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Add some docs for energy calculations

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Jan Petykiewicz 2 months ago
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8ac0d52cd1
  1. 144
      meanas/fdtd/energy.py

144
meanas/fdtd/energy.py

@ -13,7 +13,62 @@ def poynting(e: fdfield_t,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Calculate the poynting vector
Calculate the poynting vector `S` ($S$).
This is the energy transfer rate (amount of energy `U` per `dt` transferred
between adjacent cells) in each direction that happens during the half-step
bounded by the two provided fields.
The returned vector field `S` is the energy flow across +x, +y, and +z
boundaries of the corresponding `U` cell. For example,
```
mx = numpy.roll(mask, -1, axis=0)
my = numpy.roll(mask, -1, axis=1)
mz = numpy.roll(mask, -1, axis=2)
u_hstep = fdtd.energy_hstep(e0=es[ii - 1], h1=hs[ii], e2=es[ii], **args)
u_estep = fdtd.energy_estep(h0=hs[ii], e1=es[ii], h2=hs[ii + 1], **args)
delta_j_B = fdtd.delta_energy_j(j0=js[ii], e1=es[ii], dxes=dxes)
du_half_h2e = u_estep - u_hstep - delta_j_B
s_h2e = -fdtd.poynting(e=es[ii], h=hs[ii], dxes=dxes) * dt
planes = [s_h2e[0, mask].sum(), -s_h2e[0, mx].sum(),
s_h2e[1, mask].sum(), -s_h2e[1, my].sum(),
s_h2e[2, mask].sum(), -s_h2e[2, mz].sum()]
assert_close(sum(planes), du_half_h2e[mask])
```
(see `meanas.tests.test_fdtd.test_poynting_planes`)
The full relationship is
$$
\\begin{aligned}
(U_{l+\\frac{1}{2}} - U_l) / \\Delta_t
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l + \\frac{1}{2}} \\ \\
- \\hat{H}_{l+\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l+\\frac{1}{2}} \\\\
(U_l - U_{l-\\frac{1}{2}}) / \\Delta_t
&= -\\hat{\\nabla} \\cdot \\tilde{S}_{l, l - \\frac{1}{2}} \\ \\
- \\hat{H}_{l-\\frac{1}{2}} \\cdot \\hat{M}_l \\ \\
- \\tilde{E}_l \\cdot \\tilde{J}_{l-\\frac{1}{2}} \\\\
\\end{aligned}
$$
These equalities are exact and should practically hold to within numerical precision.
No time- or spatial-averaging is necessary. (See `meanas.fdtd` docs for derivation.)
Args:
e: E-field
h: H-field (one half-timestep before or after `e`)
dxes: Grid description; see `meanas.fdmath`.
Returns:
s: Vector field. Components indicate the energy transfer rate from the
corresponding energy cell into its +x, +y, and +z neighbors during
the half-step from the time of the earlier input field until the
time of later input field.
"""
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))
@ -39,7 +94,24 @@ def poynting_divergence(s: Optional[fdfield_t] = None,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Calculate the divergence of the poynting vector
Calculate the divergence of the poynting vector.
This is the net energy flow for each cell, i.e. the change in energy `U`
per `dt` caused by transfer of energy to nearby cells (rather than
absorption/emission by currents `J` or `M`).
See `poynting` and `meanas.fdtd` for more details.
Args:
s: Poynting vector, as calculated with `poynting`. Optional; caller
can provide `e` and `h` instead.
e: E-field (optional; need either `s` or both `e` and `h`)
h: H-field (optional; need either `s` or both `e` and `h`)
dxes: Grid description; see `meanas.fdmath`.
Returns:
ds: Divergence of the poynting vector.
Entries indicate the net energy flow out of the corresponding
energy cell.
"""
if s is None:
assert(e is not None)
@ -59,6 +131,22 @@ def energy_hstep(e0: fdfield_t,
mu: Optional[fdfield_t] = None,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Calculate energy `U` at the time of the provided H-field `h1`.
TODO: Figure out what this means spatially.
Args:
e0: E-field one half-timestep before the energy.
h1: H-field (at the same timestep as the energy).
e2: E-field one half-timestep after the energy.
epsilon: Dielectric constant distribution.
mu: Magnetic permeability distribution.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Energy, at the time of the H-field `h1`.
"""
u = dxmul(e0 * e2, h1 * h1, epsilon, mu, dxes)
return u
@ -70,6 +158,22 @@ def energy_estep(h0: fdfield_t,
mu: Optional[fdfield_t] = None,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Calculate energy `U` at the time of the provided E-field `e1`.
TODO: Figure out what this means spatially.
Args:
h0: H-field one half-timestep before the energy.
e1: E-field (at the same timestep as the energy).
h2: H-field one half-timestep after the energy.
epsilon: Dielectric constant distribution.
mu: Magnetic permeability distribution.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Energy, at the time of the E-field `e1`.
"""
u = dxmul(e1 * e1, h0 * h2, epsilon, mu, dxes)
return u
@ -84,7 +188,21 @@ def delta_energy_h2e(dt: float,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Change in energy during the half-step from `h1` to `e2`.
This is just from (e2 * e2 + h3 * h1) - (h1 * h1 + e0 * e2)
Args:
e0: E-field one half-timestep before the start of the energy delta.
h1: H-field at the start of the energy delta.
e2: E-field at the end of the energy delta (one half-timestep after `h1`).
h3: H-field one half-timestep after the end of the energy delta.
epsilon: Dielectric constant distribution.
mu: Magnetic permeability distribution.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Change in energy from the time of `h1` to the time of `e2`.
"""
de = e2 * (e2 - e0) / dt
dh = h1 * (h3 - h1) / dt
@ -102,7 +220,21 @@ def delta_energy_e2h(dt: float,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Change in energy during the half-step from `e1` to `h2`.
This is just from (h2 * h2 + e3 * e1) - (e1 * e1 + h0 * h2)
Args:
h0: E-field one half-timestep before the start of the energy delta.
e1: H-field at the start of the energy delta.
h2: E-field at the end of the energy delta (one half-timestep after `e1`).
e3: H-field one half-timestep after the end of the energy delta.
epsilon: Dielectric constant distribution.
mu: Magnetic permeability distribution.
dxes: Grid description; see `meanas.fdmath`.
Returns:
Change in energy from the time of `e1` to the time of `h2`.
"""
de = e1 * (e3 - e1) / dt
dh = h2 * (h2 - h0) / dt
@ -114,6 +246,14 @@ def delta_energy_j(j0: fdfield_t,
e1: fdfield_t,
dxes: Optional[dx_lists_t] = None,
) -> fdfield_t:
"""
Calculate
Note that each value of $J$ contributes to the energy twice (i.e. once per field update)
despite only causing the value of $E$ to change once (same for $M$ and $H$).
"""
if dxes is None:
dxes = tuple(tuple(numpy.ones(1) for _ in range(3)) for _ in range(2))

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