def step(self):
r"""Take one step of the conjugate gradient iteration"""
q = self.solution
s = self.search_direction
z = self.residual
δz = self._delta_residual
α = self.residual_energy / self.search_direction_energy
Gs = self.operator_product
dJ = self._rhs
delta_energy = α * (self._assemble(action(Gs, q)) +
0.5 * α * self.search_direction_energy)
self._energy += delta_energy
self._objective += delta_energy + α * self._assemble(action(dJ, s))
q.assign(q + Constant(α) * s)
z.assign(z - Constant(α) * δz)
M = self.preconditioner
residual_energy = self._assemble(firedrake.energy_norm(M, z))
β = residual_energy / self.residual_energy
s.assign(Constant(β) * s + z)
self.update_state()
self._residual_energy = residual_energy
Gs = self.operator_product
self._search_direction_energy = self._assemble(action(Gs, s))
self._iteration += 1
def gauss_newton_energy_norm(self, q):
r"""Compute the energy norm of a field w.r.t. the Gauss-Newton operator
The energy norm of a field :math:`q` w.r.t. the Gauss-Newton operator
:math:`H` can be computed using one fewer linear solve than if we were
to calculate the action of :math:`H\cdot q` on :math:`q`. This saves
computation when using the conjugate gradient method to solve for the
search direction.
"""
u, p = self.state, self.parameter
dE = derivative(self._E, u)
dR = derivative(self._R, p)
dF_du, dF_dp = self._dF_du, derivative(self._F, p)
v = firedrake.Function(u.function_space())
firedrake.solve(dF_du == action(dF_dp, q),
v,
self._bc,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
return self._assemble(
firedrake.energy_norm(derivative(dE, u), v) +
firedrake.energy_norm(derivative(dR, p), q))
def __init__(self, state, V, direction=[1,2], params=None):
super(InteriorPenalty, self).__init__(state)
dt = state.timestepping.dt
kappa = params['kappa']
mu = params['mu']
gamma = TestFunction(V)
phi = TrialFunction(V)
self.phi1 = Function(V)
n = FacetNormal(state.mesh)
a = inner(gamma,phi)*dx + dt*inner(grad(gamma), grad(phi)*kappa)*dx
def get_flux_form(dS, M):
fluxes = (-inner(2*avg(outer(phi, n)), avg(grad(gamma)*M))
- inner(avg(grad(phi)*M), 2*avg(outer(gamma, n)))
+ mu*inner(2*avg(outer(phi, n)), 2*avg(outer(gamma, n)*kappa)))*dS
return fluxes
if 1 in direction:
a += dt*get_flux_form(dS_v, kappa)
if 2 in direction:
a += dt*get_flux_form(dS_h, kappa)
L = inner(gamma,phi)*dx
problem = LinearVariationalProblem(a, action(L,self.phi1), self.phi1)
self.solver = LinearVariationalSolver(problem)
def _newton_solve(z, E, scale, tolerance=1e-6, armijo=1e-4, max_iterations=50):
F = derivative(E, z)
H = derivative(F, z)
Q = z.function_space()
bc = firedrake.DirichletBC(Q, 0, 'on_boundary')
p = firedrake.Function(Q)
for iteration in range(max_iterations):
firedrake.solve(H == -F, p, bc,
solver_parameters={'ksp_type': 'preonly', 'pc_type': 'lu'})
dE_dp = assemble(action(F, p))
α = 1.0
E0 = assemble(E)
Ez = assemble(replace(E, {z: z + firedrake.Constant(α) * p}))
while (Ez > E0 + armijo * α * dE_dp) or np.isnan(Ez):
α /= 2
Ez = assemble(replace(E, {z: z + firedrake.Constant(α) * p}))
z.assign(z + firedrake.Constant(α) * p)
if abs(dE_dp) < tolerance * assemble(scale):
return z
raise ValueError("Newton solver failed to converge after {0} iterations"
.format(max_iterations))
def __init__(self, state, V, kappa, mu, bcs=None):
super(InteriorPenalty, self).__init__(state)
dt = state.timestepping.dt
gamma = TestFunction(V)
phi = TrialFunction(V)
self.phi1 = Function(V)
