The escalating need for reliability analysis (RA) and reliability-based design optimization (RBDO) within engineering challenges has prompted the advancement of saddlepoint approximation methods (SAM) tailored for such problems. This article offers a detailed overview of the general SAM and summarizes the method characteristics first. Subsequently, recent enhancements in the SAM theoretical framework are assessed. Notably, the mean value first-order saddlepoint approximation (MVFOSA) bears resemblance to the conceptual framework of the mean value second-order saddlepoint approximation (MVSOSA); the latter serves as an auxiliary approach to the former. Their distinction is rooted in the varying expansion orders of the performance function as implemented through the Taylor method. Both the saddlepoint approximation and third-moment (SATM) and saddlepoint approximation and fourth-moment (SAFM) strategies model the cumulant generating function (CGF) by leveraging the initial random moments of the function. Although their optimal application domains diverge, each method consistently ensures superior relative precision, enhanced efficiency, and sustained stability. Every method elucidated is exemplified through pertinent RA or RBDO scenarios. By juxtaposing them against alternative strategies, the efficacy of these methods becomes evident. The outcomes proffered are subsequently employed as a foundation for contemplating prospective theoretical and practical research endeavors concerning SAMs. The main purpose and value of this article is to review the SAM and reliability-related issues, which can provide some reference and inspiration for future research scholars in this field.

A saddlepoint refers to a stationary point of a non-local extreme point, which is a singular point that is stable in one direction and unstable in the other direction [

Saddlepoint approximation arises from the approximate statistics of the probability density function (PDF) and cumulative distribution function (CDF), particularly the latter [

Particularly in reliability analysis (RA) and reliability-based design optimization (RBDO) domains, SAM can provide an important support. Yuan et al. [

The concept of reliability emerged in the 19th century, aligning with the advancements of the Industrial Revolution. The growing momentum of this era demanded a quantitative approach to reliability, specifically in factory production and the execution of engineering projects [

Let

Numerous researchers have extensively investigated and formulated various reliability methods tailored for practical engineering applications [

RA also holds significant importance in ensuring the smooth operation of entire projects across diverse engineering fields [

The first-order reliability method (FORM) is one of the crucial and commonly utilized approaches within most probable point (MPP)-based methods [

However, FORM overlooks the nonlinearity of the original limit-state function, often resulting in substantial errors in many cases. To address this, the second-order reliability method (SORM) was introduced. SORM involves a quadratic expansion of the limit-state function at the point nearest to the origin in the standard normal coordinate system. By considering the nonlinearity, SORM approximates the limit-state function using a parabolic surface, providing more accurate reliability calculations than FORM [

In the context of RA, there are many analysis methods that are closely related to saddlepoint approximation [

Reliability metrics have led to advancements in RBDO methods [

This article offers an overview of the utilization and progression of SAMs in RA and RBDO. It can serve as a valuable reference and guide for subsequent research in this domain. The article is structured into five sections. The second section presents the fundamental algorithmic logic of the general SAM. The third section presents the core algorithmic logic and distinct features of recently enhanced SAMs. The fourth section discusses the application and performance of various SAMs in RA and RBDO engineering examples, summarizing the merits of each approach. Finally, the last section presents analytical conclusions and outlines potential directions for future studies.

SAM is an effective approximation tool, particularly suited for tackling high-dimensional integration challenges. This section introduces the general process of SAM. Its fundamental logic lies in obtaining the cumulant generating function (CGF) of a random variable, which enables approximations of the associated CDF and PDF. Calculating CDFs and PDFs is a crucial aspect of engineering analysis. Therefore, the primary objective of the general SAM is to facilitate more convenient computations of CDFs and PDFs.

Distribution | CGF | |
---|---|---|

Normal | ||

Uniform | ||

Gamma | ||

Gumbel | ||

Exponential | ||

Let

The CGF of

For

Let

and

Under normal circumstances, we choose the maximum-likelihood point

According to the linear relationship above, the CGFs of

The approximation of the CDF of

This section offers a comprehensive review of the conventional SAM. The resultant equations are highly nonlinear, and the CGFs of

The general SAM features limitations in its application. Utilizing SAM necessitates acquiring the saddlepoint equation, which relies on tractable variables having existing CGFs. In cases in which CGFs are nonexistent, the general SAM cannot be applied. Additionally, for most probability distribution types, the corresponding CGFs tend to be complex, leading to potentially highly nonlinear saddlepoint equations [

MVFOSA demonstrates efficiency and robustness comparable to those of the general SAM but offers superior accuracy. MVFOSA utilizes the first-order Taylor method to linearly expand the performance function

In MVFOSA,

The CGF of

Property 1. When

Property 2. Let

For example, if

According to the abovementioned two properties, the CGF of

When both the CGF of

As previously highlighted, MVFOSA utilizes comprehensive distribution information, necessitating only a saddlepoint identification process. Owing to these characteristics, MVFOSA achieves enhanced accuracy.

