!* $Id$ !############################################################## MODULE math !############################################################## use prec, only: pReal,pInt implicit none real(pReal), parameter :: pi = 3.14159265358979323846264338327950288419716939937510_pReal real(pReal), parameter :: inDeg = 180.0_pReal/pi real(pReal), parameter :: inRad = pi/180.0_pReal real(pReal), parameter :: NaN = 0.0_pReal/0.0_pReal ! Not a number ! *** 3x3 Identity *** 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 /),(/3,3/)) ! *** Mandel notation *** integer(pInt), dimension (2,6), parameter :: mapMandel = & reshape((/& 1,1, & 2,2, & 3,3, & 1,2, & 2,3, & 1,3 & /),(/2,6/)) real(pReal), dimension(6), parameter :: nrmMandel = & (/1.0_pReal,1.0_pReal,1.0_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal/) real(pReal), dimension(6), parameter :: invnrmMandel = & (/1.0_pReal,1.0_pReal,1.0_pReal,0.7071067811865476_pReal,0.7071067811865476_pReal,0.7071067811865476_pReal/) ! *** Voigt notation *** integer(pInt), dimension (2,6), parameter :: mapVoigt = & reshape((/& 1,1, & 2,2, & 3,3, & 2,3, & 1,3, & 1,2 & /),(/2,6/)) real(pReal), dimension(6), parameter :: nrmVoigt = & (/1.0_pReal,1.0_pReal,1.0_pReal, 1.0_pReal, 1.0_pReal, 1.0_pReal/) real(pReal), dimension(6), parameter :: invnrmVoigt = & (/1.0_pReal,1.0_pReal,1.0_pReal, 1.0_pReal, 1.0_pReal, 1.0_pReal/) ! *** Plain notation *** integer(pInt), dimension (2,9), parameter :: mapPlain = & reshape((/& 1,1, & 1,2, & 1,3, & 2,1, & 2,2, & 2,3, & 3,1, & 3,2, & 3,3 & /),(/2,9/)) ! Symmetry operations as quaternions ! 24 for cubic, 12 for hexagonal = 36 integer(pInt), dimension(2), parameter :: math_NsymOperations = (/24,12/) real(pReal), dimension(4,36), parameter :: math_symOperations = & reshape((/& 1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! cubic symmetry operations 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, & ! 2-fold symmetry 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, & 0.0_pReal, -0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, 0.0_pReal, & 0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & ! 3-fold symmetry -0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & 0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, & -0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, & 0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, & -0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, & 0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, & -0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, & 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, & ! 4-fold symmetry 0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & -0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, & 0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, & -0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal, & -0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! hexagonal symmetry operations 0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & ! 2-fold symmetry 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, & 0.0_pReal, 0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, & 0.0_pReal, -0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, & 0.0_pReal, 0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, & 0.0_pReal, -0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, & 0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, & ! 6-fold symmetry -0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, & 0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, & -0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, & 0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal & /),(/4,36/)) CONTAINS !************************************************************************** ! initialization of module !************************************************************************** SUBROUTINE math_init () use prec, only: pReal,pInt,tol_math_check use numerics, only: fixedSeed use IO, only: IO_error implicit none real(pReal), dimension(3,3) :: R,R2 real(pReal), dimension(3) :: Eulers real(pReal), dimension(4) :: q,q2,axisangle real(pReal), dimension(2) :: rnd integer(pInt), dimension(1) :: randInit write(6,*) write(6,*) '<<<+- math init -+>>>' write(6,*) '$Id$' write(6,*) if (fixedSeed > 0_pInt) then randInit = fixedSeed call random_seed(put=randInit) else call random_seed() endif call random_seed(get=randInit) write(6,*) 'random seed: ',randInit(1) write(6,*) call halton_seed_set(randInit(1)) call halton_ndim_set(3) ! --- check rotation dictionary --- ! +++ q -> a -> q +++ q = math_qRnd(); axisangle = math_QuaternionToAxisAngle(q); q2 = math_AxisAngleToQuaternion(axisangle(1:3),axisangle(4)) if ( any(abs( q-q2) > tol_math_check ) .and. & any(abs(-q-q2) > tol_math_check ) ) & call IO_error(670) ! +++ q -> R -> q +++ R = math_QuaternionToR(q); q2 = math_RToQuaternion(R) if ( any(abs( q-q2) > tol_math_check ) .and. & any(abs(-q-q2) > tol_math_check ) ) & call IO_error(671) ! +++ q -> euler -> q +++ Eulers = math_QuaternionToEuler(q); q2 = math_EulerToQuaternion(Eulers) if ( any(abs( q-q2) > tol_math_check ) .and. & any(abs(-q-q2) > tol_math_check ) ) & call IO_error(672) ! +++ R -> euler -> R +++ Eulers = math_RToEuler(R); R2 = math_EulerToR(Eulers) if ( any(abs( R-R2) > tol_math_check ) ) & call IO_error(673) ENDSUBROUTINE !************************************************************************** ! Quicksort algorithm for two-dimensional integer arrays ! ! Sorting is done with respect to array(1,:) ! and keeps array(2:N,:) linked to it. !************************************************************************** RECURSIVE SUBROUTINE qsort(a, istart, iend) implicit none integer(pInt), dimension(:,:) :: a integer(pInt) :: istart,iend,ipivot if (istart < iend) then ipivot = math_partition(a,istart, iend) call qsort(a, istart, ipivot-1) call qsort(a, ipivot+1, iend) endif return ENDSUBROUTINE !************************************************************************** ! Partitioning required for quicksort !************************************************************************** integer(pInt) function math_partition(a, istart, iend) implicit none integer(pInt), dimension(:,:) :: a integer(pInt) :: istart,iend,d,i,j,k,x,tmp d = size(a,1) ! number of linked data ! set the starting and ending points, and the pivot point i = istart j = iend x = a(1,istart) do ! find the first element on the right side less than or equal to the pivot point do j = j, istart, -1 if (a(1,j) <= x) exit enddo ! find the first element on the left side greater than the pivot point do i = i, iend if (a(1,i) > x) exit enddo if (i < j ) then ! if the indexes do not cross, exchange values do k = 1,d tmp = a(k,i) a(k,i) = a(k,j) a(k,j) = tmp enddo else ! if they do cross, exchange left value with pivot and return with the partition index do k = 1,d tmp = a(k,istart) a(k,istart) = a(k,j) a(k,j) = tmp enddo math_partition = j return endif enddo endfunction !************************************************************************** ! range of integers starting at one !************************************************************************** pure function math_range(N) use prec, only: pInt implicit none integer(pInt), intent(in) :: N integer(pInt) i integer(pInt), dimension(N) :: math_range forall (i=1:N) math_range(i) = i return endfunction !************************************************************************** ! second rank identity tensor of specified dimension !************************************************************************** pure function math_identity2nd(dimen) use prec, only: pReal, pInt implicit none integer(pInt), intent(in) :: dimen integer(pInt) i real(pReal), dimension(dimen,dimen) :: math_identity2nd math_identity2nd = 0.0_pReal forall (i=1:dimen) math_identity2nd(i,i) = 1.0_pReal return endfunction !************************************************************************** ! 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 !************************************************************************** pure function math_civita(i,j,k) ! change its name from math_permut ! to math_civita <<>> use prec, only: pReal, pInt implicit none integer(pInt), intent(in) :: i,j,k real(pReal) math_civita math_civita = 0.0_pReal if (((i == 1).and.(j == 2).and.(k == 3)) .or. & ((i == 2).and.(j == 3).and.(k == 1)) .or. & ((i == 3).and.(j == 1).and.(k == 2))) math_civita = 1.0_pReal if (((i == 1).and.