n = FacetNormal(state.mesh)
a = inner(gamma, phi) * dx + dt * inner(grad(gamma),
grad(phi) * kappa) * dx
def get_flux_form(dS, M):
fluxes = (-inner(2 * avg(outer(phi, n)), avg(grad(gamma) * M)) -
inner(avg(grad(phi) * M), 2 * avg(outer(gamma, n))) +
mu * inner(2 * avg(outer(phi, n)),
2 * avg(outer(gamma, n) * kappa))) * dS
return fluxes
a += dt * get_flux_form(dS_v, kappa)
a += dt * get_flux_form(dS_h, kappa)
L = inner(gamma, phi) * dx
problem = LinearVariationalProblem(a,
action(L, self.phi1),
self.phi1,
bcs=bcs)
self.solver = LinearVariationalSolver(problem)
def __init__(self, problem, tolerance, solver_parameters=None, **kwargs):
r"""Solve a MinimizationProblem using Newton's method with backtracking
line search
Parameters
----------
problem : MinimizationProblem
The particular problem instance to solve
tolerance : float
dimensionless tolerance for when to stop iterating, measured with
with respect to the problem's scale functional
solver_parameters : dict (optional)
Linear solve parameters for computing the search direction
armijo : float (optional)
Parameter in the Armijo condition for line search; defaults to
1e-4, see Nocedal and Wright
contraction : float (optional)
shrinking factor for backtracking line search; defaults to .5
max_iterations : int (optional)
maximum number of outer-level Newton iterations; defaults to 50
"""
self.problem = problem
self.tolerance = tolerance
if solver_parameters is None:
solver_parameters = default_solver_parameters
self.armijo = kwargs.pop("armijo", 1e-4)
self.contraction = kwargs.pop("contraction", 0.5)
self.max_iterations = kwargs.pop("max_iterations", 50)
u = self.problem.u
V = u.function_space()
v = firedrake.Function(V)
self.v = v
E = self.problem.E
self.F = firedrake.derivative(E, u)
self.J = firedrake.derivative(self.F, u)
self.dE_dv = firedrake.action(self.F, v)
bcs = None
if self.problem.bcs:
bcs = firedrake.homogenize(self.problem.bcs)
problem = firedrake.LinearVariationalProblem(
self.J,
-self.F,
v,
bcs,
constant_jacobian=False,
form_compiler_parameters=self.problem.form_compiler_parameters,
)
self.search_direction_solver = firedrake.LinearVariationalSolver(
problem, solver_parameters=solver_parameters
)
self.search_direction_solver.solve()
self.t = firedrake.Constant(0.0)
self.iteration = 0
def callback(solver):
q = solver.search_direction
dJ = solver.gradient
dJ_dq = firedrake.action(dJ, q)
decrement = firedrake.assemble(dJ_dq, **fc_params) / area
error = firedrake.assemble(solver.objective) / area
penalty = firedrake.assemble(solver.regularization) / area
print(f'{error:10.4g} | {penalty:10.4g} | {decrement:10.4g}')
def evaluate_adj_component(self,
inputs,
adj_inputs,
block_variable,
idx,
prepared=None):
if not self.linear and self.func == block_variable.output:
# We are not able to calculate derivatives wrt initial guess.
return None
F_form = prepared["form"]
adj_sol = prepared["adj_sol"]
adj_sol_bdy = prepared["adj_sol_bdy"]
c = block_variable.output
c_rep = block_variable.saved_output
if isinstance(c, firedrake.Function):
trial_function = firedrake.TrialFunction(c.function_space())
elif isinstance(c, firedrake.Constant):
mesh = self.compat.extract_mesh_from_form(F_form)
trial_function = firedrake.TrialFunction(
c._ad_function_space(mesh))
elif isinstance(c, firedrake.DirichletBC):
tmp_bc = self.compat.create_bc(
c,
value=self.compat.extract_subfunction(adj_sol_bdy,
c.function_space()))
return [tmp_bc]
elif isinstance(c, self.compat.MeshType):
