def test_scalar_heat_disk(Nr, Nphi, dtype):
# Bases
dealias = 1
c, d, b, phi, r, x, y = build_disk(Nphi, Nr, dealias=dealias, dtype=dtype)
xr = radius_disk * np.cos(phi)
yr = radius_disk * np.sin(phi)
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu = field.Field(name='u', dist=d, bases=(b.S1_basis(), ), dtype=dtype)
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
f['g'] = 6 * x - 2
g = field.Field(dist=d, bases=(b.S1_basis(), ), dtype=dtype)
g['g'] = xr**3 - yr**2
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((Lap(u) + Lift(τu), f))
problem.add_equation((u(r=radius_disk), g))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
u_true = x**3 - y**2
assert np.allclose(u['g'], u_true)
def test_vector_heat_disk_dirichlet(Nr, Nphi, dtype):
# Bases
dealias = 1
c, d, b, phi, r, x, y = build_disk(Nphi, Nr, dealias=dealias, dtype=dtype)
# Fields
u = field.Field(name='u',
dist=d,
bases=(b, ),
tensorsig=(c, ),
dtype=dtype)
τu = field.Field(name='u',
dist=d,
bases=(b.S1_basis(), ),
tensorsig=(c, ),
dtype=dtype)
v = field.Field(name='u',
dist=d,
bases=(b, ),
tensorsig=(c, ),
dtype=dtype)
ex = np.array([-np.sin(phi), np.cos(phi)])
ey = np.array([np.cos(phi), np.sin(phi)])
v['g'] = (x + 4 * y) * ex
vr = operators.RadialComponent(v(r=radius_disk))
vph = operators.AzimuthalComponent(v(r=radius_disk))
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((Lap(u) + Lift(τu), 0))
problem.add_equation((u(r=radius_disk), v(r=radius_disk)))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
assert np.allclose(u['g'], v['g'])
def test_heat_ball(Nmax, Lmax, dtype):
# Bases
dealias = 1
c, d, b, phi, theta, r, x, y, z = build_ball(2 * (Lmax + 1),
Lmax + 1,
Nmax + 1,
dealias=dealias,
dtype=dtype)
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu = field.Field(name='u', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
F = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
F['g'] = 6
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((Lap(u) + Lift(τu), F))
problem.add_equation((u(r=radius_ball), 0))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
u_true = r**2 - 1
assert np.allclose(u['g'], u_true)
def test_disk_bessel_zeros(Nphi, Nr, m, radius, dtype):
# Bases
c = coords.PolarCoordinates('phi', 'r')
d = distributor.Distributor((c, ))
b = basis.DiskBasis(c, (Nphi, Nr), radius=radius, dtype=dtype)
b_S1 = b.S1_basis()
phi, r = b.local_grids((1, 1))
# Fields
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
τ_f = field.Field(dist=d, bases=(b_S1, ), 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)
print(solver.subproblems[0].group)
for sp in solver.subproblems:
if sp.group[0] == m:
break
else:
raise ValueError("Could not find subproblem with m = %i" % m)
solver.solve_dense(sp)
# Compare eigenvalues
n_compare = 5
selected_eigenvalues = np.sort(solver.eigenvalues)[:n_compare]
analytic_eigenvalues = (spec.jn_zeros(m, n_compare) / radius)**2
assert np.allclose(selected_eigenvalues, analytic_eigenvalues)
def test_heat_ball_cart(Nmax, Lmax, dtype):
# Bases
dealias = 1
c, d, b, phi, theta, r, x, y, z = build_ball(2 * (Lmax + 1),
Lmax + 1,
Nmax + 1,
dealias=dealias,
dtype=dtype)
