def test_save_and_read_function_timeseries(tempdir):
filename = os.path.join(tempdir, "function.h5")
mesh = UnitSquareMesh(MPI.comm_world, 10, 10)
Q = FunctionSpace(mesh, ("CG", 3))
F0 = Function(Q)
F1 = Function(Q)
E = Expression("t*x[0]", t=0.0, degree=1)
F0.interpolate(E)
# Save to HDF5 File
hdf5_file = HDF5File(mesh.mpi_comm(), filename, "w")
for t in range(10):
E.t = t
F0.interpolate(E)
hdf5_file.write(F0, "/function", t)
hdf5_file.close()
# Read back from file
hdf5_file = HDF5File(mesh.mpi_comm(), filename, "r")
for t in range(10):
E.t = t
F1.interpolate(E)
vec_name = "/function/vector_%d" % t
F0 = hdf5_file.read_function(Q, vec_name)
# timestamp = hdf5_file.attributes(vec_name)["timestamp"]
# assert timestamp == t
F0.vector().axpy(-1.0, F1.vector())
assert F0.vector().norm(cpp.la.Norm.l2) < 1.0e-12
hdf5_file.close()
def _check_space_order(problem, method):
mesh_generator, solution, weak_F = problem()
# Translate data into FEniCS expressions.
fenics_sol = Expression(smp.prining.ccode(solution['value']),
degree=solution['degree'],
t=0.0
)
# Create initial solution.
theta0 = Expression(fenics_sol.cppcode,
degree=solution['degree'],
t=0.0,
cell=triangle
)
# Estimate the error component in space.
# Leave out too rough discretizations to avoid showing spurious errors.
N = [2 ** k for k in range(2, 8)]
dt = 1.0e-8
Err = []
H = []
for n in N:
mesh = mesh_generator(n)
H.append(MPI.max(mesh.hmax()))
V = FunctionSpace(mesh, 'CG', 3)
# Create boundary conditions.
fenics_sol.t = dt
#bcs = DirichletBC(V, fenics_sol, 'on_boundary')
# Create initial state.
theta_approx = method(V,
weak_F,
theta0,
0.0, dt,
bcs=[solution],
tol=1.0e-12,
verbose=True
)
# Compute the error.
fenics_sol.t = dt
Err.append(errornorm(fenics_sol, theta_approx)
/ norm(fenics_sol, mesh=mesh)
)
print('n: %d error: %e' % (n, Err[-1]))
from matplotlib import pyplot as pp
# Compare with order 1, 2, 3 curves.
for o in [2, 3, 4]:
pp.loglog([H[0], H[-1]],
[Err[0], Err[0] * (H[-1] / H[0]) ** o],
color='0.5'
)
# Finally, the actual data.
pp.loglog(H, Err, '-o')
pp.xlabel('h_max')
pp.ylabel('||u-u_h|| / ||u||')
pp.show()
return
def _check_spatial_order(problem, method):
mesh_generator, solution, weak_F = problem()
# Translate data into FEniCS expressions.
fenics_sol = Expression(sympy.printing.ccode(solution['value']),
degree=solution['degree'],
t=0.0)
# Create initial solution.
theta0 = Expression(fenics_sol.cppcode,
degree=solution['degree'],
t=0.0,
cell=triangle)
# Estimate the error component in space.
# Leave out too rough discretizations to avoid showing spurious errors.
N = [2**k for k in range(2, 8)]
dt = 1.0e-8
Err = []
H = []
for n in N:
mesh = mesh_generator(n)
H.append(MPI.max(mesh.hmax()))
V = FunctionSpace(mesh, 'CG', 5)
# Create boundary conditions.
fenics_sol.t = dt
# bcs = DirichletBC(V, fenics_sol, 'on_boundary')
# Create initial state.
theta_approx = method(V,
weak_F,
theta0,
0.0,
dt,
bcs=[solution],
tol=1.0e-12,
verbose=True)
# Compute the error.
fenics_sol.t = dt
Err.append(
errornorm(fenics_sol, theta_approx) / norm(fenics_sol, mesh=mesh))
print('n: %d error: %e' % (n, Err[-1]))
