!--------------------------------------------------------------------------------------------------
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @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
!> N_constituents is defined as p x q x r (cluster)
!--------------------------------------------------------------------------------------------------
submodule(homogenization:mechanical) RGC
use rotations
use lattice
type :: tParameters
integer, dimension(:), allocatable :: &
N_constituents
real(pReal) :: &
xi_alpha, &
c_Alpha
real(pReal), dimension(:), allocatable :: &
D_alpha, &
a_g
character(len=pStringLen), allocatable, dimension(:) :: &
output
end type tParameters
type :: tRGCstate
real(pReal), pointer, dimension(:,:) :: &
relaxationVector
end type tRGCstate
type :: tRGCdependentState
real(pReal), allocatable, dimension(:) :: &
volumeDiscrepancy, &
relaxationRate_avg, &
relaxationRate_max
real(pReal), allocatable, dimension(:,:) :: &
mismatch
real(pReal), allocatable, dimension(:,:,:) :: &
orientation
end type tRGCdependentState
type :: tNumerics_RGC
real(pReal) :: &
atol, & !< absolute tolerance of RGC residuum
rtol, & !< relative tolerance of RGC residuum
absMax, & !< absolute maximum of RGC residuum
relMax, & !< relative maximum of RGC residuum
pPert, & !< perturbation for computing RGC penalty tangent
xSmoo, & !< RGC penalty smoothing parameter (hyperbolic tangent)
viscPower, & !< power (sensitivity rate) of numerical viscosity in RGC scheme, Default 1.0e0: Newton viscosity (linear model)
viscModus, & !< stress modulus of RGC numerical viscosity, Default 0.0e0: No viscosity is applied
refRelaxRate, & !< reference relaxation rate in RGC viscosity
maxdRelax, & !< threshold of maximum relaxation vector increment (if exceed this then cutback)
maxVolDiscr, & !< threshold of maximum volume discrepancy allowed
volDiscrMod, & !< stiffness of RGC volume discrepancy (zero = without volume discrepancy constraint)
volDiscrPow !< powerlaw penalty for volume discrepancy
end type tNumerics_RGC
type(tparameters), dimension(:), allocatable :: &
param
type(tRGCstate), dimension(:), allocatable :: &
state, &
state0
type(tRGCdependentState), dimension(:), allocatable :: &
dependentState
type(tNumerics_RGC) :: &
num ! numerics parameters. Better name?
contains
!--------------------------------------------------------------------------------------------------
!> @brief allocates all necessary fields, reads information from material configuration file
!--------------------------------------------------------------------------------------------------
module subroutine mechanical_RGC_init(num_homogMech)
class(tNode), pointer, intent(in) :: &
num_homogMech !< pointer to mechanical homogenization numerics data
integer :: &
ho, &
Nmaterialpoints, &
sizeState, nIntFaceTot
class (tNode), pointer :: &
num_RGC, & ! pointer to RGC numerics data
material_homogenization, &
homog, &
homogMech
print'(/,a)', ' <<<+- homogenization:mechanical:RGC init -+>>>'
print'(a,i2)', ' # instances: ',count(homogenization_type == HOMOGENIZATION_RGC_ID); flush(IO_STDOUT)
print*, 'D.D. Tjahjanto et al., International Journal of Material Forming 2(1):939–942, 2009'
print*, 'https://doi.org/10.1007/s12289-009-0619-1'//IO_EOL
print*, 'D.D. Tjahjanto et al., Modelling and Simulation in Materials Science and Engineering 18:015006, 2010'
print*, 'https://doi.org/10.1088/0965-0393/18/1/015006'//IO_EOL
material_homogenization => config_material%get('homogenization')
allocate(param(material_homogenization%length))
allocate(state(material_homogenization%length))
allocate(state0(material_homogenization%length))
allocate(dependentState(material_homogenization%length))
num_RGC => num_homogMech%get('RGC',defaultVal=emptyDict)
num%atol = num_RGC%get_asFloat('atol', defaultVal=1.0e+4_pReal)
num%rtol = num_RGC%get_asFloat('rtol', defaultVal=1.0e-3_pReal)
num%absMax = num_RGC%get_asFloat('amax', defaultVal=1.0e+10_pReal)
num%relMax = num_RGC%get_asFloat('rmax', defaultVal=1.0e+2_pReal)
num%pPert = num_RGC%get_asFloat('perturbpenalty', defaultVal=1.0e-7_pReal)
num%xSmoo = num_RGC%get_asFloat('relvantmismatch', defaultVal=1.0e-5_pReal)
num%viscPower = num_RGC%get_asFloat('viscositypower', defaultVal=1.0e+0_pReal)
num%viscModus = num_RGC%get_asFloat('viscositymodulus', defaultVal=0.0e+0_pReal)
num%refRelaxRate = num_RGC%get_asFloat('refrelaxationrate', defaultVal=1.0e-3_pReal)
num%maxdRelax = num_RGC%get_asFloat('maxrelaxationrate', defaultVal=1.0e+0_pReal)
num%maxVolDiscr = num_RGC%get_asFloat('maxvoldiscrepancy', defaultVal=1.0e-5_pReal)
