relaxationVector(intFace,homID, ip, el)                               ! get the relaxation vectors for each interface from global relaxation vector array

     nVect = homogenization_RGC_interfaceNormal(intFace,ip,el)                                      ! get the normal of each interface
     forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) &
     F(i,j,iGrain) = F(i,j,iGrain) + aVect(i)*nVect(j)                                              ! calculating deformation relaxations due to interface relaxation
   enddo
   F(1:3,1:3,iGrain) = F(1:3,1:3,iGrain) + avgF                                                     ! resulting relaxed deformation gradient

!--------------------------------------------------------------------------------------------------
! debugging the grain deformation gradients
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a32,1x,i3)')'Deformation gradient of grain: ',iGrain
     do i = 1_pInt,3_pInt
       write(6,'(1x,3(e15.8,1x))')(F(i,j,iGrain), j = 1_pInt,3_pInt)
     enddo
     write(6,*)' '
     flush(6)
     !$OMP END CRITICAL (write2out)
   endif

 enddo

end subroutine homogenization_RGC_partitionDeformation


!--------------------------------------------------------------------------------------------------
!> @brief update the internal state of the homogenization scheme and tell whether "done" and 
! "happy" with result
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_updateState(P,F,F0,avgF,dt,dPdF,ip,el)
 use prec, only: &
   dEq0
 use debug, only: &
   debug_level, &
   debug_homogenization,&
   debug_levelExtensive, &
   debug_e, &
   debug_i
 use math, only: &
   math_invert
 use mesh, only: &
   mesh_element
 use material, only: &
   homogenization_maxNgrains, &
   homogenization_typeInstance, &
   homogState, &
   mappingHomogenization, &   
   homogenization_Ngrains
 use numerics, only: &
   absTol_RGC, &
   relTol_RGC, &
   absMax_RGC, &
   relMax_RGC, &
   pPert_RGC, &
   maxdRelax_RGC, &
   viscPower_RGC, &
   viscModus_RGC, &
   refRelaxRate_RGC

 implicit none

 real(pReal), dimension (3,3,homogenization_maxNgrains),     intent(in)    :: & 
   P,&                                                                                              !< array of P
   F,&                                                                                              !< array of F
   F0                                                                                               !< array of initial F
 real(pReal), dimension (3,3,3,3,homogenization_maxNgrains), intent(in) :: dPdF                     !< array of current grain stiffness
 real(pReal), dimension (3,3),                               intent(in) :: avgF                     !< average F
 real(pReal),                                                intent(in) :: dt                       !< time increment
 integer(pInt),                                              intent(in) :: &
   ip, &                                                                                            !< integration point number
   el                                                                                               !< element number
 logical, dimension(2)        :: homogenization_RGC_updateState
 integer(pInt), dimension (4) :: intFaceN,intFaceP,faceID
 integer(pInt), dimension (3) :: nGDim,iGr3N,iGr3P,stresLoc
 integer(pInt), dimension (2) :: residLoc
 integer(pInt) homID,iNum,i,j,nIntFaceTot,iGrN,iGrP,iMun,iFace,k,l,ipert,iGrain,nGrain
 real(pReal), dimension (3,3,homogenization_maxNgrains) :: R,pF,pR,D,pD
 real(pReal), dimension (3,homogenization_maxNgrains)   :: NN,pNN
 real(pReal), dimension (3)                             :: normP,normN,mornP,mornN
 real(pReal) :: residMax,stresMax,constitutiveWork,penaltyEnergy,volDiscrep
 logical error

 integer(pInt), parameter :: nFace = 6_pInt

 real(pReal), dimension(:,:), allocatable :: tract,jmatrix,jnverse,smatrix,pmatrix,rmatrix
 real(pReal), dimension(:), allocatable   :: resid,relax,p_relax,p_resid,drelax
 
 zeroTimeStep: if(dEq0(dt)) then
   homogenization_RGC_updateState = .true.                                                          ! pretend everything is fine and return
   return                                                                    
 endif zeroTimeStep

!--------------------------------------------------------------------------------------------------
! get the dimension of the cluster (grains and interfaces)
 homID  = homogenization_typeInstance(mesh_element(3,el))
 nGDim  = homogenization_RGC_Ngrains(1:3,homID)
 nGrain = homogenization_Ngrains(mesh_element(3,el))
 nIntFaceTot = (nGDim(1)-1_pInt)*nGDim(2)*nGDim(3) + nGDim(1)*(nGDim(2)-1_pInt)*nGDim(3) &
               + nGDim(1)*nGDim(2)*(nGDim(3)-1_pInt)

!--------------------------------------------------------------------------------------------------
! allocate the size of the global relaxation arrays/jacobian matrices depending on the size of the cluster
 allocate(resid(3_pInt*nIntFaceTot),   source=0.0_pReal)
 allocate(tract(nIntFaceTot,3),        source=0.0_pReal)
 allocate(relax(3_pInt*nIntFaceTot));   relax= homogState(mappingHomogenization(2,ip,el))% &
                                                state(1:3_pInt*nIntFaceTot,mappingHomogenization(1,ip,el))
 allocate(drelax(3_pInt*nIntFaceTot)); drelax= homogState(mappingHomogenization(2,ip,el))% &
                                                state(1:3_pInt*nIntFaceTot,mappingHomogenization(1,ip,el)) - &
                                               homogState(mappingHomogenization(2,ip,el))% &
                                                state0(1:3_pInt*nIntFaceTot,mappingHomogenization(1,ip,el))
!--------------------------------------------------------------------------------------------------
! debugging the obtained state
 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Obtained state: '
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,2(e15.8,1x))')homogState(mappingHomogenization(2,ip,el))%state(i,mappingHomogenization(1,ip,el))
   enddo
   write(6,*)' '
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! computing interface mismatch and stress penalty tensor for all interfaces of all grains
 call homogenization_RGC_stressPenalty(R,NN,avgF,F,ip,el,homID)

!--------------------------------------------------------------------------------------------------
! calculating volume discrepancy and stress penalty related to overall volume discrepancy 
 call homogenization_RGC_volumePenalty(D,volDiscrep,F,avgF,ip,el)

!--------------------------------------------------------------------------------------------------
! debugging the mismatch, stress and penalties of grains
 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   do iGrain = 1_pInt,nGrain
     write(6,'(1x,a30,1x,i3,1x,a4,3(1x,e15.8))')'Mismatch magnitude of grain(',iGrain,') :',&
       NN(1,iGrain),NN(2,iGrain),NN(3,iGrain)
     write(6,'(/,1x,a30,1x,i3)')'Stress and penalties of grain: ',iGrain
     do i = 1_pInt,3_pInt
       write(6,'(1x,3(e15.8,1x),1x,3(e15.8,1x),1x,3(e15.8,1x))')(P(i,j,iGrain), j = 1_pInt,3_pInt), &
                                                          (R(i,j,iGrain), j = 1_pInt,3_pInt), &
                                                          (D(i,j,iGrain), j = 1_pInt,3_pInt)
     enddo
     write(6,*)' '
   enddo
   !$OMP END CRITICAL (write2out)
 endif

!------------------------------------------------------------------------------------------------
! computing the residual stress from the balance of traction at all (interior) interfaces
 do iNum = 1_pInt,nIntFaceTot
   faceID = homogenization_RGC_interface1to4(iNum,homID)                                            ! identifying the interface ID in local coordinate system (4-dimensional index)

!--------------------------------------------------------------------------------------------------
! identify the left/bottom/back grain (-|N)
   iGr3N = faceID(2:4)                                                                              ! identifying the grain ID in local coordinate system (3-dimensional index)
   iGrN = homogenization_RGC_grain3to1(iGr3N,homID)                                                 ! translate the local grain ID into global coordinate system (1-dimensional index)
   intFaceN = homogenization_RGC_getInterface(2_pInt*faceID(1),iGr3N)
   normN = homogenization_RGC_interfaceNormal(intFaceN,ip,el)                                       ! get the interface normal
   
