!-------------------------------------------------------------------------------------------------- !> @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 use IO use numerics use LAPACK_interface implicit none public #if __INTEL_COMPILER >= 1900 ! do not make use associated entities available to other modules private :: & prec, & IO, & numerics #endif real(pReal), parameter :: PI = acos(-1.0_pReal) !< ratio of a circle's circumference to its diameter real(pReal), parameter :: INDEG = 180.0_pReal/PI !< conversion from radian into degree real(pReal), parameter :: INRAD = PI/180.0_pReal !< conversion from degree into radian complex(pReal), parameter :: TWOPIIMG = cmplx(0.0_pReal,2.0_pReal*PI) !< Re(0.0), Im(2xPi) real(pReal), dimension(3,3), parameter :: & 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 & ],shape(math_I3)) !< 3x3 Identity real(pReal), dimension(*), parameter, private :: & NRMMANDEL = [1.0_pReal, 1.0_pReal,1.0_pReal, sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal)] !< forward weighting for Mandel notation real(pReal), dimension(*), parameter, private :: & INVNRMMANDEL = 1.0_pReal/NRMMANDEL !< backward weighting for Mandel notation integer, dimension (2,6), parameter, private :: & MAPNYE = reshape([& 1,1, & 2,2, & 3,3, & 1,2, & 2,3, & 1,3 & ],shape(MAPNYE)) !< arrangement in Nye notation. integer, dimension (2,6), parameter, private :: & MAPVOIGT = reshape([& 1,1, & 2,2, & 3,3, & 2,3, & 1,3, & 1,2 & ],shape(MAPVOIGT)) !< arrangement in Voigt notation integer, dimension (2,9), parameter, private :: & MAPPLAIN = reshape([& 1,1, & 1,2, & 1,3, & 2,1, & 2,2, & 2,3, & 3,1, & 3,2, & 3,3 & ],shape(MAPPLAIN)) !< arrangement in Plain notation interface math_eye module procedure math_identity2nd end interface math_eye !--------------------------------------------------------------------------------------------------- private :: & unitTest contains !-------------------------------------------------------------------------------------------------- !> @brief initialization of random seed generator and internal checks !-------------------------------------------------------------------------------------------------- subroutine math_init real(pReal), dimension(4) :: randTest integer :: randSize integer, dimension(:), allocatable :: randInit write(6,'(/,a)') ' <<<+- math init -+>>>'; flush(6) call random_seed(size=randSize) allocate(randInit(randSize)) if (randomSeed > 0) then randInit = randomSeed else call random_seed() call random_seed(get = randInit) randInit(2:randSize) = randInit(1) endif call random_seed(put = randInit) call random_number(randTest) write(6,'(a,i2)') ' size of random seed: ', randSize write(6,'(a,i0)') ' value of random seed: ', randInit(1) write(6,'(a,4(/,26x,f17.14),/)') ' start of random sequence: ', randTest call random_seed(put = randInit) call unitTest end subroutine math_init !-------------------------------------------------------------------------------------------------- !> @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_sort(a, istart, iend, sortDim) integer, dimension(:,:), intent(inout) :: a integer, intent(in),optional :: istart,iend, sortDim integer :: 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_sort(a, s, ipivot-1, d) call math_sort(a, ipivot+1, e, d) endif contains !------------------------------------------------------------------------------------------------- !> @brief Partitioning required for quicksort !------------------------------------------------------------------------------------------------- integer function qsort_partition(a, istart, iend, sort) integer, dimension(:,:), intent(inout) :: a integer, intent(in) :: istart,iend,sort integer, dimension(size(a,1)) :: tmp integer :: i,j do ! find the first element on the right side less than or equal to the pivot point do j = iend, istart, -1 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 ! 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 ! exchange values tmp = a(:,i) a(:,i) = a(:,j) a(:,j) = tmp endif cross enddo end function qsort_partition end subroutine math_sort !