!-------------------------------------------------------------------------------------------------- !> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH !> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH !> @author Christoph Kords, Max-Planck-Institut für Eisenforschung GmbH !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @brief Mathematical library, including random number generation and tensor represenations !-------------------------------------------------------------------------------------------------- module math use, intrinsic :: iso_c_binding use prec, only: & pReal, & pInt implicit none private real(pReal), parameter, public :: PI = 3.141592653589793_pReal !< ratio of a circle's circumference to its diameter real(pReal), parameter, public :: INDEG = 180.0_pReal/PI !< conversion from radian into degree real(pReal), parameter, public :: INRAD = PI/180.0_pReal !< conversion from degree into radian complex(pReal), parameter, public :: TWOPIIMG = (0.0_pReal,2.0_pReal)*(PI,0.0_pReal) !< Re(0.0), Im(2xPi) real(pReal), dimension(3,3), parameter, public :: & MATH_I3 = reshape([& 1.0_pReal,0.0_pReal,0.0_pReal, & 0.0_pReal,1.0_pReal,0.0_pReal, & 0.0_pReal,0.0_pReal,1.0_pReal & ],[3,3]) !< 3x3 Identity integer(pInt), dimension (2,6), parameter, private :: & mapMandel = reshape([& 1_pInt,1_pInt, & 2_pInt,2_pInt, & 3_pInt,3_pInt, & 1_pInt,2_pInt, & 2_pInt,3_pInt, & 1_pInt,3_pInt & ],[2,6]) !< arrangement in Mandel notation real(pReal), dimension(6), parameter, private :: & nrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal,& 1.414213562373095_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal ] !< weighting for Mandel notation (forward) real(pReal), dimension(6), parameter , public :: & invnrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal,& 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal ] !< weighting for Mandel notation (backward) integer(pInt), dimension (2,6), parameter, private :: & mapVoigt = reshape([& 1_pInt,1_pInt, & 2_pInt,2_pInt, & 3_pInt,3_pInt, & 2_pInt,3_pInt, & 1_pInt,3_pInt, & 1_pInt,2_pInt & ],[2,6]) !< arrangement in Voigt notation real(pReal), dimension(6), parameter, private :: & nrmVoigt = 1.0_pReal, & !< weighting for Voigt notation (forward) invnrmVoigt = 1.0_pReal !< weighting for Voigt notation (backward) integer(pInt), dimension (2,9), parameter, private :: & mapPlain = reshape([& 1_pInt,1_pInt, & 1_pInt,2_pInt, & 1_pInt,3_pInt, & 2_pInt,1_pInt, & 2_pInt,2_pInt, & 2_pInt,3_pInt, & 3_pInt,1_pInt, & 3_pInt,2_pInt, & 3_pInt,3_pInt & ],[2,9]) !< arrangement in Plain notation public :: & math_init, & math_qsort, & math_range, & math_identity2nd, & math_identity4th, & math_civita, & math_delta, & math_crossproduct, & math_tensorproduct33, & math_mul3x3, & math_mul6x6, & math_mul33xx33, & math_mul3333xx33, & math_mul3333xx3333, & math_mul33x33, & math_mul66x66, & math_mul99x99, & math_mul33x3, & math_mul33x3_complex, & math_mul66x6 , & math_exp33 , & math_transpose33, & math_inv33, & math_invert33, & math_invSym3333, & math_invert, & math_symmetric33, & math_symmetric66, & math_skew33, & math_spherical33, & math_deviatoric33, & math_equivStrain33, & math_equivStress33, & math_trace33, & math_det33, & math_Plain33to9, & math_Plain9to33, & math_Mandel33to6, & math_Mandel6to33, & math_Plain3333to99, & math_Plain99to3333, & math_Mandel66toPlain66, & math_Plain66toMandel66, & math_Mandel3333to66, & math_Mandel66to3333, & math_Voigt66to3333, & math_qRand, & math_qMul, & math_qDot, & math_qConj, & math_qInv, & math_qRot, & math_RtoEuler, & math_RtoQ, & math_EulerToR, & math_EulerToQ, & math_EulerAxisAngleToR, & math_axisAngleToR, & math_EulerAxisAngleToQ, & math_axisAngleToQ, & math_qToRodrig, & math_qToEuler, & math_qToEulerAxisAngle, & math_qToAxisAngle, & math_qToR, & math_EulerMisorientation, & math_sampleRandomOri, & math_sampleGaussOri, & math_sampleFiberOri, & math_sampleGaussVar, & math_symmetricEulers, & math_eigenvectorBasisSym33, & math_eigenvectorBasisSym, & math_eigenValuesVectorsSym33, & math_eigenValuesVectorsSym, & math_rotationalPart33, & math_invariantsSym33, & math_eigenvaluesSym33, & math_factorial, & math_binomial, & math_multinomial, & math_volTetrahedron, & math_areaTriangle, & math_rotate_forward33, & math_rotate_backward33, & math_rotate_forward3333 private :: & math_partition, & halton, & halton_memory, & halton_ndim_set, & halton_seed_set, & i_to_halton, & prime contains !-------------------------------------------------------------------------------------------------- !> @brief initialization of random seed generator !-------------------------------------------------------------------------------------------------- subroutine math_init use, intrinsic :: iso_fortran_env ! to get compiler_version and compiler_options (at least for gfortran 4.6 at the moment) use prec, only: tol_math_check use numerics, only: & worldrank, & fixedSeed use IO, only: IO_error, IO_timeStamp implicit none integer(pInt) :: i real(pReal), dimension(3,3) :: R,R2 real(pReal), dimension(3) :: Eulers,v real(pReal), dimension(4) :: q,q2,axisangle,randTest ! the following variables are system dependend and shound NOT be pInt integer :: randSize ! gfortran requires a variable length to compile integer, dimension(:), allocatable :: randInit ! if recalculations of former randomness (with given seed) is necessary ! comment the first random_seed call out, set randSize to 1, and use ifort character(len=64) :: error_msg write(6,'(/,a)') ' <<<+- math init -+>>>' write(6,'(a15,a)') ' Current time: ',IO_timeStamp() #include "compilation_info.f90" call random_seed(size=randSize) if (allocated(randInit)) deallocate(randInit) allocate(randInit(randSize)) if (fixedSeed > 0_pInt) then randInit(1:randSize) = int(fixedSeed) ! fixedSeed is of type pInt, randInit not call random_seed(put=randInit) else call random_seed() call random_seed(get = randInit) randInit(2:randSize) = randInit(1) call random_seed(put = randInit) endif do i = 1_pInt, 4_pInt call random_number(randTest(i)) enddo write(6,'(a,I)') 'size of random seed: ', randSize do i =1, randSize write(6,'(a,I,I)') 'value of random seed: ', i, randInit(i) enddo write(6,'(a,4(/,26x,f17.14),/)') ' start of random sequence: ', randTest call random_seed(put = randInit) call halton_seed_set(int(randInit(1), pInt)) call halton_ndim_set(3_pInt) ! --- check rotation dictionary --- q = math_qRand() ! random quaternion ! +++ q -> a -> q +++ axisangle = math_qToAxisAngle(q) q2 = math_axisAngleToQ(axisangle(1:3),axisangle(4)) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(401_pInt,ext_msg=error_msg) endif ! +++ q -> R -> q +++ R = math_qToR(q) q2 = math_RtoQ(R) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(402_pInt,ext_msg=error_msg) endif ! +++ q -> euler -> q +++ Eulers = math_qToEuler(q) q2 = math_EulerToQ(Eulers) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(403_pInt,ext_msg=error_msg) endif ! +++ R -> euler -> R +++ Eulers = math_RtoEuler(R) R2 = math_EulerToR(Eulers) if ( any(abs( R-R2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',maxval(abs( R-R2)) call IO_error(404_pInt,ext_msg=error_msg) endif ! +++ check rotation sense of q and R +++ q = math_qRand() ! random quaternion call halton(3_pInt,v) ! random vector R = math_qToR(q) if (any(abs(math_mul33x3(R,v) - math_qRot(q,v)) > tol_math_check)) then write(6,'(a,4(f8.3,1x))') 'q',q call IO_error(409_pInt) endif end subroutine math_init !-------------------------------------------------------------------------------------------------- !> @brief Quicksort algorithm for two-dimensional integer arrays ! Sorting is done with respect to array(1,:) ! and keeps array(2:N,:) linked to it. !-------------------------------------------------------------------------------------------------- recursive subroutine math_qsort(a, istart, iend) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in) :: istart,iend integer(pInt) :: ipivot if (istart < iend) then ipivot = math_partition(a,istart, iend) call math_qsort(a, istart, ipivot-1_pInt) call math_qsort(a, ipivot+1_pInt, iend) endif end subroutine math_qsort !-------------------------------------------------------------------------------------------------- !> @brief Partitioning required for quicksort !-------------------------------------------------------------------------------------------------- integer(pInt) function math_partition(a, istart, iend) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in) :: istart,iend integer(pInt) :: d,i,j,k,x,tmp d = int(size(a,1_pInt), pInt) ! number of linked data ! set the starting and ending points, and the pivot point i = istart j = iend x = a(1,istart) do ! find the first element on the right side less than or equal to the pivot point do j = j, istart, -1_pInt if (a(1,j) <= x) exit enddo ! find the first element on the left side greater than the pivot point do i = i, iend if (a(1,i) > x) exit enddo if (i < j) then ! if the indexes do not cross, exchange values do k = 1_pInt,d tmp = a(k,i) a(k,i) = a(k,j) a(k,j) = tmp enddo else ! if they do cross, exchange left value with pivot and return with the partition index do k = 1_pInt,d tmp = a(k,istart) a(k,istart) = a(k,j) a(k,j) = tmp enddo math_partition = j return endif enddo end function math_partition !-------------------------------------------------------------------------------------------------- !> @brief range of integers starting at one !-------------------------------------------------------------------------------------------------- pure function math_range(N) implicit none integer(pInt), intent(in) :: N !< length of range integer(pInt) :: i integer(pInt), dimension(N) :: math_range math_range = [(i,i=1_pInt,N)] end function math_range !-------------------------------------------------------------------------------------------------- !> @brief second rank identity tensor of specified dimension !-------------------------------------------------------------------------------------------------- pure function math_identity2nd(dimen) implicit none integer(pInt), intent(in) :: dimen !< tensor dimension integer(pInt) :: i real(pReal), dimension(dimen,dimen) :: math_identity2nd math_identity2nd = 0.0_pReal forall (i=1_pInt:dimen) math_identity2nd(i,i) = 1.0_pReal end function math_identity2nd !-------------------------------------------------------------------------------------------------- !> @brief symmetric fourth rank identity tensor of specified dimension ! from http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivative_of_a_second-order_tensor_with_respect_to_itself !-------------------------------------------------------------------------------------------------- pure function math_identity4th(dimen) implicit none integer(pInt), intent(in) :: dimen !< tensor dimension integer(pInt) :: i,j,k,l real(pReal), dimension(dimen,dimen,dimen,dimen) :: math_identity4th forall (i=1_pInt:dimen,j=1_pInt:dimen,k=1_pInt:dimen,l=1_pInt:dimen) math_identity4th(i,j,k,l) = & 0.5_pReal*(math_I3(i,k)*math_I3(j,l)+math_I3(i,l)*math_I3(j,k)) end function math_identity4th !-------------------------------------------------------------------------------------------------- !> @brief permutation tensor e_ijk used for computing cross product of two tensors ! e_ijk = 1 if even permutation of ijk ! e_ijk = -1 if odd permutation of ijk ! e_ijk = 0 otherwise !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_civita(i,j,k) implicit none integer(pInt), intent(in) :: i,j,k math_civita = 0.0_pReal if (((i == 1_pInt).and.(j == 2_pInt).and.(k == 3_pInt)) .or. & ((i == 2_pInt).and.(j == 3_pInt).and.(k == 1_pInt)) .or. & ((i == 3_pInt).and.(j == 1_pInt).and.(k == 2_pInt))) math_civita = 1.0_pReal if (((i == 1_pInt).and.(j == 3_pInt).and.(k == 2_pInt)) .or. & ((i == 2_pInt).and.(j == 1_pInt).and.(k == 3_pInt)) .or. & ((i == 3_pInt).and.(j == 2_pInt).and.(k == 1_pInt))) math_civita = -1.0_pReal end function math_civita !-------------------------------------------------------------------------------------------------- !> @brief kronecker delta function d_ij ! d_ij = 1 if i = j ! d_ij = 0 otherwise ! inspired by http://fortraninacworld.blogspot.de/2012/12/ternary-operator.html !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_delta(i,j) implicit none integer(pInt), intent (in) :: i,j math_delta = merge(0.0_pReal, 1.0_pReal, i /= j) end function math_delta !-------------------------------------------------------------------------------------------------- !> @brief cross product a x b !