The present investigation examines the multibar truss optimization problem in the context of a general class of unconstrained optimization procedures in conjunction with various types of penalty function transformations. Specifically, the problem is transformed into a series of unconstrained minimization problems using the penalty function techniques of Heaviside and SUMT. These are solved using the methods of Rosenbrock (orthogonal directions), Powell (conjugate directions) and Nelder-Mead (Simplex). This resulted in many cases in substantial improvements being recorded over previously reported data. The paper includes a comparative study of the synthesis based on these procedures. In this context, the penalty function approach seems to offer greater possibilities to problems with non-linear merit functions.Other studies of penalty function based structural optimization problems have been made by, amongst others, Schmit and co-worker~,~ Marcal and GellatlyY8 Wollnerll and Pappas.12 In particular, MOe,lS has made extensive use of various penalty function techniques for the design of ship structures, where he has incorporated practical improvements relating to the derivation of feasible starting points, scaling, program parameter selection, convergence criteria, efficient line searches, etc. This paper is intended to be a follow-up and consolidation of this work by attempting a more systematic evaluation of the above penalty function based unconstrained minimization procedures as applied to the structural optimization area.

This paper attempts a comparative study of some numerical methods for the optimal control design of turbine blades whose vibration characteristics are approximated by Timoshenko beam idealizations with shear and incorporating simple boundary conditions. The blade was synthesized using the following methods (1) conjugate gradient minimization of the system Hamiltonian in function space incorporating penalty function transformations (2) projection operator methods in a function space which includes the frequencies of vibration and the control function (3) ε-technique penalty function transformation resulting in a highly nonlinear programming problem (4) finite difference discretization of the state equations again resulting in a nonlinear program (5) second variation methods with complex state differential equations to include damping effects resulting in systems of inhomogeneous matrix Riccatti equations some of which are stiff (6) quasi-linear methods based on iterative linearization of the state and adjoint equation. The paper includes a discussion of some substantial computational difficulties encountered in the implementation of these techniques together with a resume of work presently in progress using a differential dynamic programming approach.

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SUMMARYThis paper describes the computational aspects of an earlier analytical of the problem of maximizing a linear combination of the natural frequencies of vibration of a turbine disc idealization subject to weight and fabrication constraints. The problem was formulated as an optimal control problem and solutions to the adjoint equations of the Pontryagin formalism obtained using perturbation techniques. These techniques transformed the problem to a non-linear program. This paper gives detailed computational solutions to the non-linear program which is characterized by pockets of relative optima. In addition, the dual problem of minimizing the weight subject to a frequency constraint is considered within the optimal control framework. The duality results are confirmed computationally for the two non-linear programs.A description of some practical procedures for improving the convergence characteristics of these highly non-linear programs is included.