MVSOSA represents an alternative method to MVFOSA [

Then,

With the assumption that the random design variables

According to the two properties of CGF mentioned in

Furthermore, the first derivative of

Then, the PDF of

MVSOSA can also provide higher accuracy than MVFOSA and can be applied in a wide range of practical projects.

SATM is a method developed from general saddlepoint approximation [

When

Therefore, the first three derivatives of

In addition, the standardized form of

Considering that the challenges of solving the saddlepoint equation and obtaining the CGF may impact the utilization of the saddlepoint method, a simplified version of the CGF

_{1}, _{2}, and _{3} are determined constants, and

According to

If

The saddlepoint equation

If the value of _{S} can be approximately expressed in terms of

The CDF of _{S} is similar to

According to

Furthermore, the PDF of _{S} and

Then, the CDF of

According to the above equations, the CGF of

Then, the CDF of

As mentioned above,

In contrast to SATM, SAFM is an enhanced high-order moment-based SAM [

According to SATM, _{S}; then, the fourth derivative of

Then,

If

According to

The saddlepoint

Then, substituting

According to

When

Under the same condition,

In SAFM, the first four stochastic moments form the foundation for approximating the CGF. Employing the general SAM enables the assessment of the failure probability in a random structure using the approximated CGF. Notably, SAFM offers enhanced accuracy compared with the conventional SAM.

As previously mentioned, the demand for reliability in modern engineering design is constantly increasing. To provide readers with a better understanding of the specific engineering applications and the value of the aforementioned method, this section presents three engineering-related examples. These examples encompass all of the methods in

The wellhead platform represents an economical solution for offshore oilfield development, providing a provisional structure that safeguards the wellhead situated on the seabed.

In the given context, wave force-induced damage is the primary factor influencing RA. The core principle of RBDO involves constructing a probabilistic model to encapsulate inherent uncertainties in engineering structures. The RBDO methodology allows for addressing prevalent uncertainties in engineering applications, ensuring that the design aligns with the project’s specified reliability criteria. ^{3}, which is 9% smaller than that obtained using the initial scheme, and it is the best result. These results indicate that RBDO with improved SAMs generates acceptable optimization schemes. Comparative data reveals that RBDO-MVSOSA tends to produce more conservative results than RBDO-MVFOSA, aligning closer with the results from RBDO-Monte Carlo simulation (MCS).

Parameter | |||
---|---|---|---|

Description | Wave height | Flow speed | Wind speed |

Distribution | Log-normal | Log-normal | Log-normal |

Mean | 15.7 m | 1.3 m/s | 73.0 m/s |

Parameter | |||

Description | Parameter of drag force | Parameter of mass force | Yield strength |

Distribution | Normal | Normal | Log-normal |

Mean | 1.90 | 3.07 | 1.70 × 10^{8} |

Design variable | ||||||
---|---|---|---|---|---|---|

RBDO-MVFOSA | 0.027 | 0.033 | 0.025 | 0.023 | 0.026 | 2.31 |

RBDO-MVSOSA | 0.026 | 0.031 | 0.028 | 0.027 | 0.028 | 2.27 |

RBDO-MCS | 0.026 | 0.030 | 0.027 | 0.026 | 0.027 | 2.29 |

Design with accurate method | 0.025 | 0.025 | 0.040 | 0.040 | 0.040 | 2.00 |

Design variable | ||||||

RBDO-MVFOSA | 3.67 | 4.41 | 11.3 | 17.7 | 25 | 38 |

RBDO-MVSOSA | 3.56 | 4.43 | 10.6 | 17.1 | 27 | 39 |

RBDO-MCS | 3.55 | 4.44 | 11.0 | 16.9 | 28 | 39 |

Design with accurate method | 3.00 | 5.00 | 7.0 | 9.0 | 34 | 37 |

This example demonstrates that RBDO-MVSOSA offers superior accuracy and that enhanced SAMs positively influence reliability-related issues.

Reinforced concrete beams, pivotal load-bearing elements in engineering structures such as buildings and bridges, have widespread applications [

The performance function of the concrete beam model can be expressed as

The distributions of input random variables are shown in

Variables’ information | ||||
---|---|---|---|---|

Mean | 4.08 | 44 | 3.12 | 2052 |

Standard deviation | 0.612 | 6.6 | 0.468 | 307.8 |

Distribution | Normal | Lognormal | Lognormal | Weibull |

The first four moments of the state variable are shown in

The results of the failure probability estimate using the different improved SAMs and MCS are shown in

Method | SATM | SAFM | MCS |
---|---|---|---|

2.087 | 2.076 | 2.076 | |

1.843 | 1.894 | 1.896 | |

2.80 | 0.12 | - |

According to the aforementioned results, both SATM and SAFM offer failure probability estimates that closely align with the results of MCS. This suggests that the refined approximation methods employed in this study yield high precision.

