DAMASK_EICMD/src/math.f90

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!--------------------------------------------------------------------------------------------------
!> @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
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use prec
use future
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use IO
use debug
use numerics
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implicit none
public
#if __INTEL_COMPILER >= 1900
! do not make use associated entities available to other modules
private :: &
prec, &
future
#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 :: &
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MATH_I3 = reshape([&
1.0_pReal,0.0_pReal,0.0_pReal, &
0.0_pReal,1.0_pReal,0.0_pReal, &
0.0_pReal,0.0_pReal,1.0_pReal &
],[3,3]) !< 3x3 Identity
real(pReal), dimension(6), parameter, private :: &
nrmMandel = [&
1.0_pReal, 1.0_pReal, 1.0_pReal, &
sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal) ] !< weighting for Mandel notation (forward)
real(pReal), dimension(6), parameter , private :: &
invnrmMandel = [&
1.0_pReal, 1.0_pReal, 1.0_pReal, &
1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal), 1.0_pReal/sqrt(2.0_pReal) ] !< weighting for Mandel notation (backward)
integer, dimension (2,6), parameter, private :: &
mapNye = reshape([&
1,1, &
2,2, &
3,3, &
1,2, &
2,3, &
1,3 &
],[2,6]) !< arrangement in Nye notation.
integer, dimension (2,6), parameter, private :: &
mapVoigt = reshape([&
1,1, &
2,2, &
3,3, &
2,3, &
1,3, &
1,2 &
],[2,6]) !< 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 &
],[2,9]) !< arrangement in Plain notation
!--------------------------------------------------------------------------------------------------
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! Provide deprecated name for compatibility
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interface math_mul3x3
module procedure math_inner
end interface math_mul3x3
public :: &
math_mul3x3
!---------------------------------------------------------------------------------------------------
private :: &
unitTest
contains
!--------------------------------------------------------------------------------------------------
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!> @brief initialization of random seed generator and internal checks
!--------------------------------------------------------------------------------------------------
subroutine math_init
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integer :: i
real(pReal), dimension(4) :: randTest
integer :: randSize
integer, dimension(:), allocatable :: randInit
write(6,'(/,a)') ' <<<+- math init -+>>>'
call random_seed(size=randSize)
allocate(randInit(randSize))
if (randomSeed > 0) then
randInit = randomSeed
call random_seed(put=randInit)
else
call random_seed()
call random_seed(get = randInit)
randInit(2:randSize) = randInit(1)
call random_seed(put = randInit)
endif
do i = 1, 4
call random_number(randTest(i))
enddo
write(6,'(a,I2)') ' size of random seed: ', randSize
do i = 1,randSize
write(6,'(a,I2,I14)') ' value of random seed: ', i, randInit(i)
enddo
write(6,'(a,4(/,26x,f17.14),/)') ' start of random sequence: ', randTest
call random_seed(put = randInit)
call unitTest
end subroutine math_init
!--------------------------------------------------------------------------------------------------
!> @brief check correctness of (some) math functions
!--------------------------------------------------------------------------------------------------
subroutine unitTest
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character(len=64) :: error_msg
! +++ check vector expansion +++
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,3.0_pReal,3.0_pReal,3.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1,2,3,0])) > tol_math_check)) then
write (error_msg, '(a)' ) 'math_expand [1,2,3] by [1,2,3,0] => [1,2,2,3,3,3]'
call IO_error(401,ext_msg=error_msg)
endif
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1,2])) > tol_math_check)) then
write (error_msg, '(a)' ) 'math_expand [1,2,3] by [1,2] => [1,2,2]'
call IO_error(401,ext_msg=error_msg)
endif
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,1.0_pReal,1.0_pReal,1.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal],[1,2,3])) > tol_math_check)) then
write (error_msg, '(a)' ) 'math_expand [1,2] by [1,2,3] => [1,2,2,1,1,1]'
call IO_error(401,ext_msg=error_msg)
endif
end subroutine unitTest
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!--------------------------------------------------------------------------------------------------
!> @brief Quicksort algorithm for two-dimensional integer arrays
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! 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)
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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
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if(present(sortDim)) then
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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
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contains
!-------------------------------------------------------------------------------------------------
!> @brief Partitioning required for quicksort
!-------------------------------------------------------------------------------------------------
integer function qsort_partition(a, istart, iend, sort)
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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 ! if the indices cross, exchange left value with pivot and return with the partition index
tmp = a(:,istart)
a(:,istart) = a(:,j)
a(:,j) = tmp
qsort_partition = j
return
else cross ! if they do not cross, exchange values
tmp = a(:,i)
a(:,i) = a(:,j)
a(:,j) = tmp
endif cross
enddo
end function qsort_partition
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end subroutine math_sort
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!--------------------------------------------------------------------------------------------------
!> @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)
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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)
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integer, intent(in) :: N !< length of range
integer :: i
integer, dimension(N) :: math_range
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math_range = [(i,i=1,N)]
end function math_range
!--------------------------------------------------------------------------------------------------
!> @brief second rank identity tensor of specified dimension
!--------------------------------------------------------------------------------------------------
pure function math_identity2nd(dimen)
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integer, intent(in) :: dimen !< tensor dimension
integer :: i
real(pReal), dimension(dimen,dimen) :: math_identity2nd
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math_identity2nd = 0.0_pReal
do i=1, dimen
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(dimen)
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integer, intent(in) :: dimen !< tensor dimension
integer :: i,j,k,l
real(pReal), dimension(dimen,dimen,dimen,dimen) :: math_identity4th
real(pReal), dimension(dimen,dimen) :: identity2nd
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identity2nd = math_identity2nd(dimen)
forall(i=1:dimen,j=1:dimen,k=1:dimen,l=1:dimen) &
math_identity4th(i,j,k,l) = 0.5_pReal*(identity2nd(i,k)*identity2nd(j,l)+identity2nd(i,l)*identity2nd(j,k))
end function math_identity4th
!--------------------------------------------------------------------------------------------------
!> @brief permutation tensor e_ijk used for computing cross product of two tensors
! e_ijk = 1 if even permutation of ijk
! e_ijk = -1 if odd permutation of ijk
! e_ijk = 0 otherwise
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_civita(i,j,k)
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integer, intent(in) :: i,j,k
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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
