Finite Element Programming in MATLAB (Linear and Non-linear Analysis)

Course Description: This course provides a comprehensive, hands-on introduction to programming the Finite Element Method (FEM) using MATLAB. It is specifically designed for beginners with little experience in FEM or programming. We will start with the fundamental concepts of FEM and the basics of MATLAB programming. Students will learn to write their own FEM code from scratch to solve 2D and 3D engineering problems. The course progresses from simple linear static analysis to the more advanced techniques required for non-linear problems, equipping students with the skills to develop and analyze their own custom FE models.

Target Audience:

• Undergraduate and graduate students in engineering, physics, and applied mathematics.
• Researchers and practicing engineers who want to understand the inner workings of FEA software.
• Individuals with a strong interest in computational mechanics and simulation.

Prerequisites:

• Basic knowledge of calculus (derivatives and integrals).
• Familiarity with linear algebra (vectors, matrices, solving systems of linear equations).
• Basic knowledge of programming.

Learning Objectives: Upon successful completion of this course, students will be able to:

• Explain the fundamental theory and workflow of the Finite Element Method.
• Write structured and efficient MATLAB code for numerical analysis.
• Develop a complete 2D and 3D linear FEM solver from scratch.
• Understand the sources of non-linearity in engineering problems.
• Implement iterative solution strategies (e.g., Newton-Raphson) for non-linear FEM.
• Analyze and visualize the results of both linear and non-linear simulations.

Course Outline

Module 1: Foundations of FEM and MATLAB

This module establishes the core theoretical concepts and the essential programming tools needed for the course.

• 1.1 What is the Finite Element Method?
  • Motivation: The limitations of analytical solutions for complex problems.
  • The Core Idea: Discretization of a continuum into finite pieces.
  • Basic Concepts: Nodes, Elements, Mesh, and Degrees of Freedom.
  • Applications of FEM across engineering and science.
• 1.2 The Standard FEM Workflow
  • Preprocessing: Defining geometry, meshing, and assigning material properties.
  • Processing: Assembling the global system of equations and applying boundary conditions.
  • Post-processing: Solving the system and interpreting/visualizing the results.
• 1.3 Introduction to MATLAB for FEM
  • Basic MATLAB syntax, environment, and script files.
  • Working with matrices and vectors: The foundation of FEM programming.
  • Essential functions for FEM (e.g., inv, \, plotting functions).

Module 2: Programming Linear Finite Element Analysis

This module is focused on the hands-on implementation of a linear FEM solver. We will translate theory directly into functional code.

• 2.1 The Mathematics of Linear Elements
  • The “Weak Form” of differential equations (e.g., structural mechanics).
  • Derivation of element stiffness matrices ( ) and force vectors ( ).
  • The concept of Shape Functions.
• 2.2 1D FEM Implementation in MATLAB
  • Programming the element stiffness matrix for a 1D bar element.
  • The Assembly Process: Looping through elements to build the global stiffness matrix ( ) and force vector ( ).
  • Applying boundary conditions (loads and supports).
  • Solving the global system: .
  • Post-processing: Calculating stress, strain, and reaction forces.
• 2.3 2D FEM Implementation in MATLAB
  • Introduction to 2D elements.
  • Deriving shape functions and the element stiffness matrix for a 2D element.
  • Data structures for managing 2D mesh information (nodes, connectivity).
  • Assembly and solution process for 2D problems.
• 2.4 3D FEM Implementation in MATLAB
  • Introduction to 3D elements.
  • Deriving shape functions and the element stiffness matrix for a 3D element.
  • Assembly and solution process for 3D problems.

Module 3: Introduction to Non-linear Finite Element Analysis

This module introduces the complexities of non-linear behavior and the iterative techniques required to solve it.

• 3.1 What Makes a Problem Non-linear?
  • Geometric non-linearity.
  • Material non-linearity.
  • Why the linear approach ( ) fails.
  • TL and UL formulations.
  • Hyperelasticity.
• 3.2 Solution Strategies for Non-linear Problems
  • The concept of an iterative solution: Guess, check, and correct.
  • Load and displacement control strategies.
• 3.3 Programming Project: A plain strain hyperelastic structure subjected to traction load

Module 4: Advanced Topics

This final module provides a look at more advanced topics and allows students to synthesize their knowledge in a final project.

• 4.1 Introduction to Dynamic Analysis
  • A brief overview of including mass and damping effects (mass and damping matrices).
  • Concepts of time-dependent problems.
• 4.2 Hybrid elements

Software:

• MATLAB (Student Version is sufficient).