!--------------------------------------------------------------------------------------------------
! identify the right/up/front grain (+|P)
   iGr3P = iGr3N
   iGr3P(faceID(1)) = iGr3N(faceID(1))+1_pInt                                                       ! identifying the grain ID in local coordinate system (3-dimensional index)
   iGrP = homogenization_RGC_grain3to1(iGr3P,homID)                                                 ! translate the local grain ID into global coordinate system (1-dimensional index)
   intFaceP = homogenization_RGC_getInterface(2_pInt*faceID(1)-1_pInt,iGr3P)
   normP = homogenization_RGC_interfaceNormal(intFaceP,ip,el)                                       ! get the interface normal

!--------------------------------------------------------------------------------------------------
! compute the residual of traction at the interface (in local system, 4-dimensional index)
   do i = 1_pInt,3_pInt
     tract(iNum,i) = sign(viscModus_RGC*(abs(drelax(i+3*(iNum-1_pInt)))/(refRelaxRate_RGC*dt))**viscPower_RGC, &
                          drelax(i+3*(iNum-1_pInt)))                                                ! contribution from the relaxation viscosity
     do j = 1_pInt,3_pInt
       tract(iNum,i) = tract(iNum,i) + (P(i,j,iGrP) + R(i,j,iGrP) + D(i,j,iGrP))*normP(j) &         ! contribution from material stress P, mismatch penalty R, and volume penalty D projected into the interface
                                     + (P(i,j,iGrN) + R(i,j,iGrN) + D(i,j,iGrN))*normN(j)
       resid(i+3_pInt*(iNum-1_pInt)) = tract(iNum,i)                                                ! translate the local residual into global 1-dimensional residual array
     enddo
   enddo
   
!--------------------------------------------------------------------------------------------------
! debugging the residual stress
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a30,1x,i3)')'Traction at interface: ',iNum
     write(6,'(1x,3(e15.8,1x))')(tract(iNum,j), j = 1_pInt,3_pInt)
     write(6,*)' '
     !$OMP END CRITICAL (write2out)
   endif
 enddo

!--------------------------------------------------------------------------------------------------
! convergence check for stress residual
 stresMax = maxval(abs(P))                                                                          ! get the maximum of first Piola-Kirchhoff (material) stress
 stresLoc = int(maxloc(abs(P)),pInt)                                                                ! get the location of the maximum stress
 residMax = maxval(abs(tract))                                                                      ! get the maximum of the residual
 residLoc = int(maxloc(abs(tract)),pInt)                                                            ! get the position of the maximum residual

!--------------------------------------------------------------------------------------------------
!  Debugging the convergent criteria
 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
     .and. debug_e == el .and. debug_i == ip) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a)')' '
   write(6,'(1x,a,1x,i2,1x,i4)')'RGC residual check ...',ip,el
   write(6,'(1x,a15,1x,e15.8,1x,a7,i3,1x,a12,i2,i2)')'Max stress: ',stresMax, &
              '@ grain',stresLoc(3),'in component',stresLoc(1),stresLoc(2)
   write(6,'(1x,a15,1x,e15.8,1x,a7,i3,1x,a12,i2)')'Max residual: ',residMax, &
              '@ iface',residLoc(1),'in direction',residLoc(2)
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif
 
 homogenization_RGC_updateState = .false.
 
!--------------------------------------------------------------------------------------------------
!  If convergence reached => done and happy
 if (residMax < relTol_RGC*stresMax .or. residMax < absTol_RGC) then 
   homogenization_RGC_updateState = .true.
   
    if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
        .and. debug_e == el .and. debug_i == ip) then 
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a55,/)')'... done and happy'
     flush(6)
     !$OMP END CRITICAL (write2out)
   endif

!--------------------------------------------------------------------------------------------------
! compute/update the state for postResult, i.e., all energy densities computed by time-integration
   constitutiveWork = homogState(mappingHomogenization(2,ip,el))%state(3*nIntFaceTot+1,mappingHomogenization(1,ip,el))
   penaltyEnergy    = homogState(mappingHomogenization(2,ip,el))%state(3*nIntFaceTot+5,mappingHomogenization(1,ip,el))
   do iGrain = 1_pInt,homogenization_Ngrains(mesh_element(3,el))                                    ! time-integration loop for the calculating the work and energy
     do i = 1_pInt,3_pInt
     do j = 1_pInt,3_pInt
       constitutiveWork = constitutiveWork + P(i,j,iGrain)*(F(i,j,iGrain) - F0(i,j,iGrain))/real(nGrain,pReal)
       penaltyEnergy    = penaltyEnergy    + R(i,j,iGrain)*(F(i,j,iGrain) - F0(i,j,iGrain))/real(nGrain,pReal)
     enddo
     enddo
   enddo
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+1,mappingHomogenization(1,ip,el)) = constitutiveWork                 ! the bulk mechanical/constitutive work
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+2,mappingHomogenization(1,ip,el)) = sum(NN(1,:))/real(nGrain,pReal)  ! the overall mismatch of all interface normal to e1-direction
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+3,mappingHomogenization(1,ip,el)) = sum(NN(2,:))/real(nGrain,pReal)  ! the overall mismatch of all interface normal to e2-direction
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+4,mappingHomogenization(1,ip,el)) = sum(NN(3,:))/real(nGrain,pReal)  ! the overall mismatch of all interface normal to e3-direction
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+5,mappingHomogenization(1,ip,el)) = penaltyEnergy                    ! the overall penalty energy
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+6,mappingHomogenization(1,ip,el)) = volDiscrep                       ! the overall volume discrepancy
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+7,mappingHomogenization(1,ip,el)) = &
                                        sum(abs(drelax))/dt/real(3_pInt*nIntFaceTot,pReal)                       ! the average rate of relaxation vectors
   homogState(mappingHomogenization(2,ip,el))% &
                        state(3*nIntFaceTot+8,mappingHomogenization(1,ip,el)) = maxval(abs(drelax))/dt           ! the maximum rate of relaxation vectors

   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
        .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a30,1x,e15.8)')   'Constitutive work: ',constitutiveWork
     write(6,'(1x,a30,3(1x,e15.8))')'Magnitude mismatch: ',sum(NN(1,:))/real(nGrain,pReal), &
                                                           sum(NN(2,:))/real(nGrain,pReal), &
                                                           sum(NN(3,:))/real(nGrain,pReal)
     write(6,'(1x,a30,1x,e15.8)')   'Penalty energy: ',penaltyEnergy
     write(6,'(1x,a30,1x,e15.8,/)') 'Volume discrepancy: ',volDiscrep
     write(6,'(1x,a30,1x,e15.8)')   'Maximum relaxation rate: ',maxval(abs(drelax))/dt
     write(6,'(1x,a30,1x,e15.8,/)') 'Average relaxation rate: ',sum(abs(drelax))/dt/real(3_pInt*nIntFaceTot,pReal)
     flush(6)
     !$OMP END CRITICAL (write2out)
   endif
   
   deallocate(tract,resid,relax,drelax)
   return

!--------------------------------------------------------------------------------------------------
! if residual blows-up => done but unhappy
 elseif (residMax > relMax_RGC*stresMax .or. residMax > absMax_RGC) then                            ! try to restart when residual blows up exceeding maximum bound
   homogenization_RGC_updateState = [.true.,.false.]                                                ! with direct cut-back

   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
       .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a55,/)')'... broken'
     flush(6)
     !$OMP END CRITICAL (write2out)
   endif
   
   deallocate(tract,resid,relax,drelax)
   return 
 else                                                                                               ! proceed with computing the Jacobian and state update
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
     .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a55,/)')'... not yet done'
     flush(6)
     !$OMP END CRITICAL (write2out)
   endif
   
 endif

!---------------------------------------------------------------------------------------------------
! construct the global Jacobian matrix for updating the global relaxation vector array when 
! convergence is not yet reached ...

