def weak_residual(self, x, x_test, m, e):
"""
The weak residual
"""
u, p, gamma = dl.split(x)
u_test, p_test, g_test = dl.split(x_test)
hbinv = hinv_u(self.metric, self.tg)
res_u = dl.Constant(2.)*(self.nu+self.nu_t(x, m))*dl.inner( self.strain(u), self.strain(u_test))*dl.dx \
+ dl.inner(dl.grad(u)*u, u_test)*dl.dx \
- p*dl.div(u_test)*dl.dx \
+ self.Cd*(self.nu+self.nu_t(x, m))*hbinv*dl.dot(u - self.u_ff, self.tg)*dl.dot(u_test, self.tg)*self.ds_ff \
- dl.dot( self.sigma_n(self.nu+self.nu_t(x, m), u), self.tg) * dl.dot(u_test, self.tg)*self.ds_ff \
- dl.dot( self.sigma_n(self.nu+self.nu_t(x, m), u_test), self.tg ) * dl.dot(u - self.u_ff, self.tg)*self.ds_ff
res_p = dl.div(u) * p_test * dl.dx
D = self.nu_g(x, m, e)
h_o_u = h_over_u(self.metric, u, D)
res_g = dl.inner( D*dl.grad(gamma), dl.grad(g_test) )*dl.dx \
+ h_o_u*dl.dot(u, dl.grad(gamma))*dl.dot(u, dl.grad(g_test))*dl.dx \
+ dl.dot(u, dl.grad(gamma))*g_test*dl.dx \
- dl.Constant(.5)*gamma*g_test*(dl.dot(u, self.e1)/(self.xfun + self._offset(m)))*dl.dx
return res_u + res_p + res_g
def computeVelocityField(mesh):
Xh = dl.VectorFunctionSpace(mesh, 'Lagrange', 2)
Wh = dl.FunctionSpace(mesh, 'Lagrange', 1)
XW = dl.MixedFunctionSpace([Xh, Wh])
Re = 1e2
g = dl.Expression(('0.0', '(x[0] < 1e-14) - (x[0] > 1 - 1e-14)'))
bc1 = dl.DirichletBC(XW.sub(0), g, v_boundary)
bc2 = dl.DirichletBC(XW.sub(1), dl.Constant(0), q_boundary, 'pointwise')
bcs = [bc1, bc2]
vq = dl.Function(XW)
(v, q) = dl.split(vq)
(v_test, q_test) = dl.TestFunctions(XW)
def strain(v):
return dl.sym(dl.nabla_grad(v))
F = ((2. / Re) * dl.inner(strain(v), strain(v_test)) +
dl.inner(dl.nabla_grad(v) * v, v_test) - (q * dl.div(v_test)) +
(dl.div(v) * q_test)) * dl.dx
dl.solve(F == 0,
vq,
bcs,
solver_parameters={
"newton_solver": {
"relative_tolerance": 1e-4,
"maximum_iterations": 100
}
})
return v
def computeVelocityField(mesh):
Xh = dl.VectorFunctionSpace(mesh,'Lagrange', 2)
Wh = dl.FunctionSpace(mesh, 'Lagrange', 1)
if dlversion() <= (1,6,0):
XW = dl.MixedFunctionSpace([Xh, Wh])
else:
mixed_element = dl.MixedElement([Xh.ufl_element(), Wh.ufl_element()])
XW = dl.FunctionSpace(mesh, mixed_element)
Re = 1e2
g = dl.Expression(('0.0','(x[0] < 1e-14) - (x[0] > 1 - 1e-14)'), element=Xh.ufl_element())
bc1 = dl.DirichletBC(XW.sub(0), g, v_boundary)
bc2 = dl.DirichletBC(XW.sub(1), dl.Constant(0), q_boundary, 'pointwise')
bcs = [bc1, bc2]
vq = dl.Function(XW)
(v,q) = dl.split(vq)
(v_test, q_test) = dl.TestFunctions (XW)
def strain(v):
return dl.sym(dl.nabla_grad(v))
F = ( (2./Re)*dl.inner(strain(v),strain(v_test))+ dl.inner (dl.nabla_grad(v)*v, v_test)
- (q * dl.div(v_test)) + ( dl.div(v) * q_test) ) * dl.dx
dl.solve(F == 0, vq, bcs, solver_parameters={"newton_solver":
{"relative_tolerance":1e-4, "maximum_iterations":100,
"linear_solver":"default"}})
return v
def stokes(self):
P2 = VectorElement("CG", self.mesh.ufl_cell(), 2)
P1 = FiniteElement("CG", self.mesh.ufl_cell(), 1)
TH = P2 * P1
VQ = FunctionSpace(self.mesh, TH)
mf = self.mf
self.no_slip = Constant((0., 0))
self.topflow = Expression(("-x[0] * (x[0] - 1.0) * 6.0 * m", "0.0"),
m=self.U_m,
degree=2)
bc0 = DirichletBC(VQ.sub(0), self.topflow, mf, self.bc_dict["top"])
bc1 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["left"])
bc2 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["bottom"])
bc3 = DirichletBC(VQ.sub(0), self.no_slip, mf, self.bc_dict["right"])
# bc4 = DirichletBC(VQ.sub(1), Constant(0), mf, self.bc_dict["top"])
bcs = [bc0, bc1, bc2, bc3]
vup = TestFunction(VQ)
up = TrialFunction(VQ)
# the solution will be in here:
up_ = Function(VQ)
u, p = split(up) # Trial
vu, vp = split(vup) # Test
u_, p_ = split(up_) # Function holding the solution
F = self.mu*inner(grad(vu), grad(u))*dx - inner(div(vu), p)*dx \
- inner(vp, div(u))*dx + dot(self.g*self.rho, vu)*dx
solve(lhs(F) == rhs(F), up_, bcs=bcs)
self.u_.assign(project(u_, self.V))
self.p_.assign(project(p_, self.Q))
return
def __init__(self, parameters, domain):
rho = Constant(parameters["density [kg/m3]"])
mu = Constant(parameters["viscosity [Pa*s]"])
dt = Constant(parameters["dt [s]"])
u, u_1, u_k, vu = domain.u, domain.u_1, domain.u_k, domain.vu
p_1 = domain.p_1
n = FacetNormal(domain.mesh)
acceleration = rho*inner((u-u_1)/dt, vu) * dx
convection = dot(div(rho*outer(u_k, u)), vu) * dx
convection = rho*dot(dot(u_k, nabla_grad(u)), vu) * dx
pressure = (inner(p_1, div(vu))*dx - dot(p_1*n, vu)*ds)
diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx) # good
# diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx
# + dot(mu * (grad(u) + grad(u).T)*n, vu)*ds) # very slow!
# F_impl = acceleration + convection + pressure + diffusion
# dot(u_1, nabla_grad(u_1)) works
# dot(u, nabla_grad(u_1)) does not change!
u_mid = (u + u_1) / 2.0
F_impl = rho*dot((u - u_1) / dt, vu)*dx \
+ rho*dot(dot(u_1, nabla_grad(u_1)), vu)*dx \
+ inner(sigma(u_mid, p_1, mu), epsilon(vu))*dx \
+ dot(p_1*n, vu)*ds - dot(mu*nabla_grad(u_mid)*n, vu)*ds
self.a, self.L = lhs(F_impl), rhs(F_impl)
self.domain = domain
self.A = assemble(self.a)
[bc.apply(self.A) for bc in domain.bcu]
return
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, mu_f, rho_f, k, dt, v_deg,
theta, **semimp_namespace):
J_theta = theta * J_(d_["n"]) + (1 - theta) * J_(d_["n-1"])
F_fluid_linear = rho_f / k * inner(J_theta *
(v_["n"] - v_["n-1"]), psi) * dx_f
F_fluid_nonlinear = rho_f*inner((Constant(theta)*J_(d_["n"])*grad(v_["n"])*inv(F_(d_["n"])) + \
Constant(1 - theta)*J_(d_["n-1"])*grad(v_["n-1"])*inv(F_(d_["n-1"]))) * \
(0.5*(3*v_["n-1"] - v_["n-2"]) - \
0.5/k*((3*d_["n-1"] - d_["n-2"]) - (3*d_["n-2"] - d_["n-3"]))), psi)*dx_f
# F_fluid_nonlinear = rho_f*inner((Constant(theta)*J_(d_["n"])*grad(v_["n"])*inv(F_(d_["n"])) + \
# Constant(1 - theta)*J_(d_["n-1"])*grad(v_["n-1"])*inv(F_(d_["n-1"]))) * \
# (0.5*(3*v_["n-1"] - v_["n-2"]) -\
# (d_["n"]-d_["n-1"])/k), psi)*dx_f
F_fluid_nonlinear += inner(
J_(d_["n"]) * sigma_f_p(p_["n"], d_["n"]) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_nonlinear += Constant(theta) * inner(
J_(d_["n"]) * sigma_f_u(v_["n"], d_["n"], mu_f) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_nonlinear += Constant(1 - theta) * inner(
J_(d_["n-1"]) * sigma_f_u(v_["n-1"], d_["n-1"], mu_f) *
inv(F_(d_["n-1"])).T, grad(psi)) * dx_f
F_fluid_nonlinear += Constant(theta) * inner(
div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]), gamma) * dx_f
F_fluid_nonlinear += Constant(1 - theta) * inner(
div(J_(d_["n-1"]) * inv(F_(d_["n-1"])) * v_["n-1"]), gamma) * dx_f
#F_fluid_nonlinear +=inner(div(J_(d_["n"])*inv(F_(d_["n"]))*v_["n"]), gamma)*dx_f
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def _generate_function_spaces(regularity, dim):
"""
Return a list of lambda functions, that, given a mesh and a
degree, defines a conforming finite element spaces for the given
space name
"""
if dim == 1:
vFunctionSpace = FunctionSpace
else:
vFunctionSpace = VectorFunctionSpace
spaces = {"h1": (lambda u, v: (dot(u,v) + inner(grad(u),grad(v)))*dx,
[lambda mesh, deg: vFunctionSpace(mesh, "CG", deg)]),
"hdiv": (lambda u, v: (dot(u, v) + div(u)*div(v))*dx,
[lambda mesh, deg: vFunctionSpace(mesh, "CG", deg),
lambda mesh, deg: FunctionSpace(mesh, "RT", deg),
lambda mesh, deg: FunctionSpace(mesh, "BDM", deg)]),
"hcurl": (lambda u, v: (dot(u, v) + inner(curl(u), curl(v)))*dx,
[lambda mesh, deg: vFunctionSpace(mesh, "CG", deg),
lambda mesh, deg: FunctionSpace(mesh, "N1curl", deg)]),
"l2": (lambda p, q: dot(p, q)*dx,
[lambda mesh, deg: vFunctionSpace(mesh, "CG", deg),
lambda mesh, deg: vFunctionSpace(mesh, "DG", deg)])
}
return spaces[regularity.lower()]
def main():
fsr = FunctionSubspaceRegistry()
deg = 2
mesh = dolfin.UnitSquareMesh(100, 3)
muc = mesh.ufl_cell()
el_w = dolfin.FiniteElement('DG', muc, deg - 1)
el_j = dolfin.FiniteElement('BDM', muc, deg)
el_DG0 = dolfin.FiniteElement('DG', muc, 0)
el = dolfin.MixedElement([el_w, el_j])
space = dolfin.FunctionSpace(mesh, el)
DG0 = dolfin.FunctionSpace(mesh, el_DG0)
fsr.register(space)
facet_normal = dolfin.FacetNormal(mesh)
xyz = dolfin.SpatialCoordinate(mesh)
trial = dolfin.Function(space)
test = dolfin.TestFunction(space)
w, c = dolfin.split(trial)
v, phi = dolfin.split(test)
sympy_exprs = derive_exprs()
exprs = {
k: sympy_dolfin_printer.to_ufl(sympy_exprs['R'], mesh, v)
for k, v in sympy_exprs['quantities'].items()
}
f = exprs['f']
w0 = dolfin.project(dolfin.conditional(dolfin.gt(xyz[0], 0.5), 1.0, 0.3),
DG0)
w_BC = exprs['w']
dx = dolfin.dx()
form = (+v * dolfin.div(c) * dx - v * f * dx +
dolfin.exp(w + w0) * dolfin.dot(phi, c) * dx +
dolfin.div(phi) * w * dx -
(w_BC - w0) * dolfin.dot(phi, facet_normal) * dolfin.ds() -
(w0('-') - w0('+')) * dolfin.dot(phi('+'), facet_normal('+')) *
dolfin.dS())
solver = NewtonSolver(form,
trial, [],
parameters=dict(relaxation_parameter=1.0,
maximum_iterations=15,
extra_iterations=10,