xr = radius_ball * np.cos(phi) * np.sin(theta)
yr = radius_ball * np.sin(phi) * np.sin(theta)
zr = radius_ball * np.cos(theta)
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu = field.Field(name='u', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
f = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
f['g'] = 12 * x**2 - 6 * y + 2
g = field.Field(name='a', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
g['g'] = xr**4 - yr**3 + zr**2
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((Lap(u) + Lift(τu), f))
problem.add_equation((u(r=radius_ball), g))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
u_true = x**4 - y**3 + z**2
assert np.allclose(u['g'], u_true)
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 test_heat_shell(Nmax, Lmax, dtype):
# Bases
dealias = 1
c, d, b, phi, theta, r, x, y, z = build_shell(2 * (Lmax + 1),
Lmax + 1,
Nmax + 1,
dealias=dealias,
dtype=dtype)
r0, r1 = b.radial_basis.radii
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu1 = field.Field(name='τu1', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
τu2 = field.Field(name='τu2', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
F = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
F['g'] = 6
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A, n: operators.Lift(A, b, n)
problem = problems.LBVP([u, τu1, τu2])
problem.add_equation((Lap(u) + Lift(τu1, -1) + Lift(τu2, -2), F))
problem.add_equation((u(r=r0), 0))
problem.add_equation((u(r=r1), 0))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
u_true = r**2 + 6 / r - 7
assert np.allclose(u['g'], u_true)
def test_tensor_dot_tensor(Nphi, Ntheta, Nr, basis, alpha, k_ncc, k_arg,
dealias, dtype):
c, d, b, phi, theta, r, x, y, z = basis(Nphi,
Ntheta,
Nr,
alpha,
dealias=dealias,
dtype=dtype)
b_ncc = b.radial_basis.clone_with(k=k_ncc)
b_arg = b.clone_with(k=k_arg)
T = field.Field(dist=d, bases=(b_ncc, ), tensorsig=(c, c), dtype=dtype)
U = field.Field(dist=d, bases=(b_arg, ), tensorsig=(c, c), dtype=dtype)
T['g'] = np.random.randn(*T['g'].shape)
U['g'] = np.random.randn(*U['g'].shape)
if isinstance(b, BallBasis):
k_out = k_out_ball(k_ncc, k_arg, T, U)
else:
k_out = k_out_shell(k_ncc, k_arg)
if k_out > 0 and dealias < 2:
T['c'][:, :, :, :, -k_out:] = 0
U['c'][:, :, :, :, -k_out:] = 0
vars = [U]
w0 = dot(T, U)
w1 = w0.reinitialize(ncc=True, ncc_vars=vars)
problem = problems.LBVP(vars)
problem.add_equation((operators.Laplacian(U, c), 0))
solver = solvers.LinearBoundaryValueSolver(problem, )
w1.store_ncc_matrices(vars, solver.subproblems)
w0 = w0.evaluate()
w1 = w1.evaluate_as_ncc()
assert np.allclose(w0['c'], w1['c'])
def test_laplacian_scalar(Nphi, Nr, dealias, basis, dtype):
c, d, b, phi, r, x, y = basis(Nphi, Nr, dealias, dtype)
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
f.preset_scales(b.domain.dealias)
f['g'] = x**4 + 2 * y**4
h = operators.Laplacian(f, c).evaluate()
hg = 12 * x**2 + 24 * y**2
assert np.allclose(h['g'], hg)
def test_laplacian_radial_scalar(Nr, dealias, basis, dtype):
Nphi = 1
c, d, b, phi, r, x, y = basis(Nphi, Nr, dealias, dtype=dtype)
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
f.preset_scales(b.domain.dealias)