num%volDiscrMod = num_RGC%get_asFloat('voldiscrepancymod', defaultVal=1.0e+12_pReal)
num%volDiscrPow = num_RGC%get_asFloat('dicrepancypower', defaultVal=5.0_pReal)
if (num%atol <= 0.0_pReal) call IO_error(301,ext_msg='absTol_RGC')
if (num%rtol <= 0.0_pReal) call IO_error(301,ext_msg='relTol_RGC')
if (num%absMax <= 0.0_pReal) call IO_error(301,ext_msg='absMax_RGC')
if (num%relMax <= 0.0_pReal) call IO_error(301,ext_msg='relMax_RGC')
if (num%pPert <= 0.0_pReal) call IO_error(301,ext_msg='pPert_RGC')
if (num%xSmoo <= 0.0_pReal) call IO_error(301,ext_msg='xSmoo_RGC')
if (num%viscPower < 0.0_pReal) call IO_error(301,ext_msg='viscPower_RGC')
if (num%viscModus < 0.0_pReal) call IO_error(301,ext_msg='viscModus_RGC')
if (num%refRelaxRate <= 0.0_pReal) call IO_error(301,ext_msg='refRelaxRate_RGC')
if (num%maxdRelax <= 0.0_pReal) call IO_error(301,ext_msg='maxdRelax_RGC')
if (num%maxVolDiscr <= 0.0_pReal) call IO_error(301,ext_msg='maxVolDiscr_RGC')
if (num%volDiscrMod < 0.0_pReal) call IO_error(301,ext_msg='volDiscrMod_RGC')
if (num%volDiscrPow <= 0.0_pReal) call IO_error(301,ext_msg='volDiscrPw_RGC')
do ho = 1, size(homogenization_type)
if (homogenization_type(ho) /= HOMOGENIZATION_RGC_ID) cycle
homog => material_homogenization%get(ho)
homogMech => homog%get('mechanics')
associate(prm => param(ho), &
stt => state(ho), &
st0 => state0(ho), &
dst => dependentState(ho))
#if defined (__GFORTRAN__)
prm%output = output_asStrings(homogMech)
#else
prm%output = homogMech%get_asStrings('output',defaultVal=emptyStringArray)
#endif
prm%N_constituents = homogMech%get_asInts('cluster_size',requiredSize=3)
if (homogenization_Nconstituents(ho) /= product(prm%N_constituents)) &
call IO_error(211,ext_msg='N_constituents (mechanical_RGC)')
prm%xi_alpha = homogMech%get_asFloat('xi_alpha')
prm%c_alpha = homogMech%get_asFloat('c_alpha')
prm%D_alpha = homogMech%get_asFloats('D_alpha', requiredSize=3)
prm%a_g = homogMech%get_asFloats('a_g', requiredSize=3)
Nmaterialpoints = count(material_homogenizationAt == ho)
nIntFaceTot = 3*( (prm%N_constituents(1)-1)*prm%N_constituents(2)*prm%N_constituents(3) &
+ prm%N_constituents(1)*(prm%N_constituents(2)-1)*prm%N_constituents(3) &
+ prm%N_constituents(1)*prm%N_constituents(2)*(prm%N_constituents(3)-1))
sizeState = nIntFaceTot
homogState(ho)%sizeState = sizeState
allocate(homogState(ho)%state0 (sizeState,Nmaterialpoints), source=0.0_pReal)
allocate(homogState(ho)%state (sizeState,Nmaterialpoints), source=0.0_pReal)
stt%relaxationVector => homogState(ho)%state(1:nIntFaceTot,:)
st0%relaxationVector => homogState(ho)%state0(1:nIntFaceTot,:)
allocate(dst%volumeDiscrepancy( Nmaterialpoints), source=0.0_pReal)
allocate(dst%relaxationRate_avg( Nmaterialpoints), source=0.0_pReal)
allocate(dst%relaxationRate_max( Nmaterialpoints), source=0.0_pReal)
allocate(dst%mismatch( 3,Nmaterialpoints), source=0.0_pReal)
!--------------------------------------------------------------------------------------------------
! assigning cluster orientations
dependentState(ho)%orientation = spread(eu2om(prm%a_g*inRad),3,Nmaterialpoints)
!dst%orientation = spread(eu2om(prm%a_g*inRad),3,Nmaterialpoints) ifort version 18.0.1 crashes (for whatever reason)
end associate
enddo
end subroutine mechanical_RGC_init
!--------------------------------------------------------------------------------------------------
!> @brief partitions the deformation gradient onto the constituents
!--------------------------------------------------------------------------------------------------
module subroutine mechanical_RGC_partitionDeformation(F,avgF,ce)
real(pReal), dimension (:,:,:), intent(out) :: F !< partitioned F per grain
real(pReal), dimension (3,3), intent(in) :: avgF !< averaged F
integer, intent(in) :: &
ce
real(pReal), dimension(3) :: aVect,nVect
integer, dimension(4) :: intFace
integer, dimension(3) :: iGrain3
integer :: iGrain,iFace,i,j,ho,me
associate(prm => param(material_homogenizationAt2(ce)))
ho = material_homogenizationAt2(ce)
me = material_homogenizationMemberAt2(ce)
!--------------------------------------------------------------------------------------------------
! compute the deformation gradient of individual grains due to relaxations
F = 0.0_pReal
do iGrain = 1,product(prm%N_constituents)
iGrain3 = grain1to3(iGrain,prm%N_constituents)
do iFace = 1,6
intFace = getInterface(iFace,iGrain3) ! identifying 6 interfaces of each grain
aVect = relaxationVector(intFace,ho,me) ! get the relaxation vectors for each interface from global relaxation vector array
nVect = interfaceNormal(intFace,ho,me)
forall (i=1:3,j=1:3) &
