A Curved Surface Integral Method for Reliability Analysis of Multiple Failure Modes System With Nonoverlapping Failure Domains

where $fX(x)$ is the joint probability density function to describe the uncertainty of variable $x=(x1,x2,…xn)$⁠, $g(x)$ is the limit state function (LSF), generally, $g(x)<0$ is defined as the failure domain.

General reliability problems often involve only a single failure mode, that is, only one LSF. In fact, most engineering problems are complex and consist of different components [5]. Under the action of complex loads, different components may have different failure states. The system contains different failure modes, that is, it has multiple LSFs. The solution methods of the multiple failure problem, which can be broadly categorized into two types: direct methods and indirect methods. For complex structural systems, their LSFs are generally not directly expressed analytically. It is effective to generate surrogate models that approximate alternative implicit LSFs and then combine them with other methods to solve the problem. The surrogate-based methods include response surface (RS) methods, support vector machines (SVM), neural networks, polynomial chaos expansion, and Kriging [6–10]. Zhang et al. [6] presented a time-dependent reliability analysis method based on RS model, which obtained using this iterative method can effectively evaluate complex implicit LSFs. Wu et al. [7] developed an adaptive surrogate modeling method. The method combines Gaussian process surrogate models with the composite expected feasibility function to effectively estimate the Mean Time to Failure of the system. Song et al. [8] presented a virtual SVM based on sequence sampling. This method has better performance for high-dimensional problems compared to traditional SVM methods. Roy et al. [9] improved the SVM model by incorporating the problem of minimizing the mean square error value. The new model constructed has a better approximation of the response values and the results are closer to the exact values. Wu et al. [10] presented a new method for dealing with system reliability problems based on dependent Kriging. By considering the autocorrelated Kriging model, an algorithm for selecting candidate points is developed to improve the computational accuracy and efficiency.

The direct method has evolved over the years and can be categorized into three types: the first-order reliability method (FORM), the second-order reliability method (SORM), and the Monte Carlo Simulation (MCS) [11–13]. MCS is a numerical test method that generates a large number of random variables based on known probability distributions and then obtains an approximate solution by calculating the percentage of sample points in the failure domain in the total. This method produces exact solutions when the sample size approaches infinity. Despite the high accuracy of its results, the time-consuming calculation process leads to the inability to analyze large-scale projects. Melchers et al. [14] use a multimodal sampling function for importance sampling to achieve the desired effect. Schuëller et al. [15] introduce the line sampling method, which determines the important direction of each failure mode in U-space. The failure probability of the system can be effectively calculated through random sampling.

In first-order reliability method, the LSF is approximated as a linear function at the Most Probable Point of failure (MPP). Since the concept of selecting the MPP on a limit state surface has been introduced [16], FORM has been heavily studied. In order to solve non-Gaussian problems, some transformations that convert the original coordinate space into the standard normal space (⁠$U$-space) have been introduced to FORM. The Hasofer Lind-Rackwitz Fiessler (HL-RF) method is widely used because of its fast convergence speed, however, it has flaws in the stability of convergence. Santos et al. [17] proposed a new step optimization algorithm that uses the Wolfe criterion to achieve local convergence of the function. This method has better efficiency and stability than HL-RF. Chen et al. [18] used the integral method to calculate the maximum error of FORM under nonlinear functions. The interval range of reliable probability is obtained more efficiently. Wang et al. [19] proposed an approximate first-order method based on the adaptive factor. The adaptive factor adjusts the double parameters to change the iteration step size. The method is more accurate and more robust in dealing with nonlinear problems. Gong and Frangopol [20] presented a time-dependent system reliability method in conjunction with FORM. The method has better results in dealing with system problems with dependent outcrossing events.

Similarly, in SORM, the LSF is approximated as a quadratic function at the MPP. An approximate function for calculating the probability of failure can be constructed using the reliability index $β$ and the second-order derivative of the LSF. This method was proposed by Breitung et al. [21–23] and has been extensively studied. Zhao et al. [24] proposed an approximate method to calculate the second-order reliability index and obtain its relationship with the number of random variables and the first-order reliability index. This method is very efficient because it does not require excessive processing of the Hessian matrix. In previous studies, the information of the Hessian matrix was not fully considered, so Mansour et al. [25] used nonchi-square distribution to propose an approximate expansion expression, which reduced the error of the traditional methods in special cases. Hu and Du [26] applied the second-order saddle-point approximation method (SOSPA) and used the cumulant generating function to find the saddle point to obtain higher-precision results. Since this method is essentially the same as SORM, it also has the disadvantage of large errors in highly nonlinear situations. Wu and Du [27] extended SOSPA to the system reliability problem, achieving higher accuracy compared to SORM. However, the accuracy is also constrained by the correlation of the estimated components. Following this, Wu et al. [28] developed a time-dependent system reliability method using envelope functions. Since SOSPA needs to deal with second-order derivatives, it is less efficient than FORM but higher accuracy is obtained.

