def test_residual(self):
"""
This test loads a SurfaceXYZFourier that interpolates the xyz coordinates
of a surface in the NCSX configuration that was computed on a previous
branch of pyplasmaopt. Here, we verify that the Boozer residual at these
interpolation points is small.
"""
s = get_exact_surface()
coils, currents, ma = get_ncsx_data()
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
bs_tf = BiotSavart(stellarator.coils, stellarator.currents)
weight = 1.
tf = ToroidalFlux(s, bs_tf)
# these data are obtained from `boozer` branch of pyplamsaopt
tf_target = 0.41431152
iota = -0.44856192
boozer_surface = BoozerSurface(bs, s, tf, tf_target)
x = np.concatenate((s.get_dofs(), [iota]))
r0 = boozer_surface.boozer_penalty_constraints(
x,
derivatives=0,
constraint_weight=weight,
optimize_G=False,
scalarize=False)
# the residual should be close to zero for all entries apart from the y
# and z coordinate at phi=0 and theta=0 (and the corresponding rotations)
ignores_idxs = np.zeros_like(r0)
ignores_idxs[[1, 2, 693, 694, 695, 1386, 1387, 1388, -2, -1]] = 1
assert np.max(np.abs(r0[ignores_idxs < 0.5])) < 1e-8
assert np.max(np.abs(r0[-2:])) < 1e-6
def subtest_biotsavart_gradient_symmetric_and_divergence_free(self, idx):
coil = get_coil()
bs = BiotSavart([coil], [1e4])
points = np.asarray(17 * [[-1.41513202e-03, 8.99999382e-01, -3.14473221e-04]])
points += 0.001 * (np.random.rand(*points.shape)-0.5)
bs.set_points(points)
dB = bs.dB_by_dX()
assert abs(dB[idx][0, 0] + dB[idx][1, 1] + dB[idx][2, 2]) < 1e-14
assert np.allclose(dB[idx], dB[idx].T)
def subtest_boozer_penalty_constraints_gradient(self,
surfacetype,
stellsym,
optimize_G=False):
np.random.seed(1)
coils, currents, ma = get_ncsx_data()
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
bs_tf = BiotSavart(stellarator.coils, stellarator.currents)
s = get_surface(surfacetype, stellsym)
s.fit_to_curve(ma, 0.1)
weight = 11.1232
tf = ToroidalFlux(s, bs_tf)
tf_target = 0.1
boozer_surface = BoozerSurface(bs, s, tf, tf_target)
iota = -0.3
x = np.concatenate((s.get_dofs(), [iota]))
if optimize_G:
x = np.concatenate((x, [
2. * np.pi * np.sum(np.abs(bs.coil_currents)) *
(4 * np.pi * 10**(-7) / (2 * np.pi))
]))
f0, J0 = boozer_surface.boozer_penalty_constraints(
x, derivatives=1, constraint_weight=weight, optimize_G=optimize_G)
h = np.random.uniform(size=x.shape) - 0.5
Jex = J0 @ h
err_old = 1e9
epsilons = np.power(2., -np.asarray(range(7, 20)))
print(
"################################################################################"
)
for eps in epsilons:
f1 = boozer_surface.boozer_penalty_constraints(
x + eps * h,
derivatives=0,
constraint_weight=weight,
optimize_G=optimize_G)
Jfd = (f1 - f0) / eps
err = np.linalg.norm(Jfd - Jex) / np.linalg.norm(Jex)
print(err / err_old, f0, f1)
assert err < err_old * 0.55
err_old = err
print(
"################################################################################"
)
def subtest_boozer_constrained_jacobian(self,
surfacetype,
stellsym,
optimize_G=False):
np.random.seed(1)
coils, currents, ma = get_ncsx_data()
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
bs_tf = BiotSavart(stellarator.coils, stellarator.currents)
s = get_surface(surfacetype, stellsym)
s.fit_to_curve(ma, 0.1)
tf = ToroidalFlux(s, bs_tf)
tf_target = 0.1
boozer_surface = BoozerSurface(bs, s, tf, tf_target)
iota = -0.3
lm = [0., 0.]
