PSI - Issue 48

Behrooz Keshtegar et al. / Procedia Structural Integrity 48 (2023) 348–355 Author name / Structural Integrity Procedia 00 (2019) 000 – 000

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where, , , , [0,1] 1 2 3  r r r r are random numbers uniform between 0 and 1, HMCR is the harmony memory considering rate, here taken as 0.95, PAR is the pitch adjusting rate , here taken as 0.4 and bw is the bandwidth, here taken as σ i /100; i i L i x K     is the lower bound and i i U i K x     is the upper bounds ( K is a scalar factor taken as K =5, ) for random variable i x , with mean i  and standard deviation i  . j inew x is the new harmony memory for variable i and harmony element j. The new elements of harmony are adjusted based on the bandwidth with probability of PAR×HMCR and they are selected from the design domain with probability of 1- HMCR . The new random variables in the new harmony memory are replaced in the old memory when the best condition is obtained based on the new adjusting process. 4. Comparative results Three examples are used to illustrate the convergence performances of HL-RF without any controlling factor, DSTM with parameters of C = I and  =0.1, PSO and HS with number of particles/ harmony size of 5. The convergence stopping criterion for gradient and non- gradient methods are respectively given as 6 1 || 10 |    k k |U -U and g( X )<0.001 . Example 1: A highly nonlinear performance function is given by the following relation (Zhao and Ono 2004): This problem involves two normal random variables with μ 1,2 =10 and σ 1,2 = 5. The reliability index by MCS with 10 7 samples is computed as β = 1.98. The reliability index values using DSTM, HL-RF, PSO and HS are presented in Fig.1. It turns out that the MPP =[0,10] with reliability index of 2.00 using the PSO with 100 total iterations, whereas the HS provides a reliability index of 1.79. The results of Fig.1 indicate that the gradient-based FORM methods using DSTM and HL-RF are unstably converged to chaotic reliability index, whereas the non-gradient approaches – based PSO and HS provide stable results. The PSO is more in agreement with the MCS reliability index than the HS, accuracy in reliability index improved of about 10% compared to HS. The modified version of analytical reliability analysis using DSTM cannot guarantee the stability of the results on this nonlinear performance function although its convergence performance is better than HL-RF. The non-gradient methods can provide stable results but their accuracies are dependent on the parameters and formulations of the optimization process. 1 1 3 1 2 1 / x x x 18/ ( ) x g    X (12)

Fig. 1. Convergence reliability index using PSO, HS, DSTM and HL-RF for Example 1

Example 2: A roof truss showed in Fig. 2 is considered with the following performance function (Li et al. 2016):