2889 lines
86 KiB
Fortran
2889 lines
86 KiB
Fortran
!* $Id$
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!##############################################################
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MODULE math
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!##############################################################
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use prec, only: pReal,pInt
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implicit none
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real(pReal), parameter :: pi = 3.14159265358979323846264338327950288419716939937510_pReal
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real(pReal), parameter :: inDeg = 180.0_pReal/pi
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real(pReal), parameter :: inRad = pi/180.0_pReal
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real(pReal), parameter :: NaN = 0.0_pReal/0.0_pReal ! Not a number
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! *** 3x3 Identity ***
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real(pReal), dimension(3,3), parameter :: math_I3 = &
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reshape( (/ &
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1.0_pReal,0.0_pReal,0.0_pReal, &
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0.0_pReal,1.0_pReal,0.0_pReal, &
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0.0_pReal,0.0_pReal,1.0_pReal /),(/3,3/))
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! *** Mandel notation ***
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integer(pInt), dimension (2,6), parameter :: mapMandel = &
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reshape((/&
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1,1, &
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2,2, &
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3,3, &
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1,2, &
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2,3, &
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1,3 &
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/),(/2,6/))
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real(pReal), dimension(6), parameter :: nrmMandel = &
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(/1.0_pReal,1.0_pReal,1.0_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal/)
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real(pReal), dimension(6), parameter :: invnrmMandel = &
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(/1.0_pReal,1.0_pReal,1.0_pReal,0.7071067811865476_pReal,0.7071067811865476_pReal,0.7071067811865476_pReal/)
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! *** Voigt notation ***
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integer(pInt), dimension (2,6), parameter :: mapVoigt = &
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reshape((/&
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1,1, &
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2,2, &
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3,3, &
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2,3, &
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1,3, &
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1,2 &
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/),(/2,6/))
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real(pReal), dimension(6), parameter :: nrmVoigt = &
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(/1.0_pReal,1.0_pReal,1.0_pReal, 1.0_pReal, 1.0_pReal, 1.0_pReal/)
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real(pReal), dimension(6), parameter :: invnrmVoigt = &
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(/1.0_pReal,1.0_pReal,1.0_pReal, 1.0_pReal, 1.0_pReal, 1.0_pReal/)
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! *** Plain notation ***
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integer(pInt), dimension (2,9), parameter :: mapPlain = &
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reshape((/&
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1,1, &
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1,2, &
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1,3, &
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2,1, &
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2,2, &
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2,3, &
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3,1, &
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3,2, &
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3,3 &
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/),(/2,9/))
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! Symmetry operations as quaternions
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! 24 for cubic, 12 for hexagonal = 36
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integer(pInt), dimension(2), parameter :: math_NsymOperations = (/24,12/)
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real(pReal), dimension(4,36), parameter :: math_symOperations = &
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reshape((/&
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1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! cubic symmetry operations
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0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, & ! 2-fold symmetry
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0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, &
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0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, &
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0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, &
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0.0_pReal, -0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, &
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0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, 0.0_pReal, &
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0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & ! 3-fold symmetry
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-0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, &
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0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, &
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-0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, &
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0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, &
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-0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, &
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0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, &
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-0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, &
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0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, & ! 4-fold symmetry
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0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, &
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-0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, &
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0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, &
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0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, &
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-0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, &
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0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, &
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0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal, &
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-0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, &
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1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! hexagonal symmetry operations
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0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & ! 2-fold symmetry
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0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, &
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0.0_pReal, 0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, &
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0.0_pReal, -0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, &
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0.0_pReal, 0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, &
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0.0_pReal, -0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, &
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0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, & ! 6-fold symmetry
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-0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, &
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0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, &
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-0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, &
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0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal &
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/),(/4,36/))
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CONTAINS
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!**************************************************************************
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! initialization of module
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!**************************************************************************
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SUBROUTINE math_init ()
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use prec, only: pReal,pInt,tol_math_check
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use numerics, only: fixedSeed
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use IO, only: IO_error
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implicit none
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real(pReal), dimension(3,3) :: R,R2
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real(pReal), dimension(3) :: Eulers
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real(pReal), dimension(4) :: q,q2,axisangle
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real(pReal), dimension(2) :: rnd
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integer(pInt), dimension(1) :: randInit
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write(6,*)
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write(6,*) '<<<+- math init -+>>>'
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write(6,*) '$Id$'
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write(6,*)
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if (fixedSeed > 0_pInt) then
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randInit = fixedSeed
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call random_seed(put=randInit)
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else
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call random_seed()
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endif
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call random_seed(get=randInit)
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write(6,*) 'random seed: ',randInit(1)
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write(6,*)
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call halton_seed_set(randInit(1))
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call halton_ndim_set(3)
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! --- check rotation dictionary ---
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! +++ q -> a -> q +++
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q = math_qRnd();
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axisangle = math_QuaternionToAxisAngle(q);
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q2 = math_AxisAngleToQuaternion(axisangle(1:3),axisangle(4))
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if ( any(abs( q-q2) > tol_math_check ) .and. &
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any(abs(-q-q2) > tol_math_check ) ) &
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call IO_error(670)
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! +++ q -> R -> q +++
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R = math_QuaternionToR(q);
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q2 = math_RToQuaternion(R)
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if ( any(abs( q-q2) > tol_math_check ) .and. &
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any(abs(-q-q2) > tol_math_check ) ) &
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call IO_error(671)
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! +++ q -> euler -> q +++
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Eulers = math_QuaternionToEuler(q);
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q2 = math_EulerToQuaternion(Eulers)
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if ( any(abs( q-q2) > tol_math_check ) .and. &
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any(abs(-q-q2) > tol_math_check ) ) &
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call IO_error(672)
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! +++ R -> euler -> R +++
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Eulers = math_RToEuler(R);
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R2 = math_EulerToR(Eulers)
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if ( any(abs( R-R2) > tol_math_check ) ) &
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call IO_error(673)
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ENDSUBROUTINE
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!**************************************************************************
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! Quicksort algorithm for two-dimensional integer arrays
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!
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! Sorting is done with respect to array(1,:)
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! and keeps array(2:N,:) linked to it.
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!**************************************************************************
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RECURSIVE SUBROUTINE qsort(a, istart, iend)
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implicit none
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integer(pInt), dimension(:,:) :: a
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integer(pInt) :: istart,iend,ipivot
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if (istart < iend) then
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ipivot = math_partition(a,istart, iend)
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call qsort(a, istart, ipivot-1)
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call qsort(a, ipivot+1, iend)
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endif
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return
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ENDSUBROUTINE
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!**************************************************************************
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! Partitioning required for quicksort
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!**************************************************************************
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integer(pInt) function math_partition(a, istart, iend)
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implicit none
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integer(pInt), dimension(:,:) :: a
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integer(pInt) :: istart,iend,d,i,j,k,x,tmp
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d = size(a,1) ! number of linked data
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! set the starting and ending points, and the pivot point
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i = istart
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j = iend
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x = a(1,istart)
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do
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! find the first element on the right side less than or equal to the pivot point
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do j = j, istart, -1
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if (a(1,j) <= x) exit
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enddo
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! find the first element on the left side greater than the pivot point
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do i = i, iend
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if (a(1,i) > x) exit
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enddo
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if (i < j ) then ! if the indexes do not cross, exchange values
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do k = 1,d
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tmp = a(k,i)
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a(k,i) = a(k,j)
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a(k,j) = tmp
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enddo
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else ! if they do cross, exchange left value with pivot and return with the partition index
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do k = 1,d
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tmp = a(k,istart)
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a(k,istart) = a(k,j)
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a(k,j) = tmp
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enddo
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math_partition = j
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return
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endif
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enddo
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endfunction
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!**************************************************************************
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! range of integers starting at one
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!**************************************************************************
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pure function math_range(N)
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use prec, only: pInt
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implicit none
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integer(pInt), intent(in) :: N
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integer(pInt) i
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integer(pInt), dimension(N) :: math_range
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forall (i=1:N) math_range(i) = i
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return
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endfunction
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!**************************************************************************
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! second rank identity tensor of specified dimension
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!**************************************************************************
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pure function math_identity2nd(dimen)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt), intent(in) :: dimen
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integer(pInt) i
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real(pReal), dimension(dimen,dimen) :: math_identity2nd
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math_identity2nd = 0.0_pReal
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forall (i=1:dimen) math_identity2nd(i,i) = 1.0_pReal
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return
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endfunction
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!**************************************************************************
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! permutation tensor e_ijk used for computing cross product of two tensors
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! e_ijk = 1 if even permutation of ijk
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! e_ijk = -1 if odd permutation of ijk
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! e_ijk = 0 otherwise
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!**************************************************************************
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pure function math_civita(i,j,k) ! change its name from math_permut
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! to math_civita <<<updated 31.07.2009>>>
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use prec, only: pReal, pInt
