def solve_thermal(self):
"""
Solve the thermal diffusion problem.
Both ice and solution.
"""
### Solve heat equation
alphalog_i = dolfin.project(dolfin.Expression('astar*exp(-2.*x[0])',degree=1,astar=self.astar_i),self.ice_V)
# Set up the variational form for the current mesh location
F_i = (self.u_i-self.u0_i)*self.v_i*dolfin.dx + self.dt*dolfin.inner(dolfin.grad(self.u_i), dolfin.grad(alphalog_i*self.v_i))*dolfin.dx
a_i = dolfin.lhs(F_i)
L_i = dolfin.rhs(F_i)
# Solve ice temperature
dolfin.solve(a_i==L_i,self.T_i,[self.bc_inf,self.bc_iWall])
# Update previous profile to current
self.u0_i.assign(self.T_i)
diff_ratio = (self.ks*const.rhoi*const.ci)/(const.ki*self.rhos*self.cs)
alphalog_s = dolfin.project(dolfin.Expression('astar*exp(-2.*x[0])',degree=1,astar=self.astar_i),self.sol_V)
# Set up the variational form for the current mesh location
F_s = (self.u_s-self.u0_s)*self.v_s*dolfin.dx + self.dt*dolfin.inner(dolfin.grad(self.u_s), dolfin.grad(alphalog_s*diff_ratio*self.v_s))*dolfin.dx
#- self.dt*(self.Q_sol*self.t0/abs(self.T_inf)/(const.rhoi*const.ci))*self.v_s*dolfin.dx #TODO: check this solution source term
# TODO: Center heat flux
#F_s -= (self.Q_center/(self.ks*diff_ratio*2.*np.pi*abs(self.T_inf)))*self.v_s*self.sds(2)
a_s = dolfin.lhs(F_s)
L_s = dolfin.rhs(F_s)
# Solve solution temperature
dolfin.solve(a_s==L_s,self.T_s,self.bc_sWall)
# Update previous profile to current
self.u0_s.assign(self.T_s)
def __init__(self, t_end=None, func=None):
Problem_verification.__init__(self, t_end, func)
self.u0_expr = dol.Constant(1)
self.V = dol.FunctionSpace(dol.UnitIntervalMesh(1), 'DG', 0)
# rhs
self.f1 = dol.Expression('pow(t, 2)', t=0, degree=2)
self.f2 = dol.Expression('pow(t, 2)', t=0, degree=2)
Problem_FE.__init__(self)
# CN, higher order
F = (dol.inner(
self.v, self.utrial - self.uold + self.dt * 0.5 *
(self.uold + self.utrial)) * dol.dx -
0.5 * self.dt * dol.inner(self.f1 + self.f2, self.v) * dol.dx)
prob = dol.LinearVariationalProblem(dol.lhs(F), dol.rhs(F), self.unew)
self.solver = dol.LinearVariationalSolver(prob)
# IE, lower order
Flow = (dol.inner(
self.v, self.utrial - self.uold + self.dt * self.utrial) * dol.dx -
self.dt * dol.inner(self.f2, self.v) * dol.dx)
problow = dol.LinearVariationalProblem(dol.lhs(Flow), dol.rhs(Flow),
self.ulow)
self.solver_low = dol.LinearVariationalSolver(problow)
def solve_initial_pressure(w_NSp, p, q, u, v, bcs_NSp, M_, g_, phi_, rho_,
rho_e_, V_, drho, sigma_bar, eps, grav, dveps,
enable_PF, enable_EC):
V = u.function_space()
grad_p = df.TrialFunction(V)
grad_p_out = df.Function(V)
F_grad_p = (df.dot(grad_p, v) * df.dx - rho_ * df.dot(grav, v) * df.dx)
if enable_PF:
F_grad_p += -drho * M_ * df.inner(df.grad(u), df.outer(df.grad(g_),
v)) * df.dx
F_grad_p += -sigma_bar * eps * df.inner(
df.outer(df.grad(phi_), df.grad(phi_)), df.grad(v)) * df.dx
if enable_EC and rho_e_ != 0:
F_grad_p += rho_e_ * df.dot(df.grad(V_), v) * df.dx
if enable_PF and enable_EC:
F_grad_p += dveps * df.dot(df.grad(phi_), v) * df.dot(
df.grad(V_), df.grad(V_)) * df.dx
info_red("Solving initial grad_p...")
df.solve(df.lhs(F_grad_p) == df.rhs(F_grad_p), grad_p_out)
F_p = (df.dot(df.grad(q), df.grad(p)) * df.dx -
df.dot(df.grad(q), grad_p_out) * df.dx)
info_red("Solving initial p...")
