2012-04-12

Step by Step Methods

Today's class was devoted to the numerical methods for the integration of the equation of motion in the time domain, that we collectively name Step by Step Methods.

We introduced the concept of the state of a linear or non linear system, and the idea of computing the state of the system at the end of a short time step as a mean to compute, step by step, the response of a SDOF system in an approximate manner.

The specific qualities and defects of SbS methods were discussed at some length, especially the cause of errors and the problem of numerical instability.

First of all, we discussed the Central Differences Method, an unstable method that appeals the numerical analyst with its semplicity. The method gives also a taste of the structure of step by step solutions, where the end state is expressed in terms of the initial state and the values of the loadings at the extremes of the step.

The central differences is an explicit method, next we introduced implicit methods, that can be collected under the name of Newmark's Beta Methods, and we studied in particular the Constant and the Linear Acceleration Methods, the Constant Acceleration being remarkable as it is a stable method of integration.

We ended introducing the modified Newton-Raphson method for the solution of non linear problems, with the discussion of an elasto-plastic system.

The homework assigned was about an elastoplastic system subjected to an impulsive loading.

Material available for this class:
  • the slides i used for my lesson
  • a ready to print version of the slides,

  • also something about your homework
  • a Python script to do analytical integration of the EOM for the homework example,
  • the short paper, that was shown in class, that documents the above script
  • the notebooks that i used in class, both the executables and a pdf version that can be read onscreen or printed without the software the is needed for the live notebooks
    • exact response for piecewise linear excitation, notebook and PDF,
    • constant acceleration algorithm, notebook and PDF,
    • linear acceleration algorithm, notebook and PDF,
    • exact integration of the EOM for the EP system of your exercise, notebook and PDF.
    In case you're interested in notebooks, they are an extension to this very nice interactive python shell, while this is the central source of info about the python programming language. Many people are confused by the number of python versions (even different major versions) that are available. If you're on a Mac or a Linux system, odds are that python is already installed on your system, if you're on Windows odds are reversed... In any case you should probably install a bunch of libraries (numpy, scipy, matplotlib, ipython) so maybe your position as a Windows'user is better, as you can install the heartily recommended Enthought Python Distribution that comprises everything I mentioned and many others scientific and engineering modules and is free (gratis) for students.
Giacomo Boffi