## 1 Introduction  
Traditional structural analysis process generally starts with the establishment of reasonable mechanical models, and then obtains the responses based on the deterministic structural parameters and certain external loads [1; 2]. However, a number of uncertainties of involved parameters are unneglectable in practical engineering, which may lead the results of reliability analysis to considerable deviations [3; 4; 5]. To assess these uncertainties, the probability-based reliability analysis methods spring up and then are maturely applied to a number of sectors and societies [6; 7; 8; 9]. The main work in structural reliability analysis is to obtain the failure probability by solving the following multi-dimensional integration:  
\[P_{f}=\Pr[g(\mathbf{X})<0]=\int\limits_{g(\mathbf{X})<0}f_{X}(\mathbf{X})dX \tag{1}\]  
where \(\Pr[\cdot]\) is the probability, \(\mathbf{X}=(X_{1},X_{2},\cdots,X_{n})\) are the basic random variables, \(g(\mathbf{X})\) is the limit state function (LSF) and \(f_{X}(X)\) is the joint probability density function (PDF) of \(\mathbf{X}\). Generally, it is arduous to calculate the multiple integral in Equation (1) directly. The commonly used methods are approximate methods and simulation methods [10]. The approximate methods include the first-order reliability method (FORM) [11; 12] and the second-order reliability method (SORM) [13; 14], which are defined under the concept of the first and second-order Taylor expansion of the LSF, respectively. The simulation methods, such as direct Monte-Carlo simulation (MCS) [15; 16], line sampling [17], and importance sampling [18; 19] evaluate the reliability through a considerable amount of stochastic simulations. In engineering practice, the FORM is used the most because of its simplicity and the good balance between accuracy and efficiency. The Hasofer-Lind and Rakwitz-Fiessler (HL-RF) algorithm, which is proposed by [PERSON] and [PERSON] and then is extended by [PERSON] and [PERSON] [11; 12], is a classical and widely used algorithm in the FORM. However, it may converge in a low speed or even fail to converge any more under the situation that highly nonlinear LSFs are involved. This limitation inspires researchers to devote their efforts to develop algorithms both in academia and the engineering practice sectors. [PERSON] et al. [20] developed a modified HL-RF (MHL-RF) algorithm by introducing a stepsize parameter. The merit function, based on the augmented Lagrange method, is used to find the proposed stepsize. [PERSON] et al. [21] constructed a simpler merit function; the proposed algorithm named as iHL-RF is superior to the MHL-RF algorithm in efficiency. [PERSON] et al. [22] established a differentiable merit function, and the Wolfe condition is implemented to search the stepsize. [PERSON] et al. [23] presented a stepsize search method for the MHL-RF algorithm based on the variant of [PERSON]'s rule. These stepsize determinations have increased a lot the computational effort. [PERSON] [24] employed chaotic dynamics analysis to construct a stability transformation method (STM) for removing the chaos phenomenon of the HL-RF algorithm. The convergence rate, however, proved to be slow. [PERSON] [25] proposed the chaotic conjugate STM algorithm based on the conjugate gradient method. [PERSON] et al. [26] established a directional STM algorithm using the chaos feedback control approach. [PERSON] et al. [27] presented a relaxed HL-RF (RHL-RF) algorithm by introducing the relaxed coefficient, which is dynamically calculated by the relaxed approach. As a continuation of the research, [PERSON] et al. [28] developed the adaptive STM and enhanced adaptive STM algorithms by transforming the calculation of the most probable failure points into finding the most probable target points. An inappropriate control factor may lead to inefficiency of these algorithms.