def test_clenshaw_vector(N, regtotal_in, k1, k2, ell):
dtype = np.complex128
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c,))
b = basis.BallBasis(c, (N, N, N), dtype=dtype, k=k1, radius=1)
basis_in = basis.BallBasis(c, (N, N, N), dtype=dtype, k=k2, radius=1)
phi, theta, r = b.local_grids((1, 1, 1))
x = r * np.sin(theta) * np.cos(phi)
y = r * np.sin(theta) * np.sin(phi)
z = r * np.cos(theta)
ncc = field.Field(dist=d, bases=(b.radial_basis,), tensorsig=(c,), dtype=dtype)
ncc['g'][2] = r*(2*r**2-1)
ncc_basis = ncc.domain.bases[0]
a_ncc = ncc_basis.alpha + ncc_basis.k
regtotal_ncc = 1
b_ncc = 1/2 + regtotal_ncc
n_size = b.n_size(ell)
coeffs_filter = ncc['c'][1,0,0,:n_size]
J = basis_in.operator_matrix('Z', ell, regtotal_in)
I = basis_in.operator_matrix('Id', ell, regtotal_in)
A, B = clenshaw.jacobi_recursion(n_size, a_ncc, b_ncc, J)
f0 = dedalus_sphere.zernike.polynomials(3, 1, a_ncc, regtotal_ncc, 1)[0] * sparse.identity(n_size)
matrix = clenshaw.matrix_clenshaw(coeffs_filter, A, B, f0, cutoff=1e-6)
assert np.allclose(J.todense(), matrix.todense())
def test_heat_ball_nlbvp(Nr, dtype, dealias):
radius = 2
ncc_cutoff = 1e-10
tolerance = 1e-10
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c,))
b = basis.BallBasis(c, (1, 1, Nr), radius=radius, dtype=dtype, dealias=dealias)
bs = b.S2_basis(radius=radius)
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
u = field.Field(name='u', dist=d, bases=(b,), dtype=dtype)
τ = field.Field(name='τ', dist=d, bases=(bs,), dtype=dtype)
F = field.Field(name='F', dist=d, bases=(b,), dtype=dtype) # todo: make this constant
F['g'] = 6
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.NLBVP([u, τ])
problem.add_equation((Lap(u) + Lift(τ), F))
problem.add_equation((u(r=radius), 0))
# Solver
solver = solvers.NonlinearBoundaryValueSolver(problem, ncc_cutoff=ncc_cutoff)
# Initial guess
u['g'] = 1
# Iterations
def error(perts):
return np.sum([np.sum(np.abs(pert['c'])) for pert in perts])
err = np.inf
while err > tolerance:
solver.newton_iteration()
err = error(solver.perturbations)
u_true = r**2 - radius**2
u.change_scales(1)
assert np.allclose(u['g'], u_true)
def build_shell(Nphi, Ntheta, Nr, dealias, dtype):
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c,))
b = basis.ShellBasis(c, (Nphi, Ntheta, Nr), radii=radii_shell, dealias=(dealias, dealias, dealias), dtype=dtype)
phi, theta, r = b.local_grids(b.domain.dealias)
x, y, z = c.cartesian(phi, theta, r)
return c, d, b, phi, theta, r, x, y, z
def build_sphere_3d(Nphi, Ntheta, radius, dealias, dtype):
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.SphereBasis(c, (Nphi, Ntheta),
radius=radius,
dealias=(dealias, dealias),
dtype=dtype)
phi, theta = b.local_grids((dealias, dealias, dealias))
return c, d, b, phi, theta
def test_ball_diffusion(Lmax, Nmax, Leig, radius, bc, dtype):
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.BallBasis(c, (2 * (Lmax + 1), Lmax + 1, Nmax + 1),
radius=radius,
dtype=dtype)
b_S2 = b.S2_basis()
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
A = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
