! Copyright 2011 Max-Planck-Institut für Eisenforschung GmbH ! ! This file is part of DAMASK, ! the Düsseldorf Advanced MAterial Simulation Kit. ! ! DAMASK is free software: you can redistribute it and/or modify ! it under the terms of the GNU General Public License as published by ! the Free Software Foundation, either version 3 of the License, or ! (at your option) any later version. ! ! DAMASK is distributed in the hope that it will be useful, ! but WITHOUT ANY WARRANTY; without even the implied warranty of ! MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the ! GNU General Public License for more details. ! ! You should have received a copy of the GNU General Public License ! along with DAMASK. If not, see . ! !-------------------------------------------------------------------------------------------------- !* $Id$ !-------------------------------------------------------------------------------------------------- !> @author Franz Roters, Max-Planck-Institut für Eisenforschung GmbH !> @author Philip Eisenlohr, Max-Planck-Institut für Eisenforschung GmbH !> @author Christoph Kords, Max-Planck-Institut für Eisenforschung GmbH !> @brief Mathematical library, including random number generation and tensor represenations !-------------------------------------------------------------------------------------------------- #ifdef Spectral #include "kdtree2.f90" #endif module math use, intrinsic :: iso_c_binding use prec, only: pReal,pInt implicit none real(pReal), parameter, public :: PI = 3.14159265358979323846264338327950288419716939937510_pReal real(pReal), parameter, public :: INDEG = 180.0_pReal/pi real(pReal), parameter, public :: INRAD = pi/180.0_pReal complex(pReal), parameter, public :: TWOPIIMG = (0.0_pReal,2.0_pReal)* pi real(pReal), dimension(3,3), parameter, public :: & math_I3 = reshape([& 1.0_pReal,0.0_pReal,0.0_pReal, & 0.0_pReal,1.0_pReal,0.0_pReal, & 0.0_pReal,0.0_pReal,1.0_pReal & ],[3,3]) !< 3x3 Identity integer(pInt), dimension (2,6), parameter, private :: & mapMandel = reshape([& 1_pInt,1_pInt, & 2_pInt,2_pInt, & 3_pInt,3_pInt, & 1_pInt,2_pInt, & 2_pInt,3_pInt, & 1_pInt,3_pInt & ],[2,6]) !< Mandel notation real(pReal), dimension(6), parameter, private :: & nrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal,& 1.414213562373095_pReal, 1.414213562373095_pReal, 1.414213562373095_pReal] real(pReal), dimension(6), parameter , public :: & invnrmMandel = [& 1.0_pReal, 1.0_pReal, 1.0_pReal,& 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal] integer(pInt), dimension (2,6), parameter, private :: & mapVoigt = reshape([& 1_pInt,1_pInt, & 2_pInt,2_pInt, & 3_pInt,3_pInt, & 2_pInt,3_pInt, & 1_pInt,3_pInt, & 1_pInt,2_pInt & ],[2,6]) !< Voigt notation real(pReal), dimension(6), parameter, private :: & nrmVoigt = 1.0_pReal, & invnrmVoigt = 1.0_pReal ! *** Plain notation *** integer(pInt), dimension (2,9), parameter, private :: & mapPlain = reshape([& 1_pInt,1_pInt, & 1_pInt,2_pInt, & 1_pInt,3_pInt, & 2_pInt,1_pInt, & 2_pInt,2_pInt, & 2_pInt,3_pInt, & 3_pInt,1_pInt, & 3_pInt,2_pInt, & 3_pInt,3_pInt & ],[2,9]) ! Symmetry operations as quaternions ! 24 for cubic, 12 for hexagonal = 36 integer(pInt), dimension(2), parameter, private :: & math_NsymOperations = [24_pInt,12_pInt] real(pReal), dimension(4,36), parameter, private :: & math_symOperations = reshape([& 1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! cubic symmetry operations 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, & ! 2-fold symmetry 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, & 0.0_pReal, -0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.7071067811865476_pReal, -0.7071067811865476_pReal, 0.0_pReal, & 0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & ! 3-fold symmetry -0.5_pReal, 0.5_pReal, 0.5_pReal, 0.5_pReal, & 0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, & -0.5_pReal, -0.5_pReal, 0.5_pReal, 0.5_pReal, & 0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, & -0.5_pReal, 0.5_pReal, -0.5_pReal, 0.5_pReal, & 0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, & -0.5_pReal, 0.5_pReal, 0.5_pReal, -0.5_pReal, & 0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, & ! 4-fold symmetry 0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & -0.7071067811865476_pReal, 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, & 0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, & -0.7071067811865476_pReal, 0.0_pReal, 0.7071067811865476_pReal, 0.0_pReal, & 0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal, & -0.7071067811865476_pReal, 0.0_pReal, 0.0_pReal, 0.7071067811865476_pReal, & 1.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal, & ! hexagonal symmetry operations 0.0_pReal, 1.0_pReal, 0.0_pReal, 0.0_pReal, & ! 2-fold symmetry 0.0_pReal, 0.0_pReal, 1.0_pReal, 0.0_pReal, & 0.0_pReal, 0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, & 0.0_pReal, -0.5_pReal, 0.866025403784439_pReal, 0.0_pReal, & 0.0_pReal, 0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, & 0.0_pReal, -0.866025403784439_pReal, 0.5_pReal, 0.0_pReal, & 0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, & ! 6-fold symmetry -0.866025403784439_pReal, 0.0_pReal, 0.0_pReal, 0.5_pReal, & 0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, & -0.5_pReal, 0.0_pReal, 0.0_pReal, 0.866025403784439_pReal, & 0.0_pReal, 0.0_pReal, 0.0_pReal, 1.0_pReal & ],[4,36]) #ifdef Spectral include 'fftw3.f03' #endif public :: math_init, & qsort, & math_range, & math_identity2nd, & math_civita private :: math_partition, & math_delta, & Gauss contains !-------------------------------------------------------------------------------------------------- !> @brief initialization of random seed generator !-------------------------------------------------------------------------------------------------- subroutine math_init use, intrinsic :: iso_fortran_env ! to get compiler_version and compiler_options (at least for gfortran 4.6 at the moment) use prec, only: tol_math_check use numerics, only: fixedSeed use IO, only: IO_error implicit none integer(pInt) :: i real(pReal), dimension(3,3) :: R,R2 real(pReal), dimension(3) :: Eulers real(pReal), dimension(4) :: q,q2,axisangle,randTest ! the following variables are system dependend and shound NOT be pInt integer :: randSize ! gfortran requires a variable length to compile integer, dimension(:), allocatable :: randInit ! if recalculations of former randomness (with given seed) is necessary ! comment the first random_seed call out, set randSize to 1, and use ifort character(len=64) :: error_msg !$OMP CRITICAL (write2out) write(6,*) '' write(6,*) '<<<+- math init -+>>>' write(6,*) '$Id$' #include "compilation_info.f90" !$OMP END CRITICAL (write2out) call random_seed(size=randSize) allocate(randInit(randSize)) if (fixedSeed > 0_pInt) then randInit(1:randSize) = int(fixedSeed) ! fixedSeed is of type pInt, randInit not call random_seed(put=randInit) else call random_seed() endif call random_seed(get=randInit) do i = 1_pInt, 4_pInt call random_number(randTest(i)) enddo !$OMP CRITICAL (write2out) ! this critical block did cause trouble at IWM write(6,*) 'value of random seed: ', randInit(1) write(6,*) 'size of random seed: ', randSize write(6,'(a,4(/,26x,f17.14))') ' start of random sequence: ', randTest write(6,*) '' !$OMP END CRITICAL (write2out) call random_seed(put=randInit) call random_seed(get=randInit) call halton_seed_set(int(randInit(1), pInt)) call halton_ndim_set(3_pInt) ! --- check rotation dictionary --- ! +++ q -> a -> q +++ q = math_qRnd(); axisangle = math_QuaternionToAxisAngle(q); q2 = math_AxisAngleToQuaternion(axisangle(1:3),axisangle(4)) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(401_pInt,ext_msg=error_msg) endif ! +++ q -> R -> q +++ R = math_QuaternionToR(q); q2 = math_RToQuaternion(R) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(402_pInt,ext_msg=error_msg) endif ! +++ q -> euler -> q +++ Eulers = math_QuaternionToEuler(q); q2 = math_EulerToQuaternion(Eulers) if ( any(abs( q-q2) > tol_math_check) .and. & any(abs(-q-q2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',min(maxval(abs( q-q2)),maxval(abs(-q-q2))) call IO_error(403_pInt,ext_msg=error_msg) endif ! +++ R -> euler -> R +++ Eulers = math_RToEuler(R); R2 = math_EulerToR(Eulers) if ( any(abs( R-R2) > tol_math_check) ) then write (error_msg, '(a,e14.6)' ) 'maximum deviation ',maxval(abs( R-R2)) call IO_error(404_pInt,ext_msg=error_msg) endif end subroutine math_init !************************************************************************** ! Quicksort algorithm for two-dimensional integer arrays ! ! Sorting is done with respect to array(1,:) ! and keeps array(2:N,:) linked to it. !************************************************************************** recursive subroutine qsort(a, istart, iend) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in) :: istart,iend integer(pInt) :: ipivot if (istart < iend) then ipivot = math_partition(a,istart, iend) call qsort(a, istart, ipivot-1_pInt) call qsort(a, ipivot+1_pInt, iend) endif end subroutine qsort !************************************************************************** ! Partitioning required for quicksort !************************************************************************** integer(pInt) function math_partition(a, istart, iend) implicit none integer(pInt), dimension(:,:), intent(inout) :: a integer(pInt), intent(in) :: istart,iend integer(pInt) :: d,i,j,k,x,tmp d = int(size(a,1_pInt), pInt) ! number of linked data ! set the starting and ending points, and the pivot point i = istart j = iend x = a(1,istart) do ! find the first element on the right side less than or equal to the pivot point do j = j, istart, -1_pInt if (a(1,j) <= x) exit enddo ! find the first element on the left side greater than the pivot point do i = i, iend if (a(1,i) > x) exit enddo if (i < j) then ! if the indexes do not cross, exchange values do k = 1_pInt,d tmp = a(k,i) a(k,i) = a(k,j) a(k,j) = tmp enddo else ! if they do cross, exchange left value with pivot and return with the partition index do k = 1_pInt,d tmp = a(k,istart) a(k,istart) = a(k,j) a(k,j) = tmp enddo math_partition = j return endif enddo end function math_partition !************************************************************************** ! range of integers starting at one !************************************************************************** pure function math_range(N) implicit none integer(pInt), intent(in) :: N integer(pInt) :: i integer(pInt), dimension(N) :: math_range forall (i=1_pInt:N) math_range(i) = i end function math_range !************************************************************************** ! second rank identity tensor of specified dimension !************************************************************************** pure function math_identity2nd(dimen) implicit none integer(pInt), intent(in) :: dimen integer(pInt) :: i real(pReal), dimension(dimen,dimen) :: math_identity2nd math_identity2nd = 0.0_pReal forall (i=1_pInt:dimen) math_identity2nd(i,i) = 1.0_pReal end function math_identity2nd !************************************************************************** ! permutation tensor e_ijk used for computing cross product of two tensors ! e_ijk = 1 if even permutation of ijk ! e_ijk = -1 if odd permutation of ijk ! e_ijk = 0 otherwise !************************************************************************** pure function math_civita(i,j,k) implicit none integer(pInt), intent(in) :: i,j,k real(pReal) math_civita math_civita = 0.0_pReal if (((i == 1_pInt).and.(j == 2_pInt).and.(k == 3_pInt)) .or. & ((i == 2_pInt).and.(j == 3_pInt).and.(k == 1_pInt)) .or. & ((i == 3_pInt).and.(j == 1_pInt).and.(k == 2_pInt))) math_civita = 1.0_pReal if (((i == 1_pInt).and.(j == 3_pInt).and.(k == 2_pInt)) .or. & ((i == 2_pInt).and.(j == 1_pInt).and.(k == 3_pInt)) .or. & ((i == 3_pInt).and.(j == 2_pInt).and.(k == 1_pInt))) math_civita = -1.0_pReal end function math_civita !************************************************************************** ! kronecker delta function d_ij ! d_ij = 1 if i = j ! d_ij = 0 otherwise !************************************************************************** pure function math_delta(i,j) implicit none integer(pInt), intent (in) :: i,j real(pReal) :: math_delta math_delta = 0.0_pReal if (i == j) math_delta = 1.0_pReal end function math_delta !************************************************************************** ! fourth rank identity tensor of specified dimension !************************************************************************** pure function math_identity4th(dimen) implicit none integer(pInt), intent(in) :: dimen integer(pInt) :: i,j,k,l real(pReal), dimension(dimen,dimen,dimen,dimen) :: math_identity4th forall (i=1_pInt:dimen,j=1_pInt:dimen,k=1_pInt:dimen,l=1_pInt:dimen) math_identity4th(i,j,k,l) = & 0.5_pReal*(math_I3(i,k)*math_I3(j,k)+math_I3(i,l)*math_I3(j,k)) end function math_identity4th !************************************************************************** ! vector product a x b !************************************************************************** pure function math_vectorproduct(A,B) implicit none real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: math_vectorproduct math_vectorproduct(1) = A(2)*B(3)-A(3)*B(2) math_vectorproduct(2) = A(3)*B(1)-A(1)*B(3) math_vectorproduct(3) = A(1)*B(2)-A(2)*B(1) end function math_vectorproduct !************************************************************************** ! tensor product a \otimes b !************************************************************************** pure function math_tensorproduct(A,B) implicit none real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3,3) :: math_tensorproduct integer(pInt) :: i,j forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) math_tensorproduct(i,j) = A(i)*B(j) end function math_tensorproduct !************************************************************************** ! matrix multiplication 3x3 = 1 !************************************************************************** pure function math_mul3x3(A,B) implicit none integer(pInt) :: i real(pReal), dimension(3), intent(in) :: A,B real(pReal), dimension(3) :: C real(pReal) :: math_mul3x3 forall (i=1_pInt:3_pInt) C(i) = A(i)*B(i) math_mul3x3 = sum(C) end function math_mul3x3 !************************************************************************** ! matrix multiplication 6x6 = 1 !************************************************************************** pure function math_mul6x6(A,B) implicit none integer(pInt) :: i real(pReal), dimension(6), intent(in) :: A,B real(pReal), dimension(6) :: C real(pReal) :: math_mul6x6 forall (i=1_pInt:6_pInt) C(i) = A(i)*B(i) math_mul6x6 = sum(C) end function math_mul6x6 !