A FEniCS Tutorial - FEniCS Project

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x, flux_x_exact, ’b-’, legend=(’numerical (projected) flux’, ’exact flux’), title=’Flux in x-direction (at y=%g)’ % y_fixed, savefig=’flux.eps’) AsseenfromFigure1.12b, thenumericalfluxisaccurateexceptintheboundary elements. Thevisualizationconstructionsshownaboveandusedtogeneratethefigures are found in the program vcp2D.py in the stationary/poisson directory. It should be easy with the information above to transform a finite element field over a uniform rectangular or box-shaped mesh to the corresponding BoxField object and perform MATLAB-style visualizations of the whole field or the field over planes or along lines through the domain. By the transformation to a regular grid we have some more flexibility than what Viper offers. However, we remark that comprehensive tools like VisIt, MayaVi2, or ParaView also have the possibility for plotting fields along lines and extracting planes in 3D geometries, though usually with less degree of control compared to Gnuplot, MATLAB, and Matplotlib. For example, in investigations of numerical accuracy or numerical artifacts one is often interested in studying curve plots where only the nodal values sampled. This is straightforward with a structured mesh data structure, but more difficult in visualization packages utilizing unstructured grids, as hitting exactly then nodes when sampling a function along a line through the grid might be non-trivial. 1.13 Combining Dirichlet and Neumann Conditions Let us make a slight extension of our two-dimensional Poisson problem from Section 1.1 and add a Neumann boundary condition. The domain is still the unit square, but now we set the Dirichlet condition u = u 0 at the left and right sides, x = 0 and x = 1, while the Neumann condition − ∂u ∂n = g is applied to the remaining sides y = 0 and y = 1. The Neumann condition is alsoknownasanatural boundary condition (incontrasttoanessentialboundary condition). Let Γ D and Γ N denote the parts of ∂Ω where the Dirichlet and Neumann conditions apply, respectively. The complete boundary-value problem can be written as −∇ 2 u = f in Ω, (30) u = u 0 on Γ D , (31) − ∂u ∂n = g on Γ N . (32) Again we choose u = 1+x 2 +2y 2 as the exact solution and adjust f, g, and u 0 38
1 0.8 Contour plot of u 4 3.5 3 2.5 2 1.5 0.6 0.4 0.2 0 0 0.2 0.4 0.6 0.8 1 (a) 0 Flux in x-direction (at y=0.5) numerical (projected) flux exact flux -0.5 -1 -1.5 -2 -2.5 -3 0 0.2 0.4 0.6 0.8 1 (b) Figure 5: Examples of plots created by transforming the finite element field to a field on a uniform, structured 2D grid: (a) contour plot of the solution; (b) curve plot of the exact flux −p∂u/∂x against the corresponding projected numerical flux. 39
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: Other functions for visualizing 2D
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 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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