# Plot order curves for comparison.
for order in [2, 3, 4]:
plt.loglog([H[0], H[-1]], [Err[0], Err[0] * (H[-1] / H[0])**order],
color='0.5')
# Finally, the actual data.
plt.loglog(H, Err, '-o')
plt.xlabel('h_max')
plt.ylabel('||u-u_h|| / ||u||')
plt.show()
return
def test_interpolation_rank0(V):
@function.expression.numba_eval
def expr_eval(values, x, t):
values[:, 0] = 1.0 * t
f = Expression(expr_eval, shape=())
f.t = 1.0
w = interpolate(f, V)
with w.vector().localForm() as x:
assert (x[:] == 1.0).all()
f.t = 2.0
w = interpolate(f, V)
with w.vector().localForm() as x:
assert (x[:] == 2.0).all()
def _compute_time_errors(problem, method, mesh_sizes, Dt, plot_error=False):
mesh_generator, solution, ProblemClass, cell_type = problem()
# Translate data into FEniCS expressions.
fenics_sol = Expression(smp.printing.ccode(solution['value']),
degree=solution['degree'],
t=0.0,
cell=cell_type
)
# Compute the problem
errors = {'theta': numpy.empty((len(mesh_sizes), len(Dt)))}
# Create initial state.
# Deepcopy the expression into theta0. Specify the cell to allow for
# more involved operations with it (e.g., grad()).
theta0 = Expression(fenics_sol.cppcode,
degree=solution['degree'],
t=0.0,
cell=cell_type
)
for k, mesh_size in enumerate(mesh_sizes):
mesh = mesh_generator(mesh_size)
V = FunctionSpace(mesh, 'CG', 1)
theta_approx = Function(V)
theta0p = project(theta0, V)
stepper = method(ProblemClass(V))
if plot_error:
error = Function(V)
for j, dt in enumerate(Dt):
# TODO We are facing a little bit of a problem here, being the
# fact that the time stepper only accept elements from V as u0.
# In principle, though, this isn't necessary or required. We
# could allow for arbitrary expressions here, but then the API
# would need changing for problem.lhs(t, u).
# Think about this.
stepper.step(theta_approx, theta0p,
0.0, dt,
tol=1.0e-12,
verbose=False
)
fenics_sol.t = dt
#
# NOTE
# When using errornorm(), it is quite likely to see a good part
# of the error being due to the spatial discretization. Some
# analyses "get rid" of this effect by (sometimes implicitly)
# projecting the exact solution onto the discrete function
# space.
errors['theta'][k][j] = errornorm(fenics_sol, theta_approx)
if plot_error:
error.assign(project(fenics_sol - theta_approx, V))
plot(error, title='error (dt=%e)' % dt)
interactive()
return errors, stepper.name, stepper.order
def _compute_time_errors(problem, method, mesh_sizes, Dt, plot_error=False):
mesh_generator, solution, ProblemClass, _ = problem()
# Translate data into FEniCS expressions.
fenics_sol = Expression(sympy.printing.ccode(solution),
degree=MAX_DEGREE,
t=0.0)
# Compute the problem
errors = numpy.empty((len(mesh_sizes), len(Dt)))
# Create initial state.
# Deepcopy the expression into theta0. Specify the cell to allow for
# more involved operations with it (e.g., grad()).
theta0 = Expression(fenics_sol.cppcode, degree=MAX_DEGREE, t=0.0)
for k, mesh_size in enumerate(mesh_sizes):
mesh = mesh_generator(mesh_size)
