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Jan Petykiewicz 2019-11-26 01:47:52 -08:00
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meanas/fdmath/__init__.py Normal file
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"""
Basic discrete calculus for finite difference (fd) simulations.
This documentation and approach is roughly based on W.C. Chew's excellent
"Electromagnetic Theory on a Lattice" (doi:10.1063/1.355770),
which covers a superset of this material with similar notation and more detail.
Define the discrete forward derivative as
Dx_forward(f)[i] = (f[i + 1] - f[i]) / dx[i]
or
$$ [\\tilde{\\partial}_x f ]_{m + \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m + 1} - f_m) $$
Likewise, discrete reverse derivative is
Dx_back(f)[i] = (f[i] - f[i - 1]) / dx[i]
or
$$ [\\hat{\\partial}_x f ]_{m - \\frac{1}{2}} = \\frac{1}{\\Delta_{x, m}} (f_{m} - f_{m - 1}) $$
The derivatives are shifted by a half-cell relative to the original function:
_________________________
| | | | |
| f0 | f1 | f2 | f3 |
|_____|_____|_____|_____|
| | | |
| Df0 | Df1 | Df2 | Df3
___|_____|_____|_____|____
Periodic boundaries are used unless otherwise noted.
Expanding to three dimensions, we can define two gradients
$$ [\\tilde{\\nabla} f]_{n,m,p} = \\vec{x} [\\tilde{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\tilde{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\tilde{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
$$ [\\hat{\\nabla} f]_{m,n,p} = \\vec{x} [\\hat{\\partial}_x f]_{m + \\frac{1}{2},n,p} +
\\vec{y} [\\hat{\\partial}_y f]_{m,n + \\frac{1}{2},p} +
\\vec{z} [\\hat{\\partial}_z f]_{m,n,p + \\frac{1}{2}} $$
The three derivatives in the gradient cause shifts in different
directions, so the x/y/z components of the resulting "vector" are defined
at different points: the x-component is shifted in the x-direction,
y in y, and z in z.
We call the resulting object a "fore-vector" or "back-vector", depending
on the direction of the shift. We write it as
$$ \\tilde{g}_{m,n,p} = \\vec{x} g^x_{m + \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n + \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p + \\frac{1}{2}} $$
$$ \\hat{g}_{m,n,p} = \\vec{x} g^x_{m - \\frac{1}{2},n,p} +
\\vec{y} g^y_{m,n - \\frac{1}{2},p} +
\\vec{z} g^z_{m,n,p - \\frac{1}{2}} $$
There are also two divergences,
$$ d_{n,m,p} = [\\tilde{\\nabla} \\cdot \\hat{g}]_{n,m,p}
= [\\tilde{\\partial}_x g^x]_{m,n,p} +
[\\tilde{\\partial}_y g^y]_{m,n,p} +
[\\tilde{\\partial}_z g^z]_{m,n,p} $$
$$ d_{n,m,p} = [\\hat{\\nabla} \\cdot \\tilde{g}]_{n,m,p}
= [\\hat{\\partial}_x g^x]_{m,n,p} +
[\\hat{\\partial}_y g^y]_{m,n,p} +
[\\hat{\\partial}_z g^z]_{m,n,p} $$
Since we applied the forward divergence to the back-vector (and vice-versa), the resulting scalar value
is defined at the back-vector's (fore-vectors) location \\( (m,n,p) \\) and not at the locations of its components
\\( (m \\pm \\frac{1}{2},n,p) \\) etc.
The two curls are then
$$ \\begin{align}
\\hat{h}_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &= \\\\
[\\tilde{\\nabla} \\times \\tilde{g}]_{m + \\frac{1}{2}, n + \\frac{1}{2}, p + \\frac{1}{2}} &=
\\vec{x} (\\tilde{\\partial}_y g^z_{m,n,p + \\frac{1}{2}} - \\tilde{\\partial}_z g^y_{m,n + \\frac{1}{2},p}) \\\\
&+ \\vec{y} (\\tilde{\\partial}_z g^x_{m + \\frac{1}{2},n,p} - \\tilde{\\partial}_x g^z_{m,n,p + \\frac{1}{2}}) \\\\
&+ \\vec{z} (\\tilde{\\partial}_x g^y_{m,n + \\frac{1}{2},p} - \\tilde{\\partial}_x g^z_{m + \\frac{1}{2},n,p})
\\end{align}$$
and
$$ \\tilde{h}_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} =
[\\hat{\\nabla} \\times \\hat{g}]_{m - \\frac{1}{2}, n - \\frac{1}{2}, p - \\frac{1}{2}} $$
where \\( \\hat{g} \\) and \\( \\tilde{g} \\) are located at \\((m,n,p)\\)
with components at \\( (m \\pm \\frac{1}{2},n,p) \\) etc.,
while \\( \\hat{h} \\) and \\( \\tilde{h} \\) are located at \\((m \pm \\frac{1}{2}, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\)
with components at \\((m, n \\pm \\frac{1}{2}, p \\pm \\frac{1}{2})\\) etc.
"""

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@ -6,19 +6,34 @@ from typing import List, Callable
# Field types
field_t = numpy.ndarray
"""vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)"""
# TODO: figure out a better way to set the docstrings without creating actual subclasses?
# Probably not a big issue since they're only used for type hinting
class field_t(numpy.ndarray):
"""
Vector field with shape (3, X, Y, Z) (e.g. `[E_x, E_y, E_z]`)
This is actually is just an unaltered `numpy.ndarray`
"""
pass
class vfield_t(numpy.ndarray):
"""
Linearized vector field (single vector of length 3*X*Y*Z)
This is actually just an unaltered `numpy.ndarray`
"""
pass
vfield_t = numpy.ndarray
"""Linearized vector field (vector of length 3*X*Y*Z)"""
dx_lists_t = List[List[numpy.ndarray]]
'''
'dxes' datastructure which contains grid cell width information in the following format:
`[[[dx_e_0, dx_e_1, ...], [dy_e_0, ...], [dz_e_0, ...]],
[[dx_h_0, dx_h_1, ...], [dy_h_0, ...], [dz_h_0, ...]]]`
where `dx_e_0` is the x-width of the `x=0` cells, as used when calculating dE/dx,
and `dy_h_0` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
[[[dx_e_0, dx_e_1, ...], [dy_e_0, ...], [dz_e_0, ...]],
[[dx_h_0, dx_h_1, ...], [dy_h_0, ...], [dz_h_0, ...]]]
where `dx_e_0` is the x-width of the `x=0` cells, as used when calculating dE/dx,
and `dy_h_0` is the y-width of the `y=0` cells, as used when calculating dH/dy, etc.
'''