Structural and Multidisciplinary Optimization

, Volume 52, Issue 2, pp 337–352 | Cite as

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

  • D. Chernikov
  • P. Krokhmal
  • O. I. Zhupanska
  • C. L. Pasiliao


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.


PDE-Constrained optimization Two-stage stochastic optimization Electro-magneto-mechanical coupling Composite materials 



Olesya Zhupanska and Dmitry Chernikov would like to acknowledge the support of DARPA, N66001-11-1-4133 (Disclaimer: Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the views of DARPA). Dmitry Chernikov is grateful for the support from the AFRL Mathematical Modeling and Optimization Institute during Summer 2013 and Summer 2014. Pavlo Krokhmal would like to acknowledge AFOSR grant FA9550-12-1-0142 and the U.S. Dept of Air Force grant FA8651-12-2-0010.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2015

Authors and Affiliations

  • D. Chernikov
    • 1
  • P. Krokhmal
    • 1
  • O. I. Zhupanska
    • 1
  • C. L. Pasiliao
    • 2
  1. 1.Department of Mechanical and Industrial EngineeringUniversity of IowaIowa CityUSA
  2. 2.Air Force Research LabEglin AFBValparaisoUSA

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