Structural reliability analysis with epistemic and aleatory uncertainties via AK-MCS with a new learning function

Kriging meta-model

The Kriging meta-model, often categorized as a GPR, serves as a regression algorithm designed for spatial modeling and prediction (interpolation) of stochastic processes or random fields, relying on covariance functions^[39]. As outlined in^[40], the Gaussian process is frequently applied to achieve Kriging predictions. This involves the assumption of a relationship between the real response of the system and the input variables, characterized by:

where the deterministic function $$ {\bf Y}\left( {{\bf{x}},\mathit{\boldsymbol{\beta}} } \right) $$ serves as an estimate of the model response's mean, and its general regression model is typically expressed as $$ {\bf Y}\left( {{\bf{x}},\mathit{\boldsymbol{\beta}}} \right) = {\bf{y}}{\left( {\bf{x}} \right)^T}{\mathit{\boldsymbol{\beta}}} $$. In this formulation, $$ {\bf{y}}{\left( {\bf{x}} \right)^T} = \left\{ {{y_1}\left( {\bf{x}} \right),{y_2}\left( {\bf{x}} \right), \ldots ,{y_k}\left( {\bf{x}} \right)} \right\} $$ represents the regression function, while $$ {{\mathit{\boldsymbol{\beta}}}^T} = \left\{ {{\beta _1},{\beta _2}, \ldots ,{\beta _k}} \right\} $$ corresponds to the associated regression coefficients. The residual or noise, denoted as $$ {\mathit{\boldsymbol{\varepsilon}}} $$, is assumed to adhere to a zero-mean normal distribution with independent and identical distribution, given by $$ f\left( {\mathit{\boldsymbol{\varepsilon}}} \right) = N\left( {{\mathit{\boldsymbol{\varepsilon}}}\left| {0,\sigma _{\mathit{\boldsymbol{\varepsilon}}}^2R\left( {{\bf{x}},{\bf{w}}} \right)} \right.} \right) $$. Here, $$ {\sigma _{\mathit{\boldsymbol{\varepsilon}}}^2} $$ signifies the variance of the Gaussian process, and $$ {R\left( {{\bf{x}},{\bf{w}}} \right)} $$ represents the correlation function between points $$ \bf{x} $$ and $$ \bf{w} $$ within the spatial domain. Besides, the Squared Exponential Kernel function, which incorporates distinct length scales for each predictor^{[41, 42]}, is chosen to model $$ {R\left( {{\bf{x}},{\bf{w}}} \right)} $$ in this paper.

The Kriging meta-model can be conceptualized as a process in which a Gaussian prior model is combined with observations to derive a posterior model. Given a training set $$ { \bf{T}} = \left\{ { {\bf{X}},{ \bf{Z}}} \right\} $$, where $$ {\bf{X}} = {\left\{ {{{\bf{x}}_1},{{\bf{x}}_2}, \ldots ,{{\bf{x}}_p}} \right\}^T} $$ represents the realizations of input variables, and $$ {{\bf{Z}}^T} = \left\{ {{z_1},{z_2}, \ldots ,{z_p}} \right\} $$ corresponds to the associated output values of the random system. The parameters $$ \beta $$ and $$ {\sigma _{\mathit{\boldsymbol{\varepsilon}}}^2} $$ are then estimated using the approach described in^[43]:

where $$ {{\bf{R}}_{ij}} = R\left( {{{\bf{x}}_i},{{\bf{x}}_j}} \right) $$ is the correlation matrix between the initial sample points in the training set and $$ \bf{1} $$ is a 1-vector of length $$ p $$. Obviously, the values of $$ \beta $$ and $$ \sigma _{\mathit{\boldsymbol{\varepsilon}}}^2 $$ depend on the correlation parameter in correlation function $$ \bf{R} $$. Cross-validation or maximum likelihood estimation can be applied when the correlation parameter is unknown^[44].

