# Adaptive stability transformation method of chaos control for first order reliability method

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## Abstract

The efficiency and robustness are two key performance indexes for first-order reliability method (FORM). In this study, two different algorithms, including the adaptive stability transformation method (ASTM) and enhanced adaptive stability transformation method (EASTM), are proposed to improve the efficiency and robustness of FORM. In both ASTM and EASTM, the computation of most probable failure point (MPFP) is converted to find a series of most probable target points (MPTPs), in which the chaos control factor is properly selected by proposing two different algorithms. Four benchmark examples with normal and non-normal random variables and one practical engineering application example are tested to verify the effectiveness of the proposed algorithms. The results illustrate that the proposed algorithms not only are more efficient than other advanced FORM algorithms, but also are very robust.

### Keywords

First-order reliability method Stability transformation method Most probable failure point Most probable target point## 1 Introduction

Uncertainty propagation extensively exists in structural and mechanical engineering system, which should be addressed to guarantee the structural safety with uncertainty factors [1, 2, 3]. To satisfy the reliability requirement at the product design phase, its reliability should be computed to handle these uncertainties by formulating the performance function with random variables [4, 5]. There are many existing methods to evaluate the structure reliability, in which the most probable failure point (MPFP) based methods are one popular type [6]. This type of method evaluates the failure of probability at MPFP by solving the formulation of first-order reliability method (FORM). Actually, FORM is one of the most commonly utilized method in the domains of reliability assessment and reliability-based design optimization (RBDO) because of the high performance of the applicability, efficiency and robustness [7, 8, 9, 10]. Besides, performance measure approach (PMA) is also extensively used in RBDO by computing the most probable target point (MPTP) due to its efficiency and robustness [11, 12].

Until now, a series of MPFP algorithms have been put forward to search MPFP. Hasofer and Lind [13] and Rackwitz and Fiessler [14] introduced the HL–RF algorithm to compute the MPFP. However, it often meets the non-convergence problem and shows the bifurcation, periodic oscillation and chaotic phenomena for nonlinear problem. To address the issue, a series of improved algorithms were developed consecutively, such as *i*HL–RF, *n*HL–RF, full characterization method. [15, 16, 17]. Among them, *i*HL–RF and *n*HL–RF are the two robust methods by given a proper iterative step, but they are inefficient for solving nonlinear problems [15, 18]. Yang et al. [19] indicated that the non-convergence problem of HL–RF algorithm not only is impacted by the curvature value and nonlinearity of limit state function, but also is related to the system property. Then, the stability transformation method (STM) based on chaos theory is used to search the MPFP stably. However, it also shows low convergence rate.

More recently, Meng et al. [20] found that the iteration point of HL–RF algorithm has the directional property, and then the efficiency of STM is improved significantly by proposing directional stability transformation method (DSTM). Moreover, other advanced iterative algorithms, such as conjugate finite-step length method [21], chaotic conjugate stability transformation method [22] and limited Fletcher–Reeves (LFR) method [23], are developed to improve the robustness of MPFP search. All these improved algorithms calculate the MPFP directly by solving a complex nonlinear constraint [24]. On the other hand, it is commonly acknowledged that PMA is more efficient and robust than MPFP search method by solving a simple constraint [11, 12]. Thus, if the concept of PMA can be applied to compute the MPFP, the performance of these algorithms can be enhanced.

This paper is dedicated to improve the efficiency and robustness of MPFP search algorithm through proposing two novel algorithms: adaptive stability transformation method (ASTM) and enhanced adaptive stability transformation method (EASTM). The basic idea of the proposed algorithms is that the MPFP search model is converted into MPTP search model. Based on the DSTM, the reliability analysis model is further simplified to the unconstraint optimization model, and then the MPFP is searched efficiently and robustly. Furthermore, EASTM is developed by estimating the approximate MPTP, so the efficiency is further improved without sacrificing its robustness.

