!-------------------------------------------------------------------------------------------------- !> @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 representations !-------------------------------------------------------------------------------------------------- module math use prec, only: & pReal, & pInt implicit none private real(pReal), parameter, public :: PI = acos(-1.0_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 real(pReal), dimension(6), parameter, private :: & nrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal, & sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal) ] !< weighting for Mandel notation (forward) real(pReal), dimension(6), parameter , private :: & invnrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal, & 1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal) ] !< weighting for Mandel notation (backward) integer(pInt), dimension (2,6), parameter, private :: & mapNye = 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 Nye notation. 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 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 !-------------------------------------------------------------------------------------------------- ! Provide deprecated names for compatibility interface math_cross module procedure math_crossproduct end interface math_cross ! ToDo MD: Our naming scheme was a little bit odd: We use essentially the re-ordering according to Nye ! (convenient because Abaqus and Marc want to have 12 on position 4) ! but weight the shear components according to Mandel (convenient for matrix multiplications) interface math_Plain33to9 module procedure math_33to9 end interface math_Plain33to9 interface math_Plain9to33 module procedure math_9to33 end interface math_Plain9to33 interface math_Mandel33to6 module procedure math_sym33to6 end interface math_Mandel33to6 interface math_Mandel6to33 module procedure math_6toSym33 end interface math_Mandel6to33 interface math_Plain3333to99 module procedure math_3333to99 end interface math_Plain3333to99 interface math_Plain99to3333 module procedure math_99to3333 end interface math_Plain99to3333 public :: & math_Plain33to9, & math_Plain9to33, & math_Mandel33to6, & math_Mandel6to33, & math_Plain3333to99, & math_Plain99to3333 !--------------------------------------------------------------------------------------------------- public :: & #if defined(__PGI) norm2, & #endif math_init, & math_qsort, & math_expand, & math_range, & math_identity2nd, & math_identity4th, & math_civita, & math_delta, & math_cross, & 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_invert2, & math_symmetric33, & math_symmetric66, & math_skew33, & math_spherical33, & math_deviatoric33, & math_equivStrain33, & math_equivStress33, & math_trace33, & math_det33, & math_33to9, & math_9to33, & math_sym33to6, & math_6toSym33, & math_3333to99, & math_99to3333, & math_sym3333to66, & math_66toSym3333, & 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_eigenvectorBasisSym33_log, & 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, & math_clip private :: & math_check, & halton contains !-------------------------------------------------------------------------------------------------- !> @brief initialization of random seed generator !-------------------------------------------------------------------------------------------------- subroutine math_init #if defined(__GFORTRAN__) || __INTEL_COMPILER >= 1800 use, intrinsic :: iso_fortran_env, only: & compiler_version, & compiler_options #endif use numerics, only: randomSeed use IO, only: IO_timeStamp implicit none integer(pInt) :: i real(pReal), dimension(4) :: 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 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 (randomSeed > 0_pInt) then randInit(1:randSize) = int(randomSeed) ! randomSeed 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,I2)') ' size of random seed: ', randSize do i = 1_pInt,randSize write(6,'(a,I2,I14)') ' 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 math_check() end subroutine math_init !-------------------------------------------------------------------------------------------------- !> @brief check correctness of (some) math functions !-------------------------------------------------------------------------------------------------- subroutine math_check use prec, only: tol_math_check use IO, only: IO_error implicit none character(len=64) :: error_msg real(pReal), dimension(3,3) :: R,R2 real(pReal), dimension(3) :: Eulers,v real(pReal), dimension(4) :: q,q2,axisangle ! --- 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)' ) & 'quat -> axisAngle -> quat 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)' ) & 'quat -> R -> quat maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(401_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)' ) & 'quat -> euler -> quat maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(401_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)' ) & 'R -> euler -> R maximum deviation ',maxval(abs( R-R2)) call IO_error(401_pInt,ext_msg=error_msg) endif ! +++ check rotation sense of q and R +++ v = halton([2_pInt,8_pInt,5_pInt]) ! random vector R = math_qToR(q) if (any(abs(math_mul33x3(R,v) - math_qRot(q,v)) > tol_math_check)) then write (error_msg, '(a)' ) 'R(q)*v has different sense than q*v' call IO_error(401_pInt,ext_msg=error_msg) endif ! +++ check vector expansion +++ if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,3.0_pReal,3.0_pReal,3.0_pReal] - & math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1_pInt,2_pInt,3_pInt,0_pInt])) > tol_math_check)) then write (error_msg, '(a)' ) 'math_expand [1,2,3] by [1,2,3,0] => [1,2,2,3,3,3]' call IO_error(401_pInt,ext_msg=error_msg) endif if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal] - & math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1_pInt,2_pInt])) > tol_math_check)) then write (error_msg, '(a)' ) 'math_expand [1,2,3] by [1,2] => [1,2,2]' call IO_error(401_pInt,ext_msg=error_msg) endif if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,1.0_pReal,1.0_pReal,1.0_pReal] - & math_expand([1.0_pReal,2.0_pReal],[1_pInt,2_pInt,3_pInt])) > tol_math_check)) then write (error_msg, '(a)' ) 'math_expand [1,2] by [1,2,3] => [1,2,2,1,1,1]' call IO_error(401_pInt,ext_msg=error_msg) endif end subroutine math_check !-------------------------------------------------------------------------------------------------- !> @brief Quicksort algorithm for two-dimensional integer arrays ! Sorting is done with respect to array(sort,:) and keeps array(/=sort,:) linked to it. ! default: sort=1 !-------------------------------------------------------------------------------------------------- recursive subroutine math_qsort(a, istart, iend, sortDim) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in),optional :: istart,iend, sortDim integer(pInt) :: ipivot,s,e,d if(present(istart)) then s = istart else s = lbound(a,2) endif if(present(iend)) then e = iend else e = ubound(a,2) endif if(present(sortDim)) then d = sortDim else d = 1 endif if (s < e) then ipivot = qsort_partition(a,s, e, d) call math_qsort(a, s, ipivot-1_pInt, d) call math_qsort(a, ipivot+1_pInt, e, d) endif !-------------------------------------------------------------------------------------------------- contains !------------------------------------------------------------------------------------------------- !> @brief Partitioning required for quicksort !------------------------------------------------------------------------------------------------- integer(pInt) function qsort_partition(a, istart, iend, sort) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in) :: istart,iend,sort integer(pInt), dimension(size(a,1)) :: tmp integer(pInt) :: i,j do ! find the first element on the right side less than or equal to the pivot point do j = iend, istart, -1_pInt if (a(sort,j) <= a(sort,istart)) exit enddo ! find the first element on the left side greater than the pivot point do i = istart, iend if (a(sort,i) > a(sort,istart)) exit enddo cross: if (i >= j) then ! if the indices cross, exchange left value with pivot and return with the partition index tmp = a(:,istart) a(:,istart) = a(:,j) a(:,j) = tmp qsort_partition = j return else cross ! if they do not cross, exchange values tmp = a(:,i) a(:,i) = a(:,j) a(:,j) = tmp endif cross enddo end function qsort_partition end subroutine math_qsort !-------------------------------------------------------------------------------------------------- !> @brief vector expansion !> @details takes a set of numbers (a,b,c,...) and corresponding multiples (x,y,z,...) !> to return a vector of x times a, y times b, z times c, ... !-------------------------------------------------------------------------------------------------- pure function math_expand(what,how) implicit none real(pReal), dimension(:), intent(in) :: what integer(pInt), dimension(:), intent(in) :: how real(pReal), dimension(sum(how)) :: math_expand integer(pInt) :: i if (sum(how) == 0_pInt) & return do i = 1_pInt, size(how) math_expand(sum(how(1:i-1))+1:sum(how(1:i))) = what(mod(i-1_pInt,size(what))+1_pInt) enddo end function math_expand !-------------------------------------------------------------------------------------------------- !> @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 real(pReal), dimension(dimen,dimen) :: identity2nd identity2nd = math_identity2nd(dimen) 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*(identity2nd(i,k)*identity2nd(j,l)+identity2nd(i,l)*identity2nd(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 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 B = math_I3 ! init invFac = 1.0_pReal ! 0! math_exp33 = B ! A^0 = eye2 do i = 1_pInt, merge(n,5_pInt,present(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) !> @details Direct Cramer inversion of matrix A. Returns all zeroes if not possible, i.e. ! if determinant is 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) !> @details Direct Cramer inversion of matrix A. Also returns determinant ! Returns an error if not possible, i.e. if determinant is close to zero ! ToDo: Output arguments should be first !-------------------------------------------------------------------------------------------------- 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_sym3333to66(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_66toSym3333(temp66_real) else call IO_error(400_pInt, ext_msg = 'math_invSym3333') endif end function math_invSym3333 !-------------------------------------------------------------------------------------------------- !