(j == 3).and.(k == 2)) .or. & ((i == 2).and.(j == 1).and.(k == 3)) .or. & ((i == 3).and.(j == 2).and.(k == 1))) math_civita = -1.0_pReal return endfunction !************************************************************************** ! kronecker delta function d_ij ! d_ij = 1 if i = j ! d_ij = 0 otherwise !************************************************************************** pure function math_delta(i,j) use prec, only: pReal, pInt implicit none integer(pInt), intent (in) :: i,j real(pReal) math_delta math_delta = 0.0_pReal if (i == j) math_delta = 1.0_pReal return endfunction !************************************************************************** ! fourth rank identity tensor of specified dimension !************************************************************************** pure function math_identity4th(dimen) use prec, only: pReal, pInt implicit none integer(pInt), intent(in) :: dimen integer(pInt) i,j,k,l real(pReal), dimension(dimen,dimen,dimen,dimen) :: math_identity4th forall (i=1:dimen,j=1:dimen,k=1:dimen,l=1:dimen) math_identity4th(i,j,k,l) = & 0.5_pReal*(math_I3(i,k)*math_I3(j,k)+math_I3(i,l)*math_I3(j,k)) return endfunction !************************************************************************** ! vector product a x b !************************************************************************** pure function math_vectorproduct(A,B) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: math_vectorproduct math_vectorproduct(1) = A(2)*B(3)-A(3)*B(2) math_vectorproduct(2) = A(3)*B(1)-A(1)*B(3) math_vectorproduct(3) = A(1)*B(2)-A(2)*B(1) return endfunction !************************************************************************** ! tensor product a \otimes b !************************************************************************** pure function math_tensorproduct(A,B) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3,3) :: math_tensorproduct integer(pInt) i,j forall (i=1:3,j=1:3) math_tensorproduct(i,j) = A(i)*B(j) return endfunction !************************************************************************** ! matrix multiplication 3x3 = 1 !************************************************************************** pure function math_mul3x3(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: C real(pReal) math_mul3x3 forall (i=1:3) C(i) = A(i)*B(i) math_mul3x3 = sum(C) return endfunction !************************************************************************** ! matrix multiplication 6x6 = 1 !************************************************************************** pure function math_mul6x6(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i real(pReal), dimension(6), intent(in) :: A,B real(pReal), dimension(6) :: C real(pReal) math_mul6x6 forall (i=1:6) C(i) = A(i)*B(i) math_mul6x6 = sum(C) return endfunction !************************************************************************** ! matrix multiplication 33x33 = 1 (double contraction --> ij * ij) !************************************************************************** pure function math_mul33xx33(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i,j real(pReal), dimension(3,3), intent(in) :: A,B real(pReal), dimension(3,3) :: C real(pReal) math_mul33xx33 forall (i=1:3,j=1:3) C(i,j) = A(i,j) * B(i,j) math_mul33xx33 = sum(C) return endfunction !************************************************************************** ! matrix multiplication 3333x33 = 33 (double contraction --> ijkl *kl = ij) !************************************************************************** pure function math_mul3333xx33(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i,j real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3), intent(in) :: B real(pReal), dimension(3,3) :: C,math_mul3333xx33 do i = 1,3 do j = 1,3 math_mul3333xx33(i,j) = sum(A(i,j,:,:)*B(:,:)) enddo; enddo return endfunction !************************************************************************** ! matrix multiplication 33x33 = 3x3 !************************************************************************** pure function math_mul33x33(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i,j real(pReal), dimension(3,3), intent(in) :: A,B real(pReal), dimension(3,3) :: math_mul33x33 forall (i=1:3,j=1:3) math_mul33x33(i,j) = & A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) return endfunction !************************************************************************** ! matrix multiplication 66x66 = 6x6 !************************************************************************** pure function math_mul66x66(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i,j real(pReal), dimension(6,6), intent(in) :: A,B real(pReal), dimension(6,6) :: math_mul66x66 forall (i=1:6,j=1:6) 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) return endfunction !************************************************************************** ! matrix multiplication 99x99 = 9x9 !************************************************************************** pure function math_mul99x99(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i,j real(pReal), dimension(9,9), intent(in) :: A,B real(pReal), dimension(9,9) :: math_mul99x99 forall (i=1:9,j=1:9) 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) return endfunction !************************************************************************** ! matrix multiplication 33x3 = 3 !************************************************************************** pure function math_mul33x3(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i real(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3), intent(in) :: B real(pReal), dimension(3) :: math_mul33x3 forall (i=1:3) math_mul33x3(i) = A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3) return endfunction !************************************************************************** ! matrix multiplication complex(33) x real(3) = complex(3) !************************************************************************** pure function math_mul33x3_complex(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i complex(pReal), dimension(3,3), intent(in) :: A real(pReal), dimension(3), intent(in) :: B complex(pReal), dimension(3) :: math_mul33x3_complex forall (i=1:3) math_mul33x3_complex(i) = A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3) return endfunction !************************************************************************** ! matrix multiplication 66x6 = 6 !************************************************************************** pure function math_mul66x6(A,B) use prec, only: pReal, pInt implicit none integer(pInt) i real(pReal), dimension(6,6), intent(in) :: A real(pReal), dimension(6), intent(in) :: B real(pReal), dimension(6) :: math_mul66x6 forall (i=1:6) 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) return endfunction !************************************************************************** ! random quaternion !************************************************************************** function math_qRnd() use prec, only: pReal, pInt implicit none real(pReal), dimension(4) :: math_qRnd real(pReal), dimension(3) :: rnd call halton(3,rnd) math_qRnd(1) = cos(2.0_pReal*pi*rnd(1))*sqrt(rnd(3)) math_qRnd(2) = sin(2.0_pReal*pi*rnd(2))*sqrt(1.0_pReal-rnd(3)) math_qRnd(3) = cos(2.0_pReal*pi*rnd(2))*sqrt(1.0_pReal-rnd(3)) math_qRnd(4) = sin(2.0_pReal*pi*rnd(1))*sqrt(rnd(3)) endfunction !************************************************************************** ! quaternion multiplication q1xq2 = q12 !************************************************************************** pure function math_qMul(A,B) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: A, B real(pReal), dimension(4) :: math_qMul math_qMul(1) = A(1)*B(1) - A(2)*B(2) - A(3)*B(3) - A(4)*B(4) math_qMul(2) = A(1)*B(2) + A(2)*B(1) + A(3)*B(4) - A(4)*B(3) math_qMul(3) = A(1)*B(3) - A(2)*B(4) + A(3)*B(1) + A(4)*B(2) math_qMul(4) = A(1)*B(4) + A(2)*B(3) - A(3)*B(2) + A(4)*B(1) endfunction !************************************************************************** ! quaternion dotproduct !************************************************************************** pure function math_qDot(A,B) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: A, B real(pReal) math_qDot math_qDot = A(1)*B(1) + A(2)*B(2) + A(3)*B(3) + A(4)*B(4) endfunction !************************************************************************** ! quaternion conjugation !