# Using CoordianteDerivative requires us to do action before
# differentiating, might change in the future.
F_form_tmp = firedrake.action(F_form, adj_sol)
X = firedrake.SpatialCoordinate(c_rep)
dFdm = firedrake.derivative(
-F_form_tmp, X, firedrake.TestFunction(c._ad_function_space()))
dFdm = self.compat.assemble_adjoint_value(dFdm,
**self.assemble_kwargs)
return dFdm
# dFdm_cache works with original variables, not block saved outputs.
if c in self._dFdm_cache:
dFdm = self._dFdm_cache[c]
else:
dFdm = -firedrake.derivative(self.lhs, c, trial_function)
dFdm = firedrake.adjoint(dFdm)
self._dFdm_cache[c] = dFdm
# Replace the form coefficients with checkpointed values.
replace_map = self._replace_map(dFdm)
replace_map[self.func] = self.get_outputs()[0].saved_output
dFdm = replace(dFdm, replace_map)
dFdm = dFdm * adj_sol
dFdm = self.compat.assemble_adjoint_value(dFdm, **self.assemble_kwargs)
return dFdm
def update_search_direction(self):
r"""Solve the Gauss-Newton system for the new search direction using
the preconditioned conjugate gradient method"""
p, q, dJ = self.parameter, self.search_direction, self.gradient
dR = derivative(self.regularization, self.parameter)
Q = q.function_space()
M = firedrake.TrialFunction(Q) * firedrake.TestFunction(Q) * dx + \
derivative(dR, p)
# Compute the preconditioned residual
z = firedrake.Function(Q)
firedrake.solve(M == -dJ,
z,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
# This variable is a search direction for a search direction, which
# is definitely not confusing at all.
s = z.copy(deepcopy=True)
q *= 0.0
old_cost = np.inf
while True:
z_mnorm = self._assemble(firedrake.energy_norm(M, z))
s_hnorm = self.gauss_newton_energy_norm(s)
α = z_mnorm / s_hnorm
δz = firedrake.Function(Q)
g = self.gauss_newton_mult(s)
firedrake.solve(M == g,
δz,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
q += α * s
z -= α * δz
β = self._assemble(firedrake.energy_norm(M, z)) / z_mnorm
s *= β
s += z
energy_norm = self.gauss_newton_energy_norm(q)
cost = 0.5 * energy_norm + self._assemble(action(dJ, q))
if (abs(old_cost - cost) /
(0.5 * energy_norm) < self._search_tolerance):
return
old_cost = cost
def gauss_newton_mult(self, q):
"""Multiply a field by the Gauss-Newton operator"""
u, p = self.state, self.parameter
dE = derivative(self._E, u)
dR = derivative(self._R, p)
dF_du, dF_dp = self._dF_du, derivative(self._F, p)
w = firedrake.Function(u.function_space())
firedrake.solve(dF_du == action(dF_dp, q),
w,
self._bc,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
v = firedrake.Function(u.function_space())
firedrake.solve(adjoint(dF_du) == derivative(dE, u, w),
v,
self._bc,
solver_parameters=self._solver_params,
form_compiler_parameters=self._fc_params)
return action(adjoint(dF_dp), v) + derivative(dR, p, q)
def _setup(self, problem, callback=(lambda s: None)):
self._problem = problem
self._callback = callback
self._p = problem.parameter.copy(deepcopy=True)
self._u = problem.state.copy(deepcopy=True)
self._model_args = dict(**problem.model_args,
dirichlet_ids=problem.dirichlet_ids)
u_name, p_name = problem.state_name, problem.parameter_name
args = dict(**self._model_args, **{u_name: self._u, p_name: self._p})
# Make the form compiler use a reasonable number of quadrature points
degree = problem.model.quadrature_degree(**args)
self._fc_params = {'quadrature_degree': degree}
# Create the error, regularization, and barrier functionals
self._E = problem.objective(self._u)
self._R = problem.regularization(self._p)
self._J = self._E + self._R
# Create the weak form of the forward model, the adjoint state, and
# the derivative of the objective functional
self._F = derivative(problem.model.action(**args), self._u)
self._dF_du = derivative(self._F, self._u)
# Create a search direction
dR = derivative(self._R, self._p)
self._solver_params = {'ksp_type': 'preonly', 'pc_type': 'lu'}
Q = self._p.function_space()
self._q = firedrake.Function(Q)
# Create the adjoint state variable
V = self.state.function_space()
self._λ = firedrake.Function(V)
dF_dp = derivative(self._F, self._p)
# Create Dirichlet BCs where they apply for the adjoint solve
rank = self._λ.ufl_element().num_sub_elements()
if rank == 0:
zero = firedrake.Constant(0)
else:
zero = firedrake.as_vector((0, ) * rank)
self._bc = firedrake.DirichletBC(V, zero, problem.dirichlet_ids)
# Create the derivative of the objective functional
self._dE = derivative(self._E, self._u)
dR = derivative(self._R, self._p)
self._dJ = (action(adjoint(dF_dp), self._λ) + dR)
def __init__(self, state, V, direction=[], supg_params=None):
super(SUPGAdvection, self).__init__(state)
dt = state.timestepping.dt
params = supg_params.copy() if supg_params else {}
params.setdefault('a0', dt/sqrt(15.))
