A FEniCS Tutorial - FEniCS Project

More documents

Recommendations

Info

dt = period/20 # 20 time steps per period theta = 1 T = TrialFunction(V) v = TestFunction(V) f = Constant(0) a = rho*c*T*v*dx + theta*dt*kappa*\ inner(nabla_grad(T), nabla_grad(v))*dx L = (rho*c*T_prev*v + dt*f*v - (1-theta)*dt*kappa*inner(nabla_grad(T), nabla_grad(v)))*dx A = assemble(a) b = None # variable used for memory savings in assemble calls T = Function(V) # unknown at the current time level We could, alternatively, breakaandLup in subexpressions and assemble a mass matrix and stiffness matrix, as exemplified in Section 3.3, to avoid assembly of b at every time level. This modification is straightforward and left as an exercise. The speed-up can be significant in 3D problems. The time loop is very similar to what we have displayed in Section 3.2: T = Function(V) # unknown at the current time level t = dt while t
def T_exact(x): a = sqrt(omega*rho*c/(2*kappa_0)) return T_R + T_A*numpy.exp(a*x)*numpy.sin(omega*t + a*x) The reader is encouraged to play around with the code and test out various parameter sets: 1. T R = 0, T A = 1, κ 0 = κ 1 = 0.2, ̺ = c = 1, ω = 2π 2. T R = 0, T A = 1, κ 0 = 0.2, κ 1 = 0.01, ̺ = c = 1, ω = 2π 3. T R = 0, T A = 1, κ 0 = 0.2, κ 1 = 0.001, ̺ = c = 1, ω = 2π 4. T R = 10 C, T A = 10 C, κ 0 = 2.3 K −1 Ns −1 , κ 1 = 100 K −1 Ns −1 , ̺ = 1500 kg/m 3 , c = 1480 Nmkg −1 K −1 , ω = 2π/24 1/h = 7.27 · 10 −5 1/s, D = 1.5 m 5. As above, but κ 0 = 12.3 K −1 Ns −1 and κ 1 = 10 4 K −1 Ns −1 Data set number 4 is relevant for real temperature variations in the ground (not necessarily the large value of κ 1 ), while data set number 5 exaggerates the effect of a large heat conduction contrast so that it becomes clearly visible in an animation. 4 Creating More Complex Domains Up to now we have been very fond of the unit square as domain, which is an appropriate choice for initial versions of a PDE solver. The strength of the finite element method, however, is its ease of handling domains with complex shapes. This section shows some methods that can be used to create different types of domains and meshes. Domains of complex shape must normally be constructed in separate preprocessor programs. Two relevant preprocessors are Triangle for 2D domains and NETGEN for 3D domains. 4.1 Built-In Mesh Generation Tools DOLFIN has a few tools for creating various types of meshes over domains with simple shape: UnitInterval, UnitSquare, UnitCube, Interval, Rectangle, Box, UnitCircle, and UnitSphere. Some of these names have been briefly met in previous sections. The hopefully self-explanatory code snippet below summarizes typical constructions of meshes with the aid of these tools: # 1D domains mesh = UnitInterval(20) # 20 cells, 21 vertices mesh = Interval(20, -1, 1) # domain [-1,1] # 2D domains (6x10 divisions, 120 cells, 77 vertices) mesh = UnitSquare(6, 10) # ’right’ diagonal is default 71
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 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
Page 69: straightforward approach is to defi
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
show all

A FEniCS Tutorial - FEniCS Project

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?