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DAMASK_EICMD
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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
use prec
use IO
use config
use YAML_types
use LAPACK_interface
implicit none
public
#if __INTEL_COMPILER >= 1900
! do not make use associated entities available to other modules
private :: &
IO, &
config
#endif
real(pReal), parameter :: PI = acos(-1.0_pReal) !< ratio of a circle's circumference to its diameter
real(pReal), parameter :: INDEG = 180.0_pReal/PI !< conversion from radian into degree
real(pReal), parameter :: INRAD = PI/180.0_pReal !< conversion from degree into radian
complex(pReal), parameter :: TWOPIIMG = cmplx(0.0_pReal,2.0_pReal*PI) !< Re(0.0), Im(2xPi)
real(pReal), dimension(3,3), parameter :: &
math_I3 = reshape([&
1.0_pReal,0.0_pReal,0.0_pReal, &
0.0_pReal,1.0_pReal,0.0_pReal, &
0.0_pReal,0.0_pReal,1.0_pReal &
],shape(math_I3)) !< 3x3 Identity
real(pReal), dimension(*), parameter, private :: &
NRMMANDEL = [1.0_pReal, 1.0_pReal,1.0_pReal, sqrt(2.0_pReal), sqrt(2.0_pReal), sqrt(2.0_pReal)] !< forward weighting for Mandel notation
real(pReal), dimension(*), parameter, private :: &
INVNRMMANDEL = 1.0_pReal/NRMMANDEL !< backward weighting for Mandel notation
integer, dimension (2,6), parameter, private :: &
MAPNYE = reshape([&
1,1, &
2,2, &
3,3, &
1,2, &
2,3, &
1,3 &
],shape(MAPNYE)) !< arrangement in Nye notation.
integer, dimension (2,6), parameter, private :: &
MAPVOIGT = reshape([&
1,1, &
2,2, &
3,3, &
2,3, &
1,3, &
1,2 &
],shape(MAPVOIGT)) !< arrangement in Voigt notation
integer, dimension (2,9), parameter, private :: &
MAPPLAIN = reshape([&
1,1, &
1,2, &
1,3, &
2,1, &
2,2, &
2,3, &
3,1, &
3,2, &
3,3 &
],shape(MAPPLAIN)) !< arrangement in Plain notation
!---------------------------------------------------------------------------------------------------
private :: &
selfTest
contains
!--------------------------------------------------------------------------------------------------
!> @brief initialization of random seed generator and internal checks
!--------------------------------------------------------------------------------------------------
subroutine math_init
real(pReal), dimension(4) :: randTest
integer :: &
randSize, &
randomSeed !< fixed seeding for pseudo-random number generator, Default 0: use random seed
integer, dimension(:), allocatable :: randInit
class(tNode), pointer :: &
num_generic
print'(/,a)', ' <<<+- math init -+>>>'; flush(IO_STDOUT)
num_generic => config_numerics%get('generic',defaultVal=emptyDict)
randomSeed = num_generic%get_asInt('random_seed', defaultVal = 0)
call random_seed(size=randSize)
allocate(randInit(randSize))
if (randomSeed > 0) then
randInit = randomSeed
else
call random_seed()
call random_seed(get = randInit)
randInit(2:randSize) = randInit(1)
endif
call random_seed(put = randInit)
call random_number(randTest)
print'(a,i2)', ' size of random seed: ', randSize
print'(a,i0)', ' value of random seed: ', randInit(1)
print'(a,4(/,26x,f17.14),/)', ' start of random sequence: ', randTest
call random_seed(put = randInit)
call selfTest
end subroutine math_init
!--------------------------------------------------------------------------------------------------
!> @brief Sorting of two-dimensional integer arrays
!> @details Based on quicksort.
! Sorting is done with respect to array(sortDim,:) and keeps array(/=sortDim,:) linked to it.
! Default: sortDim=1
!--------------------------------------------------------------------------------------------------
pure recursive subroutine math_sort(a, istart, iend, sortDim)
integer, dimension(:,:), intent(inout) :: a
integer, intent(in),optional :: istart,iend, sortDim
integer :: ipivot,s,e,d
if(present(istart)) then
s = istart
else
s = lbound(a,2)
endif
if(present(iend)) then
e = iend
else
e = ubound(a,2)
endif
if(present(sortDim)) then
d = sortDim
else
d = 1
endif
if (s < e) then
call qsort_partition(a,ipivot, s,e, d)
call math_sort(a, s, ipivot-1, d)
call math_sort(a, ipivot+1, e, d)
endif
contains
!-------------------------------------------------------------------------------------------------
!> @brief Partitioning required for quicksort
!-------------------------------------------------------------------------------------------------
pure subroutine qsort_partition(a,p, istart, iend, sort)
integer, dimension(:,:), intent(inout) :: a
integer, intent(out) :: p ! Pivot element
integer, intent(in) :: istart,iend,sort
integer, dimension(size(a,1)) :: tmp
integer :: i,j
do
! find the first element on the right side less than or equal to the pivot point
do j = iend, istart, -1
if (a(sort,j) <= a(sort,istart)) exit
enddo
! find the first element on the left side greater than the pivot point
do i = istart, iend
if (a(sort,i) > a(sort,istart)) exit
enddo
cross: if (i >= j) then ! exchange left value with pivot and return with the partition index
tmp = a(:,istart)
a(:,istart) = a(:,j)
a(:,j) = tmp
p = j
return
else cross ! exchange values
tmp = a(:,i)
a(:,i) = a(:,j)
a(:,j) = tmp
endif cross
enddo
end subroutine qsort_partition
end subroutine math_sort
!--------------------------------------------------------------------------------------------------
!> @brief vector expansion
!> @details takes a set of numbers (a,b,c,...) and corresponding multiples (x,y,z,...)
