def fig2():
'''
Plot histogram and solve characteristics of probability_cut_nooverlapsability, that we cut fibers.
Compare with binomial distribution.
Set n_sim >> 1
'''
figure(2)
subplots_adjust(wspace = 0., hspace = 0, bottom = .2)
delta = 0.
p = probability_cut_nooverlaps(spec.l_x, fib.lf, delta)
# probability for extrusion
#p = fib.lf / 2. / spec.l_x
rvb = binom(fib.n, p)
rvp = poisson(fib.n * p)
rvn = norm(fib.n * p, sqrt(fib.n * p * (1 - p)))
graph_from = floor(bin_mean - 4 * bin_stdv)
graph_to = floor(bin_mean + 4 * bin_stdv) + 1
x = arange(graph_from , graph_to)
plot(x, n_sim * rvb.pmf(x), color = 'black', linestyle = '-', linewidth = 2, label = 'Binomial')
plot(x, n_sim * rvp.pmf(x), color = 'black', linestyle = ':', linewidth = 1.5, label = 'Poisson')
plot(x, n_sim * rvn.pdf(x), color = 'black', linestyle = '--', linewidth = 1.5, label = 'Normal')
#plot( x, 20 * rv.pmf( x ) )
pdf, bins, patches = hist(v, n_sim, normed = 0, facecolor = 'grey') #, alpha=1
#set_xlim( bin_mean - 2 * bin_stdv, bin_mean + 2 * bin_stdv )
#plot( sx, sy, 'rx' ) # centroids
#print sum( pdf * diff( bins ) )
xticks(position = (0, -0.01))
xlabel('number of fibers')
ylabel('frequency')
legend()
draw()
def cut_fiber_distrib(L, l, phi):
if L >= 2 * l:
return l * cos(phi) * sin(phi) / (L - l * cos(phi)) / probability_cut_nooverlaps(L, l, 0)
#if phi > 0 and phi < arccos( L / l / 2. ): #L > l and L < 2 * l * cos( phi ):
else:
return (l * cos(phi) * sin(phi) / (L - l * cos(phi)) / probability_cut_nooverlaps(L, l, 0)) * Heaviside(phi - arccos(L / 2. / l)) * Heaviside(pi / 2. - phi) + sin(phi) / probability_cut_nooverlaps(L, l, 0) * Heaviside(phi) * Heaviside(arccos(L / 2. / l) - phi)
theta = fib.theta
phi_y = fib.phi_y
phi_z = fib.phi_z
vec_cut = fib.cut_func(sx, lx, sec)
v.append(sum(vec_cut))
#area = cut_area_func( fib.df, fib.lf, sx, phi_x, sec )
#coef_ = area / ( pi * fib.df ** 2 / 4. )
#coef.append( mean( coef_[coef_ > 0] ) )
#print 'number of cut fibers', v
p = probability_cut_nooverlaps(spec.l_x, fib.lf, sec)
bin_mean = fib.n * p
bin_stdv = sqrt(bin_mean * (1 - p))
bin_skew = (1 - 2 * p) / bin_stdv
poiss_lambda = fib.n * p
poiss_mean = poiss_lambda
poiss_stdv = poiss_lambda ** (1 / 2.)
poiss_skew = poiss_lambda ** (-1 / 2.)
# Volume fraction
Vf = fib._get_volume()
Vc = spec._get_volume()
vol_frac_V = Vf / Vc * 100 * fib.n
#vol_frac_A = mean( v_vol_frac_A ) / Ac * 100
def cut_fibers():
'''
Plot cut fibers (only ellipses), solve mean value of area and embedded length of cut fibers.
Plot histogram of embedded length.