> @brief invert quadratic matrix of arbitrary dimension ! ToDo: replaces math_invert !-------------------------------------------------------------------------------------------------- subroutine math_invert2(InvA, error, A) implicit none real(pReal), dimension(:,:), intent(in) :: A real(pReal), dimension(size(A,1),size(A,1)), intent(out) :: invA logical, intent(out) :: error call math_invert(size(A,1), A, InvA, error) end subroutine math_invert2 !-------------------------------------------------------------------------------------------------- !> @brief invert matrix of arbitrary dimension ! ToDo: Wrong order of arguments and superfluous myDim argument. ! Use math_invert2 instead !-------------------------------------------------------------------------------------------------- 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) 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 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 = 2.0_pReal/3.0_pReal & * (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_33to9(m33) implicit none real(pReal), dimension(9) :: math_33to9 real(pReal), dimension(3,3), intent(in) :: m33 integer(pInt) :: i forall(i=1_pInt:9_pInt) math_33to9(i) = m33(mapPlain(1,i),mapPlain(2,i)) end function math_33to9 !-------------------------------------------------------------------------------------------------- !> @brief convert 9 vector into 33 matrix !-------------------------------------------------------------------------------------------------- pure function math_9to33(v9) implicit none real(pReal), dimension(3,3) :: math_9to33 real(pReal), dimension(9), intent(in) :: v9 integer(pInt) :: i forall(i=1_pInt:9_pInt) math_9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i) end function math_9to33 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 33 matrix into 6 vector !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only changes order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_sym33to6(m33,weighted) implicit none real(pReal), dimension(6) :: math_sym33to6 real(pReal), dimension(3,3), intent(in) :: m33 logical, optional, intent(in) :: weighted real(pReal), dimension(6) :: w integer(pInt) :: i if(present(weighted)) then w = merge(nrmMandel,1.0_pReal,weighted) else w = nrmMandel endif forall(i=1_pInt:6_pInt) math_sym33to6(i) = w(i)*m33(mapNye(1,i),mapNye(2,i)) end function math_sym33to6 !-------------------------------------------------------------------------------------------------- !> @brief convert 6 vector into symmetric 33 matrix !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only changes order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_6toSym33(v6,weighted) implicit none real(pReal), dimension(3,3) :: math_6toSym33 real(pReal), dimension(6), intent(in) :: v6 logical, optional, intent(in) :: weighted real(pReal), dimension(6) :: w integer(pInt) :: i if(present(weighted)) then w = merge(invnrmMandel,1.0_pReal,weighted) else w = invnrmMandel endif do i=1_pInt,6_pInt math_6toSym33(mapNye(1,i),mapNye(2,i)) = w(i)*v6(i) math_6toSym33(mapNye(2,i),mapNye(1,i)) = w(i)*v6(i) enddo end function math_6toSym33 !-------------------------------------------------------------------------------------------------- !> @brief convert 3333 matrix into 99 matrix !-------------------------------------------------------------------------------------------------- pure function math_3333to99(m3333) implicit none real(pReal), dimension(9,9) :: math_3333to99 real(pReal), dimension(3,3,3,3), intent(in) :: m3333 integer(pInt) :: i,j forall(i=1_pInt:9_pInt,j=1_pInt:9_pInt) & math_3333to99(i,j) = m3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j)) end function math_3333to99 !-------------------------------------------------------------------------------------------------- !> @brief convert 99 matrix into 3333 matrix !-------------------------------------------------------------------------------------------------- pure function math_99to3333(m99) implicit none real(pReal), dimension(3,3,3,3) :: math_99to3333 real(pReal), dimension(9,9), intent(in) :: m99 integer(pInt) :: i,j forall(i=1_pInt:9_pInt,j=1_pInt:9_pInt) & math_99to3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j)) = m99(i,j) end function math_99to3333 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 3333 matrix into 66 matrix !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only changes order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_sym3333to66(m3333,weighted) implicit none real(pReal), dimension(6,6) :: math_sym3333to66 real(pReal), dimension(3,3,3,3), intent(in) :: m3333 logical, optional, intent(in) :: weighted real(pReal), dimension(6) :: w integer(pInt) :: i,j if(present(weighted)) then w = merge(nrmMandel,1.0_pReal,weighted) else w = nrmMandel endif forall(i=1_pInt:6_pInt,j=1_pInt:6_pInt) & math_sym3333to66(i,j) = w(i)*w(j)*m3333(mapNye(1,i),mapNye(2,i),mapNye(1,j),mapNye(2,j)) end function math_sym3333to66 !-------------------------------------------------------------------------------------------------- !> @brief convert 66 matrix into symmetric 3333 matrix !