-------------------------------------------------------------------------------------------------- pure function math_crossproduct(A,B) implicit none real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: math_crossproduct math_crossproduct = [ A(2)*B(3) -A(3)*B(2), & A(3)*B(1) -A(1)*B(3), & A(1)*B(2) -A(2)*B(1) ] end function math_crossproduct !-------------------------------------------------------------------------------------------------- !> @brief tensor product A \otimes B of arbitrary sized vectors A and B !-------------------------------------------------------------------------------------------------- pure function math_tensorproduct(A,B) implicit none real(pReal), dimension(:), intent(in) :: A,B real(pReal), dimension(size(A,1),size(B,1)) :: math_tensorproduct integer(pInt) :: i,j forall (i=1_pInt:size(A,1),j=1_pInt:size(B,1)) math_tensorproduct(i,j) = A(i)*B(j) end function math_tensorproduct !-------------------------------------------------------------------------------------------------- !> @brief tensor product A \otimes B of leght-3 vectors A and B !-------------------------------------------------------------------------------------------------- pure function math_tensorproduct33(A,B) implicit none real(pReal), dimension(3,3) :: math_tensorproduct33 real(pReal), dimension(3), intent(in) :: A,B integer(pInt) :: i,j forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) math_tensorproduct33(i,j) = A(i)*B(j) end function math_tensorproduct33 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 3x3 = 1 !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_mul3x3(A,B) implicit none real(pReal), dimension(3), intent(in) :: A,B math_mul3x3 = sum(A*B) end function math_mul3x3 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 6x6 = 1 !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_mul6x6(A,B) implicit none real(pReal), dimension(6), intent(in) :: A,B math_mul6x6 = sum(A*B) end function math_mul6x6 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 33xx33 = 1 (double contraction --> ij * ij) !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_mul33xx33(A,B) implicit none real(pReal), dimension(3,3), intent(in) :: A,B integer(pInt) :: i,j real(pReal), dimension(3,3) :: C forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) C(i,j) = A(i,j) * B(i,j) math_mul33xx33 = sum(C) end function math_mul33xx33 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 3333x33 = 33 (double contraction --> ijkl *kl = ij) !-------------------------------------------------------------------------------------------------- pure function math_mul3333xx33(A,B) implicit none real(pReal), dimension(3,3) :: math_mul3333xx33 real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3), intent(in) :: B integer(pInt) :: i,j forall(i = 1_pInt:3_pInt,j = 1_pInt:3_pInt) & math_mul3333xx33(i,j) = sum(A(i,j,1:3,1:3)*B(1:3,1:3)) end function math_mul3333xx33 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 3333x3333 = 3333 (ijkl *klmn = ijmn) !-------------------------------------------------------------------------------------------------- pure function math_mul3333xx3333(A,B) implicit none integer(pInt) :: i,j,k,l real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3,3,3), intent(in) :: B real(pReal), dimension(3,3,3,3) :: math_mul3333xx3333 forall(i = 1_pInt:3_pInt,j = 1_pInt:3_pInt, k = 1_pInt:3_pInt, l= 1_pInt:3_pInt) & math_mul3333xx3333(i,j,k,l) = sum(A(i,j,1:3,1:3)*B(1:3,1:3,k,l)) end function math_mul3333xx3333 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 33x33 = 33 !-------------------------------------------------------------------------------------------------- pure function math_mul33x33(A,B) implicit none real(pReal), dimension(3,3) :: math_mul33x33 real(pReal), dimension(3,3), intent(in) :: A,B integer(pInt) :: i,j forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) & math_mul33x33(i,j) = A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) end function math_mul33x33 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 66x66 = 66 !-------------------------------------------------------------------------------------------------- pure function math_mul66x66(A,B) implicit none real(pReal), dimension(6,6) :: math_mul66x66 real(pReal), dimension(6,6), intent(in) :: A,B integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) math_mul66x66(i,j) = & A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) + & A(i,4)*B(4,j) + A(i,5)*B(5,j) + A(i,6)*B(6,j) end function math_mul66x66 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 99x99 = 99 !-------------------------------------------------------------------------------------------------- pure function math_mul99x99(A,B) implicit none real(pReal), dimension(9,9) :: math_mul99x99 real(pReal), dimension(9,9), intent(in) :: A,B integer(pInt) i,j forall (i=1_pInt:9_pInt,j=1_pInt:9_pInt) math_mul99x99(i,j) = & A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) + & A(i,4)*B(4,j) + A(i,5)*B(5,j) + A(i,6)*B(6,j) + & A(i,7)*B(7,j) + A(i,8)*B(8,j) + A(i,9)*B(9,j) end function math_mul99x99 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 33x3 = 3 !-------------------------------------------------------------------------------------------------- pure function math_mul33x3(A,B) implicit none real(pReal), dimension(3) :: math_mul33x3 real(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3), intent(in) :: B integer(pInt) :: i forall (i=1_pInt:3_pInt) math_mul33x3(i) = sum(A(i,1:3)*B) end function math_mul33x3 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication complex(33) x real(3) = complex(3) !-------------------------------------------------------------------------------------------------- pure function math_mul33x3_complex(A,B) implicit none complex(pReal), dimension(3) :: math_mul33x3_complex complex(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3), intent(in) :: B integer(pInt) :: i forall (i=1_pInt:3_pInt) math_mul33x3_complex(i) = sum(A(i,1:3)*cmplx(B,0.0_pReal,pReal)) end function math_mul33x3_complex !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 66x6 = 6 !-------------------------------------------------------------------------------------------------- pure function math_mul66x6(A,B) implicit none real(pReal), dimension(6) :: math_mul66x6 real(pReal), dimension(6,6), intent(in) :: A real(pReal), dimension(6), intent(in) :: B integer(pInt) :: i forall (i=1_pInt:6_pInt) math_mul66x6(i) = & A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3) + & A(i,4)*B(4) + A(i,5)*B(5) + A(i,6)*B(6) end function math_mul66x6 !-------------------------------------------------------------------------------------------------- !> @brief 3x3 matrix exponential up to series approximation order n (default 5) !-------------------------------------------------------------------------------------------------- pure function math_exp33(A,n) implicit none integer(pInt) :: i,order integer(pInt), intent(in), optional :: n real(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3,3) :: B,math_exp33 real(pReal) :: invfac order = merge(n,5_pInt,present(n)) B = math_I3 ! init invfac = 1.0_pReal ! 0! math_exp33 = B ! A^0 = eye2 do i = 1_pInt,n invfac = invfac/real(i,pReal) ! invfac = 1/i! B = math_mul33x33(B,A) math_exp33 = math_exp33 + invfac*B ! exp = SUM (A^i)/i! enddo end function math_exp33 !-------------------------------------------------------------------------------------------------- !> @brief transposition of a 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_transpose33(A) implicit none real(pReal),dimension(3,3) :: math_transpose33 real(pReal),dimension(3,3),intent(in) :: A integer(pInt) :: i,j forall(i=1_pInt:3_pInt, j=1_pInt:3_pInt) math_transpose33(i,j) = A(j,i) end function math_transpose33 !-------------------------------------------------------------------------------------------------- !> @brief Cramer inversion of 33 matrix (function) ! direct Cramer inversion of matrix A. ! returns all zeroes if not possible, i.e. if det close to zero !-------------------------------------------------------------------------------------------------- pure function math_inv33(A) use prec, only: & dNeq0 implicit none real(pReal),dimension(3,3),intent(in) :: A real(pReal) :: DetA real(pReal),dimension(3,3) :: math_inv33 math_inv33(1,1) = A(2,2) * A(3,3) - A(2,3) * A(3,2) math_inv33(2,1) = -A(2,1) * A(3,3) + A(2,3) * A(3,1) math_inv33(3,1) = A(2,1) * A(3,2) - A(2,2) * A(3,1) DetA = A(1,1) * math_inv33(1,1) + A(1,2) * math_inv33(2,1) + A(1,3) * math_inv33(3,1) if (dNeq0(DetA)) then math_inv33(1,2) = -A(1,2) * A(3,3) + A(1,3) * A(3,2) math_inv33(2,2) = A(1,1) * A(3,3) - A(1,3) * A(3,1) math_inv33(3,2) = -A(1,1) * A(3,2) + A(1,2) * A(3,1) math_inv33(1,3) = A(1,2) * A(2,3) - A(1,3) * A(2,2) math_inv33(2,3) = -A(1,1) * A(2,3) + A(1,3) * A(2,1) math_inv33(3,3) = A(1,1) * A(2,2) - A(1,2) * A(2,1) math_inv33 = math_inv33/DetA else math_inv33 = 0.0_pReal endif end function math_inv33 !-------------------------------------------------------------------------------------------------- !> @brief Cramer inversion of 33 matrix (subroutine) ! direct Cramer inversion of matrix A. ! also returns determinant ! returns error if not possible, i.e. if det close to zero !-------------------------------------------------------------------------------------------------- pure subroutine math_invert33(A, InvA, DetA, error) use prec, only: & dEq0 implicit none logical, intent(out) :: error real(pReal),dimension(3,3),intent(in) :: A real(pReal),dimension(3,3),intent(out) :: InvA real(pReal), intent(out) :: DetA InvA(1,1) = A(2,2) * A(3,3) - A(2,3) * A(3,2) InvA(2,1) = -A(2,1) * A(3,3) + A(2,3) * A(3,1) InvA(3,1) = A(2,1) * A(3,2) - A(2,2) * A(3,1) DetA = A(1,1) * InvA(1,1) + A(1,2) * InvA(2,1) + A(1,3) * InvA(3,1) if (dEq0(DetA)) then InvA = 0.0_pReal error = .true. else InvA(1,2) = -A(1,2) * A(3,3) + A(1,3) * A(3,2) InvA(2,2) = A(1,1) * A(3,3) - A(1,3) * A(3,1) InvA(3,2) = -A(1,1) * A(3,2) + A(1,2) * A(3,1) InvA(1,3) = A(1,2) * A(2,3) - A(1,3) * A(2,2) InvA(2,3) = -A(1,1) * A(2,3) + A(1,3) * A(2,1) InvA(3,3) = A(1,1) * A(2,2) - A(1,2) * A(2,1) InvA = InvA/DetA error = .false. endif end subroutine math_invert33 !-------------------------------------------------------------------------------------------------- !> @brief Inversion of symmetriced 3x3x3x3 tensor. !-------------------------------------------------------------------------------------------------- function math_invSym3333(A) use IO, only: & IO_error implicit none real(pReal),dimension(3,3,3,3) :: math_invSym3333 real(pReal),dimension(3,3,3,3),intent(in) :: A integer(pInt) :: ierr integer(pInt), dimension(6) :: ipiv6 real(pReal), dimension(6,6) :: temp66_Real real(pReal), dimension(6) :: work6 external :: & dgetrf, & dgetri temp66_real = math_Mandel3333to66(A) call dgetrf(6,6,temp66_real,6,ipiv6,ierr) call dgetri(6,temp66_real,6,ipiv6,work6,6,ierr) if (ierr == 0_pInt) then math_invSym3333 = math_Mandel66to3333(temp66_real) else call IO_error(400_pInt, ext_msg = 'math_invSym3333') endif end function math_invSym3333 !-------------------------------------------------------------------------------------------------- !> @brief invert matrix of arbitrary dimension !-------------------------------------------------------------------------------------------------- subroutine math_invert(myDim,A, InvA, error) implicit none integer(pInt), intent(in) :: myDim real(pReal), dimension(myDim,myDim), intent(in) :: A integer(pInt) :: ierr integer(pInt), dimension(myDim) :: ipiv real(pReal), dimension(myDim) :: work real(pReal), dimension(myDim,myDim), intent(out) :: invA logical, intent(out) :: error external :: & dgetrf, & dgetri invA = A call dgetrf(myDim,myDim,invA,myDim,ipiv,ierr) call dgetri(myDim,InvA,myDim,ipiv,work,myDim,ierr) error = merge(.true.,.false., ierr /= 0_pInt) ! http://fortraninacworld.blogspot.de/2012/12/ternary-operator.html end subroutine math_invert !-------------------------------------------------------------------------------------------------- !> @brief symmetrize a 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_symmetric33(m) implicit none real(pReal), dimension(3,3) :: math_symmetric33 real(pReal), dimension(3,3), intent(in) :: m math_symmetric33 = 0.5_pReal * (m + transpose(m)) end function math_symmetric33 !-------------------------------------------------------------------------------------------------- !> @brief symmetrize a 66 matrix !-------------------------------------------------------------------------------------------------- pure function math_symmetric66(m) implicit none real(pReal), dimension(6,6) :: math_symmetric66 real(pReal), dimension(6,6), intent(in) :: m math_symmetric66 = 0.5_pReal * (m + transpose(m)) end function math_symmetric66 !