!--------------------------------------------------------------------------------------------------
! ... of the constitutive stress tangent, assembled from dPdF or material constitutive model "smatrix"
 allocate(smatrix(3*nIntFaceTot,3*nIntFaceTot), source=0.0_pReal)
 do iNum = 1_pInt,nIntFaceTot
   faceID = homogenization_RGC_interface1to4(iNum,homID)                                            ! assembling of local dPdF into global Jacobian matrix
   
!--------------------------------------------------------------------------------------------------
! identify the left/bottom/back grain (-|N)
   iGr3N = faceID(2:4)                                                                              ! identifying the grain ID in local coordinate sytem
   iGrN = homogenization_RGC_grain3to1(iGr3N,homID)                                                 ! translate into global grain ID
   intFaceN = homogenization_RGC_getInterface(2_pInt*faceID(1),iGr3N)                               ! identifying the connecting interface in local coordinate system
   normN = homogenization_RGC_interfaceNormal(intFaceN,ip,el)                                       ! get the interface normal
   do iFace = 1_pInt,nFace
     intFaceN = homogenization_RGC_getInterface(iFace,iGr3N)                                        ! identifying all interfaces that influence relaxation of the above interface
     mornN = homogenization_RGC_interfaceNormal(intFaceN,ip,el)                                     ! get normal of the interfaces
     iMun = homogenization_RGC_interface4to1(intFaceN,homID)                                        ! translate the interfaces ID into local 4-dimensional index
     if (iMun > 0) then                                                                             ! get the corresponding tangent
       do i=1_pInt,3_pInt; do j=1_pInt,3_pInt; do k=1_pInt,3_pInt; do l=1_pInt,3_pInt
         smatrix(3*(iNum-1)+i,3*(iMun-1)+j) = smatrix(3*(iNum-1)+i,3*(iMun-1)+j) + dPdF(i,k,j,l,iGrN)*normN(k)*mornN(l)
       enddo;enddo;enddo;enddo
! projecting the material tangent dPdF into the interface
! to obtain the Jacobian matrix contribution of dPdF
     endif
   enddo
   
!--------------------------------------------------------------------------------------------------
! identify the right/up/front grain (+|P)
   iGr3P = iGr3N
   iGr3P(faceID(1)) = iGr3N(faceID(1))+1_pInt                                                       ! identifying the grain ID in local coordinate sytem
   iGrP = homogenization_RGC_grain3to1(iGr3P,homID)                                                 ! translate into global grain ID
   intFaceP = homogenization_RGC_getInterface(2_pInt*faceID(1)-1_pInt,iGr3P)                        ! identifying the connecting interface in local coordinate system
   normP = homogenization_RGC_interfaceNormal(intFaceP,ip,el)                                       ! get the interface normal
   do iFace = 1_pInt,nFace
     intFaceP = homogenization_RGC_getInterface(iFace,iGr3P)                                        ! identifying all interfaces that influence relaxation of the above interface
     mornP = homogenization_RGC_interfaceNormal(intFaceP,ip,el)                                     ! get normal of the interfaces
     iMun = homogenization_RGC_interface4to1(intFaceP,homID)                                        ! translate the interfaces ID into local 4-dimensional index
     if (iMun > 0_pInt) then                                                                        ! get the corresponding tangent
       do i=1_pInt,3_pInt; do j=1_pInt,3_pInt; do k=1_pInt,3_pInt; do l=1_pInt,3_pInt
         smatrix(3*(iNum-1)+i,3*(iMun-1)+j) = smatrix(3*(iNum-1)+i,3*(iMun-1)+j) + dPdF(i,k,j,l,iGrP)*normP(k)*mornP(l)
       enddo;enddo;enddo;enddo
     endif
   enddo
 enddo
 
!--------------------------------------------------------------------------------------------------
! debugging the global Jacobian matrix of stress tangent
 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Jacobian matrix of stress'
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,100(e11.4,1x))')(smatrix(i,j), j = 1_pInt,3_pInt*nIntFaceTot)
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif
 
!--------------------------------------------------------------------------------------------------
! ... of the stress penalty tangent (mismatch penalty and volume penalty, computed using numerical 
! perturbation method) "pmatrix"
 allocate(pmatrix(3*nIntFaceTot,3*nIntFaceTot), source=0.0_pReal)
 allocate(p_relax(3*nIntFaceTot),               source=0.0_pReal)
 allocate(p_resid(3*nIntFaceTot),               source=0.0_pReal)
 do ipert = 1_pInt,3_pInt*nIntFaceTot
   p_relax = relax
   p_relax(ipert) = relax(ipert) + pPert_RGC                                                        ! perturb the relaxation vector
   homogState(mappingHomogenization(2,ip,el))%state(1:3*nIntFaceTot,mappingHomogenization(1,ip,el)) = p_relax
   call homogenization_RGC_grainDeformation(pF,avgF,ip,el)                                    ! compute the grains deformation from perturbed state
   call homogenization_RGC_stressPenalty(pR,pNN,avgF,pF,ip,el,homID)                                ! compute stress penalty due to interface mismatch from perturbed state
   call homogenization_RGC_volumePenalty(pD,volDiscrep,pF,avgF,ip,el)                               ! compute stress penalty due to volume discrepancy from perturbed state

!--------------------------------------------------------------------------------------------------
! computing the global stress residual array from the perturbed state
   p_resid = 0.0_pReal
   do iNum = 1_pInt,nIntFaceTot
     faceID = homogenization_RGC_interface1to4(iNum,homID)                                          ! identifying the interface ID in local coordinate system (4-dimensional index)

!--------------------------------------------------------------------------------------------------
! identify the left/bottom/back grain (-|N)
     iGr3N = faceID(2:4)                                                                            ! identifying the grain ID in local coordinate system (3-dimensional index)
     iGrN = homogenization_RGC_grain3to1(iGr3N,homID)                                               ! translate the local grain ID into global coordinate system (1-dimensional index)
     intFaceN = homogenization_RGC_getInterface(2_pInt*faceID(1),iGr3N)                             ! identifying the interface ID of the grain
     normN = homogenization_RGC_interfaceNormal(intFaceN,ip,el)                                     ! get the corresponding interface normal
 
!--------------------------------------------------------------------------------------------------
! identify the right/up/front grain (+|P)
     iGr3P = iGr3N
     iGr3P(faceID(1)) = iGr3N(faceID(1))+1_pInt                                                     ! identifying the grain ID in local coordinate system (3-dimensional index)
     iGrP = homogenization_RGC_grain3to1(iGr3P,homID)                                               ! translate the local grain ID into global coordinate system (1-dimensional index)
     intFaceP = homogenization_RGC_getInterface(2_pInt*faceID(1)-1_pInt,iGr3P)                      ! identifying the interface ID of the grain
     normP = homogenization_RGC_interfaceNormal(intFaceP,ip,el)                                     ! get the corresponding normal
 
!--------------------------------------------------------------------------------------------------
! compute the residual stress (contribution of mismatch and volume penalties) from perturbed state 
! at all interfaces
     do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt
       p_resid(i+3*(iNum-1)) = p_resid(i+3*(iNum-1)) + (pR(i,j,iGrP) - R(i,j,iGrP))*normP(j) &
                                                     + (pR(i,j,iGrN) - R(i,j,iGrN))*normN(j) &
                                                     + (pD(i,j,iGrP) - D(i,j,iGrP))*normP(j) &
                                                     + (pD(i,j,iGrN) - D(i,j,iGrN))*normN(j)
     enddo; enddo
   enddo
   pmatrix(:,ipert) = p_resid/pPert_RGC
 enddo
 
!--------------------------------------------------------------------------------------------------
! debugging the global Jacobian matrix of penalty tangent
 if (iand(debug_level(debug_homogenization), debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Jacobian matrix of penalty'
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,100(e11.4,1x))')(pmatrix(i,j), j = 1_pInt,3_pInt*nIntFaceTot)
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif
 
