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 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 _setup_solver(self):
self.EMsolver = d.LinearVariationalSolver(self.EMproblem)
self.Tsolver = d.LinearVariationalSolver(self.Tproblem)
solver_info = return_solver(self.data)
if solver_info is not None:
for i in solver_info:
self.logger.debug("Setting " +
str(self.EMsolver.parameters[i]) + " as " +
str(solver_info[i]))
self.EMsolver.parameters[i] = solver_info[i]
self.Tsolver.parameters[i] = solver_info[i]
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 linear_solver(self, phi_k):
# cast params as constant functions so that, if they are set to 0, FEniCS still understand
# what is being integrated
mu, M = Constant(self.physics.mu), Constant(self.physics.M)
lam = Constant(self.physics.lam)
mn, mf = Constant(self.mn), Constant(self.mf)
D = self.physics.D
# boundary condition
Dirichlet_bc = self.get_Dirichlet_bc()
# trial and test function
phi = d.TrialFunction(self.fem.S)
v = d.TestFunction(self.fem.S)
# r^(D-1)
rD = Expression('pow(x[0],D-1)', D=D, degree=self.fem.func_degree)
# bilinear form a and linear form L
a = - inner( grad(phi), grad(v) ) * rD * dx + ( (mu/mn)**2 \
- 3.*lam*(mf/mn)**2*phi_k**2 ) * phi * v * rD * dx
L = ((mn**(D - 2.) / (mf * M)) * self.source.rho - 2. * lam *
(mf / mn)**2 * phi_k**3) * v * rD * dx
# define a vector with the solution
sol = d.Function(self.fem.S)
# solve linearised system
pde = d.LinearVariationalProblem(a, L, sol, Dirichlet_bc)
solver = d.LinearVariationalSolver(pde)
solver.solve()
return sol
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( self ):
"""
Overrides the base-class nonlinear solver with a linear solver for the Poisson
equation Eq. :eq:`Eq_Poisson`.
"""
Dirichlet_bc = self.get_Dirichlet_bc()
# trial and test function
u = d.TrialFunction( self.fem.S )
v = d.TestFunction( self.fem.S )
PhiN = d.Function( self.fem.S )
# define equation
Mn, Mp = Constant( self.Mn ), Constant( self.Mp )
r2 = Expression( 'pow(x[0],2)', degree=self.fem.func_degree )
a = - inner( grad(u), grad(v) ) * r2 * dx
L = 0.5 * self.source.rho * Mn / Mp**2 * v * r2 * dx
# solve
eqn = d.LinearVariationalProblem( a, L, PhiN, Dirichlet_bc )
solver = d.LinearVariationalSolver( eqn )
solver.solve()
# set potential and force
self.PhiN = PhiN
self.Newton_force = self.grad( -PhiN, radial_units='physical' )
def initial_guess(self):
r"""
Obtains an initial guess for the Newton solver.
This is done by solving the equation of motion for :math:`\lambda=0`, which form a linear system.
All other parameters are unchanged.
"""
# cast params as constant functions so that, if they are set to 0,
# fenics still understand what it is integrating
m, M, Mp = Constant(self.fields.m), Constant(self.fields.M), Constant(
self.fields.Mp)
alpha = Constant(self.fields.alpha)
Mn, Mf1, Mf2 = Constant(self.Mn), Constant(self.Mf1), Constant(
self.Mf2)
# get the boundary conditions
Dirichlet_bc = self.get_Dirichlet_bc()
# create a vector (phi,h) with the two trial functions for the fields
u = d.TrialFunction(self.V)
# ... and split it into phi and h
phi, h, y, z = d.split(u)
# define test functions over the function space
v1, v2, v3, v4 = d.TestFunctions(self.V)
# r^2
r2 = Expression('pow(x[0],2)', degree=self.fem.func_degree)
# define bilinear form
# Eq.1
a1 = y * v1 * r2 * dx - ( m / Mn )**2 * phi * v1 * r2 * dx \
- alpha * ( Mf2/Mf1 ) * z * v1 * r2 * dx
# Eq.2
a2 = z * v2 * r2 * dx - ( M / Mn )**2 * h * v2 * r2 * dx \
- alpha * ( Mf1/Mf2 ) * y * v2 * r2 * dx
a3 = -inner(grad(phi), grad(v3)) * r2 * dx - y * v3 * r2 * dx
a4 = -inner(grad(h), grad(v4)) * r2 * dx - z * v4 * r2 * dx
# both equations
a = a1 + a2 + a3 + a4
# define linear form
L = self.source.rho / Mp * Mn / Mf1 * v1 * r2 * dx
# define a vector with the solution
sol = d.Function(self.V)
# solve linearised system
pde = d.LinearVariationalProblem(a, L, sol, Dirichlet_bc)
solver = d.LinearVariationalSolver(pde)
solver.solve()
return sol
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 __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 dTt_dolfin(T, t):
"""
Finds dT/dt using dolfin.
