def setup_NS(w_NS, u, p, v, q, p0, q0,
dx, ds, normal,
dirichlet_bcs_NS, neumann_bcs, boundary_to_mark,
u_1, rho_, rho_1, mu_, c_1, grad_g_c_,
dt, grav,
enable_EC,
trial_functions,
use_iterative_solvers,
p_lagrange,
mesh,
q_rhs,
density_per_concentration,
K,
**namespace):
""" Set up the Navier-Stokes subproblem. """
mom_1 = rho_1 * u_1
if enable_EC and density_per_concentration is not None:
for drhodci, ci_1, grad_g_ci_, Ki in zip(
density_per_concentration, c_1, grad_g_c_, K):
if drhodci > 0.:
mom_1 += -drhodci*Ki*ci_1*grad_g_ci_
F = (1./dt * rho_1 * df.dot(u - u_1, v) * dx
+ df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx
+ 2*mu_*df.inner(df.sym(df.nabla_grad(u)),
df.sym(df.nabla_grad(v))) * dx
+ 0.5*(
1./dt * (rho_ - rho_1) * df.inner(u, v)
- df.inner(mom_1, df.nabla_grad(df.dot(u, v)))) * dx
- p * df.div(v) * dx
- q * df.div(u) * dx
- rho_ * df.dot(grav, v) * dx)
for boundary_name, pressure in neumann_bcs["p"].iteritems():
F += pressure * df.inner(
normal, v) * ds(boundary_to_mark[boundary_name])
if enable_EC:
F += sum([ci_1*df.dot(grad_g_ci_, v)*dx
for ci_1, grad_g_ci_ in zip(c_1, grad_g_c_)])
if p_lagrange:
F += (p*q0 + q*p0)*dx
if "u" in q_rhs:
F += -df.dot(q_rhs["u"], v)*dx
a, L = df.lhs(F), df.rhs(F)
if not use_iterative_solvers:
problem = df.LinearVariationalProblem(a, L, w_NS, dirichlet_bcs_NS)
solver = df.LinearVariationalSolver(problem)
else:
solver = df.LUSolver("mumps")
# solver.set_operator(A)
return solver, a, L, dirichlet_bcs_NS
return solver
def setup_NSu(w_NSu, u, v, dx, ds, normal, dirichlet_bcs_NSu, neumann_bcs,
boundary_to_mark, u_, p_, u_1, p_1, rho_, rho_1, mu_, c_1,
grad_g_c_, dt, grav, enable_EC, trial_functions,
use_iterative_solvers, mesh, density_per_concentration,
viscosity_per_concentration, K, **namespace):
""" Set up the Navier-Stokes velocity subproblem. """
solvers = dict()
mom_1 = rho_1 * u_1
if enable_EC and density_per_concentration is not None:
for drhodci, ci_1, grad_g_ci_, Ki in zip(density_per_concentration,
c_1, grad_g_c_, K):
if drhodci > 0.:
mom_1 += -drhodci * Ki * ci_1 * grad_g_ci_
F_predict = (
1. / dt * rho_1 * df.dot(u - u_1, v) * dx +
df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx + 2 * mu_ *
df.inner(df.sym(df.nabla_grad(u)), df.sym(df.nabla_grad(v))) * dx +
0.5 * (1. / dt * (rho_ - rho_1) * df.dot(u, v) -
df.inner(mom_1, df.grad(df.dot(u, v)))) * dx -
p_1 * df.div(v) * dx - rho_ * df.dot(grav, v) * dx)
for boundary_name, pressure in neumann_bcs["p"].iteritems():
F_predict += pressure * df.inner(normal, v) * ds(
boundary_to_mark[boundary_name])
if enable_EC:
F_predict += sum([
ci_1 * df.dot(grad_g_ci_, v) * dx
for ci_1, grad_g_ci_ in zip(c_1, grad_g_c_)
])
a_predict, L_predict = df.lhs(F_predict), df.rhs(F_predict)
# if not use_iterative_solvers:
problem_predict = df.LinearVariationalProblem(a_predict, L_predict, w_NSu,
dirichlet_bcs_NSu)
solvers["predict"] = df.LinearVariationalSolver(problem_predict)
if use_iterative_solvers:
solvers["predict"].parameters["linear_solver"] = "bicgstab"
solvers["predict"].parameters["preconditioner"] = "amg"
F_correct = (rho_ * df.inner(u - u_, v) * dx - dt *
(p_ - p_1) * df.div(v) * dx)
a_correct, L_correct = df.lhs(F_correct), df.rhs(F_correct)
problem_correct = df.LinearVariationalProblem(a_correct, L_correct, w_NSu,
dirichlet_bcs_NSu)
solvers["correct"] = df.LinearVariationalSolver(problem_correct)
if use_iterative_solvers:
solvers["correct"].parameters["linear_solver"] = "bicgstab"
solvers["correct"].parameters["preconditioner"] = "amg"
#else:
# solver = df.LUSolver("mumps")
# # solver.set_operator(A)
# return solver, a, L, dirichlet_bcs_NS
return solvers
def setup_NSu(u, v, u_, p_, bcs_NSu, u_1, p_1, phi_, rho_, rho_1, g_, M_, nu_,
rho_e_, V_, dt, drho, sigma_bar, eps, dveps, grav, enable_PF,
enable_EC):
""" Set up the Navier-Stokes subproblem. """
# Crank-Nicolson velocity
# u_CN = 0.5*(u_1 + u)
F_predict = (
1. / dt * df.sqrt(rho_) *
df.dot(df.sqrt(rho_) * u - df.sqrt(rho_1) * u_1, v) * df.dx
# + rho_*df.inner(df.grad(u), df.outer(u_1, v))*df.dx
# + 2*nu_*df.inner(df.sym(df.grad(u)), df.grad(v))*df.dx
# - p_1 * df.div(v)*df.dx
# + df.div(u)*q*df.dx
+ rho_ * df.dot(df.dot(u_1, df.nabla_grad(u)), v) * df.dx +
2 * nu_ * df.inner(df.sym(df.grad(u)), df.sym(df.grad(v))) * df.dx -
p_1 * df.div(v) * df.dx - df.dot(rho_ * grav, v) * df.dx)
phi_filtered = unit_interval_filter(phi_)
if enable_PF:
F_predict += -drho * M_ * df.dot(
df.dot(df.nabla_grad(g_), df.nabla_grad(u)), v) * df.dx
F_predict += -sigma_bar * eps * df.inner(
df.outer(df.grad(phi_filtered), df.grad(phi_filtered)),
df.grad(v)) * df.dx
if enable_EC and rho_e_ != 0:
F_predict += rho_e_ * df.dot(df.grad(V_), v) * df.dx
if enable_PF and enable_EC:
F_predict += dveps * df.dot(df.grad(phi_filtered), v) * df.dot(
df.grad(V_), df.grad(V_)) * df.dx
# a1, L1 = df.lhs(F_predict), df.rhs(F_predict)
F_correct = (df.inner(u - u_, v) * df.dx +
dt / rho_ * df.inner(df.grad(p_ - p_1), v) * df.dx)
# a3 = df.dot(u, v)*df.dx
# L3 = df.dot(u_, v)*df.dx - dt*df.dot(df.grad(p_), v)*df.dx
# a3, L3 = df.lhs(F_correct), df.rhs(F_correct)
solver = dict()
# solver["a1"] = a1
# solver["L1"] = L1
solver["Fu"] = F_predict
solver["Fu_corr"] = F_correct
# solver["a3"] = a3
# solver["L3"] = L3
solver["bcs"] = bcs_NSu
return solver
def les_setup(u_, mesh, Smagorinsky, CG1Function, nut_krylov_solver, bcs,
**NS_namespace):
"""
Set up for solving Smagorinsky-Lilly LES model.