-------------------------------------------------------------------------------------------------- !> @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) real(pReal), dimension(:), intent(in) :: what integer, dimension(:), intent(in) :: how real(pReal), dimension(sum(how)) :: math_expand integer :: i if (sum(how) == 0) return do i = 1, size(how) math_expand(sum(how(1:i-1))+1:sum(how(1:i))) = what(mod(i-1,size(what))+1) enddo end function math_expand !-------------------------------------------------------------------------------------------------- !> @brief range of integers starting at one !-------------------------------------------------------------------------------------------------- pure function math_range(N) integer, intent(in) :: N !< length of range integer :: i integer, dimension(N) :: math_range math_range = [(i,i=1,N)] end function math_range !-------------------------------------------------------------------------------------------------- !> @brief second rank identity tensor of specified dimension !-------------------------------------------------------------------------------------------------- pure function math_identity2nd(d) integer, intent(in) :: d !< tensor dimension integer :: i real(pReal), dimension(d,d) :: math_identity2nd math_identity2nd = 0.0_pReal do i=1,d math_identity2nd(i,i) = 1.0_pReal enddo 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(d) integer, intent(in) :: d !< tensor dimension integer :: i,j,k,l real(pReal), dimension(d,d,d,d) :: math_identity4th real(pReal), dimension(d,d) :: identity2nd identity2nd = math_identity2nd(d) do i=1,d; do j=1,d; do k=1,d; do l=1,d math_identity4th(i,j,k,l) = 0.5_pReal & *(identity2nd(i,k)*identity2nd(j,l)+identity2nd(i,l)*identity2nd(j,k)) enddo; enddo; enddo; enddo end function math_identity4th !-------------------------------------------------------------------------------------------------- !> @brief permutation tensor e_ijk ! 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_LeviCivita(i,j,k) integer, intent(in) :: i,j,k if (all([i,j,k] == [1,2,3]) .or. all([i,j,k] == [2,3,1]) .or. all([i,j,k] == [3,1,2])) then math_LeviCivita = +1.0_pReal elseif (all([i,j,k] == [3,2,1]) .or. all([i,j,k] == [2,1,3]) .or. all([i,j,k] == [1,3,2])) then math_LeviCivita = -1.0_pReal else math_LeviCivita = 0.0_pReal endif end function math_LeviCivita !-------------------------------------------------------------------------------------------------- !> @brief kronecker delta function d_ij ! d_ij = 1 if i = j ! d_ij = 0 otherwise !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_delta(i,j) integer, 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_cross(A,B) real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: math_cross math_cross = [ 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_cross !-------------------------------------------------------------------------------------------------- !> @brief outer product of arbitrary sized vectors (A ⊗ B / i,j) !-------------------------------------------------------------------------------------------------- pure function math_outer(A,B) real(pReal), dimension(:), intent(in) :: A,B real(pReal), dimension(size(A,1),size(B,1)) :: math_outer integer :: i,j do i=1,size(A,1); do j=1,size(B,1) math_outer(i,j) = A(i)*B(j) enddo; enddo end function math_outer !-------------------------------------------------------------------------------------------------- !> @brief inner product of arbitrary sized vectors (A · B / i,i) !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_inner(A,B) real(pReal), dimension(:), intent(in) :: A real(pReal), dimension(size(A,1)), intent(in) :: B math_inner = sum(A*B) end function math_inner !-------------------------------------------------------------------------------------------------- !> @brief double contraction of 3x3 matrices (A : B / ij,ij) !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_tensordot(A,B) real(pReal), dimension(3,3), intent(in) :: A,B math_tensordot = sum(A*B) end function math_tensordot !