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implicit none
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integer(pInt), intent(in) :: i,j,k
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real(pReal) math_civita
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math_civita = 0.0_pReal
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if (((i == 1).and.(j == 2).and.(k == 3)) .or. &
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((i == 2).and.(j == 3).and.(k == 1)) .or. &
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((i == 3).and.(j == 1).and.(k == 2))) math_civita = 1.0_pReal
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if (((i == 1).and.(j == 3).and.(k == 2)) .or. &
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((i == 2).and.(j == 1).and.(k == 3)) .or. &
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((i == 3).and.(j == 2).and.(k == 1))) math_civita = -1.0_pReal
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return
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endfunction
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!**************************************************************************
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! kronecker delta function d_ij
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! d_ij = 1 if i = j
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! d_ij = 0 otherwise
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!**************************************************************************
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pure function math_delta(i,j)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt), intent (in) :: i,j
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real(pReal) math_delta
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math_delta = 0.0_pReal
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if (i == j) math_delta = 1.0_pReal
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return
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endfunction
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!**************************************************************************
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! fourth rank identity tensor of specified dimension
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!**************************************************************************
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pure function math_identity4th(dimen)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt), intent(in) :: dimen
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integer(pInt) i,j,k,l
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real(pReal), dimension(dimen,dimen,dimen,dimen) :: math_identity4th
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forall (i=1:dimen,j=1:dimen,k=1:dimen,l=1:dimen) math_identity4th(i,j,k,l) = &
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0.5_pReal*(math_I3(i,k)*math_I3(j,k)+math_I3(i,l)*math_I3(j,k))
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return
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endfunction
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!**************************************************************************
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! vector product a x b
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!**************************************************************************
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pure function math_vectorproduct(A,B)
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use prec, only: pReal, pInt
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implicit none
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real(pReal), dimension(3), intent(in) :: A,B
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real(pReal), dimension(3) :: math_vectorproduct
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math_vectorproduct(1) = A(2)*B(3)-A(3)*B(2)
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math_vectorproduct(2) = A(3)*B(1)-A(1)*B(3)
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math_vectorproduct(3) = A(1)*B(2)-A(2)*B(1)
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return
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endfunction
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!**************************************************************************
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! tensor product a \otimes b
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!**************************************************************************
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pure function math_tensorproduct(A,B)
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use prec, only: pReal, pInt
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implicit none
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real(pReal), dimension(3), intent(in) :: A,B
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real(pReal), dimension(3,3) :: math_tensorproduct
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integer(pInt) i,j
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forall (i=1:3,j=1:3) math_tensorproduct(i,j) = A(i)*B(j)
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return
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endfunction
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!**************************************************************************
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! matrix multiplication 3x3 = 1
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!**************************************************************************
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pure function math_mul3x3(A,B)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt) i
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real(pReal), dimension(3), intent(in) :: A,B
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real(pReal), dimension(3) :: C
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real(pReal) math_mul3x3
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forall (i=1:3) C(i) = A(i)*B(i)
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math_mul3x3 = sum(C)
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return
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endfunction
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!**************************************************************************
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! matrix multiplication 6x6 = 1
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!**************************************************************************
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pure function math_mul6x6(A,B)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt) i
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real(pReal), dimension(6), intent(in) :: A,B
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real(pReal), dimension(6) :: C
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real(pReal) math_mul6x6
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forall (i=1:6) C(i) = A(i)*B(i)
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math_mul6x6 = sum(C)
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return
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endfunction
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!**************************************************************************
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! matrix multiplication 33x33 = 1 (double contraction --> ij * ij)
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!**************************************************************************
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pure function math_mul33xx33(A,B)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt) i,j
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real(pReal), dimension(3,3), intent(in) :: A,B
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real(pReal), dimension(3,3) :: C
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real(pReal) math_mul33xx33
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forall (i=1:3,j=1:3) C(i,j) = A(i,j) * B(i,j)
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math_mul33xx33 = sum(C)
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return
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endfunction
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!**************************************************************************
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! matrix multiplication 3333x33 = 33 (double contraction --> ijkl *kl = ij)
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!**************************************************************************
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pure function math_mul3333xx33(A,B)
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use prec, only: pReal, pInt
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implicit none
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integer(pInt) i,j
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real(pReal), dimension(3,3,3,3), intent(in) :: A
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real(pReal), dimension(3,3), intent(in) :: B
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real(pReal), dimension(3,3) :: C,math_mul3333xx33
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|
|
do i = 1,3
|
|
do j = 1,3
|
|
math_mul3333xx33(i,j) = sum(A(i,j,:,:)*B(:,:))
|
|
enddo; enddo
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication 33x33 = 3x3
|
|
!**************************************************************************
|
|
pure function math_mul33x33(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i,j
|
|
real(pReal), dimension(3,3), intent(in) :: A,B
|
|
real(pReal), dimension(3,3) :: math_mul33x33
|
|
|
|
forall (i=1:3,j=1:3) math_mul33x33(i,j) = &
|
|
A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication 66x66 = 6x6
|
|
!**************************************************************************
|
|
pure function math_mul66x66(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i,j
|
|
real(pReal), dimension(6,6), intent(in) :: A,B
|
|
real(pReal), dimension(6,6) :: math_mul66x66
|
|
|
|
forall (i=1:6,j=1:6) math_mul66x66(i,j) = &
|
|
A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) + &
|
|
A(i,4)*B(4,j) + A(i,5)*B(5,j) + A(i,6)*B(6,j)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication 99x99 = 9x9
|
|
!**************************************************************************
|
|
pure function math_mul99x99(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i,j
|
|
real(pReal), dimension(9,9), intent(in) :: A,B
|
|
|
|
real(pReal), dimension(9,9) :: math_mul99x99
|
|
|
|
|
|
forall (i=1:9,j=1:9) math_mul99x99(i,j) = &
|
|
A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) + &
|
|
A(i,4)*B(4,j) + A(i,5)*B(5,j) + A(i,6)*B(6,j) + &
|
|
A(i,7)*B(7,j) + A(i,8)*B(8,j) + A(i,9)*B(9,j)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication 33x3 = 3
|
|
!**************************************************************************
|
|
pure function math_mul33x3(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i
|
|
real(pReal), dimension(3,3), intent(in) :: A
|
|
real(pReal), dimension(3), intent(in) :: B
|
|
real(pReal), dimension(3) :: math_mul33x3
|
|
|
|
forall (i=1:3) math_mul33x3(i) = A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3)
|
|
return
|
|
|
|
endfunction
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication complex(33) x real(3) = complex(3)
|
|
!**************************************************************************
|
|
pure function math_mul33x3_complex(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i
|
|
complex(pReal), dimension(3,3), intent(in) :: A
|
|
real(pReal), dimension(3), intent(in) :: B
|
|
complex(pReal), dimension(3) :: math_mul33x3_complex
|
|
|
|
forall (i=1:3) math_mul33x3_complex(i) = A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! matrix multiplication 66x6 = 6
|
|
!**************************************************************************
|
|
pure function math_mul66x6(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) i
|
|
real(pReal), dimension(6,6), intent(in) :: A
|
|
real(pReal), dimension(6), intent(in) :: B
|
|
real(pReal), dimension(6) :: math_mul66x6
|
|
|
|
forall (i=1:6) math_mul66x6(i) = &
|
|
A(i,1)*B(1) + A(i,2)*B(2) + A(i,3)*B(3) + &
|
|
A(i,4)*B(4) + A(i,5)*B(5) + A(i,6)*B(6)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! random quaternion
|
|
!**************************************************************************
|
|
function math_qRnd()
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4) :: math_qRnd
|
|
real(pReal), dimension(3) :: rnd
|
|
|
|
call halton(3,rnd)
|
|
math_qRnd(1) = cos(2.0_pReal*pi*rnd(1))*sqrt(rnd(3))
|
|
math_qRnd(2) = sin(2.0_pReal*pi*rnd(2))*sqrt(1.0_pReal-rnd(3))
|
|
math_qRnd(3) = cos(2.0_pReal*pi*rnd(2))*sqrt(1.0_pReal-rnd(3))
|
|
math_qRnd(4) = sin(2.0_pReal*pi*rnd(1))*sqrt(rnd(3))
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! quaternion multiplication q1xq2 = q12
|
|
!**************************************************************************
|
|
pure function math_qMul(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: A, B
|
|
real(pReal), dimension(4) :: math_qMul
|
|
|
|
math_qMul(1) = A(1)*B(1) - A(2)*B(2) - A(3)*B(3) - A(4)*B(4)
|
|
math_qMul(2) = A(1)*B(2) + A(2)*B(1) + A(3)*B(4) - A(4)*B(3)
|
|
math_qMul(3) = A(1)*B(3) - A(2)*B(4) + A(3)*B(1) + A(4)*B(2)
|
|
math_qMul(4) = A(1)*B(4) + A(2)*B(3) - A(3)*B(2) + A(4)*B(1)
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! quaternion dotproduct
|
|
!**************************************************************************
|
|
pure function math_qDot(A,B)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: A, B
|
|
real(pReal) math_qDot
|
|
|
|
math_qDot = A(1)*B(1) + A(2)*B(2) + A(3)*B(3) + A(4)*B(4)
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! quaternion conjugation
|
|
!**************************************************************************
|
|
pure function math_qConj(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(4) :: math_qConj
|
|
|
|
math_qConj(1) = Q(1)
|
|
math_qConj(2:4) = -Q(2:4)
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! quaternion norm
|
|
!**************************************************************************
|
|
pure function math_qNorm(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal) math_qNorm
|
|
|
|
math_qNorm = sqrt(max(0.0_pReal, Q(1)*Q(1) + Q(2)*Q(2) + Q(3)*Q(3) + Q(4)*Q(4)))
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! quaternion inversion
|
|
!**************************************************************************
|
|
pure function math_qInv(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(4) :: math_qInv
|
|
real(pReal) squareNorm
|
|
|
|
math_qInv = 0.0_pReal
|
|
|
|
squareNorm = math_qDot(Q,Q)
|
|
if (squareNorm > tiny(squareNorm)) &
|
|
math_qInv = math_qConj(Q) / squareNorm
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! action of a quaternion on a vector (rotate vector v with Q)
|
|
!**************************************************************************
|
|
pure function math_qRot(Q,v)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(3), intent(in) :: v
|
|
real(pReal), dimension(3) :: math_qRot
|
|
real(pReal), dimension(4,4) :: T
|
|
integer(pInt) i, j
|
|
|
|
do i = 1,4
|
|
do j = 1,i
|
|
T(i,j) = Q(i) * Q(j)
|
|
enddo
|
|
enddo
|
|
|
|
math_qRot(1) = -v(1)*(T(3,3)+T(4,4)) + v(2)*(T(3,2)-T(4,1)) + v(3)*(T(4,2)+T(3,1))
|
|
math_qRot(2) = v(1)*(T(3,2)+T(4,1)) - v(2)*(T(2,2)+T(4,4)) + v(3)*(T(4,3)-T(2,1))
|
|
math_qRot(3) = v(1)*(T(4,2)-T(3,1)) + v(2)*(T(4,3)+T(2,1)) - v(3)*(T(2,2)+T(3,3))
|
|
|
|
math_qRot = 2.0_pReal * math_qRot + v
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! transposition of a 3x3 matrix
|
|
!**************************************************************************
|
|
pure function math_transpose3x3(A)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal),dimension(3,3),intent(in) :: A
|
|
real(pReal),dimension(3,3) :: math_transpose3x3
|
|
integer(pInt) i,j
|
|
|
|
forall(i=1:3, j=1:3) math_transpose3x3(i,j) = A(j,i)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! Cramer inversion of 3x3 matrix (function)
|
|
!**************************************************************************
|
|
pure function math_inv3x3(A)
|
|
|
|
! direct Cramer inversion of matrix A.
|
|
! returns all zeroes if not possible, i.e. if det close to zero
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal),dimension(3,3),intent(in) :: A
|
|
real(pReal) DetA
|
|
|
|
real(pReal),dimension(3,3) :: math_inv3x3
|
|
|
|
math_inv3x3 = 0.0_pReal
|
|
|
|
DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )&
|
|
- A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )&
|
|
+ A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) )
|
|
|
|
if (DetA > tiny(DetA)) then
|
|
math_inv3x3(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA
|
|
math_inv3x3(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA
|
|
math_inv3x3(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA
|
|
|
|
math_inv3x3(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA
|
|
math_inv3x3(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA
|
|
math_inv3x3(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA
|
|
|
|
math_inv3x3(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA
|
|
math_inv3x3(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA
|
|
math_inv3x3(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA
|
|
endif
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!**************************************************************************
|
|
! Cramer inversion of 3x3 matrix (subroutine)
|
|
!**************************************************************************
|
|
PURE SUBROUTINE math_invert3x3(A, InvA, DetA, error)
|
|
|
|
! Bestimmung der Determinanten und Inversen einer 3x3-Matrix
|
|
! A = Matrix A
|
|
! InvA = Inverse of A
|
|
! DetA = Determinant of A
|
|
! error = logical
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
logical, intent(out) :: error
|
|
|
|
real(pReal),dimension(3,3),intent(in) :: A
|
|
real(pReal),dimension(3,3),intent(out) :: InvA
|
|
real(pReal), intent(out) :: DetA
|
|
|
|
DetA = A(1,1) * ( A(2,2) * A(3,3) - A(2,3) * A(3,2) )&
|
|
- A(1,2) * ( A(2,1) * A(3,3) - A(2,3) * A(3,1) )&
|
|
+ A(1,3) * ( A(2,1) * A(3,2) - A(2,2) * A(3,1) )
|
|
|
|
if (DetA <= tiny(DetA)) then
|
|
error = .true.
|
|
else
|
|
InvA(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2) ) / DetA
|
|
InvA(2,1) = ( -A(2,1) * A(3,3) + A(2,3) * A(3,1) ) / DetA
|
|
InvA(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1) ) / DetA
|
|
|
|
InvA(1,2) = ( -A(1,2) * A(3,3) + A(1,3) * A(3,2) ) / DetA
|
|
InvA(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1) ) / DetA
|
|
InvA(3,2) = ( -A(1,1) * A(3,2) + A(1,2) * A(3,1) ) / DetA
|
|
|
|
InvA(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2) ) / DetA
|
|
InvA(2,3) = ( -A(1,1) * A(2,3) + A(1,3) * A(2,1) ) / DetA
|
|
InvA(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1) ) / DetA
|
|
|
|
error = .false.
|
|
endif
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
|
|
!**************************************************************************
|
|
! Gauss elimination to invert 6x6 matrix
|
|
!**************************************************************************
|
|
PURE SUBROUTINE math_invert(dimen,A, InvA, AnzNegEW, error)
|
|
|
|
! Invertieren einer dimen x dimen - Matrix
|
|
! A = Matrix A
|
|
! InvA = Inverse von A
|
|
! AnzNegEW = Anzahl der negativen Eigenwerte von A
|
|
! error = logical
|
|
! = false: Inversion wurde durchgefuehrt.
|
|
! = true: Die Inversion in SymGauss wurde wegen eines verschwindenen
|
|
! Pivotelement abgebrochen.
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
integer(pInt), intent(in) :: dimen
|
|
real(pReal),dimension(dimen,dimen), intent(in) :: A
|
|
real(pReal),dimension(dimen,dimen), intent(out) :: InvA
|
|
integer(pInt), intent(out) :: AnzNegEW
|
|
logical, intent(out) :: error
|
|
real(pReal) LogAbsDetA
|
|
real(pReal),dimension(dimen,dimen) :: B
|
|
|
|
InvA = math_identity2nd(dimen)
|
|
B = A
|
|
CALL Gauss(dimen,B,InvA,LogAbsDetA,AnzNegEW,error)
|
|
RETURN
|
|
|
|
ENDSUBROUTINE math_invert
|
|
|
|
|
|
|
|
! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
|
|
! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++
|
|
PURE SUBROUTINE Gauss (dimen,A,B,LogAbsDetA,NegHDK,error)
|
|
|
|
! Loesung eines linearen Gleichungsssystem A * X = B mit Hilfe des
|
|
! GAUSS-Algorithmus
|
|
! Zur numerischen Stabilisierung wird eine Zeilen- und Spaltenpivotsuche
|
|
! durchgefuehrt.
|
|
|
|
! Eingabeparameter:
|
|
!
|
|
! A(dimen,dimen) = Koeffizientenmatrix A
|
|
! B(dimen,dimen) = rechte Seiten B
|
|
!
|
|
! Ausgabeparameter:
|
|
!
|
|
! B(dimen,dimen) = Matrix der Unbekanntenvektoren X
|
|
! LogAbsDetA = 10-Logarithmus des Betrages der Determinanten von A
|
|
! NegHDK = Anzahl der negativen Hauptdiagonalkoeffizienten nach der
|
|
! Vorwaertszerlegung
|
|
! error = logical
|
|
! = false: Das Gleichungssystem wurde geloest.
|
|
! = true : Matrix A ist singulaer.
|
|
|
|
! A und B werden veraendert!
