928 lines
49 KiB
Fortran
928 lines
49 KiB
Fortran
!--------------------------------------------------------------------------------------------------
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!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
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!> @author Denny Tjahjanto, Max-Planck-Institut für Eisenforschung GmbH
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!> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH
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!> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH
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!> @brief Relaxed grain cluster (RGC) homogenization scheme
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!> N_constituents is defined as p x q x r (cluster)
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!--------------------------------------------------------------------------------------------------
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submodule(homogenization:mechanical) RGC
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use rotations
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use crystal
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type :: tParameters
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integer, dimension(:), allocatable :: &
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N_constituents
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real(pREAL) :: &
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xi_alpha, &
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c_Alpha
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real(pREAL), dimension(:), allocatable :: &
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D_alpha, &
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a_g
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character(len=pSTRLEN), allocatable, dimension(:) :: &
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output
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end type tParameters
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type :: tRGCstate
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real(pREAL), pointer, dimension(:,:) :: &
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relaxationVector
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end type tRGCstate
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type :: tRGCdependentState
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real(pREAL), allocatable, dimension(:) :: &
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volumeDiscrepancy, &
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relaxationRate_avg, &
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relaxationRate_max
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real(pREAL), allocatable, dimension(:,:) :: &
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mismatch
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real(pREAL), allocatable, dimension(:,:,:) :: &
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orientation
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end type tRGCdependentState
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type :: tNumerics_RGC
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real(pREAL) :: &
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atol, & !< absolute tolerance of RGC residuum
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rtol, & !< relative tolerance of RGC residuum
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absMax, & !< absolute maximum of RGC residuum
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relMax, & !< relative maximum of RGC residuum
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pPert, & !< perturbation for computing RGC penalty tangent
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xSmoo, & !< RGC penalty smoothing parameter (hyperbolic tangent)
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viscPower, & !< power (sensitivity rate) of numerical viscosity in RGC scheme, Default 1.0e0: Newton viscosity (linear model)
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viscModus, & !< stress modulus of RGC numerical viscosity, Default 0.0e0: No viscosity is applied
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refRelaxRate, & !< reference relaxation rate in RGC viscosity
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maxdRelax, & !< threshold of maximum relaxation vector increment (if exceed this then cutback)
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maxVolDiscr, & !< threshold of maximum volume discrepancy allowed
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volDiscrMod, & !< stiffness of RGC volume discrepancy (zero = without volume discrepancy constraint)
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volDiscrPow !< powerlaw penalty for volume discrepancy
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end type tNumerics_RGC
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type(tparameters), dimension(:), allocatable :: &
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param
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type(tRGCstate), dimension(:), allocatable :: &
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state, &
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state0
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type(tRGCdependentState), dimension(:), allocatable :: &
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dependentState
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type(tNumerics_RGC) :: &
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num ! numerics parameters. Better name?
