def __init__(self, V, fixed_dims=[], direct_solve=False):
if isinstance(fixed_dims, int):
fixed_dims = [fixed_dims]
self.V = V
self.fixed_dims = fixed_dims
self.direct_solve = direct_solve
self.zero = fd.Constant(V.mesh().topological_dimension() * (0, ))
u = fd.TrialFunction(V)
v = fd.TestFunction(V)
self.zero_fun = fd.Function(V)
self.a = 1e-2 * \
fd.inner(u, v) * fd.dx + fd.inner(fd.sym(fd.grad(u)),
fd.sym(fd.grad(v))) * fd.dx
self.bc_fun = fd.Function(V)
if len(self.fixed_dims) == 0:
bcs = [fd.DirichletBC(self.V, self.bc_fun, "on_boundary")]
else:
bcs = []
for i in range(self.V.mesh().topological_dimension()):
if i in self.fixed_dims:
bcs.append(fd.DirichletBC(self.V.sub(i), 0, "on_boundary"))
else:
bcs.append(
fd.DirichletBC(self.V.sub(i), self.bc_fun.sub(i),
"on_boundary"))
self.A_ext = fd.assemble(self.a, bcs=bcs, mat_type="aij")
self.ls_ext = fd.LinearSolver(self.A_ext,
solver_parameters=self.get_params())
self.A_adj = fd.assemble(self.a,
bcs=fd.DirichletBC(self.V, self.zero,
"on_boundary"),
mat_type="aij")
self.ls_adj = fd.LinearSolver(self.A_adj,
solver_parameters=self.get_params())
def dirichlet_boundary_conditions(sim):
W = sim.solution_space
return [
fe.DirichletBC(W.sub(0), fe.Constant((0., 0.)), (1, 2, 3)),
fe.DirichletBC(W.sub(0), fe.Constant((1., 0.)), 4)
]
def dirichlet_boundary_conditions(self):
return [
fe.DirichletBC(self.solution_subspaces["u"], fe.Constant((0., 0.)),
(1, 2, 3)),
fe.DirichletBC(self.solution_subspaces["u"], fe.Constant((1., 0.)),
4)
]
def dirichlet_boundary_conditions(sim):
W = sim.function_space
return [
fe.DirichletBC(W.sub(1), (0., 0.), "on_boundary"),
fe.DirichletBC(W.sub(2), sim.hot_wall_temperature, 1),
fe.DirichletBC(W.sub(2), sim.cold_wall_temperature, 2)
]
def dirichlet_boundary_conditions(sim):
W = sim.solution_space
return [
fe.DirichletBC(sim.solution_subspaces["u"], fe.Constant((0, 0)),
(1, 2, 3)),
fe.DirichletBC(sim.solution_subspaces["u"], fe.Constant((1, 0)), 4)
]
def dirichlet_boundary_conditions(self):
W = self.solution_space
return [
fe.DirichletBC(W.sub(1), (0., 0.), "on_boundary"),
fe.DirichletBC(W.sub(2), 0.5, 1),
fe.DirichletBC(W.sub(2), -0.5, 2)
]
def temperature_solver(T, t, beta, PeInv, solver_parameters, *, t1, INLET1, t2,
INLET2):
F_T = AdvectionDiffusionGLS(T, beta, t, PeInv=PeInv)
bc1 = fd.DirichletBC(T, t1, INLET1)
bc2 = fd.DirichletBC(T, t2, INLET2)
bcs = [bc1, bc2]
problem_T = fd.NonlinearVariationalProblem(F_T, t, bcs=bcs)
return fd.NonlinearVariationalSolver(problem_T,
solver_parameters=solver_parameters)
def dirichlet_boundary_conditions(sim, manufactured_solution):
"""Apply velocity and temperature Dirichlet BC's on every boundary.
Do not apply Dirichlet BC's on the pressure.
