from LagrangianParticles import LagrangianParticles
from particle_generators import RandomCircle
import matplotlib.pyplot as plt
from dolfin import VectorFunctionSpace, interpolate, RectangleMesh, Expression, Point
from mpi4py import MPI as pyMPI
comm = pyMPI.COMM_WORLD
mesh = RectangleMesh(Point(0, 0), Point(1, 1), 10, 10)
particle_positions = RandomCircle([0.5, 0.75], 0.15).generate([100, 100])
V = VectorFunctionSpace(mesh, 'CG', 1)
u = interpolate(
Expression(("-2*sin(pi*x[1])*cos(pi*x[1])*pow(sin(pi*x[0]),2)",
"2*sin(pi*x[0])*cos(pi*x[0])*pow(sin(pi*x[1]),2)")), V)
lp = LagrangianParticles(V)
lp.add_particles(particle_positions)
fig = plt.figure()
lp.scatter(fig)
fig.suptitle('Initial')
if comm.Get_rank() == 0:
fig.show()
plt.ion()
save = False
dt = 0.01
for step in range(500):
def unstructured_mesh_2d():
domain = Rectangle(Point(0., 0.), Point(1., 1.))
return generate_mesh(domain, 5)
def test_eim_approximation_17(expression_type, basis_generation):
"""
This test is similar to test 15. However, in contrast to test 15, the solution is not split at all.
* EIM: unsplit solution is used in the definition of the parametrized expression, similarly to test 11.
* DEIM: unsplit solution is used in the definition of the parametrized tensor. This results in a single coefficient
of type Function, which however is stored internally by UFL as an Indexed of Function and a mute index. This test
requires the FEniCS backend to properly differentiate between Indexed objects with a fixed index (such as a component
of the solution as in test 13) and Indexed objects with a mute index, which should be treated has if the entire solution
was required.
"""
@StoreMapFromProblemNameToProblem
@StoreMapFromProblemToTrainingStatus
@StoreMapFromSolutionToProblem
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
# Call parent
ParametrizedProblem.__init__(
self,
os.path.join("test_eim_approximation_17_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a ParametrizedDifferentialProblem
self.V = V
self._solution = Function(V)
self.components = ["u", "s", "p"]
# Parametrized function to be interpolated
x = SpatialCoordinate(V.mesh())
mu = SymbolicParameters(self, V, (-1., -1.))
self.f00 = 1. / sqrt(
pow(x[0] - mu[0], 2) + pow(x[1] - mu[1], 2) + 0.01)
self.f01 = 1. / sqrt(
pow(x[0] - mu[0], 4) + pow(x[1] - mu[1], 4) + 0.01)
# Inner product
f = TrialFunction(self.V)
g = TestFunction(self.V)
self.inner_product = assemble(inner(f, g) * dx)
# Collapsed vector and space
self.V0 = V.sub(0).collapse()
self.V00 = V.sub(0).sub(0).collapse()
self.V1 = V.sub(1).collapse()
def name(self):
return "MockProblem_17_" + expression_type + "_" + basis_generation
def init(self):
pass
def solve(self):
assert not hasattr(self, "_is_solving")
self._is_solving = True
f00 = project(self.f00, self.V00)
f01 = project(self.f01, self.V00)
assign(self._solution.sub(0).sub(0), f00)
assign(self._solution.sub(0).sub(1), f01)
delattr(self, "_is_solving")
return self._solution
@StoreMapFromProblemToReductionMethod
@UpdateMapFromProblemToTrainingStatus
class MockReductionMethod(ReductionMethod):
def __init__(self, truth_problem, **kwargs):
# Call parent
ReductionMethod.__init__(
self,
os.path.join("test_eim_approximation_17_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a DifferentialProblemReductionMethod
self.truth_problem = truth_problem
self.reduced_problem = None
# I/O
self.folder["basis"] = os.path.join(
self.truth_problem.folder_prefix, "basis")
# Gram Schmidt
self.GS = GramSchmidt(self.truth_problem.inner_product)
def initialize_training_set(self,
ntrain,
enable_import=True,
sampling=None,
**kwargs):
return ReductionMethod.initialize_training_set(
self, self.truth_problem.mu_range, ntrain, enable_import,
sampling, **kwargs)
def initialize_testing_set(self,
ntest,
enable_import=False,
sampling=None,
**kwargs):
return ReductionMethod.initialize_testing_set(
self, self.truth_problem.mu_range, ntest, enable_import,
sampling, **kwargs)
def offline(self):
self.reduced_problem = MockReducedProblem(self.truth_problem)
if self.folder["basis"].create(
): # basis folder was not available yet
for (index, mu) in enumerate(self.training_set):
self.truth_problem.set_mu(mu)
print("solving mock problem at mu =",
self.truth_problem.mu)
f = self.truth_problem.solve()
self.update_basis_matrix((index, f))
self.reduced_problem.basis_functions.save(
self.folder["basis"], "basis")
else:
self.reduced_problem.basis_functions.load(
self.folder["basis"], "basis")
self._finalize_offline()
return self.reduced_problem
def update_basis_matrix(self, index_and_snapshot):
(index, snapshot) = index_and_snapshot
component = "u" if index % 2 == 0 else "s"
self.reduced_problem.basis_functions.enrich(snapshot, component)
self.GS.apply(self.reduced_problem.basis_functions[component], 0)
def error_analysis(self, N=None, **kwargs):
pass
def speedup_analysis(self, N=None, **kwargs):
pass
@StoreMapFromProblemToReducedProblem
class MockReducedProblem(ParametrizedProblem):
@sync_setters("truth_problem", "set_mu", "mu")
@sync_setters("truth_problem", "set_mu_range", "mu_range")
def __init__(self, truth_problem, **kwargs):
# Call parent
ParametrizedProblem.__init__(
self,
os.path.join("test_eim_approximation_17_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a ParametrizedReducedDifferentialProblem
self.truth_problem = truth_problem
self.basis_functions = BasisFunctionsMatrix(self.truth_problem.V)
self.basis_functions.init(self.truth_problem.components)
self._solution = None
def solve(self):
print("solving mock reduced problem at mu =", self.mu)
assert not hasattr(self, "_is_solving")
self._is_solving = True
f = self.truth_problem.solve()
f_N = transpose(
self.basis_functions) * self.truth_problem.inner_product * f
# Return the reduced solution
self._solution = OnlineFunction(f_N)
delattr(self, "_is_solving")
return self._solution
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, truth_problem, expression_type, basis_generation):
self.V = truth_problem.V
#
folder_prefix = os.path.join("test_eim_approximation_17_tempdir",
expression_type, basis_generation)
assert expression_type in ("Function", "Vector", "Matrix")
if expression_type == "Function":
# Call Parent constructor
EIMApproximation.__init__(
self, truth_problem,
ParametrizedExpressionFactory(truth_problem._solution),
folder_prefix, basis_generation)
elif expression_type == "Vector":
v = TestFunction(self.V)
form = inner(truth_problem._solution, v) * dx
# Call Parent constructor
EIMApproximation.__init__(self, truth_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
u = TrialFunction(self.V)
v = TestFunction(self.V)
form = inner(truth_problem._solution, u) * v[0] * dx
# Call Parent constructor
EIMApproximation.__init__(self, truth_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
element_0 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
element_1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
element = MixedElement(element_0, element_1)
V = FunctionSpace(mesh, element, components=[["u", "s"], "p"])
# 3. Create a parametrized problem
problem = MockProblem(V)
mu_range = [(-1., -0.01), (-1., -0.01)]
problem.set_mu_range(mu_range)
# 4. Create a reduction method and run the offline phase to generate the corresponding
# reduced problem
reduction_method = MockReductionMethod(problem)
reduction_method.initialize_training_set(16,
sampling=EquispacedDistribution())
reduction_method.offline()
# 5. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(
problem, expression_type, basis_generation)
parametrized_function_approximation.set_mu_range(mu_range)
# 6. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(
parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(16)
parametrized_function_reduction_method.set_tolerance(0.)