"""
DG = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
# Compute cell size and put in delta
dim = mesh.geometry().dim()
delta = Function(DG)
delta.vector().zero()
delta.vector().set_local(
assemble(TestFunction(DG) * dx).array()**(1. / dim))
delta.vector().apply('insert')
# Set up Smagorinsky form
Sij = sym(grad(u_))
magS = sqrt(2 * inner(Sij, Sij))
nut_form = Smagorinsky['Cs']**2 * delta**2 * magS
bcs_nut = derived_bcs(CG1, bcs['u0'], u_)
nut_ = CG1Function(nut_form,
mesh,
method=nut_krylov_solver,
bcs=bcs_nut,
bounded=True,
name="nut")
return dict(Sij=Sij, nut_=nut_, delta=delta, bcs_nut=bcs_nut)
def les_update(nut_, nut_form, A_mass, At, u_, dt, bc_ksgs, bt, ksgs_sol,
KineticEnergySGS, CG1, ksgs, delta, **NS_namespace):
p, q = TrialFunction(CG1), TestFunction(CG1)
Ck = KineticEnergySGS["Ck"]
Ce = KineticEnergySGS["Ce"]
Sij = sym(grad(u_))
assemble(dt * inner(dot(u_, 0.5 * grad(p)), q) * dx + inner(
(dt * Ce * sqrt(ksgs) / delta) * 0.5 * p, q) * dx +
inner(dt * Ck * sqrt(ksgs) * delta * grad(0.5 * p), grad(q)) * dx,
tensor=At)
assemble(dt * 2 * Ck * delta * sqrt(ksgs) * inner(Sij, grad(u_)) * q * dx,
tensor=bt)
bt.axpy(1.0, A_mass * ksgs.vector())
bt.axpy(-1.0, At * ksgs.vector())
At.axpy(1.0, A_mass, True)
# Solve for ksgs
bc_ksgs.apply(At, bt)
ksgs_sol.solve(At, ksgs.vector(), bt)
ksgs.vector().set_local(ksgs.vector().array().clip(min=1e-7))
ksgs.vector().apply("insert")
# Update nut_
nut_()
def les_update(nut_, nut_form, A_mass, At, u_, dt, bc_ksgs, bt, ksgs_sol,
KineticEnergySGS, CG1, ksgs, delta, **NS_namespace):
p, q = TrialFunction(CG1), TestFunction(CG1)
Ck = KineticEnergySGS["Ck"]
Ce = KineticEnergySGS["Ce"]
Sij = sym(grad(u_))
assemble(dt*inner(dot(u_, 0.5*grad(p)), q)*dx
+ inner((dt*Ce*sqrt(ksgs)/delta)*0.5*p, q)*dx
+ inner(dt*Ck*sqrt(ksgs)*delta*grad(0.5*p), grad(q))*dx, tensor=At)
assemble(dt*2*Ck*delta*sqrt(ksgs)*inner(Sij,grad(u_))*q*dx, tensor=bt)
bt.axpy(1.0, A_mass*ksgs.vector())
bt.axpy(-1.0, At*ksgs.vector())
At.axpy(1.0, A_mass, True)
# Solve for ksgs
bc_ksgs.apply(At, bt)
ksgs_sol.solve(At, ksgs.vector(), bt)
ksgs.vector().set_local(ksgs.vector().array().clip(min=1e-7))
ksgs.vector().apply("insert")
# Update nut_
nut_()
def les_setup(u_, mesh, Wale, bcs, CG1Function, nut_krylov_solver, **NS_namespace):
"""Set up for solving Wale LES model
"""
DG = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
delta = Function(DG)
delta.vector().zero()
delta.vector().axpy(1.0, assemble(TestFunction(DG)*dx))
Gij = grad(u_)
Sij = sym(Gij)
Skk = tr(Sij)
dim = mesh.geometry().dim()
Sd = sym(Gij*Gij) - 1./3.*Identity(mesh.geometry().dim())*Skk*Skk
nut_form = Wale['Cw']**2 * pow(delta, 2./dim) * pow(inner(Sd, Sd), 1.5) / (Max(pow(inner(Sij, Sij), 2.5) + pow(inner(Sd, Sd), 1.25), 1e-6))
ff = FacetFunction("size_t", mesh, 0)
bcs_nut = derived_bcs(CG1, bcs['u0'], u_)
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver, bcs=bcs_nut, name='nut', bounded=True)
return dict(Sij=Sij, Sd=Sd, Skk=Skk, nut_=nut_, delta=delta, bcs_nut=bcs_nut)
def neumann_elasticity_data():
'''
Return:
a bilinear form in the neumann elasticity problem
L linear form in therein
V function space, where a, L are defined
bc homog. dirichlet conditions for case where we want pos. def problem
z list of orthonormal vectors in the nullspace of A that form basis
of ker(A)
'''
mesh = UnitSquareMesh(40, 40)
V = VectorFunctionSpace(mesh, 'CG', 1)
u = TrialFunction(V)
v = TestFunction(V)
f = Expression(('sin(pi*x[0])', 'cos(pi*x[1])'))
epsilon = lambda u: sym(grad(u))
# Material properties
E, nu = 10.0, 0.3
mu, lmbda = Constant(E/(2*(1 + nu))), Constant(E*nu/((1 + nu)*(1 - 2*nu)))
sigma = lambda u: 2*mu*epsilon(u) + lmbda*tr(epsilon(u))*Identity(2)
a = inner(sigma(u), epsilon(v))*dx
L = inner(f, v)*dx # Zero stress
bc = DirichletBC(V, Constant((0, 0)), DomainBoundary())
z0 = interpolate(Constant((1, 0)), V).vector()
normalize(z0, 'l2')
z1 = interpolate(Constant((0, 1)), V).vector()
normalize(z1, 'l2')
X = mesh.coordinates().reshape((-1, 2))
c0, c1 = np.sum(X, axis=0)/len(X)
z2 = interpolate(Expression(('x[1]-c1',
'-(x[0]-c0)'), c0=c0, c1=c1), V).vector()
normalize(z2, 'l2')
z = [z0, z1, z2]
# Check that this is orthonormal basis
I = np.zeros((3, 3))
for i, zi in enumerate(z):
for j, zj in enumerate(z):
I[i, j] = zi.inner(zj)
print I
print la.norm(I-np.eye(3))
assert la.norm(I-np.eye(3)) < 1E-13
return a, L, V, bc, z
def les_setup(u_, mesh, Wale, bcs, CG1Function, nut_krylov_solver, **NS_namespace):
"""Set up for solving Wale LES model
"""
DG = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
# Compute cell size and put in delta
delta = Function(DG)
delta.vector().zero()
delta.vector().axpy(1.0, assemble(TestFunction(DG)*dx))
# Set up Wale form
Gij = grad(u_)
Sij = sym(Gij)
Skk = tr(Sij)
dim = mesh.geometry().dim()
Sd = sym(Gij*Gij) - 1./3.*Identity(mesh.geometry().dim())*Skk*Skk
nut_form = Wale['Cw']**2 * pow(delta, 2./dim) * pow(inner(Sd, Sd), 1.5) / (Max(pow(inner(Sij, Sij), 2.5) + pow(inner(Sd, Sd), 1.25), 1e-6))
ff = FacetFunction("size_t", mesh, 0)
bcs_nut = derived_bcs(CG1, bcs['u0'], u_)
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver, bcs=bcs_nut, name='nut', bounded=True)
return dict(Sij=Sij, Sd=Sd, Skk=Skk, nut_=nut_, delta=delta, bcs_nut=bcs_nut)
def eps(self, u):
e = df.sym(df.grad(u))
dim = self.mesh.geometric_dimension()
if dim == 1:
return df.as_vector([e[0, 0]])
if dim == 2:
return df.as_vector([e[0, 0], e[1, 1], 2 * e[0, 1]])
if dim == 3:
return df.as_vector([
e[0, 0], e[1, 1], e[2, 2], 2 * e[1, 2], 2 * e[0, 2],
2 * e[0, 1]
])
def create_forms(W, rho, nu, g_a, boundary_markers, gamma=0.0):
v, p = df.TrialFunctions(W)
v_t, p_t = df.TestFunctions(W)
a = (
2.0 * nu * df.inner(df.sym(df.grad(v)), df.grad(v_t)) -
p * df.div(v_t) - df.div(v) * p_t
#- nu * df.div(v) * p_t
) * df.dx
L = rho * df.inner(df.Constant((0.0, -g_a)), v_t) * df.dx
# Grad-div stabilization
a += df.Constant(gamma) * df.div(v) * df.div(v_t) * df.dx
return a, L
def strain_rate_tensor(self, u):
r"""Define the symmetric strain-rate tensor
:param u: The velocity field (2D).
:return: The strain rate tensor.
"""
r = dolfin.SpatialCoordinate(self._mesh)[0]
ur = u[0]
uz = u[1]
ur_r = ur.dx(0)
ur_z = ur.dx(1)
uz_r = uz.dx(0)
uz_z = uz.dx(1)
return dolfin.sym(
dolfin.as_tensor([[ur_r, 0, 0.5 * (uz_r + ur_z)], [0, ur / r, 0],
[0.5 * (uz_r + ur_z), 0, uz_z]]))
def les_setup(u_, mesh, Smagorinsky, CG1Function, nut_krylov_solver, bcs, **NS_namespace):
"""
Set up for solving Smagorinsky-Lilly LES model.
"""
DG = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
dim = mesh.geometry().dim()
delta = Function(DG)
delta.vector().zero()
delta.vector().axpy(1.0, assemble(TestFunction(DG)*dx))
delta.vector().apply('insert')
Sij = sym(grad(u_))
magS = sqrt(2*inner(Sij,Sij))
nut_form = Smagorinsky['Cs']**2 * delta**2 * magS
bcs_nut = derived_bcs(CG1, bcs['u0'], u_)
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver, bcs=bcs_nut, bounded=True, name="nut")
return dict(Sij=Sij, nut_=nut_, delta=delta, bcs_nut=bcs_nut)
def create_forms(W, rho, nu, F, g_a, p_h, boundary_markers):
v, p = df.TrialFunctions(W)
v_t, p_t = df.TestFunctions(W)
a = (
2.0 * nu * df.inner(df.sym(df.grad(v)), df.grad(v_t)) -
p * df.div(v_t) - df.div(v) * p_t
#- nu * df.div(v) * p_t
) * df.dx
L = rho * df.inner(df.Constant((0.0, -g_a)), v_t) * df.dx
n = df.FacetNormal(W.mesh())
ds = df.Measure("ds", subdomain_data=boundary_markers)
L += df.inner(df.Constant((F, 0.0)), v_t) * ds(3) # driving force
L -= p_h * df.inner(n, v_t) * (ds(2) + ds(4)) # hydrostatic balance
return a, L
def les_setup(u_, mesh, Smagorinsky, CG1Function, nut_krylov_solver, bcs, **NS_namespace):
"""
Set up for solving Smagorinsky-Lilly LES model.
