A two-stage stochastic PDE-constrained optimization approach to vibration control of an electrically conductive composite plate subjected to mechanical and electromagnetic loads

Abstract

A new two-stage stochastic partial differential equation (PDE)-constrained optimization methodology is developed for the active vibration control of structures in the presence of uncertainties in mechanical loads. The methodology relies on the two-stage stochastic optimization formulation with an embedded first-order black-box PDE-constrained optimization procedure. The PDE-constrained optimization procedure utilizes a first-order active-set algorithm with a conjugate gradient method. The objective function is determined through solution of the governing PDEs and its gradient is computed using automatic differentiation with hyper-dual numbers. The developed optimization methodology is applied to the problem of post-impact vibration control (via applied electromagnetic field) of an electrically conductive carbon fiber reinforced composite plate subjected to an uncertain, or stochastic, impact load. The corresponding governing PDEs consist of a nonlinear coupled system of equations of motion and Maxwell's equations. The conducted computational study shows that the obtained two-stage optimization solution allows for a significant suppression of vibrations caused by the randomized impact load in all impact load scenarios. Also, the effectiveness of the developed methodology is illustrated in the case of a deterministic impact load, where the two-stage strategy enables one to practically eliminate post-impact vibrations.