def updateCoefficients(self):
# Init coefficient matrix
x, y = SpatialCoordinate(self.mesh)
self.a = 0.5 * as_matrix(
[[cos(self.gamma[0]), sin(self.gamma[0])],
[- sin(self.gamma[0]), cos(self.gamma[0])]]) \
* as_matrix([[20, 1], [1, 0.1]]) \
* as_matrix(
[[cos(self.gamma[0]), - sin(self.gamma[0])],
[sin(self.gamma[0]), cos(self.gamma[0])]])
self.b = as_vector([Constant(0.0), Constant(1.0)])
self.c = Constant(-10.0)
self.u_ = (2*x-1.) \
* (exp(1 - abs(2*x-1.)) - 1) \
* (y + (1 - exp(y/self.delta))/(exp(1/self.delta)-1))
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_))) \
+ inner(self.b, grad(self.u_)) \
+ self.c * self.u_
# Set boundary conditions
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# r = .1*sqrt((x-np.pi)**2 + (y-np.pi)**2)
# self.u_ = pow(r,1.5) * sin(x) * sin(y)
# Set up explicit solution
self.u_ = sin(x) * sin(y)
# Init coefficient matrix
self.a1 = as_matrix([[2, .5], [
0.5, 1.5
]]) + conditional(x * y > 0, 1., -1.) * as_matrix([[1., .5], [.5, .5]])
self.a2 = as_matrix([[1.5, .5], [
0.5, 2.
]]) + conditional(x * y > 0, 1., -1.) * as_matrix([[.5, .5], [.5, 1.]])
self.a = self.gamma[0] * self.a1 + (1. - self.gamma[0]) * self.a2
# Init right-hand side
self.f1 = inner(self.a1, grad(grad(self.u_)))
self.f2 = inner(self.a2, grad(grad(self.u_)))
self.fcond = conditional(x * y > 0, 0., 1.)
# self.f = self.fcond * self.f1 + (1.-self.fcond) * self.f2
self.f = self.gamma[0] * self.f1 + (1. - self.gamma[0]) * self.f2
# conditional(x*x*x-y>0.0,2.0,1.0)]])
# self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = Constant(0.0)
def eval(self, values, x):
# Discretize control set
n_alpha = 200
alpha_space = np.linspace(self.amin, self.amax, n_alpha + 1)
alpha_space = alpha_space[:-1]
vxx = np.ndarray((1, ))
vxy = np.ndarray((1, ))
vyx = np.ndarray((1, ))
vyy = np.ndarray((1, ))
self.dxx.eval(vxx, x)
self.dxy.eval(vxy, x)
self.dyx.eval(vyx, x)
self.dyy.eval(vyy, x)
maxres = -1e10
# print(x)
for alpha in alpha_space:
R = np.array([[cos(alpha), -sin(alpha)], [sin(alpha), cos(alpha)]])
A = 0.5 * R.T @ np.array([[20, 1], [1, 0.1]]) @ R
res = A[0, 0] * vxx \
+ A[0, 1] * vxy \
+ A[1, 0] * vyx \
+ A[1, 1] * vyy
if res > maxres:
maxres = res
values[:] = alpha
def get_N_form(self):
try:
return self.N_form
except AttributeError:
pass
# Set up magnetic field and equivalent electric current forms
H_r = -curl(self.E_i)/(self.k0*Z0)
H_i = curl(self.E_r)/(self.k0*Z0)
J_r = cross(self.n, H_r)
J_i = cross(self.n, H_i)
#------------------------------
# Set up form for far field potential N
theta_hat = self.theta_hat
phi_hat = self.phi_hat
phase = self.phase
N_r = J_r*dolfin.cos(phase) - J_i*dolfin.sin(phase)
N_i = J_r*dolfin.sin(phase) + J_i*dolfin.cos(phase)
# UFL does not seem to like vector valued functionals, so we split the
# final functional from into theta and phi components
self.N_form = dict(
r_theta=dot(theta_hat, N_r)*ds,
r_phi=dot(phi_hat, N_r)*ds,
i_theta=dot(theta_hat, N_i)*ds,
i_phi=dot(phi_hat, N_i)*ds)
return self.N_form
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
self.a = as_matrix([[1., 0.], [0., 1.]])
