A FEniCS Tutorial - FEniCS Project

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FEniCS is quite limited. And Python is an easy-to-learn language that you certainly will love and use far beyond FEniCS programming.) Section 7.9 lists some relevant Python books. The listed FEniCS program defines a finite element mesh, the discrete function spaces V and ˆV correspondingto this mesh and the element type, boundary conditions for u (the function u 0 ), a(u,v), and L(v). Thereafter, the unknown trial function u is computed. Then we can investigate u visually or analyze the computed values. The first line in the program, from dolfin import * imports the key classes UnitSquare, FunctionSpace, Function, and so forth, from the DOLFIN library. All FEniCS programs for solving PDEs by the finite element method normally start with this line. DOLFIN is a software library with efficient and convenient C++ classes for finite element computing, and dolfin is a Python package providing access to this C++ library from Python programs. You can think of FEniCS as an umbrella, or project name, for a set of computational components, where DOLFIN is one important component for writing finite element programs. The from dolfin import * statement imports other components too, but newcomers to FEniCS programming do not need to care about this. The statement mesh = UnitSquare(6, 4) defines a uniform finite element mesh over the unit square [0,1] × [0,1]. The mesh consists of cells, which are triangles with straight sides. The parameters 6 and 4 tell that the square is first divided into 6 × 4 rectangles, and then each rectangle is divided into two triangles. The total number of triangles then becomes 48. The total number of vertices in this mesh is 7·5 = 35. DOLFIN offers some classes for creating meshes over very simple geometries. For domains of more complicated shape one needs to use a separate preprocessor program to create the mesh. The FEniCS program will then read the mesh from file. Having a mesh, we can define a discrete function space V over this mesh: V = FunctionSpace(mesh, ’Lagrange’, 1) The second argument reflects the type of element, while the third argument is the degree of the basis functions on the element. The type of element is here ”Lagrange”, implying thestandard Lagrangefamily of elements. (Some FEniCS programs use ’CG’, for Continuous Galerkin, as a synonym for ’Lagrange’.) With degree 1, we simply get the standard linear Lagrange element, which is a triangle with nodes at the three vertices. Some finite element practitioners refer to this element as the ”linear triangle”. The computed u will be continuous and linearly varying in x and y over each cell in the mesh. Higher-degree 10
polynomial approximations over each cell are trivially obtained by increasing the third parameter in FunctionSpace. Changing the second parameter to ’DG’ creates a function space for discontinuous Galerkin methods. In mathematics, we distinguish between the trial and test spaces V and ˆV. The only difference in the present problem is the boundary conditions. In FEniCS we do not specify the boundary conditions as part of the function space, so it is sufficient to work with one common space V for the and trial and test functions in the program: u = TrialFunction(V) v = TestFunction(V) The next step is to specify the boundary condition: u = u 0 on ∂Ω. This is done by bc = DirichletBC(V, u0, u0_boundary) where u0 is an instance holding the u 0 values, and u0_boundary is a function (or object) describing whether a point lies on the boundary where u is specified. Boundary conditions of the type u = u 0 are known as Dirichlet conditions, and also as essential boundary conditions in a finite element context. Naturally, thenameoftheDOLFINclassholdingtheinformationaboutDirichletboundary conditions is DirichletBC. The u0 variable refers to an Expression object, which is used to represent a mathematical function. The typical construction is u0 = Expression(formula) whereformula is a string containing the mathematical expression. This formula is written with C++ syntax (the expression is automatically turned into an efficient, compiled C++ function, see Section 7.3 for details on the syntax). The independent variables in the function expression are supposed to be available as a point vector x, where the first element x[0] corresponds to the x coordinate, the second element x[1] to the y coordinate, and (in a three-dimensional problem) x[2] to the z coordinate. With our choice of u 0 (x,y) = 1+x 2 +2y 2 , the formula string must be written as 1 + x[0]*x[0] + 2*x[1]*x[1]: u0 = Expression(’1 + x[0]*x[0] + 2*x[1]*x[1]’) Theinformationaboutwheretoapplytheu0functionasboundarycondition is coded in a function u0_boundary: def u0_boundary(x, on_boundary): return on_boundary A function like u0_boundary for marking the boundary must return a boolean value: True if the given point x lies on the Dirichlet boundary and False 11
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 and 8: The proper statement of our variati
Page 9: """ FEniCS tutorial demo program: P
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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