relative_tolerance=1e-6,
absolute_tolerance=1e-7))
solver.solve()
with closing(XdmfPlot("out/qflop_test.xdmf", fsr)) as X:
CG1 = dolfin.FunctionSpace(mesh, dolfin.FiniteElement('CG', muc, 1))
X.add('w0', 1, w0, CG1)
X.add('w_c', 1, w + w0, CG1)
X.add('w_e', 1, exprs['w'], CG1)
X.add('f', 1, f, CG1)
X.add('cx_c', 1, c[0], CG1)
X.add('cx_e', 1, exprs['c'][0], CG1)
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 _test_eigen_solver_sparse(callback_type):
from rbnics.backends.dolfin import EigenSolver
# Define mesh
mesh = UnitSquareMesh(10, 10)
# Define function space
V_element = VectorElement("Lagrange", mesh.ufl_cell(), 2)
Q_element = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
W_element = MixedElement(V_element, Q_element)
W = FunctionSpace(mesh, W_element)
# Create boundaries
class Wall(SubDomain):
def inside(self, x, on_boundary):
return on_boundary and (x[1] < 0 + DOLFIN_EPS
or x[1] > 1 - DOLFIN_EPS)
boundaries = MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
wall = Wall()
wall.mark(boundaries, 1)
# Define variational problem
vq = TestFunction(W)
(v, q) = split(vq)
up = TrialFunction(W)
(u, p) = split(up)
lhs = inner(grad(u), grad(v)) * dx - div(v) * p * dx - div(u) * q * dx
rhs = -inner(p, q) * dx
# Define boundary condition
bc = [DirichletBC(W.sub(0), Constant((0., 0.)), boundaries, 1)]
# Define eigensolver depending on callback type
assert callback_type in ("form callbacks", "tensor callbacks")
if callback_type == "form callbacks":
solver = EigenSolver(W, lhs, rhs, bc)
elif callback_type == "tensor callbacks":
LHS = assemble(lhs)
RHS = assemble(rhs)
solver = EigenSolver(W, LHS, RHS, bc)
# Solve the eigenproblem
solver.set_parameters({
"linear_solver": "mumps",
"problem_type": "gen_non_hermitian",
"spectrum": "target real",
"spectral_transform": "shift-and-invert",
"spectral_shift": 1.e-5
})
solver.solve(1)
r, c = solver.get_eigenvalue(0)
assert abs(c) < 1.e-10
assert r > 0., "r = " + str(r) + " is not positive"
print("Sparse inf-sup constant: ", sqrt(r))
return (sqrt(r), solver.condensed_A, solver.condensed_B)
def flux_derivative(self, u, coeff):
"""First derivative of flux."""
lmbda, mu = coeff
Du = self.differential_op(u)
I = Identity(u.cell().d)
Dsigma = 2.0 * mu * div(Du) + dot(nabla_grad(lmbda), tr(Du) * I)
if element_degree(u) >= 2:
Dsigma += lmbda * div(tr(Du) * I)
return Dsigma
def solve(self):
"""
Solve Stokes problem with current mesh
"""
(u, p) = TrialFunctions(self.VQ)
(v, q) = TestFunctions(self.VQ)
a = inner(grad(u), grad(v))*dx - div(u)*q*dx - div(v)*p*dx
l = inner(self.f, v)*dx
solve(a==l, self.w, bcs=self.bcs)
self.u, self.p = self.w.split()
def strong_residual(self, x, m):
"""
The strong residual
"""
u, p = dl.split(x)
res_u = -dl.div(
dl.Constant(2.) * (self.nu + self.nu_t(m)) *
self.strain(u)) + dl.grad(u) * u + dl.grad(p)
res_p = dl.div(u)
return [res_u, res_p]
def get_dwr_est(self, solns, adj_coarse, adj_fine):
times_coarse = list(adj_coarse.keys())
times_coarse.sort()
times_fine = list(adj_fine.keys())
times_fine.sort()
ntimesteps = len(times_coarse) - 1
est = np.zeros(ntimesteps)
u0 = solns[times_coarse[0]]
div0 = self.heat_cond * dol.div(dol.grad(u0))
adv0 = self.direction * self.velo * dol.dot(dol.grad(u0),
self.velocity)
zc0 = adj_coarse[times_coarse[0]]
f0 = dol.Function(self.V)
self.f1.t = times_coarse[0]
f0.interpolate(self.f1)
f0 = f0 * self.src_weight
zf0 = adj_fine[times_fine[0]]
for t in range(ntimesteps):
u1 = solns[times_coarse[t + 1]]
div1 = self.heat_cond * dol.div(dol.grad(u1))
adv1 = self.direction * self.velo * dol.dot(
dol.grad(u1), self.velocity)
f1 = dol.Function(self.V)
self.f1.t = times_coarse[t + 1]
f1.interpolate(self.f1)
f1 = f1 * self.src_weight
zc1 = adj_coarse[times_coarse[t + 1]]
zf1 = adj_fine[times_fine[2 * t + 1]]
zf2 = adj_fine[times_fine[2 * t + 2]]
dt = dol.Constant(times_coarse[t + 1] - times_coarse[t])
ut = (u1 - u0) / dt
res_left = ut - div0 + adv0 - f0
res_right = ut - div1 + adv1 - f1
res_mid = (res_left + res_right) / 2
zz_left = zf0 - zc0
zz_right = zf2 - zc1
zz_mid = zf1 - (zc0 + zc1) / 2
part_left = dol.assemble(res_left * zz_left * dol.dx)
part_right = dol.assemble(res_right * zz_right * dol.dx)
part_mid = dol.assemble(res_mid * zz_mid * dol.dx)
est[t] = abs(dt / 4 * (part_left + 2 * part_mid + part_right))
u0, div0, zc0, f0, zf0 = u1, div1, zc1, f1, zf2
return est
def fluid_setup(v_, p_, n, psi, gamma, dx, mu_f, rho_f, k, dt, theta,
**semimp_namespace):
F_fluid_linear = rho_f/k*inner(v_["n"] - v_["n-1"], psi)*dx \
+ Constant(theta)*inner(sigma_f(p_["n"], v_["n"], mu_f), grad(psi))*dx \
+ Constant(1 - theta)*inner(sigma_f(p_["n-1"], v_["n-1"], mu_f), grad(psi))*dx \
+ Constant(theta)*inner(div(v_["n"]), gamma)*dx \
+ Constant(1 -theta)*inner(div(v_["n-1"]), gamma)*dx
F_fluid_nonlinear = Constant(theta)*rho_f*inner(dot(grad(v_["n"]), v_["n"]), psi)*dx \
+ Constant(1 - theta)*rho_f*inner(dot(grad(v_["n-1"]), v_["n-1"]), psi)*dx
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def test_is_zero_with_nabla():
mesh = UnitSquareMesh(4, 4)
V1 = FunctionSpace(mesh, 'CG', 1)
V2 = VectorFunctionSpace(mesh, 'CG', 1)
v1 = TestFunction(V1)
v2 = TrialFunction(V2)
n = Constant([1, 1])
nn = dolfin.outer(n, n)
vv = dolfin.outer(v2, v2)
check_is_zero(dot(n, grad(v1)), 1)
check_is_zero(dolfin.div(v2), 1)
check_is_zero(dot(dolfin.div(nn), n), 0)
check_is_zero(dot(dolfin.div(vv), n), 1)
check_is_zero(dolfin.inner(nn, grad(v2)), 1)
def integrateFluidStress(v, p, nu, space_dim):
if space_dim == 2:
cd_integral = -20. * (nu * v_ref * (1. / l_ref**2) * inner(
dlfn.nabla_grad(v), dlfn.nabla_grad(vd)) + v_ref**2 *
(1. / l_ref) * inner(grad(v) * v, vd) - p_ref *
(1. / l_ref) * p * dlfn.div(vd)) * dV
cd = l_ref**2 * dlfn.assemble(cd_integral)
cl_integral = -20. * (nu * v_ref * (1. / l_ref**2) * inner(
dlfn.nabla_grad(v), dlfn.nabla_grad(vl)) + v_ref**2 *
(1. / l_ref) * inner(grad(v) * v, vl) - p_ref *
(1. / l_ref) * p * dlfn.div(vl)) * dV
cl = l_ref**2 * dlfn.assemble(cl_integral)
else:
raise ValueError("space dimension unequal 2")
return cd, cl
def delta_dg(mesh, expr):
CG = df.FunctionSpace(mesh, "CG", 1)
DG = df.FunctionSpace(mesh, "DG", 0)
CG3 = df.VectorFunctionSpace(mesh, "CG", 1)
DG3 = df.VectorFunctionSpace(mesh, "DG", 0)
m = df.interpolate(expr, DG3)
n = df.FacetNormal(mesh)
u_dg = df.TrialFunction(DG)
v_dg = df.TestFunction(DG)
u3 = df.TrialFunction(CG3)
v3 = df.TestFunction(CG3)
a = u_dg * df.inner(v3, n) * df.ds - u_dg * df.div(v3) * df.dx
mm = m.vector().array()
mm.shape = (3, -1)
K = df.assemble(a).array()
L3 = df.assemble(df.dot(v3, df.Constant([1, 1, 1])) * df.dx).array()
f = np.dot(K, mm[0]) / L3
fun1 = df.Function(CG3)
fun1.vector().set_local(f)
df.plot(fun1)
a = df.div(u3) * v_dg * df.dx
A = df.assemble(a).array()
L = df.assemble(v_dg * df.dx).array()
h = np.dot(A, f) / L
fun = df.Function(DG)
fun.vector().set_local(h)
df.plot(fun)
res = []
for x in xs:
res.append(fun(x, 5, 0.5))
"""
fun2 =df.interpolate(fun, df.VectorFunctionSpace(mesh, "CG", 1))
file = df.File('field2.pvd')
file << fun2
"""
return res
def compute_velocity(mesh, Vh, a, u):
#export the velocity field v = - exp( a ) \grad u: then we solve ( exp(-a) v, w) = ( u, div w)
Vv = dl.FunctionSpace(mesh, 'RT', 1)
v = dl.Function(Vv, name="velocity")
vtrial = dl.TrialFunction(Vv)
vtest = dl.TestFunction(Vv)
afun = dl.Function(Vh[PARAMETER], a)
ufun = dl.Function(Vh[STATE], u)
Mv = dl.exp(-afun) * dl.inner(vtrial, vtest) * dl.dx
n = dl.FacetNormal(mesh)
class TopBoundary(dl.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and x[1] > 1 - dl.DOLFIN_EPS
Gamma_M = TopBoundary()
boundaries = dl.FacetFunction("size_t", mesh)
boundaries.set_all(0)
Gamma_M.mark(boundaries, 1)
dss = dl.Measure("ds")[boundaries]
rhs = ufun * dl.div(vtest) * dl.dx - dl.dot(vtest, n) * dss(1)
bcv = dl.DirichletBC(Vv, dl.Expression(("0.0", "0.0")), v_boundary)
dl.solve(Mv == rhs, v, bcv)
return v
def Coupled_setup(d_, v_, p_, psi_v, psi_p, psi_d, dS2, n, mu_f, body_force,
dx_s, dx_f, mu_s, rho_s, lamda_s, k, mesh_file, rho_f,
**semimp_namespace):
F_coupled_linear = rho_s / k**2 * inner(
d_["n"] - 2 * d_["n-1"] + d_["n-2"], psi_d) * dx_s
F_coupled_linear -= rho_s * inner(body_force, psi_d) * dx_s
F_coupled_linear = rho_f / k * J_(d_["tilde"]) * inner(
v_["n"] - v_["tilde"], psi_v) * dx_f
F_coupled_linear += J_(d_["tilde"]) * inner(
inv(F_(d_["tilde"])).T * grad(p_["n"]), psi_v) * dx_f
F_coupled_linear += inner(
div(J_(d_["tilde"]) * inv(F_(d_["tilde"])) * v_["n"]), psi_p) * dx_f
# Non-linear part
F_coupled_nonlinear = Constant(0.5) * inner(Piola1(d_["n"], lamda_s, mu_s), grad(psi_d))*dx_s \
+ Constant(0.5) * inner(Piola1(d_["n-2"], lamda_s, mu_s), grad(psi_d))*dx_s
# Impose BC weakly
F_coupled_nonlinear -= inner(J_(d_["tilde"]("+"))*sigma_f(p_["n"]("+"), v_["tilde"]("+"),
d_["tilde"]("+"), mu_f) \
* inv(F_(d_["tilde"]("+"))).T*n("+"), psi_d("+"))*dS2(5)