!--------------------------------------------------------------------------------------------------
! ... of the numerical viscosity traction "rmatrix"
 allocate(rmatrix(3*nIntFaceTot,3*nIntFaceTot),source=0.0_pReal)
 forall (i=1_pInt:3_pInt*nIntFaceTot) &
   rmatrix(i,i) = viscModus_RGC*viscPower_RGC/(refRelaxRate_RGC*dt)* &                              ! tangent due to numerical viscosity traction appears
                  (abs(drelax(i))/(refRelaxRate_RGC*dt))**(viscPower_RGC - 1.0_pReal)               ! only in the main diagonal term 
                                                                           
                                                                           

!--------------------------------------------------------------------------------------------------
! debugging the global Jacobian matrix of numerical viscosity tangent
 if (iand(debug_level(debug_homogenization), debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Jacobian matrix of penalty'
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,100(e11.4,1x))')(rmatrix(i,j), j = 1_pInt,3_pInt*nIntFaceTot)
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! The overall Jacobian matrix summarizing contributions of smatrix, pmatrix, rmatrix
 allocate(jmatrix(3*nIntFaceTot,3*nIntFaceTot)); jmatrix = smatrix + pmatrix + rmatrix
 
 if (iand(debug_level(debug_homogenization), debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Jacobian matrix (total)'
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,100(e11.4,1x))')(jmatrix(i,j), j = 1_pInt,3_pInt*nIntFaceTot)
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! computing the update of the state variable (relaxation vectors) using the Jacobian matrix
 allocate(jnverse(3_pInt*nIntFaceTot,3_pInt*nIntFaceTot),source=0.0_pReal)
 call math_invert(size(jmatrix,1),jmatrix,jnverse,error)                                            ! Compute the inverse of the overall Jacobian matrix
 
!--------------------------------------------------------------------------------------------------
! debugging the inverse Jacobian matrix
 if (iand(debug_level(debug_homogenization), debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Jacobian inverse'
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,100(e11.4,1x))')(jnverse(i,j), j = 1_pInt,3_pInt*nIntFaceTot)
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! calculate the state update (global relaxation vectors) for the next Newton-Raphson iteration
 drelax = 0.0_pReal
 do i = 1_pInt,3_pInt*nIntFaceTot
   do j = 1_pInt,3_pInt*nIntFaceTot
     drelax(i) = drelax(i) - jnverse(i,j)*resid(j)                                                  ! Calculate the correction for the state variable
   enddo
 enddo
 relax = relax + drelax                                                                             ! Updateing the state variable for the next iteration
 homogState(mappingHomogenization(2,ip,el))%state(1:3*nIntFaceTot,mappingHomogenization(1,ip,el)) = relax
 if (any(abs(drelax) > maxdRelax_RGC)) then                                                         ! Forcing cutback when the incremental change of relaxation vector becomes too large
   homogenization_RGC_updateState = [.true.,.false.]
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a,1x,i3,1x,a,1x,i3,1x,a)')'RGC_updateState: ip',ip,'| el',el,'enforces cutback'
   write(6,'(1x,a,1x,e15.8)')'due to large relaxation change =',maxval(abs(drelax))
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! debugging the return state
 if (iand(debug_homogenization, debug_levelExtensive) > 0_pInt) then
   !$OMP CRITICAL (write2out)
   write(6,'(1x,a30)')'Returned state: '
   do i = 1_pInt,3_pInt*nIntFaceTot
     write(6,'(1x,2(e15.8,1x))')homogState(mappingHomogenization(2,ip,el))%state(i,mappingHomogenization(1,ip,el))
   enddo
   write(6,*)' '
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif

 deallocate(tract,resid,jmatrix,jnverse,relax,drelax,pmatrix,smatrix,p_relax,p_resid)
 
end function homogenization_RGC_updateState


!--------------------------------------------------------------------------------------------------
!> @brief derive average stress and stiffness from constituent quantities 
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_averageStressAndItsTangent(avgP,dAvgPdAvgF,P,dPdF,el)
 use debug, only: &
  debug_level, &
  debug_homogenization,&
  debug_levelExtensive
 use mesh,  only: mesh_element
 use material, only: &
  homogenization_maxNgrains, &
  homogenization_Ngrains, &
  homogenization_typeInstance 
 use math, only: math_Plain3333to99
 
 implicit none
 real(pReal), dimension (3,3),                               intent(out) :: avgP                    !< average stress at material point
 real(pReal), dimension (3,3,3,3),                           intent(out) :: dAvgPdAvgF              !< average stiffness at material point
 real(pReal), dimension (3,3,homogenization_maxNgrains),     intent(in)  :: P                       !< array of current grain stresses
 real(pReal), dimension (3,3,3,3,homogenization_maxNgrains), intent(in)  :: dPdF                    !< array of current grain stiffnesses
 integer(pInt),                                              intent(in)  :: el                      !< element number
 real(pReal), dimension (9,9) :: dPdF99

 integer(pInt) :: homID, i, j, Ngrains, iGrain

 homID = homogenization_typeInstance(mesh_element(3,el))
 Ngrains = homogenization_Ngrains(mesh_element(3,el))

!--------------------------------------------------------------------------------------------------
! debugging the grain tangent
 if (iand(debug_level(debug_homogenization), debug_levelExtensive) /= 0_pInt) then
   !$OMP CRITICAL (write2out)
   do iGrain = 1_pInt,Ngrains
     dPdF99 = math_Plain3333to99(dPdF(1:3,1:3,1:3,1:3,iGrain))
     write(6,'(1x,a30,1x,i3)')'Stress tangent of grain: ',iGrain
     do i = 1_pInt,9_pInt
       write(6,'(1x,(e15.8,1x))') (dPdF99(i,j), j = 1_pInt,9_pInt)
     enddo
     write(6,*)' '
   enddo
   flush(6)
   !$OMP END CRITICAL (write2out)
 endif
 
!--------------------------------------------------------------------------------------------------
! computing the average first Piola-Kirchhoff stress P and the average tangent dPdF
 avgP = sum(P,3)/real(Ngrains,pReal)
 dAvgPdAvgF = sum(dPdF,5)/real(Ngrains,pReal)

end subroutine homogenization_RGC_averageStressAndItsTangent


!--------------------------------------------------------------------------------------------------
!> @brief return array of homogenization results for post file inclusion 
!--------------------------------------------------------------------------------------------------
pure function homogenization_RGC_postResults(ip,el,avgP,avgF)
 use mesh, only: &
   mesh_element, &
   mesh_ipCoordinates
 use material, only: &
   homogenization_typeInstance,&
   homogState, &
   mappingHomogenization, &  
   homogenization_Noutput
 
 implicit none
 integer(pInt), intent(in) :: &
   ip, &                                                                                            !< integration point number
   el                                                                                               !< element number
 real(pReal), dimension(3,3), intent(in) :: &
   avgP, &                                                                                          !< average stress at material point
   avgF                                                                                             !< average deformation gradient at material point

 integer(pInt) homID,o,c,nIntFaceTot
 real(pReal), dimension(homogenization_RGC_sizePostResults(homogenization_typeInstance(mesh_element(3,el)))) :: &
   homogenization_RGC_postResults

 homID = homogenization_typeInstance(mesh_element(3,el))
 nIntFaceTot=(homogenization_RGC_Ngrains(1,homID)-1_pInt)*homogenization_RGC_Ngrains(2,homID)*homogenization_RGC_Ngrains(3,homID)& 
            + homogenization_RGC_Ngrains(1,homID)*(homogenization_RGC_Ngrains(2,homID)-1_pInt)*homogenization_RGC_Ngrains(3,homID)&
            + homogenization_RGC_Ngrains(1,homID)*homogenization_RGC_Ngrains(2,homID)*(homogenization_RGC_Ngrains(3,homID)-1_pInt)

