Proper update_fields interpolation scheme

[smartalecH]

The subpixel smoothing paper describes the update equation using the following figure:

which is interpolation method 26(c) in the original Werner and Cary paper.

This method performs an interpolation, then the chi1inv multiplication, followed by another interpolation step. What's important is that this particular chi1inv component does not seem to be located on a typical grid point that it's originally stored (i.e. in the figure, the fields are interpolated to a "node" that doesn't store ε⁻¹ₓy).

The code, however, seems to just grab two ε points, one from a "leading corner" for each side that is being interpolated:

meep/src/step_generic.cpp

Lines 381 to 382 in b6166b1

	#define OFFDIAG(u, g, sx) \
	(0.25 * ((g[i] + g[i - sx]) * u[i] + (g[i + s] + g[(i + s) - sx]) * u[i + s]))

Note that u[i] aligns with g[i] and u[i + s] aligns with u[i + s], leaving g[i-sx] and g[i-isx+s] "unbalanced". (Note that multiplying each point by it's corresponding chi1inv component is a different interpolation scheme altogether and not always stable).

These two operations don't seem to be the same to me, but maybe I'm missing something with how chi1inv is stored (I thought an entire row of the tensor was stored at each corresponding yee point).

[smartalecH]

I think I see what's happening...

Even though we are indexing chi1inv at the same "location" as the fields, it actually doesn't correspond to the same location in space. Rather, if it's an off-diagonal element, it's been evaluated at a slightly off-center location (to line up with the proper node, I'm assuming):

meep/src/anisotropic_averaging.cpp

Lines 238 to 253 in 3eabcef

	IVEC_LOOP_ILOC(gv, here);
	medium.eff_chi1inv_row(c, chi1invrow, gv.dV(here, smoothing_diameter), tol, maxeval);
	medium.eff_chi1inv_row(c, chi1invrow_offdiag, gv.dV(here - shift1, smoothing_diameter), tol,
	maxeval);
	if (chi1inv[c][d0]) {
	chi1inv[c][d0][i] = (d0 == dc) ? chi1invrow[0] : chi1invrow_offdiag[0];
	trivial[0] = trivial[0] && (chi1inv[c][d0][i] == trivial_val[0]);
	}
	if (chi1inv[c][d1]) {
	chi1inv[c][d1][i] = (d1 == dc) ? chi1invrow[1] : chi1invrow_offdiag[1];
	trivial[1] = trivial[1] && (chi1inv[c][d1][i] == trivial_val[1]);
	}
	if (chi1inv[c][d2]) {
	chi1inv[c][d2][i] = (d2 == dc) ? chi1invrow[2] : chi1invrow_offdiag[2];
	trivial[2] = trivial[2] && (chi1inv[c][d2][i] == trivial_val[2]);
	}

So hopefully shift1 corresponds to the distance from the Dy points and the point ε⁻¹ₓy is evaluated/indexed.

[stevengj]

The index in the array doesn't correspond exactly to the location in space — for different field components, the same index corresponds to different spatial points, because different field components are stored on different Yee grids.

For example, in the diagram above, all three of these spatial points correspond to the same array index i in their respective arrays:

[smartalecH]

Awesome, thanks!

For (my own) reference, this is the method we need to use to properly recombine the forward and adjoint fields.