def runNextStep(self):
self.coupled_problem = df.NonlinearVariationalProblem(self.R, self.U, bcs=[self.dbc0, self.dbc1, self.dbc3],
J=self.J)
self.coupled_problem.set_bounds(self.l_bound, self.u_bound)
self.coupled_solver = df.NonlinearVariationalSolver(self.coupled_problem)
# Optimizations in fenics optimizations
set_solver_options(self.coupled_solver)
try:
self.coupled_solver.solve(set_solver_options())
except:
self.coupled_solver.parameters['snes_solver']['error_on_nonconvergence'] = False
self.assigner.assign(self.U, [self.zero_sol, self.zero_sol, self.H0])
self.coupled_solver.solve()
self.coupled_solver.parameters['snes_solver']['error_on_nonconvergence'] = True
self.assigner_inv.assign([self.un, self.u2n, self.H0], self.U)
self.times.append(self.t)
self.BB.append(df.project(self.strs.B).compute_vertex_values())
self.HH.append(self.strs.H0.compute_vertex_values())
self.TD.append(df.project(self.strs.tau_d_plot).compute_vertex_values())
self.TB.append(df.project(self.strs.tau_b_plot).compute_vertex_values())
self.TX.append(df.project(self.strs.tau_xx_plot).compute_vertex_values())
self.TY.append(df.project(self.strs.tau_xy_plot).compute_vertex_values())
self.TZ.append(df.project(self.strs.tau_xz_plot).compute_vertex_values())
self.us.append(df.project(self.strs.u(0)).compute_vertex_values())
self.ub.append(df.project(self.strs.u(1)).compute_vertex_values())
self.t += self.dtFloat
return self.BB[-1], self.HH[-1], self.TD[-1], self.TB[-1], self.TX[-1], self.TY[-1], self.TZ[-1], self.us[-1], \
self.ub[-1]
def setup_NSPFEC(w_NSPFEC, w_1NSPFEC, bcs_NSPFEC, trial_func_NSPFEC, v, q, psi,
h, b, U, u_, p_, phi_, g_, c_, V_, u_1, p_1, phi_1, g_1, c_1,
V_1, M_, nu_, veps_, rho_, K_, rho_e_, M_1, nu_1, veps_1,
rho_1, K_1, rho_e_1, dbeta, dveps, drho, per_tau, sigma_bar,
eps, grav, z, enable_NS, enable_PF, enable_EC,
use_iterative_solvers):
""" The full problem of electrohydrodynamics in two pahase.
Note that it is possioble to trun off the dirffent parts at will.
"""
F_imp = NSPFEC_action(u_, u_1, phi_, phi_1, c_, c_1, u_, p_, phi_, g_, c_,
V_, v, q, psi, h, b, U, rho_, M_, nu_, rho_e_, K_,
veps_, drho, grav, sigma_bar, eps, dveps, dbeta, z,
per_tau, enable_NS, enable_PF, enable_EC)
F_exp = NSPFEC_action(u_, u_1, phi_, phi_1, c_, c_1, u_1, p_1, phi_1, g_1,
c_1, V_1, v, q, psi, h, b, U, rho_1, M_1, nu_1,
rho_e_1, K_1, veps_1, drho, grav, sigma_bar, eps,
dveps, dbeta, z, per_tau, enable_NS, enable_PF,
enable_EC)
F = 0.5 * (F_imp + F_exp)
J = df.derivative(F, w_NSPFEC)
problem_NSPFEC = df.NonlinearVariationalProblem(F, w_NSPFEC, bcs_NSPFEC, J)
solver_NSPFEC = df.NonlinearVariationalSolver(problem_NSPFEC)
if use_iterative_solvers:
solver_NSPFEC.parameters['newton_solver']['linear_solver'] = 'gmres'
solver_NSPFEC.parameters['newton_solver']['preconditioner'] = 'ilu'
return solver_NSPFEC
def equilibrium_EC(w_, test_functions,
solutes,
permittivity,
dx, ds, normal,
dirichlet_bcs, neumann_bcs, boundary_to_mark,
use_iterative_solvers,
V_lagrange,
**namespace):
""" Electrochemistry equilibrium solver. Nonlinear! """
num_solutes = len(solutes)
cV = df.split(w_["EC"])
c, V = cV[:num_solutes], cV[num_solutes]
if V_lagrange:
V0 = cV[-1]
b = test_functions["EC"][:num_solutes]
U = test_functions["EC"][num_solutes]
if V_lagrange:
U0 = test_functions["EC"][-1]
z = [] # Charge z[species]
K = [] # Diffusivity K[species]
for solute in solutes:
z.append(solute[1])
K.append(solute[2])
rho_e = sum([c_e*z_e for c_e, z_e in zip(c, z)])
veps = permittivity[0]
F_c = []
for ci, bi, Ki, zi in zip(c, b, K, z):
grad_g_ci = df.grad(alpha_c(ci) + zi*V)
F_ci = Ki*max_value(ci, 0.)*df.dot(grad_g_ci, df.grad(bi))*dx
F_c.append(F_ci)
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
F = sum(F_c) + F_V
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"
solver.solve()
def solve_molecular(self,rho_solute=const.rhom):
"""
Solve the molecular diffusion problem
if 'solve_sol_mol' is not in the flags list this will be an instantaneous mixing problem.
"""
# Solve solution concentration
# Diffusivity
ramp = dolfin.Expression('x[0] > minR ? 100. : 1.',minR=np.log(self.Rstar)-0.5,degree=1)
self.Dstar = dolfin.project(dolfin.Expression('ramp*diff_ratio*exp(-2.*x[0])',degree=1,ramp=ramp,diff_ratio=self.astar_i/self.Lewis),self.sol_V)
# calculate solute flux (this is like the Stefan condition for the molecular diffusion problem
Dwall = self.Dstar(self.sol_coords[self.sol_idx_wall])
self.solFlux = -(self.C_wall/Dwall)*(self.dR/self.dt)
# Set the concentration for points that moved out of the grid to match the solute flux
u0_c_hold = self.u0_c.vector()[:].copy()
u0_c_hold[~self.sol_idx_extrapolate] = self.C_wall+self.solFlux*(np.exp(self.sol_coords[~self.sol_idx_extrapolate,0])-self.Rstar)
u0_c_hold[u0_c_hold<0.]=0.