df.solve(df.lhs(F_p) == df.rhs(F_p), w_NSp, bcs_NSp)
def setup_NSu(w_NSu, u, v, dx, ds, normal, dirichlet_bcs_NSu, neumann_bcs,
boundary_to_mark, u_, p_, u_1, p_1, rho_, rho_1, mu_, c_1,
grad_g_c_, dt, grav, enable_EC, trial_functions,
use_iterative_solvers, mesh, density_per_concentration,
viscosity_per_concentration, K, **namespace):
""" Set up the Navier-Stokes velocity subproblem. """
solvers = dict()
mom_1 = rho_1 * u_1
if enable_EC and density_per_concentration is not None:
for drhodci, ci_1, grad_g_ci_, Ki in zip(density_per_concentration,
c_1, grad_g_c_, K):
if drhodci > 0.:
mom_1 += -drhodci * Ki * ci_1 * grad_g_ci_
F_predict = (
1. / dt * rho_1 * df.dot(u - u_1, v) * dx +
df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx + 2 * mu_ *
df.inner(df.sym(df.nabla_grad(u)), df.sym(df.nabla_grad(v))) * dx +
0.5 * (1. / dt * (rho_ - rho_1) * df.dot(u, v) -
df.inner(mom_1, df.grad(df.dot(u, v)))) * dx -
p_1 * df.div(v) * dx - rho_ * df.dot(grav, v) * dx)
for boundary_name, pressure in neumann_bcs["p"].iteritems():
F_predict += pressure * df.inner(normal, v) * ds(
boundary_to_mark[boundary_name])
if enable_EC:
F_predict += sum([
ci_1 * df.dot(grad_g_ci_, v) * dx
for ci_1, grad_g_ci_ in zip(c_1, grad_g_c_)
])
a_predict, L_predict = df.lhs(F_predict), df.rhs(F_predict)
# if not use_iterative_solvers:
problem_predict = df.LinearVariationalProblem(a_predict, L_predict, w_NSu,
dirichlet_bcs_NSu)
solvers["predict"] = df.LinearVariationalSolver(problem_predict)
if use_iterative_solvers:
solvers["predict"].parameters["linear_solver"] = "bicgstab"
solvers["predict"].parameters["preconditioner"] = "amg"
F_correct = (rho_ * df.inner(u - u_, v) * dx - dt *
(p_ - p_1) * df.div(v) * dx)
a_correct, L_correct = df.lhs(F_correct), df.rhs(F_correct)
problem_correct = df.LinearVariationalProblem(a_correct, L_correct, w_NSu,
dirichlet_bcs_NSu)
solvers["correct"] = df.LinearVariationalSolver(problem_correct)
if use_iterative_solvers:
solvers["correct"].parameters["linear_solver"] = "bicgstab"
solvers["correct"].parameters["preconditioner"] = "amg"
#else:
# solver = df.LUSolver("mumps")
# # solver.set_operator(A)
# return solver, a, L, dirichlet_bcs_NS
return solvers
def __init__(self, t_end = None, func = None, para_stiff = None, adjoint = False):
Problem_Basic.__init__(self, t_end = t_end, func = func, para_stiff = para_stiff)
self.k = dol.Constant(self.k)
self.u0_expr = dol.Constant(self.u0) # initial value
mesh = dol.UnitIntervalMesh(1)
R_elem = dol.FiniteElement("R", mesh.ufl_cell(), 0)
V_elem = dol.MixedElement([R_elem, R_elem])
self.V = dol.FunctionSpace(mesh, V_elem)
if adjoint:
self.z0_expr = dol.Constant(np.array([0., 0.])) # initial value adj
Problem_FE.__init__(self, adjoint)
(self.v1 , self.v2) = dol.split(self.v)
(self.u1trial, self.u2trial) = dol.split(self.utrial)
(self.u1old , self.u2old) = dol.split(self.uold)
(self.u1low , self.u2low) = dol.split(self.ulow)
## Crank nicolson weak formulation
F = (dol.inner(self.v1, self.u1trial - self.u1old + self.dt * (0.5 * (self.u1old + self.u1trial) - 0.5 * (self.u2old + self.u2trial)))*dol.dx
+ dol.inner(self.v2, self.u2trial - self.u2old - self.k * self.dt * 0.5 * (self.u2old + self.u2trial))*dol.dx)
prob = dol.LinearVariationalProblem(dol.lhs(F), dol.rhs(F), self.unew)
self.solver = dol.LinearVariationalSolver(prob)
## Implicit Euler weak formulation for error estimation
Flow = (dol.inner(self.v1, self.u1trial - self.u1old + self.dt * (self.u1trial - self.u2trial))*dol.dx
+ dol.inner(self.v2, self.u2trial - self.u2old - self.k * self.dt * self.u2trial)*dol.dx)
problow = dol.LinearVariationalProblem(dol.lhs(Flow), dol.rhs(Flow), self.ulow)
self.solver_low = dol.LinearVariationalSolver(problow)
if adjoint:
(self.z1old , self.z2old) = dol.split(self.zold)
(self.z1trial, self.z2trial) = dol.split(self.ztrial)
if self.func not in [1, 2]:
raise ValueError('DWR not (yet) implemented for this functional')
adj_src = dol.Function(self.V)
if self.func == 1: adj_src.interpolate(dol.Constant((1, 0)))
elif self.func == 2: adj_src.interpolate(dol.Constant((0, 1)))
src1, src2 = dol.split(adj_src)
Fadj = (dol.inner(self.z1trial - self.z1old + 0.5 * self.dt * (-self.z1trial - self.z1old + 2*src1), self.v1)*dol.dx +
dol.inner(self.z2trial - self.z2old + 0.5 * self.dt * ( self.z1trial + self.z1old + self.k*(self.z2trial + self.z2old) + 2*src2), self.v2)*dol.dx)
prob_adj = dol.LinearVariationalProblem(dol.lhs(Fadj), dol.rhs(Fadj), self.znew)
self.solver_adj = dol.LinearVariationalSolver(prob_adj)
def solveFwd(self, state, x):
""" Solve the possibly nonlinear forward problem:
Given :math:`m`, find :math:`u` such that
.. math:: \\delta_p F(u, m, p;\\hat{p}) = 0,\\quad \\forall \\hat{p}."""