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contains
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!--------------------------------------------------------------------------------------------------
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!> @brief allocates all necessary fields, reads information from material configuration file
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!--------------------------------------------------------------------------------------------------
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module subroutine RGC_init()
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integer :: &
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ho, &
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Nmembers, &
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sizeState, nIntFaceTot
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class(tDict), pointer :: &
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num_homogenization, &
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num_mechanical, &
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num_RGC, & ! pointer to RGC numerics data
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material_homogenization, &
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homog, &
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homogMech
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print'(/,1x,a)', '<<<+- homogenization:mechanical:RGC init -+>>>'
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print'(/,a,i0)', ' # homogenizations: ',count(mechanical_type == MECHANICAL_RGC_ID)
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flush(IO_STDOUT)
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print'(/,1x,a)', 'D.D. Tjahjanto et al., International Journal of Material Forming 2(1):939–942, 2009'
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print'( 1x,a)', 'https://doi.org/10.1007/s12289-009-0619-1'//IO_EOL
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print'(/,1x,a)', 'D.D. Tjahjanto et al., Modelling and Simulation in Materials Science and Engineering 18:015006, 2010'
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print'( 1x,a)', 'https://doi.org/10.1088/0965-0393/18/1/015006'//IO_EOL
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material_homogenization => config_material%get_dict('homogenization')
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allocate(param(material_homogenization%length))
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allocate(state(material_homogenization%length))
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allocate(state0(material_homogenization%length))
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allocate(dependentState(material_homogenization%length))
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num_homogenization => config_numerics%get_dict('homogenization',defaultVal=emptyDict)
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num_mechanical => num_homogenization%get_dict('mechanical',defaultVal=emptyDict)
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num_RGC => num_mechanical%get_dict('RGC',defaultVal=emptyDict)
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num%atol = num_RGC%get_asReal('eps_abs_P', defaultVal=1.0e+4_pREAL)
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num%rtol = num_RGC%get_asReal('eps_rel_P', defaultVal=1.0e-3_pREAL)
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num%absMax = num_RGC%get_asReal('eps_abs_max', defaultVal=1.0e+10_pREAL)
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num%relMax = num_RGC%get_asReal('eps_rel_max', defaultVal=1.0e+2_pREAL)
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num%pPert = num_RGC%get_asReal('Delta_a', defaultVal=1.0e-7_pREAL)
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num%xSmoo = num_RGC%get_asReal('relevant_mismatch', defaultVal=1.0e-5_pREAL)
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num%viscPower = num_RGC%get_asReal('viscosity_exponent', defaultVal=1.0e+0_pREAL)
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num%viscModus = num_RGC%get_asReal('viscosity_modulus', defaultVal=0.0e+0_pREAL)
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num%refRelaxRate = num_RGC%get_asReal('dot_a_ref', defaultVal=1.0e-3_pREAL)
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num%maxdRelax = num_RGC%get_asReal('dot_a_max', defaultVal=1.0e+0_pREAL)
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num%maxVolDiscr = num_RGC%get_asReal('Delta_V_max', defaultVal=1.0e-5_pREAL)
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num%volDiscrMod = num_RGC%get_asReal('Delta_V_modulus', defaultVal=1.0e+12_pREAL)
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num%volDiscrPow = num_RGC%get_asReal('Delta_V_exponent', defaultVal=5.0_pREAL)
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if (num%atol <= 0.0_pREAL) call IO_error(301,ext_msg='eps_abs_P')
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if (num%rtol <= 0.0_pREAL) call IO_error(301,ext_msg='eps_rel_P')
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if (num%absMax <= 0.0_pREAL) call IO_error(301,ext_msg='eps_abs_max')
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if (num%relMax <= 0.0_pREAL) call IO_error(301,ext_msg='eps_rel_max')
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if (num%pPert <= 0.0_pREAL) call IO_error(301,ext_msg='Delta_a')
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if (num%xSmoo <= 0.0_pREAL) call IO_error(301,ext_msg='relevant_mismatch')
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if (num%viscPower < 0.0_pREAL) call IO_error(301,ext_msg='viscosity_exponent')
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if (num%viscModus < 0.0_pREAL) call IO_error(301,ext_msg='viscosity_modulus')
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if (num%refRelaxRate <= 0.0_pREAL) call IO_error(301,ext_msg='dot_a_ref')
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if (num%maxdRelax <= 0.0_pREAL) call IO_error(301,ext_msg='dot_a_max')
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if (num%maxVolDiscr <= 0.0_pREAL) call IO_error(301,ext_msg='Delta_V_max')
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if (num%volDiscrMod < 0.0_pREAL) call IO_error(301,ext_msg='Delta_V_modulus')
