By Anders Logg

ISBN-10: 3642230997

ISBN-13: 9783642230998

This booklet is an instructional written by way of researchers and builders in 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 purposes. Theoretical facets are complemented with computing device code that's on hand as free/open resource software program. The booklet starts off with a unique introductory instructional for novices. Following are chapters partly I addressing primary points of the method of automating the construction of finite point solvers. Chapters partially II handle the layout and implementation of the FEnicS software program. Chapters partly III current the applying of FEniCS to a variety of functions, together with fluid stream, sturdy mechanics, electromagnetics and geophysics.

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The matrix A and vector b are first assembled without incorporating essential (Dirichlet) boundary conditions. apply(A, b) performs the necessary modifications of the linear system. apply(A, b) There is an alternative function assemble_system, which can assemble the system and take boundary conditions into account in one call: Python code A, b = assemble_system(a, L, bcs) The assemble_system function incorporates the boundary conditions in the element matrices and vectors, prior to assembly. The conditions are also incorporated in a symmetric way to preserve eventual symmetry of the coefficient matrix.

5: Examples of plots created by transforming the finite element field to a field on a uniform, structured 2D grid: (a) contour plot of the solution; (b) curve plot of the exact flux − p∂u/∂x against the corresponding projected numerical flux. 5a. The contour function needs arrays with the x and y coordinates expanded to 2D arrays (in the same way as demanded when making vectorized numpy calculations of arithmetic expressions over all grid points). coorv. The above call to contour creates 5 equally spaced contour lines, and with clabels="on" the contour values can be seen in the plot.

78) Note the important feature in Newton’s method that the previous solution uk replaces u in the formulas when computing the matrix ∂Fi /∂Uj and vector Fi for the linear system in each Newton iteration. Chapter 1. A FEniCS tutorial 43 We now turn to the implementation. To obtain a good initial guess u0 , we can solve a simplified, linear problem, typically with q(u) = 1, which yields the standard Laplace equation Δu0 = 0. 13. vector() solve(A, U_k, b) Here, u_k denotes the solution function for the previous iteration, so that the solution after each Newton iteration is u = u_k + omega*du.

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Automated solution of differential equations by the finite element method : the FEniCS book by Anders Logg

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