From 8ac0d52cd17b3101ea5cff0395e1f77aa94e713a Mon Sep 17 00:00:00 2001 From: Jan Petykiewicz Date: Sun, 11 Jul 2021 17:27:02 -0700 Subject: [PATCH] Add some docs for energy calculations --- meanas/fdtd/energy.py | 144 +++++++++++++++++++++++++++++++++++++++++- 1 file changed, 142 insertions(+), 2 deletions(-) diff --git a/meanas/fdtd/energy.py b/meanas/fdtd/energy.py index 121c4f6..93eedf0 100644 --- a/meanas/fdtd/energy.py +++ b/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))