A FEniCS Tutorial - FEniCS Project

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surf plot of u 4 4 3.5 3 2.5 2 1.5 1 3.5 3 2.5 2 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 1.5 1 (a) mesh plot of u 4 3.5 3 2.5 2 1.5 1 1 0.8 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 (b) Figure 6: Examples of plots created by transforming the finite element field to a field on a uniform, structured 2D grid: (a) a surface plot of the solution; (b) lifted mesh plot of the solution. 40
accordingly: f = −6, { −4, y = 1 g = 0, y = 0 u 0 = 1+x 2 +2y 2 . For ease of programming we may introduce a g function defined over the whole of Ω such that g takes on the right values at y = 0 and y = 1. One possible extension is g(x,y) = −4y. The first task is to derive the variational problem. This time we cannot omit the boundary term arising from the integration by parts, because v is only zero on Γ D . We have ∫ ∫ ∫ − (∇ 2 ∂u u)vdx = ∇u·∇vdx− Ω Ω ∂Ω ∂n vds, and since v = 0 on Γ D , ∫ − ∂Ω ∫ ∂u ∂n vds = − Γ N ∂u ∂n vds = ∫ Γ N gvds, by applying the boundary condition on Γ N . The resulting weak form reads ∫ ∫ ∫ ∇u·∇vdx+ gvds = fvdx. (33) Ω Γ N Expressing this equation in the standard notation a(u,v) = L(v) is straightforward with ∫ a(u,v) = ∇u·∇vdx, (34) Ω ∫ ∫ L(v) = fvdx− gvds. (35) Γ N Ω How does the Neumann condition impact the implementation? Starting with any of the previous files d*_p2D.py, say d4_p2D.py, we realize that the statements remain almost the same. Only two adjustments are necessary: • The function describing the boundary where Dirichlet conditions apply must be modified. • The new boundary term must be added to the expression in L. Step 1 can be coded as Ω 41
Page 1 and 2: A FEniCS Tutorial Hans Petter Langt
Page 3 and 4: 7 Miscellaneous Topics 82 7.1 Gloss
Page 5 and 6: that changing the PDE and boundary
Page 7 and 8: The proper statement of our variati
Page 9 and 10: """ FEniCS tutorial demo program: P
Page 11 and 12: polynomial approximations over each
Page 13 and 14: 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: 1 0.8 Contour plot of u 4 3.5 3 2.5
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 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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A FEniCS Tutorial - FEniCS Project

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