params.setdefault('a1', dt/sqrt(15.))
gamma = TestFunction(V)
theta = TrialFunction(V)
self.theta0 = Function(V)
# make SUPG test function
taus = [params["a0"], params["a1"]]
for i in direction:
taus[i] = 0.0
tau = Constant(((taus[0], 0.), (0., taus[1])))
dgamma = dot(dot(self.ubar, tau), grad(gamma))
gammaSU = gamma + dgamma
n = FacetNormal(state.mesh)
un = 0.5*(dot(self.ubar, n) + abs(dot(self.ubar, n)))
a_mass = gammaSU*theta*dx
arhs = a_mass - dt*gammaSU*dot(self.ubar, grad(theta))*dx
if 1 in direction:
arhs -= (
dt*dot(jump(gammaSU), (un('+')*theta('+')
- un('-')*theta('-')))*dS_v
- dt*(gammaSU('+')*dot(self.ubar('+'), n('+'))*theta('+')
+ gammaSU('-')*dot(self.ubar('-'), n('-'))*theta('-'))*dS_v
)
if 2 in direction:
arhs -= (
dt*dot(jump(gammaSU), (un('+')*theta('+')
- un('-')*theta('-')))*dS_h
- dt*(gammaSU('+')*dot(self.ubar('+'), n('+'))*theta('+')
+ gammaSU('-')*dot(self.ubar('-'), n('-'))*theta('-'))*dS_h
)
self.theta1 = Function(V)
self.dtheta = Function(V)
problem = LinearVariationalProblem(a_mass, action(arhs,self.theta1), self.dtheta)
self.solver = LinearVariationalSolver(problem,
options_prefix='SUPGAdvection')
def complexity(form, parameters, action=False):
if action:
coef = firedrake.Function(form.arguments()[0].function_space())
form = firedrake.action(form, coef)
impero_kernel, index_names = tsfc.driver.compile_form(
form, parameters=parameters)[0]
indices = IndexDict(
{idx: sympy.symbols(name)
for idx, name in index_names})
expr = expression(impero_kernel.tree,
impero_kernel.temporaries,
indices,
top=True)
p1 = sympy.symbols("p") + 1
'''Currently assume p+1 quad points in each direction.'''
return expr.subs([(i, p1) for i in indices.values()]).expand()
def reinit(self):
r"""Restart the solution to 0"""
self._iteration = 0
M = self._preconditioner
z = self.residual
s = self.search_direction
Gs = self.operator_product
Q = z.function_space()
self.solution.assign(firedrake.Function(Q))
self._residual_solver.solve()
s.assign(z)
self._residual_energy = self._assemble(firedrake.energy_norm(M, z))
self.update_state()
self._search_direction_energy = self._assemble(action(Gs, s))
self._energy = 0.
self._objective = 0.
