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Merge pull request #296 from majsylw/feature/relaxation

浏览代码 
[GSoC 2021] Modelling complex materials
这个提交包含在：
Craig Warren
2021-10-19 14:30:25 +01:00
提交者 GitHub
父节点 fe22d72bdd 1c7c2a99c9
当前提交 55333df656
共有 19 个文件被更改，包括 3827 次插入 和 4 次删除
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5
gprMax/cmds_geometry/add_surface_roughness.py
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@@ -179,7 +179,10 @@ class AddSurfaceRoughness(UserObjectGeometry):
surface.surfaceID = requestedsurface
surface.fractalrange = fractalrange
surface.operatingonID = volume.ID
surface.seed = int(seed)
if seed is not None:
surface.seed = int(seed)
else:
surface.seed = seed
surface.weighting = weighting
# List of existing surfaces IDs

7
gprMax/cmds_geometry/fractal_box.py
查看文件

@@ -66,7 +66,7 @@ class FractalBox(UserObjectGeometry):
rot_pts = rotate_2point_object(pts, self.axis, self.angle, self.origin)
self.kwargs['p1'] = tuple(rot_pts[0, :])
self.kwargs['p2'] = tuple(rot_pts[1, :])
def create(self, grid, uip):
try:
p1 = self.kwargs['p1']
@@ -140,7 +140,10 @@ class FractalBox(UserObjectGeometry):
volume.ID = ID
volume.operatingonID = mixing_model_id
volume.nbins = nbins
volume.seed = int(seed)
if seed is not None:
volume.seed = int(seed)
else:
volume.seed = seed
volume.weighting = weighting
volume.averaging = averagefractalbox
volume.mixingmodel = mixingmodel

4
gprMax/hash_cmds_file.py
查看文件

@@ -209,7 +209,9 @@ def check_cmd_names(processedlines, checkessential=True):
# Commands that there can be multiple instances of in a model
# - these will be lists within the dictionary
multiplecmds = {key: [] for key in ['#geometry_view',
'#geometry_objects_write', '#material',
'#geometry_objects_write', '#material',
'#havriliak_negami', '#jonscher',
'#crim', '#raw_data',
'#soil_peplinski',
'#add_dispersion_debye',
'#add_dispersion_lorentz',

103
gprMax/hash_cmds_multiuse.py
查看文件

@@ -17,6 +17,10 @@
# along with gprMax. If not, see <http://www.gnu.org/licenses/>.
import logging
import os
os.path.join(os.path.dirname(__file__), '..', 'user_libs', 'DebyeFit')
from user_libs.DebyeFit import (HavriliakNegami, Jonscher, Crim, Rawdata)
from .cmds_multiuse import (PMLCFS, AddDebyeDispersion, AddDrudeDispersion,
AddLorentzDispersion, GeometryObjectsWrite,
@@ -170,6 +174,105 @@ def process_multicmds(multicmds):
scene_objects.append(snapshot)
cmdname = '#havriliak_negami'
if multicmds[cmdname] is not None:
for cmdinstance in multicmds[cmdname]:
tmp = cmdinstance.split()
if len(tmp) != 12 and len(tmp) != 13:
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' requires either twelve or thirteen parameters')
raise ValueError
seed = None
if len(tmp) == 13:
seed = int(tmp[12])
setup = HavriliakNegami(f_min=float(tmp[0]), f_max=float(tmp[1]),
alpha=float(tmp[2]), beta=float(tmp[3]),
e_inf=float(tmp[4]), de=float(tmp[5]), tau_0=float(tmp[6]),
sigma=float(tmp[7]), mu=float(tmp[8]), mu_sigma=float(tmp[9]),
number_of_debye_poles=int(tmp[10]), material_name=tmp[11],
optimizer_options={'seed': seed})
_, properties = setup.run()
multicmds['#material'].append(properties[0].split(':')[1].strip(' \t\n'))
multicmds['#add_dispersion_debye'].append(properties[1].split(':')[1].strip(' \t\n'))
cmdname = '#jonscher'
if multicmds[cmdname] is not None:
for cmdinstance in multicmds[cmdname]:
tmp = cmdinstance.split()
if len(tmp) != 11 and len(tmp) != 12:
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' requires either eleven or twelve parameters')
raise ValueError
seed = None
if len(tmp) == 12:
seed = int(tmp[11])
setup = Jonscher(f_min=float(tmp[0]), f_max=float(tmp[1]),
e_inf=float(tmp[2]), a_p=float(tmp[3]),
omega_p=float(tmp[4]), n_p=float(tmp[5]),
sigma=float(tmp[6]), mu=float(tmp[7]), mu_sigma=float(tmp[8]),
number_of_debye_poles=int(tmp[9]), material_name=tmp[10],
optimizer_options={'seed': seed})
_, properties = setup.run()
multicmds['#material'].append(properties[0].split(':')[1].strip(' \t\n'))
multicmds['#add_dispersion_debye'].append(properties[1].split(':')[1].strip(' \t\n'))
cmdname = '#crim'
if multicmds[cmdname] is not None:
for cmdinstance in multicmds[cmdname]:
tmp = cmdinstance.split()
if len(tmp) != 10 and len(tmp) != 11:
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' requires either ten or eleven parameters')
raise ValueError
seed = None
if len(tmp) == 11:
seed = int(tmp[10])
if (tmp[3][0] != '[' and tmp[3][-1] != ']') or (tmp[4][0] != '[' and tmp[4][-1] != ']'):
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' requires list at 6th and 7th position')
raise ValueError
vol_frac = [float(i) for i in tmp[3].strip('[]').split(',')]
material = [float(i) for i in tmp[4].strip('[]').split(',')]
if len(material) % 3 != 0:
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' each material requires three parameters: e_inf, de, tau_0')
raise ValueError
materials = [material[n:n+3] for n in range(0, len(material), 3)]
setup = Crim(f_min=float(tmp[0]), f_max=float(tmp[1]), a=float(tmp[2]),
volumetric_fractions=vol_frac, materials=materials,
sigma=float(tmp[5]), mu=float(tmp[6]), mu_sigma=float(tmp[7]),
number_of_debye_poles=int(tmp[8]), material_name=tmp[9],
optimizer_options={'seed': seed})
_, properties = setup.run()
multicmds['#material'].append(properties[0].split(':')[1].strip(' \t\n'))
multicmds['#add_dispersion_debye'].append(properties[1].split(':')[1].strip(' \t\n'))
cmdname = '#raw_data'
if multicmds[cmdname] is not None:
for cmdinstance in multicmds[cmdname]:
tmp = cmdinstance.split()
if len(tmp) != 6 and len(tmp) != 7:
logger.exception("'" + cmdname + ': ' + ' '.join(tmp) + "'" + ' requires either six or seven parameters')
raise ValueError
seed = None
if len(tmp) == 7:
seed = int(tmp[6])
setup = Rawdata(filename=tmp[0], sigma=float(tmp[1]),
mu=float(tmp[2]), mu_sigma=float(tmp[3]),
number_of_debye_poles=int(tmp[4]), material_name=tmp[5],
optimizer_options={'seed': seed})
_, properties = setup.run()
multicmds['#material'].append(properties[0].split(':')[1].strip(' \t\n'))
multicmds['#add_dispersion_debye'].append(properties[1].split(':')[1].strip(' \t\n'))
cmdname = '#material'
if multicmds[cmdname] is not None:
for cmdinstance in multicmds[cmdname]:

710
user_libs/DebyeFit/Debye_Fit.py 普通文件
查看文件

@@ -0,0 +1,710 @@
# Authors: Iraklis Giannakis, and Sylwia Majchrowska
# E-mail: i.giannakis@ed.ac.uk
#
# This file is part of gprMax.
#
# gprMax 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.
#
# gprMax 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 gprMax. If not, see <http://www.gnu.org/licenses/>.
import numpy as np
import os
from matplotlib import pylab as plt
import matplotlib.gridspec as gridspec
from pathlib import Path
import sys
import scipy.interpolate
import warnings
from optimization import PSO_DLS, DA_DLS, DE_DLS
class Relaxation(object):
""" Create Relaxation function object for complex material.
:param sigma: The conductivity (Siemens/metre).
:type sigma: float, non-optional
:param mu: The relative permeability.
:type mu: float, non-optional
:param mu_sigma: The magnetic loss.
:type mu_sigma: float, non-optional
:param material_name: A string containing the given name of
the material (e.g. "Clay").
:type material_name: str, non-optional
:param: number_of_debye_poles: Number of Debye functions used to
approximate the given electric
permittivity.
:type number_of_debye_poles: int, optional
:param: fn: Number of frequency points in frequency grid.
:type fn: int, optional (Default: 50)
:param plot: if True will plot the actual and the approximated
permittivity at the end (neglected as default: False).
:type plot: bool, optional, default:False
:param save: if True will save approximated material parameters
(not neglected as default: True).
:type save: bool, optional, default:True
:param optimizer: chosen optimization method:
Hybrid Particle Swarm-Damped Least-Squares (PSO_DLS),
Dual Annealing (DA) or Differential Evolution (DE)
(Default: PSO_DLS).
:type optimizer: Optimizer class, optional
:param optimizer_options: Additional keyword arguments passed to
optimizer class (Default: empty dict).
:type optimizer_options: dict, optional, default: empty dict
"""
def __init__(self, sigma, mu, mu_sigma,
material_name, f_n=50,
number_of_debye_poles=-1,
plot=True, save=False,
optimizer=PSO_DLS,
optimizer_options={}):
self.name = 'Relaxation function'
self.params = {}
self.number_of_debye_poles = number_of_debye_poles
self.f_n = f_n
self.sigma = sigma
self.mu = mu
self.mu_sigma = mu_sigma
self.material_name = material_name
self.plot = plot
self.save = save
self.optimizer = optimizer(**optimizer_options)
def set_freq(self, f_min, f_max, f_n=50):
""" Interpolate frequency vector using n equally logarithmicaly spaced frequencies.
