def __init__(self, *args, **kwargs):
"Create Dirichlet boundary condition"
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.DirichletBC):
cpp.dolfin_error("bcs.py",
"create DirichletBC",
"Expecting a DirichleBC as only argument"\
" for copy constructor")
# Initialize base class
cpp.DirichletBC.__init__(self, args[0])
return
# Special case for value specified as float, tuple or similar
if len(args) >= 2 and not isinstance(args[1], cpp.GenericFunction):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0])
else:
expr = Constant(args[1]) # let Constant handle all problems
args = args[:1] + (expr,) + args[2:]
# Special case for sub domain specified as a function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
sub_domain = AutoSubDomain(args[2])
args = args[:2] + (sub_domain,) + args[3:]
# Special case for sub domain specified as a string
if len(args) >= 3 and isinstance(args[2], str):
sub_domain = compile_subdomains(args[2])
args = args[:2] + (sub_domain,) + args[3:]
# Store Expression to avoid scoping issue with SWIG directors
if isinstance(args[1], cpp.Expression):
self.function_arg = args[1]
# Store SubDomain to avoid scoping issue with SWIG directors
self.domain_args = args[2:]
# FIXME: Handling of multiple default arguments does not
# really work between Python and C++ when C++ requires them to
# be ordered and cannot accept the latter without the
# former...
# Add keyword arguments (in correct order...)
allowed_kwargs = ["method", "check_midpoint"]
for key in allowed_kwargs:
if key in kwargs:
args = tuple(list(args) + [kwargs[key]])
# Check for other keyword arguments
for key in kwargs:
if not key in allowed_kwargs:
cpp.dolfin_error("bcs.py",
"create boundary condition",
"Unknown keyword argument \"%s\"" % key)
# Initialize base class
cpp.DirichletBC.__init__(self, *args)
def __init__(self, *args, **kwargs):
"Create Dirichlet boundary condition"
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.DirichletBC):
cpp.dolfin_error("bcs.py",
"create DirichletBC",
"Expecting a DirichleBC as only argument"\
" for copy constructor")
# Initialize base class
cpp.DirichletBC.__init__(self, args[0])
self.domain_args = args[0].domain_args
return
# Special case for value specified as float, tuple or similar
if len(args) >= 2 and not isinstance(args[1], cpp.GenericFunction):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0])
else:
expr = Constant(args[1]) # let Constant handle all problems
args = args[:1] + (expr, ) + args[2:]
# Special case for sub domain specified as a function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
sub_domain = AutoSubDomain(args[2])
args = args[:2] + (sub_domain, ) + args[3:]
# Special case for sub domain specified as a string
if len(args) >= 3 and isinstance(args[2], str):
sub_domain = CompiledSubDomain(args[2])
args = args[:2] + (sub_domain, ) + args[3:]
# Store Expression to avoid scoping issue with SWIG directors
if isinstance(args[1], cpp.Expression):
self.function_arg = args[1]
# Store SubDomain to avoid scoping issue with SWIG directors
self.domain_args = args[2:]
# FIXME: Handling of multiple default arguments does not
# really work between Python and C++ when C++ requires them to
# be ordered and cannot accept the latter without the
# former...
# Add keyword arguments (in correct order...)
allowed_kwargs = ["method", "check_midpoint"]
for key in allowed_kwargs:
if key in kwargs:
args = tuple(list(args) + [kwargs[key]])
# Check for other keyword arguments
for key in kwargs:
if not key in allowed_kwargs:
cpp.dolfin_error("bcs.py", "create boundary condition",
"Unknown keyword argument \"%s\"" % key)
# Initialize base class
cpp.DirichletBC.__init__(self, *args)
def compute_projection_error(mesh_resolution=20, p_order=1):
# Define domain and mesh
a, b = 0, 1
mesh = IntervalMesh(mesh_resolution, a, b)
# Define finite element function space
V = FunctionSpace(mesh, "CG", p_order)
# Extract vertices of the mesh
x = V.tabulate_dof_coordinates()
# Note: Not the same as x = mesh.coordinates() (apparently has been re-ordered)
# Express the analytical function
u = Expression("1 + 4 * x[0] * x[0] - 5 * x[0] * x[0] * x[0]", degree=5)
# Project u onto V and extract the values in the mesh nodes
Pu = project(u, V)
Pua = Pu.vector().array()
# Create a function in the finite element function space V
Eu = Function(V)
Eua = Eu.vector().array()
# Evaluate function in the mesh nodes
for i in range(len(x)):
Eua[i] = 1 + 4 * x[i]**2 - 5 * x[i]**3
# Compute sum of projection error in the nodes
e = Eua - Pua
error = 0
for i in range(len(e)):
error += abs(e[i])
return error
def compute_projection_error(mesh_resolution=20, p_order=1):