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if (num%volDiscrPow <= 0.0_pREAL) call IO_error(301,ext_msg='Delta_V_exponent')
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do ho = 1, size(mechanical_type)
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if (mechanical_type(ho) /= MECHANICAL_RGC_ID) cycle
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homog => material_homogenization%get_dict(ho)
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homogMech => homog%get_dict('mechanical')
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associate(prm => param(ho), &
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stt => state(ho), &
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st0 => state0(ho), &
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dst => dependentState(ho))
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#if defined (__GFORTRAN__)
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prm%output = output_as1dStr(homogMech)
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#else
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prm%output = homogMech%get_as1dStr('output',defaultVal=emptyStrArray)
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#endif
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prm%N_constituents = homogMech%get_as1dInt('cluster_size',requiredSize=3)
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if (homogenization_Nconstituents(ho) /= product(prm%N_constituents)) &
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call IO_error(211,ext_msg='N_constituents (RGC)')
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prm%xi_alpha = homogMech%get_asReal('xi_alpha')
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prm%c_alpha = homogMech%get_asReal('c_alpha')
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prm%D_alpha = homogMech%get_as1dReal('D_alpha', requiredSize=3)
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prm%a_g = homogMech%get_as1dReal('a_g', requiredSize=3)
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Nmembers = count(material_ID_homogenization == ho)
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nIntFaceTot = 3*( (prm%N_constituents(1)-1)*prm%N_constituents(2)*prm%N_constituents(3) &
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+ prm%N_constituents(1)*(prm%N_constituents(2)-1)*prm%N_constituents(3) &
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+ prm%N_constituents(1)*prm%N_constituents(2)*(prm%N_constituents(3)-1))
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sizeState = nIntFaceTot
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homogState(ho)%sizeState = sizeState
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allocate(homogState(ho)%state0 (sizeState,Nmembers), source=0.0_pREAL)
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allocate(homogState(ho)%state (sizeState,Nmembers), source=0.0_pREAL)
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stt%relaxationVector => homogState(ho)%state(1:nIntFaceTot,:)
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st0%relaxationVector => homogState(ho)%state0(1:nIntFaceTot,:)
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allocate(dst%volumeDiscrepancy( Nmembers), source=0.0_pREAL)
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allocate(dst%relaxationRate_avg( Nmembers), source=0.0_pREAL)
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allocate(dst%relaxationRate_max( Nmembers), source=0.0_pREAL)
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allocate(dst%mismatch( 3,Nmembers), source=0.0_pREAL)
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!--------------------------------------------------------------------------------------------------
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! assigning cluster orientations
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dependentState(ho)%orientation = spread(eu2om(prm%a_g*inRad),3,Nmembers)
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!dst%orientation = spread(eu2om(prm%a_g*inRad),3,Nmembers) ifort version 18.0.1 crashes (for whatever reason)
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end associate
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end do
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end subroutine RGC_init
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!--------------------------------------------------------------------------------------------------
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!> @brief partitions the deformation gradient onto the constituents
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!--------------------------------------------------------------------------------------------------
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module subroutine RGC_partitionDeformation(F,avgF,ce)
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real(pREAL), dimension (:,:,:), intent(out) :: F !< partitioned F per grain
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real(pREAL), dimension (3,3), intent(in) :: avgF !< averaged F
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integer, intent(in) :: &
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ce
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real(pREAL), dimension(3) :: aVect,nVect
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integer, dimension(4) :: intFace
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integer, dimension(3) :: iGrain3
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integer :: iGrain,iFace,i,j,ho,en
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associate(prm => param(material_ID_homogenization(ce)))
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ho = material_ID_homogenization(ce)
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en = material_entry_homogenization(ce)
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!--------------------------------------------------------------------------------------------------
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! compute the deformation gradient of individual grains due to relaxations
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F = 0.0_pREAL
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do iGrain = 1,product(prm%N_constituents)
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iGrain3 = grain1to3(iGrain,prm%N_constituents)