-------------------------------------------------------------------------------------------------- !> @brief matrix double contraction 3333x33 = 33 (ijkl,kl) !-------------------------------------------------------------------------------------------------- pure function math_mul3333xx33(A,B) real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3), intent(in) :: B real(pReal), dimension(3,3) :: math_mul3333xx33 integer :: i,j do i=1,3; do j=1,3 math_mul3333xx33(i,j) = sum(A(i,j,1:3,1:3)*B(1:3,1:3)) enddo; enddo end function math_mul3333xx33 !-------------------------------------------------------------------------------------------------- !> @brief matrix multiplication 3333x3333 = 3333 (ijkl,klmn) !-------------------------------------------------------------------------------------------------- pure function math_mul3333xx3333(A,B) integer :: 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 do i=1,3; do j=1,3; do k=1,3; do l=1,3 math_mul3333xx3333(i,j,k,l) = sum(A(i,j,1:3,1:3)*B(1:3,1:3,k,l)) enddo; enddo; enddo; enddo end function math_mul3333xx3333 !-------------------------------------------------------------------------------------------------- !> @brief 3x3 matrix exponential up to series approximation order n (default 5) !-------------------------------------------------------------------------------------------------- pure function math_exp33(A,n) real(pReal), dimension(3,3), intent(in) :: A integer, intent(in), optional :: n real(pReal), dimension(3,3) :: B, math_exp33 real(pReal) :: invFac integer :: n_,i if (present(n)) then n_ = n else n_ = 5 endif invFac = 1.0_pReal ! 0! B = math_I3 math_exp33 = math_I3 ! A^0 = I do i = 1, n_ invFac = invFac/real(i,pReal) ! invfac = 1/(i!) B = matmul(B,A) math_exp33 = math_exp33 + invFac*B ! exp = SUM (A^i)/(i!) enddo end function math_exp33 !-------------------------------------------------------------------------------------------------- !> @brief Cramer inversion of 3x3 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) real(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3,3) :: math_inv33 real(pReal) :: DetA logical :: error call math_invert33(math_inv33,DetA,error,A) if(error) math_inv33 = 0.0_pReal end function math_inv33 !-------------------------------------------------------------------------------------------------- !> @brief Cramer inversion of 3x3 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 !-------------------------------------------------------------------------------------------------- pure subroutine math_invert33(InvA, DetA, error, A) real(pReal), dimension(3,3), intent(out) :: InvA real(pReal), intent(out) :: DetA logical, intent(out) :: error real(pReal), dimension(3,3), intent(in) :: A 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 matrix !-------------------------------------------------------------------------------------------------- function math_invSym3333(A) real(pReal),dimension(3,3,3,3) :: math_invSym3333 real(pReal),dimension(3,3,3,3),intent(in) :: A integer, dimension(6) :: ipiv6 real(pReal), dimension(6,6) :: temp66 real(pReal), dimension(6*6) :: work integer :: ierr_i, ierr_f temp66 = math_sym3333to66(A) call dgetrf(6,6,temp66,6,ipiv6,ierr_i) call dgetri(6,temp66,6,ipiv6,work,size(work,1),ierr_f) if (ierr_i /= 0 .or. ierr_f /= 0) then call IO_error(400, ext_msg = 'math_invSym3333') else math_invSym3333 = math_66toSym3333(temp66) endif end function math_invSym3333 !-------------------------------------------------------------------------------------------------- !> @brief invert quadratic matrix of arbitrary dimension !-------------------------------------------------------------------------------------------------- subroutine math_invert(InvA, error, A) real(pReal), dimension(:,:), intent(in) :: A real(pReal), dimension(size(A,1),size(A,1)), intent(out) :: invA logical, intent(out) :: error integer, dimension(size(A,1)) :: ipiv real(pReal), dimension(size(A,1)**2) :: work integer :: ierr invA = A call dgetrf(size(A,1),size(A,1),invA,size(A,1),ipiv,ierr) error = (ierr /= 0) call dgetri(size(A,1),InvA,size(A,1),ipiv,work,size(work,1),ierr) error = error .or. (ierr /= 0) end subroutine math_invert !