# 7. Perform EIM offline phase
parametrized_function_reduction_method.initialize_training_set(
64, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline(
)
# 8. Perform EIM online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 9. Perform EIM error analysis
parametrized_function_reduction_method.initialize_testing_set(100)
parametrized_function_reduction_method.error_analysis()
def average_matrix(V, TV, shape):
'''
Averaging matrix for reduction of g in V to TV by integration over shape.
'''
# We build a matrix representation of u in V -> Pi(u) in TV where
#
# Pi(u)(s) = |L(s)|^-1*\int_{L(s)}u(t) dx(s)
#
# Here L is the shape over which u is integrated for reduction.
# Its measure is |L(s)|.
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
mesh = V.mesh()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
line_mesh = TV.mesh()
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent (normal of the plane which cuts the virtual
# surface to yield the bdry curve
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Avg point here has the role of 'height' coordinate
quadrature = shape.quadrature(avg_point, n)
integration_points = quadrature.points
wq = quadrature.weights
curve_measure = sum(wq)
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
cs = tree.compute_entity_collisions(Point(*ip))
# assert False
for c in cs[:1]:
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
basis_values[:] = Vel.evaluate_basis_all(
ip, vertex_coordinates, cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(
cols_ip, values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value / curve_measure
else:
data[col] = value / curve_measure
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(list(data.keys()), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return mat
def test_advect_periodic(advection_scheme):
# FIXME: this unit test is sensitive to the ordering of the particle
# array, i.e. xp0_root and xpE_root may contain exactly the same entries
# but only in a different order. This will return an error right now
xmin, xmax = 0.0, 1.0
ymin, ymax = 0.0, 1.0
pres = 3
mesh = RectangleMesh(Point(xmin, ymin), Point(xmax, ymax), 10, 10)
lims = np.array([
[xmin, xmin, ymin, ymax],
[xmax, xmax, ymin, ymax],
[xmin, xmax, ymin, ymin],
[xmin, xmax, ymax, ymax],
])
vexpr = Constant((1.0, 1.0))
V = VectorFunctionSpace(mesh, "CG", 1)
x = RandomRectangle(Point(0.05, 0.05), Point(0.15,
0.15)).generate([pres, pres])
x = comm.bcast(x, root=0)
dt = 0.05
v = Function(V)
v.assign(vexpr)
p = particles(x, [x * 0, x**2], mesh)
if advection_scheme == "euler":
ap = advect_particles(p, V, v, "periodic", lims.flatten())
elif advection_scheme == "rk2":
ap = advect_rk2(p, V, v, "periodic", lims.flatten())
elif advection_scheme == "rk3":
ap = advect_rk3(p, V, v, "periodic", lims.flatten())
else:
assert False
xp0 = p.positions()
t = 0.0
while t < 1.0 - 1e-12:
ap.do_step(dt)
t += dt
xpE = p.positions()
# Check if position correct
xp0_root = comm.gather(xp0, root=0)
xpE_root = comm.gather(xpE, root=0)
num_particles = p.number_of_particles()
if comm.Get_rank() == 0:
xp0_root = np.float32(np.vstack(xp0_root))
xpE_root = np.float32(np.vstack(xpE_root))
# Sort on x positions
xp0_root = xp0_root[xp0_root[:, 0].argsort(), :]
xpE_root = xpE_root[xpE_root[:, 0].argsort(), :]
error = np.linalg.norm(xp0_root - xpE_root)
assert error < 1e-10
assert num_particles - pres**2 == 0
def test_nullspace_orthogonal(mesh, degree):
"""Test that null spaces orthogonalisation"""
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
null_space = build_elastic_nullspace(V)
assert not null_space.is_orthogonal()
assert not null_space.is_orthonormal()
null_space.orthonormalize()
assert null_space.is_orthogonal()
assert null_space.is_orthonormal()
@pytest.mark.parametrize("mesh", [
UnitSquareMesh(MPI.comm_world, 12, 13),
BoxMesh.create(MPI.comm_world, [
Point(0.8, -0.2, 1.2)._cpp_object,
Point(3.0, 11.0, -5.0)._cpp_object
], [12, 18, 25],
cell_type=CellType.Type.tetrahedron,
ghost_mode=GhostMode.none),
])
@pytest.mark.parametrize("degree", [1, 2])
def test_nullspace_check(mesh, degree):
V = VectorFunctionSpace(mesh, ('Lagrange', degree))
u, v = TrialFunction(V), TestFunction(V)
mesh.geometry.coord_mapping = fem.create_coordinate_map(mesh)
E, nu = 2.0e2, 0.3
mu = E / (2.0 * (1.0 + nu))
lmbda = E * nu / ((1.0 + nu) * (1.0 - 2.0 * nu))
basis = VectorSpaceBasis(nullspace_basis)
basis.orthonormalize()
_x = [basis[i] for i in range(6)]
nsp = PETSc.NullSpace()
nsp.create(_x)
return nsp
# Load mesh from file
# mesh = Mesh(MPI.comm_world)
# XDMFFile(MPI.comm_world, "../pulley.xdmf").read(mesh)
# mesh = UnitCubeMesh(2, 2, 2)
mesh = BoxMesh(MPI.comm_world,
[Point(0, 0, 0)._cpp_object,
Point(2, 1, 1)._cpp_object], [12, 12, 12],
CellType.Type.tetrahedron, dolfin.cpp.mesh.GhostMode.none)
cmap = dolfin.fem.create_coordinate_map(mesh.ufl_domain())
mesh.geometry.coord_mapping = cmap
# Function to mark inner surface of pulley
# def inner_surface(x, on_boundary):
# r = 3.75 - x[2]*0.17
# return (x[0]*x[0] + x[1]*x[1]) < r*r and on_boundary
def boundary(x, on_boundary):
return np.logical_or(x[:, 0] < 10.0 * np.finfo(float).eps,
x[:, 0] > 1.0 - 10.0 * np.finfo(float).eps)
def run_with_params(Tb, mu_value, k_s, path):
run_time_init = clock()
mesh = BoxMesh(Point(0.0, 0.0, 0.0),
Point(mesh_width, mesh_width, mesh_height), nx, ny, nz)
pbc = PeriodicBoundary()
WE = VectorElement('CG', mesh.ufl_cell(), 2)
SE = FiniteElement('CG', mesh.ufl_cell(), 1)
WSSS = FunctionSpace(mesh,
MixedElement(WE, SE, SE, SE),
constrained_domain=pbc)
# W = FunctionSpace(mesh, WE, constrained_domain=pbc)
# S = FunctionSpace(mesh, SE, constrained_domain=pbc)
W = WSSS.sub(0).collapse()
S = WSSS.sub(1).collapse()