-------------------------------------------------------------------------------------------------- !> @brief skew part of a 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_skew33(m) implicit none real(pReal), dimension(3,3) :: math_skew33 real(pReal), dimension(3,3), intent(in) :: m math_skew33 = m - math_symmetric33(m) end function math_skew33 !-------------------------------------------------------------------------------------------------- !> @brief hydrostatic part of a 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_spherical33(m) implicit none real(pReal), dimension(3,3) :: math_spherical33 real(pReal), dimension(3,3), intent(in) :: m math_spherical33 = math_I3 * math_trace33(m)/3.0_pReal end function math_spherical33 !-------------------------------------------------------------------------------------------------- !> @brief deviatoric part of a 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_deviatoric33(m) implicit none real(pReal), dimension(3,3) :: math_deviatoric33 real(pReal), dimension(3,3), intent(in) :: m math_deviatoric33 = m - math_spherical33(m) end function math_deviatoric33 !-------------------------------------------------------------------------------------------------- !> @brief equivalent scalar quantity of a full symmetric strain tensor !-------------------------------------------------------------------------------------------------- pure function math_equivStrain33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal), dimension(3) :: e,s real(pReal) :: math_equivStrain33 real(pReal), parameter :: TWOTHIRD = 2.0_pReal/3.0_pReal e = [2.0_pReal*m(1,1)-m(2,2)-m(3,3), & 2.0_pReal*m(2,2)-m(3,3)-m(1,1), & 2.0_pReal*m(3,3)-m(1,1)-m(2,2)]/3.0_pReal s = [m(1,2),m(2,3),m(1,3)]*2.0_pReal math_equivStrain33 = TWOTHIRD*(1.50_pReal*(sum(e**2.0_pReal)) + & 0.75_pReal*(sum(s**2.0_pReal)))**(0.5_pReal) end function math_equivStrain33 !-------------------------------------------------------------------------------------------------- !> @brief von Mises equivalent of a full symmetric stress tensor !-------------------------------------------------------------------------------------------------- pure function math_equivStress33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal) :: math_equivStress33 math_equivStress33 =( ( (m(1,1)-m(2,2))**2.0_pReal + & (m(2,2)-m(3,3))**2.0_pReal + & (m(3,3)-m(1,1))**2.0_pReal + & 6.0_pReal*( m(1,2)**2.0_pReal + & m(2,3)**2.0_pReal + & m(1,3)**2.0_pReal & ) & )**0.5_pReal & )/sqrt(2.0_pReal) end function math_equivStress33 !-------------------------------------------------------------------------------------------------- !> @brief trace of a 33 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_trace33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m math_trace33 = m(1,1) + m(2,2) + m(3,3) end function math_trace33 !-------------------------------------------------------------------------------------------------- !> @brief determinant of a 33 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_det33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m math_det33 = m(1,1)* (m(2,2)*m(3,3)-m(2,3)*m(3,2)) & - m(1,2)* (m(2,1)*m(3,3)-m(2,3)*m(3,1)) & + m(1,3)* (m(2,1)*m(3,2)-m(2,2)*m(3,1)) end function math_det33 !-------------------------------------------------------------------------------------------------- !> @brief determinant of a symmetric 33 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_detSym33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m math_detSym33 = -(m(1,1)*m(2,3)**2_pInt + m(2,2)*m(1,3)**2_pInt + m(3,3)*m(1,2)**2_pInt) & + m(1,1)*m(2,2)*m(3,3) - 2.0_pReal * m(1,2)*m(1,3)*m(2,3) end function math_detSym33 !-------------------------------------------------------------------------------------------------- !> @brief convert 33 matrix into vector 9 !-------------------------------------------------------------------------------------------------- pure function math_Plain33to9(m33) implicit none real(pReal), dimension(9) :: math_Plain33to9 real(pReal), dimension(3,3), intent(in) :: m33 integer(pInt) :: i forall (i=1_pInt:9_pInt) math_Plain33to9(i) = m33(mapPlain(1,i),mapPlain(2,i)) end function math_Plain33to9 !-------------------------------------------------------------------------------------------------- !> @brief convert Plain 9 back to 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_Plain9to33(v9) implicit none real(pReal), dimension(3,3) :: math_Plain9to33 real(pReal), dimension(9), intent(in) :: v9 integer(pInt) :: i forall (i=1_pInt:9_pInt) math_Plain9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i) end function math_Plain9to33 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 33 matrix into Mandel vector 6 !-------------------------------------------------------------------------------------------------- pure function math_Mandel33to6(m33) implicit none real(pReal), dimension(6) :: math_Mandel33to6 real(pReal), dimension(3,3), intent(in) :: m33 integer(pInt) :: i forall (i=1_pInt:6_pInt) math_Mandel33to6(i) = nrmMandel(i)*m33(mapMandel(1,i),mapMandel(2,i)) end function math_Mandel33to6 !-------------------------------------------------------------------------------------------------- !> @brief convert Mandel 6 back to symmetric 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_Mandel6to33(v6) implicit none real(pReal), dimension(6), intent(in) :: v6 real(pReal), dimension(3,3) :: math_Mandel6to33 integer(pInt) :: i forall (i=1_pInt:6_pInt) math_Mandel6to33(mapMandel(1,i),mapMandel(2,i)) = invnrmMandel(i)*v6(i) math_Mandel6to33(mapMandel(2,i),mapMandel(1,i)) = invnrmMandel(i)*v6(i) end forall end function math_Mandel6to33 !-------------------------------------------------------------------------------------------------- !> @brief convert 3333 tensor into plain matrix 99 !-------------------------------------------------------------------------------------------------- pure function math_Plain3333to99(m3333) implicit none real(pReal), dimension(3,3,3,3), intent(in) :: m3333 real(pReal), dimension(9,9) :: math_Plain3333to99 integer(pInt) :: i,j forall (i=1_pInt:9_pInt,j=1_pInt:9_pInt) math_Plain3333to99(i,j) = & m3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j)) end function math_Plain3333to99 !-------------------------------------------------------------------------------------------------- !> @brief plain matrix 99 into 3333 tensor !-------------------------------------------------------------------------------------------------- pure function math_Plain99to3333(m99) implicit none real(pReal), dimension(9,9), intent(in) :: m99 real(pReal), dimension(3,3,3,3) :: math_Plain99to3333 integer(pInt) :: i,j forall (i=1_pInt:9_pInt,j=1_pInt:9_pInt) math_Plain99to3333(mapPlain(1,i),mapPlain(2,i),& mapPlain(1,j),mapPlain(2,j)) = m99(i,j) end function math_Plain99to3333 !-------------------------------------------------------------------------------------------------- !> @brief convert Mandel matrix 66 into Plain matrix 66 !-------------------------------------------------------------------------------------------------- pure function math_Mandel66toPlain66(m66) implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(6,6) :: math_Mandel66toPlain66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) & math_Mandel66toPlain66(i,j) = invnrmMandel(i) * invnrmMandel(j) * m66(i,j) end function math_Mandel66toPlain66 !-------------------------------------------------------------------------------------------------- !> @brief convert Plain matrix 66 into Mandel matrix 66 !-------------------------------------------------------------------------------------------------- pure function math_Plain66toMandel66(m66) implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(6,6) :: math_Plain66toMandel66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) & math_Plain66toMandel66(i,j) = nrmMandel(i) * nrmMandel(j) * m66(i,j) end function math_Plain66toMandel66 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 3333 tensor into Mandel matrix 66 !-------------------------------------------------------------------------------------------------- pure function math_Mandel3333to66(m3333) implicit none real(pReal), dimension(3,3,3,3), intent(in) :: m3333 real(pReal), dimension(6,6) :: math_Mandel3333to66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) math_Mandel3333to66(i,j) = & nrmMandel(i)*nrmMandel(j)*m3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) end function math_Mandel3333to66 !-------------------------------------------------------------------------------------------------- !> @brief convert Mandel matrix 66 back to symmetric 3333 tensor !-------------------------------------------------------------------------------------------------- pure function math_Mandel66to3333(m66) implicit none real(pReal), dimension(3,3,3,3) :: math_Mandel66to3333 real(pReal), dimension(6,6), intent(in) :: m66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) = & invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(1,j),mapMandel(2,j)) = & invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(2,j),mapMandel(1,j)) = & invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(2,j),mapMandel(1,j)) = & invnrmMandel(i)*invnrmMandel(j)*m66(i,j) end forall end function math_Mandel66to3333 !-------------------------------------------------------------------------------------------------- !> @brief convert Voigt matrix 66 back to symmetric 3333 tensor !-------------------------------------------------------------------------------------------------- pure function math_Voigt66to3333(m66) implicit none real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333 real(pReal), dimension(6,6), intent(in) :: m66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(1,j),mapVoigt(2,j)) = & invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(1,j),mapVoigt(2,j)) = & invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(2,j),mapVoigt(1,j)) = & invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(2,j),mapVoigt(1,j)) = & invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) end forall end function math_Voigt66to3333 !-------------------------------------------------------------------------------------------------- !> @brief random quaternion !-------------------------------------------------------------------------------------------------- function math_qRand() implicit none real(pReal), dimension(4) :: math_qRand real(pReal), dimension(3) :: rnd call halton(3_pInt,rnd) math_qRand = [cos(2.0_pReal*PI*rnd(1))*sqrt(rnd(3)), sin(2.0_pReal*PI*rnd(2))*sqrt(1.0_pReal-rnd(3)), cos(2.0_pReal*PI*rnd(2))*sqrt(1.0_pReal-rnd(3)), sin(2.0_pReal*PI*rnd(1))*sqrt(rnd(3))] end function math_qRand !-------------------------------------------------------------------------------------------------- !> @brief quaternion multiplication q1xq2 = q12 !-------------------------------------------------------------------------------------------------- pure function math_qMul(A,B) implicit none real(pReal), dimension(4) :: math_qMul real(pReal), dimension(4), intent(in) :: A, B math_qMul = [ A(1)*B(1) - A(2)*B(2) - A(3)*B(3) - A(4)*B(4), & A(1)*B(2) + A(2)*B(1) + A(3)*B(4) - A(4)*B(3), & A(1)*B(3) - A(2)*B(4) + A(3)*B(1) + A(4)*B(2), & A(1)*B(4) + A(2)*B(3) - A(3)*B(2) + A(4)*B(1) ] end function math_qMul !-------------------------------------------------------------------------------------------------- !> @brief quaternion dotproduct !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_qDot(A,B) implicit none real(pReal), dimension(4), intent(in) :: A, B math_qDot = sum(A*B) end function math_qDot !-------------------------------------------------------------------------------------------------- !> @brief quaternion conjugation !-------------------------------------------------------------------------------------------------- pure function math_qConj(Q) implicit none real(pReal), dimension(4) :: math_qConj real(pReal), dimension(4), intent(in) :: Q math_qConj = [Q(1), -Q(2:4)] end function math_qConj !-------------------------------------------------------------------------------------------------- !> @brief quaternion norm !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_qNorm(Q) implicit none real(pReal), dimension(4), intent(in) :: Q math_qNorm = norm2(Q) end function math_qNorm !-------------------------------------------------------------------------------------------------- !> @brief quaternion inversion !-------------------------------------------------------------------------------------------------- pure function math_qInv(Q) use prec, only: & dNeq0 implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(4) :: math_qInv real(pReal) :: squareNorm math_qInv = 0.0_pReal squareNorm = math_qDot(Q,Q) if (dNeq0(squareNorm)) math_qInv = math_qConj(Q) / squareNorm end function math_qInv !