φ = field.Field(dist=d, bases=(b, ), dtype=dtype)
τ_A = field.Field(dist=d, bases=(b_S2, ), tensorsig=(c, ), dtype=dtype)
λ = field.Field(name='λ', dist=d, dtype=dtype)
# Parameters and operators
div = lambda A: operators.Divergence(A)
grad = lambda A: operators.Gradient(A, c)
curl = lambda A: operators.Curl(A)
lap = lambda A: operators.Laplacian(A, c)
trans = lambda A: operators.TransposeComponents(A)
radial = lambda A, index: operators.RadialComponent(A, index=index)
angular = lambda A, index: operators.AngularComponent(A, index=index)
Lift = lambda A: operators.Lift(A, b, -1)
# Problem
problem = problems.EVP([φ, A, τ_A], λ)
problem.add_equation((div(A), 0))
problem.add_equation((-λ * A + grad(φ) - lap(A) + Lift(τ_A), 0))
if bc == 'no-slip':
problem.add_equation((A(r=radius), 0))
elif bc == 'stress-free':
E = 1 / 2 * (grad(A) + trans(grad(A)))
problem.add_equation((radial(A(r=radius), 0), 0))
problem.add_equation((radial(angular(E(r=radius), 0), 1), 0))
elif bc == 'potential':
ell_func = lambda ell: ell + 1
ell_1 = lambda A: operators.SphericalEllProduct(A, c, ell_func)
problem.add_equation(
(radial(grad(A)(r=radius), 0) + ell_1(A)(r=radius) / radius, 0))
elif bc == 'conducting':
problem.add_equation((φ(r=radius), 0))
problem.add_equation((angular(A(r=radius), 0), 0))
elif bc == 'pseudo':
problem.add_equation((radial(A(r=radius), 0), 0))
problem.add_equation((angular(curl(A)(r=radius), 0), 0))
# Solver
solver = solvers.EigenvalueSolver(problem)
if not solver.subproblems[Leig].group[1] == Leig:
raise ValueError("subproblems indexed in a strange way")
solver.solve_dense(solver.subproblems[Leig])
i_sort = np.argsort(solver.eigenvalues)
solver.eigenvalues = solver.eigenvalues[i_sort]
λ_analytic = analytic_eigenvalues(Leig, Nmax + 1, bc, r0=radius)
if (bc == 'stress-free' and Leig == 1):
# add null space solution
λ_analytic = np.append(0, λ_analytic)
assert np.allclose(solver.eigenvalues[:Nmax // 4], λ_analytic[:Nmax // 4])
def build_ball(Nphi, Ntheta, Nr, dtype, dealias, radius=1):
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.BallBasis(c, (Nphi, Ntheta, Nr),
radius=radius,
dtype=dtype,
dealias=(dealias, dealias, dealias))
phi, theta, r = b.local_grids()
x, y, z = c.cartesian(phi, theta, r)
return c, d, b, phi, theta, r, x, y, z
def build_shell(Nphi, Ntheta, Nr, radii_shell, dealias, dtype, grid_scale=1):
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
dealias_tuple = (dealias, dealias, dealias)
b = basis.ShellBasis(c, (Nphi, Ntheta, Nr),
radii=radii_shell,
dealias=dealias_tuple,
dtype=dtype)
grid_scale_tuple = (grid_scale, grid_scale, grid_scale)
phi, theta, r = b.local_grids(grid_scale_tuple)
x, y, z = c.cartesian(phi, theta, r)
return c, d, b, phi, theta, r, x, y, z
def build_ball(Nphi, Ntheta, Nr, radius, alpha, k, dealias, dtype):
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.BallBasis(c, (Nphi, Ntheta, Nr),
radius=radius,
alpha=alpha,
k=k,
dealias=(dealias, dealias, dealias),
dtype=dtype)
phi, theta, r = b.local_grids((dealias, dealias, dealias))
x = r * np.sin(theta) * np.cos(phi)
y = r * np.sin(theta) * np.sin(phi)
z = r * np.cos(theta)
