def solveFwd(self, state, x):
""" Solve the possibly nonlinear forward problem:
Given :math:`m`, find :math:`u` such that
.. math:: \\delta_p F(u, m, p;\\hat{p}) = 0,\\quad \\forall \\hat{p}."""
self.n_calls["forward"] += 1
if self.solver is None:
self.solver = self._createLUSolver()
if self.is_fwd_linear:
u = dl.TrialFunction(self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
A_form = ufl.lhs(res_form)
b_form = ufl.rhs(res_form)
A, b = dl.assemble_system(A_form, b_form, bcs=self.bc)
self.solver.set_operator(A)
self.solver.solve(state, b)
else:
u = vector2Function(x[STATE], self.Vh[STATE])
m = vector2Function(x[PARAMETER], self.Vh[PARAMETER])
p = dl.TestFunction(self.Vh[ADJOINT])
res_form = self.varf_handler(u, m, p)
dl.solve(res_form == 0, u, self.bc)
state.zero()
state.axpy(1., u.vector())
def test_lhs_rhs_simple():
"""Test taking lhs/rhs of DOLFIN specific forms (constants
without cell). """
mesh = RectangleMesh(MPI.comm_world, [numpy.array([0.0, 0.0, 0.0]),
numpy.array([2.0, 1.0, 0.0])],
[3, 5], CellType.triangle)
V = FunctionSpace(mesh, "CG", 1)
f = 2.0
g = 3.0
v = TestFunction(V)
u = TrialFunction(V)
F = inner(g * grad(f * v), grad(u)) * dx + f * v * dx
a, L = system(F)
Fl = lhs(F)
Fr = rhs(F)
assert(Fr)
a0 = inner(grad(v), grad(u)) * dx
n = assemble(a).norm("frobenius") # noqa
nl = assemble(Fl).norm("frobenius") # noqa
n0 = 6.0 * assemble(a0).norm("frobenius") # noqa
assert round(n - n0, 7) == 0
assert round(n - nl, 7) == 0
def forward(x):
u = fn.TrialFunction(V)
w = fn.TestFunction(V)
sigma = lmbda * tr(sym(grad(u))) * Identity(2) + 2 * G * sym(
grad(u)) # Stress
R = simp(x) * inner(sigma, grad(w)) * dx - dot(b, w) * dx
a, L = ufl.lhs(R), ufl.rhs(R)
u = fa.Function(V)
fa.solve(a == L, u, bcs)
return u
def jvp_solve_eval(
fenics_function: Callable,
fenics_templates: Iterable[FenicsVariable],
primals: Tuple[np.array],
tangents: Tuple[np.array],
) -> Tuple[np.array]:
"""Computes the tangent linear model
"""
(
numpy_output_primal,
fenics_solution_primal,
residual_form,
fenics_primals,
bcs,
) = solve_eval(fenics_function, fenics_templates, *primals)
# Now tangent evaluation!
F = residual_form
u = fenics_solution_primal
V = u.function_space()
if len(bcs) != 0:
for bc in bcs:
bc.homogenize()
hbcs = bcs
fenics_tangents = convert_all_to_fenics(fenics_primals, *tangents)
fenics_output_tangents = []
for fp, ft in zip(fenics_primals, fenics_tangents):
dFdu = fenics.derivative(F, u)
dFdm = fenics.derivative(F, fp, ft)
u_tlm = fenics.Function(V)
tlm_F = ufl.action(dFdu, u_tlm) + dFdm
tlm_F = ufl.replace(tlm_F, {u_tlm: fenics.TrialFunction(V)})
fenics.solve(ufl.lhs(tlm_F) == ufl.rhs(tlm_F), u_tlm, bcs=hbcs)
fenics_output_tangents.append(u_tlm)
jax_output_tangents = (fenics_to_numpy(ft)
for ft in fenics_output_tangents)
jax_output_tangent = sum(jax_output_tangents)
return numpy_output_primal, jax_output_tangent
amplitude_list[i],
center=sources_positions[i])
