This repository contains two major projects developed using the Finite Element Method (FEM) to solve complex engineering problems in structural mechanics and fluid dynamics.
This study analyzes the structural behavior of two cantilever concrete beam designs under a uniform distributed load (
Analyzed Designs:
-
Design 1: Two circular holes (
$d = 0.15 \text{ m}$ ). -
Design 2: Two rotated square holes (
$d = 0.15 \text{ m}$ ).
- Software: Abaqus/Standard.
- Element Type: 3-node triangular elements under plane stress conditions.
- Mesh Density: Comparison between a Coarse Mesh (~1120 elements) and a Fine Mesh (~2000 elements).
-
Validation: Numerical results were compared against the Euler-Bernoulli beam theory (
$\sigma_{xx} = -My/I$ ).
-
Displacement: Design 1 (circular) showed slightly higher maximum deflection (
$13.62 \mu\text{m}$ ) than Design 2 ($13.22 \mu\text{m}$ ), indicating the square hole configuration provided slightly higher stiffness in this specific setup. -
Stress Concentration: The highest Von Mises stresses occurred near the fixed supports (
$19.62 \text{ Pa}$ for Design 1;$19.27 \text{ Pa}$ for Design 2). - Convergence: Fine mesh refinement was critical to capturing local stress gradients near the hole edges that theoretical models typically overlook.
This project involves the numerical simulation of a 2D steady-state potential airflow around a NACA 4424 airfoil confined between parallel plates. The fluid is assumed to be ideal (inviscid, incompressible, and irrotational).
This project demonstrates a hybrid workflow between commercial software and custom programming:
- Preprocessing: Mesh generation and nodal/element data extraction performed in Abaqus.
-
Solver: A custom MATLAB-based FEM solver developed to solve the Laplace equation (
$\nabla^2\phi = 0$ ) for the velocity potential. -
Postprocessing: Velocity components (
$u, v$ ) and pressure fields (via Bernoulli’s Equation) were calculated and visualized in MATLAB. - Advanced Features: Implementation of isoparametric elements and Gauss quadrature integration for numerical consistency.
- Flow Visualization: Successfully identified stagnation points at the leading and trailing edges.
- Pressure Field: Captured high-pressure zones at stagnation points and low-pressure zones along the airfoil surfaces, matching theoretical potential flow expectations.
- Validation: MATLAB results showed excellent agreement with Abaqus native heat transfer analogies (NT11) used for verification.
- Finite Element Method (FEM): Linear static analysis, potential flow theory, weak form derivation.
- Software: Abaqus/CAE (Standard/Explicit).
- Programming: MATLAB (Custom solver development, matrix assembly, postprocessing).
- Numerical Methods: Isoparametric mapping, Gauss quadrature, bandwidth optimization (AHBW).
These projects were completed for the course ENMEE3332: Finite Elements at Columbia University, The Fu Foundation School of Engineering and Applied Science (Fall 2025), under the supervision of Professor H. Waisman.