return c, d, b, phi, theta, r, x, y, z
def test_lane_emden_floating_R(Nr, dtype, dealias):
n = 3.0
ncc_cutoff = 1e-10
tolerance = 1e-10
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c,))
b = basis.BallBasis(c, (1, 1, Nr), radius=1, dtype=dtype, dealias=dealias)
bs = b.S2_basis(radius=1)
bs0 = b.S2_basis(radius=0)
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
f = field.Field(dist=d, bases=(b,), dtype=dtype, name='f')
R = field.Field(dist=d, dtype=dtype, name='R')
τ = field.Field(dist=d, bases=(bs,), dtype=dtype, name='τ')
one = field.Field(dist=d, bases=(bs0,), dtype=dtype)
one['g'] = 1
# Problem
lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.NLBVP([f, R, τ])
problem.add_equation((lap(f) + Lift(τ), - R**2 * f**n))
problem.add_equation((f(r=0), one))
problem.add_equation((f(r=1), 0))
# Solver
solver = solvers.NonlinearBoundaryValueSolver(problem, ncc_cutoff=ncc_cutoff)
# Initial guess
f['g'] = np.cos(np.pi/2 * r)**2
R['g'] = 5
# Iterations
def error(perts):
return np.sum([np.sum(np.abs(pert['c'])) for pert in perts])
err = np.inf
while err > tolerance:
solver.newton_iteration()
err = error(solver.perturbations)
# Compare to reference solutions from Boyd
R_ref = {0.0: np.sqrt(6),
0.5: 2.752698054065,
1.0: np.pi,
1.5: 3.65375373621912608,
2.0: 4.3528745959461246769735700,
2.5: 5.355275459010779,
3.0: 6.896848619376960375454528,
3.25: 8.018937527,
3.5: 9.535805344244850444,
4.0: 14.971546348838095097611066,
4.5: 31.836463244694285264}
assert np.allclose(R['g'].ravel(), R_ref[n])
def test_ball_bessel_eigenfunction(Lmax, Nmax, Leig, Neig, radius, dtype):
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.BallBasis(c, (2 * (Lmax + 1), Lmax + 1, Nmax + 1),
radius=radius,
dtype=dtype)
b_S2 = b.S2_basis()
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
τ_f = field.Field(dist=d, bases=(b_S2, ), dtype=dtype)
k2 = field.Field(name='k2', dist=d, dtype=dtype)
# Parameters and operators
lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
# Bessel equation: k^2*f + lap(f) = 0
problem = problems.EVP([f, τ_f], k2)
problem.add_equation((k2 * f + lap(f) + Lift(τ_f), 0))
problem.add_equation((f(r=radius), 0))
# Solver
solver = solvers.EigenvalueSolver(problem)
if not solver.subproblems[Leig].group[1] == Leig:
raise ValueError("subproblems indexed in a strange way")
solver.solve_dense(solver.subproblems[Leig])
i_sort = np.argsort(solver.eigenvalues)
solver.eigenvalues = solver.eigenvalues[i_sort]
solver.eigenvectors = solver.eigenvectors[:, i_sort]
solver.set_state(Neig,
solver.subproblems[Leig].subsystems[0]) # m = 0 mode
f.change_layout(d.layouts[1])
local_m, local_ell, local_n = f.layout.local_group_arrays(
f.domain, f.scales)
radial_eigenfunction = f.data[(local_m == 0) * (local_ell == Leig)]
i_max = np.argmax(np.abs(radial_eigenfunction))
radial_eigenfunction /= radial_eigenfunction[i_max]
k = np.sqrt(solver.eigenvalues[Neig])
sol = spec.jv(Leig + 1 / 2, k * r) / np.sqrt(k * r)
sol = sol.ravel()
sol /= sol[i_max]
assert np.allclose(radial_eigenfunction, sol)
radius = 1
Lmax = 15
L_dealias = 1
Nmax = 15
N_dealias = 1
Om = 20.
