def __init__(self,
form,
Space,
bcs=[],
name="x",
matvec=[None, None],
method="default",
solver_type="cg",
preconditioner_type="default"):
Function.__init__(self, Space, name=name)
self.form = form
self.method = method
self.bcs = bcs
self.matvec = matvec
self.trial = trial = TrialFunction(Space)
self.test = test = TestFunction(Space)
Mass = inner(trial, test) * dx()
self.bf = inner(form, test) * dx()
self.rhs = Vector(self.vector())
if method.lower() == "default":
self.A = A_cache[(Mass, tuple(bcs))]
self.sol = Solver_cache[(Mass, tuple(bcs), solver_type,
preconditioner_type)]
elif method.lower() == "lumping":
assert Space.ufl_element().degree() < 2
self.A = A_cache[(Mass, tuple(bcs))]
ones = Function(Space)
ones.vector()[:] = 1.
self.ML = self.A * ones.vector()
self.ML.set_local(1. / self.ML.array())
def solve(self, dt):
self.u = Function(self.V0)
self.w = TestFunction(self.V0)
self.du = TrialFunction(self.V0)
x = SpatialCoordinate(self.mesh0)
L = inner( self.S(), self.eps(self.w) )*dx(degree=4)\
- inner( self.b, self.w )*dx(degree=4)\
- inner( self.h, self.w )*ds(degree=4)\
+ inner( 1e-6*self.u, self.w )*ds(degree=4)\
- inner( min_value(x[2]+self.ut[2]+self.u[2], 0) * Constant((0,0,-1.0)), self.w )*ds(degree=4)
a = derivative(L, self.u, self.du)
problem = NonlinearVariationalProblem(L, self.u, bcs=[], J=a)
solver = NonlinearVariationalSolver(problem)
solver.solve()
self.ut.vector()[:] = self.ut.vector()[:] + self.u.vector()[:]
ALE.move(self.mesh, Function(self.V, self.u.vector()))
self.v.vector()[:] = self.u.vector()[:] / dt
self.n = FacetNormal(self.mesh)
def __new__(cls, mesh, tolerance=1e-12):
dim = mesh.geometry().dim()
X = MeshPool._X[dim - 1]
test = assemble(dot(X, X) * CellVolume(mesh)**(0.25) *
dx()) * assemble(Constant(1) * dx(domain=mesh))
assert test > 0.0
assert test < np.inf
# Do a garbage collect to collect any garbage references
# Needed for full parallel compatibility
gc.collect()
keys = np.array(MeshPool._existing.keys())
self = None
if len(keys) > 0:
diff = np.abs(keys - test) / np.abs(test)
idx = np.argmin(np.abs(keys - test))
if diff[idx] <= tolerance and isinstance(
mesh, type(MeshPool._existing[keys[idx]])):
self = MeshPool._existing[keys[idx]]
if self is None:
self = mesh
MeshPool._existing[test] = self
return self
def __init__(self, p_, Space, i=0, bcs=[], name="grad", method={}):
assert len(p_.ufl_shape) == 0
assert i >= 0 and i < Space.mesh().geometry().dim()
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, p_.dx(i), Space, bcs=bcs, name=name,
method=solver_method, solver_type=solver_type,
preconditioner_type=preconditioner_type)
self.i = i
Source = p_.function_space()
if not low_memory_version:
self.matvec = [
A_cache[(self.test * TrialFunction(Source).dx(i) * dx, ())], p_]
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)
dP = assemble(TrialFunction(p_.function_space()).dx(i)
* TestFunction(DG) * dx())
self.WGM = compiled_gradient_module.compute_weighted_gradient_matrix(
G, dP, dg)
def __init__(self, form, Space, bcs=[],
name="x",
matvec=[None, None],
method="default",
solver_type="cg",
preconditioner_type="default"):
Function.__init__(self, Space, name=name)
self.form = form
self.method = method
self.bcs = bcs
self.matvec = matvec
self.trial = trial = TrialFunction(Space)
self.test = test = TestFunction(Space)
Mass = inner(trial, test) * dx()
self.bf = inner(form, test) * dx()
self.rhs = Vector(self.vector())
if method.lower() == "default":
self.A = A_cache[(Mass, tuple(bcs))]
self.sol = Solver_cache[(Mass, tuple(
bcs), solver_type, preconditioner_type)]
elif method.lower() == "lumping":
assert Space.ufl_element().degree() < 2
self.A = A_cache[(Mass, tuple(bcs))]
ones = Function(Space)
ones.vector()[:] = 1.
self.ML = self.A * ones.vector()
self.ML.set_local(1. / self.ML.array())
def __init__(self, form, Space, bcs=[],
name="x",
matvec=[None, None],
method="default",
solver_type="cg",
preconditioner_type="default"):
Function.__init__(self, Space, name=name)
self.form = form
self.method = method
self.bcs = bcs
self.matvec = matvec
self.trial = trial = TrialFunction(Space)
self.test = test = TestFunction(Space)
Mass = inner(trial, test)*dx()
self.bf = inner(form, test)*dx()
self.rhs = Vector(self.vector())
if method.lower() == "default":
self.A = A_cache[(Mass, tuple(bcs))]
self.sol = KrylovSolver(solver_type, preconditioner_type)
self.sol.parameters["preconditioner"]["structure"] = "same"
self.sol.parameters["error_on_nonconvergence"] = False
self.sol.parameters["monitor_convergence"] = False
self.sol.parameters["report"] = False
elif method.lower() == "lumping":
assert Space.ufl_element().degree() < 2
self.A = A_cache[(Mass, tuple(bcs))]
ones = Function(Space)
ones.vector()[:] = 1.
self.ML = self.A * ones.vector()
self.ML.set_local(1. / self.ML.array())
def __init__(self,
form,
mesh,
bcs=[],
name="CG1",
method={},
bounded=False):
solver_type = method.get('solver_type', 'cg')
preconditioner_type = method.get('preconditioner_type', 'default')
solver_method = method.get('method', 'default')
self.bounded = bounded
Space = FunctionSpace(mesh, "CG", 1)
OasisFunction.__init__(self,
form,
Space,
bcs=bcs,
name=name,
method=solver_method,
solver_type=solver_type,
preconditioner_type=preconditioner_type)
if solver_method.lower() == "weightedaverage":
from fenicstools import compiled_gradient_module
DG = FunctionSpace(mesh, 'DG', 0)
# Cannot use cache. Matrix will be modified
self.A = assemble(TrialFunction(DG) * self.test * dx())
self.dg = dg = Function(DG)
compiled_gradient_module.compute_DG0_to_CG_weight_matrix(
self.A, dg)
self.bf_dg = inner(form, TestFunction(DG)) * dx()
def err_est_fun(mesh_name2, hol_cyl, T_sol1, T_sol2, deg_choice2, hol_cyl2,
count_it):
'''
Error estimate.