# Define domain and mesh
a, b = 0, 1
mesh = IntervalMesh(mesh_resolution, a, b)
# Define finite element function space
V = FunctionSpace(mesh, "CG", p_order)
# Extract vertices of the mesh
x = V.dofmap().tabulate_all_coordinates(mesh)
indices = np.argsort(x)
# Express the analytical function
u = Expression("1 + 4 * x[0] * x[0] - 5 * x[0] * x[0] * x[0]", degree=5)
# Project u onto V and extract the values in the mesh nodes
Pu = project(u, V)
Pua = Pu.vector().array()
# Create a function in the finite element function space V
Eu = Function(V)
Eua = Eu.vector().array()
# Evaluate function in the mesh nodes
for j in indices:
Eua[j] = 1 + 4 * x[j] * x[j] - 5 * x[j] * x[j] * x[j]
# Compute sum of projection error in the nodes
e = Eua - Pua
error = 0
for i in range(len(e)):
error += abs(e[i])
return error
def compare_errors(p_order=1):
# Express the analytical function
u = Expression("1 + sin(10*x[0])", degree=5)
mesh_resolutions = [10, 50, 100, 200]
projection_errors = []
interpolation_errors = []
for mesh_resolution in mesh_resolutions:
# Define mesh
mesh = UnitIntervalMesh(mesh_resolution)
# Define finite element function space
V = FunctionSpace(mesh, "CG", p_order)
# Compute projection
up = project(u, V)
# Compute interpolation
ui = interpolate(u, V)
# Compute errors
projection_errors.append(compute_error(u, up))
interpolation_errors.append(compute_error(u, ui))
print(projection_errors)
print(interpolation_errors)
def computeError4level(self, meanApprox4lvl, lastSpace, meanExact = None):
if meanExact == None:
solMesh = lastSpace.mesh()
solFamily = lastSpace.ufl_element().family()
solDegree = lastSpace.ufl_element().degree()
refSpace = FunctionSpace(refine(refine(solMesh)), solFamily, solDegree)
meanExact = self.solveMean(refSpace)
refSpace = meanExact.function_space()
error4lvl = list(
np.sqrt(assemble(
(project(meanApprox, refSpace) - interpolate(meanExact, refSpace)) ** 2 * dx
)) for meanApprox in meanApprox4lvl)
return error4lvl
def __init__(self, *args, **kwargs):
"Create Dirichlet boundary condition"
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.DirichletBC):
cpp.dolfin_error("bcs.py",
"create DirichletBC",
"Expecting a DirichleBC as only argument"\
" for copy constructor")
# Initialize base class
cpp.DirichletBC.__init__(self, args[0])
return
# Special case for value specified as float, tuple or similar
if len(args) >= 2 and not isinstance(args[1], cpp.GenericFunction):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0])
else:
expr = Constant(args[1]) # let Constant handle all problems
args = args[:1] + (expr,) + args[2:]
# Special case for sub domain specified as a function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
sub_domain = AutoSubDomain(args[2])
args = args[:2] + (sub_domain,) + args[3:]
# Special case for sub domain specified as a string
if len(args) >= 3 and isinstance(args[2], str):
sub_domain = compile_subdomains(args[2])
args = args[:2] + (sub_domain,) + args[3:]
# Store Expression to avoid scoping issue with SWIG directors
if isinstance(args[1], cpp.Expression):
self.function_arg = args[1]
# Store SubDomain to avoid scoping issue with SWIG directors
self.domain_args = args[2:]
# Add method argument if it's given
if "method" in kwargs:
args = tuple(list(args) + [kwargs["method"]])
# Initialize base class
cpp.DirichletBC.__init__(self, *args)
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC> or a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4,4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0 # scale the warping/glyphs
title = "Fancy plot" # Set your own title
)
"""
# Plot element
if isinstance(object, ufl.FiniteElementBase):
if os.environ.get("DOLFIN_NOPLOT", "0") != "0":
return
import ffc
return ffc.plot(object, *args, **kwargs)
# Get mesh from explicit mesh kwarg, only positional arg, or via object
mesh = kwargs.pop('mesh', None)
if isinstance(object, cpp.Mesh):
if mesh is not None and mesh.id() != object.id():
cpp.dolfin_error(
"plotting.py", "plot mesh",
"Got different mesh in plot object and keyword argument")
mesh = object
if mesh is None:
if isinstance(object, cpp.Function):
mesh = object.function_space().mesh()
elif hasattr(object, "mesh"):
mesh = object.mesh()
# Expressions do not carry their own mesh
if isinstance(object, cpp.Expression) and mesh is None:
cpp.dolfin_error("plotting.py", "plot expression",
"Expecting a mesh as keyword argument")
# Try to project if object is not a standard plottable type
if not isinstance(object, _plottable_types):
from dolfin.fem.projection import project
try:
cpp.info("Object cannot be plotted directly, projecting to"\
" piecewise linears.")
