def a(phi):
# phi = 1e5*phi
n = ufl.grad(phi) / ufl.sqrt(ufl.dot(ufl.grad(phi), ufl.grad(phi)) + 1e-5)
theta = ufl.atan_2(n[1], n[0])
n = 1e5 * n + ufl.as_vector([1e-5, 1e-5])
xx = n[1] / n[0]
# theta = ufl.asin(xx/ ufl.sqrt(1 + xx**2))
theta = ufl.atan(xx)
return 1.0 + epsilon_4 * ufl.cos(m * (theta - theta_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 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 eigenstate(A):
"""Eigenvalues and eigenprojectors of the 3x3 (real-valued) tensor A.
Provides the spectral decomposition A = sum_{a=0}^{2} λ_a * E_a
with (ordered) eigenvalues λ_a and their associated eigenprojectors E_a = n_a^R x n_a^L.
Note: Tensor A must not have complex eigenvalues!
"""
if ufl.shape(A) != (3, 3):
raise RuntimeError(
f"Tensor A of shape {ufl.shape(A)} != (3, 3) is not supported!")
#
eps = 3.0e-16 # slightly above 2**-(53 - 1), see https://en.wikipedia.org/wiki/IEEE_754
#
A = ufl.variable(A)
#
# --- determine eigenvalues λ0, λ1, λ2
#
I1, I2, I3 = invariants_principal(A)
dq = 2 * I1**3 - 9 * I1 * I2 + 27 * I3
#
Δx = [
A[0, 1] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 1],
A[0, 1]**2 * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 1] +
A[0, 1] * A[0, 2] * A[2, 2] - A[0, 2]**2 * A[2, 1],
A[0, 0] * A[0, 1] * A[2, 1] - A[0, 1]**2 * A[2, 0] -
A[0, 1] * A[2, 1] * A[2, 2] + A[0, 2] * A[2, 1]**2,
A[0, 0] * A[0, 2] * A[1, 2] + A[0, 1] * A[1, 2]**2 -
A[0, 2]**2 * A[1, 0] - A[0, 2] * A[1, 1] * A[1, 2],
A[0, 0] * A[0, 1] * A[1, 2] - A[0, 1] * A[0, 2] * A[1, 0] -
A[0, 1] * A[1, 2] * A[2, 2] +
A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501
A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1] * A[0, 2] * A[2, 0] +
A[0, 1] * A[1, 2] * A[2, 1] -
A[0, 2] * A[1, 1] * A[2, 1], # noqa: E501
A[0, 1] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0] * A[1, 1] +
A[0, 2] * A[1, 0] * A[2, 2] -
A[0, 2] * A[1, 2] * A[2, 0], # noqa: E501
A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] -
A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] +
A[0, 1] * A[1, 0] * A[1, 2] + A[0, 2] * A[1, 0] * A[2, 2] +
A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501
A[0, 0]**2 * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 0] -
A[0, 0] * A[1, 1] * A[1, 2] - A[0, 0] * A[1, 2] * A[2, 2] +
A[0, 2] * A[1, 0] * A[1, 1] + A[0, 2] * A[1, 2] * A[2, 0] +
A[1, 1] * A[1, 2] * A[2, 2] - A[1, 2]**2 * A[2, 1], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] -
A[0, 1]**2 * A[1, 0] + A[0, 1] * A[0, 2] * A[2, 0] -
A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[2, 2]**2 +
A[0, 2] * A[1, 1] * A[2, 1] -
A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 1] - A[0, 0] * A[0, 1] * A[2, 2] +
A[0, 0] * A[0, 2] * A[2, 1] - A[0, 1]**2 * A[1, 0] -
A[0, 1] * A[1, 1] * A[2, 2] + A[0, 1] * A[1, 2] * A[2, 1] +
A[0, 1] * A[2, 2]**2 - A[0, 2] * A[2, 1] * A[2, 2], # noqa: E501
A[0, 0] * A[0, 1] * A[1, 2] - A[0, 0] * A[0, 2] * A[1, 1] +
A[0, 0] * A[0, 2] * A[2, 2] - A[0, 1] * A[1, 1] * A[1, 2] -
A[0, 2]**2 * A[2, 0] + A[0, 2] * A[1, 1]**2 -