Consider a collection of unobserved points denoted as $$ {{\bf{x}}^*} $$, where the associated system output value is $$ \hat g \left( {{{\bf{x}}^*}} \right) $$. It is noteworthy that $$ \hat g \left( {{{\bf{x}}^*}} \right) $$ adheres to a Gaussian distribution characterized by its mean value and variance:

where $$ u\left( {{{\bf{x}}^ * }} \right) = {{\bf{1}}^T}{R^{ - 1}}{\bf{r}}\left( {{{\bf{x}}^ * }} \right) - 1 $$ and $$ {\bf{r}}\left( {{{\bf{x}}^ * }} \right) = {\left\{ {R\left( {{{\bf{x}}^ * },{{\bf{x}}_i}} \right)} \right\}_{i = 1,2, \ldots ,p}} $$. In summary, the Kriging meta-model stands as an exact interpolation model. For an experimental design point $$ \bf{x} $$, the attributes $$ \hat g\left( {\bf{x}} \right) = g\left( {\bf{x}} \right) $$ and $$ \sigma _{\hat g}^2\left( {\bf{x}} \right) = 0 $$ hold true, a depiction of which can be observed in Figure 2. The distinctive feature of the Kriging meta-model, namely its ability to quantify local uncertainty, has led to its widespread adoption within the realm of active learning in recent years.

Figure 2. Kriging meta-model prediction.

The proposed learning function REF

In the context of the Kriging meta-model integrated with active learning, the selection of the next best point is significantly influenced by the learning function^[38]. An effective learning function has the capacity to accelerate the refinement process of the Kriging meta-model while also ensuring result accuracy. This is particularly advantageous for intricate finite element models that demand substantial computational resources, as it can markedly enhance computational efficiency. Recent years have witnessed the emergence of several learning functions tailored specifically for reliability analysis.

Based on the Effcient Global Optimization (EGO)^[43], the Effcient Global Optimization (EFF) was proposed for efficient global reliability analysis^[45]. The EFF elucidates the degree to which the actual value of the performance function at a specific point $$ \bf x $$ satisfies the equality constraint $$ g\left( {\bf{x}} \right) = y $$ within the designated domain $$ \left[ {y - \xi ,y + \xi } \right] $$. Furthermore, it is assumed that the output value $$ \hat g\left( {{\bf{x}}} \right) $$ of the system adheres to a Gaussian distribution within the context of Kriging prediction. In other words, $$ \hat g\left( {\bf{x}} \right) $$ is modeled as being $$ \sim N\left( {{\mu _{{{\hat g}}}},\sigma _{{{\hat g}}}^2} \right) $$.

The general formulation of EFF reads:

where $$ \varPhi \left( \cdot \right) $$ and $$ \varphi \left( \cdot \right) $$ denote the CDF and PDF of standard normal distribution, respectively.

In the context of reliability analysis, significant attention is directed toward the limit state of the performance function. As a consequence, the value of $$ y $$ is typically set to 0. Additionally, the parameter $$ \xi $$ is often defined as $$ {2{\sigma _{\hat g}}} $$. In accordance with Equation (7), the EFF value provides insights into both the level of uncertainty at the point $$ \bf x $$ and the relative proximity of point $$ \bf x $$ to the limit state. Points exhibiting maximal EFF values are particularly valuable for constructing a more accurate Kriging meta-model. To facilitate the selection of points in close proximity to the limit state while also displaying high uncertainty, the U learning function has been introduced for reliability analysis^[24]:

In recent advancements, a novel learning function rooted in information entropy has been introduced, as shown in^[46]. It is worth noting that information entropy serves as a gauge of the uncertainty level of the data: higher entropy values signify greater uncertainty, while lower values indicate reduced uncertainty.

The information entropy is defined as^[47]

where $$ f\left( {{{\hat g}}} \right) $$ denotes the PDF of $$ \hat g\left( {\bf{x}} \right) $$. Equation (9) shows the information entropy function $$ H\left( {\hat g\left( {\bf{x}} \right)} \right) $$ can be used to quantitatively judge the uncertainty of $$ {\hat g\left( {\bf{x}} \right)} $$^[46].

The learning function H is proposed to be applied to AK prediction^[46].

The detailed derivation of Equation (10) {can be found} in Appendix A.

The learning criteria and stopping criteria of the above three learning functions are compared in Table 1.

As previously mentioned, information entropy serves as a means to quantify the uncertainty associated with a random variable, rendering it a suitable tool for gauging the confidence in the model response value during Kriging prediction. In the active learning procedure, the realizations of random variables that affect the system with large uncertainty are used to construct a GPR, and then the Kriging prediction will be close to the true response of the system. This ensures that the Kriging prediction closely approximates the real system response, particularly within critical regions such as limit state areas. Subsequently, based on the stability criterion, a new learning function rooted in the Relative Entropy (RE) will be put forward.

