EGM 6351 - FINITE ELEMENT METHODS
Spring 1998
NOTE: As I often update the materials on the web concerning the course, you should "reload" all web pages related to the course, each time you want to browse or to download some documents, to make sure that you have the latest version. Also, watch for the rotating icon "new" and the corresponding date stamp.
Graduate Catalog Description:
Prerequisite: Consent of instructor (see below).
Classification of second-order partial differential equations.
Methods of weighted residuals.
Classical finite element method: Weak form (Principle of
Virtual Power, calculus of variations),
basis functions (linear independence, choice),
Galerkin projection, element viewpoint, computer coding (data
structure, assembly operator). Transient problems.
Applications
to electromagnetics, heat conduction, solid and fluid mechanics.
Textbook: T.J.R. Hughes [1987], The Finite Element Method--Linear Static and Dynamic Finite Element Analysis, Prentice Hall.
This book is currently out-of-print; you can get a copy of the first four chapters of the book at University Copy Center, open 24 hours, 1620 W. University Ave., Gainesville, FL 32603, Tel: 352-372-7436. The book's author is working to republish it in a Dover edition.
References:
Many other references are listed in the
Detailed Course Contents
(Postscript, 309KB).
(For those of you who cannot read or download PostScript documents from
my web page, see my note at the bottom of this web page.)
Prerequisites by topics: Undergraduate courses on linear algebra, and on ordinary and partial differential equations (ODE's and PDE's). Have knowledge in Fortran, or C, or Matlab, and have had an undergraduate course on numerical methods.
Goals: Provide a solid foundation for the traditional Galerkin method and the classical FEM. Develop (i) the basic concepts of the (classical) Galerkin Finite Element Methods (FEM), (ii) the skills to formulate FE solution to solve the PDE's governing physical phenomena, (iii) the skills in implementing FE formulation in computer codes (Unix system, data structure of FE codes, etc.). The present course is self-contained, and will focus on elliptic, self-adjoint PDE's. Prepare students for the follow-up, advanced course EGM 6352 Advanced FEM (non-classical FEM) to be offered in Fall '98, and a course on Nonlinear FEM to be offered in Spring '99. Prepare engineering students for reading the rich mathematics literature on the subject.
Topics: Classification of second order PDE's and boundary conditions. Weak form (or Principle of Virtual Power) of elliptic PDE's, abstract formulation of classical FEM. Relation to variational principles (Rayleigh-Ritz method), minimization of energy. Linear independence of functions, Gram matrix. Basis functions. Approximation: Polynomials, trigonometric functions, FE basis functions (linear and higher order), 1-D Lagrange and Hermite basis functions. Computer coding: Finite element matrices, data structure, assemblage of element matrices into global matrices. Abstract interpretation: FEM as an orthogonal projection, best approximation. Classical boundary value problems. Transient problems via separation of variables, Sturm-Liouville eigenvalue problems. 2-D Lagrange elements: Isoparametric mapping, variable 4-to-9 node elements. Theory of Gauss-Legendre quadrature, numerical integration of element matrices.
Projects: To be announced in class.
Computer support required: Access to a computer with Fortran, or C, or Matlab, and plotting software.
Engineering applications: Various areas (electromagnetics, heat conduction, fluids, solids) of interest to students in all engineering departments.
Weekly plans: Lecture contents, homework assignments, any other announcements related to the course. See also the Detailed Course Contents (Postscript, 309KB) to have a general idea about the overall contents of the course in some detail. I generally vary its contents each time I teach this course. You should look at the weekly plans for the actual contents.
Cooperative learning techniques (PostScript): A teaching approach that is used in the course. For more information, see Purdue's experience with Coorperative Learning .
Grading policy:
10% class participation, 60% homework, 30% final exam.