fixed random Gaussian sampling
sampling needs to be performed from unfiform misorientation, NOT uniformly distributed rotations for Fiber, compute uniform tilt of Fiber axis
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Martin Diehl
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src/math.f90
97
src/math.f90
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@ -1765,14 +1765,8 @@ end function math_sampleRandomOri
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!--------------------------------------------------------------------------------------------------
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!> @brief draw a sample from an Gaussian distribution around given orientation and Full Width
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! at Half Maximum (FWHM)
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!> @details: for very small FWHM values the given orientation is returned.
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!> @details: for intermediate FWHM values, an orientation is picked from uniformly distributed
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!> @details: oreintations around the nominal orientation with maximum misorientation of 2*FWHM
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!> @deatils: according to https://math.stackexchange.com/questions/13133 followed by
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!> @details: the application of a Gaussian filter.
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!> @details: for large FWHM values, a random orientation from a uniform distribution is picked
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!> @details: followed by tge aookucatuib if a Gaussian filter. Additionally, the misorientation is
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!> @details: limited to 2*FWHM,
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!> @details: A uniform misorientation (limited to 2*FWHM) is sampled followed by convolution with
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! a Gausian distribution
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!--------------------------------------------------------------------------------------------------
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function math_sampleGaussOri(center,FWHM)
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@ -1781,35 +1775,25 @@ function math_sampleGaussOri(center,FWHM)
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real(pReal), dimension(3), intent(in) :: center
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real(pReal) :: angle
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real(pReal), dimension(3) :: math_sampleGaussOri, axis
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real(pReal), dimension(2) :: rnd
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real(pReal), dimension(4) :: rnd
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real(pReal), dimension(3,3) :: R
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noScatter: if (FWHM < 0.1_pReal*INRAD) then
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math_sampleGaussOri = center
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else noScatter
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if (FWHM < 0.1_pReal*INRAD) then
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R = math_I3
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else
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GaussConvolution: do
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selectiveSampling: if (FWHM*INRAD < 90.0_pReal) then
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rnd = halton([3_pInt,6_pInt])
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rnd = halton([8_pInt,3_pInt,6_pInt,11_pInt])
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axis(1) = rnd(1)*2.0_pReal-1.0_pReal ! uniform on [-1,1]
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axis(2:3) = [sqrt(1.0-axis(1)**2.0_pReal)*cos(rnd(2)*2.0*PI),&
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sqrt(1.0-axis(1)**2.0_pReal)*sin(rnd(2)*2.0*PI)] ! random axis
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do
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rnd = halton([14_pInt,10_pInt])
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angle = (rnd(1)*(2.0_pReal*FWHM)**3.0_pReal)**(1.0_pReal/3.0_pReal) ! maximum misorientation of 2*FWHM
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if (rnd(2) < sin(angle)**2.0_pReal/angle**2.0_pReal) exit ! rejection sampling
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enddo
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angle = (rnd(3)-0.5_pReal)*4.0_pReal*FWHM ! rotation by [0, +-2 FWHM]
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R = math_axisAngleToR(axis,angle)
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else selectiveSampling
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R = math_EulerToR(math_sampleRandomOri())
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endif selectiveSampling
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rnd = halton([8_pInt,11_pInt])
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angle = math_EulerMisorientation([0.0_pReal,0.0_pReal,0.0_pReal],math_RtoEuler(R))
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if (rnd(1) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2_pReal) .and. & ! rejection sampling (Gaussian)
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angle < 2.0_pReal * FWHM) exit ! limit (in case of non-selective orientation selection
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if (rnd(4) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2_pReal)) exit ! rejection sampling (Gaussian)
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enddo GaussConvolution
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endif
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math_sampleGaussOri = math_RtoEuler(math_mul33x33(R,math_EulerToR(center)))
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endif noScatter
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end function math_sampleGaussOri
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@ -1817,8 +1801,6 @@ end function math_sampleGaussOri
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!--------------------------------------------------------------------------------------------------
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!> @brief draw a sample from an Gaussian distribution around given fiber texture and Full Width
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! at Half Maximum (FWHM)
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!>@details: vector in cone around axis is uniformly distributed according to
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! https://math.stackexchange.com/questions/56784