-------------------------------------------------------------------------------------------------- !> @brief symmetrize a 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_symmetric33(m) 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 6x6 matrix !-------------------------------------------------------------------------------------------------- pure function math_symmetric66(m) 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 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_skew33(m) 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 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_spherical33(m) 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 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_deviatoric33(m) 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 trace of a 3x3 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_trace33(m) 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 3x3 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_det33(m) 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 3x3 matrix !-------------------------------------------------------------------------------------------------- real(pReal) pure function math_detSym33(m) real(pReal), dimension(3,3), intent(in) :: m math_detSym33 = -(m(1,1)*m(2,3)**2 + m(2,2)*m(1,3)**2 + m(3,3)*m(1,2)**2) & + 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 3x3 matrix into vector 9 !-------------------------------------------------------------------------------------------------- pure function math_33to9(m33) real(pReal), dimension(9) :: math_33to9 real(pReal), dimension(3,3), intent(in) :: m33 integer :: i do i = 1, 9 math_33to9(i) = m33(MAPPLAIN(1,i),MAPPLAIN(2,i)) enddo end function math_33to9 !-------------------------------------------------------------------------------------------------- !> @brief convert 9 vector into 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_9to33(v9) real(pReal), dimension(3,3) :: math_9to33 real(pReal), dimension(9), intent(in) :: v9 integer :: i do i = 1, 9 math_9to33(MAPPLAIN(1,i),MAPPLAIN(2,i)) = v9(i) enddo end function math_9to33 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 3x3 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) real(pReal), dimension(6) :: math_sym33to6 real(pReal), dimension(3,3), intent(in) :: m33 !< symmetric 3x3 matrix (no internal check) logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default) real(pReal), dimension(6) :: w integer :: i if(present(weighted)) then w = merge(NRMMANDEL,1.0_pReal,weighted) else w = NRMMANDEL endif do i = 1, 6 math_sym33to6(i) = w(i)*m33(MAPNYE(1,i),MAPNYE(2,i)) enddo end function math_sym33to6 !-------------------------------------------------------------------------------------------------- !> @brief convert 6 vector into symmetric 3x3 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) real(pReal), dimension(3,3) :: math_6toSym33 real(pReal), dimension(6), intent(in) :: v6 !< 6 vector logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default) real(pReal), dimension(6) :: w integer :: i if(present(weighted)) then w = merge(INVNRMMANDEL,1.0_pReal,weighted) else w = INVNRMMANDEL endif do i=1,6 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 3x3x3x3 matrix into 9x9 matrix !-------------------------------------------------------------------------------------------------- pure function math_3333to99(m3333) real(pReal), dimension(9,9) :: math_3333to99 real(pReal), dimension(3,3,3,3), intent(in) :: m3333 integer :: i,j do i=1,9; do j=1,9 math_3333to99(i,j) = m3333(MAPPLAIN(1,i),MAPPLAIN(2,i),MAPPLAIN(1,j),MAPPLAIN(2,j)) enddo; enddo end function math_3333to99 !-------------------------------------------------------------------------------------------------- !> @brief convert 9x9 matrix into 3x3x3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_99to3333(m99) real(pReal), dimension(3,3,3,3) :: math_99to3333 real(pReal), dimension(9,9), intent(in) :: m99 integer :: i,j do i=1,9; do j=1,9 math_99to3333(MAPPLAIN(1,i),MAPPLAIN(2,i),MAPPLAIN(1,j),MAPPLAIN(2,j)) = m99(i,j) enddo; enddo end function math_99to3333 !