def SetupUnitMeshHelper(mesh, V=None, Vc=None):
if V is None:
V = df.FunctionSpace(mesh, 'CG', 1)
if Vc is None:
Vc = df.FunctionSpace(mesh, 'DG', 0)
boundaries = dict()
boundaries['left'] = df.CompiledSubDomain("near(x[0], 0.0) && on_boundary")
boundaries['bottom'] = df.CompiledSubDomain(
"near(x[1], 0.0) && on_boundary")
boundaries['top'] = df.CompiledSubDomain("near(x[1], 1.0) && on_boundary")
boundaries['right'] = df.CompiledSubDomain(
"near(x[0], 1.0) && on_boundary")
boundarymarkers = df.MeshFunction('size_t', mesh,
mesh.topology().dim() - 1, 0)
boundarymarkers.set_all(0)
domainmarkers = df.MeshFunction('size_t', mesh, mesh.topology().dim(), 0)
boundaries['left'].mark(boundarymarkers, 1)
boundaries['bottom'].mark(boundarymarkers, 2)
boundaries['right'].mark(boundarymarkers, 3)
boundaries['top'].mark(boundarymarkers, 4)
ds = df.Measure('ds', domain=mesh, subdomain_data=boundarymarkers)
dx = df.Measure('dx', domain=mesh, subdomain_data=domainmarkers)
return boundaries, boundarymarkers, domainmarkers, dx, ds, V, Vc
def calTrueSol(para):
nx, ny = para['mesh_N'][0], para['mesh_N'][1]
mesh = fe.UnitSquareMesh(nx, ny)
Vu = fe.FunctionSpace(mesh, 'P', para['P'])
Vc = fe.FunctionSpace(mesh, 'P', para['P'])
al = fe.Constant(para['alpha'])
f = fe.Expression(para['f'], degree=5)
q1 = fe.interpolate(fe.Expression(para['q1'], degree=5), Vc)
q2 = fe.interpolate(fe.Expression(para['q2'], degree=5), Vc)
q3 = fe.interpolate(fe.Expression(para['q3'], degree=5), Vc)
theta = fe.interpolate(fe.Expression(para['q4'], degree=5), Vc)
class BoundaryX0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 0.0)
class BoundaryX1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 1.0)
class BoundaryY0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 0.0)
class BoundaryY1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 1.0)
boundaries = fe.MeshFunction('size_t', mesh, mesh.topology().dim() - 1)
boundaries.set_all(0)
bc0, bc1, bc2, bc3 = BoundaryX0(), BoundaryX1(), BoundaryY0(), BoundaryY1()
bc0.mark(boundaries, 1)
bc1.mark(boundaries, 2)
bc2.mark(boundaries, 3)
bc3.mark(boundaries, 4)
domains = fe.MeshFunction("size_t", mesh, mesh.topology().dim())
domains.set_all(0)
bcD = fe.DirichletBC(Vu, theta, boundaries, 4)
dx = fe.Measure('dx', domain=mesh, subdomain_data=domains)
ds = fe.Measure('ds', domain=mesh, subdomain_data=boundaries)
u_trial, u_test = fe.TrialFunction(Vu), fe.TestFunction(Vu)
u = fe.Function(Vu)
left = fe.inner(al * fe.nabla_grad(u_trial), fe.nabla_grad(u_test)) * dx
right = f * u_test * dx + (q1 * u_test * ds(1) + q2 * u_test * ds(2) +
q3 * u_test * ds(3))
left_m, right_m = fe.assemble_system(left, right, bcD)
fe.solve(left_m, u.vector(), right_m)
return u
def get_measures(self):
"""
Create the measurements from the generated subdomains and boundaries
objects. The defined measurements are:
- dx: Area measure.
- ds: External boundaries measure.
- dS: Internal boundaries measure.
Returns
-------
None.
"""
self.dx = fn.Measure('dx')(subdomain_data=self.subdomains)
self.ds = fn.Measure('ds')(subdomain_data=self.boundaries)
self.dS = fn.Measure('dS')(subdomain_data=self.boundaries)
def set_boundaries(self):
self.boundaries = fe.MeshFunction("size_t", self.mesh,
self.mesh.topology().dim() - 1)
self.boundaries.set_all(0)
self.ds = fe.Measure("ds")(subdomain_data=self.boundaries)
self.I = fe.Identity(self.mesh.topology().dim())
self.normal = fe.FacetNormal(self.mesh)
def readCellExpression(self,
group_value_dict,
value_type="scalar",
overlap=lambda x: x[0],
*args,
**kwargs):
"""
Reads cell expression and returns it.
"""
value_type_dictionary = {
"scalar": ScalarCellExpressionFromXDMF,
"vector2d": Vector2DCellExpressionFromXDMF,
"vector3d": Vector3DCellExpressionFromXDMF
}
self.readMesh()
xdmffile = fenics.XDMFFile(self.xdmffilename)
cf = value_type_dictionary[value_type.lower()](group_value_dict,
overlap=overlap,
*args,
**kwargs)
cf.init()
for (key, value) in cf.group_value_dict.items():
cf.markers[key] = fenics.MeshFunction("size_t", self.mesh,
self.mesh.topology().dim())
xdmffile.read(cf.markers[key], key)
cf.dx[key] = fenics.Measure("dx",
domain=self.mesh,
subdomain_data=cf.markers[key])
xdmffile.close()
return cf
def _make_variational_problem(V):
"""
Formulate the variational problem a(u, v) = L(v).
Parameters
----------
V: FEniCS.FunctionSpace
Returns
-------
a, L: FEniCS.Expression
Variational forms.
"""
# Define trial and test functions
u = F.TrialFunction(V)
v = F.TestFunction(V)
# Collect Neumann conditions
ds = F.Measure("ds", domain=mesh, subdomain_data=boundary_markers)
integrals_N = []
for i in boundary_conditions:
if isinstance(boundary_conditions[i], dict):
if "Neumann" in boundary_conditions[i]:
if boundary_conditions[i]["Neumann"] != 0:
g = boundary_conditions[i]["Neumann"]
integrals_N.append(g[0] * v * ds(i))
# Define variational problem
a = F.inner(u, v) * F.dx + (DT**2) * (C**2) * F.inner(
F.grad(u), F.grad(v)) * F.dx
L = ((DT**2) * f[0] + 2 * u_nm1 -
u_nm2) * v * F.dx + (DT**2) * (C**2) * sum(integrals_N)
return a, L
def get_measures(self):
"""
Get measures of the domain given the required data.
The defined measurements are:
- dx: Area measure.
- ds: External boundaries measure.
- dS: Internal boundaries measure.
Returns:
"""
self.dx = fn.Measure('dx')(
subdomain_data=self.subdomains) # Area measure.
self.ds = fn.Measure('ds')(
subdomain_data=self.boundaries) # External boundaries measures.
self.dS = fn.Measure('dS')(
subdomain_data=self.boundaries
) # Internal boundaries measures (interface boundary).
def useSternLayerCellBC(self):
"""
Interfaces at left hand side and right hand side, species-wise
number conservation within interval."""
self.boundary_conditions = []
# Introduce a Lagrange multiplier per species anderson
# rebuild discretization scheme (function spaces)
self.constraints = 0
self.K = self.M
self.discretize()
# Potential Dirichlet BC
self.u0 = self.delta_u_scaled / 2.
self.u1 = -self.delta_u_scaled / 2.