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
enddo
end associate
end subroutine mechanical_RGC_partitionDeformation
!--------------------------------------------------------------------------------------------------
!> @brief update the internal state of the homogenization scheme and tell whether "done" and
! "happy" with result
!--------------------------------------------------------------------------------------------------
module function mechanical_RGC_updateState(P,F,avgF,dt,dPdF,ce) result(doneAndHappy)
logical, dimension(2) :: doneAndHappy
real(pReal), dimension(:,:,:), intent(in) :: &
P,& !< partitioned stresses
F !< partitioned deformation gradients
real(pReal), dimension(:,:,:,:,:), intent(in) :: dPdF !< partitioned stiffnesses
real(pReal), dimension(3,3), intent(in) :: avgF !< average F
real(pReal), intent(in) :: dt !< time increment
integer, intent(in) :: &
ce !< cell
integer, dimension(4) :: intFaceN,intFaceP,faceID
integer, dimension(3) :: nGDim,iGr3N,iGr3P
integer :: ho,iNum,i,j,nIntFaceTot,iGrN,iGrP,iMun,iFace,k,l,ipert,iGrain,nGrain, me
real(pReal), dimension(3,3,size(P,3)) :: R,pF,pR,D,pD
real(pReal), dimension(3,size(P,3)) :: NN,devNull
real(pReal), dimension(3) :: normP,normN,mornP,mornN
real(pReal) :: residMax,stresMax
logical :: error
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
doneAndHappy = .true. ! pretend everything is fine and return
return
endif zeroTimeStep
ho = material_homogenizationAt2(ce)
me = material_homogenizationMemberAt2(ce)
associate(stt => state(ho), st0 => state0(ho), dst => dependentState(ho), prm => param(ho))
!--------------------------------------------------------------------------------------------------
! get the dimension of the cluster (grains and interfaces)
nGDim = prm%N_constituents
nGrain = product(nGDim)
nIntFaceTot = (nGDim(1)-1)*nGDim(2)*nGDim(3) &
+ nGDim(1)*(nGDim(2)-1)*nGDim(3) &
+ nGDim(1)*nGDim(2)*(nGDim(3)-1)
!--------------------------------------------------------------------------------------------------
! allocate the size of the global relaxation arrays/jacobian matrices depending on the size of the cluster
allocate(resid(3*nIntFaceTot), source=0.0_pReal)
allocate(tract(nIntFaceTot,3), source=0.0_pReal)
relax = stt%relaxationVector(:,me)
drelax = stt%relaxationVector(:,me) - st0%relaxationVector(:,me)
!--------------------------------------------------------------------------------------------------
! computing interface mismatch and stress penalty tensor for all interfaces of all grains
call stressPenalty(R,NN,avgF,F,ho,me)
!--------------------------------------------------------------------------------------------------
! calculating volume discrepancy and stress penalty related to overall volume discrepancy
call volumePenalty(D,dst%volumeDiscrepancy(me),avgF,F,nGrain)
!------------------------------------------------------------------------------------------------
! computing the residual stress from the balance of traction at all (interior) interfaces
do iNum = 1,nIntFaceTot
faceID = interface1to4(iNum,param(ho)%N_constituents) ! 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 = grain3to1(iGr3N,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
intFaceN = getInterface(2*faceID(1),iGr3N)
normN = interfaceNormal(intFaceN,ho,me)
!--------------------------------------------------------------------------------------------------
! identify the right/up/front grain (+|P)
iGr3P = iGr3N
iGr3P(faceID(1)) = iGr3N(faceID(1))+1 ! identifying the grain ID in local coordinate system (3-dimensional index)
iGrP = grain3to1(iGr3P,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
intFaceP = getInterface(2*faceID(1)-1,iGr3P)
normP = interfaceNormal(intFaceP,ho,me)
!--------------------------------------------------------------------------------------------------
! compute the residual of traction at the interface (in local system, 4-dimensional index)
do i = 1,3
tract(iNum,i) = sign(num%viscModus*(abs(drelax(i+3*(iNum-1)))/(num%refRelaxRate*dt))**num%viscPower, &
drelax(i+3*(iNum-1))) ! contribution from the relaxation viscosity
do j = 1,3
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*(iNum-1)) = tract(iNum,i) ! translate the local residual into global 1-dimensional residual array
enddo
enddo
enddo
!--------------------------------------------------------------------------------------------------
! convergence check for stress residual
stresMax = maxval(abs(P)) ! get the maximum of first Piola-Kirchhoff (material) stress
residMax = maxval(abs(tract)) ! get the maximum of the residual
doneAndHappy = .false.