# TODO: This might be better strongly...
F_coupled_linear += J_(d_["tilde"]("+")) * inner(
dot(v_["n"]("+") - -(d_["n"]("-") - d_["n-1"]("-")) / k, n("+")),
psi_p("+")) * dS2(5)
return dict(F_coupled_linear=F_coupled_linear,
F_coupled_nonlinear=F_coupled_nonlinear)
def __init__(self, domain):
rho, mu, dt, g = domain.rho, domain.mu, domain.dt, domain.g
u, u_1, vu = domain.u, domain.u_1, domain.vu
p_1 = domain.p_1
n = FacetNormal(domain.mesh)
acceleration = rho * inner((u - u_1) / dt, vu) * dx
pressure = inner(p_1, div(vu)) * dx - dot(p_1 * n, vu) * ds
body_force = dot(g*rho, vu)*dx \
+ dot(Constant((0.0, 0.0)), vu) * ds
# diffusion = (-inner(mu * (grad(u_1) + grad(u_1).T), grad(vu))*dx
# + dot(mu * (grad(u_1) + grad(u_1).T)*n, vu)*ds) # just fine
# diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu))*dx) # just fine
# just fine, but horribly slow in combination with ??? -> not reproducable
diffusion = (-inner(mu * (grad(u) + grad(u).T), grad(vu)) * dx +
dot(mu * (grad(u) + grad(u).T) * n, vu) * ds)
# convection = rho*dot(dot(u, nabla_grad(u_1)), vu) * dx # no vortices
convection = rho * dot(dot(u_1, nabla_grad(u)), vu) * dx
# stabilization = -gamma*psi_p*p_1
# convection = dot(div(rho * outer(u_1, u_1)), vu) * dx # not stable!
# convection = rho * dot(dot(u_1, nabla_grad(u_1)), vu) * dx # just fine
F_impl = -acceleration - convection + diffusion + pressure + body_force
self.a, self.L = lhs(F_impl), rhs(F_impl)
self.domain = domain
self.A = assemble(self.a)
[bc.apply(self.A) for bc in domain.bcu]
return
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, dS, mu_f, rho_f, k, v_deg,
**semimp_namespace):
F_fluid_linear = (rho_f / k) * inner(
J_(d_["n"]) * (v_["n"] - v_["n-1"]), psi) * dx_f
F_fluid_linear -= inner(div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]),
gamma) * dx_f
F_fluid_linear += inner(
J_(d_["n"]) * sigma_f_new(v_["n"], p_["n"], d_["n"], mu_f) *
inv(F_(d_["n"])).T, grad(psi)) * dx_f
F_fluid_linear -= inner(dot(J_(d_["n"]("-"))*\
sigma_f_new(v_["n"]("-"), p_["n"]("-"), d_["n"]("-"), mu_f)*inv(F_(d_["n"]("-"))).T, n("+")) , psi("-") )*dS(5)
F_fluid_nonlinear = rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) *
(v_["n"] - ((d_["n"] - d_["n-1"]) / k)), psi) * dx_f
if v_deg == 1:
F_fluid -= beta * h * h * inner(
J_(d_["n"]) * inv(F_(d_["n"]).T) * grad(p), grad(gamma)) * dx_f
print "v_deg", v_deg
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f1 = ParametrizedExpression(
mock_problem,
"1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)",
mu=(-1., -1.),
element=V.sub(1).ufl_element())
#
folder_prefix = os.path.join("test_eim_approximation_08_tempdir",
expression_type, basis_generation)
assert expression_type in ("Vector", "Matrix")
if expression_type == "Vector":
q = TestFunction(V.sub(1).collapse())
form = f1 * q * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
up = TrialFunction(V)
q = TestFunction(V.sub(1).collapse())
(u, p) = split(up)
form = f1 * q * div(u) * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
def divergence_matrix(mesh):
CR = VectorFunctionSpace(mesh, 'CR', 1)
DG = FunctionSpace(mesh, 'DG', 0)
A = cg1_cr_interpolation_matrix(mesh)
M = assemble(dot(div(TrialFunction(CR)), TestFunction(DG))*dx())
C = compiled_cr_module.cr_divergence_matrix(M, A, DG, CR)
return C
def tail(self, t, it, logger):
cc = self.cc
pv = self.pv
# Get div(v) locally
div_v = self.div_v
if div_v is not None:
div_v.assign(
df.project(df.div(pv["v"]), div_v.function_space()))
# Compute required functionals
keys = ["t", "E_kin", "Psi", "mean_p"]
vals = {}
vals[keys[0]] = t
vals[keys[1]] = df.assemble(0.5 * cc["rho"] *
df.inner(pv["v"], pv["v"]) * df.dx)
vals[keys[2]] = df.assemble((
0.25 * cc["a"] * cc["eps"] *\
df.inner(df.dot(cc["LA"], df.grad(pv["phi"])), df.grad(pv["phi"]))
+ (cc["b"] / cc["eps"]) * cc["F"]
) * df.dx)
vals[keys[3]] = df.assemble(pv["p"] * df.dx)
if it % self.mod == 0:
for key in keys:
self.functionals[key].append(vals[key])
# Logging and reporting
df.info("")
df.begin("Reported functionals:")
for key in keys[1:]:
desc = "{:6s} = %g".format(key)
logger.info(desc, (vals[key], ), (key, ), t)
df.end()
df.info("")
def divergence_matrix(mesh):
CR = VectorFunctionSpace(mesh, 'CR', 1)
DG = FunctionSpace(mesh, 'DG', 0)
A = cg1_cr_interpolation_matrix(mesh)
M = assemble(dot(div(TrialFunction(CR)), TestFunction(DG)) * dx())
C = compiled_cr_module.cr_divergence_matrix(M, A, DG, CR)
return C
def incompressibilityCondition(self, v):
"""
Return the incompressibility function, :math:`f(\mathbf{v})`,
for fluids such that :math:`f(\mathbf{v}) = 0`. The default is
.. math::
f(\mathbf{v}) = \\text{div}(\mathbf{v}) = 0,
where :math:`\mathbf{v}` is the velocity vector field. This can
be redefined by subclasses if a different constraint function is
desired.
Parameters
----------
v : dolfin.Function, ufl.tensors.ListTensor
The velocity vector.
Returns
-------
f : ufl.differentiation.Div
UFL object defining the incompressibility condition.
