def compute_errors(u_e, u):
"""Compute various measures of the error u - u_e, where
u is a finite element Function and u_e is an Expression.
Adapted from https://fenicsproject.org/pub/tutorial/html/._ftut1020.html
"""
print('u_e', u_e.ufl_element().degree())
# Get function space
V = u.function_space()
# Explicit computation of L2 norm
error = (u - u_e)**2 * dl.dx
E1 = np.sqrt(abs(dla.assemble(error)))
# Explicit interpolation of u_e onto the same space as u
u_e_ = dla.interpolate(u_e, V)
error = (u - u_e_)**2 * dl.dx
E2 = np.sqrt(abs(dla.assemble(error)))
# Explicit interpolation of u_e to higher-order elements.
# u will also be interpolated to the space Ve before integration
Ve = dl.FunctionSpace(V.mesh(), 'P', 5)
u_e_ = dla.interpolate(u_e, Ve)
error = (u - u_e)**2 * dl.dx
E3 = np.sqrt(abs(dla.assemble(error)))
# Infinity norm based on nodal values
u_e_ = dla.interpolate(u_e, V)
E4 = abs(u_e_.vector().get_local() - u.vector().get_local()).max()
# L2 norm
E5 = dl.errornorm(u_e, u, norm_type='L2', degree_rise=3)
# H1 seminorm
E6 = dl.errornorm(u_e, u, norm_type='H10', degree_rise=3)
# Collect error measures in a dictionary with self-explanatory keys
errors = {
'u - u_e': E1,
'u - interpolate(u_e, V)': E2,
'interpolate(u, Ve) - interpolate(u_e, Ve)': E3,
'infinity norm (of dofs)': E4,
'L2 norm': E5,
'H10 seminorm': E6
}
return errors
def run_model(function_space,
kappa,
forcing,
init_condition,
dt,
final_time,
boundary_conditions=None,
second_order_timestepping=False,
exact_sol=None,
velocity=None,
point_sources=None,
intermediate_times=None):
"""
Use implicit euler to solve transient advection diffusion equation
du/dt = grad (k* grad u) - vel*grad u + f
WARNING: when point sources solution changes significantly when mesh is
varied
"""
mesh = function_space.mesh()
time_independent_boundaries = False
if boundary_conditions is None:
bndry_obj = dl.CompiledSubDomain("on_boundary")
boundary_conditions = [['dirichlet', bndry_obj, dla.Constant(0)]]
time_independent_boundaries = True
num_bndrys = len(boundary_conditions)
boundaries = mark_boundaries(mesh, boundary_conditions)
dirichlet_bcs = collect_dirichlet_boundaries(function_space,
boundary_conditions,
boundaries)
# To express integrals over the boundary parts using ds(i), we must first
# redefine the measure ds in terms of our boundary markers:
ds = dl.Measure('ds', domain=mesh, subdomain_data=boundaries)
dx = dl.Measure('dx', domain=mesh)
# Variational problem at each time
u = dl.TrialFunction(function_space)
v = dl.TestFunction(function_space)
# Previous solution
if hasattr(init_condition, 't'):
assert init_condition.t == 0
u_1 = dla.interpolate(init_condition, function_space)
if not second_order_timestepping:
theta = 1
else:
theta = 0.5
if hasattr(forcing, 't'):
forcing_1 = copy_expression(forcing)
else:
forcing_1 = forcing
def steady_state_form(u, v, f):
F = kappa * dl.inner(dl.grad(u), dl.grad(v)) * dx
F -= f * v * dx
if velocity is not None:
F += dl.dot(velocity, dl.grad(u)) * v * dx
return F
F = u*v*dx-u_1*v*dx + dt*theta*steady_state_form(u, v, forcing) + \
dt*(1.-theta)*steady_state_form(u_1, v, forcing_1)
a, L = dl.lhs(F), dl.rhs(F)
# a = u*v*dx + theta*dt*kappa*dl.inner(dl.grad(u), dl.grad(v))*dx
# L = (u_1 + dt*theta*forcing)*v*dx
# if velocity is not None:
# a += theta*dt*v*dl.dot(velocity,dl.grad(u))*dx
# if second_order_timestepping:
# L -= (1-theta)*dt*dl.inner(kappa*dl.grad(u_1), dl.grad(v))*dx
# L += (1-theta)*dt*forcing_1*v*dx
# if velocity is not None:
# L -= (1-theta)*dt*(v*dl.dot(velocity,dl.grad(u_1)))*dx
beta_1_list = []
alpha_1_list = []
for ii in range(num_bndrys):
if (boundary_conditions[ii][0] == 'robin'):
alpha = boundary_conditions[ii][3]
a += theta * dt * alpha * u * v * ds(ii)
if second_order_timestepping:
if hasattr(alpha, 't'):
alpha_1 = copy_expression(alpha)
alpha_1_list.append(alpha_1)
else:
alpha_1 = alpha
L -= (1 - theta) * dt * alpha_1 * u_1 * v * ds(ii)
if ((boundary_conditions[ii][0] == 'robin')
or (boundary_conditions[ii][0] == 'neumann')):
beta = boundary_conditions[ii][2]
L -= theta * dt * beta * v * ds(ii)
if second_order_timestepping:
if hasattr(beta, 't'):
beta_1 = copy_expression(beta)
beta_1_list.append(beta_1)
else:
# boundary condition is constant in time
beta_1 = beta
L -= (1 - theta) * dt * beta_1 * v * ds(ii)
if time_independent_boundaries:
# TODO this can be used if dirichlet and robin conditions are not
# time dependent.
A = dla.assemble(a)
for bc in dirichlet_bcs:
bc.apply(A)
solver = dla.LUSolver(A)
#solver.parameters["reuse_factorization"] = True
else:
solver = None
u_2 = dla.Function(function_space)
u_2.assign(u_1)
t = 0.0
dt_tol = 1e-12
n_time_steps = 0
if intermediate_times is not None:
intermediate_u = []
intermediate_cnt = 0
# assert in chronological order
assert np.allclose(intermediate_times, np.array(intermediate_times))
assert np.all(intermediate_times < final_time)
while t < final_time - dt_tol:
# Update current time
prev_t = t
forcing_1.t = prev_t
t += dt
t = min(t, final_time)
forcing.t = t
# set current time for time varying boundary conditions
for ii in range(num_bndrys):
if hasattr(boundary_conditions[ii][2], 't'):
boundary_conditions[ii][2].t = t
# set previous time for time varying boundary conditions when
# using second order timestepping. lists will be empty if using
# first order timestepping
for jj in range(len(beta_1_list)):
beta_1_list[jj].t = prev_t
for jj in range(len(alpha_1_list)):
alpha_1_list[jj].t = prev_t
#A, b = dl.assemble_system(a, L, dirichlet_bcs)
# for bc in dirichlet_bcs:
# bc.apply(A,b)
if boundary_conditions is not None:
A = dla.assemble(a)
for bc in dirichlet_bcs:
bc.apply(A)
b = dla.assemble(L)
for bc in dirichlet_bcs:
bc.apply(b)
if point_sources is not None:
ps_list = []
for ii in range(len(point_sources)):
point, expr = point_sources[ii]
ps_list.append((dl.Point(point[0], point[1]), expr(t)))
ps = dla.PointSource(function_space, ps_list)
ps.apply(b)
if solver is None:
dla.solve(A, u_2.vector(), b)
else:
solver.solve(u_2.vector(), b)
# tape = dla.get_working_tape()
# tape.visualise()
#print ("t =", t, "end t=", final_time)
# Update previous solution
u_1.assign(u_2)
# import matplotlib.pyplot as plt
# plt.subplot(131)
# pp=dl.plot(u_1)
# plt.subplot(132)
# dl.plot(forcing,mesh=mesh)
# plt.subplot(133)
# dl.plot(forcing_1,mesh=mesh)
# plt.colorbar(pp)
# plt.show()
# compute error
if exact_sol is not None:
exact_sol.t = t
error = dl.errornorm(exact_sol, u_2)
print('t = %.2f: error = %.3g' % (t, error))
# dl.plot(exact_sol,mesh=mesh)
# plt.show()
if (intermediate_times is not None
and intermediate_cnt < intermediate_times.shape[0]
and t >= intermediate_times[intermediate_cnt]):
# save solution closest to intermediate time
u_t = dla.Function(function_space)
u_t.assign(u_2)
intermediate_u.append(u_t)
intermediate_cnt += 1
n_time_steps += 1
# print ("t =", t, "end t=", final_time,"# time steps", n_time_steps)
if intermediate_times is None:
return u_2
else:
return intermediate_u + [u_2]
def test_fenics_forward():
numpy_output, _, _, _, = evaluate_primal(assemble_fenics, templates, *inputs)
u1 = fa.interpolate(fa.Constant(1.0), V)
J = assemble_fenics(u1, fa.Constant(0.5), fa.Constant(0.6))
assert np.isclose(numpy_output, J)
def run_model(function_space, time_step, final_time, forcing,
boundary_conditions, init_condition, nonlinear_diffusion,
second_order_timestepping=False, nlsparam=dict(),
positivity_tol=0):
if boundary_conditions is None:
bndry_obj = dl.CompiledSubDomain("on_boundary")
boundary_conditions = [['dirichlet', bndry_obj, dla.Constant(0)]]
