def __init__(self, model, particle, variation, kernel, options, comm):
self.model = model
self.particle = particle # number of particles
self.variation = variation
self.kernel = kernel
self.options = options
self.comm = comm
self.rank = comm.Get_rank()
self.nproc = comm.Get_size()
self.save_kernel = options["save_kernel"]
self.type_Hessian = options["type_Hessian"]
self.ncalls = 0
self.gls = GlobalLocalSwitch(self.model, self.particle, self.options,
self.comm)
if not self.options["is_projection"]:
if self.particle.number_particles_all == 1:
z_trial = dl.TrialFunction(self.gls.Vh_Local)
z_test = dl.TestFunction(self.gls.Vh_Local)
self.Hessian = dl.assemble(z_trial * z_test * dl.dx)
else:
z_vec = [None] * particle.number_particles_all
c_vec = [None] * particle.number_particles_all
for n in range(particle.number_particles_all):
z_trial = dl.TrialFunction(self.gls.Vh_Global.sub(0))
z_test = dl.TestFunction(self.gls.Vh_Global.sub(0))
z_vec[n] = z_trial
c_vec[n] = z_test
z_vec = dl.as_vector(z_vec)
c_vec = dl.as_vector(c_vec)
self.Hessian = dl.assemble(dl.dot(c_vec, z_vec) * dl.dx)
def bulk_electrostriction(E, e_r, p, direction, q_b):
""" calculate the bulk electrostriction force """
pp = as_matrix(p) # photoelestic tensor is stored as as numpy array
if direction == 'backward':
EE_6vec_r = as_vector([
E[0] * E[0], E[1] * E[1], -E[2] * E[2], 0.0, 0.0, 2.0 * E[0] * E[1]
])
EE_6vec_i = as_vector(
[0.0, 0.0, 0.0, 2.0 * E[1] * E[2], 2.0 * E[0] * E[2], 0.0])
sigma_r = -0.5 * epsilon_0 * e_r**2 * pp * EE_6vec_r
sigma_i = -0.5 * epsilon_0 * e_r**2 * pp * EE_6vec_i
f_r = f_elst_r(sigma_r, sigma_i, q_b)
f_i = f_elst_i(sigma_r, sigma_i, q_b)
elif direction == 'forward':
EE_6vec_r = as_vector([
E[0] * E[0], E[1] * E[1], E[2] * E[2], 0.0, 0.0, 2.0 * E[0] * E[1]
])
# EE_6vec_i, sigma_i, q_b is zero
sigma_r = -0.5 * epsilon_0 * e_r**2 * pp * EE_6vec_r
# no need to multiply zeros ...
f_r = as_vector([
-Dx(sigma_r[0], 0) - Dx(sigma_r[5], 1),
-Dx(sigma_r[5], 0) - Dx(sigma_r[1], 1),
-Dx(sigma_r[4], 0) - Dx(sigma_r[3], 1)
])
f_i = Constant((0.0, 0.0, 0.0), cell=triangle)
#f_i = as_vector([ - q_b*sigma_r[4],- q_b*sigma_r[3],- q_b*sigma_r[2]])
else:
raise ValueError('Specify scattering direction as forward or backward')
return (f_r, f_i)
def test_bdm_identity_transform(gdim):
N = 5
k = 2
# Create mesh and define a divergence free field
if gdim == 2:
mesh = dolfin.UnitSquareMesh(N, N)
field = ['-sin(pi*x[1])*cos(pi*x[0])', 'sin(pi*x[0])*cos(pi*x[1])']
elif gdim == 3:
mesh = dolfin.UnitCubeMesh(N, N, N)
field = [
'-sin(pi*x[0])*cos(pi*x[1])*sin(pi*x[2])',
'sin(pi*x[0])*cos(pi*x[1])*sin(pi*x[2])',
'-cos(pi*x[2])*cos(pi*(x[0]-x[1]))',
]
V = dolfin.FunctionSpace(mesh, 'DG', k)
u = dolfin.as_vector([dolfin.Function(V) for _ in range(gdim)])
a = dolfin.as_vector([dolfin.Function(V) for _ in range(gdim)])
# Make a divergence free field
for i, cpp in enumerate(field):
ei = dolfin.Expression(cpp, degree=k)
u[i].interpolate(ei)
a[i].assign(u[i])
# Run the projection into BDM and then assign this to u
bdm = VelocityBDMProjection(simulation=Simulation(), w=u, use_bcs=False)
bdm.run()
# Check the changes made
for ui, ai in zip(u, a):
error = dolfin.errornorm(ai, ui, degree_rise=0)
assert error < 1e-15
def permeability_tensor(self, K):
FS = self.geometry.f0.function_space()
TS = TensorFunctionSpace(self.geometry.mesh, 'P', 1)
d = self.geometry.dim()
fibers = Function(FS)
fibers.vector()[:] = self.geometry.f0.vector().get_local()
fibers.vector()[:] /= df.norm(self.geometry.f0)
if self.geometry.s0 is not None:
# normalize vectors
sheet = Function(FS)
sheet.vector()[:] = self.geometry.s0.vector().get_local()
sheet.vector()[:] /= df.norm(self.geometry.s0)
if d == 3:
csheet = Function(FS)
csheet.vector()[:] = self.geometry.n0.vector().get_local()
csheet.vector()[:] /= df.norm(self.geometry.n0)
else:
return Constant(1)
from ufl import diag
factor = 10
if d == 3:
ftensor = df.as_matrix(((fibers[0], sheet[0], csheet[0]),
(fibers[1], sheet[1], csheet[1]),
(fibers[2], sheet[2], csheet[2])))
ktensor = diag(df.as_vector([K, K / factor, K / factor]))
else:
ftensor = df.as_matrix(
((fibers[0], sheet[0]), (fibers[1], sheet[1])))
ktensor = diag(df.as_vector([K, K / factor]))
permeability = df.project(
df.dot(df.dot(ftensor, ktensor), df.inv(ftensor)), TS)
return permeability
def Displacement2Epsilon0(self, U):
"""Converti le localisateur en deplacement en champ de predeformation
a appliquer au probleme auxiliaire suivant"""
Epsilon0 = []
for i in range(len(U)):
Epsilon0 = Epsilon0 + [
fe.as_vector((
U[i][0],
fe.interpolate(fe.Constant(0.0), self.X),
U[i][1] / fe.sqrt(2),
))
]
# zero_ = fe.interpolate(fe.Constant(0.0), self.X)
# # Possibilité de mettre directement fe.Constant(0.0) ?
# prestrain_ = (U[i][0], zero_, U[i][1] / fe.sqrt(2))
# prestrain_ = fe.as_vector(prestrain_)
# Epsilon0.append(prestrain_)
for i in range(len(U)):
Epsilon0 = Epsilon0 + [
fe.as_vector((
fe.interpolate(fe.Constant(0.0), self.X),
U[i][1],
U[i][0] / fe.sqrt(2),
))
]
