Automated Solution of Differential Equations by the Finite by Hans Petter Langtangen (auth.), Anders Logg, Kent-Andre

By Hans Petter Langtangen (auth.), Anders Logg, Kent-Andre Mardal, Garth Wells (eds.)

This publication is an academic written via researchers and builders in the back of the FEniCS undertaking and explores a complicated, expressive method of the advance of mathematical software program. The presentation spans mathematical historical past, software program layout and using FEniCS in purposes. Theoretical features are complemented with machine code that is on hand as free/open resource software program. The e-book starts with a distinct introductory educational for rookies. Following are chapters partially I addressing primary points of the method of automating the production of finite aspect solvers. Chapters partly II tackle the layout and implementation of the FEnicS software program. Chapters partially III current the appliance of FEniCS to quite a lot of functions, together with fluid circulate, strong mechanics, electromagnetics and geophysics.

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7. Convergence estimation. 34) on the unit square. This choice implies f ( x, y) = 2ω 2 π 2 u( x, y ). With ω restricted to an integer it follows that u0 = 0. 0 u_e = Expression("sin(omega*pi*x[0])*sin(omega*pi*x[1])", omega=omega) f = 2*pi**2*omega**2*u_e Chapter 1. 31) can be done by Python code error = (u - u_e)**2*dx E = sqrt(assemble(error)) Here, u_e will be interpolated onto the function space V. This implies that the exact solution used in the integral will vary linearly over the cells, and not as a sine function, if V corresponds to linear Lagrange elements.

It should be easy with the information above to transform a finite element field over a uniform rectangular or box-shaped mesh to the corresponding BoxField object and perform MATLAB-style visualizations of the whole field or the field over planes or along lines through the domain. By the transformation to a regular grid we have some more flexibility than what Viper offers. However, we must remark that comprehensive tools like VisIt, MayaVi2, or ParaView also have the possibility for plotting fields along lines and extracting planes in 3D geometries, though usually with less degree of control compared to Gnuplot, MATLAB, and matplotlib.

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. 6 exemplifies the surfaces arising from the two plotting commands above. easyviz in a terminal window to get a full tutorial. easyviz offers function names like plot and mesh, which clash with plot from dolfin and the mesh variable in our programs. Therefore, we recommend the ev prefix. 30 Chapter 1. 6: Examples of plots created by transforming the finite element field to a field on a uniform, structured 2D grid: (a) a surface plot of the solution; (b) lifted mesh plot of the solution.

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