!--------------------------------------------------------------------------------------------------
! If convergence reached => done and happy
if (residMax < num%rtol*stresMax .or. residMax < num%atol) then
doneAndHappy = .true.
dst%mismatch(1:3,me) = sum(NN,2)/real(nGrain,pReal)
dst%relaxationRate_avg(me) = sum(abs(drelax))/dt/real(3*nIntFaceTot,pReal)
dst%relaxationRate_max(me) = maxval(abs(drelax))/dt
return
!--------------------------------------------------------------------------------------------------
! if residual blows-up => done but unhappy
elseif (residMax > num%relMax*stresMax .or. residMax > num%absMax) then ! try to restart when residual blows up exceeding maximum bound
doneAndHappy = [.true.,.false.] ! with direct cut-back
return
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,nIntFaceTot
faceID = interface1to4(iNum,param(ho)%N_constituents) ! 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 = grain3to1(iGr3N,param(ho)%N_constituents) ! translate into global grain ID
intFaceN = getInterface(2*faceID(1),iGr3N) ! identifying the connecting interface in local coordinate system
normN = interfaceNormal(intFaceN,ho,me)
do iFace = 1,6
intFaceN = getInterface(iFace,iGr3N) ! identifying all interfaces that influence relaxation of the above interface
mornN = interfaceNormal(intFaceN,ho,me)
iMun = interface4to1(intFaceN,param(ho)%N_constituents) ! translate the interfaces ID into local 4-dimensional index
if (iMun > 0) then ! get the corresponding tangent
do i=1,3; do j=1,3; do k=1,3; do l=1,3
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 ! identifying the grain ID in local coordinate sytem
iGrP = grain3to1(iGr3P,param(ho)%N_constituents) ! translate into global grain ID
intFaceP = getInterface(2*faceID(1)-1,iGr3P) ! identifying the connecting interface in local coordinate system
normP = interfaceNormal(intFaceP,ho,me)
do iFace = 1,6
intFaceP = getInterface(iFace,iGr3P) ! identifying all interfaces that influence relaxation of the above interface
mornP = interfaceNormal(intFaceP,ho,me)
iMun = interface4to1(intFaceP,param(ho)%N_constituents) ! translate the interfaces ID into local 4-dimensional index
if (iMun > 0) then ! get the corresponding tangent
do i=1,3; do j=1,3; do k=1,3; do l=1,3
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
!--------------------------------------------------------------------------------------------------
! ... 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 = 1,3*nIntFaceTot
p_relax = relax
p_relax(ipert) = relax(ipert) + num%pPert ! perturb the relaxation vector
stt%relaxationVector(:,me) = p_relax
call grainDeformation(pF,avgF,ho,me) ! rain deformation from perturbed state
call stressPenalty(pR,DevNull, avgF,pF,ho,me) ! stress penalty due to interface mismatch from perturbed state
call volumePenalty(pD,devNull(1,1), avgF,pF,nGrain) ! 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,nIntFaceTot
faceID = interface1to4(iNum,param(ho)%N_constituents) ! identifying the interface ID in local coordinate system (4-dimensional index)
!--------------------------------------------------------------------------------------------------
! identify the left/bottom/back grain (-|N)
iGr3N = faceID(2:4) ! identify the grain ID in local coordinate system (3-dimensional index)
iGrN = grain3to1(iGr3N,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
intFaceN = getInterface(2*faceID(1),iGr3N) ! identify the interface ID of the grain
normN = interfaceNormal(intFaceN,ho,me)
!--------------------------------------------------------------------------------------------------
! identify the right/up/front grain (+|P)
iGr3P = iGr3N
iGr3P(faceID(1)) = iGr3N(faceID(1))+1 ! identify the grain ID in local coordinate system (3-dimensional index)
iGrP = grain3to1(iGr3P,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
intFaceP = getInterface(2*faceID(1)-1,iGr3P) ! identify the interface ID of the grain
normP = interfaceNormal(intFaceP,ho,me)
!--------------------------------------------------------------------------------------------------
! compute the residual stress (contribution of mismatch and volume penalties) from perturbed state
! at all interfaces
do i = 1,3; do j = 1,3
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/num%pPert
enddo
!--------------------------------------------------------------------------------------------------
! ... of the numerical viscosity traction "rmatrix"
allocate(rmatrix(3*nIntFaceTot,3*nIntFaceTot),source=0.0_pReal)
do i=1,3*nIntFaceTot
rmatrix(i,i) = num%viscModus*num%viscPower/(num%refRelaxRate*dt)* & ! tangent due to numerical viscosity traction appears
(abs(drelax(i))/(num%refRelaxRate*dt))**(num%viscPower - 1.0_pReal) ! only in the main diagonal term
enddo
!--------------------------------------------------------------------------------------------------
! The overall Jacobian matrix summarizing contributions of smatrix, pmatrix, rmatrix
allocate(jmatrix(3*nIntFaceTot,3*nIntFaceTot)); jmatrix = smatrix + pmatrix + rmatrix
!--------------------------------------------------------------------------------------------------
! computing the update of the state variable (relaxation vectors) using the Jacobian matrix
allocate(jnverse(3*nIntFaceTot,3*nIntFaceTot),source=0.0_pReal)
call math_invert(jnverse,error,jmatrix)
!--------------------------------------------------------------------------------------------------
! calculate the state update (global relaxation vectors) for the next Newton-Raphson iteration
drelax = 0.0_pReal
do i = 1,3*nIntFaceTot;do j = 1,3*nIntFaceTot
drelax(i) = drelax(i) - jnverse(i,j)*resid(j) ! Calculate the correction for the state variable
enddo; enddo
stt%relaxationVector(:,me) = relax + drelax ! Updateing the state variable for the next iteration
if (any(abs(drelax) > num%maxdRelax)) then ! Forcing cutback when the incremental change of relaxation vector becomes too large
doneAndHappy = [.true.,.false.]
