The Workshop will be held at the Department of Mathematics of  UW-Eau Claire. The three main seminars, and the one-hour Colloquium talk will be presented by:

 Professor Dongming Wei,
Mathematics Department,
University of New Orleans,
2000 Lakeshore Drive
New Orleans, LA 70148

 We will introduce linear elements for one dimensional problems and linear triangular and  bilinear rectangular elements for a general 2-d second order elliptic field equation.  Galerkin's method  is applied to the governing equations with piecewise constant coefficients and mixed boundary conditions.  Applications of the field equation include  2-d and 3-d axisymmetric steady state heat transfer by conduction and convection, ideal Newtonian flows and flows in porous medium  in bounded polygonal domains with steady sources and sinks. We will introduce  the  Ritz Galerkin's method and the variational formulation of second  order elliptic problems.   Some iterative methods such as the method of conjugate gradient will be used. The corresponding isoparametric elements and numerical integrations follows to allow variable equation coefficients etc.  Some computer codes will be discussed.

Lecture 2. Introduction to Nonlinear Finite Element Analysis (7/26/05 at 1:30-2:30 pm and 3:00-4:30 pm)

 In the second lecture, nonlinear elements for some one dimensional problems including heat transfer with radiation, non-Newtonian flows between parallel plates. Newton’s method and some other iterative numerical methods such as nonlinear conjugate method  will be demonstrated for the numerical solutions of these problems. For two or three dimensional problems, Newton’s method and axi-symmetric linear triangular elements are demonstrated for modeling of temperature distribution in a Poiseuille power-law flow with viscous dissipation.

Lecture 3.  Finite Element Analysis of Some Non-Newtonian Flows (7/27/05 at 1:30-2:30 pm and 3:00-4:30 pm)

In this third lecture, we will introduce the penalty finite element method for Stokes equations. The Stokes equations can be used to model isothermal viscous Newtonian and power-law non-Newtonian flows at small Reynolds numbers.  Ritz-Galerkin’s  method and linear triangular elements are used for finite element formulations.  Some nonlinear conjugate gradient schemes are used for the numerical solutions including some computer codes and examples.

 Colloquium talk (7/29/05 at 1:30-2:30 pm)

 Abstract: In this talk a class of nonlinear wave equations of p-Laplacian type are presented. These equations can be used to model vibration of rods, beams, and plates made of heat treated metals that satisfy the nonlinear stress-strain power-law. The examples include  some rods and Euler-type beams in the form of a single stainless steel fibre of the hybrid stainless steel assembly used in transportation industry for lighter and more crashworthy vehicles as well as applications in biomechanical engineering structures. These metals are nonlinear strain-hardening elastic-plastic materials  which is a special case of the Hencky plastics materials. Finite element and finite difference schemes are used to fully discretize the wave equations and obtain numerical approximations. Linear and cubic finite elements, and some iterative finite difference schemes such as Newmark, Runge-Kutta are used. The numerical results are analyzed and compared with some analytical solutions. It is also demonstrated that these type of equations admit solitary wave solutions.

Preliminary schedule for other Seminars on Related Topics and Applications

(Titles, abstracts presenters and rooms to be announced later.)

 7/25/05 11:30-12:30 (EDI)

7/25/05 11:30-12:30 R. W. Langer (Lecture 1)

7/26/05 11:30-12:30 (EDI)

7/27/05 11:30-12:30 R. W. Langer (Lecture 2)

7/28/05 11:30-12:30 (EDI)

             1:30-2:30  

              3:00-4:00

7/29/05 11:30-12:30

             3:00-4:00 M. Penkava

Several undergraduate student talks (20 minutes each) and graduate students talks (one hour each) will also be offered.  The times for students talks will be:

7/26/05 10:30-11:30  A. M. Elgindi, University of Chicago)

7/27/05 10:30-11:30  Tyler Birkel, UWEC

7/28/05 10:30-11:30   Carolyn Otto, UWEC 

7/29/05 10:30-11:30  Tarek Elgindi  & Darin Mohr, UWEC

For further information please contact:

Professor Mohamed B. Elgindi
Department of Mathematics
UW-Eau Claire,
Eau Claire, WI 54702-4004
Tel. (715) 836-2768
elgindmb@uwec.edu