A homotopy method for optimal design using an envelope function

Abstract

Philip Y. Shin and Henry A. Castillo ** Naval Postgraduate School An efficient and easy-to-implement procedure for the globally convergent homotopy method for nonlinear optimization is discussed in this paper. An envelope function is used to augment multiple inequality constraints so that difficult active set strategy can be avoided. The procedure is applied for a simple constrained minimization problem, and its convergency is tested for different values of an envelope function parameter. The results indicates that for a large value of the parameter, fast convergence is achieved, but it could result in large errors in the solution. The paper also demonstrates application to optimal design of one-ring stiffened shells. A sequence of optimal designs has been obtained as a function of external hydrostatic pressure. It is seen from the results that the single ring-stiffener works more effectively at lower levels of pressure. INTRODUCTION columns and stiffened composite plates. The multimodal characteristics which occur at the exact optimum are also discussed in the papers. One of the advantages of the homotopy method over other conventional optimization techniques is that instead of obtaining a simple optimum, which is typical of other methods, the homotopy technique generates, in a single computer execution, an entire family of optimum designs for a given parameter. So the method can be applied to the types of problems in which a designer's interest is in comparing different optimal designs for certain values of a parameter. Also, the method shows guaranteed convergence to a desired optimum design. However, the method is hampered when there are several inequality constraints. When the system of equations, derived from the Lagrange multipliers technique, are solved by a homotopy tracing algorithm, the solution path has several branches due to changes in the active constraint set. At a branching point, the Jacobian matrix looses its full ranks, so numerical ill-conditioning The objective of this paper is to devise a ------------------technique to eliminate this branching point Assistant Professor, Department of Mechanical Engistrategy. Since the homo to^^ optimization has neering, Member A I M , ASME. practical difficulties in dealing with several inequality constraints, special treatments of Graduate Student, Lieutenant, U.S. Navy. augmenting all the inequalities by using the Kreisselmeier-Steinhauser [7] smooth envelope functions are discussed. This approach is first This paper is declared a work of the U.S. Government and tested for a simple constrained function is not subject to copyright protection in the United States. minimization, then, it is applied for optimal design of stiffened cylindrical shells under hydrostatic pressure.