if self.solver is None:
self.solver = self._createLUSolver()
if self.is_fwd_linear:
u = dl.TrialFunction(self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
A_form = dl.lhs(res_form)
b_form = dl.rhs(res_form)
A, b = dl.assemble_system(A_form, b_form, bcs=self.bc)
self.solver.set_operator(A)
self.solver.solve(state, b)
else:
u = vector2Function(x[STATE], self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
dl.solve(res_form == 0, u, self.bc)
state.zero()
state.axpy(1., u.vector())
def stokes(self):
P2 = VectorElement("CG", self.mesh.ufl_cell(), 2)
P1 = FiniteElement("CG", self.mesh.ufl_cell(), 1)
TH = P2 * P1
VQ = FunctionSpace(self.mesh, TH)
mf = self.mf
self.no_slip = Constant((0., 0))
self.topflow = Expression(("-x[0] * (x[0] - 1.0) * 6.0 * m", "0.0"),
m=self.U_m,
degree=2)
bc0 = DirichletBC(VQ.sub(0), self.topflow, mf, self.bc_dict["top"])
bc1 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["left"])
bc2 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["bottom"])
bc3 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["right"])
# bc4 = DirichletBC(VQ.sub(1), Constant(0), mf, self.bc_dict["top"])
bcs = [bc0, bc1, bc2, bc3]
vup = TestFunction(VQ)
up = TrialFunction(VQ)
# the solution will be in here:
up_ = Function(VQ)
u, p = split(up) # Trial
vu, vp = split(vup) # Test
u_, p_ = split(up_) # Function holding the solution
F = self.mu*inner(grad(vu), grad(u))*dx - inner(div(vu), p)*dx \
- inner(vp, div(u))*dx + dot(self.g*self.rho, vu)*dx
solve(lhs(F) == rhs(F), up_, bcs=bcs)
self.u_.assign(project(u_, self.V))
self.p_.assign(project(p_, self.Q))
return
def EvaluateImpl(self, inputs):
"""
"""
m = dl.Function(self.V)
m.vector().set_local(inputs[0])
p_n = dl.interpolate(m, self.V)
p_nm1 = dl.interpolate(m, self.V)
p_trial = self.p_trial
v = self.v
F = (self.c**2) * (self.dt**2) * dl.inner(
dl.grad(p_trial), dl.grad(v)
) * dl.dx - 2. * p_n * v * dl.dx + p_trial * v * dl.dx + p_nm1 * v * dl.dx
a, L = dl.lhs(F), dl.rhs(F)
# Time-stepping
p = dl.Function(self.V)
t = 0
for n in range(self.numSteps):
# Update current timtime
t += self.dt
# Compute solution
dl.solve(a == L, p)
# Update previous solution
p_nm1.assign(p_n)
p_n.assign(p)
out = p.vector().array()[:]
self.outputs = [out]
def stokes(self):
P2 = df.VectorElement("CG", self.mesh.ufl_cell(), 2)
P1 = df.FiniteElement("CG", self.mesh.ufl_cell(), 1)
TH = P2 * P1
VQ = df.FunctionSpace(self.mesh, TH)
mf = self.mf
self.no_slip = df.Constant((0., 0))
U0_str = "4.*U_m*x[1]*(.41-x[1])/(.41*.41)"
U0 = df.Expression((U0_str, "0"), U_m=self.U_m, degree=2)
bc0 = df.DirichletBC(VQ.sub(0), df.Constant((0, 0)), mf,
self.bc_dict["obstacle"])
bc1 = df.DirichletBC(VQ.sub(0), df.Constant((0, 0)), mf,
self.bc_dict["channel_walls"])
bc2 = df.DirichletBC(VQ.sub(0), U0, mf, self.bc_dict["inlet"])
bc3 = df.DirichletBC(VQ.sub(1), df.Constant(0), mf,
self.bc_dict["outlet"])
bcs = [bc0, bc1, bc2, bc3]
vup = df.TestFunction(VQ)
up = df.TrialFunction(VQ)
up_ = df.Function(VQ)
u, p = df.split(up) # Trial
vu, vp = df.split(vup) # Test
u_, p_ = df.split(up_) # Function holding the solution
inner, grad, dx, div = df.inner, df.grad, df.dx, df.div
F = self.mu*inner(grad(vu), grad(u))*dx - inner(div(vu), p)*dx \
- inner(vp, div(u))*dx
df.solve(df.lhs(F) == df.rhs(F), up_, bcs=bcs)
self.u_.assign(df.project(u_, self.V))
self.p_.assign(df.project(p_, self.Q))
return
def setup_EC(w_EC, c, V, b, U, rho_e,
dx, ds,
dirichlet_bcs, neumann_bcs, boundary_to_mark,
c_1, u_1, K_, veps_, phi_,
per_tau, z, dbeta,
enable_NS, enable_PF,
use_iterative_solvers):
""" Set up electrochemistry subproblem. """
F_V = veps_*df.dot(df.grad(V), df.grad(U))*dx
for boundary_name, sigma_e in neumann_bcs["V"].iteritems():
F_V += -sigma_e*U*ds(boundary_to_mark[boundary_name])
if rho_e != 0:
F_V += -V*U*dx
F = F_V
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_EC, dirichlet_bcs)
solver = df.LinearVariationalSolver(problem)
for ci, ci_1, bi, Ki_, zi, dbetai in zip(c, c_1, b, K_, z, dbeta):
??
def solve_petsc(self):
"""
Solves the linear problem using PETSc interface, and manipulates with PETSc matrices.
:return: list[state vectors]
"""
form = self.forms._rhs_forms(
shift=self.frequency) + self.forms._lhs_forms()
w = dolf.Function(self.forms.function_space)
lhs_matrix = dolf.PETScMatrix()
lhs_matrix = dolf.assemble(dolf.lhs(form), tensor=lhs_matrix)
for bc in self.forms.dirichlet_boundary_conditions():
bc.apply(lhs_matrix)
lhs_matrix = lhs_matrix.mat()
averaged_boundary_terms = self.forms.boundary_averaged_velocity()
if averaged_boundary_terms:
lhs_matrix.axpy(-1.0, averaged_boundary_terms)
solver = self.create_ksp_solver(lhs_matrix)
rhs_vector = dolf.assemble(dolf.rhs(form))
for bc in self.forms.dirichlet_boundary_conditions():
bc.apply(rhs_vector)
solver.solve(
dolf.as_backend_type(rhs_vector).vec(),
dolf.as_backend_type(w.vector()).vec(),
)
state = w.split(True)
return state
def test_lhs_rhs_simple():
"""Test taking lhs/rhs of DOLFIN specific forms (constants
without cell). """
mesh = RectangleMesh.create(MPI.comm_world,
[Point(0, 0), Point(2, 1)], [3, 5],
CellType.Type.triangle)
V = FunctionSpace(mesh, "CG", 1)
f = 2.0
g = 3.0
v = TestFunction(V)
u = TrialFunction(V)
F = inner(g * grad(f * v), grad(u)) * dx + f * v * dx
a, L = system(F)
Fl = lhs(F)
Fr = rhs(F)
assert (Fr)
a0 = inner(grad(v), grad(u)) * dx
n = assemble(a).norm("frobenius") # noqa
nl = assemble(Fl).norm("frobenius") # noqa
n0 = 6.0 * assemble(a0).norm("frobenius") # noqa
assert round(n - n0, 7) == 0
assert round(n - nl, 7) == 0
def __init__(self, domain):
rho, mu, dt, g = domain.rho, domain.mu, domain.dt, domain.g
u, u_1, vu = domain.u, domain.u_1, domain.vu
p_1 = domain.p_1
n = FacetNormal(domain.mesh)
acceleration = rho * inner((u - u_1) / dt, vu) * dx
pressure = inner(p_1, div(vu)) * dx - dot(p_1 * n, vu) * ds
body_force = dot(g*rho, vu)*dx \
+ dot(Constant((0.0, 0.0)), vu) * ds
# diffusion = (-inner(mu * (grad(u_1) + grad(u_1).T), grad(vu))*dx
# + dot(mu * (grad(u_1) + grad(u_1).T)*n, vu)*ds) # just fine
# diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx) # just fine
# just fine, but horribly slow in combination with ??? -> not reproducable
diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu)) * dx +
dot(mu * (grad(u) + grad(u).T) * n, vu) * ds)