'''
#comm1 = MPI.COMM_WORLD
#rank1 = comm1.Get_rank()
#e1 = sqrt(int_Omega (T - T_ex)**2. dx)/sqrt(int_Omega T**2. dx)
#do.assemble yields integration
e1 = do.sqrt(do.assemble(pow(T_sol2 - T_sol1, 2.) * do.dx(domain = hol_cyl))) / \
do.sqrt(do.assemble(pow(T_sol1, 2.) * do.dx(domain = hol_cyl)))
#sometimes e1 does not work properly (it might be unstable) -> e2
#the degree of piecewise polynomials used to approximate T_th and T...
#...will be the degree of T + degree_rise
e2 = do.errornorm(T_sol1, T_sol2, norm_type = 'l2', degree_rise = 2, mesh = hol_cyl2) / \
do.sqrt(do.assemble(pow(T_sol1, 2.) * do.dx(domain = hol_cyl)))
print 'err: count_t = {}, error_1 = {}, error_2 = {}'.format(
count_it, e1, e2)
#print 'rank = ', rank1, ',
print 'max(T_sol) = ', max(T_sol2.vector().array())
#print 'rank = ', rank1, ',
print 'min(T_sol) = ', min(T_sol2.vector().array()), '\n'
return e1, e2
def __init__(self, aneurysm, near_vessel, *args, **kwargs):
Field.__init__(self, *args, **kwargs)
self.aneurysm = aneurysm
self.near_vessel = near_vessel
self.Vnv = assemble(Constant(1)*dx(near_vessel[1], domain=near_vessel[0].mesh(), subdomain_data=near_vessel[0]))
self.Va = assemble(Constant(1)*dx(aneurysm[1], domain=aneurysm[0].mesh(), subdomain_data=aneurysm[0]))
def _setup_energy(self, rho, vel, gvec, x0):
"""
Calculate kinetic and potential energy
"""
x = df.SpatialCoordinate(self.mesh)
self._form_E_k = Form(1 / 2 * rho * dot(vel, vel) *
dx(domain=self.mesh))
self._form_E_p = Form(rho * dot(-gvec, x - x0) * dx(domain=self.mesh))
def __init__(self, aneurysm, near_vessel, *args, **kwargs):
Field.__init__(self, *args, **kwargs)
self.aneurysm = aneurysm
self.near_vessel = near_vessel
self.Vnv = assemble(
Constant(1) * dx(near_vessel[1],
domain=near_vessel[0].mesh(),
subdomain_data=near_vessel[0]))
self.Va = assemble(
Constant(1) * dx(aneurysm[1],
domain=aneurysm[0].mesh(),
subdomain_data=aneurysm[0]))
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())
C = compiled_cr_module.cr_divergence_matrix(M, A, DG, CR)
return C
def dx(self, domain = "all", dx = None):
"""
Convenience wrapper for integral-cell measure. If the mesh does not contain
any cell domains, the measure for the whole mesh is returned.
*Arguments*
domain (:class:`string` / :class:`int`)
name or ID of domain
dx (:class:`dolfin:Measure`)
alternative measure
*Returns*
:class:`dolfin:Measure`
the measure
# Compute volume of magnetic domain
assemble(Constant(1.0) * state.dx('magnetic'))
# Compute volume of domain with ID 3
assemble(Constant(1.0) * state.dx(3))
"""
# return measure for whole mesh if no domains are defined
if not self.mesh_has_domains(): return self._dx
domain_ids = self.domain_ids(domain)
# XXX fall back to zero measure
if len(domain_ids) == 0: domain_ids = (999999,)
if dx == None: dx = self._dx
return reduce(lambda x,y: x+y, map(lambda i: dx(i), domain_ids))
def __init__(
self,
mesh,
field,
strain_tensor="E",
dmu=None,
approx="original",
F_ref=None,
isochoric=True,
displacement_space="CG_2",
interpolation_space="CG_1",
description="",
):
ModelObservation.__init__(self,
mesh,
target_space="R_0",
description=description)
# Check that the given field is a vector of the same dim as the mesh
dim = mesh.geometry().dim()
# assert isinstance(field, (dolfin.Function, dolfin.Constant)
assert field.ufl_shape[0] == dim
approxs = ["project", "interpolate", "original"]
msg = 'Expected "approx" for be one of {}, got {}'.format(
approxs, approx)
self.field = field
assert approx in approxs, msg
self._approx = approx
self._F_ref = F_ref if F_ref is not None else dolfin.Identity(dim)
self._isochoric = isochoric
tensors = ["gradu", "E", "almansi"]
msg = "Expected strain tensor to be one of {}, got {}".format(
tensors, strain_tensor)
assert strain_tensor in tensors, msg
self._tensor = strain_tensor
if dmu is None:
dmu = dolfin.dx(domain=mesh)
assert isinstance(dmu, dolfin.Measure)
self._dmu = dmu
self._vol = dolfin_adjoint.Constant(
dolfin.assemble(dolfin.Constant(1.0) * dmu))
# These spaces are only used if you want to project
# or interpolate the displacement before assigning it
# Space for interpolating the displacement if needed
family, degree = interpolation_space.split("_")
self._interpolation_space = dolfin.VectorFunctionSpace(
mesh, family, int(degree))
# Displacement space
family, degree = displacement_space.split("_")
self._displacement_space = dolfin.VectorFunctionSpace(
mesh, family, int(degree))
def test_to_DG0(self):
subdomains = (df.CompiledSubDomain('near(x[0], 0.5)'), df.DomainBoundary())
for subd in subdomains:
mesh = df.UnitCubeMesh(4, 4, 4)
facet_f = df.MeshFunction('size_t', mesh, 2, 0)
subd.mark(facet_f, 1)
submesh = EmbeddedMesh(facet_f, 1)
transfer = SubMeshTransfer(mesh, submesh)
V = df.FunctionSpace(mesh, 'Discontinuous Lagrange Trace', 0)