temperature_vals = [27.0 + 273, Tb + 273, 1300.0 + 273, 1305.0 + 273]
temp_prof = TemperatureProfile(temperature_vals, element=S.ufl_element())
mu_a = mu_value # this was taken from the Blankenbach paper, can change
Ep = b / temp_prof.delta
mu_bot = exp(-Ep *
(temp_prof.bottom * temp_prof.delta - 1573.0) + cc) * mu_a
# TODO: verify exponentiation
Ra = rho_0 * alpha * g * temp_prof.delta * h**3 / (kappa_0 * mu_a)
w0 = rho_0 * alpha * g * temp_prof.delta * h**2 / mu_a
tau = h / w0
p0 = mu_a * w0 / h
log(mu_a, mu_bot, Ra, w0, p0)
slip_vx = 1.6E-09 / w0 # Non-dimensional
slip_velocity = Constant((slip_vx, 0.0, 0.0))
zero_slip = Constant((0.0, 0.0, 0.0))
time_step = 3.0E11 / tau * 2
dt = Constant(time_step)
t_end = 3.0E15 / tau / 5.0 # Non-dimensional times
u = Function(WSSS)
# Instead of TrialFunctions, we use split(u) for our non-linear problem
v, p, T, Tf = split(u)
v_t, p_t, T_t, Tf_t = TestFunctions(WSSS)
T0 = interpolate(temp_prof, S)
mu_exp = Expression(
'exp(-Ep * (T_val * dTemp - 1573.0) + cc * x[2] / mesh_height)',
Ep=Ep,
dTemp=temp_prof.delta,
cc=cc,
mesh_height=mesh_height,
T_val=T0,
element=S.ufl_element())
Tf0 = interpolate(temp_prof, S)
mu = Function(S)
v0 = Function(W)
v_theta = (1.0 - theta) * v0 + theta * v
T_theta = (1.0 - theta) * T0 + theta * T
Tf_theta = (1.0 - theta) * Tf0 + theta * Tf
# TODO: Verify forms
r_v = (inner(sym(grad(v_t)), 2.0 * mu * sym(grad(v))) - div(v_t) * p -
T * v_t[2]) * dx
r_p = p_t * div(v) * dx
heat_transfer = Constant(k_s) * (Tf_theta - T_theta) * dt
r_T = (
T_t *
((T - T0) + dt * inner(v_theta, grad(T_theta))) # TODO: Inner vs dot
+
(dt / Ra) * inner(grad(T_t), grad(T_theta)) - T_t * heat_transfer) * dx
v_melt = Function(W)
z_hat = Constant((0.0, 0.0, 1.0))
# TODO: inner -> dot, take out Tf_t
r_Tf = (Tf_t * ((Tf - Tf0) + dt * inner(v_melt, grad(Tf_theta))) +
Tf_t * heat_transfer) * dx
r = r_v + r_p + r_T + r_Tf
bcv0 = DirichletBC(WSSS.sub(0), zero_slip, top)
bcv1 = DirichletBC(WSSS.sub(0), slip_velocity, bottom)
bcv2 = DirichletBC(WSSS.sub(0).sub(1), Constant(0.0), back)
bcv3 = DirichletBC(WSSS.sub(0).sub(1), Constant(0.0), front)
bcp0 = DirichletBC(WSSS.sub(1), Constant(0.0), bottom)
bct0 = DirichletBC(WSSS.sub(2), Constant(temp_prof.surface), top)
bct1 = DirichletBC(WSSS.sub(2), Constant(temp_prof.bottom), bottom)
bctf1 = DirichletBC(WSSS.sub(3), Constant(temp_prof.bottom), bottom)
bcs = [bcv0, bcv1, bcv2, bcv3, bcp0, bct0, bct1, bctf1]
t = 0
count = 0
files = DefaultDictByKey(partial(create_xdmf, path))
while t < t_end:
mu.interpolate(mu_exp)
rhosolid = rho_0 * (1.0 - alpha * (T0 * temp_prof.delta - 1573.0))
deltarho = rhosolid - rho_melt
# TODO: project (accuracy) vs interpolate
assign(
v_melt,
project(
v0 - darcy * (grad(p) * p0 / h - deltarho * z_hat * g) / w0,
W))
# TODO: Written out one step later?
# v_melt.assign(v0 - darcy * (grad(p) * p0 / h - deltarho * yvec * g) / w0)
# TODO: use nP after to avoid projection?
solve(r == 0, u, bcs)
nV, nP, nT, nTf = u.split() # TODO: write with Tf, ... etc
if count % output_every == 0:
time_left(count, t_end / time_step,
run_time_init) # TODO: timestep vs dt
# TODO: Make sure all writes are to the same function for each time step
files['T_fluid'].write(nTf, t)
files['p'].write(nP, t)
files['v_solid'].write(nV, t)
files['T_solid'].write(nT, t)
files['mu'].write(mu, t)
files['v_melt'].write(v_melt, t)
files['gradp'].write(project(grad(nP), W), t)
files['rho'].write(project(rhosolid, S), t)
files['Tf_grad'].write(project(grad(Tf), W), t)
files['advect'].write(project(dt * dot(v_melt, grad(nTf))), t)
files['ht'].write(project(heat_transfer, S), t)
assign(T0, nT)
assign(v0, nV)
assign(Tf0, nTf)
t += time_step
count += 1
log('Case mu={}, Tb={}, k={} complete. Run time = {:.2f} minutes'.format(
mu_a, Tb, k_s, (clock() - run_time_init) / 60.0))
def list2point(self, list_in):
""" Turn a list of coord into a Fenics Point
Inputs:
list_in = list containing coordinates of the Point """
dim = np.size(list_in)
return Point(dim, np.array(list_in, dtype=float))
def test_eim_approximation_02(expression_type, basis_generation):
"""
This is a second basic test for EIM/DEIM, based on the test case of section 3.3.2 of
S. Chaturantabut and D. C. Sorensen
Nonlinear Model Reduction via Discrete Empirical Interpolation
SIAM Journal on Scientific Computing 2010 32:5, 2737-2764
* EIM: test interpolation of a scalar function
* DEIM: test interpolation of form with scalar integrand on a scalar space
"""
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
ParametrizedProblem.__init__(self, "")
self.V = V
def name(self):
return "MockProblem_02_" + expression_type + "_" + basis_generation
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f = ParametrizedExpression(
mock_problem,
"1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)",
mu=(-1., -1.),
element=V.ufl_element())
#
folder_prefix = os.path.join("test_eim_approximation_02_tempdir",
expression_type, basis_generation)
assert expression_type in ("Function", "Vector", "Matrix")
if expression_type == "Function":
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedExpressionFactory(f),
folder_prefix, basis_generation)
elif expression_type == "Vector":
v = TestFunction(V)
form = f * v * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
u = TrialFunction(V)
v = TestFunction(V)
form = f * u * v * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
V = FunctionSpace(mesh, "Lagrange", 1)
# 3. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(
V, expression_type, basis_generation)
mu_range = [(-1., -0.01), (-1., -0.01)]
parametrized_function_approximation.set_mu_range(mu_range)
# 4. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(
parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(50)
parametrized_function_reduction_method.set_tolerance(0.)