self.u_ = sin(pi * x) * sin(pi * y)
self.f = inner(self.a, grad(grad(self.u_)))
self.g = Constant(0.0)
def updateCoefficients(self):
# Init coefficient matrix
x, y = SpatialCoordinate(self.mesh)
self.a = 0.5 * as_matrix(
[[cos(self.gamma[0]), sin(self.gamma[0])],
[- sin(self.gamma[0]), cos(self.gamma[0])]]) \
* as_matrix([[1, sin(self.gamma[1])], [0, cos(self.gamma[1])]]) \
* as_matrix([[1, 0], [sin(self.gamma[1]), cos(self.gamma[1])]]) \
* as_matrix(
[[cos(self.gamma[0]), - sin(self.gamma[0])],
[sin(self.gamma[0]), cos(self.gamma[0])]])
self.b = as_vector([Constant(0.0), Constant(0.0)])
self.c = -pi**2
self.u_ = exp(x * y) * sin(pi * x) * sin(pi * y)
# Init right-hand side
self.f = - sqrt(3) * (sin(self.gamma[1])/pi)**2 \
+ 111111
# TODO work here
# Set boundary conditions
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[1.0, self.kappa], [self.kappa, 1.0]])
# Init right-hand side
self.f = (1 + x**2) * sin(pi * x) * sin(2 * pi * y)
# Set boundary conditions to exact solution
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
self.a = as_matrix([[Constant(1.), Constant(self.kappa)],
[Constant(self.kappa),
Constant(1.)]])
self.u_ = sin(2 * pi * x) * sin(2 * pi * y)
self.f = inner(self.a, grad(grad(self.u_)))
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
z = pow(sqrt(pow(x, 2) + pow(y, 2)), .5)
# Init coefficient matrix
self.a = as_matrix([[z + 1., -z], [-z, 5 * z + 1.]])
self.u_ = sin(2 * pi * x) * sin(2 * pi * y) * exp(x * cos(y))
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def __init__(self, V, **kwargs):
# Call parent
ParametrizedProblem.__init__(
self,
os.path.join("test_eim_approximation_14_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a ParametrizedDifferentialProblem
self.V = V
self._solution = Function(V)
self.components = ["u", "s", "p"]
# Parametrized function to be interpolated
x = SpatialCoordinate(V.mesh())
mu = SymbolicParameters(self, V, (1., ))
self.f00 = (1 - x[0]) * cos(3 * pi * mu[0] *
(1 + x[0])) * exp(-mu[0] * (1 + x[0]))
self.f01 = (1 - x[0]) * sin(3 * pi * mu[0] *
(1 + x[0])) * exp(-mu[0] * (1 + x[0]))
# Inner product
f = TrialFunction(self.V)
g = TestFunction(self.V)
self.inner_product = assemble(inner(f, g) * dx)
# Collapsed vector and space
self.V0 = V.sub(0).collapse()
self.V00 = V.sub(0).sub(0).collapse()
self.V1 = V.sub(1).collapse()
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[
2.0,
sin(pi * (20 * x * y + .5)) * conditional(x * y > 0, 1, -1)
], [sin(pi * (20 * x * y + .5)) * conditional(x * y > 0, 1, -1), 2.0]])
self.u_ = x * y * sin(2 * pi * x) * \
sin(3 * pi * y) / (x ** 2 + y ** 2 + 1)
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix(
[[1., 0.], [0.,
2. + atan(self.K * (pow(x, 2) + pow(y, 2) - 1.0))]])
# Init exact solution
self.u_ = sin(pi * x) * sin(pi * y)
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
mu = 1
# Init coefficient matrix
self.a = as_matrix([[1., Constant(0.)], [Constant(0.), 1.]])
self.u_ = exp(-mu *
(self.T[1] - self.t)) * sin(2 * pi * x) * sin(1 * pi * y)
self.u_T = ufl.replace(self.u_, {self.t: self.T[1]})
# Init right-hand side
self.f = mu * self.u_ + inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = Constant(0.0)
def updateCoefficients(self):
x = SpatialCoordinate(self.mesh)[0]
# Init coefficient matrix
self.a = as_matrix([[Constant(1.)]])