Args:
f_min (float): First bound of the frequency range
used to approximate the given function (Hz).
f_max (float): Second bound of the frequency range
used to approximate the given function (Hz).
f_n (int): Number of frequency points in frequency grid
(Default: 50).
Note:
f_min and f_max must satisfied f_min < f_max
"""
if abs(f_min - f_max) > 1e12:
warnings.warn(f'The chosen range is realy big. '
f'Consider setting greater number of points '
f'on the frequency grid!')
self.freq = np.logspace(np.log10(f_min),
np.log10(f_max),
int(f_n))
def check_inputs(self):
""" Check the validity of the inputs. """
try:
d = [float(i) for i in
[self.number_of_debye_poles,
self.sigma, self.mu, self.mu_sigma]]
except ValueError:
sys.exit("The inputs should be numeric.")
if not isinstance(self.number_of_debye_poles, int):
sys.exit("The number of Debye poles must be integer.")
if (np.array(d[1:]) < 0).sum() != 0:
sys.exit("The inputs should be positive.")
def calculation(self):
""" Approximate the given relaxation function
(Havriliak-Negami function, Crim, Jonscher) or based on raw data.
"""
raise NotImplementedError()
def print_info(self):
"""Readable string of parameters for given approximation settings.
Returns:
s (str): Info about chosen function and its parameters.
"""
print(f"Approximating {self.name}"
f" using {self.number_of_debye_poles} Debye poles")
print(f"{self.name} parameters: ")
s = ''
for k, v in self.params.items():
s += f"{k:10s} = {v}\n"
print(s)
return f'{self.name}:\n{s}'
def optimize(self):
""" Calling the main optimisation module with defined lower and upper boundaries of search.
Returns:
tau (ndarray): The optimised relaxation times.
weights (ndarray): Resulting optimised weights for the given relaxation times.
ee (float): Average error between the actual and the approximated real part.
rl (ndarray): Real parts of chosen relaxation function
for given frequency points.
im (ndarray): Imaginary parts of chosen relaxation function
for given frequency points.
"""
# Define the lower and upper boundaries of search
lb = np.full(self.number_of_debye_poles,
-np.log10(np.max(self.freq)) - 3)
ub = np.full(self.number_of_debye_poles,
-np.log10(np.min(self.freq)) + 3)
# Call optimizer to minimize the cost function
tau, weights, ee, rl, im = self.optimizer.fit(func=self.optimizer.cost_function,
lb=lb, ub=ub,
funckwargs={'rl': self.rl,
'im': self.im,
'freq': self.freq}
)
return tau, weights, ee, rl, im
def run(self):
""" Solve the problem described by the given relaxation function
(Havriliak-Negami function, Crim, Jonscher)
or data given from a text file.
Returns:
avg_err (float): average fractional error
for relative permittivity (sum)
properties (list(str)): Given material nad Debye expnasion parameters
in a gprMax format.
"""
# Check the validity of the inputs
self.check_inputs()
# Print information about chosen approximation settings
self.print_info()
# Calculate both real and imaginary parts
# for the frequencies included in the vector freq
q = self.calculation()
# Set the real and the imaginary part of the relaxation function
self.rl, self.im = q.real, q.imag
if self.number_of_debye_poles == -1:
print("\n#########",
"Try to automaticaly fit number of Debye poles, up to 20!",
"##########\n", sep="")
error = np.infty # artificial best error starting value
self.number_of_debye_poles = 1
iteration = 1
# stop increasing number of Debye poles if error is smaller then 5%
# or 20 debye poles is reached
while error > 5 and iteration < 21:
# Calling the main optimisation module
tau, weights, ee, rl, im = self.optimize()
err_real, err_imag = self.error(rl + ee, im)
error = err_real + err_imag
self.number_of_debye_poles += 1
iteration += 1
else:
# Calling the main optimisation module
# for choosen number of debye poles
# if one of the weights is negative increase the stabiliser
# and repeat the optimisation
tau, weights, ee, rl, im = self.optimize()
err_real, err_imag = self.error(rl + ee, im)
# Print the results in gprMax format style
properties = self.print_output(tau, weights, ee)
print(f'The average fractional error for:\n'
f'- real part: {err_real}\n'
f'- imaginary part: {err_imag}\n')
if self.save:
self.save_result(properties)
# Plot the actual and the approximate dielectric properties
if self.plot:
self.plot_result(rl + ee, im)
return err_real + err_imag, properties
def print_output(self, tau, weights, ee):
""" Print out the resulting Debye parameters in a gprMax format.
Args:
tau (ndarray): The best known position form optimization module
(optimal design).
weights (ndarray): Resulting optimised weights for the given relaxation times.
ee (float): Average error between the actual and the approximated real part.
Returns:
material_prop (list(str)): Given material nad Debye expnasion parameters
in a gprMax format.
"""
print("Debye expansion parameters: ")
print(f" |{'e_inf':^14s}|{'De':^14s}|{'log(tau_0)':^25s}|")
print("_" * 65)
for i in range(0, len(tau)):
print("Debye {0:}|{1:^14.5f}|{2:^14.5f}|{3:^25.5f}|"
.format(i + 1, ee/len(tau), weights[i],
tau[i]))
print("_" * 65)
# Print the Debye expnasion in a gprMax format
material_prop = []
material_prop.append("#material: {} {} {} {} {}\n".format(ee, self.sigma,
self.mu,
self.mu_sigma,
self.material_name))
print(material_prop[0], end="")
dispersion_prop = "#add_dispersion_debye: {}".format(len(tau))
for i in range(len(tau)):
dispersion_prop += " {} {}".format(weights[i], 10**tau[i])
dispersion_prop += " {}".format(self.material_name)
print(dispersion_prop)
material_prop.append(dispersion_prop + '\n')
return material_prop
def plot_result(self, rl_exp, im_exp):
""" Plot the actual and the approximated electric permittivity,
along with relative error for real and imaginary parts
using a semilogarithm X axes.
Args:
rl_exp (ndarray): Real parts of optimised Debye expansion
for given frequency points (plus average error).
im_exp (ndarray): Imaginary parts of optimised Debye expansion
for given frequency points.
"""
plt.close("all")
fig = plt.figure(figsize=(16, 8), tight_layout=True)
gs = gridspec.GridSpec(2, 1)
ax = fig.add_subplot(gs[0])
ax.grid(b=True, which="major", linewidth=0.2, linestyle="--")
ax.semilogx(self.freq * 1e-6, rl_exp, "b-", linewidth=2.0,
label="Debye Expansion: Real part")
ax.semilogx(self.freq * 1e-6, -im_exp, "k-", linewidth=2.0,
label="Debye Expansion: Imaginary part")
ax.semilogx(self.freq * 1e-6, self.rl, "r.",
linewidth=2.0, label=f"{self.name}: Real part")
ax.semilogx(self.freq * 1e-6, -self.im, "g.", linewidth=2.0,
label=f"{self.name}: Imaginary part")
ax.set_ylim([-1, np.max(np.concatenate([self.rl, -self.im])) + 1])
ax.legend()
ax.set_xlabel("Frequency (MHz)")
ax.set_ylabel("Relative permittivity")
ax = fig.add_subplot(gs[1])
ax.grid(b=True, which="major", linewidth=0.2, linestyle="--")
ax.semilogx(self.freq * 1e-6, (rl_exp - self.rl)/(self.rl + 1), "b-", linewidth=2.0,
label="Real part")
ax.semilogx(self.freq * 1e-6, (-im_exp + self.im)/(self.im + 1), "k-", linewidth=2.0,
label="Imaginary part")
ax.legend()
ax.set_xlabel("Frequency (MHz)")
ax.set_ylabel("Relative approximation error")
plt.show()
def error(self, rl_exp, im_exp):
""" Calculate the average fractional error separately for
relative permittivity (real part) and conductivity (imaginary part)
Args:
rl_exp (ndarray): Real parts of optimised Debye expansion
for given frequency points (plus average error).
im_exp (ndarray): Imaginary parts of optimised Debye expansion
for given frequency points.
Returns:
avg_err_real (float): average fractional error
for relative permittivity (real part)
avg_err_imag (float): average fractional error
for conductivity (imaginary part)
"""
avg_err_real = np.sum(np.abs((rl_exp - self.rl)/(self.rl + 1)) * 100)/len(rl_exp)
avg_err_imag = np.sum(np.abs((-im_exp + self.im)/(self.im + 1)) * 100)/len(im_exp)
return avg_err_real, avg_err_imag
@staticmethod
def save_result(output, fdir="../materials"):
""" Save the resulting Debye parameters in a gprMax format.
Args:
output (list(str)): Material and resulting Debye parameters
in a gprMax format.
fdir (str): Path to saving directory.
"""
if fdir != "../materials" and os.path.isdir(fdir):
file_path = os.path.join(fdir, "my_materials.txt")
elif os.path.isdir("../materials"):
file_path = os.path.join("../materials",
"my_materials.txt")
elif os.path.isdir("materials"):
file_path = os.path.join("materials",
"my_materials.txt")
elif os.path.isdir("user_libs/materials"):
file_path = os.path.join("user_libs", "materials",
"my_materials.txt")
else:
sys.exit("Cannot save material properties "
f"in {os.path.join(fdir, 'my_materials.txt')}!")
fileH = open(file_path, "a")
fileH.write(f"## {output[0].split(' ')[-1]}")
fileH.writelines(output)
fileH.write("\n")
fileH.close()
print(f"Material properties save at: {file_path}")
class HavriliakNegami(Relaxation):
""" Approximate a given Havriliak-Negami function
Havriliak-Negami function = ε_∞ + Δ‎ε / (1 + (2πfjτ)**α)**β,
where f is the frequency in Hz.
:param f_min: First bound of the frequency range
used to approximate the given function (Hz).
:type f_min: float
:param f_max: Second bound of the frequency range
used to approximate the given function (Hz).
:type f_max: float
:param e_inf: The real relative permittivity at infinity frequency
:type e_inf: float
:param alpha: Real positive float number which varies 0 < alpha < 1.
For alpha = 1 and beta !=0 & beta !=1 Havriliak-Negami
transforms to Cole-Davidson function.
