PSI - Issue 52

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

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Dong Xiao et al. / Structural Integrity Procedia 00 (2023) 000–000

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With optimal hyperparameter θ , the response of a new point x ∗ be estimated based on equation 1 and 2. It turns out to be the best linear unbiased predictor of response y ( x ∗ ) is: ˆ y ( x ∗ ) = ˆ µ + r ′ R − 1 ( y − 1 ˆ µ ) (6) and the estimation variance is given by: ˆ s 2 ( x ∗ ) = ˆ σ 2 1 − r ′ R − 1 r + ( 1 − 1 ′ R − 1 r ) 2 1 ′ R − 1 1 (7) where r is the correlation vector of point x ∗ points x . It can be deduced that the estimation variance of point x ∗ ∈ x is zero. And a large estimation variance indicates x ∗ is far away from the training points x . E ffi cient global optimisation relies on the establishment of a Kriging meta-model between design variables and the target minimization function. The Kriging meta-model provides estimates of the mean and variance of the target function, which can be utilized to explore the design space. By infill sampling and updating the Kriging meta-model, the minimum of the meta-model can be identified iteratively, and the minimization process gradually converges. In this study, the generalized expected improvement criterion is adopted for infill sampling, as it strikes a balance between exploration and exploitation in the optimisation process. For a stationary Gaussian process, the target function Y at a point x ∗ follows a Gaussian distribution Y ( x ∗ ) ∼ N (ˆ y , ˆ s 2 ) with a probability density function ϕ ( · ). To minimize the target function on the Kriging surrogate, the expected improvement, which represents the probability that the target function Y at point x ∗ is less than current best function value y min , can be expressed as an integral: E [ I ( x ∗ )] = E max ( y min − Y , 0) = y min −∞ ( y min − y ) ϕ ( y ) dy (8) By introducing the cumulative density function Φ , and letting u = ( y min − ˆ y ) / ˆ s , the integration can be expressed in closed form using integration by parts: E [ I ( x ∗ )] = ( y min − ˆ y ) Φ ( u ) + ˆ s ϕ ( u ) , ˆ s > 0 . 0 , ˆ s = 0 . (9) The expected improvement criterion provides a useful tool for infill sampling. However, Schonlau Schonlau pointed out that the classical expected improvement criterion overly emphasizes local search near the optima of the fitted surrogate, neglecting the global search. To address this issue, he proposed the generalized expected improvement criterion by introducing a parameter g . The generalized expected improvement at point x ∗ is expressed as E [ I ( x ∗ ) g ] = ˆ s g g k = 0 ( − 1) k g ! k !( g − k )! u g − k T k (10) where T k represent statistical quantities which can be recursively estimated. The recursive relation is given by: T 0 =Φ ( u ) , T 1 = − ϕ ( u ) , T k = − u k − 1 ϕ u + ( k − 1) T k − 2 (11) With the increase of g , the generalized expected improvement criterion becomes more and more emphasized on global search. By varying the value of g , for example, from 1 to 10, there will be 10 infilled samples generated which reach a balance between local exploitation and global exploration. To estimate the impact force using model-based methods, it is necessary to parameterize the force. In practical scenarios, impact force histories resulting from large-mass impacts that can cause damage often exhibit a shape similar to a half-period sine function. To reduce the dimensionality of the parameterization, the force parameterization method proposed by Yan and Zhou [2009] is employed. This method represents the force using two sine functions with quarter periods. The impact force history can be expressed as follows: p ( t ) = p max sin (2 π f l t ) , 0 ≤ t ≤ 1 / 4 f l . p max cos 2 π f u t − π f u 2 f l , 1 / 4 f l ≤ t ≤ 1 / 4 f l + 1 / 4 f u . (12) 2.3. E ffi cient global optimisation 2.4. Impact force parameterization