u0 = np.sqrt(3 / (2 * np.pi))
nu = 1e-2
dt = 0.02
t_end = 20
ts = timesteppers.SBDF2
dtype = np.float64
Nphi = 2 * (Lmax + 1)
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c, ))
b = basis.BallBasis(c, (Nphi, Lmax + 1, Nmax + 1), radius=radius, dtype=dtype)
b_S2 = b.S2_basis()
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
u = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
p = field.Field(dist=d, bases=(b, ), dtype=dtype)
tau = field.Field(dist=d, bases=(b_S2, ), tensorsig=(c, ), dtype=dtype)
# Boundary conditions
u_BC = field.Field(dist=d, bases=(b_S2, ), tensorsig=(c, ), dtype=dtype)
u_BC['g'][2] = 0. # u_r = 0
u_BC['g'][1] = -u0 * np.cos(theta) * np.cos(phi)
u_BC['g'][0] = u0 * np.sin(phi)
def test_lane_emden_first_order(Nr, dtype, dealias):
n = 3.0
ncc_cutoff = 1e-10
tolerance = 1e-10
# Bases
c = coords.SphericalCoordinates('phi', 'theta', 'r')
d = distributor.Distributor((c,))
b = basis.BallBasis(c, (1, 1, Nr), radius=1, dtype=dtype, dealias=dealias)
br = b.radial_basis
phi, theta, r = b.local_grids((1, 1, 1))
# Fields
p = field.Field(dist=d, bases=(br,), dtype=dtype, name='p')
ρ = field.Field(dist=d, bases=(br,), dtype=dtype, name='ρ')
φ = field.Field(dist=d, bases=(br,), dtype=dtype, name='φ')
τ = field.Field(dist=d, dtype=dtype, name='τ')
τ2 = field.Field(dist=d, dtype=dtype, name='τ2')
rf = field.Field(dist=d, bases=(br,), tensorsig=(c,), dtype=dtype, name='r')
rf['g'][2] = r
# Problem
lap = lambda A: operators.Laplacian(A, c)
grad = lambda A: operators.Gradient(A, c)
div = lambda A: operators.Divergence(A)
Lift = lambda A: operators.Lift(A, br, -1)
dot = lambda A, B: arithmetic.DotProduct(A, B)
rdr = lambda A: dot(rf, grad(A))
problem = problems.NLBVP([p, ρ, φ, τ, τ2])
problem.add_equation((p, ρ**(1+1/n)))
problem.add_equation((lap(φ) + Lift(τ), ρ))
problem.add_equation((φ(r=1), 0))
# This works
# problem.add_equation((-φ, (n+1) * ρ**(1/n)))
# problem.add_equation((τ2, 0))
# Also works when near correct solution
# problem.add_equation((-φ**n, (n+1)**n * ρ))
# problem.add_equation((τ2, 0))
# Doesn't work well
problem.add_equation((rdr(p) + Lift(τ2), -ρ*rdr(φ)))
problem.add_equation((p(r=1), 0))
# Also doesn't work well
# problem.add_equation((lap(p) + Lift(τ2), -div(ρ*grad(φ))))
# problem.add_equation((p(r=1), 0))
# Solver
solver = solvers.NonlinearBoundaryValueSolver(problem, ncc_cutoff=ncc_cutoff)
# Initial guess
#φ['g'] = - 55 * np.cos(np.pi/2 * r)
#φ['g'] = - 50 * (1 - r) * (1 + r)
φ['g'] = np.array([[[-5.49184941e+01-2.10742982e-38j,
-5.41628923e+01-5.32970546e-38j,
-5.29461420e+01-5.04522267e-38j,
-5.13265949e+01-2.97780743e-38j,
-4.93761552e+01-2.61880274e-38j,
-4.71730013e+01-3.43967627e-38j,
-4.47948939e+01-3.04186813e-38j,
