subpixel averaging for boundaries of variable-material objects

[stevengj]

This PR implements subpixel averaging for the boundaries of objects with variable materials (e.g. user-defined material functions). I realized that we can still handle this case accurately and efficiently, as long as the material function is continuous inside the object.

For example, here is a simulation of transmitted power through a cylinder of linearly-varying ε:

def linear_eps(p):
    return 1 + 11 * (p.x + 0.4) / 0.8

def flux2(res):
    sim = mp.Simulation(cell_size=(7,1), resolution=res, boundary_layers=[mp.PML(direction=mp.X, thickness=0.5)],
                    k_point=mp.Vector3(0,0), symmetries=[mp.Mirror(direction=mp.Y)],
                    sources=[mp.Source(mp.GaussianSource(frequency=0.7, fwidth=0.2, is_integrated=True),
                                       component=mp.Ez, center=(-3,0), size=(0,1))],
                    geometry=[mp.Cylinder(radius=0.4, height=mp.inf, epsilon_func=linear_eps)])
    tran = sim.add_flux(0.7, 0.2, 1, mp.FluxRegion(center=(3,0), size=(0,1)))
    sim.run(until=200)
    return tran.flux()[0]

If I compute and plot the convergence with respect to resolution using this code:

import matplotlib.pyplot as plt
import numpy as np
r = [10,20,40,80,160,320,640]
f2 = np.asarray([flux2(r) for r in r])
r = np.asarray(r)
plt.loglog(r[0:-1], np.abs(f2[0:-1] - f2[-1]) / f2[-1], "ro-")
plt.loglog(r[0:-1], 30 / r[0:-1]**2, "k--")
plt.loglog(r[0:-1], 1 / r[0:-1], "b:")
plt.legend(["flux2 error", "$1/r^2$ reference", "$1/r$ reference"])
plt.xlabel("resolution $r$")
plt.ylabel("fractional error")

then before this PR I get first-order convergence due to the lack of subpixel averaging at the boundaries of the cylinder:

but after this PR I get second-order convergence:

[stevengj]

In #1539, this will branch to the new averaging procedure if mat is a variable material.

[stevengj]

Note that we added p_front and p_behind here — this is so that we have a point to evaluate the material at if it is a variable materials (grid or user).