RE finds its application in quantifying the disparity between two probability distributions. Within active learning algorithms, the probability distribution of the system response $$ \hat g\left( {{\bf{x}}} \right) $$ is subject to ongoing updates. Prior to introducing a new experimental design point, the system response is denoted as $$ {{\hat g}_p}\left( {\bf{x}} \right) $$ with $$ {{\hat g}_p}\left( {\bf{x}} \right)\sim N\left( {{\mu _{{{\hat g}_p}}},\sigma _{{{\hat g}_p}}^2} \right) $$, and its corresponding probability distribution is indicated by $$ f\left( {{{\hat g}_p}} \right) $$. The notation $$ {{\hat g}_q}\left( {\bf{x}} \right) $$ signifies the response after the incorporation of a new experimental design point, characterized by $$ {{\hat g}_q}\left( {\bf{x}} \right)\sim N\left( {{\mu _{{{\hat g}_q}}},\sigma _{{{\hat g}_q}}^2} \right) $$, and its associated probability distribution is denoted as $$ f\left( {{{\hat g}_q}} \right) $$. It is evident that $$ {{\hat g}_q}\left( {\bf{x}} \right) $$ is contingent upon $$ {{\hat g}_p}\left( {\bf{x}} \right) $$. In accordance with the definition of RE, the following relationships hold:

The RE serves as a metric for assessing the disparity between $$ {{\hat g}_p}\left( {\bf{x}} \right) $$ and $$ {{\hat g}_q}\left( {\bf{x}} \right) $$. A diminishing RE implies that the constructed Kriging meta-model closely mirrors the characteristics of the original stochastic system. For the purpose of pinpointing optimal points within the stochastic system, the learning function can be defined as follows:

where $$ {\hat g_p^ + } $$ and $$ {\hat g_p^ - } $$ are defined by $$ {2{\sigma _{{{\hat g}_p}}}} $$ and $$ {-2{\sigma _{{{\hat g}_p}}}} $$ to ensure the accuracy in this paper. The detailed derivation of Equation (12) can be found in Appendix B.

The REF learning function plays a pivotal role in gauging the stability of the estimated system response $$ \hat g\left( {\bf{x}} \right) $$ at a specific point $$ \bf{x} $$ and facilitates the construction of a meta-model that closely aligns with the stochastic characteristics of the system, especially in proximity to the limit state. To strike a balance between computational efficiency and precision, a termination criterion of $$ RE{F_{\max }} \le {10^{ - 3}} $$ is adopted for the learning function in this paper, based on our prior computational experiences.

To summarize, the proposed AK-MCS-REF framework entails a comprehensive set of seven steps for conducting structural reliability analysis in a precise probability sense:

Step 1: Generate a sample pool $$ S $$ comprising $$ n_{mc} $$ points using quasi-MCS methods such as Sobol sampling.

Step 2: Initialize the initial design of experiments (DoE). Select the first $$ n_0 $$ points from $$ S $$, evaluate them using the LSF, and establish a preliminary Kriging meta-model utilizing the MATLAB function fitrgp. An empirical value of $$ n_0=12 $$ is suggested based on computational considerations.

Step 3: Compute the predicted value $$ {\hat G_0} $$ of the preparation Kriging meta-model across the entire sample pool.

Step 4: Define the formal DoE using the first $$ n_1 $$ points from $$ S $$ and subsequently build the formal Kriging meta-model. A recommended value of $$ n_1=13 $$ is provided.

Step 5: Construct the formal Kriging meta-model based on the formal DoE. Compute the predicted value $$ {\hat g} $$ of the formal Kriging meta-model for the entire sample pool, thereby estimating the failure probability $$ {\hat p_f} $$.

Step 6: Implement the learning function $$ REF $$ and assess the convergence criterion. Should the termination threshold be met, proceed to the subsequent step. Otherwise, identify the optimal point using Equation (13):

Evaluate $$ g\left( {{{\bf{x}}^ * }} \right) $$, add $$ {\bf{x}}^ * $$ and $$ g\left( {{{\bf{x}}^ * }} \right) $$ to the formal DoE, and then return to Step 5 to formulate a novel formal Kriging meta-model incorporating $$ n_0+1 $$ points.