f['g'] = r**2
h = operators.Laplacian(f, c).evaluate()
hg = 4
assert np.allclose(h['g'], hg)
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 test_convert_scalar(Nphi, Nr, k, dealias, basis, dtype, layout):
c, d, b, phi, r, x, y = basis(Nphi, Nr, k, dealias, dtype)
f = field.Field(dist=d, bases=(b, ), dtype=dtype)
f.preset_scales(b.domain.dealias)
f['g'] = 3 * x**2 + 2 * y
g = operators.Laplacian(f, c).evaluate()
f.change_layout(layout)
g.change_layout(layout)
h = (f + g).evaluate()
assert np.allclose(h['g'], f['g'] + g['g'])
def test_laplacian_radial_vector(Nr, dealias, basis, dtype):
Nphi = 1
c, d, b, phi, r, x, y = basis(Nphi, Nr, dealias, dtype=dtype)
u = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
u.preset_scales(b.domain.dealias)
u['g'][1] = 4 * r**3
v = operators.Laplacian(u, c).evaluate()
vg = 0 * v['g']
vg[1] = 32 * r
assert np.allclose(v['g'], vg)
def test_laplacian_vector(Nphi, Nr, dealias, basis, dtype):
c, d, b, phi, r, x, y = basis(Nphi, Nr, dealias, dtype)
v = field.Field(dist=d, tensorsig=(c, ), bases=(b, ), dtype=dtype)
v.preset_scales(b.domain.dealias)
ex = np.array([-np.sin(phi) + 0. * r, np.cos(phi) + 0. * r])
ey = np.array([np.cos(phi) + 0. * r, np.sin(phi) + 0. * r])
v['g'] = 4 * x**3 * ey + 3 * y**2 * ey
U = operators.Laplacian(v, c).evaluate()
Ug = (24 * x + 6) * ey
assert np.allclose(U['g'], Ug)
def test_convert_vector(Nphi, Nr, k, dealias, basis, dtype, layout):
c, d, b, phi, r, x, y = basis(Nphi, Nr, k, dealias, dtype)
u = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
u.preset_scales(b.dealias)
ex = np.array([-np.sin(phi) + 0. * r, np.cos(phi) + 0. * r])
ey = np.array([np.cos(phi) + 0. * r, np.sin(phi) + 0. * r])
u['g'] = 4 * x**3 * ey + 3 * y**2 * ey
v = operators.Laplacian(u, c).evaluate()
u.change_layout(layout)
v.change_layout(layout)
w = (u + v).evaluate()
assert np.allclose(w['g'], u['g'] + v['g'])
def test_laplacian_scalar_explicit(Nphi, Ntheta, dealias, dtype):
c, d, b, phi, theta = build_sphere(Nphi, Ntheta, dealias, dtype)
# Spherical harmonic input
m, l = 6, 10
f = d.Field(bases=b)
f.preset_scales(b.domain.dealias)
if np.iscomplexobj(dtype()):
f['g'] = sph_harm(m, l, phi, theta)
else:
f['g'] = sph_harm(m, l, phi, theta).real
# Evaluate Laplacian
u = operators.Laplacian(f).evaluate()
assert np.allclose(u['g'], -f['g'] * (l * (l + 1)) / radius**2)
def test_laplacian_vector(Nphi, Ntheta, Nr, dealias, basis, dtype):
c, d, b, phi, theta, r, x, y, z = basis(Nphi, Ntheta, Nr, dealias, dtype)
u = field.Field(dist=d, bases=(b,), tensorsig=(c,), dtype=dtype)
u.preset_scales(b.domain.dealias)
ct, st, cp, sp = np.cos(theta), np.sin(theta), np.cos(phi), np.sin(phi)
u['g'][2] = r**2*st*(2*ct**2*cp-r*ct**3*sp+r**3*cp**3*st**5*sp**3+r*ct*st**2*(cp**3+sp**3))
u['g'][1] = r**2*(2*ct**3*cp-r*cp**3*st**4+r**3*ct*cp**3*st**5*sp**3-1/16*r*np.sin(2*theta)**2*(-7*sp+np.sin(3*phi)))
u['g'][0] = r**2*sp*(-2*ct**2+r*ct*cp*st**2*sp-r**3*cp**2*st**5*sp**3)
v = operators.Laplacian(u, c).evaluate()
vg = 0 * v['g']
vg[2] = 2*(2+3*r*ct)*cp*st+1/2*r**3*st**4*(4*np.sin(2*phi)+np.sin(4*phi))
vg[1] = 2*r*(-3*cp*st**2+sp)+1/2*ct*(8*cp+r**3*st**3*(4*np.sin(2*phi)+np.sin(4*phi)))