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end function math_civita
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!--------------------------------------------------------------------------------------------------
!> @brief kronecker delta function d_ij
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! d_ij = 1 if i = j
! d_ij = 0 otherwise
! inspired by http://fortraninacworld.blogspot.de/2012/12/ternary-operator.html
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_delta(i,j)
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integer, intent (in) :: i,j
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math_delta = merge(0.0_pReal, 1.0_pReal, i /= j)
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end function math_delta
!--------------------------------------------------------------------------------------------------
!> @brief cross product a x b
!--------------------------------------------------------------------------------------------------
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pure function math_cross(A,B)
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real(pReal), dimension(3), intent(in) :: A,B
real(pReal), dimension(3) :: math_cross
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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) ]
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end function math_cross
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!--------------------------------------------------------------------------------------------------
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!> @brief outer product A \otimes B of arbitrary sized vectors A and B
!--------------------------------------------------------------------------------------------------
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pure function math_outer(A,B)
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real(pReal), dimension(:), intent(in) :: A,B
real(pReal), dimension(size(A,1),size(B,1)) :: math_outer
integer :: i,j
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forall(i=1:size(A,1),j=1:size(B,1)) math_outer(i,j) = A(i)*B(j)
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end function math_outer
!--------------------------------------------------------------------------------------------------
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!> @brief outer product A \otimes B of arbitrary sized vectors A and B
!--------------------------------------------------------------------------------------------------
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real(pReal) pure function math_inner(A,B)
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real(pReal), dimension(:), intent(in) :: A
real(pReal), dimension(size(A,1)), intent(in) :: B
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math_inner = sum(A*B)
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end function math_inner
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!--------------------------------------------------------------------------------------------------
!> @brief matrix multiplication 33xx33 = 1 (double contraction --> ij * ij)
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_mul33xx33(A,B)
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real(pReal), dimension(3,3), intent(in) :: A,B
integer :: i,j
real(pReal), dimension(3,3) :: C
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forall(i=1:3,j=1:3) C(i,j) = A(i,j) * B(i,j)
math_mul33xx33 = sum(C)
end function math_mul33xx33
!--------------------------------------------------------------------------------------------------
!> @brief matrix multiplication 3333x33 = 33 (double contraction --> ijkl *kl = ij)
!--------------------------------------------------------------------------------------------------
pure function math_mul3333xx33(A,B)
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real(pReal), dimension(3,3) :: math_mul3333xx33
real(pReal), dimension(3,3,3,3), intent(in) :: A
real(pReal), dimension(3,3), intent(in) :: B
integer :: i,j
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forall(i = 1:3,j = 1:3) math_mul3333xx33(i,j) = sum(A(i,j,1:3,1:3)*B(1:3,1:3))
end function math_mul3333xx33
!--------------------------------------------------------------------------------------------------
!> @brief matrix multiplication 3333x3333 = 3333 (ijkl *klmn = ijmn)
!--------------------------------------------------------------------------------------------------
pure function math_mul3333xx3333(A,B)
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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
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forall(i = 1:3,j = 1:3, k = 1:3, l= 1:3) &
math_mul3333xx3333(i,j,k,l) = sum(A(i,j,1:3,1:3)*B(1:3,1:3,k,l))
end function math_mul3333xx3333
!--------------------------------------------------------------------------------------------------
!> @brief 3x3 matrix exponential up to series approximation order n (default 5)
!--------------------------------------------------------------------------------------------------
pure function math_exp33(A,n)
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integer :: i
integer, intent(in), optional :: n
real(pReal), dimension(3,3), intent(in) :: A
real(pReal), dimension(3,3) :: B, math_exp33
real(pReal) :: invFac
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B = math_I3 ! init
invFac = 1.0_pReal ! 0!
math_exp33 = B ! A^0 = eye2
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do i = 1, merge(n,5,present(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 33 matrix (function)
!> @details Direct Cramer inversion of matrix A. Returns all zeroes if not possible, i.e.
! if determinant is close to zero
!--------------------------------------------------------------------------------------------------
pure function math_inv33(A)
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real(pReal),dimension(3,3),intent(in) :: A
real(pReal) :: DetA
real(pReal),dimension(3,3) :: math_inv33
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math_inv33(1,1) = A(2,2) * A(3,3) - A(2,3) * A(3,2)
math_inv33(2,1) = -A(2,1) * A(3,3) + A(2,3) * A(3,1)
math_inv33(3,1) = A(2,1) * A(3,2) - A(2,2) * A(3,1)
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DetA = A(1,1) * math_inv33(1,1) + A(1,2) * math_inv33(2,1) + A(1,3) * math_inv33(3,1)
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if (dNeq0(DetA)) then
math_inv33(1,2) = -A(1,2) * A(3,3) + A(1,3) * A(3,2)
math_inv33(2,2) = A(1,1) * A(3,3) - A(1,3) * A(3,1)
math_inv33(3,2) = -A(1,1) * A(3,2) + A(1,2) * A(3,1)
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math_inv33(1,3) = A(1,2) * A(2,3) - A(1,3) * A(2,2)
math_inv33(2,3) = -A(1,1) * A(2,3) + A(1,3) * A(2,1)
math_inv33(3,3) = A(1,1) * A(2,2) - A(1,2) * A(2,1)
math_inv33 = math_inv33/DetA
else
math_inv33 = 0.0_pReal
endif
end function math_inv33
!--------------------------------------------------------------------------------------------------
!> @brief Cramer inversion of 33 matrix (subroutine)
!> @details Direct Cramer inversion of matrix A. Also returns determinant
! Returns an error if not possible, i.e. if determinant is close to zero
! ToDo: Output arguments should be first
!--------------------------------------------------------------------------------------------------
pure subroutine math_invert33(A, InvA, DetA, error)
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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
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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)
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DetA = A(1,1) * InvA(1,1) + A(1,2) * InvA(2,1) + A(1,3) * InvA(3,1)
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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)
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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)
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InvA = InvA/DetA
error = .false.
endif
end subroutine math_invert33
!--------------------------------------------------------------------------------------------------
!> @brief Inversion of symmetriced 3x3x3x3 tensor.