!> to return a vector of x times a, y times b, z times c, ...
!--------------------------------------------------------------------------------------------------
pure function math_expand(what,how)
real(pReal), dimension(:), intent(in) :: what
integer, dimension(:), intent(in) :: how
real(pReal), dimension(sum(how)) :: math_expand
integer :: i
if (sum(how) == 0) return
do i = 1, size(how)
math_expand(sum(how(1:i-1))+1:sum(how(1:i))) = what(mod(i-1,size(what))+1)
enddo
end function math_expand
!--------------------------------------------------------------------------------------------------
!> @brief range of integers starting at one
!--------------------------------------------------------------------------------------------------
pure function math_range(N)
integer, intent(in) :: N !< length of range
integer :: i
integer, dimension(N) :: math_range
math_range = [(i,i=1,N)]
end function math_range
!--------------------------------------------------------------------------------------------------
!> @brief second rank identity tensor of specified dimension
!--------------------------------------------------------------------------------------------------
pure function math_eye(d)
integer, intent(in) :: d !< tensor dimension
integer :: i
real(pReal), dimension(d,d) :: math_eye
math_eye = 0.0_pReal
do i=1,d
math_eye(i,i) = 1.0_pReal
enddo
end function math_eye
!--------------------------------------------------------------------------------------------------
!> @brief symmetric fourth rank identity tensor of specified dimension
! from http://en.wikipedia.org/wiki/Tensor_derivative_(continuum_mechanics)#Derivative_of_a_second-order_tensor_with_respect_to_itself
!--------------------------------------------------------------------------------------------------
pure function math_identity4th(d)
integer, intent(in) :: d !< tensor dimension
integer :: i,j,k,l
real(pReal), dimension(d,d,d,d) :: math_identity4th
real(pReal), dimension(d,d) :: identity2nd
identity2nd = math_eye(d)
do i=1,d; do j=1,d; do k=1,d; do l=1,d
math_identity4th(i,j,k,l) = 0.5_pReal &
*(identity2nd(i,k)*identity2nd(j,l)+identity2nd(i,l)*identity2nd(j,k))
enddo; enddo; enddo; enddo
end function math_identity4th
!--------------------------------------------------------------------------------------------------
!> @brief permutation tensor e_ijk
! e_ijk = 1 if even permutation of ijk
! e_ijk = -1 if odd permutation of ijk
! e_ijk = 0 otherwise
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_LeviCivita(i,j,k)
integer, intent(in) :: i,j,k
integer :: o
if (any([(all(cshift([i,j,k],o) == [1,2,3]),o=0,2)])) then
math_LeviCivita = +1.0_pReal
elseif (any([(all(cshift([i,j,k],o) == [3,2,1]),o=0,2)])) then
math_LeviCivita = -1.0_pReal
else
math_LeviCivita = 0.0_pReal
endif
end function math_LeviCivita
!--------------------------------------------------------------------------------------------------
!> @brief kronecker delta function d_ij
! d_ij = 1 if i = j
! d_ij = 0 otherwise
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_delta(i,j)
integer, intent (in) :: i,j
math_delta = merge(0.0_pReal, 1.0_pReal, i /= j)
end function math_delta
!--------------------------------------------------------------------------------------------------
!> @brief cross product a x b
!--------------------------------------------------------------------------------------------------
pure function math_cross(A,B)
real(pReal), dimension(3), intent(in) :: A,B
real(pReal), dimension(3) :: math_cross
math_cross = [ A(2)*B(3) -A(3)*B(2), &
A(3)*B(1) -A(1)*B(3), &
A(1)*B(2) -A(2)*B(1) ]
end function math_cross
!--------------------------------------------------------------------------------------------------
!> @brief outer product of arbitrary sized vectors (A ⊗ B / i,j)
!--------------------------------------------------------------------------------------------------
pure function math_outer(A,B)
real(pReal), dimension(:), intent(in) :: A,B
real(pReal), dimension(size(A,1),size(B,1)) :: math_outer
integer :: i,j
do i=1,size(A,1); do j=1,size(B,1)
math_outer(i,j) = A(i)*B(j)
enddo; enddo
end function math_outer
!--------------------------------------------------------------------------------------------------
!> @brief inner product of arbitrary sized vectors (A · B / i,i)
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_inner(A,B)
real(pReal), dimension(:), intent(in) :: A
real(pReal), dimension(size(A,1)), intent(in) :: B
math_inner = sum(A*B)
end function math_inner
!--------------------------------------------------------------------------------------------------
!> @brief double contraction of 3x3 matrices (A : B / ij,ij)
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_tensordot(A,B)
real(pReal), dimension(3,3), intent(in) :: A,B
math_tensordot = sum(A*B)
end function math_tensordot
!--------------------------------------------------------------------------------------------------
!> @brief matrix double contraction 3333x33 = 33 (ijkl,kl)
!--------------------------------------------------------------------------------------------------
pure function math_mul3333xx33(A,B)
real(pReal), dimension(3,3,3,3), intent(in) :: A
real(pReal), dimension(3,3), intent(in) :: B
real(pReal), dimension(3,3) :: math_mul3333xx33
integer :: i,j
do i=1,3; do j=1,3
math_mul3333xx33(i,j) = sum(A(i,j,1:3,1:3)*B(1:3,1:3))
enddo; enddo
end function math_mul3333xx33
!--------------------------------------------------------------------------------------------------
!> @brief matrix multiplication 3333x3333 = 3333 (ijkl,klmn)
!--------------------------------------------------------------------------------------------------
pure function math_mul3333xx3333(A,B)
integer :: i,j,k,l
real(pReal), dimension(3,3,3,3), intent(in) :: A
real(pReal), dimension(3,3,3,3), intent(in) :: B
real(pReal), dimension(3,3,3,3) :: math_mul3333xx3333
do i=1,3; do j=1,3; do k=1,3; do l=1,3
math_mul3333xx3333(i,j,k,l) = sum(A(i,j,1:3,1:3)*B(1:3,1:3,k,l))
enddo; enddo; enddo; enddo
end function math_mul3333xx3333
!--------------------------------------------------------------------------------------------------
!> @brief 3x3 matrix exponential up to series approximation order n (default 5)
!--------------------------------------------------------------------------------------------------
pure function math_exp33(A,n)
real(pReal), dimension(3,3), intent(in) :: A
integer, intent(in), optional :: n
real(pReal), dimension(3,3) :: B, math_exp33
real(pReal) :: invFac
integer :: n_,i
if (present(n)) then
n_ = n
else
n_ = 5
endif
invFac = 1.0_pReal ! 0!