vg[0] = 2*r*ct*cp+2*sp*(-2-r**3*(2+np.cos(2*phi))*st**3*sp)
assert np.allclose(v['g'], vg)
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_heat_ncc_shell(Nmax, Lmax, ncc_exponent, ncc_location, ncc_scale,
dtype):
# Bases
dealias = 1
c, d, b, phi, theta, r, x, y, z = build_shell(2 * (Lmax + 1),
Lmax + 1,
Nmax + 1,
dealias=dealias,
dtype=dtype)
r0, r1 = b.radial_basis.radii
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu1 = field.Field(name='τu1', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
τu2 = field.Field(name='τu2', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
ncc = field.Field(name='ncc',
dist=d,
bases=(b.radial_basis, ),
dtype=dtype)
F = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
# Test Parameters
F_value = {0: 6, 1: 2, 2: 1, 3 / 2: 3 / 4}
analytic = {
0: r**2 + 6 / r - 7,
1: r + 2 / r - 3,
2: np.log(r) + np.log(4) / r - np.log(4),
3 / 2: np.sqrt(r) + 0.82842712 / r - 1.82842712
}
if ncc_location == 'RHS':
ncc['g'] = 1
F['g'] = F_value[ncc_exponent] * r**(-ncc_exponent)
else:
ncc['g'] = r**ncc_exponent
F['g'] = F_value[ncc_exponent]
u_true = analytic[ncc_exponent]
ncc.change_scales(ncc_scale)
ncc['g'] # force transform
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A, n: operators.Lift(A, b, n)
problem = problems.LBVP([u, τu1, τu2])
problem.add_equation((ncc * Lap(u) + Lift(τu1, -1) + Lift(τu2, -2), F))
problem.add_equation((u(r=r0), 0))
problem.add_equation((u(r=r1), 0))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
assert np.allclose(u['g'], u_true)
def test_convert_vector(Nphi, Ntheta, Nr, k, dealias, basis, dtype, layout):
c, d, b, phi, theta, r, x, y, z = basis(Nphi, Ntheta, Nr, k, dealias,
dtype)
ct, st, cp, sp = np.cos(theta), np.sin(theta), np.cos(phi), np.sin(phi)
u = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
u.preset_scales(b.domain.dealias)
u['g'][2] = r**2 * st * (2 * ct**2 * cp - r * ct**3 * sp +
r**3 * cp**3 * st**5 * sp**3 + r * ct * st**2 *
(cp**3 + sp**3))
u['g'][1] = r**2 * (2 * ct**3 * cp - r * cp**3 * st**4 +
r**3 * ct * cp**3 * st**5 * sp**3 -
1 / 16 * r * np.sin(2 * theta)**2 *
(-7 * sp + np.sin(3 * phi)))
u['g'][0] = r**2 * sp * (-2 * ct**2 + r * ct * cp * st**2 * sp -
r**3 * cp**2 * st**5 * sp**3)
v = operators.Laplacian(u, c).evaluate()
u.change_layout(layout)
v.change_layout(layout)
w = (u + v).evaluate()
assert np.allclose(w['g'], u['g'] + v['g'])
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)
def test_scalar_heat_disk_axisymm(Nr, Nphi, dtype):
# Bases
dealias = 1
c, d, b, phi, r, x, y = build_disk(Nphi, Nr, dealias=dealias, dtype=dtype)
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu = field.Field(name='u', dist=d, bases=(b.S1_basis(), ), dtype=dtype)
F = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
F['g'] = 4
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((Lap(u) + Lift(τu), F))
problem.add_equation((u(r=radius_disk), 0))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
u_true = r**2 - 1
assert np.allclose(u['g'], u_true)
def test_heat_ncc_cos_ball(Nmax, Lmax, ncc_scale, dtype):
# Bases
dealias = 1
c, d, b, phi, theta, r, x, y, z = build_ball(2 * (Lmax + 1),
Lmax + 1,
Nmax + 1,
dealias=dealias,