"""
DG = FunctionSpace(mesh, "DG", 0)
CG1 = FunctionSpace(mesh, "CG", 1)
# Compute cell size and put in delta
delta = Function(DG)
delta.vector().zero()
delta.vector().axpy(1.0, assemble(TestFunction(DG) * dx))
delta.vector().apply("insert")
# Set up Smagorinsky form
Sij = sym(grad(u_))
magS = sqrt(2 * inner(Sij, Sij))
nut_form = Smagorinsky["Cs"] ** 2 * delta ** 2 * magS
bcs_nut = derived_bcs(CG1, bcs["u0"], u_)
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver, bcs=bcs_nut, bounded=True, name="nut")
return dict(Sij=Sij, nut_=nut_, delta=delta, bcs_nut=bcs_nut)
def stress_tensor(self, L, p):
"""
Return the Cauchy stress tensor for an incompressible Newtonian fluid,
namely
.. math::
\mathbf{T} = -p\mathbf{I} + 2\mu\\text{Sym}(\mathbf{L}),
where :math:`\mathbf{L} = \\text{grad}(\mathbf{v})`, :math:`\mathbf{I}`
is the identity tensor, :math:`p` is the hydrostatic pressure, and
:math:`\mu` is the dynamic viscosity.
Parameters
----------
L : ufl.differentiation.Grad
The velocity gradient.
p : dolfin.Function, ufl.indexed.Indexed
The hydrostatic pressure.
Returns
-------
T : ufl.algebra.Sum
The Cauchy stress tensor defined above.
"""
params = self._parameters
dim = ufl.domain.find_geometric_dimension(L)
mu = dlf.Constant(params['mu'], name='mu')
I = dlf.Identity(dim)
D = dlf.sym(L)
return -p * I + dlf.Constant(2.0) * mu * D
def nn_setup(u_, mesh, ModifiedCross, CG1Function, nu_nn_krylov_solver, bcs,
**NS_namespace):
"""
Set up for solving Modified-Cross non-Newtonian model.
"""
CG1 = FunctionSpace(mesh, "CG", 1)
mu_inf = ModifiedCross['mu_inf']
mu_o = ModifiedCross['mu_o']
rho = ModifiedCross['rho']
lam = ModifiedCross['lam']
m_param = ModifiedCross['m_param']
a_param = ModifiedCross['a_param']
# Set up Modified Cross form
Sij = sym(grad(u_))
SII = sqrt(2 * inner(Sij, Sij))
nu_nn_form = ((mu_o - mu_inf) / ((1 +
(lam * SII)**m_param)**a_param)) / rho
#bcs_nu_nn = derived_bcs(CG1, bcs['u0'], u_)
nunn_ = CG1Function(nu_nn_form,
mesh,
method=nu_nn_krylov_solver,
bounded=True,
name="nu_nn")
return dict(Sij=Sij, nunn_=nunn_, bcs_nu_nn=[])
def initialize_with_field(self, u):
super().initialize_with_field(u)
I = Identity(len(u))
eps = sym(grad(u))
for m in self.material_parameters:
E = m.get('E', None)
nu = m.get('nu', None)
mu = m.get('mu', None)
lm = m.get('lm', None)
if mu is None:
if E is None or nu is None:
raise RuntimeError(
'Material model requires parameter "mu"; '
'otherwise, require parameters "E" and "nu".')
mu = E / (2 * (1 + nu))
if lm is None:
if E is None or nu is None:
raise RuntimeError(
'Material model requires parameter "lm"; '
'otherwise, require parameters "E" and "nu".')
lm = E * nu / ((1 + nu) * (1 - 2 * nu))
sig = 2 * mu * eps + lm * tr(eps) * I
psi = 0.5 * inner(sig, eps)
self.psi.append(psi)
self.pk1.append(sig)
self.pk2.append(sig)
def les_setup(u_, mesh, assemble_matrix, CG1Function, nut_krylov_solver, bcs, **NS_namespace):
"""
Set up for solving the Germano Dynamic LES model applying
Lagrangian Averaging.
"""
# Create function spaces
CG1 = FunctionSpace(mesh, "CG", 1)
p, q = TrialFunction(CG1), TestFunction(CG1)
dim = mesh.geometry().dim()
# Define delta and project delta**2 to CG1
delta = pow(CellVolume(mesh), 1. / dim)
delta_CG1_sq = project(delta, CG1)
delta_CG1_sq.vector().set_local(delta_CG1_sq.vector().array()**2)
delta_CG1_sq.vector().apply("insert")
# Define nut_
Sij = sym(grad(u_))
magS = sqrt(2 * inner(Sij, Sij))
Cs = Function(CG1)
nut_form = Cs**2 * delta**2 * magS
# Create nut_ BCs
ff = MeshFunction("size_t", mesh, mesh.topology().dim() - 1, 0)
bcs_nut = []
for i, bc in enumerate(bcs['u0']):
bc.apply(u_[0].vector()) # Need to initialize bc
m = bc.markers() # Get facet indices of boundary
ff.array()[m] = i + 1
bcs_nut.append(DirichletBC(CG1, Constant(0), ff, i + 1))
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver,
bcs=bcs_nut, bounded=True, name="nut")
# Create functions for holding the different velocities
u_CG1 = as_vector([Function(CG1) for i in range(dim)])
u_filtered = as_vector([Function(CG1) for i in range(dim)])
dummy = Function(CG1)
ll = LagrangeInterpolator()
# Assemble required filter matrices and functions
G_under = Function(CG1, assemble(TestFunction(CG1) * dx))
G_under.vector().set_local(1. / G_under.vector().array())
G_under.vector().apply("insert")
G_matr = assemble(inner(p, q) * dx)
# Set up functions for Lij and Mij
Lij = [Function(CG1) for i in range(dim * dim)]
Mij = [Function(CG1) for i in range(dim * dim)]
# Check if case is 2D or 3D and set up uiuj product pairs and
# Sij forms, assemble required matrices
Sijcomps = [Function(CG1) for i in range(dim * dim)]
Sijfcomps = [Function(CG1) for i in range(dim * dim)]
# Assemble some required matrices for solving for rate of strain terms
Sijmats = [assemble_matrix(p.dx(i) * q * dx) for i in range(dim)]
if dim == 3:
tensdim = 6
uiuj_pairs = ((0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2))
else:
tensdim = 3
uiuj_pairs = ((0, 0), (0, 1), (1, 1))
# Set up Lagrange functions
JLM = Function(CG1)
JLM.vector()[:] += 1E-32
JMM = Function(CG1)
JMM.vector()[:] += 1
return dict(Sij=Sij, nut_form=nut_form, nut_=nut_, delta=delta, bcs_nut=bcs_nut,
delta_CG1_sq=delta_CG1_sq, CG1=CG1, Cs=Cs, u_CG1=u_CG1,
u_filtered=u_filtered, ll=ll, Lij=Lij, Mij=Mij, Sijcomps=Sijcomps,
Sijfcomps=Sijfcomps, Sijmats=Sijmats, JLM=JLM, JMM=JMM, dim=dim,
tensdim=tensdim, G_matr=G_matr, G_under=G_under, dummy=dummy,
uiuj_pairs=uiuj_pairs)
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
def strain(u):
return dolfin.sym(dolfin.grad(u))
def strain(self, u):
"""
Strain tensor: s = .5( u_{i,j} + u_{j,i})
"""
return dl.sym(dl.grad(u))
def run_with_params(Tb, mu_value, k_s, path):
run_time_init = clock()
mesh = BoxMesh(Point(0.0, 0.0, 0.0), Point(mesh_width, mesh_width, mesh_height), nx, ny, nz)
pbc = PeriodicBoundary()
WE = VectorElement('CG', mesh.ufl_cell(), 2)
SE = FiniteElement('CG', mesh.ufl_cell(), 1)
WSSS = FunctionSpace(mesh, MixedElement(WE, SE, SE, SE), constrained_domain=pbc)
# W = FunctionSpace(mesh, WE, constrained_domain=pbc)
# S = FunctionSpace(mesh, SE, constrained_domain=pbc)
W = WSSS.sub(0).collapse()
S = WSSS.sub(1).collapse()
temperature_vals = [27.0 + 273, Tb + 273, 1300.0 + 273, 1305.0 + 273]
temp_prof = TemperatureProfile(temperature_vals, element=S.ufl_element())
mu_a = mu_value # this was taken from the Blankenbach paper, can change
Ep = b / temp_prof.delta