# zero_ = fe.interpolate(fe.Constant(0.0), self.X)
# # Possibilité de mettre directement fe.Constant(0.0) ?
# prestrain_ = (zero_, U[i][1], U[i][0] / fe.sqrt(2))
# prestrain_ = fe.as_vector(prestrain_)
# Epsilon0.append(prestrain_)
return Epsilon0
def test_is_zero_simple_vector_expressions():
mesh = UnitSquareMesh(4, 4)
V = FunctionSpace(mesh, 'CG', 1)
v = TestFunction(V)
u = TrialFunction(V)
check_is_zero(dot(as_vector([Zero(), u]), as_vector([Zero(), v])), 1)
check_is_zero(dot(as_vector([Zero(), u]), as_vector([v, Zero()])), 0)
def test_I5y(F_dict, F_str):
F = F_dict[F_str]
if "2D" in F_str:
y = df.as_vector([0, 1])
else:
y = df.as_vector([0, 1, 0])
assert abs(df.assemble(kinematics.I5(F, y) * df.dx) - B**4) < 1e-12
def test_I5x(F_dict, F_str):
F = F_dict[F_str]
if "2D" in F_str:
x = df.as_vector([1, 0])
else:
x = df.as_vector([1, 0, 0])
assert abs(df.assemble(kinematics.I5(F, x) * df.dx) - A**4) < 1e-12
def create_functions(self):
"""
Create functions to hold solutions
"""
sim = self.simulation
# Function spaces
Vu = sim.data['Vu']
Vp = sim.data['Vp']
cd = sim.data['constrained_domain']
# Create coupled mixed function space and mixed function to hold results
func_spaces = [Vu] * sim.ndim + [Vp]
self.subspace_names = ['u%d' % d for d in range(sim.ndim)] + ['p']
# Create stress tensor space
P = Vu.ufl_element().degree()
Vs = dolfin.FunctionSpace(sim.data['mesh'],
'DG',
P,
constrained_domain=cd)
for i in range(sim.ndim**2):
stress_name = 'stress_%d' % i
sim.data[stress_name] = dolfin.Function(Vs)
func_spaces.append(Vs)
self.subspace_names.append(stress_name)
# Create mixed space
e_mixed = dolfin.MixedElement([fs.ufl_element() for fs in func_spaces])
Vcoupled = dolfin.FunctionSpace(sim.data['mesh'], e_mixed)
sim.data['Vcoupled'] = Vcoupled
# Create function assigner
Nspace = len(func_spaces)
self.subspaces = [Vcoupled.sub(i) for i in range(Nspace)]
sim.data['coupled'] = self.coupled_func = dolfin.Function(Vcoupled)
self.assigner = dolfin.FunctionAssigner(func_spaces, Vcoupled)
# Create segregated functions on component and vector form
u_list, up_list, upp_list, u_conv = [], [], [], []
for d in range(sim.ndim):
sim.data['u%d' % d] = u = dolfin.Function(Vu)
sim.data['up%d' % d] = up = dolfin.Function(Vu)
sim.data['upp%d' % d] = upp = dolfin.Function(Vu)
sim.data['u_conv%d' % d] = uc = dolfin.Function(Vu)
u_list.append(u)
up_list.append(up)
upp_list.append(upp)
u_conv.append(uc)
sim.data['u'] = dolfin.as_vector(u_list)
sim.data['up'] = dolfin.as_vector(up_list)
sim.data['upp'] = dolfin.as_vector(upp_list)
sim.data['u_conv'] = dolfin.as_vector(u_conv)
sim.data['p'] = dolfin.Function(Vp)
def test_form_splitter_matrices(shape):
mesh = UnitSquareMesh(dolfin.MPI.comm_self, 3, 3)
Vu = FunctionSpace(mesh, 'DG', 2)
Vp = FunctionSpace(mesh, 'DG', 1)
def define_eq(u, v, p, q):
"A simple Stokes-like coupled weak form"
eq = Constant(2) * dot(u, v) * dx
eq += dot(grad(p), v) * dx
eq -= dot(Constant([1, 1]), v) * dx
eq += dot(grad(q), u) * dx
eq -= dot(Constant(0.3), q) * dx
return eq
if shape == (2, 2):
eu = MixedElement([Vu.ufl_element(), Vu.ufl_element()])
ew = MixedElement([eu, Vp.ufl_element()])
W = FunctionSpace(mesh, ew)
u, p = TrialFunctions(W)
v, q = TestFunctions(W)
u = as_vector([u[0], u[1]])
v = as_vector([v[0], v[1]])
elif shape == (3, 3):
ew = MixedElement(
[Vu.ufl_element(),
Vu.ufl_element(),
Vp.ufl_element()])
W = FunctionSpace(mesh, ew)
u0, u1, p = TrialFunctions(W)
v0, v1, q = TestFunctions(W)
u = as_vector([u0, u1])
v = as_vector([v0, v1])
eq = define_eq(u, v, p, q)
# Define coupled problem
eq = define_eq(u, v, p, q)
# Split the weak form into a saddle point block matrix system
mat, vec = split_form_into_matrix(eq, W, W)
# Check shape and that appropriate elements are none
assert mat.shape == shape
assert mat[-1, -1] is None
for i in range(shape[0] - 1):
for j in range(shape[1] - 1):
if i != j:
assert mat[i, j] is None
# Check that the split matrices are identical with the
# coupled matrix when reassembled into a single matrix
compare_split_matrices(eq, mat, vec, W, W)
def fit_primitives(vec, deepcopy=False, indexed=True):
assert not (deepcopy and indexed)
N = self.parameters["N"] # 'self' is visible from here
if indexed:
ws = split(vec[0])
pv = [as_vector(ws[:N - 1]), as_vector(ws[N - 1:])]
pv += list(split(vec[1]))
else:
ws = vec[0].split(deepcopy)
pv = [as_vector(ws[:N - 1]), as_vector(ws[N - 1:])]
pv += list(vec[1].split(deepcopy))
return tuple(pv)
def total_flux(Mo, rho_mat, chi):
"""
:returns: total flux
:rtype: :py:class:`ufl.core.expr.Expr`
"""
N = len(rho_mat)
rho_mat = list(map(Constant, rho_mat))
rho_diff = as_vector(rho_mat[:-1]) - as_vector((N - 1) * [
rho_mat[-1],
])
J = -Mo * dot(grad(chi).T, rho_diff)
return J
def test_I8xy(F_dict, F_str):
F = F_dict[F_str]
if "2D" in F_str:
x = df.as_vector([1, 0])
y = df.as_vector([0, 1])
else:
x = df.as_vector([1, 0, 0])
y = df.as_vector([0, 1, 0])
assert (
abs(df.assemble(kinematics.I8(F, x, y) * df.dx) -
(A**2 * 0 + 0 * B)) < 1e-12)
def define_coupled_equation(self):
"""
Setup the coupled Navier-Stokes equation
This implementation assembles the full LHS and RHS each time they are needed
"""
Vcoupled = self.simulation.data['Vcoupled']
# Unpack the coupled trial and test functions
uc = dolfin.TrialFunction(Vcoupled)
vc = dolfin.TestFunction(Vcoupled)
ulist = []
vlist = []
ndim = self.simulation.ndim
for d in range(ndim):
ulist.append(uc[d])
vlist.append(vc[d])
u = dolfin.as_vector(ulist)
v = dolfin.as_vector(vlist)
p = uc[ndim]
q = vc[ndim]
lm_trial = lm_test = None
if self.use_lagrange_multiplicator:
lm_trial = uc[ndim + 1]
lm_test = vc[ndim + 1]
assert self.flux_type == UPWIND
eq = define_dg_equations(
u,
v,
p,
q,
lm_trial,
lm_test,
self.simulation,
include_hydrostatic_pressure=self.include_hydrostatic_pressure,
incompressibility_flux_type=self.incompressibility_flux_type,
use_grad_q_form=self.use_grad_q_form,
use_grad_p_form=self.use_grad_p_form,
use_stress_divergence_form=self.use_stress_divergence_form,
velocity_continuity_factor_D12=self.velocity_continuity_factor_D12,
pressure_continuity_factor=self.pressure_continuity_factor,
)
a, L = dolfin.system(eq)
self.form_lhs = a
self.form_rhs = L
self.tensor_lhs = None
self.tensor_rhs = None
def eps(self, u):
e = df.sym(df.grad(u))
dim = self.mesh.geometric_dimension()
if dim == 1:
return df.as_vector([e[0, 0]])
if dim == 2:
return df.as_vector([e[0, 0], e[1, 1], 2 * e[0, 1]])
if dim == 3:
return df.as_vector([
e[0, 0], e[1, 1], e[2, 2], 2 * e[1, 2], 2 * e[0, 2],
2 * e[0, 1]
])
def fit_primitives(vec, deepcopy=False, indexed=True):
assert not (deepcopy and indexed)
N = self.parameters["N"] # 'self' is visible from here
if indexed:
ws = split(vec[0])
# FIXME: Remove the following hack
#ws_ch = split(ws[0]) # Not working with older versions of DOLFIN
ws_ch = tuple([ws[0][i] for i in range(len(ws[0]))])
else:
ws = vec[0].split(deepcopy)
ws_ch = ws[0].split(deepcopy)
pv = [as_vector(ws_ch[:N - 1]), as_vector(ws_ch[N - 1:])]
pv += ws[1:]
return tuple(pv)
def set_function_space(self):
"""set_function_space."""