"""
return dlf.div(v)
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 __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f1 = ParametrizedExpression(
mock_problem, "1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)", mu=(-1., -1.),
element=V.sub(1).ufl_element())
f2 = ParametrizedExpression(
mock_problem, "exp( - 2*pow(x[0]-mu[0], 2) - 2*pow(x[1]-mu[1], 2) )", mu=(-1., -1.),
element=V.sub(1).ufl_element())
f3 = ParametrizedExpression(
mock_problem, "(1-x[0])*cos(3*pi*(pi+mu[1])*(1+x[1]))*exp(-(pi+mu[0])*(1+x[0]))", mu=(-1., -1.),
element=V.sub(1).ufl_element())
#
folder_prefix = os.path.join("test_eim_approximation_05_tempdir", expression_type, basis_generation)
assert expression_type in ("Vector", "Matrix")
if expression_type == "Vector":
vq = TestFunction(V)
(v, q) = split(vq)
form = f1 * v[0] * dx + f2 * v[1] * dx + f3 * q * dx
# Call Parent constructor
EIMApproximation.__init__(
self, mock_problem, ParametrizedTensorFactory(form), folder_prefix, basis_generation)
elif expression_type == "Matrix":
up = TrialFunction(V)
vq = TestFunction(V)
(u, p) = split(up)
(v, q) = split(vq)
form = f1 * inner(grad(u), grad(v)) * dx + f2 * p * div(v) * dx + f3 * q * div(u) * dx
# Call Parent constructor
EIMApproximation.__init__(
self, mock_problem, ParametrizedTensorFactory(form), folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
def get_residual_form(u, v, rho_e, E = 1, method='SIMP'):
if method =='SIMP':
C = rho_e**3
else:
C = rho_e/(1 + 8. * (1. - rho_e))
E = 1. * C # C is the design variable, its values is from 0 to 1
nu = 0.3 # Poisson's ratio
lambda_ = E * nu/(1. + nu)/(1 - 2 * nu)
mu = E / 2 / (1 + nu) #lame's parameters
w_ij = 0.5 * (df.grad(u) + df.grad(u).T)
v_ij = 0.5 * (df.grad(v) + df.grad(v).T)
d = len(u)
sigm = lambda_*df.div(u)*df.Identity(d) + 2*mu*w_ij
a = df.inner(sigm, v_ij) * df.dx
return a
def flux_derivative(self, u, coeff):
a = coeff
Du = self.differential_op(u)
Dsigma = dot(nabla_grad(a), Du)
if element_degree(u) >= 2:
Dsigma += a * div(Du)
return Dsigma
def fluid_setup(v_, p_, d_, n, psi, gamma, dx_f, ds, mu_f, rho_f, k, dt, v_deg,
theta, **semimp_namespace):
#J_theta = theta*J_(d_["n"]) + (1 - theta)*J_(d_["n-1"])
#F_fluid_linear = rho_f/k*inner(J_theta*(v_["n"] - v_["n-1"]), psi)*dx_f
#F_fluid_nonlinear = rho_f/k*inner(J_theta*(v_["n"] - v_["n-1"]), psi)*dx_f
J_theta = theta * J_(d_["n"]) + (1 - theta) * J_(d_["n-1"])
F_fluid_nonlinear = rho_f / k * inner(
J_theta * (v_["n"]) - theta * J_(d_["n"]) * v_["n-1"], psi) * dx_f
F_fluid_linear = rho_f / k * inner(
Constant(1 - theta) * J_(d_["n-1"]) * -(v_["n-1"]), psi) * dx_f
F_fluid_nonlinear = Constant(theta) * rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) * v_["n"], psi) * dx_f
F_fluid_nonlinear += inner(
J_(d_["n"]) * sigma_f_p(p_["n"], d_["n"]) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_nonlinear += Constant(theta) * inner(
J_(d_["n"]) * sigma_f_u(v_["n"], d_["n"], mu_f) * inv(F_(d_["n"])).T,
grad(psi)) * dx_f
F_fluid_linear += Constant(1 - theta) * inner(
J_(d_["n-1"]) * sigma_f_u(v_["n-1"], d_["n-1"], mu_f) *
inv(F_(d_["n-1"])).T, grad(psi)) * dx_f
F_fluid_nonlinear += inner(div(J_(d_["n"]) * inv(F_(d_["n"])) * v_["n"]),
gamma) * dx_f
F_fluid_linear += Constant(1 - theta) * rho_f * inner(
J_(d_["n-1"]) * grad(v_["n-1"]) * inv(F_(d_["n-1"])) * v_["n-1"],
psi) * dx_f
F_fluid_nonlinear -= rho_f * inner(
J_(d_["n"]) * grad(v_["n"]) * inv(F_(d_["n"])) *
((d_["n"] - d_["n-1"]) / k), psi) * dx_f
return dict(F_fluid_linear=F_fluid_linear,
F_fluid_nonlinear=F_fluid_nonlinear)
def get_residual_form(u, v, rho_e, T, T_hat, KAPPA, k, alpha, mode='plane_stress', method='RAMP', T_r=df.Constant(20.)):
if method=='RAMP':
p =8
C = rho_e/(1 + p * (1. - rho_e))
else:
C = rho_e**3
E = k * C
# C is the design variable, its values is from 0 to 1
nu = 0.3 # Poisson's ratio
lambda_ = E * nu/(1. + nu)/(1 - 2 * nu)
mu = E / 2 / (1 + nu) #lame's parameters
if mode == 'plane_stress':
lambda_ = 2*mu*lambda_/(lambda_+2*mu)
# Th = df.Constant(7)
I = df.Identity(len(u))
T_0 = df.Constant(20.)
w_ij = 0.5 * (df.grad(u) + df.grad(u).T) - C * alpha * I * (T-T_0)
v_ij = 0.5 * (df.grad(v) + df.grad(v).T)
d = len(u)
sigm = lambda_*df.div(u)*df.Identity(d) + 2*mu*w_ij
a = df.inner(sigm, v_ij) * df.dx + \
df.dot(C*KAPPA* df.grad(T), df.grad(T_hat)) * df.dx
print("get a-------")
return a
def __init__(self, u_, Space, bcs=[], name="div", method={}):
solver_type = method.get('solver_type', 'cg')
preconditioner_type = method.get('preconditioner_type', 'default')
solver_method = method.get('method', 'default')
low_memory_version = method.get('low_memory_version', False)
OasisFunction.__init__(self, div(u_), Space, bcs=bcs, name=name,
method=solver_method, solver_type=solver_type,
preconditioner_type=preconditioner_type)
Source = u_[0].function_space()
if not low_memory_version:
self.matvec = [[A_cache[(self.test * TrialFunction(Source).dx(i) * dx, ())], u_[i]]
for i in range(Space.mesh().geometry().dim())]
if solver_method.lower() == "gradient_matrix":
from fenicstools import compiled_gradient_module
DG = FunctionSpace(Space.mesh(), 'DG', 0)
G = assemble(TrialFunction(DG) * self.test * dx())
dg = Function(DG)
self.WGM = []
st = TrialFunction(Source)
for i in range(Space.mesh().geometry().dim()):
dP = assemble(st.dx(i) * TestFunction(DG) * dx)
A = Matrix(G)
self.WGM.append(compiled_gradient_module.compute_weighted_gradient_matrix(A, dP, dg))
def divergence_matrix(mesh):
CR = VectorFunctionSpace(mesh, 'CR', 1)
DG = FunctionSpace(mesh, 'DG', 0)
A = cg1_cr_interpolation_matrix(mesh)
M = assemble(dot(div(TrialFunction(CR)), TestFunction(DG))*dx())
compiled_cr_module.cr_divergence_matrix(M, A, DG, CR)
M_mat = as_backend_type(M).mat()
M_mat.matMult(A.mat())
return M
def comp_cont_error(v, fpbc, Q):
"""Compute the L2 norm of the residual of the continuity equation
Bv = g
"""
q = TrialFunction(Q)
divv = assemble(q*div(v)*dx)
conRhs = Function(Q)
conRhs.vector().set_local(fpbc)
ContEr = norm(conRhs.vector() - divv)
return ContEr
def _residual_strong(dx, v, phi, mu, sigma, omega, conv, voltages):
'''Get the residual in strong form, projected onto V.
'''
r = Expression('x[0]', degree=1, cell=triangle)
R = [zero() * dx(0),
zero() * dx(0)]
subdomain_indices = mu.keys()
for i in subdomain_indices:
# diffusion, reaction
R_r = - div(1 / (mu[i] * r) * grad(r * phi[0])) \
- sigma[i] * omega * phi[1]
R_i = - div(1 / (mu[i] * r) * grad(r * phi[1])) \
+ sigma[i] * omega * phi[0]
# convection
if i in conv:
R_r += dot(conv[i], 1 / r * grad(r * phi[0]))
R_i += dot(conv[i], 1 / r * grad(r * phi[1]))
# right-hand side
if i in voltages:
R_r -= sigma[i] * voltages[i].real / (2 * pi * r)
R_i -= sigma[i] * voltages[i].imag / (2 * pi * r)
R[0] += R_r * v * dx(i)
R[1] += R_i * v * dx(i)
return R
def adaptive(self, mesh, eigv, eigf):
"""Refine mesh based on residual errors."""
fraction = 0.1
C = FunctionSpace(mesh, "DG", 0) # constants on triangles
w = TestFunction(C)
h = CellSize(mesh)
n = FacetNormal(mesh)
marker = CellFunction("bool", mesh)
print len(marker)
indicators = np.zeros(len(marker))
for e, u in zip(eigv, eigf):
errform = avg(h) * jump(grad(u), n) ** 2 * avg(w) * dS \
+ h * (inner(grad(u), n) - Constant(e) * u) ** 2 * w * ds
if self.degree > 1:
errform += h ** 2 * div(grad(u)) ** 2 * w * dx
indicators[:] += assemble(errform).array() # errors for each cell
print "Residual error: ", sqrt(sum(indicators) / len(eigv))
cutoff = sorted(
indicators, reverse=True)[
int(len(indicators) * fraction) - 1]
marker.array()[:] = indicators > cutoff # mark worst errors
mesh = refine(mesh, marker)
return mesh
def ab2tr_step(W, P, dt0, dt_1,
mu, rho,
u0, u_1, u_bcs,
dudt0, dudt_1, dudt_bcs,
p_1, p_bcs,
f0, f1,
tol=1.0e-12,
verbose=True
):
# General AB2/TR step.
#
# Steps are labeled in the following way:
#
# * u_1: previous step.
# * u0: current step.
# * u1: next step.
#
# The same scheme applies to all other entities.
#
WP = W * P
# Make sure the boundary conditions fit with the space.
u_bcs_new = []
for u_bc in u_bcs:
u_bcs_new.append(DirichletBC(WP.sub(0),
u_bc.value(),
u_bc.user_sub_domain()))
p_bcs_new = []
for p_bc in p_bcs:
p_bcs_new.append(DirichletBC(WP.sub(1),
p_bc.value(),
p_bc.user_sub_domain()))
# Predict velocity.
if dudt_1:
u_pred = u0 \
+ 0.5*dt0*((2 + dt0/dt_1) * dudt0 - (dt0/dt_1) * dudt_1)
else:
# Simple linear extrapolation.
u_pred = u0 + dt0 * dudt0
uu = TrialFunctions(WP)
vv = TestFunctions(WP)
# Assign up[1] with u_pred and up[1] with p_1.
# As of now (2013/09/05), there is no proper subfunction assignment in
# Dolfin, cf.
# <https://bitbucket.org/fenics-project/dolfin/issue/84/subfunction-assignment>.
# Hence, we need to be creative here.
# TODO proper subfunction assignment
#
# up1.assign(0, u_pred)
# up1.assign(1, p_1)
#
up1 = Function(WP)
a = (dot(uu[0], vv[0]) + uu[1] * vv[1]) * dx
L = dot(u_pred, vv[0]) * dx
if p_1:
L += p_1 * vv[1] * dx
solve(a == L, up1,
bcs=u_bcs_new + p_bcs_new
)
# Split up1 for easier access.
# This is not as easy as it may seem at first, see
# <http://fenicsproject.org/qa/1123/nonlinear-solves-with-mixed-function-spaces>.
# Note in particular that
# u1, p1 = up1.split()
# doesn't work here.
#
u1, p1 = split(up1)
# Form the nonlinear equation system (3.16-235) in Gresho/Sani.
# Left-hand side of the nonlinear equation system.
F = 2.0/dt0 * rho * dot(u1, vv[0]) * dx \
+ mu * inner(grad(u1), grad(vv[0])) * dx \
+ rho * 0.5 * (inner(grad(u1)*u1, vv[0])