self.u0_c.vector()[:] = u0_c_hold[:]
# Variational Problem
F_c = (self.u_s-self.u0_c)*self.v_s*dolfin.dx + \
self.dt*dolfin.inner(dolfin.grad(self.u_s), dolfin.grad(self.Dstar*self.v_s))*dolfin.dx - \
self.dt*self.solFlux*self.v_s*self.sds(1)
F_c = dolfin.action(F_c,self.C)
# First derivative
J = dolfin.derivative(F_c, self.C, self.u_s)
# handle the bounds
lower = dolfin.project(dolfin.Constant(0.0),self.sol_V)
upper = dolfin.project(dolfin.Constant(rho_solute),self.sol_V)
# set bounds and solve
snes_solver_parameters = {"nonlinear_solver": "snes",
"snes_solver": {"linear_solver": "lu",
"maximum_iterations": 20,
"report": True,
"error_on_nonconvergence": False}}
problem = dolfin.NonlinearVariationalProblem(F_c, self.C, J=J)
problem.set_bounds(lower, upper)
solver = dolfin.NonlinearVariationalSolver(problem)
solver.parameters.update(snes_solver_parameters)
dolfin.info(solver.parameters, True)
(iter, converged) = solver.solve()
self.u0_c.assign(self.C)
# Recalculate the freezing temperature
self.Tf_last = self.Tf
self.Tf = Tf_depression(self.C.vector()[self.sol_idx_wall,0],linear=True)
# Get the updated solution properties
self.rhos = dolfin.project(dolfin.Expression('C + rhow*(1.-C/rho_solute)',
degree=1,C=self.C,rhow=const.rhow,rho_solute=rho_solute),self.sol_V)
self.cs = dolfin.project(dolfin.Expression('ce*(C/rho_solute) + cw*(1.-C/rho_solute)',
degree=1,C=self.C,cw=const.cw,ce=const.ce,rho_solute=rho_solute),self.sol_V)
self.ks = dolfin.project(dolfin.Expression('ke*(C/rho_solute) + kw*(1.-C/rho_solute)',
degree=1,C=self.C,kw=const.kw,ke=const.ke,rho_solute=rho_solute),self.sol_V)
self.rhos_wall = self.rhos.vector()[self.sol_idx_wall]
self.cs_wall = self.cs.vector()[self.sol_idx_wall]
self.ks_wall = self.ks.vector()[self.sol_idx_wall]
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 get_solver(self, bcs):
df.parameters["form_compiler"]["quadrature_degree"] = 2
v = df.TestFunction(self.W)
F = df.inner(self.eps(v), (1.0 - omega(self.kappa())) * self.sigma()) * df.dx
J = df.derivative(F, self.d)
problem = df.NonlinearVariationalProblem(F, self.d, bcs, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters["nonlinear_solver"] = "snes"
solver.parameters["symmetric"] = True
solver.parameters["snes_solver"]["error_on_nonconvergence"] = False
solver.parameters["snes_solver"]["line_search"] = "bt"
solver.parameters["snes_solver"]["linear_solver"] = "mumps"
solver.parameters["snes_solver"]["maximum_iterations"] = 10
solver.parameters["snes_solver"]["report"] = False
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 solve_system(self, rhs, factor, u0, t):
"""
Dolfin's linear solver for (M-factor*A)u = rhs
Args:
rhs (dtype_f): right-hand side for the nonlinear system
factor (float): abbrev. for the node-to-node stepsize (or any other factor required)
u0 (dtype_u): initial guess for the iterative solver (not used here so far)
t (float): current time
Returns:
dtype_u: solution as mesh
"""
sol = self.dtype_u(self.V)
self.w.assign(sol.values)
# fixme: is this really necessary to do each time?
q1, q2 = df.TestFunctions(self.V)
w1, w2 = df.split(self.w)
r1, r2 = df.split(rhs.values)
F1 = w1 * q1 * df.dx - factor * self.F1 - r1 * q1 * df.dx
F2 = w2 * q2 * df.dx - factor * self.F2 - r2 * q2 * df.dx
F = F1 + F2
du = df.TrialFunction(self.V)
J = df.derivative(F, self.w, du)
problem = df.NonlinearVariationalProblem(F, self.w, [], J)
solver = df.NonlinearVariationalSolver(problem)
prm = solver.parameters
prm['newton_solver']['absolute_tolerance'] = 1E-09
prm['newton_solver']['relative_tolerance'] = 1E-08
prm['newton_solver']['maximum_iterations'] = 100
prm['newton_solver']['relaxation_parameter'] = 1.0
solver.solve()
sol.values.assign(self.w)
return sol
def solve_system(self, rhs, factor, u0, t):
"""
Dolfin's weak solver for (M-dtA)u = rhs
Args:
rhs (dtype_f): right-hand side for the nonlinear system
factor (float): abbrev. for the node-to-node stepsize (or any other factor required)
u0 (dtype_u_: initial guess for the iterative solver (not used here so far)
t (float): current time
Returns:
dtype_u: solution as mesh
"""
sol = self.dtype_u(self.V)
self.g.t = t
self.w.assign(sol.values)
v = df.TestFunction(self.V)
F = self.w * v * df.dx - factor * self.a_K - rhs.values * v * df.dx
du = df.TrialFunction(self.V)
J = df.derivative(F, self.w, du)
problem = df.NonlinearVariationalProblem(F, self.w, self.bc, J)
solver = df.NonlinearVariationalSolver(problem)
prm = solver.parameters
prm['newton_solver']['absolute_tolerance'] = 1E-12
prm['newton_solver']['relative_tolerance'] = 1E-12
prm['newton_solver']['maximum_iterations'] = 25
prm['newton_solver']['relaxation_parameter'] = 1.0
# df.set_log_level(df.PROGRESS)
solver.solve()
sol.values.assign(self.w)
return sol
def setup_PFEC(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
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_1 * df.dot(u_proj, df.grad(psi)) * dx
F_g = (g * h * dx -
sigma_bar * eps * df.dot(df.grad(phi), df.grad(h)) * dx -
sigma_bar / eps *
(diff_pf_potential_c(phi) - diff_pf_potential_e(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)
J = df.derivative(F, w_PF)
problem = df.NonlinearVariationalProblem(F, w_PF, dirichlet_bcs_PF, J)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"]["relative_tolerance"] = 1e-5
if use_iterative_solvers:
solver.parameters["newton_solver"][
"linear_solver"] = "bicgstab" # "gmres"
solver.parameters["newton_solver"]["preconditioner"] = "jacobi" #"amg"
# solver.parameters["preconditioner"] = "hypre_euclid"
return solver
def stefan_form_cao():
# Temperature transforming model (Cao,1991)
(T, bcs, theta, _theta, theta_) = stefan_function_spaces()
theta_k = dolfin.project(theta_analytic,
T,
solver_type="cg",
preconditioner_type="hypre_amg")
q_form, q_form_k, bc_form = stefan_boundary_values(theta_, bcs)
# Cao formulation source term
def s(theta, theta0=prm.theta_m, eps=em.EPS):
return dolfin.conditional(
abs(theta - theta0) < eps,
prm.cp_m * eps + prm.L_m / 2,
dolfin.conditional(theta > theta0,
prm.cp_s * eps + prm.L_m,
prm.cp_s * eps))
# Fully implicit time discretization scheme
F = k_eff(theta, deg='C0') * dolfin.inner(
dolfin.grad(theta),
dolfin.grad(theta_)) * dx + prm.rho / dt * (
c_p_eff(theta, deg='disC') * (theta - prm.theta_m) +
s(theta) - c_p_eff(theta_k, deg='disC') *
(theta_k - prm.theta_m) -
s(theta_k)) * theta_ * dx - sum(q_form)
# Full THETA time dicretization scheme
# F = prm.rho/dt*(c_p_eff(theta,deg='disC')*(theta - prm.theta_m) + s(theta) - c_p_eff(theta_k,deg='disC')*(theta_k - prm.theta_m) - s(theta_k))*theta_*dx + (THETA*(k_eff(theta,deg='C0')*dolfin.inner(dolfin.grad(theta),dolfin.grad(theta_))*dx - sum(q_form)) + (1-THETA)*(k_eff(theta_k,deg='C0')*dolfin.inner(dolfin.grad(theta_k),dolfin.grad(theta_))*dx - sum(q_form_k)))
problem = dolfin.NonlinearVariationalProblem(