object = project(object, mesh=mesh)
except Exception as e:
msg = "Don't know how to plot given object:\n %s\n" \
"and projection failed:\n %s" % (str(object), str(e))
cpp.dolfin_error("plotting.py", "plot object", msg)
# Select backend
backend = cpp.parameters["plotting_backend"]
if backend == "vtk":
return _plot_cpp(object, mesh, kwargs)
elif backend == "matplotlib":
return _plot_matplotlib(object, mesh, kwargs)
### pvdFileU = File(outputFolder+"/u.pvd", "compressed") pvdFileV = File(outputFolder+"/v.pvd", "compressed") pvdFileStress = File(outputFolder+"/stress.pvd", "compressed") ## Save sub domains to VTK files #bndFile = File(outputFolder+"/subdomains.pvd") #bndFile << neumannSubdomains ### ### Solver ### # Class representing the initial conditions u0 = project(physics.u0, V) v0 = project(physics.v0, V) # Initial condition tk = physics.t0 u = u0 v = v0 # Source physics.f.t = tk timef = False fk = assemble(inner(physics.f, w)*dx) if isinstance(physics.f, Expression): if 't' in physics.f.user_parameters: timef = True # Neumann boundary condition
def solve_navier_stokes_equation(interior_circle=True, num_mesh_refinements=0):
"""
Solve the Navier-Stokes equation on a hard-coded mesh with hard-coded initial and boundary conditions
"""
mesh, om, im, nm, ymax, sub_domains = setup_geometry(
interior_circle, num_mesh_refinements)
dsi = Measure("ds", domain=mesh, subdomain_data=sub_domains)
# Setup FEM function spaces
# Function space for the velocity
V = VectorFunctionSpace(mesh, "CG", 1)
# Function space for the pressure
Q = FunctionSpace(mesh, "CG", 1)
# Mixed function space for velocity and pressure
W = V * Q
# Setup FEM functions
v, q = TestFunctions(W)
w = Function(W)
(u, p) = (as_vector((w[0], w[1])), w[2])
u0 = Function(V)
# Inlet velocity
uin = Expression(("4*(x[1]*(YMAX-x[1]))/(YMAX*YMAX)", "0."),
YMAX=ymax,
degree=1)
# Viscosity and stabilization parameters
nu = 1e-6
h = CellSize(mesh)
d = 0.2 * h**(3.0 / 2.0)
# Time parameters
time_step = 0.1
t_start, t_end = 0.0, 10.0
# Penalty parameter
gamma = 10 / h
# Time stepping
t = t_start
step = 0
while t < t_end:
# Time discretization (Crank–Nicolson method)
um = 0.5 * u + 0.5 * u0
# Navier-Stokes equations in weak residual form (stabilized FEM)
# Basic residual
r = (inner((u - u0) / time_step + grad(p) + grad(um) * um, v) +
nu * inner(grad(um), grad(v)) + div(um) * q) * dx
# Weak boundary conditions
r += gamma * (om * p * q + im * inner(u - uin, v) +
nm * inner(u, v)) * ds
# Stabilization
r += d * (inner(grad(p) + grad(um) * um,
grad(q) + grad(um) * v) + inner(div(um), div(v))) * dx
# Solve the Navier-Stokes equation (one time step)
solve(r == 0, w)
if step % 5 == 0:
# Plot norm of velocity at current time step
nov = project(sqrt(inner(u, u)), Q)
fig = plt.figure()
plot(nov, fig=fig)
plt.show()
# Compute drag force on circle
n = FacetNormal(mesh)
drag_force_measure = p * n[0] * dsi(1) # Drag (only pressure)
drag_force = assemble(drag_force_measure)
print("Drag force = " + str(drag_force))
# Shift to next time step
t += time_step
step += 1
u0 = project(u, V)
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC>, a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4, 4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0, # scale the warping/glyphs
title = "Fancy plot", # set your own title
)
"""
# Return if plotting is disables
if os.environ.get("DOLFIN_NOPLOT", "0") != "0":
return
# Return if Matplotlib is not available
if not _has_matplotlib():
cpp.log.info("Matplotlib is required to plot from Python.")
return
# Plot element
if isinstance(object, ufl.FiniteElementBase):
import ffc
return ffc.plot(object, *args, **kwargs)
# For dolfin.function.Function, extract cpp_object
if hasattr(object, "cpp_object"):
object = object.cpp_object()
# Get mesh from explicit mesh kwarg, only positional arg, or via
# object
mesh = kwargs.pop('mesh', None)
if isinstance(object, cpp.mesh.Mesh):
if mesh is not None and mesh.id() != object.id():
raise RuntimeError(
"Got different mesh in plot object and keyword argument")
mesh = object
if mesh is None:
if isinstance(object, cpp.function.Function):
mesh = object.function_space().mesh()
elif hasattr(object, "mesh"):
mesh = object.mesh()
# Expressions do not carry their own mesh
if isinstance(object, cpp.function.Expression) and mesh is None:
raise RuntimeError("Expecting a mesh as keyword argument")
backend = kwargs.pop("backend", "matplotlib")
if backend not in ("matplotlib", "x3dom"):
raise RuntimeError("Plotting backend %s not recognised" % backend)
# Try to project if object is not a standard plottable type
if not isinstance(object, _all_plottable_types):
from dolfin.fem.projection import project
try:
cpp.log.info("Object cannot be plotted directly, projecting to "
"piecewise linears.")