# Choose the function space such that the exact solution can be
# represented as well as possible.
V = FunctionSpace(mesh, 'CG', 4)
theta_approx = Function(V)
theta0p = project(theta0, V)
stepper = method(ProblemClass(V))
if plot_error:
error = Function(V)
for j, dt in enumerate(Dt):
# TODO We are facing a little bit of a problem here, being the fact
# that the time stepper only accept elements from V as u0. In
# principle, though, this isn't necessary or required. We could
# allow for arbitrary expressions here, but then the API would need
# changing for problem.lhs(t, u). Think about this.
theta_approx.assign(stepper.step(theta0p, 0.0, dt))
fenics_sol.t = dt
# NOTE
# When using errornorm(), it is quite likely to see a good part of
# the error being due to the spatial discretization. Some analyses
# "get rid" of this effect by (sometimes implicitly) projecting the
# exact solution onto the discrete function space.
errors[k][j] = errornorm(fenics_sol, theta_approx)
if plot_error:
error.assign(project(fenics_sol - theta_approx, V))
plot(error, title='error (dt={:e})'.format(dt))
plt.show()
return errors
def test_unsteady_stokes():
nx, ny = 15, 15
k = 1
nu = Constant(1.0e-0)
dt = Constant(2.5e-2)
num_steps = 20
theta0 = 1.0 # Initial theta value
theta1 = 0.5 # Theta after 1 step
theta = Constant(theta0)
mesh = UnitSquareMesh(nx, ny)
# The 'unsteady version' of the benchmark in the 2012 paper by Labeur&Wells
u_exact = Expression(
(
"sin(t) * x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
-6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-sin(t)* x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
- 6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
t=0,
degree=7,
domain=mesh,
)
p_exact = Expression("sin(t) * x[0]*(1.0 - x[0])",
t=0,
degree=7,
domain=mesh)
du_exact = Expression(
(
"cos(t) * x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
- 6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-cos(t)* x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
-6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
t=0,
degree=7,
domain=mesh,
)
ux_exact = Expression(
(
"x[0]*x[0]*(1.0 - x[0])*(1.0 - x[0])*(2.0*x[1] \
- 6.0*x[1]*x[1] + 4.0*x[1]*x[1]*x[1])",
"-x[1]*x[1]*(1.0 - x[1])*(1.0 - x[1])*(2.0*x[0] \
- 6.0*x[0]*x[0] + 4.0*x[0]*x[0]*x[0])",
),
degree=7,
domain=mesh,
)
px_exact = Expression("x[0]*(1.0 - x[0])", degree=7, domain=mesh)
sin_ext = Expression("sin(t)", t=0, degree=7, domain=mesh)
f = du_exact + sin_ext * div(px_exact * Identity(2) -
2 * sym(grad(ux_exact)))
Vhigh = VectorFunctionSpace(mesh, "DG", 7)
Phigh = FunctionSpace(mesh, "DG", 7)
# New syntax:
V = VectorElement("DG", mesh.ufl_cell(), k)
Q = FiniteElement("DG", mesh.ufl_cell(), k - 1)
Vbar = VectorElement("DGT", mesh.ufl_cell(), k)
Qbar = FiniteElement("DGT", mesh.ufl_cell(), k)
mixedL = FunctionSpace(mesh, MixedElement([V, Q]))
mixedG = FunctionSpace(mesh, MixedElement([Vbar, Qbar]))
V2 = FunctionSpace(mesh, V)
Uh = Function(mixedL)
Uhbar = Function(mixedG)
U0 = Function(mixedL)
Uhbar0 = Function(mixedG)
