Combined Subset Simulation and Comprehensive Learning Particle Swarm Optimization in Reliability-Based Structural Optimization

Reliability-based design optimization (RBDO) addresses the cost-effective integrity design of structures in the presence of inherent uncertain parameters. Processing this class of problem is challenging from the computational burden to determine the failure probability of structures violating the limit-state function. This paper proposes an efficient decoupling RBDO method that advantageously couples a comprehensive learning particle swarm optimization (CLPSO) algorithm with a subset simulation (SS), termed as SS-CLPSO approach. In essence, the proposed method iteratively performs the CLPSO assuming deterministic parameters based on the most probable point underpinning limit-state functions updated within the reliability evaluation process. Based on the CLPSO design data, the SS approximates the spectrum of limit-state functions under uncertain parameters, and hence enables the significant reduction of Monte-Carlo simulations for the failure probability prediction. The SS map outs the failure probability from the conditional samples constructed at each intermediate event. The proposed SS-CLPSO terminates the optimal solution to the RBDO problem as when the resulting failure probability converges to the permissible threshold. The applications of the present approach are illustrated through the steel truss design under probabilistic uncertain parameters and constraints.


Introduction
Deterministic optimization has been extensively applied in engineering structures to improve the design performance with minimum resources. The design solution computed by the deterministic optimization becomes unreliable in some cases, especially when the influences of uncertainties inheriting structural dimensions, material properties, loading and operating conditions are significant and cannot be eliminated. By addressing the performance and reliability of the structure together, the structural reliability-based design optimization (RBDO) has been considered as the alternative approach in recent years. More explicitly, the RBDO problem minimizes the cost function, denoted as C, and satisfies the certain deterministic and probabilistic constraints, as state by the following generic mathematical formulations [1]: The vector x is characterized by the joint probability density function (PDF) f(x) in the space Ω. The two L s and U s denote the lower and upper bounds on the variables s. The functions G(s,x) respectively the performance and limit-state design expressions considered, where z is a constant threshold of the failure domain specified for the design variables ( i.e.
, where a P is the allowable threshold of Pf.

CLPSO Algorithm
Whilst the general PSO algorithm provides the good convergence rate to the optimal solutions, the method is often trapped into the local optimal. To enable the diversity of design particles and overcome the premature convergence, Liang and Qin [2] Where i and d are the array indices of the particles for   . exp / exp( ) 10 1 0 05 0 45 1 10 1 1 (4)

Subset Simulation
Subset simulation (SS) is a well-known, efficient Monte Carlo technique for variance reduction in the structural reliability community. It exploits the concept of conditional probability and advanced Markov Chain Monte Carlo technique [4]. The formulation for SS optimization is given by b) Define the th j coordinate of the candidate sample by accepting or rejecting j  : where NMCS is the number of random samples generated within the space Ω, and [.] denotes the indicator function, namely

Combined SS-CLPSO Algorithm
Th proposed SS-CLPSO method for the optimal solution of the RBDO in Eq. (1) is summarized: Step 1: Initialize the random variables in the original RBDO Eq. (1).
Step 2: Perform the deterministic optimization using the CLPSO algorithm to obtain the optimal design solutions.
Step 3: Import s j into SS using   s , j  N 0 1 , number of nc samples and ns cycles, and then sort the results G0(s, x) in descending order first. Store the value G0 (nc) (s, x), then select nc sample to create the next level.
Step 4: The new sample group must be less than G0 (nc) (s, x), the number of nc samples and ns cycles, and then sort the results G1(s, x) in ascending order first. Collect the value G1 (nc) (s, x), then select nc for example and calculate Pf, If Pf > a P , repeat this Step 4 until xMMP is achieved.
Step 5: Bring xMMP repeat Steps 2 to 5. If the estimated failure probability Pf converges, terminate, where nc = np is the number of Markov chains, ns = n/nc is the number of states in each chain, n is the number of samples per level, and p= [0.1,0.2] is the level probability  A 10-bar truss in Fig. 1 was considered [1]. The RBDO problem in Eq. (1)  with the probabilistic properties listed in Table 1. The vertical displacement at node 3, denoted as  3 , was considered as the response performance of interest ( y   3 ), whose probability of exceeding the allowable value of z    3 4 10 m. was less than or equal to a P =6.21×10 -3 . Thus, the specific RBDO Eq. (1) was written as follows: In the first instance, the initial random variables were assigned to take the mean values given in Table 1. The deterministic counterpart of the problem in Eq. (11) was solved using the CLPSO algorithm. The parameters adopted: the total number of particle populations of ps = 30; c =1.5; the inertial weight of w linearly declining from 0.9 to 0.4; and the maximum number of iterations of 1,000 as per each particle set. The parameters employed in SS were samples per level n=1,000 and level probability p =0.1 set. The coupling CLPSO and SS procedures were iterated until the estimated probability of failure (viz., Pf = 6.07×10 −3 ) was converged and well complied with the limit of a P = 6.21×10 −3 . The proposed SS-CLPSO method was encoded in Python, and run using the computer hardware with RYZEN 7 4800 HS CPU @ 2.9 GHz and 16 GB RAM. The optimal solutions reported in Table 2 were successfully computed and agreed very well with the benchmark references [1,6,7].

Conclusion
The combined SS-CLPSO method determine the optimal solution of the RBDO problem. It is based the cooperative procedures between CLPSO (the sizing design of members) and SS (the efficient failure probability approximations). A number of steel truss design examples and benchmarks (one of which has been provided herein) illustrate the robustness and accuracy of the proposed RBDO solution approach.