Systematic occurrence of repeated eigenvalues in structural optimization

Examples of finite-dimensional structural design problems with constraints on natural frequency and buckling are used to demonstrate that repeated eigenvalues may be expected to occur when a structure is optimized. It is shown that a repeated eigenvalue is not generally differentiable with respect to design variables. Directional derivatives are shown to exist, and a method of calculating directional derivatives is given. Necessary conditions of optimality are derived and applied to a vibration optimization problem. Extensions of the theory to distributed-parameter structures and numerical methods are outlined.This research was supported by NSF Grant No. CMS-80-05677