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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3. Stochastic Spectral Embedding Stochastic spectral embedding is a novel surrogate model proposed by Marelli et al. (2021) that is a combination of spectral representation and adaptive domain decomposition. The flowchart for the reliability analysis using stochastic spectral embedding based PDEM is shown in Fig. 1. Consider a computational model ( )  θ which contains a set of random parameters, θ = [ θ 1 , θ 2 ,…, θ n ] T , with joint probability density function θ ~ f Θ ( θ ). Therefore, the output response of the model ( )  θ can be expressed in the form of spectral representation, which is given by: ( ) ( ) 1 ∞ = = = Ψ ∑  i i i Y a θ θ (12) where a i is the coefficient of the orthonormal basis function. In practice, the output response cannot be decomposed into an infinite number of series as in Eq. 12. Therefore, a truncation is needed, which is expressed as: ( )  ( ) ( ) 1 = = ≈ = Ψ ∑   P i i i Y a θ θ θ (13) where P is the truncation order of the polynomial expansion. The residual is given by: ( ) ( )  ( ) = −    θ θ θ (14)

Fig. 1. Flowchart for reliability analysis using PDEM

The basis function ψ has support on the entire domain Ω θ in polynomial chaos expansion (PCE). Due to that, sometimes PCE is unable to capture the nonlinearity present in the model. The entire domain is divided into smaller subdomains in stochastic spectral embedding (i.e., , Θ Θ Ω ⊆ Ω m n , where m is the level of expansion and n is the subdomain index within the level). When m = 0 i.e., a single domain is considered, 0,1 Θ Θ Ω = Ω , then residual is the same as for PCE, in Eq. 14. For multi-level of expansion, Eq. 13 is re-written as:  ( ) ( ) ( ) , , 0 1 Θ Ω = = = ∑∑   m m n M N m n m n I θ θ θ (15)