!$OMP CRITICAL (write2out)
print'(a,i3,a,i3,a)',' RGC_updateState: enforces cutback'
print'(a,e15.8)',' due to large relaxation change = ',maxval(abs(drelax))
flush(IO_STDOUT)
!$OMP END CRITICAL (write2out)
endif
end associate
contains
!------------------------------------------------------------------------------------------------
!> @brief calculate stress-like penalty due to deformation mismatch
!------------------------------------------------------------------------------------------------
subroutine stressPenalty(rPen,nMis,avgF,fDef,ho,me)
real(pReal), dimension (:,:,:), intent(out) :: rPen !< stress-like penalty
real(pReal), dimension (:,:), intent(out) :: nMis !< total amount of mismatch
real(pReal), dimension (:,:,:), intent(in) :: fDef !< deformation gradients
real(pReal), dimension (3,3), intent(in) :: avgF !< initial effective stretch tensor
integer, intent(in) :: ho, me
integer, dimension (4) :: intFace
integer, dimension (3) :: iGrain3,iGNghb3,nGDim
real(pReal), dimension (3,3) :: gDef,nDef
real(pReal), dimension (3) :: nVect,surfCorr
integer :: iGrain,iGNghb,iFace,i,j,k,l
real(pReal) :: muGrain,muGNghb,nDefNorm
real(pReal), parameter :: &
nDefToler = 1.0e-10_pReal, &
b = 2.5e-10_pReal ! Length of Burgers vector
nGDim = param(ho)%N_constituents
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 = surfaceCorrection(avgF,ho,me)
associate(prm => param(ho))
!-----------------------------------------------------------------------------------------------
! computing the mismatch and penalty stress tensor of all grains
grainLoop: do iGrain = 1,product(prm%N_constituents)
muGrain = equivalentMu(iGrain,ce)
iGrain3 = grain1to3(iGrain,prm%N_constituents) ! get the grain ID in local 3-dimensional index (x,y,z)-position
interfaceLoop: do iFace = 1,6
intFace = getInterface(iFace,iGrain3) ! get the 4-dimensional index of the interface in local numbering system of the grain
nVect = interfaceNormal(intFace,ho,me)
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))
where(iGNghb3 < 1) iGNghb3 = nGDim
where(iGNghb3 >nGDim) iGNghb3 = 1
iGNghb = grain3to1(iGNghb3,prm%N_constituents) ! get the ID of the neighboring grain
muGNghb = equivalentMu(iGNghb,ce)
gDef = 0.5_pReal*(fDef(1:3,1:3,iGNghb) - fDef(1:3,1:3,iGrain)) ! 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 = 1,3; do j = 1,3
do k = 1,3; do l = 1,3
nDef(i,j) = nDef(i,j) - nVect(k)*gDef(i,l)*math_LeviCivita(j,k,l) ! compute the interface mismatch tensor from the jump of deformation gradient
enddo; enddo
nDefNorm = nDefNorm + nDef(i,j)**2.0_pReal ! 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)
!-------------------------------------------------------------------------------------------
! compute the stress penalty of all interfaces
do i = 1,3; do j = 1,3; do k = 1,3; do l = 1,3
rPen(i,j,iGrain) = rPen(i,j,iGrain) + 0.5_pReal*(muGrain*b + muGNghb*b)*prm%xi_alpha &
*surfCorr(abs(intFace(1)))/prm%D_alpha(abs(intFace(1))) &
*cosh(prm%c_alpha*nDefNorm) &
*0.5_pReal*nVect(l)*nDef(i,k)/nDefNorm*math_LeviCivita(k,l,j) &
*tanh(nDefNorm/num%xSmoo)
enddo; enddo;enddo; enddo
enddo interfaceLoop
enddo grainLoop
end associate
end subroutine stressPenalty
!------------------------------------------------------------------------------------------------
!> @brief calculate stress-like penalty due to volume discrepancy
!------------------------------------------------------------------------------------------------
subroutine volumePenalty(vPen,vDiscrep,fAvg,fDef,nGrain)
real(pReal), dimension (:,:,:), intent(out) :: vPen ! stress-like penalty due to volume
real(pReal), intent(out) :: vDiscrep ! total volume discrepancy
real(pReal), dimension (:,:,:), intent(in) :: fDef ! deformation gradients
real(pReal), dimension (3,3), intent(in) :: fAvg ! overall deformation gradient
integer, intent(in) :: &
Ngrain
real(pReal), dimension(size(vPen,3)) :: gVol
integer :: i
!----------------------------------------------------------------------------------------------
! compute the volumes of grains and of cluster
vDiscrep = math_det33(fAvg) ! compute the volume of the cluster
do i = 1,nGrain
gVol(i) = math_det33(fDef(1:3,1:3,i)) ! compute the volume of individual grains
vDiscrep = vDiscrep - gVol(i)/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 i = 1,nGrain
