A FEniCS Tutorial - FEniCS Project

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alpha = 3; beta = 1.2 u0 = Expression(’1 + x[0]*x[0] + alpha*x[1]*x[1] + beta*t’, {’alpha’: alpha, ’beta’: beta}) u0.t = 0 Thisfunctionexpressionhasthecomponentsofxasindependentvariables, while alpha, beta, and t are parameters. The parameters can either be set through a dictionaryatconstructiontime,asdemonstratedforalphaandbeta,oranytime through attributes in the function object, as shown for the t parameter. The essential boundary conditions, along the whole boundary in this case, are set in the usual way, def boundary(x, on_boundary): # define the Dirichlet boundary return on_boundary bc = DirichletBC(V, u0, boundary) We shall use u for the unknown u at the new time level and u_1 for u at the previous time level. The initial value of u_1, implied by the initial condition on u, can be computed by either projecting or interpolating I. The I(x,y) function is available in the program through u0, as long as u0.t is zero. We can then do u_1 = interpolate(u0, V) # or u_1 = project(u0, V) Note that we could, as an equivalent alternative to using project, define a 0 and L 0 as we did in Section 1.9 and form the associated variational problem. To actually recover the exact solution (76) to machine precision, it is important not to compute the discrete initial condition by projecting I, but by interpolating I so that the nodal values are exact at t = 0 (projection results in approximative values at the nodes). The definition of a and L goes as follows: dt = 0.3 # time step u = TrialFunction(V) v = TestFunction(V) f = Constant(beta - 2 - 2*alpha) a = u*v*dx + dt*inner(nabla_grad(u), nabla_grad(v))*dx L = (u_1 + dt*f)*v*dx A = assemble(a) # assemble only once, before the time stepping Finally, we perform the time stepping in a loop: u = Function(V) T = 2 t = dt # the unknown at a new time level # total simulation time 62
while t 1, but for the values of N typically used on smaller computers, the assembly step may still represent a considerable part of the total work at each time level. Avoiding repeated assembly can therefore contribute to a significant speed-up of a finite element code in time-dependent problems. 63
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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
Page 7 and 8:
The proper statement of our variati
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""" 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 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: • if u 1 is to be computed by pro
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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