:type alpha: float
:param beta: Real positive float number which varies 0 < beta < 1.
For beta = 1 and alpha !=0 & alpha !=1 Havriliak-Negami
transforms to Cole-Cole function.
:type beta: float
:param de: The difference of relative permittivity at infinite frequency
and the relative permittivity at zero frequency.
:type de: float
:param tau_0: Real positive float number, tau_0 is the relaxation time.
:type tau_0: float
"""
def __init__(self, f_min, f_max,
alpha, beta, e_inf, de, tau_0,
sigma, mu, mu_sigma, material_name,
number_of_debye_poles=-1, f_n=50,
plot=False, save=False,
optimizer=PSO_DLS,
optimizer_options={}):
super(HavriliakNegami, self).__init__(sigma=sigma, mu=mu, mu_sigma=mu_sigma,
material_name=material_name, f_n=f_n,
number_of_debye_poles=number_of_debye_poles,
plot=plot, save=save,
optimizer=optimizer,
optimizer_options=optimizer_options)
self.name = 'Havriliak-Negami function'
# Place the lower frequency bound at f_min and the upper frequency bound at f_max
if f_min > f_max:
self.f_min, self.f_max = f_max, f_min
else:
self.f_min, self.f_max = f_min, f_max
# Choosing n frequencies logarithmicaly equally spaced between the bounds given
self.set_freq(self.f_min, self.f_max, self.f_n)
self.e_inf, self.alpha, self.beta, self.de, self.tau_0 = e_inf, alpha, beta, de, tau_0
self.params = {'f_min': self.f_min, 'f_max': self.f_max,
'eps_inf': self.e_inf, 'Delta_eps': self.de, 'tau_0': self.tau_0,
'alpha': self.alpha, 'beta': self.beta}
def check_inputs(self):
""" Check the validity of the Havriliak Negami model's inputs. """
super(HavriliakNegami, self).check_inputs()
try:
d = [float(i) for i in self.params.values()]
except ValueError:
sys.exit("The inputs should be numeric.")
if (np.array(d) < 0).sum() != 0:
sys.exit("The inputs should be positive.")
if self.alpha > 1:
sys.exit("Alpha value must range between 0-1 (0 <= alpha <= 1)")
if self.beta > 1:
sys.exit("Beta value must range between 0-1 (0 <= beta <= 1)")
if self.f_min == self.f_max:
sys.exit("Null frequency range")
def calculation(self):
"""Calculates the Havriliak-Negami function for
the given parameters."""
return self.e_inf + self.de / (
1 + (1j * 2 * np.pi *
self.freq * self.tau_0)**self.alpha
)**self.beta
class Jonscher(Relaxation):
""" Approximate a given Jonsher function
Jonscher function = ε_∞ - ap * (-1j * 2πf / omegap)**n_p,
where f is the frequency in Hz
:param f_min: First bound of the frequency range
used to approximate the given function (Hz).
:type f_min: float
:param f_max: Second bound of the frequency range
used to approximate the given function (Hz).
:type f_max: float
:params e_inf: The real relative permittivity at infinity frequency.
:type e_inf: float, non-optional
:params a_p: Jonscher parameter. Real positive float number.
:type a_p: float, non-optional
:params omega_p: Jonscher parameter. Real positive float number.
:type omega_p: float, non-optional
:params n_p: Jonscher parameter, 0 < n_p < 1.
:type n_p: float, non-optional
"""
def __init__(self, f_min, f_max,
e_inf, a_p, omega_p, n_p,
sigma, mu, mu_sigma,
material_name, number_of_debye_poles=-1,
f_n=50, plot=False, save=False,
optimizer=PSO_DLS,
optimizer_options={}):
super(Jonscher, self).__init__(sigma=sigma, mu=mu, mu_sigma=mu_sigma,
material_name=material_name, f_n=f_n,
number_of_debye_poles=number_of_debye_poles,
plot=plot, save=save,
optimizer=optimizer,
optimizer_options=optimizer_options)
self.name = 'Jonsher function'
# Place the lower frequency bound at f_min and the upper frequency bound at f_max
if f_min > f_max:
self.f_min, self.f_max = f_max, f_min
else:
self.f_min, self.f_max = f_min, f_max
# Choosing n frequencies logarithmicaly equally spaced between the bounds given
self.set_freq(self.f_min, self.f_max, self.f_n)
self.e_inf, self.a_p, self.omega_p, self.n_p = e_inf, a_p, omega_p, n_p
self.params = {'f_min': self.f_min, 'f_max': self.f_max,
'eps_inf': self.e_inf, 'n_p': self.n_p,
'omega_p': self.omega_p, 'a_p': self.a_p}
def check_inputs(self):
""" Check the validity of the inputs. """
super(Jonscher, self).check_inputs()
try:
d = [float(i) for i in self.params.values()]
except ValueError:
sys.exit("The inputs should be numeric.")
if (np.array(d) < 0).sum() != 0:
sys.exit("The inputs should be positive.")
if self.n_p > 1:
sys.exit("n_p value must range between 0-1 (0 <= n_p <= 1)")
if self.f_min == self.f_max:
sys.exit("Error: Null frequency range!")
def calculation(self):
"""Calculates the Q function for the given parameters"""
return self.e_inf + (self.a_p * (2 * np.pi *
self.freq / self.omega_p)**(self.n_p-1)) * (
1 - 1j / np.tan(self.n_p * np.pi/2))
class Crim(Relaxation):
""" Approximate a given CRIM function
CRIM = (Σ frac_i * (‎ε_∞_i + Δ‎ε_i/(1 + 2πfj*τ_i))^a)^(1/a)
:param f_min: First bound of the frequency range
used to approximate the given function (Hz).
:type f_min: float
:param f_max: Second bound of the frequency range
used to approximate the given function (Hz).
:type f_max: float
:param a: Shape factor.
:type a: float, non-optional
:param: volumetric_fractions: Volumetric fraction for each material.
:type volumetric_fractions: ndarray, non-optional
:param materials: Arrays of materials properties, for each material [e_inf, de, tau_0].
:type materials: ndarray, non-optional
"""
def __init__(self, f_min, f_max, a, volumetric_fractions,
materials, sigma, mu, mu_sigma, material_name,
number_of_debye_poles=-1, f_n=50,
plot=False, save=False,
optimizer=PSO_DLS,
optimizer_options={}):
super(Crim, self).__init__(sigma=sigma, mu=mu, mu_sigma=mu_sigma,
material_name=material_name, f_n=f_n,
number_of_debye_poles=number_of_debye_poles,
plot=plot, save=save,
optimizer=optimizer,
optimizer_options=optimizer_options)
self.name = 'CRIM function'
# Place the lower frequency bound at f_min and the upper frequency bound at f_max
if f_min > f_max:
self.f_min, self.f_max = f_max, f_min
else:
self.f_min, self.f_max = f_min, f_max
# Choosing n frequencies logarithmicaly equally spaced between the bounds given
self.set_freq(self.f_min, self.f_max, self.f_n)
self.a = a
self.volumetric_fractions = np.array(volumetric_fractions)
self.materials = np.array(materials)
self.params = {'f_min': self.f_min, 'f_max': self.f_max,
'a': self.a, 'volumetric_fractions': self.volumetric_fractions,
'materials': self.materials}
def check_inputs(self):
""" Check the validity of the inputs. """
super(Crim, self).check_inputs()
try:
d = [float(i) for i in
[self.f_min, self.f_max, self.a]]
except ValueError:
sys.exit("The inputs should be numeric.")
if (np.array(d) < 0).sum() != 0:
sys.exit("The inputs should be positive.")
if len(self.volumetric_fractions) != len(self.materials):
sys.exit("Number of volumetric volumes does not match the dielectric properties")
# Check if the materials are at least two
if len(self.volumetric_fractions) < 2:
sys.exit("The materials should be at least 2")
# Check if the frequency range is null
if self.f_min == self.f_max:
sys.exit("Null frequency range")
# Check if the inputs are positive
f = [i for i in self.volumetric_fractions if i < 0]
if len(f) != 0:
sys.exit("Error: The inputs should be positive")
for i in range(len(self.volumetric_fractions)):
f = [i for i in self.materials[i][:] if i < 0]
if len(f) != 0:
sys.exit("Error: The inputs should be positive")
# Check if the summation of the volumetric fractions equal to one
if np.sum(self.volumetric_fractions) != 1:
sys.exit("Error: The summation of volumetric volumes should be equal to 1")
def print_info(self):
""" Print information about chosen approximation settings """
print(f"Approximating Complex Refractive Index Model (CRIM)"
f" using {self.number_of_debye_poles} Debye poles")
print("CRIM parameters: ")
for i in range(len(self.volumetric_fractions)):
print("Material {}.:".format(i+1))
print("---------------------------------")
print(f"{'Vol. fraction':>27s} = {self.volumetric_fractions[i]}")
print(f"{'e_inf':>27s} = {self.materials[i][0]}")
print(f"{'De':>27s} = {self.materials[i][1]}")
print(f"{'log(tau_0)':>27s} = {np.log10(self.materials[i][2])}")
def calculation(self):
"""Calculates the Crim function for the given parameters"""
return np.sum(np.repeat(self.volumetric_fractions, len(self.freq)
).reshape((-1, len(self.materials)))*(
self.materials[:, 0] + self.materials[:, 1] / (
1 + 1j * 2 * np.pi * np.repeat(self.freq, len(self.materials)
).reshape((-1, len(self.materials))) * self.materials[:, 2]))**self.a,
axis=1)**(1 / self.a)
class Rawdata(Relaxation):
""" Interpolate data given from a text file.
:param filename: text file which contains three columns:
frequency (Hz),Real,Imaginary (separated by comma).