mu_bot = exp(-Ep * (temp_prof.bottom * temp_prof.delta - 1573.0) + cc) * mu_a
# TODO: verify exponentiation
Ra = rho_0 * alpha * g * temp_prof.delta * h ** 3 / (kappa_0 * mu_a)
w0 = rho_0 * alpha * g * temp_prof.delta * h ** 2 / mu_a
tau = h / w0
p0 = mu_a * w0 / h
log(mu_a, mu_bot, Ra, w0, p0)
slip_vx = 1.6E-09 / w0 # Non-dimensional
slip_velocity = Constant((slip_vx, 0.0, 0.0))
zero_slip = Constant((0.0, 0.0, 0.0))
time_step = 3.0E11 / tau * 2
dt = Constant(time_step)
t_end = 3.0E15 / tau / 5.0 # Non-dimensional times
u = Function(WSSS)
# Instead of TrialFunctions, we use split(u) for our non-linear problem
v, p, T, Tf = split(u)
v_t, p_t, T_t, Tf_t = TestFunctions(WSSS)
T0 = interpolate(temp_prof, S)
mu_exp = Expression('exp(-Ep * (T_val * dTemp - 1573.0) + cc * x[2] / mesh_height)',
Ep=Ep, dTemp=temp_prof.delta, cc=cc, mesh_height=mesh_height, T_val=T0,
element=S.ufl_element())
Tf0 = interpolate(temp_prof, S)
mu = Function(S)
v0 = Function(W)
v_theta = (1.0 - theta) * v0 + theta * v
T_theta = (1.0 - theta) * T0 + theta * T
Tf_theta = (1.0 - theta) * Tf0 + theta * Tf
# TODO: Verify forms
r_v = (inner(sym(grad(v_t)), 2.0 * mu * sym(grad(v)))
- div(v_t) * p
- T * v_t[2]) * dx
r_p = p_t * div(v) * dx
heat_transfer = Constant(k_s) * (Tf_theta - T_theta) * dt
r_T = (T_t * ((T - T0) + dt * inner(v_theta, grad(T_theta))) # TODO: Inner vs dot
+ (dt / Ra) * inner(grad(T_t), grad(T_theta))
- T_t * heat_transfer) * dx
v_melt = Function(W)
z_hat = Constant((0.0, 0.0, 1.0))
# TODO: inner -> dot, take out Tf_t
r_Tf = (Tf_t * ((Tf - Tf0) + dt * inner(v_melt, grad(Tf_theta)))
+ Tf_t * heat_transfer) * dx
r = r_v + r_p + r_T + r_Tf
bcv0 = DirichletBC(WSSS.sub(0), zero_slip, top)
bcv1 = DirichletBC(WSSS.sub(0), slip_velocity, bottom)
bcv2 = DirichletBC(WSSS.sub(0).sub(1), Constant(0.0), back)
bcv3 = DirichletBC(WSSS.sub(0).sub(1), Constant(0.0), front)
bcp0 = DirichletBC(WSSS.sub(1), Constant(0.0), bottom)
bct0 = DirichletBC(WSSS.sub(2), Constant(temp_prof.surface), top)
bct1 = DirichletBC(WSSS.sub(2), Constant(temp_prof.bottom), bottom)
bctf1 = DirichletBC(WSSS.sub(3), Constant(temp_prof.bottom), bottom)
bcs = [bcv0, bcv1, bcv2, bcv3, bcp0, bct0, bct1, bctf1]
t = 0
count = 0
files = DefaultDictByKey(partial(create_xdmf, path))
while t < t_end:
mu.interpolate(mu_exp)
rhosolid = rho_0 * (1.0 - alpha * (T0 * temp_prof.delta - 1573.0))
deltarho = rhosolid - rho_melt
# TODO: project (accuracy) vs interpolate
assign(v_melt, project(v0 - darcy * (grad(p) * p0 / h - deltarho * z_hat * g) / w0, W))
# TODO: Written out one step later?
# v_melt.assign(v0 - darcy * (grad(p) * p0 / h - deltarho * yvec * g) / w0)
# TODO: use nP after to avoid projection?
solve(r == 0, u, bcs)
nV, nP, nT, nTf = u.split() # TODO: write with Tf, ... etc
if count % output_every == 0:
time_left(count, t_end / time_step, run_time_init) # TODO: timestep vs dt
# TODO: Make sure all writes are to the same function for each time step
files['T_fluid'].write(nTf, t)
files['p'].write(nP, t)
files['v_solid'].write(nV, t)
files['T_solid'].write(nT, t)
files['mu'].write(mu, t)
files['v_melt'].write(v_melt, t)
files['gradp'].write(project(grad(nP), W), t)
files['rho'].write(project(rhosolid, S), t)
files['Tf_grad'].write(project(grad(Tf), W), t)
files['advect'].write(project(dt * dot(v_melt, grad(nTf))), t)
files['ht'].write(project(heat_transfer, S), t)
assign(T0, nT)
assign(v0, nV)
assign(Tf0, nTf)
t += time_step
count += 1
log('Case mu={}, Tb={}, k={} complete. Run time = {:.2f} minutes'.format(mu_a, Tb, k_s, (clock() - run_time_init) / 60.0))
def epsilon(u):
return df.sym(df.nabla_grad(u))
def test_steady_stokes(k):
# Polynomial order and mesh resolution
nx_list = [4, 8, 16]
nu = Constant(1)
if comm.Get_rank() == 0:
print('{:=^72}'.format('Computing for polynomial order ' + str(k)))
# Error listst
error_u, error_p, error_div = [], [], []
for nx in nx_list:
if comm.Get_rank() == 0:
print('# Resolution ' + str(nx))
mesh = UnitSquareMesh(nx, nx)
# Get forcing from exact solutions
u_exact, p_exact = exact_solution(mesh)
f = div(p_exact * Identity(2) - 2 * nu * sym(grad(u_exact)))
# Define FunctionSpaces and functions
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]))
Uh = Function(mixedL)
Uhbar = Function(mixedG)
# Set forms
alpha = Constant(6 * k * k)
forms_stokes = FormsStokes(mesh, mixedL, mixedG,
alpha).forms_steady(nu, f)
# No-slip boundary conditions, set pressure in one of the corners
bc0 = DirichletBC(mixedG.sub(0), Constant((0, 0)), Gamma)
bc1 = DirichletBC(mixedG.sub(1), Constant(0), Corner, "pointwise")
bcs = [bc0, bc1]
# Initialize static condensation class
ssc = StokesStaticCondensation(mesh, forms_stokes['A_S'],
forms_stokes['G_S'],
forms_stokes['B_S'],
forms_stokes['Q_S'],
forms_stokes['S_S'], bcs)
# Assemble global system and incorporates bcs
ssc.assemble_global_system(True)
# Solve using mumps
ssc.solve_problem(Uhbar, Uh, "mumps", "default")
# Compute velocity/pressure/local div error
uh, ph = Uh.split()
e_u = np.sqrt(np.abs(assemble(dot(uh - u_exact, uh - u_exact) * dx)))
e_p = np.sqrt(np.abs(assemble((ph - p_exact) * (ph - p_exact) * dx)))
e_d = np.sqrt(np.abs(assemble(div(uh) * div(uh) * dx)))
if comm.rank == 0:
error_u.append(e_u)
error_p.append(e_p)
error_div.append(e_d)
print('Error in velocity ' + str(error_u[-1]))
print('Error in pressure ' + str(error_p[-1]))
print('Local mass error ' + str(error_div[-1]))
if comm.rank == 0:
iterator_list = [1. / float(nx) for nx in nx_list]
conv_u = compute_convergence(iterator_list, error_u)
conv_p = compute_convergence(iterator_list, error_p)
assert any(conv > k + 0.75 for conv in conv_u)
assert any(conv > (k - 1) + 0.75 for conv in conv_p)
def RunJob(Tb, mu_value, path):
runtimeInit = clock()
tfile = File(path + '/t6t.pvd')
mufile = File(path + "/mu.pvd")
ufile = File(path + '/velocity.pvd')
gradpfile = File(path + '/gradp.pvd')
pfile = File(path + '/pstar.pvd')
parameters = open(path + '/parameters', 'w', 0)
vmeltfile = File(path + '/vmelt.pvd')
rhofile = File(path + '/rhosolid.pvd')
for name in dir():
ev = str(eval(name))
if name[0] != '_' and ev[0] != '<':
parameters.write(name + ' = ' + ev + '\n')
temp_values = [27. + 273, Tb + 273, 1300. + 273, 1305. + 273]
dTemp = temp_values[3] - temp_values[0]
temp_values = [x / dTemp for x in temp_values] # non dimensionalising temp
mu_a = mu_value # this was taken from the blankenbach paper, can change..