************************************************************************** ! matrix multiplication 33x33 = 1 (double contraction --> ij * ij) !************************************************************************** pure function math_mul33xx33(A,B) implicit none integer(pInt) :: i,j real(pReal), dimension(3,3), intent(in) :: A,B real(pReal), dimension(3,3) :: C real(pReal) :: math_mul33xx33 forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) C(i,j) = A(i,j) * B(i,j) math_mul33xx33 = sum(C) end function math_mul33xx33 !************************************************************************** ! matrix multiplication 3333x33 = 33 (double contraction --> ijkl *kl = ij) !************************************************************************** pure function math_mul3333xx33(A,B) implicit none integer(pInt) :: i,j real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3), intent(in) :: B real(pReal), dimension(3,3) :: math_mul3333xx33 forall(i = 1_pInt:3_pInt,j = 1_pInt:3_pInt)& math_mul3333xx33(i,j) = sum(A(i,j,1:3,1:3)*B(1:3,1:3)) end function math_mul3333xx33 !************************************************************************** ! matrix multiplication 3333x3333 = 3333 (ijkl *klmn = ijmn) !************************************************************************** pure function math_mul3333xx3333(A,B) implicit none integer(pInt) :: i,j,k,l real(pReal), dimension(3,3,3,3), intent(in) :: A real(pReal), dimension(3,3,3,3), intent(in) :: B real(pReal), dimension(3,3,3,3) :: math_mul3333xx3333 do i = 1_pInt,3_pInt do j = 1_pInt,3_pInt do k = 1_pInt,3_pInt do l = 1_pInt,3_pInt math_mul3333xx3333(i,j,k,l) = sum(A(i,j,1:3,1:3)*B(1:3,1:3,k,l)) enddo; enddo; enddo; enddo end function math_mul3333xx3333 !************************************************************************** ! matrix multiplication 33x33 = 33 !************************************************************************** pure function math_mul33x33(A,B) 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_pInt:3_pInt,j=1_pInt:3_pInt) math_mul33x33(i,j) = & A(i,1)*B(1,j) + A(i,2)*B(2,j) + A(i,3)*B(3,j) end function math_mul33x33 !************************************************************************** ! matrix multiplication 66x66 = 66 !************************************************************************** pure function math_mul66x66(A,B) 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_pInt:6_pInt,j=1_pInt:6_pInt) 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) end function math_mul66x66 !************************************************************************** ! matrix multiplication 99x99 = 99 !************************************************************************** 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_pInt:9_pInt,j=1_pInt:9_pInt) 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) end function math_mul99x99 !************************************************************************** ! matrix multiplication 33x3 = 3 !************************************************************************** pure function math_mul33x3(A,B) 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_pInt:3_pInt) math_mul33x3(i) = sum(A(i,1:3)*B) end function math_mul33x3 !************************************************************************** ! matrix multiplication complex(33) x real(3) = complex(3) !************************************************************************** pure function math_mul33x3_complex(A,B) 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_pInt:3_pInt) math_mul33x3_complex(i) = sum(A(i,1:3)*cmplx(B,0.0_pReal,pReal)) end function math_mul33x3_complex !************************************************************************** ! matrix multiplication 66x6 = 6 !************************************************************************** pure function math_mul66x6(A,B) 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_pInt:6_pInt) 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) end function math_mul66x6 !************************************************************************** ! random quaternion !************************************************************************** function math_qRnd() implicit none real(pReal), dimension(4) :: math_qRnd real(pReal), dimension(3) :: rnd call halton(3_pInt,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)) end function math_qRnd !************************************************************************** ! quaternion multiplication q1xq2 = q12 !************************************************************************** pure function math_qMul(A,B) 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) end function math_qMul !************************************************************************** ! quaternion dotproduct !************************************************************************** pure function math_qDot(A,B) 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) end function math_qDot !************************************************************************** ! quaternion conjugation !************************************************************************** pure function math_qConj(Q) 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) end function math_qConj !************************************************************************** ! quaternion norm !************************************************************************** pure function math_qNorm(Q) 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))) end function math_qNorm !************************************************************************** ! quaternion inversion !************************************************************************** pure function math_qInv(Q) 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 end function math_qInv !************************************************************************** ! action of a quaternion on a vector (rotate vector v with Q) !************************************************************************** pure function math_qRot(Q,v) 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_pInt,4_pInt do j = 1_pInt,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 end function math_qRot !************************************************************************** ! transposition of a 33 matrix !************************************************************************** pure function math_transpose33(A) implicit none real(pReal),dimension(3,3),intent(in) :: A real(pReal),dimension(3,3) :: math_transpose33 integer(pInt) :: i,j forall(i=1_pInt:3_pInt, j=1_pInt:3_pInt) math_transpose33(i,j) = A(j,i) end function math_transpose33 !************************************************************************** ! Cramer inversion of 33 matrix (function) !************************************************************************** pure function math_inv33(A) ! direct Cramer inversion of matrix A. ! returns all zeroes if not possible, i.e. if det close to zero implicit none real(pReal),dimension(3,3),intent(in) :: A real(pReal) :: DetA real(pReal),dimension(3,3) :: math_inv33 math_inv33 = 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 (abs(DetA) > tiny(abs(DetA))) then math_inv33(1,1) = ( A(2,2) * A(3,3) - A(2,3) * A(3,2)) / DetA math_inv33(2,1) = (-A(2,1) * A(3,3) + A(2,3) * A(3,1)) / DetA math_inv33(3,1) = ( A(2,1) * A(3,2) - A(2,2) * A(3,1)) / DetA math_inv33(1,2) = (-A(1,2) * A(3,3) + A(1,3) * A(3,2)) / DetA math_inv33(2,2) = ( A(1,1) * A(3,3) - A(1,3) * A(3,1)) / DetA math_inv33(3,2) = (-A(1,1) * A(3,2) + A(1,2) * A(3,1)) / DetA math_inv33(1,3) = ( A(1,2) * A(2,3) - A(1,3) * A(2,2)) / DetA math_inv33(2,3) = (-A(1,1) * A(2,3) + A(1,3) * A(2,1)) / DetA math_inv33(3,3) = ( A(1,1) * A(2,2) - A(1,2) * A(2,1)) / DetA endif end function math_inv33 !************************************************************************** ! Cramer inversion of 33 matrix (subroutine) !************************************************************************** pure subroutine math_invert33(A, InvA, DetA, error) ! Bestimmung der Determinanten und Inversen einer 33-Matrix ! A = Matrix A ! InvA = Inverse of A ! DetA = Determinant of A ! error = logical 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 (abs(DetA) <= tiny(abs(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 end subroutine math_invert33 !************************************************************************** ! Inversion of symmetriced 3x3x3x3 tensor. !************************************************************************** function math_invSym3333(A) use IO, only: IO_error implicit none real(pReal),dimension(3,3,3,3) :: math_invSym3333 real(pReal),dimension(3,3,3,3),intent(in) :: A integer(pInt) :: ierr1, ierr2 integer(pInt), dimension(6) :: ipiv6 real(pReal), dimension(6,6) :: temp66_Real real(pReal), dimension(6) :: work6 temp66_real = math_Mandel3333to66(A) call dgetrf(6,6,temp66_real,6,ipiv6,ierr1) call dgetri(6,temp66_real,6,ipiv6,work6,6,ierr2) if (ierr1*ierr2 == 0_pInt) then math_invSym3333 = math_Mandel66to3333(temp66_real) else call IO_error(400_pInt, ext_msg = 'math_invSym3333') endif end function math_invSym3333 !************************************************************************** ! Gauss elimination to invert matrix of arbitrary dimension !************************************************************************** pure subroutine math_invert(dimen,A, InvA, AnzNegEW, error) ! Invertieren einer dimen x dimen - Matrix ! A = Matrix A ! InvA = Inverse of A ! AnzNegEW = Number of negative Eigenvalues of A ! error = false: Inversion done. ! = true: Inversion stopped in SymGauss because of dimishing ! Pivotelement 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) end subroutine math_invert ! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! ++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ pure subroutine Gauss (dimen,A,B,LogAbsDetA,NegHDK,error) ! Solves a linear EQS A * X = B with the GAUSS-Algorithm ! For numerical stabilization using a pivot search in rows and columns ! ! input parameters ! A(dimen,dimen) = matrix A ! B(dimen,dimen) = right side B ! ! output parameters ! B(dimen,dimen) = Matrix containing unknown vectors X ! LogAbsDetA = 10-Logarithm of absolute value of determinatns of A ! NegHDK = Number of negative Maindiagonal coefficients resulting ! Vorwaertszerlegung ! error = false: EQS is solved ! = true : Matrix A is singular. ! ! A and B will be changed! implicit none logical, intent(out) :: error integer(pInt), intent(in) :: dimen integer(pInt), intent(out) :: NegHDK real(pReal), intent(out) :: LogAbsDetA real(pReal), intent(inout), dimension(dimen,dimen) :: A, B logical :: SortX integer(pInt) :: PivotZeile, PivotSpalte, StoreI, I, IP1, J, K, L integer(pInt), dimension(dimen) :: XNr real(pReal) :: AbsA, PivotWert, EpsAbs, Quote real(pReal), dimension(dimen) :: StoreA, StoreB error = .true.; NegHDK = 1_pInt; SortX = .false. ! Unbekanntennumerierung DO I = 1_pInt, dimen XNr(I) = I ENDDO ! Genauigkeitsschranke und Bestimmung des groessten Pivotelementes PivotWert = ABS(A(1,1)) PivotZeile = 1_pInt PivotSpalte = 1_pInt do I = 1_pInt, dimen; do J = 1_pInt, 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_pReal) 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_pInt, dimen - 1_pInt ! 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_pInt, dimen Quote = A(J,I) / A(I,I) DO K = I + 1_pInt, dimen A(J,K) = A(J,K) - Quote * A(I,K) ENDDO DO K = 1_pInt, dimen B(J,K) = B(J,K) - Quote * B(I,K) ENDDO ENDDO ! Bestimmung des groessten Pivotelementes IP1 = I + 1_pInt 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_pInt, -1_pInt DO L = 1_pInt, dimen DO J = I + 1_pInt, 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_pInt, dimen StoreA(1:dimen) = B(1:dimen,L) DO I = 1_pInt, dimen J = XNr(I) B(J,L) = StoreA(I) ENDDO ENDDO ENDIF ! Determinante LogAbsDetA = 0.0_pReal NegHDK = 0_pInt DO I = 1_pInt, dimen IF (A(I,I) .LT. 0.0_pReal) NegHDK = NegHDK + 1_pInt AbsA = ABS(A(I,I)) LogAbsDetA = LogAbsDetA + LOG10(AbsA) ENDDO error = .false. end subroutine Gauss !******************************************************************** ! symmetrize a 33 matrix !******************************************************************** function math_symmetric33(m) implicit none real(pReal), dimension(3,3) :: math_symmetric33 real(pReal), dimension(3,3), intent(in) :: m integer(pInt) :: i,j forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) math_symmetric33(i,j) = 0.5_pReal * (m(i,j) + m(j,i)) end function math_symmetric33 !******************************************************************** ! symmetrize a 66 matrix !******************************************************************** pure function math_symmetric66(m) implicit none integer(pInt) :: i,j real(pReal), dimension(6,6), intent(in) :: m real(pReal), dimension(6,6) :: math_symmetric66 forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) math_symmetric66(i,j) = 0.5_pReal * (m(i,j) + m(j,i)) end function math_symmetric66 !******************************************************************** ! skew part of a 33 matrix !******************************************************************** pure function math_skew33(m) implicit none real(pReal), dimension(3,3) :: math_skew33 real(pReal), dimension(3,3), intent(in) :: m integer(pInt) :: i,j forall (i=1_pInt:3_pInt,j=1_pInt:3_pInt) math_skew33(i,j) = m(i,j) - 0.5_pReal * (m(i,j) + m(j,i)) end function math_skew33 !******************************************************************** ! deviatoric part of a 33 matrix !******************************************************************** pure function math_deviatoric33(m) implicit none real(pReal), dimension(3,3) :: math_deviatoric33 real(pReal), dimension(3,3), intent(in) :: m integer(pInt) :: i real(pReal) :: hydrostatic hydrostatic = (m(1,1) + m(2,2) + m(3,3)) / 3.0_pReal math_deviatoric33 = m forall (i=1_pInt:3_pInt) math_deviatoric33(i,i) = m(i,i) - hydrostatic end function math_deviatoric33 !******************************************************************** ! equivalent scalar quantity of a full strain tensor !******************************************************************** pure function math_equivStrain33(m) 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 end function math_equivStrain33 !******************************************************************** subroutine math_equivStrain33_field(res,tensor,vm) !******************************************************************** !calculate von Mises equivalent of tensor field ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) :: tensor ! output variables real(pReal), intent(out), dimension(res(1),res(2),res(3)) :: vm ! other variables integer(pInt) :: i, j, k real(pReal), dimension(3,3) :: deviator, delta = 0.0_pReal real(pReal) :: J_2 delta(1,1) = 1.0_pReal delta(2,2) = 1.0_pReal delta(3,3) = 1.0_pReal do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) deviator = tensor(i,j,k,1:3,1:3) - 1.0_pReal/3.0_pReal*tensor(i,j,k,1,1)*tensor(i,j,k,2,2)*tensor(i,j,k,3,3)*delta J_2 = deviator(1,1)*deviator(2,2)& + deviator(2,2)*deviator(3,3)& + deviator(1,1)*deviator(3,3)& - (deviator(1,2))**2.0_pReal& - (deviator(2,3))**2.0_pReal& - (deviator(1,3))**2.0_pReal vm(i,j,k) = sqrt(3.0_pReal*J_2) enddo; enddo; enddo end subroutine math_equivStrain33_field !