A FEniCS Tutorial - FEniCS Project

def T_exact(x):
a = sqrt(omega*rho*c/(2*kappa_0))
return T_R + T_A*numpy.exp(a*x)*numpy.sin(omega*t + a*x)
The reader is encouraged to play around with the code and test out various
parameter sets:
1. T R = 0, T A = 1, κ 0 = κ 1 = 0.2, ̺ = c = 1, ω = 2π
2. T R = 0, T A = 1, κ 0 = 0.2, κ 1 = 0.01, ̺ = c = 1, ω = 2π
3. T R = 0, T A = 1, κ 0 = 0.2, κ 1 = 0.001, ̺ = c = 1, ω = 2π
4. T R = 10 C, T A = 10 C, κ 0 = 2.3 K −1 Ns −1 , κ 1 = 100 K −1 Ns −1 , ̺ =
1500 kg/m 3 , c = 1480 Nmkg −1 K −1 , ω = 2π/24 1/h = 7.27 · 10 −5 1/s,
D = 1.5 m
5. As above, but κ 0 = 12.3 K −1 Ns −1 and κ 1 = 10 4 K −1 Ns −1
Data set number 4 is relevant for real temperature variations in the ground
(not necessarily the large value of κ 1 ), while data set number 5 exaggerates the
effect of a large heat conduction contrast so that it becomes clearly visible in
an animation.
4 Creating More Complex Domains
Up to now we have been very fond of the unit square as domain, which is an
appropriate choice for initial versions of a PDE solver. The strength of the finite
element method, however, is its ease of handling domains with complex shapes.
This section shows some methods that can be used to create different types of
domains and meshes.
Domains of complex shape must normally be constructed in separate preprocessor
programs. Two relevant preprocessors are Triangle for 2D domains
and NETGEN for 3D domains.
4.1 Built-In Mesh Generation Tools
DOLFIN has a few tools for creating various types of meshes over domains with
simple shape: UnitInterval, UnitSquare, UnitCube, Interval, Rectangle,
Box, UnitCircle, and UnitSphere. Some of these names have been briefly
met in previous sections. The hopefully self-explanatory code snippet below
summarizes typical constructions of meshes with the aid of these tools:
# 1D domains
mesh = UnitInterval(20) # 20 cells, 21 vertices
mesh = Interval(20, -1, 1) # domain [-1,1]
# 2D domains (6x10 divisions, 120 cells, 77 vertices)
mesh = UnitSquare(6, 10) # ’right’ diagonal is default
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