A FEniCS Tutorial - FEniCS Project

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In the present problem we have that ∫ a(u,v) = ∫ L(v) = Ω Ω ∇u·∇vdx, (10) fvdx. (11) From the mathematics literature, a(u,v) is known as a bilinear form and L(v) as a linear form. We shall in every linear problem we solve identify the terms with the unknown u and collect them in a(u,v), and similarly collect all terms with only known functions in L(v). The formulas for a and L are then coded directly in the program. Tosummarize, beforemaking aFEniCS program for solving aPDE, wemust first perform two steps: • Turn the PDE problem into a discrete variational problem: find u ∈ V such that a(u,v) = L(v) ∀v ∈ ˆV. • Specify the choice of spaces (V and ˆV), which means specifying the mesh and type of finite elements. 1.3 Implementation The test problem so far has a general domain Ω and general functions u 0 and f. For our first implementation we must decide on specific choices of Ω, u 0 , and f. It will be wise to construct a specific problem where we can easily check that the computed solution is correct. Let us start with specifying an exact solution u e (x,y) = 1+x 2 +2y 2 (12) on some 2D domain. By inserting (12) in our Poisson problem, we find that u e (x,y) is a solution if f(x,y) = −6, u 0 (x,y) = u e (x,y) = 1+x 2 +2y 2 , regardlessoftheshapeofthedomain. Wechoosehere, forsimplicity, thedomain to be the unit square, Ω = [0,1]×[0,1]. The reason for specifying the solution (12) is that the finite element method, with a rectangular domain uniformly partitioned into linear triangular elements, will exactly reproduce a second-order polynomial at the vertices of the cells, regardless of the size of the elements. This property allows us to verify the implementation by comparing the computed solution, called u in this document (except when setting up the PDE problem), with the exact solution, denoted by u e : u should equal u to machine precision at the nodes. Test problems with this property will be frequently constructed throughout this tutorial. A FEniCS program for solving the Poisson equation in 2D with the given choices of u 0 , f, and Ω may look as follows: 8
""" FEniCS tutorial demo program: Poisson equation with Dirichlet conditions. Simplest example of computation and visualization with FEniCS. -Laplace(u) = f on the unit square. u = u0 on the boundary. u0 = u = 1 + x^2 + 2y^2, f = -6. """ from dolfin import * # Create mesh and define function space mesh = UnitSquare(6, 4) #mesh = UnitCube(6, 4, 5) V = FunctionSpace(mesh, ’Lagrange’, 1) # Define boundary conditions u0 = Expression(’1 + x[0]*x[0] + 2*x[1]*x[1]’) def u0_boundary(x, on_boundary): return on_boundary bc = DirichletBC(V, u0, u0_boundary) # Define variational problem u = TrialFunction(V) v = TestFunction(V) f = Constant(-6.0) a = inner(nabla_grad(u), nabla_grad(v))*dx L = f*v*dx # Compute solution u = Function(V) solve(a == L, u, bc) # Plot solution and mesh plot(u) plot(mesh) # Dump solution to file in VTK format file = File(’poisson.pvd’) file
Page 1 and 2: A FEniCS Tutorial Hans Petter Langt
Page 3 and 4: 7 Miscellaneous Topics 82 7.1 Gloss
Page 5 and 6: that changing the PDE and boundary
Page 7: The proper statement of our variati
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
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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