:type filename: str, non-optional
:param delimiter: separator for three data columns
:type delimiter: str, optional (Deafult: ',')
"""
def __init__(self, filename,
sigma, mu, mu_sigma,
material_name, number_of_debye_poles=-1,
f_n=50, delimiter=',',
plot=False, save=False,
optimizer=PSO_DLS,
optimizer_options={}):
super(Rawdata, self).__init__(sigma=sigma, mu=mu, mu_sigma=mu_sigma,
material_name=material_name, f_n=f_n,
number_of_debye_poles=number_of_debye_poles,
plot=plot, save=save,
optimizer=optimizer,
optimizer_options=optimizer_options)
self.delimiter = delimiter
self.filename = Path(filename).absolute()
self.params = {'filename': self.filename}
def check_inputs(self):
""" Check the validity of the inputs. """
super(Rawdata, self).check_inputs()
if not os.path.isfile(self.filename):
sys.exit("File doesn't exists!")
def calculation(self):
""" Interpolate real and imaginary part from data.
Column framework of the input file three columns comma-separated
Frequency(Hz),Real,Imaginary
"""
# Read the file
with open(self.filename) as f:
try:
array = np.array(
[[float(x) for x in line.split(self.delimiter)] for line in f]
)
except ValueError:
sys.exit("Error: The inputs should be numeric")
self.set_freq(min(array[:, 0]), max(array[:, 0]), self.f_n)
rl_interp = scipy.interpolate.interp1d(array[:, 0], array[:, 1],
fill_value="extrapolate")
im_interp = scipy.interpolate.interp1d(array[:, 0], array[:, 2],
fill_value="extrapolate")
return rl_interp(self.freq) - 1j * im_interp(self.freq)
if __name__ == "__main__":
# Kelley et al. parameters
setup = HavriliakNegami(f_min=1e7, f_max=1e11,
alpha=0.91, beta=0.45,
e_inf=2.7, de=8.6-2.7, tau_0=9.4e-10,
sigma=0, mu=0, mu_sigma=0,
material_name="Kelley", f_n=100,
number_of_debye_poles=6,
plot=True, save=False,
optimizer_options={'swarmsize': 30,
'maxiter': 100,
'omega': 0.5,
'phip': 1.4,
'phig': 1.4,
'minstep': 1e-8,
'minfun': 1e-8,
'seed': 111,
'pflag': True})
setup.run()
setup = HavriliakNegami(f_min=1e7, f_max=1e11,
alpha=0.91, beta=0.45,
e_inf=2.7, de=8.6-2.7, tau_0=9.4e-10,
sigma=0, mu=0, mu_sigma=0,
material_name="Kelley", f_n=100,
number_of_debye_poles=6,
plot=True, save=False,
optimizer=DA_DLS,
optimizer_options={'seed': 111})
setup.run()
setup = HavriliakNegami(f_min=1e7, f_max=1e11,
alpha=0.91, beta=0.45,
e_inf=2.7, de=8.6-2.7, tau_0=9.4e-10,
sigma=0, mu=0, mu_sigma=0,
material_name="Kelley", f_n=100,
number_of_debye_poles=6,
plot=True, save=False,
optimizer=DE_DLS,
optimizer_options={'seed': 111})
setup.run()
# Testing setup
setup = Rawdata("examples/Test.txt", 0.1, 1, 0.1, "M1",
number_of_debye_poles=3, plot=True,
optimizer_options={'seed': 111})
setup.run()
np.random.seed(111)
setup = HavriliakNegami(1e12, 1e-3, 0.5, 1, 10, 5,
1e-6, 0.1, 1, 0, "M2",
number_of_debye_poles=6,
plot=True)
setup.run()
setup = Jonscher(1e6, 1e-5, 50, 1, 1e5, 0.7,
0.1, 1, 0.1, "M3",
number_of_debye_poles=4,
plot=True)
setup.run()
f = np.array([0.5, 0.5])
material1 = [3, 25, 1e6]
material2 = [3, 0, 1e3]
materials = np.array([material1, material2])
setup = Crim(1*1e-1, 1e-9, 0.5, f, materials, 0.1,
1, 0, "M4", number_of_debye_poles=2,
plot=True)
setup.run()

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Fitting multi-pole Debye model to dielectric data
=================================================
All electromagnetic phenomena are governed by the Maxwell's equations, which describing how electric and magnetic fields are distributed due to charges and currents,
and how they are changing in time. gprMax is open source software that simulates electromagnetic wave propagation by using
Yee's algorithm to solve (3+1)D Maxwell’s equations with Finite-Difference Time-Domain (FDTD) method.
The behavior of the electromagnetic wave is closely dependent on the material in which it propagates.
Some dispersive media have quite complex electromagnetic properties depending on the wavelength.
This, for example, means that for different frequencies the wave can propagate with a different speed in different materials.
This significantly affects the solver’s output. The main goal of the GSoC 2021 project was to enhance series of scripts,
which modelled electromagnetic properties of the variety range of materials.
Multi-pole Debye model
======================
Electric permittivity is a complex function with both real and imaginary parts.
In general, as a hard and fast rule, the real part dictates the velocity of the medium while the imaginary part is related to the electromagnetic losses.
The generic form of dispersive media takes a form of
.. math::
\epsilon(\omega) = \epsilon^{'}(\omega) - j\epsilon^{''}(\omega),
where :math:`\omega` is the angular frequency, :math:`\epsilon^{'}` and :math:`\epsilon^{''}` are the real and imaginary parts of the permittivity respectively.
In the ``user_libs`` sub-package is a module called ``DebyeFit`` which can be used to fit a multi-Debye expansion to dielectric data, defined as
.. math::
\epsilon(\omega) = \epsilon_{\infty} + \sum_{i=1}^{N}\frac{\Delta\epsilon_{i}}{1+j\omega t_{0,i}},
where :math:`\epsilon(\omega)` is frequency dependent dielectric permittivity, :math:`\Delta\epsilon` - difference between the real permittivity at zero and infinity frequency.
:math:`\tau_{0}` is relaxation time (s), :math:`\epsilon_{\infty}` - real part of relative permittivity at infinity frequency, and :math:`N` is number of the Debye poles.
The user can choose between Havriliak-Negami, Jonscher, Complex Refractive Index Mixing models, and arbitrary dielectric data derived experimentally
or calculated using some other function to fit the data to a multi-Debye expansion.
<div align="center">
<img src="docs/epsilon.png" width="600"/>
</div>
License
=======
``DebyeFit`` sub-package is currently released under the `GNU General Public License v3 or higher <http://www.gnu.org/copyleft/gpl.html>`_.
Code structure
==============
The ``DebyeFit`` sub-package contains two main scripts:
* ```Debye_fit.py``` with definition of all Relaxation functions classes,
* ```optimization.py``` with definition of three choosen global optimization methods.
Class Relaxation
################
This class is designed for modelling different relaxation functions, like Havriliak-Negami (```Class HavriliakNegami```), Jonscher (```Class Jonscher```), Complex Refractive Index Mixing (```Class CRIM```) models, and arbitrary dielectric data derived experimentally
or calculated using some other function (```Class Rawdata```).
More about ``Class Relaxation`` structure can be found in [relaxation.md](docs/relaxation.md).
Havriliak-Negami Function
*************************
The Havriliak–Negami relaxation is an empirical modification of the Debye relaxation model in electromagnetism, which in additionto the Debye equation has two exponential parameters
.. math::
\epsilon(\omega) = \epsilon_{\infty} + \frac{\Delta\epsilon}{\left(1+\left(j\omega t_{0}\right)^{a}\right)^{b}}
The ``HavriliakNegami`` class has the following structure:
.. code-block:: none
HavriliakNegami(f_min, f_max,
alpha, beta, e_inf, de, tau_0,
sigma, mu, mu_sigma, material_name,
number_of_debye_poles=-1, f_n=50,
plot=False, save=True,
optimizer=PSO_DLS,
optimizer_options={})
* ``f_min`` is first bound of the frequency range used to approximate the given function (Hz),
* ``f_max`` is second bound of the frequency range used to approximate the given function (Hz),
* ``alpha`` is real positive float number which varies 0 < $\alpha$ < 1,
* ``beta`` is real positive float number which varies 0 < $\beta$ < 1,
* ``e_inf`` is a real part of relative permittivity at infinity frequency,
* ``de`` is a difference between the real permittivity at zero and infinity frequency,
* ``tau_0`` is a relaxation time (s),
* ``sigma`` is a conductivity (Siemens/metre),
* ``mu`` is a relative permeability,
* ``mu_sigma`` is a magnetic loss (Ohms/metre),
* ``material_name`` is definition of material name,
* ``number_of_debye_poles`` is choosen number of Debye poles,
* ``f_n`` is choosen number of frequences,
* ``plot`` is a switch to turn on the plotting,
* ``save`` is a switch to turn on the saving final material properties,
* ``optimizer`` is a choosen optimizer to fit model to dielectric data,
* ``optimizer_options`` is a dict for options of choosen optimizer.
Jonscher Function
****************
Jonscher function is mainly used to describe the dielectric properties of concrete and soils. The frequency domain expression of Jonscher
function is given by
.. math::
\epsilon(\omega) = \epsilon_{\infty} + a_{p}*\left( -j*\frac{\omega}{\omega_{p}} \right)^{n}
The ``Jonscher`` class has the following structure:
.. code-block:: none
Jonscher(f_min, f_max,
e_inf, a_p, omega_p, n_p,
sigma, mu, mu_sigma,
material_name, number_of_debye_poles=-1,
f_n=50, plot=False, save=True,
optimizer=PSO_DLS,
optimizer_options={})
* ``f_min`` is first bound of the frequency range used to approximate the given function (Hz),
* ``f_max`` is second bound of the frequency range used to approximate the given function (Hz),
* ``e_inf`` is a real part of relative permittivity at infinity frequency,
* ``a_p``` is a Jonscher parameter. Real positive float number,
* ``omega_p`` is a Jonscher parameter. Real positive float number,
* ``n_p`` Jonscher parameter, 0 < n_p < 1.
