Adaptive Surrogate Modeling for Time-Dependent Multidisciplinary Reliability Analysis

Multidisciplinary analysis (MDA) has been intensively studied during the past two decades [1,2]. Current MDA methods can be roughly classified into three groups, namely, the field elimination method, the monolithic method, and the partitioned method [3,4]. Based on these MDA methods, approaches have been developed for the reliability analysis of multidisciplinary systems by considering the uncertainty sources [5–7]. While most of the current multidisciplinary reliability analysis (MDRA) methods have only considered aleatory uncertainty, a couple of studies have also included model uncertainty [2,8,9]. The above MDRA methods have only considered time-independent reliability analysis, whereas time-dependent behavior and uncertainty sources are of interest in many practical multidisciplinary systems [10].

In the past decades, reliability analysis methods have been investigated for engineering systems with time-dependent uncertainty sources [11–14]. These methods, however, are developed for single disciplinary systems and cannot be directly applied to multidisciplinary systems, due to the complicated couplings between different disciplinary simulations. Motivated by solving this problem, Zhu et al. [15] recently developed a time-dependent reliability analysis method for a multidisciplinary system using the first-order reliability method (FORM). Since FORM could have large errors for problems with nonlinear limit-state functions, the method presented in Ref. [15] can only be applied to multidisciplinary systems where FORM is accurate [15].

This work aims to develop a general time-dependent MDRA method by exploring the surrogate modeling approach, where expensive physics simulation models are replaced by surrogate models. Monte Carlo simulation (MCS) is then performed on the surrogate models to estimate the failure probability. To guarantee the accuracy of reliability analysis, two types of strategies have been pursued for surrogate model refinement, namely, uncertainty minimization and region of interest (RoI). In the former, the surrogate model is refined by minimizing either the bias or variance in the prediction [16]. In the latter, the surrogate model is refined mainly in the RoI (i.e., the limit state), which is important for reliability analysis [17]. The limit-state surrogate modeling methods have been developed for both time-independent [18–20] and time-dependent reliability analysis [11,21], and have shown excellent performance. However, current limit-state surrogate methods focus only on single disciplinary systems and are inapplicable to multidisciplinary systems.

In this paper, we are interested in pursuing the RoI strategy to improve the efficiency and accuracy of the surrogate model in time-dependent MDRA. Several challenges as detailed in Sec. 2.4 are encountered. The main challenge comes from the fact that the surrogate models built for individual disciplinary systems are not directly connected with the quantities of interest (QoIs) with respect to reliability analysis, which brings challenges in determining the RoI. Some questions that need to be answered include when to refine the surrogate models, which disciplinary surrogate models to refine, and where to refine the individual disciplinary surrogate models. This work solves these challenges through the development of a systematic adaptive surrogate modeling framework. In this framework, approaches are proposed to determine whether the surrogate models need to be refined and at what time instant to refine. Along with that, a three-level global sensitivity analysis (GSA) method is developed to adaptively allocate the computational resources to refine the individual disciplinary surrogate models. The developed framework is generic and applicable to both time-dependent and time-independent MDRA. Two numerical examples are used to demonstrate the effectiveness of the proposed framework.

The remainder of the paper is organized as follows: Section 2 provides background concepts on MDRA under time-dependent uncertainty. Section 3 describes the proposed adaptive surrogate modeling framework. Section 4 illustrates the application of the proposed methodology with two numerical examples. Concluding remarks are provided in Sec. 5.

In this section, we first introduce motivation problems of the proposed research. We then briefly review the differences between time-dependent and time-independent multidisciplinary analyses, and the general procedure of RoI-based surrogate modeling methods for reliability analysis. Based on that, we summarize the challenges in the adaptive surrogate modeling for time-dependent MDRA.

Coupled multidisciplinary analysis is very common in the analysis of many engineering systems. The coupling of individual disciplinary analysis is represented as the exchanging physical quantities between different computational models. The required computational effort is very large when sophisticated simulation models are used in the analysis, even for deterministic MDA [22]. One example is the aero-elastic analysis of an aircraft wing, which involves coupled finite element analysis (FEA) and computational fluid dynamics simulation as shown in Fig. 1 [4]. Since time-dependent MDRA requires repeated MDAs under different realizations of uncertainty sources (i.e., random variables and stochastic processes), it is computationally unaffordable when the original simulation models are used.

