A novel binomial expansion method for evaluating a Neumann series for the response of a perturbed system

Abstract

This paper presents a new method for evaluating the Neumann series approximation for the response of a dynamic system perturbed by two parameters. The proposed method will be valuable for iterative model updating methods, where major computational savings are gained when the approximation is evaluated multiple times. The method is based on the binomial expansion theory and requires the initial computational investment of formulating expansion vectors which will in turn accelerate the evaluation of the approximation. The proposed method is compared to the standard evaluation method which utilizes the recursive nature of the power series when computing the terms in the series. In comparison to the proposed method, the standard method requires no initial investment but suffers from a larger computational cost of evaluating the approximation. This paper investigates the efficiency of the two methods for evaluating the Neumann series approximation with respect to the number of evaluations, order of approximation, and size of the system. Central processing unit timing tests are performed in order to compare the two methods when used to evaluate the approximate displacement response of vibrating systems. In an analytical example, the standard and proposed evaluation method, as well as the traditional direct solve are applied to an iterative model updating method where the response of a perturbed finite element model is evaluated to compute the objective function. In the example, the proposed method performs up to 11 times faster than the standard evaluation method and 22 times faster than the traditional direct solve.