Step 7: Compute the Coefficient Of Variation (C.O.V.) for the failure probability using Equation (14):

If $$ CoV\left( {{{\hat p}_f}} \right) \le {\varepsilon _{{p_f}}} $$, the procedure terminates, and $$ {\hat p}_f $$ is deemed the definitive result for the failure probability. Conversely, if $$ CoV\left( {{{\hat p}_f}} \right) > {\varepsilon _{{p_f}}} $$ upon utilizing all points in the sample pool, $$ {\hat p}_f $$ lacks credibility, necessitating the generation of a new sample pool $$ S $$ with an increased $$ n_{mc} $$. Subsequently, return to Step 2.

Imprecise structural reliability analysis for parametric p-boxes

In the context of parametric p-boxes, the distributions of the random vector $$ \bf{X} $$ are dependent upon their corresponding distribution parameters. Consequently, the determination of the conditional failure probability $$ {p_{f\left| \mathit{\boldsymbol{\beta}} \right.}} $$ and the establishment of the failure probability range $$ \left[ {{{\underline p }_f},{{\overline p }_f}} \right] $$ become pivotal components of imprecise structural reliability analysis. The conditional failure probability is formally defined as follows:

where $$ {{{\bf{X}}_\mathit{\boldsymbol{\beta}} }} $$ represents a random variable generated using $$ {F_{{{\bf{X}}_\theta }}} = {F_{\bf{X}}}\left( {{\bf{x}}\left| \mathit{\boldsymbol{\beta}} \right.} \right) $$ based on a specific set of distribution parameters $$ \mathit{\boldsymbol{\beta}} $$. As a result, the parametric p-boxes exhibit a characteristic hierarchical model structure. The approach of nested algorithms, as outlined in^[48], can be employed in imprecise structural reliability analysis to address the complexities posed by parametric p-boxes. Within this hierarchical framework, the parametric p-boxes can be segregated into two distinct components. The AK-MCS with REF learning function, as detailed above, is well-suited for addressing the task of evaluating the conditional failure probability $$ {p_{f\left| \mathit{\boldsymbol{\beta}} \right.}} $$. Since the failure probability is dependent upon the distribution parameters, the EGO algorithm^[43] can be utilized to optimize $$ \mathit{\boldsymbol{\beta}} $$ and thereby deduce the bounds of the failure probability. This integrated approach enables a comprehensive treatment of the uncertainties associated with parametric p-boxes.

Figure 3 depicts the flowchart, illustrating the process of imprecise structural reliability analysis coupled with the innovative REF learning function, denoted as AK-MCS-REF-EGO. The overall procedure encompasses the following sequential steps:

Figure 3. Flowchart of AK-MCS-REF-EGO.

Step 1: Generate a sample pool for distribution parameters denoted as $$ {{\bf D}_{\mathit{\boldsymbol{\theta}}}} $$, comprising $$ {n_{\mathit{\boldsymbol{\theta}}}} $$ points. This can be accomplished through the utilization of Latin-Hypercube Sampling (LHS), wherein each point corresponds to a distinct set of distribution parameters, i.e., $$ {{\mathit{\boldsymbol{\theta}}}_j} \in {{\bf D}_{\mathit{\boldsymbol{\theta}}}}\left( {j = 1,2, \ldots ,{n_{\mathit{\boldsymbol{\theta}}}}} \right) $$. It is noteworthy that a sample size of $$ {n_{\mathit{\boldsymbol{\theta}}}}=10^6 $$ is typically deemed adequate for this purpose.

Step 2: Define the DoE for the distribution parameters. To achieve this, $$ {n{}_{\mathit{\boldsymbol{\theta}}}^0} $$ samples are selected at random from the sample pool $$ {{\bf D}_{\mathit{\boldsymbol{ θ}} }} $$, and these selected samples are recorded as $$ {\bf D} = \left\{ {{{\mathit{\boldsymbol{\theta}}}^{\left( 1 \right)}}, \ldots {{\mathit{\boldsymbol{\theta}}}^{\left( {n{}_{\mathit{\boldsymbol{\theta}}}^0} \right)}}} \right\} $$. For the purposes of this paper, the value of $$ {n{}_{\mathit{\boldsymbol{\theta}}}^0} $$ is set to 6.

Step 3: Compute the conditional failure probability $$ p_f^{\left( 1 \right)} $$ of $$ {{{\mathit{\boldsymbol{\theta}}}^{\left( 1 \right)}}} $$ by the procedure of AK-MCS-REF.