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do iFace = 1,6
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intFace = getInterface(iFace,iGrain3) ! identifying 6 interfaces of each grain
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aVect = relaxationVector(intFace,ho,en) ! get the relaxation vectors for each interface from global relaxation vector array
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nVect = interfaceNormal(intFace,ho,en)
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forall (i=1:3,j=1:3) &
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F(i,j,iGrain) = F(i,j,iGrain) + aVect(i)*nVect(j) ! calculating deformation relaxations due to interface relaxation
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end do
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F(1:3,1:3,iGrain) = F(1:3,1:3,iGrain) + avgF ! resulting relaxed deformation gradient
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end do
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end associate
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end subroutine RGC_partitionDeformation
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!--------------------------------------------------------------------------------------------------
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!> @brief update the internal state of the homogenization scheme and tell whether "done" and
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! "happy" with result
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!--------------------------------------------------------------------------------------------------
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module function RGC_updateState(P,F,avgF,dt,dPdF,ce) result(doneAndHappy)
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logical, dimension(2) :: doneAndHappy
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real(pREAL), dimension(:,:,:), intent(in) :: &
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P,& !< partitioned stresses
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F !< partitioned deformation gradients
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real(pREAL), dimension(:,:,:,:,:), intent(in) :: dPdF !< partitioned stiffnesses
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real(pREAL), dimension(3,3), intent(in) :: avgF !< average F
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real(pREAL), intent(in) :: dt !< time increment
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integer, intent(in) :: &
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ce !< cell
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integer, dimension(4) :: intFaceN,intFaceP,faceID
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integer, dimension(3) :: nGDim,iGr3N,iGr3P
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integer :: ho,iNum,i,j,nIntFaceTot,iGrN,iGrP,iMun,iFace,k,l,ipert,nGrain, en
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real(pREAL), dimension(3,3,size(P,3)) :: R,pF,pR,D,pD
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real(pREAL), dimension(3,size(P,3)) :: NN,devNull
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real(pREAL), dimension(3) :: normP,normN,mornP,mornN
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real(pREAL) :: residMax,stresMax
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logical :: error
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real(pREAL), dimension(:,:), allocatable :: tract,jmatrix,jnverse,smatrix,pmatrix,rmatrix
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real(pREAL), dimension(:), allocatable :: resid,relax,p_relax,p_resid,drelax
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zeroTimeStep: if (dEq0(dt)) then
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doneAndHappy = .true. ! pretend everything is fine and return
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return
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end if zeroTimeStep
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ho = material_ID_homogenization(ce)
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en = material_entry_homogenization(ce)
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associate(stt => state(ho), st0 => state0(ho), dst => dependentState(ho), prm => param(ho))
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!--------------------------------------------------------------------------------------------------
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! get the dimension of the cluster (grains and interfaces)
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nGDim = prm%N_constituents
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nGrain = product(nGDim)
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nIntFaceTot = (nGDim(1)-1)*nGDim(2)*nGDim(3) &
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+ nGDim(1)*(nGDim(2)-1)*nGDim(3) &
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+ nGDim(1)*nGDim(2)*(nGDim(3)-1)
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!--------------------------------------------------------------------------------------------------
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! allocate the size of the global relaxation arrays/jacobian matrices depending on the size of the cluster
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allocate(resid(3*nIntFaceTot), source=0.0_pREAL)
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allocate(tract(nIntFaceTot,3), source=0.0_pREAL)
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relax = stt%relaxationVector(:,en)
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drelax = stt%relaxationVector(:,en) - st0%relaxationVector(:,en)
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!--------------------------------------------------------------------------------------------------
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! computing interface mismatch and stress penalty tensor for all interfaces of all grains
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call stressPenalty(R,NN,avgF,F,ho,en)
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!--------------------------------------------------------------------------------------------------
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! calculating volume discrepancy and stress penalty related to overall volume discrepancy
