MEF1D

A unidimensional finite element method (FEM) implemented in MATLAB to solve variational problems subject to boundary conditions.

About

The finite element method (FEM) is a way to solve numerically differential equations by discretizing the function space where the solution lies. This particular implementation follows the approach presented in the chapter 5 of La méthode des éléments finis: de la théorie à la pratique (A. Fortin & A. Garon). This code was made during an internship in applied maths.

Usage

This code was made entirely for educational purposes, and in no means does it pretend to be fully optimized or efficient. The function solve_mef1d() was made to solve for u in unidimensional ODEs of order 2 with this particular form: where p(x),q(x),R(x) are parameters depending on the problem. Dirichlet conditions (essential) are imposed on the boundaries of the domain (a,b). Neumann conditions (natural) and mixed conditions (Dirichlet-Neumann) are also supported. Linear and quadratic shape functions (Lagrange) are available for the discretization. You can plot the resulting approximate solution with plot_mef1d(). Also, if you have computed the analytic solution to your problem and you just want to compare it with the numerical solution, solve_mef1d() can give you the L² error on your approximate solution, and plot_mef1d() can plot both on the same graph for a visual comparison.

Validation

Code validity was ensured in 2 ways:

1. The Method of Manufactured Solution (MMS) was applied on a linear elasticity problem. Given an arbitrary analytical solution, the source term of the differential equation was computed by hand. By comparing the numerical solution (obtained with the hand-calculated source term) with the initial manufactured solution, we converged according to L² norm. It was also verified that linear solutions were exact while using order 1 and 2 elements, and that quadratic solutions were exact with order 2 elements.
2. Results from an example in the previously mentioned book were reproduced as expected.

Future Functionalities

• An option to add "point loads" (Diracs) on specific nodes will be implemented eventually;
• Comments will be translated to english;
• Other example problems will be added;
• Maybe some refactoring of the code (dividing the code into smaller modules).