object = project(object, mesh=mesh)
mesh = object.function_space().mesh()
object = object._cpp_object
except Exception as e:
msg = "Don't know how to plot given object:\n %s\n" \
"and projection failed:\n %s" % (str(object), str(e))
raise RuntimeError(msg)
# Plot
if backend == "matplotlib":
return _plot_matplotlib(object, mesh, kwargs)
elif backend == "x3dom":
return _plot_x3dom(object, kwargs)
else:
assert False, "This code should not be reached."
def solve_heat_equation(k, time_stepping_method):
"""
Solve the heat equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param k: Thermal conductivity
:param time_stepping_method: Time stepping method. Can be one of ["forward_euler", "backward_euler", "trapezoidal"]
"""
mesh, boundary = setup_geometry()
# Exact solution (Gauss curve)
ue = Expression("exp(-(x[0]*x[0]+x[1]*x[1])/(4*a*t))/(4*pi*a*t)",
a=k,
t=1e-7,
domain=mesh,
degree=1)
# Polynomial degree
r = 1
# Setup FEM function space
V = FunctionSpace(mesh, "CG", r)
# Create boundary condition
bc = DirichletBC(V, ue, boundary)
# Setup FEM functions
v = TestFunction(V)
u = Function(V)
# Time parameters
time_step = 0.001
t_start, t_end = 0.0, 20.0
# Time stepping
t = t_start
if time_stepping_method == "forward_euler":
theta = 0.0
if time_stepping_method == "backward_euler":
theta = 1.0
if time_stepping_method == "trapezoidal":
theta = 0.5
u0 = ue
step = 0
while t < t_end:
# Intermediate value for u (depending on the chosen time stepping method)
um = (1.0 - theta) * u0 + theta * u
# Weak form of the heat equation
a = (u - u0) / time_step * v * dx + k * inner(grad(um), grad(v)) * dx
# Solve the heat equation (one time step)
solve(a == 0, u, bc)
# Advance time in exact solution
t += time_step
ue.t = t
if step % 100 == 0:
# Compute error in L2 norm
error_L2 = errornorm(ue, u, 'L2')
# or equivalently
# sqrt(assemble((ue - u) * (ue - u) * dx))
# Compute norm of exact solution
nue = norm(ue)
# Print relative error
print("Relative error = {}".format(error_L2 / nue))
# Shift to next time step
u0 = project(u, V)
step += 1
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC> or a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4,4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0 # scale the warping/glyphs
title = "Fancy plot" # Set your own title
)
"""
mesh = kwargs.get('mesh')
p = cpp.Parameters()
for key in kwargs:
# If there is a "mesh" kwarg it should not be added to the parameters
if key != "mesh":
try:
p.add(key, kwargs[key])
except TypeError:
cpp.warning("Incompatible type for keyword argument \"%s\". Ignoring." % key)
# Plot element
if isinstance(object, ufl.FiniteElementBase):
if os.environ.get("DOLFIN_NOPLOT", "0") != "0": return
import ffc
return ffc.plot(object, *args, **kwargs)
if mesh is None and len(args) == 1 and isinstance(args[0], cpp.Mesh):
mesh = args[0]
# Plot expression
if isinstance(object, cpp.Expression):
if mesh is None:
raise TypeError, "expected a mesh when plotting an expression."
return cpp.plot(object, mesh, p)
# Try to project if object is not a standard plottable type
if not isinstance(object, (cpp.Function, cpp.Expression, cpp.Mesh,
cpp.DirichletBC, cpp.MeshFunction, cpp.MeshFunctionBool,
cpp.MeshFunctionInt, cpp.MeshFunctionDouble,
cpp.MeshFunctionSizet, cpp.DirichletBC, cpp.CSGGeometry)):
from dolfin.fem.projection import project
try:
cpp.info("Object cannot be plotted directly, projecting to"\
" piecewise linears.")
object = project(object, mesh=mesh)
except Exception as e:
msg = ("Don't know how to plot given object:\n %s\n"\
"and projection failed:\n %s") % (str(object), str(e))
#raise RuntimeError(msg)
raise
plot_object = cpp.plot(object, p)