'''
mask = matrix(fib.cut_func(sx, lx, sec).astype('bool'))
sx_cut = sx[ mask ]
sy_cut = sy[ mask ]
sz_cut = sz[ mask ]
phi_x_cut = phi_x[ mask ]
theta_cut = theta[ mask ]
def plot_cross_cut():
#plot cut plane with crossection of cut fibers
fig55 = figure(55)
title('Cut fibers area (ellipses)')
ax55 = Axes(fig55, [.1, .1, .8, .8])
fig55.add_axes(ax55)
for i in range(0, len(phi_x_cut)):
sy_c = sy_cut[i] + (sec - sx_cut[i]) / cos(phi_x_cut[i]) * sin(phi_x_cut[i]) * cos(theta_cut[i])
sz_c = sz_cut[i] + (sec - sx_cut[i]) / cos(phi_x_cut[i]) * sin(phi_x_cut[i]) * sin(theta_cut[i])
if (0 < phi_x_cut[i] < arctan((fib.lf) / fib.df)) and (sx_cut[i] - fib.lf / 2. * cos(phi_x_cut[i]) + fib.df / 2. * sin(phi_x_cut[i]) < sec < sx_cut[i] + fib.lf / 2. * cos(phi_x_cut[i]) - fib.df / 2. * sin(phi_x_cut[i])):
patch = Ellipse([ sy_c, sz_c ] , fib.df, fib.df / cos(phi_x_cut[i]), theta_cut[i] * 180 / pi , color = 'black')
ax55.add_artist(patch)
ax55.set_xlim(-spec.l_y / 2., spec.l_y / 2.)
ax55.set_ylim(-spec.l_z / 2., spec.l_z / 2.)
#plot_cross_cut() #plot cut plane
A_cut = sum(cut_area_func(fib.df, fib.lf, sx_cut, phi_x_cut, sec))
le_cut = le_sim(phi_x_cut, abs(sx_cut - sec * ones_like(sx_cut)), fib.lf)
le_cut_null = le_sim(phi_x, abs(sx - sec * ones_like(sx)), fib.lf) * fib.cut_func(sx, lx, sec)
fig6 = figure(6)
ax6 = Axes(fig6, [.1, .1, .8, .8])
hist(le_cut, nbins, normed = 0, color = '#cccccc')
#hist( le_cut_null[0], 50, normed=0 )
title('Embedded length l_e')
figure(8)
n_mask = 10
title('Histograms of $\phi$ fraction = $\pi/%i$' % n_mask)
for i in range(0, n_mask):
mask1 = phi_x_cut > i * pi / 2. / n_mask
mask2 = phi_x_cut < (i + 1) * pi / 2. / n_mask
#pdf, bins, patches = hist( le_cut[mask1 * mask2], nbins , histtype = 'step', label = '%i' % i )
legend()
#print mean( pdf )
figure(9)
title('Histogram of cut fibers angle')
xx = linspace(0, pi / 2., 100)
hist(phi_x_cut, nbins, normed = 1, color = '#cccccc')
if fib.overlaps:
plot(xx, sin(2 * xx), color = 'k', linewidth = 4, label = 'overlaps')
plot(xx, cut_fiber_distrib(spec.l_x, fib.lf, xx), 'k--', linewidth = 1, label = 'nooverlaps')
else:
plot(xx, sin(2 * xx), 'k--', linewidth = 1, label = 'overlaps')
plot(xx, cut_fiber_distrib(spec.l_x, fib.lf, xx), color = 'k', linewidth = 4, label = 'nooverlaps')
legend()
fig10 = figure(10)
title('2D histogram of le and phi_x')
ax10 = Axes3D(fig10)
bin = 20
H, xedges, yedges = histogram2d(le_cut, phi_x_cut, (bin, bin), normed = 1)
#extent = [xedges[0] * 100, xedges[-1] * 100, yedges[0], yedges[-1]] #[xedges[0], xedges[-1], yedges[0], yedges[-1]]
#im = imshow( H )#'binary', cmap='jet' , extent=extent
#im.set_interpolation( 'bilinear' )
#colorbar()
x = (xedges[range(0, len(xedges) - 1)] + xedges[range(1, len(xedges))]) / 2.
y = (yedges[range(0, len(yedges) - 1)] + yedges[range(1, len(yedges))]) / 2.