F, theta, bcs=bc_form, J=dolfin.derivative(F, theta))
solver = dolfin.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"] = NEWTON_PARAMS
return solver, theta, theta_k
def inbuilt_newton_solver(self):
V = self.my_function_space_cls.V
F = self.my_var_prob_cls.F
self.u = self.my_var_prob_cls.u
du = df.TrialFunction(V) #TrialFunctions --> wrong, without 's'
Jac = df.derivative(F, self.u, du)
#user given solver parameters
abs_tol = self.my_input_cls['newton_abs_tol']
rel_tol = self.my_input_cls['newton_rel_tol']
max_itr = self.my_input_cls['newton_max_itr']
step_size = self.my_input_cls['newton_relaxation_parameter']
problem = df.NonlinearVariationalProblem(F, self.u, [], Jac)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['newton_solver']['linear_solver'] = 'mumps'
solver.parameters['newton_solver']['absolute_tolerance'] = abs_tol
solver.parameters['newton_solver']['relative_tolerance'] = rel_tol
solver.parameters['newton_solver']['maximum_iterations'] = max_itr
solver.parameters['newton_solver']['relaxation_parameter'] = step_size
#df.solve(F == 0, self.u, [], J=Jac, solver_parameters={'newton_solver' : {'linear_solver' : 'mumps'}})
solver.solve()
def stefan_form_ehc():
# Define functions and spaces:
(T, bcs, theta, _theta, theta_) = stefan_function_spaces()
# Set initial condition:
theta_k = dolfin.project(theta_analytic,
T,
solver_type="cg",
preconditioner_type="hypre_amg")
# Set boundary terms
q_form, q_form_k, bc_form = stefan_boundary_values(theta_, bcs)
# Partial THETA time discretization scheme:
F = (k_eff(theta, deg='C0') * dolfin.inner(
dolfin.grad(theta), dolfin.grad(theta_)) * dx +
prm.rho / dt *
(THETA * c_p_eff(theta, deg='C0') +
(1 - THETA) * c_p_eff(theta_k, deg='C0')) *
(dolfin.inner(theta, theta_) -
dolfin.inner(theta_k, theta_)) * dx - sum(q_form))
# Full THETA time discretization scheme:
# F = (THETA*(k_eff(theta, deg = 'C0')*dolfin.inner(dolfin.grad(theta), dolfin.grad(theta_))*dx +
# prm.rho/dt*c_p_eff(theta,deg='C0')*(dolfin.inner(theta,theta_)-dolfin.inner(theta_k,theta_))*dx - sum(q_form)) +
# (1-THETA)*(k_eff(theta_k, deg = 'C0')*dolfin.inner(dolfin.grad(theta_k), dolfin.grad(theta_))*dx +
# prm.rho/dt*c_p_eff(theta_k,deg='C0')*(dolfin.inner(theta, theta_) - dolfin.inner(theta_k,theta_))*dx - sum(q_form_k)))
problem = dolfin.NonlinearVariationalProblem(
F, theta, bcs=bc_form, J=dolfin.derivative(F, theta))
solver = dolfin.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"] = NEWTON_PARAMS
return solver, theta, theta_k
def equilibrium_EC(w_, x_, test_functions, solutes, permittivity, mesh, dx, ds,
normal, dirichlet_bcs, neumann_bcs, boundary_to_mark,
use_iterative_solvers, c_lagrange, V_lagrange, **namespace):
""" Electrochemistry equilibrium solver. Nonlinear! """
num_solutes = len(solutes)
cV = df.split(w_["EC"])
c, V = cV[:num_solutes], cV[num_solutes]
if c_lagrange:
c0, V0 = cV[num_solutes + 1:2 * num_solutes + 1], cV[2 * num_solutes +
1]
if V_lagrange:
V0 = cV[-1]
b = test_functions["EC"][:num_solutes]
U = test_functions["EC"][num_solutes]
if c_lagrange:
b0, U0 = cV[num_solutes + 1:2 * num_solutes + 1], cV[2 * num_solutes +
1]
if V_lagrange:
U0 = test_functions["EC"][-1]
phi = x_["phi"]
q = []
sum_zx = sum([solute[1] * xj for solute, xj in zip(solutes, composition)])
for solute, xj in zip(solutes, composition):
q.append(-xj * Q / (area * sum_zx))
z = [] # Charge z[species]
K = [] # Diffusivity K[species]
beta = []
for solute in solutes:
z.append(solute[1])
K.append(ramp(phi, solute[2:4]))
beta.append(ramp(phi, solute[4:6]))
rho_e = sum([c_e * z_e for c_e, z_e in zip(c, z)])
veps = ramp(phi, permittivity)
F_c = []
for ci, bi, c0i, b0i, solute, qi, betai, Ki in zip(c, b, c0, b0, solutes,
q, beta, K):
zi = solute[1]
F_ci = Ki * (df.dot(
df.nabla_grad(bi),
df.nabla_grad(ci) + df.nabla_grad(betai) +
zi * ci * df.nabla_grad(V))) * dx
if c_lagrange:
F_ci += b0i * (ci - df.Constant(qi)) * dx + c0i * bi * dx
F_V = veps * df.dot(df.nabla_grad(U), df.nabla_grad(V)) * 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_lagrange:
F_V += V0 * U * dx + V * U0 * dx
F = sum(F_c) + F_V
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"
solver.solve()
def __init__(self, fileName, timeEnd, timeStep, average=False):
fc.set_log_active(False)
self.times = []
self.BB = []
self.HH = []
self.TD = []
self.TB = []
self.TX = []
self.TY = []
self.TZ = []
self.us = []
self.ub = []
##########################################################
################ MESH #################
##########################################################
# TODO: Probably do not have to save then open mesh
self.mesh = df.Mesh()
self.inFile = fc.HDF5File(self.mesh.mpi_comm(), fileName, "r")
self.inFile.read(self.mesh, "/mesh", False)
#########################################################
################# FUNCTION SPACES #####################
#########################################################
self.E_Q = df.FiniteElement("CG", self.mesh.ufl_cell(), 1)
self.Q = df.FunctionSpace(self.mesh, self.E_Q)
self.E_V = df.MixedElement(self.E_Q, self.E_Q, self.E_Q)
self.V = df.FunctionSpace(self.mesh, self.E_V)
self.assigner_inv = fc.FunctionAssigner([self.Q, self.Q, self.Q], self.V)
self.assigner = fc.FunctionAssigner(self.V, [self.Q, self.Q, self.Q])
self.U = df.Function(self.V)
self.dU = df.TrialFunction(self.V)
self.Phi = df.TestFunction(self.V)
self.u, self.u2, self.H = df.split(self.U)
self.phi, self.phi1, self.xsi = df.split(self.Phi)
self.un = df.Function(self.Q)
self.u2n = df.Function(self.Q)
self.zero_sol = df.Function(self.Q)
self.Bhat = df.Function(self.Q)
self.H0 = df.Function(self.Q)
self.A = df.Function(self.Q)
if average:
self.inFile.read(self.Bhat.vector(), "/bedAvg", True)
self.inFile.read(self.A.vector(), "/smbAvg", True)
self.inFile.read(self.H0.vector(), "/thicknessAvg", True)
else:
self.inFile.read(self.Bhat.vector(), "/bed", True)
self.inFile.read(self.A.vector(), "/smb", True)
self.inFile.read(self.H0.vector(), "/thickness", True)
self.Hmid = theta * self.H + (1 - theta) * self.H0
self.B = softplus(self.Bhat, -rho / rho_w * self.Hmid, alpha=0.2) # Is not the bed, it is the lower surface
self.S = self.B + self.Hmid
self.width = df.interpolate(Width(degree=2), self.Q)
self.strs = Stresses(self.U, self.Hmid, self.H0, self.H, self.width, self.B, self.S, self.Phi)
self.R = -(self.strs.tau_xx + self.strs.tau_xz + self.strs.tau_b + self.strs.tau_d + self.strs.tau_xy) * df.dx
#############################################################################
######################## MASS CONSERVATION ################################
#############################################################################
self.h = df.CellSize(self.mesh)
self.D = self.h * abs(self.U[0]) / 2.