Arguments:
T: Array representing the temperature at a specific time.
t: Single value of time at which to find the derivative.
Returns:
The derivative of T with respect to t as an array.
"""
# Convert T to dolfin function from array.
TOld = df.Function(funcSpace)
TOld.vector()[:] = T
# Solve the "linear" problem to find dT/dt.
# This 'a' represents the unknown, which contains no new information, but
# will eventually contribute to the solution later.
TNew = df.TrialFunction(funcSpace)
v = df.TestFunction(funcSpace)
a = TNew * v * df.dx
# 'f' here represents an expression (but not a dolfin expression) which
# describes the mathematical function that calculates dT/dt from T.
f = TOld * df.Constant(-0.9) # df.inner(df.grad(TOld), df.grad(TOld)) # <!> Failure here?
# This 'L' represents what we know, and will be used to calculate our
# solution eventually.
L = f * v
L *= df.dx
# This is not actually the solution, but it is where the solution will end
# up eventually, once the solver has done its work.
solutionEventually = df.Function(funcSpace)
# The problem defines what we want to know, what we do know, and where to
# put the solution. The solution argument is not actually the solution
# (yet), but it's where the solution will end up eventually.
problem = df.LinearVariationalProblem(a, L, solutionEventually)
# The solver solves the problem eventually.
solver = df.LinearVariationalSolver(problem)
# Now we solve the problem. solutionEventually is now populated with the
# solution to the problem.
solver.solve()
# Convert and return our solution.
return solutionEventually.vector().array()
def _setup_solver(self):
self.solver = d.LinearVariationalSolver(self.problem)
solver_info = return_solver(self.data)
if solver_info is not None:
for i in solver_info:
if isinstance(solver_info[i], dict):
for j in solver_info[i]:
self.logger.debug("Setting [" + str(i) + "][" +
str(j) + "] as " +
str(solver_info[i][j]))
self.solver.parameters[i][j] = solver_info[i][j]
else:
self.logger.debug("Setting [" + str(i) + "] as " +
str(solver_info[i]))
self.solver.parameters[i] = solver_info[i]
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 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 setup_PF(w_PF, phi, g, psi, h, dx, ds, dirichlet_bcs_PF, neumann_bcs,
boundary_to_mark, phi_1, u_1, M_, M_1, c_1, V_1, rho_1, dt,
sigma_bar, eps, drho, dbeta, dveps, grav, enable_NS, enable_EC,
use_iterative_solvers, **namespace):
""" Set up phase field subproblem. """
# Projected velocity (for energy stability)
if enable_NS:
u_proj = u_1 # - dt*phi_1*df.grad(g)/rho_1
phi_adv = phi # phi_1
F_phi = (1. / dt * (phi - phi_1) * psi * dx +
M_1 * df.dot(df.grad(g), df.grad(psi)) * dx)
if enable_NS:
F_phi += -phi_adv * df.dot(u_proj, df.grad(psi)) * dx
F_g = (
g * h * dx
# - sigma_bar/eps * (phi - phi_1) * h * dx
# Damping term (makes the system elliptic)
- sigma_bar * eps * df.dot(df.grad(phi), df.grad(h)) * dx
# - sigma_bar/eps * diff_pf_potential(phi_1)*h*dx
- sigma_bar / eps * diff_pf_potential_linearised(phi, phi_1) * h * dx)
# Add gravity
# F_g += drho * df.dot(grav, x) * 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.grad(V_1), df.grad(V_1)) * h * dx)
F = F_phi + F_g
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_PF, dirichlet_bcs_PF)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres" # "bicgstab" # "gmres"
solver.parameters["preconditioner"] = "jacobi" #"amg"
# solver.parameters["preconditioner"] = "hypre_euclid"
return solver
def setup_EC(w_EC, c, V, b, U, rho_e, dx, ds, dirichlet_bcs_EC, neumann_bcs,
boundary_to_mark, c_1, u_1, K_, veps_, phi_, rho_1, dt, z, dbeta,
enable_NS, enable_PF, use_iterative_solvers, **namespace):
""" Set up electrochemistry subproblem. """
F_c = []
for ci, ci_1, bi, Ki_, zi, dbetai in zip(c, c_1, b, K_, z, dbeta):
u_proj_i = u_1 # - dt/rho_1*df.grad(ci)
ci_adv = ci # ci_1
F_ci = (1. / dt * (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
# u_proj_i += -dt/rho_1 * zi * ci_1 * df.grad(V)
if enable_PF:
F_ci += Ki_ * ci * dbetai * df.dot(df.nabla_grad(phi_),
df.nabla_grad(bi)) * dx
# u_proj_i += -dt/rho_1 * ci_1 * dbetai*df.grad(phi_)
if enable_NS:
F_ci += -ci_adv * df.dot(u_proj_i, df.nabla_grad(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
F = sum(F_c) + F_V
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_EC, dirichlet_bcs_EC)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres"
solver.parameters["preconditioner"] = "amg"
return solver
def compute_yukawa_force( self ):
# solve an equation of motion that will give you
# the force from a scalar without nonlinearities
# cast params as constant functions so that, if they are set to 0, FEniCS still understand
# what is being integrated
mu = Constant( self.physics.mu )
mn = Constant( self.mn )
# trial and test function
phi = d.TrialFunction( self.fem.S )
v = d.TestFunction( self.fem.S )
# boundary condition - always zero
phiD = d.Constant( 0. )
# define 'infinity' boundary: the rightmost mesh point - within machine precision
def boundary(x):
return self.fem.mesh.r_max - x[0] < d.DOLFIN_EPS
Dirichlet_bc = d.DirichletBC( self.fem.S, phiD, boundary, method='pointwise' )
# r^(D-1)
rD = Expression('pow(x[0],D-1)', D=self.physics.D, degree=self.fem.func_degree)
# m^2 = 2.
a = - inner( grad(phi), grad(v) ) * rD * dx - 2. * (mu/mn)**2 * phi * v * rD * dx
L = mn**(self.physics.D-2.)*self.source.rho/(self.physics.M*self.mf) * v * rD * dx
# the Yukawa potential has linear matter coupling even when
# the symmetron has quadratic matter coupling
yukawa = d.Function( self.fem.S )
pde = d.LinearVariationalProblem( a, L, yukawa, Dirichlet_bc )
solver = d.LinearVariationalSolver( pde )
solver.solve()
self.yukawa = d.Function( self.fem.S )
self.yukawa.vector()[:] = self.mf * yukawa.vector()[:]
def setup_S(w_S, u, p, v, q, p0, q0, dx, ds, normal, dirichlet_bcs,
neumann_bcs, boundary_to_mark, u_1, phi_, rho_, rho_1, g_, M_, mu_,
rho_e_, c_, V_, c_1, V_1, dbeta, solutes, per_tau, drho, sigma_bar,
eps, dveps, grav, fric, u_comoving, enable_PF, enable_EC,
use_iterative_solvers, use_pressure_stabilization, p_lagrange,
q_rhs):
""" Set up Stokes subproblem """
F = (per_tau * rho_1 * df.dot(u - u_1, v) * dx +
mu_ * df.inner(df.grad(u), df.grad(v)) * dx - p * df.div(v) * dx +
q * df.div(u) * dx - rho_ * df.dot(grav, v) * dx)
print("Linear system size", w_S.function_space().dim())
a, L = df.system(F)
problem = df.LinearVariationalProblem(a, L, w_S, dirichlet_bcs)
solver = df.LinearVariationalSolver(problem)
solver.parameters["linear_solver"] = "mumps"
if use_iterative_solvers:
solver.parameters["linear_solver"] = "gmres"
solver.parameters["preconditioner"] = "amg"
return solver
def __init__(self, gridsize=32, t_end=6, direction=1, adjoint=False):
## end time
self.t_end = t_end
self.u0_expr = dol.Expression('1 + 0.2*sin(pi*x[1])*sin(pi*x[0]/3)',
degree=1)
if adjoint:
self.z0_expr = dol.Expression("0", degree=0)
## heat source term for weak formulation for Crank Nicolson, IE uses only f1
self.f1 = dol.Expression('5*pow(t,3)', t=0, degree=2)
self.f2 = dol.Expression('5*pow(t,3)', t=0, degree=2)
## Size of grid, number of cells = (2*gridsize)*gridsize*2
self.gridsize = gridsize
## Mesh
mesh = dol.RectangleMesh(dol.Point(0, 0), dol.Point(3, 1),
3 * self.gridsize, self.gridsize)
## Function space for solution vector
self.V = dol.FunctionSpace(mesh, "CG", 1)
Problem_FE.__init__(self, adjoint)
## Function space for velocity
V_velo = dol.VectorFunctionSpace(mesh, "CG", 2)
## direction of velocity
self.direction = direction
# create dol.Measures for relevant subdomains
colours = dol.MeshFunction(
"size_t", mesh,
dim=2) # Colouring for mesh to identify various subdomains
colours.set_all(0) # default colour
# heat source
heat_src = dol.CompiledSubDomain(
"x[0] >= 0.25 && x[0] <= 0.75 && x[1] >= 0.25 && x[1] <= 0.75"
) # identify subdomain by condition
heat_src.mark(colours, 2) # change colour of domain
self.dx_src = dol.Measure("dx", subdomain_data=colours)(
2) # define domain in the needed way for function definition
# Area of interest
subdom = dol.CompiledSubDomain(