## 2 The concept of FORM

## 3 The iterative algorithms of FORM

### 3.1 HL–RF algorithm

*β*, and MPFP is the closest point from origin to the limit state surface [1]. The probability of failure can be evaluated by reliability index as follows:

*Ф*is the cumulative distribution function in standard normal space (U-space). Assume the performance function is

*and*

**μ***, respectively. If the random variable vector*

**σ***follows the normal distribution, then it can be calculated by*

**X***=*

**X***+*

**μ***. Otherwise, when random variable vector follows the non-normal distribution, the random variables should be transformed to standard normal variables. As shown in Fig. 2a, if HL–RF is used for solving nonlinear problems, the iterative point appears oscillation or non-convergence phenomenon. Thus, other improved algorithms, including*

**σU***i*HL–RF and CC methods, are developed to overcome this problem.

### 3.2 *i*HL–RF iterative algorithm

*i*HL–RF iterative algorithm is widely used for evaluating the reliability due to its efficiency and stability [6]. The iterative formulation is constructed based on the merit function.

**d**^{ k }is the search direction and is computed by

*c*is the constant. Here,

*c*is selected as \(c=\frac{{\left\| {\varvec{U}} \right\|}}{{\left\| {\nabla G({\varvec{U}})} \right\|}}+10,\) according to the reference [6].

## 4 Iterative algorithms based on the chaos control theory

### 4.1 Stability transformation method

*is the vector of iterative function.*

**f**

**U**^{ k }is the

*m*-dimensional random variables at the

*k*th iterative step.

**C**is the

*m*×

*m*-dimensional involutory matrix, and the elements in each row and each column have only one element with value of 1 or − 1 and the other elements are 0. So the total number of involutory matrix is 2

^{ m }

*m*!. The control factor

*λ*

^{ k }is the range [0, 1] and is determined by the spectral radius of the original system’s Jacobian matrix. The larger the spectral radius of Jacobian matrix is, the smaller the control factor

*λ*should be selected to achieve stability. Specially, when

**C**=

**I**, Eq. (9) becomes

A two-dimensional example is shown in Fig. 2b. It is found that STM achieves the stability by reducing the iterative step size. Since every step size is controlled strictly, it is inefficient for MPFP search. To address this issue, the DSTM is developed by adopting the directional chaos control strategy, which will be introduced in the next section.

### 4.2 Directional stability transformation method

*k*th iterative step. Since all parameters of DSTM can be obtained by original STM, it is very convenient for engineering application. However, a proper control factor

*λ*

^{ k }should be provided to guarantee the robustness and efficiency of DSTM. If the control factor is too large, the results of DSTM may lead to non-convergence. Otherwise, if the control factor is too small, it requires too much computational cost. Therefore, the improved MPFP search methods should be suggested, which is introduced in Sect. 5.

## 5 Adaptive stability transformation method

Although DSTM improves the efficiency of STM significantly, it must select a proper chaos control factor. If the chaos control factor is too large, the iterative point may lead to non-convergence. On the contrary, if the chaos control factor is too small, it needs to much computational cost. Therefore, how to select a proper chaos control factor is crucial.

**Property 1**

*For DSTM, the iterative point at the k* + 1*th iterative step is located at hyper-sphere with radius equal to* \({\beta ^k}\).

*Proof*

Then, the radius of hyper-sphere \(\left\| {{{\varvec{U}}^{k+1}}} \right\|={\beta ^k}.\) Thus, Property 1 has been proved.

From the definition of reliability analysis model and PMA, it is observed that the direction \({\varvec{U}}_{{{\text{MPFP}}}}^{{}} - {\varvec{U}}_{{{\text{MPTP}}}}^{{k+1}}\) represents the steepest descent direction at the hyper-sphere with radius \(\left\| {{\varvec{U}}_{{{\text{MPTP}}}}^{{k+1}}} \right\|\), as shown in Fig. 3. Especially, when \(\beta\) computed by Eq. (1) is equal to \({\beta ^t}\) in Eq. (2), the MPFP and MPTP become the same point. Then, the reliability analysis model can be converted into the optimization model of PMA. From Property 1, the best chaos control factor at the *k*th iterative step that can be estimated by solving the following optimization formulation:

**C**is set as the unit matrix

**I**, the Eq. (15) becomes

*k*th step. \({\text{sgn}}\left( \cdot \right)\) is the sign function. As shown in Fig. 4, if \(\cos {\theta ^{k+1}}\) is less than zero, it means that the vectors \({{\varvec{U}}^{k+1}} - {{\varvec{U}}^k}\) and \({{\varvec{U}}^k} - {{\varvec{U}}^{k - 1}}\) have the same descent direction and the iterative point does not oscillate during the iterative process. If \(\cos {\theta ^{k+1}}\) is larger or equal to zero, the vectors \({{\varvec{U}}^{k+1}} - {{\varvec{U}}^k}\) and \({{\varvec{U}}^k} - {{\varvec{U}}^{k - 1}}\) are in the opposite descent direction and the iterative point is oscillatory during the iterative process. The formulations of Eqs. (12) and (13) are applied to search MPFP and a middle value 0.5 is selected as the initial value of chaos control factor to achieve a better efficiency. Otherwise, the HL–RF iterative algorithm is employed. In general, the flowchart of ASTM is shown in Fig. 5a and the procedures are as follows:

- Step 1:
Initialize the random variables \({{\varvec{X}}^0}\).

- Step 2:
Transform the random variables \({{\varvec{X}}^k}\) to standard normal variables \({{\varvec{U}}^k}\).

- Step 3:
Identify the oscillation of the random variables by Eq. (18).

- Step 4:
If the iterative point does not oscillate during the iterative process, the random variables are updated using HL–RF algorithm of Eq. (5). Once the iterative point oscillates in the iterative process, the chaos control factor is updated by Eqs. (14) and (17). The random variables are calculated by DSTM using Eq. (12).

- Step 5:
Transform the standard normal variables \({{\varvec{U}}^k}\) to random variables \({{\varvec{X}}^k}\).

- Step 6:
If convergent, stop; otherwise, go to step 2.

## 6 Enhanced adaptive stability transformation method

In EASTM, the approximate MPFP is computed to avoid solving the optimization formulation of Eq. (14). Therefore, the computational cost of MPFP search can be further reduced. Because the EASTM is only utilized when the iterative point is oscillatory, a middle value 0.5 is selected as initial value of chaos control factor. In general, both ASTM and EASTM can compute the MPTP efficiently. The flowchart is shown in Fig. 5b. The procedures of EASTM are identical to ASTM except that the chaos control factor is computed by Eqs. (19) and (20).

## 7 Illustrative examples

In this section, five examples with nonlinear performance function are carried on by the proposed ASTM and EASTM and are compared by other five popular iterative algorithms, including HL–RF, *i*HL–RF, LFR, STM and DSTM algorithms. The chaos control factors of both STM and DSTM are set to be 0.05. The initial point is the mean of the random variables, and the stopping criterion is 10^{− 4} \(\left( {{{\left\| {{{\mathbf{X}}^k} - {{\mathbf{X}}^{k - 1}}} \right\|} \mathord{\left/ {\vphantom {{\left\| {{{\mathbf{X}}^k} - {{\mathbf{X}}^{k - 1}}} \right\|} {\left\| {{{\mathbf{X}}^k}} \right\|}}} \right. \kern-0pt} {\left\| {{{\mathbf{X}}^k}} \right\|}} \leqslant {{10}^{ - 4}}} \right)\) for all these algorithms.

*Example 1*

\({g_1}=X_{1}^{3}+X_{1}^{2}{X_2}+X_{2}^{3} - 18,\) in which *X* _{1} and *X* _{2} represent the random variables with normal distribution, *μ* _{1} = 10, *μ* _{2} = 9.9, *σ* _{1} = *σ* _{2} = 5 [24].

The standard deviation *σ* _{2} is deemed as the control parameter that is used to demonstrate the convergence of HL–RF algorithm. The bifurcation plot of reliability index of HL–RF algorithm and two Lyapunov exponents are given in Figs. 6 and 7, respectively, which are identical to reference [20]. If all Lyapunov exponents are less than 0, it means that the HL–RF algorithm has periodic or fixed solutions. Otherwise, if the maximum Lyapunov exponent is less than 0, the solution is unstable and generates chaotic solutions. From Figs. 6 and 7, it is seen that the reliability index *β* shows the bifurcation, chaos and periodic oscillation phenomena as the change of parameter *σ* _{2}, and the Lyapunov exponents reflect these phenomena strictly.