- inner(grad(vv[0]) * u1, u1)) * dx \
+ dot(grad(p1), vv[0]) * dx \
+ div(u1) * vv[1] * dx
# Subtract the right-hand side.
F -= dot(rho*(2.0/dt0*u0 + dudt0) + f1, vv[0]) * dx
#J = derivative(F, up1)
# Solve nonlinear system for u1, p1.
solve(
F == 0, up1,
bcs=u_bcs_new + p_bcs_new,
#J = J,
solver_parameters={
#'nonlinear_solver': 'snes',
'nonlinear_solver': 'newton',
'newton_solver': {'maximum_iterations': 5,
'report': True,
'absolute_tolerance': tol,
'relative_tolerance': 0.0
},
'linear_solver': 'direct',
#'linear_solver': 'iterative',
## The nonlinear term makes the problem
## generally nonsymmetric.
#'symmetric': False,
## If the nonsymmetry is too strong, e.g., if
## u_1 is large, then AMG preconditioning
## might not work very well.
#'preconditioner': 'ilu',
##'preconditioner': 'hypre_amg',
#'krylov_solver': {'relative_tolerance': tol,
# 'absolute_tolerance': 0.0,
# 'maximum_iterations': 100,
# 'monitor_convergence': verbose}
})
## Simpler access to the solution.
#u1, p1 = up1.split()
# Invert trapezoidal rule for next du/dt.
dudt1 = 2 * (u1 - u0)/dt0 - dudt0
# Get next dt.
if dt_1:
# Compute local trunction error (LTE) estimate.
d = (u1 - u_pred) / (3*(1.0 + dt_1 / dt))
# There are other ways of estimating the LTE norm.
norm_d = numpy.sqrt(inner(d, d) / u_max**2)
# Get next step size.
dt1 = dt0 * (eps / norm_d)**(1.0/3.0)
else:
dt1 = dt0
return u1, p1, dudt1, dt1
def F(
u,
v,
kappa,
rho,
cp,
convection,
source,
r,
neumann_bcs,
robin_bcs,
my_dx,
my_ds,
stabilization,
):
"""
Compute
.. math::
F(u) =
\\int_\\Omega \\kappa r
\\langle\\nabla u, \\nabla \\frac{v}{\\rho c_p}\\rangle
\\, 2\\pi \\, \\text{d}x
+ \\int_\\Omega \\langle c, \\nabla u\\rangle v
\\, 2\\pi r\\,\\text{d}x
- \\int_\\Omega \\frac{1}{\\rho c_p} f v
\\, 2\\pi r \\,\\text{d}x\\\\
- \\int_\\Gamma r \\kappa \\langle n, \\nabla T\\rangle v
\\frac{1}{\\rho c_p} 2\\pi \\,\\text{d}s
- \\int_\\Gamma r \\kappa \\alpha (u - u_0) v
\\frac{1}{\\rho c_p} \\, 2\\pi \\,\\text{d}s,
used for time-stepping
.. math::
u' = F(u).
"""
rho_cp = rho * cp
F0 = kappa * r * dot(grad(u), grad(v / rho_cp)) * 2 * pi * my_dx
# F -= dot(b, grad(u)) * v * 2*pi*r * dx_workpiece(0)
if convection is not None:
c = as_vector([convection[0], convection[1]])
F0 += dot(c, grad(u)) * v * 2 * pi * r * my_dx
# Joule heat
F0 -= source * v / rho_cp * 2 * pi * r * my_dx
# Neumann boundary conditions
for k, n_grad_T in neumann_bcs.items():
F0 -= r * kappa * n_grad_T * v / rho_cp * 2 * pi * my_ds(k)
# Robin boundary conditions
for k, value in robin_bcs.items():
alpha, u0 = value
F0 -= r * kappa * alpha * (u - u0) * v / rho_cp * 2 * pi * my_ds(k)
if stabilization == "supg":
# Add SUPG stabilization.
assert convection is not None
# TODO u_t?
R = (
-div(kappa * r * grad(u)) / rho_cp * 2 * pi
+ dot(c, grad(u)) * 2 * pi * r
- source / rho_cp * 2 * pi * r
)
mesh = v.function_space().mesh()
element_degree = v.ufl_element().degree()
tau = stab.supg(mesh, convection, kappa, element_degree)
F0 += R * tau * dot(convection, grad(v)) * my_dx
else:
assert stabilization is None
return F0
def ab2tr_step0(u0,
P,
f, # right-hand side
rho,
mu,
dudt_bcs=[],
p_bcs=[],
eps=1.0e-4, # relative error tolerance
verbose=True
):
# Make sure that the initial velocity is divergence-free.
alpha = norm(u0, 'Hdiv0')
if abs(alpha) > DOLFIN_EPS:
warn('Initial velocity not divergence-free (||u||_div = %e).'
% alpha
)
# Get the initial u0' and p0 by solving the linear equation system
#
# [M C] [u0'] [f0 - (K+N(u0)u0)]
# [C^T 0] [p0 ] = [ g0' ],
#
# i.e.,
#
# rho u0' + nabla(p0) = f0 + mu\Delta(u0) - rho u0.nabla(u0),
# div(u0') = 0.
#
W = u0.function_space()
WP = W*P
# Translate the boundary conditions into product space. See
# <http://fenicsproject.org/qa/703/boundary-conditions-in-product-space>.
dudt_bcs_new = []
for dudt_bc in dudt_bcs:
dudt_bcs_new.append(DirichletBC(WP.sub(0),
dudt_bc.value(),
dudt_bc.user_sub_domain()))
p_bcs_new = []
for p_bc in p_bcs:
p_bcs_new.append(DirichletBC(WP.sub(1),
p_bc.value(),
p_bc.user_sub_domain()))
new_bcs = dudt_bcs_new + p_bcs_new
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
#a = rho * dot(u, v) * dx + dot(grad(p), v) * dx \
a = rho * inner(u, v) * dx - p * div(v) * dx \
- div(u) * q * dx
L = _rhs_weak(u0, v, f, rho, mu)
A, b = assemble_system(a, L, new_bcs)
# Similar preconditioner as for the Stokes problem.
# TODO implement something better!
prec = rho * inner(u, v) * dx \
- p*q*dx
M, _ = assemble_system(prec, L, new_bcs)
solver = KrylovSolver('gmres', 'amg')
solver.parameters['monitor_convergence'] = verbose
solver.parameters['report'] = verbose
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = 1.0e-6
solver.parameters['maximum_iterations'] = 10000
# Associate operator (A) and preconditioner matrix (M)
solver.set_operators(A, M)
#solver.set_operator(A)
# Solve
up = Function(WP)
solver.solve(up.vector(), b)
# Get sub-functions
dudt0, p0 = up.split()
# Choosing the first step size for the trapezoidal rule can be tricky.
# Chapters 2.7.4a, 2.7.4e of the book
#
# Incompressible flow and the finite element method,
# volume 1: advection-diffusion;
# P.M. Gresho, R.L. Sani,
#
# give some hints.
#
# eps ... relative error tolerance
# tau ... estimate of the initial 'time constant'
tau = None
if tau:
dt0 = tau * eps**(1.0/3.0)
else:
# Choose something 'reasonably small'.
dt0 = 1.0e-3
# Alternative:
# Use a dissipative scheme like backward Euler or BDF2 for the first
# couple of steps. This makes sure that noisy initial data is damped
# out.
return dudt0, p0, dt0
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 _evaluateLocalEstimator(cls, mu, w, coeff_field, pde, f, quadrature_degree, epsilon=1e-5):
"""Evaluation of patch local equilibrated estimator."""
# prepare numerical flux and f
sigma_mu, f_mu = evaluate_numerical_flux(w, mu, coeff_field, f)
# ###################
# ## MIXED PROBLEM ##
# ###################
# get setup data for mixed problem
V = w[mu]._fefunc.function_space()
mesh = V.mesh()
mesh.init()
degree = element_degree(w[mu]._fefunc)
# data for nodal bases
V_dm = V.dofmap()
V_dofs = dict([(i, V_dm.cell_dofs(i)) for i in range(mesh.num_cells())])
V1 = FunctionSpace(mesh, 'CG', 1) # V1 is to define nodal base functions
phi_z = Function(V1)
phi_coeffs = np.ndarray(V1.dim())
vertex_dof_map = V1.dofmap().vertex_to_dof_map(mesh)
# vertex_dof_map = vertex_to_dof_map(V1)
dof_list = vertex_dof_map.tolist()
# DG0 localisation
DG0 = FunctionSpace(mesh, 'DG', 0)
DG0_dofs = dict([(c.index(),DG0.dofmap().cell_dofs(c.index())[0]) for c in cells(mesh)])
dg0 = TestFunction(DG0)
# characteristic function of patch
xi_z = Function(DG0)
xi_coeffs = np.ndarray(DG0.dim())
# mesh data
h = CellSize(mesh)
n = FacetNormal(mesh)
cf = CellFunction('size_t', mesh)
# setup error estimator vector
eq_est = np.zeros(DG0.dim())
# setup global equilibrated flux vector
DG = VectorFunctionSpace(mesh, "DG", degree)
DG_dofmap = DG.dofmap()
# define form functions
tau = TrialFunction(DG)
v = TestFunction(DG)
# define global tau
tau_global = Function(DG)
tau_global.vector()[:] = 0.0
# iterate vertices
for vertex in vertices(mesh):
# get patch cell indices
vid = vertex.index()
patch_cid, FF_inner, FF_boundary = get_vertex_patch(vid, mesh, layers=1)
# set nodal base function
phi_coeffs[:] = 0
phi_coeffs[dof_list.index(vid)] = 1
phi_z.vector()[:] = phi_coeffs
# set characteristic function and mark patch
cf.set_all(0)
xi_coeffs[:] = 0
for cid in patch_cid:
xi_coeffs[DG0_dofs[int(cid)]] = 1
cf[int(cid)] = 1
xi_z.vector()[:] = xi_coeffs
# determine local dofs
lDG_cell_dofs = dict([(cid, DG_dofmap.cell_dofs(cid)) for cid in patch_cid])
lDG_dofs = [cd.tolist() for cd in lDG_cell_dofs.values()]
lDG_dofs = list(iter.chain(*lDG_dofs))
# print "\nlocal DG subspace has dimension", len(lDG_dofs), "degree", degree, "cells", len(patch_cid), patch_cid
# print "local DG_cell_dofs", lDG_cell_dofs
# print "local DG_dofs", lDG_dofs
# create patch measures
dx = Measure('dx')[cf]
dS = Measure('dS')[FF_inner]
# define forms
alpha = Constant(1 / epsilon) / h
a = inner(tau,v) * phi_z * dx(1) + alpha * div(tau) * div(v) * dx(1) + avg(alpha) * jump(tau,n) * jump(v,n) * dS(1)\
+ avg(alpha) * jump(xi_z * tau,n) * jump(v,n) * dS(2)
L = -alpha * (div(sigma_mu) + f) * div(v) * phi_z * dx(1)\
- avg(alpha) * jump(sigma_mu,n) * jump(v,n) * avg(phi_z)*dS(1)
# print "L2 f + div(sigma)", assemble((f + div(sigma)) * (f + div(sigma)) * dx(0))
# assemble forms
lhs = assemble(a, form_compiler_parameters={'quadrature_degree': quadrature_degree})
rhs = assemble(L, form_compiler_parameters={'quadrature_degree': quadrature_degree})
# convert DOLFIN representation to scipy sparse arrays
rows, cols, values = lhs.data()
lhsA = sps.csr_matrix((values, cols, rows)).tocoo()
# slice sparse matrix and solve linear problem
lhsA = coo_submatrix_pull(lhsA, lDG_dofs, lDG_dofs)
lx = spsolve(lhsA, rhs.array()[lDG_dofs])
# print ">>> local solution lx", type(lx), lx
local_tau = Function(DG)
local_tau.vector()[lDG_dofs] = lx
# print "div(tau)", assemble(inner(div(local_tau),div(local_tau))*dx(1))
# add up local fluxes
tau_global.vector()[lDG_dofs] += lx
# evaluate estimator
# maybe TODO: re-define measure dx
eq_est = assemble( inner(tau_global, tau_global) * dg0 * (dx(0)+dx(1)),\
form_compiler_parameters={'quadrature_degree': quadrature_degree})
# reorder according to cell ids
eq_est = eq_est[DG0_dofs.values()].array()
global_est = np.sqrt(np.sum(eq_est))
# eq_est_global = assemble( inner(tau_global, tau_global) * (dx(0)+dx(1)), form_compiler_parameters={'quadrature_degree': quadrature_degree} )
# global_est2 = np.sqrt(np.sum(eq_est_global))
return global_est, FlatVector(np.sqrt(eq_est))#, tau_global
def _evaluateGlobalMixedEstimator(cls, mu, w, coeff_field, pde, f, quadrature_degree, vectorspace_type='BDM'):
"""Evaluation of global mixed equilibrated estimator."""