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only changes order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_66toSym3333(m66,weighted) implicit none real(pReal), dimension(3,3,3,3) :: math_66toSym3333 real(pReal), dimension(6,6), intent(in) :: m66 logical, optional, intent(in) :: weighted real(pReal), dimension(6) :: w integer(pInt) :: i,j if(present(weighted)) then w = merge(invnrmMandel,1.0_pReal,weighted) else w = invnrmMandel endif do i=1_pInt,6_pInt; do j=1_pInt, 6_pInt math_66toSym3333(mapNye(1,i),mapNye(2,i),mapNye(1,j),mapNye(2,j)) = w(i)*w(j)*m66(i,j) math_66toSym3333(mapNye(2,i),mapNye(1,i),mapNye(1,j),mapNye(2,j)) = w(i)*w(j)*m66(i,j) math_66toSym3333(mapNye(1,i),mapNye(2,i),mapNye(2,j),mapNye(1,j)) = w(i)*w(j)*m66(i,j) math_66toSym3333(mapNye(2,i),mapNye(1,i),mapNye(2,j),mapNye(1,j)) = w(i)*w(j)*m66(i,j) enddo; enddo end function math_66toSym3333 !-------------------------------------------------------------------------------------------------- !> @brief convert 66 Voigt matrix into symmetric 3333 matrix !-------------------------------------------------------------------------------------------------- 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 do i=1_pInt,6_pInt; do j=1_pInt, 6_pInt math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(1,j),mapVoigt(2,j)) = m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(1,j),mapVoigt(2,j)) = m66(i,j) math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(2,j),mapVoigt(1,j)) = m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(2,j),mapVoigt(1,j)) = m66(i,j) enddo; enddo end function math_Voigt66to3333 !-------------------------------------------------------------------------------------------------- !> @brief random quaternion ! http://math.stackexchange.com/questions/131336/uniform-random-quaternion-in-a-restricted-angle-range ! K. Shoemake. Uniform random rotations. In D. Kirk, editor, Graphics Gems III, pages 124-132. ! Academic, New York, 1992. !-------------------------------------------------------------------------------------------------- function math_qRand() implicit none real(pReal), dimension(4) :: math_qRand real(pReal), dimension(3) :: rnd rnd = halton([8_pInt,4_pInt,9_pInt]) 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_clip(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_clip(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_clip(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_clip(-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 Bunge-Euler (3-1-3) angles (in radians) !> @details rotation matrix is meant to represent a PASSIVE rotation, composed of INTRINSIC !> @details rotations around the axes of the details rotating reference frame. !> @details similar to eu2om from "D Rowenhorst et al. Consistent representations of and conversions !> @details between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)", but R is transposed !-------------------------------------------------------------------------------------------------- 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*C*s2 math_EulerToR(1,2) = -c1*s2 -s1*C*c2 math_EulerToR(1,3) = s1*S math_EulerToR(2,1) = s1*c2 +c1*C*s2 math_EulerToR(2,2) = -s1*s2 +c1*C*c2 math_EulerToR(2,3) = -c1*S math_EulerToR(3,1) = S*s2 math_EulerToR(3,2) = S*c2 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 Bunge-Euler (3-1-3) angles (in radians) !> @details rotation matrix is meant to represent a PASSIVE rotation, composed of INTRINSIC !> @details rotations around the axes of the details rotating reference frame. !> @details similar to eu2qu from "D Rowenhorst et al. Consistent representations of and !> @details conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)", but !> @details Q is conjucated and Q is not reversed for Q(0) < 0. !-------------------------------------------------------------------------------------------------- pure function math_EulerToQ(eulerangles) implicit none real(pReal), dimension(3), intent(in) :: eulerangles real(pReal), dimension(4) :: math_EulerToQ real(pReal) :: c, s, sigma, delta c = cos(0.5_pReal * eulerangles(2)) s = sin(0.5_pReal * eulerangles(2)) sigma = 0.5_pReal * (eulerangles(1)+eulerangles(3)) delta = 0.5_pReal * (eulerangles(1)-eulerangles(3)) math_EulerToQ= [c * cos(sigma), & s * cos(delta), & s * sin(delta), & c * sin(sigma) ] 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 !> @details equivalent to eu2om (P=-1) from "D Rowenhorst et al. Consistent representations of and !> @details conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)" !-------------------------------------------------------------------------------------------------- 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) :: n real(pReal) :: norm,s,c,c1 norm = norm2(axis) wellDefined: if (norm > 1.0e-8_pReal) then n = axis/norm ! normalize axis to be sure s = sin(omega) c = cos(omega) c1 = 1.0_pReal - c math_axisAngleToR(1,1) = c + c1*n(1)**2.0_pReal math_axisAngleToR(1,2) = c1*n(1)*n(2) - s*n(3) math_axisAngleToR(1,3) = c1*n(1)*n(3) + s*n(2) math_axisAngleToR(2,1) = c1*n(1)*n(2) + s*n(3) math_axisAngleToR(2,2) = c + c1*n(2)**2.0_pReal math_axisAngleToR(2,3) = c1*n(2)*n(3) - s*n(1) math_axisAngleToR(3,1) = c1*n(1)*n(3) - s*n(2) math_axisAngleToR(3,2) = c1*n(2)*n(3) + s*n(1) math_axisAngleToR(3,3) = c + c1*n(3)**2.0_pReal else wellDefined math_axisAngleToR = math_I3 endif wellDefined 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) !> @details eq-uivalent to eu2qu (P=+1) from "D Rowenhorst et al. Consistent representations of and !