************************************************************************** pure function math_qConj(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(4) :: math_qConj math_qConj(1) = Q(1) math_qConj(2:4) = -Q(2:4) endfunction !************************************************************************** ! quaternion norm !************************************************************************** pure function math_qNorm(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal) math_qNorm math_qNorm = sqrt(max(0.0_pReal, Q(1)*Q(1) + Q(2)*Q(2) + Q(3)*Q(3) + Q(4)*Q(4))) endfunction !************************************************************************** ! quaternion inversion !************************************************************************** pure function math_qInv(Q) use prec, only: pReal, pInt 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 (squareNorm > tiny(squareNorm)) & math_qInv = math_qConj(Q) / squareNorm endfunction !************************************************************************** ! action of a quaternion on a vector (rotate vector v with Q) !************************************************************************** pure function math_qRot(Q,v) use prec, only: pReal, pInt 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 = 1,4 do j = 1,i T(i,j) = Q(i) * Q(j) enddo enddo math_qRot(1) = -v(1)*(T(3,3)+T(4,4)) + v(2)*(T(3,2)-T(4,1)) + v(3)*(T(4,2)+T(3,1)) math_qRot(2) = v(1)*(T(3,2)+T(4,1)) - v(2)*(T(2,2)+T(4,4)) + v(3)*(T(4,3)-T(2,1)) math_qRot(3) = 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 endfunction !************************************************************************** ! transposition of a 3x3 matrix !************************************************************************** pure function math_transpose3x3(A) use prec, only: pReal,pInt implicit none real(pReal),dimension(3,3),intent(in) :: A real(pReal),dimension(3,3) :: math_transpose3x3 integer(pInt) i,j forall(i=1:3, j=1:3) math_transpose3x3(i,j) = A(j,i) return endfunction !************************************************************************** ! Cramer inversion of 3x3 matrix (function) !************************************************************************** pure function math_inv3x3(A) ! direct Cramer inversion of matrix A. ! returns all zeroes if not possible, i.e. if det close to zero use prec, only: pReal,pInt implicit none real(pReal),dimension(3,3),intent(in) :: A real(pReal) DetA real(pReal),dimension(3,3) :: math_inv3x3 math_inv3x3 = 0.0_pReal DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )& - A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )& + A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) if (DetA > tiny(DetA)) then math_inv3x3(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA math_inv3x3(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA math_inv3x3(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA math_inv3x3(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA math_inv3x3(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA math_inv3x3(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA math_inv3x3(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA math_inv3x3(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA math_inv3x3(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA endif return endfunction !************************************************************************** ! Cramer inversion of 3x3 matrix (subroutine) !************************************************************************** PURE SUBROUTINE math_invert3x3(A, InvA, DetA, error) ! Bestimmung der Determinanten und Inversen einer 3x3-Matrix ! A = Matrix A ! InvA = Inverse of A ! DetA = Determinant of A ! error = logical use prec, only: pReal,pInt 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 DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )& - A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )& + A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) if (DetA <= tiny(DetA)) then error = .true. else InvA(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA InvA(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA InvA(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA InvA(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA InvA(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA InvA(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA InvA(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA InvA(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA InvA(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA error = .false. endif return ENDSUBROUTINE !************************************************************************** ! Gauss elimination to invert 6x6 matrix !************************************************************************** PURE SUBROUTINE math_invert(dimen,A, InvA, AnzNegEW, error) ! Invertieren einer dimen x dimen - Matrix ! A = Matrix A ! InvA = Inverse von A ! AnzNegEW = Anzahl der negativen Eigenwerte von A ! error = logical ! = false: Inversion wurde durchgefuehrt. ! = true: Die Inversion in SymGauss wurde wegen eines verschwindenen ! Pivotelement abgebrochen. use prec, only: pReal,pInt implicit none integer(pInt), intent(in) :: dimen real(pReal),dimension(dimen,dimen), intent(in) :: A real(pReal),dimension(dimen,dimen), intent(out) :: InvA integer(pInt), intent(out) :: AnzNegEW logical, intent(out) :: error real(pReal) LogAbsDetA real(pReal),dimension(dimen,dimen) :: B InvA = math_identity2nd(dimen) B = A CALL Gauss(dimen,B,InvA,LogAbsDetA,AnzNegEW,error) RETURN ENDSUBROUTINE math_invert ! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ PURE SUBROUTINE Gauss (dimen,A,B,LogAbsDetA,NegHDK,error) ! Loesung eines linearen Gleichungsssystem A * X = B mit Hilfe des ! GAUSS-Algorithmus ! Zur numerischen Stabilisierung wird eine Zeilen- und Spaltenpivotsuche ! durchgefuehrt. ! Eingabeparameter: ! ! A(dimen,dimen) = Koeffizientenmatrix A ! B(dimen,dimen) = rechte Seiten B ! ! Ausgabeparameter: ! ! B(dimen,dimen) = Matrix der Unbekanntenvektoren X ! LogAbsDetA = 10-Logarithmus des Betrages der Determinanten von A ! NegHDK = Anzahl der negativen Hauptdiagonalkoeffizienten nach der ! Vorwaertszerlegung ! error = logical ! = false: Das Gleichungssystem wurde geloest. ! = true : Matrix A ist singulaer. ! A und B werden veraendert! use prec, only: pReal,pInt implicit none logical error integer (pInt) dimen,NegHDK real(pReal) LogAbsDetA real(pReal) A(dimen,dimen), B(dimen,dimen) INTENT (IN) dimen INTENT (OUT) LogAbsDetA, NegHDK, error INTENT (INOUT) A, B LOGICAL SortX integer (pInt) PivotZeile, PivotSpalte, StoreI, I, IP1, J, K, L integer (pInt) XNr(dimen) real(pReal) AbsA, PivotWert, EpsAbs, Quote real(pReal) StoreA(dimen), StoreB(dimen) error = .true. NegHDK = 1 SortX = .FALSE. ! Unbekanntennumerierung DO I = 1, dimen XNr(I) = I ENDDO ! Genauigkeitsschranke und Bestimmung des groessten Pivotelementes PivotWert = ABS(A(1,1)) PivotZeile = 1 PivotSpalte = 1 DO I = 1, dimen DO J = 1, dimen AbsA = ABS(A(I,J)) IF (AbsA .GT. PivotWert) THEN PivotWert = AbsA PivotZeile = I PivotSpalte = J ENDIF ENDDO ENDDO IF (PivotWert .LT. 0.0000001) RETURN ! Pivotelement = 0? EpsAbs = PivotWert * 0.1_pReal ** PRECISION(1.0_pReal) ! V O R W A E R T S T R I A N G U L A T I O N DO I = 1, dimen - 1 ! Zeilentausch? IF (PivotZeile .NE. I) THEN StoreA(I:dimen) = A(I,I:dimen) A(I,I:dimen) = A(PivotZeile,I:dimen) A(PivotZeile,I:dimen) = StoreA(I:dimen) StoreB(1:dimen) = B(I,1:dimen) B(I,1:dimen) = B(PivotZeile,1:dimen) B(PivotZeile,1:dimen) = StoreB(1:dimen) SortX = .TRUE. ENDIF ! Spaltentausch? IF (PivotSpalte .NE. I) THEN StoreA(1:dimen) = A(1:dimen,I) A(1:dimen,I) = A(1:dimen,PivotSpalte) A(1:dimen,PivotSpalte) = StoreA(1:dimen) StoreI = XNr(I) XNr(I) = XNr(PivotSpalte) XNr(PivotSpalte) = StoreI SortX = .TRUE. ENDIF ! Triangulation DO J = I + 1, dimen Quote = A(J,I) / A(I,I) DO K = I + 1, dimen A(J,K) = A(J,K) - Quote * A(I,K) ENDDO DO K = 1, dimen B(J,K) = B(J,K) - Quote * B(I,K) ENDDO ENDDO ! Bestimmung des groessten Pivotelementes IP1 = I + 1 PivotWert = ABS(A(IP1,IP1)) PivotZeile = IP1 PivotSpalte = IP1 DO J = IP1, dimen DO K = IP1, dimen AbsA = ABS(A(J,K)) IF (AbsA .GT. PivotWert) THEN PivotWert = AbsA PivotZeile = J PivotSpalte = K ENDIF ENDDO ENDDO IF (PivotWert .LT. EpsAbs) RETURN ! Pivotelement = 0? ENDDO ! R U E C K W A E R T S A U F L O E S U N G DO I = dimen, 1, -1 DO L = 1, dimen DO J = I + 1, dimen B(I,L) = B(I,L) - A(I,J) * B(J,L) ENDDO B(I,L) = B(I,L) / A(I,I) ENDDO ENDDO ! Sortieren der Unbekanntenvektoren? IF (SortX) THEN DO L = 1, dimen StoreA(1:dimen) = B(1:dimen,L) DO I = 1, dimen J = XNr(I) B(J,L) = StoreA(I) ENDDO ENDDO ENDIF ! Determinante LogAbsDetA = 0.0_pReal NegHDK = 0 DO I = 1, dimen IF (A(I,I) .LT. 0.0_pReal) NegHDK = NegHDK + 1 AbsA = ABS(A(I,I)) LogAbsDetA = LogAbsDetA + LOG10(AbsA) ENDDO error = .false. RETURN ENDSUBROUTINE Gauss !