Finite Element Method AM4613B/9595:
We will cover the basics of the Finite Element Method (FEM). FEM is a very powerful and general method, and we will focus on the general aspects and the background of the method. After gaining the necessary theoretical background during the first half of the course, in the second half we put the ideas and concepts into practise and study the methods from the computational point of view. This second half will contain a good amount of programming, and it is essential that the students have a good background in C, C++, Java, Fortran or comparable. The course also contains a project work which is of computational nature.- Lecturer: Dr. Mikko Karttunen. Email: mkarttu [at] uwo.ca.
IMPORTANT: When emailing, please include 'AM4613B-2009' on the subject line. - TA: Roman Naryshkin. Email: rnarysh at uwo dot ca
- Practical matters:
- Brief course outline is available here as a [PDF file]
- Textbook: Finite Elements - a Gentle Introduction by David J. Henwood and Javier Bonet.
- Lectures: Mon, Wed, Thu: 08:30-09:30. Middlesex College, room MC17
- First lecture: Monday Jan. 7, 2009.
- Office hours: Mondays and Wednesdays 9:30-10:30 am
- Assessment: Three in-class exams (15% each), problem sets (35%), and a project work (20%).
- In-class exams: Thursday Jan, 29, Thursday Feb. 26., and Thursday Mar. 19, 2009
- Quiz 1 (Jan. 29): Lecture notes up to Monday Jan. 26. Assignments 1-2. Textbook Ch. 1 (including examples in there).
- Final project: Deadline (strict): Apr. 9, 2009. Project work: Write (=program + verify that it works and provide documentation as detailed in lecture notes) a general FEM solver for 2-dimensional problems.
- In-class exams: Thursday Jan, 29, Thursday Feb. 26., and Thursday Mar. 19, 2009
Weekly lecture schedule (with on-line material)
- Week 1 (Jan. 7 & Jan. 8): Practical matters and level test.
- Week 2 (Jan. 12-Jan. 16): Basic concepts of FEM, truss analysis, elasticity, equilibrium. Textbook: Chapter 1
- Week 3 (Jan. 19-Jan. 23): Coordinate systems, local, global and neutral and relations between them. Using the direct Finite Element formulation to solve 1-dimensional problems. Textbook: Chapter 1
- Week 4 (Mon Jan. 26, Wed. Jan. 28): Construction the full solution (elastic trusses) in two dimensions. Applying local coordinates.
- Quiz 1: Thursday Jan, 29: Be there on time! (overtime is not possible due to a class starting right after ours). Material: Lecture notes up to Monday Jan. 26. Assignments 1-2. Textbook Ch. 1 (including all the examples in there).[Solutions to quiz 1]
- Week 5 (Feb 2.-5.): Calculus of variations. Basic definitions. Euler equation.
- Week 6 (Feb 9.-13.): Calculus of variations. Several dependent and independent variables.Rayleigh-Ritz formulation
- Reading break: Feb. 16-20 2009.
- Week 7: Mon Feb 23: Rayleigh-Ritz formulation, trial functions, examples.
- Quiz 2: Thursday Feb. 26, 2009. Material: Lecture notes up to Monday Feb. 23. Assignments 3-5. [Solutions to quiz 2]
- Week 8: Mar. 2-5: Some important theorems, Green, Gauss, Stokes. Numerical interpolation.
- Week 9: Mar. 9-12: Numerical integration, Gaussian quadrature, FEM equations from variational formulation
- Quiz 3: Thursday Mar. 19, 2009. Textbook. Ch.2, Assignment 6. Lecture notes up to Thursday Mar. 12.
- The term ends Apr. 9, 2009.
Problem sets
- Problem set 1. [Solution set 1 (requires password)] Due: Thu. Jan. 15, 2009 by 9:30 am. Please see the instructions (week 1) about submitting answers and deadlines.