"""
_, u, T = manufactured_solution
return [
fe.DirichletBC(sim.solution_subspaces["u"], u, "on_boundary"),
fe.DirichletBC(sim.solution_subspaces["T"], T, "on_boundary")]
def dirichlet_boundary_conditions(self):
W = self.solution_space
return [
fe.DirichletBC(W.sub(1), (0., ) * self.mesh.geometric_dimension(),
"on_boundary"),
fe.DirichletBC(W.sub(2), self.hotwall_temperature, 1),
fe.DirichletBC(W.sub(2), self.initial_temperature, 2)
]
def dirichlet_boundary_conditions(sim):
W = sim.function_space
dim = sim.mesh.geometric_dimension()
return [
fe.DirichletBC(W.sub(1), (0., ) * dim, "on_boundary"),
fe.DirichletBC(W.sub(2), sim.hot_wall_temperature, 1),
fe.DirichletBC(W.sub(2), sim.cold_wall_temperature, 2)
]
def dirichlet_boundary_conditions(self):
W = self.solution.function_space()
d = self.solution.function_space().mesh().geometric_dimension()
return [
fe.DirichletBC(W.sub(1), (0., ) * d, "on_boundary"),
fe.DirichletBC(W.sub(2), self.hotwall_temperature,
self.hotwall_id),
fe.DirichletBC(W.sub(2), self.coldwall_temperature,
self.coldwall_id)
]
def setup_bcs(self):
x, y = fd.SpatialCoordinate(self.mesh)
bcu = [
fd.DirichletBC(self.V['u'], fd.Constant((0, 0)),
(1, 4)), # top-bottom and cylinder
fd.DirichletBC(self.V['u'],
((4.0 * 1.5 * y * (0.41 - y) / 0.41**2), 0), 2)
] # inflow
bcp = [fd.DirichletBC(self.V['p'], fd.Constant(0), 3)] # outflow
self.bc['u'][0] = [bcu, None, None, None, 'fixed']
self.bc['p'] = [[bcp, None, None, None, 'fixed']]
def setup_bcs(self):
x, y = fd.SpatialCoordinate(self.mesh)
bcu = [
fd.DirichletBC(self.V['u'], fd.Constant((0, 0)),
(10, 12)), # top-bottom
fd.DirichletBC(self.V['u'],
((1.0 * (y - 1) * (2 - y)) / (0.5**2), 0), 9)
] # inflow
bcp = [fd.DirichletBC(self.V['p'], fd.Constant(0), 11)] # outflow
self.bc['u'][0] = [bcu, None, None, None, 'fixed']
self.bc['p'] = [[bcp, None, None, None, 'fixed']]
def __init__(self, mesh_m, viscosity):
super().__init__()
self.mesh_m = mesh_m
self.failed_to_solve = False # when self.solver.solve() fail
# Setup problem, Taylor-Hood finite elements
self.V = fd.VectorFunctionSpace(self.mesh_m, "CG", 2) \
* fd.FunctionSpace(self.mesh_m, "CG", 1)
# Preallocate solution variables for state equation
self.solution = fd.Function(self.V, name="State")
self.testfunction = fd.TestFunction(self.V)
# Define viscosity parameter
self.viscosity = viscosity
# Weak form of incompressible Navier-Stokes equations
z = self.solution
u, p = fd.split(z)
test = self.testfunction
v, q = fd.split(test)
nu = self.viscosity # shorten notation
self.F = nu*fd.inner(fd.grad(u), fd.grad(v))*fd.dx - p*fd.div(v)*fd.dx\
+ fd.inner(fd.dot(fd.grad(u), u), v)*fd.dx + fd.div(u)*q*fd.dx
# Dirichlet Boundary conditions
X = fd.SpatialCoordinate(self.mesh_m)
dim = self.mesh_m.topological_dimension()
if dim == 2:
uin = 4 * fd.as_vector([(1 - X[1]) * X[1], 0])
elif dim == 3:
rsq = X[0]**2 + X[1]**2 # squared radius = 0.5**2 = 1/4
uin = fd.as_vector([0, 0, 1 - 4 * rsq])
else:
raise NotImplementedError
self.bcs = [
fd.DirichletBC(self.V.sub(0), 0., [12, 13]),
fd.DirichletBC(self.V.sub(0), uin, [10])
]
# PDE-solver parameters
self.nsp = None
self.params = {
"snes_max_it": 10,
"mat_type": "aij",
"pc_type": "lu",
"pc_factor_mat_solver_type": "superlu_dist",
# "snes_monitor": None, "ksp_monitor": None,
}
def setup(self, **kwargs):
for name, field in kwargs.items():
if name in self._fields.keys():
self._fields[name].assign(field)
else:
self._fields[name] = utilities.copy(field)
# Create homogeneous BCs for the Dirichlet part of the boundary
u = self._fields.get('velocity', self._fields.get('u'))
V = u.function_space()
bcs = firedrake.DirichletBC(V, u, self._dirichlet_ids)
if not self._dirichlet_ids:
bcs = None
# Find the numeric IDs for the ice front
boundary_ids = u.ufl_domain().exterior_facets.unique_markers
ice_front_ids_comp = set(self._dirichlet_ids + self._side_wall_ids)
ice_front_ids = list(set(boundary_ids) - ice_front_ids_comp)
# Create the action and scale functionals
_kwargs = {
'side_wall_ids': self._side_wall_ids,
'ice_front_ids': ice_front_ids
}
action = self._model.action(**self._fields, **_kwargs)
F = firedrake.derivative(action, u)
degree = self._model.quadrature_degree(**self._fields)
params = {'form_compiler_parameters': {'quadrature_degree': degree}}
problem = firedrake.NonlinearVariationalProblem(F, u, bcs, **params)
self._solver = firedrake.NonlinearVariationalSolver(
problem, solver_parameters=self._solver_parameters)
def __init__(self, mesh_m):
super().__init__()
self.mesh_m = mesh_m
# Setup problem
self.V = fd.FunctionSpace(self.mesh_m, "CG", 1)
# Preallocate solution variables for state and adjoint equations
self.solution = fd.Function(self.V, name="State")
# Weak form of Poisson problem
u = self.solution
v = fd.TestFunction(self.V)
self.f = fd.Constant(4.)