"x[0] >= 2.25 && x[0] <= 2.75 && x[1] >= 0.25 && x[1] <= 0.75")
subdom.mark(colours, 1) # change colour of domain
self.dx_obs = dol.Measure("dx", subdomain_data=colours)(
1) # define domain in the needed way for function definition
self.dx_cont = self.dx_obs
## u used in Crank Nicolson
self.ucn = 0.5 * (self.utrial + self.uold)
if self.direction == 1:
self.velocity = dol.interpolate(
dol.Expression((' 1', '0'), degree=0), V_velo)
elif self.direction == -1:
self.velocity = dol.interpolate(
dol.Expression(('-1', '0'), degree=0), V_velo)
else:
raise ValueError('invalid advection direction, please try again')
self.heat_cond = dol.Constant(0.01)
self.outflow = dol.Constant(0.15)
self.velo = dol.Constant(0.5)
## Variational formulation for Crank Nicolson
F = (dol.inner(self.v, self.utrial - self.uold) * dol.dx +
self.dt * self.heat_cond *
dol.inner(dol.grad(self.ucn), dol.grad(self.v)) * dol.dx - 0.5 *
self.dt * dol.inner(self.f1 + self.f2, self.v) * self.dx_src +
self.velo * self.dt * dol.dot(dol.grad(self.ucn), self.velocity) *
self.v * dol.dx + self.dt * self.heat_cond * self.outflow *
dol.inner(self.ucn, self.v) * dol.ds)
prob = dol.LinearVariationalProblem(dol.lhs(F), dol.rhs(F), self.unew)
self.solver = dol.LinearVariationalSolver(prob)
## Variational formulation for Implicit Euler
Flow = (dol.inner(self.v, self.utrial - self.uold) * dol.dx +
self.dt * self.heat_cond *
dol.inner(dol.grad(self.utrial), dol.grad(self.v)) * dol.dx -
self.dt * dol.inner(self.f2, self.v) * self.dx_src +
self.velo * self.dt * dol.dot(dol.grad(
self.utrial), self.velocity) * self.v * dol.dx +
self.dt * self.heat_cond * self.outflow *
dol.inner(self.utrial, self.v) * dol.ds)
problow = dol.LinearVariationalProblem(dol.lhs(Flow), dol.rhs(Flow),
self.ulow)
self.solver_low = dol.LinearVariationalSolver(problow)
if adjoint:
adj_src = dol.Function(self.V)
adj_src.interpolate(dol.Constant(1 / self.t_end))
self.src_weight = dol.Expression(
"((0.25 < x[0]) and (0.75 > x[0]) and (0.25 < x[1]) and (0.75 > x[1]))? 1 : 0",
degree=0)
class LeftBoundary(dol.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and dol.near(x[0], 0)
class RightBoundary(dol.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and dol.near(x[0], 3)
left = LeftBoundary()
right = RightBoundary()
boundaries = dol.MeshFunction("size_t", mesh, dim=1)
boundaries.set_all(0)
left.mark(boundaries, 1)
right.mark(boundaries, 3)
ds_self = dol.Measure('ds', domain=mesh, subdomain_data=boundaries)
zcn = 0.5 * (self.ztrial + self.zold)
Fadj = (dol.inner(self.ztrial - self.zold, self.v) * dol.dx -
self.dt * self.heat_cond *
dol.inner(dol.grad(zcn), dol.grad(self.v)) * dol.dx +
self.dt * dol.inner(adj_src, self.v) * self.dx_obs +
self.direction * self.dt * self.velo *
dol.dot(self.velocity, dol.grad(zcn)) * self.v * dol.dx -
self.velo * self.dt * dol.inner(zcn, self.v) * ds_self(1) -
self.velo * self.dt * dol.inner(zcn, self.v) * ds_self(3) -
self.outflow * self.heat_cond * self.dt *
dol.inner(zcn, self.v) * dol.ds)
probadj = dol.LinearVariationalProblem(dol.lhs(Fadj),
dol.rhs(Fadj), self.znew)
self.solver_adj = dol.LinearVariationalSolver(probadj)
n = df.FacetNormal(mesh)
chi = df.TrialFunction(S)
chi_ = df.Function(S, name="chi")
psi = df.TestFunction(S)
ds = df.Measure("ds", domain=mesh, subdomain_data=subd)
F_chi = (n[0]*psi*ds(1)
+ df.inner(df.grad(chi), df.grad(psi))*df.dx)
a_chi, L_chi = df.lhs(F_chi), df.rhs(F_chi)
problem_chi2 = df.LinearVariationalProblem(a_chi, L_chi, chi_, bcs=[])
solver_chi2 = df.LinearVariationalSolver(problem_chi2)
solver_chi2.parameters["krylov_solver"]["absolute_tolerance"] = 1e-15
solver_chi2.solve()
with df.XDMFFile(mesh.mpi_comm(),
"chi_Pe0_b{}.xdmf".format(b)) as xdmff:
xdmff.write(chi_)
integral = (2*df.assemble(chi_.dx(0)*df.dx)/V_Omega
+ df.assemble(df.inner(df.grad(chi_),
df.grad(chi_))*df.dx)/V_Omega)
if rank == 0:
print("b = {}, D_eff/D = {}".format(b, 1+integral))
def step(self, t0: float, t1: float) -> None:
"""Solve on the given time interval (t0, t1).