 c = 0_pInt
 homogenization_RGC_postResults = 0.0_pReal
 do o = 1_pInt,homogenization_Noutput(mesh_element(3,el))
   select case(homogenization_RGC_outputID(o,homID))
     case (avgdefgrad_ID)
       homogenization_RGC_postResults(c+1_pInt:c+9_pInt) = reshape(avgF,[9])
       c = c + 9_pInt
     case (avgfirstpiola_ID)
       homogenization_RGC_postResults(c+1_pInt:c+9_pInt) = reshape(avgP,[9])
       c = c + 9_pInt
     case (ipcoords_ID)
       homogenization_RGC_postResults(c+1_pInt:c+3_pInt) = mesh_ipCoordinates(1:3,ip,el)             ! current ip coordinates
       c = c + 3_pInt
     case (constitutivework_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                                              state(3*nIntFaceTot+1,mappingHomogenization(1,ip,el))
       c = c + 1_pInt
     case (magnitudemismatch_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+2,mappingHomogenization(1,ip,el))
       homogenization_RGC_postResults(c+2) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+3,mappingHomogenization(1,ip,el))
       homogenization_RGC_postResults(c+3) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+4,mappingHomogenization(1,ip,el))
       c = c + 3_pInt
     case (penaltyenergy_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+5,mappingHomogenization(1,ip,el))
       c = c + 1_pInt
     case (volumediscrepancy_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+6,mappingHomogenization(1,ip,el))
       c = c + 1_pInt
     case (averagerelaxrate_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+7,mappingHomogenization(1,ip,el))
       c = c + 1_pInt
     case (maximumrelaxrate_ID)
       homogenization_RGC_postResults(c+1) = homogState(mappingHomogenization(2,ip,el))% &
                                           state(3*nIntFaceTot+8,mappingHomogenization(1,ip,el))
       c = c + 1_pInt
   end select
 enddo

end function homogenization_RGC_postResults


!--------------------------------------------------------------------------------------------------
!> @brief calculate stress-like penalty due to deformation mismatch
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_stressPenalty(rPen,nMis,avgF,fDef,ip,el,homID)
 use debug, only: &
   debug_level, &
   debug_homogenization,&
   debug_levelExtensive, &
   debug_e, &
   debug_i
 use mesh, only: &
   mesh_element
 use constitutive, only: &
   constitutive_homogenizedC
 use math, only: &
   math_civita
 use material, only: &
   homogenization_maxNgrains,&
   homogenization_Ngrains
 use numerics, only: &
   xSmoo_RGC
 
 implicit none
 real(pReal),   dimension (3,3,homogenization_maxNgrains), intent(out) :: rPen                      !< stress-like penalty
 real(pReal),   dimension (3,homogenization_maxNgrains),   intent(out) :: nMis                      !< total amount of mismatch
 real(pReal),   dimension (3,3,homogenization_maxNgrains), intent(in)  :: fDef                      !< deformation gradients
 real(pReal),   dimension (3,3),                           intent(in)  :: avgF                      !< initial effective stretch tensor
 integer(pInt),                                            intent(in)  :: ip,el
 integer(pInt), dimension (4)   :: intFace
 integer(pInt), dimension (3)   :: iGrain3,iGNghb3,nGDim
 real(pReal),   dimension (3,3) :: gDef,nDef
 real(pReal),   dimension (3)   :: nVect,surfCorr
 real(pReal),   dimension (2)   :: Gmoduli
 integer(pInt) :: homID,iGrain,iGNghb,iFace,i,j,k,l
 real(pReal)   :: muGrain,muGNghb,nDefNorm,bgGrain,bgGNghb
 
 integer(pInt),                                             parameter  :: nFace = 6_pInt
 real(pReal),                                               parameter  :: nDefToler = 1.0e-10_pReal

 nGDim = homogenization_RGC_Ngrains(1:3,homID)
 rPen = 0.0_pReal
 nMis = 0.0_pReal

!--------------------------------------------------------------------------------------------------
! get the correction factor the modulus of penalty stress representing the evolution of area of 
! the interfaces due to deformations
 surfCorr = homogenization_RGC_surfaceCorrection(avgF,ip,el)

!--------------------------------------------------------------------------------------------------
! debugging the surface correction factor
 if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
     .and. debug_e == el .and. debug_i == ip) then
   !$OMP CRITICAL (write2out)
     write(6,'(1x,a20,2(1x,i3))')'Correction factor: ',ip,el
     write(6,'(1x,3(e11.4,1x))')(surfCorr(i), i = 1,3)
   !$OMP END CRITICAL (write2out)
 endif

!--------------------------------------------------------------------------------------------------
! computing the mismatch and penalty stress tensor of all grains 
 do iGrain = 1_pInt,homogenization_Ngrains(mesh_element(3,el))
   Gmoduli = homogenization_RGC_equivalentModuli(iGrain,ip,el)
   muGrain = Gmoduli(1)                                                                              ! collecting the equivalent shear modulus of grain
   bgGrain = Gmoduli(2)                                                                              ! and the lengthh of Burgers vector
   iGrain3 = homogenization_RGC_grain1to3(iGrain,homID)                                              ! get the grain ID in local 3-dimensional index (x,y,z)-position

!* Looping over all six interfaces of each grain
   do iFace = 1_pInt,nFace
     intFace = homogenization_RGC_getInterface(iFace,iGrain3)                                        ! get the 4-dimensional index of the interface in local numbering system of the grain
     nVect = homogenization_RGC_interfaceNormal(intFace,ip,el)                                       ! get the interface normal
     iGNghb3 = iGrain3                                                                               ! identify the neighboring grain across the interface
     iGNghb3(abs(intFace(1))) = iGNghb3(abs(intFace(1))) + int(real(intFace(1),pReal)/real(abs(intFace(1)),pReal),pInt)
     if (iGNghb3(1) < 1)        iGNghb3(1) = nGDim(1)                                               ! with periodicity along e1 direction
     if (iGNghb3(1) > nGDim(1)) iGNghb3(1) = 1_pInt
     if (iGNghb3(2) < 1)        iGNghb3(2) = nGDim(2)                                               ! with periodicity along e2 direction
     if (iGNghb3(2) > nGDim(2)) iGNghb3(2) = 1_pInt
     if (iGNghb3(3) < 1)        iGNghb3(3) = nGDim(3)                                               ! with periodicity along e3 direction
     if (iGNghb3(3) > nGDim(3)) iGNghb3(3) = 1_pInt
     iGNghb  = homogenization_RGC_grain3to1(iGNghb3,homID)                                          ! get the ID of the neighboring grain
     Gmoduli = homogenization_RGC_equivalentModuli(iGNghb,ip,el)                                    ! collecting the shear modulus and Burgers vector of the neighbor
     muGNghb = Gmoduli(1)
     bgGNghb = Gmoduli(2)
     gDef = 0.5_pReal*(fDef(1:3,1:3,iGNghb) - fDef(1:3,1:3,iGrain))                                 ! compute the difference/jump in deformation gradeint across the neighbor

!--------------------------------------------------------------------------------------------------
! compute the mismatch tensor of all interfaces
     nDefNorm = 0.0_pReal
     nDef = 0.0_pReal
     do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt
       do k = 1_pInt,3_pInt; do l = 1_pInt,3_pInt
         nDef(i,j) = nDef(i,j) - nVect(k)*gDef(i,l)*math_civita(j,k,l)                              ! compute the interface mismatch tensor from the jump of deformation gradient
       enddo; enddo
       nDefNorm = nDefNorm + nDef(i,j)*nDef(i,j)                                                    ! compute the norm of the mismatch tensor
     enddo; enddo
     nDefNorm = max(nDefToler,sqrt(nDefNorm))                                                       ! approximation to zero mismatch if mismatch is zero (singularity)
     nMis(abs(intFace(1)),iGrain) = nMis(abs(intFace(1)),iGrain) + nDefNorm                         ! total amount of mismatch experienced by the grain (at all six interfaces)