Ep = b / dTemp
mu_bot = exp(-Ep * (temp_values[3] * dTemp - 1573) + cc) * mu_a
Ra = rho_0 * alpha * g * dTemp * h**3 / (kappa_0 * mu_a)
w0 = rho_0 * alpha * g * dTemp * h**2 / mu_a
tau = h / w0
p0 = mu_a * w0 / h
print(mu_a, mu_bot, Ra, w0, p0)
vslipx = 1.6e-09 / w0
vslip = Constant((vslipx, 0.0)) # nondimensional
noslip = Constant((0.0, 0.0))
dt = 3.E11 / tau
tEnd = 3.E13 / tau # non-dimensionalising times
class PeriodicBoundary(SubDomain):
def inside(self, x, on_boundary):
return left(x, on_boundary)
def map(self, x, y):
y[0] = x[0] - MeshWidth
y[1] = x[1]
pbc = PeriodicBoundary()
class TempExp(Expression):
def eval(self, value, x):
if x[1] >= LAB(x):
value[0] = temp_values[0] + (temp_values[1] - temp_values[0]) * (MeshHeight - x[1]) / (MeshHeight - LAB(x))
else:
value[0] = temp_values[3] - (temp_values[3] - temp_values[2]) * (x[1]) / (LAB(x))
class FluidTemp(Expression):
def eval(self, value, x):
if value[0] < 1295:
value[0] = 1295
mesh = RectangleMesh(Point(0.0, 0.0), Point(MeshWidth, MeshHeight), nx, ny)
Svel = VectorFunctionSpace(mesh, 'CG', 2, constrained_domain=pbc)
Spre = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Stemp = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Smu = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Sgradp = VectorFunctionSpace(mesh, 'CG', 2, constrained_domain=pbc)
Srho = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
S0 = MixedFunctionSpace([Svel, Spre, Stemp])
u = Function(S0)
v, p, T = split(u)
v_t, p_t, T_t = TestFunctions(S0)
T0 = interpolate(TempExp(), Stemp)
muExp = Expression('exp(-Ep * (T_val * dTemp - 1573) + cc * x[2] / meshHeight)', Smu.ufl_element(),
Ep=Ep, dTemp=dTemp, cc=cc, meshHeight=MeshHeight, T_val=T0)
mu = interpolate(muExp, Smu)
rhosolid = Function(Srho)
deltarho = Function(Srho)
v0 = Function(Svel)
vmelt = Function(Svel)
v_theta = (1. - theta)*v0 + theta*v
T_theta = (1. - theta)*T + theta*T0
r_v = (inner(sym(grad(v_t)), 2.*mu*sym(grad(v))) \
- div(v_t)*p \
- T*v_t[1] )*dx
r_p = p_t*div(v)*dx
r_T = (T_t*((T - T0) \
+ dt*inner(v_theta, grad(T_theta))) \
+ (dt/Ra)*inner(grad(T_t), grad(T_theta)) )*dx
# + k_s*(Tf-T_theta)*dt
Tf = T0.interpolate(FluidTemp())
# Tf = T0.interpolate(Expression('value[0] >= 1295.0 ? value[0] : 1295.0'))
# Tf.interpolate(Expression('value[0] >= 1295 ? value[0] : 1295'))
# project(Expression('value[0] >= 1295 ? value[0] : 1295'), Tf)
# Alex, a question for you:
# can you see if there is a way to set Tf = T in regions where T >=1295 celsius
#
# 1295 celsius is my arbitrary choice for the LAB isotherm. In regions
# where T < 1295 C, set Tf to be some constant for now, such as 1295 C.
# Once we do this, then we can add in a term like that last line above where
# it will only be non-zero when the solid temperature, T, is cooler than 1295
# can you do this? After this is done, we will then worry about a calculation
# where we solve for Tf as a function of time in the regions cooler than 1295 C
# Makes sense? If not, we can skype soon -- email me with questions
# 3/19/16
r = r_v + r_p + r_T
bcv0 = DirichletBC(S0.sub(0), noslip, top)
bcv1 = DirichletBC(S0.sub(0), vslip, bottom)
bcp0 = DirichletBC(S0.sub(1), Constant(0.0), bottom)
bct0 = DirichletBC(S0.sub(2), Constant(temp_values[0]), top)
bct1 = DirichletBC(S0.sub(2), Constant(temp_values[3]), bottom)
bcs = [bcv0, bcv1, bcp0, bct0, bct1]
t = 0
count = 0
while (t < tEnd):
solve(r == 0, u, bcs)
t += dt
nV, nP, nT = u.split()
gp = grad(nP)
rhosolid = rho_0 * (1 - alpha * (nT * dTemp - 1573))
deltarho = rhosolid - rhomelt
yvec = Constant((0.0, 1.0))
vmelt = nV * w0 - darcy * (gp * p0 / h - deltarho * yvec * g)
if (count % 100 == 0):
pfile << nP
ufile << nV
tfile << nT
mufile << mu
gradpfile << project(grad(nP), Sgradp)
mufile << project(mu * mu_a, Smu)
rhofile << project(rhosolid, Srho)
vmeltfile << project(vmelt, Svel)
count += 1
assign(T0, nT)
assign(v0, nV)
mu.interpolate(muExp)
print('Case mu=%g, Tb=%g complete.' % (mu_a, Tb), ' Run time =', clock() - runtimeInit, 's')
def setup_NS(w_NS, u, p, v, q, p0, q0, dx, ds, normal, dirichlet_bcs,
neumann_bcs, boundary_to_mark, u_1, phi_, rho_, rho_1, g_, M_,
mu_, rho_e_, c_, V_, c_1, V_1, dbeta, solutes, per_tau, drho,
sigma_bar, eps, dveps, grav, fric, u_comoving, enable_PF,
enable_EC, use_iterative_solvers, use_pressure_stabilization,
p_lagrange, q_rhs):
""" Set up the Navier-Stokes subproblem. """
# F = (
# per_tau * rho_ * df.dot(u - u_1, v)*dx
# + rho_*df.inner(df.grad(u), df.outer(u_1, v))*dx
# + 2*mu_*df.inner(df.sym(df.grad(u)), df.grad(v))*dx
# - p * df.div(v)*dx
# + df.div(u)*q*dx
# - df.dot(rho_*grav, v)*dx
# )
mom_1 = rho_1 * (u_1 + u_comoving)
if enable_PF:
mom_1 += -M_ * drho * df.nabla_grad(g_)
F = (per_tau * rho_1 * df.dot(u - u_1, v) * dx +
fric * mu_ * df.dot(u + u_comoving, v) * dx + 2 * mu_ *
df.inner(df.sym(df.nabla_grad(u)), df.sym(df.nabla_grad(v))) * dx -
p * df.div(v) * dx + q * df.div(u) * dx +
df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx + 0.5 *
(per_tau * (rho_ - rho_1) * df.dot(u, v) -
df.dot(mom_1, df.nabla_grad(df.dot(u, v)))) * dx -
rho_ * df.dot(grav, v) * dx -
mu_ * df.dot(df.nabla_grad(u) * normal, v) * df.ds)
for boundary_name, slip_length in neumann_bcs["u"].items():
F += 1./slip_length * \
df.dot(u, v) * ds(boundary_to_mark[boundary_name])
for boundary_name, pressure in neumann_bcs["p"].items():
F += pressure * df.dot(normal, v) * ds(boundary_to_mark[boundary_name])
# - 2*mu_*df.dot(df.dot(df.sym(df.nabla_grad(u)), v),
# normal)) * ds(boundary_to_mark[boundary_name])
if enable_PF:
F += phi_ * df.dot(df.nabla_grad(g_), v) * dx
if enable_EC:
for ci_, ci_1, dbetai, solute in zip(c_, c_1, dbeta, solutes):
zi = solute[1]
F += df.dot(df.grad(ci_), v)*dx \
+ zi*ci_1*df.dot(df.grad(V_), v)*dx
if enable_PF:
F += ci_ * dbetai * df.dot(df.grad(phi_), v) * dx
if p_lagrange:
F += (p * q0 + q * p0) * dx
if "u" in q_rhs:
F += -df.dot(q_rhs["u"], v) * dx
a, L = df.lhs(F), df.rhs(F)
problem = df.LinearVariationalProblem(a, L, w_NS, dirichlet_bcs)
solver = df.LinearVariationalSolver(problem)
if use_iterative_solvers and use_pressure_stabilization:
solver.parameters["linear_solver"] = "gmres"
#solver.parameters["preconditioner"] = "jacobi"
#solver.parameters["preconditioner"] = "ilu"
return solver
def RunJob(Tb, mu_value, path):
runtimeInit = clock()
tfile = File(path + '/t6t.pvd')
mufile = File(path + "/mu.pvd")
ufile = File(path + '/velocity.pvd')
gradpfile = File(path + '/gradp.pvd')
pfile = File(path + '/pstar.pvd')
parameters = open(path + '/parameters', 'w', 0)
vmeltfile = File(path + '/vmelt.pvd')
rhofile = File(path + '/rhosolid.pvd')
for name in dir():
ev = str(eval(name))
if name[0] != '_' and ev[0] != '<':
parameters.write(name + ' = ' + ev + '\n')
temp_values = [27. + 273, Tb + 273, 1300. + 273, 1305. + 273]
dTemp = temp_values[3] - temp_values[0]
temp_values = [x / dTemp for x in temp_values] # non dimensionalising temp
mu_a = mu_value # this was taken from the blankenbach paper, can change..