Vsub = df.FunctionSpace(submesh, 'DG', 0)
to_Vsub = transfer.compute_map(Vsub, V, strict=False)
# Set degree 0 to get the quad order right
f = df.Expression('x[0] + 2*x[1] - x[2]', degree=0)
fV = df.interpolate(f, V)
fsub = df.Function(Vsub)
to_Vsub(fsub, fV)
error = df.inner(fsub - f, fsub - f)*df.dx(domain=submesh)
error = df.sqrt(abs(df.assemble(error)))
self.assertTrue(error < 1E-13)
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 test_mat_without_dictionnary():
"""FenicsPart instance initialized with only one instance of Material"""
L_x, L_y = 1, 1
mesh = fe.RectangleMesh(fe.Point(0.0, 0.0), fe.Point(L_x, L_y), 10, 10)
dimensions = np.array(((L_x, 0.0), (0.0, L_y)))
E, nu = 1, 0.3
material = mat.Material(E, nu, "cp")
rect_part = part.FenicsPart(
mesh,
materials=material,
subdomains=None,
global_dimensions=dimensions,
facet_regions=None,
)
elem_type = "CG"
degree = 2
strain_fspace = fe.FunctionSpace(
mesh,
fe.VectorElement(elem_type, mesh.ufl_cell(), degree, dim=3),
)
strain = fe.project(fe.Expression(("1.0+x[0]*x[0]", "0", "1.0"), degree=2),
strain_fspace)
stress = mat.sigma(rect_part.elasticity_tensor, strain)
energy = fe.assemble(fe.inner(stress, strain) * fe.dx(rect_part.mesh))
energy_theo = E / (1 + nu) * (1 + 28 / (15 * (1 - nu)))
assert energy == approx(energy_theo, rel=1e-13)
def verify_function_slice(func_slice, u, cpp_2d, expected_area):
u_2D = func_slice.get_slice(u)
# The 2D solution we want to obtain
analytical = dolfin.Expression(cpp_2d, degree=2 + 3)
comm = dolfin.MPI.comm_world
if comm.rank == 0:
error = dolfin.errornorm(analytical, u_2D)
mesh = func_slice.slice_function_space.mesh()
area = dolfin.assemble(1 * dolfin.dx(domain=mesh))
if True:
import matplotlib
matplotlib.use('Agg')
from matplotlib import pyplot
fig = pyplot.figure()
dolfin.plot(u_2D)
pyplot.gca().view_init(90, 270)
pyplot.gca().set_proj_type('ortho')
pyplot.gca().set_xlabel('x')
pyplot.gca().set_ylabel('y')
fig.savefig('test_func_slice.png')
else:
error = 0.0
area = 0.0
assert dolfin.MPI.max(comm, error) < 0.015
area = dolfin.MPI.sum(comm, area)
assert abs(area - expected_area) < 1e-8
def forward(control, annotate=True):
self.reset_problem()
annotation.annotate = annotate
states = []
functional_values = []
functional = self.create_functional()
functionals_time = []
gen = self.iteration(control)
for count in gen:
# Collect stuff
states.append(dolfin.Vector(self.problem.state.vector()))
functional_values.append(functional)
functionals_time.append(functional)
return forward_result(
functional=dolfin_adjoint.assemble(
list_sum(functionals_time) / self._meshvol *
dolfin.dx(domain=self.problem.geometry.mesh)),
converged=True,
)
def get_residual_form(u, v, rho_e, method='RAMP'):
df.dx = df.dx(metadata={"quadrature_degree": 4})
# stiffness = rho_e/(1 + 8. * (1. - rho_e))
if method == 'SIMP':
stiffness = rho_e**3
else: #RAMP
stiffness = rho_e / (1 + 8. * (1. - rho_e))
# print('the value of stiffness is:', rho_e.vector().get_local())
# Kinematics
k = 3e1
E = k * stiffness
nu = 0.3
mu, lmbda = (E / (2 * (1 + nu))), (E * nu / ((1 + nu) * (1 - 2 * nu)))
d = len(u)
I = df.Identity(d) # Identity tensor
F = I + df.grad(u) # Deformation gradient
C = F.T * F # Right Cauchy-Green tensor
E_ = 0.5 * (C - I) # Green--Lagrange strain
S = 2.0 * mu * E_ + lmbda * df.tr(E_) * df.Identity(
d) # stress tensor (C:eps)
psi = 0.5 * df.inner(S, E_) # 0.5*eps:C:eps
# Total potential energy
Pi = psi * df.dx
# Solve weak problem obtained by differentiating Pi:
res = df.derivative(Pi, u, v)
return res
def test_2_materials():
"""FenicsPart instance initialized with only one instance of Material in the materials dictionnary"""
L_x, L_y = 1, 1
mesh = fe.RectangleMesh(fe.Point(-L_x, -L_y), fe.Point(L_x, L_y), 20, 20)
dimensions = np.array(((2 * L_x, 0.0), (0.0, 2 * L_y)))
subdomains = fe.MeshFunction("size_t", mesh, 2)
class Right_part(fe.SubDomain):
def inside(self, x, on_boundary):
return x[0] >= 0 - fe.DOLFIN_EPS
subdomain_right = Right_part()
subdomains.set_all(0)
subdomain_right.mark(subdomains, 1)
E_1, E_2, nu = 1, 3, 0.3
materials = {0: mat.Material(1, 0.3, "cp"), 1: mat.Material(3, 0.3, "cp")}
rect_part = part.FenicsPart(mesh, materials, subdomains, dimensions)
elem_type = "CG"
degree = 2
strain_fspace = fe.FunctionSpace(
mesh,
fe.VectorElement(elem_type, mesh.ufl_cell(), degree, dim=3),
)
strain = fe.project(fe.Expression(("1.0+x[0]*x[0]", "0", "1.0"), degree=2),
strain_fspace)
stress = mat.sigma(rect_part.elasticity_tensor, strain)
energy = fe.assemble(fe.inner(stress, strain) * fe.dx(rect_part.mesh))
energy_theo = 2 * ((E_1 + E_2) / (1 + nu) * (1 + 28 / (15 * (1 - nu))))
assert energy == approx(energy_theo, rel=1e-13)
def strain_cross_energy(sig, eps, mesh, area):
"""
Calcul de l'energie croisée des champs de contrainte sig et de deformation eps.
"""