-------------------------------------------------------------------------------------------------- !> @brief action of a quaternion on a vector (rotate vector v with Q) !-------------------------------------------------------------------------------------------------- pure function math_qRot(Q,v) implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3), intent(in) :: v real(pReal), dimension(3) :: math_qRot real(pReal), dimension(4,4) :: T integer(pInt) :: i, j do i = 1_pInt,4_pInt do j = 1_pInt,i T(i,j) = Q(i) * Q(j) enddo enddo math_qRot = [-v(1)*(T(3,3)+T(4,4)) + v(2)*(T(3,2)-T(4,1)) + v(3)*(T(4,2)+T(3,1)), & v(1)*(T(3,2)+T(4,1)) - v(2)*(T(2,2)+T(4,4)) + v(3)*(T(4,3)-T(2,1)), & v(1)*(T(4,2)-T(3,1)) + v(2)*(T(4,3)+T(2,1)) - v(3)*(T(2,2)+T(3,3))] math_qRot = 2.0_pReal * math_qRot + v end function math_qRot !-------------------------------------------------------------------------------------------------- !> @brief Euler angles (in radians) from rotation matrix !> @details rotation matrix is meant to represent a PASSIVE rotation, !> composed of INTRINSIC rotations around the axes of the !> rotating reference frame !> (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_RtoEuler(R) implicit none real(pReal), dimension (3,3), intent(in) :: R real(pReal), dimension(3) :: math_RtoEuler real(pReal) :: sqhkl, squvw, sqhk sqhkl=sqrt(R(1,3)*R(1,3)+R(2,3)*R(2,3)+R(3,3)*R(3,3)) squvw=sqrt(R(1,1)*R(1,1)+R(2,1)*R(2,1)+R(3,1)*R(3,1)) sqhk =sqrt(R(1,3)*R(1,3)+R(2,3)*R(2,3)) ! calculate PHI math_RtoEuler(2) = acos(math_limit(R(3,3)/sqhkl,-1.0_pReal, 1.0_pReal)) if((math_RtoEuler(2) < 1.0e-8_pReal) .or. (pi-math_RtoEuler(2) < 1.0e-8_pReal)) then math_RtoEuler(3) = 0.0_pReal math_RtoEuler(1) = acos(math_limit(R(1,1)/squvw, -1.0_pReal, 1.0_pReal)) if(R(2,1) > 0.0_pReal) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) else math_RtoEuler(3) = acos(math_limit(R(2,3)/sqhk, -1.0_pReal, 1.0_pReal)) if(R(1,3) < 0.0) math_RtoEuler(3) = 2.0_pReal*pi-math_RtoEuler(3) math_RtoEuler(1) = acos(math_limit(-R(3,2)/sin(math_RtoEuler(2)), -1.0_pReal, 1.0_pReal)) if(R(3,1) < 0.0) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) end if end function math_RtoEuler !-------------------------------------------------------------------------------------------------- !> @brief converts a rotation matrix into a quaternion (w+ix+jy+kz) !> @details math adopted from http://arxiv.org/pdf/math/0701759v1.pdf !-------------------------------------------------------------------------------------------------- pure function math_RtoQ(R) implicit none real(pReal), dimension(3,3), intent(in) :: R real(pReal), dimension(4) :: absQ, math_RtoQ real(pReal) :: max_absQ integer, dimension(1) :: largest !no pInt, maxloc returns integer default math_RtoQ = 0.0_pReal absQ = [+ R(1,1) + R(2,2) + R(3,3), & + R(1,1) - R(2,2) - R(3,3), & - R(1,1) + R(2,2) - R(3,3), & - R(1,1) - R(2,2) + R(3,3)] + 1.0_pReal largest = maxloc(absQ) largestComponent: select case(largest(1)) case (1) largestComponent !1---------------------------------- math_RtoQ(2) = R(3,2) - R(2,3) math_RtoQ(3) = R(1,3) - R(3,1) math_RtoQ(4) = R(2,1) - R(1,2) case (2) largestComponent math_RtoQ(1) = R(3,2) - R(2,3) !2---------------------------------- math_RtoQ(3) = R(2,1) + R(1,2) math_RtoQ(4) = R(1,3) + R(3,1) case (3) largestComponent math_RtoQ(1) = R(1,3) - R(3,1) math_RtoQ(2) = R(2,1) + R(1,2) !3---------------------------------- math_RtoQ(4) = R(3,2) + R(2,3) case (4) largestComponent math_RtoQ(1) = R(2,1) - R(1,2) math_RtoQ(2) = R(1,3) + R(3,1) math_RtoQ(3) = R(2,3) + R(3,2) !4---------------------------------- end select largestComponent max_absQ = 0.5_pReal * sqrt(absQ(largest(1))) math_RtoQ = math_RtoQ * 0.25_pReal / max_absQ math_RtoQ(largest(1)) = max_absQ end function math_RtoQ !-------------------------------------------------------------------------------------------------- !> @brief rotation matrix from Euler angles (in radians) !> @details rotation matrix is meant to represent a PASSIVE rotation, !> @details composed of INTRINSIC rotations around the axes of the !> @details rotating reference frame !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_EulerToR(Euler) implicit none real(pReal), dimension(3), intent(in) :: Euler real(pReal), dimension(3,3) :: math_EulerToR real(pReal) c1, c, c2, s1, s, s2 C1 = cos(Euler(1)) C = cos(Euler(2)) C2 = cos(Euler(3)) S1 = sin(Euler(1)) S = sin(Euler(2)) S2 = sin(Euler(3)) math_EulerToR(1,1)=C1*C2-S1*S2*C math_EulerToR(1,2)=-C1*S2-S1*C2*C math_EulerToR(1,3)=S1*S math_EulerToR(2,1)=S1*C2+C1*S2*C math_EulerToR(2,2)=-S1*S2+C1*C2*C math_EulerToR(2,3)=-C1*S math_EulerToR(3,1)=S2*S math_EulerToR(3,2)=C2*S math_EulerToR(3,3)=C math_EulerToR = transpose(math_EulerToR) ! convert to passive rotation end function math_EulerToR !-------------------------------------------------------------------------------------------------- !> @brief quaternion (w+ix+jy+kz) from 3-1-3 Euler angles (in radians) !> @details quaternion is meant to represent a PASSIVE rotation, !> @details composed of INTRINSIC rotations around the axes of the !> @details rotating reference frame !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_EulerToQ(eulerangles) implicit none real(pReal), dimension(3), intent(in) :: eulerangles real(pReal), dimension(4) :: math_EulerToQ real(pReal), dimension(3) :: halfangles real(pReal) :: c, s halfangles = 0.5_pReal * eulerangles c = cos(halfangles(2)) s = sin(halfangles(2)) math_EulerToQ= [cos(halfangles(1)+halfangles(3)) * c, & cos(halfangles(1)-halfangles(3)) * s, & sin(halfangles(1)-halfangles(3)) * s, & sin(halfangles(1)+halfangles(3)) * c ] math_EulerToQ = math_qConj(math_EulerToQ) ! convert to passive rotation end function math_EulerToQ !-------------------------------------------------------------------------------------------------- !> @brief rotation matrix from axis and angle (in radians) !> @details rotation matrix is meant to represent a ACTIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !> @details formula for active rotation taken from http://mathworld.wolfram.com/RodriguesRotationFormula.html !-------------------------------------------------------------------------------------------------- pure function math_axisAngleToR(axis,omega) implicit none real(pReal), dimension(3,3) :: math_axisAngleToR real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega real(pReal), dimension(3) :: axisNrm real(pReal) :: norm,s,c,c1 norm = norm2(axis) if (norm > 1.0e-8_pReal) then ! non-zero rotation axisNrm = axis/norm ! normalize axis to be sure s = sin(omega) c = cos(omega) c1 = 1.0_pReal - c math_axisAngleToR(1,1) = c + c1*axisNrm(1)**2.0_pReal math_axisAngleToR(1,2) = -s*axisNrm(3) + c1*axisNrm(1)*axisNrm(2) math_axisAngleToR(1,3) = s*axisNrm(2) + c1*axisNrm(1)*axisNrm(3) math_axisAngleToR(2,1) = s*axisNrm(3) + c1*axisNrm(2)*axisNrm(1) math_axisAngleToR(2,2) = c + c1*axisNrm(2)**2.0_pReal math_axisAngleToR(2,3) = -s*axisNrm(1) + c1*axisNrm(2)*axisNrm(3) math_axisAngleToR(3,1) = -s*axisNrm(2) + c1*axisNrm(3)*axisNrm(1) math_axisAngleToR(3,2) = s*axisNrm(1) + c1*axisNrm(3)*axisNrm(2) math_axisAngleToR(3,3) = c + c1*axisNrm(3)**2.0_pReal else math_axisAngleToR = math_I3 endif end function math_axisAngleToR !-------------------------------------------------------------------------------------------------- !> @brief rotation matrix from axis and angle (in radians) !> @details rotation matrix is meant to represent a PASSIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_EulerAxisAngleToR(axis,omega) implicit none real(pReal), dimension(3,3) :: math_EulerAxisAngleToR real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega math_EulerAxisAngleToR = transpose(math_axisAngleToR(axis,omega)) ! convert to passive rotation end function math_EulerAxisAngleToR !-------------------------------------------------------------------------------------------------- !> @brief quaternion (w+ix+jy+kz) from Euler axis and angle (in radians) !> @details quaternion is meant to represent a PASSIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !> @details formula for active rotation taken from !> @details http://en.wikipedia.org/wiki/Rotation_representation_%28mathematics%29#Rodrigues_parameters !-------------------------------------------------------------------------------------------------- pure function math_EulerAxisAngleToQ(axis,omega) implicit none real(pReal), dimension(4) :: math_EulerAxisAngleToQ real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega math_EulerAxisAngleToQ = math_qConj(math_axisAngleToQ(axis,omega)) ! convert to passive rotation end function math_EulerAxisAngleToQ !-------------------------------------------------------------------------------------------------- !> @brief quaternion (w+ix+jy+kz) from axis and angle (in radians) !> @details quaternion is meant to represent an ACTIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !> @details formula for active rotation taken from !> @details http://en.wikipedia.org/wiki/Rotation_representation_%28mathematics%29#Rodrigues_parameters !-------------------------------------------------------------------------------------------------- pure function math_axisAngleToQ(axis,omega) implicit none real(pReal), dimension(4) :: math_axisAngleToQ real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega real(pReal), dimension(3) :: axisNrm real(pReal) :: norm norm = sqrt(math_mul3x3(axis,axis)) rotation: if (norm > 1.0e-8_pReal) then axisNrm = axis/norm ! normalize axis to be sure math_axisAngleToQ = [cos(0.5_pReal*omega), sin(0.5_pReal*omega) * axisNrm(1:3)] else rotation math_axisAngleToQ = [1.0_pReal,0.0_pReal,0.0_pReal,0.0_pReal] endif rotation end function math_axisAngleToQ !-------------------------------------------------------------------------------------------------- !> @brief orientation matrix from quaternion (w+ix+jy+kz) !> @details taken from http://arxiv.org/pdf/math/0701759v1.pdf !> @details see also http://en.wikipedia.org/wiki/Rotation_formalisms_in_three_dimensions !-------------------------------------------------------------------------------------------------- pure function math_qToR(q) implicit none real(pReal), dimension(4), intent(in) :: q real(pReal), dimension(3,3) :: math_qToR, T,S integer(pInt) :: i, j forall (i = 1_pInt:3_pInt, j = 1_pInt:3_pInt) & T(i,j) = q(i+1_pInt) * q(j+1_pInt) S = reshape( [0.0_pReal, -q(4), q(3), & q(4), 0.0_pReal, -q(2), & -q(3), q(2), 0.0_pReal],[3,3]) ! notation is transposed math_qToR = (2.0_pReal * q(1)*q(1) - 1.0_pReal) * math_I3 & + 2.0_pReal * T - 2.0_pReal * q(1) * S end function math_qToR !-------------------------------------------------------------------------------------------------- !> @brief 3-1-3 Euler angles (in radians) from quaternion (w+ix+jy+kz) !> @details quaternion is meant to represent a PASSIVE rotation, !> @details composed of INTRINSIC rotations around the axes of the !> @details rotating reference frame !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_qToEuler(qPassive) implicit none real(pReal), dimension(4), intent(in) :: qPassive real(pReal), dimension(4) :: q real(pReal), dimension(3) :: math_qToEuler q = math_qConj(qPassive) ! convert to active rotation, since formulas are defined for active rotations math_qToEuler(2) = acos(1.0_pReal-2.0_pReal*(q(2)**2+q(3)**2)) if (abs(math_qToEuler(2)) < 1.0e-6_pReal) then math_qToEuler(1) = sign(2.0_pReal*acos(math_limit(q(1),-1.0_pReal, 1.0_pReal)),q(4)) math_qToEuler(3) = 0.0_pReal else math_qToEuler(1) = atan2(+q(1)*q(3)+q(2)*q(4), q(1)*q(2)-q(3)*q(4)) math_qToEuler(3) = atan2(-q(1)*q(3)+q(2)*q(4), q(1)*q(2)+q(3)*q(4)) endif math_qToEuler = merge(math_qToEuler + [2.0_pReal*PI, PI, 2.0_pReal*PI], & ! ensure correct range math_qToEuler, math_qToEuler<0.0_pReal) end function math_qToEuler !-------------------------------------------------------------------------------------------------- !> @brief axis-angle (x, y, z, ang in radians) from quaternion (w+ix+jy+kz) !> @details quaternion is meant to represent an ACTIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !> @details formula for active rotation taken from !> @details http://en.wikipedia.org/wiki/Rotation_representation_%28mathematics%29#Rodrigues_parameters !