A FEniCS Tutorial - FEniCS Project

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7.5 Using a Backend-Specific Solver The linear algebra backend determines the specific data structures that are used in the Matrix and Vector classes. For example, with the PETSc backend, Matrix encapsulates a PETSc matrix storage structure, and Vector encapsulates a PETSc vector storage structure. Sometimes one wants to perform operations directly on (say) the underlying PETSc objects. These can be fetched by A_PETSc = down_cast(A).mat() b_PETSc = down_cast(b).vec() U_PETSc = down_cast(u.vector()).vec() Here, u is a Function, A is a Matrix, and b is a Vector. The same syntax applies if we want to fetch the underlying Epetra, uBLAS, or MTL4 matrices and vectors. Sometimes one wants to implement tailored solution algorithms, using special features of the underlying numerical packages. Here is an example where we create an ML preconditioned Conjugate Gradient solver by programming with Trilinos-specific objects directly. Given a linear system AU = b, represented by a Matrix object A, and two Vector objects U and b in a Python program, the purpose is to set up a solver using the Aztec Conjugate Gradient method fromTrilinos’Azteclibraryandcombinethatsolverwiththealgebraicmultigrid preconditioner ML from the ML library in Trilinos. Since the various parts of Trilinos are mirrored in Python through the PyTrilinos package, we can operate directly on Trilinos-specific objects. try: from PyTrilinos import Epetra, AztecOO, TriUtils, ML except: print ’’’You Need to have PyTrilinos with’ Epetra, AztecOO, TriUtils and ML installed for this demo to run’’’ exit() from dolfin import * if not has_la_backend(’Epetra’): print ’Warning: Dolfin is not compiled with Trilinos’ exit() parameters[’linear_algebra_backend’] = ’Epetra’ # create matrix A and vector b in the usual way # u is a Function # Fetch underlying Epetra objects A_epetra = down_cast(A).mat() b_epetra = down_cast(b).vec() U_epetra = down_cast(u.vector()).vec() # Sets up the parameters for ML using a python dictionary ML_param = {"max levels" : 3, "output" : 10, 86
} "smoother: type" : "ML symmetric Gauss-Seidel", "aggregation: type" : "Uncoupled", "ML validate parameter list" : False # Create the preconditioner prec = ML.MultiLevelPreconditioner(A_epetra, False) prec.SetParameterList(ML_param) prec.ComputePreconditioner() # Create solver and solve system solver = AztecOO.AztecOO(A_epetra, U_epetra, b_epetra) solver.SetPrecOperator(prec) solver.SetAztecOption(AztecOO.AZ_solver, AztecOO.AZ_cg) solver.SetAztecOption(AztecOO.AZ_output, 16) solver.Iterate(MaxIters=1550, Tolerance=1e-5) plot(u) 7.6 Installing FEniCS The FEniCS software components are available for Linux, Windows and Mac OS X platforms. Detailed information on how to get FEniCS running on such machines are available at the fenicsproject.org website. Here are just some quick descriptions and recommendations by the author. To make the installation of FEniCS as painless and reliable as possible, the reader is strongly recommended to use Ubuntu Linux. (Even though Mac users now can get FEniCS by a one-click install, I recommend using Ubuntu on Mac, unless you have high Unix competence and much experience with compiling and linking C++ libraries on Mac OS X.) Any standard PC can easily be equipped with Ubuntu Linux, which may live side by side with either Windows or Mac OS X or another Linux installation. Basically, you download Ubuntu from www.ubuntu.com/getubuntu/download, burn the file on a CD or copy it to a memory stick, reboot the machine with the CD or memory stick, and answer some usually straightforward questions (if necessary). On Windows, Wubi is a tool that automatically installs Ubuntu on the machine. Just give a user name and password for the Ubuntu installation, and Wubi performs the rest. The graphical user interface (GUI) of Ubuntu is quite similar to both Windows 7 and Mac OS X, but to be efficient when doing science with FEniCS this author recommends to run programs in a terminal window and write them in a text editor like Emacs or Vim. You can employ an integrated development environment such as Eclipse, but intensive FEniCS developers and users tend to find terminal windows and plain text editors more user friendly. Instead of making it possible to boot your machine with the Linux Ubuntu operating system, you can run Ubuntu in a separate window in your existing operation system. There are several solutions to chose among: the free VirtualBox and VMWare Player, or the commercial tools VMWare Fusion and Parallels (just search for the names to download the programs). 87
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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().
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 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: Parameters in expression strings mu
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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