Step 4: Calculate the conditional failure probabilities $$ {\bf P} = \left\{ {p_f^{(2)}, \ldots ,p_f^{(n_{\mathit{\boldsymbol{\theta}}}^0)}} \right\} $$ for $$ {\bf D} = \left\{ {{{\mathit{\boldsymbol{\theta}}}^{\left( 2 \right)}}, \ldots ,{{\mathit{\boldsymbol{\theta}}}^{\left( {n_{\mathit{\boldsymbol{\theta}}}^0} \right)}}} \right\} $$ using the proposed AK-MCS-REF approach. Beginning with $$ {\mathit{\boldsymbol{\theta}}}^{\left( 2 \right)} $$, the initial DoE needed for constructing the Kriging meta-model does not have to be selected exclusively from the Sobol sample pool associated with $$ {\mathit{\boldsymbol{\theta}}}^{\left( 2 \right)} $$. Instead, it can be composed of the sample points that have already been evaluated in the preceding AK-MCS-REF iterations. In essence, the initial DoE for $$ {\mathit{\boldsymbol{\theta}}}^{\left( i+1 \right)} $$ comprises all the LSF evaluations of $$ {\mathit{\boldsymbol{\theta}}}^{\left( i \right)} $$. This implies that all the previously acquired sample points, along with their corresponding outputs, can be recycled to construct a new Kriging meta-model.

Step 5: Build a second-level Kriging meta-model, utilizing the information from $$ {\bf P} $$ and $$ {\bf D} $$, to approximate the conditional failure probabilities $$ \bf{\hat P} $$ of the random system.

Step 6: Determine the lower and upper bounds of failure probabilities, $$ {\underline p _f} $$ and $$ {\overline p _f} $$, by identifying the current minimum failure probability $$ {\bf {\hat P}}_{\rm min} $$ and maximum failure probability $$ {\bf {\hat P}}_{\rm max} $$ from the approximated $$ \bf{\hat P} $$. To search for optimal distribution parameters, the expected improvement (EI) function is employed^[43]. Specifically, for $$ {\underline p _f} $$, the EI function is defined as follows:

For $$ {\overline p _f} $$, there is:

where $$ EI_{\min } $$ and $$ EI_{\max } $$ represent the EIs for minimizing and maximizing the failure probability, respectively.

Step 7: Determine the optimal distribution parameters using $$ EI_{\min} $$ and $$ EI_{\max} $$:

Step 8: Utilize the convergence criterion proposed in^[43] to estimate the bounds of failure probability.

where $$ {\varepsilon _{EI}} $$ is the threshold value, {which is} $$ {\varepsilon _{EI}}=10^{-5} $$ in this paper. If the criterion is satisfied, the EGO algorithm terminates and outputs the minimum ($$ {\underline p_f}=\min \left[ {\hat {\bf P}} \right] $$) and maximum ($$ {\overline p_f}=\max \left[ {\hat {\bf P}} \right] $$) failure probabilities. If not met, proceed to Step 9 for further iterations.

Step 9: Incorporate the optimal next distribution parameters $$ {\mathit{\boldsymbol{\theta}}}_{\min }^ * $$ ($$ {\mathit{\boldsymbol{\theta}}}_{\max }^ * $$) into $$ \bf D $$ and return to Step 4. Repeat this cycle until the convergence criterion specified in Step 8 is met, at which point the optimization algorithm stops.

Example 1

A classic series system with four branches is first employed to validate the effectiveness of the proposed method^[24]. The LSF is defined as follows:

To validate the efficacy of the proposed learning function, a precise probability level analysis is initially conducted, where the variables $$ x_1 $$ and $$ x_2 $$ are assumed to be independent standard normal random variables. The learning process of the proposed REF learning function at the precise probability level is depicted in Figure 4. The optimal points selected by the REF learning function are consistently situated near the limit state surface, and a majority of them satisfy the constraints of the limit state equation. This characteristic facilitates the capability of AK-MCS-REF to accurately predict the failure domain of the LSF. The results presented in Table 2 underscore that the proposed AK-MCS-REF method can offer a highly accurate prediction of the reliability index at the precise probability level while demanding minimal computational effort.

Figure 4. Experimental design in precise probability sense.