> @details conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)" !-------------------------------------------------------------------------------------------------- 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 !> @details equivalent to eu2qu (P=+1) from "D Rowenhorst et al. Consistent representations of and !> @details conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)" !-------------------------------------------------------------------------------------------------- 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 = norm2(axis) wellDefined: 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 wellDefined math_axisAngleToQ = [1.0_pReal,0.0_pReal,0.0_pReal,0.0_pReal] endif wellDefined 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_clip(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_clip(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, intrinsic :: & IEEE_arithmetic use prec, only: & 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),IEEE_value(1.0_pReal,IEEE_quiet_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) :: cosTheta cosTheta = (math_trace33(math_mul33x33(math_EulerToR(EulerB), & transpose(math_EulerToR(EulerA)))) - 1.0_pReal) * 0.5_pReal math_EulerMisorientation = acos(math_clip(cosTheta,-1.0_pReal,1.0_pReal)) end function math_EulerMisorientation !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Euler space !-------------------------------------------------------------------------------------------------- function math_sampleRandomOri() implicit none real(pReal), dimension(3) :: math_sampleRandomOri, rnd rnd = halton([1_pInt,7_pInt,3_pInt]) 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 sample from an Gaussian distribution around given orientation and Full Width ! at Half Maximum (FWHM) !> @details: A uniform misorientation (limited to 2*FWHM) is sampled followed by convolution with ! a Gausian distribution !-------------------------------------------------------------------------------------------------- function math_sampleGaussOri(center,FWHM) implicit none real(pReal), intent(in) :: FWHM real(pReal), dimension(3), intent(in) :: center real(pReal) :: angle real(pReal), dimension(3) :: math_sampleGaussOri, axis real(pReal), dimension(4) :: rnd real(pReal), dimension(3,3) :: R if (FWHM < 0.1_pReal*INRAD) then math_sampleGaussOri = center else GaussConvolution: do rnd = halton([8_pInt,3_pInt,6_pInt,11_pInt]) axis(1) = rnd(1)*2.0_pReal-1.0_pReal ! uniform on [-1,1] axis(2:3) = [sqrt(1.0-axis(1)**2.0_pReal)*cos(rnd(2)*2.0*PI),& sqrt(1.0-axis(1)**2.0_pReal)*sin(rnd(2)*2.0*PI)] ! random axis angle = (rnd(3)-0.5_pReal)*4.0_pReal*FWHM ! rotation by [0, +-2 FWHM] R = math_axisAngleToR(axis,angle) angle = math_EulerMisorientation([0.0_pReal,0.0_pReal,0.0_pReal],math_RtoEuler(R)) if (rnd(4) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2_pReal)) exit ! rejection sampling (Gaussian) enddo GaussConvolution math_sampleGaussOri = math_RtoEuler(math_mul33x33(R,math_EulerToR(center))) endif end function math_sampleGaussOri !-------------------------------------------------------------------------------------------------- !> @brief draw a sample from an Gaussian distribution around given fiber texture and Full Width ! at Half Maximum (FWHM) !------------------------------------------------------------------------------------------------- function math_sampleFiberOri(alpha,beta,FWHM) implicit none real(pReal), dimension(2), intent(in) :: alpha,beta real(pReal), intent(in) :: FWHM real(pReal), dimension(3) :: math_sampleFiberOri, & fInC,& !< fiber axis in crystal coordinate system fInS,& !< fiber axis in sample coordinate system u real(pReal), dimension(3) :: rnd real(pReal), dimension(:),allocatable :: a !< 2D vector to tilt integer(pInt), dimension(:),allocatable :: idx !< components of 2D vector real(pReal), dimension(3,3) :: R !< Rotation matrix (composed of three components) real(pReal):: angle,c integer(pInt):: j,& !< index of smallest component i allocate(a(0)) allocate(idx(0)) fInC = [sin(alpha(1))*cos(alpha(2)), sin(alpha(1))*sin(alpha(2)), cos(alpha(1))] fInS = [sin(beta(1))*cos(beta(2)), sin(beta(1))*sin(beta(2)), cos(beta(1))] R = math_EulerAxisAngleToR(math_crossproduct(fInC,fInS),-acos(dot_product(fInC,fInS))) !< rotation to align fiber axis in crystal and sample system rnd = halton([7_pInt,10_pInt,3_pInt]) R = math_mul33x33(R,math_EulerAxisAngleToR(fInS,rnd(1)*2.0_pReal*PI)) !< additional rotation (0..360deg) perpendicular to fiber axis if (FWHM > 0.1_pReal*INRAD) then reducedTo2D: do i=1_pInt,3_pInt if (i /= minloc(abs(fInS),1)) then a=[a,fInS(i)] idx=[idx,i] else j = i endif enddo reducedTo2D GaussConvolution: do angle = (rnd(2)-0.5_pReal)*4.0_pReal*FWHM ! rotation by [0, +-2 FWHM] ! solve cos(angle) = dot_product(fInS,u) under the assumption that their smallest component is the same c = cos(angle)-fInS(j)**2 u(idx(2)) = -(2.0_pReal*c*a(2) + sqrt(4*((c*a(2))**2-sum(a**2)*(c**2-a(1)**2*(1-fInS(j)**2)))))/& (2*sum(a**2)) u(idx(1)) = sqrt(1-u(idx(2))**2-fInS(j)**2) u(j) = fInS(j) rejectionSampling: if (rnd(3) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2_pReal)) then R = math_mul33x33(R,math_EulerAxisAngleToR(math_crossproduct(u,fInS),angle)) ! tilt around direction of smallest component exit endif rejectionSampling rnd = halton([7_pInt,10_pInt,3_pInt]) enddo GaussConvolution endif math_sampleFiberOri = math_RtoEuler(R) 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 else myWidth = merge(width,3.0_pReal,present(width)) ! use +-3*sigma as default value for scatter if not given do rnd = halton([6_pInt,2_pInt]) 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 endif end function math_sampleGaussVar !