By Anders Logg
This ebook is an instructional written by way of researchers and builders at the back of the FEniCS undertaking and explores a complicated, expressive method of the advance of mathematical software program. The presentation spans mathematical heritage, software program layout and using FEniCS in purposes. Theoretical features are complemented with laptop code that's on hand as free/open resource software program. The booklet starts with a unique introductory educational for rookies. Following are chapters partly I addressing primary points of the method of automating the construction of finite aspect solvers. Chapters partially II deal with the layout and implementation of the FEnicS software program. Chapters partly III current the applying of FEniCS to a variety of purposes, together with fluid circulation, good mechanics, electromagnetics and geophysics.
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Additional info for Automated solution of differential equations by the finite element method : the FEniCS book
16 Parameterizing the number of space dimensions FEniCS makes it is easy to write a unified simulation code that can operate in 1D, 2D, and 3D. We will conveniently make use of this feature in forthcoming examples. py in the stationary/poisson directory and change the mesh construction from UnitSquare(6, 4) to UnitCube(6, 4, 5). Now the domain is the unit cube with 6 × 4 × 5 cells. ). The visualization allows you rotate to the cube and observe the function values as colors on the boundary. The forthcoming material introduces some convenient technicalities such that the same program can run in 1D, 2D, or 3D without any modifications.
UN ) ≡ N ∑ j =1 Ω N q ∑Uφ =1 ∇φj Uj · ∇φˆ i dx = 0, i = 1, . . , N. 70) 42 Chapter 1. A FEniCS tutorial Newton’s method for the system Fi (U1 , . . , Uj ) = 0, i = 1, . . , N can be formulated as N ∂ ∑ ∂Uj Fi (U1k, . . , UNk )δUj = − Fi (U1k, . . , UNk ), j =1 Ujk+1 = Ujk + ωδUj , i = 1, . . , N, j = 1, . . 72) where ω ∈ [0, 1] is a relaxation parameter, and k is an iteration index. An initial guess u0 must be provided to start the algorithm. The original Newton method has ω = 1, but in problems where it is difficult to obtain convergence, so-called under-relaxation with ω < 1 may help.
Matrix processing in Python or MATLAB/Octave is only feasible for small PDE problems since the numpy arrays or matrices in MATLAB file format are dense matrices. DOLFIN also has an interface to the eigensolver package SLEPc, which is a preferred tool for computing the eigenvalues of large, sparse matrices of the type encountered in PDE problems (see demo/la/eigenvalue in the DOLFIN source code tree for a demo). py in the directory stationary/poisson. 14). For small linear systems, the program writes out A and b before and after incorporation of essential boundary conditions and illustrates the difference between assemble and assemble_system.
Automated solution of differential equations by the finite element method : the FEniCS book by Anders Logg