vPen(:,:,i) = -1.0_pReal/real(nGrain,pReal)*num%volDiscrMod*num%volDiscrPow/num%maxVolDiscr* &
sign((abs(vDiscrep)/num%maxVolDiscr)**(num%volDiscrPow - 1.0),vDiscrep)* &
gVol(i)*transpose(math_inv33(fDef(:,:,i)))
enddo
end subroutine volumePenalty
!--------------------------------------------------------------------------------------------------
!> @brief compute the correction factor accouted for surface evolution (area change) due to
! deformation
!--------------------------------------------------------------------------------------------------
function surfaceCorrection(avgF,ho,me)
real(pReal), dimension(3) :: surfaceCorrection
real(pReal), dimension(3,3), intent(in) :: avgF !< average F
integer, intent(in) :: &
ho, &
me
real(pReal), dimension(3,3) :: invC
real(pReal), dimension(3) :: nVect
real(pReal) :: detF
integer :: i,j,iBase
logical :: error
call math_invert33(invC,detF,error,matmul(transpose(avgF),avgF))
surfaceCorrection = 0.0_pReal
do iBase = 1,3
nVect = interfaceNormal([iBase,1,1,1],ho,me)
do i = 1,3; do j = 1,3
surfaceCorrection(iBase) = surfaceCorrection(iBase) + invC(i,j)*nVect(i)*nVect(j) ! compute the component of (the inverse of) the stretch in the direction of the normal
enddo; enddo
surfaceCorrection(iBase) = sqrt(surfaceCorrection(iBase))*detF ! get the surface correction factor (area contraction/enlargement)
enddo
end function surfaceCorrection
!-------------------------------------------------------------------------------------------------
!> @brief compute the equivalent shear and bulk moduli from the elasticity tensor
!-------------------------------------------------------------------------------------------------
real(pReal) function equivalentMu(grainID,ce)
integer, intent(in) :: &
grainID,&
ce
real(pReal), dimension(6,6) :: C
C = phase_homogenizedC(material_phaseAt2(grainID,ce),material_phaseMemberAt2(grainID,ce))
equivalentMu = lattice_equivalent_mu(C,'voigt')
end function equivalentMu
!-------------------------------------------------------------------------------------------------
!> @brief calculating the grain deformation gradient (the same with
! homogenization_RGC_partitionDeformation, but used only for perturbation scheme)
!-------------------------------------------------------------------------------------------------
subroutine grainDeformation(F, avgF, ho, me)
real(pReal), dimension(:,:,:), intent(out) :: F !< partitioned F per grain
real(pReal), dimension(:,:), intent(in) :: avgF !< averaged F
integer, intent(in) :: &
ho, &
me
real(pReal), dimension(3) :: aVect,nVect
integer, dimension(4) :: intFace
integer, dimension(3) :: iGrain3
integer :: iGrain,iFace,i,j
!-----------------------------------------------------------------------------------------------
! compute the deformation gradient of individual grains due to relaxations
associate (prm => param(ho))
F = 0.0_pReal
do iGrain = 1,product(prm%N_constituents)
iGrain3 = grain1to3(iGrain,prm%N_constituents)
do iFace = 1,6
intFace = getInterface(iFace,iGrain3)
aVect = relaxationVector(intFace,ho,me)
nVect = interfaceNormal(intFace,ho,me)
forall (i=1:3,j=1:3) &
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 associate
end subroutine grainDeformation
end function mechanical_RGC_updateState
!--------------------------------------------------------------------------------------------------
!> @brief derive average stress and stiffness from constituent quantities
!--------------------------------------------------------------------------------------------------
module subroutine mechanical_RGC_averageStressAndItsTangent(avgP,dAvgPdAvgF,P,dPdF,ho)
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 (:,:,:), intent(in) :: P !< partitioned stresses
real(pReal), dimension (:,:,:,:,:), intent(in) :: dPdF !< partitioned stiffnesses
integer, intent(in) :: ho
avgP = sum(P,3) /real(product(param(ho)%N_constituents),pReal)
dAvgPdAvgF = sum(dPdF,5)/real(product(param(ho)%N_constituents),pReal)
end subroutine mechanical_RGC_averageStressAndItsTangent
!--------------------------------------------------------------------------------------------------
!> @brief writes results to HDF5 output file
