def test_dmi_uses_unit_length_2dmesh():
"""
Set up a helical state in two meshes (one expressed in SI units
the other expressed in nanometers) and compute energies and fields.
"""
A = 8.78e-12 # J/m
D = 1.58e-3 # J/m^2
Ms = 3.84e5 # A/m
energies = []
# unit_lengths 1e-9 and 1 are common, let's throw in an intermediate length
# just to challenge the system a little:
for unit_length in (1, 1e-4, 1e-9):
radius = 200e-9 / unit_length
maxh = 5e-9 / unit_length
helical_period = (4 * pi * A / D) / unit_length
k = 2 * pi / helical_period
# HF 27 April 2014: The next command fails in dolfin 1.3
# mesh = df.CircleMesh(df.Point(0, 0), radius, maxh)
# The actual shape of the domain shouldn't matter for the test,
# so let's use a Rectangular mesh which should work the same:
nx = ny = int(round(radius / maxh))
mesh = df.RectangleMesh(df.Point(0, 0), df.Point(radius, radius), nx,
ny)
S3 = df.VectorFunctionSpace(mesh, "CG", 1, dim=3)
m_expr = df.Expression(("0", "cos(k * x[0])", "sin(k * x[0])"),
k=k,
degree=1)
m = Field(S3, m_expr, name='m')
dmi = DMI(D)
Ms_dg = Field(df.FunctionSpace(mesh, 'DG', 0), Ms)
dmi.setup(m, Ms_dg, unit_length=unit_length)
energies.append(dmi.compute_energy())
H = df.Function(S3)
H.vector()[:] = dmi.compute_field()
print H(0.0, 0.0)
print "Using unit_length = {}.".format(unit_length)
print "Helical period {}.".format(helical_period)
print "Energy {}.".format(dmi.compute_energy())
rel_diff_energies = abs(energies[0] - energies[1]) / abs(energies[1])
print "Relative difference of energy {}.".format(rel_diff_energies)
assert rel_diff_energies < 1e-13
rel_diff_energies2 = abs(energies[0] - energies[2]) / abs(energies[2])
print "Relative difference2 of energy {}.".format(rel_diff_energies2)
assert rel_diff_energies2 < 1e-13
def test_dmi_pbc2d():
mesh = df.BoxMesh(df.Point(0, 0, 0), df.Point(1, 1, 0.1), 2, 2, 1)
pbc = PeriodicBoundary2D(mesh)
S3 = df.VectorFunctionSpace(mesh, "Lagrange", 1, constrained_domain=pbc)
m_expr = df.Expression(("0", "0", "1"), degree=1)
m = Field(S3, m_expr, name='m')
dmi = DMI(1)
dmi.setup(m, Field(df.FunctionSpace(mesh, 'DG', 0), 1))
field = dmi.compute_field()
assert np.max(field) < 1e-15
def test_DMI_energy_density_3D():
"""Same as above, on a 3D mesh."""
mesh = df.UnitCubeMesh(4, 4, 4)
V = df.VectorFunctionSpace(mesh, "CG", 1, dim=3)
M = Field(V, df.Expression(("-0.5*x[1]", "0.5*x[0]", "1"), degree=1))
Ms = 10
D = 1
dmi = DMI(D)
dmi.setup(M, Field(df.FunctionSpace(mesh, 'DG', 0), Ms))
density = dmi.energy_density()
deviation = np.abs(density - 1.0)
print "3D energy density (expect array of 1):"
print density
print "Max deviation: %g" % np.max(deviation)
assert np.all(deviation < TOL), \
"Max deviation %g, should be zero." % np.max(deviation)
def test_DMI_energy_density_2D():
"""
For a vector field (x, y, z) = 0.5 * (-y, x, c),
the curl is exactly 1.0. (HF)
"""
mesh = df.UnitSquareMesh(4, 4)
V = df.VectorFunctionSpace(mesh, "CG", 1, dim=3)
M = Field(V, value=df.Expression(("-0.5*x[1]", "0.5*x[0]", "1"), degree=1))
Ms = 1
D = 1
dmi = DMI(D)
dmi.setup(M, Field(df.FunctionSpace(mesh, 'DG', 0), Ms))
density = dmi.energy_density()
deviation = np.abs(density - 1.0)
print "2D energy density (expect array of 1):"
print density
print "Max deviation: %g" % np.max(deviation)
assert np.all(deviation < TOL), \
"Max deviation %g, should be zero." % np.max(deviation)
def test_dmi_field():
"""
Simulation 1 is computing H_dmi=dE_dM via assemble.
Simulation 2 is computing H_dmi=g*M with a suitable pre-computed matrix g.
Simulation 3 is computing g using a petsc matrix.
We show that the three methods give equivalent results (this relies
on H_dmi being linear in M).
"""
m_initial = df.Expression(
('(2*x[0]-L)/L', 'sqrt(1 - ((2*x[0]-L)/L)*((2*x[0]-L)/L))', '0'),
L=length,
degree=1)
m = Field(V)
m.set(m_initial)
dmi1 = DMI(D=5e-3, method="box-assemble")
dmi1.setup(m, Field(df.FunctionSpace(mesh, 'DG', 0), 8.6e5))
dmi2 = DMI(D=5e-3, method="box-matrix-numpy")
dmi2.setup(m, Field(df.FunctionSpace(mesh, 'DG', 0), 8.6e5))
dmi3 = DMI(D=5e-3, method="box-matrix-petsc")
dmi3.setup(m, Field(df.FunctionSpace(mesh, 'DG', 0), 8.6e5))
H_dmi1 = dmi1.compute_field()
H_dmi2 = dmi2.compute_field()
H_dmi3 = dmi3.compute_field()
diff12 = np.max(np.abs(H_dmi1 - H_dmi2))
diff13 = np.max(np.abs(H_dmi1 - H_dmi3))
print "Difference between H_dmi1 and H_dmi2: max(abs(H_dmi1-H_dmi2))=%g" % diff12
print "Max value = %g, relative error = %g " % (max(H_dmi1),
diff12 / max(H_dmi1))
print "Difference between H_dmi1 and H_dmi3: max(abs(H_dmi1-H_dmi3))=%g" % diff13
print "Max value = %g, relative error = %g " % (max(H_dmi1),
diff13 / max(H_dmi1))
assert diff12 < 5e-8
assert diff13 < 5e-8
assert diff12 / max(H_dmi1) < 1e-14
assert diff13 / max(H_dmi1) < 1e-14