On Sufficient Conditions for Optimalityin the Minimum Frequency Maximization Problemfor a Shell of Revolution at a Given Weight

Abstract

We consider shallow elastic shells with a given circular boundary and seek an axisymmetric shell shape maximizing the fundamental shell vibration frequency at a given weight. The choice of functionals considered during optimal design is part of the formulation of optimization problems. The most typical problems in the theory of optimal design of compressed structures are the problems of maximizing the critical ω₀ (ω₀ is the minimum eigenvalue) for a given weight of the structure and the problems of minimizing the weight under the constraint ω₀ ≥ μ, where μ is a given number. We study an optimal control problem described by an eigenvalue problem for a system of differential equations with variable coefficients some of which are nonintegrable near zero. To solve this problem, in this paper we derive sufficient optimality conditions under the assumption that the frequency domain functional is Fréchet differentiable and the necessary conditions in the optimization problem are satisfied. It is proved that if the necessary conditions are satisfied, then the sufficient conditions are realized as well. Using the sufficient conditions, as an application, we also determine the optimal shape f = f (r) for the case in which h(r) = h0.