# convection = rho*dot(dot(u, nabla_grad(u_1)), vu) * dx # no vortices
convection = rho * dot(dot(u_1, nabla_grad(u)), vu) * dx
# stabilization = -gamma*psi_p*p_1
# convection = dot(div(rho * outer(u_1, u_1)), vu) * dx # not stable!
# convection = rho * dot(dot(u_1, nabla_grad(u_1)), vu) * dx # just fine
F_impl = -acceleration - convection + diffusion + pressure + body_force
self.a, self.L = lhs(F_impl), rhs(F_impl)
self.domain = domain
self.A = assemble(self.a)
[bc.apply(self.A) for bc in domain.bcu]
return
def __init__(self, parameters, domain):
rho = Constant(parameters["density [kg/m3]"])
mu = Constant(parameters["viscosity [Pa*s]"])
dt = Constant(parameters["dt [s]"])
u, u_1, u_k, vu = domain.u, domain.u_1, domain.u_k, domain.vu
p_1 = domain.p_1
n = FacetNormal(domain.mesh)
acceleration = rho*inner((u-u_1)/dt, vu) * dx
convection = dot(div(rho*outer(u_k, u)), vu) * dx
convection = rho*dot(dot(u_k, nabla_grad(u)), vu) * dx
pressure = (inner(p_1, div(vu))*dx - dot(p_1*n, vu)*ds)
diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx) # good
# diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx
# + dot(mu * (grad(u) + grad(u).T)*n, vu)*ds) # very slow!
# F_impl = acceleration + convection + pressure + diffusion
# dot(u_1, nabla_grad(u_1)) works
# dot(u, nabla_grad(u_1)) does not change!
u_mid = (u + u_1) / 2.0
F_impl = rho*dot((u - u_1) / dt, vu)*dx \
+ rho*dot(dot(u_1, nabla_grad(u_1)), vu)*dx \
+ inner(sigma(u_mid, p_1, mu), epsilon(vu))*dx \
+ dot(p_1*n, vu)*ds - dot(mu*nabla_grad(u_mid)*n, vu)*ds
self.a, self.L = lhs(F_impl), rhs(F_impl)
self.domain = domain
self.A = assemble(self.a)
[bc.apply(self.A) for bc in domain.bcu]
return
def setup_NS(w_NS, u, p, v, q, p0, q0,
dx, ds, normal,
dirichlet_bcs_NS, neumann_bcs, boundary_to_mark,
u_1, rho_, rho_1, mu_, c_1, grad_g_c_,
dt, grav,
enable_EC,
trial_functions,
use_iterative_solvers,
p_lagrange,
mesh,
q_rhs,
density_per_concentration,
K,
**namespace):
""" Set up the Navier-Stokes subproblem. """
mom_1 = rho_1 * u_1
if enable_EC and density_per_concentration is not None:
for drhodci, ci_1, grad_g_ci_, Ki in zip(
density_per_concentration, c_1, grad_g_c_, K):
if drhodci > 0.:
mom_1 += -drhodci*Ki*ci_1*grad_g_ci_
F = (1./dt * rho_1 * df.dot(u - u_1, v) * dx
+ df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx
+ 2*mu_*df.inner(df.sym(df.nabla_grad(u)),
df.sym(df.nabla_grad(v))) * dx
+ 0.5*(
1./dt * (rho_ - rho_1) * df.inner(u, v)
- df.inner(mom_1, df.nabla_grad(df.dot(u, v)))) * dx
- p * df.div(v) * dx
- q * df.div(u) * dx
- rho_ * df.dot(grav, v) * dx)
for boundary_name, pressure in neumann_bcs["p"].iteritems():
F += pressure * df.inner(
normal, v) * ds(boundary_to_mark[boundary_name])
if enable_EC:
F += sum([ci_1*df.dot(grad_g_ci_, v)*dx
for ci_1, grad_g_ci_ in zip(c_1, grad_g_c_)])
if p_lagrange:
F += (p*q0 + q*p0)*dx
if "u" in q_rhs:
F += -df.dot(q_rhs["u"], v)*dx
a, L = df.lhs(F), df.rhs(F)
if not use_iterative_solvers:
problem = df.LinearVariationalProblem(a, L, w_NS, dirichlet_bcs_NS)
solver = df.LinearVariationalSolver(problem)
else:
solver = df.LUSolver("mumps")
# solver.set_operator(A)
return solver, a, L, dirichlet_bcs_NS
return solver
def solve(self):
super(backwardEuler, self).solve()
# Array for storing the stage values
X = [self.u.copy(deepcopy=True)]
# Get the variational linear/nonlinear variational forms
# and embed them in respective solver class
if self.linear:
l = self.getLinearVariationalForms(X)
p = [linearStage(d.lhs(l[0]), d.rhs(l[0]), self.bcs, self.solver)]
else:
l = self.getNonlinearVariationalForms(X)
a = [d.derivative(l[0], X[0], self.U)]
p = [nonlinearStage(a[0], l[0], self.bcs)]
# Initialize save/plot of function(s)
self.figureHandling(Init=True)
# Time stepping loop
while True:
timestepStart = time.time()
# Update time dependent functions
for i in range(len(self.tdfButcher)):
for j in range(0, 1):
self.tdfButcher[i][j].t = self.t + self.dt
# Update time dependent functions on boundary
for F in self.tdfBC:
F.t = self.t + self.dt
# Solve for implicit stages
if self.linear:
p[0].solve(X[0].vector())
else:
self.solver.solve(p[0], X[0].vector())
# Constant step size integration
self.u.vector()[:] = X[0].vector()[:]
self.t += self.dt