self.area = self.Hmid * self.width
self.mesh_min = self.mesh.coordinates().min()
self.mesh_max = self.mesh.coordinates().max()
# Define boundaries
self.ocean = df.FacetFunctionSizet(self.mesh, 0)
self.ds = fc.ds(subdomain_data=self.ocean) # THIS DS IS FROM FENICS! border integral
for f in df.facets(self.mesh):
if df.near(f.midpoint().x(), self.mesh_max):
self.ocean[f] = 1
if df.near(f.midpoint().x(), self.mesh_min):
self.ocean[f] = 2
self.R += ((self.H - self.H0) / dt * self.xsi \
- self.xsi.dx(0) * self.U[0] * self.Hmid \
+ self.D * self.xsi.dx(0) * self.Hmid.dx(0) \
- (self.A - self.U[0] * self.H / self.width * self.width.dx(0)) \
* self.xsi) * df.dx + self.U[0] * self.area * self.xsi * self.ds(1) \
- self.U[0] * self.area * self.xsi * self.ds(0)
#####################################################################
######################### SOLVER SETUP ###########################
#####################################################################
# Bounds
self.l_thick_bound = df.project(Constant(thklim), self.Q)
self.u_thick_bound = df.project(Constant(1e4), self.Q)
self.l_v_bound = df.project(-10000.0, self.Q)
self.u_v_bound = df.project(10000.0, self.Q)
self.l_bound = df.Function(self.V)
self.u_bound = df.Function(self.V)
self.assigner.assign(self.l_bound, [self.l_v_bound] * 2 + [self.l_thick_bound])
self.assigner.assign(self.u_bound, [self.u_v_bound] * 2 + [self.u_thick_bound])
# This should set the velocity at the divide (left) to zero
self.dbc0 = df.DirichletBC(self.V.sub(0), 0, lambda x, o: df.near(x[0], self.mesh_min) and o)
# Set the velocity on the right terminus to zero
self.dbc1 = df.DirichletBC(self.V.sub(0), 0, lambda x, o: df.near(x[0], self.mesh_max) and o)
# overkill?
self.dbc2 = df.DirichletBC(self.V.sub(1), 0, lambda x, o: df.near(x[0], self.mesh_max) and o)
# set the thickness on the right edge to thklim
self.dbc3 = df.DirichletBC(self.V.sub(2), thklim, lambda x, o: df.near(x[0], self.mesh_max) and o)
# Define variational solver for the mass-momentum coupled problem
self.J = df.derivative(self.R, self.U, self.dU)
self.coupled_problem = df.NonlinearVariationalProblem(self.R, self.U, bcs=[self.dbc0, self.dbc1, self.dbc3], \
J=self.J)
self.coupled_problem.set_bounds(self.l_bound, self.u_bound)
self.coupled_solver = df.NonlinearVariationalSolver(self.coupled_problem)
# Acquire the optimizations in fenics_optimizations
set_solver_options(self.coupled_solver)
self.t = 0
self.timeEnd = float(timeEnd)
self.dtFloat = float(timeStep)
self.inFile.close()
R_gradphi = df.dot(
w_vec, grad_phi) * df.dx - w_jump * df.avg(phi) * df.dS - df.dot(
w_vec, nhat) * phi * df.ds(subdomain_data=edgefunction)(2) - df.dot(
w_vec, nhat) * rho_w * g * (B_cg + H_cg) * df.ds(
subdomain_data=edgefunction)(1)
O = Max(ub * (h_r - h) / l_r, 1e-3)
C = A * h * N**n
R_h = ((h - h0) / dt - O + C) * chi * df.dx
R = R_hw + R_gradphi + R_h
J = df.derivative(R, U, dU)
# Nonlinear Problem
problem = df.NonlinearVariationalProblem(R, U, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['nonlinear_solver'] = 'newton'
solver.parameters['newton_solver']['relaxation_parameter'] = 1.0
solver.parameters['newton_solver']['relative_tolerance'] = 1e-6
solver.parameters['newton_solver']['absolute_tolerance'] = 1e-6
solver.parameters['newton_solver']['error_on_nonconvergence'] = False
solver.parameters['newton_solver']['linear_solver'] = 'mumps'
solver.parameters['newton_solver']['maximum_iterations'] = 10
solver.parameters['newton_solver']['report'] = True
assigner_inv = df.FunctionAssigner([Q_dg, Q_dg, Q_dg, Q_dg], V)
assigner = df.FunctionAssigner(V, [Q_dg, Q_dg, Q_dg, Q_dg])
df.File('k01_phi0001_steady.xml') >> U
assigner_inv.assign([h0_w, h0, phi0_x, phi0_y], U)
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)
def solver(interval, dt=0.1, theta=1):
t0, t1 = interval
mesh = get_mesh(10)
cell_function = get_cell_function(mesh)
model = get_models(cell_function)
num_global_states = model.num_states()
# Create time keepers and time step
const_dt = df.Constant(dt)
current_time = df.Constant(0)
t = t0 + theta*(t1 - t0)
current_time.assign(t)
# Create stimulus
stimulus = df.Constant(0)
# Make shared function pace
mixed_vector_function_space = df.VectorFunctionSpace(
mesh,
"DG",
0,
dim=num_global_states + 1
)
# Create previous solution and assign initial conditions
previous_solution = df.Function(mixed_vector_function_space)
# _model = xb.cellmodels.Cressman()
# previous_solution.assign(_model.initial_conditions()) # Initial condition
previous_solution.assign(model.initial_conditions()) # Initial condition
v_previous, s_previous = splat(previous_solution, num_global_states + 1)
# Create current solution
current_solution = df.Function(mixed_vector_function_space)
v_current, s_current = splat(current_solution, num_global_states + 1)
# Create test functions
test_functions = df.TestFunction(mixed_vector_function_space)
test_v, test_s = splat(test_functions, num_global_states + 1)
# Crate time derivatives
Dt_v = (v_current - v_previous)/const_dt
Dt_s = (s_current - s_previous)/const_dt
# Create midpoint evaluations following theta rule
v_mid = theta*v_current + (1.0 - theta)*v_previous
s_mid = theta*s_current + (1.0 - theta)*s_previous
if isinstance(model, xb.MultiCellModel):
# model = xb.cellmodels.Cressman()
# dy = df.Measure("dx", domain=mesh)
# F_theta = model.F(v_mid, s_mid, time=current_time)
# I_theta = -model.I(v_mid, s_mid, time=current_time)
# lhs = (Dt_v - I_theta)*test_v
# lhs += df.inner(Dt_s - F_theta, test_s)
# lhs *= dy()
dy = df.Measure("dx", domain=mesh, subdomain_data=model.markers())
domain_indices = model.keys()
lhs_list = list()
for k, model_k in enumerate(model.models()):
domain_index_k = domain_indices[k]
F_theta = model.F(v_mid, s_mid, time=current_time, index=domain_index_k)
I_theta = -model.I(v_mid, s_mid, time=current_time, index=domain_index_k)
a = (Dt_v - I_theta)*test_v
a += df.inner(Dt_s - F_theta, test_s)
a *= dy(domain_index_k)
lhs_list.append(a)
# Sum the form
lhs = sum(lhs_list)
else:
# Evaluate currents at averaged v and s. Note sign for I_theta
model = xb.cellmodels.Cressman()
dy = df.Measure("dx", domain=mesh)
F_theta = model.F(v_mid, s_mid, time=current_time)
I_theta = -model.I(v_mid, s_mid, time=current_time)
lhs = (Dt_v - I_theta)*test_v
lhs += df.inner(Dt_s - F_theta, test_s)
lhs *= dy()
rhs = stimulus*test_v*dy()
G = lhs - rhs
# Solve system
current_solution.assign(previous_solution)
pde = df.NonlinearVariationalProblem(
G,
current_solution,
J=df.derivative(G, current_solution)
)
solver = df.NonlinearVariationalSolver(pde)
solver.solve()
def equilibrium_solver(F, u, bcs, bcs_set_values, bcs_values, **kwargs):
'''Nonlinear solver for the hyper-elastic equilibrium problem.