A[0, 2] * A[1, 1] * A[2, 2] +
A[0, 2] * A[1, 2] * A[2, 1], # noqa: E501
A[0, 0] * A[0, 2] * A[1, 1] - A[0, 0] * A[0, 2] * A[2, 2] -
A[0, 1] * A[0, 2] * A[1, 0] + A[0, 1] * A[1, 1] * A[1, 2] -
A[0, 1] * A[1, 2] * A[2, 2] + A[0, 2]**2 * A[2, 0] -
A[0, 2] * A[1, 1]**2 + A[0, 2] * A[1, 1] * A[2, 2], # noqa: E501
A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] -
A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] -
A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 +
A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] +
A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] -
A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2]
] # noqa: E501
Δy = [
A[0, 2] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 2] * A[2, 0],
A[1, 0]**2 * A[2, 1] - A[1, 0] * A[1, 1] * A[2, 0] +
A[1, 0] * A[2, 0] * A[2, 2] - A[1, 2] * A[2, 0]**2,
A[0, 0] * A[1, 0] * A[1, 2] - A[0, 2] * A[1, 0]**2 -
A[1, 0] * A[1, 2] * A[2, 2] + A[1, 2]**2 * A[2, 0],
A[0, 0] * A[2, 0] * A[2, 1] - A[0, 1] * A[2, 0]**2 +
A[1, 0] * A[2, 1]**2 - A[1, 1] * A[2, 0] * A[2, 1],
A[0, 0] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 0] * A[2, 0] -
A[1, 0] * A[2, 1] * A[2, 2] +
A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0] * A[1, 2] * A[2, 0] - A[0, 2] * A[1, 0] * A[2, 0] +
A[1, 0] * A[1, 2] * A[2, 1] -
A[1, 1] * A[1, 2] * A[2, 0], # noqa: E501
A[0, 1] * A[1, 0] * A[2, 1] - A[0, 1] * A[1, 1] * A[2, 0] +
A[0, 1] * A[2, 0] * A[2, 2] -
A[0, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] -
A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] +
A[0, 1] * A[1, 0] * A[2, 1] + A[0, 1] * A[2, 0] * A[2, 2] +
A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501
A[0, 0]**2 * A[2, 1] - A[0, 0] * A[0, 1] * A[2, 0] -
A[0, 0] * A[1, 1] * A[2, 1] - A[0, 0] * A[2, 1] * A[2, 2] +
A[0, 1] * A[1, 1] * A[2, 0] + A[0, 2] * A[2, 0] * A[2, 1] +
A[1, 1] * A[2, 1] * A[2, 2] - A[1, 2] * A[2, 1]**2, # noqa: E501
A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] -
A[0, 1] * A[1, 0]**2 + A[0, 2] * A[1, 0] * A[2, 0] -
A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[2, 2]**2 +
A[1, 1] * A[1, 2] * A[2, 0] -
A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0] * A[1, 0] * A[1, 1] - A[0, 0] * A[1, 0] * A[2, 2] +
A[0, 0] * A[1, 2] * A[2, 0] - A[0, 1] * A[1, 0]**2 -
A[1, 0] * A[1, 1] * A[2, 2] + A[1, 0] * A[1, 2] * A[2, 1] +
A[1, 0] * A[2, 2]**2 - A[1, 2] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0] * A[1, 0] * A[2, 1] - A[0, 0] * A[1, 1] * A[2, 0] +
A[0, 0] * A[2, 0] * A[2, 2] - A[0, 2] * A[2, 0]**2 -
A[1, 0] * A[1, 1] * A[2, 1] + A[1, 1]**2 * A[2, 0] -
A[1, 1] * A[2, 0] * A[2, 2] +
A[1, 2] * A[2, 0] * A[2, 1], # noqa: E501
A[0, 0] * A[1, 1] * A[2, 0] - A[0, 0] * A[2, 0] * A[2, 2] -
A[0, 1] * A[1, 0] * A[2, 0] + A[0, 2] * A[2, 0]**2 +
A[1, 0] * A[1, 1] * A[2, 1] - A[1, 0] * A[2, 1] * A[2, 2] -
A[1, 1]**2 * A[2, 0] + A[1, 1] * A[2, 0] * A[2, 2], # noqa: E501
A[0, 0]**2 * A[1, 1] - A[0, 0]**2 * A[2, 2] -
A[0, 0] * A[0, 1] * A[1, 0] + A[0, 0] * A[0, 2] * A[2, 0] -