!--------------------------------------------------------------------------------------------------
module subroutine mechanical_RGC_results(ho,group)
integer, intent(in) :: ho
character(len=*), intent(in) :: group
integer :: o
associate(stt => state(ho), dst => dependentState(ho), prm => param(ho))
outputsLoop: do o = 1,size(prm%output)
select case(trim(prm%output(o)))
case('M')
call results_writeDataset(group,dst%mismatch,trim(prm%output(o)), &
'average mismatch tensor','1')
case('Delta_V')
call results_writeDataset(group,dst%volumeDiscrepancy,trim(prm%output(o)), &
'volume discrepancy','m³')
case('max_a_dot')
call results_writeDataset(group,dst%relaxationrate_max,trim(prm%output(o)), &
'maximum relaxation rate','m/s')
case('avg_a_dot')
call results_writeDataset(group,dst%relaxationrate_avg,trim(prm%output(o)), &
'average relaxation rate','m/s')
end select
enddo outputsLoop
end associate
end subroutine mechanical_RGC_results
!--------------------------------------------------------------------------------------------------
!> @brief collect relaxation vectors of an interface
!--------------------------------------------------------------------------------------------------
pure function relaxationVector(intFace,ho,me)
real(pReal), dimension (3) :: relaxationVector
integer, intent(in) :: ho,me
integer, dimension(4), intent(in) :: intFace !< set of interface ID in 4D array (normal and position)
integer :: iNum
!--------------------------------------------------------------------------------------------------
! collect the interface relaxation vector from the global state array
associate (prm => param(ho), &
stt => state(ho))
iNum = interface4to1(intFace,prm%N_constituents) ! identify the position of the interface in global state array
if (iNum > 0) then
relaxationVector = stt%relaxationVector((3*iNum-2):(3*iNum),me)
else
relaxationVector = 0.0_pReal
endif
end associate
end function relaxationVector
!--------------------------------------------------------------------------------------------------
!> @brief identify the normal of an interface
!--------------------------------------------------------------------------------------------------
pure function interfaceNormal(intFace,ho,me)
real(pReal), dimension(3) :: interfaceNormal
integer, dimension(4), intent(in) :: intFace !< interface ID in 4D array (normal and position)
integer, intent(in) :: &
ho, &
me
integer :: nPos
associate (dst => dependentState(ho))
!--------------------------------------------------------------------------------------------------
! get the normal of the interface, identified from the value of intFace(1)
interfaceNormal = 0.0_pReal
nPos = abs(intFace(1)) ! identify the position of the interface in global state array
interfaceNormal(nPos) = real(intFace(1)/abs(intFace(1)),pReal) ! get the normal vector w.r.t. cluster axis
interfaceNormal = matmul(dst%orientation(1:3,1:3,me),interfaceNormal) ! map the normal vector into sample coordinate system (basis)
end associate
end function interfaceNormal
!--------------------------------------------------------------------------------------------------
!> @brief collect six faces of a grain in 4D (normal and position)
!--------------------------------------------------------------------------------------------------
pure function getInterface(iFace,iGrain3)
integer, dimension(4) :: getInterface
integer, dimension(3), intent(in) :: iGrain3 !< grain ID in 3D array
integer, intent(in) :: iFace !< face index (1..6) mapped like (-e1,-e2,-e3,+e1,+e2,+e3) or iDir = (-1,-2,-3,1,2,3)
integer :: iDir !< direction of interface normal
iDir = (int(real(iFace-1,pReal)/2.0_pReal)+1)*(-1)**iFace
getInterface(1) = iDir
!--------------------------------------------------------------------------------------------------
! identify the interface position by the direction of its normal
getInterface(2:4) = iGrain3
if (iDir < 0) getInterface(1-iDir) = getInterface(1-iDir)-1 ! to have a correlation with coordinate/position in real space
end function getInterface
!--------------------------------------------------------------------------------------------------
!> @brief map grain ID from in 1D (global array) to in 3D (local position)
!--------------------------------------------------------------------------------------------------
