A second-order perturbation method for fuzzy eigenvalue problems

Abstract

Purpose For eigenvalue problems containing uncertain inputs characterized by fuzzy basic parameters, first-order perturbation methods have been developed to extract eigen-solutions, but either the result accuracy or the computational efficiency of these methods is less satisfactory. This paper presents an efficient method for estimation of fuzzy eigenvalues with high accuracy. Design/methodology/approach Based on the first order derivatives of eigenvalues and modes with respect to the fuzzy basic parameters, expressions of the second order derivatives of eigenvalues are formulated. Then a second-order perturbation method is introduced to provide more accurate fuzzy eigenvalue solutions. Only one eigenvalue solution is sought for the perturbed formulation, and quadratic programming is performed to simplify the alpha-level optimization. Findings Fuzzy natural frequencies and buckling loads of some structures are estimated with good accuracy, illustrating the high computational efficiency of the proposed method. Originality/value Up to the second order derivatives of the eigenvalues with respect to the basic parameters are represented in functional forms, which are used to introduce a second-order perturbation method for treatment of fuzzy eigenvalue problems. The corresponding alpha-level optimization is thus simplified into quadratic programming. The proposed method provides much more accurate interval solutions at alpha-cuts for the membership functions of fuzzy eigenvalues. Analogously, third- and higher-order perturbation methods can be developed for more stringent accuracy demands or for the treatment of stronger nonlinearity. The present work can be applied to realistic structural analysis in civil engineering, especially for those structures made of dispersed materials such as concrete and soil.

Application of the polynomial dimensional decomposition method in a class of random dynamical systems

“…The goal is to allow an estimate of dynamic responses generated by these considerations on physical parameters. There are a variety of methods in a view of uncertainties for this type of issues: such as the perturbation method [16,17], Neumann method [18,19], MCS method [20][21][22], polynomial chaos expansion (PCE) method [23][24][25], and PDD method [26][27][28]. The perturbation method is based on the expansion of random quantities into Taylor series [29], and the Neumann method is on the basis of Neumann series [30,31], they can both solve the small random fluctuations problems but do not fit for the case close to the resonant frequency.…”