A univariate Chebyshev polynomials method for structural systems with interval uncertainty. (October 2021) - Wei, Tonghui; Li, Feng; Meng, Guangwei; Li, Hongfeng
Abstract: This paper presents an effective univariate Chebyshev polynomials method (UCM) for interval bounds estimation of uncertain structures with unknown-but-bounded parameters. The interpolation points required by the conventional collocation methods to generate the surrogate model are the tensor product of each one-dimensional (1D) interpolating point. Therefore, the computational cost is expensive for uncertain structures containing more interval parameters. To deal with this issue, the univariate decomposition is derived through the higher-order Taylor expansion. The structural system is decomposed into a sum of several univariate subsystems, where each subsystem only involves one uncertain parameter and replaces the other parameters with their midpoint value. Then the Chebyshev polynomials are utilized to fit the subsystems, in which the coefficients of these subsystems are confirmed only using the linear combination of 1D interpolation points. Next, a surrogate model of the actual structural system composed of explicit univariate Chebyshev functions is established. Finally, the extremum of each univariate function that is obtained by the scanning method is substituted into the surrogate model to determine the interval ranges of the uncertain structures. Numerical analysis is conducted to validate the accuracy and effectiveness of the proposed method. Highlights: A novel univariate Chebyshev polynomials method for uncertain system is proposed. The original structural system is transformed into several univariate subsystems. Each univariate function is explicitly approximated by Chebyshev polynomials. Uncertain structures with strong nonlinearity or many parameters are well solved.