-------------------------------------------------------------------------------------------------- !> @brief convert symmetric 3x3x3x3 matrix into 6x6 matrix !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only rearranges order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_sym3333to66(m3333,weighted) real(pReal), dimension(6,6) :: math_sym3333to66 real(pReal), dimension(3,3,3,3), intent(in) :: m3333 !< symmetric 3x3x3x3 matrix (no internal check) logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default) real(pReal), dimension(6) :: w integer :: i,j if(present(weighted)) then w = merge(NRMMANDEL,1.0_pReal,weighted) else w = NRMMANDEL endif do i=1,6; do j=1,6 math_sym3333to66(i,j) = w(i)*w(j)*m3333(MAPNYE(1,i),MAPNYE(2,i),MAPNYE(1,j),MAPNYE(2,j)) enddo; enddo end function math_sym3333to66 !-------------------------------------------------------------------------------------------------- !> @brief convert 66 matrix into symmetric 3x3x3x3 matrix !> @details Weighted conversion (default) rearranges according to Nye and weights shear ! components according to Mandel. Advisable for matrix operations. ! Unweighted conversion only rearranges order according to Nye !-------------------------------------------------------------------------------------------------- pure function math_66toSym3333(m66,weighted) real(pReal), dimension(3,3,3,3) :: math_66toSym3333 real(pReal), dimension(6,6), intent(in) :: m66 !< 6x6 matrix logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default) real(pReal), dimension(6) :: w integer :: i,j if(present(weighted)) then w = merge(INVNRMMANDEL,1.0_pReal,weighted) else w = INVNRMMANDEL endif do i=1,6; do j=1,6 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 3x3x3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_Voigt66to3333(m66) real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333 real(pReal), dimension(6,6), intent(in) :: m66 !< 6x6 matrix integer :: i,j do i=1,6; do j=1, 6 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 draw a random sample from Gauss variable !-------------------------------------------------------------------------------------------------- real(pReal) function math_sampleGaussVar(meanvalue, stddev, width) 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 width_ if (abs(stddev) < tol_math_check) then math_sampleGaussVar = meanvalue else if (present(width)) then width_ = width else width_ = 3.0_pReal ! use +-3*sigma as default scatter endif do call random_number(rnd) scatter = width_ * (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 eigenvalues and eigenvectors of symmetric matrix ! ToDo: has wrong oder of arguments !-------------------------------------------------------------------------------------------------- subroutine math_eigh(m,w,v,error) real(pReal), dimension(:,:), intent(in) :: m !< quadratic matrix to compute eigenvectors and values of real(pReal), dimension(size(m,1)), intent(out) :: w !< eigenvalues real(pReal), dimension(size(m,1),size(m,1)), intent(out) :: v !< eigenvectors logical, intent(out) :: error integer :: ierr real(pReal), dimension(size(m,1)**2) :: work v = m ! copy matrix to input (doubles as output) array call dsyev('V','U',size(m,1),v,size(m,1),w,work,size(work,1),ierr) error = (ierr /= 0) end subroutine math_eigh !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues and eigenvectors of symmetric 3x3 matrix 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_eigh33(m,w,v) real(pReal), dimension(3,3),intent(in) :: m !< 3x3 matrix to compute eigenvectors and values of real(pReal), dimension(3), intent(out) :: w !< eigenvalues real(pReal), dimension(3,3),intent(out) :: v !