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171 lines
5.9 KiB
Python
171 lines
5.9 KiB
Python
"""
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Functions for creating stretched coordinate PMLs.
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"""
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from typing import List, Callable
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import numpy
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__author__ = 'Jan Petykiewicz'
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dx_lists_t = List[List[numpy.ndarray]]
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s_function_type = Callable[[float], float]
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def prepare_s_function(ln_R: float = -16,
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m: float = 4
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) -> s_function_type:
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"""
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Create an s_function to pass to the SCPML functions. This is used when you would like to
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customize the PML parameters.
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:param ln_R: Natural logarithm of the desired reflectance
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:param m: Polynomial order for the PML (imaginary part increases as distance ** m)
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:return: An s_function, which takes an ndarray (distances) and returns an ndarray (complex part
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of the cell width; needs to be divided by sqrt(epilon_effective) * real(omega))
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before use.
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"""
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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)
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return s_factor
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def uniform_grid_scpml(shape: numpy.ndarray or List[int],
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thicknesses: numpy.ndarray or List[int],
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omega: float,
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epsilon_effective: float = 1.0,
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s_function: s_function_type = None,
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) -> dx_lists_t:
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"""
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Create dx arrays for a uniform grid with a cell width of 1 and a pml.
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If you want something more fine-grained, check out stretch_with_scpml(...).
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:param shape: Shape of the grid, including the PMLs (which are 2*thicknesses thick)
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:param thicknesses: [th_x, th_y, th_z] Thickness of the PML in each direction.
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Both polarities are added.
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Each th_ of pml is applied twice, once on each edge of the grid along the given axis.
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th_* may be zero, in which case no pml is added.
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:param omega: Angular frequency for the simulation
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:param epsilon_effective: Effective epsilon of the PML. Match this to the material
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at the edge of your grid.
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Default 1.
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:param s_function: s_function created by prepare_s_function(...), allowing
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customization of pml parameters.
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Default uses prepare_s_function() with no parameters.
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:return: Complex cell widths (dx_lists)
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"""
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if s_function is None:
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s_function = prepare_s_function()
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# Normalized distance to nearest boundary
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def l(u, n, t):
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return ((t - u).clip(0) + (u - (n - t)).clip(0)) / t
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dx_a = [numpy.array(numpy.inf)] * 3
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dx_b = [numpy.array(numpy.inf)] * 3
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# divide by this to adjust for epsilon_effective and omega
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s_correction = numpy.sqrt(epsilon_effective) * numpy.real(omega)
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for k, th in enumerate(thicknesses):
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s = shape[k]
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if th > 0:
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sr = numpy.arange(s)
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dx_a[k] = 1 + 1j * s_function(l(sr, s, th)) / s_correction
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dx_b[k] = 1 + 1j * s_function(l(sr+0.5, s, th)) / s_correction
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else:
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dx_a[k] = numpy.ones((s,))
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dx_b[k] = numpy.ones((s,))
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return [dx_a, dx_b]
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def stretch_with_scpml(dxes: dx_lists_t,
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axis: int,
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polarity: int,
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omega: float,
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epsilon_effective: float = 1.0,
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thickness: int = 10,
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s_function: s_function_type = None,
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) -> dx_lists_t:
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"""
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Stretch dxes to contain a stretched-coordinate PML (SCPML) in one direction along one axis.
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:param dxes: dx_tuple with coordinates to stretch
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:param axis: axis to stretch (0=x, 1=y, 2=z)
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:param polarity: direction to stretch (-1 for -ve, +1 for +ve)
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:param omega: Angular frequency for the simulation
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:param epsilon_effective: Effective epsilon of the PML. Match this to the material at the
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edge of your grid. Default 1.
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:param thickness: number of cells to use for pml (default 10)
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:param s_function: s_function created by prepare_s_function(...), allowing customization
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of pml parameters. Default uses prepare_s_function() with no parameters.
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:return: Complex cell widths
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"""
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if s_function is None:
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s_function = prepare_s_function()
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dx_ai = dxes[0][axis].astype(complex)
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dx_bi = dxes[1][axis].astype(complex)
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pos = numpy.hstack((0, dx_ai.cumsum()))
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pos_a = (pos[:-1] + pos[1:]) / 2
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pos_b = pos[:-1]
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# divide by this to adjust for epsilon_effective and omega
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s_correction = numpy.sqrt(epsilon_effective) * numpy.real(omega)
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if polarity > 0:
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# front pml
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bound = pos[thickness]
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d = bound - pos[0]
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def l_d(x):
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return (bound - x) / (bound - pos[0])
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slc = slice(thickness)
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else:
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# back pml
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bound = pos[-thickness - 1]
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d = pos[-1] - bound
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def l_d(x):
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return (x - bound) / (pos[-1] - bound)
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if thickness == 0:
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slc = slice(None)
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else:
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slc = slice(-thickness, None)
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dx_ai[slc] *= 1 + 1j * s_function(l_d(pos_a[slc])) / d / s_correction
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dx_bi[slc] *= 1 + 1j * s_function(l_d(pos_b[slc])) / d / s_correction
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dxes[0][axis] = dx_ai
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dxes[1][axis] = dx_bi
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return dxes
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def generate_periodic_dx(pos: List[numpy.ndarray]) -> dx_lists_t:
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"""
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Given a list of 3 ndarrays cell centers, creates the cell width parameters for a periodic grid.
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:param pos: List of 3 ndarrays of cell centers
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:return: (dx_a, dx_b) cell widths (no pml)
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"""
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if len(pos) != 3:
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raise Exception('Must have len(pos) == 3')
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dx_a = [numpy.array(numpy.inf)] * 3
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dx_b = [numpy.array(numpy.inf)] * 3
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for i, p_orig in enumerate(pos):
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p = numpy.array(p_orig, dtype=float)
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if p.size != 1:
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p_shifted = numpy.hstack((p[1:], p[-1] + (p[1] - p[0])))
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dx_a[i] = numpy.diff(p)
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dx_b[i] = numpy.diff((p + p_shifted) / 2)
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return dx_a, dx_b
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