An investigation of the optimal design of the tallest unloaded column
McCarthy, C. Maeve
Cox, Steven J.
Doctor of Philosophy
The problem of the optimal design of the tallest unloaded column is revisited with a view towards clarifying the optimality of the design proposed by Keller & Niordson in 1966 (19). The first eigenvalue of a singular Sturm-Liouville problem must be maximized in order to yield the tallest column. The difficulty with the proposed design comes from the nature of the spectrum associated with it. If this spectrum were discrete, Keller & Niordson's techniques would have been appropriate. The proposed design is shown here to admit continuous spectra and hence warrants further investigation. Over a class of admissible designs admitting purely discrete spectra, it is shown that the decreasing rearrangement of the shape leads to a column at least as tall as the original. Existence of an optimal design follows from this fact. Similar methods are applied to design problems arising from other classes of Sturm-Liouville problems. The necessary conditions for optimality established by Keller & Niordson are re-established using generalized gradient methods. Using a finite element approximation to the boundary value problem, the first eigenvalue is maximized using MATLAB's Sequential Quadratic Programming implementation constr (15) to optimize and Radke's implementation of the Implicitly Restarted Arnoldi Method speig (28) to compute eigenvalues. Analysis of the convergence behavior of the higher eigenvalues leads to the conclusion that the spectrum of the Keller & Niordson design is made up of an essential spectrum and one isolated eigenvalue. This justifies the methods used by Keller & Niordson and confirms the optimality of their design. Finally, inverse spectral methods are investigated as a means by which to increase the height of a given column by adding or subtracting material appropriately.