def __init__(self, state, V, continuity=False):
super(DGAdvection, self).__init__(state)
element = V.fiat_element
assert element.entity_dofs() == element.entity_closure_dofs(), "Provided space is not discontinuous"
dt = state.timestepping.dt
if V.extruded:
surface_measure = (dS_h + dS_v)
else:
surface_measure = dS
phi = TestFunction(V)
D = TrialFunction(V)
self.D1 = Function(V)
self.dD = Function(V)
n = FacetNormal(state.mesh)
# ( dot(v, n) + |dot(v, n)| )/2.0
un = 0.5*(dot(self.ubar, n) + abs(dot(self.ubar, n)))
a_mass = inner(phi,D)*dx
if continuity:
a_int = -inner(grad(phi), outer(D, self.ubar))*dx
else:
a_int = -inner(div(outer(phi,self.ubar)),D)*dx
a_flux = (dot(jump(phi), un('+')*D('+') - un('-')*D('-')))*surface_measure
arhs = a_mass - dt*(a_int + a_flux)
DGproblem = LinearVariationalProblem(a_mass, action(arhs,self.D1),
self.dD)
self.DGsolver = LinearVariationalSolver(DGproblem,
solver_parameters={
'ksp_type':'preonly',
'pc_type':'bjacobi',
'sub_pc_type': 'ilu'},
options_prefix='DGAdvection')
def solve(self, atol=0.0, rtol=1e-6, etol=0.0, max_iterations=200):
r"""Search for a new value of the parameters, stopping once either
the objective functional gets below a threshold value or stops
improving.
Parameters
----------
atol : float
Absolute stopping tolerance; stop iterating when the objective
drops below this value
rtol : float
Relative stopping tolerance; stop iterating when the relative
decrease in the objective drops below this value
etol : float
Expectation stopping tolerance; stop iterating when the relative
expected decrease in the objective from the Newton decrement drops
below this value
max_iterations : int
Maximum number of iterations to take
"""
J_initial = np.inf
for iteration in range(max_iterations):
J = self._assemble(self._J)
q = self.search_direction
dJ_dq = self._assemble(firedrake.action(self.gradient, q))
if (
((J_initial - J) < rtol * J_initial)
or (-dJ_dq < etol * J)
or (J <= atol)
):
return iteration
J_initial = J
self.step()
return max_iterations
def __init__(self, equation, alpha):
residual = equation.residual.label_map(
lambda t: t.has_label(linearisation),
lambda t: Term(t.get(linearisation).form, t.labels), drop)
dt = equation.state.dt
W = equation.function_space
beta = dt * alpha
# Split up the rhs vector (symbolically)
self.xrhs = Function(W)
aeqn = residual.label_map(
lambda t:
(t.has_label(time_derivative) and t.has_label(linearisation)),
map_if_false=lambda t: beta * t)
Leqn = residual.label_map(
lambda t:
(t.has_label(time_derivative) and t.has_label(linearisation)),
map_if_false=drop)
# Place to put result of solver
self.dy = Function(W)
# Solver
bcs = equation.bcs['u']
problem = LinearVariationalProblem(aeqn.form,
action(Leqn.form, self.xrhs),
self.dy,
bcs=bcs)
self.solver = LinearVariationalSolver(
problem,
solver_parameters=self.solver_parameters,
options_prefix='linear_solver')
def apply_adjoint_action(self, x, out):
# fd.assemble(fd.action(self.a, x), tensor=out)
out.assign(fd.assemble(fd.action(self.a, x)))
x = ufl.SpatialCoordinate(mesh)
expr = ufl.as_vector([ufl.sin(2 * ufl.pi * x[0]), ufl.cos(2 * ufl.pi * x[1])])
u = fd.interpolate(expr, V)
u_dot = fd.Function(V)
v = fd.TestFunction(V)
nu = fd.Constant(0.0001) # for burgers
if equation == "heat":
nu = fd.Constant(0.1) # for heat
M = fd.derivative(fd.inner(u, v) * fd.dx, u)
R = -(fd.inner(fd.grad(u) * u, v) + nu * fd.inner(fd.grad(u), fd.grad(v))) * fd.dx
if equation == "heat":
R = -nu * fd.inner(fd.grad(u), fd.grad(v)) * fd.dx
F = fd.action(M, u_dot) - R
bc = fd.DirichletBC(V, (0.0, 0.0), "on_boundary")
t = 0.0
end = 0.1
tspan = (t, end)
state_out = fd.File("result/state.pvd")
def ts_monitor(ts, steps, time, X):
state_out.write(u, time=time)
problem = firedrake_ts.DAEProblem(F, u, u_dot, tspan, bcs=bc)
def __init__(self, solver):
r"""State machine for solving the Gauss-Newton subproblem via the
