Exploiting expansion basis sparsity for efficient stochastic response determination of nonlinear systems via the Wiener path integral technique

• Published in: Nonlinear dynamics Vol. 107; no. 4; pp. 3669 - 3682
• Main Authors: Zhang, Yuanjin, Kougioumtzoglou, Ioannis A., Kong, Fan
• Format: Journal Article
• Language: English
• Published: Dordrecht Springer Netherlands 01.03.2022
• Subjects: Automotive Engineering; Classical Mechanics; Control; Dynamical Systems; Engineering; Mechanical Engineering; Original Paper; Vibration; Sparse representations; Stochastic dynamics; Compressive sampling; Nonlinear systems; Path integral
• ISSN: 0924-090X; 1573-269X
• DOI: 10.1007/s11071-021-07153-0

The computational efficiency of the Wiener path integral (WPI) technique for determining the stochastic response of diverse nonlinear dynamical systems is enhanced herein by relying on advanced compressive sampling concepts and tools. Specifically, exploiting the sparsity of appropriately selected expansions for the joint response probability density function (PDF), and leveraging the localization capabilities of the WPI technique for direct evaluation of specific PDF points, yield an underdetermined linear system of equations to be solved for the PDF expansion coefficients. This is done by resorting to L p -norm ( 0 < p < 1 ) minimization formulations and algorithms, which exhibit an enhanced sparsity-promoting behavior compared to standard L 1 -norm minimization approaches. This translates into a significant reduction of the associated computational cost. In fact, for approximately the same accuracy degree, it is shown that the herein developed technique based on L p -norm ( 0 < p < 1 ) minimization requires, in some cases, even up to 40% fewer boundary value problems to be solved as part of the solution scheme than a standard L 1 -norm minimization approach. The reliability of the technique is demonstrated by comparing WPI-based response PDF estimates with pertinent Monte Carlo simulation (MCS) data (10,000 realizations). In this regard, realizations compatible with the excitation stochastic process are generated, and response time-histories are obtained by integrating numerically the nonlinear system equations of motion. Next, MCS-based PDF estimates are computed based on statistical analysis of the response time-histories. Several numerical examples are considered pertaining to various stochastically excited oscillators exhibiting diverse nonlinear behaviors. These include a Duffing oscillator, an oscillator with asymmetric nonlinearities, and a nonlinear vibro-impact oscillator.