self.V = df.VectorFunctionSpace(self.mesh,'P',1,dim=self.number_of_moment)
#Set test function(s)
v_list = df.TestFunctions(self.V)
#Convert to ufl form
self.v = df.as_vector(v_list)
#set trial function(s)
if self.problem_type == 'nonlinear':
u_list = df.Function(self.V)
elif self.problem_type == 'linear':
u_list = df.TrialFunctions(self.V)
#Convert to ufl form
self.u = df.as_vector(u_list)
def updateCoefficients(self):
# Init coefficient matrix
x = SpatialCoordinate(self.mesh)[0]
self.a = as_matrix([[.5 * (x * self.gamma[0] * self.sigmax)**2]])
self.b = as_vector([x * (self.gamma[0] * (self.mu - self.r) + self.r)])
self.c = Constant(0.0)
# Init right-hand side
self.f = Constant(0.0)
self.u_ = exp(
((self.mu - self.r)**2 / (2 * self.sigmax**2) * self.alpha /
(1 - self.alpha) + self.r * self.alpha) *
(self.T[1] - self.t)) * (x**self.alpha) / self.alpha
self.u_T = (x**self.alpha) / self.alpha
# Set boundary conditions
# self.g_t = lambda t : [(Constant(0.0), "near(x[0],0)")]
self.g = Constant(0.0)
# self.g_t = lambda t : self.u_t(t)
self.gamma_star = [
Constant((self.mu - self.r) / (self.sigmax**2 * (1 - self.alpha)))
]
self.loc = conditional(x > 0.5, conditional(x < 1.5, 1, 0), 0)
def setup(self):
"""
Create mesh velocity and deformation functions
"""
sim = self.simulation
assert self.active is False, 'Trying to setup mesh morphing twice in the same simulation'
# Store previous cell volumes
mesh = sim.data['mesh']
Vcvol = dolfin.FunctionSpace(mesh, 'DG', 0)
sim.data['cvolp'] = dolfin.Function(Vcvol)
# The function spaces for mesh velocities and displacements
Vmesh = dolfin.FunctionSpace(mesh, 'CG', 1)
Vmesh_vec = dolfin.VectorFunctionSpace(mesh, 'CG', 1)
sim.data['Vmesh'] = Vmesh
# Create mesh velocity functions
u_mesh = []
for d in range(sim.ndim):
umi = dolfin.Function(Vmesh)
sim.data['u_mesh%d' % d] = umi
u_mesh.append(umi)
u_mesh = dolfin.as_vector(u_mesh)
sim.data['u_mesh'] = u_mesh
# Create mesh displacement vector function
self.displacement = dolfin.Function(Vmesh_vec)
self.assigners = [
dolfin.FunctionAssigner(Vmesh_vec.sub(d), Vmesh)
for d in range(sim.ndim)
]
self.active = True
def sigma(self, uu): eps = self.RR*self.epsilon(uu)*self.RR.T ss = self.DD*dolfin.as_vector([eps[0, 0], eps[1, 1], eps[2, 2]]) ss_01 = 2*self.G_01*eps[0, 1] ss_02 = 2*self.G_02*eps[0, 2] ss_12 = 2*self.G_12*eps[1, 2] return self.RR.T*dolfin.as_matrix([[ss[0], ss_01, ss_02], [ss_01, ss[1], ss_12], [ss_02, ss_12, ss[2]]])*self.RR
def main():
args = get_settings()
interp = itp.Interpolation(args.mesh_file_in,
args.u_file_in,
p_filename_in=args.p_file_in)
interp.update(args.step)
u_1 = interp.u
p_1 = interp.p
mesh_2 = import_mesh(args.other_mesh_in)
S_2 = df.FunctionSpace(mesh_2, "CG", 1)
u_ = dict()
for key, val in u_1.iteritems():
u_[key] = ft.interpolate_nonmatching_mesh(val, S_2)
u__ = df.as_vector([u_[key] for key in u_.keys()])
u = AssignedVectorFunction(u__)
u()
p = ft.interpolate_nonmatching_mesh(p_1, S_2)
xdmff_u = df.XDMFFile(mesh_2.mpi_comm(), args.initfile_out + "_u.xdmf")
xdmff_u.parameters["rewrite_function_mesh"] = False
xdmff_u.parameters["flush_output"] = True
xdmff_u.write(u, 0.)
xdmff_p = df.XDMFFile(mesh_2.mpi_comm(), args.initfile_out + "_p.xdmf")
xdmff_p.parameters["rewrite_function_mesh"] = False
xdmff_p.parameters["flush_output"] = True
xdmff_p.write(p, 0.)
def steadystate(geo, phys, F, u0):
F = dolfin.as_vector([F])
bc = dict(upperb=dolfin.Constant(c0), lowerb=dolfin.Constant(c0))
pde = ConvectionDiffusionSteady(geo, phys, F=F, u0=u0, bc=bc)
pde.add_functionals([partial(current, F=F), selectivity])
pde.single_solve()
return pde
def get_collected_velocity_bcs(simulation, name):
"""
When mixed Dirichlet/Neumann BCs on the same facet (for different velocity
components) is not supported it can be convenient to get all BCs collected
by boundary region. This function returns a dictionary for Dirichlet or
Neumann BCs where the "ds" meassure is the key and a vector of boundary
values for each velocity component is the value
The "name" parameter must be 'dirichlet_bcs' or 'neumann_bcs'
"""
# Collect BCs
bc_dict = {}
for d in range(simulation.ndim):
for bc in simulation.data[name].get('u%d' % d, []):
if d == 0:
bc_dict[bc.ds()] = [bc.func()]
else:
bc_dict[bc.ds()].append(bc.func())
# Verify that all components are present and convert to vector
for ds, u_bc in bc_dict.items():
assert len(u_bc) == simulation.ndim
bc_dict[ds] = dolfin.as_vector(u_bc)
simulation.log.debug(' Found %d %s boundary regions' %
(len(bc_dict), name))
return bc_dict
def force_diff(**params):
# for random walk (with pointsize force field, no current)
setup = Setup(create_geo=False, **params)
# DEBUG
#print "active params", setup.active_params
#print "inactive params", setup.inactive_params
F = force_pointsize(**params)
if setup.phys.posDTarget:
D = diffusivity_simple(**params)
name = "diffusivity_div_simple"
if not fields.exists(name, **params):
V = D.function_space()
divD = dolfin.project(
dolfin.as_vector([dolfin.grad(D[0])[0],
dolfin.grad(D[1])[1]]), V)
fields.save_functions(name, setup.active_params, divD=divD)
fields.update()
divD, = fields.get_functions(name, "divD", **params)
else:
D0 = setup.phys.DTargetBulk
D0a = np.array([D0, D0, D0])
divDa = np.array([0., 0., 0.])
D = lambda x: D0a
divD = lambda x: divDa
return F, D, divD
def df_wrap(val, description, degree, sim):
"""
Wrap numbers as dolfin.Constant and strings as
dolfin.Expression C++ code. Lists must be ndim
long and contain either numbers or strings
"""
if isinstance(val, (int, float)):
# A real number
return dolfin.Constant(val)
elif isinstance(val, str):
# A C++ code string
return OcellarisCppExpression(sim, val, description, degree)
elif isinstance(val, (list, tuple)):
D = sim.ndim
L = len(val)
if L != D:
raise OcellarisError(
'Invalid length of list',
'BC list in "%r" must be length %d, is %d.' %
(description, D, L),
)
if all(isinstance(v, str) for v in val):
# A list of C++ code strings
return OcellarisCppExpression(sim, val, description, degree)
else:
# A mix of constants and (possibly) C++ strings
val = [
df_wrap(v, description + ' item %d' % i, degree, sim)
for i, v in enumerate(val)
]
return dolfin.as_vector(val)
def Wactive_orthotropic(Ta, C, f0, s0, n0):
"""Return active strain energy for an orthotropic
active stress
Arguments
---------
Ta : dolfin.Function or dolfin.Constant
A vector function representng the mangnitude of the
active stress in the reference configuration (firt Pioala).