F : dolfin.Form
Variational form of equilibrium, i.e. F(u;v)==0 forall v. Usually `F`
is obtained by taking the derivative of the potential energy `W`, e.g.
`F = dolfin.derivative(W, u)`
u : dolfin.Function
Displacement function (or a mixed field function).
bcs : (sequence of) dolfin.DirichletBC's
Dirichlet boundary conditions.
bcs_set_values : function
Called with elements of `bcs_values`.
bcs_values : numpy.ndarray (2D)
Sequence of displacement values.
umin: dolfin.Expression, dolfin.Function, dolfin.GenericVector
Enfores displacement lower bound: `u >= umin`.
umax: dolfin.Expression, dolfin.Function, dolfin.GenericVector
Enfores displacement upper bound: `u <= umax`.
'''
if not isinstance(bcs_values, np.ndarray) or bcs_values.ndim != 2:
raise TypeError('Parameter `bcs_values` must be a 2D `numpy.ndarray`')
try:
bcs_set_values(bcs_values[-1])
except:
logger.error('Unable to set Dirichlet boundary conditions')
raise
dFdu = dolfin.derivative(F, u)
nonlinear_problem = dolfin.NonlinearVariationalProblem(F, u, bcs, dFdu)
nonlinear_solver = dolfin.NonlinearVariationalSolver(nonlinear_problem)
update_parameters(nonlinear_solver.parameters,
config.parameters_nonlinear_solver)
umin = kwargs.get('umin')
umax = kwargs.get('umax')
if umax is not None or umin is not None:
V = u.function_space()
if isinstance(umin, dolfin.Function):
umin = umin.vector()
elif isinstance(umin, dolfin.Expression):
umin = dolfin.interpolate(umin, V).vector()
elif umin is None:
umin = dolfin.Function(V).vector()
umin[:] = -np.inf
elif not isinstance(umin, dolfin.GenericVector) or umin.size() != V.dim():
raise TypeError("Key-word argument `umin` must be an instance of: "
"`dolfin.Function`, `dolfin.Expression`, or `dolfin.GenericVector` "
"(with the correct dimension).")
if isinstance(umax, dolfin.Function):
umax = umax.vector()
elif isinstance(umax, dolfin.Expression):
umax = dolfin.interpolate(umax, V).vector()
elif umax is None:
umax = dolfin.Function(V).vector()
umax[:] = np.inf
elif not isinstance(umax, dolfin.GenericVector) or umax.size() != V.dim():
raise TypeError("Key-word argument `umax` must be an instance of: "
"`dolfin.Function`, `dolfin.Expression`, or `dolfin.GenericVector` "
"(with the correct dimension).")
if not (config.parameters_nonlinear_solver['nonlinear_solver'] == 'snes' and
config.parameters_nonlinear_solver['snes_solver']['method'] == 'vinewtonrsls'):
logger.warning("Require \"snes\" solver and method called \"vinewtonrsls\" "
"to impose displacement field bounds `umin` and `umax`.")
nonlinear_problem.set_bounds(umin, umax)
def equilibrium_solve(incremental=False):
if incremental:
nonlocal bcs_values
u.vector()[:] = 0.0
u_arr_backup = None
try:
for i, values_i in enumerate(bcs_values):
logger.info(f'Solving for load {values_i}')
bcs_set_values(values_i)
nonlinear_solver.solve()
u_arr_backup = u.vector().get_local()
except RuntimeError:
logger.error('Could not solve equilibrium problem for load '
f'{values_i}; assuming previous load value.')
if u_arr_backup is None:
raise RuntimeError('Previous load value is not available')
u.vector()[:] = u_arr_backup
bcs_values = bcs_values[:i]
else:
try:
nonlinear_solver.solve()
except RuntimeError:
logger.error('Could not solve equilibrium problem; '
'Trying to re-load incrementally.')
equilibrium_solve(incremental=True)
return equilibrium_solve
s_min = sf.cmd.mpi_min(x[:, 1])
t_min = sf.cmd.mpi_min(x[:, 0])
def poles(x, on_boundary):
return on_boundary and bool(x[1] <= s_min + df.DOLFIN_EPS_LARGE
or x[1] >= s_max - df.DOLFIN_EPS_LARGE)
u_bc = df.DirichletBC(W.sub(3).sub(1), df.Constant(0.), poles)
# p_bc = df.DirichletBC(W.sub(1), df.Constant(0.),
# "x[0] < {t_min} && x[1] < {s_min}".format(
# t_min=t_min, s_min=s_min),
# "pointwise")
problem = df.NonlinearVariationalProblem(F, w_, J=J, bcs=[u_bc])
solver = df.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"]["absolute_tolerance"] = 1e-8
solver.parameters["newton_solver"]["relative_tolerance"] = 1e-5
solver.parameters["newton_solver"]["maximum_iterations"] = 16
#solver.parameters["newton_solver"]["linear_solver"] = "gmres"
#solver.parameters["newton_solver"]["preconditioner"] = "default"
# solver.parameters["newton_solver"]["krylov_solver"]["nonzero_initial_guess"] = True
# solver.parameters["newton_solver"]["krylov_solver"]["absolute_tolerance"] = 1e-8
# solver.parameters["newton_solver"]["krylov_solver"]["monitor_convergence"] = False
#solver.parameters["newton_solver"]["krylov_solver"]["maximum_iterations"] = 1000
# solver.parameters["linear_solver"] = "gmres"
# solver.parameters["preconditioner"] = "jacobi"
df.parameters["form_compiler"]["optimize"] = True
def solve_nonlinear(self, inputs, outputs):
pde_problem = self.options['pde_problem']
state_name = self.options['state_name']
problem_type = self.options['problem_type']
visualization = self.options['visualization']
state_function = pde_problem.states_dict[state_name]['function']
for argument_name, argument_function in iteritems(
self.argument_functions_dict):
density_func = argument_function
self.itr = self.itr + 1
state_function = pde_problem.states_dict[state_name]['function']
residual_form = pde_problem.states_dict[state_name]['residual_form']
self._set_values(inputs, outputs)
self.derivative_form = df.derivative(residual_form, state_function)
df.set_log_active(True)
if problem_type == 'linear_problem':
if state_name == 'density':
print('this is a variational density filter')
df.solve(residual_form == 0,
state_function,
J=self.derivative_form,
solver_parameters={
"newton_solver": {
"maximum_iterations": 1,
"error_on_nonconvergence": False
}
})
else:
df.solve(residual_form == 0,
state_function,
bcs=pde_problem.bcs_list,
J=self.derivative_form,
solver_parameters={
"newton_solver": {
"maximum_iterations": 1,
"error_on_nonconvergence": False
}
})
elif problem_type == 'nonlinear_problem':
state_function.vector().set_local(
np.zeros((state_function.function_space().dim())))
problem = df.NonlinearVariationalProblem(residual_form,
state_function,
pde_problem.bcs_list,
self.derivative_form)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['nonlinear_solver'] = 'snes'
solver.parameters["snes_solver"]["relative_tolerance"] = 5e-100
solver.parameters["snes_solver"]["absolute_tolerance"] = 5e-50
solver.parameters["snes_solver"]["line_search"] = 'bt'
solver.parameters["snes_solver"][
"linear_solver"] = 'mumps' # "cg" "gmres"
solver.parameters["snes_solver"]["maximum_iterations"] = 500
solver.parameters["snes_solver"]["relative_tolerance"] = 5e-100
