CSI 742: The Mathematics of the Finite Element Method
General Information
- Instructor:
Professor Chi
Yang (Ext. 3-4077, cyang@gmu.edu )
- Lecture:
Tuesdays 7:20
pm – 10:00 pm
- Classroom: Research I,
Room 202
- Office Hour: Tuesdays
5:00 pm – 6:00 pm or by appointmet
- Textbook: J. N. Reddy,
An Introduction to the Finite Element Method, Third Edition, McGraw Hill,2004
Course Description
- This is a one-semester course in the general area of numerical analysis.
The finite element method is introduced as a numerical technique that employs
the philosophy of constructing piecewise approximations of solutions to problems
described by differential equations. The objective of this course is to introduce
the fundamentals of finite element analysis of solids and structures and of
heat transfer and fluids. This includes the theoretic foundations and appropriate
use of finite element methods and its applications in different fields.
- The prerequisites are MATH 446 or
685, or permission of instructor.
Course Outline
- Introduction to the background and
the basic concept of the finite element method.
- Finite element analysis
- Problem classification, modeling
and discretization
- Interpolation. Elements, nodes,
and degrees of freedom
- Example of Applications. History
of finite element analysis
- Solving a problem by finite element
analysis
- Learning and using finite element
analysis
- Review the use of vectors, matrices,
and tensors, with emphasis on those aspects that are important in finite
element analysis.
- Introduction to matrices
- Vector spaces
- Definition of tensors
- Symmetric eigenproblem
- Vector and matrix norms
- Review the basic steps in the integral
formulations and the associated approximation of various boundary problems.
- Some mathematical concepts and formulae
- Weak formulation of boundary value
problems
- Variational methods of approximation
- The formulation of finite element methods
for the analysis of one-dimensional problems
- Second-order boundary value problems
- Basic steps of finite element
analysis
- Applications to heat transfer,
fluid mechanics and solid mechanics
- Bending of beams
- Euler-Bernoulli beam element
- Plane truss and Euler-Bernoulli
frame elements
- The Timoshenko beam and frame
elements
- Inclusion of constrained equations
- Finite element error analysis
- Approximation errors
- Various measures of errors
- Convergence of solution
- Accuracy of solution
- Eigenvalue and time-dependent problems
- Eigenvalue problems
- Time-dependent
problems
- Numerical integration and computer
implementation
- Isoparametric formulations and
numerical integration
- Computer implementation
- The formulation of finite element methods
for the analysis of two-dimensional problems
- Single variable problems
- Boundary value problems
- Some comments on mesh generation
and imposition
- Applications to heat transfer,
fluid mechanics and solid mechanics
- Eigenvalue and time-dependent
problems
- Interpolation functions, numerical
integration, and modeling considerations
- Library of elements and interpolation
functions
- Numerical integration
- Modeling considerations
- Plane elasticity
- Governing equations
- Weak formulations
- Finite element model
- Evaluation of integrals
- Assembly and boundary and initial
conditions
- Flow of viscous incompressible fluids
- Governing equations
- Velocity-pressure finite element
model
- Penalty–finite element model
- Computer implementation
Grading
- Homework assignment
- Term project(s)
- Final exam