self.nAcc += 1
if self.parameters['verbose']:
self.printProgress(self.estimateRuntime)
# Update / save plots
self.figureHandling(Update=True)
# Break if this is final time step
if self.breakTimeLoop:
terminateReason = "Success"
break
self.timestepTimer = time.time() - timestepStart
self.verifyStepsize()
super(backwardEuler, self).terminateTimeLoop(terminateReason)
def create_rhs(self):
self.L = self.create_bc()
if self.moment_order == 3:
self.L += self.source_term()
if self.stab_type == 'gls':
self.L += df.rhs(self.apply_stabilization())
if self.problem_type == 'nonlinear':
self.L += self.add_nonlinearity()
self.F = self.a - self.L
def get_system(self, t):
kappa.t = t
f.t = t
n = FacetNormal(self.V.mesh())
u = TrialFunction(self.V)
v = TestFunction(self.V)
F = inner(kappa * grad(u), grad(v / self.rho_cp)) * dx \
- inner(kappa * grad(u), n) * v / self.rho_cp * ds \
- f * v / self.rho_cp * dx
return assemble(lhs(F)), assemble(rhs(F))
def __init__(self, t_end=None, func=None):
Problem_verification.__init__(self, t_end, func)
self.u0_expr = dol.Constant(1)
self.V = dol.FunctionSpace(dol.UnitIntervalMesh(1), 'DG', 0)
Problem_FE.__init__(self)
# CN, higher order
F = dol.inner(
self.v, self.utrial - self.uold + self.dt * 0.5 *
(self.uold + self.utrial)) * dol.dx
prob = dol.LinearVariationalProblem(dol.lhs(F), dol.rhs(F), self.unew)
self.solver = dol.LinearVariationalSolver(prob)
# IE, lower order
Flow = dol.inner(
self.v, self.utrial - self.uold + self.dt * self.utrial) * dol.dx
problow = dol.LinearVariationalProblem(dol.lhs(Flow), dol.rhs(Flow),
self.ulow)
self.solver_low = dol.LinearVariationalSolver(problow)
def navier_stokes_IPCS(mesh, dt, parameter):
"""
fenics code: weak form of the problem.
"""
mu, rho, nu = parameter
V = VectorFunctionSpace(mesh, 'P', 2)
Q = FunctionSpace(mesh, 'P', 1)
bc0 = DirichletBC(V, Constant((0, 0)), cylinderwall)
bc1 = DirichletBC(V, Constant((0, 0)), topandbottom)
bc2 = DirichletBC(V, U0, inlet)
bc3 = DirichletBC(Q, Constant(1), outlet)
bcs = [bc0, bc1, bc2, bc3]
# ds is needed to compute drag and lift. Not used here.
ASD1 = AutoSubDomain(topandbottom)
ASD2 = AutoSubDomain(cylinderwall)
mf = MeshFunction("size_t", mesh, 1)
mf.set_all(0)
ASD1.mark(mf, 1)
ASD2.mark(mf, 2)
ds_ = ds(subdomain_data=mf, domain=mesh)
vu, vp = TestFunction(V), TestFunction(Q) # for integration
u_, p_ = Function(V), Function(Q) # for the solution
u_1, p_1 = Function(V), Function(Q) # for the prev. solution
u, p = TrialFunction(V), TrialFunction(Q) # unknown!
bcu = [bcs[0], bcs[1], bcs[2]]
bcp = [bcs[3]]
n = FacetNormal(mesh)
u_mid = (u + u_1) / 2.0
F1 = rho*dot((u - u_1) / dt, vu)*dx \
+ rho*dot(dot(u_1, nabla_grad(u_1)), vu)*dx \
+ inner(sigma(u_mid, p_1, mu), epsilon(vu))*dx \
+ dot(p_1*n, vu)*ds - dot(mu*nabla_grad(u_mid)*n, vu)*ds
a1 = lhs(F1)
L1 = rhs(F1)
# Define variational problem for step 2
a2 = dot(nabla_grad(p), nabla_grad(vp)) * dx
L2 = dot(nabla_grad(p_1), nabla_grad(vp)) * dx - (
rho / dt) * div(u_) * vp * dx # rho missing in FEniCS tutorial
# Define variational problem for step 3
a3 = dot(u, vu) * dx
L3 = dot(u_, vu) * dx - dt * dot(nabla_grad(p_ - p_1), vu) * dx
# Assemble matrices
A1 = assemble(a1)
A2 = assemble(a2)
A3 = assemble(a3)
# Apply boundary conditions to matrices
[bc.apply(A1) for bc in bcu]
[bc.apply(A2) for bc in bcp]
return u_, p_, u_1, p_1, L1, A1, L2, A2, L3, A3, bcu, bcp
def solve(self):
form = self.forms._rhs_forms(
shift=self.frequency) + self.forms._lhs_forms()
w = dolf.Function(self.forms.function_space)
dolf.solve(
dolf.lhs(form) == dolf.rhs(form),
w,
self.forms.dirichlet_boundary_conditions(),
solver_parameters={"linear_solver": "mumps"},
)
state = w.split(True)
return state
def soln_fwd(self):
"""
Solve the forward equation.
F = 0
"""
# 5. Solve (non)linear variational problem
# df.solve(self.F==0,self.states_fwd,self.ess_bc,J=self.dFdstates)
# self.states_fwd = df.Function(self.W)
df.solve(df.lhs(self.F)==df.rhs(self.F),self.states_fwd,self.ess_bc)
# df.solve(self.a==self.L,self.states_fwd,self.ess_bc)
self.soln_count[0] += 1
u_fwd, l_fwd = df.split(self.states_fwd)
return u_fwd, l_fwd
def rhs(form):
"""
Wrapper for the DOLFIN rhs function. Correctly handles QForm s.