Complex Refractive Index Mixing (CRIM) Function
***********************************************
CRIM is the most mainstream approach for estimating the bulk permittivity of heterogeneous materials and has been widely applied for GPR applications. The function takes form of
.. math::
\epsilon(\omega)^{d} = \sum_{i=1}^{m}f_{i}\epsilon_{m,i}(\omega)^{d}
The ``CRIM`` class has the following structure:
.. code-block:: none
CRIM(f_min, f_max, a, volumetric_fractions,
materials, sigma, mu, mu_sigma, material_name,
number_of_debye_poles=-1, f_n=50,
plot=False, save=True,
optimizer=PSO_DLS,
optimizer_options={})
* ``f_min`` is first bound of the frequency range used to approximate the given function (Hz),
* ``f_max`` is second bound of the frequency range used to approximate the given function (Hz),
* ``a`` is a shape factor,
* ``volumetric_fractions`` is a volumetric fraction for each material,
* ``materials`` are arrays of materials properties, for each material [e_inf, de, tau_0].
Rawdata Class
*************
The present package has the ability to model dielectric properties obtained experimentally by fitting multi-Debye functions to data given from a file.
The format of the file should be three columns. The first column contains the frequencies (Hz) associated with the electric permittivity point.
The second column contains the real part of the relative permittivity. The third column contains the imaginary part of the relative permittivity.
The columns should separated by coma by default (is it posible to define different separator).
The ``Rawdata`` class has the following structure:
.. code-block:: none
Rawdata(self, filename,
sigma, mu, mu_sigma,
material_name, number_of_debye_poles=-1,
f_n=50, delimiter =',',
plot=False, save=True,
optimizer=PSO_DLS,
optimizer_options={})
* ``filename`` is a path to text file which contains three columns,
* ``delimiter`` is a separator for three data columns.
Class Optimizer
###############
This class supports global optimization algorithms (particle swarm, duall annealing, evolutionary algorithms) for finding an optimal set of relaxation times that minimise the error between the actual and the approximated electric permittivity and calculate optimised weights for the given relaxation times.
Code written here is mainly based on external libraries, like ```scipy``` and ```pyswarm```.
More about ``Class Optimizer`` structure can be found in [optimization.md](docs/optimization.md).
PSO_DLS Class
*************
Creation of hybrid Particle Swarm-Damped Least Squares optimisation object with predefined parameters.
The current code is a modified edition of the pyswarm package which can be found at https://pythonhosted.org/pyswarm/.
DA_DLS Class
************
Creation of Dual Annealing-Damped Least Squares optimisation object with predefined parameters. The current class is a modified edition of the scipy.optimize package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.dual_annealing.html#scipy.optimize.dual_annealing.
DE_DLS Class
************
Creation of Differential Evolution-Damped Least Squares object with predefined parameters. The current class is a modified edition of the scipy.optimize package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.differential_evolution.html#scipy.optimize.differential_evolution.
DLS function
************
Finding the weights using a non-linear least squares (LS) method, the Levenberg–Marquardt algorithm (LMA or just LM), also known as the damped least-squares (DLS) method.
Examples
########
In directory [examples](./examles), we provided jupyter notebooks, scripts and data to show how use stand alone script ```DebyeFit.py```:
* ```example_DebyeFitting.ipynb```: simple cases of using all available implemented relaxation functions,
* ```example_BiologicalTissues.ipynb```: simple cases of using Cole-Cole function for biological tissues,
* ```example_ColeCole.py```: simple cases of using Cole-Cole function in case of 3, 5 and automatically chosen number of Debye poles,
* ```Test.txt```: raw data for testing ```Rawdata Class```, file contains 3 columns: the first column contains the frequencies (Hz) associated with the value of the permittivity, second column contains the real part of the relative permittivity, and the third one the imaginary part of the relative permittivity. The columns should separated by comma.
Dispersive material commands
============================
gprMax has implemented an optimisation approach to fit a multi-Debye expansion to dielectric data.
The user can choose between Havriliak-Negami, Johnsher and Complex Refractive Index Mixing models, fit arbitrary dielectric data derived experimentally or calculated using some other function.
Notice that Havriliak-Negami is an inclusive function that holds as special cases the widely-used **Cole-Cole** and **Cole-Davidson** functions.
.. note::
The technique employed here as a default is a hybrid linear-nonlinear optimisation proposed by Kelley et. al (2007).
Their method was slightly adjusted to overcome some instability issues and thus making the process more robust and faster.
In particular, in the case of negative weights we inverse the sign in order to introduce a large penalty in the optimisation process thus indirectly constraining the weights
to be always positive. This made dumbing factors unnecessary and consequently they were removed from the algorithm. Furthermore we added the real part to the cost action
to avoid possible instabilities to arbitrary given functions that does not follow the Kramers–Kronig relationship.
.. warning::
* The fitting accuracy depends on the number of the Debye poles as well as the fitted function. It is advised to check if the resulted accuracy is sufficient for your application.
* Increasing the number of Debye poles will make the approximation more accurate but it will increase the overall computational resources of FDTD.
#havriliak_negami:
##################
Allows you to model dielectric properties by fitting multi-Debye functions to Havriliak-Negami function. The syntax of the command is:
.. code-block:: none
#havriliak_negami: f1 f2 f3 f4 f5 f6 f7 f8 f9 f10 i1 str1 [i2]
* ``f1`` is the lower frequency bound (Hz).
* ``f2`` is the upper frequency bound (Hz).
* ``f3`` is the :math:`\alpha` parameter beetwen bonds :math:`\left(0 < \alpha < 1 \right)`.
* ``f4`` is the :math:`\beta` parameter beetwen bonds :math:`\left(0 < \beta < 1 \right)`.
* ``f5`` is the real relative permittivity at infinity frequency, :math:`\epsilon_{\infty}`.
* ``f6`` is the difference between the real permittivity at zero and infinity frequency, :math:`\Delta\epsilon`.
* ``f7`` is the relaxation time (s), :math:`t_{0}`.
* ``f8`` is the conductivity (Siemens/metre), :math:`\sigma`
* ``f9`` is the relative permeability, :math:`\mu_r`
* ``f10`` is the magnetic loss (Ohms/metre), :math:`\sigma_*`
* ``i1`` is the number of Debye poles, set to -1 will be automatically calculated tends to minimize the relative absolut error.
* ``str1`` is an identifier for the material.
* ``i2`` is an optional parameter which controls the seeding of the random number generator used in stochastic global optimizator. By default (if you don't specify this parameter) the random number generator will be seeded by trying to read data from ``/dev/urandom`` (or the Windows analogue) if available or from the clock otherwise.
For example ``#havriliak_negami: 1e4 1e11 0.3 1 3.4 2.7 0.8e-10 4.5e-4 1 0 5 dry_sand`` creates a material called ``dry_sand``, and approximates using five Debye poles a Cole-Cole function with :math:`\epsilon_{\infty}=3.4`, :math:`\Delta\epsilon=2.7`, :math:`t_{0}=8^{-9}` and :math:`a=0.3`.
The resulting output is the set of gprMax commands and optional a plot with the actual and the approximated Cole-Cole function.
#jonscher:
##########
Allows you to model dielectric properties by fitting multi-Debye functions to Jonscher function. The syntax of the command is:
.. code-block:: none
#jonscher: f1 f2 f3 f4 f5 f6 f7 f8 f9 i1 str1 [i2]
* ``f1`` is the lower frequency bound (in Hz).
* ``f2`` is the upper frequency bound (in Hz).
* ``f3`` is the real relative permittivity at infinity frequency, :math:`\epsilon_{\infty}`.
* ``f4`` is the :math:`a_{p}` parameter.
* ``f5`` is the :math:`\omega_{p}` parameter.
* ``f6`` is the :math:`n_{p}` parameter.
* ``f7`` is the conductivity (Siemens/metre), :math:`\sigma`
* ``f8`` is the relative permeability, :math:`\mu_r`
* ``f9`` is the magnetic loss (Ohms/metre), :math:`\sigma_*`
* ``i1`` is the number of Debye poles, set to -1 will be automatically calculated tends to minimize the relative absolut error.
* ``str1`` is an identifier for the material.
* ``i2`` is an optional parameter which controls the seeding of the random number generator used in stochastic global optimizator. By default (if you don't specify this parameter) the random number generator will be seeded by trying to read data from ``/dev/urandom`` (or the Windows analogue) if available or from the clock otherwise.
For example ``#jonscher: 1e6 1e-5 4.39 7.49 5e-10 0.4 0.1 1 0.1 4 Material_Jonscher`` creates a material called ``Material_Jonscher``, and approximates using four Debye poles a Johnsher function with :math:`\epsilon_{\infty}=4.39`, :math:`a_{p}=7.49`, :math:`\omega_{p}=0.5\times 10^{9}` and :math:`n=0.4`.
The resulting output is the set of gprMax commands and optional a plot with the actual and the approximated Johnsher function.
#crim:
######
Allows you to model dielectric properties by fitting multi-Debye functions to CRIM function. The syntax of the command is:
.. code-block:: none
#crim: f1 f2 f3 v1 v2 f4 f5 f6 i1 str1 [i2]
* ``f1`` is the lower frequency bound (in Hz).
* ``f2`` is the upper frequency bound (in Hz).
* ``f3`` is the shape factor, :math:`a`
* ``v1`` is the vector (paramiter given in input file with `[]`) of volumetric fractions [f1, f2 .... ]. The nuber of paramiters depend on number of materials.
* ``v2`` is the vector (paramiter given in input file with `[]`) containing the materials properties [:math:`\epsilon_{1\infty}`, :math:`\Delta\epsilon_{1}`, :math:`t_{0}_{1}`, :math:`\epsilon_{2\infty}`, :math:`\Delta\epsilon_{2}`, :math:`t_{0}_{2}` .... ]. The number of material vector must be divisible by three.
* ``f4`` is the conductivity (Siemens/metre), :math:`\sigma`
* ``f5`` is the relative permeability, :math:`\mu_r`
* ``f6`` is the magnetic loss (Ohms/metre), :math:`\sigma_*`
* ``i1`` is the number of Debye poles, set to -1 will be automatically calculated tends to minimize the relative absolut error.
* ``str1`` is an identifier for the material.