#ax10.scatter3D( xedges.ravel(), yedges.ravel(), H.ravel() )
xx = outer(x, ones(len(y)))
yy = outer(ones(len(x)), y)
z = outer(ones(len(y)), sin(2 * y)) / fib.lf * 2# * probability_cut_nooverlaps( spec.l_x, fib.lf, 0 )#* fib.n
zz = outer(ones(len(y)), cut_fiber_distrib(spec.l_x, fib.lf, y) / fib.lf * 2)
ax10.plot_wireframe(xx, yy, H)#, rstride=1, cstride=1
ax10.plot_wireframe(xx, yy, z , color = 'red', label = 'overlaps')
ax10.plot_wireframe(xx, yy, zz , color = 'green', label = 'nooverlaps')
legend()
print '%%%%%%%%%%%%%%%%%%%'
print '%%Embedded length%%'
print '%%%%%%%%%%%%%%%%%%%'
print 'probability cut fibers', probability_cut_nooverlaps(spec.l_x, fib.lf, 0.)
print 'le_mean cut fibers (int/p) \t', le(spec.l_x, fib.lf) / probability_cut_nooverlaps(spec.l_x, fib.lf, 0.)
print 'le_mean of cut fibers (sim) \t', mean(le_cut), '\t |\t standard deviation \t', std(le_cut)
print 'include nulls (sim) \t\t', mean(le_cut_null)
print 'E[le] (int) \t\t\t', le(spec.l_x, fib.lf)
print 'cut fibers area', A_cut, 'Ac', spec._get_cross_area(), 'af', A_cut / spec._get_cross_area() * 100, '%'
def Simulation():
from traits.api import HasTraits, HasStrictTraits, Float, Property, cached_property, \
Instance, List, on_trait_change, Int, Tuple, Bool, \
DelegatesTo, Event, Enum, implements, Button, File, CFloat
from traitsui.api import \
View, Item, Tabbed, VGroup, HGroup, ModelView, HSplit, VSplit
from traitsui.menu import OKButton
from math import e, sqrt, log, pi, floor
from matplotlib.figure import Figure
from matplotlib.pyplot import plot, hist, show
from numpy import array, linspace, frompyfunc, zeros, column_stack, \
log as ln, append, logspace, hstack, sign, trapz, sin, cos, sqrt, \
ogrid, sort, nonzero, tanh, broadcast, ones_like, ones, arange, ndarray, diff, outer, \
copy, mean, exp, std, average, arctan, histogram2d, meshgrid, savetxt, transpose, var
from pylab import savefig, plot, show, imshow, draw, colorbar, pcolor, subplot, title, legend
from scipy.interpolate import interp1d
from scipy.optimize import brentq, newton
import scipy.interpolate
from matplotlib.figure import Figure
from matplotlib.axes import Axes
from matplotlib.lines import Line2D
from matplotlib.patches import Ellipse
from matplotlib.backends.backend_agg import FigureCanvasAgg
from numpy.random import rand
from numpy import arccos, matrix, sum, arange
from scipy.stats import binom, norm, skew, poisson
from mpl_toolkits.mplot3d import Axes3D
from mpl_toolkits.mplot3d.art3d import Line3D
from matplotlib.pyplot import figure
from specimen_3D.fibers import Fibers, Specimen
from specimen_3D.probability import probability_cut_nooverlaps
from mayavi.tools.helper_functions import plot3d
def Heaviside( x ):
return ( sign( x ) + 1.0 ) / 2.0
# @todo: other types of crosssectional area
def cut_area( df, lf, ss, phi, sec ):
'''
Solve area of cut fibers (elliptic crosssection)
'''
if ( 0 < phi < arctan( ( lf ) / df ) ) and ( ss - lf / 2. * cos( phi ) + df / 2. * sin( phi ) < sec < ss + lf / 2. * cos( phi ) - df / 2. * sin( phi ) ):