B = math_I3
math_exp33 = math_I3 ! A^0 = I
do i = 1, n_
invFac = invFac/real(i,pReal) ! invfac = 1/(i!)
B = matmul(B,A)
math_exp33 = math_exp33 + invFac*B ! exp = SUM (A^i)/(i!)
enddo
end function math_exp33
!--------------------------------------------------------------------------------------------------
!> @brief Cramer inversion of 3x3 matrix (function)
!> @details Direct Cramer inversion of matrix A. Returns all zeroes if not possible, i.e.
! if determinant is close to zero
!--------------------------------------------------------------------------------------------------
pure function math_inv33(A)
real(pReal), dimension(3,3), intent(in) :: A
real(pReal), dimension(3,3) :: math_inv33
real(pReal) :: DetA
logical :: error
call math_invert33(math_inv33,DetA,error,A)
if(error) math_inv33 = 0.0_pReal
end function math_inv33
!--------------------------------------------------------------------------------------------------
!> @brief Cramer inversion of 3x3 matrix (subroutine)
!> @details Direct Cramer inversion of matrix A. Also returns determinant
! Returns an error if not possible, i.e. if determinant is close to zero
!--------------------------------------------------------------------------------------------------
pure subroutine math_invert33(InvA, DetA, error, A)
real(pReal), dimension(3,3), intent(out) :: InvA
real(pReal), intent(out) :: DetA
logical, intent(out) :: error
real(pReal), dimension(3,3), intent(in) :: A
InvA(1,1) = A(2,2) * A(3,3) - A(2,3) * A(3,2)
InvA(2,1) = -A(2,1) * A(3,3) + A(2,3) * A(3,1)
InvA(3,1) = A(2,1) * A(3,2) - A(2,2) * A(3,1)
DetA = A(1,1) * InvA(1,1) + A(1,2) * InvA(2,1) + A(1,3) * InvA(3,1)
if (dEq0(DetA)) then
InvA = 0.0_pReal
error = .true.
else
InvA(1,2) = -A(1,2) * A(3,3) + A(1,3) * A(3,2)
InvA(2,2) = A(1,1) * A(3,3) - A(1,3) * A(3,1)
InvA(3,2) = -A(1,1) * A(3,2) + A(1,2) * A(3,1)
InvA(1,3) = A(1,2) * A(2,3) - A(1,3) * A(2,2)
InvA(2,3) = -A(1,1) * A(2,3) + A(1,3) * A(2,1)
InvA(3,3) = A(1,1) * A(2,2) - A(1,2) * A(2,1)
InvA = InvA/DetA
error = .false.
endif
end subroutine math_invert33
!--------------------------------------------------------------------------------------------------
!> @brief Inversion of symmetriced 3x3x3x3 matrix
!--------------------------------------------------------------------------------------------------
function math_invSym3333(A)
real(pReal),dimension(3,3,3,3) :: math_invSym3333
real(pReal),dimension(3,3,3,3),intent(in) :: A
integer, dimension(6) :: ipiv6
real(pReal), dimension(6,6) :: temp66
real(pReal), dimension(6*6) :: work
integer :: ierr_i, ierr_f
temp66 = math_sym3333to66(A)
call dgetrf(6,6,temp66,6,ipiv6,ierr_i)
call dgetri(6,temp66,6,ipiv6,work,size(work,1),ierr_f)
if (ierr_i /= 0 .or. ierr_f /= 0) then
error stop 'matrix inversion error'
else
math_invSym3333 = math_66toSym3333(temp66)
endif
end function math_invSym3333
!--------------------------------------------------------------------------------------------------
!> @brief invert quadratic matrix of arbitrary dimension
!--------------------------------------------------------------------------------------------------
subroutine math_invert(InvA, error, A)
real(pReal), dimension(:,:), intent(in) :: A
real(pReal), dimension(size(A,1),size(A,1)), intent(out) :: invA
logical, intent(out) :: error
integer, dimension(size(A,1)) :: ipiv
real(pReal), dimension(size(A,1)**2) :: work
integer :: ierr
invA = A
call dgetrf(size(A,1),size(A,1),invA,size(A,1),ipiv,ierr)
error = (ierr /= 0)
call dgetri(size(A,1),InvA,size(A,1),ipiv,work,size(work,1),ierr)
error = error .or. (ierr /= 0)
end subroutine math_invert
!--------------------------------------------------------------------------------------------------
!> @brief symmetrize a 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_symmetric33(m)
real(pReal), dimension(3,3) :: math_symmetric33
real(pReal), dimension(3,3), intent(in) :: m
math_symmetric33 = 0.5_pReal * (m + transpose(m))
end function math_symmetric33
!--------------------------------------------------------------------------------------------------
!> @brief symmetrize a 6x6 matrix
!--------------------------------------------------------------------------------------------------
pure function math_symmetric66(m)
real(pReal), dimension(6,6) :: math_symmetric66
real(pReal), dimension(6,6), intent(in) :: m
math_symmetric66 = 0.5_pReal * (m + transpose(m))
end function math_symmetric66
!--------------------------------------------------------------------------------------------------
!> @brief skew part of a 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_skew33(m)
real(pReal), dimension(3,3) :: math_skew33
real(pReal), dimension(3,3), intent(in) :: m
math_skew33 = m - math_symmetric33(m)
end function math_skew33
!--------------------------------------------------------------------------------------------------
!> @brief hydrostatic part of a 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_spherical33(m)
real(pReal), dimension(3,3) :: math_spherical33
real(pReal), dimension(3,3), intent(in) :: m
math_spherical33 = math_I3 * math_trace33(m)/3.0_pReal
end function math_spherical33
!--------------------------------------------------------------------------------------------------
!> @brief deviatoric part of a 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_deviatoric33(m)
real(pReal), dimension(3,3) :: math_deviatoric33
real(pReal), dimension(3,3), intent(in) :: m
math_deviatoric33 = m - math_spherical33(m)
end function math_deviatoric33
!--------------------------------------------------------------------------------------------------