The results of all different algorithms are listed in Table 1, and the *F* evaluations denotes the number of function calls. In addition, Monte Carlo simulation (MSC) is used to verify outcomes of different methods with a 10-million sample size. It is found that all these FORM methods have some errors for this highly nonlinear limit state function, and the second-order reliability method (SORM) can be further utilized to improve the accuracy. From Table 1, all these methods except HL–RF iterative algorithm converge to the same optimum. *i*HL–RF and STM algorithms are robust but not efficient. The DSTM is more efficient than LFR, STM and *i*HL–RF using the directional chaos control strategy. Comparing with other algorithms, the efficiency of ASTM and EASTM is improved significantly by taking advantage of MPTP search. Although the number of iterations of ASTM is less than that of EASTM, ASTM needs calling performance function 28 times to obtain the chaos control factor. So EASTM is the more efficient than ASTM because it avoids entire optimization model for determining chaos control factor. In addition, the impact of different minimum angle ratios *η* _{ θθ } and initial chaos control factor *λ* _{0} for EASTM is investigated, and the results are listed in Table 2. When large values of *η* _{ θθ } and *λ* _{0} are related, the EASTM meets the convergence problem. On the contrary, when the values of two parameters are too small, the EASTM is inefficient. As evident from Table 2, the middle values *λ* _{0} = 0.5 and \({\eta _\theta }=0.4\) are very promising for EASTM.

The number of function evaluations of different algorithms for Example 1

Methods | MPFP | Iterations | | | |
---|---|---|---|---|---|

HL–RF | – | – | – | – | 2.5234 |

| (1.6855, 1.9679) | 31 | 280 | 2.2983 | 2.5234 |

STM | (1.6855, 1.9679) | 202 | 606 | 2.2983 | 2.5234 |

DSTM | (1.6855, 1.9679) | 25 | 75 | 2.2983 | 2.5234 |

LFR | (1.6855, 1.9679) | 20 | 60 | 2.2983 | 2.5234 |

ASTM | (1.6855, 1.9679) | 8 | 38 | 2.2983 | 2.5234 |

EASTM | (1.6855, 1.9679) | 9 | 27 | 2.2983 | 2.5234 |

The impact of different minimum angle ratios *η* _{ θ } and initial chaos control factor *λ* _{0}

| \({\eta _\theta }=0.1\) | \({\eta _\theta }=0.2\) | \({\eta _\theta }=0.4\) | \({\eta _\theta }=0.5\) | \({\eta _\theta }=0.6\) | \({\eta _\theta }=0.8\) | \({\eta _\theta }=0.9\) |
---|---|---|---|---|---|---|---|

0.1 | 2.2982 (186) | 2.2983 (114) | 2.2983 (69) | 2.2983 (57) | 2.2983 (51) | 2.2983 (39) | 2.2983 (36) |

0.2 | 2.2983 (114) | 2.2983 (69) | 2.2983 (39) | 2.2983 (33) | 2.2983 (27) | 2.2983 (27) | 2.2983 (27) |

0.5 | 2.2983 (57) | 2.2983 (33) | 2.2983 (27) | 2.2983 (48) | 2.2983 (144) | 2.2983 (42) | – |

0.7 | – | 2.2983 (30) | 2.2983 (81) | – | – | – | – |

0.9 | – | 2.2983 (27) | – | – | – | – | – |

*Example 2*

\({g_2}={{\text{e}}^{1+\alpha {X_1} - {X_2}}}+{{\text{e}}^{5 - 5\alpha {X_1} - {X_2}}} - 1,\) in which *α* = 0.4, both *X* _{1} and *X* _{2} represent the random variables with normal distribution, *μ* _{1} = 0, *μ* _{2} = 0, *σ* _{1} = *σ* _{2} = 1 [6].

The standard deviation *σ* _{1} is deemed as the control parameter, which is used to demonstrate the convergence of HL–RF algorithm. The bifurcation plot of reliability index of HL–RF algorithm and two Lyapunov exponents is, respectively, given in Figs. 8 and 9, which is identical to reference [20]. It is seen that the reliability index *β* shows the bifurcation, chaos and periodic oscillation phenomena as the change of standard deviation *σ* _{1}, and the Lyapunov exponents reflect these phenomena strictly. In addition, MCS is used to verify outcomes of different methods with a ten-million sample size, and the results are listed in Table 3, and the reliability index is 3.0707.