boundary_markers = fn.MeshFunction('size_t', self.mesh,
self.mesh.topology().dim() - 1)
bx = [
Boundary(x0=self.x0_scaled, tol=self.bctol),
Boundary(x0=self.x1_scaled, tol=self.bctol)
]
# Boundary.mark crashes the kernel if Boundary is internal class
for i, b in enumerate(bx):
b.mark(boundary_markers, i)
boundary_conditions = {
0: {
'Robin': (1. / self.lambda_S_scaled, self.u0)
},
1: {
'Robin': (1. / self.lambda_S_scaled, self.u1)
},
}
ds = fn.Measure('ds',
domain=self.mesh,
subdomain_data=boundary_markers)
integrals_R = []
for i in boundary_conditions:
if 'Robin' in boundary_conditions[i]:
r, s = boundary_conditions[i]['Robin']
integrals_R.append(r * (self.u - s) * self.v * ds(i))
self.constraints += sum(integrals_R)
# Number conservation constraints
N0 = self.L_scaled * self.c_scaled # total amount of species in cell
for k in range(self.M):
self.logger.info('{:>{lwidth}s} N0 = {:<8.4g}'.format(
'Ion species {:02d} number conservation constraint'.format(k),
N0[k],
lwidth=self.label_width))
self.applyNumberConservationConstraint(k, self.c_scaled[k])
def eva(self):
# construct solutions corresponding to the basis functions
class BoundaryX0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 0.0)
class BoundaryX1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[0], 1.0)
class BoundaryY0(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 0.0)
class BoundaryY1(fe.SubDomain):
def inside(self, x, on_boundary):
return on_boundary and fe.near(x[1], 1.0)
boundaries = fe.MeshFunction('size_t', self.mesh,
self.mesh.topology().dim() - 1)
boundaries.set_all(0)
bc0, bc1, bc2, bc3 = BoundaryX0(), BoundaryX1(), BoundaryY0(
), BoundaryY1()
bc0.mark(boundaries, 1)
bc1.mark(boundaries, 2)
bc2.mark(boundaries, 3)
bc3.mark(boundaries, 4)
domains = fe.MeshFunction("size_t", self.mesh,
self.mesh.topology().dim())
domains.set_all(0)
dx = fe.Measure('dx', domain=self.mesh, subdomain_data=domains)
ds = fe.Measure('ds', domain=self.mesh, subdomain_data=boundaries)
u_trial, u_test = fe.TrialFunction(self.Vu), fe.TestFunction(self.Vu)
left = fe.inner(fe.nabla_grad(u_trial), fe.nabla_grad(u_test)) * dx
right = self.f * u_test * dx + (self.q1 * u_test * ds(1) +
self.q2 * u_test * ds(2) +
self.q3 * u_test * ds(3))
bcD = fe.DirichletBC(self.Vu, self.theta, boundaries, 4)
left_m, right_m = fe.assemble_system(left, right, bcD)
fe.solve(left_m, self.u.vector(), right_m)
def BCsDefinition(Dimensions, Mesh, V, u_0, u_1, LoadCase, BCsType=False):
# Define geometric spaces
class LowerSide(fe.SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14
return on_boundary and fe.near(x[2], -Dimensions[2] / 2, tol)
class UpperSide(fe.SubDomain):
def inside(self, x, on_boundary):
tol = 1E-14
return on_boundary and fe.near(x[2], Dimensions[2] / 2, tol)
# Define integration over subdpmains
Domains_Facets = fe.MeshFunction('size_t', Mesh,
Mesh.geometric_dimension() - 1)
ds = fe.Measure('ds', domain=Mesh, subdomain_data=Domains_Facets)
# Mark all domain facets with 0
Domains_Facets.set_all(0)
# Mark bottom facets with 1
bottom = LowerSide()
bottom.mark(Domains_Facets, 1)
# Mark upper facets with 2
upper = UpperSide()
upper.mark(Domains_Facets, 2)
# Apply boundary conditions
if BCsType == 'Ideal':
if LoadCase == 'SimpleShear':
bcl = fe.DirichletBC(V, u_0, Domains_Facets, 1)
bcu = fe.DirichletBC(V, u_1, Domains_Facets, 2)
else:
bcl = fe.DirichletBC(V.sub(2), u_0, Domains_Facets, 1)
bcu = fe.DirichletBC(V.sub(2), u_1, Domains_Facets, 2)
else:
bcl = fe.DirichletBC(V, u_0, Domains_Facets, 1)
bcu = fe.DirichletBC(V, u_1, Domains_Facets, 2)
# Set of boundary conditions
BoundaryConditions = [bcl, bcu]
return [BoundaryConditions, ds]
def load_2d_muscle_geo(filename='../geo/muscle_2d.xml', L0=1e-2):
mesh = fe.Mesh(filename)
coords = mesh.coordinates()
coords *= L0
mesh.bounding_box_tree().build(mesh)
# Define Boundaries
bottom = fe.CompiledSubDomain("near(x[1], side, 0.01) && on_boundary", side=-20.0 * L0)
top = fe.CompiledSubDomain("near(x[1], side, 0.01) && on_boundary", side=20.0 * L0)
# Initialize mesh function for boundary domains
boundaries = fe.MeshFunction('size_t', mesh, 2)
boundaries.set_all(0)
bottom.mark(boundaries, 1)
top.mark(boundaries, 2)
# Define new measures associated with the interior domains and
# exterior boundaries
dx = fe.Measure('dx', domain=mesh)
ds = fe.Measure('ds', domain=mesh, subdomain_data=boundaries)
return mesh, dx, ds, {"top": top, "bottom": bottom}
def readFacetFunction(self, group_value_dict, *args, **kwargs):
"""
Reads facet function and returns it.
"""
self.readMesh()
xdmffile = fenics.XDMFFile(self.xdmffilename)
ff = FacetFunctionFromXDMF(group_value_dict, *args, **kwargs)
ff.init()
for (key, value) in ff.group_value_dict.items():
ff.markers[key] = fenics.MeshFunction("size_t", self.mesh, self.mesh.topology().dim() - 1)
xdmffile.read(ff.markers[key], key)
ff.marked[key] = value.get("marked", 1)
ff.ds[key] = fenics.Measure("ds", domain=self.mesh, subdomain_data=ff.markers[key])
ff.bcs[key] = value
xdmffile.close()
return ff
def run(self) -> None:
mesh = self.mesh_creator.get_mesh()
self.spaces.initialize(mesh=mesh)
self.boundary_markers.mark_boundaries(mesh=mesh)
bc = self.bc_creator.apply(
vector_space=self.spaces.vector_space,
boundary_markers=self.boundary_markers.value)
ds = fenics.Measure('ds',
domain=mesh,
subdomain_data=self.boundary_markers.value)
self.boundary_excitation.set_ds(ds=ds)
self.fields.initialize(spaces=self.spaces)
fem_solver = get_fem_solver(fem_solver=self.fem_solver_type,
problem=self.problem,
fields=self.fields,
boundary_conditions=bc)
self.time_step_builder.set(
alpha_params=self.alpha_params,
time_params=self.time_params,
fem_solver=fem_solver,
boundary_excitation=self.boundary_excitation,
field_updates=self.field_updates,
fields=self.fields,
mesh=mesh,
spaces=self.spaces)
time_step = self.time_step_builder.build()
mins = []
maxs = []
for (i, t) in enumerate(self.time_params.linear_time_space[1:]):
# for i in range(3):
print("Time: ", t)
time_step.run(i)
# v = fenics.elem_div(self.problem.constitutive_relation.get_new_value(self.fields.u_new), fenics.sym(fenics.grad(self.fields.u_new)))
# x = fenics.project(v, self.spaces.tensor_space).vector()[:]
# mins.append(min(x))
# maxs.append(max(x))
# # print('min: {}, max: {}, median, {}'.format(min(x), max(x), np.nanmedian(np.asarray(x))))
# print(x)
# print()
# print(fenics.project(self.problem.constitutive_relation.get_new_value(self.fields.u_new), self.spaces.tensor_space).vector()[:])
# print(fenics.project(fenics.sym(fenics.grad(self.fields.u_new)), self.spaces.tensor_space).vector()[:])
if time_step.halt:
break
time_step.close()
print('total: min: {}, max: {}'.format(min(mins), max(maxs)))
def wrapper2D(par, meshName, N_PN, boundaryFilePrefix, solutionFolder):
#-------- load mesh --------
mesh = fe.Mesh(meshName + '.xml')
boundaryMarkers = fe.MeshFunction('size_t', mesh, meshName + '_facet_region.xml')
ds = fe.Measure('ds', domain=mesh, subdomain_data=boundaryMarkers)
u = solvePN2nd(par, N_PN, mesh, boundaryFilePrefix, ds)
#fe.plot(u[0])
# first moment: u0 = int_{4pi} I * b0 dOmega, b0 = sqrt(1 / 4 / pi)
# radiative energy: phi = int_{4pi} I dOmega
radiativeEnergy = np.sqrt(4 * np.pi) * u[0].compute_vertex_values()
createFolder(solutionFolder)
sio.savemat('%sradiativeEnergy_P%d2nd_%s.mat'%(solutionFolder, N_PN, meshName.split('/')[-1]),
{'radiativeEnergy': radiativeEnergy, 'points': mesh.coordinates(),
'connectivityList': mesh.cells()})
return u
def readXDMFFile(self, xdmffilename, group_value_dict):
xdmffile = fenics.XDMFFile(xdmffilename)
self.group_value_dict = group_value_dict
self.mesh = fenics.Mesh()
xdmffile.read(self.mesh)
self.markers = {}
self.marked = {}
self.ds = {}
self.bcs = {}
for (key, value) in self.group_value_dict.iteritems():
# Fenics interface here: create facet function of type size_t (positive int) for every group
# TODO: examine whether size_t is appropriate or this class could be generalized
self.markers[key] = fenics.FacetFunction("size_t", self.mesh)
xdmffile.read(self.markers[key], key)
self.marked[key] = value.get("marked", 1)
self.ds[key] = fenics.Measure("ds", domain=self.mesh, subdomain_data=self.markers[key])
self.bcs[key] = value
xdmffile.close()
def avg_condition_number(mesh):
"""Computes average mesh quality based on the condition number of the reference mapping.