When multiple LSFs of a system do not interfere with each other, i.e., the system has multiple nonoverlapping failure domains. Since the concept of the nonoverlapping failure domains problem is relatively simple and still belongs to the system reliability problem, most of the references do not discuss its solution process separately. For the nonoverlapping failure domains problem, the existing methods still use the system reliability method for calculation, which causes redundancy in the calculation process. Since the failure probabilities of different LSFs are different and may even differ by many orders of magnitude, it is not necessary to calculate the sum of the failure probabilities of all LSFs. This paper proposes a curved surface integral method (CSIMMS) for solving the system reliability problems of multiple nonoverlapping failure domains. The Breitung is used to construct paraboloids at different MPPs, and the curved surface integral method (CSI) is used to calculate the intersection areas of each paraboloid and the corresponding hypersphere, thus estimating the failure probability of each failure mode. In the following Sec. 2, the reliability analysis situation of FORM, SORM, and CSI is briefly reviewed. Evaluation metrics for failure modes are presented in Sec. 3, which classifies the probability of failure into different orders of magnitude. In Sec. 4, three examples are used to demonstrate the effectiveness of the proposed method. Section 5 gives the summary of this study.

where $∇g$ is the gradient vector of the LSF, and $u*$ is the point closest to the origin of the LSF in $U$-space, which is defined as the MPP.

The concept of FORM is simple, the calculation is convenient, and it still maintains high efficiency in dealing with multivariable reliability problems. When the LSF is approximately linear near the MPP, using FORM can meet the accuracy requirements. However, when the LSF exhibits high nonlinearity, it becomes incapable of providing reasonable failure probability results, and more accurate approximate models need to be used.

where $λi$ is the coefficient of the approximate parabolic surface, and $κi=−2λi$ can be obtained from the curvature calculation formula. $κi$ represents the principal curvature of the approximate parabolic surface at the MPP and is obtained by calculating the eigenvalues of the real symmetric matrix $(HTQH)n−1$⁠.

Compared to FORM, the accuracy of failure probability has been improved by considering the nonlinearity at the MPP. However, this approximation formula is not completely effective in some cases. When facing small failure probability problems with high nonlinearity, the calculation accuracy of the Breitung method is not ideal. Therefore, the CSI is proposed and applied to reliability problems with multiple failure modes.

where $0≤r≤1,0≤φ1≤π,…,0≤φn−3≤π,0≤φn−2≤2π$⁠.

where $g1,g2,…,gm$ are all the performance functions of the system.

The failure probability $Pfi$ is calculated through the single-mode reliability analysis method mentioned in Section 1. In addition, on the premise of meeting the accuracy requirements, it is possible to solve cases where the failure domains overlap but are far from the origin in the $U$-space. In other words, the failure probabilities corresponding to the overlapping regions can be ignored.

The curved surface integral method is evaluated using the parabolic LSF $g=λu12−u2+β$ as an example, where $λ$ represents the quadratic term coefficient, its relationship with the curvature $κ$ is $λ=−κ/2$⁠, it is positively related to the degree of nonlinearity of the parabola, and $β$ represents the reliability index. Figure 4 shows the comparison of calculation results of FORM, SORM, CSI, and MCS for different functions when $β=1$ and $3$⁠. Since the results of different methods at high-reliability level are small, in order to visualize the differences between different methods, logarithmic axes are used in this paper. As shown in Fig. 4, the CSI has higher accuracy in calculating failure probability than FORM and SORM, specially for situations with low-reliability index and high nonlinearity, its effect of improving calculation accuracy is more significant.

The curved surface integral method is used to calculate parabolic functions with different degrees of nonlinearity under the conditions of $β=0.1, 1, 2, 3$⁠, and $4$⁠, respectively. The changing relationship of the failure probability is shown in Fig. 5. Choosing $λ=0.001, 0.01, 0.1, 1, 5$⁠, and $10$ takes into account the changing range of the degree of nonlinearity to the greatest extent, and uses a logarithmic coordinate chart to intuitively express the impact of the reliability index and the degree of nonlinearity on the failure probability.