plot_object.write_ps = _VTKPlotter_write_ps
# Avoid premature deletion of plotted objects if they go out of scope
# before the plot window is closed. The plotter itself is safe, since it's
# created in the plot() C++ function, not directly from Python. But the
# Python plotter proxy may disappear, so we can't store the references
# there.
global _objects_referenced_from_plot_windows
_objects_referenced_from_plot_windows[plot_object.key()] = (object, mesh, p)
return plot_object
for i in range(nPsi)
]
###
### Solver
###
# Timestep
dt = (timeInterval.tf - physics.t0) / timeInterval.nt
# Dirichlet boundary condition
dirichletBC = {}
if 'dirichletBC' in dir(physics):
for i in physics.dirichletBC:
physics.dirichletBC[i].t = physics.t0
ug = project(physics.dirichletBC[i], V)
dirichletBC[i] = DirichletBC(V, ug, globalBndSubdomains, i)
dirichletBC[i].apply(M)
dirichletBC[i].apply(R)
# Preparing solvers
solverM = LUSolver(M)
solverMdtR = LUSolver(M + newmark.beta * dt * dt * R)
solverM.parameters['reuse_factorization'] = True
solverMdtR.parameters['reuse_factorization'] = True
# Psi loop
for iPsi in range(nPsi):
if verbose: print("Psi: ", iPsi)
def __init__(self, *args, **kwargs):
if len(args) == 1:
if not isinstance(args[0], cpp.fem.MultiMeshDirichletBC):
raise RuntimeError(
"Expecting a MultiMeshDirichletBC as only argument for copy constructor"
)
cpp.fem.MultiMeshDirichletBC.__init__(self, args[0])
return
if not isinstance(args[0],
(MultiMeshFunctionSpace, MultiMeshSubSpace)):
raise RuntimeError(
"Expecting MultiMeshFunctionSpace or MultiMeshSubSpace as first argument"
)
if len(args) >= 2:
# Check if we have an UFL-expression or a concrete type
if not hasattr(args[1], "_cpp_object"):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0]) # Should be interpolation
else:
expr = Constant(args[1])
args = args[:1] + (expr, 1) + args[2:]
if isinstance(args[1], (float, int)):
u = cpp.function.Constant(float(args[1]))
elif isinstance(args[1], ufl.Coefficient):
u = args[1]._cpp_object
elif isinstance(args[1], cpp.function.GenericFunction):
u = args[1]
else:
raise RuntimeError(
"Second argument must be convertiable to a GenericFucntion")
if isinstance(args[0], MultiMeshSubSpace):
args = args[:1] + (u, ) + args[2:]
else:
args = args[:1] + (u, ) + args[2:]
args = (args[0]._cpp_object, ) + args[1:]
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
raise NotImplementedError(
"User-specified subdomains not implemented")
if isinstance(args[2], cpp.mesh.SubDomain):
self.sub_domain = args[2]
args = args[:2] + (self.sub_domain, ) + args[3:]
elif isinstance(args[2], str):
raise NotImplementedError(
"User-specified subdomains not implemented")
elif isinstance(args[2], cpp.mesh.MeshFunctionSizet):
pass
else:
raise RuntimeError("Invalid argument")
# Add kwargs
if isinstance(args[-1], str):
method = args[-1]
else:
method = kwargs.pop("method", "topological")
args += (method, )
check_midpoint = kwargs.pop("check_midpoint", None)
if check_midpoint is not None:
args += (check_midpoint, )
if (len(kwargs) > 0):
raise RuntimeError("Invalid keyword arguments", kwargs)
super().__init__(*args)
def solve_wave_equation(a, symmetric=True):
"""
Solve the wave equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param a: Wave propagation factor
:param symmetric: Whether or not the problem is symmetric
"""
mesh, boundary = setup_geometry()
# Exact solution
if symmetric:
ue = Expression(
"(1-pow(a*t-x[0],2))*exp(-pow(a*t-x[0],2)) + (1-pow(a*t+x[0],2))*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
ve = Expression(
"2*a*(a*t-x[0])*(pow(a*t-x[0],2)-2)*exp(-pow(a*t-x[0],2))"
"+ 2*a*(a*t+x[0])*(pow(a*t+x[0],2)-2)*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
else:
ue = Expression("(1-pow(a*t+x[0],2))*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
ve = Expression(
"2*a*(a*t+x[0])*(pow(a*t+x[0],2)-2)*exp(-pow(a*t+x[0],2))",
a=a,
t=0,
domain=mesh,
degree=2)
# Polynomial degree
r = 1
# Setup FEM function spaces
Q = FunctionSpace(mesh, "CG", r)
W = VectorFunctionSpace(mesh, "CG", r, dim=2)
# Create boundary conditions
bcu = DirichletBC(W.sub(0), ue, boundary)
bcv = DirichletBC(W.sub(1), ve, boundary)
bcs = [bcu, bcv]
# Setup FEM functions
p, q = TestFunctions(W)
w = Function(W)
u, v = w[0], w[1]
# Time parameters
time_step = 0.05
t_start, t_end = 0.0, 5.0
# Time stepping
t = t_start
u0 = ue
v0 = ve
step = 0
while t < t_end:
# Weak form of the wave equation
um = 0.5 * (u + u0)
vm = 0.5 * (v + v0)
a1 = (u - u0) / time_step * p * dx - vm * p * dx
a2 = (v - v0) / time_step * q * dx + a**2 * inner(grad(um),
grad(q)) * dx
# Solve the wave equation (one time step)
solve(a1 + a2 == 0, w, bcs)
# Advance time in exact solution
t += time_step
ue.t = t
ve.t = t
if step % 10 == 0:
# Plot solution at current time step
fig = plt.figure()
plot(u, fig=fig)
plt.show()
# Compute max error at vertices
vertex_values_ue = ue.compute_vertex_values(mesh)
vertex_values_w = w.compute_vertex_values(mesh)
vertex_values_u = np.split(vertex_values_w, 2)[0]
error_max = np.max(np.abs(vertex_values_ue - vertex_values_u))
# Print error
print(error_max)
# Shift to next time step
u0 = project(u, Q)
v0 = project(v, Q)
step += 1
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC> or a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4,4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0 # scale the warping/glyphs
title = "Fancy plot" # Set your own title
)
"""
mesh = kwargs.get('mesh')
p = cpp.Parameters()
for key in kwargs:
# If there is a "mesh" kwarg it should not be added to the parameters
if key != "mesh":
try:
p.add(key, kwargs[key])
except TypeError:
cpp.warning(
"Incompatible type for keyword argument \"%s\". Ignoring."