!--------------------------------------------------------------------------------------------------
! debuggin the mismatch tensor
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
       .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a20,i2,1x,a20,1x,i3)')'Mismatch to face: ',intFace(1),'neighbor grain: ',iGNghb
      do i = 1,3
        write(6,'(1x,3(e11.4,1x))')(nDef(i,j), j = 1,3)
      enddo
      write(6,'(1x,a20,e11.4)')'with magnitude: ',nDefNorm
     !$OMP END CRITICAL (write2out)
   endif

!--------------------------------------------------------------------------------------------------
! compute the stress penalty of all interfaces
     do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt
       do k = 1_pInt,3_pInt; do l = 1_pInt,3_pInt
         rPen(i,j,iGrain) = rPen(i,j,iGrain) + 0.5_pReal*(muGrain*bgGrain + muGNghb*bgGNghb)*homogenization_RGC_xiAlpha(homID) &
                                               *surfCorr(abs(intFace(1)))/homogenization_RGC_dAlpha(abs(intFace(1)),homID) &
                                               *cosh(homogenization_RGC_ciAlpha(homID)*nDefNorm) &
                                               *0.5_pReal*nVect(l)*nDef(i,k)/nDefNorm*math_civita(k,l,j) &
                                               *tanh(nDefNorm/xSmoo_RGC)
       enddo; enddo
     enddo; enddo
   enddo
   
!--------------------------------------------------------------------------------------------------
! debugging the stress-like penalty
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
       .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
      write(6,'(1x,a20,i2)')'Penalty of grain: ',iGrain
      do i = 1,3
        write(6,'(1x,3(e11.4,1x))')(rPen(i,j,iGrain), j = 1,3)
      enddo
     !$OMP END CRITICAL (write2out)
   endif

 enddo

end subroutine homogenization_RGC_stressPenalty


!--------------------------------------------------------------------------------------------------
!> @brief calculate stress-like penalty due to volume discrepancy 
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_volumePenalty(vPen,vDiscrep,fDef,fAvg,ip,el) 
 use debug, only: &
   debug_level, &
   debug_homogenization,&
   debug_levelExtensive, &
   debug_e, &
   debug_i
 use mesh, only: &
   mesh_element
 use math, only: &
   math_det33, &
   math_inv33
 use material, only: &
   homogenization_maxNgrains,&
   homogenization_Ngrains
 use numerics, only: &
   maxVolDiscr_RGC,&
   volDiscrMod_RGC,&
   volDiscrPow_RGC

 implicit none
 real(pReal), dimension (3,3,homogenization_maxNgrains), intent(out) :: vPen                        ! stress-like penalty due to volume
 real(pReal), intent(out)                  :: vDiscrep                                              ! total volume discrepancy
 real(pReal), dimension (3,3,homogenization_maxNgrains), intent(in)  :: fDef                        ! deformation gradients
 real(pReal), dimension (3,3), intent(in)  :: fAvg                                                  ! overall deformation gradient
 integer(pInt), intent(in)                 :: ip,&                                                  ! integration point
   el 
 real(pReal), dimension (homogenization_maxNgrains) :: gVol
 integer(pInt) :: iGrain,nGrain,i,j

 nGrain = homogenization_Ngrains(mesh_element(3,el))

!--------------------------------------------------------------------------------------------------
! compute the volumes of grains and of cluster
 vDiscrep = math_det33(fAvg)                                                                        ! compute the volume of the cluster
 do iGrain = 1_pInt,nGrain
   gVol(iGrain) = math_det33(fDef(1:3,1:3,iGrain))                                                  ! compute the volume of individual grains
   vDiscrep     = vDiscrep - gVol(iGrain)/real(nGrain,pReal)                                        ! calculate the difference/dicrepancy between
                                                                                                    ! the volume of the cluster and the the total volume of grains
 enddo

!--------------------------------------------------------------------------------------------------
! calculate the stress and penalty due to volume discrepancy
 vPen      = 0.0_pReal
 do iGrain = 1_pInt,nGrain
   vPen(:,:,iGrain) = -1.0_pReal/real(nGrain,pReal)*volDiscrMod_RGC*volDiscrPow_RGC/maxVolDiscr_RGC* &
                      sign((abs(vDiscrep)/maxVolDiscr_RGC)**(volDiscrPow_RGC - 1.0),vDiscrep)* &
                      gVol(iGrain)*transpose(math_inv33(fDef(:,:,iGrain)))

!--------------------------------------------------------------------------------------------------
! debugging the stress-like penalty
   if (iand(debug_level(debug_homogenization),debug_levelExtensive) /= 0_pInt &
     .and. debug_e == el .and. debug_i == ip) then
     !$OMP CRITICAL (write2out)
     write(6,'(1x,a30,i2)')'Volume penalty of grain: ',iGrain
     do i = 1,3
       write(6,'(1x,3(e11.4,1x))')(vPen(i,j,iGrain), j = 1,3)
     enddo
     !$OMP END CRITICAL (write2out)
   endif
 enddo

end subroutine homogenization_RGC_volumePenalty


!--------------------------------------------------------------------------------------------------
!> @brief compute the correction factor accouted for surface evolution (area change) due to 
! deformation
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_surfaceCorrection(avgF,ip,el)
 use math, only: &
   math_invert33, &
   math_mul33x33
 
 implicit none
 real(pReal), dimension(3)               :: homogenization_RGC_surfaceCorrection
 real(pReal), dimension(3,3), intent(in) :: avgF                                                    !< average F
 integer(pInt),               intent(in) :: ip,&                                                    !< integration point number
   el                                                                                               !< element number
 real(pReal), dimension(3,3)             :: invC,avgC
 real(pReal), dimension(3)               :: nVect
 real(pReal)  :: detF
 integer(pInt), dimension(4)             :: intFace
 integer(pInt) :: i,j,iBase
 logical     ::  error

 avgC = math_mul33x33(transpose(avgF),avgF)
 call math_invert33(avgC,invC,detF,error)
 homogenization_RGC_surfaceCorrection = 0.0_pReal
 do iBase = 1_pInt,3_pInt
   intFace = [iBase,1_pInt,1_pInt,1_pInt]
   nVect = homogenization_RGC_interfaceNormal(intFace,ip,el)                                        ! get the normal of the interface
   do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt
     homogenization_RGC_surfaceCorrection(iBase) = &                                                ! compute the component of (the inverse of) the stretch in the direction of the normal
       homogenization_RGC_surfaceCorrection(iBase) + invC(i,j)*nVect(i)*nVect(j)
   enddo; enddo
   homogenization_RGC_surfaceCorrection(iBase) = &                                                  ! get the surface correction factor (area contraction/enlargement)
     sqrt(homogenization_RGC_surfaceCorrection(iBase))*detF
 enddo

end function homogenization_RGC_surfaceCorrection


!--------------------------------------------------------------------------------------------------
!> @brief compute the equivalent shear and bulk moduli from the elasticity tensor
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_equivalentModuli(grainID,ip,el)
 use constitutive, only: &
   constitutive_homogenizedC

 implicit none
 integer(pInt), intent(in)    :: &
   grainID,&
   ip, &                                                                                            !< integration point number
   el                                                                                               !< element number
 real(pReal), dimension (6,6) :: elasTens
 real(pReal), dimension(2)    :: homogenization_RGC_equivalentModuli
 real(pReal) :: &
   cEquiv_11, &
   cEquiv_12, &
   cEquiv_44

 elasTens = constitutive_homogenizedC(grainID,ip,el)