As can be seen, as the degree of nonlinearity increases, the failure probability shows a downward trend. When the reliability index is a constant value, as the quadratic term coefficient increases from 0.001 to 10, the decline range of the failure probability remains within 1 order of magnitude, that is, the maximum value and the minimum value differ by 10 times. When the reliability index increases from 0.1 to 4, the failure probability changes from the upper limit value 10⁻¹ to the lower limit value 10⁻⁶, which decreases by 5 orders of magnitude. It can be seen that changes in the reliability index are the main reason for changes in the order of magnitude of the failure probability. At the same time, changes in the degree of nonlinearity of the failure mode also have an impact on the failure probability within a certain range.

Assuming that the expression of the $i$-th failure mode is $gi=λiu12−u2+βi$⁠, since the quadratic term coefficient $λi$ and reliability index $βi$ are known, the position of the failure probability $Pfi$ can be determined according to Fig. 5, thereby estimating its order of magnitude index $δi$⁠. Since the failure probability is between 0 and 1, the order of magnitude exponent takes a negative integer (⁠$δi=−1,−2,−3,−4,−5,−6$⁠). Through the above method, the failure probability of each failure mode can be evaluated to obtain the corresponding $δi$⁠. If there is a difference of two orders of magnitude or more between the individual failure probability and the maximum failure probability, its impact on the system failure probability can be ignored, and it will not be calculated in the process of solving the system failure probability. Simplify the calculation process and improve calculation efficiency while meeting accuracy requirements.

Based on the above derivation, the flowchart of the proposed method is shown in Fig. 6 and its steps are summarized as follows

1. Apply FORM to calculate the failure probability for each failure mode.

2. Evaluate the order of magnitude of each failure probability and determine the largest failure probability $Pfmax$ in the system.

3. Find the small probability event $Pfi$ whose difference with the order of magnitude of $Pfmax$ is greater than 2. After excluding $Pfi$⁠, CSI is calculated and summed for the remaining failure modes.

4. If $Pfi$ does not exist, CSI is used to calculate each failure probability separately and sum them all to obtain failure probability of the system.

The calculation process of the proposed method is introduced through a demonstration, which is a simple system reliability problem of three failure mode combinations. In $U$-space as shown in Fig. 7, the failure modes are $g1=0.2u12−u2+1.5$⁠, $g2=5u22−u1+3$⁠, and $g3=u1+u2+22$⁠. Since the failure probability of the parabolic function does not change after it is rotated with the reliability index as the radius, for the convenience of introduction, two parabolas that are symmetrical about the axis are selected. Among them, $λ1=0.2$⁠, $λ2=5$ and $β1=1.5$⁠, $β2=3$⁠. For the linear function, its quadratic term coefficient is infinitely close to zero, where it can be approximated as 0.001. The reliability index $β3=2$ is the shortest distance from the origin to the line.

In Fig. 5, the order of magnitude index of the three failure probabilities can be obtained as $δ1=−2$⁠, $δ2=−4$ and $δ3=−2$⁠. It can be seen that the $Pf2$ is the smallest, which is two orders of magnitude different from the other two failure probabilities. Therefore, it is only necessary to use the CSI to calculate the $g1$and $g3$⁠, and the failure probability of the system is $Pf=Pf1+Pf3$⁠. FORM and SORM are calculated according to Eq. (15), and the comparison results with MSC are shown in Table 1. The results show that the method proposed in this paper can still maintain good accuracy while ignoring the influence of individual low failure probability modes.

In this section, several examples including two numerical examples and an engineering problem are selected to verify the accuracy and stability of CSIMMS. These examples involve some common situations in reliability analysis, such as normal probability distribution, linear and nonlinear functions, etc. These examples are also analyzed with FORM, SORM, and MCS for comparison.

where $k$ is a constant that controls the degree of nonlinearity in the system, $x1$ and $x2$ are independent random variables and obey the standard normal distribution, with mean values $μx1=μx2=0$ and standard deviations $σx1=σx2=1$⁠.

As shown in Table 2, when $k=0.2$ and $k=0.5$⁠, the difference in magnitude of the two failure modes is 0 and 1 , respectively, Therefore, the failure probability of the system can be obtained by directly summing the two failure probabilities. When $k=1$⁠, the two failure modes differ by two orders of magnitude, and the impact of failure mode $g1(x)$ on the system is negligible, so the failure probability of the system can be approximated by the failure probability of failure mode $g2(x)$⁠. For the function $g1(x)$⁠, although the functional formula changes, the MPP remains unchanged, and the result of the FORM solution is always the same, resulting in a large error. SORM uses a quadratic function surface to approximate the original function at the MPP, and its accuracy will be higher than FORM, however, there are still deficiencies in solving highly nonlinear problems.