-4.23139098e+01-1.79113018e-38j,
-3.97929639e+01-1.43996160e-38j,
-3.72840673e+01-1.63817277e-38j,
-3.48280092e+01-9.99537738e-39j,
-3.24550394e+01-3.17721047e-40j,
-3.01861437e+01-5.81373831e-42j,
-2.80345785e+01-3.10228717e-39j,
-2.60074301e+01+1.28594534e-39j,
-2.41070531e+01+7.60758754e-39j,
-2.23323155e+01+7.97312927e-39j,
-2.06796271e+01+5.81693170e-39j,
-1.91437566e+01+6.56252079e-39j,
-1.77184618e+01+1.10908840e-38j,
-1.63969611e+01+1.53872437e-38j,
-1.51722763e+01+1.39129399e-38j,
-1.40374741e+01+9.43669477e-39j,
-1.29858304e+01+9.30920868e-39j,
-1.20109359e+01+1.23602737e-38j,
-1.11067589e+01+1.41710050e-38j,
-1.02676773e+01+1.60717088e-38j,
-9.48848876e+00+1.77178302e-38j,
-8.76440610e+00+1.48647842e-38j,
-8.09104289e+00+1.01146628e-38j,
-7.46439287e+00+1.11622279e-38j,
-6.88080593e+00+1.66263627e-38j,
-6.33696251e+00+1.79488585e-38j,
-5.82984767e+00+1.46579657e-38j,
-5.35672567e+00+1.34603496e-38j,
-4.91511565e+00+1.50574167e-38j,
-4.50276874e+00+1.54259944e-38j,
-4.11764669e+00+1.52307339e-38j,
-3.75790229e+00+1.61072571e-38j,
-3.42186130e+00+1.52968997e-38j,
-3.10800611e+00+1.33188351e-38j,
-2.81496085e+00+1.46531686e-38j,
-2.54147800e+00+1.65381249e-38j,
-2.28642630e+00+1.48467159e-38j,
-2.04877987e+00+1.49987605e-38j,
-1.82760852e+00+1.83704612e-38j,
-1.62206896e+00+1.68020109e-38j,
-1.43139709e+00+1.17510410e-38j,
-1.25490103e+00+1.25754442e-38j,
-1.09195489e+00+1.71504952e-38j,
-9.41993349e-01+1.76972495e-38j,
-8.04506695e-01+1.53368883e-38j,
-6.79036552e-01+1.46402303e-38j,
-5.65172039e-01+1.54974386e-38j,
-4.62546411e-01+1.60642465e-38j,
-3.70834105e-01+1.59758147e-38j,
-2.89748169e-01+1.49361039e-38j,
-2.19038018e-01+1.32679253e-38j,
-1.58487515e-01+1.40338570e-38j,
-1.07913330e-01+1.83256446e-38j,
-6.71635483e-02+2.05950514e-38j,
-3.61164846e-02+1.65676365e-38j,
-1.46794435e-02+1.02374473e-38j,
-2.78432418e-03+6.69851727e-39j]]])
ρ['g'] = (-φ['g']/(n+1))**n
p['g'] = ρ['g']**(1+1/n)
# Iterations
def error(perts):
return np.sum([np.sum(np.abs(pert['c'])) for pert in perts])
err = np.inf
while err > tolerance and solver.iteration < 20:
solver.newton_iteration()
err = error(solver.perturbations)
φcen = φ(r=0).evaluate()['g'][0,0,0]
R = -φcen / (n+1)**(3/2)
print(solver.iteration, φcen, R, err)
dH = solver.subproblems[0].dH_min
print('%.2e' %np.linalg.cond(dH.A))
if err > tolerance:
raise ValueError("Did not converge")
# Compare to reference solutions from Boyd
R_ref = {0.0: np.sqrt(6),
0.5: 2.752698054065,
1.0: np.pi,
1.5: 3.65375373621912608,
2.0: 4.3528745959461246769735700,
2.5: 5.355275459010779,
3.0: 6.896848619376960375454528,
3.25: 8.018937527,
3.5: 9.535805344244850444,
4.0: 14.971546348838095097611066,
4.5: 31.836463244694285264}
assert np.allclose(R, R_ref[n])