# TODO : Redondant. Voir si cela est nécessaire.
return fe.assemble(fe.inner(sig, eps) * fe.dx(mesh)) / area
def test_local_assembler_1D():
mesh = UnitIntervalMesh(MPI.comm_world, 20)
V = FunctionSpace(mesh, 'CG', 1)
u = TrialFunction(V)
v = TestFunction(V)
c = Cell(mesh, 0)
a_scalar = 1.0 * dx(domain=mesh)
a_vector = v * dx
a_matrix = u * v * dx
A_scalar = assemble_local(a_scalar, c)
A_vector = assemble_local(a_vector, c)
A_matrix = assemble_local(a_matrix, c)
assert isinstance(A_scalar, float)
assert numpy.isclose(A_scalar, 0.05)
assert isinstance(A_vector, numpy.ndarray)
assert A_vector.shape == (2,)
assert numpy.isclose(A_vector[0], 0.025)
assert numpy.isclose(A_vector[1], 0.025)
assert isinstance(A_matrix, numpy.ndarray)
assert A_matrix.shape == (2, 2)
assert numpy.isclose(A_matrix[0, 0], 1 / 60)
assert numpy.isclose(A_matrix[0, 1], 1 / 120)
assert numpy.isclose(A_matrix[1, 0], 1 / 120)
assert numpy.isclose(A_matrix[1, 1], 1 / 60)
def __init__(self, V, parameters=None):
# Set parameters
self.parameters = self.default_parameters()
if parameters is not None:
self.parameters.update(parameters)
# Set-up mass matrix for L^2 projection
self.V = V
self.u = df.TrialFunction(self.V)
self.v = df.TestFunction(self.V)
self.m = df.inner(self.u, self.v) * df.dx()
self.M = df.assemble(self.m)
self.b = df.Vector(V.mesh().mpi_comm(), V.dim())
solver_type = self.parameters["linear_solver_type"]
assert(solver_type == "lu" or solver_type == "cg"), \
"Expecting 'linear_solver_type' to be 'lu' or 'cg'"
if solver_type == "lu":
df.debug("Setting up direct solver for projecter")
# Customize LU solver (reuse everything)
solver = df.LUSolver(self.M)
solver.parameters["same_nonzero_pattern"] = True
solver.parameters["reuse_factorization"] = True
else:
df.debug("Setting up iterative solver for projecter")
# Customize Krylov solver (reuse everything)
solver = df.KrylovSolver("cg", "ilu")
solver.set_operator(self.M)
solver.parameters["preconditioner"]["structure"] = "same"
# solver.parameters["nonzero_initial_guess"] = True
self.solver = solver
def test_to_DG0_subdomain(self):
mesh = df.UnitSquareMesh(4, 4)
cell_f = df.MeshFunction('size_t', mesh, 2, 0)
df.CompiledSubDomain('x[0] < 0.5 + DOLFIN_EPS').mark(cell_f, 1)
submesh = EmbeddedMesh(cell_f, 1)
transfer = SubMeshTransfer(mesh, submesh)
V = df.FunctionSpace(mesh, 'DG', 0)
Vsub = df.FunctionSpace(submesh, 'DG', 0)
to_Vsub = transfer.compute_map(Vsub, V, strict=True)
# Set degree 0 to get the quad order right
f = df.Expression('x[0] + 2*x[1]', degree=0)
fV = df.interpolate(f, V)
fsub = df.Function(Vsub)
to_Vsub(fsub, fV)
error = df.inner(fsub - f, fsub - f)*df.dx(domain=submesh)
error = df.sqrt(abs(df.assemble(error)))
self.assertTrue(error < 1E-13)
def create(self):
self.metadata = {
"quadrature_degree": self.deg_q,
"quadrature_scheme": "default",
}
self.dxm = df.dx(metadata=self.metadata,
subdomain_data=self.mesh_function)
# solution field
Ed = df.VectorElement("CG", self.mesh.ufl_cell(), degree=self.deg_d)
Ee = df.FiniteElement("CG", self.mesh.ufl_cell(), degree=self.deg_d)
self.V = df.FunctionSpace(self.mesh, Ed * Ee)
self.Vd, self.Ve = self.V.split()
self.dd, self.de = df.TrialFunctions(self.V)
self.d_, self.e_ = df.TestFunctions(self.V)
self.u = df.Function(self.V, name="d-e mixed space")
self.d, self.e = df.split(self.u)
# generic quadrature function spaces
VQF, VQV, VQT = c.helper.spaces(self.mesh, self.deg_q,
c.q_dim(self.constraint))
# quadrature functions
Q = c.Q
# inputs to the model
self.q_in = OrderedDict()
self.q_in[Q.EPS] = df.Function(VQV, name="current strains")
self.q_in[Q.E] = df.Function(
VQF, name="current nonlocal equivalent strains")
self.q_in_calc = {}
self.q_in_calc[Q.EPS] = c.helper.LocalProjector(
self.eps(self.d), VQV, self.dxm)
self.q_in_calc[Q.E] = c.helper.LocalProjector(self.e, VQF, self.dxm)
# outputs of the model
self.q = {}
self.q[Q.SIGMA] = df.Function(VQV, name="current stresses")
self.q[Q.DSIGMA_DEPS] = df.Function(VQT, name="stress-strain tangent")
self.q[Q.DSIGMA_DE] = df.Function(
VQV, name="stress-nonlocal-strain tangent")
self.q[Q.EEQ] = df.Function(VQF,
name="current (local) equivalent strain")
self.q[Q.DEEQ] = df.Function(VQV,
name="equivalent-strain-strain tangent")
self.q_history = {
Q.KAPPA: df.Function(VQF, name="current history variable kappa")
}
self.n = len(self.q[Q.SIGMA].vector().get_local()) // c.q_dim(
self.constraint)
self.nq = self.n // self.mesh.num_cells()
self.ip_flags = None
if self.mesh_function is not None:
self.ip_flags = np.repeat(self.mesh_function.array(), self.nq)
def pressure_correction(self):
"""
Solve the pressure correction equation
We handle the case where only Neumann conditions are given
for the pressure by taking out the nullspace, a constant shift
of the pressure, by providing the nullspace to the solver
"""
p = self.simulation.data['p']
p_hat = self.simulation.data['p_hat']
# Temporarily store the old pressure
p_hat.vector().zero()
p_hat.vector().axpy(-1, p.vector())
# Assemble the A matrix only the first inner iteration
if self.inner_iteration == 1:
self.Ap = dolfin.as_backend_type(self.eq_pressure.assemble_lhs())
A = self.Ap
b = dolfin.as_backend_type(self.eq_pressure.assemble_rhs())
# Inform PETSc about the null space
if self.remove_null_space:
if self.pressure_null_space is None:
# Create vector that spans the null space
null_vec = dolfin.Vector(p.vector())
null_vec[:] = 1
null_vec *= 1 / null_vec.norm("l2")
# Create null space basis object
self.pressure_null_space = dolfin.VectorSpaceBasis([null_vec])
# Make sure the null space is set on the matrix
if self.inner_iteration == 1:
A.set_nullspace(self.pressure_null_space)
# Orthogonalize b with respect to the null space
self.pressure_null_space.orthogonalize(b)
# Solve for the new pressure correction
self.niters_p = self.pressure_solver.inner_solve(
A,
p.vector(),
b,
in_iter=self.inner_iteration,
co_iter=self.co_inner_iter)
# Removing the null space of the matrix system is not strictly the same as removing
# the null space of the equation, so we correct for this here
if self.remove_null_space:
dx2 = dolfin.dx(domain=p.function_space().mesh())
vol = dolfin.assemble(dolfin.Constant(1) * dx2)
pavg = dolfin.assemble(p * dx2) / vol
p.vector()[:] -= pavg
# Calculate p_hat = p_new - p_old
p_hat.vector().axpy(1, p.vector())
return p_hat.vector().norm('l2')
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 matvec(self, b):
'''Action on vector from V1'''
V, Q = self.V1, self.V0
gamma_mesh = V.mesh()
auxiliary_facet_f = self.facet_f
aux_mesh = auxiliary_facet_f.mesh()
# The problem for uh
V2 = df.FunctionSpace(aux_mesh, Q.ufl_element().family(), Q.ufl_element().degree())
u, v = df.TrialFunction(V2), df.TestFunction(V2)