-------------------------------------------------------------------------------------------------- !> @brief symmetrically equivalent Euler angles for given sample symmetry !> @detail 1 (equivalent to != 2 and !=4):triclinic, 2:monoclinic, 4:orthotropic !-------------------------------------------------------------------------------------------------- pure function math_symmetricEulers(sym,Euler) implicit none integer(pInt), intent(in) :: sym !< symmetry Class real(pReal), dimension(3), intent(in) :: Euler real(pReal), dimension(3,3) :: math_symmetricEulers math_symmetricEulers = transpose(reshape([PI+Euler(1), PI-Euler(1), 2.0_pReal*PI-Euler(1), & Euler(2), PI-Euler(2), PI -Euler(2), & Euler(3), PI+Euler(3), PI +Euler(3)],[3,3])) ! transpose is needed to have symbolic notation instead of column-major math_symmetricEulers = modulo(math_symmetricEulers,2.0_pReal*pi) select case (sym) case (4_pInt) ! orthotropic: all done case (2_pInt) ! monoclinic: return only first math_symmetricEulers(1:3,2:3) = 0.0_pReal case default ! triclinic: return blank math_symmetricEulers = 0.0_pReal end select end function math_symmetricEulers !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues and eigenvectors of symmetric matrix m ! ToDo: has wrong oder of arguments !-------------------------------------------------------------------------------------------------- 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) ! ToDo: has wrong oder of arguments !-------------------------------------------------------------------------------------------------- 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 !-------------------------------------------------------------------------------------------------- pure 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_clip(-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 logarithm eigenvector basis of symmetric 33 matrix m !-------------------------------------------------------------------------------------------------- pure function math_eigenvectorBasisSym33_log(m) implicit none real(pReal), dimension(3,3) :: math_eigenvectorBasisSym33_log 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_clip(-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_log = log(sqrt(values(1))) * EB(1:3,1:3,1) & + log(sqrt(values(2))) * EB(1:3,1:3,2) & + log(sqrt(values(3))) * EB(1:3,1:3,3) end function math_eigenvectorBasisSym33_log !-------------------------------------------------------------------------------------------------- !> @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, intrinsic :: & IEEE_arithmetic 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 = IEEE_value(1.0_pReal,IEEE_quiet_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_clip(-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 an element of a Halton sequence. !> @author John Burkardt !> @author Martin Diehl !> @details Incrementally increasing elements of the Halton sequence for given bases (> 0) !> @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. !> @details Reference for prime numbers: !> @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. !------------------------------------------------------------------------------------------------- function halton(bases) implicit none integer(pInt), intent(in), dimension(:):: & bases !< bases (prime number ID) real(pReal), dimension(size(bases)) :: & halton integer(pInt), save :: & current = 1_pInt real(pReal), dimension(size(bases)) :: & base_inv integer(pInt), dimension(size(bases)) :: & base, & t integer(pInt), dimension(0:1600), parameter :: & prime = int([& 1, & 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, & 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, & 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, & 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, & 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, & 233, 239, 241, 251, 257, 263, 269, 271, 277, 281, & 283, 293, 307, 311, 313, 317, 331, 337, 347, 349, & 353, 359, 367, 373, 379, 383, 389, 397, 401, 409, & 419, 421, 431, 433, 439, 443, 449, 457, 461, 463, & 467, 479, 487, 491, 499, 503, 509, 521, 523, 541, & ! 101:200 547, 557, 563, 569, 571, 577, 587, 593, 599, 601, & 607, 613, 617, 619, 631, 641, 643, 647, 653, 659, & 661, 673, 677, 683, 691, 701, 709, 719, 727, 733, & 739, 743, 751, 757, 761, 769, 773, 787, 797, 809, & 811, 821, 823, 827, 829, 839, 853, 857, 859, 863, & 877, 881, 883, 887, 907, 911, 919, 929, 937, 941, & 947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, & 1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, & 1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, & 1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223, & ! 