Another motivation problem is the reliability analysis of a panel structure on a conceptual hypersonic aircraft vehicle [23]. As the vehicle is subjected to hypersonic flow, an attached oblique shock is created at the forebody leading edge [24]. The resulting aerodynamic pressure causes elastic deformation of the panel, which feeds back to alter the aerodynamic pressure on the panel. The panel is also subjected to aerothermal effects (aerodynamic heating and heat transfer). The aerothermal effects are coupled with the aero-elastic effects [24]. A detailed description of this panel structure is available in Refs. [23] and [25]. Figure 2 shows the four disciplinary analysis models of the panel (aerodynamics, aerodynamic heating, heat transfer, and structural deformation) [23].

In the four-disciplinary analysis, finite element models are used in each individual disciplinary analysis to predict the response of coupling variables and response variables. A partitioned approach is employed to solve the four-disciplinary analysis over time since the partitioned method enables us to use relatively independent solution procedures for each discipline [23]. The partition sequence in the analysis is: (1) aerodynamic pressure, (2) aerodynamic heating, (3) thermal, and (4) structural [23]. Since the simulation models in each disciplinary analysis are computationally expensive, directly performing time-dependent MDRA is computationally prohibitive. New strategies are required to reduce the required computational effort in time-dependent MDRA without sacrificing the accuracy of reliability analysis.

The nodal responses of FEA or computational fluid dynamics analyses are usually used as coupling variables and response variables in MDA and MDRA. As a result, the coupling and response variables may have both spatial and temporal variability. In this paper, we only focus on coupled systems with temporal variability alone. The developed method in this paper could be integrated with the method presented in Ref. [25] to handle spatial and temporal variability simultaneously. In what follows, we first provide a generalized model of multidisciplinary analysis under time-dependent uncertainty. Based on the generalized model, we present the proposed new method.

Figure 3 shows a generalized multidisciplinary system with three individual disciplines [26]. The notations of different variables are also given in the figure. In Secs. 2–4, we use $X$ to represent input random variables and $Y(t)$ to represent input stochastic processes. For the ith discipline, the inputs include random variables (⁠$Xi, Xs$⁠), stochastic processes (⁠$Yi(t), Ys(t)$⁠), and time (⁠$t$⁠). The subscript s refers to shared inputs across multiple disciplines. There are also coupling variables (⁠$Lij(t)$⁠) between the ith and jth disciplines. The response variables (⁠$Zi(t)$⁠) are functions of the coupling variables and input variables.

In time-independent MDRA, MDA is a deterministic analysis for a given realization of uncertainty sources, and will converge after several iterations. This situation also holds for time-dependent MDRA if the time-dependent inputs only affect the system response variables but not the coupling variables [15]. However, when the time-dependent inputs affect the coupling variables, the coupling variables will change over time; thus we do not have distributions of converged values of these coupling variables and will not be able to decouple the disciplinary analyses. As a result, the computational effort becomes prohibitive in time-dependent MDRA, due to the need to run the fully coupled MDA over the entire duration of interest for each realization of the inputs. In this situation, the partitioned method of MDA needs to be employed. As mentioned previously, a partitioned approach to MDA will enable the use of separate solution procedures for each disciplinary analysis and also handle different time intervals in coupled multidisciplinary analysis, i.e., the same time-step or identical discretization for individual disciplines is not required. In addition, note that the partition sequence may affect the result of MDA; the proposed MDRA method is based on the assumption of a predetermined partition sequence from deterministic MDA.

As shown in Fig. 4, at any time instant $ti$⁠, the MDA given in Fig. 3 can be reformulated as Fig. 4. Based on the formulation given in Fig. 4, the MDA becomes a multilevel analysis problem at a given time instant and the analysis is coupled together between time instants.

where $gLkj(⋅)$ is a vector of models of coupling variables $Lkj(ti)$⁠, $gZk(⋅)$ is a vector of models of response variables $Zk(ti)$⁠, $Lk•(ti)$ is a vector of all coupling variables from the kth discipline, $Lj>k,k(ti−1)$ is a vector of coupling variables from the jth discipline to the kth discipline with j > k, and $Lj<k,k(ti)$ are coupling variables from the jth discipline to the kth discipline with j < k.

Next, we briefly review the procedure of adaptive surrogate-based reliability analysis. After that, we discuss the challenges in adaptive surrogate modeling for time-dependent MDRA.