self.F = (fd.inner(fd.grad(u), fd.grad(v)) - self.f * v) * fd.dx
self.bcs = fd.DirichletBC(self.V, 0., "on_boundary")
# PDE-solver parameters
self.params = {
"ksp_type": "cg",
"mat_type": "aij",
"pc_type": "hypre",
"pc_factor_mat_solver_package": "boomerang",
"ksp_rtol": 1e-11,
"ksp_atol": 1e-11,
"ksp_stol": 1e-15,
}
stateproblem = fd.NonlinearVariationalProblem(self.F,
self.solution,
bcs=self.bcs)
self.solver = fd.NonlinearVariationalSolver(
stateproblem, solver_parameters=self.params)
def get_mu(self, V):
W = fd.FunctionSpace(V.mesh(), "CG", 1)
bcs = []
if len(self.fixed_bids):
bcs.append(fd.DirichletBC(W, 1, self.fixed_bids))
if len(self.free_bids):
bcs.append(fd.DirichletBC(W, 10, self.free_bids))
if len(bcs) == 0:
bcs = None
u = fd.TrialFunction(W)
v = fd.TestFunction(W)
a = fd.inner(fd.grad(u), fd.grad(v)) * fd.dx
b = fd.inner(fd.Constant(0.), v) * fd.dx
mu = fd.Function(W)
fd.solve(a == b, mu, bcs=bcs)
return mu
def __init__(self, mesh_m):
super().__init__()
# Setup problem
V = fd.FunctionSpace(mesh_m, "CG", 1)
# Weak form of Poisson problem
u = fd.Function(V, name="State")
v = fd.TestFunction(V)
f = fd.Constant(4.)
self.F = (fd.inner(fd.grad(u), fd.grad(v)) - f * v) * fd.dx
self.bcs = fd.DirichletBC(V, 0., "on_boundary")
# PDE-solver parameters
self.params = {
"ksp_type": "cg",
"mat_type": "aij",
"pc_type": "hypre",
"pc_factor_mat_solver_package": "boomerang",
"ksp_rtol": 1e-11,
"ksp_atol": 1e-11,
"ksp_stol": 1e-15,
}
self.solution = u
def setup_bcs(self):
x, y = fd.SpatialCoordinate(self.mesh)
c0 = fd.exp(x * y * self.t)
bcu = [
fd.DirichletBC(self.V['u'], fd.Constant((0, 0)),
(10, 12)), # top-bottom and cylinder
fd.DirichletBC(self.V['u'],
((1.0 * (y - 1) * (2 - y)) / (0.5**2), 0), 9)
] # inflow
bcp = [fd.DirichletBC(self.V['p'], fd.Constant(0), 11)] # outflow
bcc1 = [fd.DirichletBC(self.V['c'].sub(0), c0, 'on_boundary')]
self.bc['u'][0] = [bcu, None, None, None, 'fixed']
self.bc['p'] = [[bcp, None, None, None, 'fixed']]
self.bc['c'][0] = [bcc1, c0, 'on_boundary', 0, 'update']
def _newton_solve(z, E, scale, tolerance=1e-6, armijo=1e-4, max_iterations=50):
F = derivative(E, z)
H = derivative(F, z)
Q = z.function_space()
bc = firedrake.DirichletBC(Q, 0, 'on_boundary')
p = firedrake.Function(Q)
for iteration in range(max_iterations):
firedrake.solve(H == -F, p, bc,
solver_parameters={'ksp_type': 'preonly', 'pc_type': 'lu'})
dE_dp = assemble(action(F, p))
α = 1.0
E0 = assemble(E)
Ez = assemble(replace(E, {z: z + firedrake.Constant(α) * p}))
while (Ez > E0 + armijo * α * dE_dp) or np.isnan(Ez):
α /= 2
Ez = assemble(replace(E, {z: z + firedrake.Constant(α) * p}))
z.assign(z + firedrake.Constant(α) * p)
if abs(dE_dp) < tolerance * assemble(scale):
return z
raise ValueError("Newton solver failed to converge after {0} iterations"
.format(max_iterations))
def get_boundary_conditions(self):
"""Impose Dirichlet boundary conditions."""
