PSI - Issue 48

Mohamed El Amine Ben Seghier et al. / Procedia Structural Integrity 48 (2023) 356–362 Ben Seghier et al / Structural Integrity Procedia 00 (2019) 000 – 000

360

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Ref.(Keshtegar 2016) using FORM based chaotic conjugate chaos control method is 3.259315, while the reliability index obtained using Mont Carlo simulation (MCS) with a number of simulation equal to 1.2 × 10 6 is 3.501. For 100 function call for the above LSF, the hyper-parameters of SVR as regularization coefficient C takes the values of 5000, 50000 and 500000, whereas three different values are given for the kernel parameter as σ=2, 5 and 10 with the same precision parameter ε that equal to 0.1. All SVR models combined by CFOR M for the structural reliability analysis are investigated, where the computed reliability index results are reported in Table 1. As shown in Table 1, the predicted errors from SVR models to evaluate the limit state function in Eq. (15) depend on parameter s C and σ, whereas inaccurate reliability indexes are predicted by increasing the kernel parameter, while acceptable reliability index predictions are shown at C >50000 with =0.1 and =5 for this problem.

Table. 1. Results of the reliability analysis using different SVR hyper-parameters for the studies example.

C=5000

C=50000

C=500000

SVR parameters

σ=2

σ=5

σ=10

σ=2

σ=5

σ=10

σ=2

σ=5

σ=10

Reliability index Re-Error (MCS) Re-Error (FORM)

2.9389

2.2432

1.9602

3.0262

3.4461

1.8242

3.0262

3.5211

1.7422

16.06

35.93 31.18

44.01 39.86

13.56

1.57 5.73

47.89 44.03

13.56

0.57 8.03

50.24 46.55

9.83

7.15

7.15

By given SVR hyper- parameters the following values as C=500000, ε=0.1 and σ=5, the reliability index corresponding to different training data of SVR is presented in Fig. 1. As seen, by increasing the number of train data to build the SVR model i.e. call function more than 300, the reliability index tend to stabilize at 3.345, whereas this result is more in agreement with the results of the MCS. As a result, in reliability index predictions, training data for nonlinear performance functions, as well as SVR model parameters, are equally important. The iterative reliability index using different reliability methods and their converged results are respectively presented in Fig. 2 and Table 2. The results obtained from Fig. 2 showed that the HL-RF method yielded 4 periodic solutions as β = {2.5118, 1.8550, 2.3857, 1.5966} while the proposed conjugate FORM is converged to stable results, with reliability index equal to 3.2593 after 29 iterations with 553 function call. The CFORM is more robust than the HL RF technique for this nonlinear problem, thus the predicted reliability index using the hybrid SVR combined by CFORM method provides even more stable reliability index results. According to the results, the proposed framework using SVR-CFORM produces more accurate and efficient results than CFORM. When compared to MCS, the proposed method reduces the computational burden of CFORM while producing acceptable results.

Fig. 1. The reliability index with respect to different call function for training the SVR with hyper- parameters of C=500000, ε=0.1 and σ=5.