x = np.concatenate((s.get_dofs(), [iota]))
if optimize_G:
x = np.concatenate((x, [
2. * np.pi * np.sum(np.abs(bs.coil_currents)) *
(4 * np.pi * 10**(-7) / (2 * np.pi))
]))
xl = np.concatenate((x, lm))
res0, dres0 = boozer_surface.boozer_exact_constraints(
xl, derivatives=1, optimize_G=optimize_G)
h = np.random.uniform(size=xl.shape) - 0.5
dres_exact = dres0 @ h
err_old = 1e9
epsilons = np.power(2., -np.asarray(range(7, 20)))
print(
"################################################################################"
)
for eps in epsilons:
res1 = boozer_surface.boozer_exact_constraints(
xl + eps * h, derivatives=0, optimize_G=optimize_G)
dres_fd = (res1 - res0) / eps
err = np.linalg.norm(dres_fd - dres_exact)
print(err / err_old)
assert err < err_old * 0.55
err_old = err
print(
"################################################################################"
)
def test_interpolated_field_convergence_rate(self):
R0test = 1.5
B0test = 0.8
B0 = ToroidalField(R0test, B0test)
coils, currents, _ = get_ncsx_data(Nt_coils=5, Nt_ma=10, ppp=5)
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
old_err_1 = 1e6
old_err_2 = 1e6
btotal = bs + B0
for n in [4, 8, 16]:
rmin = 1.5
rmax = 1.7
rsteps = n
phimin = 0
phimax = 2 * np.pi
phisteps = n * 16
zmin = -0.1
zmax = 0.1
zsteps = n
bsh = InterpolatedField(btotal, 2, [rmin, rmax, rsteps],
[phimin, phimax, phisteps],
[zmin, zmax, zsteps], True)
err_1 = np.mean(bsh.estimate_error_B(1000))
err_2 = np.mean(bsh.estimate_error_GradAbsB(1000))
print(err_1, err_2)
assert err_1 < 0.6**3 * old_err_1
assert err_2 < 0.6**3 * old_err_2
old_err_1 = err_1
old_err_2 = err_2
def test_poincare_plot(self):
coils, currents, ma = get_ncsx_data(Nt_coils=15)
nfp = 3
stellarator = CoilCollection(coils, currents, nfp, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
n = 10
rrange = (1.0, 1.9, n)
phirange = (0, 2*np.pi/nfp, n*2)
zrange = (0, 0.4, n)
bsh = InterpolatedField(
bs, UniformInterpolationRule(2),
rrange, phirange, zrange, True, nfp=3, stellsym=True
)
nlines = 4
r0 = np.linalg.norm(ma.gamma()[0, :2])
z0 = ma.gamma()[0, 2]
R0 = [r0 + i*0.01 for i in range(nlines)]
Z0 = [z0 for i in range(nlines)]
nphis = 4
phis = np.linspace(0, 2*np.pi/nfp, nphis, endpoint=False)
res_tys, res_phi_hits = compute_fieldlines(
bsh, R0, Z0, tmax=1000, phis=phis, stopping_criteria=[])
try:
import matplotlib # noqa
plot_poincare_data(res_phi_hits, phis, '/tmp/fieldlines.png')
except ImportError:
pass
def subtest_boozer_penalty_constraints_hessian(self,
surfacetype,
stellsym,
optimize_G=False):
np.random.seed(1)
coils, currents, ma = get_ncsx_data()
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
bs_tf = BiotSavart(stellarator.coils, stellarator.currents)
s = get_surface(surfacetype, stellsym)
s.fit_to_curve(ma, 0.1)
tf = ToroidalFlux(s, bs_tf)
tf_target = 0.1
boozer_surface = BoozerSurface(bs, s, tf, tf_target)
iota = -0.3
x = np.concatenate((s.get_dofs(), [iota]))
if optimize_G:
x = np.concatenate((x, [
2. * np.pi * np.sum(np.abs(bs.coil_currents)) *
(4 * np.pi * 10**(-7) / (2 * np.pi))