Ta = (Ta_f0, Ta_s0, Ta_n0)
C : ufl.Form
The right Cauchy-Green deformation tensor
f0 : dolfin.Function
A vector function representng the direction of the
first component
s0 : dolfin.Function
A vector function representng the direction of the
second component
n0 : dolfin.Function
A vector function representng the direction of the
third component
"""
I4f = dolfin.inner(C * f0, f0)
I4s = dolfin.inner(C * s0, s0)
I4n = dolfin.inner(C * n0, n0)
I4 = dolfin.as_vector([I4f - 1, I4s - 1, I4n - 1])
return dolfin.Constant(0.5) * dolfin.inner(Ta, I4)
def source_term(self):
if self.moment_order == 3:
F_rhs = [0.0] * self.number_of_moments
F_rhs[0] = df.Expression(
"2.0 - 1.0 * pow(sqrt(pow(x[0],2)+pow(x[1],2)),2)", degree=2)
elif self.moment_order == 'nono6':
F_rhs = [0.0] * NoV
elif self.moment_order == 'nono13':
F_rhs = [0.0] * NoV
R = Expression("sqrt(pow(x[0],2)+pow(x[1],2))", degree=2)
F_rhs[0] = Expression(
"1.0 * (1.0 - (5.0*pow(R,2))/(18.0*pow(kn,2))) * cos(phi)",
R=R,
kn=Kn,
phi=phi,
degree=2)
F_rhs[3] = Expression(
"1.0 * (1.0 - (5.0*pow(R,2))/(18.0*pow(kn,2))) * cos(phi)",
R=R,
kn=Kn,
phi=phi,
degree=2)
F_rhs[0] = Expression(
"2.0 - 1.0 * pow(sqrt(pow(x[0],2)+pow(x[1],2)),2)", degree=2)
F_rhs[3] = Expression(
"2.0 - 1.0 * pow(sqrt(pow(x[0],2)+pow(x[1],2)),2)", degree=2)
if self.moment_order == 3:
F_rhs = df.as_vector(F_rhs) # converting to ufl vector
local_source_term = df.inner(self.v, F_rhs) * df.dx
return local_source_term
def updateCoefficients(self):
# Init coefficient matrix
P, Inv = SpatialCoordinate(self.mesh)
self.a = as_matrix([[+.5 * (P * self.sigma)**2, 0], [0, 0]])
# def K(t): return self.K0 + beta_SA * sin(4*pi*(t - t_SA))
def K(t):
return self.K0
self.b = as_vector([
+self.alpha * (K(self.t) - P),
-(self.gamma[0] + self.cost(self.gamma[0]))
])
self.c = Constant(-self.r)
# Init right-hand side
self.f = -(self.gamma[0] - self.cost(self.gamma[0])) * P
self.u_T = -2 * P * ufl.Max(1000 - Inv, 0)
# self.u_T = Constant(0.0)
# Set boundary conditions
# self.g_t = lambda t : [(Constant(0.0), "near(x[0],0)")]
self.g = self.u_T
def get_variable(self, name):
zero = dolfin.Constant(0.0)
if name == 'u':
# Assume that the waves are traveling in x-direction
if self.simulation.ndim == 2:
return dolfin.as_vector([self.get_variable('uhoriz'), zero])
else:
return dolfin.as_vector(
[self.get_variable('uhoriz'), zero, zero])
elif name == 'uvert':
return zero
if name not in self._functions:
expr = self._get_expression(name)
self._functions[name] = dolfin.interpolate(expr, self.V)
return self._functions[name]
def __init__(self, simulation, u_conv):
"""
Given a velocity in DG, e.g DG2, produce a velocity in DGT0,
i.e. a constant on each facet
"""
V = u_conv[0].function_space()
V_dgt0 = dolfin.FunctionSpace(V.mesh(), 'DGT', 0)
u = dolfin.TrialFunction(V_dgt0)
v = dolfin.TestFunction(V_dgt0)
ndim = simulation.ndim
w = u_conv
w_new = dolfin.as_vector(
[dolfin.Function(V_dgt0) for _ in range(ndim)])
dot, avg, dS, ds = dolfin.dot, dolfin.avg, dolfin.dS, dolfin.ds
a = dot(avg(u), avg(v)) * dS + dot(u, v) * ds
L = []
for d in range(ndim):
L.append(avg(w[d]) * avg(v) * dS + w[d] * v * ds)
self.lhs = [dolfin.Form(Li) for Li in L]
self.A = dolfin.assemble(a)
self.solver = dolfin.PETScKrylovSolver('cg')
self.velocity = simulation.data['u_conv_dgt0'] = w_new
def __init__(
self, WPQ,
kappa, rho, rho_const, mu, cp,
g, extra_force,
heat_source,
u_bcs, p_bcs,
theta_dirichlet_bcs=None,
theta_neumann_bcs=None,
theta_robin_bcs=None,
my_dx=dx,
my_ds=ds
):
super(StokesHeat, self).__init__()
theta_dirichlet_bcs = theta_dirichlet_bcs or {}
theta_neumann_bcs = theta_neumann_bcs or {}
theta_robin_bcs = theta_robin_bcs or {}
# Translate the Dirichlet boundary conditions into the product space.
self.dirichlet_bcs = helpers.dbcs_to_productspace(
WPQ,
[u_bcs, p_bcs, theta_dirichlet_bcs]
)
self.uptheta = Function(WPQ)
u, p, theta = split(self.uptheta)
v, q, zeta = TestFunctions(WPQ)
mesh = WPQ.mesh()
r = SpatialCoordinate(mesh)[0]
# Right-hand side for momentum equation.
f = rho(theta) * g # coupling
if extra_force is not None:
f += as_vector((extra_force[0], extra_force[1], 0.0))
self.stokes_F = stokes.F(
u, p, v, q, f, r, mu, my_dx
)
self.heat_F = heat.F(
theta, zeta,
kappa=kappa, rho=rho_const, cp=cp,
convection=u, # coupling
source=heat_source,
r=r,
neumann_bcs=theta_neumann_bcs,
robin_bcs=theta_robin_bcs,
my_dx=my_dx,
my_ds=my_ds
)
self.F0 = self.stokes_F + self.heat_F
self.jacobian = derivative(self.F0, self.uptheta)
return
from src.model import Model
from pylab import zeros, linspace, sqrt
from dolfin import project, Function, File, as_vector
nx = 40
ny = 40
nz = 7
model = Model()
model.generate_uniform_mesh(nx,ny,nz,0,1,0,1,deform=False,generate_pbcs=True)
Q = model.Q
U_obs = project(dolfin.as_vector([Function(Q),Function(Q)]))
b_obs = Function(Q)
U_opt = project(as_vector([Function(Q),Function(Q),Function(Q)]))
b_opt = Function(Q)
rcParams['text.usetex']=True
rcParams['font.size'] = 12
rcParams['font.family'] = 'serif'
for L in [10000]:
File('./results/U_obs.xml') >> U_obs
File('./results/U_opt.xml') >> U_opt
File('./results/beta2_obs.xml') >> b_obs
File('./results/beta2_opt.xml') >> b_opt
U_b = zeros(100)
U_p = zeros(100)
def grad(f):
g = dolfin.grad(f)
if len(g.shape()) == 0:
return dolfin.as_vector([g])
else:
return g
def solve_fixed_point(
mesh,
W_element, P_element, Q_element,
u0, p0, theta0,
kappa, rho, mu, cp,
g, extra_force,
heat_source,
u_bcs, p_bcs,
theta_dirichlet_bcs,
theta_neumann_bcs,
my_dx, my_ds,
max_iter,
tol
):
# Solve the coupled heat-Stokes equation approximately. Do this
# iteratively by solving the heat equation, then solving Stokes with the
# updated heat, the heat equation with the updated velocity and so forth
# until the change is 'small'.
WP = FunctionSpace(mesh, MixedElement([W_element, P_element]))
Q = FunctionSpace(mesh, Q_element)
# Initialize functions.
up0 = Function(WP)
u0, p0 = up0.split()
theta1 = Function(Q)
for _ in range(max_iter):
heat_problem = heat.Heat(
Q,
kappa=kappa,
rho=rho(theta0),
cp=cp,
convection=u0,
source=heat_source,
dirichlet_bcs=theta_dirichlet_bcs,
neumann_bcs=theta_neumann_bcs,
my_dx=my_dx,
my_ds=my_ds
)
theta1.assign(heat_problem.solve_stationary())
# Solve problem for velocity, pressure.
f = rho(theta0) * g # coupling
if extra_force:
f += as_vector((extra_force[0], extra_force[1], 0.0))
# up1 = up0.copy()
stokes.stokes_solve(
up0,
mu,
u_bcs, p_bcs,
f,
my_dx=my_dx,
tol=1.0e-10,
verbose=False,
maxiter=1000
)
# from dolfin import plot
# plot(u0)
# plot(theta0)
theta_diff = errornorm(theta0, theta1)
info('||theta - theta0|| = {:e}'.format(theta_diff))
# info('||u - u0|| = {:e}'.format(u_diff))
# info('||p - p0|| = {:e}'.format(p_diff))
# diff = theta_diff + u_diff + p_diff
diff = theta_diff
info('sum = {:e}'.format(diff))
# # Show the iterates.
# plot(theta0, title='theta0')
# plot(u0, title='u0')
# interactive()
# #exit()
if diff < tol:
break
theta0.assign(theta1)
# Create a *deep* copy of u0, p0, to be able to deal with them as
# actually separate entities.
u0, p0 = up0.split(deepcopy=True)
return u0, p0, theta0
space_S3 = df.VectorFunctionSpace(mesh, 'CG', 1, dim=3)
# Create two scalar functions and a vector function
# from the previously defined spaces
f1 = df.interpolate(df.Expression("1"), space_S1)
f2 = df.Function(space_S1)
f3 = df.Function(space_S3)
# If we want a scalar function with twice
# the values of f1 (THIS WORKS)
vec = df.assemble(df.dot(2 * f1, df.TestFunction(space_S1)) * df.dP)
f2.vector().axpy(1, vec)
# Now we want to modify the vector function space
# to get the vector (2, 1, 1) in every vertex
vec = df.assemble(df.dot(df.as_vector((2 * f1, f1, f1)), df.TestFunction(space_S3)) * df.dP)
f3.vector().axpy(1, vec)
# Scalar space assign
g2 = df.Function(space_S1)
g2.assign(df.FunctionAXPY(f1, 2.))