Ep = b / dTemp
mu_bot = exp(-Ep * (temp_values[3] * dTemp - 1573) + cc) * mu_a
Ra = rho_0 * alpha * g * dTemp * h**3 / (kappa_0 * mu_a)
w0 = rho_0 * alpha * g * dTemp * h**2 / mu_a
tau = h / w0
p0 = mu_a * w0 / h
print(mu_a, mu_bot, Ra, w0, p0)
vslipx = 1.6e-09 / w0
vslip = Constant((vslipx, 0.0)) # nondimensional
noslip = Constant((0.0, 0.0))
dt = 3.E11 / tau
tEnd = 3.E13 / tau # non-dimensionalising times
class PeriodicBoundary(SubDomain):
def inside(self, x, on_boundary):
return left(x, on_boundary)
def map(self, x, y):
y[0] = x[0] - MeshWidth
y[1] = x[1]
pbc = PeriodicBoundary()
class TempExp(Expression):
def eval(self, value, x):
if x[1] >= LAB(x):
value[0] = temp_values[0] + (temp_values[1] - temp_values[0]
) * (MeshHeight -
x[1]) / (MeshHeight - LAB(x))
else:
value[0] = temp_values[3] - (
temp_values[3] - temp_values[2]) * (x[1]) / (LAB(x))
class FluidTemp(Expression):
def eval(self, value, x):
if value[0] < 1295:
value[0] = 1295
mesh = RectangleMesh(Point(0.0, 0.0), Point(MeshWidth, MeshHeight), nx, ny)
Svel = VectorFunctionSpace(mesh, 'CG', 2, constrained_domain=pbc)
Spre = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Stemp = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Smu = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
Sgradp = VectorFunctionSpace(mesh, 'CG', 2, constrained_domain=pbc)
Srho = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
S0 = MixedFunctionSpace([Svel, Spre, Stemp])
u = Function(S0)
v, p, T = split(u)
v_t, p_t, T_t = TestFunctions(S0)
T0 = interpolate(TempExp(), Stemp)
muExp = Expression(
'exp(-Ep * (T_val * dTemp - 1573) + cc * x[2] / meshHeight)',
Smu.ufl_element(),
Ep=Ep,
dTemp=dTemp,
cc=cc,
meshHeight=MeshHeight,
T_val=T0)
mu = interpolate(muExp, Smu)
rhosolid = Function(Srho)
deltarho = Function(Srho)
v0 = Function(Svel)
vmelt = Function(Svel)
v_theta = (1. - theta) * v0 + theta * v
T_theta = (1. - theta) * T + theta * T0
r_v = (inner(sym(grad(v_t)), 2.*mu*sym(grad(v))) \
- div(v_t)*p \
- T*v_t[1] )*dx
r_p = p_t * div(v) * dx
r_T = (T_t*((T - T0) \
+ dt*inner(v_theta, grad(T_theta))) \
+ (dt/Ra)*inner(grad(T_t), grad(T_theta)) )*dx
# + k_s*(Tf-T_theta)*dt
Tf = T0.interpolate(FluidTemp())
# Tf = T0.interpolate(Expression('value[0] >= 1295.0 ? value[0] : 1295.0'))
# Tf.interpolate(Expression('value[0] >= 1295 ? value[0] : 1295'))
# project(Expression('value[0] >= 1295 ? value[0] : 1295'), Tf)
# Alex, a question for you:
# can you see if there is a way to set Tf = T in regions where T >=1295 celsius
#
# 1295 celsius is my arbitrary choice for the LAB isotherm. In regions
# where T < 1295 C, set Tf to be some constant for now, such as 1295 C.
# Once we do this, then we can add in a term like that last line above where
# it will only be non-zero when the solid temperature, T, is cooler than 1295
# can you do this? After this is done, we will then worry about a calculation
# where we solve for Tf as a function of time in the regions cooler than 1295 C
# Makes sense? If not, we can skype soon -- email me with questions
# 3/19/16
r = r_v + r_p + r_T
bcv0 = DirichletBC(S0.sub(0), noslip, top)
bcv1 = DirichletBC(S0.sub(0), vslip, bottom)
bcp0 = DirichletBC(S0.sub(1), Constant(0.0), bottom)
bct0 = DirichletBC(S0.sub(2), Constant(temp_values[0]), top)
bct1 = DirichletBC(S0.sub(2), Constant(temp_values[3]), bottom)
bcs = [bcv0, bcv1, bcp0, bct0, bct1]
t = 0
count = 0
while (t < tEnd):
solve(r == 0, u, bcs)
t += dt
nV, nP, nT = u.split()
gp = grad(nP)
rhosolid = rho_0 * (1 - alpha * (nT * dTemp - 1573))
deltarho = rhosolid - rhomelt
yvec = Constant((0.0, 1.0))
vmelt = nV * w0 - darcy * (gp * p0 / h - deltarho * yvec * g)
if (count % 100 == 0):
pfile << nP
ufile << nV
tfile << nT
mufile << mu
gradpfile << project(grad(nP), Sgradp)
mufile << project(mu * mu_a, Smu)
rhofile << project(rhosolid, Srho)
vmeltfile << project(vmelt, Svel)
count += 1
assign(T0, nT)
assign(v0, nV)
mu.interpolate(muExp)
print('Case mu=%g, Tb=%g complete.' % (mu_a, Tb), ' Run time =',
clock() - runtimeInit, 's')
def strain_displacement(u):
return dolfin.sym(dolfin.grad(u))
def get_residual_form(u,
v,
rho_e,
V_density,
tractionBC,
T,
iteration_number,
additive='strain',
k=8.,
method='RAMP'):
df.dx = df.dx(metadata={"quadrature_degree": 4})
# stiffness = rho_e/(1 + 8. * (1. - rho_e))
if method == 'SIMP':
stiffness = rho_e**3
else:
stiffness = rho_e / (1 + 8. * (1. - rho_e))
# print('the value of stiffness is:', rho_e.vector().get_local())
# Kinematics
d = len(u)
I = df.Identity(d) # Identity tensor
F = I + df.grad(u) # Deformation gradient
C = F.T * F # Right Cauchy-Green tensor
# Invariants of deformation tensors
Ic = df.tr(C)
J = df.det(F)
stiffen_pow = 1.
threshold_vol = 1.
eps_star = 0.05
# print("eps_star--------")
if additive == 'strain':
print("additive == strain")
if iteration_number == 1:
print('iteration_number == 1')
eps = df.sym(df.grad(u))
eps_dev = eps - 1 / 3 * df.tr(eps) * df.Identity(2)
eps_eq = df.sqrt(2.0 / 3.0 * df.inner(eps_dev, eps_dev))
# eps_eq_proj = df.project(eps_eq, density_function_space)
ratio = eps_eq / eps_star
ratio_proj = df.project(ratio, V_density)
c1_e = k * (5.e-2) / (1 + 8. * (1. - (5.e-2))) / 6
c2_e = df.Function(V_density)
c2_e.vector().set_local(5e-4 * np.ones(V_density.dim()))
fFile = df.HDF5File(df.MPI.comm_world, "c2_e_proj.h5", "w")
fFile.write(c2_e, "/f")
fFile.close()
fFile = df.HDF5File(df.MPI.comm_world, "ratio_proj.h5", "w")
fFile.write(ratio_proj, "/f")
fFile.close()
iteration_number += 1
E = k * stiffness
phi_add = (1 - stiffness) * ((c1_e * (Ic - 3)) + (c2_e *
(Ic - 3))**2)
else:
ratio_proj = df.Function(V_density)
fFile = df.HDF5File(df.MPI.comm_world, "ratio_proj.h5", "r")
fFile.read(ratio_proj, "/f")
fFile.close()
c2_e = df.Function(V_density)
fFile = df.HDF5File(df.MPI.comm_world, "c2_e_proj.h5", "r")
fFile.read(c2_e, "/f")
fFile.close()
c1_e = k * (5.e-2) / (1 + 8. * (1. - (5.e-2))) / 6
c2_e = df.conditional(df.le(ratio_proj, eps_star),
c2_e * df.sqrt(ratio_proj),
c2_e * (ratio_proj**3))
phi_add = (1 - stiffness) * ((c1_e * (Ic - 3)) + (c2_e *
(Ic - 3))**2)
E = k * stiffness
c2_e_proj = df.project(c2_e, V_density)
print('c2_e projected -------------')
eps = df.sym(df.grad(u))
eps_dev = eps - 1 / 3 * df.tr(eps) * df.Identity(2)
eps_eq = df.sqrt(2.0 / 3.0 * df.inner(eps_dev, eps_dev))
# eps_eq_proj = df.project(eps_eq, V_density)
ratio = eps_eq / eps_star
ratio_proj = df.project(ratio, V_density)
fFile = df.HDF5File(df.MPI.comm_world, "c2_e_proj.h5", "w")
fFile.write(c2_e_proj, "/f")
fFile.close()
fFile = df.HDF5File(df.MPI.comm_world, "ratio_proj.h5", "w")
fFile.write(ratio_proj, "/f")
fFile.close()
elif additive == 'vol':
print("additive == vol")
stiffness = stiffness / (df.det(F)**stiffen_pow)
# stiffness = df.conditional(df.le(df.det(F),threshold_vol), (stiffness/(df.det(F)/threshold_vol))**stiffen_pow, stiffness)
E = k * stiffness
elif additive == 'False':
print("additive == False")
E = k * stiffness # rho_e is the design variable, its values is from 0 to 1
nu = 0.4 # Poisson's ratio
lambda_ = E * nu / (1. + nu) / (1 - 2 * nu)
mu = E / 2 / (1 + nu) #lame's parameters
# Stored strain energy density (compressible neo-Hookean model)
psi = (mu / 2) * (Ic - 3) - mu * df.ln(J) + (lambda_ / 2) * (df.ln(J))**2
# print('the length of psi is:',len(psi.vector()))
if additive == 'strain':
psi += phi_add
B = df.Constant((0.0, 0.0))
# Total potential energy
'''The first term in this equation provided this error'''
Pi = psi * df.dx - df.dot(B, u) * df.dx - df.dot(T, u) * tractionBC
res = df.derivative(Pi, u, v)
return res
def differential_op(self, u):
return sym(nabla_grad(u))
## Next make a function space
V = d.VectorFunctionSpace(mesh, 'Lagrange', 1)
# Create a test function and a trial function, and a source term:
u = d.TrialFunction(V)
w = d.TestFunction(V)
b = d.Constant((1.0,0.,0.))
