def __init__(self, gd, scale=1.0, n=1, dtype=float, k2=0.0):
"""Helmholtz for general non orthorhombic grid.
gd: GridDescriptor
Descriptor for grid.
scale: float
Scaling factor. Use scale=-0.5 for a kinetic energy operator.
n: int
Range of stencil. Stencil has O(h^(2n)) error.
dtype: float or complex
Datatype to work on.
"""
# Order the 13 neighbor grid points:
M_ic = np.indices((3, 3, 3)).reshape((3, -3)).T[-13:] - 1
u_cv = gd.h_cv / (gd.h_cv**2).sum(1)[:, np.newaxis]**0.5
u2_i = (np.dot(M_ic, u_cv)**2).sum(1)
i_d = u2_i.argsort()
m_mv = np.array([(2, 0, 0), (0, 2, 0), (0, 0, 2), (0, 1, 1), (1, 0, 1),
(1, 1, 0)])
# Try 3, 4, 5 and 6 directions:
for D in range(3, 7):
h_dv = np.dot(M_ic[i_d[:D]], gd.h_cv)
A_md = (h_dv**m_mv[:, np.newaxis, :]).prod(2)
a_d, residual, rank, s = np.linalg.lstsq(A_md, [1, 1, 1, 0, 0, 0],
rcond=-1)
if residual.sum() < 1e-14:
assert rank == D, 'You have a weird unit cell!'
# D directions was OK
break
a_d *= scale
offsets = [(0, 0, 0)]
coefs = [laplace[n][0] * a_d.sum()]
coefs[0] += k2 * scale
for d in range(D):
M_c = M_ic[i_d[d]]
offsets.extend(np.arange(1, n + 1)[:, np.newaxis] * M_c)
coefs.extend(a_d[d] * np.array(laplace[n][1:]))
offsets.extend(np.arange(-1, -n - 1, -1)[:, np.newaxis] * M_c)
coefs.extend(a_d[d] * np.array(laplace[n][1:]))
FDOperator.__init__(self, coefs, offsets, gd, dtype)
self.description = (
'%d*%d+1=%d point O(h^%d) finite-difference Helmholtz' %
((self.npoints - 1) // n, n, self.npoints, 2 * n))
def initialize_metric(self, gd):
self.gd = gd
if self.weight == 1:
self.metric = None
else:
a = 0.125 * (self.weight + 7)
b = 0.0625 * (self.weight - 1)
c = 0.03125 * (self.weight - 1)
d = 0.015625 * (self.weight - 1)
self.metric = FDOperator([a,
b, b, b, b, b, b,
c, c, c, c, c, c, c, c, c, c, c, c,
d, d, d, d, d, d, d, d],
[(0, 0, 0), # a
(-1, 0, 0), (1, 0, 0), # b
(0, -1, 0), (0, 1, 0),
(0, 0, -1), (0, 0, 1),
(1, 1, 0), (1, 0, 1), (0, 1, 1), # c
(1, -1, 0), (1, 0, -1), (0, 1, -1),
(-1, 1, 0), (-1, 0, 1), (0, -1, 1),
(-1, -1, 0), (-1, 0, -1), (0, -1, -1),
(1, 1, 1), (1, 1, -1), (1, -1, 1), # d
(-1, 1, 1), (1, -1, -1), (-1, -1, 1),
(-1, 1, -1), (-1, -1, -1)],
gd, float).apply
self.mR_G = gd.empty()
def __init__(self, gd, scale=1.0, n=1, dtype=float, k2=0.0):
"""Helmholtz for general non orthorhombic grid.
gd: GridDescriptor
Descriptor for grid.
scale: float
Scaling factor. Use scale=-0.5 for a kinetic energy operator.
n: int
Range of stencil. Stencil has O(h^(2n)) error.
dtype: float or complex
Datatype to work on.
"""
# Order the 13 neighbor grid points:
M_ic = np.indices((3, 3, 3)).reshape((3, -3)).T[-13:] - 1
u_cv = gd.h_cv / (gd.h_cv**2).sum(1)[:, np.newaxis]**0.5
u2_i = (np.dot(M_ic, u_cv)**2).sum(1)
i_d = u2_i.argsort()
m_mv = np.array([(2, 0, 0), (0, 2, 0), (0, 0, 2),
(0, 1, 1), (1, 0, 1), (1, 1, 0)])
# Try 3, 4, 5 and 6 directions:
for D in range(3, 7):
h_dv = np.dot(M_ic[i_d[:D]], gd.h_cv)
A_md = (h_dv**m_mv[:, np.newaxis, :]).prod(2)
a_d, residual, rank, s = np.linalg.lstsq(A_md, [1, 1, 1, 0, 0, 0])
if residual.sum() < 1e-14:
assert rank == D, 'You have a weird unit cell!'
# D directions was OK
break
a_d *= scale
offsets = [(0, 0, 0)]
coefs = [laplace[n][0] * a_d.sum()]
coefs[0] += k2 * scale
for d in range(D):
M_c = M_ic[i_d[d]]
offsets.extend(np.arange(1, n + 1)[:, np.newaxis] * M_c)
coefs.extend(a_d[d] * np.array(laplace[n][1:]))
offsets.extend(np.arange(-1, -n - 1, -1)[:, np.newaxis] * M_c)
coefs.extend(a_d[d] * np.array(laplace[n][1:]))
FDOperator.__init__(self, coefs, offsets, gd, dtype)
self.description = (
'%d*%d+1=%d point O(h^%d) finite-difference Helmholtz' %
((self.npoints - 1) // n, n, self.npoints, 2 * n))