!--------------------------------------------------------------------------------------------------
function math_invSym3333(A)
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real(pReal),dimension(3,3,3,3) :: math_invSym3333
real(pReal),dimension(3,3,3,3),intent(in) :: A
integer :: ierr
integer, dimension(6) :: ipiv6
real(pReal), dimension(6,6) :: temp66_Real
real(pReal), dimension(6) :: work6
external :: &
dgetrf, &
dgetri
temp66_real = math_sym3333to66(A)
call dgetrf(6,6,temp66_real,6,ipiv6,ierr)
call dgetri(6,temp66_real,6,ipiv6,work6,6,ierr)
if (ierr == 0) then
math_invSym3333 = math_66toSym3333(temp66_real)
else
call IO_error(400, ext_msg = 'math_invSym3333')
endif
end function math_invSym3333
!--------------------------------------------------------------------------------------------------
!> @brief invert quadratic matrix of arbitrary dimension
! ToDo: replaces math_invert
!--------------------------------------------------------------------------------------------------
subroutine math_invert2(InvA, error, A)
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real(pReal), dimension(:,:), intent(in) :: A
real(pReal), dimension(size(A,1),size(A,1)), intent(out) :: invA
logical, intent(out) :: error
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call math_invert(size(A,1), A, InvA, error)
end subroutine math_invert2
!--------------------------------------------------------------------------------------------------
!> @brief invert matrix of arbitrary dimension
! ToDo: Wrong order of arguments and superfluous myDim argument.
! Use math_invert2 instead
!--------------------------------------------------------------------------------------------------
subroutine math_invert(myDim,A, InvA, error)
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integer, intent(in) :: myDim
real(pReal), dimension(myDim,myDim), intent(in) :: A
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integer :: ierr
integer, dimension(myDim) :: ipiv
real(pReal), dimension(myDim) :: work
real(pReal), dimension(myDim,myDim), intent(out) :: invA
logical, intent(out) :: error
external :: &
dgetrf, &
dgetri
invA = A
call dgetrf(myDim,myDim,invA,myDim,ipiv,ierr)
call dgetri(myDim,InvA,myDim,ipiv,work,myDim,ierr)
error = merge(.true.,.false., ierr /= 0)
end subroutine math_invert
!--------------------------------------------------------------------------------------------------
!> @brief symmetrize a 33 matrix
!--------------------------------------------------------------------------------------------------
pure function math_symmetric33(m)
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real(pReal), dimension(3,3) :: math_symmetric33
real(pReal), dimension(3,3), intent(in) :: m
math_symmetric33 = 0.5_pReal * (m + transpose(m))
end function math_symmetric33
!--------------------------------------------------------------------------------------------------
!> @brief symmetrize a 66 matrix
!--------------------------------------------------------------------------------------------------
pure function math_symmetric66(m)
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real(pReal), dimension(6,6) :: math_symmetric66
real(pReal), dimension(6,6), intent(in) :: m
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math_symmetric66 = 0.5_pReal * (m + transpose(m))
end function math_symmetric66
!--------------------------------------------------------------------------------------------------
!> @brief skew part of a 33 matrix
!--------------------------------------------------------------------------------------------------
pure function math_skew33(m)
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real(pReal), dimension(3,3) :: math_skew33
real(pReal), dimension(3,3), intent(in) :: m
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math_skew33 = m - math_symmetric33(m)
end function math_skew33
!--------------------------------------------------------------------------------------------------
!> @brief hydrostatic part of a 33 matrix
!--------------------------------------------------------------------------------------------------
pure function math_spherical33(m)
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real(pReal), dimension(3,3) :: math_spherical33
real(pReal), dimension(3,3), intent(in) :: m
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math_spherical33 = math_I3 * math_trace33(m)/3.0_pReal
end function math_spherical33
!--------------------------------------------------------------------------------------------------
!> @brief deviatoric part of a 33 matrix
!--------------------------------------------------------------------------------------------------
pure function math_deviatoric33(m)
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real(pReal), dimension(3,3) :: math_deviatoric33
real(pReal), dimension(3,3), intent(in) :: m
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math_deviatoric33 = m - math_spherical33(m)
end function math_deviatoric33
!--------------------------------------------------------------------------------------------------
!> @brief equivalent scalar quantity of a full symmetric strain tensor
!--------------------------------------------------------------------------------------------------
pure function math_equivStrain33(m)
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real(pReal), dimension(3,3), intent(in) :: m
real(pReal), dimension(3) :: e,s
real(pReal) :: math_equivStrain33
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e = [2.0_pReal*m(1,1)-m(2,2)-m(3,3), &
2.0_pReal*m(2,2)-m(3,3)-m(1,1), &
2.0_pReal*m(3,3)-m(1,1)-m(2,2)]/3.0_pReal
s = [m(1,2),m(2,3),m(1,3)]*2.0_pReal
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math_equivStrain33 = 2.0_pReal/3.0_pReal &
* (1.50_pReal*(sum(e**2.0_pReal))+ 0.75_pReal*(sum(s**2.0_pReal)))**(0.5_pReal)
end function math_equivStrain33
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!--------------------------------------------------------------------------------------------------
!> @brief von Mises equivalent of a full symmetric stress tensor
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!--------------------------------------------------------------------------------------------------
pure function math_equivStress33(m)
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real(pReal), dimension(3,3), intent(in) :: m
real(pReal) :: math_equivStress33
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math_equivStress33 =( ( (m(1,1)-m(2,2))**2.0_pReal + &
(m(2,2)-m(3,3))**2.0_pReal + &
(m(3,3)-m(1,1))**2.0_pReal + &
6.0_pReal*( m(1,2)**2.0_pReal + &
m(2,3)**2.0_pReal + &
m(1,3)**2.0_pReal &
) &
)**0.5_pReal &
)/sqrt(2.0_pReal)
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end function math_equivStress33
!--------------------------------------------------------------------------------------------------
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!> @brief trace of a 33 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_trace33(m)
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real(pReal), dimension(3,3), intent(in) :: m
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math_trace33 = m(1,1) + m(2,2) + m(3,3)
end function math_trace33
!--------------------------------------------------------------------------------------------------