]))
f0, J0, H0 = boozer_surface.boozer_penalty_constraints(
x, derivatives=2, optimize_G=optimize_G)
h1 = np.random.uniform(size=x.shape) - 0.5
h2 = np.random.uniform(size=x.shape) - 0.5
d2f = h1 @ H0 @ h2
err_old = 1e9
epsilons = np.power(2., -np.asarray(range(10, 20)))
print(
"################################################################################"
)
for eps in epsilons:
fp, Jp = boozer_surface.boozer_penalty_constraints(
x + eps * h1, derivatives=1, optimize_G=optimize_G)
d2f_fd = (Jp @ h2 - J0 @ h2) / eps
err = np.abs(d2f_fd - d2f) / np.abs(d2f)
print(err / err_old)
assert err < err_old * 0.55
err_old = err
def test_get_set_points_cyl_cart(self):
coils, currents, _ = get_ncsx_data(Nt_coils=5, Nt_ma=10, ppp=5)
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
points_xyz = np.asarray([[0.5, 0.6, 0.7]])
points_rphiz = np.zeros_like(points_xyz)
points_rphiz[:, 0] = np.linalg.norm(points_xyz[:, 0:2], axis=1)
points_rphiz[:, 1] = np.mod(
np.arctan2(points_xyz[:, 1], points_xyz[:, 0]), 2 * np.pi)
points_rphiz[:, 2] = points_xyz[:, 2]
bs.set_points_cyl(points_rphiz)
# import IPython; IPython.embed()
# import sys; sys.exit()
assert np.allclose(bs.get_points_cyl(), points_rphiz)
assert np.allclose(bs.get_points_cart(), points_xyz)
bs.set_points_cart(points_xyz)
assert np.allclose(bs.get_points_cyl(), points_rphiz)
assert np.allclose(bs.get_points_cart(), points_xyz)
def test_poincare_ncsx_known(self):
coils, currents, ma = get_ncsx_data(Nt_coils=25)
nfp = 3
stellarator = CoilCollection(coils, currents, nfp, True)
currents = [-c for c in currents]
bs = BiotSavart(stellarator.coils, stellarator.currents)
R0 = [np.linalg.norm(ma.gamma()[0, :2])]
Z0 = [ma.gamma()[0, 2]]
phis = np.arctan2(ma.gamma()[:, 1], ma.gamma()[:, 0])
res_tys, res_phi_hits = compute_fieldlines(
bs, R0, Z0, tmax=10, phis=phis, stopping_criteria=[])
for i in range(len(phis)-1):
assert np.linalg.norm(ma.gamma()[i+1, :] - res_phi_hits[0][i, 2:5]) < 1e-4
def subtest_biotsavart_dBdX_taylortest(self, idx):
coil = get_coil()
bs = BiotSavart([coil], [1e4])
points = np.asarray(17 * [[-1.41513202e-03, 8.99999382e-01, -3.14473221e-04]])
points += 0.001 * (np.random.rand(*points.shape)-0.5)
bs.set_points(points)
B0 = bs.B()[idx]
dB = bs.dB_by_dX()[idx]
for direction in [np.asarray((1., 0, 0)), np.asarray((0, 1., 0)), np.asarray((0, 0, 1.))]:
deriv = dB.T.dot(direction)
err = 1e6
for i in range(5, 10):
eps = 0.5**i
bs.set_points(points + eps * direction)
Beps = bs.B()[idx]
deriv_est = (Beps-B0)/(eps)
new_err = np.linalg.norm(deriv-deriv_est)
assert new_err < 0.55 * err
err = new_err
def test_interpolated_field_close_with_symmetries(self):
R0test = 1.5
B0test = 0.8
B0 = ToroidalField(R0test, B0test)
coils, currents, _ = get_ncsx_data(Nt_coils=10, Nt_ma=10, ppp=5)
nfp = 3
stellarator = CoilCollection(coils, currents, nfp, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
btotal = bs + B0
n = 12
rmin = 1.5
rmax = 1.7
rsteps = n
phimin = 0
phimax = 2 * np.pi / nfp
phisteps = n * 32 // nfp
zmin = 0.