******************************************************************** ! symmetrize a 3x3 matrix !******************************************************************** function math_symmetric3x3(m) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3) :: math_symmetric3x3,m integer(pInt) i,j forall (i=1:3,j=1:3) math_symmetric3x3(i,j) = 0.5_pReal * (m(i,j) + m(j,i)) endfunction !******************************************************************** ! symmetrize a 6x6 matrix !******************************************************************** pure function math_symmetric6x6(m) use prec, only: pReal,pInt implicit none integer(pInt) i,j real(pReal), dimension(6,6), intent(in) :: m real(pReal), dimension(6,6) :: math_symmetric6x6 forall (i=1:6,j=1:6) math_symmetric6x6(i,j) = 0.5_pReal * (m(i,j) + m(j,i)) endfunction !******************************************************************** ! equivalent scalar quantity of a full strain tensor !******************************************************************** pure function math_equivStrain33(m) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal) math_equivStrain33,e11,e22,e33,s12,s23,s31 e11 = (2.0_pReal*m(1,1)-m(2,2)-m(3,3))/3.0_pReal e22 = (2.0_pReal*m(2,2)-m(3,3)-m(1,1))/3.0_pReal e33 = (2.0_pReal*m(3,3)-m(1,1)-m(2,2))/3.0_pReal s12 = 2.0_pReal*m(1,2) s23 = 2.0_pReal*m(2,3) s31 = 2.0_pReal*m(3,1) math_equivStrain33 = 2.0_pReal*(1.50_pReal*(e11**2.0_pReal+e22**2.0_pReal+e33**2.0_pReal) + & 0.75_pReal*(s12**2.0_pReal+s23**2.0_pReal+s31**2.0_pReal))**(0.5_pReal)/3.0_pReal return endfunction !******************************************************************** ! determinant of a 3x3 matrix !******************************************************************** pure function math_det3x3(m) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal) math_det3x3 math_det3x3 = 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)) return endfunction !******************************************************************** ! euclidic norm of a 3x1 vector !******************************************************************** pure function math_norm3(v) use prec, only: pReal,pInt implicit none real(pReal), dimension(3), intent(in) :: v real(pReal) math_norm3 math_norm3 = sqrt(v(1)*v(1) + v(2)*v(2) + v(3)*v(3)) return endfunction !******************************************************************** ! convert 3x3 matrix into vector 9x1 !******************************************************************** pure function math_Plain33to9(m33) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3), intent(in) :: m33 real(pReal), dimension(9) :: math_Plain33to9 integer(pInt) i forall (i=1:9) math_Plain33to9(i) = m33(mapPlain(1,i),mapPlain(2,i)) return endfunction !******************************************************************** ! convert Plain 9x1 back to 3x3 matrix !******************************************************************** pure function math_Plain9to33(v9) use prec, only: pReal,pInt implicit none real(pReal), dimension(9), intent(in) :: v9 real(pReal), dimension(3,3) :: math_Plain9to33 integer(pInt) i forall (i=1:9) math_Plain9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i) return endfunction !******************************************************************** ! convert symmetric 3x3 matrix into Mandel vector 6x1 !******************************************************************** pure function math_Mandel33to6(m33) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3), intent(in) :: m33 real(pReal), dimension(6) :: math_Mandel33to6 integer(pInt) i forall (i=1:6) math_Mandel33to6(i) = nrmMandel(i)*m33(mapMandel(1,i),mapMandel(2,i)) return endfunction !******************************************************************** ! convert Mandel 6x1 back to symmetric 3x3 matrix !******************************************************************** pure function math_Mandel6to33(v6) use prec, only: pReal,pInt implicit none real(pReal), dimension(6), intent(in) :: v6 real(pReal), dimension(3,3) :: math_Mandel6to33 integer(pInt) i forall (i=1:6) math_Mandel6to33(mapMandel(1,i),mapMandel(2,i)) = invnrmMandel(i)*v6(i) math_Mandel6to33(mapMandel(2,i),mapMandel(1,i)) = invnrmMandel(i)*v6(i) end forall return endfunction !******************************************************************** ! convert 3x3x3x3 tensor into plain matrix 9x9 !******************************************************************** pure function math_Plain3333to99(m3333) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3,3,3), intent(in) :: m3333 real(pReal), dimension(9,9) :: math_Plain3333to99 integer(pInt) i,j forall (i=1:9,j=1:9) math_Plain3333to99(i,j) = & m3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j)) return endfunction !******************************************************************** ! plain matrix 9x9 into 3x3x3x3 tensor !******************************************************************** pure function math_Plain99to3333(m99) use prec, only: pReal,pInt implicit none real(pReal), dimension(9,9), intent(in) :: m99 real(pReal), dimension(3,3,3,3) :: math_Plain99to3333 integer(pInt) i,j forall (i=1:9,j=1:9) math_Plain99to3333(mapPlain(1,i),mapPlain(2,i),& mapPlain(1,j),mapPlain(2,j)) = m99(i,j) return endfunction !******************************************************************** ! convert symmetric 3x3x3x3 tensor into Mandel matrix 6x6 !******************************************************************** pure function math_Mandel3333to66(m3333) use prec, only: pReal,pInt implicit none real(pReal), dimension(3,3,3,3), intent(in) :: m3333 real(pReal), dimension(6,6) :: math_Mandel3333to66 integer(pInt) i,j forall (i=1:6,j=1:6) math_Mandel3333to66(i,j) = & nrmMandel(i)*nrmMandel(j)*m3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) return endfunction !******************************************************************** ! convert Mandel matrix 6x6 back to symmetric 3x3x3x3 tensor !******************************************************************** pure function math_Mandel66to3333(m66) use prec, only: pReal,pInt implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(3,3,3,3) :: math_Mandel66to3333 integer(pInt) i,j forall (i=1:6,j=1:6) math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(1,j),mapMandel(2,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(2,j),mapMandel(1,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j) math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(2,j),mapMandel(1,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j) end forall return endfunction !******************************************************************** ! convert Voigt matrix 6x6 back to symmetric 3x3x3x3 tensor !******************************************************************** pure function math_Voigt66to3333(m66) use prec, only: pReal,pInt implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333 integer(pInt) i,j forall (i=1:6,j=1:6) math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(1,j),mapVoigt(2,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(1,j),mapVoigt(2,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(2,j),mapVoigt(1,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(2,j),mapVoigt(1,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j) end forall return endfunction !******************************************************************** ! Euler angles (in radians) from rotation matrix !