# set quadrature degree
# quadrature_degree_old = parameters["form_compiler"]["quadrature_degree"]
# parameters["form_compiler"]["quadrature_degree"] = quadrature_degree
# logger.debug("residual quadrature order = " + str(quadrature_degree))
# prepare numerical flux and f
sigma_mu, f_mu = evaluate_numerical_flux(w, mu, coeff_field, f)
# ###################
# ## MIXED PROBLEM ##
# ###################
# get setup data for mixed problem
V = w[mu]._fefunc.function_space()
mesh = V.mesh()
degree = element_degree(w[mu]._fefunc)
# create function spaces
DG0 = FunctionSpace(mesh, 'DG', 0)
DG0_dofs = [DG0.dofmap().cell_dofs(c.index())[0] for c in cells(mesh)]
RT = FunctionSpace(mesh, vectorspace_type, degree)
W = RT * DG0
# setup boundary conditions
# bcs = pde.create_dirichlet_bcs(W.sub(1))
# debug ===
# from dolfin import DOLFIN_EPS, DirichletBC
# def boundary(x):
# return x[0] < DOLFIN_EPS or x[0] > 1.0 + DOLFIN_EPS or x[1] < DOLFIN_EPS or x[1] > 1.0 + DOLFIN_EPS
# bcs = [DirichletBC(W.sub(1), Constant(0.0), boundary)]
# === debug
# create trial and test functions
(sigma, u) = TrialFunctions(W)
(tau, v) = TestFunctions(W)
# define variational form
a_eq = (dot(sigma, tau) + div(tau) * u + div(sigma) * v) * dx
L_eq = (- f_mu * v + dot(sigma_mu, tau)) * dx
# compute solution
w_eq = Function(W)
solve(a_eq == L_eq, w_eq)
(sigma_mixed, u_mixed) = w_eq.split()
# #############################
# ## EQUILIBRATION ESTIMATOR ##
# #############################
# evaluate error estimator
dg0 = TestFunction(DG0)
eta_mu = inner(sigma_mu, sigma_mu) * dg0 * dx
eta_T = assemble(eta_mu, form_compiler_parameters={'quadrature_degree': quadrature_degree})
eta_T = np.array([sqrt(e) for e in eta_T])
# evaluate global error
eta = sqrt(sum(i**2 for i in eta_T))
# reorder array entries for local estimators
eta_T = eta_T[DG0_dofs]
# restore quadrature degree
# parameters["form_compiler"]["quadrature_degree"] = quadrature_degree_old
return eta, FlatVector(eta_T)
def momentum_equation(u, v, p, f, rho, mu, stabilization=None, dx=dx):
'''Weak form of the momentum equation.
'''
assert rho > 0.0
assert mu > 0.0
# Skew-symmetric formulation.
# Don't include the boundary term
#
# - mu *inner(r*grad(u2)*n , v2 ) * 2*pi*ds.
#
# This effectively means that at all boundaries where no sufficient
# Dirichlet-conditions are posed, we assume grad(u)*n to vanish.
#
# The original term
# u2[0]/(r*r) * v2[0]
# doesn't explode iff u2[0]~r and v2[0]~r at r=0. Hence, we need to enforce
# homogeneous Dirichlet-conditions for n.u at r=0. This corresponds to no
# flow in normal direction through the symmetry axis -- makes sense.
# When using the 2*pi*r weighting, we can even be a bit lax on the
# enforcement on u2[0].
#
# For this to be well defined, u[0]/r and u[2]/r must be bounded for r=0,
# so u[0]~u[2]~r must hold. This either needs to be enforced in the
# boundary conditions (homogeneous Dirichlet for u[0], u[2] at r=0) or must
# follow from the dynamics of the system.
#
# TODO some more explanation for the following lines of code
r = Expression('x[0]', degree=1, domain=v.function_space().mesh())
F = rho * 0.5 * (dot(grad(u) * u, v) - dot(grad(v) * u, u)) * 2*pi*r*dx \
+ mu * inner(r * grad(u), grad(v)) * 2 * pi * dx \
+ mu * u[0] / r * v[0] * 2 * pi * dx \
- dot(f, v) * 2 * pi * r * dx
if p:
F += (p.dx(0) * v[0] + p.dx(1) * v[1]) * 2 * pi * r * dx
if len(u) == 3:
F += rho * (-u[2] * u[2] * v[0] + u[0] * u[2] * v[2]) \
* 2 * pi * dx
F += mu * u[2] / r * v[2] * 2 * pi * dx
if stabilization:
if stabilization == 'SUPG':
# TODO check this part of the code
#
# SUPG stabilization has the form
#
# <R, tau*grad(v)*u[-1]>
#
# with R being the residual in strong form. The choice of the tau
# is subject to research.
tau = stab.supg2(u.function_space().W.mesh(),
u,
mu / rho,
u.function_space().ufl_element().degree()
)
# We need to deal with the term
#
# \int mu * (u2[0]/r**2, 0) * dot(R, grad(v2)*b_tau) 2*pi*r*dx
#
# somehow. Unfortunately, it's not easy to construct (u2[0]/r**2,
# 0), cf. <https://answers.launchpad.net/dolfin/+question/228353>.
# Strong residual:
R = rho * grad(u) * u * 2 * pi * r \
- mu * div(r * grad(u)) * 2 * pi \
- f * 2 * pi * r
if p:
R += (p.dx(0) * v[0] + p.dx(1) * v[1]) \
* 2 * pi * r * dx
gv = tau * grad(v) * u
F += dot(R, gv) * dx
# Manually add the parts of the residual which couldn't be cleanly
# implemented above.
F += mu * u[0] / r * 2 * pi * gv[0] * dx
if u.function_space().num_sub_spaces() == 3:
F += rho * (-u[2] * u[2] * gv[0] + u[0] * u[2] * gv[2]) \
* 2 * pi * dx
F += mu * u[2] / r * gv[2] * 2 * pi * dx
else:
raise ValueError('Illegal stabilization type.')
return F
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 _pressure_poisson(self,
p1, p0,
mu, ui,
divu,
p_bcs=None,
p_n=None,
rotational_form=False,
tol=1.0e-10,
verbose=True
):
'''Solve the pressure Poisson equation
- \Delta phi = -div(u),
boundary conditions,
for
\nabla p = u.
'''
P = p1.function_space()
p = TrialFunction(P)
q = TestFunction(P)
a2 = dot(grad(p), grad(q)) * dx
L2 = -divu * q * dx
if p0:
L2 += dot(grad(p0), grad(q)) * dx
if p_n:
n = FacetNormal(P.mesh())
L2 += dot(n, p_n) * q * ds
if rotational_form:
L2 -= mu * dot(grad(div(ui)), grad(q)) * dx
if p_bcs:
solve(a2 == L2, p1,
bcs=p_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'hypre_amg',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose}
})
else:
# If we're dealing with a pure Neumann problem here (which is the
# default case), this doesn't hurt CG if the system is consistent,
# cf.
#
# Iterative Krylov methods for large linear systems,
# Henk A. van der Vorst.
#
# And indeed, it is consistent: Note that
#
# <1, rhs> = \sum_i 1 * \int div(u) v_i
# = 1 * \int div(u) \sum_i v_i
# = \int div(u).
#
# With the divergence theorem, we have
#
# \int div(u) = \int_\Gamma n.u.
#
# The latter term is 0 iff inflow and outflow are exactly the same
# at any given point in time. This corresponds with the
# incompressibility of the liquid.
#
# In turn, this hints towards penetrable boundaries to require
# Dirichlet conditions on the pressure.
#
A = assemble(a2)
b = assemble(L2)
#
# In principle, the ILU preconditioner isn't advised here since it
# might destroy the semidefiniteness needed for CG.
#
# The system is consistent, but the matrix has an eigenvalue 0.
# This does not harm the convergence of CG, but when
# preconditioning one has to take care that the preconditioner
# preserves the kernel. ILU might destroy this (and the
# semidefiniteness). With AMG, the coarse grid solves cannot be LU
# then, so try Jacobi here.
# <http://lists.mcs.anl.gov/pipermail/petsc-users/2012-February/012139.html>
#
prec = PETScPreconditioner('hypre_amg')
PETScOptions.set('pc_hypre_boomeramg_relax_type_coarse', 'jacobi')
solver = PETScKrylovSolver('cg', prec)
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 100
solver.parameters['monitor_convergence'] = verbose
# Create solver and solve system
A_petsc = as_backend_type(A)
b_petsc = as_backend_type(b)
p1_petsc = as_backend_type(p1.vector())
solver.set_operator(A_petsc)
try:
solver.solve(p1_petsc, b_petsc)
except RuntimeError as error:
info('')
# Check if the system is indeed consistent.
#
# If the right hand side is flawed (e.g., by round-off errors),
# then it may have a component b1 in the direction of the null
# space, orthogonal the image of the operator:
#
# b = b0 + b1.
#
# When starting with initial guess x0=0, the minimal achievable
# relative tolerance is then
#
# min_rel_tol = ||b1|| / ||b||.
#
# If ||b|| is very small, which is the case when ui is almost
# divergence-free, then min_rel_to may be larger than the
# prescribed relative tolerance tol.
#
# Use this as a consistency check, i.e., bail out if
#
# tol < min_rel_tol = ||b1|| / ||b||.
#
# For computing ||b1||, we use the fact that the null space is
# one-dimensional, i.e., b1 = alpha e, and
#
# e.b = e.(b0 + b1) = e.b1 = alpha ||e||^2,
#
# so alpha = e.b/||e||^2 and
#
# ||b1|| = |alpha| ||e|| = e.b / ||e||
#
e = Function(P)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
info('Linear system convergence failure.')