Arguments:
interval (:py:class:`tuple`)
The time interval (t0, t1) for the step
*Invariants*
Assuming that v\_ is in the correct state for t0, gives
self.vur in correct state at t1.
"""
timer = df.Timer("PDE step")
# Extract theta and conductivities
theta = self._parameters["theta"]
Mi = self._M_i
Me = self._M_e
# Extract interval and thus time-step
kn = df.Constant(t1 - t0)
# Define variational formulation
if self._parameters["linear_solver_type"] == "direct":
v, u, l = df.TrialFunctions(self.VUR)
w, q, lamda = df.TestFunctions(self.VUR)
else:
v, u = df.TrialFunctions(self.VUR)
w, q = df.TestFunctions(self.VUR)
# Get physical parameters
chi = self._parameters["Chi"]
capacitance = self._parameters["Cm"]
Dt_v = (v - self.v_) / kn
Dt_v *= chi * capacitance
v_mid = theta * v + (1.0 - theta) * self.v_
# Set time
t = t0 + theta * (t1 - t0)
self.time.assign(t)
# Define spatial integration domains:
dz = df.Measure("dx",
domain=self._mesh,
subdomain_data=self._cell_domains)
db = df.Measure("ds",
domain=self._mesh,
subdomain_data=self._facet_domains)
# Get domain labels
cell_tags = map(int, set(
self._cell_domains.array())) # np.int64 does not workv
facet_tags = map(int, set(self._facet_domains.array()))
# Loop overe all domain labels
G = Dt_v * w * dz()
for key in cell_tags:
G += df.inner(Mi[key] * df.grad(v_mid), df.grad(w)) * dz(key)
G += df.inner(Mi[key] * df.grad(u), df.grad(w)) * dz(key)
G += df.inner(Mi[key] * df.grad(v_mid), df.grad(q)) * dz(key)
G += df.inner(
(Mi[key] + Me[key]) * df.grad(u), df.grad(q)) * dz(key)
if self._I_s is None:
G -= chi * df.Constant(0) * w * dz(key)
else:
# _is = self._I_s.get(key, df.Constant(0))
# G -= chi*_is*w*dz(key)
G -= chi * self._I_s[key] * w * dz(key)
# If Lagrangian multiplier
if self._parameters["linear_solver_type"] == "direct":
G += (lamda * u + l * q) * dz(key)
# Add applied current as source in elliptic equation if applicable
if self._I_a:
G -= chi * self._I_a[key] * q * dz(key)
if self._ect_current is not None:
for key in facet_tags:
# Detfalt to 0 if not defined for that facet tag
# TODO: Should I include `chi` here? I do not think so
G += self._ect_current.get(key, df.Constant(0)) * q * db(key)
# Define variational problem
a, L = df.system(G)
pde = df.LinearVariationalProblem(a, L, self.vur, bcs=self._bcs)
# Set-up solver
solver = df.LinearVariationalSolver(pde)
solver.solve()
def setup_EC(w_EC, c, V, V0, b, U, U0,
rho_e, grad_g_c, c_reg, g_c,
dx, ds, normal,
dirichlet_bcs_EC, neumann_bcs, boundary_to_mark,
c_1, u_1, K, veps,
rho_, rho_1,
dt,
enable_NS,
solutes,
use_iterative_solvers,
nonlinear_EC,
V_lagrange, p_lagrange,
q_rhs,
reactions,
beta,
**namespace):
""" Set up electrochemistry subproblem. """
if enable_NS:
# Projected velocity
u_star = u_1 - dt/rho_1*sum([ci_1*grad_g_ci
for ci_1, grad_g_ci
in zip(c_1, grad_g_c)])
F_c = []
for ci, ci_1, bi, Ki, grad_g_ci, solute, ci_reg in zip(
c, c_1, b, K, grad_g_c, solutes, c_reg):
F_ci = (1./dt*(ci-ci_1)*bi*dx +
Ki*ci_reg*df.dot(grad_g_ci, df.grad(bi))*dx)
if enable_NS:
F_ci += - ci_1*df.dot(u_star, df.grad(bi))*dx
if solute[0] in q_rhs:
F_ci += - q_rhs[solute[0]]*bi*dx
if enable_NS:
for boundary_name, value in neumann_bcs[solute[0]].iteritems():
# F_ci += df.dot(u_1, normal)*bi*ci_1*ds(
# boundary_to_mark[boundary_name])
pass
F_c.append(F_ci)
for reaction_constant, nu in reactions:
g_less = []
g_more = []
for nui, ci_1, betai in zip(nu, c_1, beta):
if nui < 0:
g_less.append(-nui*(df.ln(ci_1)+betai))
elif nui > 0:
g_more.append(nui*(df.ln(ci_1)+betai))
g_less = sum(g_less)
g_more = sum(g_more)
C = reaction_constant*(
df.exp(g_less) - df.exp(g_more))/(g_less - g_more)
R = C*sum([nui*g_ci for nui, g_ci in zip(nu, g_c)])
for nui, bi in zip(nu, b):
if nui != 0:
F_c.append(nui*R*bi*dx)
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 += -rho_e*U*dx
if V_lagrange:
F_V += veps*V0*U*dx + veps*V*U0*dx
if "V" in q_rhs:
F_V += q_rhs["V"]*U*dx
F = sum(F_c) + F_V
if nonlinear_EC:
J = df.derivative(F, w_EC)
problem = df.NonlinearVariationalProblem(F, w_EC, dirichlet_bcs_EC, J)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"]["relative_tolerance"] = 1e-7
if use_iterative_solvers:
solver.parameters["newton_solver"]["linear_solver"] = "bicgstab"
if not V_lagrange:
solver.parameters["newton_solver"]["preconditioner"] = "hypre_amg"
else:
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_EC, dirichlet_bcs_EC)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers:
solver.parameters["linear_solver"] = "bicgstab"
solver.parameters["preconditioner"] = "hypre_amg"
return solver
def test_scipy_uprime_integration_with_fenics():
NB_OF_CELLS_X = NB_OF_CELLS_Y = 2
mesh = df.UnitSquare(NB_OF_CELLS_X, NB_OF_CELLS_Y)
V = df.FunctionSpace(mesh, 'CG', 1)
f = df.Expression("cos(t)",t=0)
uprime = df.TrialFunction(V)
uprev = df.interpolate(f,V)
v = df.TestFunction(V)
#ODE is du/dt= uprime = cos(t), exact solution is u(t)=sin(t)
a = uprime*v*dx
L = f*v*dx
uprime_solution = df.Function(V)
uprime_problem = df.LinearVariationalProblem(a, L, uprime_solution)
uprime_solver = df.LinearVariationalSolver(uprime_problem)
def rhs_fenics(y,t):
"""A somewhat strange case where the right hand side is constant
and thus we don't need to use the information in y."""
#print "time: ",t
uprev.vector()[:]=y
f.t = t #dolfin needs to know the current time for cos(t)
uprime_solver.solve()
return uprime_solution.vector().array()
def rhs(y,t):
"""
dy/dt = f(y,t) with y(0)=1
dy/dt = -2y -> solution y(t) = c * exp(-2*t)"""
return math.cos(t)
T_MAX=2
ts = numpy.arange(0,T_MAX+0.1,0.5)
ysfenics=scipy.integrate.odeint(rhs_fenics, uprev.vector().array(), ts)
def exact(t,y0=1):
return y0*numpy.sin(t)
print "With fenics:"
err_abs = abs(ysfenics[-1][0]-exact(ts[-1])) #use value at mesh done 0 for check
print "Error: abs=%g, rel=%g, y_exact=%g" % (err_abs,err_abs/exact(ts[-1]),exact(ts[-1]))
fenics_error=err_abs
print "Without fenics:"
ys = scipy.integrate.odeint(rhs, 1, ts)
err_abs = abs(ys[-1]-exact(ts[-1]))
print "Error: abs=%g, rel=%g, y_exact=%g" % (err_abs,err_abs/exact(ts[-1]),exact(ts[-1]))
non_fenics_error = float(err_abs)
print("Difference between fenics and non-fenics calculation: %g" % abs(fenics_error-non_fenics_error))
assert abs(fenics_error-non_fenics_error)<7e-15
#should also check that solution is the same on all mesh points
for i in range(ysfenics.shape[0]): #for all result rows
#each row contains the data at all mesh points for one t in ts
row = ysfenics[i,:]
number_range = abs(row.min()-row.max())
print "row: %d, time %f, range %g" % (i,ts[i],number_range)
assert number_range < 16e-15
#for debugging
if False:
from IPython.Shell import IPShellEmbed
ipshell = IPShellEmbed()