dim = self.mesh_m.cell_dimension()
if dim == 2:
zerovector = fd.Constant((0.0, 0.0))
elif dim == 3:
zerovector = fd.Constant((0.0, 0.0, 0.0))
bcs = []
if len(self.inflow_bids) is not None:
bcs.append(
fd.DirichletBC(self.V.sub(0), self.inflow_expr,
self.inflow_bids))
if len(self.noslip_bids) > 0:
bcs.append(
fd.DirichletBC(self.V.sub(0), zerovector, self.noslip_bids))
return bcs
def setup(self, **kwargs):
for name, field in kwargs.items():
if name in self._fields.keys():
self._fields[name].assign(field)
else:
if isinstance(field, firedrake.Constant):
self._fields[name] = firedrake.Constant(field)
elif isinstance(field, firedrake.Function):
self._fields[name] = field.copy(deepcopy=True)
else:
raise TypeError(
"Input %s field has type %s, must be Constant or Function!"
% (name, type(field))
)
# Create homogeneous BCs for the Dirichlet part of the boundary
u = self._fields["velocity"]
V = u.function_space()
# NOTE: This will have to change when we do Stokes!
if hasattr(V._ufl_element, "_sub_element"):
bcs = firedrake.DirichletBC(V, Constant((0, 0)), self._dirichlet_ids)
else:
bcs = firedrake.DirichletBC(V, Constant(0), self._dirichlet_ids)
if not self._dirichlet_ids:
bcs = None
# Find the numeric IDs for the ice front
boundary_ids = u.ufl_domain().exterior_facets.unique_markers
ice_front_ids_comp = set(self._dirichlet_ids + self._side_wall_ids)
ice_front_ids = list(set(boundary_ids) - ice_front_ids_comp)
# Create the action and scale functionals
_kwargs = {"side_wall_ids": self._side_wall_ids, "ice_front_ids": ice_front_ids}
action = self._model.action(**self._fields, **_kwargs)
scale = self._model.scale(**self._fields, **_kwargs)
# Set up a minimization problem and solver
quadrature_degree = self._model.quadrature_degree(**self._fields)
params = {"quadrature_degree": quadrature_degree}
problem = MinimizationProblem(action, scale, u, bcs, params)
self._solver = NewtonSolver(
problem,
self._tolerance,
solver_parameters=self._solver_parameters,
max_iterations=self._max_iterations,
)
def __init__(self, function_space, state, boundary):
self.state = state.clone()
self.values = fire.Function(function_space)
self.dofs = function_space.boundary_nodes(boundary, "topological")
self.dirichlet = fire.DirichletBC(function_space,
self.values,
boundary,
method="topological")
def dirichlet_boundary_conditions(sim, manufactured_solution):
"""Apply velocity Dirichlet BC's on every boundary."""
p, u = manufactured_solution
return [fe.DirichletBC(
sim.solution_subspaces["u"],
u,
"on_boundary"),]
def dirichlet_boundary_conditions(sim, manufactured_solution):
"""Apply velocity Dirichlet BC's on every boundary."""
