[21] A. Quarteroni and A. Valli. Numerical Approximation of Partial Differential Equations. Springer Series in Computational Mathematics. Springer, 1994. 92
Index alg newton np.py, 52 assemble, 44, 63 assemble system, 44 assembly of linear systems, 44 assembly, increasing efficiency, 63 attribute (class), 82 automatic differentiation, 57 boundary conditions, 77 boundary specification (class), 75 boundary specification (function), 11 Box, 71 BoxField, 35 CG finite element family, 10 class, 82 compilation problems, 88 contour plot, 36 coordinate stretching, 72 coordinate transformations, 72 d1 d2D.py, 61 d1 p2D.py, 8 d2 d2D.py, 65 d2 p2D.py, 15 d3 p2D.py, 17 d4 p2D.py, 17, 45 d5 p2D.py, 28 d6 p2D.py, 31 degree of freedom, 18 degrees of freedom array, 18, 27 degrees of freedom array (vector field), 27 derivative, 57 dimension-independent code, 46 Dirichlet boundary conditions, 11, 77 DirichletBC, 11 dn1 p2D.py, 41 dn2 p2D.py, 42 dn3 p2D.py, 45 dnr p2D.py, 79 DOLFIN, 82 DOLFIN mesh, 10 down-casting matrices and vectors, 86 energy functional, 31 Epetra, 86 error functional, 31 Expresion, 22 Expression, 11 Expression with parameters, 22 <strong>FEniCS</strong>, 82 finite element specifications, 10 flux functional, 34 functionals, 31 FunctionSpace, 10 Gateaux derivative, 56 heterogeneous media, 73 heterogeneous medium, 68 info function, 16 instance, 82 interpolate, 19 interpolation, 19, 22 Interval, 71 Jacobian, automatic computation, 57 Jacobian, manual computation, 51 KrylovSolver, 45 Lagrange finite element family, 10 linear algebra backend, 15 linear systems (in <strong>FEniCS</strong>), 44 LinearVariationalProblem, 17 LinearVariationalSolver, 17 mat2 p2D.py, 75 membrane1.py, 22 membrane1v.p, 24 membrane2.py, 31 Mesh, 10 mesh transformations, 72 method (class), 82 93
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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
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""" FEniCS program for the deflecti
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Figure 3: Plot of the deflection of
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Figure 4: Example of visualizing th
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Let us continue to use our favorite
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1.11 Computing Functionals After th
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e_Ve.vector()[:] = u_e_Ve.vector().
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It is possible to restrict the inte
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Other functions for visualizing 2D
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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 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: Applied Mathematics (SIAM), Philade
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