201:300 1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, & 1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, & 1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, & 1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, & 1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, & 1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, & 1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, & 1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, & 1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, & 1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987, & ! 301:400 1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, & 2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, & 2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, & 2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, & 2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, & 2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, & 2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, & 2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, & 2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, & 2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741, & ! 401:500 2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, & 2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, & 2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, & 3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, & 3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, & 3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, & 3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, & 3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, & 3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, & 3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571, & ! 501:600 3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, & 3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, & 3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, & 3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, & 3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, & 4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, & 4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, & 4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, & 4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, & 4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409, & ! 601:700 4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, & 4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, & 4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, & 4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, & 4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, & 4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, & 4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, & 5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, & 5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, & 5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279, & ! 701:800 5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, & 5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, & 5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, & 5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, & 5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, & 5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, & 5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, & 5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, & 5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, & 6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133, & ! 801:900 6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, & 6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, & 6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, & 6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, & 6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, & 6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, & 6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, & 6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, & 6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, & 6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997, & ! 901:1000 7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, & 7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, & 7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, & 7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, & 7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, & 7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, & 7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, & 7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, & 7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, & 7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919, & ! 1001:1100 7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, & 8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, & 8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, & 8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, & 8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, & 8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, & 8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, & 8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, & 8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, & 8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831, & ! 