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A polynomial chaos‐based approach to risk‐averse piezoelectric control of random vibrations of beams
Author(s) -
Marín F.J.,
MartínezFrutos J.,
Periago Francisco
Publication year - 2018
Publication title -
international journal for numerical methods in engineering
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.421
H-Index - 168
eISSN - 1097-0207
pISSN - 0029-5981
DOI - 10.1002/nme.5823
Subject(s) - polynomial chaos , classification of discontinuities , monte carlo method , uncertainty quantification , bernoulli's principle , mathematical optimization , risk aversion (psychology) , mathematics , optimal control , vibration , computer science , mathematical analysis , physics , expected utility hypothesis , statistics , quantum mechanics , thermodynamics
Summary This paper proposes a risk‐averse formulation for the problem of piezoelectric control of random vibrations of elastic structures. The proposed formulation, inspired by the notion of risk aversion in economy, is applied to the piezoelectric control of a Bernoulli‐Euler beam subjected to uncertainties in its input data. To address the high computational burden associated to the presence of random fields in the model and the discontinuities involved in the cost functional and its gradient, a combination of a nonintrusive anisotropic polynomial chaos approach for uncertainty propagation with a Monte Carlo sampling method is proposed. In the first part, the well‐posedness of the control problem is established by proving the existence of optimal controls. In the second part, an adaptive gradient‐based method is proposed for the numerical resolution of the problem. Several experiments illustrate the performance of the proposed approach and the significant differences that may occur between the classical deterministic formulation of the problem and its stochastic risk‐averse counterpart.