preconditioned conjugate gradient method"""
self._assemble = solver._assemble
u = solver.state
p = solver.parameter
E = solver._E
dE = derivative(E, u)
R = solver._R
dR = derivative(R, p)
F = solver._F
dF_du = derivative(F, u)
dF_dp = derivative(F, p)
# TODO: Make this an arbitrary RHS -- the solver can set it to the
# gradient if we want
dJ = solver.gradient
bc = solver._bc
V = u.function_space()
Q = p.function_space()
# Create the preconditioned residual and solver
z = firedrake.Function(Q)
s = firedrake.Function(Q)
φ, ψ = firedrake.TestFunction(Q), firedrake.TrialFunction(Q)
M = φ * ψ * dx + derivative(dR, p)
residual_problem = firedrake.LinearVariationalProblem(
M,
-dJ,
z,
form_compiler_parameters=solver._fc_params,
constant_jacobian=False)
residual_solver = firedrake.LinearVariationalSolver(
residual_problem, solver_parameters=solver._solver_params)
self._preconditioner = M
self._residual = z
self._search_direction = s
self._residual_solver = residual_solver
# Create a variable to store the current solution of the Gauss-Newton
# problem and the solutions of the auxiliary tangent sub-problems
q = firedrake.Function(Q)
v = firedrake.Function(V)
w = firedrake.Function(V)
# Create linear problem and solver objects for the auxiliary tangent
# sub-problems
tangent_linear_problem = firedrake.LinearVariationalProblem(
dF_du,
action(dF_dp, s),
w,
bc,
form_compiler_parameters=solver._fc_params,
constant_jacobian=False)
tangent_linear_solver = firedrake.LinearVariationalSolver(
tangent_linear_problem, solver_parameters=solver._solver_params)
adjoint_tangent_linear_problem = firedrake.LinearVariationalProblem(
adjoint(dF_du),
derivative(dE, u, w),
v,
bc,
form_compiler_parameters=solver._fc_params,
constant_jacobian=False)
adjoint_tangent_linear_solver = firedrake.LinearVariationalSolver(
adjoint_tangent_linear_problem,
solver_parameters=solver._solver_params)
self._rhs = dJ
self._solution = q
self._tangent_linear_solution = w
self._tangent_linear_solver = tangent_linear_solver
self._adjoint_tangent_linear_solution = v
self._adjoint_tangent_linear_solver = adjoint_tangent_linear_solver
self._product = action(adjoint(dF_dp), v) + derivative(dR, p, s)
# Create the update to the residual and the associated solver
δz = firedrake.Function(Q)
Gs = self._product
delta_residual_problem = firedrake.LinearVariationalProblem(
M,
Gs,
δz,
form_compiler_parameters=solver._fc_params,
constant_jacobian=False)
delta_residual_solver = firedrake.LinearVariationalSolver(
delta_residual_problem, solver_parameters=solver._solver_params)
self._delta_residual = δz
self._delta_residual_solver = delta_residual_solver
self._residual_energy = 0.
self._search_direction_energy = 0.
self.reinit()
def _setup(self, problem, callback=(lambda s: None)):
self._problem = problem
self._callback = callback
self._p = problem.parameter.copy(deepcopy=True)
self._u = problem.state.copy(deepcopy=True)
self._solver = self.problem.solver_type(self.problem.model,
**self.problem.solver_kwargs)
u_name, p_name = problem.state_name, problem.parameter_name
solve_kwargs = dict(**problem.diagnostic_solve_kwargs, **{
u_name: self._u,
p_name: self._p
})
# Make the form compiler use a reasonable number of quadrature points
degree = problem.model.quadrature_degree(**solve_kwargs)
self._fc_params = {'quadrature_degree': degree}
# Create the error, regularization, and barrier functionals
self._E = problem.objective(self._u)
self._R = problem.regularization(self._p)
self._J = self._E + self._R
# Create the weak form of the forward model, the adjoint state, and
# the derivative of the objective functional
A = problem.model.action(**solve_kwargs)
self._F = derivative(A, self._u)
self._dF_du = derivative(self._F, self._u)
# Create a search direction
dR = derivative(self._R, self._p)