-------------------------------------------------------------------------------------------------- pure function math_qToAxisAngle(Q) implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal) :: halfAngle, sinHalfAngle real(pReal), dimension(4) :: math_qToAxisAngle halfAngle = acos(math_limit(Q(1),-1.0_pReal,1.0_pReal)) sinHalfAngle = sin(halfAngle) smallRotation: if (sinHalfAngle <= 1.0e-4_pReal) then math_qToAxisAngle = 0.0_pReal else smallRotation math_qToAxisAngle= [ Q(2:4)/sinHalfAngle, halfAngle*2.0_pReal] endif smallRotation end function math_qToAxisAngle !-------------------------------------------------------------------------------------------------- !> @brief Euler axis-angle (x, y, z, ang in radians) from quaternion (w+ix+jy+kz) !> @details quaternion is meant to represent a PASSIVE rotation !> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions) !-------------------------------------------------------------------------------------------------- pure function math_qToEulerAxisAngle(qPassive) implicit none real(pReal), dimension(4), intent(in) :: qPassive real(pReal), dimension(4) :: q real(pReal), dimension(4) :: math_qToEulerAxisAngle q = math_qConj(qPassive) ! convert to active rotation math_qToEulerAxisAngle = math_qToAxisAngle(q) end function math_qToEulerAxisAngle !-------------------------------------------------------------------------------------------------- !> @brief Rodrigues vector (x, y, z) from unit quaternion (w+ix+jy+kz) !-------------------------------------------------------------------------------------------------- pure function math_qToRodrig(Q) use prec, only: & DAMASK_NaN, & tol_math_check implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3) :: math_qToRodrig math_qToRodrig = merge(Q(2:4)/Q(1),DAMASK_NaN,abs(Q(1)) > tol_math_check) ! NaN for 180 deg since Rodrig is unbound end function math_qToRodrig !-------------------------------------------------------------------------------------------------- !> @brief misorientation angle between two sets of Euler angles !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_EulerMisorientation(EulerA,EulerB) implicit none real(pReal), dimension(3), intent(in) :: EulerA,EulerB real(pReal), dimension(3,3) :: r real(pReal) :: tr r = math_mul33x33(math_EulerToR(EulerB),transpose(math_EulerToR(EulerA))) tr = (math_trace33(r)-1.0_pReal)*0.4999999_pReal math_EulerMisorientation = abs(0.5_pReal*PI-asin(tr)) end function math_EulerMisorientation !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Euler space !-------------------------------------------------------------------------------------------------- function math_sampleRandomOri() implicit none real(pReal), dimension(3) :: math_sampleRandomOri, rnd call halton(3_pInt,rnd) math_sampleRandomOri = [rnd(1)*2.0_pReal*PI, & acos(2.0_pReal*rnd(2)-1.0_pReal), & rnd(3)*2.0_pReal*PI] end function math_sampleRandomOri !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Gauss component with noise (in radians) half-width !-------------------------------------------------------------------------------------------------- function math_sampleGaussOri(center,noise) use prec, only: & tol_math_check implicit none real(pReal), intent(in) :: noise real(pReal), dimension(3), intent(in) :: center real(pReal) :: cosScatter,scatter real(pReal), dimension(3) :: math_sampleGaussOri, disturb real(pReal), dimension(3), parameter :: ORIGIN = [0.0_pReal,0.0_pReal,0.0_pReal] real(pReal), dimension(5) :: rnd integer(pInt) :: i if (abs(noise) < tol_math_check) then math_sampleGaussOri = center return endif ! Helming uses different distribution with Bessel functions ! therefore the gauss scatter width has to be scaled differently scatter = 0.95_pReal * noise cosScatter = cos(scatter) do call halton(5_pInt,rnd) forall (i=1_pInt:3_pInt) rnd(i) = 2.0_pReal*rnd(i)-1.0_pReal ! expand 1:3 to range [-1,+1] disturb = [ scatter * rnd(1), & ! phi1 sign(1.0_pReal,rnd(2))*acos(cosScatter+(1.0_pReal-cosScatter)*rnd(4)), & ! Phi scatter * rnd(2)] ! phi2 if (rnd(5) <= exp(-1.0_pReal*(math_EulerMisorientation(ORIGIN,disturb)/scatter)**2_pReal)) exit enddo math_sampleGaussOri = math_RtoEuler(math_mul33x33(math_EulerToR(disturb),math_EulerToR(center))) end function math_sampleGaussOri !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Fiber component with noise (in radians) !-------------------------------------------------------------------------------------------------- function math_sampleFiberOri(alpha,beta,noise) use prec, only: & tol_math_check implicit none real(pReal), dimension(3) :: math_sampleFiberOri, fiberInC,fiberInS,axis real(pReal), dimension(2), intent(in) :: alpha,beta real(pReal), dimension(6) :: rnd real(pReal), dimension(3,3) :: oRot,fRot,pRot real(pReal) :: noise, scatter, cos2Scatter, angle integer(pInt), dimension(2,3), parameter :: ROTMAP = reshape([2_pInt,3_pInt,& 3_pInt,1_pInt,& 1_pInt,2_pInt],[2,3]) integer(pInt) :: i ! Helming uses different distribution with Bessel functions ! therefore the gauss scatter width has to be scaled differently scatter = 0.95_pReal * noise cos2Scatter = cos(2.0_pReal*scatter) ! fiber axis in crystal coordinate system fiberInC = [ sin(alpha(1))*cos(alpha(2)) , & sin(alpha(1))*sin(alpha(2)), & cos(alpha(1))] ! fiber axis in sample coordinate system fiberInS = [ sin(beta(1))*cos(beta(2)), & sin(beta(1))*sin(beta(2)), & cos(beta(1))] ! ---# rotation matrix from sample to crystal system #--- angle = -acos(dot_product(fiberInC,fiberInS)) if(abs(angle) > tol_math_check) then ! rotation axis between sample and crystal system (cross product) forall(i=1_pInt:3_pInt) axis(i) = fiberInC(ROTMAP(1,i))*fiberInS(ROTMAP(2,i))-fiberInC(ROTMAP(2,i))*fiberInS(ROTMAP(1,i)) oRot = math_EulerAxisAngleToR(math_crossproduct(fiberInC,fiberInS),angle) else oRot = math_I3 end if ! ---# rotation matrix about fiber axis (random angle) #--- do call halton(6_pInt,rnd) fRot = math_EulerAxisAngleToR(fiberInS,rnd(1)*2.0_pReal*pi) ! ---# rotation about random axis perpend to fiber #--- ! random axis pependicular to fiber axis axis(1:2) = rnd(2:3) if (abs(fiberInS(3)) > tol_math_check) then axis(3)=-(axis(1)*fiberInS(1)+axis(2)*fiberInS(2))/fiberInS(3) else if(abs(fiberInS(2)) > tol_math_check) then axis(3)=axis(2) axis(2)=-(axis(1)*fiberInS(1)+axis(3)*fiberInS(3))/fiberInS(2) else if(abs(fiberInS(1)) > tol_math_check) then axis(3)=axis(1) axis(1)=-(axis(2)*fiberInS(2)+axis(3)*fiberInS(3))/fiberInS(1) end if ! scattered rotation angle if (noise > 0.0_pReal) then angle = acos(cos2Scatter+(1.0_pReal-cos2Scatter)*rnd(4)) if (rnd(5) <= exp(-1.0_pReal*(angle/scatter)**2.0_pReal)) exit else angle = 0.0_pReal exit end if enddo if (rnd(6) <= 0.5) angle = -angle pRot = math_EulerAxisAngleToR(axis,angle) ! ---# apply the three rotations #--- math_sampleFiberOri = math_RtoEuler(math_mul33x33(pRot,math_mul33x33(fRot,oRot))) end function math_sampleFiberOri !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Gauss variable !-------------------------------------------------------------------------------------------------- real(pReal) function math_sampleGaussVar(meanvalue, stddev, width) use prec, only: & tol_math_check implicit none real(pReal), intent(in) :: meanvalue, & ! meanvalue of gauss distribution stddev ! standard deviation of gauss distribution real(pReal), intent(in), optional :: width ! width of considered values as multiples of standard deviation real(pReal), dimension(2) :: rnd ! random numbers real(pReal) :: scatter, & ! normalized scatter around meanvalue myWidth if (abs(stddev) < tol_math_check) then math_sampleGaussVar = meanvalue return endif myWidth = merge(width,3.0_pReal,present(width)) ! use +-3*sigma as default value for scatter if not given do call halton(2_pInt, rnd) scatter = myWidth * (2.0_pReal * rnd(1) - 1.0_pReal) if (rnd(2) <= exp(-0.5_pReal * scatter ** 2.0_pReal)) exit ! test if scattered value is drawn enddo math_sampleGaussVar = scatter * stddev end function math_sampleGaussVar !-------------------------------------------------------------------------------------------------- !> @brief symmetrically equivalent Euler angles for given sample symmetry 1:triclinic, 2:monoclinic, 4:orthotropic !-------------------------------------------------------------------------------------------------- pure function math_symmetricEulers(sym,Euler) implicit none integer(pInt), intent(in) :: sym real(pReal), dimension(3), intent(in) :: Euler real(pReal), dimension(3,3) :: math_symmetricEulers integer(pInt) :: i,j math_symmetricEulers(1,1) = PI+Euler(1) math_symmetricEulers(2,1) = Euler(2) math_symmetricEulers(3,1) = Euler(3) math_symmetricEulers(1,2) = PI-Euler(1) math_symmetricEulers(2,2) = PI-Euler(2) math_symmetricEulers(3,2) = PI+Euler(3) math_symmetricEulers(1,3) = 2.0_pReal*PI-Euler(1) math_symmetricEulers(2,3) = PI-Euler(2) math_symmetricEulers(3,3) = PI+Euler(3) forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) math_symmetricEulers(j,i) = modulo(math_symmetricEulers(j,i),2.0_pReal*pi) select case (sym) case (4_pInt) ! all done case (2_pInt) ! return only first math_symmetricEulers(1:3,2:3) = 0.0_pReal case default ! return blank math_symmetricEulers = 0.0_pReal end select end function math_symmetricEulers !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues and eigenvectors of symmetric matrix m !-------------------------------------------------------------------------------------------------- subroutine math_eigenValuesVectorsSym(m,values,vectors,error) implicit none real(pReal), dimension(:,:), intent(in) :: m real(pReal), dimension(size(m,1)), intent(out) :: values real(pReal), dimension(size(m,1),size(m,1)), intent(out) :: vectors logical, intent(out) :: error integer(pInt) :: info real(pReal), dimension((64+2)*size(m,1)) :: work ! block size of 64 taken from http://www.netlib.org/lapack/double/dsyev.f external :: & dsyev vectors = m ! copy matrix to input (doubles as output) array call dsyev('V','U',size(m,1),vectors,size(m,1),values,work,(64+2)*size(m,1),info) error = (info == 0_pInt) end subroutine math_eigenValuesVectorsSym !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues and eigenvectors of symmetric 33 matrix m using an analytical expression !> and the general LAPACK powered version for arbritrary sized matrices as fallback !> @author Joachim Kopp, Max–Planck–Institut für Kernphysik, Heidelberg (Copyright (C) 2006) !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @details See http://arxiv.org/abs/physics/0610206 (DSYEVH3) !-------------------------------------------------------------------------------------------------- subroutine math_eigenValuesVectorsSym33(m,values,vectors) implicit none real(pReal), dimension(3,3),intent(in) :: m real(pReal), dimension(3), intent(out) :: values real(pReal), dimension(3,3),intent(out) :: vectors real(pReal) :: T, U, norm, threshold logical :: error values = math_eigenvaluesSym33(m) vectors(1:3,2) = [ m(1, 2) * m(2, 3) - m(1, 3) * m(2, 2), & m(1, 3) * m(1, 2) - m(2, 3) * m(1, 1), & m(1, 2)**2_pInt] T = maxval(abs(values)) U = max(T, T**2_pInt) threshold = sqrt(5.68e-14_pReal * U**2_pInt) ! Calculate first eigenvector by the formula v[0] = (m - lambda[0]).e1 x (m - lambda[0]).e2 vectors(1:3,1) = [ vectors(1,2) + m(1, 3) * values(1), & vectors(2,2) + m(2, 3) * values(1), & (m(1,1) - values(1)) * (m(2,2) - values(1)) - vectors(3,2)] norm = norm2(vectors(1:3, 1)) fallback1: if(norm < threshold) then call math_eigenValuesVectorsSym(m,values,vectors,error) return endif fallback1 vectors(1:3,1) = vectors(1:3, 1) / norm ! Calculate second eigenvector by the formula v[1] = (m - lambda[1]).e1 x (m - lambda[1]).e2 vectors(1:3,2) = [ vectors(1,2) + m(1, 3) * values(2), & vectors(2,2) + m(2, 3) * values(2), & (m(1,1) - values(2)) * (m(2,2) - values(2)) - vectors(3,2)] norm = norm2(vectors(1:3, 2)) fallback2: if(norm < threshold) then call math_eigenValuesVectorsSym(m,values,vectors,error) return endif fallback2 vectors(1:3,2) = vectors(1:3, 2) / norm ! Calculate third eigenvector according to v[2] = v[0] x v[1] vectors(1:3,3) = math_crossproduct(vectors(1:3,1),vectors(1:3,2)) end subroutine math_eigenValuesVectorsSym33 !-------------------------------------------------------------------------------------------------- !> @brief eigenvector basis of symmetric matrix m !-------------------------------------------------------------------------------------------------- function math_eigenvectorBasisSym(m) implicit none real(pReal), dimension(:,:), intent(in) :: m real(pReal), dimension(size(m,1)) :: values real(pReal), dimension(size(m,1),size(m,1)) :: vectors real(pReal), dimension(size(m,1),size(m,1)) :: math_eigenvectorBasisSym logical :: error integer(pInt) :: i math_eigenvectorBasisSym = 0.0_pReal call math_eigenValuesVectorsSym(m,values,vectors,error) if(error) return do i=1_pInt, size(m,1) math_eigenvectorBasisSym = math_eigenvectorBasisSym & + sqrt(values(i)) * math_tensorproduct(vectors(:,i),vectors(:,i)) enddo end function math_eigenvectorBasisSym !