By Anders Logg
This ebook is an educational written by means of researchers and builders at the back of the FEniCS venture and explores a sophisticated, expressive method of the advance of mathematical software program. The presentation spans mathematical heritage, software program layout and using FEniCS in functions. Theoretical facets are complemented with machine code that's on hand as free/open resource software program. The e-book starts with a unique introductory educational for newcomers. Following are chapters partly I addressing primary points of the method of automating the construction of finite point solvers. Chapters partially II tackle the layout and implementation of the FEnicS software program. Chapters partly III current the applying of FEniCS to quite a lot of functions, together with fluid circulation, good mechanics, electromagnetics and geophysics.
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Extra info for Automated solution of differential equations by the finite element method : the FEniCS book
6): Python code f = interpolate(f, V) Calling plot(f) will produce a plot of f . Note that the assignment to f destroys the previous Expression object f, so if it is of interest to still have access to this object another name must be used for the Function object returned by interpolate. We need some evidence that the program works, and to this end we may use the analytical solution listed above for the case σ → ∞. In scaled coordinates the solution reads w ( x, y) = 1 − x2 − y2 . Practical values for an infinite σ may be 50 or larger, and in such cases the program will report the maximum deviation between the computed w and the (approximate) exact we .
We could also interpolate/project onto a finer mesh in the higher-degree case. Such transformations to linear finite element fields are very often needed when calling up plotting packages or data analysis tools. ufl_element() method returns an object holding the element type, and this object has a method degree() for returning the element degree as an integer. The parameters nx and ny are the number of divisions in each space direction that were used when calling UnitSquare to make the mesh object.
We then only need to take care of Dirichlet conditions at two sides: Python code tol = 1E-14 # tolerance for coordinate comparisons def Dirichlet_boundary0(x, on_boundary): return on_boundary and abs(x) < tol def Dirichlet_boundary1(x, on_boundary): return on_boundary and abs(x - 1) < tol bc0 = DirichletBC(V, Constant(0), Dirichlet_boundary0) bc1 = DirichletBC(V, Constant(1), Dirichlet_boundary1) bcs = [bc0, bc1] Note that this code is independent of the number of space dimensions. py (Poisson problem in any–D).
Automated solution of differential equations by the finite element method : the FEniCS book by Anders Logg