fdfd_tools/meanas/fdfd/scpml.py

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"""
Functions for creating stretched coordinate perfectly matched layer (PML) absorbers.
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"""
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from typing import List, Callable
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import numpy
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from ..fdmath import dx_lists_t
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__author__ = 'Jan Petykiewicz'
s_function_t = Callable[[float], float]
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"""Typedef for s-functions, see `prepare_s_function()`"""
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def prepare_s_function(ln_R: float = -16,
m: float = 4
) -> s_function_t:
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"""
Create an s_function to pass to the SCPML functions. This is used when you would like to
customize the PML parameters.
Args:
ln_R: Natural logarithm of the desired reflectance
m: Polynomial order for the PML (imaginary part increases as distance ** m)
Returns:
An s_function, which takes an ndarray (distances) and returns an ndarray (complex part
of the cell width; needs to be divided by `sqrt(epilon_effective) * real(omega))`
before use.
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"""
def s_factor(distance: numpy.ndarray) -> numpy.ndarray:
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s_max = (m + 1) * ln_R / 2 # / 2 because we assume periodic boundaries
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return s_max * (distance ** m)
return s_factor
def uniform_grid_scpml(shape: numpy.ndarray or List[int],
thicknesses: numpy.ndarray or List[int],
omega: float,
epsilon_effective: float = 1.0,
s_function: s_function_t = None,
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) -> dx_lists_t:
"""
Create dx arrays for a uniform grid with a cell width of 1 and a pml.
If you want something more fine-grained, check out `stretch_with_scpml(...)`.
Args:
shape: Shape of the grid, including the PMLs (which are 2*thicknesses thick)
thicknesses: `[th_x, th_y, th_z]`
Thickness of the PML in each direction.
Both polarities are added.
Each th_ of pml is applied twice, once on each edge of the grid along the given axis.
`th_*` may be zero, in which case no pml is added.
omega: Angular frequency for the simulation
epsilon_effective: Effective epsilon of the PML. Match this to the material
at the edge of your grid.
Default 1.
s_function: created by `prepare_s_function(...)`, allowing customization of pml parameters.
Default uses `prepare_s_function()` with no parameters.
Returns:
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Complex cell widths (dx_lists_t) as discussed in `meanas.fdmath.types`.
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"""
if s_function is None:
s_function = prepare_s_function()
# Normalized distance to nearest boundary
def l(u, n, t):
return ((t - u).clip(0) + (u - (n - t)).clip(0)) / t
dx_a = [numpy.array(numpy.inf)] * 3
dx_b = [numpy.array(numpy.inf)] * 3
# divide by this to adjust for epsilon_effective and omega
s_correction = numpy.sqrt(epsilon_effective) * numpy.real(omega)
for k, th in enumerate(thicknesses):
s = shape[k]
if th > 0:
sr = numpy.arange(s)
dx_a[k] = 1 + 1j * s_function(l(sr, s, th)) / s_correction
dx_b[k] = 1 + 1j * s_function(l(sr+0.5, s, th)) / s_correction
else:
dx_a[k] = numpy.ones((s,))
dx_b[k] = numpy.ones((s,))
return [dx_a, dx_b]
def stretch_with_scpml(dxes: dx_lists_t,
axis: int,
polarity: int,
omega: float,
epsilon_effective: float = 1.0,
thickness: int = 10,
s_function: s_function_t = None,
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) -> dx_lists_t:
"""
Stretch dxes to contain a stretched-coordinate PML (SCPML) in one direction along one axis.
Args:
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dxes: Grid parameters `[dx_e, dx_h]` as described in `meanas.fdmath.types`
axis: axis to stretch (0=x, 1=y, 2=z)
polarity: direction to stretch (-1 for -ve, +1 for +ve)
omega: Angular frequency for the simulation
epsilon_effective: Effective epsilon of the PML. Match this to the material at the
edge of your grid. Default 1.
thickness: number of cells to use for pml (default 10)
s_function: Created by `prepare_s_function(...)`, allowing customization
of pml parameters. Default uses `prepare_s_function()` with no parameters.
Returns:
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Complex cell widths (dx_lists_t) as discussed in `meanas.fdmath.types`.
Multiple calls to this function may be necessary if multiple absorpbing boundaries are needed.
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"""
if s_function is None:
s_function = prepare_s_function()
dx_ai = dxes[0][axis].astype(complex)
dx_bi = dxes[1][axis].astype(complex)
pos = numpy.hstack((0, dx_ai.cumsum()))
pos_a = (pos[:-1] + pos[1:]) / 2
pos_b = pos[:-1]
# divide by this to adjust for epsilon_effective and omega
s_correction = numpy.sqrt(epsilon_effective) * numpy.real(omega)
if polarity > 0:
# front pml
bound = pos[thickness]
d = bound - pos[0]
def l_d(x):
return (bound - x) / (bound - pos[0])
slc = slice(thickness)
else:
# back pml
bound = pos[-thickness - 1]
d = pos[-1] - bound
def l_d(x):
return (x - bound) / (pos[-1] - bound)
if thickness == 0:
slc = slice(None)
else:
slc = slice(-thickness, None)
dx_ai[slc] *= 1 + 1j * s_function(l_d(pos_a[slc])) / d / s_correction
dx_bi[slc] *= 1 + 1j * s_function(l_d(pos_b[slc])) / d / s_correction
dxes[0][axis] = dx_ai
dxes[1][axis] = dx_bi
return dxes