% key)
# Plot element
if isinstance(object, ufl.FiniteElementBase):
if os.environ.get("DOLFIN_NOPLOT", "0") != "0": return
import ffc
return ffc.plot(object, *args, **kwargs)
if mesh is None and len(args) == 1 and isinstance(args[0], cpp.Mesh):
mesh = args[0]
# Plot expression
if isinstance(object, cpp.Expression):
if mesh is None:
raise TypeError, "expected a mesh when plotting an expression."
return cpp.plot(object, mesh, p)
# Try to project if object is not a standard plottable type
if not isinstance(
object, (cpp.Function, cpp.Expression, cpp.Mesh, cpp.DirichletBC,
cpp.MeshFunction, cpp.MeshFunctionBool,
cpp.MeshFunctionInt, cpp.MeshFunctionDouble,
cpp.MeshFunctionSizet, cpp.DirichletBC, cpp.CSGGeometry)):
from dolfin.fem.projection import project
try:
cpp.info("Object cannot be plotted directly, projecting to"\
" piecewise linears.")
object = project(object, mesh=mesh)
except Exception as e:
msg = ("Don't know how to plot given object:\n %s\n"\
"and projection failed:\n %s") % (str(object), str(e))
#raise RuntimeError(msg)
raise
plot_object = cpp.plot(object, p)
plot_object.write_ps = _VTKPlotter_write_ps
# Avoid premature deletion of plotted objects if they go out of scope
# before the plot window is closed. The plotter itself is safe, since it's
# created in the plot() C++ function, not directly from Python. But the
# Python plotter proxy may disappear, so we can't store the references
# there.
global _objects_referenced_from_plot_windows
_objects_referenced_from_plot_windows[plot_object.key()] = (object, mesh,
p)
return plot_object
def __init__(self, *args, **kwargs):
# FIXME: the logic in this function is really messy and
# unclear
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.fem.DirichletBC):
raise RuntimeError("Expecting a DirichleBC as only argument for copy constructor")
# Initialize base class
cpp.fem.DirichletBC.__init__(self, args[0])
return
# Get FunctionSpace
if not isinstance(args[0], FunctionSpace):
raise RuntimeError("First argument must be of type FunctionSpace")
# FIXME: correct the below comment
# Case: boundary value specified as float, tuple or similar
# if len(args) >= 2 and not isinstance(args[1], (cpp.function.GenericFunction):
if len(args) >= 2:
# Check if we have a UFL expression or a concrete type
if not hasattr(args[1], "_cpp_object"):
if isinstance(args[1], ufl.classes.Expr):
expr = project(args[1], args[0]) # FIXME: This should really be interpolaton (project is expensive)
else:
expr = Constant(args[1])
args = args[:1] + (expr,) + args[2:]
# Get boundary condition field (the condition that is applied)
if isinstance(args[1], float) or isinstance(args[1], int):
u = cpp.function.Constant(float(args[1]))
elif isinstance(args[1], ufl.Coefficient):
u = args[1].cpp_object()
elif isinstance(args[1], cpp.function.GenericFunction):
u = args[1]
else:
raise RuntimeError("Second argument must be convertiable to a GenericFunction: ",
args[1], type(args[1]))
args = args[:1] + (u,) + args[2:]
args = (args[0]._cpp_object,) + args[1:]
# Case: Special sub domain 'inside' function provided as a
# function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
# Note: using self below to avoid a problem where the user
# function attached to AutoSubDomain get prematurely
# destroyed. Maybe a pybind11 bug? Was the same with SWIG...
self.sub_domain = AutoSubDomain(args[2])
args = args[:2] + (self.sub_domain,) + args[3:]
