Finite Element Method Implementation Using MATLAB

The Finite Element Method (FEM) is a powerful numerical analysis technique widely used in engineering and scientific computations. Implementing FEM programs in MATLAB allows transforming complex physical problems into computable mathematical models.

### Core Methodology Problem Discretization: Divide continuous domains into finite elements (e.g., triangles, quadrilaterals) to form mesh structures using MATLAB's geometry toolbox or custom meshing functions. Element Stiffness Matrix: Compute local stiffness matrices for each element based on material parameters and geometric properties through Gaussian quadrature integration and shape function derivations. Global Assembly: Assemble individual element matrices into a global stiffness matrix using MATLAB's sparse matrix operations, followed by applying boundary conditions via constraint enforcement techniques. Linear System Solution: Leverage MATLAB's built-in solvers (e.g., backslash operator or iterative methods) to efficiently solve the linear system for nodal displacements or other physical quantities. Post-processing Analysis: Visualize results like displacements and stress distributions using MATLAB's plotting functions (e.g., contourf, patch) to validate computational accuracy and convergence.

### Advantages MATLAB's matrix manipulation capabilities significantly simplify stiffness matrix assembly and solution processes through vectorized operations. Built-in visualization functions enable intuitive presentation of FEM results, such as deformed meshes or stress contour plots. Script-based programming facilitates rapid parameter adjustments (e.g., mesh density, material properties) via modular code design.

FEM implementation in MATLAB serves both as an educational demonstration tool and a practical solution for research and engineering analysis applications.