computing the normal vector to an interface for a MaterialGrid

[oskooi]

The following schematic shows the analytic expressions necessary to compute the gradient of the weights array u for a MaterialGrid in 2d from which the normal vector to an interface can be obtained at any Yee grid point using bilinear interpolation. In the figure, the MaterialGrid are the green points and the Yee grid points for E_x and E_y are the blue and red points, respectively.

To set this up, we will need to modify the existing material_function::normal_vector routine which uses spherical quadrature such that for the case of a MaterialGrid it uses the analytic formulas shown above:

meep/src/anisotropic_averaging.cpp

Lines 40 to 64 in 272bd0c

	vec material_function::normal_vector(field_type ft, const volume &v) {
	vec gradient(zero_vec(v.dim));
	vec p(v.center());
	double R = v.diameter();
	int num_dirs = number_of_directions(v.dim);
	int min_iters = 1 << num_dirs;
	double chi1p1_prev = 0;
	bool break_early = true;
	for (int i = 0; i < num_sphere_quad[num_dirs - 1]; ++i) {
	double weight;
	vec pt = sphere_pt(p, R, i, weight);
	double chi1p1_val = chi1p1(ft, pt);
	if (i > 0 && i < min_iters) {
	if (chi1p1_val != chi1p1_prev) { break_early = false; }
	if (i == min_iters - 1 && break_early) {
	// Don't average regions where epsilon is uniform
	return zero_vec(v.dim);
	}
	}
	chi1p1_prev = chi1p1_val;
	gradient += (pt - p) * (weight * chi1p1_val);
	}
	return gradient;
	}

We can then verify that the new approach to computing the normal vector is working correctly by comparing with the results obtained using spherical quadrature for e.g. a test case involving a cylinder (similar to #1522 (comment)).

[stevengj]

The first thing is to implement a function matgrid_grad that computes the gradient, similar to matgrid_val. Then you can call this when computing the normal vector if you detect that that pixel is completely within a material-grid object.