def MeshDefinition(Dimensions, NumberElements, Type='Lagrange', PolynomDegree=1):
# Mesh
Mesh = fe.BoxMesh(fe.Point(-Dimensions[0]/2, -Dimensions[1]/2, -Dimensions[2]/2), fe.Point(Dimensions[0]/2, Dimensions[1]/2, Dimensions[2]/2), NumberElements, NumberElements, NumberElements)
# Functions spaces
V_ele = fe.VectorElement(Type, Mesh.ufl_cell(), PolynomDegree)
V = fe.VectorFunctionSpace(Mesh, Type, PolynomDegree)
# Finite element functions
du = fe.TrialFunction(V)
v = fe.TestFunction(V)
u = fe.Function(V)
return [Mesh, V, u, du, v]
def setup_coarse_mesh(self):
""" This creates the rectangular mesh, or rectangular prism in 3D. """
if len(self.mesh_size) == 2:
self.mesh = fenics.RectangleMesh(
fenics.mpi_comm_world(), fenics.Point(self.xmin, self.ymin),
fenics.Point(self.xmax, self.ymax), self.mesh_size[0],
self.mesh_size[1], "crossed")
elif len(self.mesh_size) == 3:
self.mesh = fenics.BoxMesh(
fenics.mpi_comm_world(),
fenics.Point(self.xmin, self.ymin, self.zmin),
fenics.Point(self.xmax, self.ymax, self.zmax),
self.mesh_size[0], self.mesh_size[1], self.mesh_size[2])
def get_mesh(length, nx, ny, nz=None):
"""获得 mesh
"""
if nz is None:
mesh = fs.RectangleMesh(fs.Point(0.0, 0.0), fs.Point(length, length),
nx, ny)
else:
mesh = fs.BoxMesh(
fs.Point(0.0, 0.0, 0.0),
fs.Point(length, length, length),
nx,
ny,
nz,
)
return mesh
def test_minexample():
import scipy.integrate
import fenics as fe
import numpy as np
import magnics
# Material parameters
Ms = 8.6e5 # saturation magnetisation (A/m)
alpha = 0.1 # Gilbert damping
gamma = 2.211e5 # gyromagnetic ratio
A = 1e-11 # exchange constant (J/m)
#A = 0
D = 0*1.58e-3 # DMI constant (FeGe: D = 1.58e-3 J/m²)
K = 0*5e3 # anisotropy constant (Co: K = 4.5e5 J/m³)
# External magnetic field.
B = 0.1 # (T)
mu0 = 4 * np.pi * 1e-7 # vacuum permeability
# Zeeman field
H = Ms / 2 * fe.Constant((0,0,1))
# easy axis
ea = fe.Constant((0,1,1))
# mesh parameters
d = 100e-9
thickness = 100e-9
nx = ny = 20
nz = 10
# create mesh
p1 = fe.Point(0, 0, 0)
p2 = fe.Point(d, d, thickness)
mesh = fe.BoxMesh(p1, p2, nx, ny, nz)
m, Heff, u, v, V = magnics.vectorspace(mesh)
def effective_field(m, volume=None):
w_Zeeman = - mu0 * Ms * fe.dot(m, H)
w_exchange = A * fe.inner(fe.grad(m), fe.grad(m))
w_DMI = D * fe.inner(m, fe.curl(m))
w_ani = - K * fe.inner(m, ea)**2
w = w_Zeeman + w_exchange + w_DMI + w_ani
return -1/(mu0*Ms) * fe.derivative(w*fe.dx, m)
# Effective field
Heff_form = effective_field(m)
# Preassemble projection Matrix
Amat = fe.assemble(fe.dot(u, v)*fe.dx)
LU = fe.LUSolver()
LU.set_operator(Amat)
def compute_dmdt(m):
"""Convenience function that does all in one go"""
# Assemble RHS
b = fe.assemble(Heff_form)
# Project onto Heff
LU.solve(Heff.vector(), b)
LLG = -gamma/(1+alpha*alpha)*fe.cross(m, Heff) - alpha*gamma/(1+alpha*alpha)*fe.cross(m, fe.cross(m, Heff))
result = fe.assemble(fe.dot(LLG, v)*fe.dP)
return result.array()
# function for integration of system of ODEs
def rhs_micromagnetic(m_vector_array, t, counter=[0]):
assert isinstance(m_vector_array, np.ndarray)
m.vector()[:] = m_vector_array[:]
dmdt = compute_dmdt(m)
return dmdt
m_init = fe.Constant((1, 0, 0))
m = fe.interpolate(m_init, V)
ts = np.linspace(0, 1e-11, 100)
# empty call of time integrator, just to get FEniCS to cache all forms etc
rhs_micromagnetic(m.vector().array(), 0)
ms = scipy.integrate.odeint(rhs_micromagnetic, y0=m.vector().array(), t=ts, rtol=1e-10, atol=1e-10)
def macrospin_analytic_solution(alpha, gamma, H, t_array):
"""
Computes the analytic solution of magnetisation x component
as a function of time for the macrospin in applied external
magnetic field H.