# Elasticity parameters:
E, nu = 10., 0.3
mu, lambda_param = E / (2. * (1.+nu)), E * nu / ((1. + nu) * (1.-2. * nu))
# Stress tensor:
# + usually of form \sigma_{ij} = \lambda e_{kk}\delta_{ij} + 2\mu e_{ij},
# + or = \lambda Tr(e_{ij})I + 2\mu e_{ij}, for e_{ij} the strain tensor.
sigma = lambda_param*d.tr(d.grad(u)) * d.Identity(w.cell().d) + 2 * mu * d.sym(d.grad(u))
# Governing balance equation:
F = d.inner(sigma, d.grad(w)) * d.dx - d.dot(b,w)*d.dx
# Extract the bi- and linear forms from F:
a = d.lhs(F)
L = d.rhs(F)
# Dirichlet BC on entire boundary:
c = d.Constant((0.,0.,0.))
bc = d.DirichletBC(V, c, d.DomainBoundary())
## Testing some new boundary definitions:
def bzo_boundary(r_vec, on_boundary):
def setup_NSu(w_NSu, u, v, dx, ds, normal, dirichlet_bcs_NSu, neumann_bcs,
boundary_to_mark, u_, u_1, p_, p_1, phi_, phi_1, rho_, rho_1, g_,
g_1, c_, c_1, M_, M_1, mu_, mu_1, rho_e_, rho_e_1, V_, dt, drho,
sigma_bar, eps, dveps, grav, dbeta, z, enable_PF, enable_EC,
use_iterative_solvers, **namespace):
""" Set up the Navier-Stokes velocity subproblem. """
mom_1 = rho_1 * u_1
if enable_PF:
mom_1 += -drho * M_1 * df.grad(g_1)
F_predict = (
1. / dt * rho_1 * df.dot(u - u_1, v) * dx +
df.inner(df.nabla_grad(u), df.outer(mom_1, v)) * dx + 2 * mu_ *
df.inner(df.sym(df.nabla_grad(u)), df.sym(df.nabla_grad(v))) * dx -
p_1 * df.div(v) * dx - rho_ * df.dot(grav, v) * dx + 0.5 *
(1. / dt *
(rho_ - rho_1) - df.inner(mom_1, df.grad(df.dot(u, v)))) * dx)
if enable_PF:
F_predict += phi_1 * df.dot(df.grad(g_), v) * dx
for boundary_name, pressure in neumann_bcs["p"].items():
F_predict += pressure * df.inner(normal, v) * ds(
boundary_to_mark[boundary_name])
if enable_EC:
for ci_, ci_1_ in zip(c_, c_1):
# F_predict += df.dot(df.nabla_grad(ci_), v) * dx
pass
#if enable_EC and rho_e_ != 0:
# F += rho_e_1 * df.dot(df.nabla_grad(V_), v) * dx
#if enable_PF and enable_EC:
# # Not clear how to discretize this term!
# # F += dveps * df.dot(df.grad(phi_), v)*df.dot(df.grad(V_),
# # df.grad(V_))*dx
# F_c = []
# for ci_, ci_1, zi, dbetai in zip(c_, c_1, z, dbeta):
# F_ci = ci_1*dbetai*df.dot(df.grad(phi_), v)*dx
# F_c.append(F_ci)
# F += sum(F_c)
solvers = dict()
a_predict, L_predict = df.lhs(F_predict), df.rhs(F_predict)
problem_predict = df.LinearVariationalProblem(a_predict, L_predict, w_NSu,
dirichlet_bcs_NSu)
solvers["predict"] = df.LinearVariationalSolver(problem_predict)
if use_iterative_solvers:
solvers["predict"].parameters["linear_solver"] = "bicgstab"
solvers["predict"].parameters["preconditioner"] = "jacobi" # "amg"
F_correct = (rho_ * df.inner(u - u_, v) * dx - dt *
(p_ - p_1) * df.div(v) * dx)
a_correct, L_correct = df.lhs(F_correct), df.rhs(F_correct)
problem_correct = df.LinearVariationalProblem(a_correct, L_correct, w_NSu,
dirichlet_bcs_NSu)
solvers["correct"] = df.LinearVariationalSolver(problem_correct)
if use_iterative_solvers:
solvers["correct"].parameters["linear_solver"] = "cg" # "bicgstab"
solvers["correct"].parameters["preconditioner"] = "jacobi" # "amg"
return solvers
def strain(v):
return dl.sym(dl.nabla_grad(v))
def epsilon(u):
# Define symmetric gradient
return sym(nabla_grad(u))
def setup_NSPFEC(w_NSPFEC, w_1NSPFEC,
dirichlet_bcs_NSPFEC,
neumann_bcs,
boundary_to_mark,
dx, ds, normal,
v, q, q0, psi, h, b, U,
u_, p_, p0_, phi_, g_, c_, V_,
u_1, p_1, p0_1, phi_1, g_1, c_1, V_1,
M_, nu_, veps_, rho_, K_, beta_, rho_e_,
dbeta, dveps, drho,
per_tau, sigma_bar, eps, grav, z,
solutes,
enable_NS, enable_PF, enable_EC,
use_iterative_solvers,
p_lagrange,
q_rhs):
""" The full problem of electrohydrodynamics in two phases.
Note that it is possible to turn off the different parts at will.
"""
# Setup of the Navier-Stokes part of F
mom_ = rho_*u_
if enable_PF:
mom_ += -M_*drho * df.nabla_grad(g_)
F = []
if enable_NS:
F_NS = (per_tau * rho_ * df.dot(u_ - u_1, v) * dx
+ df.inner(df.nabla_grad(u_), df.outer(mom_, v)) * dx
+ 2*nu_*df.inner(df.sym(df.nabla_grad(u_)),
df.sym(df.nabla_grad(v))) * dx
- p_ * df.div(v) * dx
- df.div(u_) * q * dx
- df.dot(rho_ * grav, v) * dx)
# if enable_PF:
# F_NS += - sigma_bar*eps*df.inner(
# df.outer(df.grad(phi_),
# df.grad(phi_)), df.grad(v)) * dx
# if enable_EC and rho_e_ != 0:
# F_NS += rho_e_*df.dot(df.grad(V_), v) * dx
# if enable_PF and enable_EC:
# F_NS += dveps*df.dot(
# df.grad(phi_), v)*df.dot(df.grad(V_),
# df.grad(V_)) * dx
if enable_PF:
F_NS += phi_*df.dot(df.grad(g_), v) * dx
if enable_EC:
for ci_, dbetai, solute in zip(c_, dbeta, solutes):
zi = solute[1]
F_NS += df.dot(df.grad(ci_), v) * dx \
+ ci_*dbetai*df.dot(df.grad(phi_), v) * dx \
+ zi*ci_*df.dot(df.grad(V_), v) * dx
# Slip boundary condition
for boundary_name, slip_length in neumann_bcs["u"].items():
F_NS += 1./slip_length * \
df.dot(u_, v) * ds(boundary_to_mark[boundary_name])
# Pressure boundary condition
for boundary_name, pressure in neumann_bcs["p"].items():
F_NS += pressure * df.inner(
normal, v) * ds(boundary_to_mark[boundary_name])
# Lagrange pressure
if p_lagrange:
F_NS += (p_*q0 + q*p0_)*dx
# RHS source terms
if "u" in q_rhs:
F_NS += -df.dot(q_rhs["u"], v)*dx
F.append(F_NS)
# Setup of the phase-field equations
if enable_PF:
phi_1_flt = unit_interval_filter(phi_1)
F_PF_phi = (per_tau*(phi_-phi_1_flt)*psi*df.dx +
M_*df.dot(df.grad(g_), df.grad(psi)) * dx)
if enable_NS:
F_PF_phi += df.dot(u_, df.grad(phi_)) * psi * dx
F_PF_g = (g_ * h * dx
- sigma_bar*eps*df.dot(df.grad(phi_), df.grad(h)) * dx
- sigma_bar/eps*diff_pf_potential(phi_) * h * dx)
if enable_EC:
F_PF_g += (-sum([dbeta_i * ci_ * h * dx
for dbeta_i, ci_ in zip(dbeta, c_)])
+ dveps * df.dot(df.grad(V_), df.grad(V_)) * h * dx)
# Contact angle boundary condtions
for boundary_name, costheta in neumann_bcs["phi"].items():
fw_prime = diff_pf_contact(phi_)
F_PF_g += sigma_bar * costheta * fw_prime * h * ds(
boundary_to_mark[boundary_name])
# RHS source terms
if "phi" in q_rhs:
F_PF_phi += -q_rhs["phi"]*psi*dx
F_PF = F_PF_phi + F_PF_g
F.append(F_PF)
# Setup of the electrochemistry
if enable_EC:
F_E_c = []
for ci_, ci_1, bi, Ki_, zi, dbetai, solute in zip(
c_, c_1, b, K_, z, dbeta, solutes):
ci_1_flt = max_value(ci_1, 0.)