< eigenvectors real(pReal) :: T, U, norm, threshold logical :: error w = math_eigvalsh33(m) v(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] T = maxval(abs(w)) U = max(T, T**2) threshold = sqrt(5.68e-14_pReal * U**2) v(1:3,1) = [ v(1,2) + m(1, 3) * w(1), & v(2,2) + m(2, 3) * w(1), & (m(1,1) - w(1)) * (m(2,2) - w(1)) - v(3,2)] norm = norm2(v(1:3, 1)) fallback1: if(norm < threshold) then call math_eigh(m,w,v,error) else fallback1 v(1:3,1) = v(1:3, 1) / norm v(1:3,2) = [ v(1,2) + m(1, 3) * w(2), & v(2,2) + m(2, 3) * w(2), & (m(1,1) - w(2)) * (m(2,2) - w(2)) - v(3,2)] norm = norm2(v(1:3, 2)) fallback2: if(norm < threshold) then call math_eigh(m,w,v,error) else fallback2 v(1:3,2) = v(1:3, 2) / norm v(1:3,3) = math_cross(v(1:3,1),v(1:3,2)) endif fallback2 endif fallback1 end subroutine math_eigh33 !-------------------------------------------------------------------------------------------------- !> @brief rotational part from polar decomposition of 3x3 tensor !-------------------------------------------------------------------------------------------------- function math_rotationalPart(m) real(pReal), intent(in), dimension(3,3) :: m real(pReal), dimension(3,3) :: math_rotationalPart real(pReal), dimension(3,3) :: U , Uinv U = eigenvectorBasis(matmul(transpose(m),m)) Uinv = math_inv33(U) inversionFailed: if (all(dEq0(Uinv))) then math_rotationalPart = math_I3 call IO_warning(650) else inversionFailed math_rotationalPart = matmul(m,Uinv) endif inversionFailed contains !-------------------------------------------------------------------------------------------------- !> @brief eigenvector basis of positive-definite 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function eigenvectorBasis(m) real(pReal), dimension(3,3) :: eigenvectorBasis real(pReal), dimension(3,3), intent(in) :: m !< positive-definite matrix of which the basis is computed real(pReal), dimension(3) :: I, v real(pReal) :: P, Q, rho, phi real(pReal), parameter :: TOL=1.e-14_pReal real(pReal), dimension(3,3,3) :: N, EB I = math_invariantsSym33(m) P = I(2)-I(1)**2.0_pReal/3.0_pReal Q = -2.0_pReal/27.0_pReal*I(1)**3.0_pReal+product(I(1:2))/3.0_pReal-I(3) threeSimilarEigVals: if(all(abs([P,Q]) < TOL)) then v = I(1)/3.0_pReal ! this is not really correct, but at least the basis is correct EB = 0.0_pReal EB(1,1,1)=1.0_pReal EB(2,2,2)=1.0_pReal EB(3,3,3)=1.0_pReal else threeSimilarEigVals 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)) v = 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) & ] + I(1)/3.0_pReal N(1:3,1:3,1) = m-v(1)*math_I3 N(1:3,1:3,2) = m-v(2)*math_I3 N(1:3,1:3,3) = m-v(3)*math_I3 twoSimilarEigVals: if(abs(v(1)-v(2)) < TOL) then EB(1:3,1:3,3) = matmul(N(1:3,1:3,1),N(1:3,1:3,2))/((v(3)-v(1))*(v(3)-v(2))) EB(1:3,1:3,1) = math_I3-EB(1:3,1:3,3) EB(1:3,1:3,2) = 0.0_pReal elseif (abs(v(2)-v(3)) < TOL) then twoSimilarEigVals EB(1:3,1:3,1) = matmul(N(1:3,1:3,2),N(1:3,1:3,3))/((v(1)-v(2))*(v(1)-v(3))) EB(1:3,1:3,2) = math_I3-EB(1:3,1:3,1) EB(1:3,1:3,3) = 0.0_pReal elseif (abs(v(3)-v(1)) < TOL) then twoSimilarEigVals EB(1:3,1:3,2) = matmul(N(1:3,1:3,3),N(1:3,1:3,1))/((v(2)-v(3))*(v(2)-v(1))) EB(1:3,1:3,3) = math_I3-EB(1:3,1:3,2) EB(1:3,1:3,1) = 0.0_pReal else twoSimilarEigVals EB(1:3,1:3,1) = matmul(N(1:3,1:3,2),N(1:3,1:3,3))/((v(1)-v(2))*(v(1)-v(3))) EB(1:3,1:3,2) = matmul(N(1:3,1:3,3),N(1:3,1:3,1))/((v(2)-v(3))*(v(2)-v(1))) EB(1:3,1:3,3) = matmul(N(1:3,1:3,1),N(1:3,1:3,2))/((v(3)-v(1))*(v(3)-v(2))) endif twoSimilarEigVals endif threeSimilarEigVals eigenvectorBasis = sqrt(v(1)) * EB(1:3,1:3,1) & + sqrt(v(2)) * EB(1:3,1:3,2) & + sqrt(v(3)) * EB(1:3,1:3,3) end function eigenvectorBasis end function math_rotationalPart !-------------------------------------------------------------------------------------------------- !> @brief Eigenvalues of symmetric matrix ! will return NaN on error !-------------------------------------------------------------------------------------------------- function math_eigvalsh(m) real(pReal), dimension(:,:), intent(in) :: m !< symmetric matrix to compute eigenvalues of real(pReal), dimension(size(m,1)) :: math_eigvalsh real(pReal), dimension(size(m,1),size(m,1)) :: m_ integer :: ierr real(pReal), dimension(size(m,1)**2) :: work m_= m ! copy matrix to input (will be destroyed) call dsyev('N','U',size(m,1),m_,size(m,1),math_eigvalsh,work,size(work,1),ierr) if (ierr /= 0) math_eigvalsh = IEEE_value(1.0_pReal,IEEE_quiet_NaN) end function math_eigvalsh !