def test_dirichlet_bvp_spherical_basis():
N_COMPONENTS = 25
r0, r1 = x0, x1
r2 = (r0 + r1) / 2
R0 = torch.rand(N_SAMPLES, N_COMPONENTS)
R1 = torch.rand(N_SAMPLES, N_COMPONENTS)
R2 = torch.rand(N_SAMPLES, N_COMPONENTS)
condition = DirichletBVPSphericalBasis(r_0=r0, R_0=R0, r_1=r1, R_1=R1)
net = FCNN(1, N_COMPONENTS)
r = r0 * ones
assert all_close(condition.enforce(net, r), R0), "inner Dirichlet BC not satisfied"
r = r1 * ones
assert all_close(condition.enforce(net, r), R1), "outer Dirichlet BC not satisfied"
condition = DirichletBVPSphericalBasis(r_0=r2, R_0=R2)
r = r2 * ones
assert all_close(condition.enforce(net, r), R2), "single ended BC not satisfied"
def test_electric_potential_gaussian_charged_density():
# total charge
Q = 1.
# standard deviation of gaussian
sigma = 1.
# medium permittivity
epsilon = 1.
# Coulomb constant
k = 1 / (4 * np.pi * epsilon)
# coefficient of gaussian term
gaussian_coeff = Q / (sigma**3) / np.power(2 * np.pi, 1.5)
# distribution of charge
rho_f = lambda r: gaussian_coeff * torch.exp(-r.pow(2) / (2 * sigma**2))
# analytic solution, refer to https://en.wikipedia.org/wiki/Poisson%27s_equation
analytic_solution = lambda r, th, ph: (k * Q / r) * torch.erf(r / (np.sqrt(
2) * sigma))
# interior and exterior radius
r_0, r_1 = 0.1, 3.
# values at interior and exterior boundary
v_0 = (k * Q / r_0) * erf(r_0 / (np.sqrt(2) * sigma))
v_1 = (k * Q / r_1) * erf(r_1 / (np.sqrt(2) * sigma))
def validate(solution):
generator = GeneratorSpherical(512, r_min=r_0, r_max=r_1)
rs, thetas, phis = generator.get_examples()
us = solution(rs, thetas, phis, to_numpy=True)
vs = analytic_solution(rs, thetas, phis).detach().cpu().numpy()
assert us.shape == vs.shape
# solving the problem using normal network (subject to the influence of polar singularity of laplacian operator)
pde1 = lambda u, r, th, ph: laplacian_spherical(u, r, th, ph) + rho_f(
r) / epsilon
condition1 = DirichletBVPSpherical(r_0, lambda th, ph: v_0, r_1,
lambda th, ph: v_1)
monitor1 = MonitorSpherical(r_0, r_1, check_every=50)
with pytest.warns(FutureWarning):
solution1, metrics_history = solve_spherical(
pde1,
condition1,
r_0,
r_1,
max_epochs=2,
return_best=True,
analytic_solution=analytic_solution,
monitor=monitor1,
)
validate(solution1)
# solving the problem using spherical harmonics (laplcian computation is optimized)
max_degree = 2
harmonics_fn = RealSphericalHarmonics(max_degree=max_degree)
harmonic_laplacian = HarmonicsLaplacian(max_degree=max_degree)
pde2 = lambda R, r, th, ph: harmonic_laplacian(R, r, th, ph) + rho_f(
r) / epsilon
R_0 = torch.tensor([v_0 * 2] + [0 for _ in range((max_degree + 1)**2 - 1)])
R_1 = torch.tensor([v_1 * 2] + [0 for _ in range((max_degree + 1)**2 - 1)])
def analytic_solution2(r, th, ph):
sol = torch.zeros(r.shape[0], (max_degree + 1)**2)
sol[:, 0:1] = 2 * analytic_solution(r, th, ph)
return sol
condition2 = DirichletBVPSphericalBasis(r_0=r_0,
R_0=R_0,
r_1=r_1,
R_1=R_1,
max_degree=max_degree)
monitor2 = MonitorSphericalHarmonics(r_0,
r_1,
check_every=50,
harmonics_fn=harmonics_fn)
net2 = FCNN(n_input_units=1, n_output_units=(max_degree + 1)**2)
with pytest.warns(FutureWarning):
solution2, metrics_history = solve_spherical(
pde2,
condition2,
r_0,
r_1,
net=net2,
max_epochs=2,
return_best=True,
analytic_solution=analytic_solution2,
monitor=monitor2,
harmonics_fn=harmonics_fn,
)
validate(solution2)