"""
if not isinstance(form, ufl.form.Form):
raise InvalidArgumentException("form must be a Form")
nform = dolfin.rhs(form)
if isinstance(form, QForm):
return QForm(nform, quadrature_degree = form_quadrature_degree(form))
else:
return nform
def matrix_optimisation(form):
"""
Attempt to convert a linear form into the action of a bi-linear form.
Return a (bi-linear Form, Function) pair on success, and None on failure.
"""
if not isinstance(form, ufl.form.Form):
raise InvalidArgumentException("form must be a Form")
# Find the test function
args = ufl.algorithms.extract_arguments(form)
if not len(args) == 1:
# This is not a linear form
return None
# Look for a single non-static Function dependency
tcs = extract_non_static_coefficients(form)
if not len(tcs) == 1:
# Found too many non-static coefficients
return None
elif not isinstance(tcs[0], dolfin.Function):
# The only non-static coefficient is not a Function
return None
# Found a single non-static Function dependency
fn = tcs[0]
# Look for a static bi-linear form whose action can be used to construct
# the linear form
mat_form = derivative(
form,
fn,
# Hack to work around an obscure FEniCS bug
expand=dolfin.MPI.size(dolfin.mpi_comm_world()) == 1
or (not is_r0_function_space(args[0].function_space())
and not is_r0_function(fn)))
if n_non_static_coefficients(mat_form) > 0:
# The form is non-linear
return None
try:
rhs_form = dolfin.rhs(
dolfin.replace(form,
{fn: dolfin.TrialFunction(fn.function_space())}))
except ufl.log.UFLException:
# The form might be inhomogeneous
return None
if not is_empty_form(rhs_form):
# The form might be inhomogeneous
return None
# Success
return mat_form, fn
def __init__(self, domain):
rho, mu, dt = domain.rho, domain.mu, domain.dt
u, u_1, p_1, vu = domain.u, domain.u_1, domain.p_1, domain.vu
n = FacetNormal(domain.mesh)
acceleration = rho * inner((u - u_1) / dt, vu) * dx
diffusion = (-inner(mu * (grad(u_1) + grad(u_1).T), grad(vu)) * dx +
dot(mu * (grad(u_1) + grad(u_1).T) * n, vu) * ds)
pressure = inner(p_1, div(vu)) * dx - dot(p_1 * n, vu) * ds
convection = rho * dot(dot(u_1, nabla_grad(u_1)), vu) * dx
F_impl = -acceleration - convection + diffusion + pressure
self.a, self.L = lhs(F_impl), rhs(F_impl)
self.domain = domain
self.A = assemble(self.a)
[bc.apply(self.A) for bc in domain.bcu]
return
def setup_NSp(w_NSp, p, q, dirichlet_bcs_NSp, dt, u_, p_1, rho_0,
use_iterative_solvers, **namespace):
""" Set up Navier-Stokes pressure subproblem. """
F = (df.dot(df.nabla_grad(p - p_1), df.nabla_grad(q)) * df.dx +
1. / dt * rho_0 * df.div(u_) * q * df.dx)
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_NSp, dirichlet_bcs_NSp)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "bicgstab"
solver.parameters["preconditioner"] = "amg"
return solver
def setup_NSp(w_NSp, p, q, dx, ds, dirichlet_bcs_NSp, neumann_bcs,
boundary_to_mark, u_, u_1, p_, p_1, rho_, dt, rho_min,
use_iterative_solvers, **namespace):
F = (df.dot(df.nabla_grad(p - p_1), df.nabla_grad(q)) * df.dx +
1. / dt * rho_min * df.div(u_) * q * df.dx)
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_NSp, dirichlet_bcs_NSp)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres"
solver.parameters["preconditioner"] = "hypre_amg" # "amg"
# solver.parameters["preconditioner"] = "hypre_euclid"
return solver
def setup_EC(w_EC, c, V, b, U, rho_e,
dx, ds,
dirichlet_bcs, neumann_bcs, boundary_to_mark,
c_1, u_1, K_, veps_, phi_,
solutes,
per_tau, z, dbeta,
enable_NS, enable_PF,
use_iterative_solvers,
q_rhs):
""" Set up electrochemistry subproblem. """
F_c = []
for ci, ci_1, bi, Ki_, zi, dbetai, solute in zip(c, c_1, b, K_, z, dbeta, solutes):
F_ci = (per_tau*(ci-ci_1)*bi*dx +
Ki_*df.dot(df.nabla_grad(ci), df.nabla_grad(bi))*dx)
if zi != 0:
F_ci += Ki_*zi*ci_1*df.dot(df.nabla_grad(V), df.nabla_grad(bi))*dx
if enable_PF:
F_ci += Ki_*ci*dbetai*df.dot(df.nabla_grad(phi_), df.nabla_grad(bi))*dx
if enable_NS:
# F_ci += df.div(ci*u_1)*bi*dx
F_ci += - ci*df.dot(u_1, df.grad(bi))*dx
if solute[0] in q_rhs:
F_ci += - q_rhs[solute[0]]*bi*dx
F_c.append(F_ci)
F_V = veps_*df.dot(df.nabla_grad(V), df.nabla_grad(U))*dx
for boundary_name, sigma_e in neumann_bcs["V"].items():
F_V += -sigma_e*U*ds(boundary_to_mark[boundary_name])
if rho_e != 0:
F_V += -rho_e*U*dx
if "V" in q_rhs:
F_V += q_rhs["V"]*U*dx
F = sum(F_c) + F_V
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_EC, dirichlet_bcs)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres"
# solver.parameters["preconditioner"] = "hypre_euclid"
return solver
def EvaluateImpl(self, inputs):
"""
"""
numObs = self.numObs
numSteps = self.numSteps
# Each
output = np.zeros((numObs * numSteps))
m = dl.Function(self.V)
m.vector().set_local(inputs[0])
p_n = dl.Function(self.V)
p_nm1 = dl.Function(self.V)
p_n.assign(m)
p_nm1.assign(m)
p_trial = self.p_trial
v = self.v
F = (self.c**2) * (self.dt**2) * dl.inner(
dl.grad(p_trial), dl.grad(v)
) * dl.dx - 2. * p_n * v * dl.dx + p_trial * v * dl.dx + p_nm1 * v * dl.dx
a, L = dl.lhs(F), dl.rhs(F)
# Time-stepping
p = dl.Function(self.V)
t = 0
output[0:numObs] = self.ObservationOperator(p_nm1)
output[numObs:2 * numObs] = self.ObservationOperator(p_n)
for n in range(2, self.numSteps):
# Update current timtime
t += self.dt
# Compute solution
dl.solve(a == L, p)
# nb.plot(p)
# plt.title("p")
# plt.show()
output[n * numObs:(n + 1) * numObs] = self.ObservationOperator(p)
# Update previous solution
p_nm1.assign(p_n)
p_n.assign(p)
self.outputs = [output]
def algo_fvs_to_cell_averages(self,
mesh_util,
bcdata,
fv_adjacent_cells_function,
target_unit=None):
'''Algorithm for turning BC data into cell averages.