for i in range(len(omega_p_list))
]
g = sum(pulses)
F = rho * inner(a_ * N_ddot(u, u0, a0, v0, dt, beta) \
+ b_ * N_dot(u, u0, v0, a0, dt, beta, gamma) + c_ * u, w) * dx \
+ inner(N_dot(S, U0, V0, A0, dt, beta, gamma).T * Lambda_e + S.T * Lambda_p, grad(w)) * dx \
- inner(g, w) * ds \
+ inner(compliance(a_ * N_ddot(S, U0, A0, V0, dt, beta) + b_ * N_dot(S, U0, V0, A0, dt, beta, gamma) + c_ * S, u, mu, lmbda), T) * dx \
- 0.5 * inner(grad(u) * Lambda_p + Lambda_p * grad(u).T + grad(N_dot(u, u0, v0, a0, dt, beta, gamma)) * Lambda_e \
+ Lambda_e * grad(N_dot(u, u0, v0, a0, dt, beta, gamma)).T, T) * dx \
a, L = lhs(F), rhs(F)
# Assemble rhs (once)
A = assemble(a)
# Create GMRES Krylov solver
solver = KrylovSolver(A, "gmres")
# Create solution function
S = Function(W)
experiment_count_file = open("experiment_counter", 'rb')
experiment_count = pickle.load(experiment_count_file)
experiment_count_file.close()
paraview_file_name = "experiment_{}".format(experiment_count)
k_absorb = Function(V)
k_absorb.interpolate(adiabatic_layer)
# Interpolate incident wave field onto V
ui = Function(V)
ui.interpolate(incident)
# Define variational problem
u = TrialFunction(V)
v = TestFunction(V)
F = inner(grad(u), grad(v)) * dx - k0**2 * inner(u, v) * dx - \
k_absorb * inner(u, v) * dx \
+ inner(dot(grad(ui), n), v) * ds(1)
a = lhs(F)
L = rhs(F)
''' Assemble matrix and vector and set up direct solver '''
A = dolfinx.fem.assemble_matrix(a)
A.assemble()
b = dolfinx.fem.assemble_vector(L)
b.ghostUpdate(addv=PETSc.InsertMode.ADD, mode=PETSc.ScatterMode.REVERSE)
solver = PETSc.KSP().create(mesh.mpi_comm())
opts = PETSc.Options()
opts["ksp_type"] = "preonly"
opts["pc_type"] = "lu"
opts["pc_factor_mat_solver_type"] = "mumps"
solver.setFromOptions()
solver.setOperators(A)
def forward(mu_expression,
lmbda_expression,
rho,
Lx=10,
Ly=10,
t_end=1,
omega_p=5,
amplitude=5000,
center=0,
target=False):
Lpml = Lx / 10
#c_p = cp(mu.vector(), lmbda.vector(), rho)
max_velocity = 200 #c_p.max()
stable_hx = stable_dx(max_velocity, omega_p)
nx = int(Lx / stable_hx) + 1
#nx = max(nx, 60)
ny = int(Ly * nx / Lx) + 1
mesh = mesh_generator(Lx, Ly, Lpml, nx, ny)
used_hx = Lx / nx
dt = stable_dt(used_hx, max_velocity)
cfl_ct = cfl_constant(max_velocity, dt, used_hx)
print(used_hx, stable_hx)
print(cfl_ct)
#time.sleep(10)
PE = FunctionSpace(mesh, "DG", 0)
mu = interpolate(mu_expression, PE)
lmbda = interpolate(lmbda_expression, PE)
m = 2
R = 10e-8
t = 0.0
gamma = 0.50
beta = 0.25
ff = MeshFunction("size_t", mesh, mesh.geometry().dim() - 1)
Dirichlet(Lx, Ly, Lpml).mark(ff, 1)
# Create function spaces
VE = VectorElement("CG", mesh.ufl_cell(), 1, dim=2)
TE = TensorElement("DG", mesh.ufl_cell(), 0, shape=(2, 2), symmetry=True)
W = FunctionSpace(mesh, MixedElement([VE, TE]))
F = FunctionSpace(mesh, "CG", 2)
V = W.sub(0).collapse()
M = W.sub(1).collapse()
alpha_0 = Alpha_0(m, stable_hx, R, Lpml)
