def J(f):
u_1 = Function(V)
u_1.vector()[:] = 1
a = u_1 * u * v * dx + dt * f * inner(grad(u), grad(v)) * dx
L = u_1 * v * dx
# Time loop
t = dt
while t <= T:
solve(a == L, u_, bc)
u_1.assign(u_)
t += dt
return assemble(u_1**2 * dx)
def test_solver_ident_zeros():
"""
Test using ident zeros to restrict half of the domain
"""
from fenics_adjoint import (UnitSquareMesh, Function, assemble, solve,
project, Expression, DirichletBC)
mesh = UnitSquareMesh(10, 10)
cf = MeshFunction("size_t", mesh, mesh.topology().dim(), 0)
top_half().mark(cf, 1)
ff = MeshFunction("size_t", mesh, mesh.topology().dim() - 1, 0)
top_boundary().mark(ff, 1)
dx = Measure("dx", domain=mesh, subdomain_data=cf)
V = FunctionSpace(mesh, "CG", 1)
u, v = TrialFunction(V), TestFunction(V)
a = inner(grad(u), grad(v)) * dx(1)
w = Function(V)
with stop_annotating():
w.assign(project(Expression("x[0]", degree=1), V))
rhs = w**3 * v * dx(1)
A = assemble(a, keep_diagonal=True)
A.ident_zeros()
b = assemble(rhs)
bc = DirichletBC(V, Constant(1), ff, 1)
bc.apply(A, b)
uh = Function(V)
solve(A, uh.vector(), b, "umfpack")
J = assemble(inner(uh, uh) * dx(1))
Jhat = ReducedFunctional(J, Control(w))
with stop_annotating():
w1 = project(Expression("x[0]*x[1]", degree=2), V)
results = taylor_to_dict(Jhat, w, w1)
assert (min(results["R0"]["Rate"]) > 0.95)
assert (min(results["R1"]["Rate"]) > 1.95)
assert (min(results["R2"]["Rate"]) > 2.95)