self.b = as_vector([Constant(0.2)])
self.c = Constant(0.1)
self.u_ = 0.5 * (self.t**2 + 1) * sin(2 * x * pi)
self.u_T = ufl.replace(self.u_, {self.t: self.T[1]})
# Init right-hand side
self.f = self.t * sin(2*pi*x) \
+ inner(self.a, grad(grad(self.u_))) \
+ inner(self.b, grad(self.u_)) \
+ self.c * self.u_
self.g = Constant(0.0)
def get_residual_form(u,
v,
rho_e,
phi_angle,
k,
alpha,
method='RAMP',
Th=df.Constant(1.)):
if method == 'SIMP':
C = rho_e**3
else:
C = rho_e / (1 + 8. * (1. - rho_e))
E = k
# C is the design variable, its values is from 0 to 1
nu = 0.49 # Poisson's ratio
lambda_ = E * nu / (1. + nu) / (1 - 2 * nu)
mu = E / 2 / (1 + nu) #lame's parameters
# Th = df.Constant(5e1)
# Th = df.Constant(1.)
# Th = df.Constant(5e0)
w_ij = 0.5 * (df.grad(u) + df.grad(u).T)
v_ij = 0.5 * (df.grad(v) + df.grad(v).T)
S = df.as_matrix([[-2., 0., 0.], [0., 1., 0.], [0., 0., 1.]])
L = df.as_matrix([[df.cos(phi_angle),
df.sin(phi_angle), 0.],
[-df.sin(phi_angle),
df.cos(phi_angle), 0.], [0., 0., 1.]])
alpha_e = alpha * C
eps = w_ij - alpha_e * Th * L.T * S * L
d = len(u)
sigm = lambda_ * df.div(u) * df.Identity(d) + 2 * mu * eps
a = df.inner(sigm, v_ij) * df.dx
return a
def get_L_form(self):
try:
return self.L_form
except AttributeError:
pass
# Set up equivalent magnetic current forms
M_r = -cross(self.n, self.E_r)
M_i = -cross(self.n, self.E_i)
#------------------------------
# Set up form for far field potential L
theta_hat = self.theta_hat
phi_hat = self.phi_hat
phase = self.phase
L_r = M_r * dolfin.cos(phase) - M_i * dolfin.sin(phase)
L_i = M_r * dolfin.sin(phase) + M_i * dolfin.cos(phase)
self.L_form = dict(r_theta=dot(theta_hat, L_r) * ds,
r_phi=dot(phi_hat, L_r) * ds,
i_theta=dot(theta_hat, L_i) * ds,
i_phi=dot(phi_hat, L_i) * ds)
return self.L_form
def get_L_form(self):
try:
return self.L_form
except AttributeError:
pass
# Set up equivalent magnetic current forms
M_r = -cross(self.n, self.E_r)
M_i = -cross(self.n, self.E_i)
#------------------------------
# Set up form for far field potential L
theta_hat = self.theta_hat
phi_hat = self.phi_hat
phase = self.phase
L_r = M_r*dolfin.cos(phase) - M_i*dolfin.sin(phase)
L_i = M_r*dolfin.sin(phase) + M_i*dolfin.cos(phase)
self.L_form = dict(
r_theta=dot(theta_hat, L_r)*ds,
r_phi=dot(phi_hat, L_r)*ds,
i_theta=dot(theta_hat, L_i)*ds,
i_phi=dot(phi_hat, L_i)*ds)
return self.L_form
def updateCoefficients(self):
# Init coefficient matrix
x, y = SpatialCoordinate(self.mesh)
self.a = self.gamma[0] * as_matrix([[1.0, 0.0], [0.0, 1.0]])
self.b = as_vector([Constant(0.0), Constant(0.0)])
self.c = Constant(0.0)
# Init right-hand side
# self.f_t = 529 * (sin(xxx) + 0.5 * sin(2 * xxx) + 4./10 * sin(8 * xxx))**2
# self.f_t = (sin(pi**2*(x-0.63)*(y-0.26)/0.07))**2
# self.f_t = Constant(1.)
# self.u_ = sin(pi*x) * sin(pi * y)
# self.f = -self.alpha1 * 2 * pi**2 * sin(pi*x) * sin(pi*y)
self.u_ = sin(2 * pi * x) * sin(2 * pi * y)
self.f = -conditional(
(x - 0.5) *
(y - 0.5) > 0, self.alpha1, self.alpha0) * 8 * pi**2 * self.u_
# Set boundary conditions
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix(
[[pow(abs(sin(4 * pi * (x - 0.5))), 1.0 / 5),
cos(2 * pi * x * y)],
[
cos(2 * pi * x * y),
pow(abs(sin(4 * pi * (y - 0.5))), 1.0 / 5) + 1.
