A FEniCS Tutorial - FEniCS Project

weak form has
∫
a(T,v) =
∫
L(v) =
Ω
Ω
(̺cTv +θ∆tκ∇T ·∇v) dx, (84)
(̺cT k−1 v −(1−θ)∆tκ∇T k−1 ·∇v ) dx. (85)
Observe that boundary integrals vanish because of the Neumann boundary conditions.
The size of a 3D box is taken as W × W × D, where D is the depth and
W = D/2isthewidth. Wegivethedegreeofthebasisfunctionsatthecommand
line, then D, and then the divisions of the domain in the various directions. To
make a box, rectangle, or interval of arbitrary (not unit) size, we have the
DOLFIN classes Box, Rectangle, and Interval at our disposal. The mesh and
the function space can be created by the following code:
degree = int(sys.argv[1])
D = float(sys.argv[2])
W = D/2.0
divisions = [int(arg) for arg in sys.argv[3:]]
d = len(divisions) # no of space dimensions
if d == 1:
mesh = Interval(divisions[0], -D, 0)
elif d == 2:
mesh = Rectangle(-W/2, -D, W/2, 0, divisions[0], divisions[1])
elif d == 3:
mesh = Box(-W/2, -W/2, -D, W/2, W/2, 0,
divisions[0], divisions[1], divisions[2])
V = FunctionSpace(mesh, ’Lagrange’, degree)
The Rectangle and Box objects are defined by the coordinates of the ”minimum”
and ”maximum” corners.
Setting Dirichlet conditions at the upper boundary can be done by
T_R = 0; T_A = 1.0; omega = 2*pi
T_0 = Expression(’T_R + T_A*sin(omega*t)’,
T_R=T_R, T_A=T_A, omega=omega, t=0.0)
def surface(x, on_boundary):
return on_boundary and abs(x[d-1]) < 1E-14
bc = DirichletBC(V, T_0, surface)
The κ function can be defined as a constant κ 1 inside the particular rectangular
area with a special soil composition, as indicated in Figure 7. Outside
this area κ is a constant κ 0 . The domain of the rectangular area is taken as
[−W/4,W/4]×[−W/4,W/4]×[−D/2,−D/2+D/4]
in 3D, with [−W/4,W/4]×[−D/2,−D/2+D/4] in 2D and [−D/2,−D/2+D/4]
in 1D. Since we need some testing in the definition of the κ(x) function, the most
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