!> @brief trace of a 3x3 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_trace33(m)
real(pReal), dimension(3,3), intent(in) :: m
math_trace33 = m(1,1) + m(2,2) + m(3,3)
end function math_trace33
!--------------------------------------------------------------------------------------------------
!> @brief determinant of a 3x3 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_det33(m)
real(pReal), dimension(3,3), intent(in) :: m
math_det33 = m(1,1)* (m(2,2)*m(3,3)-m(2,3)*m(3,2)) &
- m(1,2)* (m(2,1)*m(3,3)-m(2,3)*m(3,1)) &
+ m(1,3)* (m(2,1)*m(3,2)-m(2,2)*m(3,1))
end function math_det33
!--------------------------------------------------------------------------------------------------
!> @brief determinant of a symmetric 3x3 matrix
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_detSym33(m)
real(pReal), dimension(3,3), intent(in) :: m
math_detSym33 = -(m(1,1)*m(2,3)**2 + m(2,2)*m(1,3)**2 + m(3,3)*m(1,2)**2) &
+ m(1,1)*m(2,2)*m(3,3) + 2.0_pReal * m(1,2)*m(1,3)*m(2,3)
end function math_detSym33
!--------------------------------------------------------------------------------------------------
!> @brief convert 3x3 matrix into vector 9
!--------------------------------------------------------------------------------------------------
pure function math_33to9(m33)
real(pReal), dimension(9) :: math_33to9
real(pReal), dimension(3,3), intent(in) :: m33
integer :: i
math_33to9 = [(m33(MAPPLAIN(1,i),MAPPLAIN(2,i)),i=1,9)]
end function math_33to9
!--------------------------------------------------------------------------------------------------
!> @brief convert 9 vector into 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_9to33(v9)
real(pReal), dimension(3,3) :: math_9to33
real(pReal), dimension(9), intent(in) :: v9
integer :: i
do i = 1, 9
math_9to33(MAPPLAIN(1,i),MAPPLAIN(2,i)) = v9(i)
enddo
end function math_9to33
!--------------------------------------------------------------------------------------------------
!> @brief convert symmetric 3x3 matrix into 6 vector
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
pure function math_sym33to6(m33,weighted)
real(pReal), dimension(6) :: math_sym33to6
real(pReal), dimension(3,3), intent(in) :: m33 !< symmetric 3x3 matrix (no internal check)
logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default)
real(pReal), dimension(6) :: w
integer :: i
if(present(weighted)) then
w = merge(NRMMANDEL,1.0_pReal,weighted)
else
w = NRMMANDEL
endif
math_sym33to6 = [(w(i)*m33(MAPNYE(1,i),MAPNYE(2,i)),i=1,6)]
end function math_sym33to6
!--------------------------------------------------------------------------------------------------
!> @brief convert 6 vector into symmetric 3x3 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only changes order according to Nye
!--------------------------------------------------------------------------------------------------
pure function math_6toSym33(v6,weighted)
real(pReal), dimension(3,3) :: math_6toSym33
real(pReal), dimension(6), intent(in) :: v6 !< 6 vector
logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default)
real(pReal), dimension(6) :: w
integer :: i
if(present(weighted)) then
w = merge(INVNRMMANDEL,1.0_pReal,weighted)
else
w = INVNRMMANDEL
endif
do i=1,6
math_6toSym33(MAPNYE(1,i),MAPNYE(2,i)) = w(i)*v6(i)
math_6toSym33(MAPNYE(2,i),MAPNYE(1,i)) = w(i)*v6(i)
enddo
end function math_6toSym33
!--------------------------------------------------------------------------------------------------
!> @brief convert 3x3x3x3 matrix into 9x9 matrix
!--------------------------------------------------------------------------------------------------
pure function math_3333to99(m3333)
real(pReal), dimension(9,9) :: math_3333to99
real(pReal), dimension(3,3,3,3), intent(in) :: m3333
integer :: i,j
do i=1,9; do j=1,9
math_3333to99(i,j) = m3333(MAPPLAIN(1,i),MAPPLAIN(2,i),MAPPLAIN(1,j),MAPPLAIN(2,j))
enddo; enddo
end function math_3333to99
!--------------------------------------------------------------------------------------------------
!> @brief convert 9x9 matrix into 3x3x3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_99to3333(m99)
real(pReal), dimension(3,3,3,3) :: math_99to3333
real(pReal), dimension(9,9), intent(in) :: m99
integer :: i,j
do i=1,9; do j=1,9
math_99to3333(MAPPLAIN(1,i),MAPPLAIN(2,i),MAPPLAIN(1,j),MAPPLAIN(2,j)) = m99(i,j)
enddo; enddo
end function math_99to3333
!--------------------------------------------------------------------------------------------------
!> @brief convert symmetric 3x3x3x3 matrix into 6x6 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only rearranges order according to Nye
!--------------------------------------------------------------------------------------------------
pure function math_sym3333to66(m3333,weighted)
real(pReal), dimension(6,6) :: math_sym3333to66
real(pReal), dimension(3,3,3,3), intent(in) :: m3333 !< symmetric 3x3x3x3 matrix (no internal check)
logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default)
real(pReal), dimension(6) :: w
integer :: i,j
if(present(weighted)) then
w = merge(NRMMANDEL,1.0_pReal,weighted)
else
w = NRMMANDEL
endif
do i=1,6; do j=1,6
math_sym3333to66(i,j) = w(i)*w(j)*m3333(MAPNYE(1,i),MAPNYE(2,i),MAPNYE(1,j),MAPNYE(2,j))
enddo; enddo
end function math_sym3333to66
!--------------------------------------------------------------------------------------------------
!> @brief convert 66 matrix into symmetric 3x3x3x3 matrix
!> @details Weighted conversion (default) rearranges according to Nye and weights shear
! components according to Mandel. Advisable for matrix operations.