g2.vector().axpy(-1, f2.vector())
assert df.near(g2.vector().norm('l2'), 0)
# Assigner for components of the vector space
S3_assigners = [df.FunctionAssigner(space_S3.sub(i), space_S1) for i in range(space_S3.num_sub_spaces())]
g3 = df.Function(space_S3)
# Assign to components
comps = [f2, f1, f1]
import src.model import pylab import dolfin import pickle from pylab import * nx = 20 ny = 20 nz = 6 model = src.model.Model() model.generate_uniform_mesh(nx,ny,nz,0,1,0,1,deform=False,generate_pbcs=True) Q = model.Q U_opt = dolfin.project(dolfin.as_vector([dolfin.Function(Q),dolfin.Function(Q),dolfin.Function(Q)])) b_opt = dolfin.Function(Q) n = len(b_opt.compute_vertex_values()) rcParams['text.usetex']=True rcParams['font.size'] = 12 rcParams['font.family'] = 'serif' Us = zeros((50,n)) betas = zeros((50,n)) fig,axs = subplots(2,1,sharex=True) fig.set_size_inches(8,4) bed_indices = model.mesh.coordinates()[:,2]==0 surface_indices = model.mesh.coordinates()[:,2]==1
def compute_pressure(
P,
p0,
mu,
ui,
u,
my_dx,
p_bcs=None,
rotational_form=False,
tol=1.0e-10,
verbose=True,
):
"""Solve the pressure Poisson equation
.. math::
\\begin{align}
-\\frac{1}{r} \\div(r \\nabla (p_1-p_0)) =
-\\frac{1}{r} \\div(r u),\\\\
\\text{(with boundary conditions)},
\\end{align}
for :math:`\\nabla p = u`.
The pressure correction is based on the update formula
.. math::
\\frac{\\rho}{dt} (u_{n+1}-u^*)
+ \\begin{pmatrix}
\\text{d}\\phi/\\text{d}r\\\\
\\text{d}\\phi/\\text{d}z\\\\
\\frac{1}{r} \\text{d}\\phi/\\text{d}\\theta
\\end{pmatrix}
= 0
with :math:`\\phi = p_{n+1} - p^*` and
.. math::
\\frac{1}{r} \\frac{\\text{d}}{\\text{d}r} (r u_r^{(n+1)})
+ \\frac{\\text{d}}{\\text{d}z} (u_z^{(n+1)})
+ \\frac{1}{r} \\frac{\\text{d}}{\\text{d}\\theta} (u_{\\theta}^{(n+1)})
= 0
With the assumption that u does not change in the direction
:math:`\\theta`, one derives
.. math::
- \\frac{1}{r} \\div(r \\nabla \\phi) =
\\frac{1}{r} \\frac{\\rho}{dt} \\div(r (u_{n+1} - u^*))\\\\
- \\frac{1}{r} \\langle n, r \\nabla \\phi\\rangle =
\\frac{1}{r} \\frac{\\rho}{dt} \\langle n, r (u_{n+1} - u^*)\\rangle
In its weak form, this is
.. math::
\\int r \\langle\\nabla\\phi, \\nabla q\\rangle \\,2 \\pi =
- \\frac{\\rho}{dt} \\int \\div(r u^*) q \\, 2 \\pi
- \\frac{\\rho}{dt} \\int_{\\Gamma}
\\langle n, r (u_{n+1}-u^*)\\rangle q \\, 2\\pi.
(The terms :math:`1/r` cancel with the volume elements :math:`2\\pi r`.)
If the Dirichlet boundary conditions are applied to both :math:`u^*` and
:math:`u_n` (the latter in the velocity correction step), the boundary
integral vanishes.
If no Dirichlet conditions are given (which is the default case), the
system has no unique solution; one eigenvalue is 0. This however, does not
hurt CG convergence if the system is consistent, cf. :cite:`vdV03`. And
indeed it is consistent if and only if
.. math::
\\int_\\Gamma r \\langle n, u\\rangle = 0.
This condition makes clear that for incompressible Navier-Stokes, one
either needs to make sure that inflow and outflow always add up to 0, or
one has to specify pressure boundary conditions.
Note that, when using a multigrid preconditioner as is done here, the
coarse solver must be chosen such that it preserves the nullspace of the
problem.
"""
W = ui.function_space()
r = SpatialCoordinate(W.mesh())[0]
p = TrialFunction(P)
q = TestFunction(P)
a2 = dot(r * grad(p), grad(q)) * 2 * pi * my_dx
# The boundary conditions
# n.(p1-p0) = 0
# are implicitly included.
#
# L2 = -div(r*u) * q * 2*pi*my_dx
div_u = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
L2 = -div_u * q * 2 * pi * r * my_dx
if p0:
L2 += r * dot(grad(p0), grad(q)) * 2 * pi * my_dx
# In the Cartesian variant of the rotational form, one makes use of the
# fact that
#
# curl(curl(u)) = grad(div(u)) - div(grad(u)).
#
# The same equation holds true in cylindrical form. Hence, to get the
# rotational form of the splitting scheme, we need to
#
# rotational form
if rotational_form:
# If there is no dependence of the angular coordinate, what is
# div(grad(div(u))) in Cartesian coordinates becomes
#
# 1/r div(r * grad(1/r div(r*u)))
#
# in cylindrical coordinates (div and grad are in cylindrical
# coordinates). Unfortunately, we cannot write it down that
# compactly since u_phi is in the game.
# When using P2 elements, this value will be 0 anyways.
div_ui = 1 / r * (r * ui[0]).dx(0) + ui[1].dx(1)
grad_div_ui = as_vector((div_ui.dx(0), div_ui.dx(1)))
L2 -= r * mu * dot(grad_div_ui, grad(q)) * 2 * pi * my_dx
# div_grad_div_ui = 1/r * (r * grad_div_ui[0]).dx(0) \
# + (grad_div_ui[1]).dx(1)
# L2 += mu * div_grad_div_ui * q * 2*pi*r*dx
# n = FacetNormal(Q.mesh())
# L2 -= mu * (n[0] * grad_div_ui[0] + n[1] * grad_div_ui[1]) \
# * q * 2*pi*r*ds
p1 = Function(P)
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. :cite:`vdV03`. And indeed it is consistent if and only if
#
# \int_\Gamma r n.u = 0.
#
# This makes clear that for incompressible Navier-Stokes, one
# either needs to make sure that inflow and outflow always add up
# to 0, or one has to specify pressure boundary conditions.
#
# If the right-hand side is very small, round-off errors may impair
# the consistency of the system. Make sure the system we are
# solving remains consistent.
A = assemble(a2)
b = assemble(L2)
# Assert that the system is indeed consistent.
e = Function(P)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
# Assume that in every component of the vector, a round-off error
# of the magnitude DOLFIN_EPS is present. This leads to the
# criterion
# |<b,e>| / (||b||*||e||) < DOLFIN_EPS
# as a check whether to consider the system consistent up to
# round-off error.
#
# TODO think about condition here
# if abs(alpha) > normB * DOLFIN_EPS:
if abs(alpha) > normB * 1.0e-12:
# divu = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
adivu = assemble(((r * u[0]).dx(0) + u[1].dx(1)) * 2 * pi * my_dx)
info("\\int 1/r * div(r*u) * 2*pi*r = {:e}".format(adivu))
n = FacetNormal(P.mesh())
boundary_integral = assemble((n[0] * u[0] + n[1] * u[1]) * 2 * pi * r * ds)
info("\\int_Gamma n.u * 2*pi*r = {:e}".format(boundary_integral))
message = (
"System not consistent! "
"<b,e> = {:g}, ||b|| = {:g}, <b,e>/||b|| = {:e}.".format(
alpha, normB, alpha / normB
)
)
info(message)
# # Plot the stuff, and project it to a finer mesh with linear
# # elements for the purpose.
# plot(divu, title='div(u_tentative)')
# # Vp = FunctionSpace(Q.mesh(), 'CG', 2)
# # Wp = MixedFunctionSpace([Vp, Vp])
# # up = project(u, Wp)
# fine_mesh = Q.mesh()
# for k in range(1):
# fine_mesh = refine(fine_mesh)
# V = FunctionSpace(fine_mesh, 'CG', 1)
# W = V * V
# # uplot = Function(W)
# # uplot.interpolate(u)
# uplot = project(u, W)
# plot(uplot[0], title='u_tentative[0]')
# plot(uplot[1], title='u_tentative[1]')
# # plot(u, title='u_tentative')
# interactive()
# exit()
raise RuntimeError(message)
# Project out the roundoff error.
b -= alpha * evec
#
# 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 make sure 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")
from dolfin import PETScOptions
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)
solver.solve(p1_petsc, b_petsc)
return p1
def FacetNormal(mesh):
nm = dolfin.FacetNormal(mesh)
if len(nm.shape()) == 0:
return dolfin.as_vector([nm])
else:
return nm
def initialize_eqs(self):
"""
"""
# initial height
h = self.initial_h(120+0.0003*self.L, self.c, 0.4*self.L, self.L)
self.h_old = d.project(h, self.V)
# initial guess of velocity
u = d.project(d.Expression('x[0]/L', L=1000*self.L), self.V)
self.h_u_dhdx = d.project(d.as_vector((h,u,h.dx(0))), self.V3)
self.h,self.u,self.dhdx = d.split(self.h_u_dhdx)
# initialize floating conditional
class FloatBoolean(Expression):
"""
creates boolean function over length of ice, true when ice is floating.