* ``i2`` is an optional parameter which controls the seeding of the random number generator used in stochastic global optimizator. By default (if you don't specify this parameter) the random number generator will be seeded by trying to read data from ``/dev/urandom`` (or the Windows analogue) if available or from the clock otherwise.
For example ``#crim: 1e-1 1e-9 0.5 [0.5,0.1,0.4] [3,25,1e-8,3,25,1e-9,1,10,1e-10] 0 1 0 5 CRIM`` creates a material called ``CRIM``, and approximates using five Debye poles the following CRIM function
.. math::
\epsilon(\omega)^{0.5} = \sum_{i=1}^{m}f_{i}\epsilon_{m,i}(\omega)^{0.5}
.. math::
f = [0.5, 0.1, 0.4]
.. math::
\epsilon_{m,1} = 3 + \frac{25}{1+j\omega\times 10^{-8}}
.. math::
\epsilon_{m,2} = 3 + \frac{25}{1+j\omega\times 10^{-9}}
.. math::
\epsilon_{m,3} = 1 + \frac{10}{1+j\omega\times 10^{-10}}
The resulting output is the set of gprMax commands and optional a plot with the actual and the approximated CRIM function.
#raw_data:
##########
Allows you to model dielectric properties obtained experimentally by fitting multi-Debye functions to data given from a file. The syntax of the command is:
.. code-block:: none
#raw_data: file1 f1 f2 f3 i1 str1 [i2]
* ``file1`` is an path to text file with experimental data points.
* ``f1`` is the conductivity (Siemens/metre), :math:`\sigma`
* ``f2`` is the relative permeability, :math:`\mu_r`
* ``f3`` is the magnetic loss (Ohms/metre), :math:`\sigma_*`
* ``i1`` is the number of Debye poles, set to -1 will be automatically calculated tends to minimize the relative absolut error.
* ``str1`` is an identifier for the material.
* ``i2`` is an optional parameter which controls the seeding of the random number generator used in stochastic global optimizator. By default (if you don't specify this parameter) the random number generator will be seeded by trying to read data from ``/dev/urandom`` (or the Windows analogue) if available or from the clock otherwise.
For example ``#raw_data: user_libs/DebyeFit/examples/Test.txt 0.1 1 0.1 3 Experimental`` creates a material called ``Experimental`` which model dielectric properties obtained experimentally by fitting three Debye poles function to data given from a ``user_libs/DebyeFit/examples/Test.txt`` file.
The resulting output is the set of gprMax commands and optional a plot with the actual and the approximated function.

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user_libs/DebyeFit/__init__.py 普通文件
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import os, sys; sys.path.append(os.path.dirname(os.path.realpath(__file__)))
from .Debye_Fit import (HavriliakNegami,
Jonscher,
Crim,
Rawdata)
__all__ = [
'HavriliakNegami', 'Jonscher', 'Crim',
'Rawdata'
]

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user_libs/DebyeFit/docs/optimization.md 普通文件
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# Optimization methods of multi-Debye fitting
``Class Optimizer`` supports global optimization algorithms (particle swarm, duall annealing, evolutionary algorithms) for finding an optimal set of relaxation times that minimise the error between the actual and the approximated electric permittivity and calculate optimised weights for the given relaxation times.
Code written here is mainly based on external libraries, like ```scipy``` and ```pyswarm```.
Supported methods:
- [x] hybrid Particle Swarm-Damped Least Squares
- [x] hybrid Dual Annealing-Damped Least Squares
- [x] hybrid Differential Evolution-Damped Least Squares
## Methods
1. __constructor__ - it is called in all childern classes.
It takes the following arguments:
- `maxiter`: maximum number of iterations for the optimizer,
- `seed`: Seed for RandomState.
In constructor the attributes:
- `maxiter`,
- `seed`,
- `calc_weights` (used to fit weight, non-linear least squares (LS) method is used as a default)
are set.
2. __fit__ - it is inherited by all children classes. It calls the optimization function that tries to find an optimal set of relaxation times that minimise the error between the actual and the approximated electric permittivity and calculate optimised weights for the given relaxation times.
It takes the following arguments:
- `func`: objective function to be optimized,
- `lb`: the lower bounds of the design variable(s),
- `ub`: the upper bounds of the design variable(s),
- `funckwargs`: optional arguments takien by objective function.
3. __cost_function__ - it is inherited by all children classes. It calculate cost function as the average error between the actual and the approximated electric permittivity (sum of real and imaginary part).
It takes the following arguments:
- `x`: the logarithm with base 10 of relaxation times of the Debyes poles,
- `rl`: real parts of chosen relaxation function for given frequency points,
- `im`: imaginary parts of chosen relaxation function for given frequency points,
- `freq`: the frequencies vector for defined grid.
4. __calc_relaxation_times__ - it find an optimal set of relaxation times that minimise an objective function using appropriate optimization procedure.
It takes the following arguments:
- `func`: objective function to be optimized,
- `lb`: the lower bounds of the design variable(s),
- `ub`: the upper bounds of the design variable(s),
- `funckwargs`: optional arguments takien by objective function.
Each new class of optimizer should:
- define constructor with appropriate arguments,
- overload __calc_relaxation_times__ method (and optional define __calc_weights__ function in case of hybrid optimization procedure).

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user_libs/DebyeFit/docs/relaxation.md 普通文件
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# Relaxation classes for multi-Debye fitting
This class is designed for modelling different relaxation functions, like Havriliak-Negami (```Class HavriliakNegami```), Jonscher (```Class Jonscher```), Complex Refractive Index Mixing (```Class CRIM```) models, and arbitrary dielectric data derived experimentally or calculated using some other function (```Class Rawdata```).
Supported relaxation classes:
- [x] Havriliak-Negami,
- [x] Jonscher,
- [x] Complex Refractive Index Mixing,
- [x] Experimental data,
## Methods
1. __constructor__ - it is called in all childern classes, creates Relaxation function object for complex material.
It takes the following arguments:
- ``sigma`` is a conductivity (Siemens/metre),
- ``mu`` is a relative permeability,
- ``mu_sigma`` is a magnetic loss (Ohms/metre),
- ``material_name`` is definition of material name,
- ``number_of_debye_poles`` is choosen number of Debye poles,
- ``f_n`` is choosen number of frequences,
- ``plot`` is a switch to turn on the plotting,
- ``save`` is a switch to turn on the saving final material properties,
- ``optimizer`` is a choosen optimizer to fit model to dielectric data,
- ``optimizer_options`` is a dict for options of choosen optimizer.
Additional parameters:
- ``rl`` calculated real part of chosen relaxation function for given frequency points,
- ``im`` calculated imaginary part of chosen relaxation function for given frequency points.
2. __set_freq__ - it is inherited by all childern classes, interpolates frequency vector using n equally logarithmicaly spaced frequencies.
It takes the following arguments:
- `f_min_`: first bound of the frequency range used to approximate the given function (Hz),
- `f_max`: second bound of the frequency range used to approximate the given function (Hz),
- `f_n`: the number of frequency points in frequency grid (Default: 50).
3. __run__ - it is inherited by all childern classes, solves the problem described by the given relaxation function (main operational method).
It consists of following steps:
1) Check the validity of the inputs using ```check_inputs``` method.
2) Print information about chosen approximation settings using ```print_info``` method.
3) Calculate both real and imaginary parts using ```calculation``` method, and then set ```self.rl``` and ```self.im``` properties.
4) Calling the main optimisation module using ```optimize``` method and calculate error based on ```error``` method.
a) [OPTIONAL] If number of debye poles is set to -1 optimization procedure is repeted until percentage error goes smaller than 5% or 20 Debye poles is reached.
5) Print the results in gprMax format style using ```print_output``` method.
6) [OPTIONAL] Save results in gprMax style using ```save_result``` method.
7) [OPTIONAL] Plot the actual and the approximate dielectric properties using ```plot_result``` method.
4. __check_inputs__ - it is called in all childern classes, finds an optimal set of relaxation times that minimise an objective function using appropriate optimization procedure.
5. __calculation__ - it is inherited by all childern classes, should be definied in all new children classes, approximates the given relaxation function.
6. __print_info__ - it is inherited by all childern classes, prints readable string of parameters for given approximation settings.
7. __optimize__ - it is inherited by all childern classes, calls the main optimisation module with defined lower and upper boundaries of search.
8. __print_output__ - it is inherited by all childern classes, prints out the resulting Debye parameters in a gprMax format.
9. __plot_result__ - it is inherited by all childern classes, plots the actual and the approximated electric permittivity, along with relative error for real and imaginary parts using a semilogarithm X axes.
10. __save_result__ - it is inherited by all childern classes, saves the resulting Debye parameters in a gprMax format.
11. __error__ - it is inherited by all childern classes, calculates the average fractional error separately for relative permittivity (real part) and conductivity (imaginary part).
Each new class of relaxation object should:
- define constructor with appropriate arguments,
- define __check_inputs__ method to check relaxation class specific parameters,
- overload __calculation__ method.

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user_libs/DebyeFit/examples/Readme.md 普通文件
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# Getting started
Please see provided examples for the basic usage of ``DebyeFit`` module. We provide jupyter tutorials, and full guidance for quick run with existing relaxation functions and optimizers:
* ```example_DebyeFitting.ipynb```: simple cases of using all available implemented relaxation functions,
* ```example_BiologicalTissues.ipynb```: simple cases of using Cole-Cole function for biological tissues,
* ```example_ColeCole.py```: simple cases of using Cole-Cole function in case of 3, 5 and automatically chosen number of Debye poles.
Main usage of the specific relaxation fucntion based on creation of choosen relaxation model and then calling run method.