-------------------------------------------------------------------------------------------------- !> @brief eigenvalues of symmetric 3x3 matrix 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_eigvalsh33(m) real(pReal), intent(in), dimension(3,3) :: m !< 3x3 symmetric matrix to compute eigenvalues of real(pReal), dimension(3) :: math_eigvalsh33,I real(pReal) :: P, Q, rho, phi real(pReal), parameter :: TOL=1.e-14_pReal I = 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 = I(2)-I(1)**2.0_pReal/3.0_pReal ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK) Q = product(I(1:2))/3.0_pReal & - 2.0_pReal/27.0_pReal*I(1)**3.0_pReal & - I(3) ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK) if(all(abs([P,Q]) < TOL)) then math_eigvalsh33 = math_eigvalsh(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_eigvalsh33 = 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) & ] & + I(1)/3.0_pReal endif end function math_eigvalsh33 !-------------------------------------------------------------------------------------------------- !> @brief invariants of symmetrix 3x3 matrix !-------------------------------------------------------------------------------------------------- pure function math_invariantsSym33(m) 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 factorial !-------------------------------------------------------------------------------------------------- integer pure function math_factorial(n) integer, intent(in) :: n math_factorial = product(math_range(n)) end function math_factorial !-------------------------------------------------------------------------------------------------- !> @brief binomial coefficient !-------------------------------------------------------------------------------------------------- integer pure function math_binomial(n,k) integer, intent(in) :: n, k integer :: i, k_, n_ k_ = min(k,n-k) n_ = n math_binomial = merge(1,0,k_>-1) ! handling special cases k < 0 or k > n do i = 1, k_ math_binomial = (math_binomial * n_)/i n_ = n_ -1 enddo end function math_binomial !-------------------------------------------------------------------------------------------------- !> @brief multinomial coefficient !-------------------------------------------------------------------------------------------------- integer pure function math_multinomial(alpha) integer, intent(in), dimension(:) :: alpha integer :: i math_multinomial = 1 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) 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) = v1-v3 m(1:3,3) = v1-v4 math_volTetrahedron = abs(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) real(pReal), dimension (3), intent(in) :: v1,v2,v3 math_areaTriangle = 0.5_pReal * norm2(math_cross(v1-v2,v1-v3)) end function math_areaTriangle !-------------------------------------------------------------------------------------------------- !> @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) 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 !-------------------------------------------------------------------------------------------------- !> @brief check correctness of some math functions !-------------------------------------------------------------------------------------------------- subroutine unitTest integer, dimension(2,4) :: & sort_in_ = reshape([+1,+5, +5,+6, -1,-1, +3,-2],[2,4]) integer, dimension(2,4), parameter :: & sort_out_ = reshape([-1,-1, +1,+5, +5,+6, +3,-2],[2,4]) integer, dimension(5) :: range_out_ = [1,2,3,4,5] integer, dimension(3) :: ijk real(pReal) :: det real(pReal), dimension(3) :: v3_1,v3_2,v3_3,v3_4 real(pReal), dimension(6) :: v6 real(pReal), dimension(9) :: v9 real(pReal), dimension(3,3) :: t33,t33_2 real(pReal), dimension(6,6) :: t66 real(pReal), dimension(9,9) :: t99,t99_2 real(pReal), dimension(:,:), & allocatable :: txx,txx_2 real(pReal) :: r integer :: d logical :: e 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],[1,2,3,0])) > tol_math_check)) & call IO_error(0,ext_msg='math_expand [1,2,3] by [1,2,3,0] => [1,2,2,3,3,3]') if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal] - & math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1,2])) > tol_math_check)) & call IO_error(0,ext_msg='math_expand [1,2,3] by [1,2] => [1,2,2]') 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],[1,2,3])) > tol_math_check)) & call IO_error(0,ext_msg='math_expand [1,2] by [1,2,3] => [1,2,2,1,1,1]') call math_sort(sort_in_,1,3,2) if(any(sort_in_ /= sort_out_)) & call IO_error(0,ext_msg='math_sort') if(any(math_range(5) /= range_out_)) & call IO_error(0,ext_msg='math_range') if(any(dNeq(math_exp33(math_I3,0),math_I3))) & call IO_error(0,ext_msg='math_exp33(math_I3,1)') if(any(dNeq(math_exp33(math_I3,256),exp(1.0_pReal)*math_I3))) & call IO_error(0,ext_msg='math_exp33(math_I3,256)') call random_number(v9) if(any(dNeq(math_33to9(math_9to33(v9)),v9))) & call IO_error(0,ext_msg='math_33to9/math_9to33') call random_number(t99) if(any(dNeq(math_3333to99(math_99to3333(t99)),t99))) & call IO_error(0,ext_msg='math_3333to99/math_99to3333') call random_number(v6) if(any(dNeq(math_sym33to6(math_6toSym33(v6)),v6))) & call IO_error(0,ext_msg='math_sym33to6/math_6toSym33') call random_number(t66) if(any(dNeq(math_sym3333to66(math_66toSym3333(t66)),t66))) & call IO_error(0,ext_msg='math_sym3333to66/math_66toSym3333') call random_number(v6) if(any(dNeq0(math_6toSym33(v6) - math_symmetric33(math_6toSym33(v6))))) & call IO_error(0,ext_msg='math_symmetric33') call random_number(v3_1) call random_number(v3_2) call random_number(v3_3) call random_number(v3_4) if(dNeq(abs(dot_product(math_cross(v3_1-v3_4,v3_2-v3_4),v3_3-v3_4))/6.0, & math_volTetrahedron(v3_1,v3_2,v3_3,v3_4),tol=1.0e-12_pReal)) & call IO_error(0,ext_msg='math_volTetrahedron') call random_number(t33) if(dNeq(math_det33(math_symmetric33(t33)),math_detSym33(math_symmetric33(t33)),tol=1.0e-12_pReal)) & call IO_error(0,ext_msg='math_det33/math_detSym33') if(any(dNeq0(math_identity2nd(3),math_inv33(math_I3)))) & call IO_error(0,ext_msg='math_inv33(math_I3)') do while(abs(math_det33(t33))<1.0e-9_pReal) call random_number(t33) enddo if(any(dNeq0(matmul(t33,math_inv33(t33)) - math_identity2nd(3),tol=1.0e-9_pReal))) & call IO_error(0,ext_msg='math_inv33') call math_invert33(t33_2,det,e,t33) if(any(dNeq0(matmul(t33,t33_2) - math_identity2nd(3),tol=1.0e-9_pReal)) .or. e) & call IO_error(0,ext_msg='math_invert33: T:T^-1 != I') if(dNeq(det,math_det33(t33),tol=1.0e-12_pReal)) & call IO_error(0,ext_msg='math_invert33 (determinant)') call math_invert(t33_2,e,t33) if(any(dNeq0(matmul(t33,t33_2) - math_identity2nd(3),tol=1.0e-9_pReal)) .or. e) & call IO_error(0,ext_msg='math_invert t33') do while(math_det33(t33)<1.0e-2_pReal) ! O(det(F)) = 1 call random_number(t33) enddo t33_2 = math_rotationalPart(transpose(t33)) t33 = math_rotationalPart(t33) if(any(dNeq0(matmul(t33_2,t33) - math_I3,tol=1.0e-10_pReal))) & call IO_error(0,ext_msg='math_rotationalPart') call random_number(r) d = int(r*5.0_pReal) + 1 txx = math_identity2nd(d) allocate(txx_2(d,d)) call math_invert(txx_2,e,txx) if(any(dNeq0(txx_2,txx) .or. e)) & call IO_error(0,ext_msg='math_invert(txx)/math_identity2nd') call math_invert(t99_2,e,t99) ! not sure how likely it is that we get a singular matrix if(any(dNeq0(matmul(t99_2,t99)-math_identity2nd(9),tol=1.0e-9_pReal)) .or. e) & call IO_error(0,ext_msg='math_invert(t99)') if(any(dNeq(math_clip([4.0_pReal,9.0_pReal],5.0_pReal,6.5_pReal),[5.0_pReal,6.5_pReal]))) & call IO_error(0,ext_msg='math_clip') if(math_factorial(10) /= 3628800) & call IO_error(0,ext_msg='math_factorial') if(math_binomial(49,6) /= 13983816) & call IO_error(0,ext_msg='math_binomial') ijk = cshift([1,2,3],int(r*1.0e2_pReal)) if(dNeq(math_LeviCivita(ijk(1),ijk(2),ijk(3)),+1.0_pReal)) & call IO_error(0,ext_msg='math_LeviCivita(even)') ijk = cshift([3,2,1],int(r*2.0e2_pReal)) if(dNeq(math_LeviCivita(ijk(1),ijk(2),ijk(3)),-1.0_pReal)) & call IO_error(0,ext_msg='math_LeviCivita(odd)') ijk = cshift([2,2,1],int(r*2.0e2_pReal)) if(dNeq0(math_LeviCivita(ijk(1),ijk(2),ijk(3))))& call IO_error(0,ext_msg='math_LeviCivita') end subroutine unitTest end module math