def test_pde_constrained(polynomial_order, in_expression):
interpolate_expression = Expression(in_expression, degree=3)
xmin, xmax = 0., 1.
ymin, ymax = 0., 1.
property_idx = 1
dt = 1.
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
step += 1
t += float(dt)
if comm.rank == 0:
print("Step " + str(step))
# Advect particle, assemble and solve pde projection
t1 = Timer("[P] Advect particles step")
AD.do_sweep()
ap.do_step(float(dt))
AD.do_sweep_failsafe(4)
del t1
if projection_type == "PDE":
t1 = Timer("[P] Assemble PDE system")
pde_projection.assemble(True, True)
del t1
t1 = Timer("[P] Solve projection")
pde_projection.solve_problem(psibar_h, psi_h, "mumps",
"default")
del t1
else:
t1 = Timer("[P] Solve projection")
lstsq_psi.project(psi_h)
del t1
t1 = Timer("[P] Update and store")
# Update old solution
assign(psi0_h, psi_h)
# Store field
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)
ALE.move(mesh, project(umesh * dt, Vcg))
# Relocate particles as a result of mesh motion
ap.update_facets_info()
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, phih, 'mumps', 'none')
# Area on new configuration
new_area = assemble(phih*dx)
# Update solution
assign(phih0, phih)
# Global mass error, should be machine precision
print("Mass error "+str(new_area - old_area))
outfile << phih0
while step < num_steps:
step += 1
t += float(dt)
if comm.Get_rank() == 0:
print("Step " + str(step) + ', time = ' + str(t))
# Advect
t1 = Timer("[P] advect particles")
ap.do_step(float(dt))
del (t1)
# Project density and specific momentum
t1 = Timer("[P] density projection")
if projection_type == 'PDE':
pde_rho.assemble(True, True)
pde_rho.solve_problem(rhobar, rho, solver, "default")
else:
lstsq_rho.project(rho, float(rho2), float(rho1))
del (t1)
t1 = Timer("[P] momentum projection")
if projection_type == 'PDE':
pde_u.assemble(True, True)
pde_u.solve_problem(ustar_bar, ustar, solver, "default")
else:
lstsq_u.project(ustar)
del (t1)
t1 = Timer("[P] Computing conservation statements, just output!")
#
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