```python
# set Havrilak-Negami function with initial parameters
setup = HavriliakNegami(f_min=1e4, f_max=1e11,
alpha=0.3, beta=1,
e_inf=3.4, de=2.7, tau_0=.8e-10,
sigma=0.45e-3, mu=1, mu_sigma=0,
material_name="dry_sand", f_n=100,
plot=True, save=False,
number_of_debye_poles=3,
optimizer_options={'swarmsize':30,
'maxiter':100,
'omega':0.5,
'phip':1.4,
'phig':1.4,
'minstep':1e-8,
'minfun':1e-8,
'seed': 111,
'pflag': True})
# run optimization
setup.run()
```

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user_libs/DebyeFit/examples/Test.txt 普通文件
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10000000,29.9213536481435,1.25169556541143
11220184.5430196,29.9010908161542,1.40299702432563
12589254.1179417,29.8756399586587,1.57217537410568
14125375.4462276,29.8436916400072,1.76117429474009
15848931.9246111,29.8036168365262,1.972079339829
17782794.1003892,29.7533952203391,2.20709802135445
19952623.1496888,29.6905309907513,2.46852367160776
22387211.3856834,29.611956285414,2.75867656083315
25118864.3150958,29.5139238169428,3.0798138913592
28183829.3126445,29.3918930811742,3.43399830398795
31622776.6016838,29.2404187150832,3.82291275991739
35481338.9233575,29.0530558254938,4.24760876499865
39810717.0553497,28.8223057349563,4.70817602888384
44668359.2150963,28.5396365567361,5.20332657814755
50118723.3627272,28.1956253280391,5.729897510637
56234132.5190349,27.7802793753847,6.28229678071347
63095734.4480193,27.283598801889,6.85194778826586
70794578.4384137,26.6964309897057,7.42683054876774
79432823.4724282,26.0116311645087,7.99126340085894
89125093.8133746,25.2254715223637,8.52610387978976
100000000,24.339136006498,9.00954486736777
112201845.430197,23.3600192468154,9.41861299029905
125892541.179417,22.302462791172,9.73132391277098
141253754.462276,21.1875720188749,9.92923323827099
158489319.246111,20.0419098344834,9.99991217790304
177827941.003892,18.8951464050085,9.93877751706085
199526231.496888,17.7770654023544,9.74979803763085
223872113.856834,16.714546101342,9.44488182455412
251188643.150958,15.7291491674702,9.04211441899954
281838293.126445,14.8357276987863,8.56331078490773
316227766.016838,14.0421664572876,8.03145189099272
354813389.233576,13.3500622522389,7.46848900052093
398107170.553497,12.7559948799797,6.89379357256362
446683592.150963,12.2530185094144,6.32331224790652
501187233.627271,11.8320839792422,5.76932819987279
562341325.190349,11.4832204965472,5.24065519659195
630957344.480194,11.1964125953287,4.74308431385235
707945784.384137,10.9621824341848,4.27993617324392
794328234.724282,10.7719266477577,3.85261757843109
891250938.133744,10.6180694756136,3.46112404221322
1000000000,10.4940904606372,3.10446192269295

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user_libs/DebyeFit/examples/example_ColeCole.py 普通文件
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# I. Giannakis, A. Giannopoulos and N. Davidson,
# "Incorporating dispersive electrical properties in FDTD GPR models
# using a general Cole-Cole dispersion function,"
# 2012 14th International Conference on Ground Penetrating Radar (GPR), 2012, pp. 232-236
import os, sys
sys.path.append(os.path.join(os.path.dirname(__file__), '..'))
from Debye_Fit import HavriliakNegami
if __name__ == "__main__":
# set Havrilak-Negami function with initial parameters
setup = HavriliakNegami(f_min=1e4, f_max=1e11,
alpha=0.3, beta=1,
e_inf=3.4, de=2.7, tau_0=.8e-10,
sigma=0.45e-3, mu=1, mu_sigma=0,
material_name="dry_sand", f_n=100,
plot=True, save=False,
optimizer_options={'swarmsize':30,
'maxiter':100,
'omega':0.5,
'phip':1.4,
'phig':1.4,
'minstep':1e-8,
'minfun':1e-8,
'seed': 111,
'pflag': True})
### Dry Sand in case of 3, 5
# and automatically set number of Debye poles (-1)
for number_of_debye_poles in [3, 5, -1]:
setup.number_of_debye_poles = number_of_debye_poles
setup.run()
### Moist sand
# set Havrilak-Negami function parameters
setup.material_name="moist_sand"
setup.alpha = 0.25
setup.beta = 1
setup.e_inf = 5.6
setup.de = 3.3
setup.tau_0 = 1.1e-10,
setup.sigma = 2e-3
# calculate for different number of Debye poles
for number_of_debye_poles in [3, 5, -1]:
setup.number_of_debye_poles = number_of_debye_poles
setup.run()

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user_libs/DebyeFit/optimization.py 普通文件
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# Authors: Iraklis Giannakis, and Sylwia Majchrowska
# E-mail: i.giannakis@ed.ac.uk
#
# This file is part of gprMax.
#
# gprMax 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.
#
# gprMax 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 gprMax. If not, see <http://www.gnu.org/licenses/>.
import numpy as np
from matplotlib import pylab as plt
import scipy.optimize
from tqdm import tqdm
class Optimizer(object):
"""
Create choosen optimisation object.
:param maxiter: The maximum number of iterations for the
optimizer (Default: 1000).
:type maxiter: int, optional
:param seed: Seed for RandomState. Must be convertible to 32 bit
unsigned integers (Default: None).
:type seed: int, NoneType, optional
"""
def __init__(self, maxiter=1000, seed=None):
self.maxiter = maxiter
self.seed = seed
self.calc_weights = None
def fit(self, func, lb, ub, funckwargs={}):
"""Call the optimization function that tries to find an optimal set
of relaxation times that minimise the error
between the actual and the approximated electric permittivity
and calculate optimised weights for the given relaxation times.
Args:
func (function): The function to be minimized.
lb (ndarray): The lower bounds of the design variable(s).
ub (ndarray): The upper bounds of the design variable(s).
funckwargs (dict): Additional keyword arguments passed to
objective and constraint function
(Default: empty dict).
Returns:
tau (ndarray): The the best relaxation times.
weights (ndarray): Resulting optimised weights for the given
relaxation times.
ee (float): Average error between the actual and the approximated
real part.
rl (ndarray): Real parts of chosen relaxation function
for given frequency points.
im (ndarray): Imaginary parts of chosen relaxation function
for given frequency points.
"""
np.random.seed(self.seed)
# find the relaxation frequencies using choosen optimization alghoritm
tau, _ = self.calc_relaxation_times(func, lb, ub, funckwargs)
# find the weights using a calc_weights method
if self.calc_weights is None:
raise NotImplementedError()
_, _, weights, ee, rl_exp, im_exp = \
self.calc_weights(tau, **funckwargs)
return tau, weights, ee, rl_exp, im_exp
def calc_relaxation_times(self):
""" Optimisation method that tries to find an optimal set
of relaxation times that minimise the error
between the actual and the approximated electric permittivity.
"""
raise NotImplementedError()
@staticmethod
def cost_function(x, rl, im, freq):
"""
The cost function is the average error between
the actual and the approximated electric permittivity.
Args:
x (ndarray): The logarithm with base 10 of relaxation times
of the Debyes poles.
rl (ndarray): Real parts of chosen relaxation function
for given frequency points.
im (ndarray): Imaginary parts of chosen relaxation function
for given frequency points.
freq (ndarray): The frequencies vector for defined grid.
Returns:
cost (float): Sum of mean absolute errors for real and
imaginary parts.
"""
cost_i, cost_r, _, _, _, _ = DLS(x, rl, im, freq)
return cost_i + cost_r
class PSO_DLS(Optimizer):
""" Create hybrid Particle Swarm-Damped Least Squares optimisation
object with predefined parameters.
:param swarmsize: The number of particles in the swarm (Default: 40).
:type swarmsize: int, optional
:param maxiter: The maximum number of iterations for the swarm
to search (Default: 50).
:type maxiter: int, optional
:param omega: Particle velocity scaling factor (Default: 0.9).
:type omega: float, optional
:param phip: Scaling factor to search away from the particle's
best known position (Default: 0.9).
:type phip: float, optional
:param phig: Scaling factor to search away from the swarm's
best known position (Default: 0.9).
:type phig: float, optional
:param minstep: The minimum stepsize of swarm's best position
before the search terminates (Default: 1e-8).
:type minstep: float, optional
:param minfun: The minimum change of swarm's best objective value
before the search terminates (Default: 1e-8)
:type minfun: float, optional
:param pflag: if True will plot the actual and the approximated
value during optimization process (Default: False).
:type pflag: bool, optional
"""
def __init__(self, swarmsize=40, maxiter=50,
omega=0.9, phip=0.9, phig=0.9,
minstep=1e-8, minfun=1e-8,
pflag=False, seed=None):
super(PSO_DLS, self).__init__(maxiter, seed)
self.swarmsize = swarmsize
self.omega = omega
self.phip = phip
self.phig = phig
self.minstep = minstep
self.minfun = minfun
self.pflag = pflag
self.calc_weights = DLS
def calc_relaxation_times(self, func, lb, ub, funckwargs={}):
"""
A particle swarm optimisation that tries to find an optimal set
of relaxation times that minimise the error
between the actual and the approximated electric permittivity.
The current code is a modified edition of the pyswarm package
which can be found at https://pythonhosted.org/pyswarm/
Args:
func (function): The function to be minimized.
lb (ndarray): The lower bounds of the design variable(s).
ub (ndarray): The upper bounds of the design variable(s).
funckwargs (dict): Additional keyword arguments passed to
objective and constraint function
(Default: empty dict).
Returns:
g (ndarray): The swarm's best known position (relaxation times).
fg (float): The objective value at ``g``.