# Wrap the vector as function ...
f = df.Function(V, b)
# ... to be used in the weak form for the Laplacian
a = df.inner(df.grad(u), df.grad(v))*df.dx + df.inner(u, v)*df.dx
L = df.inner(f, Trace(v, gamma_mesh))*df.dx(domain=gamma_mesh)
A, b = list(map(xii.assembler.xii_assembly.assemble, (a, L)))
# We have boundary conditions to apply
# bc = df.DirichletBC(V2, df.Constant(0), auxiliary_facet_f, 1)
# A, b = apply_bc(A, b, bc)
uh = df.Function(V2)
df.solve(A, uh.vector(), b)
# Now take trace of that
ext_mesh = Q.mesh()
# Get the trace at extended domain by projection
p, q = df.TrialFunction(Q), df.TestFunction(Q)
f = Trace(uh, ext_mesh)
a = df.inner(p, q)*df.dx
L = df.inner(f, q)*df.dx(domain=ext_mesh)
A, b = list(map(xii.assembler.xii_assembly.assemble, (a, L)))
# We have boundary conditions to apply
# FIXME: inherit from uh?
# bc = df.DirichletBC(Q, uh, 'on_boundary')
# A, b = apply_bc(A, b, bc)
qh = df.Function(Q)
df.solve(A, qh.vector(), b)
return qh.vector()
def compute(self, get):
u = get("Velocity")
assemble(dot(u[1].dx(0) - u[0].dx(1), self.q) * dx(), tensor=self.L)
self.bc.apply(self.L)
self.solver.solve(self.psi.vector(), self.L)
#solve(self.A, self.psi.vector(), self.L)
return self.psi
def test_nasty_jit_caching_bug():
# This may result in something like "matrices are not aligned"
# from FIAT if the JIT caching does not recognize that the two
# forms are different
default_parameters = parameters["form_compiler"]["representation"]
for representation in ["quadrature"]:
parameters["form_compiler"]["representation"] = representation
M1 = assemble(Constant(1.0)*dx(UnitSquareMesh(4, 4)))
M2 = assemble(Constant(1.0)*dx(UnitCubeMesh(4, 4, 4)))
assert round(M1 - 1.0, 7) == 0
assert round(M2 - 1.0, 7) == 0
parameters["form_compiler"]["representation"] = default_parameters
def __init__(self, valuename, nu, subdomain,*args, **kwargs):
Field.__init__(self, *args, **kwargs)
self.valuename = valuename
mf = subdomain[0]
idx = subdomain[1]
self.dx = dx(idx, domain=mf.mesh(), subdomain_data=mf)
self.vol = assemble(Constant(1)*self.dx)
self.nu = nu
def set_cell_domains(self, cell_domains, cell_mark_value):
"""Set cell domain mesh function
Needed if the functional is to be calculated over only part of the domain
"""
self.cell_domains = cell_domains
self.cell_mark_value = cell_mark_value
self.dx = dx(self.cell_mark_value)
self.dirty = True
def set_cell_domains(self, cell_domains, cell_mark_value):
"""Set cell domain mesh function
Needed if the functional is to be calculated over only part of the domain
"""
self.cell_domains = cell_domains
self.cell_mark_value = cell_mark_value
self.dx = dx(self.cell_mark_value)
self.dirty = True
def __call__(self, u=None, assemb_rhs=True):
if isinstance(u, Coefficient):
self.matvec[1] = u
self.bf = u.dx(self.i)*self.test*dx()
if self.method.lower() == "gradient_matrix":
self.vector()[:] = self.WGM * self.matvec[1].vector()
else:
OasisFunction.__call__(self, assemb_rhs=assemb_rhs)
def make_mass_matrix_quadrature_lumping(function_space_V):
V = function_space_V
u_trial = dl.TrialFunction(V)
v_test = dl.TestFunction(V)
# mass lumping scheme code by Jeremy Bleyer
# See: https://comet-fenics.readthedocs.io/en/latest/demo/tips_and_tricks/mass_lumping.html
mass_form = u_trial * v_test * dl.dx(scheme="lumped", degree=2)
ML = dl.assemble(mass_form)
return ML
def test_nasty_jit_caching_bug():
# This may result in something like "matrices are not aligned"
# from FIAT if the JIT caching does not recognize that the two
# forms are different
default_parameters = parameters["form_compiler"]["representation"]
for representation in ["quadrature"]:
parameters["form_compiler"]["representation"] = representation
M1 = assemble(Constant(1.0) * dx(UnitSquareMesh(4, 4)))
M2 = assemble(Constant(1.0) * dx(UnitCubeMesh(4, 4, 4)))
assert round(M1 - 1.0, 7) == 0
assert round(M2 - 1.0, 7) == 0
parameters["form_compiler"]["representation"] = default_parameters
def __init__(self, valuename, nu, subdomain, *args, **kwargs):
Field.__init__(self, *args, **kwargs)
self.valuename = valuename
mf = subdomain[0]
idx = subdomain[1]
self.dx = dx(idx, domain=mf.mesh(), subdomain_data=mf)
self.vol = assemble(Constant(1) * self.dx)
self.nu = nu
def compute(self, get):
u = get("Velocity")
assemble(dot(u[1].dx(0)-u[0].dx(1), self.q)*dx(), tensor=self.L)
self.bc.apply(self.L)
self.solver.solve(self.psi.vector(), self.L)
#solve(self.A, self.psi.vector(), self.L)
return self.psi
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 __init__(self, form, mesh, bcs=[], name="CG1", method={}, bounded=False):
solver_type = method.get('solver_type', 'cg')
preconditioner_type = method.get('preconditioner_type', 'default')
solver_method = method.get('method', 'default')
self.bounded = bounded
Space = FunctionSpace(mesh, "CG", 1)
OasisFunction.__init__(self, form, Space,
bcs=bcs, name=name,
method=solver_method, solver_type=solver_type,
preconditioner_type=preconditioner_type)
if solver_method.lower() == "weightedaverage":
from fenicstools import compiled_gradient_module
DG = FunctionSpace(mesh, 'DG', 0)