1101:1200 8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, & 8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, & 9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, & 9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, & 9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, & 9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, & 9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, & 9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, & 9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, & 9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733, & ! 1201:1300 9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, & 9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, & 9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973, 10007, & 10009, 10037, 10039, 10061, 10067, 10069, 10079, 10091, 10093, 10099, & 10103, 10111, 10133, 10139, 10141, 10151, 10159, 10163, 10169, 10177, & 10181, 10193, 10211, 10223, 10243, 10247, 10253, 10259, 10267, 10271, & 10273, 10289, 10301, 10303, 10313, 10321, 10331, 10333, 10337, 10343, & 10357, 10369, 10391, 10399, 10427, 10429, 10433, 10453, 10457, 10459, & 10463, 10477, 10487, 10499, 10501, 10513, 10529, 10531, 10559, 10567, & 10589, 10597, 10601, 10607, 10613, 10627, 10631, 10639, 10651, 10657, & ! 1301:1400 10663, 10667, 10687, 10691, 10709, 10711, 10723, 10729, 10733, 10739, & 10753, 10771, 10781, 10789, 10799, 10831, 10837, 10847, 10853, 10859, & 10861, 10867, 10883, 10889, 10891, 10903, 10909, 10937, 10939, 10949, & 10957, 10973, 10979, 10987, 10993, 11003, 11027, 11047, 11057, 11059, & 11069, 11071, 11083, 11087, 11093, 11113, 11117, 11119, 11131, 11149, & 11159, 11161, 11171, 11173, 11177, 11197, 11213, 11239, 11243, 11251, & 11257, 11261, 11273, 11279, 11287, 11299, 11311, 11317, 11321, 11329, & 11351, 11353, 11369, 11383, 11393, 11399, 11411, 11423, 11437, 11443, & 11447, 11467, 11471, 11483, 11489, 11491, 11497, 11503, 11519, 11527, & 11549, 11551, 11579, 11587, 11593, 11597, 11617, 11621, 11633, 11657, & ! 1401:1500 11677, 11681, 11689, 11699, 11701, 11717, 11719, 11731, 11743, 11777, & 11779, 11783, 11789, 11801, 11807, 11813, 11821, 11827, 11831, 11833, & 11839, 11863, 11867, 11887, 11897, 11903, 11909, 11923, 11927, 11933, & 11939, 11941, 11953, 11959, 11969, 11971, 11981, 11987, 12007, 12011, & 12037, 12041, 12043, 12049, 12071, 12073, 12097, 12101, 12107, 12109, & 12113, 12119, 12143, 12149, 12157, 12161, 12163, 12197, 12203, 12211, & 12227, 12239, 12241, 12251, 12253, 12263, 12269, 12277, 12281, 12289, & 12301, 12323, 12329, 12343, 12347, 12373, 12377, 12379, 12391, 12401, & 12409, 12413, 12421, 12433, 12437, 12451, 12457, 12473, 12479, 12487, & 12491, 12497, 12503, 12511, 12517, 12527, 12539, 12541, 12547, 12553, & ! 1501:1600 12569, 12577, 12583, 12589, 12601, 12611, 12613, 12619, 12637, 12641, & 12647, 12653, 12659, 12671, 12689, 12697, 12703, 12713, 12721, 12739, & 12743, 12757, 12763, 12781, 12791, 12799, 12809, 12821, 12823, 12829, & 12841, 12853, 12889, 12893, 12899, 12907, 12911, 12917, 12919, 12923, & 12941, 12953, 12959, 12967, 12973, 12979, 12983, 13001, 13003, 13007, & 13009, 13033, 13037, 13043, 13049, 13063, 13093, 13099, 13103, 13109, & 13121, 13127, 13147, 13151, 13159, 13163, 13171, 13177, 13183, 13187, & 13217, 13219, 13229, 13241, 13249, 13259, 13267, 13291, 13297, 13309, & 13313, 13327, 13331, 13337, 13339, 13367, 13381, 13397, 13399, 13411, & 13417, 13421, 13441, 13451, 13457, 13463, 13469, 13477, 13487, 13499],pInt) current = current + 1_pInt base = prime(bases) base_inv = 1.0_pReal/real(base,pReal) halton = 0.0_pReal t = current do while (any( t /= 0_pInt) ) halton = halton + real(mod(t,base), pReal) * base_inv base_inv = base_inv / real(base, pReal) t = t / base enddo end function halton !-------------------------------------------------------------------------------------------------- !> @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,transpose(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(transpose(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(i,m) * rot_tensor(j,n) * rot_tensor(k,o) * rot_tensor(l,p) * tensor(m,n,o,p) enddo; enddo; enddo; enddo; enddo; enddo; enddo; enddo end function math_rotate_forward3333 !-------------------------------------------------------------------------------------------------- !> @brief limits a scalar value to a certain range (either one or two sided) ! Will return NaN if left > right !-------------------------------------------------------------------------------------------------- real(pReal) pure elemental function math_clip(a, left, right) use, intrinsic :: & IEEE_arithmetic implicit none real(pReal), intent(in) :: a real(pReal), intent(in), optional :: left, right math_clip = a if (present(left)) math_clip = max(left,math_clip) if (present(right)) math_clip = min(right,math_clip) if (present(left) .and. present(right)) & math_clip = merge (IEEE_value(1.0_pReal,IEEE_quiet_NaN),math_clip, left>right) end function math_clip #if defined(__PGI) !-------------------------------------------------------------------------------------------------- !> @brief substitute for the norm2 intrinsic which is not available in PGI 18.10 !-------------------------------------------------------------------------------------------------- real(pReal) pure function norm2(v) implicit none real(pReal), intent(in), dimension(3) :: v norm2 = sqrt(sum(v**2)) end function norm2 #endif end module math