info(error.message)
message = ('Linear system not consistent! '
'<b,e> = %g, ||b|| = %g, <b,e>/||b|| = %e, tol = %e.') \
% (alpha, normB, alpha/normB, tol)
info(message)
if tol < abs(alpha) / normB:
info('\int div(u) = %e' % assemble(divu * dx))
#n = FacetNormal(Q.mesh())
#info('\int_Gamma n.u = %e' % assemble(dot(n, u)*ds))
#info('\int_Gamma u[0] = %e' % assemble(u[0]*ds))
#info('\int_Gamma u[1] = %e' % assemble(u[1]*ds))
## Now plot the faulty u on a finer mesh (to resolve the
## quadratic trial functions).
#fine_mesh = Q.mesh()
#for k in range(1):
# fine_mesh = refine(fine_mesh)
#V1 = FunctionSpace(fine_mesh, 'CG', 1)
#W1 = V1*V1
#uplot = project(u, W1)
##uplot = Function(W1)
##uplot.interpolate(u)
#plot(uplot, title='u_tentative')
#plot(uplot[0], title='u_tentative[0]')
#plot(uplot[1], title='u_tentative[1]')
plot(divu, title='div(u_tentative)')
interactive()
exit()
raise RuntimeError(message)
else:
exit()
raise RuntimeError('Linear system failed to converge.')
except:
exit()
return
def step(self,
dt,
u1, p1,
u, p0,
u_bcs, p_bcs,
f0=None, f1=None,
verbose=True,
tol=1.0e-10
):
# Some initial sanity checkups.
assert dt > 0.0
# Define trial and test functions
ui = Function(self.W)
v = TestFunction(self.W)
# Create functions
# Define coefficients
k = Constant(dt)
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
# Compute tentative velocity step:
#
# F(u) = 0,
# F(u) := rho (U0 + (u.\nabla)u) - mu \div(\nabla u) - f = 0.
#
with Message('Computing tentative velocity'):
# TODO higher-order scheme for time integration
#
# For higher-order schemes, see
#
# A comparison of time-discretization/linearization approaches
# for the incompressible Navier-Stokes equations;
# Volker John, Gunar Matthies, Joachim Rang;
# Comput. Methods Appl. Mech. Engrg. 195 (2006) 5995-6010;
# <http://www.wias-berlin.de/people/john/ELECTRONIC_PAPERS/JMR06.CMAME.pdf>.
#
F1 = self.rho * inner((ui - u[0])/k, v) * dx
if abs(self.theta) > DOLFIN_EPS:
# Implicit terms.
if f1 is None:
raise RuntimeError('Implicit schemes need right-hand side '
'at target step (f1).')
F1 -= self.theta * _rhs_weak(ui, v, f1, self.rho, self.mu)
if abs(1.0 - self.theta) > DOLFIN_EPS:
# Explicit terms.
if f0 is None:
raise RuntimeError('Explicit schemes need right-hand side '
'at current step (f0).')
F1 -= (1.0 - self.theta) \
* _rhs_weak(u[0], v, f0, self.rho, self.mu)
if p0:
F1 += dot(grad(p0), v) * dx
#if stabilization:
# tau = stab.supg2(V.mesh(),
# u_1,
# mu/rho,
# V.ufl_element().degree()
# )
# R = rho*(ui - u_1)/k
# if abs(theta) > DOLFIN_EPS:
# R -= theta * _rhs_strong(ui, f1, rho, mu)
# if abs(1.0-theta) > DOLFIN_EPS:
# R -= (1.0-theta) * _rhs_strong(u_1, f0, rho, mu)
# if p_1:
# R += grad(p_1)
# # TODO use u_1 or ui here?
# F1 += tau * dot(R, grad(v)*u_1) * dx
# Get linearization and solve nonlinear system.
# If the scheme is fully explicit (theta=0.0), then the system is
# actually linear and only one Newton iteration is performed.
J = derivative(F1, ui)
# What is a good initial guess for the Newton solve?
# Three choices come to mind:
#
# (1) the previous solution u_1,
# (2) the intermediate solution from the previous step ui_1,
# (3) the solution of the semilinear system
# (u.\nabla(u) -> u_1.\nabla(u)).
#
# Numerical experiments with the Karman vortex street show that the
# order of accuracy is (1), (3), (2). Typical norms would look like
#
# ||u - u_1 || = 1.726432e-02
# ||u - ui_1|| = 2.720805e+00
# ||u - u_e || = 5.921522e-02
#
# Hence, use u_1 as initial guess.
ui.assign(u[0])
# Wrap the solution in a try-catch block to make sure we call end()
# if necessary.
#problem = NonlinearVariationalProblem(F1, ui, u_bcs, J)
#solver = NonlinearVariationalSolver(problem)
solve(
F1 == 0, ui,
bcs=u_bcs,
J=J,
solver_parameters={
#'nonlinear_solver': 'snes',
'nonlinear_solver': 'newton',
'newton_solver': {
'maximum_iterations': 5,
'report': True,
'absolute_tolerance': tol,
'relative_tolerance': 0.0,
'linear_solver': 'iterative',
## The nonlinear term makes the problem generally
## nonsymmetric.
#'symmetric': False,
# If the nonsymmetry is too strong, e.g., if u_1 is
# large, then AMG preconditioning might not work very
# well.
'preconditioner': 'ilu',
#'preconditioner': 'hypre_amg',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 1000,
'monitor_convergence': verbose}
}})
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
with Message('Computing pressure correction'):
#
# The following is based on the update formula
#
# rho/dt (u_{n+1}-u*) + \nabla phi = 0
#
# with
#
# phi = (p_{n+1} - p*) + chi*mu*div(u*)
#
# and div(u_{n+1})=0. One derives
#
# - \nabla^2 phi = rho/dt div(u_{n+1} - u*),
# - n.\nabla phi = rho/dt n.(u_{n+1} - u*),
#
# In its weak form, this is
#
# \int \grad(phi).\grad(q)
# = - rho/dt \int div(u*) q - rho/dt \int_Gamma n.(u_{n+1}-u*) q.
#
# If Dirichlet boundary conditions are applied to both u* and
# u_{n+1} (the latter in the final step), the boundary integral
# vanishes.
#
# Assume that on the boundary
# L2 -= inner(n, rho/k (u_bcs - ui)) * q * ds
# is zero. This requires the boundary conditions to be set for
# ui as well as u_final.
# This creates some problems if the boundary conditions are
# supposed to remain 'free' for the velocity, i.e., no Dirichlet
# conditions in normal direction. In that case, one needs to
# specify Dirichlet pressure conditions.
#
rotational_form = False
self._pressure_poisson(p1, p0,
self.mu, ui,
divu=Constant(self.rho/dt)*div(ui),
p_bcs=p_bcs,
rotational_form=rotational_form,
tol=tol,
verbose=verbose
)
#plot(p - phi, title='p-phi')
#plot(ui, title='u intermediate')
#plot(f, title='f')
##plot(ui[1], title='u intermediate[1]')
#plot(div(ui), title='div(u intermediate)')
#plot(phi, title='phi')
#interactive()
# - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
# Velocity correction.
# U = U0 - dt/rho \nabla p.
with Message('Computing velocity correction'):
u2 = TrialFunction(self.W)
a3 = inner(u2, v) * dx
if p0:
phi = Function(self.P)
phi.assign(p1)
phi -= p0
else:
phi = p1
if rotational_form:
phi += self.mu * div(ui)
L3 = inner(ui, v) * dx \
- k/self.rho * inner(grad(phi), v) * dx
solve(a3 == L3, u1,
bcs=u_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': 'jacobi',
'krylov_solver': {'relative_tolerance': tol,
'absolute_tolerance': 0.0,
'maximum_iterations': 100,
'monitor_convergence': verbose}
})
#u = project(ui - k/rho * grad(phi), V)
#print '||u||_div = %e' % norm(u, 'Hdiv0')
#uu = TrialFunction(Q)
#vv = TestFunction(Q)
#div_u = Function(Q)
#solve(uu*vv*dx == div(u)*vv*dx, div_u,
# #bcs=DirichletBC(Q, 0.0, 'on_boundary')
# )
#plot(div_u, title='div(u)')
#interactive()
return
def solve(W, P,
mu,
u_bcs, p_bcs,
f,
verbose=True,
tol=1.0e-10
):
# Some initial sanity checks.
assert mu > 0.0
WP = MixedFunctionSpace([W, P])
# Translate the boundary conditions into the product space.
# This conditional loop is able to deal with conditions of the kind
#
# DirichletBC(W.sub(1), 0.0, right_boundary)
#
new_bcs = []
for k, bcs in enumerate([u_bcs, p_bcs]):
for bc in bcs:
space = bc.function_space()
C = space.component()
if len(C) == 0:
new_bcs.append(DirichletBC(WP.sub(k),
bc.value(),
bc.domain_args[0]))
elif len(C) == 1:
new_bcs.append(DirichletBC(WP.sub(k).sub(int(C[0])),
bc.value(),
bc.domain_args[0]))
else:
raise RuntimeError('Illegal number of subspace components.')
# Define variational problem
(u, p) = TrialFunctions(WP)
(v, q) = TestFunctions(WP)
# Build system.
# The sign of the div(u)-term is somewhat arbitrary since the right-hand
# side is 0 here. We can either make the system symmetric or positive-
# definite.
# On a second note, we have
#
# \int grad(p).v = - \int p * div(v) + \int_\Gamma p n.v.
#
# Since, we have either p=0 or n.v=0 on the boundary, we could as well
# replace the term dot(grad(p), v) by -p*div(v).
#
a = mu * inner(grad(u), grad(v))*dx \
- p * div(v) * dx \
- q * div(u) * dx
#a = mu * inner(grad(u), grad(v))*dx + dot(grad(p), v) * dx \
# - div(u) * q * dx
L = dot(f, v)*dx
A, b = assemble_system(a, L, new_bcs)
if has_petsc():
# For an assortment of preconditioners, see
#
# Performance and analysis of saddle point preconditioners
# for the discrete steady-state Navier-Stokes equations;
# H.C. Elman, D.J. Silvester, A.J. Wathen;
# Numer. Math. (2002) 90: 665-688;
# <http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.145.3554>.
#
# Set up field split.
W = SubSpace(WP, 0)
P = SubSpace(WP, 1)
u_dofs = W.dofmap().dofs()
p_dofs = P.dofmap().dofs()
prec = PETScPreconditioner()
prec.set_fieldsplit([u_dofs, p_dofs], ['u', 'p'])
PETScOptions.set('pc_type', 'fieldsplit')
PETScOptions.set('pc_fieldsplit_type', 'additive')
PETScOptions.set('fieldsplit_u_pc_type', 'lu')
PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
## <http://scicomp.stackexchange.com/questions/7288/which-preconditioners-and-solver-in-petsc-for-indefinite-symmetric-systems-sho>
#PETScOptions.set('pc_type', 'fieldsplit')
##PETScOptions.set('pc_fieldsplit_type', 'schur')
##PETScOptions.set('pc_fieldsplit_schur_fact_type', 'upper')
#PETScOptions.set('pc_fieldsplit_detect_saddle_point')
##PETScOptions.set('fieldsplit_u_pc_type', 'lsc')
##PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('fieldsplit_u_pc_type', 'hypre')
#PETScOptions.set('fieldsplit_u_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_p_pc_type', 'jacobi')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
## From PETSc/src/ksp/ksp/examples/tutorials/ex42-fsschur.opts:
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'SCHUR')
#PETScOptions.set('pc_fieldsplit_schur_fact_type', 'UPPER')
#PETScOptions.set('fieldsplit_p_ksp_type', 'preonly')