******************************************************************** pure function math_RtoEuler(R) use prec, only: pReal, pInt implicit none real(pReal), dimension (3,3), intent(in) :: R real(pReal), dimension(3) :: math_RtoEuler real(pReal) sqhkl, squvw, sqhk, val 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 val=R(3,3)/sqhkl if(val > 1.0_pReal) val = 1.0_pReal if(val < -1.0_pReal) val = -1.0_pReal math_RtoEuler(2) = acos(val) if(math_RtoEuler(2) < 1.0e-8_pReal) then ! calculate phi2 math_RtoEuler(3) = 0.0_pReal ! calculate phi1 val=R(1,1)/squvw if(val > 1.0_pReal) val = 1.0_pReal if(val < -1.0_pReal) val = -1.0_pReal math_RtoEuler(1) = acos(val) if(R(2,1) > 0.0_pReal) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) else ! calculate phi2 val=R(2,3)/sqhk if(val > 1.0_pReal) val = 1.0_pReal if(val < -1.0_pReal) val = -1.0_pReal math_RtoEuler(3) = acos(val) if(R(1,3) < 0.0) math_RtoEuler(3) = 2.0_pReal*pi-math_RtoEuler(3) ! calculate phi1 val=-R(3,2)/sin(math_RtoEuler(2)) if(val > 1.0_pReal) val = 1.0_pReal if(val < -1.0_pReal) val = -1.0_pReal math_RtoEuler(1) = acos(val) if(R(3,1) < 0.0) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) end if return endfunction !******************************************************************** ! quaternion (w+ix+jy+kz) from orientation matrix !******************************************************************** pure function math_RtoQuaternion(R) use prec, only: pReal, pInt implicit none real(pReal), dimension (3,3), intent(in) :: R real(pReal), dimension(4) :: absQ,math_RtoQuaternion real(pReal) max_absQ integer(pInt), dimension(1) :: largest ! math adopted from http://code.google.com/p/mtex/source/browse/trunk/geometry/geometry_tools/mat2quat.m math_RtoQuaternion = 0.0_pReal absQ(1) = 1.0_pReal+R(1,1)+R(2,2)+R(3,3) absQ(2) = 1.0_pReal+R(1,1)-R(2,2)-R(3,3) absQ(3) = 1.0_pReal-R(1,1)+R(2,2)-R(3,3) absQ(4) = 1.0_pReal-R(1,1)-R(2,2)+R(3,3) largest = maxloc(absQ) max_absQ=0.5_pReal * sqrt(absQ(largest(1))) select case(largest(1)) case (1) math_RtoQuaternion(2) = R(2,3)-R(3,2) math_RtoQuaternion(3) = R(3,1)-R(1,3) math_RtoQuaternion(4) = R(1,2)-R(2,1) case (2) math_RtoQuaternion(1) = R(2,3)-R(3,2) math_RtoQuaternion(3) = R(1,2)+R(2,1) math_RtoQuaternion(4) = R(3,1)+R(1,3) case (3) math_RtoQuaternion(1) = R(3,1)-R(1,3) math_RtoQuaternion(2) = R(1,2)+R(2,1) math_RtoQuaternion(4) = R(2,3)+R(3,2) case (4) math_RtoQuaternion (1) = R(1,2)-R(2,1) math_RtoQuaternion (2) = R(3,1)+R(1,3) math_RtoQuaternion (3) = R(3,2)+R(2,3) end select math_RtoQuaternion = math_RtoQuaternion*0.25_pReal/max_absQ math_RtoQuaternion(largest(1)) = max_absQ return endfunction !**************************************************************** ! rotation matrix from Euler angles (in radians) !**************************************************************** pure function math_EulerToR (Euler) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: Euler real(pReal), dimension(3,3) :: math_EulerToR real(pReal) c1, c, c2, s1, s, s2 C1 = cos(Euler(1)) C = cos(Euler(2)) C2 = cos(Euler(3)) S1 = sin(Euler(1)) S = sin(Euler(2)) S2 = sin(Euler(3)) math_EulerToR(1,1)=C1*C2-S1*S2*C math_EulerToR(1,2)=S1*C2+C1*S2*C math_EulerToR(1,3)=S2*S math_EulerToR(2,1)=-C1*S2-S1*C2*C math_EulerToR(2,2)=-S1*S2+C1*C2*C math_EulerToR(2,3)=C2*S math_EulerToR(3,1)=S1*S math_EulerToR(3,2)=-C1*S math_EulerToR(3,3)=C return endfunction !******************************************************************** ! quaternion (w+ix+jy+kz) from 3-1-3 Euler angles (in radians) !******************************************************************** pure function math_EulerToQuaternion(eulerangles) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: eulerangles real(pReal), dimension(4) :: math_EulerToQuaternion real(pReal), dimension(3) :: halfangles real(pReal) c, s halfangles = 0.5_pReal * eulerangles c = cos(halfangles(2)) s = sin(halfangles(2)) math_EulerToQuaternion(1) = cos(halfangles(1)+halfangles(3)) * c math_EulerToQuaternion(2) = cos(halfangles(1)-halfangles(3)) * s math_EulerToQuaternion(3) = sin(halfangles(1)-halfangles(3)) * s math_EulerToQuaternion(4) = sin(halfangles(1)+halfangles(3)) * c endfunction !**************************************************************** ! rotation matrix from axis and angle (in radians) !**************************************************************** pure function math_AxisAngleToR(axis,omega) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega real(pReal), dimension(3) :: axisNrm real(pReal), dimension(3,3) :: math_AxisAngleToR real(pReal) norm,s,c,c1 integer(pInt) i norm = sqrt(math_mul3x3(axis,axis)) if (norm > 1.0e-8_pReal) then ! non-zero rotation forall (i=1:3) axisNrm(i) = axis(i)/norm ! normalize axis to be sure s = sin(omega) c = cos(omega) c1 = 1.0_pReal - c ! formula for active rotation taken from http://mathworld.wolfram.com/RodriguesRotationFormula.html ! below is transposed form to get passive rotation math_AxisAngleToR(1,1) = c + c1*axisNrm(1)**2 math_AxisAngleToR(2,1) = -s*axisNrm(3) + c1*axisNrm(1)*axisNrm(2) math_AxisAngleToR(3,1) = s*axisNrm(2) + c1*axisNrm(1)*axisNrm(3) math_AxisAngleToR(1,2) = s*axisNrm(3) + c1*axisNrm(2)*axisNrm(1) math_AxisAngleToR(2,2) = c + c1*axisNrm(2)**2 math_AxisAngleToR(3,2) = -s*axisNrm(1) + c1*axisNrm(2)*axisNrm(3) math_AxisAngleToR(1,3) = -s*axisNrm(2) + c1*axisNrm(3)*axisNrm(1) math_AxisAngleToR(2,3) = s*axisNrm(1) + c1*axisNrm(3)*axisNrm(2) math_AxisAngleToR(3,3) = c + c1*axisNrm(3)**2 else math_AxisAngleToR = math_I3 endif return endfunction !**************************************************************** ! quaternion (w+ix+jy+kz) from axis and angle (in radians) !**************************************************************** pure function math_AxisAngleToQuaternion(axis,omega) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: axis real(pReal), intent(in) :: omega real(pReal), dimension(3) :: axisNrm real(pReal), dimension(4) :: math_AxisAngleToQuaternion real(pReal) s,c,norm integer(pInt) i norm = sqrt(math_mul3x3(axis,axis)) if (norm > 1.0e-8_pReal) then ! non-zero rotation forall (i=1:3) axisNrm(i) = axis(i)/norm ! normalize axis to be sure ! formula taken from http://en.wikipedia.org/wiki/Rotation_representation_%28mathematics%29#Rodrigues_parameters s = sin(omega/2.0_pReal) c = cos(omega/2.0_pReal) math_AxisAngleToQuaternion(1) = c math_AxisAngleToQuaternion(2:4) = s * axisNrm(1:3) else math_AxisAngleToQuaternion = (/1.0_pReal,0.0_pReal,0.0_pReal,0.0_pReal/) ! no rotation endif return endfunction !******************************************************************** ! orientation matrix from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToR(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3,3) :: math_QuaternionToR, T,S real(pReal) w2 integer(pInt) i, j forall (i = 1:3, j = 1:3) & T(i,j) = Q(i+1) * Q(j+1) 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_QuaternionToR = (2.0_pReal * Q(1)*Q(1) - 1.0_pReal) * math_I3 + & 2.0_pReal * T - & 2.0_pReal * Q(1) * S return endfunction !******************************************************************** ! 3-1-3 Euler angles (in radians) from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToEuler(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3) :: math_QuaternionToEuler real(pReal) acos_arg math_QuaternionToEuler(2) = acos(1.0_pReal-2.0_pReal*(Q(2)*Q(2)+Q(3)*Q(3))) if (abs(math_QuaternionToEuler(2)) < 1.0e-3_pReal) then acos_arg=Q(1) if(acos_arg > 1.0_pReal)acos_arg = 1.0_pReal if(acos_arg < -1.0_pReal)acos_arg = -1.0_pReal math_QuaternionToEuler(1) = 2.0_pReal*acos(acos_arg) math_QuaternionToEuler(3) = 0.0_pReal else math_QuaternionToEuler(1) = atan2(Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)-Q(3)*Q(4)) if (math_QuaternionToEuler(1) < 0.0_pReal) & math_QuaternionToEuler(1) = math_QuaternionToEuler(1) + 2.0_pReal * pi math_QuaternionToEuler(3) = atan2(-Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)+Q(3)*Q(4)) if (math_QuaternionToEuler(3) < 0.0_pReal) & math_QuaternionToEuler(3) = math_QuaternionToEuler(3) + 2.0_pReal * pi endif if (math_QuaternionToEuler(2) < 0.0_pReal) & math_QuaternionToEuler(2) = math_QuaternionToEuler(2) + pi return endfunction !******************************************************************** ! axis-angle (x, y, z, ang in radians) from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToAxisAngle(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal) halfAngle, sinHalfAngle real(pReal), dimension(4) :: math_QuaternionToAxisAngle halfAngle = acos(max(-1.0_pReal, min(1.0_pReal, Q(1)))) ! limit to [-1,1] --> 0 to 180 deg sinHalfAngle = sin(halfAngle) if (sinHalfAngle <= 1.0e-4_pReal) then ! very small rotation angle? math_QuaternionToAxisAngle = 0.0_pReal else math_QuaternionToAxisAngle(1:3) = Q(2:4)/sinHalfAngle math_QuaternionToAxisAngle(4) = halfAngle*2.0_pReal endif return endfunction !