-------------------------------------------------------------------------------------------------- !> @brief eigenvector basis of symmetric 33 matrix m !-------------------------------------------------------------------------------------------------- function math_eigenvectorBasisSym33(m) implicit none real(pReal), dimension(3,3) :: math_eigenvectorBasisSym33 real(pReal), dimension(3) :: invariants, values real(pReal), dimension(3,3), intent(in) :: m real(pReal) :: P, Q, rho, phi real(pReal), parameter :: TOL=1.e-14_pReal real(pReal), dimension(3,3,3) :: N, EB invariants = math_invariantsSym33(m) EB = 0.0_pReal P = invariants(2)-invariants(1)**2.0_pReal/3.0_pReal Q = -2.0_pReal/27.0_pReal*invariants(1)**3.0_pReal+product(invariants(1:2))/3.0_pReal-invariants(3) threeSimilarEigenvalues: if(all(abs([P,Q]) < TOL)) then values = invariants(1)/3.0_pReal ! this is not really correct, but at least the basis is correct EB(1,1,1)=1.0_pReal EB(2,2,2)=1.0_pReal EB(3,3,3)=1.0_pReal else threeSimilarEigenvalues rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal phi=acos(math_limit(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal)) values = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* & [cos(phi/3.0_pReal), & cos((phi+2.0_pReal*PI)/3.0_pReal), & cos((phi+4.0_pReal*PI)/3.0_pReal) & ] + invariants(1)/3.0_pReal N(1:3,1:3,1) = m-values(1)*math_I3 N(1:3,1:3,2) = m-values(2)*math_I3 N(1:3,1:3,3) = m-values(3)*math_I3 twoSimilarEigenvalues: if(abs(values(1)-values(2)) < TOL) then EB(1:3,1:3,3)=math_mul33x33(N(1:3,1:3,1),N(1:3,1:3,2))/ & ((values(3)-values(1))*(values(3)-values(2))) EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,3) elseif(abs(values(2)-values(3)) < TOL) then twoSimilarEigenvalues EB(1:3,1:3,1)=math_mul33x33(N(1:3,1:3,2),N(1:3,1:3,3))/ & ((values(1)-values(2))*(values(1)-values(3))) EB(1:3,1:3,2)=math_I3-EB(1:3,1:3,1) elseif(abs(values(3)-values(1)) < TOL) then twoSimilarEigenvalues EB(1:3,1:3,2)=math_mul33x33(N(1:3,1:3,1),N(1:3,1:3,3))/ & ((values(2)-values(1))*(values(2)-values(3))) EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,2) else twoSimilarEigenvalues EB(1:3,1:3,1)=math_mul33x33(N(1:3,1:3,2),N(1:3,1:3,3))/ & ((values(1)-values(2))*(values(1)-values(3))) EB(1:3,1:3,2)=math_mul33x33(N(1:3,1:3,1),N(1:3,1:3,3))/ & ((values(2)-values(1))*(values(2)-values(3))) EB(1:3,1:3,3)=math_mul33x33(N(1:3,1:3,1),N(1:3,1:3,2))/ & ((values(3)-values(1))*(values(3)-values(2))) endif twoSimilarEigenvalues endif threeSimilarEigenvalues math_eigenvectorBasisSym33 = sqrt(values(1)) * EB(1:3,1:3,1) & + sqrt(values(2)) * EB(1:3,1:3,2) & + sqrt(values(3)) * EB(1:3,1:3,3) end function math_eigenvectorBasisSym33 !-------------------------------------------------------------------------------------------------- !> @brief rotational part from polar decomposition of 33 tensor m !-------------------------------------------------------------------------------------------------- function math_rotationalPart33(m) use prec, only: & dEq0 use IO, only: & IO_warning implicit none real(pReal), intent(in), dimension(3,3) :: m real(pReal), dimension(3,3) :: math_rotationalPart33 real(pReal), dimension(3,3) :: U , Uinv U = math_eigenvectorBasisSym33(math_mul33x33(transpose(m),m)) Uinv = math_inv33(U) inversionFailed: if (all(dEq0(Uinv))) then math_rotationalPart33 = math_I3 call IO_warning(650_pInt) else inversionFailed math_rotationalPart33 = math_mul33x33(m,Uinv) endif inversionFailed end function math_rotationalPart33 !-------------------------------------------------------------------------------------------------- !> @brief Eigenvalues of symmetric matrix m ! will return NaN on error !-------------------------------------------------------------------------------------------------- function math_eigenvaluesSym(m) use prec, only: & DAMASK_NaN implicit none real(pReal), dimension(:,:), intent(in) :: m real(pReal), dimension(size(m,1)) :: math_eigenvaluesSym real(pReal), dimension(size(m,1),size(m,1)) :: vectors integer(pInt) :: info real(pReal), dimension((64+2)*size(m,1)) :: work ! block size of 64 taken from http://www.netlib.org/lapack/double/dsyev.f external :: & dsyev vectors = m ! copy matrix to input (doubles as output) array call dsyev('N','U',size(m,1),vectors,size(m,1),math_eigenvaluesSym,work,(64+2)*size(m,1),info) if (info /= 0_pInt) math_eigenvaluesSym = DAMASK_NaN end function math_eigenvaluesSym !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues of symmetric 33 matrix m using an analytical expression !> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH !> @details similar to http://arxiv.org/abs/physics/0610206 (DSYEVC3) !> but apparently more stable solution and has general LAPACK powered version for arbritrary sized !> matrices as fallback !-------------------------------------------------------------------------------------------------- function math_eigenvaluesSym33(m) implicit none real(pReal), intent(in), dimension(3,3) :: m real(pReal), dimension(3) :: math_eigenvaluesSym33,invariants real(pReal) :: P, Q, rho, phi real(pReal), parameter :: TOL=1.e-14_pReal invariants = math_invariantsSym33(m) ! invariants are coefficients in characteristic polynomial apart for the sign of c0 and c2 in http://arxiv.org/abs/physics/0610206 P = invariants(2)-invariants(1)**2.0_pReal/3.0_pReal ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK) Q = -2.0_pReal/27.0_pReal*invariants(1)**3.0_pReal+product(invariants(1:2))/3.0_pReal-invariants(3)! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK) if(all(abs([P,Q]) < TOL)) then math_eigenvaluesSym33 = math_eigenvaluesSym(m) else rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal phi=acos(math_limit(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal)) math_eigenvaluesSym33 = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* & [cos(phi/3.0_pReal), & cos((phi+2.0_pReal*PI)/3.0_pReal), & cos((phi+4.0_pReal*PI)/3.0_pReal) & ] + invariants(1)/3.0_pReal endif end function math_eigenvaluesSym33 !-------------------------------------------------------------------------------------------------- !> @brief invariants of symmetrix 33 matrix m !-------------------------------------------------------------------------------------------------- pure function math_invariantsSym33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal), dimension(3) :: math_invariantsSym33 math_invariantsSym33(1) = math_trace33(m) math_invariantsSym33(2) = m(1,1)*m(2,2) + m(1,1)*m(3,3) + m(2,2)*m(3,3) & -(m(1,2)**2 + m(1,3)**2 + m(2,3)**2) math_invariantsSym33(3) = math_detSym33(m) end function math_invariantsSym33 !-------------------------------------------------------------------------------------------------- !> @brief computes the next element in the Halton sequence. !> @author John Burkardt !-------------------------------------------------------------------------------------------------- subroutine halton(ndim, r) implicit none integer(pInt), intent(in) :: ndim !< dimension of the element real(pReal), intent(out), dimension(ndim) :: r !< next element of the current Halton sequence integer(pInt), dimension(ndim) :: base integer(pInt) :: seed integer(pInt), dimension(1) :: value_halton call halton_memory ('GET', 'SEED', 1_pInt, value_halton) seed = value_halton(1) call halton_memory ('GET', 'BASE', ndim, base) call i_to_halton (seed, base, ndim, r) value_halton(1) = 1_pInt call halton_memory ('INC', 'SEED', 1_pInt, value_halton) end subroutine halton !-------------------------------------------------------------------------------------------------- !> @brief sets or returns quantities associated with the Halton sequence. !> @details If action_halton is 'SET' and action_halton is 'BASE', then NDIM is input, and !> @details is the number of entries in value_halton to be put into BASE. !> @details If action_halton is 'SET', then on input, value_halton contains values to be assigned !> @details to the internal variable. !> @details If action_halton is 'GET', then on output, value_halton contains the values of !> @details the specified internal variable. !> @details If action_halton is 'INC', then on input, value_halton contains the increment to !> @details be added to the specified internal variable. !> @author John Burkardt !-------------------------------------------------------------------------------------------------- subroutine halton_memory (action_halton, name_halton, ndim, value_halton) implicit none character(len = *), intent(in) :: & action_halton, & !< desired action: GET the value of a particular quantity, SET the value of a particular quantity, INC the value of a particular quantity (only for SEED) name_halton !< name of the quantity: BASE: Halton base(s), NDIM: spatial dimension, SEED: current Halton seed integer(pInt), dimension(*), intent(inout) :: value_halton integer(pInt), allocatable, save, dimension(:) :: base logical, save :: first_call = .true. integer(pInt), intent(in) :: ndim !< dimension of the quantity integer(pInt):: i integer(pInt), save :: ndim_save = 0_pInt, seed = 1_pInt if (first_call) then ndim_save = 1_pInt allocate(base(ndim_save)) base(1) = 2_pInt first_call = .false. endif !-------------------------------------------------------------------------------------------------- ! Set if(action_halton(1:1) == 'S' .or. action_halton(1:1) == 's') then if(name_halton(1:1) == 'B' .or. name_halton(1:1) == 'b') then if(ndim_save /= ndim) then deallocate(base) ndim_save = ndim allocate(base(ndim_save)) endif base(1:ndim) = value_halton(1:ndim) elseif(name_halton(1:1) == 'N' .or. name_halton(1:1) == 'n') then if(ndim_save /= value_halton(1)) then deallocate(base) ndim_save = value_halton(1) allocate(base(ndim_save)) do i = 1_pInt, ndim_save base(i) = prime (i) enddo else ndim_save = value_halton(1) endif elseif(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then seed = value_halton(1) endif !-------------------------------------------------------------------------------------------------- ! Get elseif(action_halton(1:1) == 'G' .or. action_halton(1:1) == 'g') then if(name_halton(1:1) == 'B' .or. name_halton(1:1) == 'b') then if(ndim /= ndim_save) then deallocate(base) ndim_save = ndim allocate(base(ndim_save)) do i = 1_pInt, ndim_save base(i) = prime(i) enddo endif value_halton(1:ndim_save) = base(1:ndim_save) elseif(name_halton(1:1) == 'N' .or. name_halton(1:1) == 'n') then value_halton(1) = ndim_save elseif(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then value_halton(1) = seed endif !-------------------------------------------------------------------------------------------------- ! Increment elseif(action_halton(1:1) == 'I' .or. action_halton(1:1) == 'i') then if(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then seed = seed + value_halton(1) end if endif end subroutine halton_memory !-------------------------------------------------------------------------------------------------- !> @brief sets the dimension for a Halton sequence !> @author John Burkardt !-------------------------------------------------------------------------------------------------- subroutine halton_ndim_set (ndim) implicit none integer(pInt), intent(in) :: ndim !< dimension of the Halton vectors integer(pInt) :: value_halton(1) value_halton(1) = ndim call halton_memory ('SET', 'NDIM', 1_pInt, value_halton) end subroutine halton_ndim_set !-------------------------------------------------------------------------------------------------- !> @brief sets the seed for the Halton sequence. !> @details Calling HALTON repeatedly returns the elements of the Halton sequence in order, !> @details starting with element number 1. !> @details An internal counter, called SEED, keeps track of the next element to return. Each time !> @details is computed, and then SEED is incremented by 1. !> @details To restart the Halton sequence, it is only necessary to reset SEED to 1. It might also !> @details be desirable to reset SEED to some other value. This routine allows the user to specify !> @details any value of SEED. !> @details The default value of SEED is 1, which restarts the Halton sequence. !> @author John Burkardt !-------------------------------------------------------------------------------------------------- subroutine halton_seed_set(seed) implicit none integer(pInt), parameter :: NDIM = 1_pInt integer(pInt), intent(in) :: seed !< seed for the Halton sequence. integer(pInt) :: value_halton(ndim) value_halton(1) = seed call halton_memory ('SET', 'SEED', NDIM, value_halton) end subroutine halton_seed_set !-------------------------------------------------------------------------------------------------- !