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
## From
##
## Composable Linear Solvers for Multiphysics;
## J. Brown, M. Knepley, D.A. May, L.C. McInnes, B. Smith;
## <http://www.computer.org/csdl/proceedings/ispdc/2012/4805/00/4805a055-abs.html>;
## <http://www.mcs.anl.gov/uploads/cels/papers/P2017-0112.pdf>.
##
#PETScOptions.set('pc_type', 'fieldsplit')
#PETScOptions.set('pc_fieldsplit_type', 'schur')
#PETScOptions.set('pc_fieldsplit_schur_factorization_type', 'upper')
##
#PETScOptions.set('fieldsplit_u_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_u_ksp_rtol', 1.0e-6)
#PETScOptions.set('fieldsplit_u_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_u_sub_pc_type', 'cholesky')
##
#PETScOptions.set('fieldsplit_p_ksp_type', 'fgmres')
#PETScOptions.set('fieldsplit_p_ksp_constant_null_space')
#PETScOptions.set('fieldsplit_p_pc_type', 'lsc')
##
#PETScOptions.set('fieldsplit_p_lsc_ksp_type', 'cg')
#PETScOptions.set('fieldsplit_p_lsc_ksp_rtol', 1.0e-2)
#PETScOptions.set('fieldsplit_p_lsc_ksp_constant_null_space')
##PETScOptions.set('fieldsplit_p_lsc_ksp_converged_reason')
#PETScOptions.set('fieldsplit_p_lsc_pc_type', 'bjacobi')
#PETScOptions.set('fieldsplit_p_lsc_sub_pc_type', 'icc')
# Create Krylov solver with custom preconditioner.
solver = PETScKrylovSolver('gmres', prec)
solver.set_operator(A)
else:
# Use the preconditioner as recommended in
# <http://fenicsproject.org/documentation/dolfin/dev/python/demo/pde/stokes-iterative/python/documentation.html>,
#
# prec = inner(grad(u), grad(v))*dx - p*q*dx
#
# although it doesn't seem to be too efficient.
# The sign on the last term doesn't matter.
prec = mu * inner(grad(u), grad(v))*dx \
- p*q*dx
M, _ = assemble_system(prec, L, new_bcs)
#solver = KrylovSolver('tfqmr', 'amg')
solver = KrylovSolver('gmres', 'amg')
solver.set_operators(A, M)
solver.parameters['monitor_convergence'] = verbose
solver.parameters['report'] = verbose
solver.parameters['absolute_tolerance'] = 0.0
solver.parameters['relative_tolerance'] = tol
solver.parameters['maximum_iterations'] = 500
# Solve
up = Function(WP)
solver.solve(up.vector(), b)
# Get sub-functions
u, p = up.split()
return u, p
def __init__(self, mesh, Vh, t_init, t_final, t_1, dt, wind_velocity, gls_stab, Prior):
self.mesh = mesh
self.Vh = Vh
self.t_init = t_init
self.t_final = t_final
self.t_1 = t_1
self.dt = dt
self.sim_times = np.arange(self.t_init, self.t_final+.5*self.dt, self.dt)
u = dl.TrialFunction(Vh[STATE])
v = dl.TestFunction(Vh[STATE])
kappa = dl.Constant(.001)
dt_expr = dl.Constant(self.dt)
r_trial = u + dt_expr*( -dl.div(kappa*dl.nabla_grad(u))+ dl.inner(wind_velocity, dl.nabla_grad(u)) )
r_test = v + dt_expr*( -dl.div(kappa*dl.nabla_grad(v))+ dl.inner(wind_velocity, dl.nabla_grad(v)) )
h = dl.CellSize(mesh)
vnorm = dl.sqrt(dl.inner(wind_velocity, wind_velocity))
if gls_stab:
tau = dl.Min((h*h)/(dl.Constant(2.)*kappa), h/vnorm )
else:
tau = dl.Constant(0.)
self.M = dl.assemble( dl.inner(u,v)*dl.dx )
self.M_stab = dl.assemble( dl.inner(u, v+tau*r_test)*dl.dx )
self.Mt_stab = dl.assemble( dl.inner(u+tau*r_trial,v)*dl.dx )
Nvarf = (dl.inner(kappa *dl.nabla_grad(u), dl.nabla_grad(v)) + dl.inner(wind_velocity, dl.nabla_grad(u))*v )*dl.dx
Ntvarf = (dl.inner(kappa *dl.nabla_grad(v), dl.nabla_grad(u)) + dl.inner(wind_velocity, dl.nabla_grad(v))*u )*dl.dx
self.N = dl.assemble( Nvarf )
self.Nt = dl.assemble(Ntvarf)
stab = dl.assemble( tau*dl.inner(r_trial, r_test)*dl.dx)
self.L = self.M + dt*self.N + stab
self.Lt = self.M + dt*self.Nt + stab
boundaries = dl.FacetFunction("size_t", mesh)
boundaries.set_all(0)
class InsideBoundary(dl.SubDomain):
def inside(self,x,on_boundary):
x_in = x[0] > dl.DOLFIN_EPS and x[0] < 1 - dl.DOLFIN_EPS
y_in = x[1] > dl.DOLFIN_EPS and x[1] < 1 - dl.DOLFIN_EPS
return on_boundary and x_in and y_in
Gamma_M = InsideBoundary()
Gamma_M.mark(boundaries,1)
ds_marked = dl.Measure("ds", subdomain_data=boundaries)
self.Q = dl.assemble( self.dt*dl.inner(u, v) * ds_marked(1) )
self.Prior = Prior
if dlversion() <= (1,6,0):
self.solver = dl.PETScLUSolver()
else:
self.solver = dl.PETScLUSolver(self.mesh.mpi_comm())
self.solver.set_operator(self.L)
if dlversion() <= (1,6,0):
self.solvert = dl.PETScLUSolver()
else:
self.solvert = dl.PETScLUSolver(self.mesh.mpi_comm())
self.solvert.set_operator(self.Lt)
self.ud = self.generate_vector(STATE)
self.noise_variance = 0
# Part of model public API
self.gauss_newton_approx = False
def get_stokessysmats(V, Q, nu=None, bccontrol=False,
cbclist=None, cbshapefuns=None):
""" Assembles the system matrices for Stokes equation
in mixed FEM formulation, namely
.. math::
\\begin{bmatrix} A & -J' \\\\ J & 0 \\end{bmatrix}\
\\colon V \\times Q \\to V' \\times Q'
as the discrete representation of
.. math::
\\begin{bmatrix} -\\Delta & \\text{grad} \\\\ \
\\text{div} & 0 \\end{bmatrix}
plus the velocity and pressure mass matrices
for a given trial and test space W = V * Q
not considering boundary conditions.
Parameters
----------
V : dolfin.VectorFunctionSpace
Fenics VectorFunctionSpace for the velocity
Q : dolfin.FunctionSpace
Fenics FunctionSpace for the pressure
nu : float, optional
viscosity parameter - defaults to 1
bccontrol : boolean, optional
whether boundary control (via penalized Robin conditions)
is applied, defaults to `False`
cbclist : list, optional
list of dolfin's Subdomain classes describing the control boundaries
cbshapefuns : list, optional
list of spatial shape functions of the control boundaries
Returns
-------
stokesmats, dictionary
a dictionary with the following keys:
* ``M``: the mass matrix of the velocity space,
* ``A``: the stiffness matrix \
:math:`\\nu \\int_\\Omega (\\nabla \\phi_i, \\nabla \\phi_j)`
* ``JT``: the gradient matrix,
* ``J``: the divergence matrix, and
* ``MP``: the mass matrix of the pressure space
* ``Apbc``: (N, N) sparse matrix, \
the contribution of the Robin conditions to `A` \
:math:`\\nu \\int_\\Gamma (\\phi_i, \\phi_j)`
* ``Bpbc``: (N, k) sparse matrix, the input matrix of the Robin \
conditions :math:`\\nu \\int_\\Gamma (\\phi_i, g_k)`, \
where :math:`g_k` is the shape function associated with the \
j-th control boundary segment
"""
u = dolfin.TrialFunction(V)
p = dolfin.TrialFunction(Q)
v = dolfin.TestFunction(V)
q = dolfin.TestFunction(Q)
if nu is None:
nu = 1
print('No viscosity provided -- we set `nu=1`')
ma = inner(u, v) * dx
mp = inner(p, q) * dx
aa = nu * inner(grad(u), grad(v)) * dx
grada = div(v) * p * dx
diva = q * div(u) * dx
# Assemble system
M = dolfin.assemble(ma)
A = dolfin.assemble(aa)
Grad = dolfin.assemble(grada)
Div = dolfin.assemble(diva)
MP = dolfin.assemble(mp)
# Convert DOLFIN representation to scipy arrays
Ma = mat_dolfin2sparse(M)
MPa = mat_dolfin2sparse(MP)
Aa = mat_dolfin2sparse(A)
JTa = mat_dolfin2sparse(Grad)
Ja = mat_dolfin2sparse(Div)
stokesmats = {'M': Ma,
'A': Aa,
'JT': JTa,
'J': Ja,
'MP': MPa}
if bccontrol:
amatrobl, bmatrobl = [], []
mesh = V.mesh()
for bc, bcfun in zip(cbclist, cbshapefuns):
# get an instance of the subdomain class
Gamma = bc()
# bparts = dolfin.MeshFunction('size_t', mesh,
# mesh.topology().dim() - 1)
boundaries = dolfin.MeshFunction("size_t", mesh,
mesh.topology().dim()-1)
boundaries.set_all(0)
Gamma.mark(boundaries, 1)
ds = dolfin.Measure('ds', domain=mesh, subdomain_data=boundaries)
# Gamma.mark(bparts, 0)
# Robin boundary form
arob = dolfin.inner(u, v) * ds(1) # , subdomain_data=bparts)
brob = dolfin.inner(v, bcfun) * ds(1) # , subdomain_data=bparts)
amatrob = dolfin.assemble(arob) # , exterior_facet_domains=bparts)
bmatrob = dolfin.assemble(brob) # , exterior_facet_domains=bparts)
amatrob = mat_dolfin2sparse(amatrob)
amatrob.eliminate_zeros()
amatrobl.append(amatrob)
bmatrobl.append(bmatrob.get_local().reshape((V.dim(), 1)))
amatrob = amatrobl[0]
for amatadd in amatrobl[1:]:
amatrob = amatrob + amatadd
bmatrob = np.hstack(bmatrobl)
stokesmats.update({'amatrob': amatrob, 'bmatrob': bmatrob})
return stokesmats
def _rhs_strong(u, f, rho, mu):
'''Right-hand side of the Navier--Stokes momentum equation in strong form.
'''
return f \
- mu * div(grad(u)) \
- rho * (grad(u)*u + 0.5*div(u)*u)