dt = time_step
mesh = function_space.mesh()
num_bndrys = len(boundary_conditions)
assert num_bndrys > 0 # specify None for no boundaries
if (len(boundary_conditions) == 1 and
isinstance(boundary_conditions[0][2], dla.DirichletBC)):
ds = dl.Measure('ds', domain=mesh)
dirichlet_bcs = [boundary_conditions[0][2]]
else:
boundaries = mark_boundaries(mesh, boundary_conditions)
dirichlet_bcs = collect_dirichlet_boundaries(
function_space, boundary_conditions, boundaries)
ds = dl.Measure('ds', domain=mesh, subdomain_data=boundaries)
dx = dl.Measure('dx', domain=mesh)
u = dl.TrialFunction(function_space)
v = dl.TestFunction(function_space)
# Previous solution
u_1 = dla.interpolate(init_condition, function_space)
u_2 = dla.Function(function_space)
u_2.assign(u_1)
if not second_order_timestepping:
theta = 1
else:
theta = 0.5
if second_order_timestepping and hasattr(forcing, 't'):
forcing_1 = copy_expression(forcing)
else:
forcing_1 = forcing
kappa = nonlinear_diffusion(u)
a = u*v*dx + theta*dt*kappa*dl.inner(dl.grad(u), dl.grad(v))*dx
L = (u_1 + theta*dt*forcing)*v*dx
# subtract of positivity preserving part added to diffusion
if positivity_tol > 0:
a -= positivity_tol*dl.inner(dl.grad(u), dl.grad(v))*dx
if second_order_timestepping:
kappa_1 = nonlinear_diffusion(u_1)
L -= (1-theta)*dt*kappa_1*dl.inner(dl.grad(u_1), dl.grad(v))*dx
L += (1-theta)*dt*forcing_1*v*dx
beta_1_list = []
alpha_1_list = []
for ii in range(num_bndrys):
if (boundary_conditions[ii][0] == 'robin'):
alpha = boundary_conditions[ii][3]
a += theta*dt*alpha*u*v*ds(ii)
if second_order_timestepping:
if hasattr(alpha, 't'):
alpha_1 = copy_expression(alpha)
alpha_1_list.append(alpha_1)
else:
alpha_1 = alpha
L -= (1-theta)*dt*alpha_1*u_1*v*ds(ii)
if ((boundary_conditions[ii][0] == 'robin') or
(boundary_conditions[ii][0] == 'neumann')):
beta = boundary_conditions[ii][2]
print(type(theta), type(dt), type(beta), type(v))
L -= theta*dt*beta*v*ds(ii)
if second_order_timestepping:
if hasattr(beta, 't'):
beta_1 = copy_expression(beta)
beta_1_list.append(beta_1)
else:
# boundary condition is constant in time
beta_1 = beta
L -= (1-theta)*dt*beta_1*v*ds(ii)
if hasattr(init_condition, 't'):
t = init_condition.t
else:
t = 0.0
while t < final_time:
# print('TIME',t)
# Update current time
prev_t = t
forcing_1.t = prev_t
t += dt
t = min(t, final_time)
forcing.t = t
# set current time for time varying boundary conditions
for ii in range(num_bndrys):
if hasattr(boundary_conditions[ii][2], 't'):
boundary_conditions[ii][2].t = t
# set previous time for time varying boundary conditions when
# using second order timestepping. lists will be empty if using
# first order timestepping
for jj in range(len(beta_1_list)):
beta_1_list[jj].t = prev_t
for jj in range(len(alpha_1_list)):
alpha_1_list[jj].t = prev_t
# solver must be redefined at every timestep
F = a-L
F = dl.action(F, u_2)
J = dl.derivative(F, u_2, u)
dla.solve(F == 0, u_2, dirichlet_bcs, J=J, solver_parameters=nlsparam)
# import matplotlib.pyplot as plt
# pl = dl.plot(sol); plt.colorbar(pl); plt.show()
# import matplotlib.pyplot as plt
# pl = dl.plot(sol); plt.show()
# Update previous solution
u_1.assign(u_2)
return u_1