Parameters
----------
mesh_util: :py:class:`.mesh_util.MeshUtil`
Mesh utilities and data.
bcdata: dict
Mapping `{expr: {(facet_value, sign), ...}}`.
target_unit: optional
Unit to ``expr`` to in the ``bcdict`` argument.
fv_adjacent_cells_function: :py:class:`dolfin.Function`
DG0 function containing ones and zeros; ones iff the cell is
adjacent to a BC.
'''
mu = mesh_util
dless = mu.unit_registry('dimensionless')
DG0 = mu.space.DG0
dx = mu.dx
u = dolfin.TrialFunction(DG0)
v = dolfin.TestFunction(DG0)
form = DelayedForm()
for expr, fvss in bcdata.items():
expr = dless * expr.object
if target_unit is None:
target_unit = expr.units
expr_ = expr.m_as(target_unit)
for fv, sign in fvss:
# TODO: figure out what to do with the sign - relevant
# for internal boundary conditions
form += mu.ds(subdomain_id=fv) * v('+') * (u('+') - expr_)
if target_unit is None:
target_unit = dless
form += dx * u * v * (1.0 - fv_adjacent_cells_function)
form = form.delete_units().to_ufl().m
u = dolfin.Function(DG0)
dolfin.solve(dolfin.lhs(form) == dolfin.rhs(form), u, [])
return target_unit * u
def setup_PF(w_PF, phi, g, psi, h,
dx, ds,
dirichlet_bcs, neumann_bcs, boundary_to_mark,
phi_1, u_1, M_1, c_1, V_1,
per_tau, sigma_bar, eps,
dbeta, dveps,
enable_NS, enable_EC,
use_iterative_solvers,
q_rhs):
""" Set up phase field subproblem. """
F_phi = (per_tau*(phi-unit_interval_filter(phi_1))*psi*dx +
M_1*df.dot(df.grad(g), df.grad(psi))*dx)
if enable_NS:
F_phi += -phi*df.dot(u_1, df.grad(psi))*dx
# F_phi += df.div(phi*u_1)*psi*dx
F_g = (g*h*dx
- sigma_bar*eps*df.dot(df.nabla_grad(phi), df.nabla_grad(h))*dx
- sigma_bar/eps*(
diff_pf_potential_linearised(phi,
unit_interval_filter(
phi_1))*h*dx))
if enable_EC:
F_g += (-sum([dbeta_i*ci_1*h*dx
for dbeta_i, ci_1 in zip(dbeta, c_1)])
+ 0.5*dveps*df.dot(df.nabla_grad(V_1), df.nabla_grad(V_1))*h*dx)
for boundary_name, costheta in neumann_bcs["phi"].items():
fw_prime = diff_pf_contact_linearised(phi, unit_interval_filter(phi_1))
# Should be just surface tension!
F_g += sigma_bar*costheta*fw_prime*h*ds(boundary_to_mark[boundary_name])
if "phi" in q_rhs:
F_phi += -q_rhs["phi"]*psi*dx
F = F_phi + F_g
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_PF)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres"
# solver.parameters["preconditioner"] = "hypre_euclid"
return solver
def setSolverParams(self, step=None):
if self.eqn == "poisboltz":
print("PoisBoltz")
self.F = dolfin.action(self.F, self.u_s)
J = dolfin.derivative(self.F, self.u_s, self.u)
problem = dolfin.NonlinearVariationalProblem(
self.F, self.u_s, self.bcs, J)
self.solver = dolfin.NonlinearVariationalSolver(problem)
prm = self.solver.parameters
prm['newton_solver']['absolute_tolerance'] = self.max_abs_error
prm['newton_solver']['relative_tolerance'] = self.max_rel_error
prm['newton_solver']['maximum_iterations'] = 10000
prm['newton_solver']['relaxation_parameter'] = 0.1
prm['newton_solver']['report'] = True
prm['newton_solver']['linear_solver'] = self.method
prm['newton_solver']['preconditioner'] = self.preconditioner
prm['newton_solver']['krylov_solver']['maximum_iterations'] = 10000
prm['newton_solver']['krylov_solver'][
'absolute_tolerance'] = self.max_abs_error
prm['newton_solver']['krylov_solver'][
'relative_tolerance'] = self.max_rel_error
prm['newton_solver']['krylov_solver']['monitor_convergence'] = True
else:
print("Separating LHS and RHS...")