This quality criterion uses the condition number (in the Frobenius norm) of the
(linear) mapping from the elements of the mesh to the reference element. Computes
the average of the condition number over all elements.
Parameters
----------
mesh : dolfin.cpp.mesh.Mesh
The mesh, whose quality shall be computed.
Returns
-------
float
The average mesh quality based on the condition number.
"""
DG0 = fenics.FunctionSpace(mesh, 'DG', 0)
jac = Jacobian(mesh)
inv = JacobianInverse(mesh)
options = [
['ksp_type', 'preonly'],
['pc_type', 'jacobi'],
['pc_jacobi_type', 'diagonal'],
['ksp_rtol', 1e-16],
['ksp_atol', 1e-20],
['ksp_max_it', 1000]
]
ksp = PETSc.KSP().create()
_setup_petsc_options([ksp], [options])
dx = fenics.Measure('dx', mesh)
a = fenics.TrialFunction(DG0)*fenics.TestFunction(DG0)*dx
L = fenics.sqrt(fenics.inner(jac, jac))*fenics.sqrt(fenics.inner(inv, inv))*fenics.TestFunction(DG0)*dx
cond = fenics.Function(DG0)
A, b = _assemble_petsc_system(a, L)
_solve_linear_problem(ksp, A, b, cond.vector().vec(), options)
cond.vector().apply('')
return np.average(np.sqrt(mesh.geometric_dimension()) / cond.vector()[:])
def defineBoundaryMarkers1D(mesh, zMin, zMax):
"""For implementation details, see
https://fenicsproject.org/olddocs/dolfin/1.3.0/python/demo/documented/subdomains-poisson/python/documentation.html
https://fenicsproject.discourse.group/t/natural-boundary-conditions-without-facetfunction/228/3
"""
boundaries = fe.MeshFunction("size_t", mesh, mesh.topology().dim() - 1)
class Left(fe.SubDomain):
def inside(self, x, on_boundary):
return fe.near(x[0], zMin)
class Right(fe.SubDomain):
def inside(self, x, on_boundary):
return fe.near(x[0], zMax)
left = Left()
right = Right()
left.mark(boundaries, 1)
right.mark(boundaries, 2)
ds = fe.Measure("ds", domain=mesh, subdomain_data=boundaries)
return ds
def readXDMFfile(self, xdmffilename, group_value_dict):
"""
Initialization of CellExpressionXDMF by reading an XDMF file.
@param: xdmffilename: path to xdmf file
@param: group_value_dict: {"groupname":function(x)}
function(x) is a function which is evaluated at the marked positions of the cells
"""
xdmffile = fenics.XDMFFile(xdmffilename)
self.group_value_dict = group_value_dict
self.mesh = fenics.Mesh()
xdmffile.read(self.mesh)
self.markers = {}
self.dx = {}
for (key, value) in self.group_value_dict.iteritems():
# Fenics interface here: create cell function of type int for every group
# TODO: examine whether int is appropriate or this class could be generalized
self.markers[key] = fenics.CellFunction("size_t", self.mesh)
xdmffile.read(self.markers[key], key)
self.dx[key] = fenics.Measure("dx", domain=self.mesh, subdomain_data=self.markers[key])
xdmffile.close()
def _build_function_space(self):
class Exterior(fa.SubDomain):
def inside(self, x, on_boundary):
return on_boundary
self.exterior = Exterior()
self.V = fa.FunctionSpace(self.mesh, 'P', 1)
self.sub_domains = fa.MeshFunction("size_t", self.mesh,
self.mesh.topology().dim() - 1)
self.sub_domains.set_all(0)
self.normal = fa.FacetNormal(self.mesh)
self.ds = fa.Measure("ds")(subdomain_data=self.sub_domains)
# self.source = fa.Expression(("cos(pi*x[0])*cos(pi*x[1])"), degree=3)
self.source = fa.Expression(("x[0]*x[0] + x[1]*x[1]"), degree=3)
self.source = fa.interpolate(self.source, self.V)
# self.source = fa.Constant(1.)
self.bcs = []
boundary_fn = fa.Constant(0.)
boundary_bc = fa.DirichletBC(self.V, boundary_fn, self.exterior)
self.bcs = self.bcs + [boundary_bc]
def _assembly(V):
"""
Assemble the matrices used in the linear system Au = b.
Alternative representation of the variational form.
Pre-calculate to speed up.
"""
# Define trial and test functions
u = F.TrialFunction(V)
v = F.TestFunction(V)
# Assemble
a_M = F.inner(u, v) * F.dx
a_K = F.inner(F.grad(u), F.grad(v)) * F.dx
M = F.assemble(a_M)
K = F.assemble(a_K)
A = M + (dt**2) * (c**2) * K
ds = F.Measure("ds", domain=mesh, subdomain_data=boundary_markers)
M_ = F.assemble(u * v * ds(1))
return M, K, A, M_
def __init__(self, state_forms, bcs_list, cost_functional_form, states, controls, adjoints, config=None,
riesz_scalar_products=None, control_constraints=None, initial_guess=None, ksp_options=None,
adjoint_ksp_options=None, desired_weights=None):
r"""This is used to generate all classes and functionalities. First ensures
consistent input, afterwards, the solution algorithm is initialized.
Parameters
----------
state_forms : ufl.form.Form or list[ufl.form.Form]
The weak form of the state equation (user implemented). Can be either
a single UFL form, or a (ordered) list of UFL forms.
bcs_list : list[dolfin.fem.dirichletbc.DirichletBC] or list[list[dolfin.fem.dirichletbc.DirichletBC]] or dolfin.fem.dirichletbc.DirichletBC or None
The list of DirichletBC objects describing Dirichlet (essential) boundary conditions.
