System reliability analysis through active learning Kriging model with truncated candidate region

doi:10.1016/j.ress.2017.08.016

Reliability Engineering & System Safety

Volume 169, January 2018, Pages 235-241

https://doi.org/10.1016/j.ress.2017.08.016 Get rights and content

Highlights

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Existing strategies fail to identify the insignificant component(s) if large numerical difference exists.
•
A brand-new theory to circumvent the shortcoming is proposed.
•
A method based on ALK model with a truncated candidate region is proposed.
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The performance of the proposed method is compared with the existing methods.

Abstract

System reliability analysis (SRA) with multiple failure modes is researched in this paper. Active learning Kriging (ALK) model which only finely approximates the performance function in the narrow region close to the limit state has shown great potential and several strategies based on ALK model have been proposed. The key of SRA based on ALK model is to identify the components with little contribution to system failure and avoid approximating them. However, we figure out that the existing strategies fail to fulfill this task if large numerical difference exists among the values of component performance functions. Therefore, a brand-new theory on identifying the unimportant component(s) is proposed. Based on this theory, the method based on ALK model with a truncated candidate region (TCR) is proposed and it is termed as ALK-TCR. ALK-TCR is capable to recognize and avoid approximating the unimportant component(s), even if large numerical difference arises among the components. Its high performance is demonstrated by three complicated examples.

Introduction

The key point of reliability analysis is to estimate the failure probability of a system considering randomness of input variables. Complex mechanical system can have multiple failure modes. Reliability analysis simultaneously considering all the failure modes is called ‘‘system reliability analysis’ ’ (SRA) as opposed to ‘‘component reliability analysis’ ’ (CRA) where only one single failure mode is considered [1].

In practical engineering, the performance function of each failure mode usually needs to be computed by time-consuming simulation software [1], [2], [3]. Obtaining the system reliability with as few function calls as possible has motivated many researches so far. Monte Carlo simulation (MCS) method is available to SRA, but it needs N × n function evaluations for a system with n components and each component with N simulations. Variance reduction techniques can also be applied to SRA. Methods based on subset simulation and adaptive importance sampling were proposed in Refs. [4], [5], [6]. However, this kind of simulation methods still needs thousands of function calls. The first-order or second-order reliability methods (FORM/ SORM) originally proposed for CRA can be adapted to SRA [7], [8]. However, FORM or SORM behaves poorly if the performance function is highly non-linear or has multiple failure regions which are frequently confronted in SRA. Using bounding inequalities, the failure probability of the system can be bounded in an interval. To calculate the lower and upper bounds, several methods were developed, such as the first order or second-order bound method [9], the linear programming bound method [10], the so-called complementary intersection method [11], [12], the saddle-point approximation method [13], et al. However, the bounds can be pretty large in practice and it is not convenient to estimate the true value of failure probability accordingly [3].

When confronted to highly non-linear or multimodal performance functions, it is available to approximate them with advanced surrogate models such as support vector machines [14], neural networks [15], polynomial chaos expansions [16] and Kriging (or Gaussian process) model. Among them, the proposition of active learning Kriging (ALK) model is a major step forward. ALK model means the Kriging model constructed following the three steps: (I) build an initial Kriging model with a small number of training points; (II) add a new training points probably located in the region of interest and update the Kriging model; (III) if the Kriging model is not accurate enough, go to (II). Through this iterative process, large portion of the training points will be located in the region of interest rather than throughout the design space. Locally approximating the performance function is the major advantage of ALK model over other plain surrogate models. Past few years witness the booming of ALK model and many brilliant works have been done by researchers from all over the world [17], [18], [19], [20], [21]. Among them, the Active learning and Kriging-based Monte-Carlo Simulation (AK-MCS) method [18] and the Efficient Global Reliability Analysis (EGRA) method [17] are the most popular ones. In reliability analysis, only the sign of performance function affects the value of failure probability. In each iteration, AK-MCS or EGRA adds a new training point at which the sign of performance function has the largest risk to be wrongly predicted by the current Kriging model. As a result, most of the training points will be located in the vicinity of the limit state. Therefore, both of them are able to estimate the failure probability with high efficiency and accuracy.

With such advantages, EGRA and AK-MCS were adapted to SRA in Refs. [1], [3] and the adapted methods are named EGRA-SYS and AK-SYS. In SRA, some components may have little contribution to the system failure. The basic idea of EGRA-SYS or AK-SYS is to pay as little attention as possible to the insignificant components. However, we figure out that both of them fail to rightly identify the insignificant component if large numerical difference exists among the values of component performance functions. In practical engineering, the dimensions of the performance functions can be very different. The dimensions may be Kg for mass, mm for length, KN for force, MPa for stress, m/s for velocity, etc. The differences of dimensions will result in large numerical differences among the performance functions. Sometimes, the values may differ by several orders of magnitude. If all those performance functions are black-box functions, it is hard to transform their values into the same numerical magnitude. In such case, EGRA-SYS and AK-SYS cannot avoid approximating the insignificant component and wasted computational efforts will arise. In Refs. [22], a so-called Integrated Performance Measure Approach(IPMA) was proposed for SRA. IPMA built an initial Kriging model for each component and update the one with largest prediction error in each iteration. That means if a Kriging model is not accurate enough, it will be updated, even the one has little contribution to the system failure. Therefore, the IPMA strategy is not the optimal option for SRA. Similar shortcoming arises in the strategy from Ref. [23] where time-dependent SRA was researched. In Ref. [2], a new active learning algorithm based on Kriging model was proposed for SRA. Focusing directly on the accuracy of SRA and considering the dependence between Kriging predictions, the efficiency of AK-SYS was improved in Ref. [24]. However, the problem on large numerical difference was not figured out or solved by those two papers.

