A joint time-space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems非线性系统高效随机响应分析的 Wiener 路径积分技术联合时空外推方法
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Mavromatis IG, Kougioumtzoglou IA, 2024. A joint time-space extrapolation approach within the Wiener path integral technique for efficient stochastic response determination of nonlinear systems. Probabilistic Engineering Mechanics, 78: 103685.DOI: 10.1016/j.probengmech.2024.103685
对 Wiener 路径积分 (Wiener path integral, WPI) 技术开发了一种联合时空外推方法，用于高效精确地确定各类非线性动力系统的非平稳随机响应。该方法可理解为最近发展的空间域外推格式的扩展，以考虑时间维度。具体地，基于变分原理，Wiener 路径积分技术需求解一类边值问题 (boundary value problem, BVP) 以确定给定边界终止条件下的最可能路径。此外，最可能路径用于近似估计给定时刻下系统响应的联合概率密度函数 (probability density function, PDF) 值。值得注意的是，本文利用边值问题表现出的两个特性发展了一类高效联合时空外推方法。首先，响应概率密度函数空间域相邻网格点的边值问题具有同一方程，且边界条件也很相近。其次，可采用已确定的最可能路径时程信息估计任意时刻下响应概率密度函数值，而无需求解额外的边值问题。简言之，利用上述 Wiener 路径积分边值问题的优势，可确定完全非平稳响应的联合概率密度函数，首先，数值计算少量概率密度函数值，其次，在联合时空域中外推，几乎不需要额外计算成本。与 Wiener 路径积分技术的标准蛮力实现方法相比，所提外推方法将相应计算成本降低了几个数量级。为验证外推方法的可靠性，考虑了含非对称非线性和分数阶导数项的振子以及随机和确定性周期荷载组合下的非线性结构作为两个数值算例，还包括与相应蒙特卡罗模拟数据的对比。关键词: 随机动力学, 非线性系统, 路径积分, 最可能路径, 边值问题A joint time–space extrapolation approach within the Wiener path integral (WPI) technique is developed for determining, efficiently and accurately, the non-stationary stochastic response of diverse nonlinear dynamical systems. The approach can be construed as an extension of a recently developed space-domain extrapolation scheme to account also for the temporal dimension. Specifically, based on a variational principle, the WPI technique yields a boundary value problem (BVP) to be solved for determining a most probable path corresponding to specific final boundary conditions. Further, the most probable path is used for evaluating, approximately, a point of the system response joint probability density function (PDF) corresponding to a specific time instant. Remarkably, the BVP exhibits two unique features that are exploited in this paper for developing an efficient joint time–space extrapolation approach. First, the BVPs corresponding to two neighboring grid points in the spatial domain of the response PDF not only share the same equations, but also the boundary conditions differ only slightly. Second, information inherent in the time-history of an already determined most probable path can be used for evaluating points of the response PDF corresponding to arbitrary time instants, without the need for solving additional BVPs. In a nutshell, relying on the aforementioned unique and advantageous features of the WPI-based BVP, the complete non-stationary response joint PDF is determined, first, by calculating numerically a relatively small number of PDF points, and second, by extrapolating in the joint time–space domain at practically zero additional computational cost. Compared to a standard brute-force implementation of the WPI technique, the developed extrapolation approach reduces the associated computational cost by several orders of magnitude. Two numerical examples relating to an oscillator with asymmetric nonlinearities and fractional derivative elements, and to a nonlinear structure under combined stochastic and deterministic periodic loading are considered for demonstrating the reliability of the extrapolation approach. Juxtapositions with pertinent Monte Carlo simulation data are included as well.Keywords: Stochastic dynamics; Nonlinear system; Path integral; Most probable path; Boundary value problem.图 1: Wiener 路径积分技术的蛮力数值实现Fig. 1. Brute-force numerical implementation of the WPI technique: Solution of N_tN^2m boundary value problems for determining the non-stationary response joint PDF, demonstrated for a generic single-DOF oscillator (m = 1)
图 2: 减少待求边值问题数以确定完全非平稳响应的联合概率密度函数Fig. 2. Determination of the complete non-stationary response joint PDF by reducing the required number of boundary value problems to be solved from N_tN^2m to M^2m (with M << N)
图 3: 含非对称非线性与分数阶导数项振子在给定网格数下的最可能路径Fig. 3. Most probable paths referring to a grid of M^2 = 7^2 points (x_f,x_f) at t_f = 3 s (black points) corresponding to an oscillator with asymmetric nonlinearities and fractional derivative elements
图 4: 含非对称非线性与分数阶导数项振子在代表性时刻的非平稳响应联合概率密度函数: (a) Wiener 路径积分技术结合外推获得的结果; (b) 蒙特卡罗模拟数据对比Fig. 4. Non-stationary response joint PDF at representative time instants t_f = 1, 2, and 3 s corresponding to an oscillator with asymmetric nonlinearities and fractional derivative elements: (a) Results obtained by the WPI technique at M^2 = 7^2 points and extrapolated to N^2 = 101^2 points; (b) Comparison with MCS data (100,000 realizations)
图 5: 随机荷载与确定性谐波激励组合下含三次刚度与阻尼非线性两自由度系统在代表性时刻的非平稳响应联合概率密度函数: (a) Wiener 路径积分技术结合外推获得的结果; (b) 蒙特卡罗模拟数据对比Fig. 5. Non-stationary response joint PDF p(x_{1,f},x_{2,f},t_f | 0,0,0) at representative time instants t_f = 1, 2.5, and 4 s corresponding to a 2-DOF system with cubic stiffness and damping nonlinearities subjected both to stochastic loading and to a deterministic harmonic excitation component: (a) Results obtained by the WPI technique at M^4 = 7^4 points and extrapolated to N^4 = 31^4 points; (b) Comparison with MCS data (100,000 realizations)
图 6: 随机荷载与确定性谐波激励组合下含三次刚度与阻尼非线性两自由度系统在代表性时刻的非平稳响应联合概率密度函数: (a) Wiener 路径积分技术结合外推获得的结果; (b) 蒙特卡罗模拟数据对比Fig. 6. Non-stationary response joint PDF p(x_{1,f},x_{1,f},t_f | 0,0,0) at representative time instants t_f = 1, 2.5, and 4 s corresponding to a 2-DOF system with cubic stiffness and damping nonlinearities subjected both to stochastic loading and to a deterministic harmonic excitation component: (a) Results obtained by the WPI technique at M^4 = 7^4 points and extrapolated to N^4 = 31^4 points; (b) Comparison with MCS data (100,000 realizations)
作者信息 | Authors
美国哥伦比亚大学 (Columbia University) 土木工程与工程力学系
Ioannis A. Kougioumtzoglou, 通讯作者 (Corresp.)美国哥伦比亚大学 (Columbia University) 土木工程与工程力学系Email: ikougioum@columbia.edu
律梦泽 M.Z. Lyu | 编辑 (Ed)
P.D. Spanos | 审校 (Rev)
陈建兵 J.B. Chen | 审校 (Rev)
彭勇波 Y.B. Peng | 审校 (Rev)