zmax = 0.1
zsteps = n // 2
bsh = InterpolatedField(btotal,
4, [rmin, rmax, rsteps],
[phimin, phimax, phisteps],
[zmin, zmax, zsteps],
True,
nfp=nfp,
stellsym=True)
N = 1000
points = np.random.uniform(size=(N, 3))
points[:, 0] = points[:, 0] * (rmax - rmin) + rmin
points[:, 1] = points[:, 1] * (nfp * phimax - phimin) + phimin
points[:, 2] = points[:, 2] * (2 * zmax) - zmax
btotal.set_points_cyl(points)
dB = btotal.GradAbsB()
B = btotal.B()
dBc = btotal.GradAbsB_cyl()
Bc = btotal.B_cyl()
bsh.set_points_cyl(points)
Bh = bsh.B()
dBh = bsh.GradAbsB()
Bhc = bsh.B_cyl()
dBhc = bsh.GradAbsB_cyl()
assert np.allclose(B, Bh, rtol=1e-3)
assert np.allclose(dB, dBh, rtol=1e-3)
assert np.allclose(Bc, Bhc, rtol=1e-3)
assert np.allclose(dBc, dBhc, rtol=1e-3)
def test_helicalcoil_Bfield(self):
point = [[-1.41513202e-03, 8.99999382e-01, -3.14473221e-04]]
field = [[-0.00101961, 0.20767292, -0.00224908]]
derivative = [[[0.47545098, 0.01847397, 1.10223595],
[0.01847426, -2.66700072, 0.01849548],
[1.10237535, 0.01847085, 2.19154973]]]
coils = [CurveHelical(100, 2, 5, 2, 1., 0.3) for i in range(2)]
coils[0].set_dofs(np.concatenate(([0, 0], [0, 0])))
coils[1].set_dofs(np.concatenate(([np.pi / 2, 0], [0, 0])))
currents = [-3.07e5, 3.07e5]
Bhelical = BiotSavart(coils, currents)
Bhelical.set_points(point)
assert np.allclose(Bhelical.B(), field)
assert np.allclose(Bhelical.dB_by_dX(), derivative)
def test_interpolated_field_close_no_sym(self):
R0test = 1.5
B0test = 0.8
B0 = ToroidalField(R0test, B0test)
coils, currents, _ = get_ncsx_data(Nt_coils=5, Nt_ma=10, ppp=5)
stellarator = CoilCollection(coils, currents, 3, True)
bs = BiotSavart(stellarator.coils, stellarator.currents)
btotal = bs + B0
n = 8
rmin = 1.5
rmax = 1.7
rsteps = n
phimin = 0
phimax = 2 * np.pi
phisteps = n * 16
zmin = -0.1
zmax = 0.1
zsteps = n
bsh = InterpolatedField(btotal, 4, [rmin, rmax, rsteps],
[phimin, phimax, phisteps],
[zmin, zmax, zsteps], True)
N = 100
points = np.random.uniform(size=(N, 3))
points[:, 0] = points[:, 0] * (rmax - rmin) + rmin
points[:, 1] = points[:, 1] * (phimax - phimin) + phimin
points[:, 2] = points[:, 2] * (zmax - zmin) + zmin
btotal.set_points_cyl(points)
dB = btotal.GradAbsB()
B = btotal.B()
dBc = btotal.GradAbsB_cyl()
Bc = btotal.B_cyl()
bsh.set_points_cyl(points)
Bh = bsh.B()
dBh = bsh.GradAbsB()
Bhc = bsh.B_cyl()
dBhc = bsh.GradAbsB_cyl()
assert np.allclose(B, Bh, rtol=1e-2)
assert np.allclose(dB, dBh, rtol=1e-2)
assert np.allclose(Bc, Bhc, rtol=1e-2)
assert np.allclose(dBc, dBhc, rtol=1e-2)
def test_poincare_caryhanson(self):