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
logical error
|
|
integer (pInt) dimen,NegHDK
|
|
real(pReal) LogAbsDetA
|
|
real(pReal) A(dimen,dimen), B(dimen,dimen)
|
|
|
|
INTENT (IN) dimen
|
|
INTENT (OUT) LogAbsDetA, NegHDK, error
|
|
INTENT (INOUT) A, B
|
|
|
|
LOGICAL SortX
|
|
integer (pInt) PivotZeile, PivotSpalte, StoreI, I, IP1, J, K, L
|
|
integer (pInt) XNr(dimen)
|
|
real(pReal) AbsA, PivotWert, EpsAbs, Quote
|
|
real(pReal) StoreA(dimen), StoreB(dimen)
|
|
|
|
error = .true.
|
|
NegHDK = 1
|
|
SortX = .FALSE.
|
|
|
|
! Unbekanntennumerierung
|
|
|
|
DO I = 1, dimen
|
|
XNr(I) = I
|
|
ENDDO
|
|
|
|
! Genauigkeitsschranke und Bestimmung des groessten Pivotelementes
|
|
|
|
PivotWert = ABS(A(1,1))
|
|
PivotZeile = 1
|
|
PivotSpalte = 1
|
|
|
|
DO I = 1, dimen
|
|
DO J = 1, dimen
|
|
AbsA = ABS(A(I,J))
|
|
IF (AbsA .GT. PivotWert) THEN
|
|
PivotWert = AbsA
|
|
PivotZeile = I
|
|
PivotSpalte = J
|
|
ENDIF
|
|
ENDDO
|
|
ENDDO
|
|
|
|
IF (PivotWert .LT. 0.0000001) RETURN ! Pivotelement = 0?
|
|
|
|
EpsAbs = PivotWert * 0.1_pReal ** PRECISION(1.0_pReal)
|
|
|
|
! V O R W A E R T S T R I A N G U L A T I O N
|
|
|
|
DO I = 1, dimen - 1
|
|
! Zeilentausch?
|
|
IF (PivotZeile .NE. I) THEN
|
|
StoreA(I:dimen) = A(I,I:dimen)
|
|
A(I,I:dimen) = A(PivotZeile,I:dimen)
|
|
A(PivotZeile,I:dimen) = StoreA(I:dimen)
|
|
StoreB(1:dimen) = B(I,1:dimen)
|
|
B(I,1:dimen) = B(PivotZeile,1:dimen)
|
|
B(PivotZeile,1:dimen) = StoreB(1:dimen)
|
|
SortX = .TRUE.
|
|
ENDIF
|
|
! Spaltentausch?
|
|
IF (PivotSpalte .NE. I) THEN
|
|
StoreA(1:dimen) = A(1:dimen,I)
|
|
A(1:dimen,I) = A(1:dimen,PivotSpalte)
|
|
A(1:dimen,PivotSpalte) = StoreA(1:dimen)
|
|
StoreI = XNr(I)
|
|
XNr(I) = XNr(PivotSpalte)
|
|
XNr(PivotSpalte) = StoreI
|
|
SortX = .TRUE.
|
|
ENDIF
|
|
! Triangulation
|
|
DO J = I + 1, dimen
|
|
Quote = A(J,I) / A(I,I)
|
|
DO K = I + 1, dimen
|
|
A(J,K) = A(J,K) - Quote * A(I,K)
|
|
ENDDO
|
|
DO K = 1, dimen
|
|
B(J,K) = B(J,K) - Quote * B(I,K)
|
|
ENDDO
|
|
ENDDO
|
|
! Bestimmung des groessten Pivotelementes
|
|
IP1 = I + 1
|
|
PivotWert = ABS(A(IP1,IP1))
|
|
PivotZeile = IP1
|
|
PivotSpalte = IP1
|
|
DO J = IP1, dimen
|
|
DO K = IP1, dimen
|
|
AbsA = ABS(A(J,K))
|
|
IF (AbsA .GT. PivotWert) THEN
|
|
PivotWert = AbsA
|
|
PivotZeile = J
|
|
PivotSpalte = K
|
|
ENDIF
|
|
ENDDO
|
|
ENDDO
|
|
|
|
IF (PivotWert .LT. EpsAbs) RETURN ! Pivotelement = 0?
|
|
|
|
ENDDO
|
|
|
|
! R U E C K W A E R T S A U F L O E S U N G
|
|
|
|
DO I = dimen, 1, -1
|
|
DO L = 1, dimen
|
|
DO J = I + 1, dimen
|
|
B(I,L) = B(I,L) - A(I,J) * B(J,L)
|
|
ENDDO
|
|
B(I,L) = B(I,L) / A(I,I)
|
|
ENDDO
|
|
ENDDO
|
|
|
|
! Sortieren der Unbekanntenvektoren?
|
|
|
|
IF (SortX) THEN
|
|
DO L = 1, dimen
|
|
StoreA(1:dimen) = B(1:dimen,L)
|
|
DO I = 1, dimen
|
|
J = XNr(I)
|
|
B(J,L) = StoreA(I)
|
|
ENDDO
|
|
ENDDO
|
|
ENDIF
|
|
|
|
! Determinante
|
|
|
|
LogAbsDetA = 0.0_pReal
|
|
NegHDK = 0
|
|
|
|
DO I = 1, dimen
|
|
IF (A(I,I) .LT. 0.0_pReal) NegHDK = NegHDK + 1
|
|
AbsA = ABS(A(I,I))
|
|
LogAbsDetA = LogAbsDetA + LOG10(AbsA)
|
|
ENDDO
|
|
|
|
error = .false.
|
|
|
|
RETURN
|
|
|
|
ENDSUBROUTINE Gauss
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! symmetrize a 3x3 matrix
|
|
!********************************************************************
|
|
function math_symmetric3x3(m)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3) :: math_symmetric3x3,m
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:3,j=1:3) math_symmetric3x3(i,j) = 0.5_pReal * (m(i,j) + m(j,i))
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! symmetrize a 6x6 matrix
|
|
!********************************************************************
|
|
pure function math_symmetric6x6(m)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
integer(pInt) i,j
|
|
real(pReal), dimension(6,6), intent(in) :: m
|
|
real(pReal), dimension(6,6) :: math_symmetric6x6
|
|
|
|
forall (i=1:6,j=1:6) math_symmetric6x6(i,j) = 0.5_pReal * (m(i,j) + m(j,i))
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! equivalent scalar quantity of a full strain tensor
|
|
!********************************************************************
|
|
pure function math_equivStrain33(m)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3), intent(in) :: m
|
|
real(pReal) math_equivStrain33,e11,e22,e33,s12,s23,s31
|
|
|
|
e11 = (2.0_pReal*m(1,1)-m(2,2)-m(3,3))/3.0_pReal
|
|
e22 = (2.0_pReal*m(2,2)-m(3,3)-m(1,1))/3.0_pReal
|
|
e33 = (2.0_pReal*m(3,3)-m(1,1)-m(2,2))/3.0_pReal
|
|
s12 = 2.0_pReal*m(1,2)
|
|
s23 = 2.0_pReal*m(2,3)
|
|
s31 = 2.0_pReal*m(3,1)
|
|
|
|
math_equivStrain33 = 2.0_pReal*(1.50_pReal*(e11**2.0_pReal+e22**2.0_pReal+e33**2.0_pReal) + &
|
|
0.75_pReal*(s12**2.0_pReal+s23**2.0_pReal+s31**2.0_pReal))**(0.5_pReal)/3.0_pReal
|
|
return
|
|
|
|
endfunction
|
|
|
|
!********************************************************************
|
|
! determinant of a 3x3 matrix
|
|
!********************************************************************
|
|
pure function math_det3x3(m)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3), intent(in) :: m
|
|
real(pReal) math_det3x3
|
|
|
|
math_det3x3 = m(1,1)*(m(2,2)*m(3,3)-m(2,3)*m(3,2)) &
|
|
-m(1,2)*(m(2,1)*m(3,3)-m(2,3)*m(3,1)) &
|
|
+m(1,3)*(m(2,1)*m(3,2)-m(2,2)*m(3,1))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! euclidic norm of a 3x1 vector
|
|
!********************************************************************
|
|
pure function math_norm3(v)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: v
|
|
real(pReal) math_norm3
|
|
|
|
math_norm3 = sqrt(v(1)*v(1) + v(2)*v(2) + v(3)*v(3))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! convert 3x3 matrix into vector 9x1
|
|
!********************************************************************
|
|
pure function math_Plain33to9(m33)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3), intent(in) :: m33
|
|
real(pReal), dimension(9) :: math_Plain33to9
|
|
integer(pInt) i
|
|
|
|
forall (i=1:9) math_Plain33to9(i) = m33(mapPlain(1,i),mapPlain(2,i))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! convert Plain 9x1 back to 3x3 matrix
|
|
!********************************************************************
|
|
pure function math_Plain9to33(v9)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(9), intent(in) :: v9
|
|
real(pReal), dimension(3,3) :: math_Plain9to33
|
|
integer(pInt) i
|
|
|
|
forall (i=1:9) math_Plain9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i)
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! convert symmetric 3x3 matrix into Mandel vector 6x1
|
|
!********************************************************************
|
|
pure function math_Mandel33to6(m33)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3), intent(in) :: m33
|
|
real(pReal), dimension(6) :: math_Mandel33to6
|
|
integer(pInt) i
|
|
|
|
forall (i=1:6) math_Mandel33to6(i) = nrmMandel(i)*m33(mapMandel(1,i),mapMandel(2,i))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! convert Mandel 6x1 back to symmetric 3x3 matrix
|
|
!********************************************************************
|
|
pure function math_Mandel6to33(v6)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(6), intent(in) :: v6
|
|
real(pReal), dimension(3,3) :: math_Mandel6to33
|
|
integer(pInt) i
|
|
|
|
forall (i=1:6)
|
|
math_Mandel6to33(mapMandel(1,i),mapMandel(2,i)) = invnrmMandel(i)*v6(i)
|
|
math_Mandel6to33(mapMandel(2,i),mapMandel(1,i)) = invnrmMandel(i)*v6(i)
|
|
end forall
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! convert 3x3x3x3 tensor into plain matrix 9x9
|
|
!********************************************************************
|
|
pure function math_Plain3333to99(m3333)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3,3,3), intent(in) :: m3333
|
|
real(pReal), dimension(9,9) :: math_Plain3333to99
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:9,j=1:9) math_Plain3333to99(i,j) = &
|
|
m3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j))
|
|
return
|
|
|
|
endfunction
|
|
|
|
!********************************************************************
|
|
! plain matrix 9x9 into 3x3x3x3 tensor
|
|
!********************************************************************
|
|
pure function math_Plain99to3333(m99)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(9,9), intent(in) :: m99
|
|
real(pReal), dimension(3,3,3,3) :: math_Plain99to3333
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:9,j=1:9) math_Plain99to3333(mapPlain(1,i),mapPlain(2,i),&
|
|
mapPlain(1,j),mapPlain(2,j)) = m99(i,j)
|
|
return
|
|
|
|
endfunction
|
|
|
|
!********************************************************************
|
|
! convert symmetric 3x3x3x3 tensor into Mandel matrix 6x6
|
|
!********************************************************************
|
|
pure function math_Mandel3333to66(m3333)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3,3,3,3), intent(in) :: m3333
|
|
real(pReal), dimension(6,6) :: math_Mandel3333to66
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:6,j=1:6) math_Mandel3333to66(i,j) = &
|
|
nrmMandel(i)*nrmMandel(j)*m3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! convert Mandel matrix 6x6 back to symmetric 3x3x3x3 tensor
|
|
!********************************************************************
|
|
pure function math_Mandel66to3333(m66)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(6,6), intent(in) :: m66
|
|
real(pReal), dimension(3,3,3,3) :: math_Mandel66to3333
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:6,j=1:6)
|
|
math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j)
|
|
math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(1,j),mapMandel(2,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j)
|
|
math_Mandel66to3333(mapMandel(1,i),mapMandel(2,i),mapMandel(2,j),mapMandel(1,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j)
|
|
math_Mandel66to3333(mapMandel(2,i),mapMandel(1,i),mapMandel(2,j),mapMandel(1,j)) = invnrmMandel(i)*invnrmMandel(j)*m66(i,j)
|
|
end forall
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! convert Voigt matrix 6x6 back to symmetric 3x3x3x3 tensor
|
|
!********************************************************************
|
|
pure function math_Voigt66to3333(m66)
|
|
|
|
use prec, only: pReal,pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(6,6), intent(in) :: m66
|
|
real(pReal), dimension(3,3,3,3) :: math_Voigt66to3333
|
|
integer(pInt) i,j
|
|
|
|
forall (i=1:6,j=1:6)
|