"""
np.random.seed(self.seed)
# check input parameters
assert len(lb) == len(ub), \
'Lower- and upper-bounds must be the same length'
assert hasattr(func, '__call__'), 'Invalid function handle'
lb = np.array(lb)
ub = np.array(ub)
assert np.all(ub > lb), \
'All upper-bound values must be greater than lower-bound values'
vhigh = np.abs(ub - lb)
vlow = -vhigh
# Initialize objective function
obj = lambda x: func(x=x, **funckwargs)
# Initialize the particle swarm
d = len(lb) # the number of dimensions each particle has
x = np.random.rand(self.swarmsize, d) # particle positions
v = np.zeros_like(x) # particle velocities
p = np.zeros_like(x) # best particle positions
fp = np.zeros(self.swarmsize) # best particle function values
g = [] # best swarm position
fg = np.inf # artificial best swarm position starting value
for i in range(self.swarmsize):
# Initialize the particle's position
x[i, :] = lb + x[i, :] * (ub - lb)
# Initialize the particle's best known position
p[i, :] = x[i, :]
# Calculate the objective's value at the current particle's
fp[i] = obj(p[i, :])
# At the start, there may not be any feasible starting point,
# so just
# give it a temporary "best" point since it's likely to change
if i == 0:
g = p[0, :].copy()
# If the current particle's position is better than the swarm's,
# update the best swarm position
if fp[i] < fg:
fg = fp[i]
g = p[i, :].copy()
# Initialize the particle's velocity
v[i, :] = vlow + np.random.rand(d) * (vhigh - vlow)
# Iterate until termination criterion met
for it in tqdm(range(self.maxiter), desc='Debye fitting'):
rp = np.random.uniform(size=(self.swarmsize, d))
rg = np.random.uniform(size=(self.swarmsize, d))
for i in range(self.swarmsize):
# Update the particle's velocity
v[i, :] = self.omega * v[i, :] + self.phip * rp[i, :] * \
(p[i, :] - x[i, :]) + \
self.phig * rg[i, :] * (g - x[i, :])
# Update the particle's position,
# correcting lower and upper bound
# violations, then update the objective function value
x[i, :] = x[i, :] + v[i, :]
mark1 = x[i, :] < lb
mark2 = x[i, :] > ub
x[i, mark1] = lb[mark1]
x[i, mark2] = ub[mark2]
fx = obj(x[i, :])
# Compare particle's best position
if fx < fp[i]:
p[i, :] = x[i, :].copy()
fp[i] = fx
# Compare swarm's best position to current
# particle's position
if fx < fg:
tmp = x[i, :].copy()
stepsize = np.sqrt(np.sum((g - tmp) ** 2))
if np.abs(fg - fx) <= self.minfun:
print(f'Stopping search: Swarm best objective '
f'change less than {self.minfun}')
return tmp, fx
elif stepsize <= self.minstep:
print(f'Stopping search: Swarm best position '
f'change less than {self.minstep}')
return tmp, fx
else:
g = tmp.copy()
fg = fx
# Dynamically plot the error as the optimisation takes place
if self.pflag:
if it == 0:
xpp = [it]
ypp = [fg]
else:
xpp.append(it)
ypp.append(fg)
PSO_DLS.plot(xpp, ypp)
return g, fg
@staticmethod
def plot(x, y):
"""
Dynamically plot the error as the optimisation takes place.
Args:
x (array): The number of current iterations.
y (array): The objective value at for all x points.
"""
# it clears an axes
plt.cla()
plt.plot(x, y, "b-", linewidth=1.0)
plt.ylim(min(y) - 0.1 * min(y),
max(y) + 0.1 * max(y))
plt.xlim(min(x) - 0.1, max(x) + 0.1)
plt.grid(b=True, which="major", color="k",
linewidth=0.2, linestyle="--")
plt.suptitle("Debye fitting process")
plt.xlabel("Iteration")
plt.ylabel("Average Error")
plt.pause(0.0001)
class DA_DLS(Optimizer):
""" Create Dual Annealing object with predefined parameters.
The current class is a modified edition of the scipy.optimize
package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/
scipy.optimize.dual_annealing.html#scipy.optimize.dual_annealing
"""
def __init__(self, maxiter=1000,
local_search_options={}, initial_temp=5230.0,
restart_temp_ratio=2e-05, visit=2.62, accept=-5.0,
maxfun=1e7, no_local_search=False,
callback=None, x0=None, seed=None):
super(DA_DLS, self).__init__(maxiter, seed)
self.local_search_options = local_search_options
self.initial_temp = initial_temp
self.restart_temp_ratio = restart_temp_ratio
self.visit = visit
self.accept = accept
self.maxfun = maxfun
self.no_local_search = no_local_search
self.callback = callback
self.x0 = x0
self.calc_weights = DLS
def calc_relaxation_times(self, func, lb, ub, funckwargs={}):
"""
Find the global minimum of a function using Dual Annealing.
The current class is a modified edition of the scipy.optimize
package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/
scipy.optimize.dual_annealing.html#scipy.optimize.dual_annealing
Args:
func (function): The function to be minimized
lb (ndarray): The lower bounds of the design variable(s)
ub (ndarray): The upper bounds of the design variable(s)
funckwargs (dict): Additional keyword arguments passed to
objective and constraint function
(Default: empty dict)
Returns:
x (ndarray): The solution array (relaxation times).
fun (float): The objective value at the best solution.
"""
np.random.seed(self.seed)
result = scipy.optimize.dual_annealing(
func,
bounds=list(zip(lb, ub)),
args=funckwargs.values(),
maxiter=self.maxiter,
local_search_options=self.local_search_options,
initial_temp=self.initial_temp,
restart_temp_ratio=self.restart_temp_ratio,
visit=self.visit,
accept=self.accept,
maxfun=self.maxfun,
no_local_search=self.no_local_search,
callback=self.callback,
x0=self.x0)
print(result.message)
return result.x, result.fun
class DE_DLS(Optimizer):
"""
Create Differential Evolution-Damped Least Squares object
with predefined parameters.
The current class is a modified edition of the scipy.optimize
package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/
scipy.optimize.differential_evolution.html#scipy.optimize.differential_evolution
"""
def __init__(self, maxiter=1000,
strategy='best1bin', popsize=15, tol=0.01, mutation=(0.5, 1),
recombination=0.7, callback=None, disp=False, polish=True,
init='latinhypercube', atol=0,
updating='immediate', workers=1,
constraints=(), seed=None):
super(DE_DLS, self).__init__(maxiter, seed)
self.strategy = strategy
self.popsize = popsize
self.tol = tol
self.mutation = mutation
self.recombination = recombination
self.callback = callback
self.disp = disp
self.polish = polish
self.init = init
self.atol = atol
self.updating = updating
self.workers = workers
self.constraints = constraints
self.calc_weights = DLS
def calc_relaxation_times(self, func, lb, ub, funckwargs={}):
"""
Find the global minimum of a function using Differential Evolution.
The current class is a modified edition of the scipy.optimize
package which can be found at:
https://docs.scipy.org/doc/scipy/reference/generated/
scipy.optimize.differential_evolution.html#scipy.optimize.differential_evolution
Args:
func (function): The function to be minimized
lb (ndarray): The lower bounds of the design variable(s)
ub (ndarray): The upper bounds of the design variable(s)
funckwargs (dict): Additional keyword arguments passed to
objective and constraint function
(Default: empty dict)
Returns:
x (ndarray): The solution array (relaxation times).
fun (float): The objective value at the best solution.
"""
np.random.seed(self.seed)
result = scipy.optimize.differential_evolution(
func,
bounds=list(zip(lb, ub)),
args=funckwargs.values(),
strategy=self.strategy,
popsize=self.popsize,
tol=self.tol,
mutation=self.mutation,
recombination=self.recombination,
callback=self.callback,
disp=self.disp,
polish=self.polish,
init=self.init,
atol=self.atol,
updating=self.updating,
workers=self.workers,
constraints=self.constraints)
print(result.message)
return result.x, result.fun
def DLS(logt, rl, im, freq):
"""
Find the weights using a non-linear least squares (LS) method,
the Levenberg–Marquardt algorithm (LMA or just LM),
also known as the damped least-squares (DLS) method.
Args:
logt (ndarray): The best known position form optimization module
(optimal design),
the logarithm with base 10 of relaxation times
of the Debyes poles.
rl (ndarray): Real parts of chosen relaxation function
for given frequency points.
im (ndarray): Imaginary parts of chosen relaxation function
for given frequency points.
freq (ndarray): The frequencies vector for defined grid.
Returns:
cost_i (float): Mean absolute error between the actual and
the approximated imaginary part.
cost_r (float): Mean absolute error between the actual and
the approximated real part (plus average error).
x (ndarray): Resulting optimised weights for the given
relaxation times.
ee (float): Average error between the actual and the approximated
real part.
rp (ndarray): The real part of the permittivity for the optimised
relaxation times and weights for the frequnecies included
in freq.
ip (ndarray): The imaginary part of the permittivity for the optimised
relaxation times and weights for the frequnecies included
in freq.
"""
# The relaxation time of the Debyes are given at as logarithms
# logt=log10(t0) for efficiency during the optimisation
# Here they are transformed back t0=10**logt
tt = 10**logt
# y = Ax, here the A matrix for the real and the imaginary part is builded
d = 1 / (1 + 1j * 2 * np.pi * np.repeat(
freq, len(tt)).reshape((-1, len(tt))) * tt)
# Adding dumping (Levenberg–Marquardt algorithm)
# Solving the overdetermined system y=Ax
x = np.abs(np.linalg.lstsq(d.imag, im, rcond=None)[0])
# x - absolute damped least-squares solution
rp, ip = np.matmul(d.real, x[np.newaxis].T).T[0], np.matmul(
d.imag, x[np.newaxis].T).T[0]
cost_i = np.sum(np.abs(ip-im))/len(im)
ee = np.mean(rl - rp)
if ee < 1:
ee = 1
cost_r = np.sum(np.abs(rp + ee - rl))/len(im)
return cost_i, cost_r, x, ee, rp, ip

4
user_libs/DebyeFit/requirements.txt 普通文件
查看文件

@@ -0,0 +1,4 @@
matplotlib==3.3.4
numpy==1.20.2
scipy==1.6.2
tqdm==4.61.1

7
user_models/CRIM_test.in 普通文件
查看文件

@@ -0,0 +1,7 @@
#domain: 0.7 0.5 0.004
#dx_dy_dz: 0.004 0.004 0.004
#time_window: 3000
#crim: 1e-1 1e-9 0.5 [0.5,0.5] [3,25,1e6,3,0,1e3] 0.1 1 0 2 M3
#box: 0 0 0 0.07 0.4 0.004 M3