# Separate left and right hand sides of equation
self.a, self.L = dolfin.lhs(self.F), dolfin.rhs(self.F)
self.problem = dolfin.LinearVariationalProblem(
self.a, self.L, self.u_s, self.bcs)
self.solver = dolfin.LinearVariationalSolver(self.problem)
dolfin.parameters['form_compiler']['optimize'] = True
self.solver.parameters['linear_solver'] = self.method
self.solver.parameters['preconditioner'] = self.preconditioner
spec_param = self.solver.parameters['krylov_solver']
#These only accessible after spec_param available.
if self.init_guess == "prev":
if step == 0:
spec_param['nonzero_initial_guess'] = False
else:
spec_param['nonzero_initial_guess'] = True
elif self.init_guess == "zero":
spec_param['nonzero_initial_guess'] = False
spec_param['absolute_tolerance'] = self.max_abs_error
spec_param['relative_tolerance'] = self.max_rel_error
spec_param['maximum_iterations'] = self.max_linear_iters
def __init__(
self,
Q,
kappa,
rho,
cp,
convection,
source,
dirichlet_bcs=None,
neumann_bcs=None,
robin_bcs=None,
my_dx=dx,
my_ds=ds,
stabilization=None,
):
super(Heat, self).__init__()
self.Q = Q
dirichlet_bcs = dirichlet_bcs or []
neumann_bcs = neumann_bcs or {}
robin_bcs = robin_bcs or {}
self.convection = convection
u = TrialFunction(Q)
v = TestFunction(Q)
# If there are sharp temperature gradients, numerical oscillations may
# occur. This happens because the resulting matrix is not an M-matrix,
# caused by the fact that A1 puts positive elements in places other
# than the main diagonal. To prevent that, it is suggested by
# Großmann/Roos to use a vertex-centered discretization for the mass
# matrix part.
# Check
# https://bitbucket.org/fenics-project/ffc/issues/145/uflacs-error-for-vertex-quadrature-scheme
#
self.M = assemble(
u * v * dx,
form_compiler_parameters={
"representation": "quadrature",
"quadrature_rule": "vertex",
},
)
mesh = Q.mesh()
r = SpatialCoordinate(mesh)[0]
self.F0 = F(
u,
v,
kappa,
rho,
cp,
convection,
source,
r,
neumann_bcs,
robin_bcs,
my_dx,
my_ds,
stabilization,
)
self.dirichlet_bcs = dirichlet_bcs
self.A, self.b = assemble_system(-lhs(self.F0), rhs(self.F0))
return
def get_system(self, t):
# Don't use assemble_system()! See bugs
# <https://bitbucket.org/fenics-project/dolfin/issue/257/system_assembler-bilinear-and-linear-forms>,
# <https://bitbucket.org/fenics-project/dolfin/issue/78/systemassembler-problem-with-subdomains-on>.
return assemble(lhs(self.F0)), assemble(rhs(self.F0))
def stokes_solve(up_out, mu, u_bcs, p_bcs, f, my_dx=dx):
# Some initial sanity checks.
assert mu > 0.0
WP = up_out.function_space()
# Translate the boundary conditions into the product space.
new_bcs = helpers.dbcs_to_productspace(WP, [u_bcs, p_bcs])
# TODO define p*=-1 and reverse sign in the end to get symmetric system?
# Define variational problem
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
mesh = WP.mesh()
r = SpatialCoordinate(mesh)[0]
# build system
f = F(u, p, v, q, f, r, mu, my_dx)
a = lhs(f)
L = rhs(f)
A, b = assemble_system(a, L, new_bcs)
mode = "lu"
assert mode == "lu"
solve(A, up_out.vector(), b, "lu")
# TODO Krylov solver for Stokes
# assert mode == 'gmres'
# # For preconditioners for the Stokes system, see
# #
# # Fast iterative solvers for discrete Stokes equations;
# # J. Peters, V. Reichelt, A. Reusken.
# #
# prec = mu * inner(r * grad(u), grad(v)) * 2 * pi * my_dx \
# - p * q * 2 * pi * r * my_dx
# P, _ = assemble_system(prec, L, new_bcs)
# solver = KrylovSolver('tfqmr', 'hypre_amg')
# # solver = KrylovSolver('gmres', 'hypre_amg')
# solver.set_operators(A, P)
# solver.parameters['monitor_convergence'] = verbose
# solver.parameters['report'] = verbose
# solver.parameters['absolute_tolerance'] = 0.0
# solver.parameters['relative_tolerance'] = tol
# solver.parameters['maximum_iterations'] = maxiter
# # Solve
# solver.solve(up_out.vector(), b)
# elif mode == 'fieldsplit':
# # For an assortment of preconditioners, see
# #
# # Performance and analysis of saddle point preconditioners
# # for the discrete steady-state Navier-Stokes equations;
# # H.C. Elman, D.J. Silvester, A.J. Wathen;
# # Numer. Math. (2002) 90: 665-688;
# # <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.3554>.
# #
# # Set up field split.
# W = SubSpace(WP, 0)
# P = SubSpace(WP, 1)
# u_dofs = W.dofmap().dofs()
# p_dofs = P.dofmap().dofs()
# prec = PETScPreconditioner()
# prec.set_fieldsplit([u_dofs, p_dofs], ['u', 'p'])
# PETScOptions.set('pc_type', 'fieldsplit')
# PETScOptions.set('pc_fieldsplit_type', 'additive')
# PETScOptions.set('fieldsplit_u_pc_type', 'lu')
# PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
# # Create Krylov solver with custom preconditioner.
# solver = PETScKrylovSolver('gmres', prec)
# solver.set_operator(A)
return
# Elasticity parameters:
E, nu = 10., 0.3
mu, lambda_param = E / (2. * (1.+nu)), E * nu / ((1. + nu) * (1.-2. * nu))
# Stress tensor:
# + usually of form \sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij},
# + or = \lambda Tr(e_{ij})I + 2\mu e_{ij}, for e_{ij} the strain tensor.
sigma = lambda_param*d.tr(d.grad(u)) * d.Identity(w.cell().d) + 2 * mu * d.sym(d.grad(u))
# Governing balance equation:
F = d.inner(sigma, d.grad(w)) * d.dx - d.dot(b,w)*d.dx
# Extract the bi- and linear forms from F:
a = d.lhs(F)
L = d.rhs(F)
# Dirichlet BC on entire boundary:
c = d.Constant((0.,0.,0.))
bc = d.DirichletBC(V, c, d.DomainBoundary())
## Testing some new boundary definitions:
def bzo_boundary(r_vec, on_boundary):
# NB: input r_vec is a position vector
r,theta,z = cart_to_cyl(x) # NB: check the function cart_to_cyl can take non-tuple collecs.
return d.near(r,bzo_radius)
## Testing some differing-boundary BCs:
bzo = d.DirichletBC(V, u_bzo, bzo_boundary)
ybco = d.DirichletBC(V, u_ybco, ybco_boundary)