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call volumePenalty(D,dst%volumeDiscrepancy(en),avgF,F,nGrain)
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!------------------------------------------------------------------------------------------------
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! computing the residual stress from the balance of traction at all (interior) interfaces
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do iNum = 1,nIntFaceTot
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faceID = interface1to4(iNum,param(ho)%N_constituents) ! identifying the interface ID in local coordinate system (4-dimensional index)
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!--------------------------------------------------------------------------------------------------
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! identify the left/bottom/back grain (-|N)
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iGr3N = faceID(2:4) ! identifying the grain ID in local coordinate system (3-dimensional index)
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iGrN = grain3to1(iGr3N,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
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intFaceN = getInterface(2*faceID(1),iGr3N)
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normN = interfaceNormal(intFaceN,ho,en)
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!--------------------------------------------------------------------------------------------------
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! identify the right/up/front grain (+|P)
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iGr3P = iGr3N
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iGr3P(faceID(1)) = iGr3N(faceID(1))+1 ! identifying the grain ID in local coordinate system (3-dimensional index)
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iGrP = grain3to1(iGr3P,param(ho)%N_constituents) ! translate the local grain ID into global coordinate system (1-dimensional index)
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intFaceP = getInterface(2*faceID(1)-1,iGr3P)
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normP = interfaceNormal(intFaceP,ho,en)
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!--------------------------------------------------------------------------------------------------
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! compute the residual of traction at the interface (in local system, 4-dimensional index)
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do i = 1,3
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tract(iNum,i) = sign(num%viscModus*(abs(drelax(i+3*(iNum-1)))/(num%refRelaxRate*dt))**num%viscPower, &
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drelax(i+3*(iNum-1))) ! contribution from the relaxation viscosity
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do j = 1,3
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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
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+ (P(i,j,iGrN) + R(i,j,iGrN) + D(i,j,iGrN))*normN(j)
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resid(i+3*(iNum-1)) = tract(iNum,i) ! translate the local residual into global 1-dimensional residual array
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end do
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end do
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end do
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!--------------------------------------------------------------------------------------------------
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! convergence check for stress residual
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stresMax = maxval(abs(P)) ! get the maximum of first Piola-Kirchhoff (material) stress
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residMax = maxval(abs(tract)) ! get the maximum of the residual
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doneAndHappy = .false.
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!--------------------------------------------------------------------------------------------------
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! If convergence reached => done and happy
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if (residMax < num%rtol*stresMax .or. residMax < num%atol) then
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doneAndHappy = .true.
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dst%mismatch(1:3,en) = sum(NN,2)/real(nGrain,pREAL)
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dst%relaxationRate_avg(en) = sum(abs(drelax))/dt/real(3*nIntFaceTot,pREAL)
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dst%relaxationRate_max(en) = maxval(abs(drelax))/dt
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return
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!--------------------------------------------------------------------------------------------------
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! if residual blows-up => done but unhappy
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elseif (residMax > num%relMax*stresMax .or. residMax > num%absMax) then ! try to restart when residual blows up exceeding maximum bound
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doneAndHappy = [.true.,.false.] ! with direct cut-back
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return
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end if
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!---------------------------------------------------------------------------------------------------
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! construct the global Jacobian matrix for updating the global relaxation vector array when
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! convergence is not yet reached ...
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!--------------------------------------------------------------------------------------------------
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! ... of the constitutive stress tangent, assembled from dPdF or material constitutive model "smatrix"
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allocate(smatrix(3*nIntFaceTot,3*nIntFaceTot), source=0.0_pREAL)
