def get_frequency_set(qm, qu, dim):
"""
Returns set of unique frequencies in Fourier series
Parameters
----------
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
q_set: float, array_like
Set of unique frequencies
q2_set: float, array_like
Set of unique frequencies to bin coefficients to
"""
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
q, q2 = calculate_frequencies(u_array[indices], v_array[indices], dim)
q_set = np.unique(q)[1:]
q2_set = np.unique(q2)[1:]
return q_set, q2_set
def xi_var(coeff, qm, qu):
"""Calculate average variance of surface heights
Parameters
----------
coeff: float, array_like; shape=(n_frame, n_waves**2)
Optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
qu: int
Upper limit of wave frequencies in Fourier Sum
representing intrinsic surface
Returns
-------
calc_var: float
Variance of surface heights across whole surface
"""
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
Psi = vcheck(u_array[indices], v_array[indices]) / 4.
coeff_filter = coeff[:, :, indices]
mid_point = len(indices) / 2
av_coeff = np.mean(coeff_filter[:, :, mid_point], axis=0)
av_coeff_2 = np.mean(coeff_filter**2, axis=(0, 1)) * Psi
calc_var = np.sum(av_coeff_2) - np.mean(av_coeff**2, axis=0)
return calc_var
def dxy_dxi(x, y, coeff, qm, qu, dim):
"""
Function returning derivatives of intrinsic surface at
position (x,y) wrt x and y
Parameters
----------
x: float
Coordinate in x dimension
y: float
Coordinate in y dimension
coeff: float, array_like; shape=(n_waves**2)
Optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
qu: int
Upper limit of wave frequencies in Fourier Sum
representing intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
dx_dxi: float
Derivative of intrinsic surface in x dimension
dy_dxi: float
Derivative of intrinsic surface in y dimension
"""
if np.isscalar(x):
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
wave_x = wave_function_array(x, u_array[indices], dim[0])
wave_y = wave_function_array(y, v_array[indices], dim[1])
wave_dx = d_wave_function_array(x, u_array[indices], dim[0])
wave_dy = d_wave_function_array(y, v_array[indices], dim[1])
dx_dxi = np.sum(wave_dx * wave_y * coeff[indices])
dy_dxi = np.sum(wave_x * wave_dy * coeff[indices])
else:
dx_dxi = np.zeros(x.shape)
dy_dxi = np.zeros(x.shape)
for u in range(-qu, qu + 1):
for v in range(-qu, qu + 1):
j = (2 * qm + 1) * (u + qm) + (v + qm)
wave_x = wave_function(x, u, dim[0])
wave_y = wave_function(y, v, dim[1])
wave_dx = d_wave_function(x, u, dim[0])
wave_dy = d_wave_function(y, v, dim[1])
dx_dxi += wave_dx * wave_y * coeff[j]
dy_dxi += wave_x * wave_dy * coeff[j]
return dx_dxi, dy_dxi
def surface_tension_coeff(coeff_2, qm, qu, dim, T):
"""
Returns spectrum of surface tension, corresponding to
the frequencies in q2_set
Parameters
----------
coeff_2: float, array_like; shape=(n_waves**2)
Square of optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
qu: int
Upper limit of wave frequencies in Fourier Sum
representing intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
T: float
Average temperature of simulation (K)
Returns
-------
unique_q: float, array_like
Set of frequencies for power spectrum histogram
av_gamma: float, array_like
Surface tension histogram of Fourier series frequencies
"""
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
q, q2 = calculate_frequencies(u_array[indices], v_array[indices], dim)
int_A = dim[0] * dim[1] * q2 * coeff_2[indices] * vcheck(
u_array[indices], v_array[indices]) / 4
gamma = con.k * T * 1E23 / int_A
unique_q, av_gamma = filter_frequencies(q, gamma)
return unique_q, av_gamma
def power_spectrum_coeff(coeff_2, qm, qu, dim):
"""
Returns power spectrum of average surface coefficients,
corresponding to the frequencies in q2_set
Parameters
----------
coeff_2: float, array_like; shape=(n_waves**2)
Square of optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
qu: int
Upper limit of wave frequencies in Fourier Sum
representing intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
unique_q: float, array_like
Set of frequencies for power spectrum
histogram
av_fourier: float, array_like
Power spectrum histogram of Fourier
series coefficients
"""
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
q, q2 = calculate_frequencies(u_array[indices], v_array[indices], dim)
fourier = coeff_2[indices] / 4 * vcheck(u_array[indices], v_array[indices])
# Remove redundant frequencies
unique_q, av_fourier = filter_frequencies(q, fourier)
return unique_q, av_fourier
def intrinsic_area(coeff, qm, qu, dim):
"""
Calculate the intrinsic surface area from coefficients
at resolution qu
Parameters
----------
coeff: float, array_like; shape=(n_waves**2)
Optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
qu: int
Upper limit of wave frequencies in Fourier Sum
representing intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
int_A: float
Relative size of intrinsic surface area, compared
to cell cross section XY
"""
u_array, v_array = wave_arrays(qm)
indices = wave_indices(qu, u_array, v_array)
q2 = np.pi**2 * (u_array[indices]**2 / dim[0]**2 +
v_array[indices]**2 / dim[1]**2)
int_A = q2 * coeff[indices]**2 * vcheck(u_array[indices], v_array[indices])
int_A = 1 + 0.5 * np.sum(int_A)
return int_A