ipshell()
if False:
import pylab
pylab.plot(ts,ys,'o')
pylab.plot(ts,numpy.exp(-2*ts),'-')
pylab.show()
def setup_NS(w_NS, u, p, v, q, p0, q0, dx, ds, normal, dirichlet_bcs,
neumann_bcs, boundary_to_mark, u_1, phi_, rho_, rho_1, g_, M_,
mu_, rho_e_, c_, V_, c_1, V_1, dbeta, solutes, per_tau, drho,
sigma_bar, eps, dveps, grav, fric, u_comoving, enable_PF,
enable_EC, use_iterative_solvers, use_pressure_stabilization,
p_lagrange, q_rhs):
""" Set up the Navier-Stokes subproblem. """
# F = (
# per_tau * rho_ * df.dot(u - u_1, v)*dx
# + rho_*df.inner(df.grad(u), df.outer(u_1, v))*dx
# + 2*mu_*df.inner(df.sym(df.grad(u)), df.grad(v))*dx
# - p * df.div(v)*dx
# + df.div(u)*q*dx
# - df.dot(rho_*grav, v)*dx
# )
mom_1 = rho_1 * (u_1 + u_comoving)
if enable_PF:
mom_1 += -M_ * drho * df.nabla_grad(g_)
F = (per_tau * rho_1 * df.dot(u - u_1, v) * dx +
fric * mu_ * df.dot(u + u_comoving, v) * dx + 2 * mu_ *
df.inner(df.sym(df.nabla_grad(u)), df.sym(df.nabla_grad(v))) * dx -
p * df.div(v) * dx + q * df.div(u) * dx +
df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx + 0.5 *
(per_tau * (rho_ - rho_1) * df.dot(u, v) -
df.dot(mom_1, df.nabla_grad(df.dot(u, v)))) * dx -
rho_ * df.dot(grav, v) * dx -
mu_ * df.dot(df.nabla_grad(u) * normal, v) * df.ds)
for boundary_name, slip_length in neumann_bcs["u"].items():
F += 1./slip_length * \
df.dot(u, v) * ds(boundary_to_mark[boundary_name])
for boundary_name, pressure in neumann_bcs["p"].items():
F += pressure * df.dot(normal, v) * ds(boundary_to_mark[boundary_name])
# - 2*mu_*df.dot(df.dot(df.sym(df.nabla_grad(u)), v),
# normal)) * ds(boundary_to_mark[boundary_name])
if enable_PF:
F += phi_ * df.dot(df.nabla_grad(g_), v) * dx
if enable_EC:
for ci_, ci_1, dbetai, solute in zip(c_, c_1, dbeta, solutes):
zi = solute[1]
F += df.dot(df.grad(ci_), v)*dx \
+ zi*ci_1*df.dot(df.grad(V_), v)*dx
if enable_PF:
F += ci_ * dbetai * df.dot(df.grad(phi_), v) * dx
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)
problem = df.LinearVariationalProblem(a, L, w_NS, dirichlet_bcs)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers and use_pressure_stabilization:
solver.parameters["linear_solver"] = "gmres"
#solver.parameters["preconditioner"] = "jacobi"
#solver.parameters["preconditioner"] = "ilu"
return solver
mu = dolfin.Constant(E/2./(1.+nu))
lambda_ = dolfin.Constant(E*nu/(1.+nu)/(1.-2.*nu))
def strain_displacement(u):
return dolfin.sym(dolfin.grad(u))
def stress_strain(eps):
return lambda_*dolfin.tr(eps)*I2+2*mu*eps
bilinear_form = dolfin.inner(stress_strain(strain_displacement(u_trial)),
strain_displacement(v))*dolfin.dx
traction = dolfin.Expression(('x[0] < x_right ? 0.0 : -12.*M/d/d/d*x[1]', '0.0'),
x_right=ell-atol, M=1.0, d=d, degree=degree)
linear_form = dolfin.dot(traction, v)*dolfin.ds
u_prescribed = dolfin.Constant((0., 0.))
boundary_condition_left = dolfin.DirichletBC(V, u_prescribed, left)
boundary_conditions = [boundary_condition_left]
problem = dolfin.LinearVariationalProblem(bilinear_form, linear_form, u, boundary_conditions)
solver = dolfin.LinearVariationalSolver(problem)
solver.solve()
dolfin.plot(u/u.vector().max(),mode="displacement")
plt.savefig("u.png")
with dolfin.XDMFFile(mesh.mpi_comm(), "output/u.xdmf") as file:
file.write(u, 0)
def _setup_problem(self):
"""
Method setting up solver objects of the stationary problem.