Source: PhD Thesis Matteo Franchin,
http://eprints.soton.ac.uk/161207/1.hasCoversheetVersion/thesis.pdf,
Appendix B, page 127
"""
t0 = 1 / (gamma * alpha * H) * np.log(np.sin(np.pi / 2) / \
(1 + np.cos(np.pi / 2)))
mx_analytic = []
for t in t_array:
phi = gamma * H * t # (B16)
costheta = np.tanh(gamma * alpha * H * (t - t0)) # (B17)
sintheta = 1 / np.cosh(gamma * alpha * H * (t - t0)) # (B18)
mx_analytic.append(sintheta * np.cos(phi))
return np.array(mx_analytic)
mx_analytic = macrospin_analytic_solution(alpha, gamma, Ms/2, ts)
tmp2 = ms[:,0:1] # might be m_x, m_y, m_z of first vector
difference = tmp2[:,0] - mx_analytic
print("max deviation: {}".format(max(abs(difference))))
# add quick test
eps = 0.01
#ref = 0.0655908475021 # for tmax = 1e-10
ref = 0.0422068879247 # for tmax = 5e-11
ref = 0.00765261606231 # for tmax = 1e-11
assert ref - eps < max(abs(difference)) < ref + eps
def regular_box_mesh(n=10,
S_x=0.0,
S_y=0.0,
S_z=None,
E_x=1.0,
E_y=1.0,
E_z=None):
r"""Creates a mesh corresponding to a rectangle or cube.
This function creates a uniform mesh of either a rectangle
or a cube, with specified start (``S_``) and end points (``E_``).
The resulting mesh uses ``n`` elements along the shortest direction
and accordingly many along the longer ones. The resulting domain is
.. math::
\begin{alignedat}{2}
&[S_x, E_x] \times [S_y, E_y] \quad &&\text{ in } 2D, \\
&[S_x, E_x] \times [S_y, E_y] \times [S_z, E_z] \quad &&\text{ in } 3D.
\end{alignedat}
The boundary markers are ordered as follows:
- 1 corresponds to :math:`x=S_x`.
- 2 corresponds to :math:`x=E_x`.
- 3 corresponds to :math:`y=S_y`.
- 4 corresponds to :math:`y=E_y`.
- 5 corresponds to :math:`z=S_z` (only in 3D).
- 6 corresponds to :math:`z=E_z` (only in 3D).
Parameters
----------
n : int
Number of elements in the shortest coordinate direction.
S_x : float
Start of the x-interval.
S_y : float
Start of the y-interval.
S_z : float or None, optional
Start of the z-interval, mesh is 2D if this is ``None``
(default is ``None``).
E_x : float
End of the x-interval.
E_y : float
End of the y-interval.
E_z : float or None, optional
End of the z-interval, mesh is 2D if this is ``None``
(default is ``None``).
Returns
-------
mesh : dolfin.cpp.mesh.Mesh
The computational mesh.
subdomains : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the subdomains.
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the boundaries.
dx : ufl.measure.Measure
The volume measure of the mesh corresponding to subdomains.
ds : ufl.measure.Measure
The surface measure of the mesh corresponding to boundaries.
dS : ufl.measure.Measure
The interior facet measure of the mesh corresponding to boundaries.
"""
n = int(n)
if not n > 0:
raise InputError('cashocs.geometry.regular_box_mesh', 'n',
'This needs to be positive.')
if not S_x < E_x:
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_x',
'Incorrect input for the x-coordinate. S_x has to be smaller than E_x.'