If this is ``None``, then no Dirichlet boundary conditions are imposed.
cost_functional_form : ufl.form.Form or list[ufl.form.Form]
UFL form of the cost functional.
states : dolfin.function.function.Function or list[dolfin.function.function.Function]
The state variable(s), can either be a :py:class:`fenics.Function`, or a list of these.
controls : dolfin.function.function.Function or list[dolfin.function.function.Function]
The control variable(s), can either be a :py:class:`fenics.Function`, or a list of these.
adjoints : dolfin.function.function.Function or list[dolfin.function.function.Function]
The adjoint variable(s), can either be a :py:class:`fenics.Function`, or a (ordered) list of these.
config : configparser.ConfigParser or None
The config file for the problem, generated via :py:func:`cashocs.create_config`.
Alternatively, this can also be ``None``, in which case the default configurations
are used, except for the optimization algorithm. This has then to be specified
in the :py:meth:`solve <cashocs.OptimalControlProblem.solve>` method. The
default is ``None``.
riesz_scalar_products : None or ufl.form.Form or list[ufl.form.Form], optional
The scalar products of the control space. Can either be None, a single UFL form, or a
(ordered) list of UFL forms. If ``None``, the :math:`L^2(\Omega)` product is used.
(default is ``None``).
control_constraints : None or list[dolfin.function.function.Function] or list[float] or list[list[dolfin.function.function.Function]] or list[list[float]], optional
Box constraints posed on the control, ``None`` means that there are none (default is ``None``).
The (inner) lists should contain two elements of the form ``[u_a, u_b]``, where ``u_a`` is the lower,
and ``u_b`` the upper bound.
initial_guess : list[dolfin.function.function.Function], optional
List of functions that act as initial guess for the state variables, should be valid input for :py:func:`fenics.assign`.
Defaults to ``None``, which means a zero initial guess.
ksp_options : list[list[str]] or list[list[list[str]]] or None, optional
A list of strings corresponding to command line options for PETSc,
used to solve the state systems. If this is ``None``, then the direct solver
mumps is used (default is ``None``).
adjoint_ksp_options : list[list[str]] or list[list[list[str]]] or None
A list of strings corresponding to command line options for PETSc,
used to solve the adjoint systems. If this is ``None``, then the same options
as for the state systems are used (default is ``None``).
Examples
--------
Examples how to use this class can be found in the :ref:`tutorial <tutorial_index>`.
"""
OptimizationProblem.__init__(self, state_forms, bcs_list, cost_functional_form, states, adjoints, config, initial_guess, ksp_options, adjoint_ksp_options, desired_weights)
### Overloading, such that we do not have to use lists for a single state and a single control
### controls
try:
if type(controls) == list and len(controls) > 0:
for i in range(len(controls)):
if controls[i].__module__ == 'dolfin.function.function' and type(controls[i]).__name__ == 'Function':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'controls have to be fenics Functions.')
self.controls = controls
elif controls.__module__ == 'dolfin.function.function' and type(controls).__name__ == 'Function':
self.controls = [controls]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'Type of controls is wrong.')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls', 'Type of controls is wrong.')
self.control_dim = len(self.controls)
### riesz_scalar_products
if riesz_scalar_products is None:
dx = fenics.Measure('dx', self.controls[0].function_space().mesh())
self.riesz_scalar_products = [fenics.inner(fenics.TrialFunction(self.controls[i].function_space()), fenics.TestFunction(self.controls[i].function_space())) * dx
for i in range(len(self.controls))]
else:
try:
if type(riesz_scalar_products)==list and len(riesz_scalar_products) > 0:
for i in range(len(riesz_scalar_products)):
if riesz_scalar_products[i].__module__== 'ufl.form' and type(riesz_scalar_products[i]).__name__== 'Form':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
self.riesz_scalar_products = riesz_scalar_products
elif riesz_scalar_products.__module__== 'ufl.form' and type(riesz_scalar_products).__name__== 'Form':
self.riesz_scalar_products = [riesz_scalar_products]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'riesz_scalar_products have to be ufl forms')
### control_constraints
if control_constraints is None:
self.control_constraints = []
for control in self.controls:
u_a = fenics.Function(control.function_space())
u_a.vector()[:] = float('-inf')
u_b = fenics.Function(control.function_space())
u_b.vector()[:] = float('inf')
self.control_constraints.append([u_a, u_b])
else:
try:
if type(control_constraints) == list and len(control_constraints) > 0:
if type(control_constraints[0]) == list:
for i in range(len(control_constraints)):
if type(control_constraints[i]) == list and len(control_constraints[i]) == 2:
for j in range(2):
if type(control_constraints[i][j]) in [float, int]:
pass
elif control_constraints[i][j].__module__ == 'dolfin.function.function' and type(control_constraints[i][j]).__name__ == 'Function':
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
self.control_constraints = control_constraints
elif (type(control_constraints[0]) in [float, int] or (control_constraints[0].__module__ == 'dolfin.function.function' and type(control_constraints[0]).__name__=='Function')) \
and (type(control_constraints[1]) in [float, int] or (control_constraints[1].__module__ == 'dolfin.function.function' and type(control_constraints[1]).__name__=='Function')):
self.control_constraints = [control_constraints]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
except:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'control_constraints has to be a list containing upper and lower bounds')
# recast floats into functions for compatibility
temp_constraints = self.control_constraints[:]
self.control_constraints = []
for idx, pair in enumerate(temp_constraints):
if type(pair[0]) in [float, int]:
lower_bound = fenics.Function(self.controls[idx].function_space())
lower_bound.vector()[:] = pair[0]
elif pair[0].__module__ == 'dolfin.function.function' and type(pair[0]).__name__ == 'Function':
lower_bound = pair[0]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Wrong type for the control constraints')
if type(pair[1]) in [float, int]:
upper_bound = fenics.Function(self.controls[idx].function_space())
upper_bound.vector()[:] = pair[1]
elif pair[1].__module__ == 'dolfin.function.function' and type(pair[1]).__name__ == 'Function':
upper_bound = pair[1]
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Wrong type for the control constraints')
self.control_constraints.append([lower_bound, upper_bound])
### Check whether the control constraints are feasible, and whether they are actually present
self.require_control_constraints = [False for i in range(self.control_dim)]
for idx, pair in enumerate(self.control_constraints):
if not np.alltrue(pair[0].vector()[:] < pair[1].vector()[:]):
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints',
'The lower bound must always be smaller than the upper bound for the control_constraints.')
if np.max(pair[0].vector()[:]) == float('-inf') and np.min(pair[1].vector()[:]) == float('inf'):
# no control constraint for this component
pass
else:
self.require_control_constraints[idx] = True
control_element = self.controls[idx].ufl_element()
if control_element.family() == 'Mixed':
for j in range(control_element.value_size()):
sub_elem = control_element.extract_component(j)[1]
if sub_elem.family() == 'Real' or (sub_elem.family() == 'Lagrange' and sub_elem.degree() == 1) \
or (sub_elem.family() == 'Discontinuous Lagrange' and sub_elem.degree() == 0):
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls',
'Control constraints are only implemented for linear Lagrange, constant Discontinuous Lagrange, and Real elements.')
else:
if control_element.family() == 'Real' or (control_element.family() == 'Lagrange' and control_element.degree() == 1) \
or (control_element.family() == 'Discontinuous Lagrange' and control_element.degree() == 0):
pass
else:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'controls',
'Control constraints are only implemented for linear Lagrange, constant Discontinuous Lagrange, and Real elements.')