!--------------------------------------------------------------------------------------------------
! compute the equivalent shear modulus after Turterltaub and Suiker, JMPS (2005)
 cEquiv_11 = (elasTens(1,1) + elasTens(2,2) + elasTens(3,3))/3.0_pReal
 cEquiv_12 = (elasTens(1,2) + elasTens(2,3) + elasTens(3,1) + &
              elasTens(1,3) + elasTens(2,1) + elasTens(3,2))/6.0_pReal
 cEquiv_44 = (elasTens(4,4) + elasTens(5,5) + elasTens(6,6))/3.0_pReal
 homogenization_RGC_equivalentModuli(1) = 0.2_pReal*(cEquiv_11 - cEquiv_12) + 0.6_pReal*cEquiv_44

!--------------------------------------------------------------------------------------------------
! obtain the length of Burgers vector (could be model dependend)
 homogenization_RGC_equivalentModuli(2) = 2.5e-10_pReal

end function homogenization_RGC_equivalentModuli


!--------------------------------------------------------------------------------------------------
!> @brief collect relaxation vectors of an interface
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_relaxationVector(intFace,homID, ip, el)
  use material, only: &
   homogState, &
   mappingHomogenization

 implicit none
 integer(pInt),                intent(in) :: ip, el 
 real(pReal),   dimension (3)             :: homogenization_RGC_relaxationVector
 integer(pInt), dimension (4), intent(in) :: intFace                                                !< set of interface ID in 4D array (normal and position)
 integer(pInt), dimension (3) ::             nGDim
 integer(pInt) :: &
   iNum, &
   homID                                                                                            !< homogenization ID

!--------------------------------------------------------------------------------------------------
! collect the interface relaxation vector from the global state array
 homogenization_RGC_relaxationVector = 0.0_pReal
 nGDim = homogenization_RGC_Ngrains(1:3,homID)
 iNum = homogenization_RGC_interface4to1(intFace,homID)                                             ! identify the position of the interface in global state array
 if (iNum > 0_pInt) homogenization_RGC_relaxationVector = homogState(mappingHomogenization(2,ip,el))% &
                                                            state((3*iNum-2):(3*iNum),mappingHomogenization(1,ip,el))             ! get the corresponding entries

end function homogenization_RGC_relaxationVector


!--------------------------------------------------------------------------------------------------
!> @brief identify the normal of an interface 
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_interfaceNormal(intFace,ip,el)
 use debug, only: &
   debug_homogenization,&
   debug_levelExtensive
 use math, only: &
   math_mul33x3
 
 implicit none
 real(pReal),   dimension (3)  ::            homogenization_RGC_interfaceNormal
 integer(pInt), dimension (4), intent(in) :: intFace                                                !< interface ID in 4D array (normal and position)
 integer(pInt),                intent(in) :: &
   ip, &                                                                                            !< integration point number
   el                                                                                               !< element number
 integer(pInt) ::                            nPos

!--------------------------------------------------------------------------------------------------
! get the normal of the interface, identified from the value of intFace(1)
 homogenization_RGC_interfaceNormal = 0.0_pReal
 nPos = abs(intFace(1))                                                                             ! identify the position of the interface in global state array
 homogenization_RGC_interfaceNormal(nPos) = real(intFace(1)/abs(intFace(1)),pReal)                  ! get the normal vector w.r.t. cluster axis

 homogenization_RGC_interfaceNormal = &
   math_mul33x3(homogenization_RGC_orientation(1:3,1:3,ip,el),homogenization_RGC_interfaceNormal)
                                                                                                    ! map the normal vector into sample coordinate system (basis)

end function homogenization_RGC_interfaceNormal


!--------------------------------------------------------------------------------------------------
!> @brief collect six faces of a grain in 4D (normal and position)
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_getInterface(iFace,iGrain3)

 implicit none
 integer(pInt), dimension (4) ::             homogenization_RGC_getInterface
 integer(pInt), dimension (3), intent(in) :: iGrain3                                                !< grain ID in 3D array
 integer(pInt),                intent(in) :: iFace                                                  !< face index (1..6) mapped like (-e1,-e2,-e3,+e1,+e2,+e3) or iDir = (-1,-2,-3,1,2,3)
 integer(pInt) ::                            iDir
 
!* Direction of interface normal
 iDir = (int(real(iFace-1_pInt,pReal)/2.0_pReal,pInt)+1_pInt)*(-1_pInt)**iFace
 homogenization_RGC_getInterface(1) = iDir
 
!--------------------------------------------------------------------------------------------------
! identify the interface position by the direction of its normal
 homogenization_RGC_getInterface(2:4) = iGrain3
 if (iDir < 0_pInt) &                                                                               ! to have a correlation with coordinate/position in real space
   homogenization_RGC_getInterface(1_pInt-iDir) = homogenization_RGC_getInterface(1_pInt-iDir)-1_pInt

end function homogenization_RGC_getInterface

!--------------------------------------------------------------------------------------------------
!> @brief map grain ID from in 1D (global array) to in 3D (local position)
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_grain1to3(grain1,homID)
 
 implicit none
 integer(pInt), dimension (3) ::            homogenization_RGC_grain1to3
 integer(pInt),               intent(in) :: &
   grain1,&                                                                                         !< grain ID in 1D array
   homID                                                                                            !< homogenization ID
 integer(pInt), dimension (3) ::            nGDim

!--------------------------------------------------------------------------------------------------
! get the grain position
 nGDim = homogenization_RGC_Ngrains(1:3,homID)
 homogenization_RGC_grain1to3(3) = 1_pInt+(grain1-1_pInt)/(nGDim(1)*nGDim(2))
 homogenization_RGC_grain1to3(2) = 1_pInt+mod((grain1-1_pInt)/nGDim(1),nGDim(2))
 homogenization_RGC_grain1to3(1) = 1_pInt+mod((grain1-1_pInt),nGDim(1))

end function homogenization_RGC_grain1to3


!--------------------------------------------------------------------------------------------------
!> @brief map grain ID from in 3D (local position) to in 1D (global array)
!--------------------------------------------------------------------------------------------------
pure function homogenization_RGC_grain3to1(grain3,homID)

 implicit none
 integer(pInt), dimension (3), intent(in) :: grain3                                                 !< grain ID in 3D array (pos.x,pos.y,pos.z)
 integer(pInt)                :: homogenization_RGC_grain3to1                                      
 integer(pInt), dimension (3) :: nGDim
 integer(pInt), intent(in) :: homID                                                                 ! homogenization ID

!--------------------------------------------------------------------------------------------------
! get the grain ID
 nGDim = homogenization_RGC_Ngrains(1:3,homID)
 homogenization_RGC_grain3to1 = grain3(1) + nGDim(1)*(grain3(2)-1_pInt) + nGDim(1)*nGDim(2)*(grain3(3)-1_pInt)

end function homogenization_RGC_grain3to1


!--------------------------------------------------------------------------------------------------
!> @brief maps interface ID from 4D (normal and local position) into 1D (global array)
!--------------------------------------------------------------------------------------------------
integer(pInt) pure function homogenization_RGC_interface4to1(iFace4D, homID)
 
 implicit none
 integer(pInt), dimension (4), intent(in) :: iFace4D                                                !< interface ID in 4D array (n.dir,pos.x,pos.y,pos.z)
 integer(pInt), dimension (3) :: nGDim,nIntFace
 integer(pInt), intent(in) :: homID                                                                 !< homogenization ID

 nGDim = homogenization_RGC_Ngrains(1:3,homID)
 
