A FEniCS Tutorial - FEniCS Project

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Figure 8: Hollow cylinder generated by mapping a rectangular mesh, stretched toward the left side. 5.1 Working with Two Subdomains Suppose we want to solve ∇·[k(x,y)∇u(x,y)] = 0, (88) in a domain Ω consisting of two subdomains where k takes on a different value in each subdomain. For simplicity, yet without loss of generality, we choose for the current implementation the domain Ω = [0,1]×[0,1] and divide it into two equal subdomains, as depicted in Figure 5.1, Ω 0 = [0,1]×[0,1/2], Ω 1 = [0,1]×(1/2,1]. We define k(x,y) = k 0 in Ω 0 and k(x,y) = k 1 in Ω 1 , where k 0 > 0 and k 1 > 0 are given constants. As boundary conditions, we choose u = 0 at y = 0, u = 1 at y = 1, and ∂u/∂n = 0 at x = 0 and x = 1. One can show that the exact solution is now given by u(x,y) = { 2yk1 k 0+k 1 , y ≤ 1/2 (2y−1)k 0+k 1 k 0+k 1 , y ≥ 1/2 (89) As long as the element boundaries coincide with the internal boundary y = 1/2, this piecewise linear solution should be exactly recovered by Lagrange elements of any degree. We use this property to verify the implementation. Physically, the present problem may correspond to heat conduction, where the heat conduction in Ω 1 is ten times more efficient than in Ω 0 . An alternative 74
y ✻ u = 1 Ω 1 ∂u ∂n = 0 ∂u ∂n = 0 Ω 0 u = 0 ✲ x Figure9: SketchofaPoissonproblemwithavariablecoefficientthatisconstant in each of the two subdomains Ω 0 and Ω 1 . interpretation is flow in porous media with two geological layers, where the layers’ ability to transport the fluid differs by a factor of 10. 5.2 Implementation The new functionality in this subsection regards how to define the subdomains Ω 0 and Ω 1 . For this purpose we need to use subclasses of class SubDomain, not only plain functions as we have used so far for specifying boundaries. Consider the boundary function def boundary(x, on_boundary): tol = 1E-14 return on_boundary and abs(x[0]) < tol fordefiningtheboundaryx = 0. Insteadofusingsuchastand-alonefunction,we can create an instance (or object) of a subclass of SubDomain, which implements the inside method as an alternative to the boundary function: class Boundary(SubDomain): def inside(self, x, on_boundary): tol = 1E-14 return on_boundary and abs(x[0]) < tol boundary = Boundary() bc = DirichletBC(V, Constant(0), boundary) A word about computer science terminology may be used here: The term instance meansaPythonobjectofaparticulartype(suchasSubDomain,Function 75
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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
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solution during program development
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The demo program d2_p2D.py in the s
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All mesh objects are of type Mesh s
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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 and 66: if we identify the matrix M with en
Page 67 and 68: y T 0 (t) = T R + T A sin(ωt) x D
Page 69 and 70: straightforward approach is to defi
Page 71 and 72: def T_exact(x): a = sqrt(omega*rho*
Page 73: Theta = pi/2 a, b = 1, 5.0 nr = 10
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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A FEniCS Tutorial - FEniCS Project

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