pure function grain1to3(grain1,nGDim)
integer, dimension(3) :: grain1to3
integer, intent(in) :: grain1 !< grain ID in 1D array
integer, dimension(3), intent(in) :: nGDim
grain1to3 = 1 + [mod((grain1-1), nGDim(1)), &
mod((grain1-1)/ nGDim(1),nGDim(2)), &
(grain1-1)/(nGDim(1)*nGDim(2))]
end function grain1to3
!--------------------------------------------------------------------------------------------------
!> @brief map grain ID from in 3D (local position) to in 1D (global array)
!--------------------------------------------------------------------------------------------------
integer pure function grain3to1(grain3,nGDim)
integer, dimension(3), intent(in) :: grain3 !< grain ID in 3D array (pos.x,pos.y,pos.z)
integer, dimension(3), intent(in) :: nGDim
grain3to1 = grain3(1) &
+ nGDim(1)*(grain3(2)-1) &
+ nGDim(1)*nGDim(2)*(grain3(3)-1)
end function grain3to1
!--------------------------------------------------------------------------------------------------
!> @brief maps interface ID from 4D (normal and local position) into 1D (global array)
!--------------------------------------------------------------------------------------------------
integer pure function interface4to1(iFace4D, nGDim)
integer, dimension(4), intent(in) :: iFace4D !< interface ID in 4D array (n.dir,pos.x,pos.y,pos.z)
integer, dimension(3), intent(in) :: nGDim
select case(abs(iFace4D(1)))
case(1)
if ((iFace4D(2) == 0) .or. (iFace4D(2) == nGDim(1))) then
interface4to1 = 0
else
interface4to1 = iFace4D(3) + nGDim(2)*(iFace4D(4)-1) &
+ nGDim(2)*nGDim(3)*(iFace4D(2)-1)
endif
case(2)
if ((iFace4D(3) == 0) .or. (iFace4D(3) == nGDim(2))) then
interface4to1 = 0
else
interface4to1 = iFace4D(4) + nGDim(3)*(iFace4D(2)-1) &
+ nGDim(3)*nGDim(1)*(iFace4D(3)-1) &
+ (nGDim(1)-1)*nGDim(2)*nGDim(3) ! total # of interfaces normal || e1
endif
case(3)
if ((iFace4D(4) == 0) .or. (iFace4D(4) == nGDim(3))) then
interface4to1 = 0
else
interface4to1 = iFace4D(2) + nGDim(1)*(iFace4D(3)-1) &
+ nGDim(1)*nGDim(2)*(iFace4D(4)-1) &
+ (nGDim(1)-1)*nGDim(2)*nGDim(3) & ! total # of interfaces normal || e1
+ nGDim(1)*(nGDim(2)-1)*nGDim(3) ! total # of interfaces normal || e2
endif
case default
interface4to1 = -1
end select
end function interface4to1
!--------------------------------------------------------------------------------------------------
!> @brief maps interface ID from 1D (global array) into 4D (normal and local position)
!--------------------------------------------------------------------------------------------------
pure function interface1to4(iFace1D, nGDim)
integer, dimension(4) :: interface1to4
integer, intent(in) :: iFace1D !< interface ID in 1D array
integer, dimension(3), intent(in) :: nGDim
integer, dimension(3) :: nIntFace
!--------------------------------------------------------------------------------------------------
! compute the total number of interfaces, which ...
nIntFace = [(nGDim(1)-1)*nGDim(2)*nGDim(3), & ! ... normal || e1
nGDim(1)*(nGDim(2)-1)*nGDim(3), & ! ... normal || e2
nGDim(1)*nGDim(2)*(nGDim(3)-1)] ! ... 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
interface1to4(1) = 1
interface1to4(3) = mod((iFace1D-1),nGDim(2))+1
interface1to4(4) = mod(int(real(iFace1D-1,pReal)/real(nGDim(2),pReal)),nGDim(3))+1
interface1to4(2) = int(real(iFace1D-1,pReal)/real(nGDim(2),pReal)/real(nGDim(3),pReal))+1
elseif (iFace1D > nIntFace(1) .and. iFace1D <= (nIntFace(2) + nIntFace(1))) then ! interface with normal || e2
interface1to4(1) = 2
interface1to4(4) = mod((iFace1D-nIntFace(1)-1),nGDim(3))+1
interface1to4(2) = mod(int(real(iFace1D-nIntFace(1)-1,pReal)/real(nGDim(3),pReal)),nGDim(1))+1
interface1to4(3) = int(real(iFace1D-nIntFace(1)-1,pReal)/real(nGDim(3),pReal)/real(nGDim(1),pReal))+1
elseif (iFace1D > nIntFace(2) + nIntFace(1) .and. iFace1D <= (nIntFace(3) + nIntFace(2) + nIntFace(1))) then ! interface with normal || e3
interface1to4(1) = 3
interface1to4(2) = mod((iFace1D-nIntFace(2)-nIntFace(1)-1),nGDim(1))+1
interface1to4(3) = mod(int(real(iFace1D-nIntFace(2)-nIntFace(1)-1,pReal)/real(nGDim(1),pReal)),nGDim(2))+1
interface1to4(4) = int(real(iFace1D-nIntFace(2)-nIntFace(1)-1,pReal)/real(nGDim(1),pReal)/real(nGDim(2),pReal))+1
endif
end function interface1to4
end submodule RGC