|
math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(1,j),mapVoigt(2,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j)
|
|
math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(1,j),mapVoigt(2,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j)
|
|
math_Voigt66to3333(mapVoigt(1,i),mapVoigt(2,i),mapVoigt(2,j),mapVoigt(1,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j)
|
|
math_Voigt66to3333(mapVoigt(2,i),mapVoigt(1,i),mapVoigt(2,j),mapVoigt(1,j)) = invnrmVoigt(i)*invnrmVoigt(j)*m66(i,j)
|
|
end forall
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! Euler angles (in radians) from rotation matrix
|
|
!********************************************************************
|
|
pure function math_RtoEuler(R)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension (3,3), intent(in) :: R
|
|
real(pReal), dimension(3) :: math_RtoEuler
|
|
real(pReal) sqhkl, squvw, sqhk, val
|
|
|
|
sqhkl=sqrt(R(1,3)*R(1,3)+R(2,3)*R(2,3)+R(3,3)*R(3,3))
|
|
squvw=sqrt(R(1,1)*R(1,1)+R(2,1)*R(2,1)+R(3,1)*R(3,1))
|
|
sqhk=sqrt(R(1,3)*R(1,3)+R(2,3)*R(2,3))
|
|
! calculate PHI
|
|
val=R(3,3)/sqhkl
|
|
|
|
if(val > 1.0_pReal) val = 1.0_pReal
|
|
if(val < -1.0_pReal) val = -1.0_pReal
|
|
|
|
math_RtoEuler(2) = acos(val)
|
|
|
|
if(math_RtoEuler(2) < 1.0e-8_pReal) then
|
|
! calculate phi2
|
|
math_RtoEuler(3) = 0.0_pReal
|
|
! calculate phi1
|
|
val=R(1,1)/squvw
|
|
if(val > 1.0_pReal) val = 1.0_pReal
|
|
if(val < -1.0_pReal) val = -1.0_pReal
|
|
|
|
math_RtoEuler(1) = acos(val)
|
|
if(R(2,1) > 0.0_pReal) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1)
|
|
else
|
|
! calculate phi2
|
|
val=R(2,3)/sqhk
|
|
if(val > 1.0_pReal) val = 1.0_pReal
|
|
if(val < -1.0_pReal) val = -1.0_pReal
|
|
|
|
math_RtoEuler(3) = acos(val)
|
|
if(R(1,3) < 0.0) math_RtoEuler(3) = 2.0_pReal*pi-math_RtoEuler(3)
|
|
! calculate phi1
|
|
val=-R(3,2)/sin(math_RtoEuler(2))
|
|
if(val > 1.0_pReal) val = 1.0_pReal
|
|
if(val < -1.0_pReal) val = -1.0_pReal
|
|
|
|
math_RtoEuler(1) = acos(val)
|
|
if(R(3,1) < 0.0) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1)
|
|
end if
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! quaternion (w+ix+jy+kz) from orientation matrix
|
|
!********************************************************************
|
|
pure function math_RtoQuaternion(R)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension (3,3), intent(in) :: R
|
|
real(pReal), dimension(4) :: absQ,math_RtoQuaternion
|
|
integer(pInt), dimension(1) :: largest
|
|
|
|
! math adopted from http://code.google.com/p/mtex/source/browse/trunk/geometry/geometry_tools/mat2quat.m
|
|
|
|
math_RtoQuaternion = 0.0_pReal
|
|
|
|
absQ(1) = 0.5_pReal * sqrt(1.0_pReal+R(1,1)+R(2,2)+R(3,3))
|
|
absQ(2) = 0.5_pReal * sqrt(1.0_pReal+R(1,1)-R(2,2)-R(3,3))
|
|
absQ(3) = 0.5_pReal * sqrt(1.0_pReal-R(1,1)+R(2,2)-R(3,3))
|
|
absQ(4) = 0.5_pReal * sqrt(1.0_pReal-R(1,1)-R(2,2)+R(3,3))
|
|
|
|
largest = maxloc(absQ)
|
|
select case(largest(1))
|
|
case (1)
|
|
|
|
math_RtoQuaternion(2) = R(2,3)-R(3,2)
|
|
math_RtoQuaternion(3) = R(3,1)-R(1,3)
|
|
math_RtoQuaternion(4) = R(1,2)-R(2,1)
|
|
|
|
case (2)
|
|
math_RtoQuaternion(1) = R(2,3)-R(3,2)
|
|
|
|
math_RtoQuaternion(3) = R(1,2)+R(2,1)
|
|
math_RtoQuaternion(4) = R(3,1)+R(1,3)
|
|
|
|
case (3)
|
|
math_RtoQuaternion(1) = R(3,1)-R(1,3)
|
|
math_RtoQuaternion(2) = R(1,2)+R(2,1)
|
|
|
|
math_RtoQuaternion(4) = R(2,3)+R(3,2)
|
|
|
|
case (4)
|
|
math_RtoQuaternion (1) = R(1,2)-R(2,1)
|
|
math_RtoQuaternion (2) = R(3,1)+R(1,3)
|
|
math_RtoQuaternion (3) = R(3,2)+R(2,3)
|
|
|
|
end select
|
|
|
|
math_RtoQuaternion = math_RtoQuaternion*0.25_pReal/maxval(absQ)
|
|
math_RtoQuaternion(largest(1)) = maxval(absQ)
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!****************************************************************
|
|
! rotation matrix from Euler angles (in radians)
|
|
!****************************************************************
|
|
pure function math_EulerToR (Euler)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: Euler
|
|
real(pReal), dimension(3,3) :: math_EulerToR
|
|
real(pReal) c1, c, c2, s1, s, s2
|
|
|
|
C1 = cos(Euler(1))
|
|
C = cos(Euler(2))
|
|
C2 = cos(Euler(3))
|
|
S1 = sin(Euler(1))
|
|
S = sin(Euler(2))
|
|
S2 = sin(Euler(3))
|
|
math_EulerToR(1,1)=C1*C2-S1*S2*C
|
|
math_EulerToR(1,2)=S1*C2+C1*S2*C
|
|
math_EulerToR(1,3)=S2*S
|
|
math_EulerToR(2,1)=-C1*S2-S1*C2*C
|
|
math_EulerToR(2,2)=-S1*S2+C1*C2*C
|
|
math_EulerToR(2,3)=C2*S
|
|
math_EulerToR(3,1)=S1*S
|
|
math_EulerToR(3,2)=-C1*S
|
|
math_EulerToR(3,3)=C
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! quaternion (w+ix+jy+kz) from 3-1-3 Euler angles (in radians)
|
|
!********************************************************************
|
|
pure function math_EulerToQuaternion(eulerangles)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: eulerangles
|
|
real(pReal), dimension(4) :: math_EulerToQuaternion
|
|
real(pReal), dimension(3) :: halfangles
|
|
real(pReal) c, s
|
|
|
|
halfangles = 0.5_pReal * eulerangles
|
|
|
|
c = cos(halfangles(2))
|
|
s = sin(halfangles(2))
|
|
|
|
math_EulerToQuaternion(1) = cos(halfangles(1)+halfangles(3)) * c
|
|
math_EulerToQuaternion(2) = cos(halfangles(1)-halfangles(3)) * s
|
|
math_EulerToQuaternion(3) = sin(halfangles(1)-halfangles(3)) * s
|
|
math_EulerToQuaternion(4) = sin(halfangles(1)+halfangles(3)) * c
|
|
|
|
endfunction
|
|
|
|
|
|
!****************************************************************
|
|
! rotation matrix from axis and angle (in radians)
|
|
!****************************************************************
|
|
pure function math_AxisAngleToR(axis,omega)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: axis
|
|
real(pReal), intent(in) :: omega
|
|
real(pReal), dimension(3) :: axisNrm
|
|
real(pReal), dimension(3,3) :: math_AxisAngleToR
|
|
real(pReal) norm,s,c,c1
|
|
integer(pInt) i
|
|
|
|
norm = sqrt(math_mul3x3(axis,axis))
|
|
if (norm > 1.0e-8_pReal) then ! non-zero rotation
|
|
forall (i=1:3) axisNrm(i) = axis(i)/norm ! normalize axis to be sure
|
|
|
|
s = sin(omega)
|
|
c = cos(omega)
|
|
c1 = 1.0_pReal - c
|
|
|
|
! formula for active rotation taken from http://mathworld.wolfram.com/RodriguesRotationFormula.html
|
|
! below is transposed form to get passive rotation
|
|
|
|
math_AxisAngleToR(1,1) = c + c1*axisNrm(1)**2
|
|
math_AxisAngleToR(2,1) = -s*axisNrm(3) + c1*axisNrm(1)*axisNrm(2)
|
|
math_AxisAngleToR(3,1) = s*axisNrm(2) + c1*axisNrm(1)*axisNrm(3)
|
|
|
|
math_AxisAngleToR(1,2) = s*axisNrm(3) + c1*axisNrm(2)*axisNrm(1)
|
|
math_AxisAngleToR(2,2) = c + c1*axisNrm(2)**2
|
|
math_AxisAngleToR(3,2) = -s*axisNrm(1) + c1*axisNrm(2)*axisNrm(3)
|
|
|
|
math_AxisAngleToR(1,3) = -s*axisNrm(2) + c1*axisNrm(3)*axisNrm(1)
|
|
math_AxisAngleToR(2,3) = s*axisNrm(1) + c1*axisNrm(3)*axisNrm(2)
|
|
math_AxisAngleToR(3,3) = c + c1*axisNrm(3)**2
|
|
else
|
|
math_AxisAngleToR = math_I3
|
|
endif
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!****************************************************************
|
|
! quaternion (w+ix+jy+kz) from axis and angle (in radians)
|
|
!****************************************************************
|
|
pure function math_AxisAngleToQuaternion(axis,omega)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: axis
|
|
real(pReal), intent(in) :: omega
|
|
real(pReal), dimension(3) :: axisNrm
|
|
real(pReal), dimension(4) :: math_AxisAngleToQuaternion
|
|
real(pReal) s,c,norm
|
|
integer(pInt) i
|
|
|
|
norm = sqrt(math_mul3x3(axis,axis))
|
|
if (norm > 1.0e-8_pReal) then ! non-zero rotation
|
|
forall (i=1:3) axisNrm(i) = axis(i)/norm ! normalize axis to be sure
|
|
! formula taken from http://en.wikipedia.org/wiki/Rotation_representation_%28mathematics%29#Rodrigues_parameters
|
|
s = sin(omega/2.0_pReal)
|
|
c = cos(omega/2.0_pReal)
|
|
math_AxisAngleToQuaternion(1) = c
|
|
math_AxisAngleToQuaternion(2:4) = s * axisNrm(1:3)
|
|
else
|
|
math_AxisAngleToQuaternion = (/1.0_pReal,0.0_pReal,0.0_pReal,0.0_pReal/) ! no rotation
|
|
endif
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! orientation matrix from quaternion (w+ix+jy+kz)
|
|
!********************************************************************
|
|
pure function math_QuaternionToR(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(3,3) :: math_QuaternionToR, T,S
|
|
real(pReal) w2
|
|
integer(pInt) i, j
|
|
|
|
forall (i = 1:3, j = 1:3) &
|
|
T(i,j) = Q(i+1) * Q(j+1)
|
|
S = reshape( (/0.0_pReal, Q(4), -Q(3), &
|
|
-Q(4),0.0_pReal, +Q(2), &
|
|
Q(3), -Q(2),0.0_pReal/),(/3,3/)) ! notation is transposed!
|
|
|
|
math_QuaternionToR = (2.0_pReal * Q(1)*Q(1) - 1.0_pReal) * math_I3 + &
|
|
2.0_pReal * T - &
|
|
2.0_pReal * Q(1) * S
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! 3-1-3 Euler angles (in radians) from quaternion (w+ix+jy+kz)
|
|
!********************************************************************
|
|
pure function math_QuaternionToEuler(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(3) :: math_QuaternionToEuler
|
|
|
|
math_QuaternionToEuler(2) = acos(1.0_pReal-2.0_pReal*(Q(2)*Q(2)+Q(3)*Q(3)))
|
|
|
|
if (abs(math_QuaternionToEuler(2)) < 1.0e-3_pReal) then
|
|
math_QuaternionToEuler(1) = 2.0_pReal*acos(Q(1))
|
|
math_QuaternionToEuler(3) = 0.0_pReal
|
|
else
|
|
math_QuaternionToEuler(1) = atan2(Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)-Q(3)*Q(4))
|
|
if (math_QuaternionToEuler(1) < 0.0_pReal) &
|
|
math_QuaternionToEuler(1) = math_QuaternionToEuler(1) + 2.0_pReal * pi
|
|
|
|
math_QuaternionToEuler(3) = atan2(-Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)+Q(3)*Q(4))
|
|
if (math_QuaternionToEuler(3) < 0.0_pReal) &
|
|
math_QuaternionToEuler(3) = math_QuaternionToEuler(3) + 2.0_pReal * pi
|
|
endif
|
|
|
|
if (math_QuaternionToEuler(2) < 0.0_pReal) &
|
|
math_QuaternionToEuler(2) = math_QuaternionToEuler(2) + pi
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! axis-angle (x, y, z, ang in radians) from quaternion (w+ix+jy+kz)
|
|
!********************************************************************
|
|
pure function math_QuaternionToAxisAngle(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal) halfAngle, sinHalfAngle
|
|
real(pReal), dimension(4) :: math_QuaternionToAxisAngle
|
|
|
|
halfAngle = acos(max(-1.0_pReal, min(1.0_pReal, Q(1)))) ! limit to [-1,1] --> 0 to 180 deg
|
|
sinHalfAngle = sin(halfAngle)
|
|
|
|
if (sinHalfAngle <= 1.0e-4_pReal) then ! very small rotation angle?