> @brief computes an element of a Halton sequence. !> @details Only the absolute value of SEED is considered. SEED = 0 is allowed, and returns R = 0. !> @details Halton Bases should be distinct prime numbers. This routine only checks that each base !> @details is greater than 1. !> @details Reference: !> @details J.H. Halton: On the efficiency of certain quasi-random sequences of points in evaluating !> @details multi-dimensional integrals, Numerische Mathematik, Volume 2, pages 84-90, 1960. !> @author John Burkardt !-------------------------------------------------------------------------------------------------- subroutine i_to_halton (seed, base, ndim, r) use IO, only: & IO_error implicit none integer(pInt), intent(in) :: ndim !< dimension of the sequence integer(pInt), intent(in), dimension(ndim) :: base !< Halton bases real(pReal), dimension(ndim) :: base_inv integer(pInt), dimension(ndim) :: digit real(pReal), dimension(ndim), intent(out) ::r !< the SEED-th element of the Halton sequence for the given bases integer(pInt) , intent(in):: seed !< index of the desired element integer(pInt), dimension(ndim) :: seed2 seed2(1:ndim) = abs(seed) r(1:ndim) = 0.0_pReal if (any (base(1:ndim) <= 1_pInt)) call IO_error(error_ID=405_pInt) base_inv(1:ndim) = 1.0_pReal / real (base(1:ndim), pReal) do while ( any ( seed2(1:ndim) /= 0_pInt) ) digit(1:ndim) = mod ( seed2(1:ndim), base(1:ndim)) r(1:ndim) = r(1:ndim) + real ( digit(1:ndim), pReal) * base_inv(1:ndim) base_inv(1:ndim) = base_inv(1:ndim) / real ( base(1:ndim), pReal) seed2(1:ndim) = seed2(1:ndim) / base(1:ndim) enddo end subroutine i_to_halton !-------------------------------------------------------------------------------------------------- !> @brief returns any of the first 1500 prime numbers. !> @details n <= 0 returns 1500, the index of the largest prime (12553) available. !> @details n = 0 is legal, returning PRIME = 1. !> @details Reference: !> @details Milton Abramowitz and Irene Stegun: Handbook of Mathematical Functions, !> @details US Department of Commerce, 1964, pages 870-873. !> @details Daniel Zwillinger: CRC Standard Mathematical Tables and Formulae, !> @details 30th Edition, CRC Press, 1996, pages 95-98. !> @author John Burkardt !-------------------------------------------------------------------------------------------------- integer(pInt) function prime(n) use IO, only: & IO_error implicit none integer(pInt), intent(in) :: n !< index of the desired prime number integer(pInt), parameter :: PRIME_MAX = 1500_pInt integer(pInt), save :: icall = 0_pInt integer(pInt), save, dimension(PRIME_MAX) :: npvec if (icall == 0_pInt) then icall = 1_pInt npvec = [& 2_pInt, 3_pInt, 5_pInt, 7_pInt, 11_pInt, 13_pInt, 17_pInt, 19_pInt, 23_pInt, 29_pInt, & 31_pInt, 37_pInt, 41_pInt, 43_pInt, 47_pInt, 53_pInt, 59_pInt, 61_pInt, 67_pInt, 71_pInt, & 73_pInt, 79_pInt, 83_pInt, 89_pInt, 97_pInt, 101_pInt, 103_pInt, 107_pInt, 109_pInt, 113_pInt, & 127_pInt, 131_pInt, 137_pInt, 139_pInt, 149_pInt, 151_pInt, 157_pInt, 163_pInt, 167_pInt, 173_pInt, & 179_pInt, 181_pInt, 191_pInt, 193_pInt, 197_pInt, 199_pInt, 211_pInt, 223_pInt, 227_pInt, 229_pInt, & 233_pInt, 239_pInt, 241_pInt, 251_pInt, 257_pInt, 263_pInt, 269_pInt, 271_pInt, 277_pInt, 281_pInt, & 283_pInt, 293_pInt, 307_pInt, 311_pInt, 313_pInt, 317_pInt, 331_pInt, 337_pInt, 347_pInt, 349_pInt, & 353_pInt, 359_pInt, 367_pInt, 373_pInt, 379_pInt, 383_pInt, 389_pInt, 397_pInt, 401_pInt, 409_pInt, & 419_pInt, 421_pInt, 431_pInt, 433_pInt, 439_pInt, 443_pInt, 449_pInt, 457_pInt, 461_pInt, 463_pInt, & 467_pInt, 479_pInt, 487_pInt, 491_pInt, 499_pInt, 503_pInt, 509_pInt, 521_pInt, 523_pInt, 541_pInt, & ! 101:200 547_pInt, 557_pInt, 563_pInt, 569_pInt, 571_pInt, 577_pInt, 587_pInt, 593_pInt, 599_pInt, 601_pInt, & 607_pInt, 613_pInt, 617_pInt, 619_pInt, 631_pInt, 641_pInt, 643_pInt, 647_pInt, 653_pInt, 659_pInt, & 661_pInt, 673_pInt, 677_pInt, 683_pInt, 691_pInt, 701_pInt, 709_pInt, 719_pInt, 727_pInt, 733_pInt, & 739_pInt, 743_pInt, 751_pInt, 757_pInt, 761_pInt, 769_pInt, 773_pInt, 787_pInt, 797_pInt, 809_pInt, & 811_pInt, 821_pInt, 823_pInt, 827_pInt, 829_pInt, 839_pInt, 853_pInt, 857_pInt, 859_pInt, 863_pInt, & 877_pInt, 881_pInt, 883_pInt, 887_pInt, 907_pInt, 911_pInt, 919_pInt, 929_pInt, 937_pInt, 941_pInt, & 947_pInt, 953_pInt, 967_pInt, 971_pInt, 977_pInt, 983_pInt, 991_pInt, 997_pInt, 1009_pInt, 1013_pInt, & 1019_pInt, 1021_pInt, 1031_pInt, 1033_pInt, 1039_pInt, 1049_pInt, 1051_pInt, 1061_pInt, 1063_pInt, 1069_pInt, & 1087_pInt, 1091_pInt, 1093_pInt, 1097_pInt, 1103_pInt, 1109_pInt, 1117_pInt, 1123_pInt, 1129_pInt, 1151_pInt, & 1153_pInt, 1163_pInt, 1171_pInt, 1181_pInt, 1187_pInt, 1193_pInt, 1201_pInt, 1213_pInt, 1217_pInt, 1223_pInt, & ! 201:300 1229_pInt, 1231_pInt, 1237_pInt, 1249_pInt, 1259_pInt, 1277_pInt, 1279_pInt, 1283_pInt, 1289_pInt, 1291_pInt, & 1297_pInt, 1301_pInt, 1303_pInt, 1307_pInt, 1319_pInt, 1321_pInt, 1327_pInt, 1361_pInt, 1367_pInt, 1373_pInt, & 1381_pInt, 1399_pInt, 1409_pInt, 1423_pInt, 1427_pInt, 1429_pInt, 1433_pInt, 1439_pInt, 1447_pInt, 1451_pInt, & 1453_pInt, 1459_pInt, 1471_pInt, 1481_pInt, 1483_pInt, 1487_pInt, 1489_pInt, 1493_pInt, 1499_pInt, 1511_pInt, & 1523_pInt, 1531_pInt, 1543_pInt, 1549_pInt, 1553_pInt, 1559_pInt, 1567_pInt, 1571_pInt, 1579_pInt, 1583_pInt, & 1597_pInt, 1601_pInt, 1607_pInt, 1609_pInt, 1613_pInt, 1619_pInt, 1621_pInt, 1627_pInt, 1637_pInt, 1657_pInt, & 1663_pInt, 1667_pInt, 1669_pInt, 1693_pInt, 1697_pInt, 1699_pInt, 1709_pInt, 1721_pInt, 1723_pInt, 1733_pInt, & 1741_pInt, 1747_pInt, 1753_pInt, 1759_pInt, 1777_pInt, 1783_pInt, 1787_pInt, 1789_pInt, 1801_pInt, 1811_pInt, & 1823_pInt, 1831_pInt, 1847_pInt, 1861_pInt, 1867_pInt, 1871_pInt, 1873_pInt, 1877_pInt, 1879_pInt, 1889_pInt, & 1901_pInt, 1907_pInt, 1913_pInt, 1931_pInt, 1933_pInt, 1949_pInt, 1951_pInt, 1973_pInt, 1979_pInt, 1987_pInt, & ! 301:400 1993_pInt, 1997_pInt, 1999_pInt, 2003_pInt, 2011_pInt, 2017_pInt, 2027_pInt, 2029_pInt, 2039_pInt, 2053_pInt, & 2063_pInt, 2069_pInt, 2081_pInt, 2083_pInt, 2087_pInt, 2089_pInt, 2099_pInt, 2111_pInt, 2113_pInt, 2129_pInt, & 2131_pInt, 2137_pInt, 2141_pInt, 2143_pInt, 2153_pInt, 2161_pInt, 2179_pInt, 2203_pInt, 2207_pInt, 2213_pInt, & 2221_pInt, 2237_pInt, 2239_pInt, 2243_pInt, 2251_pInt, 2267_pInt, 2269_pInt, 2273_pInt, 2281_pInt, 2287_pInt, & 2293_pInt, 2297_pInt, 2309_pInt, 2311_pInt, 2333_pInt, 2339_pInt, 2341_pInt, 2347_pInt, 2351_pInt, 2357_pInt, & 2371_pInt, 2377_pInt, 2381_pInt, 2383_pInt, 2389_pInt, 2393_pInt, 2399_pInt, 2411_pInt, 2417_pInt, 2423_pInt, & 2437_pInt, 2441_pInt, 2447_pInt, 2459_pInt, 2467_pInt, 2473_pInt, 2477_pInt, 2503_pInt, 2521_pInt, 2531_pInt, & 2539_pInt, 2543_pInt, 2549_pInt, 2551_pInt, 2557_pInt, 2579_pInt, 2591_pInt, 2593_pInt, 2609_pInt, 2617_pInt, & 2621_pInt, 2633_pInt, 2647_pInt, 2657_pInt, 2659_pInt, 2663_pInt, 2671_pInt, 2677_pInt, 2683_pInt, 2687_pInt, & 2689_pInt, 2693_pInt, 2699_pInt, 2707_pInt, 2711_pInt, 2713_pInt, 2719_pInt, 2729_pInt, 2731_pInt, 2741_pInt, & ! 401:500 2749_pInt, 2753_pInt, 2767_pInt, 2777_pInt, 2789_pInt, 2791_pInt, 2797_pInt, 2801_pInt, 2803_pInt, 2819_pInt, & 2833_pInt, 2837_pInt, 2843_pInt, 2851_pInt, 2857_pInt, 2861_pInt, 2879_pInt, 2887_pInt, 2897_pInt, 2903_pInt, & 2909_pInt, 2917_pInt, 2927_pInt, 2939_pInt, 2953_pInt, 2957_pInt, 2963_pInt, 2969_pInt, 2971_pInt, 2999_pInt, & 3001_pInt, 3011_pInt, 3019_pInt, 3023_pInt, 3037_pInt, 3041_pInt, 3049_pInt, 3061_pInt, 3067_pInt, 3079_pInt, & 3083_pInt, 3089_pInt, 3109_pInt, 3119_pInt, 3121_pInt, 3137_pInt, 3163_pInt, 3167_pInt, 3169_pInt, 3181_pInt, & 3187_pInt, 3191_pInt, 3203_pInt, 3209_pInt, 3217_pInt, 3221_pInt, 3229_pInt, 3251_pInt, 3253_pInt, 3257_pInt, & 3259_pInt, 3271_pInt, 3299_pInt, 3301_pInt, 3307_pInt, 3313_pInt, 3319_pInt, 3323_pInt, 3329_pInt, 3331_pInt, & 3343_pInt, 3347_pInt, 3359_pInt, 3361_pInt, 3371_pInt, 3373_pInt, 3389_pInt, 3391_pInt, 3407_pInt, 3413_pInt, & 3433_pInt, 3449_pInt, 3457_pInt, 3461_pInt, 3463_pInt, 3467_pInt, 3469_pInt, 3491_pInt, 3499_pInt, 3511_pInt, & 3517_pInt, 3527_pInt, 3529_pInt, 3533_pInt, 3539_pInt, 3541_pInt, 3547_pInt, 3557_pInt, 3559_pInt, 3571_pInt, & ! 501:600 3581_pInt, 3583_pInt, 3593_pInt, 3607_pInt, 3613_pInt, 3617_pInt, 3623_pInt, 3631_pInt, 3637_pInt, 3643_pInt, & 3659_pInt, 3671_pInt, 3673_pInt, 3677_pInt, 3691_pInt, 3697_pInt, 3701_pInt, 3709_pInt, 3719_pInt, 3727_pInt, & 3733_pInt, 3739_pInt, 3761_pInt, 3767_pInt, 3769_pInt, 3779_pInt, 3793_pInt, 3797_pInt, 3803_pInt, 3821_pInt, & 3823_pInt, 3833_pInt, 3847_pInt, 3851_pInt, 3853_pInt, 3863_pInt, 3877_pInt, 3881_pInt, 3889_pInt, 3907_pInt, & 3911_pInt, 3917_pInt, 3919_pInt, 3923_pInt, 3929_pInt, 3931_pInt, 3943_pInt, 3947_pInt, 3967_pInt, 3989_pInt, & 4001_pInt, 4003_pInt, 4007_pInt, 4013_pInt, 4019_pInt, 4021_pInt, 4027_pInt, 4049_pInt, 4051_pInt, 4057_pInt, & 4073_pInt, 4079_pInt, 4091_pInt, 4093_pInt, 4099_pInt, 4111_pInt, 4127_pInt, 4129_pInt, 4133_pInt, 4139_pInt, & 4153_pInt, 4157_pInt, 4159_pInt, 4177_pInt, 4201_pInt, 4211_pInt, 4217_pInt, 4219_pInt, 4229_pInt, 4231_pInt, & 4241_pInt, 4243_pInt, 4253_pInt, 4259_pInt, 4261_pInt, 4271_pInt, 4273_pInt, 4283_pInt, 4289_pInt, 4297_pInt, & 4327_pInt, 4337_pInt, 4339_pInt, 4349_pInt, 4357_pInt, 4363_pInt, 4373_pInt, 4391_pInt, 4397_pInt, 4409_pInt, & ! 601:700 4421_pInt, 4423_pInt, 4441_pInt, 4447_pInt, 4451_pInt, 4457_pInt, 4463_pInt, 4481_pInt, 4483_pInt, 4493_pInt, & 4507_pInt, 4513_pInt, 4517_pInt, 4519_pInt, 4523_pInt, 4547_pInt, 4549_pInt, 4561_pInt, 4567_pInt, 4583_pInt, & 4591_pInt, 4597_pInt, 4603_pInt, 4621_pInt, 4637_pInt, 4639_pInt, 4643_pInt, 4649_pInt, 4651_pInt, 4657_pInt, & 4663_pInt, 4673_pInt, 4679_pInt, 4691_pInt, 4703_pInt, 4721_pInt, 4723_pInt, 4729_pInt, 4733_pInt, 4751_pInt, & 4759_pInt, 4783_pInt, 4787_pInt, 4789_pInt, 4793_pInt, 4799_pInt, 4801_pInt, 4813_pInt, 4817_pInt, 4831_pInt, & 4861_pInt, 4871_pInt, 4877_pInt, 4889_pInt, 4903_pInt, 4909_pInt, 4919_pInt, 4931_pInt, 4933_pInt, 4937_pInt, & 4943_pInt, 4951_pInt, 4957_pInt, 4967_pInt, 4969_pInt, 4973_pInt, 4987_pInt, 4993_pInt, 4999_pInt, 5003_pInt, & 5009_pInt, 5011_pInt, 5021_pInt, 5023_pInt, 5039_pInt, 5051_pInt, 5059_pInt, 5077_pInt, 5081_pInt, 5087_pInt, & 5099_pInt, 5101_pInt, 5107_pInt, 5113_pInt, 5119_pInt, 5147_pInt, 5153_pInt, 5167_pInt, 5171_pInt, 5179_pInt, & 5189_pInt, 5197_pInt, 5209_pInt, 5227_pInt, 5231_pInt, 5233_pInt, 5237_pInt, 5261_pInt, 5273_pInt, 5279_pInt, & ! 701:800 5281_pInt, 5297_pInt, 5303_pInt, 5309_pInt, 5323_pInt, 5333_pInt, 5347_pInt, 5351_pInt, 5381_pInt, 5387_pInt, & 5393_pInt, 5399_pInt, 5407_pInt, 5413_pInt, 5417_pInt, 5419_pInt, 5431_pInt, 5437_pInt, 5441_pInt, 5443_pInt, & 5449_pInt, 5471_pInt, 5477_pInt, 5479_pInt, 5483_pInt, 5501_pInt, 5503_pInt, 5507_pInt, 5519_pInt, 5521_pInt, & 5527_pInt, 5531_pInt, 5557_pInt, 5563_pInt, 5569_pInt, 5573_pInt, 5581_pInt, 5591_pInt, 5623_pInt, 5639_pInt, & 5641_pInt, 5647_pInt, 5651_pInt, 5653_pInt, 5657_pInt, 5659_pInt, 5669_pInt, 5683_pInt, 5689_pInt, 5693_pInt, & 5701_pInt, 5711_pInt, 5717_pInt, 5737_pInt, 5741_pInt, 5743_pInt, 5749_pInt, 5779_pInt, 5783_pInt, 5791_pInt, & 5801_pInt, 5807_pInt, 5813_pInt, 5821_pInt, 5827_pInt, 5839_pInt, 5843_pInt, 5849_pInt, 5851_pInt, 5857_pInt, & 5861_pInt, 5867_pInt, 5869_pInt, 5879_pInt, 5881_pInt, 5897_pInt, 5903_pInt, 5923_pInt, 5927_pInt, 5939_pInt, & 5953_pInt, 5981_pInt, 5987_pInt, 6007_pInt, 6011_pInt, 6029_pInt, 6037_pInt, 6043_pInt, 6047_pInt, 6053_pInt, & 6067_pInt, 6073_pInt, 6079_pInt, 6089_pInt, 6091_pInt, 6101_pInt, 6113_pInt, 6121_pInt, 6131_pInt, 6133_pInt, & ! 801:900 6143_pInt, 6151_pInt, 6163_pInt, 6173_pInt, 6197_pInt, 6199_pInt, 6203_pInt, 6211_pInt, 6217_pInt, 6221_pInt, & 6229_pInt, 6247_pInt, 6257_pInt, 6263_pInt, 6269_pInt, 6271_pInt, 6277_pInt, 6287_pInt, 6299_pInt, 6301_pInt, & 6311_pInt, 6317_pInt, 6323_pInt, 6329_pInt, 6337_pInt, 6343_pInt, 6353_pInt, 6359_pInt, 6361_pInt, 6367_pInt, & 6373_pInt, 6379_pInt, 6389_pInt, 6397_pInt, 6421_pInt, 6427_pInt, 6449_pInt, 6451_pInt, 6469_pInt, 6473_pInt, & 6481_pInt, 6491_pInt, 6521_pInt, 6529_pInt, 6547_pInt, 6551_pInt, 6553_pInt, 6563_pInt, 6569_pInt, 6571_pInt, & 6577_pInt, 6581_pInt, 6599_pInt, 6607_pInt, 6619_pInt, 6637_pInt, 6653_pInt, 6659_pInt, 6661_pInt, 6673_pInt, & 6679_pInt, 6689_pInt, 6691_pInt, 6701_pInt, 6703_pInt, 6709_pInt, 6719_pInt, 6733_pInt, 6737_pInt, 6761_pInt, & 6763_pInt, 6779_pInt, 6781_pInt, 6791_pInt, 6793_pInt, 6803_pInt, 6823_pInt, 6827_pInt, 6829_pInt, 6833_pInt, & 6841_pInt, 6857_pInt, 6863_pInt, 6869_pInt, 6871_pInt, 6883_pInt, 6899_pInt, 6907_pInt, 6911_pInt, 6917_pInt, & 6947_pInt, 6949_pInt, 6959_pInt, 6961_pInt, 6967_pInt, 6971_pInt, 6977_pInt, 6983_pInt, 6991_pInt, 6997_pInt, & ! 