An Efficient Wiener Path Integral Technique Formulation for Stochastic Response Determination of Nonlinear MDOF Systems

2015 ◽  
Vol 82 (10) ◽  
Author(s):  
Ioannis A. Kougioumtzoglou ◽  
Alberto Di Matteo ◽  
Pol D. Spanos ◽  
Antonina Pirrotta ◽  
Mario Di Paola
Keyword(s):  
Path Integral ◽  
Computational Cost ◽  
High Dimensional ◽  
System Response ◽  
Stochastic Response ◽  
Mdof Systems ◽  
Low Dimensional ◽  
Response Determination ◽  
Efficiency And Reliability

The recently developed approximate Wiener path integral (WPI) technique for determining the stochastic response of nonlinear/hysteretic multi-degree-of-freedom (MDOF) systems has proven to be reliable and significantly more efficient than a Monte Carlo simulation (MCS) treatment of the problem for low-dimensional systems. Nevertheless, the standard implementation of the WPI technique can be computationally cumbersome for relatively high-dimensional MDOF systems. In this paper, a novel WPI technique formulation/implementation is developed by combining the “localization” capabilities of the WPI solution framework with an appropriately chosen expansion for approximating the system response PDF. It is shown that, for the case of relatively high-dimensional systems, the herein proposed implementation can drastically decrease the associated computational cost by several orders of magnitude, as compared to both the standard WPI technique and an MCS approach. Several numerical examples are included, whereas comparisons with pertinent MCS data demonstrate the efficiency and reliability of the technique.

Stochastic response determination of nonlinear oscillators with fractional derivatives elements via the Wiener path integral

2014 ◽  
Vol 38 ◽  
pp. 127-135 ◽  
Author(s):  
Alberto Di Matteo ◽  
Ioannis A. Kougioumtzoglou ◽  
Antonina Pirrotta ◽  
Pol D. Spanos ◽  
Mario Di Paola
Keyword(s):  
Path Integral ◽  
Fractional Derivatives ◽  
Nonlinear Oscillators ◽  
Stochastic Response ◽  
Response Determination

Nonlinear Oscillator Stochastic Response and Survival Probability Determination via the Wiener Path Integral

2015 ◽  
Vol 1 (2) ◽  
Author(s):  
Yuanjin Zhang ◽  
Ioannis A. Kougioumtzoglou
Keyword(s):  
Closed Form ◽  
Path Integral ◽  
Survival Probability ◽  
Nonlinear Oscillator ◽  
Transition Probability ◽  
Computational Cost ◽  
Discrete Version ◽  
Stochastic Response ◽  
Transition Probability Density ◽  
Approximate Analytical Expression

A Wiener path integral (WPI) technique based on a variational formulation is developed for nonlinear oscillator stochastic response determination and reliability assessment. This is done in conjunction with a stochastic averaging/linearization treatment of the problem. Specifically, first, the nonlinear oscillator is cast into an equivalent linear one with time-varying stiffness and damping elements. Next, relying on the concept of the most probable path, a closed-form approximate analytical expression for the oscillator joint transition probability density function (PDF) is derived for small time intervals. Finally, the transition PDF in conjunction with a discrete version of the Chapman–Kolmogorov (C–K) equation is utilized for advancing the solution in short-time steps. In this manner, not only the nonstationary response PDF but also the oscillator survival probability and first-passage PDF are determined. In comparison with existing numerical path integral schemes, a significant advantage of the proposed WPI technique is that closed-form analytical expressions are derived for the involved multidimensional integrals; thus, the computational cost is kept at a minimum level. The hardening Duffing and the bilinear hysteretic oscillators are considered as numerical examples. Comparisons with pertinent Monte Carlo simulation (MCS) data demonstrate the reliability of the developed technique.

Stochastic Response of Hysteresis System Under Combined Periodic and Stochastic Excitation Via the Statistical Linearization Method

2021 ◽  
Vol 88 (5) ◽  
Author(s):  
Fan Kong ◽  
Pol D. Spanos
Keyword(s):  
Harmonic Balance Method ◽  
Linearization Method ◽  
System Response ◽  
Statistical Linearization ◽  
Algebraic Equations ◽  
Stochastic Response ◽  
Single Degree Of Freedom ◽  
Stochastic Component ◽  
Linear Differential

Abstract A statistical linearization approach is proposed for determining the response of the single-degree-of-freedom of the classical Bouc–Wen hysteretic system subjected to excitation both with harmonic and stochastic components. The method is based on representing the system response as a combination of a harmonic and of a zero-mean stochastic component. Specifically, first, the equation of motion is decomposed into a set of two coupled non-linear differential equations in terms of the unknown deterministic and stochastic response components. Next, the harmonic balance method and the statistical linearization method are used for the determination of the Fourier coefficients of the deterministic component, and the variance of the stochastic component, respectively. This yields a set of coupled algebraic equations which can be solved by any of the standard apropos algorithms. Pertinent numerical examples demonstrate the applicability, and reliability of the proposed method.

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