"""
def eval(s,value,x):
if self.h_u_dhdx(x[0])[0] <= -h_b(x[0])*(self.rho_w/self.rho-1):
value[0] = 1
else:
value[0] = 0
# floating conditional
fc = self.smooth_step_conditional(FloatBoolean())
# A useful expression for the effective pressure term
reduction = fc*(self.rho_w/self.rho)*self.h_b
# H (ice thickness) conditional on floating
if self.floating:
# archimedes principle, H (ice thickness)
self.H = fc * (h/(1-self.rho/self.rho_w)) + (1-fc) * (self.h-self.h_b)
else:
# ice thickness = surface elevation - bed elevation
self.H = self.h - self.h_b
H = self.H
h = self.h
W = self.W
u = self.u
n = self.n
# basal drag
self.basal_drag = self.mu*self.A_s*(self.H+reduction)**self.p*u**(1/n)
if self.floating:
self.basal_drag = (1-fc) * self.basal_drag
# define variational problem
testFunction = TestFunction(self.V3)
trialFunction = TrialFunction(self.V3)
phi1, phi2, phi3 = split(testFunction)
self.driving_stress = self.rho*self.g*H*self.dhdx
self.lateral_drag = (self.B_s*H/W)*((n+2)*u/(2*W))**(1/n)
force_balance = (self.driving_stress+self.basal_drag+self.lateral_drag)*phi2
####????####################################################################
mass_conservation = ((h-self.h_old)/self.dt + (H*u*W).dx(0)/W - M)*phi1
####????####################################################################
F = (mass_conservation + force_balance)*d.dx
###????#####################################################################
invariant_squared = ((n+2.)/(n+1.)*u/W)**2+u.dx(0)**2
self.longitudinal_stress = \
2*self.B_s*H*invariant_squared**((1-n)/(2*n))*u.dx(0)
F += self.longitudinal_stress * phi2.dx(0) * d.dx
F -= self.longitudinal_stress * phi2 * self.ds(self.TERMINUS)
F += (self.h.dx(0)-self.dhdx)*phi3*d.dx
###????#####################################################################
J = derivative(F, self.h_u_dhdx, trialFunction)
problem =\
NonlinearVariationalProblem(F, self.h_u_dhdx, self.boundary_conditions, J)
self.solver = NonlinearVariationalSolver(problem)
nz = 10
m = UnitCubeMesh(nx,ny,nz)
Q = FunctionSpace(m,"CG",1)
u = Function(Q)
v = Function(Q)
w = Function(Q)
S = Function(Q)
for L in [5000,10000,20000,40000,80000,160000]:
File('./results_stokes/'+str(L)+'/u.xml') >> u
File('./results_stokes/'+str(L)+'/v.xml') >> v
File('./results_stokes/'+str(L)+'/w.xml') >> w
U = zeros(100)
profile = linspace(0,1,100)
for ii,x in enumerate(profile):
uu = u(x,0.25,0.99999)
vv = v(x,0.25,0.99999)
ww = w(x,0.25,0.99999)
U[ii] = sqrt(uu**2 + vv**2)
#plot(profile,U)
#data = zip(profile,U)
#dump(data,open("djb1a{0:03d}.p".format(L/1000),'w'))
U = project(as_vector([u,v,w]))
File('./results_stokes/'+str(L)+'/U.pvd') << U
def _compute_boussinesq(
problem, u0, p0, theta0, lorentz, joule, target_time=0.1, show=False
):
# Define a facet measure on the boundaries. See discussion on
# <https://bitbucket.org/fenics-project/dolfin/issue/249/facet-specification-doesnt-work-on-ds>.
ds_workpiece = Measure("ds", subdomain_data=problem.wp_boundaries)
submesh_workpiece = problem.W.mesh()
# Start time, time step.
t = 0.0
dt = 1.0e-3
dt_max = 1.0e-1
# Standard gravity, <https://en.wikipedia.org/wiki/Standard_gravity>.
grav = 9.80665
assert problem.W.num_sub_spaces() == 3
g = Constant((0.0, -grav, 0.0))
# Compute a few mesh characteristics.
wpi_area = assemble(1.0 * dx(submesh_workpiece))
# mesh.hmax() is a local function; get the global hmax.
hmax_workpiece = MPI.max(submesh_workpiece.mpi_comm(), submesh_workpiece.hmax())
# Take the maximum length in x-direction as characteristic length of the
# domain.
coords = submesh_workpiece.coordinates()
char_length = max(coords[:, 0]) - min(coords[:, 0])
# Prepare some parameters for the Navier-Stokes simulation in the workpiece
m = problem.subdomain_materials[problem.wpi]
k_wpi = m.thermal_conductivity
cp_wpi = m.specific_heat_capacity
rho_wpi = m.density
mu_wpi = m.dynamic_viscosity
theta_average = average(theta0)
# show_total_force = True
# if show_total_force:
# f = rho_wpi(theta0) * g
# if lorentz:
# f += as_vector((lorentz[0], lorentz[1], 0.0))
# tri = plot(f, mesh=submesh_workpiece, title='Total external force')
# plt.colorbar(tri)
# plt.show()
with XDMFFile(submesh_workpiece.mpi_comm(), "all.xdmf") as outfile:
outfile.parameters["flush_output"] = True
outfile.parameters["rewrite_function_mesh"] = False
_store(outfile, u0, p0, theta0, t)
if show:
_plot(p0, theta0)
plt.show()
successful_steps = 0
failed_steps = 0
while t < target_time + DOLFIN_EPS:
info(
"Successful steps: {} (failed: {}, total: {})".format(
successful_steps, failed_steps, successful_steps + failed_steps
)
)
with Message("Time step {:e} -> {:e}...".format(t, t + dt)):
# Do one heat time step.
with Message("Computing heat..."):
# Redefine the heat problem with the new u0.
heat_problem = cyl_heat.Heat(
problem.Q,
kappa=k_wpi,
rho=rho_wpi(theta_average),
cp=cp_wpi,
convection=u0,
source=joule,
dirichlet_bcs=problem.theta_bcs_d,
neumann_bcs=problem.theta_bcs_n,
my_dx=dx(submesh_workpiece),
my_ds=ds_workpiece,
)