)
if not S_y < E_y:
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_y',
'Incorrect input for the y-coordinate. S_y has to be smaller than E_y.'
)
if not ((S_z is None and E_z is None) or (S_z < E_z)):
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_z',
'Incorrect input for the z-coordinate. S_z has to be smaller than E_z, or only one of them is specified.'
)
if S_z is None:
lx = E_x - S_x
ly = E_y - S_y
sizes = [lx, ly]
dim = 2
else:
lx = E_x - S_x
ly = E_y - S_y
lz = E_z - S_z
sizes = [lx, ly, lz]
dim = 3
size_min = np.min(sizes)
num_points = [int(np.round(length / size_min * n)) for length in sizes]
if S_z is None:
mesh = fenics.RectangleMesh(fenics.Point(S_x, S_y),
fenics.Point(E_x, E_y), num_points[0],
num_points[1])
else:
mesh = fenics.BoxMesh(fenics.Point(S_x, S_y, S_z),
fenics.Point(E_x, E_y, E_z), num_points[0],
num_points[1], num_points[2])
subdomains = fenics.MeshFunction('size_t', mesh, dim=dim)
boundaries = fenics.MeshFunction('size_t', mesh, dim=dim - 1)
x_min = fenics.CompiledSubDomain('on_boundary && near(x[0], sx, tol)',
tol=fenics.DOLFIN_EPS,
sx=S_x)
x_max = fenics.CompiledSubDomain('on_boundary && near(x[0], ex, tol)',
tol=fenics.DOLFIN_EPS,
ex=E_x)
x_min.mark(boundaries, 1)
x_max.mark(boundaries, 2)
y_min = fenics.CompiledSubDomain('on_boundary && near(x[1], sy, tol)',
tol=fenics.DOLFIN_EPS,
sy=S_y)
y_max = fenics.CompiledSubDomain('on_boundary && near(x[1], ey, tol)',
tol=fenics.DOLFIN_EPS,
ey=E_y)
y_min.mark(boundaries, 3)
y_max.mark(boundaries, 4)
if S_z is not None:
z_min = fenics.CompiledSubDomain('on_boundary && near(x[2], sz, tol)',
tol=fenics.DOLFIN_EPS,
sz=S_z)
z_max = fenics.CompiledSubDomain('on_boundary && near(x[2], ez, tol)',
tol=fenics.DOLFIN_EPS,
ez=E_z)
z_min.mark(boundaries, 5)
z_max.mark(boundaries, 6)
dx = fenics.Measure('dx', mesh, subdomain_data=subdomains)
ds = fenics.Measure('ds', mesh, subdomain_data=boundaries)
dS = fenics.Measure('dS', mesh)
return mesh, subdomains, boundaries, dx, ds, dS
def regular_mesh(n=10, L_x=1.0, L_y=1.0, L_z=None):
r"""Creates a mesh corresponding to a rectangle or cube.
This function creates a uniform mesh of either a rectangle
or a cube, starting at the origin and having length specified
in ``L_x``, ``L_y``, and ``L_z``. The resulting mesh uses ``n`` elements along the
shortest direction and accordingly many along the longer ones.
The resulting domain is
.. math::
\begin{alignedat}{2}
&[0, L_x] \times [0, L_y] \quad &&\text{ in } 2D, \\
&[0, L_x] \times [0, L_y] \times [0, L_z] \quad &&\text{ in } 3D.
\end{alignedat}
The boundary markers are ordered as follows:
- 1 corresponds to :math:`x=0`.
- 2 corresponds to :math:`x=L_x`.
- 3 corresponds to :math:`y=0`.
- 4 corresponds to :math:`y=L_y`.
- 5 corresponds to :math:`z=0` (only in 3D).
- 6 corresponds to :math:`z=L_z` (only in 3D).
Parameters
----------
n : int
Number of elements in the shortest coordinate direction.
L_x : float
Length in x-direction.
L_y : float
Length in y-direction.
L_z : float or None, optional
Length in z-direction, if this is ``None``, then the geometry
will be two-dimensional (default is ``None``).
