A FEniCS Tutorial - FEniCS Project

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the abstract mathematical formulation that FEniCS builds upon. Those who have a weak background in differential equations in general should consult a more fundamental book, and Eriksson et al. [6] is a very good choice. On the otherhand, FEniCSuserswithastrongbackgroundinmathematicsandinterest in the mathematical properties of the finite element method, will appreciate the texts by Brenner and Scott [3], Braess [2], Ern and Guermond [7], Quarteroni and Valli [21], or Ciarlet [4]. 7.9 Books on Python Two very popular introductory books on Python are ”Learning Python” by Lutz [16] and ”Practical Python” by Hetland [9]. More advanced and comprehensive books include ”Programming Python” by Lutz [15], and ”Python Cookbook” [18] and ”Python in a Nutshell” [17] by Martelli. The web page http://wiki.python.org/moin/PythonBookslistsnumerousadditionalbooks. Very few texts teach Python in a mathematical and numerical context, but the references [12, 13, 11] are exceptions. 7.10 Acknowledgments The author is very thankful to Johan Hake, Anders Logg, Kent-Andre Mardal, and Kristian Valen-Sendstad for promptly answering all my questions about FEniCS functionality and for implementing all my requests. I will in particular thank Professor Douglas Arnold for very valuable feedback on the text. Øystein Sørensen pointed out a lot of typos and contributed with many helpful comments. Many errors and typos were also reported by Mauricio Angeles, Ida Drøsdal, Hans Ekkehard Plesser, and Marie Rognes. Ekkehard Ellmann as well as two anonymous reviewers provided a series of suggestions and improvements. 8 Bibliography References [1] W. B. Bickford. A First Course in the Finite Element Method. Irwin, 2nd edition, 1994. [2] Dietrich Braess. Finite elements. Cambridge University Press, Cambridge, third edition, 2007. [3] Susanne C. Brenner and L. Ridgway Scott. The mathematical theory of finite element methods, volume15ofTexts in Applied Mathematics. Springer, New York, third edition, 2008. [4] Philippe G. Ciarlet. The finite element method for elliptic problems, volume 40 of Classics in Applied Mathematics. Society for Industrial and 90
Applied Mathematics (SIAM), Philadelphia, PA, 2002. Reprint of the 1978 original [North-Holland, Amsterdam; MR0520174 (58 #25001)]. [5] J.DoneaandA.Huerta. Finite Element Methods for Flow Problems. Wiley, 2003. [6] K. Eriksson, D. Estep, P. Hansbo, and C. Johnson. Computational Differential Equations. Cambridge University Press, 1996. [7] A. Ern and J.-L. Guermond. Theory and Practice of Finite Elements. Springer, 2004. [8] M. Gockenbach. Understanding and Implementing the Finite Element Method. SIAM, 2006. [9] M. L. Hetland. Practical Python. APress, 2002. [10] T. J. R. Hughes. The Finite Element Method: Linear Static and Dynamic Finite Element Analysis. Prentice-Hall, 1987. [11] J. Kiusalaas. Numerical Methods in Engineering with Python. Cambridge, 2005. [12] H. P. Langtangen. Python Scripting for Computational Science. Springer, third edition, 2009. [13] H. P. Langtangen. A Primer on Scientific Programming with Python. Texts in Computational Science and Engineering, vol 6. Springer, second edition, 2011. [14] A. Logg, K.-A. Mardal, and G. N. Wells, editors. Automated Solution of Partial Differential Equations by the Finite Element Method. Springer, 2011. [15] M. Lutz. Programming Python. O’Reilly, third edition, 2006. [16] M. Lutz. Learning Python. O’Reilly, third edition, 2007. [17] A. Martelli. Python in a Nutshell. O’Reilly, second edition, 2006. [18] A. Martelli and D. Ascher. Python Cookbook. O’Reilly, second edition, 2005. [19] M. Mortensen, H. P. Langtangen, and J. Myre. cbc.rans – a new flexible, programmable software framework for computational fluid dynamics. In H. I. Andersson and B. Skallerud, editors, Sixth National Conference on Computational Mechanics (MekIT’11). Tapir, 2011. [20] M. Mortensen, H. P. Langtangen, and G. N. Wells. A FEniCS-based programming framework for modeling turbulent flow by the Reynoldsaveraged Navier-Stokes equations. Advances in Water Resources, 2011. DOI: 10.1016/j.advwatres.2011.02.013. 91
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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
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 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: 2. forgetting that the spatial coor
Page 93 and 94: Index alg newton np.py, 52 assemble
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A FEniCS Tutorial - FEniCS Project

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