901:1000 7001_pInt, 7013_pInt, 7019_pInt, 7027_pInt, 7039_pInt, 7043_pInt, 7057_pInt, 7069_pInt, 7079_pInt, 7103_pInt, & 7109_pInt, 7121_pInt, 7127_pInt, 7129_pInt, 7151_pInt, 7159_pInt, 7177_pInt, 7187_pInt, 7193_pInt, 7207_pInt, & 7211_pInt, 7213_pInt, 7219_pInt, 7229_pInt, 7237_pInt, 7243_pInt, 7247_pInt, 7253_pInt, 7283_pInt, 7297_pInt, & 7307_pInt, 7309_pInt, 7321_pInt, 7331_pInt, 7333_pInt, 7349_pInt, 7351_pInt, 7369_pInt, 7393_pInt, 7411_pInt, & 7417_pInt, 7433_pInt, 7451_pInt, 7457_pInt, 7459_pInt, 7477_pInt, 7481_pInt, 7487_pInt, 7489_pInt, 7499_pInt, & 7507_pInt, 7517_pInt, 7523_pInt, 7529_pInt, 7537_pInt, 7541_pInt, 7547_pInt, 7549_pInt, 7559_pInt, 7561_pInt, & 7573_pInt, 7577_pInt, 7583_pInt, 7589_pInt, 7591_pInt, 7603_pInt, 7607_pInt, 7621_pInt, 7639_pInt, 7643_pInt, & 7649_pInt, 7669_pInt, 7673_pInt, 7681_pInt, 7687_pInt, 7691_pInt, 7699_pInt, 7703_pInt, 7717_pInt, 7723_pInt, & 7727_pInt, 7741_pInt, 7753_pInt, 7757_pInt, 7759_pInt, 7789_pInt, 7793_pInt, 7817_pInt, 7823_pInt, 7829_pInt, & 7841_pInt, 7853_pInt, 7867_pInt, 7873_pInt, 7877_pInt, 7879_pInt, 7883_pInt, 7901_pInt, 7907_pInt, 7919_pInt, & ! 1001:1100 7927_pInt, 7933_pInt, 7937_pInt, 7949_pInt, 7951_pInt, 7963_pInt, 7993_pInt, 8009_pInt, 8011_pInt, 8017_pInt, & 8039_pInt, 8053_pInt, 8059_pInt, 8069_pInt, 8081_pInt, 8087_pInt, 8089_pInt, 8093_pInt, 8101_pInt, 8111_pInt, & 8117_pInt, 8123_pInt, 8147_pInt, 8161_pInt, 8167_pInt, 8171_pInt, 8179_pInt, 8191_pInt, 8209_pInt, 8219_pInt, & 8221_pInt, 8231_pInt, 8233_pInt, 8237_pInt, 8243_pInt, 8263_pInt, 8269_pInt, 8273_pInt, 8287_pInt, 8291_pInt, & 8293_pInt, 8297_pInt, 8311_pInt, 8317_pInt, 8329_pInt, 8353_pInt, 8363_pInt, 8369_pInt, 8377_pInt, 8387_pInt, & 8389_pInt, 8419_pInt, 8423_pInt, 8429_pInt, 8431_pInt, 8443_pInt, 8447_pInt, 8461_pInt, 8467_pInt, 8501_pInt, & 8513_pInt, 8521_pInt, 8527_pInt, 8537_pInt, 8539_pInt, 8543_pInt, 8563_pInt, 8573_pInt, 8581_pInt, 8597_pInt, & 8599_pInt, 8609_pInt, 8623_pInt, 8627_pInt, 8629_pInt, 8641_pInt, 8647_pInt, 8663_pInt, 8669_pInt, 8677_pInt, & 8681_pInt, 8689_pInt, 8693_pInt, 8699_pInt, 8707_pInt, 8713_pInt, 8719_pInt, 8731_pInt, 8737_pInt, 8741_pInt, & 8747_pInt, 8753_pInt, 8761_pInt, 8779_pInt, 8783_pInt, 8803_pInt, 8807_pInt, 8819_pInt, 8821_pInt, 8831_pInt, & ! 1101:1200 8837_pInt, 8839_pInt, 8849_pInt, 8861_pInt, 8863_pInt, 8867_pInt, 8887_pInt, 8893_pInt, 8923_pInt, 8929_pInt, & 8933_pInt, 8941_pInt, 8951_pInt, 8963_pInt, 8969_pInt, 8971_pInt, 8999_pInt, 9001_pInt, 9007_pInt, 9011_pInt, & 9013_pInt, 9029_pInt, 9041_pInt, 9043_pInt, 9049_pInt, 9059_pInt, 9067_pInt, 9091_pInt, 9103_pInt, 9109_pInt, & 9127_pInt, 9133_pInt, 9137_pInt, 9151_pInt, 9157_pInt, 9161_pInt, 9173_pInt, 9181_pInt, 9187_pInt, 9199_pInt, & 9203_pInt, 9209_pInt, 9221_pInt, 9227_pInt, 9239_pInt, 9241_pInt, 9257_pInt, 9277_pInt, 9281_pInt, 9283_pInt, & 9293_pInt, 9311_pInt, 9319_pInt, 9323_pInt, 9337_pInt, 9341_pInt, 9343_pInt, 9349_pInt, 9371_pInt, 9377_pInt, & 9391_pInt, 9397_pInt, 9403_pInt, 9413_pInt, 9419_pInt, 9421_pInt, 9431_pInt, 9433_pInt, 9437_pInt, 9439_pInt, & 9461_pInt, 9463_pInt, 9467_pInt, 9473_pInt, 9479_pInt, 9491_pInt, 9497_pInt, 9511_pInt, 9521_pInt, 9533_pInt, & 9539_pInt, 9547_pInt, 9551_pInt, 9587_pInt, 9601_pInt, 9613_pInt, 9619_pInt, 9623_pInt, 9629_pInt, 9631_pInt, & 9643_pInt, 9649_pInt, 9661_pInt, 9677_pInt, 9679_pInt, 9689_pInt, 9697_pInt, 9719_pInt, 9721_pInt, 9733_pInt, & ! 1201:1300 9739_pInt, 9743_pInt, 9749_pInt, 9767_pInt, 9769_pInt, 9781_pInt, 9787_pInt, 9791_pInt, 9803_pInt, 9811_pInt, & 9817_pInt, 9829_pInt, 9833_pInt, 9839_pInt, 9851_pInt, 9857_pInt, 9859_pInt, 9871_pInt, 9883_pInt, 9887_pInt, & 9901_pInt, 9907_pInt, 9923_pInt, 9929_pInt, 9931_pInt, 9941_pInt, 9949_pInt, 9967_pInt, 9973_pInt,10007_pInt, & 10009_pInt,10037_pInt,10039_pInt,10061_pInt,10067_pInt,10069_pInt,10079_pInt,10091_pInt,10093_pInt,10099_pInt, & 10103_pInt,10111_pInt,10133_pInt,10139_pInt,10141_pInt,10151_pInt,10159_pInt,10163_pInt,10169_pInt,10177_pInt, & 10181_pInt,10193_pInt,10211_pInt,10223_pInt,10243_pInt,10247_pInt,10253_pInt,10259_pInt,10267_pInt,10271_pInt, & 10273_pInt,10289_pInt,10301_pInt,10303_pInt,10313_pInt,10321_pInt,10331_pInt,10333_pInt,10337_pInt,10343_pInt, & 10357_pInt,10369_pInt,10391_pInt,10399_pInt,10427_pInt,10429_pInt,10433_pInt,10453_pInt,10457_pInt,10459_pInt, & 10463_pInt,10477_pInt,10487_pInt,10499_pInt,10501_pInt,10513_pInt,10529_pInt,10531_pInt,10559_pInt,10567_pInt, & 10589_pInt,10597_pInt,10601_pInt,10607_pInt,10613_pInt,10627_pInt,10631_pInt,10639_pInt,10651_pInt,10657_pInt, & ! 1301:1400 10663_pInt,10667_pInt,10687_pInt,10691_pInt,10709_pInt,10711_pInt,10723_pInt,10729_pInt,10733_pInt,10739_pInt, & 10753_pInt,10771_pInt,10781_pInt,10789_pInt,10799_pInt,10831_pInt,10837_pInt,10847_pInt,10853_pInt,10859_pInt, & 10861_pInt,10867_pInt,10883_pInt,10889_pInt,10891_pInt,10903_pInt,10909_pInt,19037_pInt,10939_pInt,10949_pInt, & 10957_pInt,10973_pInt,10979_pInt,10987_pInt,10993_pInt,11003_pInt,11027_pInt,11047_pInt,11057_pInt,11059_pInt, & 11069_pInt,11071_pInt,11083_pInt,11087_pInt,11093_pInt,11113_pInt,11117_pInt,11119_pInt,11131_pInt,11149_pInt, & 11159_pInt,11161_pInt,11171_pInt,11173_pInt,11177_pInt,11197_pInt,11213_pInt,11239_pInt,11243_pInt,11251_pInt, & 11257_pInt,11261_pInt,11273_pInt,11279_pInt,11287_pInt,11299_pInt,11311_pInt,11317_pInt,11321_pInt,11329_pInt, & 11351_pInt,11353_pInt,11369_pInt,11383_pInt,11393_pInt,11399_pInt,11411_pInt,11423_pInt,11437_pInt,11443_pInt, & 11447_pInt,11467_pInt,11471_pInt,11483_pInt,11489_pInt,11491_pInt,11497_pInt,11503_pInt,11519_pInt,11527_pInt, & 11549_pInt,11551_pInt,11579_pInt,11587_pInt,11593_pInt,11597_pInt,11617_pInt,11621_pInt,11633_pInt,11657_pInt, & ! 1401:1500 11677_pInt,11681_pInt,11689_pInt,11699_pInt,11701_pInt,11717_pInt,11719_pInt,11731_pInt,11743_pInt,11777_pInt, & 11779_pInt,11783_pInt,11789_pInt,11801_pInt,11807_pInt,11813_pInt,11821_pInt,11827_pInt,11831_pInt,11833_pInt, & 11839_pInt,11863_pInt,11867_pInt,11887_pInt,11897_pInt,11903_pInt,11909_pInt,11923_pInt,11927_pInt,11933_pInt, & 11939_pInt,11941_pInt,11953_pInt,11959_pInt,11969_pInt,11971_pInt,11981_pInt,11987_pInt,12007_pInt,12011_pInt, & 12037_pInt,12041_pInt,12043_pInt,12049_pInt,12071_pInt,12073_pInt,12097_pInt,12101_pInt,12107_pInt,12109_pInt, & 12113_pInt,12119_pInt,12143_pInt,12149_pInt,12157_pInt,12161_pInt,12163_pInt,12197_pInt,12203_pInt,12211_pInt, & 12227_pInt,12239_pInt,12241_pInt,12251_pInt,12253_pInt,12263_pInt,12269_pInt,12277_pInt,12281_pInt,12289_pInt, & 12301_pInt,12323_pInt,12329_pInt,12343_pInt,12347_pInt,12373_pInt,12377_pInt,12379_pInt,12391_pInt,12401_pInt, & 12409_pInt,12413_pInt,12421_pInt,12433_pInt,12437_pInt,12451_pInt,12457_pInt,12473_pInt,12479_pInt,12487_pInt, & 12491_pInt,12497_pInt,12503_pInt,12511_pInt,12517_pInt,12527_pInt,12539_pInt,12541_pInt,12547_pInt,12553_pInt] endif if(n < 0_pInt) then prime = PRIME_MAX else if (n == 0_pInt) then prime = 1_pInt else if (n <= PRIME_MAX) then prime = npvec(n) else prime = -1_pInt call IO_error(error_ID=406_pInt) end if end function prime !-------------------------------------------------------------------------------------------------- !> @brief factorial !-------------------------------------------------------------------------------------------------- integer(pInt) pure function math_factorial(n) implicit none integer(pInt), intent(in) :: n integer(pInt) :: i math_factorial = product([(i, i=1,n)]) end function math_factorial !-------------------------------------------------------------------------------------------------- !> @brief binomial coefficient !-------------------------------------------------------------------------------------------------- integer(pInt) pure function math_binomial(n,k) implicit none integer(pInt), intent(in) :: n, k integer(pInt) :: i, j j = min(k,n-k) math_binomial = product([(i, i=n, n-j+1, -1)])/math_factorial(j) end function math_binomial !-------------------------------------------------------------------------------------------------- !> @brief multinomial coefficient !-------------------------------------------------------------------------------------------------- integer(pInt) pure function math_multinomial(alpha) implicit none integer(pInt), intent(in), dimension(:) :: alpha integer(pInt) :: i math_multinomial = 1_pInt do i = 1, size(alpha) math_multinomial = math_multinomial*math_binomial(sum(alpha(1:i)),alpha(i)) enddo end function math_multinomial !-------------------------------------------------------------------------------------------------- !> @brief volume of tetrahedron given by four vertices !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_volTetrahedron(v1,v2,v3,v4) implicit none real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4 real(pReal), dimension (3,3) :: m m(1:3,1) = v1-v2 m(1:3,2) = v2-v3 m(1:3,3) = v3-v4 math_volTetrahedron = math_det33(m)/6.0_pReal end function math_volTetrahedron !-------------------------------------------------------------------------------------------------- !> @brief area of triangle given by three vertices !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_areaTriangle(v1,v2,v3) implicit none real(pReal), dimension (3), intent(in) :: v1,v2,v3 math_areaTriangle = 0.5_pReal * norm2(math_crossproduct(v1-v2,v1-v3)) end function math_areaTriangle !-------------------------------------------------------------------------------------------------- !> @brief rotate 33 tensor forward !-------------------------------------------------------------------------------------------------- pure function math_rotate_forward33(tensor,rot_tensor) implicit none real(pReal), dimension(3,3) :: math_rotate_forward33 real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor math_rotate_forward33 = math_mul33x33(rot_tensor,& math_mul33x33(tensor,math_transpose33(rot_tensor))) end function math_rotate_forward33 !-------------------------------------------------------------------------------------------------- !> @brief rotate 33 tensor backward !-------------------------------------------------------------------------------------------------- pure function math_rotate_backward33(tensor,rot_tensor) implicit none real(pReal), dimension(3,3) :: math_rotate_backward33 real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor math_rotate_backward33 = math_mul33x33(math_transpose33(rot_tensor),& math_mul33x33(tensor,rot_tensor)) end function math_rotate_backward33 !-------------------------------------------------------------------------------------------------- !> @brief rotate 3333 tensor C'_ijkl=g_im*g_jn*g_ko*g_lp*C_mnop !-------------------------------------------------------------------------------------------------- pure function math_rotate_forward3333(tensor,rot_tensor) implicit none real(pReal), dimension(3,3,3,3) :: math_rotate_forward3333 real(pReal), dimension(3,3), intent(in) :: rot_tensor real(pReal), dimension(3,3,3,3), intent(in) :: tensor integer(pInt) :: i,j,k,l,m,n,o,p math_rotate_forward3333= 0.0_pReal do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt; do k = 1_pInt,3_pInt; do l = 1_pInt,3_pInt do m = 1_pInt,3_pInt; do n = 1_pInt,3_pInt; do o = 1_pInt,3_pInt; do p = 1_pInt,3_pInt math_rotate_forward3333(i,j,k,l) = math_rotate_forward3333(i,j,k,l) & + rot_tensor(m,i) * rot_tensor(n,j) & * rot_tensor(o,k) * rot_tensor(p,l) * tensor(m,n,o,p) enddo; enddo; enddo; enddo; enddo; enddo; enddo; enddo end function math_rotate_forward3333 !-------------------------------------------------------------------------------------------------- !> @brief calculate average of tensor field !-------------------------------------------------------------------------------------------------- function math_tensorAvg(field) implicit none real(pReal), dimension(3,3) :: math_tensorAvg real(pReal), intent(in), dimension(:,:,:,:,:) :: field real(pReal) :: wgt wgt = 1.0_pReal/real(size(field,3)*size(field,4)*size(field,5), pReal) math_tensorAvg = sum(sum(sum(field,dim=5),dim=4),dim=3)*wgt end function math_tensorAvg !-------------------------------------------------------------------------------------------------- !> @brief limits a scalar value to a certain range (either one or two sided) ! Will return NaN if left > right !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_limit(a, left, right) use prec, only: & DAMASK_NaN implicit none real(pReal), intent(in) :: a real(pReal), intent(in), optional :: left, right math_limit = min ( & max (merge(left, -huge(a), present(left)), a), & merge(right, huge(a), present(right)) & ) if (present(left) .and. present(right)) math_limit = merge (DAMASK_NaN,math_limit, left>right) end function math_limit end module math