!--------------------------------------------------------------------------------------------------
! compute the total number of interfaces, which ...
 nIntFace(1) = (nGDim(1)-1_pInt)*nGDim(2)*nGDim(3)                                                 ! ... normal //e1
 nIntFace(2) = nGDim(1)*(nGDim(2)-1_pInt)*nGDim(3)                                                 ! ... normal //e2
 nIntFace(3) = nGDim(1)*nGDim(2)*(nGDim(3)-1_pInt)                                                 ! ... normal //e3

 homogenization_RGC_interface4to1 = -1_pInt
 
!--------------------------------------------------------------------------------------------------
! get the corresponding interface ID in 1D global array
 if (abs(iFace4D(1)) == 1_pInt) then                                                                ! interface with normal //e1
   homogenization_RGC_interface4to1 = iFace4D(3) + nGDim(2)*(iFace4D(4)-1_pInt) &
                                      + nGDim(2)*nGDim(3)*(iFace4D(2)-1_pInt)
   if ((iFace4D(2) == 0_pInt) .or. (iFace4D(2) == nGDim(1))) homogenization_RGC_interface4to1 = 0_pInt
 elseif (abs(iFace4D(1)) == 2_pInt) then                                                            ! interface with normal //e2
   homogenization_RGC_interface4to1 = iFace4D(4) + nGDim(3)*(iFace4D(2)-1_pInt) &
                                      + nGDim(3)*nGDim(1)*(iFace4D(3)-1_pInt) + nIntFace(1)
   if ((iFace4D(3) == 0_pInt) .or. (iFace4D(3) == nGDim(2))) homogenization_RGC_interface4to1 = 0_pInt
 elseif (abs(iFace4D(1)) == 3_pInt) then                                                            ! interface with normal //e3
   homogenization_RGC_interface4to1 = iFace4D(2) + nGDim(1)*(iFace4D(3)-1_pInt) &
                                      + nGDim(1)*nGDim(2)*(iFace4D(4)-1_pInt) + nIntFace(1) + nIntFace(2)
   if ((iFace4D(4) == 0_pInt) .or. (iFace4D(4) == nGDim(3))) homogenization_RGC_interface4to1 = 0_pInt
 endif

end function homogenization_RGC_interface4to1


!--------------------------------------------------------------------------------------------------
!> @brief maps interface ID from 1D (global array) into 4D (normal and local position)
!--------------------------------------------------------------------------------------------------
function homogenization_RGC_interface1to4(iFace1D, homID)
 
 implicit none
 integer(pInt), dimension (4) :: homogenization_RGC_interface1to4
 integer(pInt), intent(in)    :: iFace1D                                                            !< interface ID in 1D array
 integer(pInt), dimension (3) :: nGDim,nIntFace
 integer(pInt), intent(in) :: homID                                                                 !< homogenization ID

 nGDim = homogenization_RGC_Ngrains(:,homID)

!--------------------------------------------------------------------------------------------------
! compute the total number of interfaces, which ...
 nIntFace(1) = (nGDim(1)-1_pInt)*nGDim(2)*nGDim(3)                                                  ! ... normal //e1
 nIntFace(2) = nGDim(1)*(nGDim(2)-1_pInt)*nGDim(3)                                                  ! ... normal //e2
 nIntFace(3) = nGDim(1)*nGDim(2)*(nGDim(3)-1_pInt)                                                  ! ... normal //e3

!--------------------------------------------------------------------------------------------------
! get the corresponding interface ID in 4D (normal and local position)
 if (iFace1D > 0 .and. iFace1D <= nIntFace(1)) then                                                 ! interface with normal //e1
   homogenization_RGC_interface1to4(1) = 1_pInt
   homogenization_RGC_interface1to4(3) = mod((iFace1D-1_pInt),nGDim(2))+1_pInt
   homogenization_RGC_interface1to4(4) = mod(&
                                             int(&
                                                 real(iFace1D-1_pInt,pReal)/&
                                                 real(nGDim(2),pReal)&
                                                 ,pInt)&
                                             ,nGDim(3))+1_pInt
   homogenization_RGC_interface1to4(2) = int(&
                                             real(iFace1D-1_pInt,pReal)/&
                                             real(nGDim(2),pReal)/&
                                             real(nGDim(3),pReal)&
                                             ,pInt)+1_pInt
 elseif (iFace1D > nIntFace(1) .and. iFace1D <= (nIntFace(2) + nIntFace(1))) then                   ! interface with normal //e2
   homogenization_RGC_interface1to4(1) = 2_pInt
   homogenization_RGC_interface1to4(4) = mod((iFace1D-nIntFace(1)-1_pInt),nGDim(3))+1_pInt
   homogenization_RGC_interface1to4(2) = mod(&
                                             int(&
                                                 real(iFace1D-nIntFace(1)-1_pInt,pReal)/&
                                                 real(nGDim(3),pReal)&
                                                 ,pInt)&
                                              ,nGDim(1))+1_pInt
   homogenization_RGC_interface1to4(3) = int(&
                                             real(iFace1D-nIntFace(1)-1_pInt,pReal)/&
                                             real(nGDim(3),pReal)/&
                                             real(nGDim(1),pReal)&
                                             ,pInt)+1_pInt
 elseif (iFace1D > nIntFace(2) + nIntFace(1) .and. iFace1D <= (nIntFace(3) + nIntFace(2) + nIntFace(1))) then ! interface with normal //e3
   homogenization_RGC_interface1to4(1) = 3_pInt
   homogenization_RGC_interface1to4(2) = mod((iFace1D-nIntFace(2)-nIntFace(1)-1_pInt),nGDim(1))+1_pInt
   homogenization_RGC_interface1to4(3) = mod(&
                                             int(&
                                                 real(iFace1D-nIntFace(2)-nIntFace(1)-1_pInt,pReal)/&
                                                 real(nGDim(1),pReal)&
                                                 ,pInt)&
                                             ,nGDim(2))+1_pInt
   homogenization_RGC_interface1to4(4) = int(&
                                             real(iFace1D-nIntFace(2)-nIntFace(1)-1_pInt,pReal)/&
                                             real(nGDim(1),pReal)/&
                                             real(nGDim(2),pReal)&
                                             ,pInt)+1_pInt
 endif

end function homogenization_RGC_interface1to4


!--------------------------------------------------------------------------------------------------
!> @brief calculating the grain deformation gradient (the same with 
! homogenization_RGC_partionDeformation, but used only for perturbation scheme)
!--------------------------------------------------------------------------------------------------
subroutine homogenization_RGC_grainDeformation(F, avgF, ip, el)
 use mesh, only: &
   mesh_element
 use material, only: &
   homogenization_maxNgrains,&
   homogenization_Ngrains, &
   homogenization_typeInstance
 
 implicit none
 real(pReal),   dimension (3,3,homogenization_maxNgrains), intent(out) :: F                         !< partioned F per grain
 real(pReal),   dimension (3,3),                           intent(in)  :: avgF                      !< 
 integer(pInt),                                            intent(in)  :: &
   el, &                                                                                            !< element number
   ip                                                                                               !< integration point number
 real(pReal),   dimension (3) :: aVect,nVect
 integer(pInt), dimension (4) :: intFace
 integer(pInt), dimension (3) :: iGrain3
 integer(pInt) :: homID, iGrain,iFace,i,j
 integer(pInt),                                             parameter :: nFace = 6_pInt

!--------------------------------------------------------------------------------------------------
! compute the deformation gradient of individual grains due to relaxations
 homID = homogenization_typeInstance(mesh_element(3,el))
 F = 0.0_pReal
 do iGrain = 1_pInt,homogenization_Ngrains(mesh_element(3,el))
   iGrain3 = homogenization_RGC_grain1to3(iGrain,homID)
   do iFace = 1_pInt,nFace
     intFace = homogenization_RGC_getInterface(iFace,iGrain3)
     aVect = homogenization_RGC_relaxationVector(intFace,homID, ip, el)
     nVect = homogenization_RGC_interfaceNormal(intFace,ip,el)
     forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) &
     F(i,j,iGrain) = F(i,j,iGrain) + aVect(i)*nVect(j)                                              ! effective relaxations
   enddo
   F(1:3,1:3,iGrain) = F(1:3,1:3,iGrain) + avgF                                                     ! relaxed deformation gradient
 enddo

end subroutine homogenization_RGC_grainDeformation

end module homogenization_RGC