In the context of imprecise probability, the variables $$ x_i $$ are all characterized by parametric p-boxes, wherein the distribution parameters are selected as interval variables in this numerical illustration. The specific forms of the parametric p-boxes are presented below:

The LHS technique is employed to generate the parameter sample pool within the domains of $$ \mu \in [-0.1, 0.1] $$ and $$ \sigma \in [0.95, 1.05] $$, with a total of $$ 10^6 $$ samples. Among these samples, an initial set of six parameter samples are selected. Subsequently, the failure probabilities associated with these initial parameter sample points are computed using the AK-MCS-REF method, and the results are presented in Table 3. The initial sample points and their associated failure probabilities are employed to establish a second-level AK meta-model to address the imprecise probability scenario. The focus is on establishing the bounds of failure probabilities, achieved through the application of the $$ EI $$ criterion to locate extreme values. The evolution of the search for reliability index bounds $$ \left[ {\beta ^L, \beta ^U} \right] $$ is depicted in Figure 5, with summarized results provided in Table 4. Notably, the double-loop MCS utilizes $$ 10^5 $$ samples for the outer loop and $$ 10^6 $$ samples for the inner loop, resulting in a combined sample size of $$ 10^5 \times 10^6 $$ for this scenario. The upper and lower bounds of the reliability index are accurately determined by the proposed method, exhibiting a maximum error of just $$ 0.55% $$ when compared to double-loop MCS results. In order to visualize the required number of model calculations involved in the analysis, the number of model calculations is actually divided into three parts. Specifically, 88 computations correspond to generating failure probabilities for initial parameter sample points, while ten model evaluations are conducted to identify the minimum reliability index within the parameter sample pool. An additional ten evaluations are dedicated to locating the maximum reliability index within the parameter sample pool. This accumulation results in a total of $$ N_{call} = 80 + 10 + 10 = 108 $$ calculations. Consequently, the proposed method yields highly accurate results with notably fewer model calculations compared to the double-loop MCS approach. Observing Figure 5, it becomes evident that the EGO algorithm mandates four iterations to locate the maximum reliability index, while a mere two iterations suffice for the minimum reliability index search. Paradoxically, Table 4 presents a contrary perspective: regardless of aiming for the maximum or minimum reliability index, the essential number of LSF calls remains constant at 10. This divergence between the number of iterations and LSF calls underscores a nonlinear relationship between the two factors.

Figure 5. Evolution of the bounds of reliability index (Example 1).

Example 2

The second example employs a nonlinear LSF characterized by independent random variables, as detailed in^[52]. This LSF involves ten imprecise random variables and is represented as follows:

Likewise, the variables $$ x_i $$, where $$ i = 1,2, \ldots ,10 $$, are modeled using parametric p-boxes. The statistical details for these variables are provided in Table 5.

As depicted in Figure 6, both the maximum and minimum reliability indexes necessitate four iterations. Notably, Table 6 reveals that the maximum reliability index necessitates only one LSF evaluation, while the minimum reliability index requires 11 LSF evaluation calls. In this regard, the proposed method achieves rapid convergence to the bounds of the reliability index, requiring a total of 123 LSF evaluations. Clearly, the maximum error remains merely $$ 0.99% $$. These results serve as further validation of the accuracy and efficiency of the proposed method for conducting structural reliability analysis in the presence of dual sources of uncertainties.

Figure 6. Evolution of the bounds of reliability index (Example 2).

Example 3

The applicability of the proposed method for non-monotonic functions is demonstrated through the utilization of a highly nonlinear undamped single-degree-of-freedom system [Figure 7], as elaborated in^[24]. The LSF for this case is expressed as follows:

Figure 7. Single-degree-of-freedom system.

The statistical characteristics of the input random variables are presented in Table 7. In this context, the spring stiffness parameters ($$ k_1 $$ and $$ k_2 $$), mass ($$ m $$), and the displacement at which the secondary spring yields ($$ r $$) are modeled with precise probabilities. Conversely, the amplitude of the applied force ($$ F_1 $$) and the duration of the load ($$ t_1 $$) possess uncertain information and are represented using parametric p-boxes. Notably, the means and standard deviations of these variables are denoted as interval variables within the parametric p-boxes.

As depicted in Figure 8 and summarized in Table 8, the proposed method requires five iterations to successfully converge to the maximum reliability index. In this process, a total of 22 evaluations of the LSF are conducted, yielding an impressively low relative error of only $$ 0.17% $$. In comparison, for the minimum reliability index, the proposed method utilizes two iterations and performs four LSF evaluations, resulting in a slight increase in error to $$ 1.2% $$. Remarkably, the total number of LSF evaluations for the proposed method is 60+22+4=86, which is significantly fewer compared to the total number of evaluations required for the double-loop MCS, which amounts to $$ 10^5 \times 10^7 $$. For the double-loop MCS, the number of samples for the outer loop is $$ 10^5 $$, and the sample size for the inner loop is $$ 10^7 $$. Then, the total number of sample sizes in the double-loop MCS is $$ 10^5 \times 10^7 $$ in this example. These results underscore the efficacy of the proposed method in achieving both high efficiency and accuracy in structural reliability analysis, particularly when dealing with scenarios encompassing both epistemic and aleatory uncertainties.