!> @brief determinant of a 33 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_det33(m)
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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
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!--------------------------------------------------------------------------------------------------
!> @brief determinant of a symmetric 33 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_detSym33(m)
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real(pReal), dimension(3,3), intent(in) :: m
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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)
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end function math_detSym33
!--------------------------------------------------------------------------------------------------
!> @brief convert 33 matrix into vector 9
!--------------------------------------------------------------------------------------------------
pure function math_33to9(m33)
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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 33 matrix
!--------------------------------------------------------------------------------------------------
pure function math_9to33(v9)
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real(pReal), dimension(3,3) :: math_9to33
real(pReal), dimension(9), intent(in) :: v9
integer :: i
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do i = 1, 9
math_9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i)
enddo
end function math_9to33
!--------------------------------------------------------------------------------------------------
!> @brief convert symmetric 33 matrix into 6 vector
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
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pure function math_sym33to6(m33,weighted)
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real(pReal), dimension(6) :: math_sym33to6
real(pReal), dimension(3,3), intent(in) :: m33
logical, optional, intent(in) :: weighted
real(pReal), dimension(6) :: w
integer :: i
if(present(weighted)) then
w = merge(nrmMandel,1.0_pReal,weighted)
else
w = nrmMandel
endif
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do i = 1, 6
math_sym33to6(i) = w(i)*m33(mapNye(1,i),mapNye(2,i))
enddo
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end function math_sym33to6
!--------------------------------------------------------------------------------------------------
!> @brief convert 6 vector into symmetric 33 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
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pure function math_6toSym33(v6,weighted)
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real(pReal), dimension(3,3) :: math_6toSym33
real(pReal), dimension(6), intent(in) :: v6
logical, optional, intent(in) :: weighted
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real(pReal), dimension(6) :: w
integer :: i
if(present(weighted)) then
w = merge(invnrmMandel,1.0_pReal,weighted)
else
w = invnrmMandel
endif
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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
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end function math_6toSym33
!--------------------------------------------------------------------------------------------------
!> @brief convert 3333 matrix into 99 matrix
!--------------------------------------------------------------------------------------------------
pure function math_3333to99(m3333)
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real(pReal), dimension(9,9) :: math_3333to99
real(pReal), dimension(3,3,3,3), intent(in) :: m3333
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integer :: i,j
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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 99 matrix into 3333 matrix
!--------------------------------------------------------------------------------------------------
pure function math_99to3333(m99)
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real(pReal), dimension(3,3,3,3) :: math_99to3333
real(pReal), dimension(9,9), intent(in) :: m99
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integer :: i,j
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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 3333 matrix into 66 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
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pure function math_sym3333to66(m3333,weighted)
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real(pReal), dimension(6,6) :: math_sym3333to66
real(pReal), dimension(3,3,3,3), intent(in) :: m3333
logical, optional, intent(in) :: weighted
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real(pReal), dimension(6) :: w
integer :: i,j
if(present(weighted)) then
w = merge(nrmMandel,1.0_pReal,weighted)
else
w = nrmMandel
endif
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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
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end function math_sym3333to66
!--------------------------------------------------------------------------------------------------
!> @brief convert 66 matrix into symmetric 3333 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
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pure function math_66toSym3333(m66,weighted)
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real(pReal), dimension(3,3,3,3) :: math_66toSym3333
real(pReal), dimension(6,6), intent(in) :: m66
logical, optional, intent(in) :: weighted
real(pReal), dimension(6) :: w
integer :: 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
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end function math_66toSym3333
!--------------------------------------------------------------------------------------------------
!> @brief convert 66 Voigt matrix into symmetric 3333 matrix
!--------------------------------------------------------------------------------------------------
pure function math_Voigt66to3333(m66)
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real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333
real(pReal), dimension(6,6), intent(in) :: m66
integer :: i,j
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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 quaternion multiplication q1xq2 = q12
!--------------------------------------------------------------------------------------------------
pure function math_qMul(A,B)
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real(pReal), dimension(4) :: math_qMul
real(pReal), dimension(4), intent(in) :: A, B
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math_qMul = [ A(1)*B(1) - A(2)*B(2) - A(3)*B(3) - A(4)*B(4), &
A(1)*B(2) + A(2)*B(1) + A(3)*B(4) - A(4)*B(3), &
A(1)*B(3) - A(2)*B(4) + A(3)*B(1) + A(4)*B(2), &
A(1)*B(4) + A(2)*B(3) - A(3)*B(2) + A(4)*B(1) ]
end function math_qMul
!--------------------------------------------------------------------------------------------------
!> @brief quaternion conjugation
!--------------------------------------------------------------------------------------------------
pure function math_qConj(Q)
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real(pReal), dimension(4) :: math_qConj