F_E_ci = (per_tau*(ci_-ci_1_flt)*bi*df.dx
+ Ki_*df.dot(df.grad(ci_), df.grad(bi))*df.dx)
if zi != 0:
F_E_ci += Ki_*zi*ci_*df.dot(df.grad(V_),
df.grad(bi))*df.dx
if enable_NS:
F_E_ci += df.dot(u_, df.grad(ci_))*bi*df.dx
if enable_PF:
F_E_ci += Ki_*ci_*dbetai*df.dot(df.grad(phi_),
df.grad(bi)) * dx
if solute[0] in q_rhs:
F_E_ci += - q_rhs[solute[0]] * bi * dx
F_E_c.append(F_E_ci)
F_E_V = veps_*df.dot(df.grad(V_), df.grad(U))*df.dx
if rho_e_ != 0:
F_E_V += -rho_e_*U*df.dx
# Surface charge boundary condition
for boundary_name, sigma_e in neumann_bcs["V"].items():
F_E_V += -sigma_e*U*ds(boundary_to_mark[boundary_name])
# RHS source terms
if "V" in q_rhs:
F_E_V += q_rhs["V"]*U*dx
F_E = sum(F_E_c) + F_E_V
F.append(F_E)
F = sum(F)
J = df.derivative(F, w_NSPFEC)
problem_NSPFEC = df.NonlinearVariationalProblem(F, w_NSPFEC, dirichlet_bcs_NSPFEC, J)
solver_NSPFEC = df.NonlinearVariationalSolver(problem_NSPFEC)
if use_iterative_solvers:
solver_NSPFEC.parameters['newton_solver']['linear_solver'] = 'gmres'
solver_NSPFEC.parameters['newton_solver']['preconditioner'] = 'ilu'
return solver_NSPFEC
def epsilon(self, uu): return dolfin.sym(dolfin.grad(uu))
def les_setup(u_, mesh, assemble_matrix, CG1Function, nut_krylov_solver, bcs,
**NS_namespace):
"""
Set up for solving the Germano Dynamic LES model applying
Lagrangian Averaging.
"""
# Create function spaces
CG1 = FunctionSpace(mesh, "CG", 1)
p, q = TrialFunction(CG1), TestFunction(CG1)
dim = mesh.geometry().dim()
# Define delta and project delta**2 to CG1
delta = pow(CellVolume(mesh), 1. / dim)
delta_CG1_sq = project(delta, CG1)
delta_CG1_sq.vector().set_local(delta_CG1_sq.vector().array()**2)
delta_CG1_sq.vector().apply("insert")
# Define nut_
Sij = sym(grad(u_))
magS = sqrt(2 * inner(Sij, Sij))
Cs = Function(CG1)
nut_form = Cs**2 * delta**2 * magS
# Create nut_ BCs
ff = FacetFunction("size_t", mesh, 0)
bcs_nut = []
for i, bc in enumerate(bcs['u0']):
bc.apply(u_[0].vector()) # Need to initialize bc
m = bc.markers() # Get facet indices of boundary
ff.array()[m] = i + 1
bcs_nut.append(DirichletBC(CG1, Constant(0), ff, i + 1))
nut_ = CG1Function(nut_form,
mesh,
method=nut_krylov_solver,
bcs=bcs_nut,
bounded=True,
name="nut")
# Create functions for holding the different velocities
u_CG1 = as_vector([Function(CG1) for i in range(dim)])
u_filtered = as_vector([Function(CG1) for i in range(dim)])
dummy = Function(CG1)
ll = LagrangeInterpolator()
# Assemble required filter matrices and functions
G_under = Function(CG1, assemble(TestFunction(CG1) * dx))
G_under.vector().set_local(1. / G_under.vector().array())
G_under.vector().apply("insert")
G_matr = assemble(inner(p, q) * dx)
# Set up functions for Lij and Mij
Lij = [Function(CG1) for i in range(dim * dim)]
Mij = [Function(CG1) for i in range(dim * dim)]
# Check if case is 2D or 3D and set up uiuj product pairs and
# Sij forms, assemble required matrices
Sijcomps = [Function(CG1) for i in range(dim * dim)]
Sijfcomps = [Function(CG1) for i in range(dim * dim)]
# Assemble some required matrices for solving for rate of strain terms
Sijmats = [assemble_matrix(p.dx(i) * q * dx) for i in range(dim)]
if dim == 3:
tensdim = 6
uiuj_pairs = ((0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2))
else:
tensdim = 3
uiuj_pairs = ((0, 0), (0, 1), (1, 1))
# Set up Lagrange functions
JLM = Function(CG1)
JLM.vector()[:] += 1E-32
JMM = Function(CG1)
JMM.vector()[:] += 1
return dict(Sij=Sij,
nut_form=nut_form,
nut_=nut_,
delta=delta,
bcs_nut=bcs_nut,
delta_CG1_sq=delta_CG1_sq,
CG1=CG1,
Cs=Cs,
u_CG1=u_CG1,
u_filtered=u_filtered,
ll=ll,
Lij=Lij,
Mij=Mij,
Sijcomps=Sijcomps,
Sijfcomps=Sijfcomps,
Sijmats=Sijmats,
JLM=JLM,
JMM=JMM,
dim=dim,
tensdim=tensdim,
G_matr=G_matr,
G_under=G_under,
dummy=dummy,
uiuj_pairs=uiuj_pairs)
def epsilon(u):
# Define symmetric gradient
return dolfin.sym(dolfin.nabla_grad(u))
def strain(v):
return dl.sym(dl.nabla_grad(v))
def __ufl_forms(self, nu, f):
(w, q, wbar, qbar) = self.__test_functions()
(u, p, ubar, pbar) = self.__trial_functions()
# Infer geometric dimension
zero_vec = np.zeros(self.gdim)
ds = self.ds
n = self.n
he = self.he
alpha = self.alpha
h_d = self.h_d
beta_stab = self.beta_stab
facet_integral = self.facet_integral
pI = p * Identity(
self.mixedL.sub(1).ufl_cell().topological_dimension())
pbI = pbar * \
Identity(self.mixedL.sub(1).ufl_cell().topological_dimension())
# Upper left block
# Contribution comes from local momentum balance
AB = inner(2*nu*sym(grad(u)), grad(w))*dx \
+ facet_integral(dot(-2*nu*sym(grad(u))*n
+ (2*nu*alpha/he)*u, w)) \
+ facet_integral(dot(-2*nu*u, sym(grad(w))*n)) \
- inner(pI, grad(w))*dx
# Contribution comes from local mass balance
BtF = -dot(q, div(u))*dx - \
facet_integral(beta_stab*he/(nu+1)*dot(p, q))
A_S = AB + BtF
# Upper right block
# Contribution from local momentum
CD = facet_integral(-alpha/he*2*nu*inner(ubar, w)) \
+ facet_integral(2*nu*inner(ubar, sym(grad(w))*n)) \
+ facet_integral(dot(pbI*n, w))
H = facet_integral(beta_stab * he / (nu + 1) * dot(pbar, q))
G_S = CD + H
# Transpose block
CDT = facet_integral(- alpha/he*2*nu*inner(wbar, u)) \
+ facet_integral(2*nu*inner(wbar, sym(grad(u))*n)) \
+ facet_integral(qbar * dot(u, n))
HT = facet_integral(beta_stab * he / (nu + 1) * dot(p, qbar))
G_ST = CDT + HT
# Lower right block, penalty on ds(98) approximates free-slip
KL = facet_integral(alpha/he * 2 * nu*dot(ubar, wbar)) \
- facet_integral(dot(pbar*n, wbar)) \
+ Constant(1E12)/he * inner(outer(ubar, wbar), outer(n, n)) * ds(98)
LtP = - facet_integral(dot(ubar, n)*qbar) \
- facet_integral(beta_stab*he/(nu+1) * pbar * qbar)
B_S = KL + LtP
# Righthandside
Q_S = dot(f, w) * dx
S_S = facet_integral(dot(Constant(zero_vec), wbar))
S_S += dot(h_d[0], wbar) * ds(99) + dot(h_d[1], wbar) * ds(
100) #Neumann BC
return {
'A_S': A_S,
'G_S': G_S,
'G_ST': G_ST,
'B_S': B_S,
'Q_S': Q_S,
'S_S': S_S
}
