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 update_A_b(xmol, ymol, zmol, dim, qm, new_pivot):
"""
Update A matrix and b vector for new pivot selection
Parameters
----------
xmol: float, array_like; shape=(nmol)
Molecular coordinates in x dimension
ymol: float, array_like; shape=(nmol)
Molecular coordinates in y dimension
zmol: float, array_like; shape=(nmol)
Molecular coordinates in z dimension
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
qm: int
Maximum number of wave frequencies in Fouier Sum
representing intrinsic surface
n_waves: int
Number of coefficients / waves in surface
new_pivot: int, array_like
Indices of new pivot molecules for both surfaces
Returns
-------
A: float, array_like; shape=(2, n_waves**2, n_waves**2)
Matrix containing wave product weightings
f(x, u1, Lx).f(y, v1, Ly).f(x, u2, Lx).f(y, v2, Ly)
for each coefficient in the linear algebra equation
Ax = b for both surfaces
b: float, array_like; shape=(2, n_waves**2)
Vector containing solutions z.f(x, u, Lx).f(y, v, Ly)
to the linear algebra equation Ax = b
for both surfaces
"""
n_waves = 2 * qm + 1
u_array, v_array = wave_arrays(qm)
A = np.zeros((2, n_waves**2, n_waves**2))
b = np.zeros((2, n_waves**2))
fuv = np.zeros((2, n_waves**2, len(new_pivot[0])))
for surf in range(2):
for index in range(n_waves**2):
wave_x = wave_function(xmol[new_pivot[surf]], u_array[index],
dim[0])
wave_y = wave_function(ymol[new_pivot[surf]], v_array[index],
dim[1])
fuv[surf][index] = wave_x * wave_y
b[surf][index] += np.sum(zmol[new_pivot[index]] * fuv[surf][index])
A[surf] += np.dot(fuv[surf], fuv[surf].T)
return A, b, fuv
def H_xy(x, y, coeff, qm, qu, dim):
"""
H_xy(x, y, coeff, qm, qu, dim)
Calculation of mean curvature at position (x,y) at resolution qu
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 Fouier Sum representing
intrinsic surface
qu: int
Upper limit of wave frequencies in Fouier Sum representing
intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
H: float
Mean curvature of intrinsic surface at point x,y
"""
n_waves = 2 * qm + 1
if np.isscalar(x) and np.isscalar(y):
u_array = np.array(np.arange(n_waves**2) / n_waves, dtype=int) - qm
v_array = np.array(np.arange(n_waves**2) % n_waves, dtype=int) - qm
wave_check = ((u_array >= -qu) * (u_array <= qu) * (v_array >= -qu) *
(v_array <= qu))
indices = np.argwhere(wave_check).flatten()
fuv = wave_function_array(x, u_array[indices], dim[0])
fuv *= wave_function_array(y, v_array[indices], dim[1])
H = -4 * np.pi**2 * np.sum(
(u_array[indices]**2 / dim[0]**2 + v_array[indices]**2 / dim[1]**2)
* fuv * coeff[indices])
else:
H_array = 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)
H_array += (wave_function(x, u, dim[0]) *
wave_function(y, v, dim[1]) *
(u**2 / dim[0]**2 + v**2 / dim[1]**2) * coeff[j])
H = -4 * np.pi**2 * H_array
return H
def test_wave_function(self):
self.assertEqual(
1.0, wave_function(0, 0, self.lx))
self.assertEqual(
1.0, wave_function(0, self.lx, self.lx))
self.assertEqual(
1.0, wave_function(self.lx, 0, self.lx))
self.assertEqual(
1.0, wave_function(self.lx, self.lx, self.lx))
self.assertEqual(
-1.0, wave_function(1, self.lx / 2, self.lx))
self.assertEqual(
1.0, wave_function(2, self.lx / 2, self.lx))
def make_pos_dxdy(xmol, ymol, coeff, nmol, dim, qm):
"""
Calculate distances and derivatives at each molecular position with
respect to intrinsic surface
Parameters
----------
xmol: float, array_like; shape=(nmol)
Molecular coordinates in x dimension
ymol: float, array_like; shape=(nmol)
Molecular coordinates in y dimension
coeff: float, array_like; shape=(n_waves**2)
Optimised surface coefficients
nmol: int
Number of molecules in simulation
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
qm: int
Maximum number of wave frequencies in Fourier Sum
representing intrinsic surface
Returns
-------
int_z_mol: array_like (float); shape=(nframe, 2, qm+1, nmol)
Molecular distances from intrinsic surface
int_dxdy_mol: array_like (float); shape=(nframe, 4, qm+1, nmol)
First derivatives of intrinsic surface wrt x and y at xmol, ymol
int_ddxddy_mol: array_like (float); shape=(nframe, 4, qm+1, nmol)
Second derivatives of intrinsic surface wrt x and y at xmol, ymol
"""
int_z_mol = np.zeros((qm + 1, 2, nmol))
int_dxdy_mol = np.zeros((qm + 1, 4, nmol))
