Finite elements

Welcome to this learning project about Finite elements!

• Project code:
• Suggested Prerequisites:
  • Linear algebra
  • Partial differential equations
• Time investment: 6 months
• Portal:Engineering and Technology
• School:Engineering
• Department:Mechanical Engineering
• Stream:Applied Mechanics
• Level: Upper-division undergraduate/First year graduate

This course will introduce you to the topic of finite element analysis. The course will cover linear finite elements and the analysis of simple solid mechanics and heat transfer problems.

This course aims to:

• Introduce you to the finite element method
• Show you how finite element formulations are arrived at
• Show you how some engineering problems are solved numerically

1. Mathematical Preliminaries
   1. Set notation
   2. Functions
   3. Vectors
   4. Matrices
   5. Tensors
   6. Partial differential equations
   7. Variational calculus
2. Tensor exercises and Physical Principles in Differential Form
   1. Quick indicial notation review
   2. Conservation of mass
   3. Conservation of linear momentum
   4. Conservation of angular momentum
   5. Conservation of energy
3. Finite element basics in one-dimension
   1. An example: Axially loaded bar
      • Strong form: governing differential equation
      • Weak form: integral equation
      • Approximate solution: Galerkin method
      • Approximate solution: Finite element method
      • Approximate solution: Finite element method as a lower bound solution
   2. More examples: Some model problems
      • Weak form: Weighted residual methods
      • Choosing a weight function
      • Bubnov Galerkin methods
      • Finite element basis functions
      • Example of a finite element approximation
   3. A time-dependent problem: the heat equation
      • Steady state heat conduction
      • Weak form of the steady state heat equation
      • Weak form of the time-dependent heat equation
      • Finite element approximation of Poisson equation
      • Finite element approximation of the heat equation
      • Time integration of the heat equation.

• Homework 1 Problem set
  • Solutions
• Homework 2 Problem set
  • Solutions
• Homework 3 Problem set
  • Solutions
• Homework 4 Problem set
  • Solutions
• Homework 5 Problem set
  • Solutions
• Homework 6 Problem set
  • Solutions
• Homework 7 Problem set
  • Solutions
• Homework 8 Problem set
  • Solutions
• Homework 9 Problem set
  • Solutions
• Homework 10 Problem set
  • Solutions
• Homework 11 Problem set
  • Solutions

• Quiz 1
  • Solutions

Textbooks

• The Finite Element Method: Linear Static and Dynamic Finite Element Analysis by T. J. R. Hughes, Dover Publications, 2000.
• K-J. Bathe (1996), Finite Element Procedures, Prentice-Hall. Useful repository of information on nonlinear finite elements.
• J. N. Reddy (1993), An Introduction to the Finite Element Method, McGraw-Hill. This book is referred to a number of times in one of the texts.
• O. C. Zienkiewicz and R. L. Taylor (2000), The Finite Element Method: Volume 2 Solid Mechanics, Butterworth-Heinemann. Another excellent repository of information of nonlinear finite elements geared toward the Civil Engineers.

References

• Mathematics:
  • R. M. Brannon (2004), Elementary Vector and Tensor Analysis for Engineers. This free online book is the best introduction I have seen for vector and tensor analysis for nonlinear mechanics.
  • B. Daya Reddy (1998), Introductory Functional Analysis: With applications to boundary value problems and finite elements. , Springer-Verlag. Excellent book for engineers who want to understand the terminology used in the finite element literature and how error analysis is done.
  • A.P.S. Selvadurai (2000), Partial Differential Equations in Mechanics 1,2. Springer. Excellent introductory text on partial differential equations with engineers in mind.