# 5. Perform the offline phase
parametrized_function_reduction_method.initialize_training_set(
225, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline(
)
# 6. Perform an online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 7. Perform an error analysis
parametrized_function_reduction_method.initialize_testing_set(225)
parametrized_function_reduction_method.error_analysis()
def average_matrix(V, TV, radius, quad_degree):
'''Averaging matrix'''
mesh = V.mesh()
line_mesh = TV.mesh()
# We are going to perform the integration with Gauss quadrature at
# the end (PI u)(x):
# A cell of mesh (an edge) defines a normal vector. Let P be the plane
# that is defined by the normal vector n and some point x on Gamma. Let L
# be the circle that is the intersect of P and S. The value of q (in Q) at x
# is defined as
#
# q(x) = (1/|L|)*\int_{L}g(x)*dL
#
# which simplifies to g(x) = (1/(2*pi*R))*\int_{-pi}^{pi}u(L)*R*d(theta) and
# or = (1/2) * \int_{-1}^{1} u (L(pi*s)) * ds
# This can be integrated no problemo once we figure out L. To this end, let
# t_1 and t_2 be two unit mutually orthogonal vectors that are orthogonal to
# n. Then L(pi*s) = p + R*t_1*cos(pi*s) + R*t_2*sin(pi*s) can be seen to be
# such that i) |x-p| = R and ii) x.n = 0 [i.e. this the suitable
# parametrization]
# Clearly we can scale the weights as well as precompute
# cos and sin terms.
xq, wq = leggauss(quad_degree)
wq *= 0.5
cos_xq = np.cos(np.pi * xq).reshape((-1, 1))
sin_xq = np.sin(np.pi * xq).reshape((-1, 1))
if is_number(radius):
radius = lambda x, radius=radius: radius
mesh_x = TV.mesh().coordinates()
# The idea for point evaluation/computing dofs of TV is to minimize
# the number of evaluation. I mean a vector dof if done naively would
# have to evaluate at same x number of component times.
value_size = TV.ufl_element().value_size()
# Eval at points will require serch
tree = mesh.bounding_box_tree()
limit = mesh.num_cells()
TV_coordinates = TV.tabulate_dof_coordinates().reshape((TV.dim(), -1))
TV_dm = TV.dofmap()
V_dm = V.dofmap()
# For non scalar we plan to make compoenents by shift
if value_size > 1:
TV_dm = TV.sub(0).dofmap()
Vel = V.element()
basis_values = np.zeros(V.element().space_dimension() * value_size)
with petsc_serial_matrix(TV, V) as mat:
for line_cell in cells(line_mesh):
# Get the tangent => orthogonal tangent vectors
v0, v1 = mesh_x[line_cell.entities(0)]
n = v0 - v1
t1 = np.array([n[1] - n[2], n[2] - n[0], n[0] - n[1]])
t2 = np.cross(n, t1)
t1 /= np.linalg.norm(t1)
t2 = t2 / np.linalg.norm(t2)
# The idea is now to minimize the point evaluation
scalar_dofs = TV_dm.cell_dofs(line_cell.index())
scalar_dofs_x = TV_coordinates[scalar_dofs]
for scalar_row, avg_point in zip(scalar_dofs, scalar_dofs_x):
# Get radius and integration points
rad = radius(avg_point)
integration_points = avg_point + rad * t1 * sin_xq + rad * t2 * cos_xq
data = {}
for index, ip in enumerate(integration_points):
c = tree.compute_first_entity_collision(Point(*ip))
if c >= limit: continue
Vcell = Cell(mesh, c)
vertex_coordinates = Vcell.get_vertex_coordinates()
cell_orientation = Vcell.orientation()
Vel.evaluate_basis_all(basis_values, ip,
vertex_coordinates,
cell_orientation)
cols_ip = V_dm.cell_dofs(c)
values_ip = basis_values * wq[index]
# Add
for col, value in zip(cols_ip,
values_ip.reshape((-1, value_size))):
if col in data:
data[col] += value
else:
data[col] = value
# The thing now that with data we can assign to several
# rows of the matrix
column_indices = np.array(data.keys(), dtype='int32')
for shift in range(value_size):
row = scalar_row + shift
column_values = np.array(
[data[col][shift] for col in column_indices])
mat.setValues([row], column_indices, column_values,
PETSc.InsertMode.INSERT_VALUES)
# On to next avg point
# On to next cell
return PETScMatrix(mat)
def test_eim_approximation_06(expression_type, basis_generation):
"""
The aim of this test is to check the interpolation of vector valued functions.
* EIM: test interpolation of a scalar function
* DEIM: test interpolation of form with integrand given by the inner product of a vector valued function
and some derivative of a test/trial functions of a scalar space
"""
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
ParametrizedProblem.__init__(self, "")
self.V = V
def name(self):
return "MockProblem_06_" + expression_type + "_" + basis_generation
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f_expression = (
"1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)",
"10.*(1-x[0])*cos(3*pi*(pi+mu[1])*(1+x[1]))*exp(-(pi+mu[0])*(1+x[0]))"
)
vector_element = VectorElement(V.ufl_element())
f = ParametrizedExpression(mock_problem,
f_expression,
mu=(-1., -1.),
element=vector_element)
#
folder_prefix = os.path.join("test_eim_approximation_06_tempdir",
expression_type, basis_generation)
assert expression_type in ("Function", "Vector", "Matrix")
if expression_type == "Function":
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedExpressionFactory(f),
folder_prefix, basis_generation)
elif expression_type == "Vector":
v = TestFunction(V)
form = inner(f, grad(v)) * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
u = TrialFunction(V)
v = TestFunction(V)
form = inner(f, grad(u)) * v * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
V = FunctionSpace(mesh, "Lagrange", 1)
# 3. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(
V, expression_type, basis_generation)
mu_range = [(-1., -0.01), (-1., -0.01)]
parametrized_function_approximation.set_mu_range(mu_range)
# 4. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(
parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(50)
parametrized_function_reduction_method.set_tolerance(0.)
# 5. Perform the offline phase
parametrized_function_reduction_method.initialize_training_set(
225, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline(
)
# 6. Perform an online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 7. Perform an error analysis
parametrized_function_reduction_method.initialize_testing_set(225)
parametrized_function_reduction_method.error_analysis()
def Regular(n, size=50):
"""Build mesh for a regular polygon with n sides."""
points = [(np.cos(2 * np.pi * i / n), np.sin(2 * np.pi * i / n))
for i in range(n)]
polygon = Polygon([Point(*p) for p in points])
return generate_mesh(polygon, size)
degree=7,
U=float(1.0),
nu=float(nu),
t=float(0.0),
mode=mode)
P_exact = "-rho*0.25*exp(-4*nu*pow(pi,2)*t)*(cos(2*pi*(x[0])) + cos(2*pi*(x[1]))-2.0)"
p_exact = Expression(P_exact, degree=3, rho=1.0, nu=float(nu), t=float(0.0))
f = Constant((0.0, 0.0))
# Create mesh
xmin, ymin = geometry["xmin"], geometry["ymin"]
xmax, ymax = geometry["xmax"], geometry["ymax"]
mesh = RectangleMesh(MPI.comm_world, Point(xmin, ymin), Point(xmax, ymax), nx,
ny)
pbc = PeriodicBoundary(geometry)
# xdmf output
xdmf_u = XDMFFile(mesh.mpi_comm(), outdir_base + "u.xdmf")
xdmf_p = XDMFFile(mesh.mpi_comm(), outdir_base + "p.xdmf")
# Required elements
W_E_2 = VectorElement("DG", mesh.ufl_cell(), k)
T_E_2 = VectorElement("DG", mesh.ufl_cell(), 0)
Wbar_E_2 = VectorElement("DGT", mesh.ufl_cell(), kbar)
Wbar_E_2_H12 = VectorElement("CG", mesh.ufl_cell(), kbar)["facet"]
Q_E = FiniteElement("DG", mesh.ufl_cell(), k - 1)
Qbar_E = FiniteElement("DGT", mesh.ufl_cell(), k)
def test_eim_approximation_18(expression_type, basis_generation):
"""
This test is the version of test 17 where high fidelity solution is used in place of reduced order one.