u0, p0 = split(U0)
ubar0, pbar0 = split(Uhbar0)
ustar = Function(V2)
# Then the boundary conditions
bc0 = DirichletBC(mixedG.sub(0), Constant((0, 0)), Gamma)
bc1 = DirichletBC(mixedG.sub(1), Constant(0), Corner, "pointwise")
bcs = [bc0, bc1]
alpha = Constant(6 * k * k)
forms_stokes = FormsStokes(mesh, mixedL, mixedG,
alpha).forms_unsteady(ustar, dt, nu, f)
ssc = StokesStaticCondensation(
mesh,
forms_stokes["A_S"],
forms_stokes["G_S"],
forms_stokes["G_ST"],
forms_stokes["B_S"],
forms_stokes["Q_S"],
forms_stokes["S_S"],
)
t = 0.0
step = 0
for step in range(num_steps):
step += 1
t += float(dt)
if comm.Get_rank() == 0:
print("Step " + str(step) + " Time " + str(t))
# Set time level in exact solution
u_exact.t = t
p_exact.t = t
du_exact.t = t - (1 - float(theta)) * float(dt)
sin_ext.t = t - (1 - float(theta)) * float(dt)
ssc.assemble_global_lhs()
ssc.assemble_global_rhs()
for bc in bcs:
ssc.apply_boundary(bc)
ssc.solve_problem(Uhbar, Uh, "none", "default")
assign(U0, Uh)
assign(ustar, U0.sub(0))
assign(Uhbar0, Uhbar)
if step == 1:
theta.assign(theta1)
udiv_e = sqrt(assemble(div(Uh.sub(0)) * div(Uh.sub(0)) * dx))
u_ex_h = interpolate(u_exact, Vhigh)
p_ex_h = interpolate(p_exact, Phigh)
u_error = sqrt(assemble(dot(Uh.sub(0) - u_ex_h, Uh.sub(0) - u_ex_h) * dx))
p_error = sqrt(assemble(dot(Uh.sub(1) - p_ex_h, Uh.sub(1) - p_ex_h) * dx))
assert udiv_e < 1e-12
assert u_error < 1.5e-4
assert p_error < 1e-2
v = TestFunction(V)
f = Constant(beta - 2 - 2 * alpha)
F = u * v * dx + dt * dot(grad(u), grad(v)) * dx - (u_n + dt * f) * v * dx
a, L = lhs(F), rhs(F)
# Time-stepping
u = Function(V)
t = 0
vtkfile = File('heat/solution_3D.pvd')
for n in range(num_steps):
# Update current time
t += dt
u_D.t = t
# Compute solution
solve(a == L, u, bc)
# Plot solution
# plot(u)
# axi.triplot(u)
vtkfile << u
# Compute error at vertices
# u_e = interpolate(u_D, V)
# error = np.abs(u_e.vector().array() - u.vector().array()).max()
# print('t = %.2f: error = %.3g' % (t, error))
# Update previous solution
pde_projection = PDEStaticCondensation(mesh, p,
forms_pde['N_a'], forms_pde['G_a'], forms_pde['L_a'],
forms_pde['H_a'],
forms_pde['B_a'],
forms_pde['Q_a'], forms_pde['R_a'], forms_pde['S_a'],
[], 1)
ap = advect_rk3(p, V, uh, 'open')
# Initialize the initial condition at mesh by an l2 projection
lstsq_rho = l2projection(p, Q_Rho, 1)
lstsq_rho.project(phih0)
outfile << phih0
for step in range(num_steps):
u_expr.t = step * float(dt)
u_expre_neg.t = step * float(dt)
uh.assign(u_expr)
# Compute area at old configuration
old_area = assemble(phih0*dx)
# Pre-assemble rhs
pde_projection.assemble_state_rhs()
# Advect the particles
ap.do_step(float(dt))
# Move mesh
umesh.assign(u_expre_neg)
def _test_time_stepping_1_sparse(callback_type, integrator_type):
# Create mesh and define function space
mesh = IntervalMesh(132, 0, 2*pi)
V = FunctionSpace(mesh, "Lagrange", 1)
# Define Dirichlet boundary (x = 0 or x = 2*pi)
def boundary(x):
return x[0] < 0 + DOLFIN_EPS or x[0] > 2*pi - 10*DOLFIN_EPS
# Define time step
dt = 0.01
T = 1.