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do iNum = 1,nIntFaceTot
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faceID = interface1to4(iNum,param(ho)%N_constituents) ! assembling of local dPdF into global Jacobian matrix
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!--------------------------------------------------------------------------------------------------
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! identify the left/bottom/back grain (-|N)
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iGr3N = faceID(2:4) ! identifying the grain ID in local coordinate sytem
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iGrN = grain3to1(iGr3N,param(ho)%N_constituents) ! translate into global grain ID
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intFaceN = getInterface(2*faceID(1),iGr3N) ! identifying the connecting interface in local coordinate system
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normN = interfaceNormal(intFaceN,ho,en)
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do iFace = 1,6
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intFaceN = getInterface(iFace,iGr3N) ! identifying all interfaces that influence relaxation of the above interface
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mornN = interfaceNormal(intFaceN,ho,en)
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iMun = interface4to1(intFaceN,param(ho)%N_constituents) ! translate the interfaces ID into local 4-dimensional index
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if (iMun > 0) then ! get the corresponding tangent
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do i=1,3; do j=1,3; do k=1,3; do l=1,3
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smatrix(3*(iNum-1)+i,3*(iMun-1)+j) = smatrix(3*(iNum-1)+i,3*(iMun-1)+j) &
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+ dPdF(i,k,j,l,iGrN)*normN(k)*mornN(l)
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end do;end do;end do;end do
|
||
! projecting the material tangent dPdF into the interface
|
||
! to obtain the Jacobian matrix contribution of dPdF
|
||
end if
|
||
end do
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! 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,en)
|
||
do iFace = 1,6
|
||
intFaceP = getInterface(iFace,iGr3P) ! identifying all interfaces that influence relaxation of the above interface
|
||
mornP = interfaceNormal(intFaceP,ho,en)
|
||
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)
|
||
end do;end do;end do;end do
|
||
end if
|
||
end do
|
||
end do
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! ... 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(:,en) = p_relax
|
||
call grainDeformation(pF,avgF,ho,en) ! rain deformation from perturbed state
|
||
call stressPenalty(pR,DevNull, avgF,pF,ho,en) ! 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,en)
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! 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,en)
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! 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)
|
||
end do; end do
|
||
end do
|
||
pmatrix(:,ipert) = p_resid/num%pPert
|
||
end do
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! ... 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
|
||
end do
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
! 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
|
||
end do; end do
|
||
stt%relaxationVector(:,en) = 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)
|
||
end if
|
||
|
||
end associate
|
||
|
||
contains
|
||
!------------------------------------------------------------------------------------------------
|
||
!> @brief calculate stress-like penalty due to deformation mismatch
|
||
!------------------------------------------------------------------------------------------------
|
||
subroutine stressPenalty(rPen,nMis,avgF,fDef,ho,en)
|
||
|
||
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, en
|
||
|
||
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,en)
|
||
|
||
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,en)
|
||
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
|
||
end do; end do
|
||
nDefNorm = nDefNorm + nDef(i,j)**2 ! compute the norm of the mismatch tensor
|
||
end do; end do
|
||
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)
|
||
end do; end do;end do; end do
|
||
end do interfaceLoop
|
||
|
||
|
||
end do 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
|
||
end do
|
||
|
||
!----------------------------------------------------------------------------------------------
|
||
! calculate the stress and penalty due to volume discrepancy
|
||
vPen = 0.0_pREAL
|
||
do i = 1,nGrain
|
||
vPen(:,:,i) = -real(nGrain,pREAL)**(-1)*num%volDiscrMod*num%volDiscrPow/num%maxVolDiscr &
|
||
* sign((abs(vDiscrep)/num%maxVolDiscr)**(num%volDiscrPow - 1.0_pREAL),vDiscrep) &
|
||
* gVol(i)*transpose(math_inv33(fDef(:,:,i)))
|
||
end do
|
||
|
||
end subroutine volumePenalty
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
!> @brief compute the correction factor accouted for surface evolution (area change) due to
|
||
! deformation
|
||
!--------------------------------------------------------------------------------------------------
|
||
function surfaceCorrection(avgF,ho,en)
|
||
|
||
real(pREAL), dimension(3) :: surfaceCorrection
|
||
|
||
real(pREAL), dimension(3,3), intent(in) :: avgF !< average F
|
||
integer, intent(in) :: &
|
||
ho, &
|
||
en
|
||
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,en)
|
||
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
|
||
end do; end do
|
||
surfaceCorrection(iBase) = sqrt(surfaceCorrection(iBase))*detF ! get the surface correction factor (area contraction/enlargement)
|
||
end do
|
||
|
||
end function surfaceCorrection
|
||
|
||
|
||
!-------------------------------------------------------------------------------------------------
|
||
!> @brief compute the equivalent shear and bulk moduli from the elasticity tensor
|
||
!-------------------------------------------------------------------------------------------------
|
||
real(pREAL) function equivalentMu(co,ce)
|
||
|
||
integer, intent(in) :: &
|
||
co,&
|
||
ce
|
||
|
||
real(pREAL), dimension(6,6) :: C
|
||
|
||
C = phase_homogenizedC66(material_ID_phase(co,ce),material_entry_phase(co,ce)) ! damage not included!
