A FEniCS Tutorial - FEniCS Project

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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 66
y T 0 (t) = T R + T A sin(ωt) x D κ ≪ κ 0 ∂u/∂n = 0 ∂u/∂n = 0 ̺, c, κ 0 ∂u/∂n = 0 W Figure 7: Sketch of a (2D) problem involving heating and cooling of the ground due to an oscillating surface temperature. 67
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A FEniCS Tutorial Hans Petter Langt
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7 Miscellaneous Topics 82 7.1 Gloss
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that changing the PDE and boundary
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The proper statement of our variati
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""" FEniCS tutorial demo program: P
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polynomial approximations over each
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Instead of nabla_grad we could also
Page 15 and 16: solution during program development
Page 17 and 18: The demo program d2_p2D.py in the s
Page 19 and 20: All mesh objects are of type Mesh s
Page 21 and 22: Here, T is the tension in the membr
Page 23 and 24: """ FEniCS program for the deflecti
Page 25 and 26: Figure 3: Plot of the deflection of
Page 27 and 28: Figure 4: Example of visualizing th
Page 29 and 30: Let us continue to use our favorite
Page 31 and 32: 1.11 Computing Functionals After th
Page 33 and 34: e_Ve.vector()[:] = u_e_Ve.vector().
Page 35 and 36: It is possible to restrict the inte
Page 37 and 38: Other functions for visualizing 2D
Page 39 and 40: 1 0.8 Contour plot of u 4 3.5 3 2.5
Page 41 and 42: accordingly: f = −6, { −4, y =
Page 43 and 44: Here, Γ 0 is the boundary x = 0, w
Page 45 and 46: The object A is of type Matrix, whi
Page 47 and 48: with exact solution u(x) = x 2 . Ou
Page 49 and 50: The variational formulation of our
Page 51 and 52: iter += 1 solve(a == L, u, bcs) dif
Page 53 and 54: tol = 1E-14 def left_boundary(x, on
Page 55 and 56: We may collect the terms with the u
Page 57 and 58: as ǫ → 0. This last expression i
Page 59 and 60: 3.1 A Diffusion Problem and Its Dis
Page 61 and 62: • if u 1 is to be computed by pro
Page 63 and 64: while t 1, but for the values of N
Page 65: if we identify the matrix M with en
Page 69 and 70: straightforward approach is to defi
Page 71 and 72: def T_exact(x): a = sqrt(omega*rho*
Page 73 and 74: Theta = pi/2 a, b = 1, 5.0 nr = 10
Page 75 and 76: y ✻ u = 1 Ω 1 ∂u ∂n = 0 ∂u
Page 77 and 78: subdomain1 = Omega1() subdomain1.ma
Page 79 and 80: The weak form then becomes ∫ ∫
Page 81 and 82: A = assemble(a, exterior_facet_doma
Page 83 and 84: 7.2 Overview of Objects and Functio
Page 85 and 86: Parameters in expression strings mu
Page 87 and 88: } "smoother: type" : "ML symmetric
Page 89 and 90: 2. forgetting that the spatial coor
Page 91 and 92: Applied Mathematics (SIAM), Philade
Page 93 and 94: Index alg newton np.py, 52 assemble
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