if not len(self.riesz_scalar_products) == self.control_dim:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'riesz_scalar_products', 'Length of controls does not match')
if not len(self.control_constraints) == self.control_dim:
raise InputError('cashocs._optimal_control.optimal_control_problem.OptimalControlProblem', 'control_constraints', 'Length of controls does not match')
### end overloading
self.form_handler = ControlFormHandler(self.lagrangian, self.bcs_list, self.states, self.controls, self.adjoints, self.config,
self.riesz_scalar_products, self.control_constraints, self.ksp_options, self.adjoint_ksp_options,
self.require_control_constraints)
self.state_spaces = self.form_handler.state_spaces
self.control_spaces = self.form_handler.control_spaces
self.adjoint_spaces = self.form_handler.adjoint_spaces
self.projected_difference = [fenics.Function(V) for V in self.control_spaces]
self.state_problem = StateProblem(self.form_handler, self.initial_guess)
self.adjoint_problem = AdjointProblem(self.form_handler, self.state_problem)
self.gradient_problem = GradientProblem(self.form_handler, self.state_problem, self.adjoint_problem)
self.algorithm = _optimization_algorithm_configuration(self.config)
self.reduced_cost_functional = ReducedCostFunctional(self.form_handler, self.state_problem)
self.gradients = self.gradient_problem.gradients
self.objective_value = 1.0
def strain(v):
return fn.sym(fn.grad(v))
# boundary conditions
bdry = fn.MeshFunction("size_t", mesh, 1)
bdry.set_all(0)
FooT = fn.CompiledSubDomain(
"(x[0] >= -0.2) && (x[0] <= 0.2) && near(x[1], 0.75) && on_boundary")
GammaU = fn.CompiledSubDomain(
"( near(x[0], -0.5) || near(x[1], 0.0) ) && on_boundary")
GammaU.mark(bdry, 31)
FooT.mark(bdry, 32)
ds = fn.Measure("ds", subdomain_data=bdry)
bcU = fn.DirichletBC(Hh.sub(0), u_g, bdry, 31)
bcP = fn.DirichletBC(Hh.sub(2), p_g, bdry, 32)
bcs = [bcU, bcP]
# ******** Weak forms ********** #
PLeft = 2*mu*fn.inner(strain(u),strain(v)) * fn.dx \
- fn.div(v) * phi * fn.dx \
+ (c0/alpha + 1.0/lmbda)* p * q * fn.dx \
+ kappa/(alpha*nu) * fn.dot(fn.grad(p),fn.grad(q)) * fn.dx \
- 1.0/lmbda * phi * q * fn.dx \
- fn.div(u) * psi * fn.dx \
+ 1.0/lmbda * psi * p * fn.dx \
- 1.0/lmbda * phi * psi * fn.dx
def __init__(self, lagrangian, bcs_list, states, adjoints, config,
ksp_options, adjoint_ksp_options):
"""Initializes the form handler.
Parameters
----------
lagrangian : cashocs._forms.Lagrangian
The lagrangian of the optimization problem.
bcs_list : list[list[dolfin.fem.dirichletbc.DirichletBC]]
The list of DirichletBCs for the state equation.
states : list[dolfin.function.function.Function]
The function that acts as the state variable.
adjoints : list[dolfin.function.function.Function]
The function that acts as the adjoint variable.
config : configparser.ConfigParser
The configparser object of the config file.
ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
state systems.
adjoint_ksp_options : list[list[list[str]]]
The list of command line options for the KSP for the
adjoint systems.
"""
# Initialize the attributes from the arguments
self.lagrangian = lagrangian
self.bcs_list = bcs_list
self.states = states
self.adjoints = adjoints
self.config = config
self.state_ksp_options = ksp_options
self.adjoint_ksp_options = adjoint_ksp_options
# Further initializations
self.cost_functional_form = self.lagrangian.cost_functional_form
self.state_forms = self.lagrangian.state_forms
self.state_dim = len(self.states)
self.state_spaces = [x.function_space() for x in self.states]
self.adjoint_spaces = [x.function_space() for x in self.adjoints]
# Test if state_spaces coincide with adjoint_spaces
if self.state_spaces == self.adjoint_spaces:
self.state_adjoint_equal_spaces = True
else:
self.state_adjoint_equal_spaces = False
self.mesh = self.state_spaces[0].mesh()
self.dx = fenics.Measure('dx', self.mesh)
self.trial_functions_state = [
fenics.TrialFunction(V) for V in self.state_spaces
]
self.test_functions_state = [
fenics.TestFunction(V) for V in self.state_spaces
]
self.trial_functions_adjoint = [
fenics.TrialFunction(V) for V in self.adjoint_spaces
]
self.test_functions_adjoint = [
fenics.TestFunction(V) for V in self.adjoint_spaces
]
self.state_is_linear = self.config.getboolean('StateSystem',
'is_linear',
fallback=False)
self.state_is_picard = self.config.getboolean('StateSystem',
'picard_iteration',
fallback=False)
self.opt_algo = _optimization_algorithm_configuration(config)
if self.opt_algo == 'pdas':
self.inner_pdas = self.config.get('AlgoPDAS', 'inner_pdas')
self.__compute_state_equations()
self.__compute_adjoint_equations()
def __init__(self, form_handler):
"""Initializes the regularization
Parameters
----------
form_handler : cashocs._forms.ShapeFormHandler
the corresponding shape form handler object
"""
self.form_handler = form_handler
self.config = self.form_handler.config
self.dx = fenics.Measure('dx', self.form_handler.mesh)
self.ds = fenics.Measure('ds', self.form_handler.mesh)
self.spatial_coordinate = fenics.SpatialCoordinate(
self.form_handler.mesh)
self.measure_hole = self.config.getboolean('Regularization',
'measure_hole',
fallback=False)
if self.measure_hole:
self.x_start = self.config.getfloat('Regularization',
'x_start',
fallback=0.0)
self.x_end = self.config.getfloat('Regularization',
'x_end',
fallback=1.0)
if not self.x_end >= self.x_start:
raise ConfigError('Regularization', 'x_end',
'x_end must not be smaller than x_start.')
self.delta_x = self.x_end - self.x_start
self.y_start = self.config.getfloat('Regularization',
'y_start',
fallback=0.0)
self.y_end = self.config.getfloat('Regularization',
'y_end',
fallback=1.0)
if not self.y_end >= self.y_start:
raise ConfigError('Regularization', 'y_end',
'y_end must not be smaller than y_start.')
self.delta_y = self.y_end - self.y_start
self.z_start = self.config.getfloat('Regularization',
'z_start',
fallback=0.0)
self.z_end = self.config.getfloat('Regularization',
'z_end',
fallback=1.0)
if not self.z_end >= self.z_start:
raise ConfigError('Regularization', 'z_end',
'z_end must not be smaller than z_start.')
self.delta_z = self.z_end - self.z_start
if self.form_handler.mesh.geometric_dimension() == 2:
self.delta_z = 1.0
self.mu_volume = self.config.getfloat('Regularization',
'factor_volume',
fallback=0.0)
self.target_volume = self.config.getfloat('Regularization',
'target_volume',
fallback=0.0)
if self.config.getboolean('Regularization',
'use_initial_volume',
fallback=False):
if not self.measure_hole:
self.target_volume = fenics.assemble(Constant(1) * self.dx)
else:
self.target_volume = self.delta_x * self.delta_y * self.delta_z - fenics.assemble(
Constant(1.0) * self.dx)
self.mu_surface = self.config.getfloat('Regularization',
'factor_surface',
fallback=0.0)
self.target_surface = self.config.getfloat('Regularization',
'target_surface',
fallback=0.0)
if self.config.getboolean('Regularization',
'use_initial_surface',
fallback=False):
self.target_surface = fenics.assemble(Constant(1) * self.ds)
self.mu_barycenter = self.config.getfloat('Regularization',
'factor_barycenter',
fallback=0.0)
self.target_barycenter_list = json.loads(
self.config.get('Regularization',
'target_barycenter',
fallback='[0,0,0]'))
if not type(self.target_barycenter_list) == list:
raise ConfigError('Regularization', 'target_barycenter',
'This has to be a list.')