A[0, 0] * A[1, 1]**2 + A[0, 0] * A[2, 2]**2 +
A[0, 1] * A[1, 0] * A[1, 1] - A[0, 2] * A[2, 0] * A[2, 2] +
A[1, 1]**2 * A[2, 2] - A[1, 1] * A[1, 2] * A[2, 1] -
A[1, 1] * A[2, 2]**2 + A[1, 2] * A[2, 1] * A[2, 2]
] # noqa: E501
Δd = [9, 6, 6, 6, 8, 8, 8, 2, 2, 2, 2, 2, 2, 1]
Δ = 0
for i in range(len(Δd)):
Δ += Δx[i] * Δd[i] * Δy[i]
Δxp = [
A[1, 0], A[2, 0], A[2, 1], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2],
-A[1, 1] + A[2, 2]
]
Δyp = [
A[0, 1], A[0, 2], A[1, 2], -A[0, 0] + A[1, 1], -A[0, 0] + A[2, 2],
-A[1, 1] + A[2, 2]
]
Δdp = [6, 6, 6, 1, 1, 1]
dp = 0
for i in range(len(Δdp)):
dp += 1 / 2 * Δxp[i] * Δdp[i] * Δyp[i]
# Avoid dp = 0 and disc = 0, both are known with absolute error of ~eps**2
# Required to avoid sqrt(0) derivatives and negative square roots
dp += eps**2
Δ += eps**2
phi3 = ufl.atan_2(ufl.sqrt(27) * ufl.sqrt(Δ), dq)
# sorted eigenvalues: λ0 <= λ1 <= λ2
λ = [(I1 + 2 * ufl.sqrt(dp) * ufl.cos((phi3 + 2 * ufl.pi * k) / 3)) / 3
for k in range(1, 4)]
#
# --- determine eigenprojectors E0, E1, E2
#
E = [ufl.diff(λk, A).T for λk in λ]
return λ, E
# and
#
# $D_n$ is the anisotropic diffusion using the previous solution $\textbf{u}_n$ to compute the entries.
#
# First we put in the right hand side which only contains explicit data.
# <codecell>
from ufl import inner, dx
a_ex = (inner(u_h_n, v) - inner(u_h_n[0], v[1])) * dx
# <markdowncell>
# For the left hand side we have the spatial derivatives and the implicit parts.
# <codecell>
from ufl import pi, atan, atan_2, tan, grad, as_vector, inner, dot
psi = pi / 8.0 + atan_2(grad(u_h_n[0])[1], (grad(u_h_n[0])[0]))
Phi = tan(N / 2.0 * psi)
beta = (1.0 - Phi * Phi) / (1.0 + Phi * Phi)
dbeta_dPhi = -2.0 * N * Phi / (1.0 + Phi * Phi)
fac = 1.0 + c * beta
diag = fac * fac
offdiag = -fac * c * dbeta_dPhi
d0 = as_vector([diag, offdiag])
d1 = as_vector([-offdiag, diag])
m = u[0] - 0.5 - kappa1 / pi * atan(kappa2 * u[1])
s = as_vector([dt / tau * u[0] * (1.0 - u[0]) * m, u[0]])
a_im = (alpha * alpha * dt / tau *
(inner(dot(d0, grad(u[0])),
grad(v[0])[0]) + inner(dot(d1, grad(u[0])),
grad(v[0])[1])) + 2.25 * dt *
inner(grad(u[1]), grad(v[1])) + inner(u, v) - inner(s, v)) * dx
def eval(self, values, x):
values[0] = ufl.atan_2(x[1], x[0])
from dolfin import *
import matplotlib.pyplot as plt
import ufl
mesh = RectangleMesh(Point(-1.0, -1.0), Point(1.0, 1.0), 10, 10)
el = FiniteElement("Lagrange", mesh.ufl_cell(), degree=1)
V = FunctionSpace(mesh, el)
x = SpatialCoordinate(mesh)
# Approach 1: Fails with project
phi = project(ufl.atan_2(x[1], x[0]), V)
# phi = interpolate(ufl.atan_2(x[1], x[0]), V) # fails, no eval() method..
plt.subplot(221)
p1 = plot(phi, title="phi1")
plt.colorbar(p1)
file = File("phi1.pvd")
file.write(phi)
# Approach 2: Works with interpolate and UserExpression (has eval method)
class RadialAngle(UserExpression):
def eval(self, values, x):
values[0] = ufl.atan_2(x[1], x[0])
phi2 = interpolate(RadialAngle(), V)
plt.subplot(222)
p2 = plot(phi2, title="phi2")
plt.colorbar(p2)
file2 = File("phi2.pvd")