According to the results of the comparison, the closer the tail line in the figure is to MCS, the higher the accuracy of the method. According to the data in

This example demonstrates the high practicality of both SATM and SAFM in addressing reliability issues in engineering.

The schematic diagram of a typical roof structure in civil construction is shown in

The upper boom and compression bars are made of concrete, while the lower boom and tension bars are constructed from steel. The vertical deflection of the rooftop’s apex node

There are three limit-state functions generated by different conditions of the structure:

The first failure mode occurs when the perpendicular deflection

Method | SOSA | SORM | FORM | MCS |
---|---|---|---|---|

3.6983 | 3.6642 | 3.5714 | 3.7110 | |

0.34 | 1.26 | 3.76 | - | |

Total function calls | 243 | 243 | 135 |

With the MCS solution as the benchmark, SOSA demonstrates superior accuracy and efficiency, emphasizing the applicability of saddlepoint approximation in engineering contexts.

This study comprehensively reviews SAM, explores its enhancement techniques, and provides examples showcasing each augmented SAM in RA. Following these examples, the merits of each method are critically evaluated. The primary goal of SAM is to provide efficient and precise fitting results in RA. Its foundational principles establish a groundwork for effective fitting, adaptable to further refinement based on specific operational needs. RA often involves extensive datasets, posing significant challenges. Remarkably, the results of SAM with minimal sample sizes closely resemble those obtained via MCS with larger sample sizes, indicating SAM’s capability to provide superior approximations even with limited data. Analyzing large-scale samples significantly increases time and financial investments. SAM’s efficiency alleviates substantial operational demands and expedites problem-solving in reliability analysis. However, with the evolving landscape of contemporary engineering, systems requiring RA are growing in size and complexity. An increasing number of uncertainties are being incorporated to ensure the reliability and robustness of extensive mechanical systems. Consequently, achieving a balance between sampling efficiency and reliability is becoming progressively challenging.

In RBDO, challenges arise when the dimensionality of the input information becomes excessively high. Integrating RBDO with saddlepoint approximation removes the need for the spatial transformation of random variables. This approach mitigates potential inaccuracies in reliability assessment caused by amplified nonlinearity in the limit-state function, thereby enhancing the attractiveness of SAM. Presently, the engineering field associated with reliability increasingly tends to adopt machine learning and similar techniques. Utilizing machine learning algorithms enables engineers to construct models derived from data, thereby facilitating predictive analyses and informed decision-making.

The theoretical foundation and validation of SAM remain a dynamic field for future exploration. Its mathematical properties, convergence, and method stability offer significant research value. Additionally, integrating saddlepoint approximation with machine learning models can improve solution accuracy and efficacy. SAM holds potential for utilization in deep neural network (DNN) training. The training processes of DNNs are susceptible to encountering numerous local minima interspersed with saddlepoints, complicating the convergence to a global optimum. SAM can address these saddlepoint challenges during training. SAM involves an effective search in the saddlepoint area to find the saddlepoint and continuous optimization until the global minimum is found. Consequently, SAM can more effectively train DNNs, improving both model accuracy and generalization capability. Integrating saddlepoint approximation into conventional machine learning algorithms, such as support vector machines, logistic regression, and decision trees, enhances model optimization and predictive performance. The amalgamation of SAMs with genetic algorithms is a promising research area. Genetic algorithms, rooted in evolutionary principles, leverage mechanisms such as crossover, mutation, and selection for optimization. Their adeptness at global searching compensates for SAM’s tendency to favor local optimization in high-dimensional spaces. Moreover, genetic algorithms can fine-tune relevant parameters within SAM, enhancing its convergence rate and stability. In essence, combining SAM with these methodologies amplifies its effectiveness in addressing challenges within RA and RBDO.

The authors extend their sincere gratitude to the reviewers for their valuable suggestions, which have significantly enhanced the quality of this paper. We also express our heartfelt thanks to the editors for their patience, amiable guidance, and diligent efforts in refining the manuscript.

This research was funded by the National Natural Science Foundation of China under Grant No. 52175130, the Sichuan Science and Technology Program under Grants Nos. 2022YFQ0087 and 2022JDJQ0024, the Guangdong Basic and Applied Basic Research Foundation under Grant No. 2022A1515240010, and the Students Go Abroad for Scientific Research and Internship Funding Program of University of Electronic Science and Technology of China.

The authors confirm contribution to the paper as follows: Study conception and design: Debiao Meng, Shiyuan Yang, Yipeng Guo; data collection and analysis: Yihe Xu, Yongqiang Guo; collection and arrangement of references: Lidong Pan, Xinkai Guo; draft manuscript preparation: Debiao Meng, Yipeng Guo. All authors reviewed the results and approved the final version of the manuscript.

All data presented in this paper can reasonably be obtained from the relevant references or by contacting the corresponding author of this paper.

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this article.