|
|
math_QuaternionToAxisAngle = 0.0_pReal
|
|
else
|
|
math_QuaternionToAxisAngle(1:3) = Q(2:4)/sinHalfAngle
|
|
math_QuaternionToAxisAngle(4) = halfAngle*2.0_pReal
|
|
endif
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! Rodrigues vector (x, y, z) from unit quaternion (w+ix+jy+kz)
|
|
!********************************************************************
|
|
pure function math_QuaternionToRodrig(Q)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(4), intent(in) :: Q
|
|
real(pReal), dimension(3) :: math_QuaternionToRodrig
|
|
|
|
if (Q(1) /= 0.0_pReal) then ! unless rotation by 180 deg
|
|
math_QuaternionToRodrig = Q(2:4)/Q(1)
|
|
else
|
|
math_QuaternionToRodrig = NaN ! Rodrig is unbound for 180 deg...
|
|
endif
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! misorientation angle between two sets of Euler angles
|
|
!**************************************************************************
|
|
pure function math_EulerMisorientation(EulerA,EulerB)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3), intent(in) :: EulerA,EulerB
|
|
real(pReal), dimension(3,3) :: r
|
|
real(pReal) math_EulerMisorientation, tr
|
|
|
|
r = math_mul33x33(math_EulerToR(EulerB),transpose(math_EulerToR(EulerA)))
|
|
|
|
tr = (r(1,1)+r(2,2)+r(3,3)-1.0_pReal)*0.4999999_pReal
|
|
math_EulerMisorientation = abs(0.5_pReal*pi-asin(tr))
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! figures whether unit quat falls into stereographic standard triangle
|
|
!**************************************************************************
|
|
pure function math_QuaternionInSST(Q, symmetryType)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
!*** input variables
|
|
real(pReal), dimension(4), intent(in) :: Q ! orientation
|
|
integer(pInt), intent(in) :: symmetryType ! Type of crystal symmetry; 1:cubic, 2:hexagonal
|
|
|
|
!*** output variables
|
|
logical math_QuaternionInSST
|
|
|
|
!*** local variables
|
|
real(pReal), dimension(3) :: Rodrig ! Rodrigues vector of Q
|
|
|
|
Rodrig = math_QuaternionToRodrig(Q)
|
|
select case (symmetryType)
|
|
case (1)
|
|
math_QuaternionInSST = Rodrig(1) > Rodrig(2) .and. &
|
|
Rodrig(2) > Rodrig(3) .and. &
|
|
Rodrig(3) > 0.0_pReal
|
|
case (2)
|
|
math_QuaternionInSST = Rodrig(1) > sqrt(3.0_pReal)*Rodrig(2) .and. &
|
|
Rodrig(2) > 0.0_pReal .and. &
|
|
Rodrig(3) > 0.0_pReal
|
|
case default
|
|
math_QuaternionInSST = .true.
|
|
end select
|
|
|
|
endfunction
|
|
|
|
|
|
!**************************************************************************
|
|
! calculates the disorientation for 2 unit quaternions
|
|
!**************************************************************************
|
|
function math_QuaternionDisorientation(Q1, Q2, symmetryType)
|
|
|
|
use prec, only: pReal, pInt
|
|
use IO, only: IO_error
|
|
implicit none
|
|
|
|
!*** input variables
|
|
real(pReal), dimension(4), intent(in) :: Q1, & ! 1st orientation
|
|
Q2 ! 2nd orientation
|
|
integer(pInt), intent(in) :: symmetryType ! Type of crystal symmetry; 1:cubic, 2:hexagonal
|
|
|
|
!*** output variables
|
|
real(pReal), dimension(4) :: math_QuaternionDisorientation ! disorientation
|
|
|
|
!*** local variables
|
|
real(pReal), dimension(4) :: dQ,dQsymA,mis
|
|
integer(pInt) i,j,k,s
|
|
|
|
dQ = math_qMul(math_qConj(Q1),Q2)
|
|
math_QuaternionDisorientation = dQ
|
|
|
|
select case (symmetryType)
|
|
case (0)
|
|
if (math_QuaternionDisorientation(1) < 0.0_pReal) &
|
|
math_QuaternionDisorientation = -math_QuaternionDisorientation ! keep omega within 0 to 180 deg
|
|
|
|
case (1,2)
|
|
s = sum(math_NsymOperations(1:symmetryType-1))
|
|
do i = 1,2
|
|
dQ = math_qConj(dQ) ! switch order of "from -- to"
|
|
do j = 1,math_NsymOperations(symmetryType) ! run through first crystal's symmetries
|
|
dQsymA = math_qMul(math_symOperations(:,s+j),dQ) ! apply sym
|
|
do k = 1,math_NsymOperations(symmetryType) ! run through 2nd crystal's symmetries
|
|
mis = math_qMul(dQsymA,math_symOperations(:,s+k)) ! apply sym
|
|
if (mis(1) < 0.0_pReal) & ! want positive angle
|
|
mis = -mis
|
|
if (mis(1)-math_QuaternionDisorientation(1) > -1e-8_pReal .and. &
|
|
math_QuaternionInSST(mis,symmetryType)) &
|
|
math_QuaternionDisorientation = mis ! found better one
|
|
enddo; enddo; enddo
|
|
|
|
case default
|
|
call IO_error(550,symmetryType) ! complain about unknown symmetry
|
|
end select
|
|
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! draw a random sample from Euler space
|
|
!********************************************************************
|
|
function math_sampleRandomOri()
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3) :: math_sampleRandomOri, rnd
|
|
|
|
call halton(3,rnd)
|
|
math_sampleRandomOri(1) = rnd(1)*2.0_pReal*pi
|
|
math_sampleRandomOri(2) = acos(2.0_pReal*rnd(2)-1.0_pReal)
|
|
math_sampleRandomOri(3) = rnd(3)*2.0_pReal*pi
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! draw a random sample from Gauss component
|
|
! with noise (in radians) half-width
|
|
!********************************************************************
|
|
function math_sampleGaussOri(center,noise)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3) :: math_sampleGaussOri, center, disturb
|
|
real(pReal), dimension(3), parameter :: origin = (/0.0_pReal,0.0_pReal,0.0_pReal/)
|
|
real(pReal), dimension(5) :: rnd
|
|
real(pReal) noise,scatter,cosScatter
|
|
integer(pInt) i
|
|
|
|
if (noise==0.0) then
|
|
math_sampleGaussOri = center
|
|
return
|
|
endif
|
|
|
|
! Helming uses different distribution with Bessel functions
|
|
! therefore the gauss scatter width has to be scaled differently
|
|
scatter = 0.95_pReal * noise
|
|
cosScatter = cos(scatter)
|
|
|
|
do
|
|
call halton(5,rnd)
|
|
forall (i=1:3) rnd(i) = 2.0_pReal*rnd(i)-1.0_pReal ! expand 1:3 to range [-1,+1]
|
|
disturb(1) = scatter * rnd(1) ! phi1
|
|
disturb(2) = sign(1.0_pReal,rnd(2))*acos(cosScatter+(1.0_pReal-cosScatter)*rnd(4)) ! Phi
|
|
disturb(3) = scatter * rnd(2) ! phi2
|
|
if (rnd(5) <= exp(-1.0_pReal*(math_EulerMisorientation(origin,disturb)/scatter)**2)) exit
|
|
enddo
|
|
|
|
math_sampleGaussOri = math_RtoEuler(math_mul33x33(math_EulerToR(disturb),math_EulerToR(center)))
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
!********************************************************************
|
|
! draw a random sample from Fiber component
|
|
! with noise (in radians)
|
|
!********************************************************************
|
|
function math_sampleFiberOri(alpha,beta,noise)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), dimension(3) :: math_sampleFiberOri, fiberInC,fiberInS,axis
|
|
real(pReal), dimension(2) :: alpha,beta, rnd
|
|
real(pReal), dimension(3,3) :: oRot,fRot,pRot
|
|
real(pReal) noise, scatter, cos2Scatter, angle
|
|
integer(pInt), dimension(2,3), parameter :: rotMap = reshape((/2,3, 3,1, 1,2/),(/2,3/))
|
|
integer(pInt) i
|
|
|
|
! Helming uses different distribution with Bessel functions
|
|
! therefore the gauss scatter width has to be scaled differently
|
|
scatter = 0.95_pReal * noise
|
|
cos2Scatter = cos(2.0_pReal*scatter)
|
|
|
|
! fiber axis in crystal coordinate system
|
|
fiberInC(1)=sin(alpha(1))*cos(alpha(2))
|
|
fiberInC(2)=sin(alpha(1))*sin(alpha(2))
|
|
fiberInC(3)=cos(alpha(1))
|
|
! fiber axis in sample coordinate system
|
|
fiberInS(1)=sin(beta(1))*cos(beta(2))
|
|
fiberInS(2)=sin(beta(1))*sin(beta(2))
|
|
fiberInS(3)=cos(beta(1))
|
|
|
|
! ---# rotation matrix from sample to crystal system #---
|
|
angle = -acos(dot_product(fiberInC,fiberInS))
|
|
if(angle /= 0.0_pReal) then
|
|
! rotation axis between sample and crystal system (cross product)
|
|
forall(i=1:3) axis(i) = fiberInC(rotMap(1,i))*fiberInS(rotMap(2,i))-fiberInC(rotMap(2,i))*fiberInS(rotMap(1,i))
|
|
oRot = math_AxisAngleToR(math_vectorproduct(fiberInC,fiberInS),angle)
|
|
else
|
|
oRot = math_I3
|
|
end if
|
|
|
|
! ---# rotation matrix about fiber axis (random angle) #---
|
|
call halton(1,rnd)
|
|
fRot = math_AxisAngleToR(fiberInS,rnd(1)*2.0_pReal*pi)
|
|
|
|
! ---# rotation about random axis perpend to fiber #---
|
|
! random axis pependicular to fiber axis
|
|
call halton(2,axis)
|
|
if (fiberInS(3) /= 0.0_pReal) then
|
|
axis(3)=-(axis(1)*fiberInS(1)+axis(2)*fiberInS(2))/fiberInS(3)
|
|
else if(fiberInS(2) /= 0.0_pReal) then
|
|
axis(3)=axis(2)
|
|
axis(2)=-(axis(1)*fiberInS(1)+axis(3)*fiberInS(3))/fiberInS(2)
|
|
else if(fiberInS(1) /= 0.0_pReal) then
|
|
axis(3)=axis(1)
|
|
axis(1)=-(axis(2)*fiberInS(2)+axis(3)*fiberInS(3))/fiberInS(1)
|
|
end if
|
|
|
|
! scattered rotation angle
|
|
do
|
|
call halton(2,rnd)
|
|
angle = acos(cos2Scatter+(1.0_pReal-cos2Scatter)*rnd(1))
|
|
if (rnd(2) <= exp(-1.0_pReal*(angle/scatter)**2)) exit
|
|
enddo
|
|
call halton(1,rnd)
|
|
if (rnd(1) <= 0.5) angle = -angle
|
|
pRot = math_AxisAngleToR(axis,angle)
|
|
|
|
! ---# apply the three rotations #---
|
|
math_sampleFiberOri = math_RtoEuler(math_mul33x33(pRot,math_mul33x33(fRot,oRot)))
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! symmetric Euler angles for given symmetry string
|
|
! 'triclinic' or '', 'monoclinic', 'orthotropic'
|
|
!********************************************************************
|
|
pure function math_symmetricEulers(sym,Euler)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt), intent(in) :: sym
|
|
real(pReal), dimension(3), intent(in) :: Euler
|
|
real(pReal), dimension(3,3) :: math_symmetricEulers
|
|
integer(pInt) i,j
|
|
|
|
math_symmetricEulers(1,1) = pi+Euler(1)
|
|
math_symmetricEulers(2,1) = Euler(2)
|
|
math_symmetricEulers(3,1) = Euler(3)
|
|
|
|
math_symmetricEulers(1,2) = pi-Euler(1)
|
|
math_symmetricEulers(2,2) = pi-Euler(2)
|
|
math_symmetricEulers(3,2) = pi+Euler(3)
|
|
|
|
math_symmetricEulers(1,3) = 2.0_pReal*pi-Euler(1)
|
|
math_symmetricEulers(2,3) = pi-Euler(2)
|
|
math_symmetricEulers(3,3) = pi+Euler(3)
|
|
|
|
forall (i=1:3,j=1:3) math_symmetricEulers(j,i) = modulo(math_symmetricEulers(j,i),2.0_pReal*pi)
|
|
|
|
select case (sym)
|
|
case (4) ! all done
|
|
|
|
case (2) ! return only first
|
|
math_symmetricEulers(:,2:3) = 0.0_pReal
|
|
|
|
case default ! return blank
|
|
math_symmetricEulers = 0.0_pReal
|
|
end select
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!********************************************************************
|
|
! draw a random sample from Gauss variable
|
|
!********************************************************************
|
|
function math_sampleGaussVar(meanvalue, stddev, width)
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
!*** input variables
|
|
real(pReal), intent(in) :: meanvalue, & ! meanvalue of gauss distribution
|
|
stddev ! standard deviation of gauss distribution
|
|
real(pReal), intent(in), optional :: width ! width of considered values as multiples of standard deviation
|
|
|
|
!*** output variables
|
|
real(pReal) math_sampleGaussVar
|
|
|
|
!*** local variables
|
|
real(pReal), dimension(2) :: rnd ! random numbers
|
|
real(pReal) scatter, & ! normalized scatter around meanvalue
|
|
myWidth
|
|
|
|
if (stddev == 0.0) then
|
|
math_sampleGaussVar = meanvalue
|
|
return
|
|
endif
|
|
|
|
if (present(width)) then
|
|
myWidth = width
|
|
else
|
|
myWidth = 3.0_pReal ! use +-3*sigma as default value for scatter
|
|
endif
|
|
|
|
do
|
|
call halton(2, rnd)
|
|
scatter = myWidth * (2.0_pReal * rnd(1) - 1.0_pReal)
|
|
if (rnd(2) <= exp(-0.5_pReal * scatter ** 2.0_pReal)) & ! test if scattered value is drawn
|
|
exit
|
|
enddo
|
|
|
|
math_sampleGaussVar = scatter * stddev
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
!****************************************************************
|
|
pure subroutine math_pDecomposition(FE,U,R,error)
|
|
!-----FE = R.U
|
|
!****************************************************************
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), intent(in) :: FE(3,3)
|
|
real(pReal), intent(out) :: R(3,3), U(3,3)
|
|
logical, intent(out) :: error
|
|
real(pReal) CE(3,3),EW1,EW2,EW3,EB1(3,3),EB2(3,3),EB3(3,3),UI(3,3),det
|
|
|
|
error = .false.