# For time-stepping in buoyancy-driven flows, see
#
# Numerical solution of buoyancy-driven flows;
# Einar Rossebø Christensen;
# Master's thesis;
# <http://www.diva-portal.org/smash/get/diva2:348831/FULLTEXT01.pdf>.
#
# Similar to the present approach, one first solves for
# velocity and pressure, then for temperature.
#
heat_stepper = parabolic.ImplicitEuler(heat_problem)
ns_stepper = cyl_ns.IPCS(time_step_method="backward euler")
theta1 = heat_stepper.step(theta0, t, dt)
theta0_average = average(theta0)
try:
# Do one Navier-Stokes time step.
with Message("Computing flux and pressure..."):
# Include proper temperature-dependence here to account
# for the Boussinesq effect.
f0 = rho_wpi(theta0) * g
f1 = rho_wpi(theta1) * g
if lorentz is not None:
f = as_vector((lorentz[0], lorentz[1], 0.0))
f0 += f
f1 += f
u1, p1 = ns_stepper.step(
Constant(dt),
{0: u0},
p0,
problem.W,
problem.P,
problem.u_bcs,
problem.p_bcs,
# Make constant TODO
Constant(rho_wpi(theta0_average)),
Constant(mu_wpi(theta0_average)),
f={0: f0, 1: f1},
tol=1.0e-10,
my_dx=dx(submesh_workpiece),
)
except RuntimeError as e:
info(e.args[0])
info(
"Navier--Stokes solver failed to converge. "
"Decrease time step from {:e} to {:e} and try again.".format(
dt, 0.5 * dt
)
)
dt *= 0.5
failed_steps += 1
continue
successful_steps += 1
# Assignments and plotting.
theta0.assign(theta1)
u0.assign(u1)
p0.assign(p1)
_store(outfile, u0, p0, theta0, t + dt)
if show:
_plot(p0, theta0)
plt.show()
t += dt
with Message("Diagnostics..."):
# Print some general info on the flow in the crucible.
umax = get_umax(u0)
_print_diagnostics(
theta0,
umax,
submesh_workpiece,
wpi_area,
problem.subdomain_materials,
problem.wpi,
rho_wpi,
mu_wpi,
char_length,
grav,
)
info("")
with Message("Step size adaptation..."):
# Some smooth step-size adaption.
target_dt = 0.2 * hmax_workpiece / umax
info("previous dt: {:e}".format(dt))
info("target dt: {:e}".format(target_dt))
# agg is the aggressiveness factor. The distance between
# the current step size and the target step size is reduced
# by |1-agg|. Hence, if agg==1 then dt_next==target_dt.
# Otherwise target_dt is approached more slowly.
agg = 0.5
dt = min(
dt_max,
# At most double the step size from step to step.
dt * min(2.0, 1.0 + agg * (target_dt - dt) / dt),
)
info("new dt: {:e}".format(dt))
info("")
info("")
return u0, p0, theta0
def les_setup(u_, mesh, assemble_matrix, CG1Function, nut_krylov_solver, bcs, **NS_namespace):
"""
Set up for solving the Germano Dynamic LES model applying
Lagrangian Averaging.
"""
# Create function spaces
CG1 = FunctionSpace(mesh, "CG", 1)
p, q = TrialFunction(CG1), TestFunction(CG1)
dim = mesh.geometry().dim()
# Define delta and project delta**2 to CG1
delta = pow(CellVolume(mesh), 1. / dim)
delta_CG1_sq = project(delta, CG1)
delta_CG1_sq.vector().set_local(delta_CG1_sq.vector().array()**2)
delta_CG1_sq.vector().apply("insert")
# Define nut_
Sij = sym(grad(u_))
magS = sqrt(2 * inner(Sij, Sij))
Cs = Function(CG1)
nut_form = Cs**2 * delta**2 * magS
# Create nut_ BCs
ff = MeshFunction("size_t", mesh, mesh.topology().dim() - 1, 0)
bcs_nut = []
for i, bc in enumerate(bcs['u0']):
bc.apply(u_[0].vector()) # Need to initialize bc
m = bc.markers() # Get facet indices of boundary
ff.array()[m] = i + 1
bcs_nut.append(DirichletBC(CG1, Constant(0), ff, i + 1))
nut_ = CG1Function(nut_form, mesh, method=nut_krylov_solver,
bcs=bcs_nut, bounded=True, name="nut")
# Create functions for holding the different velocities
u_CG1 = as_vector([Function(CG1) for i in range(dim)])
u_filtered = as_vector([Function(CG1) for i in range(dim)])
dummy = Function(CG1)
ll = LagrangeInterpolator()
# Assemble required filter matrices and functions
G_under = Function(CG1, assemble(TestFunction(CG1) * dx))
G_under.vector().set_local(1. / G_under.vector().array())
G_under.vector().apply("insert")
G_matr = assemble(inner(p, q) * dx)
# Set up functions for Lij and Mij
Lij = [Function(CG1) for i in range(dim * dim)]
Mij = [Function(CG1) for i in range(dim * dim)]
# Check if case is 2D or 3D and set up uiuj product pairs and
# Sij forms, assemble required matrices
Sijcomps = [Function(CG1) for i in range(dim * dim)]
Sijfcomps = [Function(CG1) for i in range(dim * dim)]
# Assemble some required matrices for solving for rate of strain terms
Sijmats = [assemble_matrix(p.dx(i) * q * dx) for i in range(dim)]
if dim == 3:
tensdim = 6
uiuj_pairs = ((0, 0), (0, 1), (0, 2), (1, 1), (1, 2), (2, 2))
else:
tensdim = 3
uiuj_pairs = ((0, 0), (0, 1), (1, 1))
# Set up Lagrange functions
JLM = Function(CG1)
JLM.vector()[:] += 1E-32
JMM = Function(CG1)
JMM.vector()[:] += 1
return dict(Sij=Sij, nut_form=nut_form, nut_=nut_, delta=delta, bcs_nut=bcs_nut,
delta_CG1_sq=delta_CG1_sq, CG1=CG1, Cs=Cs, u_CG1=u_CG1,
u_filtered=u_filtered, ll=ll, Lij=Lij, Mij=Mij, Sijcomps=Sijcomps,
Sijfcomps=Sijfcomps, Sijmats=Sijmats, JLM=JLM, JMM=JMM, dim=dim,
tensdim=tensdim, G_matr=G_matr, G_under=G_under, dummy=dummy,
uiuj_pairs=uiuj_pairs)
nz = 10
m = dolfin.UnitCubeMesh(nx,ny,nz)
Q = dolfin.FunctionSpace(m,"CG",1)
u = dolfin.Function(Q)
v = dolfin.Function(Q)
w = dolfin.Function(Q)
S = dolfin.Function(Q)
for L in [5000,10000,20000,40000,80000,160000]:
dolfin.File('./results_stokes/'+str(L)+'/u.xml') >> u
dolfin.File('./results_stokes/'+str(L)+'/v.xml') >> v
dolfin.File('./results_stokes/'+str(L)+'/w.xml') >> w
U = pylab.zeros(100)
profile = pylab.linspace(0,1,100)
for ii,x in enumerate(profile):
uu = u(x,0.25,0.99999)
vv = v(x,0.25,0.99999)
ww = w(x,0.25,0.99999)
U[ii] = pylab.sqrt(uu**2 + vv**2)
#pylab.plot(profile,U)
#data = zip(profile,U)
#pickle.dump(data,open("djb1a{0:03d}.p".format(L/1000),'w'))
U = dolfin.project(dolfin.as_vector([u,v,w]))
dolfin.File('./results_stokes/'+str(L)+'/U.pvd') << U
def _pressure_poisson(self, p1, p0,
mu, ui,
u,
p_bcs=None,
rotational_form=False,
tol=1.0e-10,
verbose=True
):
'''Solve the pressure Poisson equation
-1/r \div(r \nabla (p1-p0)) = -1/r div(r*u),
boundary conditions,
for
\nabla p = u.
'''
r = Expression('x[0]', degree=1, domain=self.W.mesh())
Q = p1.function_space()
p = TrialFunction(Q)
q = TestFunction(Q)
a2 = dot(r * grad(p), grad(q)) * 2 * pi * dx
# The boundary conditions
# n.(p1-p0) = 0
# are implicitly included.
#
# L2 = -div(r*u) * q * 2*pi*dx
div_u = 1/r * (r * u[0]).dx(0) + u[1].dx(1)
L2 = -div_u * q * 2*pi*r*dx
if p0:
L2 += r * dot(grad(p0), grad(q)) * 2*pi*dx
# In the Cartesian variant of the rotational form, one makes use of the
# fact that
#
# curl(curl(u)) = grad(div(u)) - div(grad(u)).
#
# The same equation holds true in cylindrical form. Hence, to get the
# rotational form of the splitting scheme, we need to
#
# rotational form
if rotational_form:
# If there is no dependence of the angular coordinate, what is
# div(grad(div(u))) in Cartesian coordinates becomes
#
# 1/r div(r * grad(1/r div(r*u)))
#
# in cylindrical coordinates (div and grad are in cylindrical
# coordinates). Unfortunately, we cannot write it down that
# compactly since u_phi is in the game.
# When using P2 elements, this value will be 0 anyways.
div_ui = 1/r * (r * ui[0]).dx(0) + ui[1].dx(1)
grad_div_ui = as_vector((div_ui.dx(0), div_ui.dx(1)))
L2 -= r * mu * dot(grad_div_ui, grad(q)) * 2*pi*dx
#div_grad_div_ui = 1/r * (r * grad_div_ui[0]).dx(0) \
# + (grad_div_ui[1]).dx(1)
#L2 += mu * div_grad_div_ui * q * 2*pi*r*dx
#n = FacetNormal(Q.mesh())
#L2 -= mu * (n[0] * grad_div_ui[0] + n[1] * grad_div_ui[1]) \
# * q * 2*pi*r*ds
if p_bcs:
solve(
a2 == L2, p1,
bcs=p_bcs,
solver_parameters={
'linear_solver': 'iterative',
'symmetric': True,
'preconditioner': '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. :cite:`vdV03`. And indeed it is consistent if and only if