self.A = assemble(TrialFunction(DG)*self.test*dx()) # Cannot use cache. Matrix will be modified
self.dg = dg = Function(DG)
compiled_gradient_module.compute_DG0_to_CG_weight_matrix(self.A, dg)
self.bf_dg = inner(form, TestFunction(DG))*dx()
def compute(self, get):
t1 = get("t")
t0 = get("t", -1)
dt = Constant(t1 - t0)
u = get("Velocity")
hF = self._hF
hK = self._hK
scaling = 1.0 / hK
assemble((dt * sqrt(u**2) / hF)*self._v*scaling*dx(),
tensor=self._cfl.vector())
return self._cfl
def _print_diagnostics(
theta0,
umax,
submesh_workpiece,
wpi_area,
subdomain_materials,
wpi,
rho,
mu,
char_length,
grav,
):
av_temperature = assemble(theta0 * dx(submesh_workpiece)) / wpi_area
vec = theta0.vector()
temperature_difference = vec.max() - vec.min()
info("Max temperature: {:e}".format(vec.max()))
info("Min temperature: {:e}".format(vec.min()))
info("Av temperature: {:e}".format(av_temperature))
info("")
info("Max velocity: {:e}".format(umax))
info("")
char_velocity = umax
melt_material = subdomain_materials[wpi]
rho_const = rho(av_temperature)
cp = melt_material.specific_heat_capacity
if not isinstance(cp, float):
cp = cp(av_temperature)
k = melt_material.thermal_conductivity
if not isinstance(k, float):
k = k(av_temperature)
mu_const = mu(av_temperature)
#
info("Prandtl number: {:e}".format(_get_prandtl(mu_const, rho_const, cp, k)))
info(
"Reynolds number: {:e}".format(
_get_reynolds(rho_const, mu_const, char_length, char_velocity)
)
)
info(
"Grashof number: {:e}".format(
_get_grashof(
rho, mu_const, grav, av_temperature, char_length, temperature_difference
)
)
)
return
def before_first_compute(self, get):
u = get("Velocity")
assert len(u) == 2, "Can only compute stream function for 2D problems"
V = u.function_space()
spaces = SpacePool(V.mesh())
degree = V.ufl_element().degree()
V = spaces.get_space(degree, 0)
psi = TrialFunction(V)
self.q = TestFunction(V)
a = dot(grad(psi), grad(self.q))*dx()
self.bc = DirichletBC(V, Constant(0), DomainBoundary())
self.A = assemble(a)
self.L = Vector()
self.bc.apply(self.A)
self.solver = KrylovSolver(self.A, "cg")
self.psi = Function(V)
def add_fields(self):
Field.start_recording()
params = self.params.copy_recursive()
if not self.params.debug:
params["save"] = False
params["plot"] = False
params.pop("debug")
params.pop("use_timeaverage")
params.pop("finalize")
T0, T1 = self.params.start_time, self.params.end_time
assert T0 != Field.default_params().start_time
assert T1 != Field.default_params().end_time
tau = Magnitude("WSS")
if self.params.use_timeaverage:
tau = TimeAverage(tau, params=params)
threshold = DomainAvg(tau, cell_domains=self.near_vessel[0], indicator=self.near_vessel[1], label="nv") + \
DomainSD(tau, cell_domains=self.aneurysm[0], indicator=self.aneurysm[1], label="aneurysm")
threshold.name = "threshold_sci_nv"
mask = Threshold(tau, threshold, dict(threshold_by="above"))
mf = self.aneurysm[0]
idx = self.aneurysm[1]
Aa = ConstantField(assemble(Constant(1)*dx(idx, domain=mf.mesh(), subdomain_data=mf)))
Aa.name = "AneurysmArea"
Fh = Aa*DomainAvg(tau*mask, cell_domains=mf, indicator=idx, params=params)
Fh.name = "HighShear"
Ah = Aa*DomainAvg(mask, cell_domains=mf, indicator=idx, params=params)+ConstantField(1e-12)
Ah.name = "HighShearArea"
Fa = Aa*DomainAvg(tau, cell_domains=mf, indicator=idx, params=params)
Fa.name = "TotalShear"
f = (Fh/Fa)/(Ah/Aa)
if not self.params.use_timeaverage:
f = TimeAverage(f, params=params)
self.valuename = f.name
fields = Field.stop_recording()
return fields
def assemble_rhs(self, u=None):
"""
Assemble right hand side trial.dx(i)*test*dx.
Possible Coefficient u may replace p_ and makes it possible
to use this Function to compute both grad(p) and grad(dp), i.e.,
the gradient of pressure correction.
"""
if isinstance(u, Coefficient):
self.matvec[1] = u
self.bf = u.dx(self.i) * self.test * dx()
if not self.matvec[0] is None:
mat, func = self.matvec
self.rhs.zero()
self.rhs.axpy(1.0, mat * func.vector())
else:
assemble(self.bf, tensor=self.rhs)
def sobolev_norm(u, p, k=1, domain=None):
"""Return Sobolev seminorm on W^{k, p} of function u. If u is
None, return infinity."""
# Special case
if u is None:
return np.infty
# Prepare exponent and measure
p = Constant(p) if p is not 2 else p
dX = dx(domain)
# Take derivative if k==1
if k == 1:
u = grad(u)
elif k == 0:
u = u
else:
raise NotImplementedError
# Assemble functional and return
return assemble(inner(u, u)**(p/2)*dX)**(1.0/float(p))
# Right hand side and boundary value for velocity
f = Expression(('sin(pi*x[1])', 'cos(pi*(x[0]+x[1]))'))
g = Constant((0, 0))
N = 20
gamma_d = 'everywhere'
Ad, ud, pd, eigs_d = dolfin_system(f=f, g=g, N=N, gamma_d=gamma_d)
Ac, uc, pc, eigs_c = cbc_block_system(f=f, g=g, N=N, gamma_d=gamma_d)
plt.figure()
plt.suptitle('DOLFIN')
plt.spy(Ad, marker='.')
plt.figure()
plt.suptitle('CBC.BLOCK')
plt.spy(Ac, marker='.')
plt.show()
mesh = UnitSquareMesh(20, 20)
print 'Velocity diff', assemble(inner(ud-uc, ud-uc)*dx(domain=mesh))
print 'Pressure diff', assemble(inner(pd-pc, pd-pc)*dx(domain=mesh))
print 'DOLFIN eigs', eigs_d
print 'CBC.BLOCK eigs', eigs_c
print 'Norm of diff', (eigs_d - eigs_c).dot(eigs_d - eigs_c)