The number of function evaluations of different algorithms for Example 2

Methods | MPFP | Iterations | | | |
---|---|---|---|---|---|

HL–RF | – | – | – | – | 3.0707 |

| (1.7113, 2.3256) | 21 | 247 | 2.8873 | 3.0707 |

STM | (1.7113, 2.3256) | 202 | 606 | 2.8873 | 3.0707 |

DSTM | (1.7113, 2.3256) | 50 | 150 | 2.8873 | 3.0707 |

LFR | (2.5518, 2.1336) | 31 | 93 | 3.3262 | 3.0707 |

ASTM | (1.7113, 2.3256) | 13 | 47 | 2.8873 | 3.0707 |

EASTM | (1.7113, 2.3256) | 13 | 39 | 2.8873 | 3.0707 |

The results of different algorithms are listed in Table 3. It is shown that FORM method has some errors for highly nonlinear limit state function. All methods except HL–RF iterative algorithm converge to the same optimum. *i*HL–RF and STM are robust but inefficient. DSTM is more efficient than *i*HL–RF, STM and LFR using the directional chaos control strategy. Comparing with STM and DSTM, the efficiency of ASTM is improved significantly by finding a suitable chaos control factor, and the number of function calls for calculating chaos control factor is eight times. EASTM is the most efficient method by updating the chaos control factor adaptively.

### Example 3

\({g_3}=X_{1}^{4}+X_{2}^{4}+X_{2}^{4} - 20,\) in which *X* _{1} and *X* _{2} represent the random variables with normal distribution, *μ* _{1} = 10, *μ* _{2} = 10, *σ* _{1} = *σ* _{2} = 5 [20].

The mean *μ* _{2} is deemed as the control parameter, which is used to demonstrate the convergence of HL–RF algorithm. The bifurcation plot of reliability index of HL–RF algorithm and two Lyapunov exponents is given in Figs. 10 and 11, respectively, which is identical to reference [20]. It is seen that the reliability index *β* shows the bifurcation, chaos and periodic oscillation phenomena as the change of mean *μ* _{2}, and the Lyapunov exponents reflect these phenomena strictly. In addition, MCS is used to verify outcomes of different methods with a 10-million sample size, and the results are listed in Table 4. It is shown that FORM method has some errors for highly nonlinear limit state function.

The number of function evaluations of different algorithms for Example 3

Methods | MPFP | Iterations | | | |
---|---|---|---|---|---|

HL–RF | – | – | – | – | 2.9038 |

| (1.8158, 1.4617) | 53 | 745 | 2.3654 | 2.9038 |

STM | (1.8158, 1.4617) | 241 | 723 | 2.3655 | 2.9038 |

DSTM | (1.8158, 1.4617) | 21 | 63 | 2.3655 | 2.9038 |

LFR | (1.8158, 1.4617) | 35 | 105 | 2.3655 | 2.9038 |

ASTM | (1.8158, 1.4617) | 12 | 92 | 2.3655 | 2.9038 |

EASTM | (1.8158, 1.4617) | 12 | 36 | 2.3655 | 2.9038 |

The results of different algorithms are listed in Table 4. It is shown that FORM method has some errors for highly nonlinear limit state function. All methods except HL–RF iterative algorithm converge to the same optimum. *i*HL–RF and STM are robust but inefficient. LFR is more efficient than *i*HL–RF and STM. DSTM shows the high efficiency, but how to find a proper chaos control factor is questionable. Both ASTM and EASTM are very efficient. Since EASTM only obtains the approximate MPTP during each iterative step, it is more efficient than ASTM.

### Example 4

*X*

_{1},

*X*

_{2}and

*X*

_{3}follow normal distribution (

*μ*

_{1}= 10,

*σ*

_{1}= 5,

*μ*

_{2}= 25,

*σ*

_{2}= 5,

*μ*

_{3}= 0.8,

*σ*

_{3}= 0.2). Random variable

*X*

_{4}follows lognormal distribution (

*μ*

_{4}= 0.0625,

*σ*

_{4}= 0.0625).

HL–RF algorithm obtains periodic solutions, and the Lyapunov exponents are (–0.1266, − 1.2681, − 1.7133, − 3.5744). So it reflects the chaotic phenomenon of HL–RF algorithm clearly. To address this issue, advanced algorithms (*i*HL–RF, STM, DSTM, LFR, ASTM and EASTM) are employed, and the results of all these algorithms are listed in Table 5. The MCS is used to validate the accuracy of different algorithms with a 10-million sample size. It is seen that all these algorithms except HL–RF can satisfy the accuracy requirement. *i*HL–RF and STM are robust but inefficient, in which *i*HL–RF algorithm requires many computational cost to find MPFP. DSTM is more efficient than *i*HL–RF and STM using the directional chaos control strategy. EASTM and LFR are efficient than other advanced algorithms. Although the number of iterations of ASTM is less than that of EASTM, it needs to call the performance function 16 times to determine the chaos control factor, and thus the computational cost of EASTM is more than that of EASM.