Returns
-------
mesh : dolfin.cpp.mesh.Mesh
The computational mesh.
subdomains : dolfin.cpp.mesh.MeshFunctionSizet
A :py:class:`fenics.MeshFunction` object containing the subdomains.
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the boundaries.
dx : ufl.measure.Measure
The volume measure of the mesh corresponding to subdomains.
ds : ufl.measure.Measure
The surface measure of the mesh corresponding to boundaries.
dS : ufl.measure.Measure
The interior facet measure of the mesh corresponding to boundaries.
"""
if not n > 0:
raise InputError('cashocs.geometry.regular_mesh', 'n',
'This needs to be positive.')
if not L_x > 0.0:
raise InputError('cashocs.geometry.regular_mesh', 'L_x',
'L_x needs to be positive')
if not L_y > 0.0:
raise InputError('cashocs.geometry.regular_mesh', 'L_y',
'L_y needs to be positive')
if not (L_z is None or L_z > 0.0):
raise InputError('cashocs.geometry.regular_mesh', 'L_z',
'L_z needs to be positive or None (for 2D mesh)')
n = int(n)
if L_z is None:
sizes = [L_x, L_y]
dim = 2
else:
sizes = [L_x, L_y, L_z]
dim = 3
size_min = np.min(sizes)
num_points = [int(np.round(length / size_min * n)) for length in sizes]
if L_z is None:
mesh = fenics.RectangleMesh(fenics.Point(0, 0), fenics.Point(sizes),
num_points[0], num_points[1])
else:
mesh = fenics.BoxMesh(fenics.Point(0, 0, 0), fenics.Point(sizes),
num_points[0], num_points[1], num_points[2])
subdomains = fenics.MeshFunction('size_t', mesh, dim=dim)
boundaries = fenics.MeshFunction('size_t', mesh, dim=dim - 1)
x_min = fenics.CompiledSubDomain('on_boundary && near(x[0], 0, tol)',
tol=fenics.DOLFIN_EPS)
x_max = fenics.CompiledSubDomain('on_boundary && near(x[0], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[0])
x_min.mark(boundaries, 1)
x_max.mark(boundaries, 2)
y_min = fenics.CompiledSubDomain('on_boundary && near(x[1], 0, tol)',
tol=fenics.DOLFIN_EPS)
y_max = fenics.CompiledSubDomain('on_boundary && near(x[1], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[1])
y_min.mark(boundaries, 3)
y_max.mark(boundaries, 4)
if L_z is not None:
z_min = fenics.CompiledSubDomain('on_boundary && near(x[2], 0, tol)',
tol=fenics.DOLFIN_EPS)
z_max = fenics.CompiledSubDomain(
'on_boundary && near(x[2], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[2])
z_min.mark(boundaries, 5)
z_max.mark(boundaries, 6)
dx = fenics.Measure('dx', mesh, subdomain_data=subdomains)
ds = fenics.Measure('ds', mesh, subdomain_data=boundaries)
dS = fenics.Measure('dS', mesh)
return mesh, subdomains, boundaries, dx, ds, dS
def box(self, p1, p2, nx, ny, nz):
return FEN.BoxMesh(FEN.Point(p1), FEN.Point(p2), nx, ny, nz)
# import
import fenics as fn
# set parameters
fn.parameters["form_compiler"]["representation"] = "uflacs"
fn.parameters["form_compiler"]["cpp_optimize"] = True
# create mesh and function spaces
L = 50.0
mesh = fn.BoxMesh(
fn.Point(-L, 0.0, -L),
fn.Point(L, 2*L, L),
20, 5, 20)
nn = fn.FacetNormal(mesh)
Vh = fn.VectorElement("CG", mesh.ufl_cell(), 2)
Zh = fn.FiniteElement("CG", mesh.ufl_cell(), 1)
Qh = fn.FiniteElement("CG", mesh.ufl_cell(), 2)
Hh = fn.FunctionSpace(mesh, fn.MixedElement([Vh, Zh, Qh]))
