50-Line MATLAB Finite Element Program for 2D Nonlinear Problems

This article introduces a concise 50-line MATLAB finite element program designed for solving two-dimensional nonlinear problems. The program, named fem2d_nonlinear.m, can be executed through straightforward steps to perform complete finite element analysis.

First, ensure MATLAB is properly installed and configured. Once set up, you can utilize fem2d_nonlinear.m to solve your engineering problems. The program's initial step involves reading required input files through file I/O operations - typically including mesh files containing nodal coordinates and connectivity, material property files defining nonlinear constitutive relationships, and boundary condition files specifying constraints and loads.

Prior to finite element analysis, the program defines model nodes and elements by processing mesh data. This involves computing shape functions for each element using isoparametric formulation (typically bilinear quadrilateral Q4 elements). These shape functions are then used to calculate element stiffness matrices and load vectors through numerical integration (Gauss quadrature) while accounting for material nonlinearity through iterative procedures like Newton-Raphson method.

After computing individual element matrices and vectors, the program assembles them into global stiffness matrix and load vector using efficient sparse matrix storage techniques. The resulting system of algebraic equations is solved using MATLAB's built-in linear solvers (backslash operator or iterative methods) to obtain unknown nodal displacements.

Finally, the computed displacements are used to derive engineering quantities of interest through postprocessing: strains are calculated using strain-displacement matrices, while stresses are obtained through constitutive relations. The program's compact 50-line structure allows easy modification and customization for specific nonlinear material models or boundary conditions, making it an excellent educational and prototyping tool.

In summary, fem2d_nonlinear.m provides an efficient framework for solving 2D nonlinear problems while demonstrating core finite element implementation concepts in minimal code. Its concise structure facilitates understanding of key algorithms including mesh processing, matrix assembly, nonlinear iteration, and results extraction.