# FIXME: for clarity, can the user provided function case be
# handled here too?
# Create SubDomain object
if isinstance(args[2], cpp.mesh.SubDomain):
self.sub_domain = args[2]
args = args[:2] + (self.sub_domain,) + args[3:]
elif isinstance(args[2], str):
self.sub_domain = CompiledSubDomain(args[2], mpi_comm=args[0].mesh().mpi_comm())
args = args[:2] + (self.sub_domain,) + args[3:]
elif isinstance(args[2], cpp.mesh.MeshFunctionSizet):
self.domain_args = args[2:]
else:
raise RuntimeError("Invalid argument")
# Add kwargs
if isinstance(args[-1], str):
method = args[-1]
else:
method = kwargs.pop("method", "topological")
args += (method,)
check_midpoint = kwargs.pop("check_midpoint", None)
if check_midpoint is not None:
args += (check_midpoint,)
if (len(kwargs) > 0):
raise RuntimeError("Invalid keyword arguments", kwargs)
super().__init__(*args)
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC> or a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4,4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0 # scale the warping/glyphs
title = "Fancy plot" # Set your own title
)
"""
# Plot element
if isinstance(object, ufl.FiniteElementBase):
if os.environ.get("DOLFIN_NOPLOT", "0") != "0":
return
import ffc
return ffc.plot(object, *args, **kwargs)
# Get mesh from explicit mesh kwarg, only positional arg, or via object
mesh = kwargs.pop('mesh', None)
if isinstance(object, cpp.Mesh):
if mesh is not None and mesh.id() != object.id():
cpp.dolfin_error("plotting.py",
"plot mesh",
"Got different mesh in plot object and keyword argument")
mesh = object
if mesh is None:
if isinstance(object, cpp.Function):
mesh = object.function_space().mesh()
elif hasattr(object, "mesh"):
mesh = object.mesh()
# Expressions do not carry their own mesh
if isinstance(object, cpp.Expression) and mesh is None:
cpp.dolfin_error("plotting.py",
"plot expression",
"Expecting a mesh as keyword argument")
# Try to project if object is not a standard plottable type
if not isinstance(object, _plottable_types):
from dolfin.fem.projection import project
try:
cpp.info("Object cannot be plotted directly, projecting to"\
" piecewise linears.")
object = project(object, mesh=mesh)
except Exception as e:
msg = "Don't know how to plot given object:\n %s\n" \
"and projection failed:\n %s" % (str(object), str(e))
cpp.dolfin_error("plotting.py", "plot object", msg)
# Select backend
backend = cpp.parameters["plotting_backend"]
if backend == "vtk":
return _plot_cpp(object, mesh, kwargs)
elif backend == "matplotlib":
return _plot_matplotlib(object, mesh, kwargs)
def plot(object, *args, **kwargs):
"""
Plot given object.
*Arguments*
object
a :py:class:`Mesh <dolfin.cpp.Mesh>`, a :py:class:`MeshFunction
<dolfin.cpp.MeshFunction>`, a :py:class:`Function
<dolfin.functions.function.Function>`, a :py:class:`Expression`
<dolfin.cpp.Expression>, a :py:class:`DirichletBC`
<dolfin.cpp.DirichletBC>, a :py:class:`FiniteElement
<ufl.FiniteElement>`.
*Examples of usage*
In the simplest case, to plot only e.g. a mesh, simply use
.. code-block:: python
mesh = UnitSquare(4, 4)
plot(mesh)
Use the ``title`` argument to specify title of the plot
.. code-block:: python
plot(mesh, tite="Finite element mesh")
It is also possible to plot an element
.. code-block:: python
element = FiniteElement("BDM", tetrahedron, 3)
plot(element)
Vector valued functions can be visualized with an alternative mode
.. code-block:: python
plot(u, mode = "glyphs")
A more advanced example
.. code-block:: python
plot(u,
wireframe = True, # use wireframe rendering
interactive = False, # do not hold plot on screen
scalarbar = False, # hide the color mapping bar
hardcopy_prefix = "myplot", # default plotfile name
scale = 2.0, # scale the warping/glyphs
title = "Fancy plot", # set your own title
)
"""
# Return if plotting is disables
if os.environ.get("DOLFIN_NOPLOT", "0") != "0":
return
# Return if Matplotlib is not available
if not _has_matplotlib():
cpp.log.info("Matplotlib is required to plot from Python.")
return
# Plot element
if isinstance(object, ufl.FiniteElementBase):
import ffc
return ffc.plot(object, *args, **kwargs)
# For dolfin.function.Function, extract cpp_object
if hasattr(object, "cpp_object"):
object = object.cpp_object()
# Get mesh from explicit mesh kwarg, only positional arg, or via
# object
mesh = kwargs.pop('mesh', None)
if isinstance(object, cpp.mesh.Mesh):
if mesh is not None and mesh.id() != object.id():
raise RuntimeError("Got different mesh in plot object and keyword argument")
mesh = object
if mesh is None:
if isinstance(object, cpp.function.Function):
mesh = object.function_space().mesh()
elif hasattr(object, "mesh"):
mesh = object.mesh()
# Expressions do not carry their own mesh
if isinstance(object, cpp.function.Expression) and mesh is None:
raise RuntimeError("Expecting a mesh as keyword argument")
backend = kwargs.pop("backend", "matplotlib")
if backend not in ("matplotlib", "x3dom"):
raise RuntimeError("Plotting backend %s not recognised" % backend)
# Try to project if object is not a standard plottable type
if not isinstance(object, _all_plottable_types):
from dolfin.fem.projection import project
try:
cpp.log.info("Object cannot be plotted directly, projecting to "
"piecewise linears.")