|
|
ce = math_mul33x33(transpose(FE),FE)
|
|
|
|
CALL math_spectral1(CE,EW1,EW2,EW3,EB1,EB2,EB3)
|
|
U=sqrt(EW1)*EB1+sqrt(EW2)*EB2+sqrt(EW3)*EB3
|
|
call math_invert3x3(U,UI,det,error)
|
|
if (.not. error) R = math_mul33x33(FE,UI)
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
!**********************************************************************
|
|
pure subroutine math_spectral1(M,EW1,EW2,EW3,EB1,EB2,EB3)
|
|
!**** EIGENWERTE UND EIGENWERTBASIS DER SYMMETRISCHEN 3X3 MATRIX M
|
|
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), intent(in) :: M(3,3)
|
|
real(pReal), intent(out) :: EB1(3,3),EB2(3,3),EB3(3,3),EW1,EW2,EW3
|
|
real(pReal) HI1M,HI2M,HI3M,TOL,R,S,T,P,Q,RHO,PHI,Y1,Y2,Y3,D1,D2,D3
|
|
real(pReal) C1,C2,C3,M1(3,3),M2(3,3),M3(3,3),arg
|
|
TOL=1.e-14_pReal
|
|
CALL math_hi(M,HI1M,HI2M,HI3M)
|
|
R=-HI1M
|
|
S= HI2M
|
|
T=-HI3M
|
|
P=S-R**2.0_pReal/3.0_pReal
|
|
Q=2.0_pReal/27.0_pReal*R**3.0_pReal-R*S/3.0_pReal+T
|
|
EB1=0.0_pReal
|
|
EB2=0.0_pReal
|
|
EB3=0.0_pReal
|
|
IF((ABS(P).LT.TOL).AND.(ABS(Q).LT.TOL))THEN
|
|
! DREI GLEICHE EIGENWERTE
|
|
EW1=HI1M/3.0_pReal
|
|
EW2=EW1
|
|
EW3=EW1
|
|
! this is not really correct, but this way U is calculated
|
|
! correctly in PDECOMPOSITION (correct is EB?=I)
|
|
EB1(1,1)=1.0_pReal
|
|
EB2(2,2)=1.0_pReal
|
|
EB3(3,3)=1.0_pReal
|
|
ELSE
|
|
RHO=sqrt(-3.0_pReal*P**3.0_pReal)/9.0_pReal
|
|
arg=-Q/RHO/2.0_pReal
|
|
if(arg.GT.1) arg=1
|
|
if(arg.LT.-1) arg=-1
|
|
PHI=acos(arg)
|
|
Y1=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal)
|
|
Y2=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal+2.0_pReal/3.0_pReal*PI)
|
|
Y3=2*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal+4.0_pReal/3.0_pReal*PI)
|
|
EW1=Y1-R/3.0_pReal
|
|
EW2=Y2-R/3.0_pReal
|
|
EW3=Y3-R/3.0_pReal
|
|
C1=ABS(EW1-EW2)
|
|
C2=ABS(EW2-EW3)
|
|
C3=ABS(EW3-EW1)
|
|
|
|
IF(C1.LT.TOL) THEN
|
|
! EW1 is equal to EW2
|
|
D3=1.0_pReal/(EW3-EW1)/(EW3-EW2)
|
|
M1=M-EW1*math_I3
|
|
M2=M-EW2*math_I3
|
|
EB3=math_mul33x33(M1,M2)*D3
|
|
|
|
EB1=math_I3-EB3
|
|
! both EB2 and EW2 are set to zero so that they do not
|
|
! contribute to U in PDECOMPOSITION
|
|
EW2=0.0_pReal
|
|
ELSE IF(C2.LT.TOL) THEN
|
|
! EW2 is equal to EW3
|
|
D1=1.0_pReal/(EW1-EW2)/(EW1-EW3)
|
|
M2=M-math_I3*EW2
|
|
M3=M-math_I3*EW3
|
|
EB1=math_mul33x33(M2,M3)*D1
|
|
EB2=math_I3-EB1
|
|
! both EB3 and EW3 are set to zero so that they do not
|
|
! contribute to U in PDECOMPOSITION
|
|
EW3=0.0_pReal
|
|
ELSE IF(C3.LT.TOL) THEN
|
|
! EW1 is equal to EW3
|
|
D2=1.0_pReal/(EW2-EW1)/(EW2-EW3)
|
|
M1=M-math_I3*EW1
|
|
M3=M-math_I3*EW3
|
|
EB2=math_mul33x33(M1,M3)*D2
|
|
EB1=math_I3-EB2
|
|
! both EB3 and EW3 are set to zero so that they do not
|
|
! contribute to U in PDECOMPOSITION
|
|
EW3=0.0_pReal
|
|
ELSE
|
|
! all three eigenvectors are different
|
|
D1=1.0_pReal/(EW1-EW2)/(EW1-EW3)
|
|
D2=1.0_pReal/(EW2-EW1)/(EW2-EW3)
|
|
D3=1.0_pReal/(EW3-EW1)/(EW3-EW2)
|
|
M1=M-EW1*math_I3
|
|
M2=M-EW2*math_I3
|
|
M3=M-EW3*math_I3
|
|
EB1=math_mul33x33(M2,M3)*D1
|
|
EB2=math_mul33x33(M1,M3)*D2
|
|
EB3=math_mul33x33(M1,M2)*D3
|
|
|
|
END IF
|
|
END IF
|
|
RETURN
|
|
ENDSUBROUTINE
|
|
|
|
|
|
!**********************************************************************
|
|
!**** HAUPTINVARIANTEN HI1M, HI2M, HI3M DER 3X3 MATRIX M
|
|
|
|
PURE SUBROUTINE math_hi(M,HI1M,HI2M,HI3M)
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
real(pReal), intent(in) :: M(3,3)
|
|
real(pReal), intent(out) :: HI1M, HI2M, HI3M
|
|
|
|
HI1M=M(1,1)+M(2,2)+M(3,3)
|
|
HI2M=HI1M**2/2.0_pReal-(M(1,1)**2+M(2,2)**2+M(3,3)**2)/2.0_pReal-M(1,2)*M(2,1)-M(1,3)*M(3,1)-M(2,3)*M(3,2)
|
|
HI3M=math_det3x3(M)
|
|
! QUESTION: is 3rd equiv det(M) ?? if yes, use function math_det !agreed on YES
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
SUBROUTINE get_seed(seed)
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! GET_SEED returns a seed for the random number generator.
|
|
!
|
|
!
|
|
! Discussion:
|
|
!
|
|
! The seed depends on the current time, and ought to be (slightly)
|
|
! different every millisecond. Once the seed is obtained, a random
|
|
! number generator should be called a few times to further process
|
|
! the seed.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 27 June 2000
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Output, integer SEED, a pseudorandom seed value.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) seed
|
|
real(pReal) temp
|
|
character ( len = 10 ) time
|
|
character ( len = 8 ) today
|
|
integer(pInt) values(8)
|
|
character ( len = 5 ) zone
|
|
|
|
call date_and_time ( today, time, zone, values )
|
|
|
|
temp = 0.0D+00
|
|
|
|
temp = temp + dble ( values(2) - 1 ) / 11.0D+00
|
|
temp = temp + dble ( values(3) - 1 ) / 30.0D+00
|
|
temp = temp + dble ( values(5) ) / 23.0D+00
|
|
temp = temp + dble ( values(6) ) / 59.0D+00
|
|
temp = temp + dble ( values(7) ) / 59.0D+00
|
|
temp = temp + dble ( values(8) ) / 999.0D+00
|
|
temp = temp / 6.0D+00
|
|
|
|
if ( temp <= 0.0D+00 ) then
|
|
temp = 1.0D+00 / 3.0D+00
|
|
else if ( 1.0D+00 <= temp ) then
|
|
temp = 2.0D+00 / 3.0D+00
|
|
end if
|
|
|
|
seed = int ( dble ( huge ( 1 ) ) * temp , pInt)
|
|
!
|
|
! Never use a seed of 0 or maximum integer.
|
|
!
|
|
if ( seed == 0 ) then
|
|
seed = 1
|
|
end if
|
|
|
|
if ( seed == huge ( 1 ) ) then
|
|
seed = seed - 1
|
|
end if
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
subroutine halton ( ndim, r )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! HALTON computes the next element in the Halton sequence.
|
|
!
|
|
!
|
|
! Modified:
|
|
!
|
|
! 09 March 2003
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, integer NDIM, the dimension of the element.
|
|
!
|
|
! Output, real R(NDIM), the next element of the current Halton
|
|
! sequence.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, ONLY: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) ndim
|
|
|
|
integer(pInt) base(ndim)
|
|
real(pReal) r(ndim)
|
|
integer(pInt) seed
|
|
integer(pInt) value(1)
|
|
|
|
call halton_memory ( 'GET', 'SEED', 1, value )
|
|
seed = value(1)
|
|
|
|
call halton_memory ( 'GET', 'BASE', ndim, base )
|
|
|
|
call i_to_halton ( seed, base, ndim, r )
|
|
|
|
value(1) = 1
|
|
call halton_memory ( 'INC', 'SEED', 1, value )
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
subroutine halton_memory ( action, name, ndim, value )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! HALTON_MEMORY sets or returns quantities associated with the Halton sequence.
|
|
!
|
|
!
|
|
! Modified:
|
|
!
|
|
! 09 March 2003
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, character ( len = * ) ACTION, the desired action.
|
|
! 'GET' means get the value of a particular quantity.
|
|
! 'SET' means set the value of a particular quantity.
|
|
! 'INC' means increment the value of a particular quantity.
|
|
! (Only the SEED can be incremented.)
|
|
!
|
|
! Input, character ( len = * ) NAME, the name of the quantity.
|
|
! 'BASE' means the Halton base or bases.
|
|
! 'NDIM' means the spatial dimension.
|
|
! 'SEED' means the current Halton seed.
|
|
!
|
|
! Input/output, integer NDIM, the dimension of the quantity.
|
|
! If ACTION is 'SET' and NAME is 'BASE', then NDIM is input, and
|
|
! is the number of entries in VALUE to be put into BASE.
|
|
!
|
|
! Input/output, integer VALUE(NDIM), contains a value.
|
|
! If ACTION is 'SET', then on input, VALUE contains values to be assigned
|
|
! to the internal variable.
|
|
! If ACTION is 'GET', then on output, VALUE contains the values of
|
|
! the specified internal variable.
|
|
! If ACTION is 'INC', then on input, VALUE contains the increment to
|
|
! be added to the specified internal variable.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
character ( len = * ) action
|
|
integer(pInt), allocatable, save :: base(:)
|
|
logical, save :: first_call = .true.
|
|
integer(pInt) i
|
|
character ( len = * ) name
|
|
integer(pInt) ndim
|
|
integer(pInt), save :: ndim_save = 0
|
|
integer(pInt), save :: seed = 1
|
|
integer(pInt) value(*)
|
|
|
|
if ( first_call ) then
|
|
ndim_save = 1
|
|
allocate ( base(ndim_save) )
|
|
base(1) = 2
|
|
first_call = .false.
|
|
end if
|
|
!
|
|
! Set
|
|
!