return pi * df ** 2 / 4. / cos( phi )
if ( phi == pi / 2. ) and ( ss - df / 2. < sec < ss + df / 2. ):
return lf * sqrt( df ** 2 - 4 * abs( sx - sec ) ** 2 )
if ( 0 < phi < arctan( ( lf ) / df ) ) and ( ss + lf / 2. * cos( phi ) - df / 2. * sin( phi ) < sec < ss + lf / 2. * cos( phi ) + df / 2. * sin( phi ) )and ( ss - lf / 2. * cos( phi ) - df / 2. * sin( phi ) < sec < ss - lf / 2. * cos( phi ) + df / 2. * sin( phi ) ):
return 0
else:
return 0#df * lf / sin( phi )
cut_area_func = frompyfunc( cut_area, 5, 1 )
def cut_fiber_distrib( L, l, phi ):
if L >= 2 * l:
return l * cos( phi ) * sin( phi ) / ( L - l * cos( phi ) ) / probability_cut_nooverlaps( L, l, 0 )
#if phi > 0 and phi < arccos( L / l / 2. ): #L > l and L < 2 * l * cos( phi ):
else:
return ( l * cos( phi ) * sin( phi ) / ( L - l * cos( phi ) ) / probability_cut_nooverlaps( L, l, 0 ) ) * Heaviside( phi - arccos( L / 2. / l ) ) * Heaviside( pi / 2. - phi ) + sin( phi ) / probability_cut_nooverlaps( L, l, 0 ) * Heaviside( phi ) * Heaviside( arccos( L / 2. / l ) - phi )
def le_sim( phi, x, lf ):
'''
Solve embedded length l_e of fiber
'''
return lf / 2. - abs( x ) / cos( phi )
# @todo: very short specimen
def le( L, l ):
'''
Solve embedded length l_e of fibes (including null values) (integral)
'''
if L < l:
#print 'very short specimen',
return L / 4.
if L < 2. * l:
#print 'short specimen',
return - L / 4. - L / 4. * ln( L / 2. ) + L / 4. * ln( L ) + 1 / 2. * l * ( 1 - L / ( 2. * l ) ) + 1 / 4. * L * ln( L / ( 2. * l ) ) + l / 4.
if L >= 2. * l:
#print 'long specimen',
return - 1 / 4. * l - 1 / 4. * L * ln( L - l ) + 1 / 4. * L * ln( L )
# Configuration
l_x = 0.1 # [m]
l_y = 0.04 # [m]
l_z = 0.04 # [m]
vf = 0.015 # [-]
lf = 0.017 # [m]
df = 0.0003#0.175e-3 # [m]
Ac = l_y * l_z
Af = df ** 2 / 4. * pi
print 'Area', Af
n = int( Ac * l_x * vf / ( Af * lf ) )
print 'number of fibers in specimen volume', n
spec = Specimen( l_x = l_x, l_y = l_y, l_z = l_z ) #
fib = Fibers( spec = spec, n = n, lf = lf, df = df, overlaps = True ) # #23e-6
p = 1 / 2. * lf / l_x # probability_cut_nooverlaps( spec.l_x, fib.lf, 0. )
#p = 1 / 2. * lf / l_x
print 'probability', p
bin_mean = fib.n * p
bin_var = bin_mean * ( 1 - p )
ek = bin_mean
dk = bin_var
std_k = sqrt( dk )
print 'E[k] ', ek, 'D[k] ', dk, 'std', std_k, 'cov', std_k / ek
ep_1 = 7.21819420846
dp_1 = 17.3747024443
std_p_1 = sqrt( dp_1 )
print 'E[P1]', ep_1, 'D[P1]', dp_1, 'std', std_p_1, 'cov', std_p_1 / ep_1
ep_k = ep_1 * ek
dp_k = ek * dp_1 + dk * ep_1 ** 2
std_p_k = sqrt( dp_k )
print 'E[Pk]', ep_k, 'D[Pk]', dp_k, 'std', std_p_k, 'cov', std_p_k / ep_k
print '#' * 50
print 'E[P1]', ep_k, 'D[Pk]', dp_k, 'std', std_p_1 * sqrt( ek ), 'cov', std_p_1 * sqrt( ek ) / ep_k
import matplotlib.pyplot as plt
x = linspace( 0, ep_k + 3 * std_p_k, 1000 )
#plt.plot( x, norm( ep_k, std_p_k ).pdf( x ) )
n_sim = 100
#sec = rand( 1, n_sec ) * l_x
v = []
# crosssectional position
sec = .0 * spec.l_x / 2.#[-l_x * 0.99 / 2. , -l_x / 3., 0., l_x / 3. , l_x * 0.99 / 2. ]#linspace( 0, l_x, n_sec )
E = 200.e9
d = 0.0003
A = d ** 2 * pi / 4.