- Problem set 2. [Solution set 2 (requires password)
] Due: Thu. Jan. 22, 2009 by 9:30 am. Please see the instructions (week 1) about submitting answers and deadlines. Notice: Problem 4 is actually two separate problems, the problem number was missing in the first posting. Now the correct numbering is there (=the numbering of problems is up to 7 but no new problems has been added, only numbering was corrected). The points from the bonus problem will simply be added to the point you get out of the regular 6 problems
(= you can get up to 42 out of 36 points).
Monday Jan. 26: Assignment 2 has been marked, you can pick them up from me. - Problem set 3. [Solution set 3 (requires password)]. Due: Fri. Jan. 31, 2009 by 4:30 pm. Notice the exceptional time due to quiz on Thursday.
- Problem set 4. [Solution set 4 (requires password).] Due: Thu. Feb. 5, 2009 by 9:30 pm.
- Problem set 5. [Solution set 5 (requires password).] Change!!: A small clarifaction to problem 3 added. -> Due: Fri. Feb. 12, 2009 by 9:30 pm.
- Problem set 6. [Solution set 6 (requires password).] Due: Fri. Feb. 27, 2009 by 9:30 pm.
- Problem set 7. Due: Mon. Mar. 16, 2009 by 9:30 pm.
- Problem set 8. Due: Mon. Mar. 23, 2009 by 9:30 pm.
- Problem set 9. Due: Wed. Apr. 8, 2009 by 6:00 pm.
More on the finite element method and related matters
- FEM Books:
- S.C. Brenner and L.R. Scott, The Mathematical Theory of Finite Element Methods, Springer-Verlag, 1994
- A. Tveito and R. Winther: Introduction to Partial Differential Equations; A computational Approach, Springer, 1998
- C. Johnson: Numerical Solution of PDEs by the Finite Element method, Cambridge University Press, 1995.
- D. Braess: Finite Elements; Theory, Fast Solvers, and Applications in Solid Mechanics, Cambridge University Press, 1997.
- O.C. Zienkiewicz and R.L. Taylor: The Finite Element Method, Vol. 1: The Basis, Butterworth-Heinemann, 2000.
- O.C. Zienkiewicz and R.L. Taylor,:The Finite Element Method, Vol. 2: Solid Mechanics, Butterworth-Heinemann, 2000.
- O.C. Zienkiewicz and R.L. Taylor: The Finite Element Method, Vol. 3: Fluid Dynamics, Butterworth-Heinemann, 2000.
- Huebener, Dewhirst, Smith and Byrom: The Finite Element Method for Engineers (Wiley)
- Solin: Partial Differential Equations and the Finite Element Method (Wiley)
- H. Grandin Jr.: Fundamentals of the Finite Element Method (MacMillan)
- Strang/ and Fix: An Analysis of the Finite Element Method (Wellesley-Cambridge)
- Matrices, linear algebra:
- Golub and van Loan: Matrix Computations (Johns Hopkins).
- Web resources related to FEM + FEM software:
- Finite Element Method Universal Resource
- Introduction to Finite Element Methods, course at U. Colorado
- Numerical Methods for Partial Differential Equations by Olivier Pironneau
- OpenFEM - Open Sources Finite Element Toolbox
- Elmer - Open Source Finite Element Software for Multiphysical Problems
- OOFEM - Object Oriented Finite Element Solver
- GrAL - Grid Algorithms Library
- List of Mesh Generation Software (free and commercial)
- Gmsh: a three-dimensional finite element mesh generator
- GRUMMP - Generation and Refinement of Unstructured, Mixed-Element Meshes in Parallel
- Finite Element Method & Mathematica
- Finite Element Method & Maple
- Programming + friends:
- Fortran programming course from Oxford
- The FORTRAN Programming Language (U. Michigan)
- Modular Programming with Fortran 90 (Liverpool)
- Programming in C (Cardiff)
- Programming in C (Wolverhampton)
- Introduction to C Programming (Leicester)
- Programming in C by Brian Kernighan
- The C++ Programming Language
- C++ Programming Language Tutorials (WUSTL)
- SciLab. A Free Scientific Software Package. Works
in Windows, Linux, OSX...
- Numerical recipes in books (C, C++, Fortran)
- The repository for free mathematical software