|
|
if ( action(1:1) == 'S' .or. action(1:1) == 's' ) then
|
|
|
|
if ( name(1:1) == 'B' .or. name(1:1) == 'b' ) then
|
|
|
|
if ( ndim_save /= ndim ) then
|
|
deallocate ( base )
|
|
ndim_save = ndim
|
|
allocate ( base(ndim_save) )
|
|
end if
|
|
|
|
base(1:ndim) = value(1:ndim)
|
|
|
|
else if ( name(1:1) == 'N' .or. name(1:1) == 'n' ) then
|
|
|
|
if ( ndim_save /= value(1) ) then
|
|
deallocate ( base )
|
|
ndim_save = value(1)
|
|
allocate ( base(ndim_save) )
|
|
do i = 1, ndim_save
|
|
base(i) = prime ( i )
|
|
enddo
|
|
else
|
|
ndim_save = value(1)
|
|
end if
|
|
else if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then
|
|
seed = value(1)
|
|
end if
|
|
!
|
|
! Get
|
|
!
|
|
else if ( action(1:1) == 'G' .or. action(1:1) == 'g' ) then
|
|
if ( name(1:1) == 'B' .or. name(1:1) == 'b' ) then
|
|
if ( ndim /= ndim_save ) then
|
|
deallocate ( base )
|
|
ndim_save = ndim
|
|
allocate ( base(ndim_save) )
|
|
do i = 1, ndim_save
|
|
base(i) = prime(i)
|
|
enddo
|
|
end if
|
|
value(1:ndim_save) = base(1:ndim_save)
|
|
else if ( name(1:1) == 'N' .or. name(1:1) == 'n' ) then
|
|
value(1) = ndim_save
|
|
else if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then
|
|
value(1) = seed
|
|
end if
|
|
!
|
|
! Increment
|
|
!
|
|
else if ( action(1:1) == 'I' .or. action(1:1) == 'i' ) then
|
|
if ( name(1:1) == 'S' .or. name(1:1) == 's' ) then
|
|
seed = seed + value(1)
|
|
end if
|
|
end if
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
subroutine halton_ndim_set ( ndim )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! HALTON_NDIM_SET sets the dimension for a Halton sequence.
|
|
!
|
|
!
|
|
! Modified:
|
|
!
|
|
! 26 February 2001
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, integer NDIM, the dimension of the Halton vectors.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) ndim
|
|
integer(pInt) value(1)
|
|
|
|
value(1) = ndim
|
|
call halton_memory ( 'SET', 'NDIM', 1, value )
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
subroutine halton_seed_set ( seed )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! HALTON_SEED_SET sets the "seed" for the Halton sequence.
|
|
!
|
|
!
|
|
! Discussion:
|
|
!
|
|
! Calling HALTON repeatedly returns the elements of the
|
|
! Halton sequence in order, starting with element number 1.
|
|
! An internal counter, called SEED, keeps track of the next element
|
|
! to return. Each time the routine is called, the SEED-th element
|
|
! is computed, and then SEED is incremented by 1.
|
|
!
|
|
! To restart the Halton sequence, it is only necessary to reset
|
|
! SEED to 1. It might also be desirable to reset SEED to some other value.
|
|
! This routine allows the user to specify any value of SEED.
|
|
!
|
|
! The default value of SEED is 1, which restarts the Halton sequence.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 26 February 2001
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, integer SEED, the seed for the Halton sequence.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt), parameter :: ndim = 1
|
|
|
|
integer(pInt) seed
|
|
integer(pInt) value(ndim)
|
|
|
|
value(1) = seed
|
|
call halton_memory ( 'SET', 'SEED', ndim, value )
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
subroutine i_to_halton ( seed, base, ndim, r )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! I_TO_HALTON computes an element of a Halton sequence.
|
|
!
|
|
!
|
|
! Reference:
|
|
!
|
|
! J H Halton,
|
|
! On the efficiency of certain quasi-random sequences of points
|
|
! in evaluating multi-dimensional integrals,
|
|
! Numerische Mathematik,
|
|
! Volume 2, pages 84-90, 1960.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 26 February 2001
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, integer SEED, the index of the desired element.
|
|
! Only the absolute value of SEED is considered. SEED = 0 is allowed,
|
|
! and returns R = 0.
|
|
!
|
|
! Input, integer BASE(NDIM), the Halton bases, which should be
|
|
! distinct prime numbers. This routine only checks that each base
|
|
! is greater than 1.
|
|
!
|
|
! Input, integer NDIM, the dimension of the sequence.
|
|
!
|
|
! Output, real R(NDIM), the SEED-th element of the Halton sequence
|
|
! for the given bases.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, ONLY: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt) ndim
|
|
|
|
integer(pInt) base(ndim)
|
|
real(pReal) base_inv(ndim)
|
|
integer(pInt) digit(ndim)
|
|
integer(pInt) i
|
|
real(pReal) r(ndim)
|
|
integer(pInt) seed
|
|
integer(pInt) seed2(ndim)
|
|
|
|
seed2(1:ndim) = abs ( seed )
|
|
|
|
r(1:ndim) = 0.0_pReal
|
|
|
|
if ( any ( base(1:ndim) <= 1 ) ) then
|
|
!$OMP CRITICAL (write2out)
|
|
write ( *, '(a)' ) ' '
|
|
write ( *, '(a)' ) 'I_TO_HALTON - Fatal error!'
|
|
write ( *, '(a)' ) ' An input base BASE is <= 1!'
|
|
do i = 1, ndim
|
|
write ( *, '(i6,i6)' ) i, base(i)
|
|
enddo
|
|
call flush(6)
|
|
!$OMP END CRITICAL (write2out)
|
|
stop
|
|
end if
|
|
|
|
base_inv(1:ndim) = 1.0_pReal / real ( base(1:ndim), pReal )
|
|
|
|
do while ( any ( seed2(1:ndim) /= 0 ) )
|
|
digit(1:ndim) = mod ( seed2(1:ndim), base(1:ndim) )
|
|
r(1:ndim) = r(1:ndim) + real ( digit(1:ndim), pReal ) * base_inv(1:ndim)
|
|
base_inv(1:ndim) = base_inv(1:ndim) / real ( base(1:ndim), pReal )
|
|
seed2(1:ndim) = seed2(1:ndim) / base(1:ndim)
|
|
enddo
|
|
|
|
return
|
|
|
|
ENDSUBROUTINE
|
|
|
|
|
|
function prime ( n )
|
|
!
|
|
!*******************************************************************************
|
|
!
|
|
!! PRIME returns any of the first PRIME_MAX prime numbers.
|
|
!
|
|
!
|
|
! Note:
|
|
!
|
|
! PRIME_MAX is 1500, and the largest prime stored is 12553.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 21 June 2002
|
|
!
|
|
! Author:
|
|
!
|
|
! John Burkardt
|
|
!
|
|
! Reference:
|
|
!
|
|
! Milton Abramowitz and Irene Stegun,
|
|
! Handbook of Mathematical Functions,
|
|
! US Department of Commerce, 1964, pages 870-873.
|
|
!
|
|
! Daniel Zwillinger,
|
|
! CRC Standard Mathematical Tables and Formulae,
|
|
! 30th Edition,
|
|
! CRC Press, 1996, pages 95-98.
|
|
!
|
|
! Parameters:
|
|
!
|
|
! Input, integer N, the index of the desired prime number.
|
|
! N = -1 returns PRIME_MAX, the index of the largest prime available.
|
|
! N = 0 is legal, returning PRIME = 1.
|
|
! It should generally be true that 0 <= N <= PRIME_MAX.
|
|
!
|
|
! Output, integer PRIME, the N-th prime. If N is out of range, PRIME
|
|
! is returned as 0.
|
|
!
|
|
! Modified:
|
|
!
|
|
! 29 April 2005
|
|
!
|
|
! Author:
|
|
!
|
|
! Franz Roters
|
|
!
|
|
use prec, only: pReal, pInt
|
|
implicit none
|
|
|
|
integer(pInt), parameter :: prime_max = 1500
|
|
|
|
integer(pInt), save :: icall = 0
|
|
integer(pInt) n
|
|
integer(pInt), save, dimension ( prime_max ) :: npvec
|
|
integer(pInt) prime
|
|
|
|
if ( icall == 0 ) then
|
|
|
|
icall = 1
|
|
|
|
npvec(1:100) = (/&
|
|
2, 3, 5, 7, 11, 13, 17, 19, 23, 29, &
|
|
31, 37, 41, 43, 47, 53, 59, 61, 67, 71, &
|
|
73, 79, 83, 89, 97, 101, 103, 107, 109, 113, &
|
|
127, 131, 137, 139, 149, 151, 157, 163, 167, 173, &
|
|
179, 181, 191, 193, 197, 199, 211, 223, 227, 229, &
|
|
233, 239, 241, 251, 257, 263, 269, 271, 277, 281, &
|
|
283, 293, 307, 311, 313, 317, 331, 337, 347, 349, &
|
|
353, 359, 367, 373, 379, 383, 389, 397, 401, 409, &
|
|
419, 421, 431, 433, 439, 443, 449, 457, 461, 463, &
|
|
467, 479, 487, 491, 499, 503, 509, 521, 523, 541 /)
|
|
|
|
npvec(101:200) = (/ &
|
|
547, 557, 563, 569, 571, 577, 587, 593, 599, 601, &
|
|
607, 613, 617, 619, 631, 641, 643, 647, 653, 659, &
|
|
661, 673, 677, 683, 691, 701, 709, 719, 727, 733, &
|
|
739, 743, 751, 757, 761, 769, 773, 787, 797, 809, &
|
|
811, 821, 823, 827, 829, 839, 853, 857, 859, 863, &
|
|
877, 881, 883, 887, 907, 911, 919, 929, 937, 941, &
|
|
947, 953, 967, 971, 977, 983, 991, 997, 1009, 1013, &
|
|
1019, 1021, 1031, 1033, 1039, 1049, 1051, 1061, 1063, 1069, &
|
|
1087, 1091, 1093, 1097, 1103, 1109, 1117, 1123, 1129, 1151, &
|
|
1153, 1163, 1171, 1181, 1187, 1193, 1201, 1213, 1217, 1223 /)
|
|
|
|
npvec(201:300) = (/ &
|
|
1229, 1231, 1237, 1249, 1259, 1277, 1279, 1283, 1289, 1291, &
|
|
1297, 1301, 1303, 1307, 1319, 1321, 1327, 1361, 1367, 1373, &
|
|
1381, 1399, 1409, 1423, 1427, 1429, 1433, 1439, 1447, 1451, &
|
|
1453, 1459, 1471, 1481, 1483, 1487, 1489, 1493, 1499, 1511, &
|
|
1523, 1531, 1543, 1549, 1553, 1559, 1567, 1571, 1579, 1583, &
|
|
1597, 1601, 1607, 1609, 1613, 1619, 1621, 1627, 1637, 1657, &
|
|
1663, 1667, 1669, 1693, 1697, 1699, 1709, 1721, 1723, 1733, &
|
|
1741, 1747, 1753, 1759, 1777, 1783, 1787, 1789, 1801, 1811, &
|
|
1823, 1831, 1847, 1861, 1867, 1871, 1873, 1877, 1879, 1889, &
|
|
1901, 1907, 1913, 1931, 1933, 1949, 1951, 1973, 1979, 1987 /)
|
|
|
|
npvec(301:400) = (/ &
|
|
1993, 1997, 1999, 2003, 2011, 2017, 2027, 2029, 2039, 2053, &
|
|
2063, 2069, 2081, 2083, 2087, 2089, 2099, 2111, 2113, 2129, &
|
|
2131, 2137, 2141, 2143, 2153, 2161, 2179, 2203, 2207, 2213, &
|
|
2221, 2237, 2239, 2243, 2251, 2267, 2269, 2273, 2281, 2287, &
|
|
2293, 2297, 2309, 2311, 2333, 2339, 2341, 2347, 2351, 2357, &
|
|