# Test with a known magnetic field - optimized Cary&Hanson configuration
# with a magnetic axis at R=0.9413. Field created using the Biot-Savart
# solver given a set of two helical coils created using the CurveHelical
# class. The total magnetic field is a superposition of a helical and
# a toroidal magnetic field.
coils = [CurveHelical(200, 2, 5, 2, 1., 0.3) for i in range(2)]
coils[0].set_dofs(np.concatenate(([np.pi/2, 0.2841], [0, 0])))
coils[1].set_dofs(np.concatenate(([0, 0], [0, 0.2933])))
currents = [3.07e5, -3.07e5]
Btoroidal = ToroidalField(1.0, 1.0)
Bhelical = BiotSavart(coils, currents)
bs = Bhelical+Btoroidal
ma = CurveXYZFourier(300, 1)
magnetic_axis_radius = 0.9413
ma.set_dofs([0, 0, magnetic_axis_radius, 0, magnetic_axis_radius, 0, 0, 0, 0])
R0 = [np.linalg.norm(ma.gamma()[0, :2])]
Z0 = [ma.gamma()[0, 2]]
phis = np.arctan2(ma.gamma()[:, 1], ma.gamma()[:, 0])
res_tys, res_phi_hits = compute_fieldlines(
bs, R0, Z0, tmax=2, phis=phis, stopping_criteria=[])
for i in range(len(res_phi_hits[0])):
assert np.linalg.norm(ma.gamma()[i+1, :] - res_phi_hits[0][i, 2:5]) < 2e-3
def test_sum_Bfields(self):
pointVar = 1e-1
npoints = 20
points = np.asarray(
npoints * [[-1.41513202e-03, 8.99999382e-01, -3.14473221e-04]])
points += pointVar * (np.random.rand(*points.shape) - 0.5)
# Set up helical field
coils = [CurveHelical(101, 2, 5, 2, 1., 0.3) for i in range(2)]
coils[0].set_dofs(np.concatenate(([np.pi / 2, 0], [0, 0])))
coils[1].set_dofs(np.concatenate(([0, 0], [0, 0])))
currents = [-2.1e5, 2.1e5]
Bhelical = BiotSavart(coils, currents)
# Set up toroidal fields
Btoroidal1 = ToroidalField(1., 1.)
Btoroidal2 = ToroidalField(1.2, 0.1)
# Set up sum of the three in two different ways
Btotal1 = MagneticFieldSum([Bhelical, Btoroidal1, Btoroidal2])
Btotal2 = Bhelical + Btoroidal1 + Btoroidal2
Btotal3 = Btoroidal1 + Btoroidal2
# Evaluate at a given point
Bhelical.set_points(points)
Btoroidal1.set_points(points)
Btoroidal2.set_points(points)
Btotal1.set_points(points)
Btotal2.set_points(points)
Btotal3.set_points(points)
# Verify
assert np.allclose(Btotal1.B(), Btotal2.B())
assert np.allclose(Bhelical.B() + Btoroidal1.B() + Btoroidal2.B(),
Btotal1.B())
assert np.allclose(Btotal1.dB_by_dX(), Btotal2.dB_by_dX())
assert np.allclose(
Bhelical.dB_by_dX() + Btoroidal1.dB_by_dX() +
Btoroidal2.dB_by_dX(), Btotal1.dB_by_dX())
assert np.allclose(Btoroidal1.d2B_by_dXdX() + Btoroidal2.d2B_by_dXdX(),
Btotal3.d2B_by_dXdX())
assert np.allclose(Btoroidal1.A() + Btoroidal2.A(), Btotal3.A())
assert np.allclose(Btoroidal1.dA_by_dX() + Btoroidal2.dA_by_dX(),
Btotal3.dA_by_dX())
assert np.allclose(Btoroidal1.d2A_by_dXdX() + Btoroidal2.d2A_by_dXdX(),
Btotal3.d2A_by_dXdX())