]])
# Init exact solution
self.u_ = x * y * sin(2 * pi * x) * sin(
3 * pi * y) / (pow(x, 2) + pow(y, 2) + 1)
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def updateCoefficients(self):
# Init coefficient matrix
x, y = SpatialCoordinate(self.mesh)
self.a = self.gamma[0] * as_matrix([[1.0, 0.0], [0.0, 1.0]])
self.b = as_vector([self.gamma[1], self.gamma[2]])
# Init right-hand side
# self.f_t = 529 * (sin(xxx)
# + 0.5 * sin(2 * xxx) + 4./10 * sin(8 * xxx))**2
# self.f_t = (sin(pi**2*(x-0.63)*(y-0.26)/0.07))**2
# self.f_t = Constant(1.)
self.f = -10 * (sin(pi**2 * (x - 0.63) *
(y - 0.26) / 0.07) + 0.5 * sin(2 * pi**2 *
(x - 0.63) *
(y - 0.26) / 0.07) +
4. / 10 * sin(8 * pi**2 * (x - 0.63) *
(y - 0.26) / 0.07))**2
# Set boundary conditions
self.g = Constant(0.0)
self.u_T = Constant(0.0)
def initControl(self):
self.controlSpace = [
FunctionSpace(self.mesh, "DG", 1),
FunctionSpace(self.mesh, "DG", 1)
]
u_x = Dx(self.u, 0)
u_y = Dx(self.u, 1)
# phi = atan(u_y/u_x) <==> sin(phi) / cos(phi) = u_y / u_x
self.gamma = []
phi = ufl.atan_2(u_y, u_x)
self.gamma.append(1. / self.alpha * (cos(phi) * u_x + sin(phi) * u_y))
self.gamma.append(phi)
def get_N_form(self):
try:
return self.N_form
except AttributeError:
pass
# Set up magnetic field and equivalent electric current forms
H_r = -curl(self.E_i) / (self.k0 * Z0)
H_i = curl(self.E_r) / (self.k0 * Z0)
J_r = cross(self.n, H_r)
J_i = cross(self.n, H_i)
#------------------------------
# Set up form for far field potential N
theta_hat = self.theta_hat
phi_hat = self.phi_hat
phase = self.phase
N_r = J_r * dolfin.cos(phase) - J_i * dolfin.sin(phase)
N_i = J_r * dolfin.sin(phase) + J_i * dolfin.cos(phase)
# UFL does not seem to like vector valued functionals, so we split the
# final functional from into theta and phi components
self.N_form = dict(r_theta=dot(theta_hat, N_r) * ds,
r_phi=dot(phi_hat, N_r) * ds,
i_theta=dot(theta_hat, N_i) * ds,
i_phi=dot(phi_hat, N_i) * ds)
return self.N_form
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[1., 0], [0, self.eps]])
self.b = as_vector([0.0, -10 * (y - 0.5) * x])
self.c = Constant(0.0)
# self.u_T = Constant(0)
self.u_ = 0.5 * (self.t**2 + 1) * sin(2 * x * pi) * sin(2 * y * pi)
self.u_T = ufl.replace(self.u_, {self.t: self.T[1]})
# Init right-hand side
self.f = +self.t * sin(2*pi*x) * sin(2*pi*y) \
+ inner(self.a[0][0], Dx(Dx(self.u_, 0), 0)) \
+ inner(self.b, grad(self.u_)) \
+ self.c * self.u_
# self.f = +self.t * sin(2*pi*x) * sin(2*pi*y) \
# + inner(self.a, grad(grad(self.u_))) \
# + inner(self.b, grad(self.u_)) \
# + self.c * self.u_
self.g = Constant(0.0)
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[1., 0.], [0.,
2.0 + sin(1.0 / (abs(x) + abs(y)))]])
# Init exact solution
self.u_ = pow(2 - pow(x, 2) - pow(y, 2), 0.5)
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def updateCoefficients(self):
x, y = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[1., Constant(0.)], [Constant(0.), 1.]])