******************************************************************** ! determinant of a 33 matrix !******************************************************************** pure function math_det33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal) :: math_det33 math_det33 = m(1,1)*(m(2,2)*m(3,3)-m(2,3)*m(3,2)) & -m(1,2)*(m(2,1)*m(3,3)-m(2,3)*m(3,1)) & +m(1,3)*(m(2,1)*m(3,2)-m(2,2)*m(3,1)) end function math_det33 !******************************************************************** ! norm of a 33 matrix !******************************************************************** pure function math_norm33(m) implicit none real(pReal), dimension(3,3), intent(in) :: m real(pReal) :: math_norm33 math_norm33 = sqrt(sum(m**2.0_pReal)) end function !******************************************************************** ! euclidic norm of a 3 vector !******************************************************************** pure function math_norm3(v) 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)) end function math_norm3 !******************************************************************** ! convert 33 matrix into vector 9 !******************************************************************** pure function math_Plain33to9(m33) implicit none real(pReal), dimension(3,3), intent(in) :: m33 real(pReal), dimension(9) :: math_Plain33to9 integer(pInt) :: i forall (i=1_pInt:9_pInt) math_Plain33to9(i) = m33(mapPlain(1,i),mapPlain(2,i)) end function math_Plain33to9 !******************************************************************** ! convert Plain 9 back to 33 matrix !******************************************************************** pure function math_Plain9to33(v9) implicit none real(pReal), dimension(9), intent(in) :: v9 real(pReal), dimension(3,3) :: math_Plain9to33 integer(pInt) :: i forall (i=1_pInt:9_pInt) math_Plain9to33(mapPlain(1,i),mapPlain(2,i)) = v9(i) end function math_Plain9to33 !******************************************************************** ! convert symmetric 33 matrix into Mandel vector 6 !******************************************************************** pure function math_Mandel33to6(m33) implicit none real(pReal), dimension(3,3), intent(in) :: m33 real(pReal), dimension(6) :: math_Mandel33to6 integer(pInt) :: i forall (i=1_pInt:6_pInt) math_Mandel33to6(i) = nrmMandel(i)*m33(mapMandel(1,i),mapMandel(2,i)) end function math_Mandel33to6 !******************************************************************** ! convert Mandel 6 back to symmetric 33 matrix !******************************************************************** pure function math_Mandel6to33(v6) implicit none real(pReal), dimension(6), intent(in) :: v6 real(pReal), dimension(3,3) :: math_Mandel6to33 integer(pInt) :: i forall (i=1_pInt:6_pInt) 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 end function math_Mandel6to33 !******************************************************************** ! convert 3333 tensor into plain matrix 99 !******************************************************************** pure function math_Plain3333to99(m3333) 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_pInt:9_pInt,j=1_pInt:9_pInt) math_Plain3333to99(i,j) = & m3333(mapPlain(1,i),mapPlain(2,i),mapPlain(1,j),mapPlain(2,j)) end function math_Plain3333to99 !******************************************************************** ! plain matrix 99 into 3333 tensor !******************************************************************** pure function math_Plain99to3333(m99) 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_pInt:9_pInt,j=1_pInt:9_pInt) math_Plain99to3333(mapPlain(1,i),mapPlain(2,i),& mapPlain(1,j),mapPlain(2,j)) = m99(i,j) end function math_Plain99to3333 !******************************************************************** ! convert Mandel matrix 66 into Plain matrix 66 !******************************************************************** pure function math_Mandel66toPlain66(m66) implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(6,6) :: math_Mandel66toPlain66 integer(pInt) :: i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) & math_Mandel66toPlain66(i,j) = invnrmMandel(i) * invnrmMandel(j) * m66(i,j) return end function !******************************************************************** ! convert Plain matrix 66 into Mandel matrix 66 !******************************************************************** pure function math_Plain66toMandel66(m66) implicit none real(pReal), dimension(6,6), intent(in) :: m66 real(pReal), dimension(6,6) :: math_Plain66toMandel66 integer(pInt) i,j forall (i=1_pInt:6_pInt,j=1_pInt:6_pInt) & math_Plain66toMandel66(i,j) = nrmMandel(i) * nrmMandel(j) * m66(i,j) return end function !******************************************************************** ! convert symmetric 3333 tensor into Mandel matrix 66 !******************************************************************** pure function math_Mandel3333to66(m3333) 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_pInt:6_pInt,j=1_pInt:6_pInt) math_Mandel3333to66(i,j) = & nrmMandel(i)*nrmMandel(j)*m3333(mapMandel(1,i),mapMandel(2,i),mapMandel(1,j),mapMandel(2,j)) end function math_Mandel3333to66 !******************************************************************** ! convert Mandel matrix 66 back to symmetric 3333 tensor !******************************************************************** pure function math_Mandel66to3333(m66) 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_pInt:6_pInt,j=1_pInt:6_pInt) 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 end function math_Mandel66to3333 !******************************************************************** ! convert Voigt matrix 66 back to symmetric 3333 tensor !******************************************************************** pure function math_Voigt66to3333(m66) 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_pInt:6_pInt,j=1_pInt:6_pInt) 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 end function math_Voigt66to3333 !******************************************************************** ! Euler angles (in radians) from rotation matrix !******************************************************************** pure function math_RtoEuler(R) implicit none real(pReal), dimension (3,3), intent(in) :: R real(pReal), dimension(3) :: math_RtoEuler real(pReal) :: sqhkl, squvw, sqhk, myVal 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 myVal=R(3,3)/sqhkl if(myVal > 1.0_pReal) myVal = 1.0_pReal if(myVal < -1.0_pReal) myVal = -1.0_pReal math_RtoEuler(2) = acos(myVal) if(math_RtoEuler(2) < 1.0e-8_pReal) then ! calculate phi2 math_RtoEuler(3) = 0.0_pReal ! calculate phi1 myVal=R(1,1)/squvw if(myVal > 1.0_pReal) myVal = 1.0_pReal if(myVal < -1.0_pReal) myVal = -1.0_pReal math_RtoEuler(1) = acos(myVal) if(R(2,1) > 0.0_pReal) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) else ! calculate phi2 myVal=R(2,3)/sqhk if(myVal > 1.0_pReal) myVal = 1.0_pReal if(myVal < -1.0_pReal) myVal = -1.0_pReal math_RtoEuler(3) = acos(myVal) if(R(1,3) < 0.0) math_RtoEuler(3) = 2.0_pReal*pi-math_RtoEuler(3) ! calculate phi1 myVal=-R(3,2)/sin(math_RtoEuler(2)) if(myVal > 1.0_pReal) myVal = 1.0_pReal if(myVal < -1.0_pReal) myVal = -1.0_pReal math_RtoEuler(1) = acos(myVal) if(R(3,1) < 0.0) math_RtoEuler(1) = 2.0_pReal*pi-math_RtoEuler(1) end if end function math_RtoEuler !******************************************************************** ! quaternion (w+ix+jy+kz) from orientation matrix !******************************************************************** ! math adopted from http://code.google.com/p/mtex/source/browse/trunk/geometry/geometry_tools/mat2quat.m pure function math_RtoQuaternion(R) implicit none real(pReal), dimension (3,3), intent(in) :: R real(pReal), dimension(4) :: absQ, math_RtoQuaternion real(pReal) :: max_absQ integer, dimension(1) :: largest !no pInt, maxloc returns integer default absQ(1) = 1.0_pReal+R(1,1)+R(2,2)+R(3,3) absQ(2) = 1.0_pReal+R(1,1)-R(2,2)-R(3,3) absQ(3) = 1.0_pReal-R(1,1)+R(2,2)-R(3,3) absQ(4) = 1.0_pReal-R(1,1)-R(2,2)+R(3,3) math_RtoQuaternion = 0.0_pReal largest = maxloc(absQ) max_absQ=0.5_pReal * sqrt(absQ(largest(1))) select case(largest(1)) case (1_pInt) !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_pInt) math_RtoQuaternion(1) = R(2,3)-R(3,2) !2---------------------------------- math_RtoQuaternion(3) = R(1,2)+R(2,1) math_RtoQuaternion(4) = R(3,1)+R(1,3) case (3_pInt) math_RtoQuaternion(1) = R(3,1)-R(1,3) math_RtoQuaternion(2) = R(1,2)+R(2,1) !3---------------------------------- math_RtoQuaternion(4) = R(2,3)+R(3,2) case (4_pInt) 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) !4---------------------------------- end select math_RtoQuaternion = math_RtoQuaternion*0.25_pReal/max_absQ math_RtoQuaternion(largest(1)) = max_absQ end function math_RtoQuaternion !**************************************************************** ! rotation matrix from Euler angles (in radians) !**************************************************************** pure function math_EulerToR(Euler) 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 end function math_EulerToR !******************************************************************** ! quaternion (w+ix+jy+kz) from 3-1-3 Euler angles (in radians) !******************************************************************** pure function math_EulerToQuaternion(eulerangles) 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 end function math_EulerToQuaternion !**************************************************************** ! rotation matrix from axis and angle (in radians) !**************************************************************** pure function math_AxisAngleToR(axis,omega) 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_pInt:3_pInt) 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)**2.0_pReal 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)**2.0_pReal 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)**2.0_pReal else math_AxisAngleToR = math_I3 endif end function math_AxisAngleToR !**************************************************************** ! quaternion (w+ix+jy+kz) from axis and angle (in radians) !**************************************************************** pure function math_AxisAngleToQuaternion(axis,omega) 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_pInt:3_pInt) 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 end function math_AxisAngleToQuaternion !******************************************************************** ! orientation matrix from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToR(Q) implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3,3) :: math_QuaternionToR, T,S integer(pInt) :: i, j forall (i = 1_pInt:3_pInt, j = 1_pInt:3_pInt) & T(i,j) = Q(i+1_pInt) * Q(j+1_pInt) 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 end function math_QuaternionToR !******************************************************************** ! 3-1-3 Euler angles (in radians) from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToEuler(Q) implicit none real(pReal), dimension(4), intent(in) :: Q real(pReal), dimension(3) :: math_QuaternionToEuler real(pReal) :: acos_arg math_QuaternionToEuler(2) = acos(1.0_pReal-2.0_pReal*(Q(2)*Q(2)+Q(3)*Q(3))) if (abs(math_QuaternionToEuler(2)) < 1.0e-3_pReal) then acos_arg=Q(1) if(acos_arg > 1.0_pReal)acos_arg = 1.0_pReal if(acos_arg < -1.0_pReal)acos_arg = -1.0_pReal math_QuaternionToEuler(1) = 2.0_pReal*acos(acos_arg) math_QuaternionToEuler(3) = 0.0_pReal else math_QuaternionToEuler(1) = atan2(Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)-Q(3)*Q(4)) if (math_QuaternionToEuler(1) < 0.0_pReal) & math_QuaternionToEuler(1) = math_QuaternionToEuler(1) + 2.0_pReal * pi math_QuaternionToEuler(3) = atan2(-Q(1)*Q(3)+Q(2)*Q(4), Q(1)*Q(2)+Q(3)*Q(4)) if (math_QuaternionToEuler(3) < 0.0_pReal) & math_QuaternionToEuler(3) = math_QuaternionToEuler(3) + 2.0_pReal * pi endif if (math_QuaternionToEuler(2) < 0.0_pReal) & math_QuaternionToEuler(2) = math_QuaternionToEuler(2) + pi end function math_QuaternionToEuler !******************************************************************** ! axis-angle (x, y, z, ang in radians) from quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToAxisAngle(Q) 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 end function math_QuaternionToAxisAngle !******************************************************************** ! Rodrigues vector (x, y, z) from unit quaternion (w+ix+jy+kz) !******************************************************************** pure function math_QuaternionToRodrig(Q) use prec, only: DAMASK_NaN 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 = DAMASK_NaN ! NaN since Rodrig is unbound for 180 deg... endif end function math_QuaternionToRodrig !************************************************************************** ! misorientation angle between two sets of Euler angles !************************************************************************** pure function math_EulerMisorientation(EulerA,EulerB) 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)) end function math_EulerMisorientation !************************************************************************** ! figures whether unit quat falls into stereographic standard triangle !************************************************************************** pure function math_QuaternionInSST(Q, symmetryType) 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) if (any(Rodrig/=Rodrig)) then math_QuaternionInSST = .false. else select case (symmetryType) case (1_pInt) math_QuaternionInSST = Rodrig(1) > Rodrig(2) .and. & Rodrig(2) > Rodrig(3) .and. & Rodrig(3) > 0.0_pReal case (2_pInt) 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 endif end function math_QuaternionInSST !************************************************************************** ! calculates the disorientation for 2 unit quaternions !************************************************************************** function math_QuaternionDisorientation(Q1, Q2, symmetryType) 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_pInt) if (math_QuaternionDisorientation(1) < 0.0_pReal) & math_QuaternionDisorientation = -math_QuaternionDisorientation ! keep omega within 0 to 180 deg case (1_pInt,2_pInt) s = sum(math_NsymOperations(1:symmetryType-1_pInt)) do i = 1_pInt,2_pInt dQ = math_qConj(dQ) ! switch order of "from -- to" do j = 1_pInt,math_NsymOperations(symmetryType) ! run through first crystal's symmetries dQsymA = math_qMul(math_symOperations(1:4,s+j),dQ) ! apply sym do k = 1_pInt,math_NsymOperations(symmetryType) ! run through 2nd crystal's symmetries mis = math_qMul(dQsymA,math_symOperations(1:4,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(450_pInt,symmetryType) ! complain about unknown symmetry end select end function math_QuaternionDisorientation !