object = project(object, mesh=mesh)
mesh = object.function_space().mesh()
object = object._cpp_object
except Exception as e:
msg = "Don't know how to plot given object:\n %s\n" \
"and projection failed:\n %s" % (str(object), str(e))
raise RuntimeError(msg)
# Plot
if backend == "matplotlib":
return _plot_matplotlib(object, mesh, kwargs)
elif backend == "x3dom":
return _plot_x3dom(object, kwargs)
else:
assert False, "This code should not be reached."
def __init__(self, *args, **kwargs):
# FIXME: the logic in this function is really messy and
# unclear
# Copy constructor
if len(args) == 1:
if not isinstance(args[0], cpp.fem.DirichletBC):
raise RuntimeError(
"Expecting a DirichleBC as only argument for copy constructor"
)
# Initialize base class
cpp.fem.DirichletBC.__init__(self, args[0])
return
# Get FunctionSpace
if not isinstance(args[0], FunctionSpace):
raise RuntimeError("First argument must be of type FunctionSpace")
# FIXME: correct the below comment
# Case: boundary value specified as float, tuple or similar
# if len(args) >= 2 and not isinstance(args[1], (cpp.function.GenericFunction):
if len(args) >= 2:
# Check if we have a UFL expression or a concrete type
if not hasattr(args[1], "_cpp_object"):
if isinstance(args[1], ufl.classes.Expr):
expr = project(
args[1], args[0]
) # FIXME: This should really be interpolaton (project is expensive)
else:
expr = Constant(args[1])
args = args[:1] + (expr, ) + args[2:]
# Get boundary condition field (the condition that is applied)
if isinstance(args[1], float) or isinstance(args[1], int):
u = cpp.function.Constant(float(args[1]))
elif isinstance(args[1], ufl.Coefficient):
u = args[1].cpp_object()
elif isinstance(args[1], cpp.function.GenericFunction):
u = args[1]
else:
raise RuntimeError(
"Second argument must be convertiable to a GenericFunction: ",
args[1], type(args[1]))
args = args[:1] + (u, ) + args[2:]
args = (args[0]._cpp_object, ) + args[1:]
# Case: Special sub domain 'inside' function provided as a
# function
if len(args) >= 3 and isinstance(args[2], types.FunctionType):
# Note: using self below to avoid a problem where the user
# function attached to AutoSubDomain get prematurely
# destroyed. Maybe a pybind11 bug? Was the same with SWIG...
self.sub_domain = AutoSubDomain(args[2])
args = args[:2] + (self.sub_domain, ) + args[3:]
# FIXME: for clarity, can the user provided function case be
# handled here too?
# Create SubDomain object
if isinstance(args[2], cpp.mesh.SubDomain):
self.sub_domain = args[2]
args = args[:2] + (self.sub_domain, ) + args[3:]
elif isinstance(args[2], str):
self.sub_domain = CompiledSubDomain(
args[2], mpi_comm=args[0].mesh().mpi_comm())
args = args[:2] + (self.sub_domain, ) + args[3:]
elif isinstance(args[2], cpp.mesh.MeshFunctionSizet):
self.domain_args = args[2:]
else:
raise RuntimeError("Invalid argument")
# Add kwargs
if isinstance(args[-1], str):
method = args[-1]
else:
method = kwargs.pop("method", "topological")
args += (method, )
check_midpoint = kwargs.pop("check_midpoint", None)
if check_midpoint is not None:
args += (check_midpoint, )
if (len(kwargs) > 0):
raise RuntimeError("Invalid keyword arguments", kwargs)
super().__init__(*args)
def solve_heat_equation(k):
"""
Solve the heat equation on a hard-coded mesh with a hard-coded initial and boundary conditions
:param k: Thermal conductivity
"""
mesh, boundary = setup_geometry()
# Exact solution (Gauss curve)
ue = Expression("exp(-(x[0]*x[0]+x[1]*x[1])/(4*a*t))/(4*pi*a*t)",
a=k,
t=1e-7,
domain=mesh,
degree=2)
# Polynomial degree
r = 1
# Setup FEM function space
V = FunctionSpace(mesh, "CG", r)
# Create boundary condition
bc = DirichletBC(V, ue, boundary)
# Setup FEM functions
v = TestFunction(V)
u = Function(V)
# Time parameters
time_step = 0.5
t_start, t_end = 0.0, 20.0
# Time stepping
t = t_start
u0 = ue
step = 0
while t < t_end:
# Weak form of the heat equation
a = (u - u0) / time_step * v * dx + k * inner(grad(u), grad(v)) * dx
# Solve the heat equation (one time step)
solve(a == 0, u, bc)
# Advance time in exact solution
t += time_step
ue.t = t
if step % 5 == 0:
# Plot solution at current time step
fig = plt.figure()
plot(u, fig=fig)
plt.show()
# Compute error in L2 norm
error_L2 = errornorm(ue, u, 'L2')
# or equivalently
# sqrt(assemble((ue - u) * (ue - u) * dx))
# Print error
print(error_L2)
# Shift to next time step
u0 = project(u, V)
step += 1