"""
@StoreMapFromProblemNameToProblem
@StoreMapFromProblemToTrainingStatus
@StoreMapFromSolutionToProblem
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
# Call parent
ParametrizedProblem.__init__(
self,
os.path.join("test_eim_approximation_18_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a ParametrizedDifferentialProblem
self.V = V
self._solution = Function(V)
self.components = ["u", "s", "p"]
# Parametrized function to be interpolated
x = SpatialCoordinate(V.mesh())
mu = SymbolicParameters(self, V, (-1., -1.))
self.f00 = 1. / sqrt(
pow(x[0] - mu[0], 2) + pow(x[1] - mu[1], 2) + 0.01)
self.f01 = 1. / sqrt(
pow(x[0] - mu[0], 4) + pow(x[1] - mu[1], 4) + 0.01)
# Inner product
f = TrialFunction(self.V)
g = TestFunction(self.V)
self.inner_product = assemble(inner(f, g) * dx)
# Collapsed vector and space
self.V0 = V.sub(0).collapse()
self.V00 = V.sub(0).sub(0).collapse()
self.V1 = V.sub(1).collapse()
def name(self):
return "MockProblem_18_" + expression_type + "_" + basis_generation
def init(self):
pass
def solve(self):
print("solving mock problem at mu =", self.mu)
assert not hasattr(self, "_is_solving")
self._is_solving = True
f00 = project(self.f00, self.V00)
f01 = project(self.f01, self.V00)
assign(self._solution.sub(0).sub(0), f00)
assign(self._solution.sub(0).sub(1), f01)
delattr(self, "_is_solving")
return self._solution
@StoreMapFromProblemToReductionMethod
class MockReductionMethod(ReductionMethod):
def __init__(self, truth_problem, **kwargs):
# Call parent
ReductionMethod.__init__(
self,
os.path.join("test_eim_approximation_18_tempdir",
expression_type, basis_generation,
"mock_problem"))
# Minimal subset of a DifferentialProblemReductionMethod
self.truth_problem = truth_problem
self.reduced_problem = None
def initialize_training_set(self,
ntrain,
enable_import=True,
sampling=None,
**kwargs):
return ReductionMethod.initialize_training_set(
self, self.truth_problem.mu_range, ntrain, enable_import,
sampling, **kwargs)
def initialize_testing_set(self,
ntest,
enable_import=False,
sampling=None,
**kwargs):
return ReductionMethod.initialize_testing_set(
self, self.truth_problem.mu_range, ntest, enable_import,
sampling, **kwargs)
def offline(self):
pass
def update_basis_matrix(self, snapshot):
pass
def error_analysis(self, N=None, **kwargs):
pass
def speedup_analysis(self, N=None, **kwargs):
pass
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, truth_problem, expression_type, basis_generation):
self.V = truth_problem.V
#
folder_prefix = os.path.join("test_eim_approximation_18_tempdir",
expression_type, basis_generation)
assert expression_type in ("Function", "Vector", "Matrix")
if expression_type == "Function":
# Call Parent constructor
EIMApproximation.__init__(
self, truth_problem,
ParametrizedExpressionFactory(truth_problem._solution),
folder_prefix, basis_generation)
elif expression_type == "Vector":
v = TestFunction(self.V)
form = inner(truth_problem._solution, v) * dx
# Call Parent constructor
EIMApproximation.__init__(self, truth_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
u = TrialFunction(self.V)
v = TestFunction(self.V)
form = inner(truth_problem._solution, u) * v[0] * dx
# Call Parent constructor
EIMApproximation.__init__(self, truth_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
element_0 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
element_1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
element = MixedElement(element_0, element_1)
V = FunctionSpace(mesh, element, components=[["u", "s"], "p"])
# 3. Create a parametrized problem
problem = MockProblem(V)
mu_range = [(-1., -0.01), (-1., -0.01)]
problem.set_mu_range(mu_range)
# 4. Create a reduction method, but postpone generation of the reduced problem
MockReductionMethod(problem)
# 5. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(
problem, expression_type, basis_generation)
parametrized_function_approximation.set_mu_range(mu_range)
# 6. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(
parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(16)
parametrized_function_reduction_method.set_tolerance(0.)
# 7. Perform EIM offline phase
parametrized_function_reduction_method.initialize_training_set(
64, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline(
)
# 8. Perform EIM online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 9. Perform EIM error analysis
parametrized_function_reduction_method.initialize_testing_set(100)
parametrized_function_reduction_method.error_analysis()
def initMesh(self, n):
# Set mesh to square on [0,1]^2
self.mesh = RectangleMesh(Point(-np.pi, -np.pi), Point(np.pi, np.pi),
n, n)
# Load meshes and mesh-functions used in the MultiMesh from file
multimesh = MultiMesh()
mfs = []
meshes = []
for i in range(2):
mesh_i = Mesh()
with XDMFFile("meshes/multimesh_%d.xdmf" % i) as infile:
infile.read(mesh_i)
mvc = MeshValueCollection("size_t", mesh_i, 1)
with XDMFFile("meshes/mf_%d.xdmf" % i) as infile:
infile.read(mvc, "name_to_read")
mfs.append(cpp.mesh.MeshFunctionSizet(mesh_i, mvc))
meshes.append(mesh_i)
multimesh.add(mesh_i)
multimesh.build()
multimesh.auto_cover(0, Point(1.25, 0.875))
def deformation_vector():
# n1 = VolumeNormal(multimesh.part(1))
x1 = SpatialCoordinate(multimesh.part(1))
n1 = as_vector((x1[1], x1[0]))
S_sm = VectorFunctionSpace(multimesh.part(1), "CG", 1)
#bcs = [DirichletBC(S_sm, Constant((0,0)), mfs[1],2),
bcs = [DirichletBC(S_sm, n1, mfs[1], 1)]
u, v = TrialFunction(S_sm), TestFunction(S_sm)
a = inner(grad(u), grad(v)) * dx
l = inner(Constant((0., 0.)), v) * dx
n = Function(S_sm)
solve(a == l, n, bcs=bcs)
D_an = Constant(1.0E-7) # m^2/s
D_ca = Constant(1.0E-7) # m^2/s
mu_an = Constant(3.9607E-6) # m^2/s*V
mu_ca = Constant(3.9607E-6) # m^2/s*V
z_an = Constant(-1.0)
z_ca = Constant(1.0)
z_fc = Constant(-1.0)
Farad = Constant(9.6487E4) # C/mol
eps0 = Constant(8.854E-12) # As/Vm
epsR = Constant(100.0)
Temp = Constant(293.0)
R = Constant(8.3143)
################################## mesh part ##################################
