A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertainty
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ORIGINAL PAPER
A polynomial chaos expansion approach for nonlinear dynamic systems with interval uncertainty Liqun Wang
. Zengtao Chen . Guolai Yang
Received: 7 May 2020 / Accepted: 12 August 2020 Springer Nature B.V. 2020
Abstract This paper proposes a non-intrusive interval uncertainty analysis method for estimation of the dynamic response bounds of nonlinear systems with uncertain-but-bounded parameters using polynomial chaos expansion. The conventional interval arithmetic and Taylor series methods usually lead to large overestimation because of the intrinsic wrapping effect, especially for the multidimensional and nonmonotonic problems. To overcome this drawback, a novel polynomial chaos inclusion function, based on the truncated polynomial chaos expansion, is proposed in the present work to evaluate interval functions. In this method, the Legendre polynomial in interval space is employed as the trial basis to expand the interval processes, and the polynomial coefficients are calculated through the collocation method. Two examples show that the polynomial chaos inclusion function is capable of determining tighter enclosures of the true solutions and effectively dealing with the wrapping effect. The response of nonlinear systems with respect to interval variables is approximated by the polynomial chaos inclusion function, through L. Wang G. Yang (&) School of Mechanical Engineering, Nanjing University of Science and Technology, Nanjing 210094, People’s Republic of China e-mail: [email protected] Z. Chen Department of Mechanical Engineering, University of Alberta, Edmonton T6G 1H9, Canada
which the supremum and infimum of the dynamic responses over all time iteration steps can be easily estimated by an appropriate numerical solver. Four dynamics examples described by ordinary differential equations demonstrate the effectiveness, feasibility, and efficiency of the proposed interval uncertainty analysis method compared with other methods. Keywords Nonlinear dynamic response Polynomial chaos expansion Interval uncertainty Ordinary differential equations Uncertainty propagation
1 Introduction Dynamic response calculation is preliminarily important in structural design and analysis for most engineering problems. However, a variety of uncertainties exist in dynamic systems, related to material properties, external loads, dimensional tolerances, boundary conditions, and environmental factors. These uncertainties will inevitably affect the final system performances, and small variations associated with uncertainties might result in significant changes in the dynamic responses. Latest scientific and engineering advances [1, 2] have recognized the importance of uncertainty and widely applied uncertainty methods to static structural response analysis [3–5], eigenvalue
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calculation [6–8], natural frequency analysis [9, 10], reliability analysis [11, 12], and uncertain design optimization [13, 14]. The probabilistic model [1, 7,10] has
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