The number of function evaluations for Example 4

Methods | MPFP | Iterations | | | |
---|---|---|---|---|---|

HL–RF | – | – | – | – | 1.3810 |

| (14.8387, 25.0441, 0.8667, 0.0466) | 17 | 314 | 1.0256 | 1.3810 |

STM | (14.7909, 25.0440, 0.8622, 0.0468) | 89 | 445 | 1.0256 | 1.3810 |

DSTM | (14.8622, 25.0925, 0.8620, 0.0479) | 36 | 180 | 1.0255 | 1.3810 |

LFR | (14.8349, 25.0452, 0.8669, 0.0468) | 9 | 45 | 1.0256 | 1.3810 |

ASTM | (14.8389, 25.0419, 0.8666, 0.0466) | 7 | 51 | 1.0256 | 1.3810 |

EASTM | (14.8389, 25.0436, 0.8666, 0.0467) | 9 | 45 | 1.0256 | 1.3810 |

### Example 5

A tower crane.

A tower crane with 87 m × 1.52 m × 1.52 m is shown in Fig. 12, which is extensively used for lifting materials in construction sites. The entire tower cranes are composed by 29 standard components, and each component is composed by 32 bars. The sectional dimension of vertical bar is 120 × 10 mm, while the sectional dimension of cross bar is 70 × 5 mm. Four random variables, including Young’s modulus, Poisson’s ratio, windward wind load and transverse wind load are considered. The means and coefficient of variations for these four random parameters are [206 Gpa, 2.8, 172.29 kN, 109.16 kN] and [0.02, 0.02, 0.1, 0.1], respectively. The maximum lifting weight is 7.6 *t*, and the performance function is defined as follows:

The results of different algorithms, including the values of MPFP, iterative numbers and the number of function calls, are given in Table 6. The MCS is performed with 1.5 × 10^{4} samples, and the reliability index is 2.2464. According to the results shown in Table 6, *i*HL–RF, STM, DSTM and LFR show inaccuracy, while other methods are converged accurately with reliability index 2.2462. ASTM is more efficient than HL–RF and is more accurate than *i*HL–RF, STM, DSTM and LFR, and it calls the performance function 41 times to search the chaos control factor. Comparing with ASTM, the efficiency of EASTM is further enhanced in this example.

The number of function evaluations for tower crane

Methods | MPFP | Iterations | | | |
---|---|---|---|---|---|

HL–RF | (1.9758 × 10 | 16 | 80 | 2.2462 | 2.2464 |

| (1.9761 × 10 | 3 | 22 | 1.9377 | 2.2464 |

STM | (1.9803 × 10 | 78 | 312 | 2.2436 | 2.2464 |

DSTM | (1.9766 × 10 | 4 | 20 | 2.2421 | 2.2464 |

LFR | (1.9765 × 10 | 41 | 205 | 2.2572 | 2.2464 |

ASTM | (1.9766 × 10 | 6 | 71 | 2.2462 | 2.2464 |

EASTM | (1.9765 × 10 | 10 | 50 | 2.2462 | 2.2464 |

Since the convergence is also impacted by stopping criterion, four different stopping criterion values, i.e., 10^{− 3}, 10^{− 5}, 10^{− 7} and 10^{− 9}, are used for all benchmark examples, and the computational results are listed in Table 7. HL–RF algorithm cannot find the correct reliability index. STM is very robust but inefficient. DSTM and LFR are very efficient; however, LFR appears oscillatory and cannot converge when the stopping criterion is too small. ASTM and EASTM are more efficient and robust than other existing methods. To show this, the iterative histories of Examples 3 and 4 with stopping criterion *ε* = 10^{− 7} are given in Fig. 13 as a representative. It is observed that HL–RF method generates periodic-2 solutions. For Example 3, the iterative points of LFR meet the convergence problem and oscillate slightly around the MPFP. ASTM and EASTM are more efficient than other methods. Since ASTM needs solving the optimization model to determine the chaos control factor, the number of iterations of ASTM is less than that of EASTM.