mesh = RectangleMesh(Point(0.0, 0.0), Point(15.0E-3, 15.0E-3), 150, 150)
#File('mesh_orig.pvd') << mesh
refinement_cycles = 5
for _ in range(refinement_cycles):
refine_cells = MeshFunction("bool", mesh, 2)
refine_cells.set_all(False)
for cell in cells(mesh):
mp = cell.midpoint()
if mp.x() > 4.8e-3 and mp.x() < 10.2e-3:
if abs(mp.y() - 10.0e-3) < 0.2e-3:
refine_cells[cell] = True
elif abs(mp.y() - 5.0e-3) < 0.2e-3:
refine_cells[cell] = True
if mp.y() > 4.8e-3 and mp.y() < 10.2e-3:
outdir + "rhop_nx" + str(nx) + ".pickle",
]
property_list = [0, 1]
conservation_data = outdir + "conservation_nx" + str(nx) + ".csv"
if comm.rank == 0:
with open(conservation_data, "w") as write_file:
writer = csv.writer(write_file)
writer.writerow(["Time", "Total mass", "Mass conservation"])
# Compute num steps till completion
num_steps = np.rint(Tend / float(dt))
# Generate mesh
mesh = RectangleMesh.create(
[Point(xmin, ymin), Point(xmax, ymax)], [nx, nx],
CellType.Type.triangle)
output_field = XDMFFile(mesh.mpi_comm(),
outdir + "psi_h" + "_nx" + str(nx) + ".xdmf")
# Velocity and initial condition
V = VectorFunctionSpace(mesh, "CG", 1)
uh = Function(V)
uh.assign(Expression((ux, vy), degree=1))
psi0_expression = SineHump(center=[0.5, 0.5],
U=[float(ux), float(vy)],
time=0.0,
degree=6)
# Generate particles
outdir + "psi_h_nx" + str(nx) + ".xdmf")
# Velocity and initial condition
V = VectorFunctionSpace(mesh, 'DG', 3)
uh = Function(V)
uh.assign(Expression(('-Uh*x[1]', 'Uh*x[0]'), Uh=Uh, degree=3))
psi0_expression = GaussianPulse(center=(xc, yc),
sigma=float(sigma),
U=[Uh, Uh],
time=0.,
height=1.,
degree=3)
# Generate particles
x = RandomCircle(Point(x0, y0), r).generate([pres, pres])
s = np.zeros((len(x), 1), dtype=np.float_)
# Initialize particles with position x and scalar property s at the mesh
p = particles(x, [s], mesh)
property_idx = 1 # Scalar quantity is stored at slot 1
# Initialize advection class, use RK3 scheme
ap = advect_rk3(p, V, uh, 'open')
# Define the variational (projection problem)
W_e = FiniteElement("DG", mesh.ufl_cell(), k)
T_e = FiniteElement("DG", mesh.ufl_cell(), 0)
Wbar_e = FiniteElement("DGT", mesh.ufl_cell(), k)
W = FunctionSpace(mesh, W_e)
RectangleMesh, SpatialCoordinate, TestFunction,
TrialFunction, solve)
from dolfin.io import XDMFFile
from ufl import ds, dx, grad, inner
# We begin by defining a mesh of the domain and a finite element
# function space :math:`V` relative to this mesh. As the unit square is
# a very standard domain, we can use a built-in mesh provided by the
# class :py:class:`UnitSquareMesh <dolfin.cpp.UnitSquareMesh>`. In order
# to create a mesh consisting of 32 x 32 squares with each square
# divided into two triangles, we do as follows ::
# Create mesh and define function space
mesh = RectangleMesh.create(
MPI.comm_world,
[Point(0, 0)._cpp_object, Point(1, 1)._cpp_object], [32, 32],
CellType.Type.triangle, dolfin.cpp.mesh.GhostMode.none)
V = FunctionSpace(mesh, ("Lagrange", 1))
cmap = dolfin.fem.create_coordinate_map(mesh.ufl_domain())
mesh.geometry.coord_mapping = cmap
# The second argument to :py:class:`FunctionSpace
# <dolfin.functions.functionspace.FunctionSpace>` is the finite element
# family, while the third argument specifies the polynomial
# degree. Thus, in this case, our space ``V`` consists of first-order,
# continuous Lagrange finite element functions (or in order words,
# continuous piecewise linear polynomials).
#
# Next, we want to consider the Dirichlet boundary condition. A simple
# Python function, returning a boolean, can be used to define the
def initMesh(self, n):
self.mesh = RectangleMesh(Point(self.pmin, self.Imin),
Point(self.pmax, self.Imax), n, n)
basis = VectorSpaceBasis(nullspace_basis)
basis.orthonormalize()
_x = [basis[i] for i in range(6)]
nsp = PETSc.NullSpace()
nsp.create(_x)
return nsp
# Load mesh from file
# mesh = Mesh(MPI.comm_world)
# XDMFFile(MPI.comm_world, "../pulley.xdmf").read(mesh)
# mesh = UnitCubeMesh(2, 2, 2)
mesh = BoxMesh.create(
MPI.comm_world, [Point(0, 0, 0)._cpp_object,
Point(2, 1, 1)._cpp_object], [12, 12, 12],
CellType.Type.tetrahedron, dolfin.cpp.mesh.GhostMode.none)
cmap = dolfin.fem.create_coordinate_map(mesh.ufl_domain())
mesh.geometry.coord_mapping = cmap
# Function to mark inner surface of pulley
# def inner_surface(x, on_boundary):
# r = 3.75 - x[2]*0.17
# return (x[0]*x[0] + x[1]*x[1]) < r*r and on_boundary
def boundary(x, on_boundary):
return np.logical_or(x[:, 0] < np.finfo(float).eps,
x[:, 0] > 1.0 - np.finfo(float).eps)
def spherical_shell(dim, radii, boundary_ids, n_refinements=0):
assert isinstance(dim, int)
assert dim == 2 or dim == 3
assert isinstance(radii, (list, tuple)) and len(radii) == 2
ri, ro = radii
assert isinstance(ri, float) and ri > 0.
assert isinstance(ro, float) and ro > 0.
assert ri < ro
assert isinstance(boundary_ids, (list, tuple)) and len(boundary_ids) == 2
inner_boundary_id, outer_boundary_id = boundary_ids
assert isinstance(inner_boundary_id, int) and inner_boundary_id >= 0
assert isinstance(outer_boundary_id, int) and outer_boundary_id >= 0
assert inner_boundary_id != outer_boundary_id
assert isinstance(n_refinements, int) and n_refinements >= 0
# mesh generation
from dolfin import Point
if dim == 2:
center = Point(0., 0.)
elif dim == 3:
center = Point(0., 0., 0.)
from mshr import Sphere, Circle, generate_mesh
if dim == 2:
domain = Circle(center, ro) \
- Circle(center, ri)
mesh = generate_mesh(domain, 75)
elif dim == 3:
domain = Sphere(center, ro) \
- Sphere(center, ri)
mesh = generate_mesh(domain, 15)
# mesh refinement
from dolfin import refine
for i in range(n_refinements):
mesh = refine(mesh)
# subdomains for boundaries
from dolfin import MeshFunctionSizet
facet_marker = MeshFunctionSizet(mesh, mesh.topology().dim() - 1)
facet_marker.set_all(0)
# size of smallest element
hmin = mesh.hmin()
# inner circle boundary
class InnerCircle(SubDomain):
def inside(self, x, on_boundary):
# tolerance: half length of smallest element
tol = hmin / 2.