# TODO: Make this customizable
self._solver_params = default_solver_parameters
Q = self._p.function_space()
self._q = firedrake.Function(Q)
# Create the adjoint state variable
V = self.state.function_space()
self._λ = firedrake.Function(V)
dF_dp = derivative(self._F, self._p)
# Create Dirichlet BCs where they apply for the adjoint solve
rank = self._λ.ufl_element().num_sub_elements()
if rank == 0:
zero = Constant(0)
else:
zero = firedrake.as_vector((0, ) * rank)
self._bc = firedrake.DirichletBC(V, zero, problem.dirichlet_ids)
# Create the derivative of the objective functional
self._dE = derivative(self._E, self._u)
dR = derivative(self._R, self._p)
self._dJ = (action(adjoint(dF_dp), self._λ) + dR)
# Create problem and solver objects for the adjoint state
L = adjoint(self._dF_du)
adjoint_problem = firedrake.LinearVariationalProblem(
L,
-self._dE,
self._λ,
self._bc,
form_compiler_parameters=self._fc_params,
constant_jacobian=False)
self._adjoint_solver = firedrake.LinearVariationalSolver(
adjoint_problem, solver_parameters=self._solver_params)
def newton_search(E, u, bc, tolerance, scale,
max_iterations=50, armijo=1e-4, contraction_factor=0.5,
form_compiler_parameters={},
solver_parameters={'ksp_type': 'preonly', 'pc_type': 'lu'}):
r"""Find the minimizer of a convex functional
Parameters
----------
E : firedrake.Form
The functional to be minimized
u0 : firedrake.Function
Initial guess for the minimizer
tolerance : float
Stopping criterion for the optimization procedure
scale : firedrake.Form
A positive scale functional by which to measure the objective
max_iterations : int, optional
Optimization procedure will stop at this many iterations regardless
of convergence
armijo : float, optional
The constant in the Armijo condition (see Nocedal and Wright)
contraction_factor : float, optional
The amount by which to backtrack in the line search if the Armijo
condition is not satisfied
form_compiler_parameters : dict, optional
Extra options to pass to the firedrake form compiler
solver_parameters : dict, optional
Extra options to pass to the linear solver
Returns
-------
firedrake.Function
The approximate minimizer of `E` to within tolerance
"""
F = firedrake.derivative(E, u)
H = firedrake.derivative(F, u)
v = firedrake.Function(u.function_space())
dE_dv = firedrake.action(F, v)
def assemble(*args, **kwargs):
return firedrake.assemble(
*args, **kwargs, form_compiler_parameters=form_compiler_parameters)
problem = firedrake.LinearVariationalProblem(H, -F, v, bc,
form_compiler_parameters=form_compiler_parameters,
constant_jacobian=False)
solver = firedrake.LinearVariationalSolver(problem,
solver_parameters=solver_parameters)
n = 0
while True:
# Compute a search direction
solver.solve()
# Compute the directional derivative, check if we're done
slope = assemble(dE_dv)
assert slope < 0
if (abs(slope) < assemble(scale) * tolerance) or (n >= max_iterations):
return u
# Backtracking search
E0 = assemble(E)
α = firedrake.Constant(1)
Eα = firedrake.replace(E, {u: u + α * v})
while assemble(Eα) > E0 + armijo * α.values()[0] * slope:
α.assign(α * contraction_factor)
u.assign(u + α * v)
n += 1
def compliance_optimization(n_iters=200):
output_dir = "cantilever/"
path = os.path.abspath(__file__)
dir_path = os.path.dirname(path)
m = fd.Mesh(f"{dir_path}/mesh_cantilever.msh")
mesh = fd.MeshHierarchy(m, 0)[-1]
# Perturb the mesh coordinates. Necessary to calculate shape derivatives
S = fd.VectorFunctionSpace(mesh, "CG", 1)
s = fd.Function(S, name="deform")
mesh.coordinates.assign(mesh.coordinates + s)
# Initial level set function
x, y = fd.SpatialCoordinate(mesh)
PHI = fd.FunctionSpace(mesh, "CG", 1)
lx = 2.0
ly = 1.0
phi_expr = (
-cos(6.0 / lx * pi * x) * cos(4.0 * pi * y)
- 0.6
+ max_value(200.0 * (0.01 - x ** 2 - (y - ly / 2) ** 2), 0.0)
+ max_value(100.0 * (x + y - lx - ly + 0.1), 0.0)
+ max_value(100.0 * (x - y - lx + 0.1), 0.0)
)