In adaptive surrogate-based reliability analysis, the original computer simulation models are substituted with surrogate models. Training points are added adaptively in the RoI (i.e., around the limit state) to guarantee the efficiency and accuracy of reliability analysis. As discussed in Sec. 1, adaptive surrogate modeling approaches were first developed for time-independent reliability analysis and have been extended to time-dependent reliability analysis of single disciplinary systems [11]. The general procedure of adaptive surrogate modeling for time-dependent reliability analysis can be summarized as follows:

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  Step 1: Build the initial surrogate model and generate random realizations of random variables and stochastic processes.

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  Step 2: Perform predictions using the surrogate model and samples generated in step 1.

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  Step 3: Check convergence of the failure probability estimate. If the convergence criterion is satisfied, obtain the failure probability estimate. Otherwise, go to the next step.

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  Step 4: Identify a new training point in the region of interest, update the surrogate model by adding the new training point into current training points, and then go to step 2.

In step 4, the identification of new training points in the RoI can be achieved based on different learning functions (e.g., the expected feasibility function defined in Ref. [17] or the U function defined in Ref. [27]) or performing global sensitivity analysis [28]. Even if the detailed implementation procedures may vary with methods, most current adaptive surrogate modeling approaches can be generalized into the above summarized procedure.

where $Zk(ti)≥0$ is the safe state of the kth disciplinary response variables, “$∀$” means “for all,” $[t0, te]$ is the time interval of interest, and $Rseries(t0, te)$ and $Rparallel(t0, te)$ are the reliability values of series and parallel system, respectively.

For systems with other configurations, the time-dependent reliability needs to be defined according to the reliability block diagram. The objective of this paper is to develop a framework to efficiently and accurately estimate the time-dependent multidisciplinary system reliability by pursuing the adaptive surrogate modeling strategy. However, the procedure summarized in Sec. 2.3 cannot be directly applied to MDRA due to the difference between multidisciplinary system and single disciplinary system. A major difference is that the QoIs in a single disciplinary system are only affected by the uncertainty sources of that discipline, whereas the QoIs in a multidisciplinary system are affected by not only the uncertainty sources in their own discipline, but also by those in the other disciplines. The uncertainty sources are present in each discipline and the uncertainty accumulates as the analysis proceeds from one model to the other and over time. This is also why current available surrogate modeling method for single disciplinary system reliability analysis [29] cannot be applied to solve the MDRA problem. To efficiently and accurately estimate the time-dependent failure probability, a question that needs to be answered is how to effectively allocate the computational resources to train the individual disciplinary surrogates. To answer this question, the following challenges need to be addressed:

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  Whether: After we replace the original simulation models with surrogate models, the first question that needs to be answered is whether the surrogate models are good enough to be used to estimate the failure probability. This question is answered by quantifying the uncertainty in the failure probability estimate and assessing whether the uncertainty is larger than a threshold.

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  Where: At what realizations of the inputs and uncertainty sources to refine the surrogate models? If the surrogate models of the individual disciplinary analyses are not directly connected with the failure probability estimate, currently available learning functions cannot be used to determine where to refine the surrogate models.

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  When: As shown in Fig. 4, the multidisciplinary system simulation under time-dependent uncertainty is performed iteratively over time. The uncertainty due to various sources will accumulate over time. At what time instant to refine the surrogate models needs to be addressed?

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  Which: After we identified when and at what realization of the uncertainty sources to refine the surrogate models, the next challenge is to determine which disciplinary surrogates to refine and which surrogate model within the discipline to refine (since each disciplinary analysis is substituted with multiple surrogate models, one for each output of the disciplinary analysis).

In Sec. 3, we develop a framework called adaptive surrogate-based multidisciplinary analysis of reliability over time (A-SMART) to address the above questions.

In this section, we first provide an overview of the proposed A-SMART framework. Following that, we explain the framework in detail.

In the proposed A-SMART framework, for the purpose of generalization, we assume that all the simulation models are expensive. We substitute individual disciplinary simulation models with surrogates. In other words, we build surrogate models for all $Lkj(t)$ and $Zk(t)$ and determine how to allocate computational resources to refine these surrogate models. Figure 5 shows the flowchart of the main steps of the proposed framework.