# variables to solve for, and test functions
u, phi, p = fn.TrialFunctions(Hh)
v, psi, q = fn.TestFunctions(Hh)
# file output
fileU = fn.File(mesh.mpi_comm(), "Output/2_5_2_Footing_3D/u.pvd")
filePHI = fn.File(mesh.mpi_comm(), "Output/2_5_2_Footing_3D/phi.pvd")
fileP = fn.File(mesh.mpi_comm(), "Output/2_5_2_Footing_3D/p.pvd")
# constants
import fenics as fs
import matplotlib.pyplot as plt
# scaled variables
L = 1
W = 0.2
mu = 1
rho = 1
delta = W / L
gamma = 0.4 * delta**2
beta = 1.25
lambda_ = beta
g = gamma
# Create mesh and define function space
mesh = fs.BoxMesh(fs.Point(0, 0, 0), fs.Point(L, W, W), 10, 3, 3)
V = fs.VectorFunctionSpace(mesh, 'P', 1)
# define boundary condition
tol = 1e-14
def clamped_boundary(x, on_boundary):
return on_boundary and x[0] < tol
bc = fs.DirichletBC(V, fs.Constant((0, 0, 0)), clamped_boundary)
# define strain and stress
def epsilon(u):
def melt_pcm_3d(m=(10, 10, 1),
dt=1.e-3,
initial_pci_refinement_cycles=2,
max_pci_refinement_cycles_per_time=6,
output_dir='output/melt_pcm_3d',
start_time=0.,
end_time=0.05,
nlp_divergence_threshold=1.e12,
nlp_max_iterations=50,
restart=False,
restart_filepath=''):
theta_hot = 1.
theta_cold = -0.1
w, mesh = phaseflow.run(
Ste=1.,
Ra=1.,
Pr=1.,
mu_s=1.e4,
mu_l=1.,
g=(0., -1., 0.),
mesh=fenics.BoxMesh(fenics.Point(0., 0., -0.2),
fenics.Point(1., 1., 0.2), m[0], m[1], m[2]),
time_step_bounds=dt,
start_time=start_time,
end_time=end_time,
stop_when_steady=True,
regularization={
'T_f': 0.1,
'r': 0.05
},
initial_pci_refinement_cycles=initial_pci_refinement_cycles,
max_pci_refinement_cycles_per_time=max_pci_refinement_cycles_per_time,
minimum_cell_diameter=0.01,
nlp_max_iterations=nlp_max_iterations,
nlp_divergence_threshold=nlp_divergence_threshold,
initial_values_expression=("0.", "0.", "0.", "0.", "(" +
str(theta_hot) + " - " + str(theta_cold) +
")*(x[0] < 0.001) + " + str(theta_cold)),
boundary_conditions=[{
'subspace': 0,
'value_expression': ("0.", "0.", "0."),
'degree': 3,
'location_expression':
"near(x[0], 0.) | near(x[0], 1.) | near(x[1], 0.) | near(x[1], 1.) | near(x[2], -0.2) | near(x[2], 0.2)",
'method': "topological"
}, {
'subspace': 2,
'value_expression': str(theta_hot),
'degree': 2,
'location_expression': "near(x[0], 0.)",
'method': "topological"
}, {
'subspace': 2,
'value_expression': str(theta_cold),
'degree': 2,
'location_expression': "near(x[0], 1.)",
'method': "topological"
}],
output_dir=output_dir,
debug=True,
restart=restart,
restart_filepath=restart_filepath)
return w, mesh
# Zeeman field
H = Ms / 2 * fe.Constant((0, 0, 1))
# easy axis
ea = fe.Constant((0, 1, 1))
# mesh parameters
d = 100e-9
thickness = 100e-9
nx = ny = 20
nz = 10
# create mesh
p1 = fe.Point(0, 0, 0)
p2 = fe.Point(d, d, thickness)
mesh = fe.BoxMesh(p1, p2, nx, ny, nz)
m, Heff, u, v, V = setup_vectorspace(mesh)
def effective_field(m, volume=None):
w_Zeeman = -mu0 * Ms * fe.dot(m, H)
w_exchange = A * fe.inner(fe.grad(m), fe.grad(m))
w_DMI = D * fe.inner(m, fe.curl(m))
w_ani = -K * fe.inner(m, ea)**2
w = w_Zeeman + w_exchange + w_DMI + w_ani
return -1 / (mu0 * Ms) * fe.derivative(w * fe.dx, m)
m_init = fe.Constant((1, 0, 0))
m = fe.interpolate(m_init, V)