******************************************************************** ! draw a random sample from Euler space !******************************************************************** function math_sampleRandomOri() implicit none real(pReal), dimension(3) :: math_sampleRandomOri, rnd call halton(3_pInt,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 end function math_sampleRandomOri !******************************************************************** ! draw a random sample from Gauss component ! with noise (in radians) half-width !******************************************************************** function math_sampleGaussOri(center,noise) 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_pReal) 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_pInt,rnd) forall (i=1_pInt:3_pInt) 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_pReal)) exit enddo math_sampleGaussOri = math_RtoEuler(math_mul33x33(math_EulerToR(disturb),math_EulerToR(center))) end function math_sampleGaussOri !-------------------------------------------------------------------------------------------------- !> @brief draw a random sample from Fiber component with noise (in radians) !-------------------------------------------------------------------------------------------------- function math_sampleFiberOri(alpha,beta,noise) implicit none real(pReal), dimension(3) :: math_sampleFiberOri, fiberInC,fiberInS,axis real(pReal), dimension(2), intent(in) :: alpha,beta real(pReal), dimension(6) :: rnd real(pReal), dimension(3,3) :: oRot,fRot,pRot real(pReal) :: noise, scatter, cos2Scatter, angle integer(pInt), dimension(2,3), parameter :: rotMap = reshape((/2_pInt,3_pInt,& 3_pInt,1_pInt,& 1_pInt,2_pInt/),(/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_pInt:3_pInt) 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) #--- do call halton(6_pInt,rnd) fRot = math_AxisAngleToR(fiberInS,rnd(1)*2.0_pReal*pi) ! ---# rotation about random axis perpend to fiber #--- ! random axis pependicular to fiber axis axis(1:2) = rnd(2:3) 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 if (noise > 0.0_pReal) then angle = acos(cos2Scatter+(1.0_pReal-cos2Scatter)*rnd(4)) if (rnd(5) <= exp(-1.0_pReal*(angle/scatter)**2.0_pReal)) exit else angle = 0.0_pReal exit end if enddo if (rnd(6) <= 0.5) angle = -angle pRot = math_AxisAngleToR(axis,angle) ! ---# apply the three rotations #--- math_sampleFiberOri = math_RtoEuler(math_mul33x33(pRot,math_mul33x33(fRot,oRot))) end function math_sampleFiberOri !******************************************************************** ! symmetric Euler angles for given symmetry string ! 'triclinic' or '', 'monoclinic', 'orthotropic' !******************************************************************** pure function math_symmetricEulers(sym,Euler) 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_pInt:3_pInt,j=1_pInt:3_pInt) math_symmetricEulers(j,i) = modulo(math_symmetricEulers(j,i),2.0_pReal*pi) select case (sym) case (4_pInt) ! all done case (2_pInt) ! return only first math_symmetricEulers(1:3,2:3) = 0.0_pReal case default ! return blank math_symmetricEulers = 0.0_pReal end select end function math_symmetricEulers !******************************************************************** ! draw a random sample from Gauss variable !******************************************************************** function math_sampleGaussVar(meanvalue, stddev, width) 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_pReal) 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_pInt, 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 end function math_sampleGaussVar !**************************************************************** subroutine math_spectralDecompositionSym33(M,values,vectors,error) !**************************************************************** implicit none real(pReal), dimension(3,3), intent(in) :: M real(pReal), dimension(3), intent(out) :: values real(pReal), dimension(3,3), intent(out) :: vectors logical, intent(out) :: error integer(pInt) :: info real(pReal), dimension((64+2)*3) :: work ! block size of 64 taken from http://www.netlib.org/lapack/double/dsyev.f vectors = M ! copy matrix to input (doubles as output) array call DSYEV('V','U',3,vectors,3,values,work,(64+2)*3,info) error = (info == 0_pInt) end subroutine !**************************************************************** pure subroutine math_pDecomposition(FE,U,R,error) !-----FE = R.U !**************************************************************** implicit none real(pReal), intent(in), dimension(3,3) :: FE real(pReal), intent(out), dimension(3,3) :: R, U logical, intent(out) :: error real(pReal), dimension(3,3) :: CE, EB1, EB2, EB3, UI real(pReal) :: EW1, EW2, EW3, det error = .false. ce = math_mul33x33(math_transpose33(FE),FE) CALL math_spectral1(CE,EW1,EW2,EW3,EB1,EB2,EB3) U=sqrt(EW1)*EB1+sqrt(EW2)*EB2+sqrt(EW3)*EB3 call math_invert33(U,UI,det,error) if (.not. error) R = math_mul33x33(FE,UI) end subroutine math_pDecomposition !********************************************************************** pure subroutine math_spectral1(M,EW1,EW2,EW3,EB1,EB2,EB3) !**** EIGENWERTE UND EIGENWERTBASIS DER SYMMETRISCHEN 3X3 MATRIX M implicit none real(pReal), dimension(3,3), intent(in) :: M real(pReal), dimension(3,3), intent(out) :: EB1, EB2, EB3 real(pReal), intent(out) :: EW1,EW2,EW3 real(pReal) HI1M, HI2M, HI3M, R, S, T, P, Q, RHO, PHI, Y1, Y2, Y3, D1, D2, D3 real(pReal), parameter :: TOL=1.e-14_pReal real(pReal), dimension(3,3) :: M1, M2, M3 real(pReal) C1,C2,C3,arg 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.1.0_pReal) arg=1.0_pReal if(arg.LT.-1.0_pReal) arg=-1.0_pReal PHI=acos(arg) Y1=2.0_pReal*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal) Y2=2.0_pReal*RHO**(1.0_pReal/3.0_pReal)*cos(PHI/3.0_pReal+2.0_pReal/3.0_pReal*PI) Y3=2.0_pReal*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 end subroutine math_spectral1 !********************************************************************** function math_eigenvalues33(M) !**** Eigenvalues of symmetric 3X3 matrix M implicit none real(pReal), intent(in), dimension(3,3) :: M real(pReal), dimension(3,3) :: EB1 = 0.0_pReal, EB2 = 0.0_pReal, EB3 = 0.0_pReal real(pReal), dimension(3) :: math_eigenvalues33 real(pReal) :: HI1M, HI2M, HI3M, R, S, T, P, Q, RHO, PHI, Y1, Y2, Y3, arg real(pReal), parameter :: 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 if((abs(P) < TOL) .and. (abs(Q) < TOL)) THEN ! three equivalent eigenvalues math_eigenvalues33(1) = HI1M/3.0_pReal math_eigenvalues33(2)=math_eigenvalues33(1) math_eigenvalues33(3)=math_eigenvalues33(1) ! 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.1.0_pReal) arg=1.0_pReal if(arg.LT.-1.0_pReal) arg=-1.0_pReal 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) math_eigenvalues33(1) = Y1-R/3.0_pReal math_eigenvalues33(2) = Y2-R/3.0_pReal math_eigenvalues33(3) = Y3-R/3.0_pReal endif end function math_eigenvalues33 !********************************************************************** !**** HAUPTINVARIANTEN HI1M, HI2M, HI3M DER 3X3 MATRIX M pure subroutine math_hi(M,HI1M,HI2M,HI3M) 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**2.0_pReal/2.0_pReal- (M(1,1)**2.0_pReal+M(2,2)**2.0_pReal+M(3,3)**2.0_pReal)& /2.0_pReal-M(1,2)*M(2,1)-M(1,3)*M(3,1)-M(2,3)*M(3,2) HI3M=math_det33(M) ! QUESTION: is 3rd equiv det(M) ?? if yes, use function math_det !agreed on YES end subroutine math_hi !******************************************************************************* ! GET_SEED returns a seed for the random number generator. ! ! 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. ! ! Parameters: ! Output, integer SEED, a pseudorandom seed value. ! ! Modified: 27 June 2000 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters ! subroutine get_seed(seed) implicit none integer(pInt) :: seed real(pReal) :: temp = 0.0_pReal 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 = temp + real(values(2)- 1_pInt, pReal) / 11.0_pReal temp = temp + real(values(3)- 1_pInt, pReal) / 30.0_pReal temp = temp + real(values(5), pReal) / 23.0_pReal temp = temp + real(values(6), pReal) / 59.0_pReal temp = temp + real(values(7), pReal) / 59.0_pReal temp = temp + real(values(8), pReal) / 999.0_pReal temp = temp / 6.0_pReal if (temp <= 0.0_pReal) then temp = 1.0_pReal / 3.0_pReal else if (1.0_pReal <= temp) then temp = 2.0_pReal / 3.0_pReal end if seed = int(real(huge(1_pInt),pReal)*temp, pInt) ! ! Never use a seed of 0 or maximum integer. ! if (seed == 0_pInt) then seed = 1_pInt end if if (seed == huge(1_pInt)) then seed = seed -1_pInt end if end subroutine get_seed !******************************************************************************* ! HALTON computes the next element in the Halton sequence. ! ! Parameters: ! Input, integer NDIM, the dimension of the element. ! Output, real R(NDIM), the next element of the current Halton sequence. ! ! Modified: 09 March 2003 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters ! subroutine halton(ndim, r) implicit none integer(pInt), intent(in) :: ndim real(pReal), intent(out), dimension(ndim) :: r integer(pInt), dimension(ndim) :: base integer(pInt) :: seed integer(pInt), dimension(1) :: value_halton call halton_memory ('GET', 'SEED', 1_pInt, value_halton) seed = value_halton(1) call halton_memory ('GET', 'BASE', ndim, base) call i_to_halton (seed, base, ndim, r) value_halton(1) = 1_pInt call halton_memory ('INC', 'SEED', 1_pInt, value_halton) end subroutine halton !******************************************************************************* ! HALTON_MEMORY sets or returns quantities associated with the Halton sequence. ! ! Parameters: ! Input, character (len = *) action_halton, 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_halton, 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_halton is 'SET' and action_halton is 'BASE', then NDIM is input, and ! is the number of entries in value_halton to be put into BASE. ! ! Input/output, integer value_halton(NDIM), contains a value. ! If action_halton is 'SET', then on input, value_halton contains values to be assigned ! to the internal variable. ! If action_halton is 'GET', then on output, value_halton contains the values of ! the specified internal variable. ! If action_halton is 'INC', then on input, value_halton contains the increment to ! be added to the specified internal variable. ! ! Modified: 09 March 2003 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters subroutine halton_memory (action_halton, name_halton, ndim, value_halton) implicit none character(len = *), intent(in) :: action_halton, name_halton integer(pInt), dimension(*), intent(inout) :: value_halton integer(pInt), allocatable, save, dimension(:) :: base logical, save :: first_call = .true. integer(pInt), intent(in) :: ndim integer(pInt):: i integer(pInt), save :: ndim_save = 0_pInt, seed = 1_pInt if (first_call) then ndim_save = 1_pInt allocate(base(ndim_save)) base(1) = 2_pInt first_call = .false. endif ! ! Set ! if(action_halton(1:1) == 'S' .or. action_halton(1:1) == 's') then if(name_halton(1:1) == 'B' .or. name_halton(1:1) == 'b') then if(ndim_save /= ndim) then deallocate(base) ndim_save = ndim allocate(base(ndim_save)) endif base(1:ndim) = value_halton(1:ndim) elseif(name_halton(1:1) == 'N' .or. name_halton(1:1) == 'n') then if(ndim_save /= value_halton(1)) then deallocate(base) ndim_save = value_halton(1) allocate(base(ndim_save)) do i = 1_pInt, ndim_save base(i) = prime (i) enddo else ndim_save = value_halton(1) endif elseif(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then seed = value_halton(1) endif ! ! Get ! elseif(action_halton(1:1) == 'G' .or. action_halton(1:1) == 'g') then if(name_halton(1:1) == 'B' .or. name_halton(1:1) == 'b') then if(ndim /= ndim_save) then deallocate(base) ndim_save = ndim allocate(base(ndim_save)) do i = 1_pInt, ndim_save base(i) = prime(i) enddo endif value_halton(1:ndim_save) = base(1:ndim_save) elseif(name_halton(1:1) == 'N' .or. name_halton(1:1) == 'n') then value_halton(1) = ndim_save elseif(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then value_halton(1) = seed endif ! ! Increment ! elseif(action_halton(1:1) == 'I' .or. action_halton(1:1) == 'i') then if(name_halton(1:1) == 'S' .or. name_halton(1:1) == 's') then seed = seed + value_halton(1) end if endif end subroutine halton_memory !******************************************************************************* ! HALTON_NDIM_SET sets the dimension for a Halton sequence. ! ! Parameters: ! Input, integer NDIM, the dimension of the Halton vectors. ! ! Modified: 26 February 2001 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters ! subroutine halton_ndim_set (ndim) implicit none integer(pInt), intent(in) :: ndim integer(pInt) :: value_halton(1) value_halton(1) = ndim call halton_memory ('SET', 'NDIM', 1_pInt, value_halton) end subroutine halton_ndim_set !******************************************************************************* ! HALTON_SEED_SET sets the "seed" for the Halton sequence. ! ! 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. ! ! Parameters: ! Input, integer SEED, the seed for the Halton sequence. ! ! Modified: 26 February 2001 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters ! subroutine halton_seed_set (seed) implicit none integer(pInt), parameter :: ndim = 1_pInt integer(pInt), intent(in) :: seed integer(pInt) :: value_halton(ndim) value_halton(1) = seed call halton_memory ('SET', 'SEED', ndim, value_halton) end subroutine halton_seed_set !******************************************************************************* ! 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. ! ! 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: 26 February 2001 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz RotersA subroutine i_to_halton (seed, base, ndim, r) use IO, only: IO_error implicit none integer(pInt), intent(in) :: ndim integer(pInt), intent(in), dimension(ndim) :: base real(pReal), dimension(ndim) :: base_inv integer(pInt), dimension(ndim) :: digit real(pReal), dimension(ndim), intent(out) ::r integer(pInt) :: seed integer(pInt), dimension(ndim) :: seed2 seed2(1:ndim) = abs(seed) r(1:ndim) = 0.0_pReal if (any (base(1:ndim) <= 1_pInt)) call IO_error(error_ID=405_pInt) base_inv(1:ndim) = 1.0_pReal / real (base(1:ndim), pReal) do while ( any ( seed2(1:ndim) /= 0_pInt) ) 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 end subroutine i_to_halton !