def fcn2min(params, x, data):
shape = params['shape'].value
scale = params['scale'].value
loc = params['loc'].value
# m = params['f_shape'].value
# sV0 = params['f_scale'].value
sV0 = 0.0076
m = 6.7
tau = RV('gamma', shape=shape, scale=scale, loc=loc)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
r = 3.5e-3
E_f = 182e3
T = 2. * tau_arr / r
# scale parameter with respect to a reference volume
s = ((T * (m + 1.) * sV0 ** m) /
(2. * E_f * pi * r ** 2)) ** (1. / (m + 1.))
ef0 = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
Gxi = 1 - np.exp(-(ef0 / s) ** (m + 1.))
mu_int = ef0 * (1 - Gxi)
sigma = mu_int * E_f
return np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000 - data
def fcn2min(shape, scale, loc, x):
# shape = params['shape'].value
# scale = params['scale'].value
# loc = params['loc'].value
# m = params['f_shape'].value
# sV0 = params['f_scale'].value
sV0 = 0.0069
m = 7.1
tau = RV('gamma', shape=shape, scale=scale, loc=loc)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
r = 3.5e-3
E_f = 180e3
# lm = 1000
#
# def cdf(e, depsf, r, lm, m, sV0):
# '''weibull_fibers_cdf_mc'''
# s = ((depsf * (m + 1.) * sV0 ** m) /
# (2. * pi * r ** 2.)) ** (1. / (m + 1.))
# a0 = (e + 1e-15) / depsf
# expfree = (e / s) ** (m + 1.)
# expfixed = a0 / \
# (lm / 2.0) * (e / s) ** (m + 1) * \
# (1. - (1. - lm / 2.0 / a0) ** (m + 1.))
# print expfree
# print expfixed
# mask = a0 < lm / 2.0
# exp = expfree * mask + \
# np.nan_to_num(expfixed * (mask == False))
# return 1. - np.exp(- exp)
#
# T = 2. * tau_arr / r + 1e-10
# ef0cb = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
# ef0lin = x[:, np.newaxis] / lm + \
# T[np.newaxis, :] * lm / 4. / E_f
# depsf = T / E_f
# a0 = ef0cb / depsf
# mask = a0 < lm / 2.0
# e = ef0cb * mask + ef0lin * (mask == False)
# Gxi = cdf(e, depsf, r, lm, m, sV0)
# mu_int = e * (1. - Gxi)
# sigma = mu_int * E_f
T = 2. * tau_arr / r
# scale parameter with respect to a reference volume
s = ((T * (m + 1.) * sV0 ** m) /
(2. * E_f * pi * r ** 2)) ** (1. / (m + 1.))
ef0 = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
Gxi = 1 - np.exp(-(ef0 / s) ** (m + 1.))
mu_int = ef0 * (1 - Gxi)
sigma = mu_int * E_f
return np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000
def fcn2min(params, x, data):
""" model decaying sine wave, subtract data"""
shape = params['shape'].value
scale = params['scale'].value
m = params['f_shape'].value
sV0 = params['f_scale'].value
# m = 8.806672387136711
# sV0 = 0.013415768576509945
# sV0 = 3243. / \
# (182e3 * (pi * 3.5e-3 ** 2 * 50.) ** (-1. / m) * gamma(1 + 1. / m))
# shape = 0.0505
# scale = 2.276
# CS = 12.
# mu_tau = 1.3 * 3.5e-3 * 3.6 * (1. - 0.01) / (2. * 0.01 * CS)
# scale = mu_tau / shape
tau = RV('gamma', shape=shape, scale=scale, loc=0.)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
r = 3.5e-3
E_f = 180e3
lm = 1000.
def cdf(e, depsf, r, lm, m, sV0):
'''weibull_fibers_cdf_mc'''
s = ((depsf * (m + 1.) * sV0 ** m) /
(2. * pi * r ** 2.)) ** (1. / (m + 1.))
a0 = (e + 1e-15) / depsf
expfree = (e / s) ** (m + 1)
expfixed = a0 / \
(lm / 2.0) * (e / s) ** (m + 1) * \
(1. - (1. - lm / 2.0 / a0) ** (m + 1.))