# Set up explicit solution
r = sqrt(x**2 + y**2)
phi = atan_2(x, y)
self.u_ = r**self.alpha * sin(2. * phi) * (1 - x) * (1 - y)
# Init right-hand side
self.f = inner(self.a, grad(grad(self.u_)))
# Set boundary conditions to exact solution
self.g = self.u_
def updateCoefficients(self):
# Init coefficient matrix
x, y = SpatialCoordinate(self.mesh)
self.a = as_matrix([[1.0, 0.0], [0.0, 1.0]])
# self.a = as_matrix([[1.0 , 0.0], [0.0, 1.0]])
self.b = -self.gamma[0] * \
as_vector([cos(self.gamma[1]), sin(self.gamma[1])])
# self.u_ = x*y*(1-exp(1-abs(x))) * (1-exp(1-abs(y)))
# Init right-hand side
self.f = -1 \
- 0.5 * self.alpha * sqrt(pow(self.gamma[0], 2))
# Set boundary conditions
self.g = Constant(0.0)
def run_dolfin():
import dolfin as df
x_array = np.linspace(-49.5, 49.5, 100)
mesh = df.IntervalMesh(100, -50, 50)
Delta = np.sqrt(A / K)
xi = 2 * A / D
Delta_s = Delta * 1e9
V = df.FunctionSpace(mesh, "Lagrange", 1)
u = df.TrialFunction(V)
v = df.TestFunction(V)
u_ = df.Function(V)
F = -df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx - \
(0.5 / Delta_s**2) * df.sin(2 * u) * v * df.dx
F = df.action(F, u_)
J = df.derivative(F, u_, u)
# the boundary condition is from equation (8)
theta0 = np.arcsin(Delta / xi)
ss = 'x[0]<0? %g: %g ' % (-theta0, theta0)
u0 = df.Expression(ss)
def u0_boundary(x, on_boundary):
return on_boundary
bc = df.DirichletBC(V, u0, u0_boundary)
problem = df.NonlinearVariationalProblem(F, u_, bcs=bc, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.solve()
u_array = u_.vector().array()
mx_df = []
for x in x_array:
mx_df.append(u_(x))
return mx_df
def run_dolfin():
import dolfin as df
x_array = np.linspace(-49.5, 49.5, 100)
mesh = df.IntervalMesh(100, -50, 50)
Delta = np.sqrt(A / K)
xi = 2 * A / D
Delta_s = Delta * 1e9
V = df.FunctionSpace(mesh, "Lagrange", 1)
u = df.TrialFunction(V)
v = df.TestFunction(V)
u_ = df.Function(V)
F = -df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx - \
(0.5 / Delta_s**2) * df.sin(2 * u) * v * df.dx
F = df.action(F, u_)
J = df.derivative(F, u_, u)
# the boundary condition is from equation (8)
theta0 = np.arcsin(Delta / xi)
ss = 'x[0]<0? %g: %g ' % (-theta0, theta0)
u0 = df.Expression(ss)
def u0_boundary(x, on_boundary):
return on_boundary
bc = df.DirichletBC(V, u0, u0_boundary)
problem = df.NonlinearVariationalProblem(F, u_, bcs=bc, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.solve()
u_array = u_.vector().array()
mx_df = []
for x in x_array:
mx_df.append(u_(x))
return mx_df
def deformation_vector():
n1 = (1 + x1[0] * sin(2 * pi * x1[1])) * VolumeNormal(multimesh.part(1))
S_sm = VectorFunctionSpace(multimesh.part(1), "CG", 1)
# Note removing 0 dirichlet at dI introduces problems with the change in
# the cut cell and overlap cell part. We do not have any way of describing
# this movement
bcs = [
DirichletBC(S_sm, n1, mfs[1], 2),
DirichletBC(S_sm, Constant((0, 0)), mfs[1], 1)
]
u, v = TrialFunction(S_sm), TestFunction(S_sm)
a = inner(grad(u), grad(v)) * dx
l = inner(Constant((0., 0.)), v) * dx
n = Function(S_sm)
solve(a == l, n, bcs=bcs)
S = MultiMeshVectorFunctionSpace(multimesh, "CG", 1)
s = MultiMeshFunction(S)
s.assign_part(1, n)