******************************************************************************* ! PRIME returns any of the first PRIME_MAX prime numbers. ! ! Note: ! PRIME_MAX is 1500, and the largest prime stored is 12553. ! 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: 21 June 2002 ! Author: John Burkardt ! ! Modified: 29 April 2005 ! Author: Franz Roters ! function prime(n) use IO, only: IO_error implicit none integer(pInt), parameter :: prime_max = 1500_pInt integer(pInt), save :: icall = 0_pInt integer(pInt), intent(in) :: n integer(pInt), save, dimension(prime_max) :: npvec integer(pInt) prime if (icall == 0_pInt) then icall = 1_pInt npvec(1:100) = (/& 2_pInt, 3_pInt, 5_pInt, 7_pInt, 11_pInt, 13_pInt, 17_pInt, 19_pInt, 23_pInt, 29_pInt, & 31_pInt, 37_pInt, 41_pInt, 43_pInt, 47_pInt, 53_pInt, 59_pInt, 61_pInt, 67_pInt, 71_pInt, & 73_pInt, 79_pInt, 83_pInt, 89_pInt, 97_pInt, 101_pInt, 103_pInt, 107_pInt, 109_pInt, 113_pInt, & 127_pInt, 131_pInt, 137_pInt, 139_pInt, 149_pInt, 151_pInt, 157_pInt, 163_pInt, 167_pInt, 173_pInt, & 179_pInt, 181_pInt, 191_pInt, 193_pInt, 197_pInt, 199_pInt, 211_pInt, 223_pInt, 227_pInt, 229_pInt, & 233_pInt, 239_pInt, 241_pInt, 251_pInt, 257_pInt, 263_pInt, 269_pInt, 271_pInt, 277_pInt, 281_pInt, & 283_pInt, 293_pInt, 307_pInt, 311_pInt, 313_pInt, 317_pInt, 331_pInt, 337_pInt, 347_pInt, 349_pInt, & 353_pInt, 359_pInt, 367_pInt, 373_pInt, 379_pInt, 383_pInt, 389_pInt, 397_pInt, 401_pInt, 409_pInt, & 419_pInt, 421_pInt, 431_pInt, 433_pInt, 439_pInt, 443_pInt, 449_pInt, 457_pInt, 461_pInt, 463_pInt, & 467_pInt, 479_pInt, 487_pInt, 491_pInt, 499_pInt, 503_pInt, 509_pInt, 521_pInt, 523_pInt, 541_pInt/) npvec(101:200) = (/ & 547_pInt, 557_pInt, 563_pInt, 569_pInt, 571_pInt, 577_pInt, 587_pInt, 593_pInt, 599_pInt, 601_pInt, & 607_pInt, 613_pInt, 617_pInt, 619_pInt, 631_pInt, 641_pInt, 643_pInt, 647_pInt, 653_pInt, 659_pInt, & 661_pInt, 673_pInt, 677_pInt, 683_pInt, 691_pInt, 701_pInt, 709_pInt, 719_pInt, 727_pInt, 733_pInt, & 739_pInt, 743_pInt, 751_pInt, 757_pInt, 761_pInt, 769_pInt, 773_pInt, 787_pInt, 797_pInt, 809_pInt, & 811_pInt, 821_pInt, 823_pInt, 827_pInt, 829_pInt, 839_pInt, 853_pInt, 857_pInt, 859_pInt, 863_pInt, & 877_pInt, 881_pInt, 883_pInt, 887_pInt, 907_pInt, 911_pInt, 919_pInt, 929_pInt, 937_pInt, 941_pInt, & 947_pInt, 953_pInt, 967_pInt, 971_pInt, 977_pInt, 983_pInt, 991_pInt, 997_pInt, 1009_pInt, 1013_pInt, & 1019_pInt, 1021_pInt, 1031_pInt, 1033_pInt, 1039_pInt, 1049_pInt, 1051_pInt, 1061_pInt, 1063_pInt, 1069_pInt, & 1087_pInt, 1091_pInt, 1093_pInt, 1097_pInt, 1103_pInt, 1109_pInt, 1117_pInt, 1123_pInt, 1129_pInt, 1151_pInt, & 1153_pInt, 1163_pInt, 1171_pInt, 1181_pInt, 1187_pInt, 1193_pInt, 1201_pInt, 1213_pInt, 1217_pInt, 1223_pInt/) npvec(201:300) = (/ & 1229_pInt, 1231_pInt, 1237_pInt, 1249_pInt, 1259_pInt, 1277_pInt, 1279_pInt, 1283_pInt, 1289_pInt, 1291_pInt, & 1297_pInt, 1301_pInt, 1303_pInt, 1307_pInt, 1319_pInt, 1321_pInt, 1327_pInt, 1361_pInt, 1367_pInt, 1373_pInt, & 1381_pInt, 1399_pInt, 1409_pInt, 1423_pInt, 1427_pInt, 1429_pInt, 1433_pInt, 1439_pInt, 1447_pInt, 1451_pInt, & 1453_pInt, 1459_pInt, 1471_pInt, 1481_pInt, 1483_pInt, 1487_pInt, 1489_pInt, 1493_pInt, 1499_pInt, 1511_pInt, & 1523_pInt, 1531_pInt, 1543_pInt, 1549_pInt, 1553_pInt, 1559_pInt, 1567_pInt, 1571_pInt, 1579_pInt, 1583_pInt, & 1597_pInt, 1601_pInt, 1607_pInt, 1609_pInt, 1613_pInt, 1619_pInt, 1621_pInt, 1627_pInt, 1637_pInt, 1657_pInt, & 1663_pInt, 1667_pInt, 1669_pInt, 1693_pInt, 1697_pInt, 1699_pInt, 1709_pInt, 1721_pInt, 1723_pInt, 1733_pInt, & 1741_pInt, 1747_pInt, 1753_pInt, 1759_pInt, 1777_pInt, 1783_pInt, 1787_pInt, 1789_pInt, 1801_pInt, 1811_pInt, & 1823_pInt, 1831_pInt, 1847_pInt, 1861_pInt, 1867_pInt, 1871_pInt, 1873_pInt, 1877_pInt, 1879_pInt, 1889_pInt, & 1901_pInt, 1907_pInt, 1913_pInt, 1931_pInt, 1933_pInt, 1949_pInt, 1951_pInt, 1973_pInt, 1979_pInt, 1987_pInt/) npvec(301:400) = (/ & 1993_pInt, 1997_pInt, 1999_pInt, 2003_pInt, 2011_pInt, 2017_pInt, 2027_pInt, 2029_pInt, 2039_pInt, 2053_pInt, & 2063_pInt, 2069_pInt, 2081_pInt, 2083_pInt, 2087_pInt, 2089_pInt, 2099_pInt, 2111_pInt, 2113_pInt, 2129_pInt, & 2131_pInt, 2137_pInt, 2141_pInt, 2143_pInt, 2153_pInt, 2161_pInt, 2179_pInt, 2203_pInt, 2207_pInt, 2213_pInt, & 2221_pInt, 2237_pInt, 2239_pInt, 2243_pInt, 2251_pInt, 2267_pInt, 2269_pInt, 2273_pInt, 2281_pInt, 2287_pInt, & 2293_pInt, 2297_pInt, 2309_pInt, 2311_pInt, 2333_pInt, 2339_pInt, 2341_pInt, 2347_pInt, 2351_pInt, 2357_pInt, & 2371_pInt, 2377_pInt, 2381_pInt, 2383_pInt, 2389_pInt, 2393_pInt, 2399_pInt, 2411_pInt, 2417_pInt, 2423_pInt, & 2437_pInt, 2441_pInt, 2447_pInt, 2459_pInt, 2467_pInt, 2473_pInt, 2477_pInt, 2503_pInt, 2521_pInt, 2531_pInt, & 2539_pInt, 2543_pInt, 2549_pInt, 2551_pInt, 2557_pInt, 2579_pInt, 2591_pInt, 2593_pInt, 2609_pInt, 2617_pInt, & 2621_pInt, 2633_pInt, 2647_pInt, 2657_pInt, 2659_pInt, 2663_pInt, 2671_pInt, 2677_pInt, 2683_pInt, 2687_pInt, & 2689_pInt, 2693_pInt, 2699_pInt, 2707_pInt, 2711_pInt, 2713_pInt, 2719_pInt, 2729_pInt, 2731_pInt, 2741_pInt/) npvec(401:500) = (/ & 2749_pInt, 2753_pInt, 2767_pInt, 2777_pInt, 2789_pInt, 2791_pInt, 2797_pInt, 2801_pInt, 2803_pInt, 2819_pInt, & 2833_pInt, 2837_pInt, 2843_pInt, 2851_pInt, 2857_pInt, 2861_pInt, 2879_pInt, 2887_pInt, 2897_pInt, 2903_pInt, & 2909_pInt, 2917_pInt, 2927_pInt, 2939_pInt, 2953_pInt, 2957_pInt, 2963_pInt, 2969_pInt, 2971_pInt, 2999_pInt, & 3001_pInt, 3011_pInt, 3019_pInt, 3023_pInt, 3037_pInt, 3041_pInt, 3049_pInt, 3061_pInt, 3067_pInt, 3079_pInt, & 3083_pInt, 3089_pInt, 3109_pInt, 3119_pInt, 3121_pInt, 3137_pInt, 3163_pInt, 3167_pInt, 3169_pInt, 3181_pInt, & 3187_pInt, 3191_pInt, 3203_pInt, 3209_pInt, 3217_pInt, 3221_pInt, 3229_pInt, 3251_pInt, 3253_pInt, 3257_pInt, & 3259_pInt, 3271_pInt, 3299_pInt, 3301_pInt, 3307_pInt, 3313_pInt, 3319_pInt, 3323_pInt, 3329_pInt, 3331_pInt, & 3343_pInt, 3347_pInt, 3359_pInt, 3361_pInt, 3371_pInt, 3373_pInt, 3389_pInt, 3391_pInt, 3407_pInt, 3413_pInt, & 3433_pInt, 3449_pInt, 3457_pInt, 3461_pInt, 3463_pInt, 3467_pInt, 3469_pInt, 3491_pInt, 3499_pInt, 3511_pInt, & 3517_pInt, 3527_pInt, 3529_pInt, 3533_pInt, 3539_pInt, 3541_pInt, 3547_pInt, 3557_pInt, 3559_pInt, 3571_pInt/) npvec(501:600) = (/ & 3581_pInt, 3583_pInt, 3593_pInt, 3607_pInt, 3613_pInt, 3617_pInt, 3623_pInt, 3631_pInt, 3637_pInt, 3643_pInt, & 3659_pInt, 3671_pInt, 3673_pInt, 3677_pInt, 3691_pInt, 3697_pInt, 3701_pInt, 3709_pInt, 3719_pInt, 3727_pInt, & 3733_pInt, 3739_pInt, 3761_pInt, 3767_pInt, 3769_pInt, 3779_pInt, 3793_pInt, 3797_pInt, 3803_pInt, 3821_pInt, & 3823_pInt, 3833_pInt, 3847_pInt, 3851_pInt, 3853_pInt, 3863_pInt, 3877_pInt, 3881_pInt, 3889_pInt, 3907_pInt, & 3911_pInt, 3917_pInt, 3919_pInt, 3923_pInt, 3929_pInt, 3931_pInt, 3943_pInt, 3947_pInt, 3967_pInt, 3989_pInt, & 4001_pInt, 4003_pInt, 4007_pInt, 4013_pInt, 4019_pInt, 4021_pInt, 4027_pInt, 4049_pInt, 4051_pInt, 4057_pInt, & 4073_pInt, 4079_pInt, 4091_pInt, 4093_pInt, 4099_pInt, 4111_pInt, 4127_pInt, 4129_pInt, 4133_pInt, 4139_pInt, & 4153_pInt, 4157_pInt, 4159_pInt, 4177_pInt, 4201_pInt, 4211_pInt, 4217_pInt, 4219_pInt, 4229_pInt, 4231_pInt, & 4241_pInt, 4243_pInt, 4253_pInt, 4259_pInt, 4261_pInt, 4271_pInt, 4273_pInt, 4283_pInt, 4289_pInt, 4297_pInt, & 4327_pInt, 4337_pInt, 4339_pInt, 4349_pInt, 4357_pInt, 4363_pInt, 4373_pInt, 4391_pInt, 4397_pInt, 4409_pInt/) npvec(601:700) = (/ & 4421_pInt, 4423_pInt, 4441_pInt, 4447_pInt, 4451_pInt, 4457_pInt, 4463_pInt, 4481_pInt, 4483_pInt, 4493_pInt, & 4507_pInt, 4513_pInt, 4517_pInt, 4519_pInt, 4523_pInt, 4547_pInt, 4549_pInt, 4561_pInt, 4567_pInt, 4583_pInt, & 4591_pInt, 4597_pInt, 4603_pInt, 4621_pInt, 4637_pInt, 4639_pInt, 4643_pInt, 4649_pInt, 4651_pInt, 4657_pInt, & 4663_pInt, 4673_pInt, 4679_pInt, 4691_pInt, 4703_pInt, 4721_pInt, 4723_pInt, 4729_pInt, 4733_pInt, 4751_pInt, & 4759_pInt, 4783_pInt, 4787_pInt, 4789_pInt, 4793_pInt, 4799_pInt, 4801_pInt, 4813_pInt, 4817_pInt, 4831_pInt, & 4861_pInt, 4871_pInt, 4877_pInt, 4889_pInt, 4903_pInt, 4909_pInt, 4919_pInt, 4931_pInt, 4933_pInt, 4937_pInt, & 4943_pInt, 4951_pInt, 4957_pInt, 4967_pInt, 4969_pInt, 4973_pInt, 4987_pInt, 4993_pInt, 4999_pInt, 5003_pInt, & 5009_pInt, 5011_pInt, 5021_pInt, 5023_pInt, 5039_pInt, 5051_pInt, 5059_pInt, 5077_pInt, 5081_pInt, 5087_pInt, & 5099_pInt, 5101_pInt, 5107_pInt, 5113_pInt, 5119_pInt, 5147_pInt, 5153_pInt, 5167_pInt, 5171_pInt, 5179_pInt, & 5189_pInt, 5197_pInt, 5209_pInt, 5227_pInt, 5231_pInt, 5233_pInt, 5237_pInt, 5261_pInt, 5273_pInt, 5279_pInt/) npvec(701:800) = (/ & 5281_pInt, 5297_pInt, 5303_pInt, 5309_pInt, 5323_pInt, 5333_pInt, 5347_pInt, 5351_pInt, 5381_pInt, 5387_pInt, & 5393_pInt, 5399_pInt, 5407_pInt, 5413_pInt, 5417_pInt, 5419_pInt, 5431_pInt, 5437_pInt, 5441_pInt, 5443_pInt, & 5449_pInt, 5471_pInt, 5477_pInt, 5479_pInt, 5483_pInt, 5501_pInt, 5503_pInt, 5507_pInt, 5519_pInt, 5521_pInt, & 5527_pInt, 5531_pInt, 5557_pInt, 5563_pInt, 5569_pInt, 5573_pInt, 5581_pInt, 5591_pInt, 5623_pInt, 5639_pInt, & 5641_pInt, 5647_pInt, 5651_pInt, 5653_pInt, 5657_pInt, 5659_pInt, 5669_pInt, 5683_pInt, 5689_pInt, 5693_pInt, & 5701_pInt, 5711_pInt, 5717_pInt, 5737_pInt, 5741_pInt, 5743_pInt, 5749_pInt, 5779_pInt, 5783_pInt, 5791_pInt, & 5801_pInt, 5807_pInt, 5813_pInt, 5821_pInt, 5827_pInt, 5839_pInt, 5843_pInt, 5849_pInt, 5851_pInt, 5857_pInt, & 5861_pInt, 5867_pInt, 5869_pInt, 5879_pInt, 5881_pInt, 5897_pInt, 5903_pInt, 5923_pInt, 5927_pInt, 5939_pInt, & 5953_pInt, 5981_pInt, 5987_pInt, 6007_pInt, 6011_pInt, 6029_pInt, 6037_pInt, 6043_pInt, 6047_pInt, 6053_pInt, & 6067_pInt, 6073_pInt, 6079_pInt, 6089_pInt, 6091_pInt, 6101_pInt, 6113_pInt, 6121_pInt, 6131_pInt, 6133_pInt/) npvec(801:900) = (/ & 6143_pInt, 6151_pInt, 6163_pInt, 6173_pInt, 6197_pInt, 6199_pInt, 6203_pInt, 6211_pInt, 6217_pInt, 6221_pInt, & 6229_pInt, 6247_pInt, 6257_pInt, 6263_pInt, 6269_pInt, 6271_pInt, 6277_pInt, 6287_pInt, 6299_pInt, 6301_pInt, & 6311_pInt, 6317_pInt, 6323_pInt, 6329_pInt, 6337_pInt, 6343_pInt, 6353_pInt, 6359_pInt, 6361_pInt, 6367_pInt, & 6373_pInt, 6379_pInt, 6389_pInt, 6397_pInt, 6421_pInt, 6427_pInt, 6449_pInt, 6451_pInt, 6469_pInt, 6473_pInt, & 6481_pInt, 6491_pInt, 6521_pInt, 6529_pInt, 6547_pInt, 6551_pInt, 6553_pInt, 6563_pInt, 6569_pInt, 6571_pInt, & 6577_pInt, 6581_pInt, 6599_pInt, 6607_pInt, 6619_pInt, 6637_pInt, 6653_pInt, 6659_pInt, 6661_pInt, 6673_pInt, & 6679_pInt, 6689_pInt, 6691_pInt, 6701_pInt, 6703_pInt, 6709_pInt, 6719_pInt, 6733_pInt, 6737_pInt, 6761_pInt, & 6763_pInt, 6779_pInt, 6781_pInt, 6791_pInt, 6793_pInt, 6803_pInt, 6823_pInt, 6827_pInt, 6829_pInt, 6833_pInt, & 6841_pInt, 6857_pInt, 6863_pInt, 6869_pInt, 6871_pInt, 6883_pInt, 6899_pInt, 6907_pInt, 6911_pInt, 6917_pInt, & 6947_pInt, 6949_pInt, 6959_pInt, 6961_pInt, 6967_pInt, 6971_pInt, 6977_pInt, 6983_pInt, 6991_pInt, 6997_pInt/) npvec(901:1000) = (/ & 7001_pInt, 7013_pInt, 7019_pInt, 7027_pInt, 7039_pInt, 7043_pInt, 7057_pInt, 7069_pInt, 7079_pInt, 7103_pInt, & 7109_pInt, 7121_pInt, 7127_pInt, 7129_pInt, 7151_pInt, 7159_pInt, 7177_pInt, 7187_pInt, 7193_pInt, 7207_pInt, & 7211_pInt, 7213_pInt, 7219_pInt, 7229_pInt, 7237_pInt, 7243_pInt, 7247_pInt, 7253_pInt, 7283_pInt, 7297_pInt, & 7307_pInt, 7309_pInt, 7321_pInt, 7331_pInt, 7333_pInt, 7349_pInt, 7351_pInt, 7369_pInt, 7393_pInt, 7411_pInt, & 7417_pInt, 7433_pInt, 7451_pInt, 7457_pInt, 7459_pInt, 7477_pInt, 7481_pInt, 7487_pInt, 7489_pInt, 7499_pInt, & 7507_pInt, 7517_pInt, 7523_pInt, 7529_pInt, 7537_pInt, 7541_pInt, 7547_pInt, 7549_pInt, 7559_pInt, 7561_pInt, & 7573_pInt, 7577_pInt, 7583_pInt, 7589_pInt, 7591_pInt, 7603_pInt, 7607_pInt, 7621_pInt, 7639_pInt, 7643_pInt, & 7649_pInt, 7669_pInt, 7673_pInt, 7681_pInt, 7687_pInt, 7691_pInt, 7699_pInt, 7703_pInt, 7717_pInt, 7723_pInt, & 7727_pInt, 7741_pInt, 7753_pInt, 7757_pInt, 7759_pInt, 7789_pInt, 7793_pInt, 7817_pInt, 7823_pInt, 7829_pInt, & 7841_pInt, 7853_pInt, 7867_pInt, 7873_pInt, 7877_pInt, 7879_pInt, 7883_pInt, 7901_pInt, 7907_pInt, 7919_pInt/) npvec(1001:1100) = (/ & 7927_pInt, 7933_pInt, 7937_pInt, 7949_pInt, 7951_pInt, 7963_pInt, 7993_pInt, 8009_pInt, 8011_pInt, 8017_pInt, & 8039_pInt, 8053_pInt, 8059_pInt, 8069_pInt, 8081_pInt, 8087_pInt, 8089_pInt, 8093_pInt, 8101_pInt, 8111_pInt, & 8117_pInt, 8123_pInt, 8147_pInt, 8161_pInt, 8167_pInt, 8171_pInt, 8179_pInt, 8191_pInt, 8209_pInt, 8219_pInt, & 8221_pInt, 8231_pInt, 8233_pInt, 8237_pInt, 8243_pInt, 8263_pInt, 8269_pInt, 8273_pInt, 8287_pInt, 8291_pInt, & 8293_pInt, 8297_pInt, 8311_pInt, 8317_pInt, 8329_pInt, 8353_pInt, 8363_pInt, 8369_pInt, 8377_pInt, 8387_pInt, & 8389_pInt, 8419_pInt, 8423_pInt, 8429_pInt, 8431_pInt, 8443_pInt, 8447_pInt, 8461_pInt, 8467_pInt, 8501_pInt, & 8513_pInt, 8521_pInt, 8527_pInt, 8537_pInt, 8539_pInt, 8543_pInt, 8563_pInt, 8573_pInt, 8581_pInt, 8597_pInt, & 8599_pInt, 8609_pInt, 8623_pInt, 8627_pInt, 8629_pInt, 8641_pInt, 8647_pInt, 8663_pInt, 8669_pInt, 8677_pInt, & 8681_pInt, 8689_pInt, 8693_pInt, 8699_pInt, 8707_pInt, 8713_pInt, 8719_pInt, 8731_pInt, 8737_pInt, 8741_pInt, & 8747_pInt, 8753_pInt, 8761_pInt, 8779_pInt, 8783_pInt, 8803_pInt, 8807_pInt, 8819_pInt, 8821_pInt, 8831_pInt/) npvec(1101:1200) = (/ & 8837_pInt, 8839_pInt, 8849_pInt, 8861_pInt, 8863_pInt, 8867_pInt, 8887_pInt, 8893_pInt, 8923_pInt, 8929_pInt, & 8933_pInt, 8941_pInt, 8951_pInt, 8963_pInt, 8969_pInt, 8971_pInt, 8999_pInt, 9001_pInt, 9007_pInt, 9011_pInt, & 9013_pInt, 9029_pInt, 9041_pInt, 9043_pInt, 9049_pInt, 9059_pInt, 9067_pInt, 9091_pInt, 9103_pInt, 9109_pInt, & 9127_pInt, 9133_pInt, 9137_pInt, 9151_pInt, 9157_pInt, 9161_pInt, 9173_pInt, 9181_pInt, 9187_pInt, 9199_pInt, & 9203_pInt, 9209_pInt, 9221_pInt, 9227_pInt, 9239_pInt, 9241_pInt, 9257_pInt, 9277_pInt, 9281_pInt, 9283_pInt, & 9293_pInt, 9311_pInt, 9319_pInt, 9323_pInt, 9337_pInt, 9341_pInt, 9343_pInt, 9349_pInt, 9371_pInt, 9377_pInt, & 9391_pInt, 9397_pInt, 9403_pInt, 9413_pInt, 9419_pInt, 9421_pInt, 9431_pInt, 9433_pInt, 9437_pInt, 9439_pInt, & 9461_pInt, 9463_pInt, 9467_pInt, 9473_pInt, 9479_pInt, 9491_pInt, 9497_pInt, 9511_pInt, 9521_pInt, 9533_pInt, & 9539_pInt, 9547_pInt, 9551_pInt, 9587_pInt, 9601_pInt, 9613_pInt, 9619_pInt, 9623_pInt, 9629_pInt, 9631_pInt, & 9643_pInt, 9649_pInt, 9661_pInt, 9677_pInt, 9679_pInt, 9689_pInt, 9697_pInt, 9719_pInt, 9721_pInt, 9733_pInt/) npvec(1201:1300) = (/ & 9739_pInt, 9743_pInt, 9749_pInt, 9767_pInt, 9769_pInt, 9781_pInt, 9787_pInt, 9791_pInt, 9803_pInt, 9811_pInt, & 9817_pInt, 9829_pInt, 9833_pInt, 9839_pInt, 9851_pInt, 9857_pInt, 9859_pInt, 9871_pInt, 9883_pInt, 9887_pInt, & 9901_pInt, 9907_pInt, 9923_pInt, 9929_pInt, 9931_pInt, 9941_pInt, 9949_pInt, 9967_pInt, 9973_pInt,10007_pInt, & 10009_pInt,10037_pInt,10039_pInt,10061_pInt,10067_pInt,10069_pInt,10079_pInt,10091_pInt,10093_pInt,10099_pInt, & 10103_pInt,10111_pInt,10133_pInt,10139_pInt,10141_pInt,10151_pInt,10159_pInt,10163_pInt,10169_pInt,10177_pInt, & 