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!-------------------------------------------------------------------------------------------------
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function math_sampleFiberOri(alpha,beta,FWHM)
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@ -1828,38 +1810,49 @@ function math_sampleFiberOri(alpha,beta,FWHM)
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real(pReal), dimension(3) :: math_sampleFiberOri, &
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fInC,& !< fiber axis in crystal coordinate system
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fInS,& !< fiber axis in sample coordinate system
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axis
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real(pReal), dimension(5) :: rnd
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real(pReal), dimension(3,3) :: &
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R_o, & !< rotation to aling fiber axis in crystal and sample system
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R_f, & !< random rotation along fiber axis [0, 2*Pi[
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R_p !< deviation of axis alingment, bound by 2*FWHM
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real(pReal) :: angle
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u
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real(pReal), dimension(3) :: rnd
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real(pReal), dimension(:),allocatable :: a !< 2D vector to tilt
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integer(pInt), dimension(:),allocatable :: idx !< components of 2D vector
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real(pReal), dimension(3,3) :: R !< Rotation matrix (composed of three components)
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real(pReal):: angle,c
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integer(pInt):: j,& !< index of smallest component
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i
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fInC = [sin(alpha(1))*cos(alpha(2)), sin(alpha(1))*sin(alpha(2)), cos(alpha(1))]
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fInS = [sin(beta(1))*cos(beta(2)), sin(beta(1))*sin(beta(2)), cos(beta(1))]
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R_o = math_EulerAxisAngleToR(math_crossproduct(fInC,fInS),-acos(dot_product(fInC,fInS)))
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R = math_EulerAxisAngleToR(math_crossproduct(fInC,fInS),-acos(dot_product(fInC,fInS))) !< rotation to align fiber axis in crystal and sample system
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if (FWHM > 0.0_pReal) then
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GaussConvolution: do
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rnd = halton(int([5,10,3,9,17],pInt))
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rnd(1:2) = [cos(FWHM*2.0_pReal),-1.0_pReal] + rnd(1:2)*[1.0_pReal - cos(FWHM*2.0_pReal),2.0_pReal]
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axis = [sqrt(1.0_pReal - rnd(2)**2.0_pReal)*sin(rnd(1)),&
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sqrt(1.0_pReal - rnd(2)**2.0_pReal)*cos(rnd(1)),rnd(2)]
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angle = acos(dot_product([0.0_pReal,0.0_pReal,1.0_pReal],axis))
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if (rnd(3) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2.0_pReal)) exit
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enddo GaussConvolution
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if (rnd(4) <= 0.5) angle = -angle
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R_p = math_EulerAxisAngleToR(math_crossproduct(axis,[0.0_pReal,0.0_pReal,1.0_pReal]),angle)
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rnd = halton([7_pInt,10_pInt,3_pInt])
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R = math_mul33x33(R,math_EulerAxisAngleToR(fInS,rnd(1)*2.0_pReal*PI)) !< additional rotation (0..360deg) perpendicular to fiber axis
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if (FWHM > 0.1_pReal*INRAD) then
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reducedTo2D: do i=1_pInt,3_pInt
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if (i /= minloc(abs(fInS),1)) then
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a=[a,fInS(i)]
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idx=[b,i]
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else
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R_p = math_I3
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rnd = halton(int([5,10,3,9,17],pInt))
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j = i
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endif
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enddo reducedTo2D
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GaussConvolution: do
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angle = (rnd(2)-0.5_pReal)*4.0_pReal*FWHM ! rotation by [0, +-2 FWHM]
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! solve cos(angle) = dot_product(fInS,u) under the assumption that their smallest component is the same
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c = cos(angle)-fInS(j)**2
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u(idx(2)) = -(2.0_pReal*c*a(2) + sqrt(4*((c*a(2))**2-sum(a**2)*(c**2-a(1)**2*(1-fInS(j)**2)))))/&
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(2*sum(a**2))
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u(idx(1)) = sqrt(1-u(idx(2))**2-fInS(j)**2)
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u(j) = fInS(j)
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R_f = math_EulerAxisAngleToR(fInS,rnd(5)*2.0_pReal*PI)
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math_sampleFiberOri = math_RtoEuler(math_mul33x33(R_p,math_mul33x33(R_f,R_o)))
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rejectionSampling: if (rnd(3) <= exp(-4.0_pReal*log(2.0_pReal)*(angle/FWHM)**2_pReal)) then
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R = math_mul33x33(R,math_EulerAxisAngleToR(math_crossproduct(u,fInS),angle)) ! tilt around direction of smallest component
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exit
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endif rejectionSampling
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rnd = halton([7_pInt,10_pInt,3_pInt])
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enddo GaussConvolution
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endif
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math_sampleFiberOri = math_RtoEuler(R)
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end function math_sampleFiberOri
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