# All okay
# Setting sigma for eigensolver to detect zero!
# Sign of a21 to get eigenvalue agreement with DOLFIN
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 __init__(self, Vh, covariance, mean=None):
"""
Constructor
Inputs:
- :code:`Vh`: Finite element space on which the prior is
defined. Must be the Real space with one global
degree of freedom
- :code:`covariance`: The covariance of the prior. Must be a
:code:`numpy.ndarray` of appropriate size
- :code:`mean`(optional): Mean of the prior distribution. Must be of
type `dolfin.Vector()`
"""
self.Vh = Vh
if Vh.dim() != covariance.shape[0] or Vh.dim() != covariance.shape[1]:
raise ValueError("Covariance incompatible with Finite Element space")
if not np.issubdtype(covariance.dtype, np.floating):
raise TypeError("Covariance matrix must be a float array")
self.covariance = covariance
#np.linalg.cholesky automatically provides more error checking,
#so use those
self.chol = np.linalg.cholesky(self.covariance)
self.chol_inv = scila.solve_triangular(
self.chol,
np.identity(Vh.dim()),
lower=True)
self.precision = np.dot(self.chol_inv.T, self.chol_inv)
trial = dl.TrialFunction(Vh)
test = dl.TestFunction(Vh)
domain_measure_inv = dl.Constant(1.0 \
/ dl.assemble(dl.Constant(1.) * dl.dx(Vh.mesh())))
#Identity mass matrix
self.M = dl.assemble(domain_measure_inv * dl.inner(trial, test) * dl.dx)
self.Msolver = Operator2Solver(self.M)
if mean:
self.mean = mean
else:
tmp = dl.Vector()
self.M.init_vector(tmp, 0)
tmp.zero()
self.mean = tmp
if Vh.dim() == 1:
trial = dl.as_matrix([[trial]])
test = dl.as_matrix([[test]])
#Create form matrices
covariance_op = dl.as_matrix(list(map(list, self.covariance)))
precision_op = dl.as_matrix(list(map(list, self.precision)))
chol_op = dl.as_matrix(list(map(list, self.chol)))
chol_inv_op = dl.as_matrix(list(map(list, self.chol_inv)))
#variational for the regularization operator, or the precision matrix
var_form_R = domain_measure_inv \
* dl.inner(test, dl.dot(precision_op, trial)) * dl.dx
#variational for the inverse regularization operator, or the covariance
#matrix
var_form_Rinv = domain_measure_inv \
* dl.inner(test, dl.dot(covariance_op, trial)) * dl.dx
#variational form for the square root of the regularization operator
var_form_R_sqrt = domain_measure_inv \
* dl.inner(test, dl.dot(chol_inv_op.T, trial)) * dl.dx
#variational form for the square root of the inverse regularization
#operator
var_form_Rinv_sqrt = domain_measure_inv \
* dl.inner(test, dl.dot(chol_op, trial)) * dl.dx
self.R = dl.assemble(var_form_R)
self.RSolverOp = dl.assemble(var_form_Rinv)
self.Rsolver = Operator2Solver(self.RSolverOp)
self.sqrtR = dl.assemble(var_form_R_sqrt)
self.sqrtRinv = dl.assemble(var_form_Rinv_sqrt)
def compute_time_errors(problem, MethodClass, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Translate data into FEniCS expressions.
sol_u = Expression((smp.printing.ccode(solution['u']['value'][0]),
smp.printing.ccode(solution['u']['value'][1])
),
degree=_truncate_degree(solution['u']['degree']),
t=0.0,
cell=cell_type
)
sol_p = Expression(smp.printing.ccode(solution['p']['value']),
degree=_truncate_degree(solution['p']['degree']),
t=0.0,
cell=cell_type
)
fenics_rhs0 = Expression((smp.printing.ccode(f['value'][0]),
smp.printing.ccode(f['value'][1])
),
degree=_truncate_degree(f['degree']),
t=0.0,
mu=mu, rho=rho,
cell=cell_type
)
# Deep-copy expression to be able to provide f0, f1 for the Dirichlet-
# boundary conditions later on.
fenics_rhs1 = Expression(fenics_rhs0.cppcode,
degree=_truncate_degree(f['degree']),
t=0.0,
mu=mu, rho=rho,
cell=cell_type
)
# Create initial states.
p0 = Expression(
sol_p.cppcode,
degree=_truncate_degree(solution['p']['degree']),
t=0.0,
cell=cell_type
)
# Compute the problem
errors = {'u': numpy.empty((len(mesh_sizes), len(Dt))),
'p': numpy.empty((len(mesh_sizes), len(Dt)))
}
for k, mesh_size in enumerate(mesh_sizes):
info('')
info('')
with Message('Computing for mesh size %r...' % mesh_size):
mesh = mesh_generator(mesh_size)
mesh_area = assemble(1.0 * dx(mesh))
W = VectorFunctionSpace(mesh, 'CG', 2)
P = FunctionSpace(mesh, 'CG', 1)
method = MethodClass(W, P,
rho, mu,
theta=1.0,
#theta=0.5,
stabilization=None
#stabilization='SUPG'
)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u = [Expression(
sol_u.cppcode,
degree=_truncate_degree(solution['u']['degree']),
t=0.0,
cell=cell_type
),
# Expression(
#sol_u.cppcode,
#degree=_truncate_degree(solution['u']['degree']),
#t=0.5*dt,
#cell=cell_type
#)
]
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, 'on_boundary')]
sol_p.t = dt
#p_bcs = [DirichletBC(P, sol_p, 'on_boundary')]
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
method.step(dt,
u1, p1,
u, p0,
u_bcs=u_bcs, p_bcs=p_bcs,
f0=fenics_rhs0, f1=fenics_rhs1,
verbose=False,
tol=1.0e-10
)
sol_u.t = dt
sol_p.t = dt
errors['u'][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = assemble(sol_p * dx(mesh)) \
- assemble(p1 * dx(mesh))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors['p'][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title='p1', mesh=mesh)
plot(sol_p, title='sol_p', mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
#plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title='p1 - sol_p', mesh=mesh)
#r = Expression('x[0]', degree=1, cell=triangle)
#divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title='div(u1)')
interactive()
return errors
def average(u):
"""Computes the average value of a function u over its domain.
"""
return assemble(u * dx) / assemble(1.0 * dx(u.function_space().mesh()))
def add_fields(self):
Field.start_recording()
params = self.params.copy_recursive()
if not self.params.debug:
params["save"] = False
params["plot"] = False
params.pop("debug")
params.pop("use_timeaverage")
params.pop("finalize")
T0, T1 = self.params.start_time, self.params.end_time
assert T0 != Field.default_params().start_time
assert T1 != Field.default_params().end_time
if self.params.use_timeaverage:
u = TimeAverage("Velocity", params=params)
velocity = u.name
else:
velocity = "Velocity"
uneck = SubFunction(velocity, self.neck, params=params, label="neck")
A = assemble(Constant(1)*dx(domain=self.neck))
Qin = Dot(ConstantField(self.necknormal), uneck, params=params)
t = Threshold(Qin, ConstantField(0), dict(threshold_by="above"))
t.params.update(params)
t.name = "threshold_neck"
Ain = A*DomainAvg(t)
Ain.name = "Ain"
Ain.params.update(params)
Qin = A*DomainAvg(Dot(Qin,t, params=params))
Qin.name = "Qin"
Qin.params.update(params)
Qpa = 0
for i, (plane, n) in enumerate(self.pa_planes):
upa = SubFunction(velocity, plane, params=params, label="pa_%d" %i)
Ai = assemble(Constant(1)*dx(domain=plane))
Q = Ai*DomainAvg(Dot(ConstantField(n), upa, params=params))
Q.name = "Qpa%d" %i
Q.params.update(params)
Qpa += Magnitude(Q)
Qpa.params.update(params)
Qpa.name = "Sum_Qpa"
f = (Qin/Qpa)/(Ain/A)
if not self.params.use_timeaverage:
f = TimeAverage(f.name, params=params)
self.valuename = f.name
self.Qin = Qin.name
self.Qpa = Qpa.name
self.Ain = Ain.name
self.A = A
fields = Field.stop_recording()
return fields
def _average(u):
'''Computes the average value of a function u over its domain.