******************************************************************** ! Rodrigues vector (x, y, z) from unit quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToRodrig(Q) use prec, only: pReal, pInt implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3) :: math_QuaternionToRodrig if (Q(1) /= 0.0_pReal) then ! unless rotation by 180 deg math_QuaternionToRodrig = Q(2:4)/Q(1) else math_QuaternionToRodrig = NaN ! Rodrig is unbound for 180 deg... endif return endfunction !************************************************************************** ! misorientation angle between two sets of Euler angles !************************************************************************** pure function math_EulerMisorientation(EulerA,EulerB) use prec, only: pReal, pInt implicit none real(pReal), dimension(3), intent(in) :: EulerA,EulerB real(pReal), dimension(3,3) :: r real(pReal) math_EulerMisorientation, tr r = math_mul33x33(math_EulerToR(EulerB),transpose(math_EulerToR(EulerA))) tr = (r(1,1)+r(2,2)+r(3,3)-1.0_pReal)*0.4999999_pReal math_EulerMisorientation = abs(0.5_pReal*pi-asin(tr)) return endfunction !************************************************************************** ! figures whether unit quat falls into stereographic standard triangle !************************************************************************** pure function math_QuaternionInSST(Q, symmetryType) use prec, only: pReal, pInt implicit none !*** input variables real(pReal), dimension(4), intent(in) :: Q ! orientation integer(pInt), intent(in) :: symmetryType ! Type of crystal symmetry; 1:cubic, 2:hexagonal !*** output variables logical math_QuaternionInSST !*** local variables real(pReal), dimension(3) :: Rodrig ! Rodrigues vector of Q Rodrig = math_QuaternionToRodrig(Q) select case (symmetryType) case (1) math_QuaternionInSST = Rodrig(1) > Rodrig(2) .and. & Rodrig(2) > Rodrig(3) .and. & Rodrig(3) > 0.0_pReal case (2) math_QuaternionInSST = Rodrig(1) > sqrt(3.0_pReal)*Rodrig(2) .and. & Rodrig(2) > 0.0_pReal .and. & Rodrig(3) > 0.0_pReal case default math_QuaternionInSST = .true. end select endfunction !************************************************************************** ! calculates the disorientation for 2 unit quaternions !************************************************************************** function math_QuaternionDisorientation(Q1, Q2, symmetryType) use prec, only: pReal, pInt use IO, only: IO_error implicit none !*** input variables real(pReal), dimension(4), intent(in) :: Q1, & ! 1st orientation Q2 ! 2nd orientation integer(pInt), intent(in) :: symmetryType ! Type of crystal symmetry; 1:cubic, 2:hexagonal !*** output variables real(pReal), dimension(4) :: math_QuaternionDisorientation ! disorientation !*** local variables real(pReal), dimension(4) :: dQ,dQsymA,mis integer(pInt) i,j,k,s dQ = math_qMul(math_qConj(Q1),Q2) math_QuaternionDisorientation = dQ select case (symmetryType) case (0) if (math_QuaternionDisorientation(1) < 0.0_pReal) & math_QuaternionDisorientation = -math_QuaternionDisorientation ! keep omega within 0 to 180 deg case (1,2) s = sum(math_NsymOperations(1:symmetryType-1)) do i = 1,2 dQ = math_qConj(dQ) ! switch order of "from -- to" do j = 1,math_NsymOperations(symmetryType) ! run through first crystal's symmetries dQsymA = math_qMul(math_symOperations(:,s+j),dQ) ! apply sym do k = 1,math_NsymOperations(symmetryType) ! run through 2nd crystal's symmetries mis = math_qMul(dQsymA,math_symOperations(:,s+k)) ! apply sym if (mis(1) < 0.0_pReal) & ! want positive angle mis = -mis if (mis(1)-math_QuaternionDisorientation(1) > -1e-8_pReal .and. & math_QuaternionInSST(mis,symmetryType)) & math_QuaternionDisorientation = mis ! found better one enddo; enddo; enddo case default call IO_error(550,symmetryType) ! complain about unknown symmetry end select endfunction !******************************************************************** ! draw a random sample from Euler space !******************************************************************** function math_sampleRandomOri() use prec, only: pReal, pInt implicit none real(pReal), dimension(3) :: math_sampleRandomOri, rnd call halton(3,rnd) math_sampleRandomOri(1) = rnd(1)*2.0_pReal*pi math_sampleRandomOri(2) = acos(2.0_pReal*rnd(2)-1.0_pReal) math_sampleRandomOri(3) = rnd(3)*2.0_pReal*pi return endfunction !******************************************************************** ! draw a random sample from Gauss component ! with noise (in radians) half-width !******************************************************************** function math_sampleGaussOri(center,noise) use prec, only: pReal, pInt implicit none real(pReal), dimension(3) :: math_sampleGaussOri, center, disturb real(pReal), dimension(3), parameter :: origin = (/0.0_pReal,0.0_pReal,0.0_pReal/) real(pReal), dimension(5) :: rnd real(pReal) noise,scatter,cosScatter integer(pInt) i if (noise==0.0) then math_sampleGaussOri = center return endif ! Helming uses different distribution with Bessel functions ! therefore the gauss scatter width has to be scaled differently scatter = 0.95_pReal * noise cosScatter = cos(scatter) do call halton(5,rnd) forall (i=1:3) rnd(i) = 2.0_pReal*rnd(i)-1.0_pReal ! expand 1:3 to range [-1,+1] disturb(1) = scatter * rnd(1) ! phi1 disturb(2) = sign(1.0_pReal,rnd(2))*acos(cosScatter+(1.0_pReal-cosScatter)*rnd(4)) ! Phi disturb(3) = scatter * rnd(2) ! phi2 if (rnd(5) <= exp(-1.0_pReal*(math_EulerMisorientation(origin,disturb)/scatter)**2)) exit enddo math_sampleGaussOri = math_RtoEuler(math_mul33x33(math_EulerToR(disturb),math_EulerToR(center))) return endfunction !******************************************************************** ! draw a random sample from Fiber component ! with noise (in radians) !******************************************************************** function math_sampleFiberOri(alpha,beta,noise) use prec, only: pReal, pInt implicit none real(pReal), dimension(3) :: math_sampleFiberOri, fiberInC,fiberInS,axis real(pReal), dimension(2) :: alpha,beta, rnd real(pReal), dimension(3,3) :: oRot,fRot,pRot real(pReal) noise, scatter, cos2Scatter, angle integer(pInt), dimension(2,3), parameter :: rotMap = reshape((/2,3, 3,1, 1,2/),(/2,3/)) integer(pInt) i ! Helming uses different distribution with Bessel functions ! therefore the gauss scatter width has to be scaled differently scatter = 0.95_pReal * noise cos2Scatter = cos(2.0_pReal*scatter) ! fiber axis in crystal coordinate system fiberInC(1)=sin(alpha(1))*cos(alpha(2)) fiberInC(2)=sin(alpha(1))*sin(alpha(2)) fiberInC(3)=cos(alpha(1)) ! fiber axis in sample coordinate system fiberInS(1)=sin(beta(1))*cos(beta(2)) fiberInS(2)=sin(beta(1))*sin(beta(2)) fiberInS(3)=cos(beta(1)) ! ---# rotation matrix from sample to crystal system #--- angle = -acos(dot_product(fiberInC,fiberInS)) if(angle /= 0.0_pReal) then ! rotation axis between sample and crystal system (cross product) forall(i=1:3) axis(i) = fiberInC(rotMap(1,i))*fiberInS(rotMap(2,i))-fiberInC(rotMap(2,i))*fiberInS(rotMap(1,i)) oRot = math_AxisAngleToR(math_vectorproduct(fiberInC,fiberInS),angle) else oRot = math_I3 end if ! ---# rotation matrix about fiber axis (random angle) #--- call halton(1,rnd) fRot = math_AxisAngleToR(fiberInS,rnd(1)*2.0_pReal*pi) ! ---# rotation about random axis perpend to fiber #--- ! random axis pependicular to fiber axis call halton(2,axis) if (fiberInS(3) /= 0.0_pReal) then axis(3)=-(axis(1)*fiberInS(1)+axis(2)*fiberInS(2))/fiberInS(3) else if(fiberInS(2) /= 0.0_pReal) then axis(3)=axis(2) axis(2)=-(axis(1)*fiberInS(1)+axis(3)*fiberInS(3))/fiberInS(2) else if(fiberInS(1) /= 0.0_pReal) then axis(3)=axis(1) axis(1)=-(axis(2)*fiberInS(2)+axis(3)*fiberInS(3))/fiberInS(1) end if ! scattered rotation angle do call halton(2,rnd) angle = acos(cos2Scatter+(1.0_pReal-cos2Scatter)*rnd(1)) if (rnd(2) <= exp(-1.0_pReal*(angle/scatter)**2)) exit enddo call halton(1,rnd) if (rnd(1) <= 0.5) angle = -angle pRot = math_AxisAngleToR(axis,angle) ! ---# apply the three rotations #--- math_sampleFiberOri = math_RtoEuler(math_mul33x33(pRot,math_mul33x33(fRot,oRot))) return endfunction !******************************************************************** ! symmetric Euler angles for given symmetry string ! 'triclinic' or '', 'monoclinic', 'orthotropic' !