On design derivatives for optimization with a critical point constraint

Abstract

Design optimization of geometrically nonlinear structures with a critical point constraint is considered. A staggered scheme is applied to the optimization problem and the reduced optimization problem is solved at the critical point. Derivatives of the objective function and constraints are defined consistently with the algorithmic steps of the staggered scheme.

It is noticed that different schemes require different design derivatives of the objective function and constraints. It is stressed that a distinction must be made between the derivative of displacements at the critical load and the derivative of critical displacements. For the sake of simplicity a nonlinear two-bar truss structure is used to show that their properties are quite different; while the first one grows to infinity when approaching the critical point and thus does not exist, the other exists at the critical point and is equal to zero.

Subsequently, two methods of computing the design derivative of critical loads are analysed, and it is demonstrated, for the truss example, that both methods yield correct results. Then, two optimization problems, i.e. the minimum volume problem and the maximum critical load problem, are formulated. Both problems are solved for the two-bar truss, and yield results that compare favourably with those obtained analytically. The solution scheme is shown to be insensitive to initial errors in the determination of the critical point.