2371, 2377, 2381, 2383, 2389, 2393, 2399, 2411, 2417, 2423, &
|
|
2437, 2441, 2447, 2459, 2467, 2473, 2477, 2503, 2521, 2531, &
|
|
2539, 2543, 2549, 2551, 2557, 2579, 2591, 2593, 2609, 2617, &
|
|
2621, 2633, 2647, 2657, 2659, 2663, 2671, 2677, 2683, 2687, &
|
|
2689, 2693, 2699, 2707, 2711, 2713, 2719, 2729, 2731, 2741 /)
|
|
|
|
npvec(401:500) = (/ &
|
|
2749, 2753, 2767, 2777, 2789, 2791, 2797, 2801, 2803, 2819, &
|
|
2833, 2837, 2843, 2851, 2857, 2861, 2879, 2887, 2897, 2903, &
|
|
2909, 2917, 2927, 2939, 2953, 2957, 2963, 2969, 2971, 2999, &
|
|
3001, 3011, 3019, 3023, 3037, 3041, 3049, 3061, 3067, 3079, &
|
|
3083, 3089, 3109, 3119, 3121, 3137, 3163, 3167, 3169, 3181, &
|
|
3187, 3191, 3203, 3209, 3217, 3221, 3229, 3251, 3253, 3257, &
|
|
3259, 3271, 3299, 3301, 3307, 3313, 3319, 3323, 3329, 3331, &
|
|
3343, 3347, 3359, 3361, 3371, 3373, 3389, 3391, 3407, 3413, &
|
|
3433, 3449, 3457, 3461, 3463, 3467, 3469, 3491, 3499, 3511, &
|
|
3517, 3527, 3529, 3533, 3539, 3541, 3547, 3557, 3559, 3571 /)
|
|
|
|
npvec(501:600) = (/ &
|
|
3581, 3583, 3593, 3607, 3613, 3617, 3623, 3631, 3637, 3643, &
|
|
3659, 3671, 3673, 3677, 3691, 3697, 3701, 3709, 3719, 3727, &
|
|
3733, 3739, 3761, 3767, 3769, 3779, 3793, 3797, 3803, 3821, &
|
|
3823, 3833, 3847, 3851, 3853, 3863, 3877, 3881, 3889, 3907, &
|
|
3911, 3917, 3919, 3923, 3929, 3931, 3943, 3947, 3967, 3989, &
|
|
4001, 4003, 4007, 4013, 4019, 4021, 4027, 4049, 4051, 4057, &
|
|
4073, 4079, 4091, 4093, 4099, 4111, 4127, 4129, 4133, 4139, &
|
|
4153, 4157, 4159, 4177, 4201, 4211, 4217, 4219, 4229, 4231, &
|
|
4241, 4243, 4253, 4259, 4261, 4271, 4273, 4283, 4289, 4297, &
|
|
4327, 4337, 4339, 4349, 4357, 4363, 4373, 4391, 4397, 4409 /)
|
|
|
|
npvec(601:700) = (/ &
|
|
4421, 4423, 4441, 4447, 4451, 4457, 4463, 4481, 4483, 4493, &
|
|
4507, 4513, 4517, 4519, 4523, 4547, 4549, 4561, 4567, 4583, &
|
|
4591, 4597, 4603, 4621, 4637, 4639, 4643, 4649, 4651, 4657, &
|
|
4663, 4673, 4679, 4691, 4703, 4721, 4723, 4729, 4733, 4751, &
|
|
4759, 4783, 4787, 4789, 4793, 4799, 4801, 4813, 4817, 4831, &
|
|
4861, 4871, 4877, 4889, 4903, 4909, 4919, 4931, 4933, 4937, &
|
|
4943, 4951, 4957, 4967, 4969, 4973, 4987, 4993, 4999, 5003, &
|
|
5009, 5011, 5021, 5023, 5039, 5051, 5059, 5077, 5081, 5087, &
|
|
5099, 5101, 5107, 5113, 5119, 5147, 5153, 5167, 5171, 5179, &
|
|
5189, 5197, 5209, 5227, 5231, 5233, 5237, 5261, 5273, 5279 /)
|
|
|
|
npvec(701:800) = (/ &
|
|
5281, 5297, 5303, 5309, 5323, 5333, 5347, 5351, 5381, 5387, &
|
|
5393, 5399, 5407, 5413, 5417, 5419, 5431, 5437, 5441, 5443, &
|
|
5449, 5471, 5477, 5479, 5483, 5501, 5503, 5507, 5519, 5521, &
|
|
5527, 5531, 5557, 5563, 5569, 5573, 5581, 5591, 5623, 5639, &
|
|
5641, 5647, 5651, 5653, 5657, 5659, 5669, 5683, 5689, 5693, &
|
|
5701, 5711, 5717, 5737, 5741, 5743, 5749, 5779, 5783, 5791, &
|
|
5801, 5807, 5813, 5821, 5827, 5839, 5843, 5849, 5851, 5857, &
|
|
5861, 5867, 5869, 5879, 5881, 5897, 5903, 5923, 5927, 5939, &
|
|
5953, 5981, 5987, 6007, 6011, 6029, 6037, 6043, 6047, 6053, &
|
|
6067, 6073, 6079, 6089, 6091, 6101, 6113, 6121, 6131, 6133 /)
|
|
|
|
npvec(801:900) = (/ &
|
|
6143, 6151, 6163, 6173, 6197, 6199, 6203, 6211, 6217, 6221, &
|
|
6229, 6247, 6257, 6263, 6269, 6271, 6277, 6287, 6299, 6301, &
|
|
6311, 6317, 6323, 6329, 6337, 6343, 6353, 6359, 6361, 6367, &
|
|
6373, 6379, 6389, 6397, 6421, 6427, 6449, 6451, 6469, 6473, &
|
|
6481, 6491, 6521, 6529, 6547, 6551, 6553, 6563, 6569, 6571, &
|
|
6577, 6581, 6599, 6607, 6619, 6637, 6653, 6659, 6661, 6673, &
|
|
6679, 6689, 6691, 6701, 6703, 6709, 6719, 6733, 6737, 6761, &
|
|
6763, 6779, 6781, 6791, 6793, 6803, 6823, 6827, 6829, 6833, &
|
|
6841, 6857, 6863, 6869, 6871, 6883, 6899, 6907, 6911, 6917, &
|
|
6947, 6949, 6959, 6961, 6967, 6971, 6977, 6983, 6991, 6997 /)
|
|
|
|
npvec(901:1000) = (/ &
|
|
7001, 7013, 7019, 7027, 7039, 7043, 7057, 7069, 7079, 7103, &
|
|
7109, 7121, 7127, 7129, 7151, 7159, 7177, 7187, 7193, 7207, &
|
|
7211, 7213, 7219, 7229, 7237, 7243, 7247, 7253, 7283, 7297, &
|
|
7307, 7309, 7321, 7331, 7333, 7349, 7351, 7369, 7393, 7411, &
|
|
7417, 7433, 7451, 7457, 7459, 7477, 7481, 7487, 7489, 7499, &
|
|
7507, 7517, 7523, 7529, 7537, 7541, 7547, 7549, 7559, 7561, &
|
|
7573, 7577, 7583, 7589, 7591, 7603, 7607, 7621, 7639, 7643, &
|
|
7649, 7669, 7673, 7681, 7687, 7691, 7699, 7703, 7717, 7723, &
|
|
7727, 7741, 7753, 7757, 7759, 7789, 7793, 7817, 7823, 7829, &
|
|
7841, 7853, 7867, 7873, 7877, 7879, 7883, 7901, 7907, 7919 /)
|
|
|
|
npvec(1001:1100) = (/ &
|
|
7927, 7933, 7937, 7949, 7951, 7963, 7993, 8009, 8011, 8017, &
|
|
8039, 8053, 8059, 8069, 8081, 8087, 8089, 8093, 8101, 8111, &
|
|
8117, 8123, 8147, 8161, 8167, 8171, 8179, 8191, 8209, 8219, &
|
|
8221, 8231, 8233, 8237, 8243, 8263, 8269, 8273, 8287, 8291, &
|
|
8293, 8297, 8311, 8317, 8329, 8353, 8363, 8369, 8377, 8387, &
|
|
8389, 8419, 8423, 8429, 8431, 8443, 8447, 8461, 8467, 8501, &
|
|
8513, 8521, 8527, 8537, 8539, 8543, 8563, 8573, 8581, 8597, &
|
|
8599, 8609, 8623, 8627, 8629, 8641, 8647, 8663, 8669, 8677, &
|
|
8681, 8689, 8693, 8699, 8707, 8713, 8719, 8731, 8737, 8741, &
|
|
8747, 8753, 8761, 8779, 8783, 8803, 8807, 8819, 8821, 8831 /)
|
|
|
|
npvec(1101:1200) = (/ &
|
|
8837, 8839, 8849, 8861, 8863, 8867, 8887, 8893, 8923, 8929, &
|
|
8933, 8941, 8951, 8963, 8969, 8971, 8999, 9001, 9007, 9011, &
|
|
9013, 9029, 9041, 9043, 9049, 9059, 9067, 9091, 9103, 9109, &
|
|
9127, 9133, 9137, 9151, 9157, 9161, 9173, 9181, 9187, 9199, &
|
|
9203, 9209, 9221, 9227, 9239, 9241, 9257, 9277, 9281, 9283, &
|
|
9293, 9311, 9319, 9323, 9337, 9341, 9343, 9349, 9371, 9377, &
|
|
9391, 9397, 9403, 9413, 9419, 9421, 9431, 9433, 9437, 9439, &
|
|
9461, 9463, 9467, 9473, 9479, 9491, 9497, 9511, 9521, 9533, &
|
|
9539, 9547, 9551, 9587, 9601, 9613, 9619, 9623, 9629, 9631, &
|
|
9643, 9649, 9661, 9677, 9679, 9689, 9697, 9719, 9721, 9733 /)
|
|
|
|
npvec(1201:1300) = (/ &
|
|
9739, 9743, 9749, 9767, 9769, 9781, 9787, 9791, 9803, 9811, &
|
|
9817, 9829, 9833, 9839, 9851, 9857, 9859, 9871, 9883, 9887, &
|
|
9901, 9907, 9923, 9929, 9931, 9941, 9949, 9967, 9973,10007, &
|
|
10009,10037,10039,10061,10067,10069,10079,10091,10093,10099, &
|
|
10103,10111,10133,10139,10141,10151,10159,10163,10169,10177, &
|
|
10181,10193,10211,10223,10243,10247,10253,10259,10267,10271, &
|
|
10273,10289,10301,10303,10313,10321,10331,10333,10337,10343, &
|
|
10357,10369,10391,10399,10427,10429,10433,10453,10457,10459, &
|
|
10463,10477,10487,10499,10501,10513,10529,10531,10559,10567, &
|
|
10589,10597,10601,10607,10613,10627,10631,10639,10651,10657 /)
|
|
|
|
npvec(1301:1400) = (/ &
|
|
10663,10667,10687,10691,10709,10711,10723,10729,10733,10739, &
|
|
10753,10771,10781,10789,10799,10831,10837,10847,10853,10859, &
|
|
10861,10867,10883,10889,10891,10903,10909,19037,10939,10949, &
|
|
10957,10973,10979,10987,10993,11003,11027,11047,11057,11059, &
|
|
11069,11071,11083,11087,11093,11113,11117,11119,11131,11149, &
|
|
11159,11161,11171,11173,11177,11197,11213,11239,11243,11251, &
|
|
11257,11261,11273,11279,11287,11299,11311,11317,11321,11329, &
|
|
11351,11353,11369,11383,11393,11399,11411,11423,11437,11443, &
|
|
11447,11467,11471,11483,11489,11491,11497,11503,11519,11527, &
|
|
11549,11551,11579,11587,11593,11597,11617,11621,11633,11657 /)
|
|
|
|
npvec(1401:1500) = (/ &
|
|
11677,11681,11689,11699,11701,11717,11719,11731,11743,11777, &
|
|
11779,11783,11789,11801,11807,11813,11821,11827,11831,11833, &
|
|
11839,11863,11867,11887,11897,11903,11909,11923,11927,11933, &
|
|
11939,11941,11953,11959,11969,11971,11981,11987,12007,12011, &
|
|
12037,12041,12043,12049,12071,12073,12097,12101,12107,12109, &
|
|
12113,12119,12143,12149,12157,12161,12163,12197,12203,12211, &
|
|
12227,12239,12241,12251,12253,12263,12269,12277,12281,12289, &
|
|
12301,12323,12329,12343,12347,12373,12377,12379,12391,12401, &
|
|
12409,12413,12421,12433,12437,12451,12457,12473,12479,12487, &
|
|
12491,12497,12503,12511,12517,12527,12539,12541,12547,12553 /)
|
|
|
|
end if
|
|
|
|
if ( n == -1 ) then
|
|
prime = prime_max
|
|
else if ( n == 0 ) then
|
|
prime = 1
|
|
else if ( n <= prime_max ) then
|
|
prime = npvec(n)
|
|
else
|
|
prime = 0
|
|
!$OMP CRITICAL (write2out)
|
|
|
|
write ( 6, '(a)' ) ' '
|
|
write ( 6, '(a)' ) 'PRIME - Fatal error!'
|
|
write ( 6, '(a,i6)' ) ' Illegal prime index N = ', n
|
|
write ( 6, '(a,i6)' ) ' N must be between 0 and PRIME_MAX =',prime_max
|
|
call flush(6)
|
|
!$OMP END CRITICAL (write2out)
|
|
|
|
stop
|
|
end if
|
|
|
|
return
|
|
|
|
endfunction
|
|
|
|
!**************************************************************************
|
|
! volume of tetrahedron given by four vertices
|
|
!**************************************************************************
|
|
pure function math_volTetrahedron(v1,v2,v3,v4)
|
|
|
|
use prec, only: pReal
|
|
implicit none
|
|
|
|
real(pReal) math_volTetrahedron
|
|
real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4
|
|
real(pReal), dimension (3,3) :: m
|
|
|
|
m(:,1) = v1-v2
|
|
m(:,2) = v2-v3
|
|
m(:,3) = v3-v4
|
|
|
|
math_volTetrahedron = math_det3x3(m)/6.0_pReal
|
|
return
|
|
|
|
endfunction
|
|
|
|
|
|
|
|
END MODULE math
|