10181_pInt,10193_pInt,10211_pInt,10223_pInt,10243_pInt,10247_pInt,10253_pInt,10259_pInt,10267_pInt,10271_pInt, & 10273_pInt,10289_pInt,10301_pInt,10303_pInt,10313_pInt,10321_pInt,10331_pInt,10333_pInt,10337_pInt,10343_pInt, & 10357_pInt,10369_pInt,10391_pInt,10399_pInt,10427_pInt,10429_pInt,10433_pInt,10453_pInt,10457_pInt,10459_pInt, & 10463_pInt,10477_pInt,10487_pInt,10499_pInt,10501_pInt,10513_pInt,10529_pInt,10531_pInt,10559_pInt,10567_pInt, & 10589_pInt,10597_pInt,10601_pInt,10607_pInt,10613_pInt,10627_pInt,10631_pInt,10639_pInt,10651_pInt,10657_pInt/) npvec(1301:1400) = (/ & 10663_pInt,10667_pInt,10687_pInt,10691_pInt,10709_pInt,10711_pInt,10723_pInt,10729_pInt,10733_pInt,10739_pInt, & 10753_pInt,10771_pInt,10781_pInt,10789_pInt,10799_pInt,10831_pInt,10837_pInt,10847_pInt,10853_pInt,10859_pInt, & 10861_pInt,10867_pInt,10883_pInt,10889_pInt,10891_pInt,10903_pInt,10909_pInt,19037_pInt,10939_pInt,10949_pInt, & 10957_pInt,10973_pInt,10979_pInt,10987_pInt,10993_pInt,11003_pInt,11027_pInt,11047_pInt,11057_pInt,11059_pInt, & 11069_pInt,11071_pInt,11083_pInt,11087_pInt,11093_pInt,11113_pInt,11117_pInt,11119_pInt,11131_pInt,11149_pInt, & 11159_pInt,11161_pInt,11171_pInt,11173_pInt,11177_pInt,11197_pInt,11213_pInt,11239_pInt,11243_pInt,11251_pInt, & 11257_pInt,11261_pInt,11273_pInt,11279_pInt,11287_pInt,11299_pInt,11311_pInt,11317_pInt,11321_pInt,11329_pInt, & 11351_pInt,11353_pInt,11369_pInt,11383_pInt,11393_pInt,11399_pInt,11411_pInt,11423_pInt,11437_pInt,11443_pInt, & 11447_pInt,11467_pInt,11471_pInt,11483_pInt,11489_pInt,11491_pInt,11497_pInt,11503_pInt,11519_pInt,11527_pInt, & 11549_pInt,11551_pInt,11579_pInt,11587_pInt,11593_pInt,11597_pInt,11617_pInt,11621_pInt,11633_pInt,11657_pInt/) npvec(1401:1500) = (/ & 11677_pInt,11681_pInt,11689_pInt,11699_pInt,11701_pInt,11717_pInt,11719_pInt,11731_pInt,11743_pInt,11777_pInt, & 11779_pInt,11783_pInt,11789_pInt,11801_pInt,11807_pInt,11813_pInt,11821_pInt,11827_pInt,11831_pInt,11833_pInt, & 11839_pInt,11863_pInt,11867_pInt,11887_pInt,11897_pInt,11903_pInt,11909_pInt,11923_pInt,11927_pInt,11933_pInt, & 11939_pInt,11941_pInt,11953_pInt,11959_pInt,11969_pInt,11971_pInt,11981_pInt,11987_pInt,12007_pInt,12011_pInt, & 12037_pInt,12041_pInt,12043_pInt,12049_pInt,12071_pInt,12073_pInt,12097_pInt,12101_pInt,12107_pInt,12109_pInt, & 12113_pInt,12119_pInt,12143_pInt,12149_pInt,12157_pInt,12161_pInt,12163_pInt,12197_pInt,12203_pInt,12211_pInt, & 12227_pInt,12239_pInt,12241_pInt,12251_pInt,12253_pInt,12263_pInt,12269_pInt,12277_pInt,12281_pInt,12289_pInt, & 12301_pInt,12323_pInt,12329_pInt,12343_pInt,12347_pInt,12373_pInt,12377_pInt,12379_pInt,12391_pInt,12401_pInt, & 12409_pInt,12413_pInt,12421_pInt,12433_pInt,12437_pInt,12451_pInt,12457_pInt,12473_pInt,12479_pInt,12487_pInt, & 12491_pInt,12497_pInt,12503_pInt,12511_pInt,12517_pInt,12527_pInt,12539_pInt,12541_pInt,12547_pInt,12553_pInt/) endif if(n == -1_pInt) then prime = prime_max else if (n == 0_pInt) then prime = 1_pInt else if (n <= prime_max) then prime = npvec(n) else call IO_error(error_ID=406_pInt) end if end function prime !************************************************************************** ! volume of tetrahedron given by four vertices !************************************************************************** pure function math_volTetrahedron(v1,v2,v3,v4) implicit none real(pReal) math_volTetrahedron real(pReal), dimension (3), intent(in) :: v1,v2,v3,v4 real(pReal), dimension (3,3) :: m m(1:3,1) = v1-v2 m(1:3,2) = v2-v3 m(1:3,3) = v3-v4 math_volTetrahedron = math_det33(m)/6.0_pReal end function math_volTetrahedron !************************************************************************** ! rotate 33 tensor forward !************************************************************************** pure function math_rotate_forward33(tensor,rot_tensor) implicit none real(pReal), dimension(3,3) :: math_rotate_forward33 real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor math_rotate_forward33 = math_mul33x33(rot_tensor,& math_mul33x33(tensor,math_transpose33(rot_tensor))) end function math_rotate_forward33 !************************************************************************** ! rotate 33 tensor backward !************************************************************************** pure function math_rotate_backward33(tensor,rot_tensor) implicit none real(pReal), dimension(3,3) :: math_rotate_backward33 real(pReal), dimension(3,3), intent(in) :: tensor, rot_tensor math_rotate_backward33 = math_mul33x33(math_transpose33(rot_tensor),& math_mul33x33(tensor,rot_tensor)) end function math_rotate_backward33 !************************************************************************** ! rotate 3333 tensor ! C'_ijkl=g_im*g_jn*g_ko*g_lp*C_mnop !************************************************************************** pure function math_rotate_forward3333(tensor,rot_tensor) implicit none real(pReal), dimension(3,3,3,3) :: math_rotate_forward3333 real(pReal), dimension(3,3), intent(in) :: rot_tensor real(pReal), dimension(3,3,3,3), intent(in) :: tensor integer(pInt) :: i,j,k,l,m,n,o,p math_rotate_forward3333= 0.0_pReal do i = 1_pInt,3_pInt; do j = 1_pInt,3_pInt; do k = 1_pInt,3_pInt; do l = 1_pInt,3_pInt do m = 1_pInt,3_pInt; do n = 1_pInt,3_pInt; do o = 1_pInt,3_pInt; do p = 1_pInt,3_pInt math_rotate_forward3333(i,j,k,l) = tensor(i,j,k,l)+rot_tensor(m,i)*rot_tensor(n,j)*& rot_tensor(o,k)*rot_tensor(p,l)*tensor(m,n,o,p) enddo; enddo; enddo; enddo; enddo; enddo; enddo; enddo end function math_rotate_forward3333 !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Functions below are taken from the old postprocessingMath.f90 ! mostly they are used in combination with f2py to build fortran !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! put the next two funtions into mesh? function mesh_location(idx,resolution) ! small helper functions for indexing ! CAREFULL, index and location runs from 0 to N-1 (python style) integer(pInt), intent(in) :: idx integer(pInt), intent(in), dimension(3) :: resolution integer(pInt), dimension(3) :: mesh_location mesh_location = (/modulo(idx/ resolution(3) / resolution(2),resolution(1)), & modulo(idx/ resolution(3), resolution(2)), & modulo(idx, resolution(3))/) end function mesh_location function mesh_index(location,resolution) ! small helper functions for indexing ! CAREFULL, index and location runs from 0 to N-1 (python style) integer(pInt), intent(in), dimension(3) :: resolution, location integer(pInt) :: mesh_index mesh_index = modulo(location(3), resolution(3)) +& (modulo(location(2), resolution(2)))*resolution(3) +& (modulo(location(1), resolution(1)))*resolution(3)*resolution(2) end function mesh_index !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine volume_compare(res,geomdim,defgrad,nodes,volume_mismatch) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate the mismatch between volume of reconstructed (compatible ! cube and determinant of defgrad at the FP use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(res(1), res(2), res(3), 3,3) :: defgrad real(pReal), intent(in), dimension(res(1)+1_pInt,res(2)+1_pInt,res(3)+1_pInt,3) :: nodes ! output variables real(pReal), intent(out), dimension(res(1), res(2), res(3)) :: volume_mismatch ! other variables real(pReal), dimension(8,3) :: coords integer(pInt) i,j,k real(pReal) vol_initial if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Calculating volume mismatch' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif vol_initial = geomdim(1)*geomdim(2)*geomdim(3)/(real(res(1)*res(2)*res(3), pReal)) do k = 1_pInt,res(3) do j = 1_pInt,res(2) do i = 1_pInt,res(1) coords(1,1:3) = nodes(i, j, k ,1:3) coords(2,1:3) = nodes(i+1_pInt,j, k ,1:3) coords(3,1:3) = nodes(i+1_pInt,j+1_pInt,k ,1:3) coords(4,1:3) = nodes(i, j+1_pInt,k ,1:3) coords(5,1:3) = nodes(i, j, k+1_pInt,1:3) coords(6,1:3) = nodes(i+1_pInt,j, k+1_pInt,1:3) coords(7,1:3) = nodes(i+1_pInt,j+1_pInt,k+1_pInt,1:3) coords(8,1:3) = nodes(i, j+1_pInt,k+1_pInt,1:3) volume_mismatch(i,j,k) = abs(math_volTetrahedron(coords(7,1:3),coords(1,1:3),coords(8,1:3),coords(4,1:3))) & + abs(math_volTetrahedron(coords(7,1:3),coords(1,1:3),coords(8,1:3),coords(5,1:3))) & + abs(math_volTetrahedron(coords(7,1:3),coords(1,1:3),coords(3,1:3),coords(4,1:3))) & + abs(math_volTetrahedron(coords(7,1:3),coords(1,1:3),coords(3,1:3),coords(2,1:3))) & + abs(math_volTetrahedron(coords(7,1:3),coords(5,1:3),coords(2,1:3),coords(6,1:3))) & + abs(math_volTetrahedron(coords(7,1:3),coords(5,1:3),coords(2,1:3),coords(1,1:3))) volume_mismatch(i,j,k) = volume_mismatch(i,j,k)/math_det33(defgrad(i,j,k,1:3,1:3)) enddo; enddo; enddo volume_mismatch = volume_mismatch/vol_initial end subroutine volume_compare !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine shape_compare(res,geomdim,defgrad,nodes,centroids,shape_mismatch) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate the mismatch between the vectors from the central point to ! the corners of reconstructed (combatible) volume element and the vectors calculated by deforming ! the initial volume element with the current deformation gradient use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(res(1), res(2), res(3), 3,3) :: defgrad real(pReal), intent(in), dimension(res(1)+1_pInt,res(2)+1_pInt,res(3)+1_pInt,3) :: nodes real(pReal), intent(in), dimension(res(1), res(2), res(3), 3) :: centroids ! output variables real(pReal), intent(out), dimension(res(1), res(2), res(3)) :: shape_mismatch ! other variables real(pReal), dimension(8,3) :: coords_initial integer(pInt) i,j,k if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Calculating shape mismatch' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif coords_initial(1,1:3) = (/-geomdim(1)/2.0_pReal/real(res(1),pReal),& -geomdim(2)/2.0_pReal/real(res(2),pReal),& -geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(2,1:3) = (/+geomdim(1)/2.0_pReal/real(res(1),pReal),& -geomdim(2)/2.0_pReal/real(res(2),pReal),& -geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(3,1:3) = (/+geomdim(1)/2.0_pReal/real(res(1),pReal),& +geomdim(2)/2.0_pReal/real(res(2),pReal),& -geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(4,1:3) = (/-geomdim(1)/2.0_pReal/real(res(1),pReal),& +geomdim(2)/2.0_pReal/real(res(2),pReal),& -geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(5,1:3) = (/-geomdim(1)/2.0_pReal/real(res(1),pReal),& -geomdim(2)/2.0_pReal/real(res(2),pReal),& +geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(6,1:3) = (/+geomdim(1)/2.0_pReal/real(res(1),pReal),& -geomdim(2)/2.0_pReal/real(res(2),pReal),& +geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(7,1:3) = (/+geomdim(1)/2.0_pReal/real(res(1),pReal),& +geomdim(2)/2.0_pReal/real(res(2),pReal),& +geomdim(3)/2.0_pReal/real(res(3),pReal)/) coords_initial(8,1:3) = (/-geomdim(1)/2.0_pReal/real(res(1),pReal),& +geomdim(2)/2.0_pReal/real(res(2),pReal),& +geomdim(3)/2.0_pReal/real(res(3),pReal)/) do i=1_pInt,8_pInt enddo do k = 1_pInt,res(3) do j = 1_pInt,res(2) do i = 1_pInt,res(1) shape_mismatch(i,j,k) = & sqrt(sum((nodes(i, j, k, 1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(1,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i+1_pInt,j, k, 1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(2,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i+1_pInt,j+1_pInt,k, 1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(3,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i, j+1_pInt,k, 1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(4,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i, j, k+1_pInt,1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(5,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i+1_pInt,j, k+1_pInt,1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(6,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i+1_pInt,j+1_pInt,k+1_pInt,1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(7,1:3)))**2.0_pReal))& + sqrt(sum((nodes(i, j+1_pInt,k+1_pInt,1:3) - centroids(i,j,k,1:3)& - matmul(defgrad(i,j,k,1:3,1:3), coords_initial(8,1:3)))**2.0_pReal)) enddo; enddo; enddo end subroutine shape_compare !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine mesh_regular_grid(res,geomdim,defgrad_av,centroids,nodes) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to build mesh of (distorted) cubes for given coordinates (= center of the cubes) ! use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(3,3) :: defgrad_av real(pReal), intent(in), dimension(res(1), res(2), res(3), 3) :: centroids ! output variables real(pReal),intent(out), dimension(res(1)+1_pInt,res(2)+1_pInt,res(3)+1_pInt,3) :: nodes ! variables with dimension depending on input real(pReal), dimension(res(1)+2_pInt,res(2)+2_pInt,res(3)+2_pInt,3) :: wrappedCentroids ! other variables integer(pInt) :: i,j,k,n integer(pInt), dimension(3), parameter :: diag = 1_pInt integer(pInt), dimension(3) :: shift = 0_pInt, lookup = 0_pInt, me = 0_pInt integer(pInt), dimension(3,8) :: neighbor = reshape((/ & 0_pInt, 0_pInt, 0_pInt, & 1_pInt, 0_pInt, 0_pInt, & 1_pInt, 1_pInt, 0_pInt, & 0_pInt, 1_pInt, 0_pInt, & 0_pInt, 0_pInt, 1_pInt, & 1_pInt, 0_pInt, 1_pInt, & 1_pInt, 1_pInt, 1_pInt, & 0_pInt, 1_pInt, 1_pInt & /), & (/3,8/)) if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Meshing cubes around centroids' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif nodes = 0.0_pReal wrappedCentroids = 0.0_pReal wrappedCentroids(2_pInt:res(1)+1_pInt,2_pInt:res(2)+1_pInt,2_pInt:res(3)+1_pInt,1:3) = centroids do k = 0_pInt,res(3)+1_pInt do j = 0_pInt,res(2)+1_pInt do i = 0_pInt,res(1)+1_pInt if (k==0_pInt .or. k==res(3)+1_pInt .or. & ! z skin j==0_pInt .or. j==res(2)+1_pInt .or. & ! y skin i==0_pInt .or. i==res(1)+1_pInt ) then ! x skin me = (/i,j,k/) ! me on skin shift = sign(abs(res+diag-2_pInt*me)/(res+diag),res+diag-2_pInt*me) lookup = me-diag+shift*res wrappedCentroids(i+1_pInt,j+1_pInt,k+1_pInt,1:3) = & centroids(lookup(1)+1_pInt,lookup(2)+1_pInt,lookup(3)+1_pInt,1:3) - & matmul(defgrad_av, shift*geomdim) endif enddo; enddo; enddo do k = 0_pInt,res(3) do j = 0_pInt,res(2) do i = 0_pInt,res(1) do n = 1_pInt,8_pInt nodes(i+1_pInt,j+1_pInt,k+1_pInt,1:3) = & nodes(i+1_pInt,j+1_pInt,k+1_pInt,1:3) + wrappedCentroids(i+1_pInt+neighbor(1_pInt,n), & j+1_pInt+neighbor(2,n), & k+1_pInt+neighbor(3,n),1:3) enddo; enddo; enddo; enddo nodes = nodes/8.0_pReal end subroutine mesh_regular_grid !