FEM resources

This is a repo for Finite Element resources. I have used this resources for learning (myself) or prototyping some FEM features before implement them in a bigger FEM software.

Right now it counts with some wxMaxima worksheets, where the CAS (Computer Algebra System) Maxima is used to compute analytically the elements.

It is a good idea (and practice) using CAS for manipulate algebraic expressions, like the ones that appear commonly in the derivation of FEM (or other numerical methods). Two examples: Prof. Carlos Felipa suggest to use CAS for it (and uses Mathematica in its Introductory FEM course), Prof. Alan Bower presents some Maple codes for FEM in its online book Applied Mechanics of Solids.

TODO: Add Python code with matrices generated "automatically" from Maxima.

Installation

For wxMaxima worksheets (.wxm files) a wxMaxima (and Maxima) installation is needed. For Windows the Maxima installer already contains wxMaxima. In Linux and Mac OS they should be installed separately: Maxima, wxMaxima. Once wxMaxima is installed the worksheets can be downloaded and executed (10 minutes Maxima, spanish version).

List of files

This will change in the future (probably?). But the current list of files is:

Maxima

• Analytical FEM.wxm: a Finite Element Method code written in Maxima language. It illustrates some of the steps in a FEM code but the use of a CAS ease the process.
• Elemental Matrices.wxm: computes stiffness and mass matrices for common types of (2D) elements.

Python

• Analytical FEM.ipynb: a Finite Element Method code written in SymPy. It illustrates some of the steps in a FEM code but the use of a CAS ease the process.
• Bilinear interpolation plots: Example of interpolation using bilinear elements.
• Circular_membrane.ipynb: Weighted residual method illustration for a circular membrane.
• FEM_4Nodes.ipynb: Notebook that computes analytically the stiffness and mass matrices for a 4 node element with undistorted shape. common procedure for (explicit) time marching algorithms.
• Hermite_interpolation.ipynb: Piecewise Hermite interpolation.
• Lumped mass FEM.ipynb: computes lumped mass matrices. A
• poly_ritz.ipynb: IPython Notebook with Vibration modes of membranes with convex polygonal shape.
• XFEM_enrichment.ipynb: Notebook for enrichment basis functions for XFEM.

License

• Text and multimedia is under Creative Commons Attribution 4.0 (CC-by 4.0) https://creativecommons.org/licenses/by/4.0/
• Code is under MIT License http://opensource.org/licenses/MIT