# Avoid recording the operation interpolate into the tape.
# Otherwise, the shape derivatives will not be correct
with fda.stop_annotating():
phi = fd.interpolate(phi_expr, PHI)
phi.rename("LevelSet")
fd.File(output_dir + "phi_initial.pvd").write(phi)
# Physics. Elasticity
rho_min = 1e-5
beta = fd.Constant(200.0)
def hs(phi, beta):
return fd.Constant(1.0) / (
fd.Constant(1.0) + exp(-beta * phi)
) + fd.Constant(rho_min)
H1_elem = fd.VectorElement("CG", mesh.ufl_cell(), 1)
W = fd.FunctionSpace(mesh, H1_elem)
u = fd.TrialFunction(W)
v = fd.TestFunction(W)
# Elasticity parameters
E, nu = 1.0, 0.3
mu, lmbda = fd.Constant(E / (2 * (1 + nu))), fd.Constant(
E * nu / ((1 + nu) * (1 - 2 * nu))
)
def epsilon(u):
return sym(nabla_grad(u))
def sigma(v):
return 2.0 * mu * epsilon(v) + lmbda * tr(epsilon(v)) * Identity(2)
a = inner(hs(-phi, beta) * sigma(u), nabla_grad(v)) * dx
t = fd.Constant((0.0, -75.0))
L = inner(t, v) * ds(2)
bc = fd.DirichletBC(W, fd.Constant((0.0, 0.0)), 1)
parameters = {
"ksp_type": "preonly",
"pc_type": "lu",
"mat_type": "aij",
"ksp_converged_reason": None,
"pc_factor_mat_solver_type": "mumps",
}
u_sol = fd.Function(W)
F = fd.action(a, u_sol) - L
problem = fd.NonlinearVariationalProblem(F, u_sol, bcs=bc)
solver = fd.NonlinearVariationalSolver(
problem, solver_parameters=parameters
)
solver.solve()
# fd.solve(
# a == L, u_sol, bcs=[bc], solver_parameters=parameters
# ) # , nullspace=nullspace)
with fda.stop_annotating():
fd.File("u_sol.pvd").write(u_sol)
# Cost function: Compliance
J = fd.assemble(
fd.Constant(1e-2)
* inner(hs(-phi, beta) * sigma(u_sol), epsilon(u_sol))
* dx
)
# Constraint: Volume
with fda.stop_annotating():
total_volume = fd.assemble(fd.Constant(1.0) * dx(domain=mesh))
VolPen = fd.assemble(hs(-phi, beta) * dx)
# Needed to track the value of the volume
VolControl = fda.Control(VolPen)
Vval = total_volume / 2.0
phi_pvd = fd.File("phi_evolution.pvd", target_continuity=fd.H1)
def deriv_cb(phi):
with fda.stop_annotating():
phi_pvd.write(phi[0])
c = fda.Control(s)
Jhat = LevelSetFunctional(J, c, phi, derivative_cb_pre=deriv_cb)
Vhat = LevelSetFunctional(VolPen, c, phi)
beta_param = 0.1
# Boundary conditions for the shape derivatives.
# They must be zero at the boundary conditions.
bcs_vel = fd.DirichletBC(S, fd.Constant((0.0, 0.0)), (1, 2))
# Regularize the shape derivatives
reg_solver = RegularizationSolver(
S,
mesh,
beta=beta_param,
gamma=1.0e5,
dx=dx,
bcs=bcs_vel,
output_dir=None,
)
# Hamilton-Jacobi equation to advect the level set
dt = 0.05
tol = 1e-5
# Optimization problem
vol_constraint = Constraint(Vhat, Vval, VolControl)
problem = InfDimProblem(Jhat, reg_solver, ineqconstraints=vol_constraint)
parameters = {
"ksp_type": "preonly",
"pc_type": "lu",
"mat_type": "aij",
"ksp_converged_reason": None,
"pc_factor_mat_solver_type": "mumps",
}
params = {
"alphaC": 3.0,
"K": 0.1,
"debug": 5,
"alphaJ": 1.0,
"dt": dt,
"maxtrials": 10,
"maxit": n_iters,
"itnormalisation": 50,
"tol": tol,
}
results = nlspace_solve(problem, params)
return results