# Define exact solution
exact_solution_expression = Expression("sin(x[0]+t)", t=0, element=V.ufl_element())
# ... and interpolate it at the final time
exact_solution_expression.t = T
exact_solution = project(exact_solution_expression, V)
# Define exact solution dot
exact_solution_dot_expression = Expression("cos(x[0]+t)", t=0, element=V.ufl_element())
# ... and interpolate it at the final time
exact_solution_dot_expression.t = T
exact_solution_dot = project(exact_solution_dot_expression, V)
# Define variational problem
du = TrialFunction(V)
du_dot = TrialFunction(V)
v = TestFunction(V)
u = Function(V)
u_dot = Function(V)
g = Expression("sin(x[0]+t) + cos(x[0]+t)", t=0., element=V.ufl_element())
r_u = inner(grad(u), grad(v))*dx
j_u = derivative(r_u, u, du)
r_u_dot = inner(u_dot, v)*dx
j_u_dot = derivative(r_u_dot, u_dot, du_dot)
r = r_u_dot + r_u - g*v*dx
x = inner(du, v)*dx
def bc(t):
exact_solution_expression.t = t
return [DirichletBC(V, exact_solution_expression, boundary)]
# Assemble inner product matrix
X = assemble(x)
# Define callback function depending on callback type
assert callback_type in ("form callbacks", "tensor callbacks")
if callback_type == "form callbacks":
def callback(arg):
return arg
elif callback_type == "tensor callbacks":
def callback(arg):
return assemble(arg)
# Define problem wrapper
class SparseProblemWrapper(TimeDependentProblem1Wrapper):
# Residual and jacobian functions
def residual_eval(self, t, solution, solution_dot):
g.t = t
return callback(r)
def jacobian_eval(self, t, solution, solution_dot, solution_dot_coefficient):
return callback(Constant(solution_dot_coefficient)*j_u_dot + j_u)
# Define boundary condition
def bc_eval(self, t):
return bc(t)
# Define initial condition
def ic_eval(self):
exact_solution_expression.t = 0.
return project(exact_solution_expression, V)
# Define custom monitor to plot the solution
def monitor(self, t, solution, solution_dot):
if matplotlib.get_backend() != "agg":
plt.subplot(1, 2, 1).clear()
plot(solution, title="u at t = " + str(t))
plt.subplot(1, 2, 2).clear()
plot(solution_dot, title="u_dot at t = " + str(t))
plt.show(block=False)
plt.pause(DOLFIN_EPS)
else:
print("||u|| at t = " + str(t) + ": " + str(solution.vector().norm("l2")))
print("||u_dot|| at t = " + str(t) + ": " + str(solution_dot.vector().norm("l2")))
# Solve the time dependent problem
sparse_problem_wrapper = SparseProblemWrapper()
(sparse_solution, sparse_solution_dot) = (u, u_dot)
sparse_solver = SparseTimeStepping(sparse_problem_wrapper, sparse_solution, sparse_solution_dot)
sparse_solver.set_parameters({
"initial_time": 0.0,
"time_step_size": dt,
"final_time": T,
"exact_final_time": "stepover",
"integrator_type": integrator_type,
"problem_type": "linear",
"linear_solver": "mumps",
"monitor": sparse_problem_wrapper.monitor,
"report": True
})
all_sparse_solutions_time, all_sparse_solutions, all_sparse_solutions_dot = sparse_solver.solve()
assert len(all_sparse_solutions_time) == int(T/dt + 1)
assert len(all_sparse_solutions) == int(T/dt + 1)
assert len(all_sparse_solutions_dot) == int(T/dt + 1)
# Compute the error
sparse_error = Function(V)
sparse_error.vector().add_local(+ sparse_solution.vector().get_local())
sparse_error.vector().add_local(- exact_solution.vector().get_local())
sparse_error.vector().apply("")
sparse_error_norm = sparse_error.vector().inner(X*sparse_error.vector())
sparse_error_dot = Function(V)
sparse_error_dot.vector().add_local(+ sparse_solution_dot.vector().get_local())
sparse_error_dot.vector().add_local(- exact_solution_dot.vector().get_local())
sparse_error_dot.vector().apply("")
sparse_error_dot_norm = sparse_error_dot.vector().inner(X*sparse_error_dot.vector())
print("SparseTimeStepping error (" + callback_type + ", " + integrator_type + "):", sparse_error_norm, sparse_error_dot_norm)
assert isclose(sparse_error_norm, 0., atol=1.e-4)
assert isclose(sparse_error_dot_norm, 0., atol=1.e-4)
return ((sparse_error_norm, sparse_error_dot_norm), V, dt, T, u, u_dot, g, r, j_u, j_u_dot, X, exact_solution_expression, exact_solution, exact_solution_dot)
def compute_time_errors(problem, method, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Translate data into FEniCS expressions.