if self.form_handler.mesh.geometric_dimension() == 2 and len(
self.target_barycenter_list) == 2:
self.target_barycenter_list.append(0.0)
if self.config.getboolean('Regularization',
'use_initial_barycenter',
fallback=False):
self.target_barycenter_list = [0.0, 0.0, 0.0]
if not self.measure_hole:
volume = fenics.assemble(Constant(1) * self.dx)
self.target_barycenter_list[0] = fenics.assemble(
self.spatial_coordinate[0] * self.dx) / volume
self.target_barycenter_list[1] = fenics.assemble(
self.spatial_coordinate[1] * self.dx) / volume
if self.form_handler.mesh.geometric_dimension() == 3:
self.target_barycenter_list[2] = fenics.assemble(
self.spatial_coordinate[2] * self.dx) / volume
else:
self.target_barycenter_list[2] = 0.0
else:
volume = self.delta_x * self.delta_y * self.delta_z - fenics.assemble(
Constant(1) * self.dx)
self.target_barycenter_list[0] = (
0.5 * (pow(self.x_end, 2) - pow(self.x_start, 2)) *
self.delta_y * self.delta_z - fenics.assemble(
self.spatial_coordinate[0] * self.dx)) / volume
self.target_barycenter_list[1] = (
0.5 * (pow(self.y_end, 2) - pow(self.y_start, 2)) *
self.delta_x * self.delta_z - fenics.assemble(
self.spatial_coordinate[1] * self.dx)) / volume
if self.form_handler.mesh.geometric_dimension() == 3:
self.target_barycenter_list[2] = (
0.5 * (pow(self.z_end, 2) - pow(self.z_start, 2)) *
self.delta_x * self.delta_y - fenics.assemble(
self.spatial_coordinate[2] * self.dx)) / volume
else:
self.target_barycenter_list[2] = 0.0
if not (self.mu_volume >= 0.0 and self.mu_surface >= 0.0
and self.mu_barycenter >= 0.0):
raise ConfigError(
'Regularization', 'mu_volume, mu_surface, or mu_barycenter',
'All regularization constants have to be nonnegative.')
if self.mu_volume > 0.0 or self.mu_surface > 0.0 or self.mu_barycenter > 0.0:
self.has_regularization = True
else:
self.has_regularization = False
# self.relative_scaling = self.config.getboolean('Regularization', 'relative_scaling')
# if self.relative_scaling and self.has_regularization:
# self.scale_weights()
self.current_volume = fenics.Expression('val', degree=0, val=1.0)
self.current_surface = fenics.Expression('val', degree=0, val=1.0)
self.current_barycenter_x = fenics.Expression('val', degree=0, val=0.0)
self.current_barycenter_y = fenics.Expression('val', degree=0, val=0.0)
self.current_barycenter_z = fenics.Expression('val', degree=0, val=0.0)
def regular_box_mesh(n=10,
S_x=0.0,
S_y=0.0,
S_z=None,
E_x=1.0,
E_y=1.0,
E_z=None):
r"""Creates a mesh corresponding to a rectangle or cube.
This function creates a uniform mesh of either a rectangle
or a cube, with specified start (``S_``) and end points (``E_``).
The resulting mesh uses ``n`` elements along the shortest direction
and accordingly many along the longer ones. The resulting domain is
.. math::
\begin{alignedat}{2}
&[S_x, E_x] \times [S_y, E_y] \quad &&\text{ in } 2D, \\
&[S_x, E_x] \times [S_y, E_y] \times [S_z, E_z] \quad &&\text{ in } 3D.
\end{alignedat}
The boundary markers are ordered as follows:
- 1 corresponds to :math:`x=S_x`.
- 2 corresponds to :math:`x=E_x`.
- 3 corresponds to :math:`y=S_y`.
- 4 corresponds to :math:`y=E_y`.
- 5 corresponds to :math:`z=S_z` (only in 3D).
- 6 corresponds to :math:`z=E_z` (only in 3D).
Parameters
----------
n : int
Number of elements in the shortest coordinate direction.
S_x : float
Start of the x-interval.
S_y : float
Start of the y-interval.
S_z : float or None, optional
Start of the z-interval, mesh is 2D if this is ``None``
(default is ``None``).
E_x : float
End of the x-interval.
E_y : float
End of the y-interval.
E_z : float or None, optional
End of the z-interval, mesh is 2D if this is ``None``
(default is ``None``).
Returns
-------
mesh : dolfin.cpp.mesh.Mesh
The computational mesh.
subdomains : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the subdomains.
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the boundaries.
dx : ufl.measure.Measure
The volume measure of the mesh corresponding to subdomains.
ds : ufl.measure.Measure
The surface measure of the mesh corresponding to boundaries.
dS : ufl.measure.Measure
The interior facet measure of the mesh corresponding to boundaries.
"""
n = int(n)
if not n > 0:
raise InputError('cashocs.geometry.regular_box_mesh', 'n',
'This needs to be positive.')
if not S_x < E_x:
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_x',
'Incorrect input for the x-coordinate. S_x has to be smaller than E_x.'
)
if not S_y < E_y:
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_y',
'Incorrect input for the y-coordinate. S_y has to be smaller than E_y.'
)
if not ((S_z is None and E_z is None) or (S_z < E_z)):
raise InputError(
'cashocs.geometry.regular_box_mesh', 'S_z',
'Incorrect input for the z-coordinate. S_z has to be smaller than E_z, or only one of them is specified.'
)
if S_z is None:
lx = E_x - S_x
ly = E_y - S_y
sizes = [lx, ly]
dim = 2
else:
lx = E_x - S_x
ly = E_y - S_y
lz = E_z - S_z
sizes = [lx, ly, lz]
dim = 3
size_min = np.min(sizes)
num_points = [int(np.round(length / size_min * n)) for length in sizes]
if S_z is None:
mesh = fenics.RectangleMesh(fenics.Point(S_x, S_y),
fenics.Point(E_x, E_y), num_points[0],
num_points[1])
else:
mesh = fenics.BoxMesh(fenics.Point(S_x, S_y, S_z),
fenics.Point(E_x, E_y, E_z), num_points[0],
num_points[1], num_points[2])
subdomains = fenics.MeshFunction('size_t', mesh, dim=dim)
boundaries = fenics.MeshFunction('size_t', mesh, dim=dim - 1)
x_min = fenics.CompiledSubDomain('on_boundary && near(x[0], sx, tol)',
tol=fenics.DOLFIN_EPS,
sx=S_x)
x_max = fenics.CompiledSubDomain('on_boundary && near(x[0], ex, tol)',
tol=fenics.DOLFIN_EPS,
ex=E_x)
x_min.mark(boundaries, 1)
x_max.mark(boundaries, 2)
y_min = fenics.CompiledSubDomain('on_boundary && near(x[1], sy, tol)',
tol=fenics.DOLFIN_EPS,
sy=S_y)
y_max = fenics.CompiledSubDomain('on_boundary && near(x[1], ey, tol)',
tol=fenics.DOLFIN_EPS,
ey=E_y)
y_min.mark(boundaries, 3)
y_max.mark(boundaries, 4)
if S_z is not None:
z_min = fenics.CompiledSubDomain('on_boundary && near(x[2], sz, tol)',
tol=fenics.DOLFIN_EPS,
sz=S_z)
z_max = fenics.CompiledSubDomain('on_boundary && near(x[2], ez, tol)',
tol=fenics.DOLFIN_EPS,
ez=E_z)
z_min.mark(boundaries, 5)
z_max.mark(boundaries, 6)
dx = fenics.Measure('dx', mesh, subdomain_data=subdomains)
ds = fenics.Measure('ds', mesh, subdomain_data=boundaries)
dS = fenics.Measure('dS', mesh)
return mesh, subdomains, boundaries, dx, ds, dS
############################### IMPORT MESH ##################################
#cprint("\nMESH PRE-PROCESSING", 'blue', attrs=['bold'])
# Import mesh and groups
#cprint("Creating gmsh mesh...", 'green')
subprocess.check_output("gmsh ./gmsh/beam.geo -3", shell=True)
#cprint("Converting mesh to DOLFIN format...", 'green')
subprocess.check_output('dolfin-convert ./gmsh/beam.msh mesh/beam.xml',
shell=True)
#cprint("Importing mesh in FEniCS...", 'green')
mesh = fe.Mesh('mesh/beam.xml')
#cprint("Generating boundaries and subdomains...", 'green')
subdomains = fe.MeshFunction("size_t", mesh, "mesh/beam_physical_region.xml")
boundaries = fe.MeshFunction("size_t", mesh, "mesh/beam_facet_region.xml")
# Redefine the integration measures
dxp = fe.Measure('dx', domain=mesh, subdomain_data=subdomains)
dsp = fe.Measure('ds', domain=mesh, subdomain_data=boundaries)
##################### FINITE ELEMENT SPACES ##################################
# Finite element spaces
W = fe.FunctionSpace(mesh, 'P', 1)
V = fe.VectorFunctionSpace(mesh, 'P', 1)
Z = fe.TensorFunctionSpace(mesh, 'P', 1)
# Finite element functions
du = fe.TrialFunction(V)
v = fe.TestFunction(V)
u = fe.Function(V)
######################### PROBLEM PARAMS ######################################
# Material properties
def regular_mesh(n=10, L_x=1.0, L_y=1.0, L_z=None):
r"""Creates a mesh corresponding to a rectangle or cube.