return s
def __init__(self, mesh, *, domains = None, facets = None, coefficients = None): super().__init__(mesh, domains = domains, facets = facets) coeffs = self.coefficient_functions(coefficients) cos = dolfin.cos(coeffs['angle']) sin = dolfin.sin(coeffs['angle']) ax_0, ax_1, ax_2 = coeffs['axis_0'], coeffs['axis_1'], coeffs['axis_2'] E_0, E_1, E_2 = coeffs['E_0'], coeffs['E_1'], coeffs['E_2'] nu_01, nu_02, nu_12 = coeffs['nu_01'], coeffs['nu_02'], coeffs['nu_12'] self.G_01, self.G_02, self.G_12 = coeffs['G_01'], coeffs['G_02'], coeffs['G_12'] self.RR = dolfin.as_matrix([ [cos + ax_0*ax_0*(1 - cos), ax_0*ax_1*(1 - cos) - ax_2*sin, ax_0*ax_2*(1 - cos) + ax_1*sin], [ax_0*ax_1*(1 - cos) + ax_2*sin, cos + ax_1*ax_1*(1 - cos), ax_1*ax_2*(1 - cos) - ax_0*sin], [ax_0*ax_2*(1 - cos) - ax_1*sin, ax_1*ax_2*(1 - cos) + ax_0*sin, cos + ax_2*ax_2*(1 - cos)] ]) nu_10 = nu_01*E_1/E_0; nu_20 = nu_02*E_2/E_0; nu_21 = nu_12*E_2/E_1 kk = 1 - nu_01*nu_10 - nu_12*nu_21 - nu_02*nu_20 - 2*nu_01*nu_12*nu_20 self.DD = dolfin.as_matrix([ [E_0*(1 - nu_12*nu_21)/kk, E_0*(nu_12*nu_20 + nu_10)/kk, E_0*(nu_21*nu_10 + nu_20)/kk], [E_1*(nu_02*nu_21 + nu_01)/kk, E_1*(1 - nu_02*nu_20)/kk, E_1*(nu_20*nu_01 + nu_21)/kk], [E_2*(nu_01*nu_12 + nu_02)/kk, E_2*(nu_10*nu_02 + nu_12)/kk, E_2*(1 - nu_01*nu_10)/kk] ])
def updateCoefficients(self):
x, y, z = SpatialCoordinate(self.mesh)
# Init coefficient matrix
self.a = as_matrix([[1, 0, 0], [0, 2, 0], [0, 0, 1]])
self.b = as_vector([0.2, -0.1, 0.1])
self.c = Constant(0.1)
self.u_ = 0.5 * \
(self.t**2+1) * sin(2*x*pi) * sin(2*y*pi) * sin(2*z*pi)
# self.u_T = ufl.replace(self.u_, {self.t: self.T[1]})
# Init right-hand side
self.f = +self.t * sin(2*pi*x) * sin(2*pi*y)*sin(2*pi*z) \
+ inner(self.a, grad(grad(self.u_))) \
+ inner(self.b, grad(self.u_)) \
+ self.c * self.u_
self.g = Constant(0.0)
def eval(self, values, x):
y_0 = -H + a0 * (exp(-((x[0]-L/2.)**2 + (x[1]-L/2.)**2) / sigma**2))
values[0] = sin(theta)/cos(theta) * (x[0] + sin(theta)*y_0) \
+ cos(theta)*y_0
def bc_eval(self, t):
return (sin(t), sin(t))
def test_prb88_184422():
mu0 = 4 * np.pi * 1e-7
Ms = 8.6e5
A = 16e-12
D = 3.6e-3
K = 510e3
mesh = CuboidMesh(nx=100, dx=1, unit_length=1e-9)
sim = Sim(mesh)
sim.driver.set_tols(rtol=1e-10, atol=1e-14)
sim.driver.alpha = 0.5
sim.driver.gamma = 2.211e5
sim.Ms = Ms
sim.do_precession = False
sim.set_m((0, 0, 1))
sim.add(UniformExchange(A))
sim.add(DMI(-D, type='interfacial'))
sim.add(UniaxialAnisotropy(K, axis=(0, 0, 1)))
sim.relax(dt=1e-13, stopping_dmdt=0.01, max_steps=5000,
save_m_steps=None, save_vtk_steps=50)
m = sim.spin
mx, my, mz = np.split(m, 3)
x_array = np.linspace(-49.5, 49.5, 100)
#plt.plot(x_array, mx)
#plt.plot(x_array, my)
#plt.plot(x_array, mz)
mesh = df.IntervalMesh(100, -50, 50)
Delta = np.sqrt(A / K)
xi = 2 * A / D
Delta_s = Delta * 1e9
V = df.FunctionSpace(mesh, "Lagrange", 1)
u = df.TrialFunction(V)
v = df.TestFunction(V)
u_ = df.Function(V)
F = -df.inner(df.nabla_grad(u), df.nabla_grad(v)) * df.dx - \
(0.5 / Delta_s**2) * df.sin(2 * u) * v * df.dx
F = df.action(F, u_)
J = df.derivative(F, u_, u)
# the boundary condition is from equation (8)
theta0 = np.arcsin(Delta / xi)