The proposed framework (as given in Fig. 5) consists of three main modules, namely, initialization, uncertainty propagation, and three-level GSA. The initialization and uncertainty propagation modules inherit the basic idea of the single loop Kriging surrogate modeling approach presented by Hu and Mahadevan [11]. In these two modules, initial surrogate models are built for the response and coupling variables. Then the surrogate model uncertainty sources are propagated through the time-dependent MDA to quantify the effects of surrogate model uncertainty on the time-dependent failure probability estimate. Based on the results of uncertainty propagation, we can quantify the uncertainty in the failure probability estimate from the current set of surrogate models and thus determine whether the surrogate models need to be refined, where to refine, and when to refine. In the three-level GSA module, we overcome the challenge that the surrogate model uncertainty sources are not directly connected to the time-dependent failure probability by performing GSA at three levels. The first two modules address the challenges of whether, where, and when summarized in Sec. 2.4. The last module solves challenge of which. A brief summary of the three modules corresponding to the procedure given in Sec. 2.3 is given as follows:

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  Module 1—Initialization: In this module, random realizations of $Xk, Xs$ and $Yk(t), Ys(t)$⁠, $k=1, 2, …, Nd$ are generated. Initial surrogate models are built for $Lkj(t)$ and $Zk(t)$⁠. This is similar to step 1 of the procedure summarized in Sec. 2.3.

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  Module 2—Uncertainty propagation: This module includes steps 2–4 shown in Fig. 5. The basic idea is similar to steps 2–3 discussed in Sec. 2.3. Due to the multilevel simulation at each time instant and the coupling between different time instants, the implementation details are quite different from that of single disciplinary analysis even if the basic idea is similar.

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  Module 3—Three-level GSA: This module covers steps 5–8 of the flowchart presented in Fig. 5. In this module, we first identify the response variable that makes the highest contribution to the uncertainty in the failure probability estimate (this is level 1 GSA) since the response variables are directly connected to the failure probability. After the response variable is identified, we identify which surrogate model (i.e., response surrogate model or coupling variable surrogate model) to refine to reduce the uncertainty in the response variable through level 2 and 3 GSAs.

In Secs. 3.2–3.4, we explain the three modules in details. For the sake of illustration, we use a multidisciplinary system with two disciplines to explain the three modules.

According to the description of multidisciplinary system given in Sec. 2.2, at a given time instant $ti$⁠, a two disciplinary system is simulated as shown in Fig. 6(a). For each discipline, there are models of coupling variables and response variables. For example, as depicted in Fig. 6(b), for the kth discipline, there are models of $Lk·(t)$ and $Zk(t)$⁠, $∀k=1, …, Nd$⁠.

In module 1, we build initial surrogate models for $Lk·(t)$ and $Zk(t)$⁠, $∀k=1, …, Nd$⁠. In this paper, the surrogate models are built using the Kriging surrogate method. For the sake of illustration, in Sec. 3.2, we use $XLk=[Xk, Xs, Yk(ti),Ys(ti), Lj<k,k(ti), Lj>k,k(ti−1)]$ to represent all the inputs of $Lk·(t)$⁠.

When we are building the surrogate models, we do not know the domains of the coupling variables. To generate training points, we need to get a rough guess of the domains by performing several deterministic MDAs over the time duration of interest with fixed random realizations of random variables and stochastic processes. Once we have a rough guess of the domains, we generate training points of $XLk$⁠, $Lk·$⁠, and t. We then evaluate the models of individual disciplines separately at these training points. Based on the training points, we build the initial surrogate models for $Lk·(t)$ and $Zk(t)$⁠, $∀k=1, …, Nd$⁠. Note that the surrogate models are built for one time-step MDA as indicated in Fig. 4. Here, the generated training points are therefore for partitioned MDA at one time-step. In addition to the generated training points, the MDA iterations used to get the initial guess of coupling variables ranges can also be added into the pool of initial training points for surrogate modeling. Even if the initial training points may not fully cover the ranges of $XLk$⁠, $Lk·$⁠, and t, more training points will be added in important regions later in the proposed method through adaptive sampling approach to improve the accuracy of reliability analysis.

where $ĝLk·$ and $ĝZk$ represent the surrogate model approximations of $gLk·$ and $gZk$⁠, respectively.