W = sim.solution_space
u, p = manufactured_solution
return [
fe.DirichletBC(W.sub(0), u, "on_boundary"),
]
def test_solver_no_flow_region():
mesh = fd.Mesh("./2D_mesh.msh")
no_flow = [2]
no_flow_markers = [1]
mesh = mark_no_flow_regions(mesh, no_flow, no_flow_markers)
P2 = fd.VectorElement("CG", mesh.ufl_cell(), 1)
P1 = fd.FiniteElement("CG", mesh.ufl_cell(), 1)
TH = P2 * P1
W = fd.FunctionSpace(mesh, TH)
(v, q) = fd.TestFunctions(W)
# Stokes 1
w_sol1 = fd.Function(W)
nu = fd.Constant(0.05)
F = NavierStokesBrinkmannForm(W, w_sol1, nu, beta_gls=2.0)
x, y = fd.SpatialCoordinate(mesh)
u_mms = fd.as_vector(
[sin(2.0 * pi * x) * sin(pi * y),
sin(pi * x) * sin(2.0 * pi * y)])
p_mms = -0.5 * (u_mms[0]**2 + u_mms[1]**2)
f_mms_u = (grad(u_mms) * u_mms + grad(p_mms) -
2.0 * nu * div(sym(grad(u_mms))))
f_mms_p = div(u_mms)
F += -inner(f_mms_u, v) * dx - f_mms_p * q * dx
bc1 = fd.DirichletBC(W.sub(0), u_mms, "on_boundary")
bc2 = fd.DirichletBC(W.sub(1), p_mms, "on_boundary")
bc_no_flow = InteriorBC(W.sub(0), fd.Constant((0.0, 0.0)), no_flow_markers)
solver_parameters = {"ksp_max_it": 500, "ksp_monitor": None}
problem1 = fd.NonlinearVariationalProblem(F,
w_sol1,
bcs=[bc1, bc2, bc_no_flow])
solver1 = NavierStokesBrinkmannSolver(
problem1,
options_prefix="navier_stokes",
solver_parameters=solver_parameters,
)
solver1.solve()
u_sol, _ = w_sol1.split()
u_mms_func = fd.interpolate(u_mms, W.sub(0))
error = fd.errornorm(u_sol, u_mms_func)
assert error < 0.07
def run_solver(r):
mesh = fd.UnitSquareMesh(2**r, 2**r)
P2 = fd.VectorElement("CG", mesh.ufl_cell(), 1)
P1 = fd.FiniteElement("CG", mesh.ufl_cell(), 1)
TH = P2 * P1
W = fd.FunctionSpace(mesh, TH)
(v, q) = fd.TestFunctions(W)
# Stokes 1
w_sol1 = fd.Function(W)
nu = fd.Constant(0.05)
F = NavierStokesBrinkmannForm(W, w_sol1, nu, beta_gls=2.0)
from firedrake import sin, grad, pi, sym, div, inner
x, y = fd.SpatialCoordinate(mesh)
u_mms = fd.as_vector(
[sin(2.0 * pi * x) * sin(pi * y),
sin(pi * x) * sin(2.0 * pi * y)])
p_mms = -0.5 * (u_mms[0]**2 + u_mms[1]**2)
f_mms_u = (grad(u_mms) * u_mms + grad(p_mms) -
2.0 * nu * div(sym(grad(u_mms))))
f_mms_p = div(u_mms)
F += -inner(f_mms_u, v) * dx - f_mms_p * q * dx
bc1 = fd.DirichletBC(W.sub(0), u_mms, "on_boundary")
bc2 = fd.DirichletBC(W.sub(1), p_mms, "on_boundary")
solver_parameters = {"ksp_max_it": 200}
problem1 = fd.NonlinearVariationalProblem(F, w_sol1, bcs=[bc1, bc2])
solver1 = NavierStokesBrinkmannSolver(
problem1,
options_prefix="navier_stokes",
solver_parameters=solver_parameters,
)
solver1.solve()
u_sol, _ = w_sol1.split()
fd.File("test_u_sol.pvd").write(u_sol)
u_mms_func = fd.interpolate(u_mms, W.sub(0))
error = fd.errornorm(u_sol, u_mms_func)
print(f"Error: {error}")
return error
def mms_dirichlet_boundary_conditions(sim, manufactured_solution):
W = sim.function_space
if type(sim.element) is fe.FiniteElement:
w = (manufactured_solution, )
else:
w = manufactured_solution
return [fe.DirichletBC(V, g, "on_boundary") for V, g in zip(W, w)]
def coarsen_bcs(fbcs):
cbcs = []
for bc in fbcs:
cV_ = cV
for index in bc._indices:
cV_ = cV_.sub(index)
cbc_value = self.coarsen_bc_value(bc, cV_)
if type(bc) == firedrake.DirichletBC:
cbcs.append(firedrake.DirichletBC(cV_, cbc_value,
bc.sub_domain))
else:
raise NotImplementedError("Unsupported BC type, please get in touch if you need this")
return cbcs
def default_mms_dirichlet_boundary_conditions(sim, manufactured_solution):
"""Apply Dirichlet BC's to every component on every boundary."""
W = sim.solution_space
if type(W.ufl_element()) is fe.FiniteElement:
w = (manufactured_solution, )
else:
w = manufactured_solution
return [fe.DirichletBC(V, g, "on_boundary") for V, g in zip(W, w)]