ss = 'x[0]<0? %g: %g ' % (-theta0, theta0)
u0 = df.Expression(ss)
def u0_boundary(x, on_boundary):
return on_boundary
bc = df.DirichletBC(V, u0, u0_boundary)
problem = df.NonlinearVariationalProblem(F, u_, bcs=bc, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.solve()
u_array = u_.vector().array()
mx_df = []
for x in x_array:
mx_df.append(u_(x))
#plt.plot(x_array, mx_df)
assert abs(np.max(mx - mx_df)) < 0.05
def linear_interpolation_two(self, image0, image1, n):
"""
Define a linear interpolation between
two states of the energy band (m0, m1) to get
an initial state. The interpolation is
done in the magnetic moments that constitute the
magnetic system.
"""
# Get the spherical coordinates dolfin functions
# for the m0 and m1 magnetisation vector fields
self.sim.set_m(self.initial_images[image0])
theta0, phi0 = self.sim._m_field.get_spherical()
self.sim.set_m(self.initial_images[image1])
theta1, phi1 = self.sim._m_field.get_spherical()
# To not depend on numpy arrays, we will assemble every
# interpolation into dolfin functions and assign their
# values to the subdomains of the MixedFunctionSpace of images
# Create a scalar Function Space
S1 = df.FunctionSpace(self.sim.m_field.functionspace.mesh(), 'CG', 1)
# Define a variable to use as vector in all the assemble instances
assemble_vector = None
# Define the interpolations step for theta
assemble_vector = df.assemble(df.dot((theta1 - theta0) / (n + 1))
* df.dP,
tensor=assemble_vector)
dtheta = df.Function(S1)
dtheta.vector().axpy(1, assemble_vector)
# The same for Phi
assemble_vector = df.assemble(df.dot((theta1 - theta0) / (n + 1))
* df.dP,
tensor=assemble_vector)
dphi = df.Function(S1)
dphi.vector().axpy(1, assemble_vector)
# Now loop for every interpolation and assign it to
# the MixedFunctionSpace
for i in xrange(n):
# Create a dolfin function from the FS, for the interpolation
interpolation_theta = df.Function(S1)
interpolation_phi = df.Function(S1)
# Compute the radius using the assemble method with dolfin dP
# (like a dirac delta to get values on every node of the mesh)
# This returns a dolfin vector
# Theta
assemble_vector = df.assemble(df.dot(theta0 + (i + 1) * dtheta,
#
df.TestFunction(S1)) * df.dP,
tensor=assemble_vector
)
# Set the vector values to the dolfin function
interpolation_theta.vector().axpy(1, assemble_vector)
# Phi
assemble_vector = df.assemble(df.dot(phi0 + (i + 1) * dphi,
#
df.TestFunction(S1)) * df.dP,
tensor=assemble_vector
)
# Set the vector values to the dolfin function
interpolation_phi.vector().axpy(1, assemble_vector)
# Now set this interpolation to the corresponding image
# Set a vector function space for the simulation
# magnetisation vector field
interpolation = df.VectorFunction(self.sim.S3)
interpolation = df.assemble(df.dot(df.as_vector((df.sin(interpolation_theta) * df.cos(interpolation_phi),
df.sin(interpolation_theta) * df.sin(interpolation_phi),
df.cos(interpolation_theta)
)),
df.TestFunction(self.sim.S3)) * df.dP
)
interpolation.vector().axpy(1, interpolation)
# Now assign the interpolation vector function values to the corresponding
# image in the MixedFunctionSpace
df.assign(self.images_fun.sub(i), interpolation)
