A FEniCS Tutorial - FEniCS Project

3.4 A Physical Example
With the basic programming techniques for time-dependent problems from Sections
3.3-3.2 we are ready to attack more physically realistic examples. The
next example concerns the question: How is the temperature in the ground
affected by day and night variations at the earth’s surface? We consider some
box-shaped domain Ω in d dimensions with coordinates x 0 ,...,x d−1 (the problem
is meaningful in 1D, 2D, and 3D). At the top of the domain, x d−1 = 0, we
have an oscillating temperature
T 0 (t) = T R +T A sin(ωt),
where T R is some reference temperature, T A is the amplitude of the temperature
variations at the surface, and ω is the frequency of the temperature oscillations.
At all other boundaries we assume that the temperature does not change anymore
when we move away from the boundary, i.e., the normal derivative is zero.
Initially, the temperature can be taken as T R everywhere. The heat conductivity
properties of the soil in the ground may vary with space so we introduce
a variable coefficient κ reflecting this property. Figure 7 shows a sketch of the
problem, with a small region where the heat conductivity is much lower.
The initial-boundary value problem for this problem reads
̺c ∂T
∂t = ∇·(κ∇T) in Ω×(0,t stop ], (80)
T = T 0 (t) on Γ 0 , (81)
∂T
∂n = 0 on ∂Ω\Γ 0, (82)
T = T R at t = 0. (83)
Here, ̺ is the density of the soil, c is the heat capacity, κ is the thermal conductivity
(heat conduction coefficient) in the soil, and Γ 0 is the surface boundary
x d−1 = 0.
We use a θ-scheme in time, i.e., the evolution equation ∂P/∂t = Q(t) is
discretized as
P k −P k−1
= θQ k +(1−θ)Q k−1 ,
∆t
where θ ∈ [0,1] is a weighting factor: θ = 1 corresponds to the backward
difference scheme, θ = 1/2 to the Crank-Nicolson scheme, and θ = 0 to a
forward difference scheme. The θ-scheme applied to our PDE results in
̺c Tk −T k−1
∆t
= θ∇·(κ∇T k) +(1−θ)∇·(k∇T k−1) .
Bringing this time-discrete PDE into weak form follows the technique shown
many times earlier in this tutorial. In the standard notation a(T,v) = L(v) the
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