u = solution['u']
sol_u = Expression((ccode(u['value'][0]), ccode(u['value'][1])),
degree=_truncate_degree(u['degree']),
t=0.0)
p = solution['p']
sol_p = Expression(ccode(p['value']),
degree=_truncate_degree(p['degree']),
t=0.0)
# Deep-copy expression to be able to provide f0, f1 for the Dirichlet-
# boundary conditions later on.
fenics_rhs0 = Expression((ccode(f['value'][0]), ccode(f['value'][1])),
degree=_truncate_degree(f['degree']),
t=0.0,
mu=mu,
rho=rho)
fenics_rhs1 = Expression(fenics_rhs0.cppcode,
element=fenics_rhs0.ufl_element(),
**fenics_rhs0.user_parameters)
# Create initial states.
p = Expression(sol_p.cppcode,
degree=_truncate_degree(solution['p']['degree']),
t=0.0,
cell=cell_type)
# Compute the problem
errors = {
'u': numpy.empty((len(mesh_sizes), len(Dt))),
'p': numpy.empty((len(mesh_sizes), len(Dt)))
}
for k, mesh_size in enumerate(mesh_sizes):
print()
print()
print('Computing for mesh size %r...' % mesh_size)
mesh = mesh_generator(mesh_size)
mesh_area = assemble(1.0 * dx(mesh))
W = VectorFunctionSpace(mesh, 'CG', 2)
P = FunctionSpace(mesh, 'CG', 1)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u_1 = Expression(sol_u.cppcode,
degree=_truncate_degree(solution['u']['degree']),
t=-dt,
cell=cell_type)
u_1 = project(u_1, W)
u0 = Expression(
sol_u.cppcode,
degree=_truncate_degree(solution['u']['degree']),
t=0.0,
# t=0.5*dt,
cell=cell_type)
u0 = project(u0, W)
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, 'on_boundary')]
sol_p.t = dt
p0 = project(p, P)
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
u1, p1 = method.step(Constant(dt), {
-1: u_1,
0: u0
},
p0,
u_bcs=u_bcs,
p_bcs=p_bcs,
rho=Constant(rho),
mu=Constant(mu),
f={
0: fenics_rhs0,
1: fenics_rhs1
},
verbose=False,
tol=1.0e-10)
# plot(sol_u, mesh=mesh, title='u_sol')
# plot(sol_p, mesh=mesh, title='p_sol')
# plot(u1, title='u')
# plot(p1, title='p')
# from dolfin import div
# plot(div(u1), title='div(u)')
# plot(p1 - sol_p, title='p_h - p')
# interactive()
sol_u.t = dt
sol_p.t = dt
errors['u'][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = (+assemble(sol_p * dx(mesh)) - assemble(p1 * dx(mesh)))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors['p'][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title='p1', mesh=mesh)
plot(sol_p, title='sol_p', mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
# plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title='p1 - sol_p', mesh=mesh)
# r = Expression('x[0]', degree=1, cell=triangle)
# divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title='div(u1)')
interactive()
return errors
def compute_time_errors(problem, MethodClass, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Translate data into FEniCS expressions.
sol_u = Expression((smp.printing.ccode(solution['u']['value'][0]),
smp.printing.ccode(solution['u']['value'][1])
),
degree=_truncate_degree(solution['u']['degree']),
t=0.0,
cell=cell_type
)
sol_p = Expression(smp.printing.ccode(solution['p']['value']),
degree=_truncate_degree(solution['p']['degree']),
t=0.0,
cell=cell_type
)
fenics_rhs0 = Expression((smp.printing.ccode(f['value'][0]),
smp.printing.ccode(f['value'][1])
),
degree=_truncate_degree(f['degree']),
t=0.0,
mu=mu, rho=rho,
cell=cell_type
)
# Deep-copy expression to be able to provide f0, f1 for the Dirichlet-
# boundary conditions later on.
fenics_rhs1 = Expression(fenics_rhs0.cppcode,
degree=_truncate_degree(f['degree']),
t=0.0,
mu=mu, rho=rho,
cell=cell_type
)
