Integrated approaches to parallelism in optimization and solution of inverse problems
Abstract
Mathematical models for mechanical design problems and development of analytical and numerical tools for their solution was studied under this grant. The mathematical problems separate into ones of rods and membranes. Regarding the former, with M. Overton, the PI provided the first rigorous study of the shape of the strongest rod. In particular, within the context of the EulerBernoulli model, we established existence, necessary conditions, regularity, and a general, though efficient, means of computing an optimal shape. Previous studies had not fully dealt with the fact that the strength of a rod (the axial load under which it commences to bucle) need not be a differentiable function of its shape. The Mathematical Intelligence solicited an expository account of this work. This article was picked up by Discover magazine, where the review appears. Engineers in offshore oil rig design at Exxon Production Research in Houston have approached the PI regarding the research. Via the rolling of thin plates they have the means to create rods with piecewise conical crosssections. With J. Maddocks, the PI has extended all of the above analytical findings to a much richer class of rods. In particular, they are able to accommodate hyperelastic rods with nonlinear bending laws and vanishing crosssections in both the interior and at the boundary. In this new framework, they are also finally able to carefully pose and settle bifurcation questions as to whether the branches of equilibria stemming from the buckling load of the optimal column are indeed supercritical, i.e., rightward.
 Publication:

Final Report
 Pub Date:
 January 1993
 Bibcode:
 1993rice.reptQ....S
 Keywords:

 Buckling;
 Design Analysis;
 Mathematical Models;
 Mechanical Engineering;
 Membranes;
 Optimization;
 Rods;
 Axial Loads;
 Bending;
 Loads (Forces);
 Nonlinearity;
 Thin Plates;
 Structural Mechanics