! Unweighted conversion only rearranges order according to Nye
!--------------------------------------------------------------------------------------------------
pure function math_66toSym3333(m66,weighted)
real(pReal), dimension(3,3,3,3) :: math_66toSym3333
real(pReal), dimension(6,6), intent(in) :: m66 !< 6x6 matrix
logical, optional, intent(in) :: weighted !< weight according to Mandel (.true. by default)
real(pReal), dimension(6) :: w
integer :: i,j
if(present(weighted)) then
w = merge(INVNRMMANDEL,1.0_pReal,weighted)
else
w = INVNRMMANDEL
endif
do i=1,6; do j=1,6
math_66toSym3333(MAPNYE(1,i),MAPNYE(2,i),MAPNYE(1,j),MAPNYE(2,j)) = w(i)*w(j)*m66(i,j)
math_66toSym3333(MAPNYE(2,i),MAPNYE(1,i),MAPNYE(1,j),MAPNYE(2,j)) = w(i)*w(j)*m66(i,j)
math_66toSym3333(MAPNYE(1,i),MAPNYE(2,i),MAPNYE(2,j),MAPNYE(1,j)) = w(i)*w(j)*m66(i,j)
math_66toSym3333(MAPNYE(2,i),MAPNYE(1,i),MAPNYE(2,j),MAPNYE(1,j)) = w(i)*w(j)*m66(i,j)
enddo; enddo
end function math_66toSym3333
!--------------------------------------------------------------------------------------------------
!> @brief convert 66 Voigt matrix into symmetric 3x3x3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_Voigt66to3333(m66)
real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333
real(pReal), dimension(6,6), intent(in) :: m66 !< 6x6 matrix
integer :: i,j
do i=1,6; do j=1, 6
math_Voigt66to3333(MAPVOIGT(1,i),MAPVOIGT(2,i),MAPVOIGT(1,j),MAPVOIGT(2,j)) = m66(i,j)
math_Voigt66to3333(MAPVOIGT(2,i),MAPVOIGT(1,i),MAPVOIGT(1,j),MAPVOIGT(2,j)) = m66(i,j)
math_Voigt66to3333(MAPVOIGT(1,i),MAPVOIGT(2,i),MAPVOIGT(2,j),MAPVOIGT(1,j)) = m66(i,j)
math_Voigt66to3333(MAPVOIGT(2,i),MAPVOIGT(1,i),MAPVOIGT(2,j),MAPVOIGT(1,j)) = m66(i,j)
enddo; enddo
end function math_Voigt66to3333
!--------------------------------------------------------------------------------------------------
!> @brief draw a random sample from Gauss variable
!--------------------------------------------------------------------------------------------------
real(pReal) function math_sampleGaussVar(mu, sigma, width)
real(pReal), intent(in) :: mu, & !< mean
sigma !< standard deviation
real(pReal), intent(in), optional :: width !< cut off as multiples of standard deviation
real(pReal), dimension(2) :: rnd ! random numbers
real(pReal) :: scatter, & ! normalized scatter around mean
width_
if (abs(sigma) < tol_math_check) then
math_sampleGaussVar = mu
else
if (present(width)) then
width_ = width
else
width_ = 3.0_pReal ! use +-3*sigma as default scatter
endif
do
call random_number(rnd)
scatter = width_ * (2.0_pReal * rnd(1) - 1.0_pReal)
if (rnd(2) <= exp(-0.5_pReal * scatter ** 2.0_pReal)) exit ! test if scattered value is drawn
enddo
math_sampleGaussVar = scatter * sigma
endif
end function math_sampleGaussVar
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues and eigenvectors of symmetric matrix
!--------------------------------------------------------------------------------------------------
subroutine math_eigh(w,v,error,m)
real(pReal), dimension(:,:), intent(in) :: m !< quadratic matrix to compute eigenvectors and values of
real(pReal), dimension(size(m,1)), intent(out) :: w !< eigenvalues
real(pReal), dimension(size(m,1),size(m,1)), intent(out) :: v !< eigenvectors
logical, intent(out) :: error
integer :: ierr
real(pReal), dimension(size(m,1)**2) :: work
v = m ! copy matrix to input (doubles as output) array
call dsyev('V','U',size(m,1),v,size(m,1),w,work,size(work,1),ierr)
error = (ierr /= 0)
end subroutine math_eigh
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues and eigenvectors of symmetric 3x3 matrix using an analytical expression
!> and the general LAPACK powered version for arbritrary sized matrices as fallback
!> @author Joachim Kopp, Max-Planck-Institut für Kernphysik, Heidelberg (Copyright (C) 2006)
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @details See http://arxiv.org/abs/physics/0610206 (DSYEVH3)
!--------------------------------------------------------------------------------------------------
subroutine math_eigh33(w,v,m)
real(pReal), dimension(3,3),intent(in) :: m !< 3x3 matrix to compute eigenvectors and values of
real(pReal), dimension(3), intent(out) :: w !< eigenvalues
real(pReal), dimension(3,3),intent(out) :: v !< eigenvectors
real(pReal) :: T, U, norm, threshold