As can be seen from Fig. 9, as the curvature increases, the calculation error of FORM remains between 30% and 50%, the error of SORM fluctuates around 5%, and the error of CSIMMS gradually approaches the horizontal axis. This paper proposes to use the CSI to accurately solve the proportion of failure areas in the system, and at the same time reduce the amount of calculation through selective calculation of failure modes of different orders of magnitude. Therefore, the obtained results are closer to the MCS solution while improving the computational efficiency, which proves the effectiveness of the method. In addition, the similarity in the trend of the error with curvature for the three methods illustrates the fact that CSIMMS is essentially the same as the other two methods, which are all based on the approximation of the original function by the MPP.

In the $U$-space shown in Fig. 10, the shaded part represents the failure domain of the system, and the blank part represents the safety domain. After MPP search and second-order approximation, the reliability index $β1=β2=3$ and quadratic term coefficient $λ1=λ2=0.2$ of the nonlinear functions $g1(u)$ and $g2(u)$⁠, the reliability index $β3=β4=3.5$ of the linear functions $g3(u)$ and $g4(u)$⁠. When the overlapping region of the failure domains is small or far from the spatial origin, its failure probability differs by orders of magnitude from that of the nonoverlapping failure domain. For this case, the failure probability of the overlapping region is negligible, and therefore, this type of problem can be approximated to be analyzed as a system reliability problem where the failure domains do not overlap each other. Using FORM, SORM, CSIMMS proposed in this article, and MSC (10⁷ samples) for analysis, the results of the failure probability are given in Table 4.

Table 4 shows that the results of the method proposed in this article are closer to the real values. The results of FORM have the largest error, followed by SORM. The error of FORM mainly comes from the nonlinear function in the system. The evaluation of the failure area of the nonlinear function by SORM is not as accurate as the CSIMMS The results of the three methods are larger than the real values because the failure probability of the overlapping area is repeatedly calculated.

This example shows that the method proposed in this paper is applicable to both linear functions and nonlinear functions of nonstandard normal distribution. In addition, the method is also applicable to the case where the intersection area of the failure domains is small and the corresponding reliability index is large. By skipping the analysis of overlapping failure domains, the complex calculation process is reduced and the efficiency is improved. In addition, the accuracy is improved by combining the CSI method for reliability analysis.

where $AC, AS, fC, fS,Ec,Es, q$⁠, and $l$ are independent random variables. Their parameters and distribution forms are given in Table 5.

The results of three different reliability methods including FORM, SORM, and the proposed method are shown in Table 6. The results indicate that compared to MCS (10⁷ samples), the proposed method in this study demonstrates high accuracy, with an error in the failure probability of only 1.02%. It is observed from the FORM results that the failure probability of the second failure mode is two orders of magnitude different from the system failure probability. Therefore, in the proposed CSIMMS, the computation of the LSF $g2$ can be skipped, which reduces the computational process while ensuring the accuracy. The example demonstrates that the proposed CSIMMS exhibits good stability when dealing with multivariate and multiple failure modes problems such as the truss structure mentioned in this study.

Based on the quadratic surface properties generated by second-order reliability analysis methods, CSI has been proposed for accurately solving the failure probability of approximate surfaces. In this paper, the computational accuracy of the CSI in the two-dimensional case is evaluated, while different failure modes are categorized based on the reliability index and the degree of nonlinearity, and CSIMMS for analyzing the reliability of systems with nonoverlapping computational failure domains is proposed.

The process of solving for system failure probability is simplified by determining the magnitude of failure probabilities of different failure modes through order of magnitude indices and ignoring the effect of individual failure probabilities of lower order of magnitude on the system. In addition, for the case of overlapping partial failure domains, when the overlap area is small and the reliability index is large, the impact of the failure probability corresponding to the overlapping part on the system can be ignored, and CSIMMS proposed in this paper is also applicable. In order to demonstrate the validity and accuracy of the method, the results of the last three examples are given in the paper. The method is currently applied to cases where the failure domains do not overlap each other and to cases where individual failure domains overlap. Future work could focus on reliability analysis when the failure domains of multiple failure modes overlap each other.

• National Natural Science Foundation of China (Grant No.52375236; Funder ID: 10.13039/501100001809).

The authors attest that all data for this study are included in the paper.

$fX(x)$ =

joint probability density function

$gi$ =

$i$th failure mode

LSF =

limit state function

MPP =

most probable point of failure

$Pfi=$ =

failure probability of $gi$

$U$ =

$n$-dimensional standard normal vector

$u*$ =

position vector of the MPP

$x$ =

$n$-dimensional random vector

$Y$ =

standard normal vector after rotational transformation

$δi$ =

order of magnitude index of $Pfi$

$κi$ =

$i$th the principal curvature of the approximate parabolic surface

$λi$ =

$i$th coefficient of the approximate parabolic surface

$β$ =

first-order reliability index