tau = 1.76e6 * d * pi
f = 0.03
def pullout2( w, le, phi ):
w = w[None, :]
le = le[:, None]
phi = phi[:, None]
d = sqrt( E * A * tau * w ) * exp( f * phi )
p = ones_like( w ) * tau * le * exp( f * phi )
return d * Heaviside( p - d ) + p * Heaviside( d - p )
p_plot = 0
def cut_fibers():
'''
Plot cut fibers (only ellipses), solve mean value of area and embedded length of cut fibers.
Plot histogram of embedded length.
'''
mask = matrix( fib.cut_func( sx, lx, sec ).astype( 'bool' ) )
sx_cut = sx[ mask ]
sy_cut = sy[ mask ]
sz_cut = sz[ mask ]
phi_x_cut = phi_x[ mask ]
theta_cut = theta[ mask ]
A_cut = sum( cut_area_func( fib.df, fib.lf, sx_cut, phi_x_cut, sec ) )
le_cut = le_sim( phi_x_cut, abs( sx_cut - sec * ones_like( sx_cut ) ), fib.lf )
le_cut_null = le_sim( phi_x, abs( sx - sec * ones_like( sx ) ), fib.lf ) * fib.cut_func( sx, lx, sec )
# fig6 = figure( 6 )
# ax6 = Axes( fig6, [.1, .1, .8, .8] )
# hist( le_cut, 50, normed=0 )
# #hist( le_cut_null[0], 50, normed=0 )
# title( 'Embedded length l_e' )
# figure( 8 )
# n_mask = 10
# title( 'Histograms of $\phi$ fraction = $\pi/%i$' % n_mask )
# for i in range( 0, n_mask ):
# mask1 = phi_x_cut > i * pi / 2. / n_mask
# mask2 = phi_x_cut < ( i + 1 ) * pi / 2. / n_mask
# pdf, bins, patches = hist( le_cut[mask1 * mask2], 50 , histtype='step', label='%i' % i )
# legend()
#print mean( pdf )