Figure 8. Evolution of the reliability index boundary (Example 3).

Example 4

A practical validation is performed using a two-bay, four-story spatial concrete frame structure, aimed at demonstrating the practical engineering applicability of the proposed method. The structure takes into account the intricate nonlinear behaviors inherent to both concrete and rebar materials. To accurately capture the behavior of the system, a nonlinear beam-column finite element representation of each member is implemented within the OpenSees software^[53]. In accordance with Figure 9, the horizontal displacement at node 8 is designated as the control index, thereby defining the implicit LSF as follows:

Figure 9. Two-bay four-story spatial concrete frame.

where $$ D_8 $$ represents the horizontal displacement at node 8, while $$ \bar D $$ signifies the allowable displacement, set at $$ \bar D=50 \rm mm $$ within the context of this specific example. To ascertain the suitability of the proposed learning function in handling time-consuming finite element models, the example is initially employed to conduct a reliability analysis in a precise probability sense. The pertinent physical interpretations and statistical properties of the involved random variables are itemized in Table 9.

Table 10 presents the results of the reliability analysis conducted in a precise probability sense. This table includes results from various methods for comparison purposes, such as Importance Sampling (IS) and Subset Simulation (SS). Moreover, the AK-MCS approach is coupled with several well-established learning functions, including U^[24], EFF^[45], and H^[46], to facilitate a comprehensive evaluation. Upon comparison, it becomes evident that the proposed learning function REF enhances computational efficiency without compromising accuracy. Furthermore, the AK-MCS-REF methodology exhibits superior performance compared to classical approaches such as IS and SS, demonstrating improved accuracy and efficiency in the context of reliability analysis in a precise probability sense.

Next, the proposed method is applied to conduct imprecise reliability analysis. In this context, the loads are treated as imprecise random variables modeled using parametric p-boxes, with interval variables specifying their standard deviations, as shown in Table 11. Given the consideration of nonlinear behaviors in both beams and columns, the corresponding LSF becomes more intricate. The findings from Table 12 reveal that a total of 515 LSF evaluations are necessary to compute the conditional failure probabilities aligned with the initial parameter sample pool. Regarding the determination of reliability index bounds, Figure 10 illustrates that 32 and 39 iterations are required. Notably, a total of {358 and 309 times} of LSF evaluations are further required to estimate the maximum and minimum values of reliability indexes, respectively. To alleviate computational demands, the double-loop MCS approach is still employed, where the IS technique is utilized for the inner loop for reliability analysis in a precise probability sense. Meanwhile, the LHS method is employed for the outer loop, utilizing a parameter sample pool with $$ 10^4 $$ samples. The combination of IS and LHS is referred to as LHS-IS, serving as reference results to validate the proposed method. The results, presented in Table 12, indicate a maximum reliability index error of merely $$ 1.52% $$. Compared to the LHS-IS approach, which requires 11, 204, 350 finite-element analyses of the structure, the proposed method significantly reduces the computational burden, necessitating only 515 + 358 + 309 = 1182 model evaluations. These results underscore the capacity of the proposed method to effectively balance computational accuracy and efficiency for practical engineering structures.

Figure 10. Evolution of the bounds of reliability index (Example 4).

Authors' contributions

Made substantial contributions to the conception and design of the study and performed data analysis and interpretation: Du Y

Performed data acquisition and provided administrative, technical, and material support: Xu J

Availability of data and materials

Some or all data, models, or codes generated or used during the study are available from the corresponding author by request.

Financial support and sponsorship

The National Natural Science Foundation of China (Nos. 52278178, 51978253), Natural Science Foundation of Hunan Province, China (No. 2022JJ20012), the Open Fund of State Key Laboratory of Disaster Prevention in Civil Engineering, China (No. SLDRCE21-04) and Fundamental Research Funds for the Central Universities, China (No.531107040224) are gratefully appreciated for the financial support of this research.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

Not applicable.

Copyright

© The Author(s) 2023.