logical :: error
w = math_eigvalsh33(m)
v(1:3,2) = [ m(1, 2) * m(2, 3) - m(1, 3) * m(2, 2), &
m(1, 3) * m(1, 2) - m(2, 3) * m(1, 1), &
m(1, 2)**2]
T = maxval(abs(w))
U = max(T, T**2)
threshold = sqrt(5.68e-14_pReal * U**2)
v(1:3,1) = [ v(1,2) + m(1, 3) * w(1), &
v(2,2) + m(2, 3) * w(1), &
(m(1,1) - w(1)) * (m(2,2) - w(1)) - v(3,2)]
norm = norm2(v(1:3, 1))
fallback1: if(norm < threshold) then
call math_eigh(w,v,error,m)
else fallback1
v(1:3,1) = v(1:3, 1) / norm
v(1:3,2) = [ v(1,2) + m(1, 3) * w(2), &
v(2,2) + m(2, 3) * w(2), &
(m(1,1) - w(2)) * (m(2,2) - w(2)) - v(3,2)]
norm = norm2(v(1:3, 2))
fallback2: if(norm < threshold) then
call math_eigh(w,v,error,m)
else fallback2
v(1:3,2) = v(1:3, 2) / norm
v(1:3,3) = math_cross(v(1:3,1),v(1:3,2))
endif fallback2
endif fallback1
end subroutine math_eigh33
!--------------------------------------------------------------------------------------------------
!> @brief Calculate rotational part of a deformation gradient
!> @details https://www.jstor.org/stable/43637254
!! https://www.jstor.org/stable/43637372
!! https://doi.org/10.1023/A:1007407802076
!--------------------------------------------------------------------------------------------------
pure function math_rotationalPart(F) result(R)
real(pReal), dimension(3,3), intent(in) :: &
F ! deformation gradient
real(pReal), dimension(3,3) :: &
C, & ! right Cauchy-Green tensor
R ! rotational part
real(pReal), dimension(3) :: &
lambda, & ! principal stretches
I_C, & ! invariants of C
I_U ! invariants of U
real(pReal), dimension(2) :: &
I_F ! first two invariants of F
real(pReal) :: x,Phi
C = matmul(transpose(F),F)
I_C = math_invariantsSym33(C)
I_F = [math_trace33(F), 0.5*(math_trace33(F)**2 - math_trace33(matmul(F,F)))]
x = math_clip(I_C(1)**2 -3.0_pReal*I_C(2),0.0_pReal)**(3.0_pReal/2.0_pReal)
if(dNeq0(x)) then
Phi = acos(math_clip((I_C(1)**3 -4.5_pReal*I_C(1)*I_C(2) +13.5_pReal*I_C(3))/x,-1.0_pReal,1.0_pReal))
lambda = I_C(1) +(2.0_pReal * sqrt(math_clip(I_C(1)**2-3.0_pReal*I_C(2),0.0_pReal))) &
*cos((Phi-2.0_pReal * PI*[1.0_pReal,2.0_pReal,3.0_pReal])/3.0_pReal)
lambda = sqrt(math_clip(lambda,0.0_pReal)/3.0_pReal)
else
lambda = sqrt(I_C(1)/3.0_pReal)
endif
I_U = [sum(lambda), lambda(1)*lambda(2)+lambda(2)*lambda(3)+lambda(3)*lambda(1), product(lambda)]
R = I_U(1)*I_F(2) * math_I3 &
+(I_U(1)**2-I_U(2)) * F &
- I_U(1)*I_F(1) * transpose(F) &
+ I_U(1) * transpose(matmul(F,F)) &
- matmul(F,C)
R = R /(I_U(1)*I_U(2)-I_U(3))
end function math_rotationalPart
!--------------------------------------------------------------------------------------------------
!> @brief Eigenvalues of symmetric matrix
! will return NaN on error
!--------------------------------------------------------------------------------------------------
function math_eigvalsh(m)
real(pReal), dimension(:,:), intent(in) :: m !< symmetric matrix to compute eigenvalues of
real(pReal), dimension(size(m,1)) :: math_eigvalsh
real(pReal), dimension(size(m,1),size(m,1)) :: m_
integer :: ierr
real(pReal), dimension(size(m,1)**2) :: work
m_= m ! copy matrix to input (will be destroyed)
call dsyev('N','U',size(m,1),m_,size(m,1),math_eigvalsh,work,size(work,1),ierr)
if (ierr /= 0) math_eigvalsh = IEEE_value(1.0_pReal,IEEE_quiet_NaN)
end function math_eigvalsh
!--------------------------------------------------------------------------------------------------
!> @brief eigenvalues of symmetric 3x3 matrix using an analytical expression
!> @author Martin Diehl, Max-Planck-Institut für Eisenforschung GmbH
!> @details similar to http://arxiv.org/abs/physics/0610206 (DSYEVC3)
!> but apparently more stable solution and has general LAPACK powered version for arbritrary sized
!> matrices as fallback
!--------------------------------------------------------------------------------------------------
function math_eigvalsh33(m)
real(pReal), intent(in), dimension(3,3) :: m !< 3x3 symmetric matrix to compute eigenvalues of
real(pReal), dimension(3) :: math_eigvalsh33,I
real(pReal) :: P, Q, rho, phi
real(pReal), parameter :: TOL=1.e-14_pReal
I = math_invariantsSym33(m) ! invariants are coefficients in characteristic polynomial apart for the sign of c0 and c2 in http://arxiv.org/abs/physics/0610206
P = I(2)-I(1)**2.0_pReal/3.0_pReal ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK)