******************************************************************** pure function math_symmetricEulers(sym,Euler) use prec, only: pReal, pInt implicit none integer(pInt), intent(in) :: sym real(pReal), dimension(3), intent(in) :: Euler real(pReal), dimension(3,3) :: math_symmetricEulers integer(pInt) i,j math_symmetricEulers(1,1) = pi+Euler(1) math_symmetricEulers(2,1) = Euler(2) math_symmetricEulers(3,1) = Euler(3) math_symmetricEulers(1,2) = pi-Euler(1) math_symmetricEulers(2,2) = pi-Euler(2) math_symmetricEulers(3,2) = pi+Euler(3) math_symmetricEulers(1,3) = 2.0_pReal*pi-Euler(1) math_symmetricEulers(2,3) = pi-Euler(2) math_symmetricEulers(3,3) = pi+Euler(3) forall (i=1:3,j=1:3) math_symmetricEulers(j,i) = modulo(math_symmetricEulers(j,i),2.0_pReal*pi) select case (sym) case (4) ! all done case (2) ! return only first math_symmetricEulers(:,2:3) = 0.0_pReal case default ! return blank math_symmetricEulers = 0.0_pReal end select return endfunction !******************************************************************** ! draw a random sample from Gauss variable !******************************************************************** function math_sampleGaussVar(meanvalue, stddev, width) use prec, only: pReal, pInt implicit none !*** input variables 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 !*** output variables real(pReal) math_sampleGaussVar !*** local variables real(pReal), dimension(2) :: rnd ! random numbers real(pReal) scatter, & ! normalized scatter around meanvalue myWidth if (stddev == 0.0) then math_sampleGaussVar = meanvalue return endif if (present(width)) then myWidth = width else myWidth = 3.0_pReal ! use +-3*sigma as default value for scatter endif do call halton(2, rnd) scatter = myWidth * (2.0_pReal * rnd(1) - 1.0_pReal) if (rnd(2) <= exp(-0.5_pReal * scatter ** 2.0_pReal)) & ! test if scattered value is drawn exit enddo math_sampleGaussVar = scatter * stddev endfunction !**************************************************************** pure subroutine math_pDecomposition(FE,U,R,error) !-----FE = R.U !**************************************************************** use prec, only: pReal, pInt implicit none real(pReal), intent(in) :: FE(3,3) real(pReal), intent(out) :: R(3,3), U(3,3) logical, intent(out) :: error real(pReal) CE(3,3),EW1,EW2,EW3,EB1(3,3),EB2(3,3),EB3(3,3),UI(3,3),det error = .false. ce = math_mul33x33(transpose(FE),FE) CALL math_spectral1(CE,EW1,EW2,EW3,EB1,EB2,EB3) U=sqrt(EW1)*EB1+sqrt(EW2)*EB2+sqrt(EW3)*EB3 call math_invert3x3(U,UI,det,error) if (.not. error) R = math_mul33x33(FE,UI) return ENDSUBROUTINE !********************************************************************** pure subroutine math_spectral1(M,EW1,EW2,EW3,EB1,EB2,EB3) !**** EIGENWERTE UND EIGENWERTBASIS DER SYMMETRISCHEN 3X3 MATRIX M use prec, only: pReal, pInt implicit none real(pReal), intent(in) :: M(3,3) real(pReal), intent(out) :: EB1(3,3),EB2(3,3),EB3(3,3),EW1,EW2,EW3 real(pReal) HI1M,HI2M,HI3M,TOL,R,S,T,P,Q,RHO,PHI,Y1,Y2,Y3,D1,D2,D3 real(pReal) C1,C2,C3,M1(3,3),M2(3,3),M3(3,3),arg TOL=1.e-14_pReal CALL math_hi(M,HI1M,HI2M,HI3M) R=-HI1M S= HI2M T=-HI3M P=S-R**2.0_pReal/3.0_pReal Q=2.0_pReal/27.0_pReal*R**3.0_pReal-R*S/3.0_pReal+T EB1=0.0_pReal EB2=0.0_pReal EB3=0.0_pReal IF((ABS(P).LT.TOL).AND.(ABS(Q).LT.TOL))THEN ! DREI GLEICHE EIGENWERTE EW1=HI1M/3.0_pReal EW2=EW1 EW3=EW1 ! this is not really correct, but this way U is calculated ! correctly in PDECOMPOSITION (correct is EB?=I) EB1(1,1)=1.0_pReal EB2(2,2)=1.0_pReal EB3(3,3)=1.0_pReal ELSE RHO=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal arg=-Q/RHO/2.0_pReal if(arg.GT.1) arg=1 if(arg.LT.-1) arg=-1 PHI=acos(arg) Y1=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal) Y2=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal+2.0_pReal/3.0_pReal*PI) Y3=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal+4.0_pReal/3.0_pReal*PI) EW1=Y1-R/3.0_pReal EW2=Y2-R/3.0_pReal EW3=Y3-R/3.0_pReal C1=ABS(EW1-EW2) C2=ABS(EW2-EW3) C3=ABS(EW3-EW1) IF(C1.LT.TOL) THEN ! EW1 is equal to EW2 D3=1.0_pReal/(EW3-EW1)/(EW3-EW2) M1=M-EW1*math_I3 M2=M-EW2*math_I3 EB3=math_mul33x33(M1,M2)*D3 EB1=math_I3-EB3 ! both EB2 and EW2 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW2=0.0_pReal ELSE IF(C2.LT.TOL) THEN ! EW2 is equal to EW3 D1=1.0_pReal/(EW1-EW2)/(EW1-EW3) M2=M-math_I3*EW2 M3=M-math_I3*EW3 EB1=math_mul33x33(M2,M3)*D1 EB2=math_I3-EB1 ! both EB3 and EW3 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW3=0.0_pReal ELSE IF(C3.LT.TOL) THEN ! EW1 is equal to EW3 D2=1.0_pReal/(EW2-EW1)/(EW2-EW3) M1=M-math_I3*EW1 M3=M-math_I3*EW3 EB2=math_mul33x33(M1,M3)*D2 EB1=math_I3-EB2 ! both EB3 and EW3 are set to zero so that they do not ! contribute to U in PDECOMPOSITION EW3=0.0_pReal ELSE ! all three eigenvectors are different D1=1.0_pReal/(EW1-EW2)/(EW1-EW3) D2=1.0_pReal/(EW2-EW1)/(EW2-EW3) D3=1.0_pReal/(EW3-EW1)/(EW3-EW2) M1=M-EW1*math_I3 M2=M-EW2*math_I3 M3=M-EW3*math_I3 EB1=math_mul33x33(M2,M3)*D1 EB2=math_mul33x33(M1,M3)*D2 EB3=math_mul33x33(M1,M2)*D3 END IF END IF RETURN ENDSUBROUTINE !********************************************************************** !**** HAUPTINVARIANTEN HI1M, HI2M, HI3M DER 3X3 MATRIX M PURE SUBROUTINE math_hi(M,HI1M,HI2M,HI3M) use prec, only: pReal, pInt implicit none real(pReal), intent(in) :: M(3,3) real(pReal), intent(out) :: HI1M, HI2M, HI3M HI1M=M(1,1)+M(2,2)+M(3,3) HI2M=HI1M**2/2.0_pReal-(M(1,1)**2+M(2,2)**2+M(3,3)**2)/2.0_pReal-M(1,2)*M(2,1)-M(1,3)*M(3,1)-M(2,3)*M(3,2) HI3M=math_det3x3(M) ! QUESTION: is 3rd equiv det(M) ?? if yes, use function math_det !agreed on YES return ENDSUBROUTINE SUBROUTINE get_seed(seed) ! !******************************************************************************* ! !! GET_SEED returns a seed for the random number generator. ! ! ! Discussion: ! ! The seed depends on the current time, and ought to be (slightly) ! different every millisecond. Once the seed is obtained, a random ! number generator should be called a few times to further process ! the seed. ! ! Modified: ! ! 27 June 2000 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Output, integer SEED, a pseudorandom seed value. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, only: pReal, pInt implicit none integer(pInt) seed real(pReal) temp character ( len = 10 ) time character ( len = 8 ) today integer(pInt) values(8) character ( len = 5 ) zone call date_and_time ( today, time, zone, values ) temp = 0.0D+00 temp = temp + dble ( values(2) - 1 ) / 11.0D+00 temp = temp + dble ( values(3) - 1 ) / 30.0D+00 temp = temp + dble ( values(5) ) / 23.0D+00 temp = temp + dble ( values(6) ) / 59.0D+00 temp = temp + dble ( values(7) ) / 59.0D+00 temp = temp + dble ( values(8) ) / 999.0D+00 temp = temp / 6.0D+00 if ( temp <= 0.0D+00 ) then temp = 1.0D+00 / 3.0D+00 else if ( 1.0D+00 <= temp ) then temp = 2.0D+00 / 3.0D+00 end if seed = int ( dble ( huge ( 1 ) ) * temp , pInt) ! ! Never use a seed of 0 or maximum integer. ! if ( seed == 0 ) then seed = 1 end if if ( seed == huge ( 1 ) ) then seed = seed - 1 end if return ENDSUBROUTINE subroutine halton ( ndim, r ) ! !******************************************************************************* ! !! HALTON computes the next element in the Halton sequence. ! ! ! Modified: ! ! 09 March 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer NDIM, the dimension of the element. ! ! Output, real R(NDIM), the next element of the current Halton ! sequence. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, ONLY: pReal, pInt implicit none integer(pInt) ndim integer(pInt) base(ndim) real(pReal) r(ndim) integer(pInt) seed integer(pInt) value(1) call halton_memory ( 'GET', 'SEED', 1, value ) seed = value(1) call halton_memory ( 'GET', 'BASE', ndim, base ) call i_to_halton ( seed, base, ndim, r ) value(1) = 1 call halton_memory ( 'INC', 'SEED', 1, value ) return ENDSUBROUTINE subroutine halton_memory ( action, name, ndim, value ) ! !******************************************************************************* ! !! HALTON_MEMORY sets or returns quantities associated with the Halton sequence. ! ! ! Modified: ! ! 09 March 2003 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, character ( len = * ) ACTION, the desired action. ! 'GET' means get the value of a particular quantity. ! 'SET' means set the value of a particular quantity. ! 'INC' means increment the value of a particular quantity. ! (Only the SEED can be incremented.) ! ! Input, character ( len = * ) NAME, the name of the quantity. ! 'BASE' means the Halton base or bases. ! 'NDIM' means the spatial dimension. ! 