real(pReal), dimension(4), intent(in) :: Q
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math_qConj = [Q(1), -Q(2:4)]
end function math_qConj
!--------------------------------------------------------------------------------------------------
!> @brief action of a quaternion on a vector (rotate vector v with Q)
!--------------------------------------------------------------------------------------------------
pure function math_qRot(Q,v)
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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 :: i, j
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do i = 1,4
do j = 1,i
T(i,j) = Q(i) * Q(j)
enddo
enddo
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math_qRot = [-v(1)*(T(3,3)+T(4,4)) + v(2)*(T(3,2)-T(4,1)) + v(3)*(T(4,2)+T(3,1)), &
v(1)*(T(3,2)+T(4,1)) - v(2)*(T(2,2)+T(4,4)) + v(3)*(T(4,3)-T(2,1)), &
v(1)*(T(4,2)-T(3,1)) + v(2)*(T(4,3)+T(2,1)) - v(3)*(T(2,2)+T(3,3))]
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math_qRot = 2.0_pReal * math_qRot + v
end function math_qRot
!--------------------------------------------------------------------------------------------------
!> @brief Euler angles (in radians) from rotation matrix
!> @details rotation matrix is meant to represent a PASSIVE rotation,
!> composed of INTRINSIC rotations around the axes of the
!> rotating reference frame
!> (see http://en.wikipedia.org/wiki/Euler_angles for definitions)
!--------------------------------------------------------------------------------------------------
pure function math_RtoEuler(R)
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real(pReal), dimension (3,3), intent(in) :: R
real(pReal), dimension(3) :: math_RtoEuler
real(pReal) :: sqhkl, squvw, sqhk
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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))
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math_RtoEuler(2) = acos(math_clip(R(3,3)/sqhkl,-1.0_pReal, 1.0_pReal))
2007-03-21 15:50:25 +05:30
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if((math_RtoEuler(2) < 1.0e-8_pReal) .or. (pi-math_RtoEuler(2) < 1.0e-8_pReal)) then
math_RtoEuler(3) = 0.0_pReal
math_RtoEuler(1) = acos(math_clip(R(1,1)/squvw, -1.0_pReal, 1.0_pReal))
if(R(2,1) > 0.0_pReal) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1)
else
math_RtoEuler(3) = acos(math_clip(R(2,3)/sqhk, -1.0_pReal, 1.0_pReal))
if(R(1,3) < 0.0) math_RtoEuler(3) = 2.0_pReal*pi-math_RtoEuler(3)
math_RtoEuler(1) = acos(math_clip(-R(3,2)/sin(math_RtoEuler(2)), -1.0_pReal, 1.0_pReal))
if(R(3,1) < 0.0) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1)
end if
end function math_RtoEuler
!--------------------------------------------------------------------------------------------------
!> @brief converts a rotation matrix into a quaternion (w+ix+jy+kz)
!> @details math adopted from http://arxiv.org/pdf/math/0701759v1.pdf
!--------------------------------------------------------------------------------------------------
pure function math_RtoQ(R)
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real(pReal), dimension(3,3), intent(in) :: R
real(pReal), dimension(4) :: absQ, math_RtoQ
real(pReal) :: max_absQ
integer, dimension(1) :: largest
math_RtoQ = 0.0_pReal
absQ = [+ R(1,1) + R(2,2) + R(3,3), &
+ R(1,1) - R(2,2) - R(3,3), &
- R(1,1) + R(2,2) - R(3,3), &
- R(1,1) - R(2,2) + R(3,3)] + 1.0_pReal
largest = maxloc(absQ)
largestComponent: select case(largest(1))
case (1) largestComponent
!1----------------------------------
math_RtoQ(2) = R(3,2) - R(2,3)
math_RtoQ(3) = R(1,3) - R(3,1)
math_RtoQ(4) = R(2,1) - R(1,2)
case (2) largestComponent
math_RtoQ(1) = R(3,2) - R(2,3)
!2----------------------------------
math_RtoQ(3) = R(2,1) + R(1,2)
math_RtoQ(4) = R(1,3) + R(3,1)
case (3) largestComponent
math_RtoQ(1) = R(1,3) - R(3,1)
math_RtoQ(2) = R(2,1) + R(1,2)
!3----------------------------------
math_RtoQ(4) = R(3,2) + R(2,3)
case (4) largestComponent
math_RtoQ(1) = R(2,1) - R(1,2)
math_RtoQ(2) = R(1,3) + R(3,1)
math_RtoQ(3) = R(2,3) + R(3,2)
!4----------------------------------
end select largestComponent
max_absQ = 0.5_pReal * sqrt(absQ(largest(1)))
math_RtoQ = math_RtoQ * 0.25_pReal / max_absQ
math_RtoQ(largest(1)) = max_absQ
end function math_RtoQ
2007-03-21 15:50:25 +05:30
!--------------------------------------------------------------------------------------------------
!> @brief rotation matrix from Bunge-Euler (3-1-3) angles (in radians)
!> @details rotation matrix is meant to represent a PASSIVE rotation, composed of INTRINSIC
!> @details rotations around the axes of the details rotating reference frame.
!> @details similar to eu2om from "D Rowenhorst et al. Consistent representations of and conversions
!> @details between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)", but R is transposed
!--------------------------------------------------------------------------------------------------
pure function math_EulerToR(Euler)
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real(pReal), dimension(3), intent(in) :: Euler
real(pReal), dimension(3,3) :: math_EulerToR
real(pReal) :: c1, C, c2, s1, S, s2
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c1 = cos(Euler(1))
C = cos(Euler(2))
c2 = cos(Euler(3))
s1 = sin(Euler(1))
S = sin(Euler(2))
s2 = sin(Euler(3))
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math_EulerToR(1,1) = c1*c2 -s1*C*s2
math_EulerToR(1,2) = -c1*s2 -s1*C*c2
math_EulerToR(1,3) = s1*S
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math_EulerToR(2,1) = s1*c2 +c1*C*s2
math_EulerToR(2,2) = -s1*s2 +c1*C*c2
math_EulerToR(2,3) = -c1*S
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math_EulerToR(3,1) = S*s2
math_EulerToR(3,2) = S*c2
math_EulerToR(3,3) = C
math_EulerToR = transpose(math_EulerToR) ! convert to passive rotation
end function math_EulerToR
!--------------------------------------------------------------------------------------------------
!> @brief rotation matrix from axis and angle (in radians)
!> @details rotation matrix is meant to represent a ACTIVE rotation
!> @details (see http://en.wikipedia.org/wiki/Euler_angles for definitions)
!> @details formula for active rotation taken from http://mathworld.wolfram.com/RodriguesRotationFormula.html
!> @details equivalent to eu2om (P=-1) from "D Rowenhorst et al. Consistent representations of and
!> @details conversions between 3D rotations, Model. Simul. Mater. Sci. Eng. 23-8 (2015)"
!--------------------------------------------------------------------------------------------------
pure function math_axisAngleToR(axis,omega)
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real(pReal), dimension(3,3) :: math_axisAngleToR
real(pReal), dimension(3), intent(in) :: axis
real(pReal), intent(in) :: omega
real(pReal), dimension(3) :: n
real(pReal) :: norm,s,c,c1
norm = norm2(axis)
wellDefined: if (norm > 1.0e-8_pReal) then
n = axis/norm ! normalize axis to be sure
s = sin(omega)
c = cos(omega)
c1 = 1.0_pReal - c
math_axisAngleToR(1,1) = c + c1*n(1)**2.0_pReal
math_axisAngleToR(1,2) = c1*n(1)*n(2) - s*n(3)
math_axisAngleToR(1,3) = c1*n(1)*n(3) + s*n(2)
math_axisAngleToR(2,1) = c1*n(1)*n(2) + s*n(3)
math_axisAngleToR(2,2) = c + c1*n(2)**2.0_pReal
math_axisAngleToR(2,3) = c1*n(2)*n(3) - s*n(1)