Q = product(I(1:2))/3.0_pReal &
- 2.0_pReal/27.0_pReal*I(1)**3.0_pReal &
- I(3) ! different from http://arxiv.org/abs/physics/0610206 (this formulation was in DAMASK)
if(all(abs([P,Q]) < TOL)) then
math_eigvalsh33 = math_eigvalsh(m)
else
rho=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal
phi=acos(math_clip(-Q/rho*0.5_pReal,-1.0_pReal,1.0_pReal))
math_eigvalsh33 = 2.0_pReal*rho**(1.0_pReal/3.0_pReal)* &
[cos( phi /3.0_pReal), &
cos((phi+2.0_pReal*PI)/3.0_pReal), &
cos((phi+4.0_pReal*PI)/3.0_pReal) &
] &
+ I(1)/3.0_pReal
endif
end function math_eigvalsh33
!--------------------------------------------------------------------------------------------------
!> @brief invariants of symmetrix 3x3 matrix
!--------------------------------------------------------------------------------------------------
pure function math_invariantsSym33(m)
real(pReal), dimension(3,3), intent(in) :: m
real(pReal), dimension(3) :: math_invariantsSym33
math_invariantsSym33(1) = math_trace33(m)
math_invariantsSym33(2) = m(1,1)*m(2,2) + m(1,1)*m(3,3) + m(2,2)*m(3,3) &
-(m(1,2)**2 + m(1,3)**2 + m(2,3)**2)
math_invariantsSym33(3) = math_detSym33(m)
end function math_invariantsSym33
!--------------------------------------------------------------------------------------------------
!> @brief factorial
!--------------------------------------------------------------------------------------------------
integer pure function math_factorial(n)
integer, intent(in) :: n
math_factorial = product(math_range(n))
end function math_factorial
!--------------------------------------------------------------------------------------------------
!> @brief binomial coefficient
!--------------------------------------------------------------------------------------------------
integer pure function math_binomial(n,k)
integer, intent(in) :: n, k
integer :: i, k_, n_
k_ = min(k,n-k)
n_ = n
math_binomial = merge(1,0,k_>-1) ! handling special cases k < 0 or k > n
do i = 1, k_
math_binomial = (math_binomial * n_)/i
n_ = n_ -1
enddo
end function math_binomial
!--------------------------------------------------------------------------------------------------
!> @brief multinomial coefficient
!--------------------------------------------------------------------------------------------------
integer pure function math_multinomial(alpha)
integer, intent(in), dimension(:) :: alpha
integer :: i
math_multinomial = 1
do i = 1, size(alpha)
math_multinomial = math_multinomial*math_binomial(sum(alpha(1:i)),alpha(i))
enddo
end function math_multinomial
!--------------------------------------------------------------------------------------------------
!> @brief volume of tetrahedron given by four vertices
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_volTetrahedron(v1,v2,v3,v4)
real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4
real(pReal), dimension (3,3) :: m
m(1:3,1) = v1-v2
m(1:3,2) = v1-v3
m(1:3,3) = v1-v4
math_volTetrahedron = abs(math_det33(m))/6.0_pReal
end function math_volTetrahedron
!--------------------------------------------------------------------------------------------------
!> @brief area of triangle given by three vertices
!--------------------------------------------------------------------------------------------------
real(pReal) pure function math_areaTriangle(v1,v2,v3)
real(pReal), dimension (3), intent(in) :: v1,v2,v3
math_areaTriangle = 0.5_pReal * norm2(math_cross(v1-v2,v1-v3))
end function math_areaTriangle
!--------------------------------------------------------------------------------------------------
!> @brief Limit a scalar value to a certain range (either one or two sided).
!--------------------------------------------------------------------------------------------------
real(pReal) pure elemental function math_clip(a, left, right)
real(pReal), intent(in) :: a
real(pReal), intent(in), optional :: left, right
math_clip = a
if (present(left)) math_clip = max(left,math_clip)
if (present(right)) math_clip = min(right,math_clip)
if (present(left) .and. present(right)) then
if(left>right) error stop 'left > right'
endif
end function math_clip
!--------------------------------------------------------------------------------------------------
!> @brief Check correctness of some math functions.