# Create initial states.
p0 = Expression(
sol_p.cppcode,
degree=_truncate_degree(solution['p']['degree']),
t=0.0,
cell=cell_type
)
# Compute the problem
errors = {'u': numpy.empty((len(mesh_sizes), len(Dt))),
'p': numpy.empty((len(mesh_sizes), len(Dt)))
}
for k, mesh_size in enumerate(mesh_sizes):
info('')
info('')
with Message('Computing for mesh size %r...' % mesh_size):
mesh = mesh_generator(mesh_size)
mesh_area = assemble(1.0 * dx(mesh))
W = VectorFunctionSpace(mesh, 'CG', 2)
P = FunctionSpace(mesh, 'CG', 1)
method = MethodClass(W, P,
rho, mu,
theta=1.0,
#theta=0.5,
stabilization=None
#stabilization='SUPG'
)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u = [Expression(
sol_u.cppcode,
degree=_truncate_degree(solution['u']['degree']),
t=0.0,
cell=cell_type
),
# Expression(
#sol_u.cppcode,
#degree=_truncate_degree(solution['u']['degree']),
#t=0.5*dt,
#cell=cell_type
#)
]
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, 'on_boundary')]
sol_p.t = dt
#p_bcs = [DirichletBC(P, sol_p, 'on_boundary')]
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
method.step(dt,
u1, p1,
u, p0,
u_bcs=u_bcs, p_bcs=p_bcs,
f0=fenics_rhs0, f1=fenics_rhs1,
verbose=False,
tol=1.0e-10
)
sol_u.t = dt
sol_p.t = dt
errors['u'][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = assemble(sol_p * dx(mesh)) \
- assemble(p1 * dx(mesh))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors['p'][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title='p1', mesh=mesh)
plot(sol_p, title='sol_p', mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
#plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title='p1 - sol_p', mesh=mesh)
#r = Expression('x[0]', degree=1, cell=triangle)
#divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title='div(u1)')
interactive()
return errors
def compute_time_errors(problem, MethodClass, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Compute the problem
errors = {
"u": numpy.empty((len(mesh_sizes), len(Dt))),
"p": numpy.empty((len(mesh_sizes), len(Dt))),
}
for k, mesh_size in enumerate(mesh_sizes):
info("")
info("")
with Message("Computing for mesh size {}...".format(mesh_size)):
mesh = mesh_generator(mesh_size)
# Define all expression with `domain`, see
# <https://bitbucket.org/fenics-project/ufl/issues/96>.
#
# Translate data into FEniCS expressions.
sol_u = Expression(
(ccode(solution["u"]["value"][0]), ccode(solution["u"]["value"][1])),
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
domain=mesh,
)
sol_p = Expression(
ccode(solution["p"]["value"]),
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
fenics_rhs0 = Expression(
(ccode(f["value"][0]), ccode(f["value"][1])),
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Deep-copy expression to be able to provide f0, f1 for the
# Dirichlet boundary conditions later on.
fenics_rhs1 = Expression(
fenics_rhs0.cppcode,
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Create initial states.
W = VectorFunctionSpace(mesh, "CG", 2)
P = FunctionSpace(mesh, "CG", 1)
p0 = Expression(
sol_p.cppcode,
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
mesh_area = assemble(1.0 * dx(mesh))
method = MethodClass(
time_step_method="backward euler",
# time_step_method='crank-nicolson',
# stabilization=None
# stabilization='SUPG'
)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u = {
0: Expression(
sol_u.cppcode,
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
cell=cell_type,
)
}
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, "on_boundary")]
sol_p.t = dt
# p_bcs = [DirichletBC(P, sol_p, 'on_boundary')]
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
u1, p1 = method.step(
Constant(dt),
u,
p0,
W,
P,
u_bcs,
p_bcs,
Constant(rho),
Constant(mu),
f={0: fenics_rhs0, 1: fenics_rhs1},
verbose=False,
tol=1.0e-10,
)
sol_u.t = dt
sol_p.t = dt
errors["u"][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = +assemble(sol_p * dx(mesh)) - assemble(p1 * dx(mesh))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors["p"][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title="p1", mesh=mesh)
plot(sol_p, title="sol_p", mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
# plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title="p1 - sol_p", mesh=mesh)
# r = SpatialCoordinate(mesh)[0]
# divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title="div(u1)")
return errors
p.increment(Udiv.cpp_object(), ustar.cpp_object(),
np.array([1, 2], dtype=np.uintp), theta_p, step)
if step == 2:
theta_L.assign(theta_next)