|
||
|
||
equivalentMu = crystal_isotropic_mu(C,'isostrain')
|
||
|
||
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, en)
|
||
|
||
real(pREAL), dimension(:,:,:), intent(out) :: F !< partitioned F per grain
|
||
|
||
real(pREAL), dimension(:,:), intent(in) :: avgF !< averaged F
|
||
integer, intent(in) :: &
|
||
ho, &
|
||
en
|
||
|
||
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,en)
|
||
nVect = interfaceNormal(intFace,ho,en)
|
||
forall (i=1:3,j=1:3) &
|
||
F(i,j,iGrain) = F(i,j,iGrain) + aVect(i)*nVect(j) ! effective relaxations
|
||
end do
|
||
F(1:3,1:3,iGrain) = F(1:3,1:3,iGrain) + avgF ! relaxed deformation gradient
|
||
end do
|
||
|
||
end associate
|
||
|
||
end subroutine grainDeformation
|
||
|
||
end function RGC_updateState
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
!> @brief writes results to HDF5 output file
|
||
!--------------------------------------------------------------------------------------------------
|
||
module subroutine RGC_result(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 result_writeDataset(dst%mismatch,group,trim(prm%output(o)), &
|
||
'average mismatch tensor','1')
|
||
case('Delta_V')
|
||
call result_writeDataset(dst%volumeDiscrepancy,group,trim(prm%output(o)), &
|
||
'volume discrepancy','m³')
|
||
case('max_dot_a')
|
||
call result_writeDataset(dst%relaxationrate_max,group,trim(prm%output(o)), &
|
||
'maximum relaxation rate','m/s')
|
||
case('avg_dot_a')
|
||
call result_writeDataset(dst%relaxationrate_avg,group,trim(prm%output(o)), &
|
||
'average relaxation rate','m/s')
|
||
end select
|
||
end do outputsLoop
|
||
end associate
|
||
|
||
end subroutine RGC_result
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
!> @brief collect relaxation vectors of an interface
|
||
!--------------------------------------------------------------------------------------------------
|
||
pure function relaxationVector(intFace,ho,en)
|
||
|
||
real(pREAL), dimension (3) :: relaxationVector
|
||
|
||
integer, intent(in) :: ho,en
|
||
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),en)
|
||
else
|
||
relaxationVector = 0.0_pREAL
|
||
end if
|
||
|
||
end associate
|
||
|
||
end function relaxationVector
|
||
|
||
|
||
!--------------------------------------------------------------------------------------------------
|
||
!> @brief identify the normal of an interface
|
||
!--------------------------------------------------------------------------------------------------
|
||
pure function interfaceNormal(intFace,ho,en) result(n)
|
||
|
||
real(pREAL), dimension(3) :: n
|
||
integer, dimension(4), intent(in) :: intFace !< interface ID in 4D array (normal and position)
|
||
integer, intent(in) :: &
|
||
ho, &
|
||
en
|
||
|
||
|
||
associate (dst => dependentState(ho))
|
||
|
||
n = 0.0_pREAL
|
||
n(abs(intFace(1))) = real(intFace(1)/abs(intFace(1)),pREAL) ! get the normal vector w.r.t. cluster axis
|
||
|
||
n = matmul(dst%orientation(1:3,1:3,en),n) ! 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) result(i)
|
||
|
||
integer, dimension(4) :: i
|
||
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
|
||
i = [iDir,iGrain3]
|
||
if (iDir < 0) i(1-iDir) = i(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)
|
||
end if
|
||
|
||
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
|
||
end if
|
||
|
||
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
|
||
end if
|
||
|
||
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
|
||
end if
|
||
|
||
end function interface1to4
|
||
|
||
|
||
end submodule RGC
|