Add some docs for energy calculations

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))