MAE541  Topics in Finite Element Methods

Welcome to the home page for the MAE 541 Class. Please stop by often!
 


 

All class announcements, homework assignments, due dates etc. will be posted here.
Administrivia
Instructor: Dr. Abani K. Patra
           605 Furnas
email :     abani@eng.buffalo.edu
Tel:            645-2593x2240

Office Hours:  T Th  10:00-11:00 a.m. or by appointment.
 
 
 
 
 
 

Text:  Finite Elements by B. Szabo and I. Babuska, John Wiley.
This book  is a little advanced and goes over the  basic concepts  quickly but its coverage of advanced concepts is excellent. Please note that I will frequently cover material outside the text. Appropriate notes and references will be provided as we go along. I will post most material on the web where possible. Please shop around a little before purchasing this -- costs range from $61 to
$140!
Reference:

  1. Introduction to FEM, Linear Static and Dynamic Analysis, T.J.R. Hughes, Prentice Hall Good mix of theory and application A good buy at only $21 in paperback!! Considering the coverage and our focus on time dependent problems -- definitely worth buying!
  2. An introduction to the Finite Element Method  by  J. N. Reddy, McGraw Hill, available at many bookstores . This is a nice book to understand basic concepts but is somewhat limited in its coverage of applications and more advanced concepts.
  3. FEM Notes, Prof. J. Flaherty at RPI. A little mathematical but nice very modern in coverage of topics.
  4. Finite Element Procedures, K.J. Bathe, Prentice Hall. Good overall coverage especially for non-linear problems
  5. Finite Elements in Design(?), J.E. Akin. Paperback and inexpensive and good coverage of errror estimation
  6. Modeling and Numerical Errors -- Kurowski and Szabo


Course Objective:
The objective of this course is to enable you to perform basic analysis of physical systems using the finite element method. At the end of this course you should have a clear understanding of the fundamentals of the method, the underlying mathematics and its application to several problems of interest. As a second level class, the focus will be on developing a deeper understanding of the method, as needed for applying it to more complex problems.

Background and Prior Preparation
The course will be taught at a graduate student level. Some backgroud in finite element methods and a normal undergraduate background in partial differential equations, linear algebra and computer programming will be assumed.
 

The topics to be covered  are listed below:

Course Outline:

Module 1:  Fundamentals

  1. Fundamental Concept of the FEM
  2. Mathematical preliminaries
  3. Integral Forms, Principle of Virtual Work and Variational Methods
Module 2:  Methodology I
  1. Element Interpolations,  Lagrange Interpolations, Hierarchical Interpolations
  2. Approximation Theory
  3. Numerical Integration
Module 3: Methodology II
  1. Assembly, boundary conditions, solution methods
  2. Trusses, Beams, Plates, Shells and Other FE crimes
Module 4: Preprocessing
  1. Problem Set-up and simplifications
  2. Boundary Conditions
  3. Mesh generation
Module 5:  Post-Processing
  1. Stress calculations and  smoothing
  2. Introduction to Error Estimation
  3. Mesh Modification and Adaptivity
  4. Sensitivity/Optimization Calculations and FEM in the Design Cycle
After the completion of the above 5 modules we will complete as many of the following optional modules as time permits.

Module 6: Time Dependent Problems

  1. Semi-discrete Galerkin Formulations
  2. Time Integration Schemes For Parabolic and Hyperbolic Problems
  3. Frequency Domain Analysis
Module 7: Non-Linear Problems
  1. Sources of non-linearity, geometric and material
  2. Total Lagrangian and Updated Lagrangian Formulationss
  3. Iterative Solution Techniques
Grading:
The grading in this class will use a mix of homework, take home tests and  group projects.
 
Homework  20 %
In-Class Tests 30 %
Projects 50 %
Total 100 %
Class Schedule:
 
 
DATE
TOPICS DISCUSSED
Homework etc.
Week 1 Introduction to Math. Modeling -- History etc. Read Ch1 and 2 in text, Read history 
section from J.F. notes
Week 1  Mathematical Preliminaries: Vector and functions 
spaces, norms Cauchy-Schwartz Inequality		  HW1 
Week 2 Generalized Formulation, Bilinear Forms
Week 2 Boundary conditions, Min. Pot. Energy, Galerkin methods and 
Best Approximation Properties		 HW 1 due 
Week 3   continued
Week 3 Numerical Integrations
Week 4 2D formulations and simple setup --  Read Ch. 5
Week 4 Extensions and convergence rates Read Ch. 4, HW2
Week 5 Extensions -- convergence rates, Wrap-up Ch. 5
Week 5 Element Types -- approximability ...
Week 6  More on elements
Week 6  HW2 due
Week 7 Solvers -- Direct, Iterative ...
Week 7
Week 8 Time dependent problems -- Generalized Trapezoidal Algorithms
Week 8 Analysis of GTA
Week 9  Test 1
START FINAL CLASS PROJECT
Week 9 Non-Linear Problems
Week 10 Non-Linear Transient Problems
Week 10 Non-Linear Transient Problems HW 3
Week 11 Error Estimation Procedures
Week 11 Error Estimation Procedures
Week 12 Plates and Shells HW3 due
Week 12 Exam 2, Project DUE