#
# \int_\Gamma r n.u = 0.
#
# This makes clear that for incompressible Navier-Stokes, one
# either needs to make sure that inflow and outflow always add up
# to 0, or one has to specify pressure boundary conditions.
#
# If the right-hand side is very small, round-off errors may impair
# the consistency of the system. Make sure the system we are
# solving remains consistent.
A = assemble(a2)
b = assemble(L2)
# Assert that the system is indeed consistent.
e = Function(Q)
e.interpolate(Constant(1.0))
evec = e.vector()
evec /= norm(evec)
alpha = b.inner(evec)
normB = norm(b)
# Assume that in every component of the vector, a round-off error
# of the magnitude DOLFIN_EPS is present. This leads to the
# criterion
# |<b,e>| / (||b||*||e||) < DOLFIN_EPS
# as a check whether to consider the system consistent up to
# round-off error.
#
# TODO think about condition here
#if abs(alpha) > normB * DOLFIN_EPS:
if abs(alpha) > normB * 1.0e-12:
divu = 1 / r * (r * u[0]).dx(0) + u[1].dx(1)
adivu = assemble(((r * u[0]).dx(0) + u[1].dx(1)) * 2 * pi * dx)
info('\int 1/r * div(r*u) * 2*pi*r = %e' % adivu)
n = FacetNormal(Q.mesh())
boundary_integral = assemble((n[0] * u[0] + n[1] * u[1])
* 2 * pi * r * ds)
info('\int_Gamma n.u * 2*pi*r = %e' % boundary_integral)
message = ('System not consistent! '
'<b,e> = %g, ||b|| = %g, <b,e>/||b|| = %e.') \
% (alpha, normB, alpha / normB)
info(message)
# Plot the stuff, and project it to a finer mesh with linear
# elements for the purpose.
plot(divu, title='div(u_tentative)')
#Vp = FunctionSpace(Q.mesh(), 'CG', 2)
#Wp = MixedFunctionSpace([Vp, Vp])
#up = project(u, Wp)
fine_mesh = Q.mesh()
for k in range(1):
fine_mesh = refine(fine_mesh)
V = FunctionSpace(fine_mesh, 'CG', 1)
W = V * V
#uplot = Function(W)
#uplot.interpolate(u)
uplot = project(u, W)
plot(uplot[0], title='u_tentative[0]')
plot(uplot[1], title='u_tentative[1]')
#plot(u, title='u_tentative')
interactive()
exit()
raise RuntimeError(message)
# Project out the roundoff error.
b -= alpha * evec
#
# 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 make sure 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')
from dolfin import PETScOptions
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)
solver.solve(p1_petsc, b_petsc)
# This would be the stump for Epetra:
#solve(A, p.vector(), b, 'cg', 'ml_amg')
return
def test(show=False):
problem = problems.Crucible()
# The voltage is defined as
#
# v(t) = Im(exp(i omega t) v)
# = Im(exp(i (omega t + arg(v)))) |v|
# = sin(omega t + arg(v)) |v|.
#
# Hence, for a lagging voltage, arg(v) needs to be negative.
voltages = [
38.0 * numpy.exp(-1j * 2 * pi * 2 * 70.0 / 360.0),
38.0 * numpy.exp(-1j * 2 * pi * 1 * 70.0 / 360.0),
38.0 * numpy.exp(-1j * 2 * pi * 0 * 70.0 / 360.0),
25.0 * numpy.exp(-1j * 2 * pi * 0 * 70.0 / 360.0),
25.0 * numpy.exp(-1j * 2 * pi * 1 * 70.0 / 360.0),
]
lorentz, joule, Phi = get_lorentz_joule(problem, voltages, show=show)
# Some assertions
ref = 1.4627674791126285e-05
assert abs(norm(Phi[0], "L2") - ref) < 1.0e-3 * ref
ref = 3.161363929287592e-05
assert abs(norm(Phi[1], "L2") - ref) < 1.0e-3 * ref
#
ref = 12.115309575057681
assert abs(norm(lorentz, "L2") - ref) < 1.0e-3 * ref
#
ref = 1406.336109054347
V = FunctionSpace(problem.submesh_workpiece, "CG", 1)
jp = project(joule, V)
jp.rename("s", "Joule heat source")
assert abs(norm(jp, "L2") - ref) < 1.0e-3 * ref
# check_currents = False
# if check_currents:
# r = SpatialCoordinate(problem.mesh)[0]
# begin('Currents computed after the fact:')
# k = 0
# with XDMFFile('currents.xdmf') as xdmf_file:
# for coil in coils:
# for ii in coil['rings']:
# J_r = sigma[ii] * (
# voltages[k].real/(2*pi*r) + problem.omega * Phi[1]
# )
# J_i = sigma[ii] * (
# voltages[k].imag/(2*pi*r) - problem.omega * Phi[0]
# )
# alpha = assemble(J_r * dx(ii))
# beta = assemble(J_i * dx(ii))
# info('J = {:e} + i {:e}'.format(alpha, beta))
# info(
# '|J|/sqrt(2) = {:e}'.format(
# numpy.sqrt(0.5 * (alpha**2 + beta**2))
# ))
# submesh = SubMesh(problem.mesh, problem.subdomains, ii)
# V1 = FunctionSpace(submesh, 'CG', 1)
# # Those projections may take *very* long.
# # TODO find out why
# j_v1 = [
# project(J_r, V1),
# project(J_i, V1)
# ]
# # show=Trueplot(j_v1[0], title='j_r')
# # plot(j_v1[1], title='j_i')
# current = project(as_vector(j_v1), V1*V1)
# current.rename('j{}'.format(ii), 'current {}'.format(ii))
# xdmf_file.write(current)
# k += 1
# end()
filename = "./maxwell.xdmf"
with XDMFFile(filename) as xdmf_file:
xdmf_file.parameters["flush_output"] = True
xdmf_file.parameters["rewrite_function_mesh"] = False
# Store phi
info("Writing out Phi to {}...".format(filename))
V = FunctionSpace(problem.mesh, "CG", 1)
phi = Function(V, name="phi")
Phi0 = project(Phi[0], V)
Phi1 = project(Phi[1], V)
omega = problem.omega
for t in numpy.linspace(0.0, 2 * pi / omega, num=100, endpoint=False):
# Im(Phi * exp(i*omega*t))
phi.vector().zero()
phi.vector().axpy(sin(problem.omega * t), Phi0.vector())
phi.vector().axpy(cos(problem.omega * t), Phi1.vector())
xdmf_file.write(phi, t)
# Show the resulting magnetic field
#
# B_r = -dphi/dz,
# B_z = 1/r d(rphi)/dr.
#
r = SpatialCoordinate(problem.mesh)[0]
g = 1.0 / r * grad(r * Phi[0])
V_element = FiniteElement("CG", V.mesh().ufl_cell(), 1)
VV = FunctionSpace(V.mesh(), V_element * V_element)
B_r = project(as_vector((-g[1], g[0])), VV)
g = 1 / r * grad(r * Phi[1])
B_i = project(as_vector((-g[1], g[0])), VV)
info("Writing out B to {}...".format(filename))
B = Function(VV)
B.rename("B", "magnetic field")
if abs(problem.omega) < DOLFIN_EPS:
B.assign(B_r)
xdmf_file.write(B)
# plot(B_r, title='Re(B)')
# plot(B_i, title='Im(B)')
else:
# Write those out to a file.
lspace = numpy.linspace(
0.0, 2 * pi / problem.omega, num=100, endpoint=False
)
for t in lspace:
# Im(B * exp(i*omega*t))
B.vector().zero()
B.vector().axpy(sin(problem.omega * t), B_r.vector())
B.vector().axpy(cos(problem.omega * t), B_i.vector())
xdmf_file.write(B, t)
filename = "./lorentz-joule.xdmf"
info("Writing out Lorentz force and Joule heat source to {}...".format(filename))
with XDMFFile(filename) as xdmf_file:
xdmf_file.write(lorentz, 0.0)
# xdmf_file.write(jp, 0.0)
return
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 eps(u):
""" Returns a vector of strains of size (3,1) in the Voigt notation
layout {eps_xx, eps_yy, gamma_xy} where gamma_xy = 2*eps_xy"""
return df.as_vector([u[i].dx(i) for i in range(2)] +
[u[i].dx(j) + u[j].dx(i) for (i,j) in [(0,1)]])