PSI - Issue 44

Sourav Das et al. / Procedia Structural Integrity 44 (2023) 1680–1687 Das and Tesfamariam/ Structural Integrity Procedia 00 (2022) 000–000

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integration method (Au and Beck 2001, Bourinet et al. 2011). This paper focuses on probability density evolution method (PDEM), belongs to the numerical integration method, proposed by Li and Chen (2008). The PDEM is formulated based on the principle of probability conservation. To estimate probability density function (PDF) of the performance function using PDEM, a proper strategy is required to generate the representative points of the random variables present in the system. Generalized F (GF) discrepancy-based method (Chen et al. 2016) is the most common and popular for estimating PDF using PDEM. To improve the PDF’s estimation accuracy, the PDEM requires large number of representative points, which becomes computationally expensive for high-fidelity models. To reduce the computational burden, surrogate model is required which approximates the response surface. Polynomial chaos expansion (PCE) is one of the popular surrogate models in which the output response is expressed in the form of a spectral representation considering the entire domain of random parameters. As a result, it sometimes does not capture the nonlinearity of the system accurately. In this study, stochastic spectral embedding method proposed by Marelli et al. (2021) which is an advanced version of PCE is used. This method combines spectral representation and adaptive domain decomposition. The entire domain is divided into a number of subdomains. This discretization occurs adaptively until the residual between original and predicted responses becomes minimal. The present study sets the following objectives: The reliability analysis of the stochastic system is carried out using first-passage reliability method where probability density function of the performance function is estimated using PDEM; Develop an efficient algorithm, stochastic spectral embedding surrogate based PDEM for reducing the computational cost involved in the traditional PDEM; and asses performance of the proposed method is compared with Monte-Carlo simulation. The outline of the study is structured as follows. Section 2 describes the complete overview of the reliability analysis of the stochastic system using PDEM. Section 3 provides a brief description of stochastic spectral embedding. The numerical applications of the proposed method are presented in Section 4. Finally, Section 5 presents the concluding remarks. 2. Reliability Estimation of Stochastic System Consider an N -degrees of freedom structural system that is subjected to external dynamic excitation. The governing equation of motion of the structural system, considering vector of random parameters Θ associated with the structural properties and external excitation, can be written as: ( ) ( ) ( , ) ( , ) Θ + Θ + Θ = Γη Θ M X C X f X   t (1) where M and C are the mass and damping matrix of the structure, respectively; f is the restoring force vector. The displacement, velocity, and acceleration vectors of the structure are denoted by X , ̇ and ̈ , respectively. The influence vector is represented by Γ , and the external excitation is represented by η ( t ). The governing equation of motion depicted in Eq. 1 can be expressed in state-space form, which is given by: ( , , ) = Θ Y A Y  t (2) 2.1. Probability Density Evolution Method (PDEM) The PDEM proposed by Li and Chen (2008) is used for the reliability analysis, which is formulated based on the principle of probability conservation. Let ( , ) represents the PDF of the i th component of Y ( t ). The change in probability due to the transition of probability through the boundaries y L and y R during the time interval [ t , t + Δ t ] is expressed as (Li and Chen 2008): ( , ) ( , ) ( , ) ( )  ∂  ∆ = + ∆ − = ∆ + ∆   ∂   ∫ ∫ ∫ R R R i i i L L L y y y Y J Y Y y y y p y t P p y t t dy p y t dy dy t o t t (3) where Y is the state vector which is expressed as [ ] 1 2    Y X X  T T T  = = … Y T N Y and [ ] T N A A . 1 2 = … A