# An Adaptive Kriging-Based Fourth-Moment Reliability Analysis Method for Engineering Structures
###### Abstract  
The fourth-moment method can accurately perform a reliability analysis when it is challenging to determine the distribution of the random variable due to limited available samples. This method only utilizes the first four moments of the random variable and constructs the fourth-moment reliability index. However, it cannot be applied in engineering cases where the state function cannot be expressed explicitly, as it becomes difficult to establish a correlation between the first four moments of the random variable and the state function. Simplifying the state function forcefully may result in significant reliability prediction errors. To address this limitation, this study proposes an adaptive Kriging-based fourth-moment method for reliability analysis under complex state equations. The proposed method demonstrates better applicability and efficiency compared to existing methods. Several numerical examples are provided to validate the effectiveness and accuracy of the proposed method. reliability analysis; fourth moment; adaptive Kriging; limited samples; state function 2024202  
based on the assumption that the stochastic parameters obey a three-parameter logarithmic distribution. A reliability index expressed using the first four orders of moments is derived based on the normalization of the random parameters [21]. The improved fourth-moment method [18] takes into account previously neglected parameters in the Taylor series of the state function and gives more accurate reliability results in the explicit state function case based on the example of multiple distribution-type input parameters. However, for complex engineering problems where the state function cannot be expressed explicitly, the improved fourth-moments method cannot be directly applied because of the difficulty in constructing the correlation of moments between the random variables and the state function [22]. Surrogate models are commonly utilized to enhance the computational efficiency of the reliability analysis by approximating the relationship between inputs and outputs [23; 24]. Popular surrogate models include the Kriging model [25; 26; 27], neural network (NN) [28; 29], polynomial chaos expansion (PCE) [30; 31], support vector machine (SVM) [32; 33], etc. Among them, the Kriging model is widely used as an exact interpolation model with the convenience of obtaining the predicted value and prediction deviation simultaneously. However, the prediction accuracy of the surrogate model constructed with limited samples may be greatly reduced, so it is necessary to plan and select the appropriate inputs to obtain the corresponding responses so as to construct the optimal surrogate model. A Kriging model adaptive updating strategy [34] based on the U function is proposed, which is able to filter and introduce samples near the limit state boundary to effectively improve the prediction accuracy of the agent model [35]. This adaptive strategy is combined with the Monte Carlo simulation to form an adaptive Kriging Monte Carlo simulation method (AK-MCS), which is able to make a small number of invocations to the state function and thus perform a reliability analysis efficiently. Then, the adaptive Kriging model is implemented in the low dimensional space identified by the activity scores [36], which reduces the time cost of the adaptive analysis process with a high input space dimension. An Enhanced Adaptive Kriging Monte Carlo Simulation (EAK-MCSI) [37] is proposed based on a candidate sample pool reduction strategy that reduces the time cost required to traverse the candidate sample pool in small failure probability problems. In this paper, an adaptive Kriging-based fourth-moment method (AK-FM) is proposed. The method establishes a mapping relationship between the moments of random variables and the moments of complex state functions using the adaptive Kriging method and performs a reliability analysis based on the fourth-moment method without the need for a large number of samples to determine the type of distribution of the random variable.