'''
return assemble(u * dx) \
/ assemble(1.0 * dx(u.function_space().mesh()))
def test_boussinesq(with_voltage, target_time=1.0e-2, show=False):
"""Simple boussinesq test; no Maxwell involved.
"""
problem = problems.Crucible()
# Solve construct initial state without Lorentz force and Joule heat.
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
g = Constant((0.0, -9.80665, 0.0))
submesh_workpiece = problem.W.mesh()
ds_workpiece = Measure("ds", subdomain_data=problem.wp_boundaries)
u0, p0, theta0 = _construct_initial_state(
submesh_workpiece,
problem.W_element,
problem.P_element,
problem.Q_element,
k_wpi,
cp_wpi,
rho_wpi,
mu_wpi,
Constant(0.0),
problem.u_bcs,
problem.p_bcs,
problem.theta_bcs_d,
problem.theta_bcs_n,
dx(submesh_workpiece),
ds_workpiece,
g,
extra_force=None,
)
if with_voltage:
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, _ = get_lorentz_joule(problem, voltages, show=show)
else:
lorentz = None
joule = Constant(0.0)
u1, _, theta1 = _compute_boussinesq(
problem,
u0,
p0,
theta0,
lorentz=lorentz,
joule=joule,
target_time=target_time,
show=show,
)
if with_voltage:
# p is only defined up to a constant
ref = 0.001069306823450495
assert abs(norm(u1, "L2") - ref) < 1.0e-3 * ref
ref = 86.96271240592199
assert abs(norm(theta1, "L2") - ref) < 1.0e-3 * ref
else:
ref = 0.001069283700624996
assert abs(norm(u1, "L2") - ref) < 1.0e-3 * ref
ref = 86.96274092197706
assert abs(norm(theta1, "L2") - ref) < 1.0e-3 * ref
return
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 compute_time_errors(problem, MethodClass, mesh_sizes, Dt):
mesh_generator, solution, f, mu, rho, cell_type = problem()
# Compute the problem
errors = {
"u": numpy.empty((len(mesh_sizes), len(Dt))),
"p": numpy.empty((len(mesh_sizes), len(Dt))),
}
for k, mesh_size in enumerate(mesh_sizes):
info("")
info("")
with Message("Computing for mesh size {}...".format(mesh_size)):
mesh = mesh_generator(mesh_size)
# Define all expression with `domain`, see
# <https://bitbucket.org/fenics-project/ufl/issues/96>.
#
# Translate data into FEniCS expressions.
sol_u = Expression(
(ccode(solution["u"]["value"][0]), ccode(solution["u"]["value"][1])),
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
domain=mesh,
)
sol_p = Expression(
ccode(solution["p"]["value"]),
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
fenics_rhs0 = Expression(
(ccode(f["value"][0]), ccode(f["value"][1])),
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Deep-copy expression to be able to provide f0, f1 for the
# Dirichlet boundary conditions later on.
fenics_rhs1 = Expression(
fenics_rhs0.cppcode,
degree=_truncate_degree(f["degree"]),
t=0.0,
mu=mu,
rho=rho,
domain=mesh,
)
# Create initial states.
W = VectorFunctionSpace(mesh, "CG", 2)
P = FunctionSpace(mesh, "CG", 1)
p0 = Expression(
sol_p.cppcode,
degree=_truncate_degree(solution["p"]["degree"]),
t=0.0,
domain=mesh,
)
mesh_area = assemble(1.0 * dx(mesh))
method = MethodClass(
time_step_method="backward euler",
# time_step_method='crank-nicolson',
# stabilization=None
# stabilization='SUPG'
)
u1 = Function(W)
p1 = Function(P)
err_p = Function(P)
divu1 = Function(P)
for j, dt in enumerate(Dt):
# Prepare previous states for multistepping.
u = {
0: Expression(
sol_u.cppcode,
degree=_truncate_degree(solution["u"]["degree"]),
t=0.0,
cell=cell_type,
)
}
sol_u.t = dt
u_bcs = [DirichletBC(W, sol_u, "on_boundary")]
sol_p.t = dt
# p_bcs = [DirichletBC(P, sol_p, 'on_boundary')]
p_bcs = []
fenics_rhs0.t = 0.0
fenics_rhs1.t = dt
u1, p1 = method.step(
Constant(dt),
u,
p0,
W,
P,
u_bcs,
p_bcs,
Constant(rho),
Constant(mu),
f={0: fenics_rhs0, 1: fenics_rhs1},
verbose=False,
tol=1.0e-10,
)
sol_u.t = dt
sol_p.t = dt
errors["u"][k][j] = errornorm(sol_u, u1)
# The pressure is only determined up to a constant which makes
# it a bit harder to define what the error is. For our
# purposes, choose an alpha_0\in\R such that
#
# alpha0 = argmin ||e - alpha||^2
#
# with e := sol_p - p.
# This alpha0 is unique and explicitly given by
#
# alpha0 = 1/(2|Omega|) \int (e + e*)
# = 1/|Omega| \int Re(e),
#
# i.e., the mean error in \Omega.
alpha = +assemble(sol_p * dx(mesh)) - assemble(p1 * dx(mesh))
alpha /= mesh_area
# We would like to perform
# p1 += alpha.
# To avoid creating a temporary function every time, assume
# that p1 lives in a function space where the coefficients
# represent actual function values. This is true for CG
# elements, for example. In that case, we can just add any
# number to the vector of p1.
p1.vector()[:] += alpha
errors["p"][k][j] = errornorm(sol_p, p1)
show_plots = False
if show_plots:
plot(p1, title="p1", mesh=mesh)
plot(sol_p, title="sol_p", mesh=mesh)
err_p.vector()[:] = p1.vector()
sol_interp = interpolate(sol_p, P)
err_p.vector()[:] -= sol_interp.vector()
# plot(sol_p - p1, title='p1 - sol_p', mesh=mesh)
plot(err_p, title="p1 - sol_p", mesh=mesh)
# r = SpatialCoordinate(mesh)[0]
# divu1 = 1 / r * (r * u1[0]).dx(0) + u1[1].dx(1)
divu1.assign(project(u1[0].dx(0) + u1[1].dx(1), P))
plot(divu1, title="div(u1)")
return errors