# figure( 9 )
# title( 'Histogram of cut fibers angle' )
# xx = linspace( 0, pi / 2., 100 )
# hist( phi_x_cut, 50, normed=1 )
# plot( xx, sin( 2 * xx ), color='red', linewidth=4, label='overlaps' )
# plot( xx, cut_fiber_distrib( spec.l_x, fib.lf, xx ), color='green', linewidth=4, label='nooverlaps' )
# legend()
# fig10 = figure( 10 )
# title( '2D histogram of le and phi_x' )
# ax10 = Axes3D( fig10 )
# bin = 20
# H, xedges, yedges = histogram2d( le_cut, phi_x_cut, ( bin, bin ), normed=1 )
# #extent = [xedges[0] * 100, xedges[-1] * 100, yedges[0], yedges[-1]] #[xedges[0], xedges[-1], yedges[0], yedges[-1]]
# #im = imshow( H )#'binary', cmap='jet' , extent=extent
# #im.set_interpolation( 'bilinear' )
# #colorbar()
# x = ( xedges[range( 0, len( xedges ) - 1 )] + xedges[range( 1, len( xedges ) )] ) / 2.
# y = ( yedges[range( 0, len( yedges ) - 1 )] + yedges[range( 1, len( yedges ) )] ) / 2.
# #ax10.scatter3D( xedges.ravel(), yedges.ravel(), H.ravel() )
# xx = outer( x, ones( len( y ) ) )
# yy = outer( ones( len( x ) ), y )
# z = outer( ones( len( y ) ), sin( 2 * y ) ) / fib.lf * 2# * probability_cut_nooverlaps( spec.l_x, fib.lf, 0 )#* fib.n
# zz = outer( ones( len( y ) ), cut_fiber_distrib( spec.l_x, fib.lf, y ) / fib.lf * 2 )
# ax10.plot_wireframe( xx, yy, H )#, rstride=1, cstride=1
# ax10.plot_wireframe( xx, yy, z , color='red', label='overlaps' )
# ax10.plot_wireframe( xx, yy, zz , color='green', label='nooverlaps' )
# legend()
# print '%%%%%%%%%%%%%%%%%%%'
# print '%%Embedded length%%'
# print '%%%%%%%%%%%%%%%%%%%'
# print 'probability cut fibers', probability_cut_nooverlaps( spec.l_x, fib.lf, 0. )
# print 'le_mean cut fibers (int/p) \t', le( spec.l_x, fib.lf ) / probability_cut_nooverlaps( spec.l_x, fib.lf, 0. )
# print 'le_mean of cut fibers (sim) \t', mean( le_cut ), '\t |\t standard deviation \t', std( le_cut )
# print 'include nulls (sim) \t\t', mean( le_cut_null )
# print 'E[le] (int) \t\t\t', le( spec.l_x, fib.lf )
# print 'cut fibers area', A_cut, 'Ac', spec._get_cross_area(), 'af', A_cut / spec._get_cross_area()*100, '%'
PP = tau * le_cut * e ** ( f * phi_x_cut )
#print 'mean PP', mean( PP )
p_plot = pullout2( w, le_cut, phi_x_cut )
P = mean( p_plot , axis = 0 )
varP = var( pullout2( w, le_cut, phi_x_cut ), axis = 0 )
#print 'mean P', P[-1], 'variance P', varP[-1]
return P, varP, p_plot
P = []
V = []
w = linspace( 0, 0.016, 500 ) / 1000.
for j in range( 0, n_sim ):
fib.sim_i = j
sx = fib.sx
lx = fib.lx
sy = fib.sy
ly = fib.ly
sz = fib.sz
lz = fib.lz
phi_x = fib.phi_x
theta = fib.theta
phi_y = fib.phi_y
phi_z = fib.phi_z
vec_cut = fib.cut_func( sx, lx, sec )
v.append( sum( vec_cut ) )
pp, vv, pplt = cut_fibers()
P.append( pp )
V.append( vv )
#plt.figure( 100 )
#plt.plot( w, array( P ).T, color='red' )
#plt.plot( w, mean( array( P ), axis=0 ), linewidth=3, color='blue' )
print 'P', mean( P ), var( P ), std( P )
p = probability_cut_nooverlaps( spec.l_x, fib.lf, sec )
bin_mean = fib.n * p
bin_stdv = sqrt( bin_mean * ( 1 - p ) )
bin_skew = ( 1 - 2 * p ) / bin_stdv
poiss_lambda = fib.n * p
poiss_mean = poiss_lambda
poiss_stdv = poiss_lambda ** ( 1 / 2. )
poiss_skew = poiss_lambda ** ( -1 / 2. )
# Volume fraction
Vf = fib._get_volume()
Vc = spec._get_volume()
vol_frac_V = Vf / Vc * 100 * fib.n
#vol_frac_A = mean( v_vol_frac_A ) / Ac * 100
print "sec:", sec # , "vec ", v
print " mean value = %.5f" % ( matrix( v ).mean() ), "\t| stnd devia = %.5f" % ( matrix( v ).std() ) , "\t| skewness ", skew( v )
print " mean_binom = %.5f" % bin_mean , "\t| stdv_binom = %.5f" % bin_stdv , "\t| skew_binom ", bin_skew
print " mean_poiss = %.5f" % poiss_mean , "\t| stdv_poiss = %.5f" % poiss_stdv , "\t| skew_poiss ", poiss_skew
print " vol_frac_V = %.3f%%" % vol_frac_V # , "\t| vol_frac_A = %.3g%%" % vol_frac_A , "\t| v_V / v_A ", vol_frac_V / vol_frac_A
#print Af
#print Af / cos_cut[cos_cut != 0]
#for i in range( 0, len( cos_cut[0] ) ):
# if cos_cut[0][i] != 0:
# print cos_cut[0][i]
return w, array( P ).T, mean( array( P ), axis = 0 ), array( V ).T, mean( array( V ), axis = 0 ), pplt