# Probably can be combined into one file?
xdmf_u.write(Uh.sub(0), t)
xdmf_p.write(Uh.sub(1), t)
del t1
timer.stop()
# Compute errors
u_exact.t = t
p_exact.t = t
u_error = sqrt(assemble(dot(Uh.sub(0) - u_exact, Uh.sub(0) - u_exact) * dx))
p_error = sqrt(assemble(dot(Uh.sub(1) - p_exact, Uh.sub(1) - p_exact) * dx))
udiv = sqrt(assemble(div(Uh.sub(0)) * div(Uh.sub(0)) * dx))
momentum = assemble((dot(Uh.sub(0), ex) + dot(Uh.sub(0), ey)) * dx)
if comm.Get_rank() == 0:
print("Velocity error " + str(u_error))
print("Pressure error " + str(p_error))
print("Momentum " + str(momentum))
print("Divergence " + str(udiv))
print("Elapsed time " + str(timer.elapsed()[0]))
def compute_time_errors(problem, MethodClass, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Compute the problem
errors = {
"u": numpy.empty((len(mesh_sizes), len(Dt))),
"p": numpy.empty((len(mesh_sizes), len(Dt))),
}
for k, mesh_size in enumerate(mesh_sizes):
info("")
info("")
with Message("Computing for mesh size {}...".format(mesh_size)):
mesh = mesh_generator(mesh_size)
# Define all expression with `domain`, see
# <https://bitbucket.org/fenics-project/ufl/issues/96>.
#
# Translate data into FEniCS expressions.
sol_u = Expression(
(ccode(solution["u"]["value"][0]),
ccode(solution["u"]["value"][1])),
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
domain=mesh,
)
sol_p = Expression(
ccode(solution["p"]["value"]),
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
fenics_rhs0 = Expression(
(ccode(f["value"][0]), ccode(f["value"][1])),
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Deep-copy expression to be able to provide f0, f1 for the
# Dirichlet boundary conditions later on.
fenics_rhs1 = Expression(
fenics_rhs0.cppcode,
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Create initial states.
W = VectorFunctionSpace(mesh, "CG", 2)
P = FunctionSpace(mesh, "CG", 1)
p0 = Expression(
sol_p.cppcode,
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
mesh_area = assemble(1.0 * dx(mesh))
method = MethodClass(
time_step_method="backward euler",
# time_step_method='crank-nicolson',
# stabilization=None
# stabilization='SUPG'
)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u = {
0:
Expression(
sol_u.cppcode,
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
cell=cell_type,
)
}
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, "on_boundary")]
sol_p.t = dt
# p_bcs = [DirichletBC(P, sol_p, 'on_boundary')]
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
u1, p1 = method.step(
Constant(dt),
u,
p0,
W,
P,
u_bcs,
p_bcs,
Constant(rho),
Constant(mu),
f={
0: fenics_rhs0,
1: fenics_rhs1
},
verbose=False,
tol=1.0e-10,
)
sol_u.t = dt
sol_p.t = dt
errors["u"][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = +assemble(sol_p * dx(mesh)) - assemble(p1 * dx(mesh))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors["p"][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title="p1", mesh=mesh)
plot(sol_p, title="sol_p", mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
# plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title="p1 - sol_p", mesh=mesh)
# r = SpatialCoordinate(mesh)[0]
# divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title="div(u1)")
return errors
from dolfin import Expression
ccode = '(t-x[0])*(((t-x[0]) < 0) ? 0 : 1)*(((t-x[0]) > tmax) ? 0 : 1)'
foo = Expression(ccode, tmax=20, t=0.0)
for t in (0.1, 0.2, 0.3):
foo.t = t
print[foo(x) for x in (-1, -2, -3)]