Different stopping criteria for all examples

Methods | Example 1 | Example 2 | Example 3 | Example 4 |
---|---|---|---|---|

| ||||

HL–RF | – | – | – | – |

| 2.2982 (203) | 2.8873 (226) | 2.3610 (539) | 1.0256 (237) |

STM | 2.2980 (468) | 2.8860 (468) | 2.3652 (582) | 1.0227 (240) |

DSTM | 2.2982 (57) | 2.8871 (111) | 2.3654 (57) | 1.0167 (70) |

LFR | 2.2983 (48) | 3.3283 (36) | 2.3655 (93) | 1.0256 (35) |

ASTM | 2.2982(38) | 2.8873 (37) | 2.3655 (92) | 1.0256 (44) |

EASTM | 2.2983 (24) | 2.8873 (33) | 2.3655 (33) | 1.0256 (35) |

| ||||

HL–RF | – | – | – | – |

| 2.2983 (301) | 2.8873 (324) | 2.3655 (850) | 1.0256 (314) |

STM | 2.2982 (741) | 2.8873 (741) | 2.3655 (858) | 1.0256 (650) |

DSTM | 2.2983 (93) | 2.8873 (189) | 2.3655 (66) | 1.0256 (400) |

LFR | 2.2983 (66) | 3.3262 (54) | 2.3655 (105) | 1.0256 (70) |

ASTM | 2.2983 (43) | 2.8873 (57) | 2.3655 (97) | 1.0256 (65) |

EASTM | 2.2983 (30) | 2.8873 (45) | 2.3655 (42) | 1.0256 (55) |

| ||||

HL–RF | – | – | – | – |

| 2.2983 (441) | 2.8873 (349) | 2.3655 (997) | 1.0256 (1092) |

STM | 2.2983 (1011) | 2.8873 (1011) | 2.3655 (1123) | 1.0256 (1105) |

DSTM | 2.2983 (132) | 2.8873 (270) | 2.3655 (75) | 1.0256 (885) |

LFR | – | – | – | 1.0256 (125) |

ASTM | 2.2983 (43) | 2.8873 (77) | 2.3655 (102) | 1.0256 (93) |

EASTM | 2.2983 (51) | 2.8873 (60) | 2.3655 (48) | 1.0256 (100) |

| ||||

HL–RF | – | – | – | – |

| 2.2983 (652) | 2.8873 (374) | 2.3655 (1317) | 1.0256 (2113) |

STM | 2.2983 (1281) | 2.8873 (1281) | 2.3655 (1395) | 1.0256 (1585) |

DSTM | 2.2983 (168) | 2.8873 (351) | 2.3655 (81) | 1.0256 (1460) |

LFR | – | – | – | 1.0256 (185) |

ASTM | 2.2983 (48) | 2.8873 (102) | 2.3655 (102) | 1.0256 (135) |

EASTM | 2.2983 (66) | 2.8873 (75) | 2.3655 (57) | 1.0256 (150) |

## 8 Conclusions

In this study, two effective iterative algorithms are developed to enhance the efficiency and robustness of most probable failure point (MPFP) search algorithm via transforming it to solving a series of MPTPs. Adaptive stability transformation method (ASTM) is proposed to improve the performance of HL–RF algorithm. Then, the proposed method is enhanced using the most probable target point (MPTP) approximation, which is named as enhanced adaptive stability transformation method (EASTM). The proposed ASTM and EASTM show high performance in terms of efficiency and robustness.

Moreover, the performances of two proposed algorithms are tested by five benchmark examples. The results of ASTM and EASTM are compared to five popular reliability algorithms, including HL–RF iterative algorithm, *i*HL–RF iterative algorithm, stability transformation algorithm (STM) and directional stability transformation algorithm (DSTM) and limited Fletcher–Reeves (LFR) algorithm. Results reveal that ASTM and EASTM can find the accurate MPFP with less computational cost. The application of ASTM and EASTM for other nonlinear engineering problems will be promising in future.

## Notes

### Acknowledgements

The supports of the National Natural Science Foundation of China (Grant Nos. 11602076 and 51605127), the Natural Science Foundation of Anhui Province (No. 1708085QA06) and the Fundamental Research Funds for the Central Universities of China (No. JZ2016HGBZ0751) are much appreciated.

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