This project is a small example of coding the finite element method. The first coded example is to determine the modal frequencies of a Timoshenko cantilever. The second solves Poisson's equation which models many physical phenomena.
The finite element method for a Timoshenko beam derives a mass and stiffness matrix which governs the behaviour of the beam. With no applied force there is a non-trival solution found from an eigenvalue problem. The coded example determines the first 5 modal frequencies of the beam from the eigenvalue problem and compares them with the theoretical modal frequencies of an Euler-Bernoulli beam.
Poisson's equation results from modeling in many physcial domains including
heat conduction and electrostatics. The example coded in poisson_fem.py is
used to determine the temperature field T(x,y) resulting from a set of
heat sources Q(x,y) on a rectangular structure with varying heat conductivity
k(x,y).
The coded example determines the temperature field over a rectangular domain due to presence of heat sources in high conductivity areas seperated by insulating areas. The solution is shown below.
Three different approaches to assembling are explored. By far the worst is creating a block diagonal sparse matrix and then applying constraints. The standard iterative method has good performance. However, an approach building the contents of the sparse matrix using vector operations was the fastest.
Matlab Codes for Finite Element Analysis; A. J. M. Ferreira; 2009; Springer;
Effecient topology optimization in MATLAB using 88 lines of code; Erik Andreassen, Anders Clausen, Mattias Schevenels, Boyan S. Lazarov, Ole Sigmund; Structural and Multidisciplinary Optimization; January 2011; Volume 43, Issue 1, pp 1–16;
The Finite Element Method for Electromagnetic Modelling; Gerard Meunier; 2008; Wiley;