A FEniCS Tutorial - FEniCS Project

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Contents 1 Fundamentals 4 1.1 The Poisson equation . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.2 Variational Formulation . . . . . . . . . . . . . . . . . . . . . . . 6 1.3 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.4 Controlling the Solution Process . . . . . . . . . . . . . . . . . . 15 1.5 Linear Variational Problem and Solver Objects . . . . . . . . . . 17 1.6 Examining the Discrete Solution . . . . . . . . . . . . . . . . . . 17 1.7 Solving a Real Physical Problem . . . . . . . . . . . . . . . . . . 20 1.8 Quick Visualization with VTK . . . . . . . . . . . . . . . . . . . 24 1.9 Computing Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 26 1.10 A Variable-Coefficient Poisson Problem . . . . . . . . . . . . . . 28 1.11 Computing Functionals . . . . . . . . . . . . . . . . . . . . . . . 31 1.12 Visualization of Structured Mesh Data . . . . . . . . . . . . . . . 35 1.13 Combining Dirichlet and Neumann Conditions . . . . . . . . . . 38 1.14 Multiple Dirichlet Conditions . . . . . . . . . . . . . . . . . . . . 42 1.15 A Linear Algebra Formulation . . . . . . . . . . . . . . . . . . . . 43 1.16 Parameterizing the Number of Space Dimensions . . . . . . . . . 46 2 Nonlinear Problems 48 2.1 Picard Iteration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.2 A Newton Method at the Algebraic Level . . . . . . . . . . . . . 51 2.3 A Newton Method at the PDE Level . . . . . . . . . . . . . . . . 54 2.4 Solving the Nonlinear Variational Problem Directly . . . . . . . . 55 3 Time-Dependent Problems 58 3.1 A Diffusion Problem and Its Discretization . . . . . . . . . . . . 59 3.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 3.3 Avoiding Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.4 A Physical Example . . . . . . . . . . . . . . . . . . . . . . . . . 66 4 Creating More Complex Domains 71 4.1 Built-In Mesh Generation Tools . . . . . . . . . . . . . . . . . . . 71 4.2 Transforming Mesh Coordinates . . . . . . . . . . . . . . . . . . . 72 5 Handling Domains with Different Materials 73 5.1 Working with Two Subdomains . . . . . . . . . . . . . . . . . . . 74 5.2 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.3 Multiple Neumann, Robin, and Dirichlet Condition . . . . . . . . 77 6 More Examples 81 2
7 Miscellaneous Topics 82 7.1 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 7.2 Overview of Objects and Functions . . . . . . . . . . . . . . . . . 83 7.3 User-Defined Functions . . . . . . . . . . . . . . . . . . . . . . . 84 7.4 Linear Solvers and Preconditioners . . . . . . . . . . . . . . . . . 85 7.5 Using a Backend-Specific Solver . . . . . . . . . . . . . . . . . . . 86 7.6 Installing <strong>FEniCS</strong> . . . . . . . . . . . . . . . . . . . . . . . . . . 87 7.7 Troubleshooting: Compilation Problems . . . . . . . . . . . . . . 88 7.8 Books on the Finite Element Method . . . . . . . . . . . . . . . . 89 7.9 Books on Python . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 7.10 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 8 Bibliography 90 3
Page 1: A FEniCS Tutorial Hans Petter Langt
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 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
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straightforward approach is to defi
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def T_exact(x): a = sqrt(omega*rho*
Page 73 and 74:
Theta = pi/2 a, b = 1, 5.0 nr = 10
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y ✻ u = 1 Ω 1 ∂u ∂n = 0 ∂u
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subdomain1 = Omega1() subdomain1.ma
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The weak form then becomes ∫ ∫
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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
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} "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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