A second-order perturbation method for fuzzy eigenvalue problems

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. The purpose of this paper is to present 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 a-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 a-level optimization is thus simplified into quadratic programming. The proposed method provides much more accurate interval solutions at a-cuts for the membership functions of fuzzy eigenvalues. Analogously, third and higher order perturbationmethods can be developed formore stringent accuracy demands or for the treatment of stronger non-linearity. The present work can be applied to realistic structural analysis in civil engineering, especially to those structures made of dispersed materials such as concrete and soil.