# Initialize forms
FuncSpace_adv = {
"FuncSpace_local": W,
"FuncSpace_lambda": T,
"FuncSpace_bar": Wbar
}
forms_pde = FormsPDEMap(mesh, FuncSpace_adv).forms_theta_linear(
psi0_h, uh, dt, Constant(1.0))
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"],
[],
property_idx,
)
# Initialize the l2 projection
lstsq_psi = l2projection(p, W, property_idx)
# Set initial condition at mesh and particles
psi0_h.interpolate(psi0_expression)
p.interpolate(psi0_h, property_idx)
step = 0
lambda_h = Function(T)
psibar_h = Function(Wbar)
# Boundary conditions
bc = DirichletBC(Wbar, Constant(0.), "on_boundary")
# Initialize forms
FuncSpace_adv = {
'FuncSpace_local': W,
'FuncSpace_lambda': T,
'FuncSpace_bar': Wbar
}
forms_pde = FormsPDEMap(mesh, FuncSpace_adv).forms_theta_linear(
psi0_h, uh, dt, Constant(1.0))
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'], [bc], property_idx)
# Set initial condition at mesh and particles
psi0_h.interpolate(psi0_expression)
p.interpolate(psi0_h.cpp_object(), property_idx)
# Initialize add/delete particle
AD = AddDelete(p, 15, 25, [psi0_h])
step = 0
t = 0.
area_0 = assemble(psi0_h * dx)
timer = Timer()
timer.start()
def test_moving_mesh():
t = 0.
dt = 0.025
num_steps = 20
xmin, ymin = 0., 0.
xmax, ymax = 2., 2.
xc, yc = 1., 1.
nx, ny = 20, 20
pres = 150
k = 1
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), nx, ny)
n = FacetNormal(mesh)
# Class for mesh motion
dU = PeriodicVelocity(xmin, xmax, dt, t, degree=1)
Qcg = VectorFunctionSpace(mesh, 'CG', 1)
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim()-1)
boundaries.set_all(0)
leftbound = Left(xmin)
leftbound.mark(boundaries, 99)
ds = Measure('ds', domain=mesh, subdomain_data=boundaries)
# Create function spaces
Q_E_Rho = FiniteElement("DG", mesh.ufl_cell(), k)
T_1 = FunctionSpace(mesh, 'DG', 0)
Qbar_E = FiniteElement("DGT", mesh.ufl_cell(), k)
Q_Rho = FunctionSpace(mesh, Q_E_Rho)
Qbar = FunctionSpace(mesh, Qbar_E)
phih, phih0 = Function(Q_Rho), Function(Q_Rho)
phibar = Function(Qbar)
# Advective velocity
uh = Function(Qcg)
uh.assign(Constant((0., 0.)))
# Mesh velocity
umesh = Function(Qcg)
# Total velocity
uadvect = uh-umesh
# Now throw in the particles
x = RandomRectangle(Point(xmin, ymin), Point(xmax, ymax)).generate([pres, pres])
s = assign_particle_values(x, GaussianPulse(center=(xc, yc), sigma=float(0.25),
U=[0, 0], time=0., height=1., degree=3))
x = comm.bcast(x, root=0)
s = comm.bcast(s, root=0)
p = particles(x, [s], mesh)
# Define projections problem
FuncSpace_adv = {'FuncSpace_local': Q_Rho, 'FuncSpace_lambda': T_1, 'FuncSpace_bar': Qbar}
FormsPDE = FormsPDEMap(mesh, FuncSpace_adv, ds=ds)
forms_pde = FormsPDE.forms_theta_linear(phih0, uadvect, dt, Constant(1.0), zeta=Constant(0.))
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)
# Initialize the initial condition at mesh by an l2 projection
lstsq_rho = l2projection(p, Q_Rho, 1)
lstsq_rho.project(phih0.cpp_object())
for step in range(num_steps):
# Compute old area at old configuration
old_area = assemble(phih0*dx)
# Pre-assemble rhs
pde_projection.assemble_state_rhs()
# Move mesh
dU.compute_ubc()
umesh.assign(project(dU, Qcg))
ALE.move(mesh, project(dU * dt, Qcg))
dU.update()
# Relocate particles as a result of mesh motion
# NOTE: if particles were advected themselve,
# we had to run update_facets_info() here as well
p.relocate()
# Assemble left-hand side on new config, but not the right-hand side
pde_projection.assemble(True, False)
pde_projection.solve_problem(phibar.cpp_object(), phih.cpp_object(),
'mumps', 'none')
# Needed to compute conservation, note that there
# is an outgoing flux at left boundary
new_area = assemble(phih*dx)
gamma = conditional(ge(dot(uadvect, n), 0), 0, 1)
bflux = assemble((1-gamma) * dot(uadvect, n) * phih * ds)
# Update solution
assign(phih0, phih)
# Put assertion on (global) mass balance, local mass balance is
# too time consuming but should pass also
assert new_area - old_area + bflux * dt < 1e-12
# Assert that max value of phih stays close to 2 and
# min value close to 0. This typically will fail if
# we do not do a correct relocate of particles
assert np.amin(phih.vector().get_local()) > -0.015
assert np.amax(phih.vector().get_local()) < 1.04
"FuncSpace_bar": Wbar
}
forms_pde = FormsPDEMap(mesh,
FuncSpace_adv).forms_theta_linear(psi_h0,
uh,
dt,
Constant(1.0),
zeta=zeta,
h=Constant(0.0))
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"],
[bc],
1,
)
# Init projection
lstsq_psi = l2projection(p, W, 1)
# Do projection to get initial field
lstsq_psi.project(psi_h0, lb, ub)
assign(psi_h00, psi_h0)
step = 0
def test_pde_constrained(polynomial_order, in_expression):
interpolate_expression = Expression(in_expression, degree=3)
xmin, xmax = 0.0, 1.0
ymin, ymax = 0.0, 1.0
property_idx = 1
dt = 1.0
k = polynomial_order
# Make mesh
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), 40, 40)
# Make function spaces and functions
W_e = FiniteElement("DG", mesh.ufl_cell(), k)
T_e = FiniteElement("DG", mesh.ufl_cell(), 0)
Wbar_e = FiniteElement("DGT", mesh.ufl_cell(), k)
W = FunctionSpace(mesh, W_e)
T = FunctionSpace(mesh, T_e)
Wbar = FunctionSpace(mesh, Wbar_e)
psi_h, psi0_h = Function(W), Function(W)
lambda_h = Function(T)
psibar_h = Function(Wbar)
uadvect = Constant((0, 0))
# Define particles
x = RandomRectangle(Point(xmin, ymin), Point(xmax, ymax)).generate([500, 500])
s = assign_particle_values(x, interpolate_expression)
psi0_h.assign(interpolate_expression)
# Just make a complicated particle, possibly with scalars and vectors mixed
p = particles(x, [s], mesh)
p.interpolate(psi0_h, 1)
# Initialize forms
FuncSpace_adv = {"FuncSpace_local": W, "FuncSpace_lambda": T, "FuncSpace_bar": Wbar}
forms_pde = FormsPDEMap(mesh, FuncSpace_adv).forms_theta_linear(
psi0_h, uadvect, dt, Constant(1.0)
)
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"],
[],
property_idx,
)
# Assemble and solve
pde_projection.assemble(True, True)
pde_projection.solve_problem(psibar_h, psi_h, lambda_h, "none", "default")
error_psih = abs(assemble((psi_h - psi0_h) * (psi_h - psi0_h) * dx))
error_lamb = abs(assemble(lambda_h * lambda_h * dx))
assert error_psih < 1e-15
assert error_lamb < 1e-15