dtype=dtype)
# Fields
u = field.Field(name='u', dist=d, bases=(b, ), dtype=dtype)
τu = field.Field(name='u', dist=d, bases=(b.S2_basis(), ), dtype=dtype)
ncc = field.Field(name='ncc',
dist=d,
bases=(b.radial_basis, ),
dtype=dtype)
F = field.Field(name='a', dist=d, bases=(b, ), dtype=dtype)
# Test Parameters
R = radius_ball
u_true = np.cos(np.pi / 2 * (r / R)**2)
g = -np.pi / 2 / R**2 * (4 * np.pi / 2 * (r / R)**2 * np.cos(np.pi / 2 *
(r / R)**2) +
6 * np.sin(np.pi / 2 * (r / R)**2))
ncc['g'] = u_true
F['g'] = u_true * g
ncc.change_scales(ncc_scale)
ncc['g'] # force transform
# Problem
Lap = lambda A: operators.Laplacian(A, c)
Lift = lambda A: operators.Lift(A, b, -1)
problem = problems.LBVP([u, τu])
problem.add_equation((ncc * Lap(u) + Lift(τu), F))
problem.add_equation((u(r=radius_ball), 0))
# Solver
solver = solvers.LinearBoundaryValueSolver(problem)
solver.solve()
# Check solution
assert np.allclose(u['g'], u_true)
def __init__(self, Nx, Ny, Lx, Ly, dtype, **kw):
self.Nx = Nx
self.Ny = Ny
self.Lx = Lx
self.Ly = Ly
self.dtype = dtype
self.N = Nx * Ny
self.R = 2 * Nx + 2 * Ny
self.M = self.N + self.R
# Bases
self.c = c = coords.CartesianCoordinates('x', 'y')
self.d = d = distributor.Distributor((c, ))
self.xb = xb = basis.ChebyshevT(c.coords[0], Nx, bounds=(0, Lx))
self.yb = yb = basis.ChebyshevT(c.coords[1], Ny, bounds=(0, Ly))
self.x = x = xb.local_grid(1)
self.y = y = yb.local_grid(1)
xb2 = xb._new_a_b(1.5, 1.5)
yb2 = yb._new_a_b(1.5, 1.5)
# Forcing
self.f = f = field.Field(name='f',
dist=d,
bases=(xb2, yb2),
dtype=dtype)
# Boundary conditions
self.uL = uL = field.Field(name='f', dist=d, bases=(yb, ), dtype=dtype)
self.uR = uR = field.Field(name='f', dist=d, bases=(yb, ), dtype=dtype)
self.uT = uT = field.Field(name='f', dist=d, bases=(xb, ), dtype=dtype)
self.uB = uB = field.Field(name='f', dist=d, bases=(xb, ), dtype=dtype)
# Fields
self.u = u = field.Field(name='u', dist=d, bases=(xb, yb), dtype=dtype)
self.tx1 = tx1 = field.Field(name='tx1',
dist=d,
bases=(xb2, ),
dtype=dtype)
self.tx2 = tx2 = field.Field(name='tx2',
dist=d,
bases=(xb2, ),
dtype=dtype)
self.ty1 = ty1 = field.Field(name='ty1',
dist=d,
bases=(yb2, ),
dtype=dtype)
self.ty2 = ty2 = field.Field(name='ty2',
dist=d,
bases=(yb2, ),
dtype=dtype)
# Problem
Lap = lambda A: operators.Laplacian(A, c)
self.problem = problem = problems.LBVP([u, tx1, tx2, ty1, ty2])
problem.add_equation((Lap(u), f))
problem.add_equation((u(y=Ly), uT))
problem.add_equation((u(x=Lx), uR))
problem.add_equation((u(y=0), uB))
problem.add_equation((u(x=0), uL))
# Solver
self.solver = solver = solvers.LinearBoundaryValueSolver(
problem,
bc_top=False,
tau_left=False,
store_expanded_matrices=False,
**kw)
# Tau entries
L = solver.subproblems[0].L_min.tolil()
# Taus
for nx in range(Nx):
L[Ny - 1 + nx * Ny, Nx * Ny + 0 * Nx + nx] = 1 # tx1 * Py1
L[Ny - 2 + nx * Ny, Nx * Ny + 1 * Nx + nx] = 1 # tx2 * Py2
for ny in range(Ny - 2):
L[(Nx - 1) * Ny + ny,
Nx * Ny + 2 * Nx + 0 * Ny + ny] = 1 # ty1 * Px1
L[(Nx - 2) * Ny + ny,
Nx * Ny + 2 * Nx + 1 * Ny + ny] = 1 # ty2 * Px2
# BC taus not resolution safe
if Nx != Ny:
raise ValueError("Current implementation requires Nx == Ny.")
