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cahnhilliard.py
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cahnhilliard.py
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import numpy as np
import matplotlib.pyplot as plt
from matplotlib import cm
from scipy.fft import fft2, ifft2
# import pyfftw
# import multiprocessing
# from pyfftw.interfaces.scipy_fftpack import fft2, ifft2
# Author: Elvis do A. Soares
# Github: @elvissoares
# Date: 2020-08-16
# Updated: 2022-03-25
# pyfftw.config.NUM_THREADS = multiprocessing.cpu_count()
# print('Number of cpu cores:',multiprocessing.cpu_count())
"""
The python script to solve the Cahn-Hilliard equation using
an implicit pseudospectral algorithm
"""
def run(c, nsteps=1000, dt=0.1, dx=1.0, M=1.0, kappa=0.5, rho=2.0, alpha=0.0, beta=1.0):
N = c.shape[0]
c_hat = np.empty((N,N), dtype=np.complex64)
dfdc_hat = np.empty((N,N), dtype=np.complex64)
L = N * dx
print('c0 = ',c.sum()*dx**2/L**2)
kx = ky = np.fft.fftfreq(N, d=dx)*2*np.pi
K = np.array(np.meshgrid(kx , ky ,indexing ='ij'), dtype=np.float32)
K2 = np.sum(K*K,axis=0, dtype=np.float32)
# The anti-aliasing factor
kmax_dealias = kx.max()*2.0/3.0 # The Nyquist mode
dealias = np.array((np.abs(K[0]) < kmax_dealias )*(np.abs(K[1]) < kmax_dealias ),dtype =bool)
"""
The interfacial free energy density f(c) = Wc^2(1-c)^2
"""
def finterf(c_hat):
return kappa*ifft2(K2*c_hat**2).real
"""
The bulk free energy density f(c) = W*c^2(1-c)^2
"""
def fbulk(c):
return rho * (c - alpha) ** 2 * (c - beta) ** 2
"""
The derivative of bulk free energy density f(c) = Wc^2(1-c)^2
"""
def dfdc(c):
return 2 * rho * (c - alpha) * (c - beta) * (2 * c - (alpha + beta))
def free_energy(c, c_hat):
c_x = ifft2(c_hat * 1j * K[0]).real
c_x_hat = fft2(c_x)
from scipy.signal import fftconvolve
conv = fftconvolve(c_x_hat, c_x_hat, mode='same')
conv_ = ifft2(conv).real * dx
other = dx * c_x**2
print(c_x.shape)
print(c_x_hat.shape)
print(conv.shape)
print(conv_.shape)
print(conv_[:5, :5])
print(other[:5, :5])
raw_input('stopped')
c_y = ifft2(c_hat * 1j * K[1]).real
return (kappa * (c_x**2 + c_y**2) / 2. + fbulk(c)).sum() * dx**2
c_hat[:] = fft2(c)
c_old = c.copy()
free_energies = []
free_energies.append(free_energy(c, c_hat))
for i in range(nsteps):
dfdc_hat[:] = fft2(dfdc(c_old)) # the FT of the derivative
dfdc_hat *= dealias # dealising
c_hat[:] = (c_hat-dt*K2*M*dfdc_hat)/(1+dt*M*kappa*K2**2) # updating in time
c_old[:] = c
c = ifft2(c_hat).real # inverse fourier transform
free_energies.append(free_energy(c, c_hat))
print('relative_error = ',np.abs(c_old.sum()-c.sum())/c.sum())
return c, free_energies
def plot(c, alpha, beta):
plt.imshow(c,cmap='RdBu_r', vmin=alpha, vmax=beta)
plt.savefig('cahn-hilliard.1f.png')
plt.show()
def plot_free_energy(fs, nsteps, dt):
import pandas
dd = pandas.read_csv('moose_psu_1a_IA.csv')
print(dd.columns)
plt.loglog(np.arange(nsteps + 1) * dt, fs)
plt.loglog(dd.time, dd.f_density)
plt.show()
def initial_conc(x, y):
return 0.5 + 0.01 * (np.cos(0.105 * x) * np.cos(0.11 * y) + (np.cos(0.13 * x) * np.cos(0.087 * y))**2 \
+ np.cos(0.025 * x - 0.15 * y) * np.cos(0.07 * x - 0.02 * y))
if __name__ == '__main__':
dt = 0.01
N = 512
Lx = 200.0
dx = Lx / N
# xx = np.linspace(dx / 2.0, Lx - dx / 2.0, N)
xx = np.linspace(0, Lx - dx, N)
x, y = np.meshgrid(xx, xx)
alpha = 0.3
beta = 0.7
c = initial_conc(x, y)
nsteps = 10000
# rng = np.random.default_rng(12345) # the seed of random numbers generator
# noise = 0.1
# c = c0 + noise * rng.standard_normal([256, 256])
c, fs = run(c, nsteps=nsteps, dx=dx, dt=dt, alpha=0.3, beta=0.7, rho=5.0, kappa=2.0, M=5)
# plot(c, alpha, beta)
plot_free_energy(fs, nsteps, dt)