Finite Element Analysis Implementation using MATLAB Code

Application of MATLAB in Finite Element Analysis

Finite Element Analysis (FEA) is a widely used numerical method in engineering and scientific computing for solving complex partial differential equations. MATLAB serves as an ideal tool for FEA implementation due to its powerful matrix operation capabilities and comprehensive toolbox support.

Core Implementation Approach

Problem Discretization The solution domain is first divided into finite elements (such as triangular or quadrilateral elements), with each element connected through nodes. The discretization process requires defining node coordinates and element connectivity using MATLAB arrays.

Element Stiffness Matrix Calculation For each element, compute the local stiffness matrix typically involving numerical integration (e.g., Gaussian quadrature) and shape function derivatives. MATLAB's matrix operations efficiently handle these calculations through vectorized code implementation.

Global Matrix Assembly Assemble all element stiffness matrices into a global stiffness matrix according to nodal degrees of freedom. For large-scale problems, using sparse matrix storage (via MATLAB's `sparse` function) significantly conserves memory while maintaining computational efficiency.

Boundary Condition Treatment Apply displacement boundary conditions by modifying the global matrix and load vector, potentially using matrix condensation or the unity method for handling fixed degrees of freedom through logical indexing in MATLAB.

Linear System Solution Solve for nodal displacements using MATLAB's solvers (such as the backslash `\` operator or `pcg` iterative method). The sparse matrix特性 enables efficient solution of large-scale systems with optimized memory usage.

Post-processing Calculations Derive strain, stress, and other results from nodal displacements, with visualization capabilities using MATLAB's `contour` or `patch` functions for generating color maps and deformation plots.

Optimization Strategies Sparsity Utilization: Enhance large model efficiency through memory pre-allocation and batch assembly using sparse matrix operations Parallel Computing: Accelerate independent element calculations using `parfor` loops in Parallel Computing Toolbox Adaptive Meshing: Dynamically refine critical region meshes by incorporating error estimation algorithms

For beginners, starting with 2D elastic problems (like plane stress analysis) provides a foundation before advancing to nonlinear or dynamic analysis. MATLAB's Symbolic Toolbox can additionally assist in deriving stiffness matrix expressions for complex element formulations through symbolic computation.