else:
# Remember L is left preconditoined but not right preconditioned
L[-7, Nx * Ny + 2 * Nx + 0 * Ny + Ny - 2] = 1 # Right -2
L[-5, Nx * Ny + 2 * Nx + 0 * Ny + Ny - 1] = 1 # Right -1
L[-3, Nx * Ny + 2 * Nx + 1 * Ny + Ny - 2] = 1 # Left -2
L[-1, Nx * Ny + 2 * Nx + 1 * Ny + Ny - 1] = 1 # Left -1
solver.subproblems[0].L_min = L.tocsr()
# Neumann operators
ux = operators.Differentiate(u, c.coords[0])
uy = operators.Differentiate(u, c.coords[1])
self.duL = -ux(x='left')
self.duR = ux(x='right')
self.duT = uy(y='right')
self.duB = -uy(y='left')
# Neumann matrix
duT_mat = self.duT.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duR_mat = self.duR.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duB_mat = self.duB.expression_matrices(solver.subproblems[0],
vars=[u])[u]
duL_mat = self.duL.expression_matrices(solver.subproblems[0],
vars=[u])[u]
self.interior_to_neumann_matrix = sp.vstack(
[duT_mat, duR_mat, duB_mat, duL_mat], format='csr')
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])
results.append(result)
print(len(results), ':', result, '(gradient of a scalar)')
# Gradient of a vector
T = operators.Gradient(u, c)
T = T.evaluate()
T.name = 'T'
Tg = np.array([[-np.sin(x) * y**5, np.cos(x) * 5 * y**4],
[np.cos(x) * 5 * y**4,
np.sin(x) * 20 * y**3]])
result = np.allclose(T['g'], Tg)
results.append(result)
print(len(results), ':', result, '(gradient of a vector)')
# Divergence of a vector
h = operators.Divergence(u)
h = h.evaluate()
h.name = 'h'
hg = -np.sin(x) * y**5 + np.sin(x) * 20 * y**3
result = np.allclose(h['g'], hg)
results.append(result)
print(len(results), ':', result, '(divergence of a vector)')
# Laplacian of a scalar
k = operators.Laplacian(f, c)
k = k.evaluate()
k.name = 'k'
result = np.allclose(k['g'], hg)
results.append(result)
print(len(results), ':', result, '(laplacian of a scalar)')
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)
# Parameters and operators
ez = field.Field(dist=d, bases=(b, ), tensorsig=(c, ), dtype=dtype)
ez['g'][1] = -np.sin(theta)
ez['g'][2] = np.cos(theta)
div = lambda A: operators.Divergence(A, index=0)
lap = lambda A: operators.Laplacian(A, c)
grad = lambda A: operators.Gradient(A, c)
dot = lambda A, B: arithmetic.DotProduct(A, B)
cross = lambda A, B: arithmetic.CrossProduct(A, B)
ddt = lambda A: operators.TimeDerivative(A)
LiftTau = lambda A: operators.LiftTau(A, b, -1)
# Problem
def eq_eval(eq_str):
return [eval(expr) for expr in split_equation(eq_str)]
problem = problems.IVP([p, u, tau])
# Equations for ell != 0
problem.add_equation(eq_eval("div(u) = 0"), condition="ntheta != 0")