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine deformed_linear(res,geomdim,defgrad_av,defgrad,coord) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate coordinates in current configuration for given defgrad ! using linear interpolation (blurres out high frequency defomation) ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(3,3) :: defgrad_av real(pReal), intent(in), dimension( res(1),res(2),res(3),3,3) :: defgrad ! output variables real(pReal), intent(out), dimension( res(1),res(2),res(3),3) :: coord ! variables with dimension depending on input real(pReal), dimension( 8,res(1),res(2),res(3),3) :: coord_avgOrder ! other variables real(pReal), dimension(3) :: myStep, fones = 1.0_pReal, parameter_coords, negative, positive, offset_coords integer(pInt), dimension(3) :: rear, init, ones = 1_pInt, oppo, me integer(pInt) i, j, k, s, o integer(pInt), dimension(3,8) :: corner = reshape([ & 0_pInt, 0_pInt, 0_pInt,& 1_pInt, 0_pInt, 0_pInt,& 1_pInt, 1_pInt, 0_pInt,& 0_pInt, 1_pInt, 0_pInt,& 1_pInt, 1_pInt, 1_pInt,& 0_pInt, 1_pInt, 1_pInt,& 0_pInt, 0_pInt, 1_pInt,& 1_pInt, 0_pInt, 1_pInt & ],[3,8]) integer(pInt), dimension(3,8) :: step = reshape([& 1_pInt, 1_pInt, 1_pInt,& -1_pInt, 1_pInt, 1_pInt,& -1_pInt,-1_pInt, 1_pInt,& 1_pInt,-1_pInt, 1_pInt,& -1_pInt,-1_pInt,-1_pInt,& 1_pInt,-1_pInt,-1_pInt,& 1_pInt, 1_pInt,-1_pInt,& -1_pInt, 1_pInt,-1_pInt & ], [3,8]) integer(pInt), dimension(3,6) :: order = reshape([ & 1_pInt, 2_pInt, 3_pInt,& 1_pInt, 3_pInt, 2_pInt,& 2_pInt, 1_pInt, 3_pInt,& 2_pInt, 3_pInt, 1_pInt,& 3_pInt, 1_pInt, 2_pInt,& 3_pInt, 2_pInt, 1_pInt & ], [3,6]) coord_avgOrder = 0.0_pReal do s = 0_pInt, 7_pInt ! corners (from 0 to 7) init = corner(:,s+1_pInt)*(res-ones) +ones oppo = corner(:,mod((s+4_pInt),8_pInt)+1_pInt)*(res-ones) +ones do o=1_pInt,6_pInt ! orders (from 1 to 6) coord = 0_pReal do k = init(order(3,o)), oppo(order(3,o)), step(order(3,o),s+1_pInt) rear(order(2,o)) = init(order(2,o)) do j = init(order(2,o)), oppo(order(2,o)), step(order(2,o),s+1_pInt) rear(order(1,o)) = init(order(1,o)) do i = init(order(1,o)), oppo(order(1,o)), step(order(1,o),s+1_pInt) me(order(1:3,o)) = [i,j,k] if ( all(me==init)) then coord(me(1),me(2),me(3),1:3) = geomdim * (matmul(defgrad_av,real(corner(1:3,s+1),pReal)) + & matmul(defgrad(me(1),me(2),me(3),1:3,1:3),0.5_pReal*real(step(1:3,s+1_pInt)/res,pReal))) else myStep = (me-rear)*geomdim/res coord(me(1),me(2),me(3),1:3) = coord(rear(1),rear(2),rear(3),1:3) + & 0.5_pReal*matmul(defgrad(me(1),me(2),me(3),1:3,1:3) + & defgrad(rear(1),rear(2),rear(3),1:3,1:3),myStep) endif rear = me enddo; enddo; enddo coord_avgOrder(s+1_pInt,1:res(1),1:res(2),1:res(3),1:3) = & coord_avgOrder(s+1_pInt,1:res(1),1:res(2),1:res(3),1:3) + coord/6.0_pReal enddo offset_coords = coord_avgOrder(s+1,1,1,1,1:3) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) coord_avgOrder(s+1,i,j,k,1:3) = coord_avgOrder(s+1,i,j,k,1:3) - offset_coords enddo; enddo; enddo enddo do k = 0_pInt, res(3)-1_pInt do j = 0_pInt, res(2)-1_pInt do i = 0_pInt, res(1)-1_pInt parameter_coords = (2.0_pReal*real([i,j,k]+1,pReal)-real(res,pReal))/(real(res,pReal)) positive = fones + parameter_coords negative = fones - parameter_coords coord(i+1_pInt,j+1_pInt,k+1_pInt,1:3)& =(coord_avgOrder(1,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *negative(1)*negative(2)*negative(3)& + coord_avgOrder(2,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *positive(1)*negative(2)*negative(3)& + coord_avgOrder(3,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *positive(1)*positive(2)*negative(3)& + coord_avgOrder(4,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *negative(1)*positive(2)*negative(3)& + coord_avgOrder(5,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *positive(1)*positive(2)*positive(3)& + coord_avgOrder(6,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *negative(1)*positive(2)*positive(3)& + coord_avgOrder(7,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *negative(1)*negative(2)*positive(3)& + coord_avgOrder(8,i+1_pInt,j+1_pInt,k+1_pInt,1:3) *positive(1)*negative(2)*positive(3))*0.125_pReal enddo; enddo; enddo offset_coords = matmul(defgrad(1,1,1,1:3,1:3),geomdim/real(res, pReal)/2.0_pReal) - coord(1,1,1,1:3) do k = 1_pInt, res(3) do j = 1_pInt, res(2) do i = 1_pInt, res(1) coord(i,j,k,1:3) = coord(i,j,k,1:3)+ offset_coords enddo; enddo; enddo end subroutine deformed_linear #ifdef Spectral !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine deformed_fft(res,geomdim,defgrad_av,scaling,defgrad,coords) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Routine to calculate coordinates in current configuration for given defgrad ! using integration in Fourier space (more accurate than deformed(...)) ! use IO, only: IO_error use numerics, only: fftw_timelimit, fftw_planner_flag use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(3,3) :: defgrad_av real(pReal), intent(in) :: scaling real(pReal), intent(in), dimension(res(1), res(2),res(3),3,3) :: defgrad ! output variables real(pReal), intent(out), dimension(res(1), res(2),res(3),3) :: coords ! allocatable arrays for fftw c routines type(C_PTR) :: fftw_forth, fftw_back type(C_PTR) :: coords_fftw, defgrad_fftw real(pReal), dimension(:,:,:,:,:), pointer :: defgrad_real complex(pReal), dimension(:,:,:,:,:), pointer :: defgrad_fourier real(pReal), dimension(:,:,:,:), pointer :: coords_real complex(pReal), dimension(:,:,:,:), pointer :: coords_fourier ! other variables integer(pInt) :: i, j, k, m, res1_red integer(pInt), dimension(3) :: k_s real(pReal), dimension(3) :: step, offset_coords, integrator integrator = geomdim / 2.0_pReal / pi ! see notes where it is used if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Restore geometry using FFT-based integration' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif res1_red = res(1)/2_pInt + 1_pInt ! size of complex array in first dimension (c2r, r2c) step = geomdim/real(res, pReal) if (pReal /= C_DOUBLE .or. pInt /= C_INT) call IO_error(error_ID=808_pInt) call fftw_set_timelimit(fftw_timelimit) defgrad_fftw = fftw_alloc_complex(int(res1_red *res(2)*res(3)*9_pInt,C_SIZE_T)) !C_SIZE_T is of type integer(8) call c_f_pointer(defgrad_fftw, defgrad_real, [res(1)+2_pInt,res(2),res(3),3_pInt,3_pInt]) call c_f_pointer(defgrad_fftw, defgrad_fourier,[res1_red ,res(2),res(3),3_pInt,3_pInt]) coords_fftw = fftw_alloc_complex(int(res1_red *res(2)*res(3)*3_pInt,C_SIZE_T)) !C_SIZE_T is of type integer(8) call c_f_pointer(coords_fftw, coords_real, [res(1)+2_pInt,res(2),res(3),3_pInt]) call c_f_pointer(coords_fftw, coords_fourier, [res1_red ,res(2),res(3),3_pInt]) fftw_forth = fftw_plan_many_dft_r2c(3_pInt,(/res(3),res(2) ,res(1)/),9_pInt,& ! dimensions , length in each dimension in reversed order defgrad_real,(/res(3),res(2) ,res(1)+2_pInt/),& ! input data , physical length in each dimension in reversed order 1_pInt, res(3)*res(2)*(res(1)+2_pInt),& ! striding , product of physical lenght in the 3 dimensions defgrad_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,fftw_planner_flag) fftw_back = fftw_plan_many_dft_c2r(3_pInt,(/res(3),res(2) ,res(1)/),3_pInt,& coords_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,& coords_real,(/res(3),res(2) ,res(1)+2_pInt/),& 1_pInt, res(3)*res(2)*(res(1)+2_pInt),fftw_planner_flag) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) defgrad_real(i,j,k,1:3,1:3) = defgrad(i,j,k,1:3,1:3) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo call fftw_execute_dft_r2c(fftw_forth, defgrad_real, defgrad_fourier) !remove highest frequency in each direction if(res(1)>1_pInt) & defgrad_fourier( res(1)/2_pInt+1_pInt,1:res(2) ,1:res(3) ,& 1:3,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(2)>1_pInt) & defgrad_fourier(1:res1_red ,res(2)/2_pInt+1_pInt,1:res(3) ,& 1:3,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(3)>1_pInt) & defgrad_fourier(1:res1_red ,1:res(2) ,res(3)/2_pInt+1_pInt,& 1:3,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) coords_fourier = cmplx(0.0_pReal,0.0_pReal,pReal) do k = 1_pInt, res(3) k_s(3) = k-1_pInt if(k > res(3)/2_pInt+1_pInt) k_s(3) = k_s(3)-res(3) do j = 1_pInt, res(2) k_s(2) = j-1_pInt if(j > res(2)/2_pInt+1_pInt) k_s(2) = k_s(2)-res(2) do i = 1_pInt, res1_red k_s(1) = i-1_pInt do m = 1_pInt,3_pInt coords_fourier(i,j,k,m) = sum(defgrad_fourier(i,j,k,m,1:3)*cmplx(0.0_pReal,real(k_s,pReal)*integrator,pReal)) enddo if (k_s(3) /= 0_pInt .or. k_s(2) /= 0_pInt .or. k_s(1) /= 0_pInt) & coords_fourier(i,j,k,1:3) = coords_fourier(i,j,k,1:3) / real(-sum(k_s*k_s),pReal) ! if(i/=1_pInt) coords_fourier(i,j,k,1:3) = coords_fourier(i,j,k,1:3)& ! substituting division by (on the fly calculated) xi * 2pi * img by multiplication with reversed img/real part ! - defgrad_fourier(i,j,k,1:3,1)*cmplx(0.0_pReal,integrator(1)/real(k_s(1),pReal),pReal) ! if(j/=1_pInt) coords_fourier(i,j,k,1:3) = coords_fourier(i,j,k,1:3)& ! - defgrad_fourier(i,j,k,1:3,2)*cmplx(0.0_pReal,integrator(2)/real(k_s(2),pReal),pReal) ! if(k/=1_pInt) coords_fourier(i,j,k,1:3) = coords_fourier(i,j,k,1:3)& ! - defgrad_fourier(i,j,k,1:3,3)*cmplx(0.0_pReal,integrator(3)/real(k_s(3),pReal),pReal) enddo; enddo; enddo call fftw_execute_dft_c2r(fftw_back,coords_fourier,coords_real) coords_real = coords_real/real(res(1)*res(2)*res(3),pReal) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) coords(i,j,k,1:3) = coords_real(i,j,k,1:3) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo offset_coords = matmul(defgrad(1,1,1,1:3,1:3),step/2.0_pReal) - scaling*coords(1,1,1,1:3) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) coords(i,j,k,1:3) = scaling*coords(i,j,k,1:3) + offset_coords + matmul(defgrad_av,& (/step(1)*real(i-1_pInt,pReal),& step(2)*real(j-1_pInt,pReal),& step(3)*real(k-1_pInt,pReal)/)) enddo; enddo; enddo call fftw_destroy_plan(fftw_forth) call fftw_destroy_plan(fftw_back) call fftw_free(defgrad_fftw) call fftw_free(coords_fftw) end subroutine deformed_fft !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine curl_fft(res,geomdim,vec_tens,field,curl) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! calculates curl field using differentation in Fourier space ! use vec_tens to decide if tensor (3) or vector (1) use IO, only: IO_error use numerics, only: fftw_timelimit, fftw_planner_flag use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim integer(pInt), intent(in) :: vec_tens real(pReal), intent(in), dimension(res(1), res(2),res(3),vec_tens,3) :: field ! output variables real(pReal), intent(out), dimension(res(1), res(2),res(3),vec_tens,3) :: curl ! variables with dimension depending on input real(pReal), dimension(res(1)/2_pInt+1_pInt,res(2),res(3),3) :: xi ! allocatable arrays for fftw c routines type(C_PTR) :: fftw_forth, fftw_back type(C_PTR) :: field_fftw, curl_fftw real(pReal), dimension(:,:,:,:,:), pointer :: field_real complex(pReal), dimension(:,:,:,:,:), pointer :: field_fourier real(pReal), dimension(:,:,:,:,:), pointer :: curl_real complex(pReal), dimension(:,:,:,:,:), pointer :: curl_fourier ! other variables integer(pInt) i, j, k, l, res1_red integer(pInt), dimension(3) :: k_s real(pReal) :: wgt if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Calculating curl of vector/tensor field' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif wgt = 1.0_pReal/real(res(1)*res(2)*res(3),pReal) res1_red = res(1)/2_pInt + 1_pInt ! size of complex array in first dimension (c2r, r2c) if (pReal /= C_DOUBLE .or. pInt /= C_INT) call IO_error(error_ID=808_pInt) call fftw_set_timelimit(fftw_timelimit) field_fftw = fftw_alloc_complex(int(res1_red *res(2)*res(3)*vec_tens*3_pInt,C_SIZE_T)) !C_SIZE_T is of type integer(8) call c_f_pointer(field_fftw, field_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(field_fftw, field_fourier,[res1_red ,res(2),res(3),vec_tens,3_pInt]) curl_fftw = fftw_alloc_complex(int(res1_red *res(2)*res(3)*vec_tens*3_pInt,C_SIZE_T)) !C_SIZE_T is of type integer(8) call c_f_pointer(curl_fftw, curl_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(curl_fftw, curl_fourier, [res1_red ,res(2),res(3),vec_tens,3_pInt]) fftw_forth = fftw_plan_many_dft_r2c(3_pInt,(/res(3),res(2) ,res(1)/),vec_tens*3_pInt,& ! dimensions , length in each dimension in reversed order field_real,(/res(3),res(2) ,res(1)+2_pInt/),& ! input data , physical length in each dimension in reversed order 1_pInt, res(3)*res(2)*(res(1)+2_pInt),& ! striding , product of physical lenght in the 3 dimensions field_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,fftw_planner_flag) fftw_back = fftw_plan_many_dft_c2r(3_pInt,(/res(3),res(2) ,res(1)/),vec_tens*3_pInt,& curl_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,& curl_real,(/res(3),res(2) ,res(1)+2_pInt/),& 1_pInt, res(3)*res(2)*(res(1)+2_pInt),fftw_planner_flag) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) field_real(i,j,k,1:vec_tens,1:3) = field(i,j,k,1:vec_tens,1:3) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo call fftw_execute_dft_r2c(fftw_forth, field_real, field_fourier) !remove highest frequency in each direction if(res(1)>1_pInt) & field_fourier( res(1)/2_pInt+1_pInt,1:res(2) ,1:res(3) ,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(2)>1_pInt) & field_fourier(1:res1_red ,res(2)/2_pInt+1_pInt,1:res(3) ,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(3)>1_pInt) & field_fourier(1:res1_red ,1:res(2) ,res(3)/2_pInt+1_pInt,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) do k = 1_pInt, res(3) ! calculation of discrete angular frequencies, ordered as in FFTW (wrap around) k_s(3) = k - 1_pInt if(k > res(3)/2_pInt + 1_pInt) k_s(3) = k_s(3) - res(3) do j = 1_pInt, res(2) k_s(2) = j - 1_pInt if(j > res(2)/2_pInt + 1_pInt) k_s(2) = k_s(2) - res(2) do i = 1_pInt, res1_red k_s(1) = i - 1_pInt xi(i,j,k,1:3) = real(k_s, pReal)/geomdim enddo; enddo; enddo do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res1_red do l = 1_pInt, vec_tens curl_fourier(i,j,k,l,1) = ( field_fourier(i,j,k,l,3)*xi(i,j,k,2)& -field_fourier(i,j,k,l,2)*xi(i,j,k,3) )*TWOPIIMG curl_fourier(i,j,k,l,2) = (-field_fourier(i,j,k,l,3)*xi(i,j,k,1)& +field_fourier(i,j,k,l,1)*xi(i,j,k,3) )*TWOPIIMG curl_fourier(i,j,k,l,3) = ( field_fourier(i,j,k,l,2)*xi(i,j,k,1)& -field_fourier(i,j,k,l,1)*xi(i,j,k,2) )*TWOPIIMG enddo enddo; enddo; enddo call fftw_execute_dft_c2r(fftw_back, curl_fourier, curl_real) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) curl(i,j,k,1:vec_tens,1:3) = curl_real(i,j,k,1:vec_tens,1:3) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo curl = curl * wgt call fftw_destroy_plan(fftw_forth); call fftw_destroy_plan(fftw_back) call c_f_pointer(C_NULL_PTR, field_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) ! let all pointers point on NULL-Type call c_f_pointer(C_NULL_PTR, field_fourier,[res1_red ,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(C_NULL_PTR, curl_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(C_NULL_PTR, curl_fourier, [res1_red ,res(2),res(3),vec_tens,3_pInt]) if(.not. (c_associated(C_LOC(field_real(1,1,1,1,1))) .and. c_associated(C_LOC(field_fourier(1,1,1,1,1)))))& ! Check if pointers are deassociated and free memory call fftw_free(field_fftw) ! This procedure ensures that optimization do not mix-up lines, because a if(.not.