These surrogate models are then used to substitute the original simulation models in time-dependent MDA. Along with the surrogate models, in the initialization module, we also generate random samples for $Xk, Xs, Yk(ti), Ys(ti)$⁠, $∀k=1, …,Nd; i=1, 2, …,Nt$⁠, where $Nt$ is the number of time instants in the time interval $[t0, te]$⁠. We denote the generated random samples as $xk(q), xs(q), yk(q)(ti), ys(q)(ti)$⁠, $k=1, …,Nd; i=1, 2, …, Nt; q=1,2, …, NMCS$⁠, where $xk(q)$ and $xs(q)$ represent the qth sample of $Xk$ and $Xs$⁠, $yk(q)(ti)$ and $ys(q)(ti)$ are the qth trajectory of $Yk(t)$ and $Ys(t)$ at time instant $ti$⁠, and $NMCS$ is the number of MCS samples.

where $NkZ$ is the number of response variables from the kth discipline, and $μZk(p)(t)$ and $σZk(p)2(t)$ are the mean and variances of the prediction of $Ẑk(p)(t)$ at time instant $t$⁠. The mean and variance are obtained from Kriging prediction.

The above equations imply that after substituting the original simulation models with surrogate models, we introduce surrogate model uncertainty into MDA. Next, we will discuss the propagation of the surrogate model uncertainty and investigate how to determine the quality of the surrogate models based on the uncertainty propagation.

In this work, to investigate the effect of surrogate model uncertainty on the failure probability estimate, we first quantify the uncertainty in the response variables due to the uncertainty of the coupling and response surrogates for any given realization of the input variables.

Taking the multidisciplinary system shown in Fig. 6(a) as an example, for the qth sample of the input variables (i.e., $xk(q), xs(q), yk(q)(ti), ys(q)(ti)$⁠, $k=1, 2; i=1, 2, …, Nt;$⁠) generated in Sec. 3.2, the MDA under uncertainty is a multimodel uncertainty quantification (UQ) problem as shown in Fig. 7. In the network, square nodes represent deterministic nodes and elliptical nodes represent random nodes. At any time instant $ti$⁠, a multimodel UQ is performed and the uncertainty quantified at $ti$ will be propagated to $ti+1$ due to the coupling between time instants. This process continues until the maximum number of iterations is reached.

where $μL21(p)(t1)|l12(w)(t1)$ and $σL21(p)2(t1)|l12(w)(t1)$ are the conditional mean and variance predicted from the Kriging surrogate model by using $[x2(q), xs(q), y2(q)(t1), ys(q)(t1), l12(w)(t1)]$ as inputs.

where $zk(p, w)(ti)$ is the wth sample of the pth response variable of the kth discipline at $ti$⁠.

For the system given in Fig. 7, $Nd=2$⁠. Next, we study how to determine whether the surrogate models need to be refined or not based on the random samples given in Eq. (11).

where $Perror(p, q, k, i)$ is the probability of making an error on the sign of the pth response of the kth discipline at time instant i for given $xk(q), xs(q), yk(q)(ti), ys(q)(ti)$⁠, and $Pr{⋅}$ stands for probability.

where $Ĩk,p(q)$ is the safety state indicator of the pth response of the kth discipline for the qth realization of random variables and stochastic processes.

In addition, if there exists such a $ti$ that $I(μZk(p)(ti))=1$ and $Perror(p, q, k, i)<0.05$⁠, we can directly assign $Perrort(p, q, k)=0$⁠. This is due to the fact that once the response is classified as failed at a time instant and the probability of making an error on the classification is low, the corresponding trajectory can be classified as failed.

The above equation indicates that if the kth disciplinary response needs to be refined, it should be refined at the input setting $xk(q*), xs(q*), yk(q*)(tik*), ys(q*)(tik*), tik*$⁠. From Eqs. (20)–(25), the questions of whether, where, and when (summarized in Sec. 2.4) have been answered. Next, we investigate which surrogate to refine through the development of a three-level GSA scheme.

As shown in Fig. 5, there are three levels of analysis: coupling variables→response variables→system safety indicator. To minimize the maximum probability of making an error $Perrorsys(q*)$ by reducing the surrogate model uncertainty, a three-level GSA scheme is developed.

where $Ai$ is the ith input variable, $A∼i$ is the vector of variables excluding variable $Ai$⁠, $Var(B)$ is the variance of quantity of interest (B), and $EA∼i(B|Ai)$ is the expectation by freezing $Ai$⁠.

where $Pr{IS(q*)=1}$ is computed using Eq. (19), $Pr{Ik,p(q*)=1}$ is computed using Eq. (16), and $Pr{IS(q*)=1|Ik,p(q*)=0}$ is computed by combining the RBD with Eq. (16).