math_axisAngleToR(3,1) = c1*n(1)*n(3) - s*n(2)
math_axisAngleToR(3,2) = c1*n(2)*n(3) + s*n(1)
math_axisAngleToR(3,3) = c + c1*n(3)**2.0_pReal
else wellDefined
math_axisAngleToR = math_I3
endif wellDefined
end function math_axisAngleToR
!--------------------------------------------------------------------------------------------------
!> @brief Rodrigues vector (x, y, z) from unit quaternion (w+ix+jy+kz)
!--------------------------------------------------------------------------------------------------
pure function math_qToRodrig(Q)
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real(pReal), dimension(4), intent(in) :: Q
real(pReal), dimension(3) :: math_qToRodrig
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math_qToRodrig = merge(Q(2:4)/Q(1),IEEE_value(1.0_pReal,IEEE_quiet_NaN),abs(Q(1)) > tol_math_check)! NaN for 180 deg since Rodrig is unbound
end function math_qToRodrig
!--------------------------------------------------------------------------------------------------
!> @brief draw a random sample from Gauss variable
!--------------------------------------------------------------------------------------------------
real(pReal) function math_sampleGaussVar(meanvalue, stddev, width)
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real(pReal), intent(in) :: meanvalue, & ! meanvalue of gauss distribution
stddev ! standard deviation of gauss distribution
real(pReal), intent(in), optional :: width ! width of considered values as multiples of standard deviation
real(pReal), dimension(2) :: rnd ! random numbers
real(pReal) :: scatter, & ! normalized scatter around meanvalue
myWidth
if (abs(stddev) < tol_math_check) then
math_sampleGaussVar = meanvalue
else
myWidth = merge(width,3.0_pReal,present(width)) ! use +-3*sigma as default value for scatter if not given
do
call random_number(rnd)
scatter = myWidth * (2.0_pReal * rnd(1) - 1.0_pReal)
if (rnd(2) <= exp(-0.5_pReal * scatter ** 2.0_pReal)) exit ! test if scattered value is drawn
enddo
math_sampleGaussVar = scatter * stddev
endif
end function math_sampleGaussVar
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues and eigenvectors of symmetric matrix m
! ToDo: has wrong oder of arguments
!--------------------------------------------------------------------------------------------------
subroutine math_eigenValuesVectorsSym(m,values,vectors,error)
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real(pReal), dimension(:,:), intent(in) :: m
real(pReal), dimension(size(m,1)), intent(out) :: values
real(pReal), dimension(size(m,1),size(m,1)), intent(out) :: vectors
logical, intent(out) :: error
integer :: info
real(pReal), dimension((64+2)*size(m,1)) :: work ! block size of 64 taken from http://www.netlib.org/lapack/double/dsyev.f
external :: &
dsyev
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vectors = m ! copy matrix to input (doubles as output) array
call dsyev('V','U',size(m,1),vectors,size(m,1),values,work,(64+2)*size(m,1),info)
error = (info == 0)
end subroutine math_eigenValuesVectorsSym
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues and eigenvectors of symmetric 33 matrix m using an analytical expression
!> and the general LAPACK powered version for arbritrary sized matrices as fallback
!> @author Joachim Kopp, Max-Planck-Institut für Kernphysik, Heidelberg (Copyright (C) 2006)
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @details See http://arxiv.org/abs/physics/0610206 (DSYEVH3)
! ToDo: has wrong oder of arguments
!--------------------------------------------------------------------------------------------------
subroutine math_eigenValuesVectorsSym33(m,values,vectors)
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real(pReal), dimension(3,3),intent(in) :: m
real(pReal), dimension(3), intent(out) :: values
real(pReal), dimension(3,3),intent(out) :: vectors
real(pReal) :: T, U, norm, threshold
logical :: error
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values = math_eigenvaluesSym33(m)
vectors(1:3,2) = [ m(1, 2) * m(2, 3) - m(1, 3) * m(2, 2), &
m(1, 3) * m(1, 2) - m(2, 3) * m(1, 1), &
m(1, 2)**2]
T = maxval(abs(values))
U = max(T, T**2)
threshold = sqrt(5.68e-14_pReal * U**2)
! Calculate first eigenvector by the formula v[0] = (m - lambda[0]).e1 x (m - lambda[0]).e2
vectors(1:3,1) = [ vectors(1,2) + m(1, 3) * values(1), &
vectors(2,2) + m(2, 3) * values(1), &
(m(1,1) - values(1)) * (m(2,2) - values(1)) - vectors(3,2)]
norm = norm2(vectors(1:3, 1))
fallback1: if(norm < threshold) then
call math_eigenValuesVectorsSym(m,values,vectors,error)
return
endif fallback1
vectors(1:3,1) = vectors(1:3, 1) / norm
! Calculate second eigenvector by the formula v[1] = (m - lambda[1]).e1 x (m - lambda[1]).e2
vectors(1:3,2) = [ vectors(1,2) + m(1, 3) * values(2), &
vectors(2,2) + m(2, 3) * values(2), &
(m(1,1) - values(2)) * (m(2,2) - values(2)) - vectors(3,2)]
norm = norm2(vectors(1:3, 2))
fallback2: if(norm < threshold) then
call math_eigenValuesVectorsSym(m,values,vectors,error)
return
endif fallback2
vectors(1:3,2) = vectors(1:3, 2) / norm
! Calculate third eigenvector according to v[2] = v[0] x v[1]
vectors(1:3,3) = math_cross(vectors(1:3,1),vectors(1:3,2))
end subroutine math_eigenValuesVectorsSym33
!--------------------------------------------------------------------------------------------------
!> @brief eigenvector basis of symmetric matrix m
!--------------------------------------------------------------------------------------------------
function math_eigenvectorBasisSym(m)
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real(pReal), dimension(:,:), intent(in) :: m
real(pReal), dimension(size(m,1)) :: values
real(pReal), dimension(size(m,1),size(m,1)) :: vectors
real(pReal), dimension(size(m,1),size(m,1)) :: math_eigenvectorBasisSym
logical :: error
integer :: i
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math_eigenvectorBasisSym = 0.0_pReal
call math_eigenValuesVectorsSym(m,values,vectors,error)
if(error) return
do i=1, size(m,1)
math_eigenvectorBasisSym = math_eigenvectorBasisSym &
+ sqrt(values(i)) * math_outer(vectors(:,i),vectors(:,i))
enddo
end function math_eigenvectorBasisSym
!--------------------------------------------------------------------------------------------------
!> @brief eigenvector basis of symmetric 33 matrix m
!--------------------------------------------------------------------------------------------------
pure function math_eigenvectorBasisSym33(m)
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real(pReal), dimension(3,3) :: math_eigenvectorBasisSym33
real(pReal), dimension(3) :: invariants, values
real(pReal), dimension(3,3), intent(in) :: m
real(pReal) :: P, Q, rho, phi
real(pReal), parameter :: TOL=1.e-14_pReal
real(pReal), dimension(3,3,3) :: N, EB
invariants = math_invariantsSym33(m)
EB = 0.0_pReal
P = invariants(2)-invariants(1)**2.0_pReal/3.0_pReal
Q = -2.0_pReal/27.0_pReal*invariants(1)**3.0_pReal+product(invariants(1:2))/3.0_pReal-invariants(3)
threeSimilarEigenvalues: if(all(abs([P,Q]) < TOL)) then
values = invariants(1)/3.0_pReal
! this is not really correct, but at least the basis is correct
EB(1,1,1)=1.0_pReal
EB(2,2,2)=1.0_pReal
EB(3,3,3)=1.0_pReal
else threeSimilarEigenvalues
rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal
phi=acos(math_clip(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal))
values = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* &
[cos(phi/3.0_pReal), &
cos((phi+2.0_pReal*PI)/3.0_pReal), &
cos((phi+4.0_pReal*PI)/3.0_pReal) &
] + invariants(1)/3.0_pReal
N(1:3,1:3,1) = m-values(1)*math_I3
N(1:3,1:3,2) = m-values(2)*math_I3
N(1:3,1:3,3) = m-values(3)*math_I3
twoSimilarEigenvalues: if(abs(values(1)-values(2)) < TOL) then
EB(1:3,1:3,3)=matmul(N(1:3,1:3,1),N(1:3,1:3,2))/ &
((values(3)-values(1))*(values(3)-values(2)))
EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,3)
elseif(abs(values(2)-values(3)) < TOL) then twoSimilarEigenvalues
EB(1:3,1:3,1)=matmul(N(1:3,1:3,2),N(1:3,1:3,3))/ &
((values(1)-values(2))*(values(1)-values(3)))
EB(1:3,1:3,2)=math_I3-EB(1:3,1:3,1)
elseif(abs(values(3)-values(1)) < TOL) then twoSimilarEigenvalues
EB(1:3,1:3,2)=matmul(N(1:3,1:3,1),N(1:3,1:3,3))/ &
((values(2)-values(1))*(values(2)-values(3)))
EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,2)
else twoSimilarEigenvalues
EB(1:3,1:3,1)=matmul(N(1:3,1:3,2),N(1:3,1:3,3))/ &
((values(1)-values(2))*(values(1)-values(3)))
EB(1:3,1:3,2)=matmul(N(1:3,1:3,1),N(1:3,1:3,3))/ &
((values(2)-values(1))*(values(2)-values(3)))
EB(1:3,1:3,3)=matmul(N(1:3,1:3,1),N(1:3,1:3,2))/ &
((values(3)-values(1))*(values(3)-values(2)))
endif twoSimilarEigenvalues
endif threeSimilarEigenvalues
math_eigenvectorBasisSym33 = sqrt(values(1)) * EB(1:3,1:3,1) &
+ sqrt(values(2)) * EB(1:3,1:3,2) &
+ sqrt(values(3)) * EB(1:3,1:3,3)
end function math_eigenvectorBasisSym33
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!--------------------------------------------------------------------------------------------------
!> @brief logarithm eigenvector basis of symmetric 33 matrix m
!--------------------------------------------------------------------------------------------------
pure function math_eigenvectorBasisSym33_log(m)
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real(pReal), dimension(3,3) :: math_eigenvectorBasisSym33_log
real(pReal), dimension(3) :: invariants, values
real(pReal), dimension(3,3), intent(in) :: m
real(pReal) :: P, Q, rho, phi
real(pReal), parameter :: TOL=1.e-14_pReal
real(pReal), dimension(3,3,3) :: N, EB
invariants = math_invariantsSym33(m)
EB = 0.0_pReal
P = invariants(2)-invariants(1)**2.0_pReal/3.0_pReal
Q = -2.0_pReal/27.0_pReal*invariants(1)**3.0_pReal+product(invariants(1:2))/3.0_pReal-invariants(3)
threeSimilarEigenvalues: if(all(abs([P,Q]) < TOL)) then
values = invariants(1)/3.0_pReal
! this is not really correct, but at least the basis is correct
EB(1,1,1)=1.0_pReal
EB(2,2,2)=1.0_pReal
EB(3,3,3)=1.0_pReal
else threeSimilarEigenvalues
rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal
phi=acos(math_clip(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal))
values = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* &
[cos(phi/3.0_pReal), &
cos((phi+2.0_pReal*PI)/3.0_pReal), &
cos((phi+4.0_pReal*PI)/3.0_pReal) &
] + invariants(1)/3.0_pReal
N(1:3,1:3,1) = m-values(1)*math_I3
N(1:3,1:3,2) = m-values(2)*math_I3
N(1:3,1:3,3) = m-values(3)*math_I3
twoSimilarEigenvalues: if(abs(values(1)-values(2)) < TOL) then
EB(1:3,1:3,3)=matmul(N(1:3,1:3,1),N(1:3,1:3,2))/ &
((values(3)-values(1))*(values(3)-values(2)))
EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,3)
elseif(abs(values(2)-values(3)) < TOL) then twoSimilarEigenvalues
EB(1:3,1:3,1)=matmul(N(1:3,1:3,2),N(1:3,1:3,3))/ &
((values(1)-values(2))*(values(1)-values(3)))
EB(1:3,1:3,2)=math_I3-EB(1:3,1:3,1)
elseif(abs(values(3)-values(1)) < TOL) then twoSimilarEigenvalues
EB(1:3,1:3,2)=matmul(N(1:3,1:3,1),N(1:3,1:3,3))/ &
((values(2)-values(1))*(values(2)-values(3)))
EB(1:3,1:3,1)=math_I3-EB(1:3,1:3,2)
else twoSimilarEigenvalues
EB(1:3,1:3,1)=matmul(N(1:3,1:3,2),N(1:3,1:3,3))/ &
((values(1)-values(2))*(values(1)-values(3)))
EB(1:3,1:3,2)=matmul(N(1:3,1:3,1),N(1:3,1:3,3))/ &
((values(2)-values(1))*(values(2)-values(3)))
EB(1:3,1:3,3)=matmul(N(1:3,1:3,1),N(1:3,1:3,2))/ &
((values(3)-values(1))*(values(3)-values(2)))
endif twoSimilarEigenvalues
endif threeSimilarEigenvalues
math_eigenvectorBasisSym33_log = log(sqrt(values(1))) * EB(1:3,1:3,1) &
+ log(sqrt(values(2))) * EB(1:3,1:3,2) &
+ log(sqrt(values(3))) * EB(1:3,1:3,3)
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end function math_eigenvectorBasisSym33_log
!--------------------------------------------------------------------------------------------------
!> @brief rotational part from polar decomposition of 33 tensor m
!--------------------------------------------------------------------------------------------------
function math_rotationalPart33(m)
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real(pReal), intent(in), dimension(3,3) :: m
real(pReal), dimension(3,3) :: math_rotationalPart33
real(pReal), dimension(3,3) :: U , Uinv
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U = math_eigenvectorBasisSym33(matmul(transpose(m),m))
Uinv = math_inv33(U)
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inversionFailed: if (all(dEq0(Uinv))) then
math_rotationalPart33 = math_I3
call IO_warning(650)
else inversionFailed
math_rotationalPart33 = matmul(m,Uinv)
endif inversionFailed
end function math_rotationalPart33
!--------------------------------------------------------------------------------------------------
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!> @brief Eigenvalues of symmetric matrix m
! will return NaN on error
!--------------------------------------------------------------------------------------------------
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function math_eigenvaluesSym(m)
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real(pReal), dimension(:,:), intent(in) :: m
real(pReal), dimension(size(m,1)) :: math_eigenvaluesSym
real(pReal), dimension(size(m,1),size(m,1)) :: vectors
integer :: info
real(pReal), dimension((64+2)*size(m,1)) :: work ! block size of 64 taken from http://www.netlib.org/lapack/double/dsyev.f
external :: &
dsyev
vectors = m ! copy matrix to input (doubles as output) array
call dsyev('N','U',size(m,1),vectors,size(m,1),math_eigenvaluesSym,work,(64+2)*size(m,1),info)
if (info /= 0) math_eigenvaluesSym = IEEE_value(1.0_pReal,IEEE_quiet_NaN)
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end function math_eigenvaluesSym
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues of symmetric 33 matrix m using an analytical expression
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @details similar to http://arxiv.org/abs/physics/0610206 (DSYEVC3)
!> but apparently more stable solution and has general LAPACK powered version for arbritrary sized
!> matrices as fallback
!--------------------------------------------------------------------------------------------------
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function math_eigenvaluesSym33(m)
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real(pReal), intent(in), dimension(3,3) :: m
real(pReal), dimension(3) :: math_eigenvaluesSym33,invariants
real(pReal) :: P, Q, rho, phi
real(pReal), parameter :: TOL=1.e-14_pReal
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invariants = math_invariantsSym33(m) ! invariants are coefficients in characteristic polynomial apart for the sign of c0 and c2 in http://arxiv.org/abs/physics/0610206