!--------------------------------------------------------------------------------------------------
subroutine selfTest
integer, dimension(2,4) :: &
sort_in_ = reshape([+1,+5, +5,+6, -1,-1, +3,-2],[2,4])
integer, dimension(2,4), parameter :: &
sort_out_ = reshape([-1,-1, +1,+5, +5,+6, +3,-2],[2,4])
integer, dimension(5) :: range_out_ = [1,2,3,4,5]
integer, dimension(3) :: ijk
real(pReal) :: det
real(pReal), dimension(3) :: v3_1,v3_2,v3_3,v3_4
real(pReal), dimension(6) :: v6
real(pReal), dimension(9) :: v9
real(pReal), dimension(3,3) :: t33,t33_2
real(pReal), dimension(6,6) :: t66
real(pReal), dimension(9,9) :: t99,t99_2
real(pReal), dimension(:,:), &
allocatable :: txx,txx_2
real(pReal) :: r
integer :: d
logical :: e
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,3.0_pReal,3.0_pReal,3.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1,2,3,0])) > tol_math_check)) &
error stop 'math_expand [1,2,3] by [1,2,3,0] => [1,2,2,3,3,3]'
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal,3.0_pReal],[1,2])) > tol_math_check)) &
error stop 'math_expand [1,2,3] by [1,2] => [1,2,2]'
if (any(abs([1.0_pReal,2.0_pReal,2.0_pReal,1.0_pReal,1.0_pReal,1.0_pReal] - &
math_expand([1.0_pReal,2.0_pReal],[1,2,3])) > tol_math_check)) &
error stop 'math_expand [1,2] by [1,2,3] => [1,2,2,1,1,1]'
call math_sort(sort_in_,1,3,2)
if(any(sort_in_ /= sort_out_)) &
error stop 'math_sort'
if(any(math_range(5) /= range_out_)) &
error stop 'math_range'
if(any(dNeq(math_exp33(math_I3,0),math_I3))) &
error stop 'math_exp33(math_I3,1)'
if(any(dNeq(math_exp33(math_I3,128),exp(1.0_pReal)*math_I3))) &
error stop 'math_exp33(math_I3,128)'
call random_number(v9)
if(any(dNeq(math_33to9(math_9to33(v9)),v9))) &
error stop 'math_33to9/math_9to33'
call random_number(t99)
if(any(dNeq(math_3333to99(math_99to3333(t99)),t99))) &
error stop 'math_3333to99/math_99to3333'
call random_number(v6)
if(any(dNeq(math_sym33to6(math_6toSym33(v6)),v6))) &
error stop 'math_sym33to6/math_6toSym33'
call random_number(t66)
if(any(dNeq(math_sym3333to66(math_66toSym3333(t66)),t66))) &
error stop 'math_sym3333to66/math_66toSym3333'
call random_number(v6)
if(any(dNeq0(math_6toSym33(v6) - math_symmetric33(math_6toSym33(v6))))) &
error stop 'math_symmetric33'
call random_number(v3_1)
call random_number(v3_2)
call random_number(v3_3)
call random_number(v3_4)
if(dNeq(abs(dot_product(math_cross(v3_1-v3_4,v3_2-v3_4),v3_3-v3_4))/6.0, &
math_volTetrahedron(v3_1,v3_2,v3_3,v3_4),tol=1.0e-12_pReal)) &
error stop 'math_volTetrahedron'
call random_number(t33)
if(dNeq(math_det33(math_symmetric33(t33)),math_detSym33(math_symmetric33(t33)),tol=1.0e-12_pReal)) &
error stop 'math_det33/math_detSym33'
if(any(dNeq0(math_eye(3),math_inv33(math_I3)))) &
error stop 'math_inv33(math_I3)'
do while(abs(math_det33(t33))<1.0e-9_pReal)
call random_number(t33)
enddo
if(any(dNeq0(matmul(t33,math_inv33(t33)) - math_eye(3),tol=1.0e-9_pReal))) &
error stop 'math_inv33'
call math_invert33(t33_2,det,e,t33)
if(any(dNeq0(matmul(t33,t33_2) - math_eye(3),tol=1.0e-9_pReal)) .or. e) &
error stop 'math_invert33: T:T^-1 != I'
if(dNeq(det,math_det33(t33),tol=1.0e-12_pReal)) &
error stop 'math_invert33 (determinant)'
call math_invert(t33_2,e,t33)
if(any(dNeq0(matmul(t33,t33_2) - math_eye(3),tol=1.0e-9_pReal)) .or. e) &
error stop 'math_invert t33'
do while(math_det33(t33)<1.0e-2_pReal) ! O(det(F)) = 1
call random_number(t33)
enddo
t33_2 = math_rotationalPart(transpose(t33))
t33 = math_rotationalPart(t33)
if(any(dNeq0(matmul(t33_2,t33) - math_I3,tol=1.0e-10_pReal))) &
error stop 'math_rotationalPart'
call random_number(r)
d = int(r*5.0_pReal) + 1
txx = math_eye(d)
allocate(txx_2(d,d))
call math_invert(txx_2,e,txx)
if(any(dNeq0(txx_2,txx) .or. e)) &
error stop 'math_invert(txx)/math_eye'
call math_invert(t99_2,e,t99) ! not sure how likely it is that we get a singular matrix
if(any(dNeq0(matmul(t99_2,t99)-math_eye(9),tol=1.0e-9_pReal)) .or. e) &
error stop 'math_invert(t99)'
if(any(dNeq(math_clip([4.0_pReal,9.0_pReal],5.0_pReal,6.5_pReal),[5.0_pReal,6.5_pReal]))) &
error stop 'math_clip'
if(math_factorial(10) /= 3628800) &
error stop 'math_factorial'
if(math_binomial(49,6) /= 13983816) &
error stop 'math_binomial'
ijk = cshift([1,2,3],int(r*1.0e2_pReal))
if(dNeq(math_LeviCivita(ijk(1),ijk(2),ijk(3)),+1.0_pReal)) &
error stop 'math_LeviCivita(even)'
ijk = cshift([3,2,1],int(r*2.0e2_pReal))
if(dNeq(math_LeviCivita(ijk(1),ijk(2),ijk(3)),-1.0_pReal)) &
error stop 'math_LeviCivita(odd)'
ijk = cshift([2,2,1],int(r*2.0e2_pReal))
if(dNeq0(math_LeviCivita(ijk(1),ijk(2),ijk(3)))) &
error stop 'math_LeviCivita'
end subroutine selfTest
end module math
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