alpha_1 = Alpha_1(alpha_0, Lx, Lpml, degree=2)
alpha_2 = Alpha_2(alpha_0, Ly, Lpml, degree=2)
beta_0 = Beta_0(m, max_velocity, R, Lpml)
beta_1 = Beta_1(beta_0, Lx, Lpml, degree=2)
beta_2 = Beta_2(beta_0, Ly, Lpml, degree=2)
alpha_1 = interpolate(alpha_1, F)
alpha_2 = interpolate(alpha_2, F)
beta_1 = interpolate(beta_1, F)
beta_2 = interpolate(beta_2, F)
a_ = alpha_1 * alpha_2
b_ = alpha_1 * beta_2 + alpha_2 * beta_1
c_ = beta_1 * beta_2
Lambda_e = as_tensor([[alpha_2, 0], [0, alpha_1]])
Lambda_p = as_tensor([[beta_2, 0], [0, beta_1]])
a_ = alpha_1 * alpha_2
b_ = alpha_1 * beta_2 + alpha_2 * beta_1
c_ = beta_1 * beta_2
Lambda_e = as_tensor([[alpha_2, 0], [0, alpha_1]])
Lambda_p = as_tensor([[beta_2, 0], [0, beta_1]])
# Set up boundary condition
bc = DirichletBC(W.sub(0), Constant(("0.0", "0.0")), ff, 1)
# Create measure for the source term
dx = Measure("dx", domain=mesh)
ds = Measure("ds", domain=mesh, subdomain_data=ff)
# Set up initial values
u0 = Function(V)
u0.set_allow_extrapolation(True)
v0 = Function(V)
a0 = Function(V)
U0 = Function(M)
V0 = Function(M)
A0 = Function(M)
# Test and trial functions
(u, S) = TrialFunctions(W)
(w, T) = TestFunctions(W)
g = ModifiedRickerPulse(0, omega_p, amplitude, center)
F = rho * inner(a_ * N_ddot(u, u0, a0, v0, dt, beta) \
+ b_ * N_dot(u, u0, v0, a0, dt, beta, gamma) + c_ * u, w) * dx \
+ inner(N_dot(S, U0, V0, A0, dt, beta, gamma).T * Lambda_e + S.T * Lambda_p, grad(w)) * dx \
- inner(g, w) * ds \
+ inner(compliance(a_ * N_ddot(S, U0, A0, V0, dt, beta) + b_ * N_dot(S, U0, V0, A0, dt, beta, gamma) + c_ * S, u, mu, lmbda), T) * dx \
- 0.5 * inner(grad(u) * Lambda_p + Lambda_p * grad(u).T + grad(N_dot(u, u0, v0, a0, dt, beta, gamma)) * Lambda_e \
+ Lambda_e * grad(N_dot(u, u0, v0, a0, dt, beta, gamma)).T, T) * dx \
a, L = lhs(F), rhs(F)
# Assemble rhs (once)
A = assemble(a)
# Create GMRES Krylov solver
solver = KrylovSolver(A, "gmres")
# Create solution function
S = Function(W)
if target:
xdmffile_u = XDMFFile("inversion_temporal_file/target/u.xdmf")
pvd = File("inversion_temporal_file/target/u.pvd")
xdmffile_u.write(u0, t)
timeseries_u = TimeSeries(
"inversion_temporal_file/target/u_timeseries")
else:
xdmffile_u = XDMFFile("inversion_temporal_file/obs/u.xdmf")
xdmffile_u.write(u0, t)
timeseries_u = TimeSeries("inversion_temporal_file/obs/u_timeseries")
rec_counter = 0
while t < t_end - 0.5 * dt:
t += float(dt)
if rec_counter % 10 == 0:
print(
'\n\rtime: {:.3f} (Progress: {:.2f}%)'.format(
t, 100 * t / t_end), )
g.t = t
# Assemble rhs and apply boundary condition
b = assemble(L)
bc.apply(A, b)
# Compute solution
solver.solve(S.vector(), b)
(u, U) = S.split(True)
# Update previous time step
update(u, u0, v0, a0, beta, gamma, dt)
update(U, U0, V0, A0, beta, gamma, dt)
xdmffile_u.write(u, t)
pvd << (u, t)
timeseries_u.store(u.vector(), t)
energy = inner(u, u) * dx
E = assemble(energy)
print("E = ", E)
print(u.vector().max())