This function creates a uniform mesh of either a rectangle
or a cube, starting at the origin and having length specified
in ``L_x``, ``L_y``, and ``L_z``. The resulting mesh uses ``n`` elements along the
shortest direction and accordingly many along the longer ones.
The resulting domain is
.. math::
\begin{alignedat}{2}
&[0, L_x] \times [0, L_y] \quad &&\text{ in } 2D, \\
&[0, L_x] \times [0, L_y] \times [0, L_z] \quad &&\text{ in } 3D.
\end{alignedat}
The boundary markers are ordered as follows:
- 1 corresponds to :math:`x=0`.
- 2 corresponds to :math:`x=L_x`.
- 3 corresponds to :math:`y=0`.
- 4 corresponds to :math:`y=L_y`.
- 5 corresponds to :math:`z=0` (only in 3D).
- 6 corresponds to :math:`z=L_z` (only in 3D).
Parameters
----------
n : int
Number of elements in the shortest coordinate direction.
L_x : float
Length in x-direction.
L_y : float
Length in y-direction.
L_z : float or None, optional
Length in z-direction, if this is ``None``, then the geometry
will be two-dimensional (default is ``None``).
Returns
-------
mesh : dolfin.cpp.mesh.Mesh
The computational mesh.
subdomains : dolfin.cpp.mesh.MeshFunctionSizet
A :py:class:`fenics.MeshFunction` object containing the subdomains.
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the boundaries.
dx : ufl.measure.Measure
The volume measure of the mesh corresponding to subdomains.
ds : ufl.measure.Measure
The surface measure of the mesh corresponding to boundaries.
dS : ufl.measure.Measure
The interior facet measure of the mesh corresponding to boundaries.
"""
if not n > 0:
raise InputError('cashocs.geometry.regular_mesh', 'n',
'This needs to be positive.')
if not L_x > 0.0:
raise InputError('cashocs.geometry.regular_mesh', 'L_x',
'L_x needs to be positive')
if not L_y > 0.0:
raise InputError('cashocs.geometry.regular_mesh', 'L_y',
'L_y needs to be positive')
if not (L_z is None or L_z > 0.0):
raise InputError('cashocs.geometry.regular_mesh', 'L_z',
'L_z needs to be positive or None (for 2D mesh)')
n = int(n)
if L_z is None:
sizes = [L_x, L_y]
dim = 2
else:
sizes = [L_x, L_y, L_z]
dim = 3
size_min = np.min(sizes)
num_points = [int(np.round(length / size_min * n)) for length in sizes]
if L_z is None:
mesh = fenics.RectangleMesh(fenics.Point(0, 0), fenics.Point(sizes),
num_points[0], num_points[1])
else:
mesh = fenics.BoxMesh(fenics.Point(0, 0, 0), fenics.Point(sizes),
num_points[0], num_points[1], num_points[2])
subdomains = fenics.MeshFunction('size_t', mesh, dim=dim)
boundaries = fenics.MeshFunction('size_t', mesh, dim=dim - 1)
x_min = fenics.CompiledSubDomain('on_boundary && near(x[0], 0, tol)',
tol=fenics.DOLFIN_EPS)
x_max = fenics.CompiledSubDomain('on_boundary && near(x[0], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[0])
x_min.mark(boundaries, 1)
x_max.mark(boundaries, 2)
y_min = fenics.CompiledSubDomain('on_boundary && near(x[1], 0, tol)',
tol=fenics.DOLFIN_EPS)
y_max = fenics.CompiledSubDomain('on_boundary && near(x[1], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[1])
y_min.mark(boundaries, 3)
y_max.mark(boundaries, 4)
if L_z is not None:
z_min = fenics.CompiledSubDomain('on_boundary && near(x[2], 0, tol)',
tol=fenics.DOLFIN_EPS)
z_max = fenics.CompiledSubDomain(
'on_boundary && near(x[2], length, tol)',
tol=fenics.DOLFIN_EPS,
length=sizes[2])
z_min.mark(boundaries, 5)
z_max.mark(boundaries, 6)
dx = fenics.Measure('dx', mesh, subdomain_data=subdomains)
ds = fenics.Measure('ds', mesh, subdomain_data=boundaries)
dS = fenics.Measure('dS', mesh)
return mesh, subdomains, boundaries, dx, ds, dS
def compute_static_deformation(self):
assert self.mesh is not None
# now we define subdomains on the mesh
bottom = fe.CompiledSubDomain('near(x[2], 0) && on_boundary')
top = fe.CompiledSubDomain('near(x[2], 1) && on_boundary')
# middle = fe.CompiledSubDomain('x[2] > 0.3 && x[2] < 0.7')
# Initialize mesh function for interior domains
self.domains = fe.MeshFunction('size_t', self.mesh, 3)
self.domains.set_all(0)
# middle.mark(self.domains, 1)
# Initialize mesh function for boundary domains
self.boundaries = fe.MeshFunction('size_t', self.mesh, 2)
self.boundaries.set_all(0)
bottom.mark(self.boundaries, 1)
top.mark(self.boundaries, 2)
# Define new measures associated with the interior domains and
# exterior boundaries
self.dx = fe.Measure('dx',
domain=self.mesh,
subdomain_data=self.domains)
self.ds = fe.Measure('ds',
domain=self.mesh,
subdomain_data=self.boundaries)
# define function spaces
V = fe.VectorFunctionSpace(self.mesh, "Lagrange", 1)
# now we define subdomains on the mesh
bottom = fe.CompiledSubDomain('near(x[2], 0) && on_boundary')
top = fe.CompiledSubDomain('near(x[2], 1) && on_boundary')
# middle = fe.CompiledSubDomain('x[2] > 0.3 && x[2] < 0.7')
d = self.mesh.geometry().dim()
# Initialize mesh function for interior domains
self.domains = fe.MeshFunction('size_t', self.mesh, d)
self.domains.set_all(0)
# middle.mark(self.domains, 1)
# Initialize mesh function for boundary domains
self.boundaries = fe.MeshFunction('size_t', self.mesh, d - 1)
self.boundaries.set_all(0)
bottom.mark(self.boundaries, 1)
top.mark(self.boundaries, 2)
# Define new measures associated with the interior domains and
# exterior boundaries
self.dx = fe.Measure('dx',
domain=self.mesh,
subdomain_data=self.domains)
self.ds = fe.Measure('ds',
domain=self.mesh,
subdomain_data=self.boundaries)
c_zero = fe.Constant((0, 0, 0))
# define boundary conditions
bc_bottom = fe.DirichletBC(V, c_zero, bottom)
bc_top = fe.DirichletBC(V, c_zero, top)
bcs = [bc_bottom] # , bc_top]
# define functions
du = TrialFunction(V)
v = TestFunction(V)
u = Function(V)
B = fe.Constant((0., 2.0, 0.))