solver.parameters["snes_solver"]["absolute_tolerance"] = 5e-50
# solver.parameters["snes_solver"]["linear_solver_"]["maximum_iterations"]=1000
solver.parameters["snes_solver"]["error_on_nonconvergence"] = False
solver.solve()
elif problem_type == 'nonlinear_problem_load_stepping':
num_steps = 4
state_function.vector().set_local(
np.zeros((state_function.function_space().dim())))
for i in range(num_steps):
v = df.TestFunction(state_function.function_space())
if i < (num_steps - 1):
residual_form = get_residual_form(
state_function,
v,
density_func,
density_func.function_space,
tractionBC,
# df.Constant((0.0, -9.e-1))
df.Constant((0.0, -9.e-1 / num_steps * (i + 1))),
'False')
else:
residual_form = get_residual_form(
state_function,
v,
density_func,
density_func.function_space,
tractionBC,
# df.Constant((0.0, -9.e-1))
df.Constant((0.0, -9.e-1 / num_steps * (i + 1))),
'vol')
problem = df.NonlinearVariationalProblem(
residual_form, state_function, pde_problem.bcs_list,
self.derivative_form)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['nonlinear_solver'] = 'snes'
solver.parameters["snes_solver"]["line_search"] = 'bt'
solver.parameters["snes_solver"][
"linear_solver"] = 'mumps' # "cg" "gmres"
solver.parameters["snes_solver"]["maximum_iterations"] = 500
solver.parameters["snes_solver"]["relative_tolerance"] = 1e-15
solver.parameters["snes_solver"]["absolute_tolerance"] = 1e-15
# solver.parameters["snes_solver"]["linear_solver_"]["maximum_iterations"]=1000
solver.parameters["snes_solver"][
"error_on_nonconvergence"] = False
solver.solve()
# option to store the visualization results
if (visualization == 'True') and (self.itr % 50 == 0):
for argument_name, argument_function in iteritems(
self.argument_functions_dict):
if argument_name == 'density':
df.File('solutions_iterations_40ramp/{}_{}.pvd'.format(
argument_name, self.itr)) << df.project(
argument_function / (1 + 8. *
(1. - argument_function)),
argument_function.function_space())
else:
df.File('solutions_iterations_40ramp/{}_{}.pvd'.format(
argument_name, self.itr)) << argument_function
df.File('solutions_iterations_40ramp/{}_{}.pvd'.format(
state_name, self.itr)) << state_function
self.L = -residual_form
outputs[state_name] = state_function.vector().get_local()
self._set_values(inputs, outputs)
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=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 NSPFEC_action(u_, u_1, phi_, phi_1, c_, c_1, u, p, phi, g, c, V, v, q, psi,
h, b, U, rho, M, nu, rho_e, K, veps, drho, grav, sigma_bar,
eps, dveps, dbeta, z, per_tau, enable_NS, enable_PF,
enable_EC):
# The setup of the Navier-Stokes part of F
F = []
if enable_NS:
F_NS = (per_tau * rho * df.dot(u_ - u_1, v) * df.dx + df.inner(
df.grad(u), df.outer(rho * u - drho * M * df.grad(g), v)) * df.dx +
2 * nu * df.inner(df.sym(df.grad(u)), df.grad(v)) * df.dx -
p * df.div(v) * df.dx + df.div(u) * q * df.dx -
df.dot(rho * grav, v) * df.dx)
if enable_PF:
F_NS += -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_NS += rho_e * df.dot(df.grad(V), v) * df.dx
if enable_PF and enable_EC:
F_NS += dveps * df.dot(df.grad(phi), v) * df.dot(
df.grad(V), df.grad(V)) * df.dx
F.append(F_NS)
# The setup of the Phase feild equations
if enable_PF:
F_PF_phi = (per_tau * (phi_ - phi_1) * psi * df.dx +
M * df.dot(df.grad(g), df.grad(psi)) * df.dx)
if enable_NS:
F_PF_phi += df.dot(u, df.grad(phi)) * psi * df.dx
F_PF_g = (g * h * df.dx -
sigma_bar * eps * df.dot(df.grad(phi), df.grad(h)) * df.dx -
sigma_bar / eps * diff_pf_potential(phi) * h * df.dx)
if enable_EC:
F_PF_g += (-sum(
[dbeta_i * ci * h * df.dx for dbeta_i, ci in zip(dbeta, c)]) +
dveps * df.dot(df.grad(V), df.grad(V)) * h * df.dx)
F_PF = F_PF_phi + F_PF_g
F.append(F_PF)
# The setup of the Electrochemistry
if enable_EC:
F_E_c = []
for ci, ci_, ci_1, bi, Ki, zi in zip(c, c_, c_1, b, K, z):
F_E_ci = (per_tau * (ci_ - ci_1) * bi * df.dx +
Ki * df.dot(df.grad(ci), df.grad(bi)) * df.dx)
if zi != 0:
F_E_ci += Ki * zi * ci * df.dot(df.grad(V),
df.grad(bi)) * df.dx
if enable_NS:
F_E_ci += df.dot(u, df.grad(ci)) * bi * df.dx
F_E_c.append(F_E_ci)
F_E_V = veps * df.dot(df.grad(V), df.grad(U)) * df.dx
if rho_e != 0:
F_E_V += -rho_e * U * df.dx
F_E = sum(F_E_c) + F_E_V
F.append(F_E)
F = sum(F)
J = df.derivative(F, w_NSPFEC)
problem_NSPFEC = df.NonlinearVariationalProblem(F, w_NSPFEC, bcs_NSPFEC, J)
solver_NSPFEC = df.NonlinearVariationalSolver(problem_NSPFEC)
if use_iterative_solvers:
solver_NSPFEC.parameters['newton_solver']['linear_solver'] = 'gmres'
solver_NSPFEC.parameters['newton_solver']['preconditioner'] = 'ilu'
return solver_NSPFEC
def solve_nonlinear(self, inputs, outputs):
pde_problem = self.options['pde_problem']
state_name = self.options['state_name']
problem_type = self.options['problem_type']
visualization = self.options['visualization']
state_function = pde_problem.states_dict[state_name]['function']
for argument_name, argument_function in iteritems(
self.argument_functions_dict):
density_func = argument_function
mesh = state_function.function_space().mesh()
sub_domains = df.MeshFunction('size_t', mesh,
mesh.topology().dim() - 1)
upper_edge = TractionBoundary()
upper_edge.mark(sub_domains, 6)
dss = df.Measure('ds')(subdomain_data=sub_domains)
tractionBC = dss(6)
self.itr = self.itr + 1
state_function = pde_problem.states_dict[state_name]['function']
residual_form = get_residual_form(
state_function,
df.TestFunction(state_function.function_space()),
density_func,
density_func.function_space(),
tractionBC,
# df.Constant((0.0, -9.e-1))
df.Constant((0.0, -9.e-1)),
int(self.itr))
self._set_values(inputs, outputs)
self.derivative_form = df.derivative(residual_form, state_function)
df.set_log_level(df.LogLevel.ERROR)
df.set_log_active(True)
# df.solve(residual_form==0, state_function, bcs=pde_problem.bcs_list, J=self.derivative_form)
if problem_type == 'linear_problem':
df.solve(residual_form == 0,
state_function,
bcs=pde_problem.bcs_list,
J=self.derivative_form,
solver_parameters={
"newton_solver": {
"maximum_iterations": 60,
"error_on_nonconvergence": False
}
})
elif problem_type == 'nonlinear_problem':
problem = df.NonlinearVariationalProblem(residual_form,
state_function,
pde_problem.bcs_list,
self.derivative_form)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['nonlinear_solver'] = 'snes'
solver.parameters["snes_solver"]["line_search"] = 'bt'
solver.parameters["snes_solver"][
"linear_solver"] = 'mumps' # "cg" "gmres"
solver.parameters["snes_solver"]["maximum_iterations"] = 500
solver.parameters["snes_solver"]["relative_tolerance"] = 5e-13
