Optimal Design for Problems in Hydroelasticity
In this chapter we discuss optimal design of elastic plates interacting with an ideal fluid. In Section 6.1 we introduce the basic relations describing small vibrations of elastic plates submerged in an ideal fluid and study some essential properties of the corresponding boundary-value problem. Section 6.2 contains the formulation and the study of problems of frequency optimization for plates vibrating in a fluid. In Section 6.3 we derive an analytic solution of the exterior hydrodynamic problem with two-dimensional plane motions of the plate and of the fluid. In Section 6.4 we derive the optimum shapes of long rectangular plates simply supported along the longer edges. In Section 6.4 we consider a specific case of a trilayered plate whose optimum shape is determined by analytic techniques. In Section 6.5 we study a two-dimensional parallel flow problem concerning divergence (i.e., the loss of stability caused by the action of hydrodynamic forces) of an elastic plate. We seek an optimum distribution of thickness for plates having a maximal rate of divergence. To determine nontrivial equilibrium shapes for rectangular plates that are clamped along one edge, we present in Section 6.6 a description of a solenoidal flow in the presence of an infinite cavity, and we study some properties of the corresponding boundary-value problem. We consider an optimization problem of finding the thickness function for plates whose nontrivial equilibrium shape coincides with the largest value of the velocity of the impacting fluid. Results presented in this chapter were partially published in Refs. 31–34.
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