University of Minnesota
Aerospace Engineering and Mechanics
Spring 1998 Seminar Series

Minimizing the Elastic Engergy Functionals for Rods and Beams Using Pontryagin's Maximum Principle

Professor Emeritus William H. Warner
Department of Aerospace Engineering
and Mechanics

University of Minnesota


In this talk I shall present the solution to the minimization of a slight generalization of the potential energy functional used previously by Fosdick, et al., (1996) for problems in mixture theory for stressed elastic rods and by me (1998) for a structural optimization analog to the mixture problem. The concentration density of one of the two materials as a function of center line position and the axial displacement are the two functions to be determined in the mixture problem; the cross-sectional area replaces the concentration as the design variable in the problem. Minimizing the elastic potential energy under inequality constraints on the concentration (or the area) and also prescribing the total volume fraction (or the total volume of Material) leads to solutions of the same mathematical form for these two problems for the parameter ranges prescribed.

The generalization contains both previous solutions and also shows when another class of optimizers, ruled out necessarily in the structures problems, can occur. A peculiar borderline parameter choice giving an infinite number of optimal solutions will also be shown. Extension of the results to Euler-Bernouilli beams will be mentioned. The method used will be Pontryagin¹s Maximum Principle, usually considered to be part of dynamical systems and control theory.

Friday, April 24, 1998
209 Akerman Hall
2:30-3:30 p.m.

Refreshments served after the seminar in 227 Akerman Hall.
Disability accomodations provided upon request.
Contact Audrey Stark-Evers, Senior Secretary, 625-8000.