solver.parameters["snes_solver"]["absolute_tolerance"] = 5e-13
# solver.parameters["snes_solver"]["linear_solver_"]["maximum_iterations"]=1000
solver.parameters["snes_solver"]["error_on_nonconvergence"] = False
solver.solve()
elif problem_type == 'nonlinear_problem_load_stepping':
num_steps = 3
state_function.vector().set_local(
np.zeros((state_function.function_space().dim())))
for i in range(num_steps):
v = df.TestFunction(state_function.function_space())
if i < (num_steps - 1):
residual_form = get_residual_form(
state_function,
v,
density_func,
density_func.function_space(),
tractionBC,
# df.Constant((0.0, -9.e-1))
df.Constant((0.0, -9.e-1 / num_steps * (i + 1))),
int(self.itr))
else:
residual_form = get_residual_form(
state_function,
v,
density_func,
density_func.function_space(),
tractionBC,
# df.Constant((0.0, -9.e-1))
df.Constant((0.0, -9.e-1 / num_steps * (i + 1))),
int(self.itr))
problem = df.NonlinearVariationalProblem(
residual_form, state_function, pde_problem.bcs_list,
self.derivative_form)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters['nonlinear_solver'] = 'snes'
solver.parameters["snes_solver"]["line_search"] = 'bt'
solver.parameters["snes_solver"][
"linear_solver"] = 'mumps' # "cg" "gmres"
solver.parameters["snes_solver"]["maximum_iterations"] = 500
solver.parameters["snes_solver"]["relative_tolerance"] = 1e-15
solver.parameters["snes_solver"]["absolute_tolerance"] = 1e-15
# solver.parameters["snes_solver"]["linear_solver_"]["maximum_iterations"]=1000
solver.parameters["snes_solver"][
"error_on_nonconvergence"] = False
solver.solve()
# option to store the visualization results
if visualization == 'True':
for argument_name, argument_function in iteritems(
self.argument_functions_dict):
df.File('solutions_iterations_3d/{}_{}.pvd'.format(
argument_name, self.itr)) << argument_function
self.L = -residual_form
self.itr = self.itr + 1
outputs[state_name] = state_function.vector().get_local()
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
####################################################################
######################### TOTAL RESIDUAL #########################
####################################################################
# We solve all these simulataneously. Add the residuals together and
# differentiate for the complete system.
R = R_stress + R_c + R_mass + R_len
J = df.derivative(R, U, dU)
#####################################################################
###################### Variational Solvers ########################
#####################################################################
# Define variational problem subject to no Dirichlet BCs, but with a
# thickness bound, plus form compiler parameters for efficiency.
mass_problem = df.NonlinearVariationalProblem(
R, U, bcs=[], J=J, form_compiler_parameters=ffc_options)
mass_problem.set_bounds(l_bound, u_bound)
# Create an instance of vinewtonrsls, PETSc's variational inequality
# solver. Not clear that this is needed, given that the ice should never be
# negative by design.
mass_solver = df.NonlinearVariationalSolver(mass_problem)
mass_solver.parameters['nonlinear_solver'] = 'snes'
mass_solver.parameters['snes_solver']['method'] = 'vinewtonrsls'
mass_solver.parameters['snes_solver']['relative_tolerance'] = 1e-3
mass_solver.parameters['snes_solver']['absolute_tolerance'] = 1e-3
mass_solver.parameters['snes_solver']['error_on_nonconvergence'] = True
mass_solver.parameters['snes_solver']['linear_solver'] = 'mumps'
mass_solver.parameters['snes_solver']['maximum_iterations'] = 10
mass_solver.parameters['snes_solver']['report'] = False
def setup_NSPFEC(w_NSPFEC, w_1NSPFEC,
dirichlet_bcs_NSPFEC,
neumann_bcs,
boundary_to_mark,
dx, ds, normal,
v, q, q0, psi, h, b, U,
u_, p_, p0_, phi_, g_, c_, V_,
u_1, p_1, p0_1, phi_1, g_1, c_1, V_1,
M_, nu_, veps_, rho_, K_, beta_, rho_e_,
dbeta, dveps, drho,
per_tau, sigma_bar, eps, grav, z,
solutes,
enable_NS, enable_PF, enable_EC,
use_iterative_solvers,
p_lagrange,
q_rhs):
""" The full problem of electrohydrodynamics in two phases.
Note that it is possible to turn off the different parts at will.
"""
# Setup of the Navier-Stokes part of F
mom_ = rho_*u_
if enable_PF:
mom_ += -M_*drho * df.nabla_grad(g_)
F = []
if enable_NS:
F_NS = (per_tau * rho_ * df.dot(u_ - u_1, v) * dx
+ df.inner(df.nabla_grad(u_), df.outer(mom_, v)) * dx
+ 2*nu_*df.inner(df.sym(df.nabla_grad(u_)),
df.sym(df.nabla_grad(v))) * dx
- p_ * df.div(v) * dx
- df.div(u_) * q * dx
- df.dot(rho_ * grav, v) * dx)
# if enable_PF:
# F_NS += - sigma_bar*eps*df.inner(
# df.outer(df.grad(phi_),
# df.grad(phi_)), df.grad(v)) * dx
# if enable_EC and rho_e_ != 0:
# F_NS += rho_e_*df.dot(df.grad(V_), v) * dx
# if enable_PF and enable_EC:
# F_NS += dveps*df.dot(
# df.grad(phi_), v)*df.dot(df.grad(V_),
# df.grad(V_)) * dx
if enable_PF:
F_NS += phi_*df.dot(df.grad(g_), v) * dx
if enable_EC:
for ci_, dbetai, solute in zip(c_, dbeta, solutes):
zi = solute[1]
F_NS += df.dot(df.grad(ci_), v) * dx \
+ ci_*dbetai*df.dot(df.grad(phi_), v) * dx \
+ zi*ci_*df.dot(df.grad(V_), v) * dx
# Slip boundary condition
for boundary_name, slip_length in neumann_bcs["u"].items():
F_NS += 1./slip_length * \
df.dot(u_, v) * ds(boundary_to_mark[boundary_name])
# Pressure boundary condition
for boundary_name, pressure in neumann_bcs["p"].items():
F_NS += pressure * df.inner(
normal, v) * ds(boundary_to_mark[boundary_name])
# Lagrange pressure
if p_lagrange:
F_NS += (p_*q0 + q*p0_)*dx
# RHS source terms
if "u" in q_rhs:
F_NS += -df.dot(q_rhs["u"], v)*dx
F.append(F_NS)
# Setup of the phase-field equations
if enable_PF:
phi_1_flt = unit_interval_filter(phi_1)
F_PF_phi = (per_tau*(phi_-phi_1_flt)*psi*df.dx +
M_*df.dot(df.grad(g_), df.grad(psi)) * dx)
if enable_NS:
F_PF_phi += df.dot(u_, df.grad(phi_)) * psi * dx
F_PF_g = (g_ * h * dx
- sigma_bar*eps*df.dot(df.grad(phi_), df.grad(h)) * dx
- sigma_bar/eps*diff_pf_potential(phi_) * h * dx)
if enable_EC:
F_PF_g += (-sum([dbeta_i * ci_ * h * dx
for dbeta_i, ci_ in zip(dbeta, c_)])
+ dveps * df.dot(df.grad(V_), df.grad(V_)) * h * dx)
# Contact angle boundary condtions
for boundary_name, costheta in neumann_bcs["phi"].items():
fw_prime = diff_pf_contact(phi_)
F_PF_g += sigma_bar * costheta * fw_prime * h * ds(
boundary_to_mark[boundary_name])
# RHS source terms
if "phi" in q_rhs:
F_PF_phi += -q_rhs["phi"]*psi*dx
F_PF = F_PF_phi + F_PF_g
F.append(F_PF)
# Setup of the electrochemistry
if enable_EC:
F_E_c = []
for ci_, ci_1, bi, Ki_, zi, dbetai, solute in zip(
c_, c_1, b, K_, z, dbeta, solutes):
ci_1_flt = max_value(ci_1, 0.)