int_ddxddy_mol = np.zeros((qm + 1, 4, nmol))
tmp_int_z_mol = np.zeros((2, nmol))
tmp_dxdy_mol = np.zeros((4, nmol))
tmp_ddxddy_mol = np.zeros((4, nmol))
for qu in range(qm + 1):
if qu == 0:
j = (2 * qm + 1) * qm + qm
f_x = wave_function(xmol, 0, dim[0])
f_y = wave_function(ymol, 0, dim[1])
tmp_int_z_mol[0] += f_x * f_y * coeff[0][j]
tmp_int_z_mol[1] += f_x * f_y * coeff[1][j]
else:
for u in [-qu, qu]:
for v in range(-qu, qu + 1):
j = (2 * qm + 1) * (u + qm) + (v + qm)
f_x = wave_function(xmol, u, dim[0])
f_y = wave_function(ymol, v, dim[1])
df_dx = d_wave_function(xmol, u, dim[0])
df_dy = d_wave_function(ymol, v, dim[1])
ddf_ddx = dd_wave_function(xmol, u, dim[0])
ddf_ddy = dd_wave_function(ymol, v, dim[1])
tmp_int_z_mol[0] += f_x * f_y * coeff[0][j]
tmp_int_z_mol[1] += f_x * f_y * coeff[1][j]
tmp_dxdy_mol[0] += df_dx * f_y * coeff[0][j]
tmp_dxdy_mol[1] += f_x * df_dy * coeff[0][j]
tmp_dxdy_mol[2] += df_dx * f_y * coeff[1][j]
tmp_dxdy_mol[3] += f_x * df_dy * coeff[1][j]
tmp_ddxddy_mol[0] += ddf_ddx * f_y * coeff[0][j]
tmp_ddxddy_mol[1] += f_x * ddf_ddy * coeff[0][j]
tmp_ddxddy_mol[2] += ddf_ddx * f_y * coeff[1][j]
tmp_ddxddy_mol[3] += f_x * ddf_ddy * coeff[1][j]
for u in range(-qu + 1, qu):
for v in [-qu, qu]:
j = (2 * qm + 1) * (u + qm) + (v + qm)
f_x = wave_function(xmol, u, dim[0])
f_y = wave_function(ymol, v, dim[1])
df_dx = d_wave_function(xmol, u, dim[0])
df_dy = d_wave_function(ymol, v, dim[1])
ddf_ddx = dd_wave_function(xmol, u, dim[0])
ddf_ddy = dd_wave_function(ymol, v, dim[1])
tmp_int_z_mol[0] += f_x * f_y * coeff[0][j]
tmp_int_z_mol[1] += f_x * f_y * coeff[1][j]
tmp_dxdy_mol[0] += df_dx * f_y * coeff[0][j]
tmp_dxdy_mol[1] += f_x * df_dy * coeff[0][j]
tmp_dxdy_mol[2] += df_dx * f_y * coeff[1][j]
tmp_dxdy_mol[3] += f_x * df_dy * coeff[1][j]
tmp_ddxddy_mol[0] += ddf_ddx * f_y * coeff[0][j]
tmp_ddxddy_mol[1] += f_x * ddf_ddy * coeff[0][j]
tmp_ddxddy_mol[2] += ddf_ddx * f_y * coeff[1][j]
tmp_ddxddy_mol[3] += f_x * ddf_ddy * coeff[1][j]
int_z_mol[qu] += tmp_int_z_mol
int_dxdy_mol[qu] += tmp_dxdy_mol
int_ddxddy_mol[qu] += tmp_ddxddy_mol
int_z_mol = np.swapaxes(int_z_mol, 0, 1)
int_dxdy_mol = np.swapaxes(int_dxdy_mol, 0, 1)
int_ddxddy_mol = np.swapaxes(int_ddxddy_mol, 0, 1)
return int_z_mol, int_dxdy_mol, int_ddxddy_mol
def H_var_mol(xmol, ymol, coeff, qm, qu, dim):
"""
Variance of mean curvature H at molecular positions determined by
coeff at resolution qu
Parameters
----------
xmol: float, array_like; shape=(nmol)
Molecular coordinates in x dimension
ymol: float, array_like; shape=(nmol)
Molecular coordinates in y dimension
coeff: float, array_like; shape=(n_waves**2)
Optimised surface coefficients
qm: int
Maximum number of wave frequencies in Fouier Sum representing
intrinsic surface
qu: int
Upper limit of wave frequencies in Fouier Sum representing
intrinsic surface
dim: float, array_like; shape=(3)
XYZ dimensions of simulation cell
Returns
-------
H_var: float
Variance of mean curvature H at pivot points
"""
if qu == 0:
return 0
n_waves = 2 * qm + 1
nmol = xmol.shape[0]
# Create arrays of wave frequency indicies u and v
u_array = np.array(np.arange(n_waves**2) / n_waves, dtype=int) - qm
v_array = np.array(np.arange(n_waves**2) % n_waves, dtype=int) - qm
wave_check = ((u_array >= -qu) * (u_array <= qu) * (v_array >= -qu) *
(v_array <= qu))
indices = np.argwhere(wave_check).flatten()
# Create matrix of wave frequency indicies (u, v)**2
u_matrix = np.tile(u_array[indices], (len([indices]), 1))
v_matrix = np.tile(v_array[indices], (len([indices]), 1))
# Make curvature diagonal terms of A matrix
curve_diag = 16 * np.pi**4 * (
u_matrix**2 * u_matrix.T**2 / dim[0]**4 +
v_matrix**2 * v_matrix.T**2 / dim[1]**4 +
(u_matrix**2 * v_matrix.T**2 + u_matrix.T**2 * v_matrix**2) /
(dim[0]**2 * dim[1]**2))
# Form the diagonal xi^2 terms and b vector solutions
fuv = np.zeros((n_waves**2, nmol))
for u in range(-qu, qu + 1):
for v in range(-qu, qu + 1):
j = (2 * qm + 1) * (u + qm) + (v + qm)
fuv[j] = (wave_function(xmol, u_array[j], dim[0]) *
wave_function(ymol, v_array[j], dim[1]))
ffuv = np.dot(fuv[indices], fuv[indices].T)
coeff_matrix = np.tile(coeff[indices], (len([indices]), 1))
H_var = np.sum(coeff_matrix * coeff_matrix.T * ffuv * curve_diag / nmol)
return H_var