PSI - Issue 52

Dong Xiao et al. / Procedia Structural Integrity 52 (2024) 667–678 Dong Xiao et al. / Structural Integrity Procedia 00 (2023) 000–000

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The pseudocode for implementing the method is outlined as follows, assuming the availability of structural re sponses for the target impact ¯ S : 1. Identify the impact duration bound ([ T l , T u ]) using continuous wavelet transfer (CWT). 2. Extract time of arrivals ( TOA ) for the target impact. 3. Initialize the impact location ( x 0 , y 0 ) and impact force ( p 0 ) of target impact based on the impact duration bound and the time of arrivals. 4. Establish FEM model for simulating impacts and evaluate the structural responses S for initialized impacts. 5. Extract time of arrivals ( TOA ), peak force ( p max ), peak response ( S max ), force gradient ( G = p max / S max ), and the relative response di ff erences ( d = S − S 2 2 / S 2 2 ) between the simulated impacts and the target impact. 6. Identify the impact with minimal response di ff erences if the di ff erences are below a predetermined threshold or the maximum number of iterations is reached; otherwise, proceed to the following steps of surrogate updating and infill sampling. 7. Establish Kriging meta-model d = F opt ( x , y , p ) to relate the impact location ( x , y ), impact force ( p ) with the response di ff erences ( d ). 8. Construct surrogates for impact location ( x , y ) = F loc ( TOA ) and impact peak force p max = S max ∗ F G ( TOA )with time of arrivals as inputs. 9. Determine the location bound ([ x l , x u ] , [ y l , y u ]) and peak force bound ([ p l max , p u max ]) of target impact based on the two surrogates. 10. Infill sampling based on the Kriging meta-model using the generalized expected improvement criterion, consid ering the bound constraints of impact location, impact peak force. New samples (combinations of impact location and impact force) are generated. 11. Update the impacts pool by enhancing the impacts corresponding to the infilled samples, and estimate the struc tural responses of the enhanced impacts using the FEM model. Then, return to step 5. The ordinary Kriging meta-model approximate the relationships between inputs and outputs as a stochastic process Jones et al. [1998]; Forrester et al. [2008]; Viana et al. [2013]; Haftka et al. [2016]; Santner et al. [2018]. This process is represented as a combination of a global mean contribution ( µ ) and a localised variation contribution expressed through a stationary Gaussian process denoted as Z ( x ). Y ( x ) = µ + Z ( x ) (1) The covariance between elements of this stochastic process is given as Cov i j = Cov Z ( x i ) , Z ( x j ) = σ 2 R i j ( θ ) (2) where σ 2 denotes process variance, R i j represents the correlation of point x i and x j . In this study, the Gaussian corre lation function is adopted, defined as an exponential decay function controlled by the hyperparameter θ : R i j ( θ ) = exp − n k = 1 θ k x i k − x j k 2 (3) where n the dimension of inputs. To estimate the parameters of the Kriging meta-model, namely µ , σ 2 and θ , the likelihood function is maximized using the training points ( x ) and their corresponding responses ( y ). Given the optimal hyperparameters θ , the value of µ and σ 2 can be estimated as: ˆ µ = (4) (5) The symmetric matrix R represents the correlation matrix of training points x , whose i-row, j-column element is R i j . y denotes the responses vector and 1 indicates the vector consisting of 1, m is the number of points in x . 2.2. Kriging meta-model 1 ′ R − 1 y 1 ′ R − 1 1 ˆ σ 2 = ( y − 1 ˆ µ ) ′ R − 1 ( y − 1 ˆ µ ) m