from math import sqrt
if dim == 2:
result = abs(sqrt(x[0]**2 + x[1]**2) - ri) < tol
elif dim == 3:
result = abs(sqrt(x[0]**2 + x[1]**2 + x[2]**2) - ri) < tol
return result and on_boundary
# outer cirlce boundary
class OuterCircle(SubDomain):
def inside(self, x, on_boundary):
# tolerance: half length of smallest element
tol = hmin / 2.
from math import sqrt
if dim == 2:
result = abs(sqrt(x[0]**2 + x[1]**2) - ro) < tol
elif dim == 3:
result = abs(sqrt(x[0]**2 + x[1]**2 + x[2]**2) - ro) < tol
return result and on_boundary
# mark boundaries
gamma_inner = InnerCircle()
gamma_inner.mark(facet_marker, inner_boundary_id)
gamma_outer = OuterCircle()
gamma_outer.mark(facet_marker, outer_boundary_id)
return mesh, facet_marker
pres = 2200
res = 'high'
dt = Constant(5.e-4)
store_step = 200
mu = 1e-2
theta_p = .5
theta_L = Constant(1.0)
probe_radius = 0.01
probe1_y = 0.003
probe2_y = 0.015
probe3_y = 0.03
probe4_y = 0.08
probe1_loc = Point(xmax - 1e-10, probe1_y)
probe2_loc = Point(xmax - 1e-10, probe2_y)
probe3_loc = Point(xmax - 1e-10, probe3_y)
probe4_loc = Point(xmax - 1e-10, probe4_y)
# Specify body force
f = Constant((0, -9.81))
geometry = {
'xmin': xmin_rho1,
'xmax': xmax_rho1,
'ymin': ymin_rho1,
'ymax': ymax_rho1
}
rho1 = Constant(1000.)
def run_with_params(Tb, mu_value, k_s, path):
run_time_init = clock()
temperature_vals = [27.0 + 273, Tb + 273, 1300.0 + 273, 1305.0 + 273]
temp_prof = TemperatureProfile(temperature_vals)
mu_a = mu_value # this was taken from the Blankenbach paper, can change
Ep = b / temp_prof.delta
mu_bot = exp(-Ep *
(temp_prof.bottom * temp_prof.delta - 1573.0) + cc) * mu_a
Ra = rho_0 * alpha * g * temp_prof.delta * h**3 / (kappa_0 * mu_a)
w0 = rho_0 * alpha * g * temp_prof.delta * h**2 / mu_a
tau = h / w0
p0 = mu_a * w0 / h
log(mu_a, mu_bot, Ra, w0, p0)
vslipx = 1.6e-09 / w0
vslip = Constant((vslipx, 0.0)) # Non-dimensional
noslip = Constant((0.0, 0.0))
time_step = 3.0E11 / tau / 10.0
dt = Constant(time_step)
tEnd = 3.0E15 / tau / 5.0 # Non-dimensionalising times
mesh = RectangleMesh(Point(0.0, 0.0), Point(mesh_width, mesh_height), nx,
ny)
pbc = PeriodicBoundary()
W = VectorFunctionSpace(mesh, 'CG', 2, constrained_domain=pbc)
S = FunctionSpace(mesh, 'CG', 1, constrained_domain=pbc)
WSSS = MixedFunctionSpace([W, S, S, S]) # WSSS -> W
u = Function(WSSS)
# Instead of TrialFunctions, we use split(u) for our non-linear problem
v, p, T, Tf = split(u)
v_t, p_t, T_t, Tf_t = TestFunctions(WSSS)
T0 = interpolate(temp_prof, S)
FluidTemp = Expression('max(T0, 1.031)', T0=T0)
muExp = Expression(
'exp(-Ep * (T_val * dTemp - 1573.0) + cc * x[1] / mesh_height)',
Ep=Ep,
dTemp=temp_prof.delta,
cc=cc,
mesh_height=mesh_height,
T_val=T0)
Tf0 = interpolate(temp_prof, S)
mu = Function(S)
v0 = Function(W)
v_theta = (1.0 - theta) * v0 + theta * v
T_theta = (1.0 - theta) * T0 + theta * T
Tf_theta = (1.0 - theta) * Tf0 + theta * Tf
r_v = (inner(sym(grad(v_t)), 2.0 * mu * sym(grad(v))) - div(v_t) * p -
T * v_t[1]) * dx
r_p = p_t * div(v) * dx
heat_transfer = k_s * (Tf_theta - T_theta) * dt
r_T = (T_t * ((T - T0) + dt * inner(v_theta, grad(T_theta))) +
(dt / Ra) * inner(grad(T_t), grad(T_theta)) -
T_t * heat_transfer) * dx
# yvec = Constant((0.0, 1.0))
# rhosolid = rho_0 * (1.0 - alpha * (T_theta * temp_prof.delta - 1573.0))
# deltarho = rhosolid - rhomelt
# v_f = v_theta - darcy * (grad(p) * p0 / h - deltarho * yvec * g) / w0
v_melt = Function(W)
yvec = Constant((0.0, 1.0))
# TODO: inner -> dot, take out Tf_t
r_Tf = (Tf_t * ((Tf - Tf0) + dt * dot(v_melt, grad(Tf_theta))) +
Tf_t * heat_transfer) * dx
r = r_v + r_p + r_T + r_Tf
bcv0 = DirichletBC(WSSS.sub(0), noslip, top)
bcv1 = DirichletBC(WSSS.sub(0), vslip, bottom)
bcp0 = DirichletBC(WSSS.sub(1), Constant(0.0), bottom)
bct0 = DirichletBC(WSSS.sub(2), Constant(temp_prof.surface), top)
bct1 = DirichletBC(WSSS.sub(2), Constant(temp_prof.bottom), bottom)
bctf1 = DirichletBC(WSSS.sub(3), Constant(temp_prof.bottom), bottom)
bcs = [bcv0, bcv1, bcp0, bct0, bct1, bctf1]
t = 0
count = 0
files = DefaultDictByKey(partial(create_xdmf, path))
while t < tEnd:
mu.interpolate(muExp)
rhosolid = rho_0 * (1.0 - alpha * (T0 * temp_prof.delta - 1573.0))
deltarho = rhosolid - rhomelt
assign(
v_melt,
project(v0 - darcy * (grad(p) * p0 / h - deltarho * yvec * g) / w0,
W))
# use nP after to avoid projection?
# pdb.set_trace()
solve(r == 0, u, bcs)
nV, nP, nT, nTf = u.split()
if count % output_every == 0:
time_left(count, tEnd / time_step, run_time_init)
# TODO: Make sure all writes are to the same function for each time step
files['T_fluid'].write(nTf)
files['p'].write(nP)
files['v_solid'].write(nV)
files['T_solid'].write(nT)
files['mu'].write(mu)
files['v_melt'].write(v_melt)
files['gradp'].write(project(grad(nP), W))
files['rho'].write(project(rhosolid, S))
files['Tf_grad'].write(project(grad(Tf), W))
files['advect'].write(project(dt * dot(v_melt, grad(nTf))))
files['ht'].write(project(heat_transfer, S))
assign(T0, nT)
assign(v0, nV)
assign(Tf0, nTf)
t += time_step
count += 1
log('Case mu=%g, Tb=%g complete. Run time = %g s' %
(mu_a, Tb, clock() - run_time_init))
def test_eim_approximation_08(expression_type, basis_generation):
"""
The aim of this script is to test that DEIM correctly handles collapsed subspaces. This is a prototype of
the restricted operators required by the right-hand side of a supremizer solve.
