50-Line MATLAB Finite Element Program for 2D Heat Transfer Problems

This documentation presents a 50-line MATLAB finite element program designed to solve two-dimensional heat transfer problems. The program, named fem2d.m, can be executed directly to initiate simulations. Despite its brevity, the code incorporates sophisticated computational algorithms that effectively model various practical heat transfer scenarios. The implementation follows standard finite element methodology with optimized matrix operations and sparse system solvers for computational efficiency.

First, let's understand the finite element method and its application to heat transfer problems. FEM is a numerical technique that decomposes complex physical systems into smaller, manageable elements. In thermal analysis, the method discretizes the domain into finite elements where heat transfer equations are solved locally before being assembled into a global solution. The program implements this through Gaussian quadrature integration and shape function calculations for each element.

The program structure consists of a main function coordinating four specialized subfunctions: mesh generation (creating triangular elements and nodes), element matrix computation (using conductivity matrices and heat flux calculations), boundary condition application (handling Dirichlet/Neumann conditions), and linear system solution (employing MATLAB's backslash operator for efficient sparse matrix solving). Each module uses vectorized operations to minimize computational overhead. For customization, users can modify subfunction parameters like material properties, mesh density, or boundary conditions to adapt to different problem configurations.

This compact program serves as an effective tool for 2D thermal analysis, demonstrating core FEA principles through practical implementation. For extended functionality, users can reference MATLAB's PDE toolbox documentation or consult computational physics resources for advanced modifications.