In this paper, we propose a brand-new theory on identifying the insignificant component(s). Based on this theory, a so-called adaptive truncating region (ATR) is defined to identify the unimportant region and the candidate region of each Kriging model is truncated by the corresponding ATR. Then each Kriging model is updated in its truncated candidate region (TCR). Following this strategy, rare training points will arise in the unimportant region and thus the unimportant component will not be finely approximated by its Kriging model. The proposed method is developed in the framework of AK-SYS and is termed as ALK-TCR. ALK-TCR inherits the basic features of AK-MCS and AK-SYS, like it does not approximate the performance function in the region with little probability density and most training points are located in the neighborhood of limit state. Moreover, ALK-TCR is capable to recognize and avoid approximating the unimportant component(s), even if large numerical difference arises among the components.

This paper is organized as follows. In Section 2, component reliability analysis with AK-MCS and EGRA is briefly reviewed and some comments are made on the two strategies. Section 3 introduces how AK-MCS and EGRA are adapted to SRA and the shortcoming resulting from large numerical difference is expressed. Our basic idea and the methodology ALK-TCR are explained in Section 4. Three complicated numerical examples are researched to demonstrate the performance of ALK-TCR in Section 5 and conclusions are made in the last section.

Section snippets

Component reliability analysis

Let us denote the performance function as g(x) and define the failure region as $F = {x | g (x) < 0}$ , with x the input random variables. The failure probability is defined as $P_{f} = P {x \in F} = \int \int \dots \int I (x \in F) f (x) d x$ where I( · ) is the indicator function of an event, with value 1 if the event is true and 0 otherwise.

Traditional methods need a large number of performance function evaluations. If the performance function needs to be evaluated by time-consuming simulation software, it will be economic to approximate the

System reliability analysis

The failure probability of a series system is defined as $P_{f} = P {\cup_{i = 1}^{p} g_{i} (x) < 0} = P {{min}_{i = 1}^{p} g_{i} (x) < 0}$ and that of a parallel system is $P_{f} = P {\cap_{i = 1}^{p} g_{i} (x) < 0} = P {{max}_{i = 1}^{p} g_{i} (x) < 0}$

Define $G (x) = \min_{i = 1}^{p} g_{i} (x)$ or $G (x) = \max_{i = 1}^{p} g_{i} (x)$ and G(x) is called the system performance function (SPF), as opposed to g_i(x) the component performance function (CPF). $G (x) = 0$ is named system limit state (SLS) as opposed to $g_{i} (x) = 0$ the component limit state (CLS).

AK-MCS or EGRA can be adapted to SRA by the flowing three strategies.

The

Basic idea

Directly comparing the values of the Kriging models cannot figure out the unimportant components when large numerical difference exists. Here we offer a brand-new point of view which can avoid such comparison.

Let us start with a series system. For the k^th Kriging model ${\hat{g}}_{k} (x)$ , there is no need to rightly predict the sign of g_k(x) in the region $T_{k} = {x | {\cup_{i = 1}^{k - 1} g_{i} (x) < 0} \cup {\cup_{i = k + 1}^{p} g_{i} (x) < 0}}$ . This proposition is obvious, because, if x ∈ T_k, no matter g_k(x) is negative or positive, it holds true that $G (x$

A parallel system with multiple failure regions

The first example is taken from Ref. [3]. The failure probability is defined by $P_{f} = P {g_{1} (x) < 0 \cap g_{2} (x) < 0 \cap g_{3} (x) < 0}$ and the CPFs are given as $\begin{matrix} {\begin{matrix} g_{1} (x) = 8 x_{2}^{2} - 8 x_{1}^{2} + {(x_{1}^{2} + x_{2}^{2})}^{2} \\ g_{2} (x) = 2 x_{1}^{2} - 2 x_{2}^{2} - {(x_{1}^{2} + x_{2}^{2})}^{2} \\ g_{3} (x) = α (8 x_{2}^{2} - 8 x_{1}^{2} - {(x_{1}^{2} + x_{2}^{2})}^{2}) \end{matrix} \end{matrix}$ in which x₁ and x₂ obey standard normal distribution, α is a coefficient controlling the value of g₃(x). Two cases, i.e. $α = 1$ and $α = 1000$ , are investigated here to demonstrate that the performance of the proposed method is proof against large numerical difference among the

Conclusion

SRA with multiple performance functions is researched in this paper. To minimize the computational cost, it is wise to identify the unimportant CPFs and pay little attention to their limit states. The existing strategies try to fulfill this task by directly comparing the predicted values of Kriging models which may fail if large numerical difference exists among the CPFs. This paper proposes a brand-new strategy which avoids such comparison by introducing the so-called ATR. ATR is the region

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 51705433), the Fundamental Research Funds for the Central Universities (Grant No. 2682017CX028), and Science and Technology Innovation Research Team of Sichuan Province (Grant No. 2017TD0017).

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System reliability analysis through active learning Kriging model with truncated candidate region

Highlights

Abstract

Introduction

Section snippets

Component reliability analysis

System reliability analysis

Basic idea

A parallel system with multiple failure regions

Conclusion

Acknowledgements

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Comput Struct

Comput Struct

Probab Eng Mech

Struct Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf

Struct Saf

Comput Methods Appl Mech Eng

Reliab Eng Syst Saf

Struct Saf

Reliab Eng Syst Saf

Reliab Eng Syst Saf