mask = a0 < lm / 2.0
exp = expfree * mask + \
np.nan_to_num(expfixed * (mask == False))
return 1. - np.exp(- exp)
T = 2. * tau_arr / r + 1e-10
ef0cb = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
ef0lin = x[:, np.newaxis] / lm + \
T[np.newaxis, :] * lm / 4. / E_f
depsf = T / E_f
a0 = ef0cb / depsf
mask = a0 < lm / 2.0
e = ef0cb * mask + ef0lin * (mask == False)
Gxi = cdf(e, depsf, r, lm, m, sV0)
mu_int = e * (1. - Gxi)
sigma = mu_int * E_f
return np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000 - data
def func1(w_arr, k):
tau = RV('gamma', shape=k, scale=scale, loc=0.)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
# sV0 = sV0
# m = m
r = 3.5e-3
E_f = 180e3
lm = 1000.
def cdf(e, depsf, r, lm, m, sV0):
'''weibull_fibers_cdf_mc'''
s = ((depsf * (m + 1.) * sV0 ** m) /
(2. * pi * r ** 2.)) ** (1. / (m + 1.))
a0 = (e + 1e-15) / depsf
expfree = (e / s) ** (m + 1.)
expfixed = a0 / \
(lm / 2.0) * (e / s) ** (m + 1.) * \
(1. - (1. - lm / 2.0 / a0) ** (m + 1.))
mask = a0 < lm / 2.0
exp = expfree * mask + \
np.nan_to_num(expfixed * (mask == False))
return 1. - np.exp(- exp)
T = 2. * tau_arr / r + 1e-10
ef0cb = np.sqrt(w_arr[:, np.newaxis] * T[np.newaxis, :] / E_f)
ef0lin = w_arr[:, np.newaxis] / lm + \
T[np.newaxis, :] * lm / 4. / E_f
depsf = T / E_f
a0 = ef0cb / depsf
mask = a0 < lm / 2.0
e = ef0cb * mask + ef0lin * (mask == False)
Gxi = cdf(e, depsf, r, lm, m, sV0)
# plt.plot(w_arr, np.average(Gxi, axis=1), label='s='+str(s)+'m='+str(m))
mu_int = e * (1. - Gxi)
sigma = mu_int * E_f
return np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000
def lackoffit(m, sV0, shape, scale, x, data):
""" model decaying sine wave, subtract data"""
tau = RV('gamma', shape=shape, scale=scale, loc=0.)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
r = 3.5e-3
E_f = 180e3
lm = 1000.
def cdf(e, depsf, r, lm, m, sV0):
'''weibull_fibers_cdf_mc'''
s = ((depsf * (m + 1.) * sV0 ** m) /
(2. * pi * r ** 2.)) ** (1. / (m + 1.))
a0 = (e + 1e-15) / depsf
expfree = (e / s) ** (m + 1.)
expfixed = a0 / \
(lm / 2.0) * (e / s) ** (m + 1.) * \
(1. - (1. - lm / 2.0 / a0) ** (m + 1.))
mask = a0 < lm / 2.0
exp = expfree * mask + \
np.nan_to_num(expfixed * (mask == False))
return 1. - np.exp(- exp)
T = 2. * tau_arr / r + 1e-10
ef0cb = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
ef0lin = x[:, np.newaxis] / lm + \
T[np.newaxis, :] * lm / 4. / E_f
depsf = T / E_f
a0 = ef0cb / depsf
mask = a0 < lm / 2.0
e = ef0cb * mask + ef0lin * (mask == False)
Gxi = cdf(e, depsf, r, lm, m, sV0)
mu_int = e * (1. - Gxi)
sigma = mu_int * E_f
residual = np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000 - data
return np.sum(residual ** 2)
def fcn2min(x):
""" model decaying sine wave, subtract data"""
shape = 0.0539
scale = 1.44
loc = 0.00126
m = 6.7
sV0 = 0.0076
tau = RV('gamma', shape=shape, scale=scale, loc=loc)
n_int = 500
p_arr = np.linspace(0.5 / n_int, 1 - 0.5 / n_int, n_int)
tau_arr = tau.ppf(p_arr) + 1e-10
r = 3.5e-3
E_f = 180e3
T = 2. * tau_arr / r
# scale parameter with respect to a reference volume
s = ((T * (m + 1.) * sV0 ** m) /
(2. * E_f * pi * r ** 2)) ** (1. / (m + 1.))
ef0 = np.sqrt(x[:, np.newaxis] * T[np.newaxis, :] / E_f)
Gxi = 1 - np.exp(-(ef0 / s) ** (m + 1.))
mu_int = ef0 * (1 - Gxi)
sigma = mu_int * E_f
return np.sum(sigma, axis=1) / n_int * (11. * 0.445) / 1000