(c_associated(C_LOC(curl_real(1,1,1,1,1))) .and. c_associated(C_LOC(curl_fourier(1,1,1,1,1)))))& ! simple fftw_free(field_fftw) could be done immediately after the last line where field_fftw appears, e.g: call fftw_free(curl_fftw) ! call c_f_pointer(field_fftw, field_fourier, [res1_red ,res(2),res(3),vec_tens,3]) end subroutine curl_fft !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine divergence_fft(res,geomdim,vec_tens,field,divergence) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! calculates divergence field using integration in Fourier space ! use vec_tens to decide if tensor (3) or vector (1) use IO, only: IO_error use numerics, only: fftw_timelimit, fftw_planner_flag use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(3) :: geomdim integer(pInt), intent(in) :: vec_tens real(pReal), intent(in), dimension(res(1), res(2),res(3),vec_tens,3) :: field ! output variables real(pReal), intent(out), dimension(res(1), res(2),res(3),vec_tens) :: divergence ! variables with dimension depending on input real(pReal), dimension(res(1)/2_pInt+1_pInt,res(2),res(3),3) :: xi ! allocatable arrays for fftw c routines type(C_PTR) :: fftw_forth, fftw_back type(C_PTR) :: field_fftw, divergence_fftw real(pReal), dimension(:,:,:,:,:), pointer :: field_real complex(pReal), dimension(:,:,:,:,:), pointer :: field_fourier real(pReal), dimension(:,:,:,:), pointer :: divergence_real complex(pReal), dimension(:,:,:,:), pointer :: divergence_fourier ! other variables integer(pInt) :: i, j, k, l, res1_red real(pReal) :: wgt integer(pInt), dimension(3) :: k_s if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print '(a)', 'Calculating divergence of tensor/vector field using FFT' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif res1_red = res(1)/2_pInt + 1_pInt ! size of complex array in first dimension (c2r, r2c) wgt = 1.0_pReal/real(res(1)*res(2)*res(3),pReal) if (pReal /= C_DOUBLE .or. pInt /= C_INT) call IO_error(error_ID=808_pInt) call fftw_set_timelimit(fftw_timelimit) field_fftw = fftw_alloc_complex(int(res1_red*res(2)*res(3)*vec_tens*3_pInt,C_SIZE_T)) !C_SIZE_T is of type integer(8) call c_f_pointer(field_fftw, field_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(field_fftw, field_fourier, [res1_red ,res(2),res(3),vec_tens,3_pInt]) divergence_fftw = fftw_alloc_complex(int(res1_red*res(2)*res(3)*vec_tens,C_SIZE_T)) call c_f_pointer(divergence_fftw, divergence_real, [res(1)+2_pInt,res(2),res(3),vec_tens]) call c_f_pointer(divergence_fftw, divergence_fourier,[res1_red ,res(2),res(3),vec_tens]) fftw_forth = fftw_plan_many_dft_r2c(3_pInt,(/res(3),res(2) ,res(1)/),vec_tens*3_pInt,& ! dimensions , length in each dimension in reversed order field_real,(/res(3),res(2) ,res(1)+2_pInt/),& ! input data , physical length in each dimension in reversed order 1_pInt, res(3)*res(2)*(res(1)+2_pInt),& ! striding , product of physical lenght in the 3 dimensions field_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,fftw_planner_flag) fftw_back = fftw_plan_many_dft_c2r(3_pInt,(/res(3),res(2) ,res(1)/),vec_tens,& divergence_fourier,(/res(3),res(2) ,res1_red/),& 1_pInt, res(3)*res(2)* res1_red,& divergence_real,(/res(3),res(2) ,res(1)+2_pInt/),& 1_pInt, res(3)*res(2)*(res(1)+2_pInt),fftw_planner_flag) ! padding do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) field_real(i,j,k,1:vec_tens,1:3) = field(i,j,k,1:vec_tens,1:3) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo call fftw_execute_dft_r2c(fftw_forth, field_real, field_fourier) do k = 1_pInt, res(3) ! calculation of discrete angular frequencies, ordered as in FFTW (wrap around) k_s(3) = k - 1_pInt if(k > res(3)/2_pInt + 1_pInt) k_s(3) = k_s(3) - res(3) do j = 1_pInt, res(2) k_s(2) = j - 1_pInt if(j > res(2)/2_pInt + 1_pInt) k_s(2) = k_s(2) - res(2) do i = 1_pInt, res1_red k_s(1) = i - 1_pInt xi(i,j,k,1:3) = real(k_s, pReal)/geomdim enddo; enddo; enddo !remove highest frequency in each direction if(res(1)>1_pInt) & field_fourier( res(1)/2_pInt+1_pInt,1:res(2) ,1:res(3) ,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(2)>1_pInt) & field_fourier(1:res1_red ,res(2)/2_pInt+1_pInt,1:res(3) ,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) if(res(3)>1_pInt) & field_fourier(1:res1_red ,1:res(2) ,res(3)/2_pInt+1_pInt,& 1:vec_tens,1:3) = cmplx(0.0_pReal,0.0_pReal,pReal) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res1_red do l = 1_pInt, vec_tens divergence_fourier(i,j,k,l)=sum(field_fourier(i,j,k,l,1:3)*cmplx(xi(i,j,k,1:3),0.0_pReal,pReal))& *TWOPIIMG enddo enddo; enddo; enddo call fftw_execute_dft_c2r(fftw_back, divergence_fourier, divergence_real) do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) divergence(i,j,k,1:vec_tens) = divergence_real(i,j,k,1:vec_tens) ! ensure that data is aligned properly (fftw_alloc) enddo; enddo; enddo divergence = divergence * wgt call fftw_destroy_plan(fftw_forth); call fftw_destroy_plan(fftw_back) call c_f_pointer(C_NULL_PTR, field_real, [res(1)+2_pInt,res(2),res(3),vec_tens,3_pInt]) ! let all pointers point on NULL-Type call c_f_pointer(C_NULL_PTR, field_fourier, [res1_red ,res(2),res(3),vec_tens,3_pInt]) call c_f_pointer(C_NULL_PTR, divergence_real, [res(1)+2_pInt,res(2),res(3),vec_tens]) call c_f_pointer(C_NULL_PTR, divergence_fourier,[res1_red ,res(2),res(3),vec_tens]) if(.not. (c_associated(C_LOC(field_real(1,1,1,1,1))) .and. c_associated(C_LOC(field_fourier(1,1,1,1,1)))))& ! Check if pointers are deassociated and free memory call fftw_free(field_fftw) ! This procedure ensures that optimization do not mix-up lines, because a if(.not.(c_associated(C_LOC(divergence_real(1,1,1,1))) .and. c_associated(C_LOC(divergence_fourier(1,1,1,1)))))& ! simple fftw_free(field_fftw) could be done immediately after the last line where field_fftw appears, e.g: call fftw_free(divergence_fftw) ! call c_f_pointer(field_fftw, field_fourier, [res1_red ,res(2),res(3),vec_tens,3]) end subroutine divergence_fft !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine divergence_fdm(res,geomdim,vec_tens,order,field,divergence) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! calculates divergence field using FDM with variable accuracy ! use vec_tes to decide if tensor (3) or vector (1) use debug, only: debug_math, & debug_level, & debug_levelBasic implicit none integer(pInt), intent(in), dimension(3) :: res integer(pInt), intent(in) :: vec_tens integer(pInt), intent(inout) :: order real(pReal), intent(in), dimension(3) :: geomdim real(pReal), intent(in), dimension(res(1),res(2),res(3),vec_tens,3) :: field ! output variables real(pReal), intent(out), dimension(res(1),res(2),res(3),vec_tens) :: divergence ! other variables integer(pInt), dimension(6,3) :: coordinates integer(pInt) i, j, k, m, l real(pReal), dimension(4,4), parameter :: FDcoefficient = reshape((/ & 1.0_pReal/2.0_pReal, 0.0_pReal, 0.0_pReal, 0.0_pReal,& !from http://en.wikipedia.org/wiki/Finite_difference_coefficients 2.0_pReal/3.0_pReal,-1.0_pReal/12.0_pReal, 0.0_pReal, 0.0_pReal,& 3.0_pReal/4.0_pReal,-3.0_pReal/20.0_pReal,1.0_pReal/ 60.0_pReal, 0.0_pReal,& 4.0_pReal/5.0_pReal,-1.0_pReal/ 5.0_pReal,4.0_pReal/105.0_pReal,-1.0_pReal/280.0_pReal/),& (/4,4/)) if (iand(debug_level(debug_math),debug_levelBasic) /= 0_pInt) then print*, 'Calculating divergence of tensor/vector field using FDM' print '(a,3(e12.5))', ' Dimension: ', geomdim print '(a,3(i5))', ' Resolution:', res endif divergence = 0.0_pReal order = order + 1_pInt do k = 0_pInt, res(3)-1_pInt; do j = 0_pInt, res(2)-1_pInt; do i = 0_pInt, res(1)-1_pInt do m = 1_pInt, order coordinates(1,1:3) = mesh_location(mesh_index((/i+m,j,k/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) coordinates(2,1:3) = mesh_location(mesh_index((/i-m,j,k/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) coordinates(3,1:3) = mesh_location(mesh_index((/i,j+m,k/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) coordinates(4,1:3) = mesh_location(mesh_index((/i,j-m,k/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) coordinates(5,1:3) = mesh_location(mesh_index((/i,j,k+m/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) coordinates(6,1:3) = mesh_location(mesh_index((/i,j,k-m/),(/res(1),res(2),res(3)/)),(/res(1),res(2),res(3)/))& + (/1_pInt,1_pInt,1_pInt/) do l = 1_pInt, vec_tens divergence(i+1_pInt,j+1_pInt,k+1_pInt,l) = divergence(i+1_pInt,j+1_pInt,k+1_pInt,l) + FDcoefficient(m,order) * & ((field(coordinates(1,1),coordinates(1,2),coordinates(1,3),l,1)- & field(coordinates(2,1),coordinates(2,2),coordinates(2,3),l,1))*real(res(1),pReal)/geomdim(1) +& (field(coordinates(3,1),coordinates(3,2),coordinates(3,3),l,2)- & field(coordinates(4,1),coordinates(4,2),coordinates(4,3),l,2))*real(res(2),pReal)/geomdim(2) +& (field(coordinates(5,1),coordinates(5,2),coordinates(5,3),l,3)- & field(coordinates(6,1),coordinates(6,2),coordinates(6,3),l,3))*real(res(3),pReal)/geomdim(3)) enddo enddo enddo; enddo; enddo end subroutine divergence_fdm #endif !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine tensor_avg(res,tensor,avg) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate average of tensor field ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) ::tensor ! output variables real(pReal), intent(out), dimension(3,3) :: avg ! other variables real(pReal) wgt integer(pInt) m,n wgt = 1.0_pReal/real(res(1)*res(2)*res(3), pReal) do m = 1_pInt,3_pInt; do n = 1_pInt,3_pInt avg(m,n) = sum(tensor(1:res(1),1:res(2),1:res(3),m,n)) * wgt enddo; enddo end subroutine tensor_avg !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine logstrain_spat(res,defgrad,logstrain_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate logarithmic strain in spatial configuration for given defgrad field ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) :: defgrad ! output variables real(pReal), intent(out), dimension(res(1),res(2),res(3),3,3) :: logstrain_field ! other variables real(pReal), dimension(3,3) :: temp33_Real, temp33_Real2 real(pReal), dimension(3,3,3) :: eigenvectorbasis real(pReal), dimension(3) :: eigenvalue integer(pInt) :: i, j, k logical :: errmatinv do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) call math_pDecomposition(defgrad(i,j,k,1:3,1:3),temp33_Real2,temp33_Real,errmatinv) !store R in temp33_Real temp33_Real2 = math_inv33(temp33_Real) temp33_Real = math_mul33x33(defgrad(i,j,k,1:3,1:3),temp33_Real2) ! v = F o inv(R), store in temp33_Real2 call math_spectral1(temp33_Real,eigenvalue(1), eigenvalue(2), eigenvalue(3),& eigenvectorbasis(1,1:3,1:3),eigenvectorbasis(2,1:3,1:3),eigenvectorbasis(3,1:3,1:3)) eigenvalue = log(sqrt(eigenvalue)) logstrain_field(i,j,k,1:3,1:3) = eigenvalue(1)*eigenvectorbasis(1,1:3,1:3)+& eigenvalue(2)*eigenvectorbasis(2,1:3,1:3)+& eigenvalue(3)*eigenvectorbasis(3,1:3,1:3) enddo; enddo; enddo end subroutine logstrain_spat !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine logstrain_mat(res,defgrad,logstrain_field) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate logarithmic strain in material configuration for given defgrad field ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) :: defgrad ! output variables real(pReal), intent(out), dimension(res(1),res(2),res(3),3,3) :: logstrain_field ! other variables real(pReal), dimension(3,3) :: temp33_Real, temp33_Real2 real(pReal), dimension(3,3,3) :: eigenvectorbasis real(pReal), dimension(3) :: eigenvalue integer(pInt) :: i, j, k logical :: errmatinv do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) call math_pDecomposition(defgrad(i,j,k,1:3,1:3),temp33_Real,temp33_Real2,errmatinv) !store U in temp33_Real call math_spectral1(temp33_Real,eigenvalue(1), eigenvalue(2), eigenvalue(3),& eigenvectorbasis(1,1:3,1:3),eigenvectorbasis(2,1:3,1:3),eigenvectorbasis(3,1:3,1:3)) eigenvalue = log(sqrt(eigenvalue)) logstrain_field(i,j,k,1:3,1:3) = eigenvalue(1)*eigenvectorbasis(1,1:3,1:3)+& eigenvalue(2)*eigenvectorbasis(2,1:3,1:3)+& eigenvalue(3)*eigenvectorbasis(3,1:3,1:3) enddo; enddo; enddo end subroutine logstrain_mat !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine calculate_cauchy(res,defgrad,p_stress,c_stress) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ !calculate cauchy stress for given PK1 stress and defgrad field ! implicit none ! input variables integer(pInt), intent(in), dimension(3) :: res real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) :: defgrad real(pReal), intent(in), dimension(res(1),res(2),res(3),3,3) :: p_stress ! output variables real(pReal), intent(out), dimension(res(1),res(2),res(3),3,3) :: c_stress ! other variables real(pReal) :: jacobi integer(pInt) :: i, j, k c_stress = 0.0_pReal do k = 1_pInt, res(3); do j = 1_pInt, res(2); do i = 1_pInt, res(1) jacobi = math_det33(defgrad(i,j,k,1:3,1:3)) c_stress(i,j,k,1:3,1:3) = matmul(p_stress(i,j,k,1:3,1:3),transpose(defgrad(i,j,k,1:3,1:3)))/jacobi enddo; enddo; enddo end subroutine calculate_cauchy #ifdef Spectral !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ subroutine math_nearestNeighborSearch(spatialDim, Favg, geomdim, queryPoints, domainPoints, querySet, domainSet, indices) !+++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++++ ! Obtain the nearest neighbor in domain set for all points in querySet ! use kdtree2_module use IO, only: & IO_error implicit none ! input variables integer(pInt), intent(in) :: spatialDim real(pReal), dimension(3,3), intent(in) :: Favg real(pReal), dimension(3), intent(in) :: geomdim integer(pInt), intent(in) :: domainPoints integer(pInt), intent(in) :: queryPoints real(pReal), dimension(spatialDim,queryPoints), intent(in) :: querySet real(pReal), dimension(spatialDim,domainPoints), intent(in) :: domainSet ! output variable integer(pInt), dimension(queryPoints), intent(out) :: indices ! other variables depending on input real(pReal), dimension(spatialDim,(3_pInt**spatialDim)*domainPoints) :: domainSetLarge ! other variables integer(pInt) :: i,j, l,m,n type(kdtree2), pointer :: tree type(kdtree2_result), dimension(1) :: Results if (size(querySet(:,1)) /= spatialDim) call IO_error(407_pInt,ext_msg='query set') if (size(domainSet(:,1)) /= spatialDim) call IO_error(407_pInt,ext_msg='domain set') i = 0_pInt if(spatialDim == 2_pInt) then do j = 1_pInt, domainPoints do l = -1_pInt, 1_pInt; do m = -1_pInt, 1_pInt i = i + 1_pInt domainSetLarge(1:2,i) = domainSet(1:2,j) +matmul(Favg(1:2,1:2),real([l,m],pReal)*geomdim(1:2)) enddo; enddo enddo else do j = 1_pInt, domainPoints do l = -1_pInt, 1_pInt; do m = -1_pInt, 1_pInt; do n = -1_pInt, 1_pInt i = i + 1_pInt domainSetLarge(1:3,i) = domainSet(1:3,j) + math_mul33x3(Favg,real([l,m,n],pReal)*geomdim) enddo; enddo; enddo enddo endif tree => kdtree2_create(domainSetLarge,sort=.true.,rearrange=.true.) do j = 1_pInt, queryPoints call kdtree2_n_nearest(tp=tree, qv=querySet(1:spatialDim,j),nn=1_pInt, results = Results) indices(j) = Results(1)%idx enddo indices = indices -1_pInt ! let them run from 0 to domainPoints -1 end subroutine math_nearestNeighborSearch #endif end module math