def set_linear_dirichlet_constant(mesh,
dt,
var_name='Temperature',
bc_temperature=1300.,
temperature=300.,
source_value=0.,
axis=1,
right=True,
tol=1e-4,
conductivity=1.,
density=1.,
specific_heat=1.):
'''
Heat equation problem where an edge/surface is set at a given temperature
and everything else at another.
Parameters
----------
axis : int
Controls boundary to which Dirichlet bc is applied.
right : bool
Dirichlet bc is applied to the planar and perpendicular face found in
the max (True) or min (False) coordinate.
'''
# function space
V = FunctionSpace(mesh, 'Lagrange', 1)
# boundary conditions
bcs = [
set_dirichlet_bc_lim(V,
value=bc_temperature,
axis=axis,
right=right,
tol=tol)
]
# initial condition
u_initial = Constant(temperature)
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# problem formulation
f = Constant(source_value)
u_n, a, L = get_formulation_constant_props(V,
u_initial,
f,
dt,
conductivity,
density,
specific_heat,
var_name=var_name)
u = Function(V, name=var_name)
return u_n, a, L, u, bcs
def get_formulation(V, u_initial, f, dt, var_name='Temperature'):
# interpolate initial condition
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# function spaces
u_h = TrialFunction(V)
v = TestFunction(V)
# formulation
a = u_h * v * dx + dt * dot(grad(u_h), grad(v)) * dx
L = (u_n + dt * f) * v * dx
return u_n, a, L
def get_formulation_constant_props(V, u_initial, f, dt, conductivity, density,
specific_heat, var_name='Temperature'):
# medium properties
k = Constant(conductivity)
rho = Constant(density)
c_p = Constant(specific_heat)
# interpolate initial condition
u_n = Function(V, name=var_name)
u_n.interpolate(u_initial)
# function spaces
u_h = TrialFunction(V)
v = TestFunction(V)
# formulation
param1 = rho * c_p
param2 = k / param1
a = u_h * v * dx + param2 * dt * dot(grad(u_h), grad(v)) * dx
L = (u_n + param1 * dt * f) * v * dx
return u_n, a, L
def interpolate(v, V):
"""Return interpolation of a given function into a given finite
element space.
*Arguments*
v
a :py:class:`Function <dolfin.functions.function.Function>` or
an :py:class:`Expression <dolfin.functions.expression.Expression>`
V
a :py:class:`FunctionSpace (standard, mixed, etc.)
<dolfin.functions.functionspace.FunctionSpace>`
*Example of usage*
.. code-block:: python
v = Expression("sin(pi*x[0])")
V = FunctionSpace(mesh, "Lagrange", 1)
Iv = interpolate(v, V)
"""
# Check arguments
# if not isinstance(V, cpp.functionFunctionSpace):
# cpp.dolfin_error("interpolation.py",
# "compute interpolation",
# "Illegal function space for interpolation, not a FunctionSpace (%s)" % str(v))
# Compute interpolation
Pv = Function(V)
if hasattr(v, "_cpp_object"):
Pv.interpolate(v._cpp_object)
else:
Pv.interpolate(v)
return Pv