F_E_ci = (per_tau*(ci_-ci_1_flt)*bi*df.dx
+ Ki_*df.dot(df.grad(ci_), df.grad(bi))*df.dx)
if zi != 0:
F_E_ci += Ki_*zi*ci_*df.dot(df.grad(V_),
df.grad(bi))*df.dx
if enable_NS:
F_E_ci += df.dot(u_, df.grad(ci_))*bi*df.dx
if enable_PF:
F_E_ci += Ki_*ci_*dbetai*df.dot(df.grad(phi_),
df.grad(bi)) * dx
if solute[0] in q_rhs:
F_E_ci += - q_rhs[solute[0]] * bi * dx
F_E_c.append(F_E_ci)
F_E_V = veps_*df.dot(df.grad(V_), df.grad(U))*df.dx
if rho_e_ != 0:
F_E_V += -rho_e_*U*df.dx
# Surface charge boundary condition
for boundary_name, sigma_e in neumann_bcs["V"].items():
F_E_V += -sigma_e*U*ds(boundary_to_mark[boundary_name])
# RHS source terms
if "V" in q_rhs:
F_E_V += q_rhs["V"]*U*dx
F_E = sum(F_E_c) + F_E_V
F.append(F_E)
F = sum(F)
J = df.derivative(F, w_NSPFEC)
problem_NSPFEC = df.NonlinearVariationalProblem(F, w_NSPFEC, dirichlet_bcs_NSPFEC, J)
solver_NSPFEC = df.NonlinearVariationalSolver(problem_NSPFEC)
if use_iterative_solvers:
solver_NSPFEC.parameters['newton_solver']['linear_solver'] = 'gmres'
solver_NSPFEC.parameters['newton_solver']['preconditioner'] = 'ilu'
return solver_NSPFEC
def _setup_implicit_problem(self):
assert hasattr(self, "_parameters")
assert hasattr(self, "_mesh")
assert hasattr(self, "_Wh")
assert hasattr(self, "_coefficients")
assert hasattr(self, "_one")
assert hasattr(self, "_omega")
assert hasattr(self, "_v0")
assert hasattr(self, "_v00")
assert hasattr(self, "_T0")
assert hasattr(self, "_T00")
print " setup implicit problem..."
#=======================================================================
# retrieve imex coefficients
a, c = self._imex_alpha, self._imex_gamma
#=======================================================================
# trial and test function
(del_v, del_p, del_T) = dlfn.TestFunctions(self._Wh)
v, p, T = dlfn.split(self._sol)
# volume element
dV = dlfn.Measure("dx", domain=self._mesh)
# reference to time step
timestep = self._timestep
#=======================================================================
from dolfin import dot, grad
# 1) momentum equation
momentum_eqn = (
dot((a[0] * v + a[1] * self._v0 + a[2] * self._v00) / timestep,
del_v) + dot(dot(grad(v), v), del_v) + self._coefficients[1] *
a_op(c[0] * v + c[1] * self._v0 + c[2] * self._v00, del_v) -
b_op(del_v, p) - b_op(v, del_p)) * dV
# 2) momentum equation: coriolis term
if self._parameters.rotation is True:
assert self._coefficients[0] != 0.0
print " adding rotation to the model..."
# add Coriolis term
if self._space_dim == 2:
momentum_eqn += self._coefficients[0] * (-v[1] * del_v[0] +
v[0] * del_v[1]) * dV
elif self._space_dim == 3:
assert hasattr(self, "_rotation_vector")
assert isinstance(self._rotation_vector,
(dlfn.Constant, dlfn.Expression))
from dolfin import cross
momentum_eqn += self._coefficients[0] * dot(
cross(self._rotation_vector, v), del_v) * dV
# 3) momentum equation: coriolis term
if self._parameters.buoyancy is True:
assert self._coefficients[2] != 0.0
assert hasattr(self, "_gravity") and isinstance(
self._gravity, (dlfn.Expression, dlfn.Constant))
print " adding buoyancy to the model..."
# add buoyancy term
momentum_eqn += self._coefficients[2] * T * dot(
self._gravity, del_v) * dV
#=======================================================================
# 4) energy equation
energy_eqn = (
(a[0] * T + a[1] * self._T0 + a[2] * self._T00) / timestep * del_T
+ dot(v, grad(T)) * del_T + self._coefficients[3] *
a_op(c[0] * T + c[1] * self._T0 + c[2] * self._T00, del_T)) * dV
#=======================================================================
# full problem
self._eqn = momentum_eqn + energy_eqn
if not hasattr(self, "_dirichlet_bcs"):
self._setup_boundary_conditons()
self._jacobian = dlfn.derivative(self._eqn, self._sol)
problem = dlfn.NonlinearVariationalProblem(self._eqn,
self._sol,
J=self._jacobian,
bcs=self._dirichlet_bcs)
self._solver = dlfn.NonlinearVariationalSolver(problem)
prm = self._solver.parameters
prm["newton_solver"]["absolute_tolerance"] = 1e-6
prm["newton_solver"]["relative_tolerance"] = 1e-9
prm["newton_solver"]["maximum_iterations"] = 10
F_nu = geo_map.form(nu_*eta + geo_map.gab[i, j]*psi_.dx(i)*eta.dx(j))
F_nuhat = geo_map.form(nuhat_*etahat
+ geo_map.Kab[i, j]*psi_.dx(i)*etahat.dx(j))
F = F_psi + F_mu + F_nu + F_nuhat
# a = df.lhs(F)
# L = df.rhs(F)
J = df.derivative(F, u_, du=u)
# SOLVER
# problem = df.LinearVariationalProblem(a, L, u_)
# solver = df.LinearVariationalSolver(problem)
problem = df.NonlinearVariationalProblem(F, u_, J=J)
solver = df.NonlinearVariationalSolver(problem)
solver.parameters["newton_solver"]["absolute_tolerance"] = 1e-8
solver.parameters["newton_solver"]["relative_tolerance"] = 1e-6
solver.parameters["newton_solver"]["maximum_iterations"] = 16
# solver.parameters["newton_solver"]["linear_solver"] = "gmres"
# solver.parameters["newton_solver"]["preconditioner"] = "default"
# solver.parameters["newton_solver"]["krylov_solver"]["nonzero_initial_guess"] = True
# solver.parameters["newton_solver"]["krylov_solver"]["absolute_tolerance"] = 1e-8
# solver.parameters["newton_solver"]["krylov_solver"]["monitor_convergence"] = False
# solver.parameters["newton_solver"]["krylov_solver"]["maximum_iterations"] = 1000
# solver.parameters["linear_solver"] = "gmres"
# solver.parameters["preconditioner"] = "jacobi"
df.parameters["form_compiler"]["optimize"] = True
# <codecell>
def boundary(h, on_boundary):
return on_boundary
bc = dfn.DirichletBC(V, dfn.Constant(0.0), boundary)
# bc = dfn.DirichletBC(V, M, boundary)
# <markdowncell>
# Setup the the problem so we can solve it.
# <codecell>
problem = dfn.NonlinearVariationalProblem(F, H_n, bc, J)
# <markdowncell>
# Construct the solver
# <codecell>
solver = dfn.NonlinearVariationalSolver(problem)
# <markdowncell>
# Set the solver parameters
# <codecell>