* EIM: not applicable.
* DEIM: define a test function on a collapsed subspace (while, in case of rank 2 forms, the trial is defined
on the full space), and integrate.
"""
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
ParametrizedProblem.__init__(self, "")
self.V = V
def name(self):
return "MockProblem_08_" + expression_type + "_" + basis_generation
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f1 = ParametrizedExpression(
mock_problem, "1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)", mu=(-1., -1.),
element=V.sub(1).ufl_element())
#
folder_prefix = os.path.join("test_eim_approximation_08_tempdir", expression_type, basis_generation)
assert expression_type in ("Vector", "Matrix")
if expression_type == "Vector":
q = TestFunction(V.sub(1).collapse())
form = f1 * q * dx
# Call Parent constructor
EIMApproximation.__init__(
self, mock_problem, ParametrizedTensorFactory(form), folder_prefix, basis_generation)
elif expression_type == "Matrix":
up = TrialFunction(V)
q = TestFunction(V.sub(1).collapse())
(u, p) = split(up)
form = f1 * q * div(u) * dx
# Call Parent constructor
EIMApproximation.__init__(
self, mock_problem, ParametrizedTensorFactory(form), folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
element_0 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
element_1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
element = MixedElement(element_0, element_1)
V = FunctionSpace(mesh, element)
# 3. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(V, expression_type, basis_generation)
mu_range = [(-1., -0.01), (-1., -0.01)]
parametrized_function_approximation.set_mu_range(mu_range)
# 4. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(20)
parametrized_function_reduction_method.set_tolerance(0.)
# 5. Perform the offline phase
parametrized_function_reduction_method.initialize_training_set(100, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline()
# 6. Perform an online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 7. Perform an error analysis
parametrized_function_reduction_method.initialize_testing_set(100)
parametrized_function_reduction_method.error_analysis()
def test_eim_approximation_05(expression_type, basis_generation):
"""
This test is combination of tests 01-03.
The aim of this script is to test that integration correctly handles several parametrized expressions.
* EIM: not applicable.
* DEIM: several parametrized expressions are combined when defining forms.
"""
class MockProblem(ParametrizedProblem):
def __init__(self, V, **kwargs):
ParametrizedProblem.__init__(self, "")
self.V = V
def name(self):
return "MockProblem_05_" + expression_type + "_" + basis_generation
class ParametrizedFunctionApproximation(EIMApproximation):
def __init__(self, V, expression_type, basis_generation):
self.V = V
# Parametrized function to be interpolated
mock_problem = MockProblem(V)
f1 = ParametrizedExpression(
mock_problem,
"1/sqrt(pow(x[0]-mu[0], 2) + pow(x[1]-mu[1], 2) + 0.01)",
mu=(-1., -1.),
element=V.sub(1).ufl_element())
f2 = ParametrizedExpression(
mock_problem,
"exp( - 2*pow(x[0]-mu[0], 2) - 2*pow(x[1]-mu[1], 2) )",
mu=(-1., -1.),
element=V.sub(1).ufl_element())
f3 = ParametrizedExpression(
mock_problem,
"(1-x[0])*cos(3*pi*(pi+mu[1])*(1+x[1]))*exp(-(pi+mu[0])*(1+x[0]))",
mu=(-1., -1.),
element=V.sub(1).ufl_element())
#
folder_prefix = os.path.join("test_eim_approximation_05_tempdir",
expression_type, basis_generation)
assert expression_type in ("Vector", "Matrix")
if expression_type == "Vector":
vq = TestFunction(V)
(v, q) = split(vq)
form = f1 * v[0] * dx + f2 * v[1] * dx + f3 * q * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
elif expression_type == "Matrix":
up = TrialFunction(V)
vq = TestFunction(V)
(u, p) = split(up)
(v, q) = split(vq)
form = f1 * inner(grad(u), grad(v)) * dx + f2 * p * div(
v) * dx + f3 * q * div(u) * dx
# Call Parent constructor
EIMApproximation.__init__(self, mock_problem,
ParametrizedTensorFactory(form),
folder_prefix, basis_generation)
else: # impossible to arrive here anyway thanks to the assert
raise AssertionError("Invalid expression_type")
# 1. Create the mesh for this test
mesh = RectangleMesh(Point(0.1, 0.1), Point(0.9, 0.9), 20, 20)
# 2. Create Finite Element space (Lagrange P1)
element_0 = VectorElement("Lagrange", mesh.ufl_cell(), 2)
element_1 = FiniteElement("Lagrange", mesh.ufl_cell(), 1)
element = MixedElement(element_0, element_1)
V = FunctionSpace(mesh, element)
# 3. Allocate an object of the ParametrizedFunctionApproximation class
parametrized_function_approximation = ParametrizedFunctionApproximation(
V, expression_type, basis_generation)
mu_range = [(-1., -0.01), (-1., -0.01)]
parametrized_function_approximation.set_mu_range(mu_range)
# 4. Prepare reduction with EIM
parametrized_function_reduction_method = EIMApproximationReductionMethod(
parametrized_function_approximation)
parametrized_function_reduction_method.set_Nmax(20)
parametrized_function_reduction_method.set_tolerance(0.)
# 5. Perform the offline phase
parametrized_function_reduction_method.initialize_training_set(
100, sampling=EquispacedDistribution())
reduced_parametrized_function_approximation = parametrized_function_reduction_method.offline(
)
# 6. Perform an online solve
online_mu = (-1., -1.)
reduced_parametrized_function_approximation.set_mu(online_mu)
reduced_parametrized_function_approximation.solve()
# 7. Perform an error analysis
parametrized_function_reduction_method.initialize_testing_set(100)
parametrized_function_reduction_method.error_analysis()
from dolfin import UnitSquareMesh, FunctionSpace, TestFunction, TrialFunction,\
Constant, Expression, assemble, dx, Point, PointSource, plot, interactive,\
inner, nabla_grad, Function, solve, MPI, mpi_comm_world
import numpy as np
from fenicstools.sourceterms import PointSources
mycomm = mpi_comm_world()
myrank = MPI.rank(mycomm)
mesh = UnitSquareMesh(2, 2)
V = FunctionSpace(mesh, 'Lagrange', 1)
trial = TrialFunction(V)
test = TestFunction(V)
f0 = Constant('0')
L0 = f0 * test * dx
b = assemble(L0)
P = Point(0.1, 0.5)
delta = PointSource(V, P, 1.0)
delta.apply(b)
myown = PointSources(V, [[0.1, 0.5], [0.9, 0.5]])
print 'p{}: max(PointSource)={}, max(PointSources[0])={}, max(PointSources[1])={}'.format(\
myrank, max(abs(b.array())), max(abs(myown[0].array())), max(abs(myown[1].array())))
def get(self):
"""Built-in mesh."""
x = self.pad(self.values)
return BoxMesh(Point(-x[0] / 2.0, -x[1] / 2.0, -x[2] / 2.0),
Point(x[0] / 2.0, x[1] / 2.0, x[2] / 2.0), x[3], x[4],
x[5])