'SEED' means the current Halton seed. ! ! Input/output, integer NDIM, the dimension of the quantity. ! If ACTION is 'SET' and NAME is 'BASE', then NDIM is input, and ! is the number of entries in VALUE to be put into BASE. ! ! Input/output, integer VALUE(NDIM), contains a value. ! If ACTION is 'SET', then on input, VALUE contains values to be assigned ! to the internal variable. ! If ACTION is 'GET', then on output, VALUE contains the values of ! the specified internal variable. ! If ACTION is 'INC', then on input, VALUE contains the increment to ! be added to the specified internal variable. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, only: pReal, pInt implicit none character ( len = * ) action integer(pInt), allocatable, save :: base(:) logical, save :: first_call = .true. integer(pInt) i character ( len = * ) name integer(pInt) ndim integer(pInt), save :: ndim_save = 0 integer(pInt), save :: seed = 1 integer(pInt) value(*) if ( first_call ) then ndim_save = 1 allocate ( base(ndim_save) ) base(1) = 2 first_call = .false. end if ! ! Set ! if ( action(1:1) == 'S' .or. action(1:1) == 's' ) then if ( name(1:1) == 'B' .or. name(1:1) == 'b' ) then if ( ndim_save /= ndim ) then deallocate ( base ) ndim_save = ndim allocate ( base(ndim_save) ) end if base(1:ndim) = value(1:ndim) else if ( name(1:1) == 'N' .or. name(1:1) == 'n' ) then if ( ndim_save /= value(1) ) then deallocate ( base ) ndim_save = value(1) allocate ( base(ndim_save) ) do i = 1, ndim_save base(i) = prime ( i ) enddo else ndim_save = value(1) end if else if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then seed = value(1) end if ! ! Get ! else if ( action(1:1) == 'G' .or. action(1:1) == 'g' ) then if ( name(1:1) == 'B' .or. name(1:1) == 'b' ) then if ( ndim /= ndim_save ) then deallocate ( base ) ndim_save = ndim allocate ( base(ndim_save) ) do i = 1, ndim_save base(i) = prime(i) enddo end if value(1:ndim_save) = base(1:ndim_save) else if ( name(1:1) == 'N' .or. name(1:1) == 'n' ) then value(1) = ndim_save else if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then value(1) = seed end if ! ! Increment ! else if ( action(1:1) == 'I' .or. action(1:1) == 'i' ) then if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then seed = seed + value(1) end if end if return ENDSUBROUTINE subroutine halton_ndim_set ( ndim ) ! !******************************************************************************* ! !! HALTON_NDIM_SET sets the dimension for a Halton sequence. ! ! ! Modified: ! ! 26 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer NDIM, the dimension of the Halton vectors. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, only: pReal, pInt implicit none integer(pInt) ndim integer(pInt) value(1) value(1) = ndim call halton_memory ( 'SET', 'NDIM', 1, value ) return ENDSUBROUTINE subroutine halton_seed_set ( seed ) ! !******************************************************************************* ! !! HALTON_SEED_SET sets the "seed" for the Halton sequence. ! ! ! Discussion: ! ! Calling HALTON repeatedly returns the elements of the ! Halton sequence in order, starting with element number 1. ! An internal counter, called SEED, keeps track of the next element ! to return. Each time the routine is called, the SEED-th element ! is computed, and then SEED is incremented by 1. ! ! To restart the Halton sequence, it is only necessary to reset ! SEED to 1. It might also be desirable to reset SEED to some other value. ! This routine allows the user to specify any value of SEED. ! ! The default value of SEED is 1, which restarts the Halton sequence. ! ! Modified: ! ! 26 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer SEED, the seed for the Halton sequence. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, only: pReal, pInt implicit none integer(pInt), parameter :: ndim = 1 integer(pInt) seed integer(pInt) value(ndim) value(1) = seed call halton_memory ( 'SET', 'SEED', ndim, value ) return ENDSUBROUTINE subroutine i_to_halton ( seed, base, ndim, r ) ! !******************************************************************************* ! !! I_TO_HALTON computes an element of a Halton sequence. ! ! ! Reference: ! ! J H Halton, ! On the efficiency of certain quasi-random sequences of points ! in evaluating multi-dimensional integrals, ! Numerische Mathematik, ! Volume 2, pages 84-90, 1960. ! ! Modified: ! ! 26 February 2001 ! ! Author: ! ! John Burkardt ! ! Parameters: ! ! Input, integer SEED, the index of the desired element. ! Only the absolute value of SEED is considered. SEED = 0 is allowed, ! and returns R = 0. ! ! Input, integer BASE(NDIM), the Halton bases, which should be ! distinct prime numbers. This routine only checks that each base ! is greater than 1. ! ! Input, integer NDIM, the dimension of the sequence. ! ! Output, real R(NDIM), the SEED-th element of the Halton sequence ! for the given bases. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, ONLY: pReal, pInt implicit none integer(pInt) ndim integer(pInt) base(ndim) real(pReal) base_inv(ndim) integer(pInt) digit(ndim) integer(pInt) i real(pReal) r(ndim) integer(pInt) seed integer(pInt) seed2(ndim) seed2(1:ndim) = abs ( seed ) r(1:ndim) = 0.0_pReal if ( any ( base(1:ndim) <= 1 ) ) then !$OMP CRITICAL (write2out) write ( *, '(a)' ) ' ' write ( *, '(a)' ) 'I_TO_HALTON - Fatal error!' write ( *, '(a)' ) ' An input base BASE is <= 1!' do i = 1, ndim write ( *, '(i6,i6)' ) i, base(i) enddo call flush(6) !$OMP END CRITICAL (write2out) stop end if base_inv(1:ndim) = 1.0_pReal / real ( base(1:ndim), pReal ) do while ( any ( seed2(1:ndim) /= 0 ) ) digit(1:ndim) = mod ( seed2(1:ndim), base(1:ndim) ) r(1:ndim) = r(1:ndim) + real ( digit(1:ndim), pReal ) * base_inv(1:ndim) base_inv(1:ndim) = base_inv(1:ndim) / real ( base(1:ndim), pReal ) seed2(1:ndim) = seed2(1:ndim) / base(1:ndim) enddo return ENDSUBROUTINE function prime ( n ) ! !******************************************************************************* ! !! PRIME returns any of the first PRIME_MAX prime numbers. ! ! ! Note: ! ! PRIME_MAX is 1500, and the largest prime stored is 12553. ! ! Modified: ! ! 21 June 2002 ! ! Author: ! ! John Burkardt ! ! Reference: ! ! Milton Abramowitz and Irene Stegun, ! Handbook of Mathematical Functions, ! US Department of Commerce, 1964, pages 870-873. ! ! Daniel Zwillinger, ! CRC Standard Mathematical Tables and Formulae, ! 30th Edition, ! CRC Press, 1996, pages 95-98. ! ! Parameters: ! ! Input, integer N, the index of the desired prime number. ! N = -1 returns PRIME_MAX, the index of the largest prime available. ! N = 0 is legal, returning PRIME = 1. ! It should generally be true that 0 <= N <= PRIME_MAX. ! ! Output, integer PRIME, the N-th prime. If N is out of range, PRIME ! is returned as 0. ! ! Modified: ! ! 29 April 2005 ! ! Author: ! ! Franz Roters ! use prec, only: pReal, pInt implicit none integer(pInt), parameter :: prime_max = 1500 integer(pInt), save :: icall = 0 integer(pInt) n integer(pInt), save, dimension ( prime_max ) :: npvec integer(pInt) prime if ( icall == 0 ) then icall = 1 npvec(1:100) = (/& 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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(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 /) npvec(1301:1400) = (/ & 10663,10667,10687,10691,10709,10711,10723,10729,10733,10739, & 10753,10771,10781,10789,10799,10831,10837,10847,10853,10859, & 10861,10867,10883,10889,10891,10903,10909,19037,10939,10949, & 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 /) npvec(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 /) end if if ( n == -1 ) then prime = prime_max else if ( n == 0 ) then prime = 1 else if ( n <= prime_max ) then prime = npvec(n) else prime = 0 !$OMP CRITICAL (write2out) write ( 6, '(a)' ) ' ' write ( 6, '(a)' ) 'PRIME - Fatal error!' write ( 6, '(a,i6)' ) ' Illegal prime index N = ', n write ( 6, '(a,i6)' ) ' N must be between 0 and PRIME_MAX =',prime_max call flush(6) !$OMP END CRITICAL (write2out) stop end if return endfunction !************************************************************************** ! volume of tetrahedron given by four vertices !************************************************************************** pure function math_volTetrahedron(v1,v2,v3,v4) use prec, only: pReal implicit none real(pReal) math_volTetrahedron real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4 real(pReal), dimension (3,3) :: m m(:,1) = v1-v2 m(:,2) = v2-v3 m(:,3) = v3-v4 math_volTetrahedron = math_det3x3(m)/6.0_pReal return endfunction END MODULE math