"""
assert hasattr(self, "_mesh")
assert hasattr(self, "_boundary_markers")
self._setup_function_spaces()
self._setup_boundary_conditions()
# creating test function
self._v = dlfn.TestFunction(self._Vh)
self._u = dlfn.TrialFunction(self._Vh)
# solution
self._solution = dlfn.Function(self._Vh)
# volume element
self._dV = dlfn.Measure("dx", domain=self._mesh)
self._dA = dlfn.Measure("ds",
domain=self._mesh,
subdomain_data=self._boundary_markers)
# setup the parameters for the elastic law
self._elastic_law.set_parameters(self._mesh, self._C)
# virtual work
if self._elastic_law.linearity_type == "Linear":
self._dw_int = self._elastic_law.dw_int(self._u,
self._v) * self._dV
elif self._elastic_law.linearity_type == "Nonlinear":
self._dw_int = self._elastic_law.dw_int(self._solution,
self._v) * self._dV
# virtual work of external forces
self._dw_ext = dlfn.dot(self._null_vector, self._v) * self._dV
# add body force term
if hasattr(self, "_body_force"):
assert hasattr(self, "_D"), "Dimensionless parameter related to" + \
"the body forces is not specified."
self._dw_ext += self._D * dot(self._body_force, self._v) * self._dV
# add boundary tractions
if hasattr(self, "_traction_bcs"):
for bc in self._traction_bcs:
# unpack values
if len(bc) == 3:
bc_type, bndry_id, traction = bc
elif len(bc) == 4:
bc_type, bndry_id, component_index, traction = bc
if bc_type is TractionBCType.constant:
assert isinstance(traction, (tuple, list))
const_function = dlfn.Constant(traction)
self._dw_ext += dot(const_function,
self._v) * self._dA(bndry_id)
elif bc_type is TractionBCType.constant_component:
assert isinstance(traction, float)
const_function = dlfn.Constant(traction)
self._dw_ext += const_function * self._v[
component_index] * self._dA(bndry_id)
elif bc_type is TractionBCType.function:
assert isinstance(traction, dlfn.Expression)
self._dw_ext += dot(traction, self._v) * self._dA(bndry_id)
elif bc_type is TractionBCType.function_component:
assert isinstance(traction, dlfn.Expression)
self._dw_ext += traction * self._v[
component_index] * self._dA(bndry_id)
if self._elastic_law.linearity_type == "Linear":
# linear variational problem
self._problem = dlfn.LinearVariationalProblem(
self._dw_int, self._dw_ext, self._solution,
self._dirichlet_bcs)
# setup linear variational solver
self._solver = dlfn.LinearVariationalSolver(self._problem)
elif self._elastic_law.linearity_type == "Nonlinear":
self._Form = self._dw_int - self._dw_ext
self._J_newton = dlfn.derivative(self._Form, self._solution)
self._problem = dlfn.NonlinearVariationalProblem(
self._Form,
self._solution,
self._dirichlet_bcs,
J=self._J_newton)
# setup linear variational solver
self._solver = dlfn.NonlinearVariationalSolver(self._problem)
- half / dt * dot(v00, del_v) * dV
#==============================================================================
pvd_velocity = dlfn.File("results_{1}/Re_{0:d}/solution_velocity_Re_{0:d}.pvd".\
format(int(reynolds), "FEniCS"))
pvd_pressure = dlfn.File("results_{1}/Re_{0:d}/solution_pressure_Re_{0:d}.pvd".\
format(int(reynolds), "FEniCS"))
#==============================================================================
# time loop
dlfn.tic()
t = 0.
cnt = 0
n_steps = 5000
t_end = 80.
while t < t_end: #and cnt <= n_steps:
problem = dlfn.LinearVariationalProblem(lhs, rhs, sol, bcs=bcs)
solver = dlfn.LinearVariationalSolver(problem)
solver.solve()
v, p = sol.split()
if cnt % 3200 == 0:
print "t = {0:6.4f}".format(t)
pvd_velocity << (v, t)
pvd_pressure << (p, t)
cd, cl = integrateFluidStress(v, p, nu, space_dim)
drag_coeff_list.append(cd)
lift_coeff_list.append(cl)
iter_list.append(cnt)
t_dim = t * t_ref
t_list.append(t_dim)
# update for next iteration
sol00.assign(sol0)
sol0.assign(sol)