T = fe.Constant((0.0, 0.0, 0.0))
d = u.geometric_dimension()
I = fe.Identity(d)
F = I + grad(u)
C = F.T * F
I_1 = tr(C)
J = det(F)
E, mu = 10., 0.3
mu, lmbda = fe.Constant(E / (2 * (1 + mu))), fe.Constant(
E * mu / ((1 + mu) * (1 - 2 * mu)))
# stored energy (comp. neo-hookean model)
psi = (mu / 2.) * (I_1 - 3) - mu * fe.ln(J) + (lmbda /
2.) * (fe.ln(J))**2
dx = self.dx
ds = self.ds
Pi = psi * fe.dx - dot(B, u) * fe.dx - dot(T, u) * fe.ds
F = fe.derivative(Pi, u, v)
J = fe.derivative(F, u, du)
fe.solve(F == 0, u, bcs, J=J)
# save results
self.u = u
# write to disk
output = fe.File("/tmp/static.pvd")
output << u
def import_mesh(arg):
"""Imports a mesh file for use with CASHOCS / FEniCS.
This function imports a mesh file that was generated by GMSH and converted to
.xdmf with the command line function :ref:`cashocs-convert <cashocs_convert>`.
If there are Physical quantities specified in the GMSH file, these are imported
to the subdomains and boundaries output of this function and can also be directly
accessed via the measures, e.g., with ``dx(1)``, ``ds(1)``, etc.
Parameters
----------
arg : str or configparser.ConfigParser
This is either a string, in which case it corresponds to the location
of the mesh file in .xdmf file format, or a config file that
has this path stored in its settings, under the section Mesh, as
parameter ``mesh_file``.
Returns
-------
mesh : dolfin.cpp.mesh.Mesh
The imported (computational) mesh.
subdomains : dolfin.cpp.mesh.MeshFunctionSizet
A :py:class:`fenics.MeshFunction` object containing the subdomains,
i.e., the Physical regions marked in the GMSH
file.
boundaries : dolfin.cpp.mesh.MeshFunctionSizet
A MeshFunction object containing the boundaries,
i.e., the Physical regions marked in the GMSH
file. Can, e.g., be used to set up boundary
conditions.
dx : ufl.measure.Measure
The volume measure of the mesh corresponding to
the subdomains (i.e. GMSH Physical region indices).
ds : ufl.measure.Measure
The surface measure of the mesh corresponding to
the boundaries (i.e. GMSH Physical region indices).
dS : ufl.measure.Measure
The interior facet measure of the mesh corresponding
to boundaries (i.e. GMSH Physical region indices).
"""
start_time = time.time()
print('Importing mesh to FEniCS')
# Check for the file format
if type(arg) == str:
mesh_file = arg
mesh_attribute = 'str'
elif type(arg) == configparser.ConfigParser:
mesh_attribute = 'config'
### overloading for remeshing
if not arg.getboolean('Mesh', 'remesh', fallback=False):
mesh_file = arg.get('Mesh', 'mesh_file')
else:
if not ('_cashocs_remesh_flag' in sys.argv):
mesh_file = arg.get('Mesh', 'mesh_file')
else:
temp_dir = sys.argv[-1]
with open(temp_dir + '/temp_dict.json', 'r') as file:
temp_dict = json.load(file)
mesh_file = temp_dict['mesh_file']
else:
raise InputError(
'cashocs.geometry.import_mesh', 'arg',
'Not a valid argument for import_mesh. Has to be either a path to a mesh file (str) or a config.'
)
if mesh_file[-5:] == '.xdmf':
file_string = mesh_file[:-5]
else:
raise InputError(
'cashocs.geometry.import_mesh', 'arg',
'Not a suitable mesh file format. Has to end in .xdmf.')
mesh = fenics.Mesh()
xdmf_file = fenics.XDMFFile(mesh.mpi_comm(), mesh_file)
xdmf_file.read(mesh)
xdmf_file.close()
subdomains_mvc = fenics.MeshValueCollection('size_t', mesh,
mesh.geometric_dimension())
boundaries_mvc = fenics.MeshValueCollection('size_t', mesh,
mesh.geometric_dimension() - 1)
if os.path.isfile(file_string + '_subdomains.xdmf'):
xdmf_subdomains = fenics.XDMFFile(mesh.mpi_comm(),
file_string + '_subdomains.xdmf')
xdmf_subdomains.read(subdomains_mvc, 'subdomains')
xdmf_subdomains.close()
if os.path.isfile(file_string + '_boundaries.xdmf'):
xdmf_boundaries = fenics.XDMFFile(mesh.mpi_comm(),
file_string + '_boundaries.xdmf')
xdmf_boundaries.read(boundaries_mvc, 'boundaries')
xdmf_boundaries.close()
subdomains = fenics.MeshFunction('size_t', mesh, subdomains_mvc)
boundaries = fenics.MeshFunction('size_t', mesh, boundaries_mvc)
dx = fenics.Measure('dx', domain=mesh, subdomain_data=subdomains)
ds = fenics.Measure('ds', domain=mesh, subdomain_data=boundaries)
dS = fenics.Measure('dS', domain=mesh, subdomain_data=boundaries)
end_time = time.time()
print('Done Importing Mesh. Elapsed Time: ' +
format(end_time - start_time, '.3e') + ' s')
print('')
# Add an attribute to the mesh to show with what procedure it was generated
mesh._cashocs_generator = mesh_attribute
return mesh, subdomains, boundaries, dx, ds, dS
# section of boundary that is free
free_boundary = fn.CompiledSubDomain(
"(near(x[1], mesh_height) || near(x[0], mesh_width/2)) && on_boundary",
mesh_height=mesh_height, mesh_width=mesh_width)
# section of boundary that is a rigid wall
wall_boundary = fn.CompiledSubDomain(
"(near(x[1], 0.0) || near(x[0], -mesh_width/2)) && on_boundary",
mesh_width=mesh_width)
# mark boundaries
free_boundary.mark(mesh_boundary, 31)
wall_boundary.mark(mesh_boundary, 32)
# ds, for integrating
ds = fn.Measure("ds", subdomain_data=mesh_boundary)
# boundary conditions
bcU = fn.DirichletBC(Hh.sub(0), u_g, mesh_boundary, 32)
bcP = fn.DirichletBC(Hh.sub(2), p_g, mesh_boundary, 31)
boundary_conditions = [bcU, bcP]
# initial conditions - poroelasticity
u_old = fn.interpolate(fn.Constant((0.0, 0.0)), Vhf)
phi_old = fn.interpolate(fn.Constant(0.0), Zhf)
p_old = fn.interpolate(fn.Constant(0.0), Qhf)
# initial conditions - reaction-diffusion
E_old = fn.interpolate(fn.Constant(0.0),Mhf)
n_old = fn.interpolate(fn.Constant(0.0),Mhf)