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P = invariants(2)-invariants(1)**2.0_pReal/3.0_pReal ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK)
Q = -2.0_pReal/27.0_pReal*invariants(1)**3.0_pReal+product(invariants(1:2))/3.0_pReal-invariants(3)! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK)
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if(all(abs([P,Q]) < TOL)) then
math_eigenvaluesSym33 = math_eigenvaluesSym(m)
else
rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal
phi=acos(math_clip(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal))
math_eigenvaluesSym33 = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* &
[cos(phi/3.0_pReal), &
cos((phi+2.0_pReal*PI)/3.0_pReal), &
cos((phi+4.0_pReal*PI)/3.0_pReal) &
] + invariants(1)/3.0_pReal
endif
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end function math_eigenvaluesSym33
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!--------------------------------------------------------------------------------------------------
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!> @brief invariants of symmetrix 33 matrix m
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!--------------------------------------------------------------------------------------------------
pure function math_invariantsSym33(m)
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real(pReal), dimension(3,3), intent(in) :: m
real(pReal), dimension(3) :: math_invariantsSym33
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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)
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end function math_invariantsSym33
!--------------------------------------------------------------------------------------------------
!> @brief factorial
!--------------------------------------------------------------------------------------------------
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integer pure function math_factorial(n)
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integer, intent(in) :: n
integer :: i
math_factorial = product([(i, i=1,n)])
end function math_factorial
!--------------------------------------------------------------------------------------------------
!> @brief binomial coefficient
!--------------------------------------------------------------------------------------------------
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integer pure function math_binomial(n,k)
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integer, intent(in) :: n, k
integer :: i, j
j = min(k,n-k)
math_binomial = product([(i, i=n, n-j+1, -1)])/math_factorial(j)
end function math_binomial
!--------------------------------------------------------------------------------------------------
!> @brief multinomial coefficient
!--------------------------------------------------------------------------------------------------
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integer pure function math_multinomial(alpha)
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integer, intent(in), dimension(:) :: alpha
integer :: i
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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)
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real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4
real(pReal), dimension (3,3) :: m
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m(1:3,1) = v1-v2
m(1:3,2) = v2-v3
m(1:3,3) = v3-v4
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math_volTetrahedron = math_det33(m)/6.0_pReal
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end function math_volTetrahedron
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!--------------------------------------------------------------------------------------------------
!> @brief area of triangle given by three vertices
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_areaTriangle(v1,v2,v3)
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real(pReal), dimension (3), intent(in) :: v1,v2,v3
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math_areaTriangle = 0.5_pReal * norm2(math_cross(v1-v2,v1-v3))
end function math_areaTriangle
!--------------------------------------------------------------------------------------------------
!> @brief rotate 33 tensor forward
!--------------------------------------------------------------------------------------------------
pure function math_rotate_forward33(tensor,rot_tensor)
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real(pReal), dimension(3,3) :: math_rotate_forward33
real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor
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math_rotate_forward33 = matmul(rot_tensor,matmul(tensor,transpose(rot_tensor)))
end function math_rotate_forward33
!--------------------------------------------------------------------------------------------------
!> @brief rotate 33 tensor backward
!--------------------------------------------------------------------------------------------------
pure function math_rotate_backward33(tensor,rot_tensor)
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real(pReal), dimension(3,3) :: math_rotate_backward33
real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor
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math_rotate_backward33 = matmul(transpose(rot_tensor),matmul(tensor,rot_tensor))
end function math_rotate_backward33
!--------------------------------------------------------------------------------------------------
!> @brief rotate 3333 tensor C'_ijkl=g_im*g_jn*g_ko*g_lp*C_mnop
!--------------------------------------------------------------------------------------------------
pure function math_rotate_forward3333(tensor,rot_tensor)
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real(pReal), dimension(3,3,3,3) :: math_rotate_forward3333
real(pReal), dimension(3,3), intent(in) :: rot_tensor
real(pReal), dimension(3,3,3,3), intent(in) :: tensor
integer :: i,j,k,l,m,n,o,p
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math_rotate_forward3333 = 0.0_pReal
do i = 1,3;do j = 1,3;do k = 1,3;do l = 1,3
do m = 1,3;do n = 1,3;do o = 1,3;do p = 1,3
math_rotate_forward3333(i,j,k,l) &
= math_rotate_forward3333(i,j,k,l) &
+ rot_tensor(i,m) * rot_tensor(j,n) * rot_tensor(k,o) * rot_tensor(l,p) * tensor(m,n,o,p)
enddo; enddo; enddo; enddo; enddo; enddo; enddo; enddo
end function math_rotate_forward3333
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!--------------------------------------------------------------------------------------------------
!> @brief limits a scalar value to a certain range (either one or two sided)
! Will return NaN if left > right
!--------------------------------------------------------------------------------------------------
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real(pReal) pure elemental function math_clip(a, left, right)
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real(pReal), intent(in) :: a
real(pReal), intent(in), optional :: left, right
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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)
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end function math_clip
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end module math