Efficient surrogate models for reliability analysis of systems with multiple failure modes
Highlights
► Extends efficient global reliability analysis to systems with multiple failure modes. ► Constructs locally accurate Gaussian process models of each response. ► Highly efficient and accurate method for assessing system reliability. ► Effectiveness is demonstrated on several test problems from the literature.
Introduction
The aim of computational reliability analysis is to assess the safety, or “reliability,” of an engineered system by considering how its performance is affected by random variations and uncertainties in demands, system properties, boundary and initial conditions, etc. The objective is to compute an estimate of the probability that the performance of the system will satisfy a given requirement, or, conversely, the probability that the system will fail. However, most systems have multiple modes in which failure can occur. The system might consist only of a single component that is subject to multiple different failure mechanisms (e.g., stress and displacement in a mechanical component), or it might be made up of multiple components. The analysis of such systems is referred to as “system reliability analysis” where multiple failure modes are considered, as opposed to “component reliability analysis” where only a single failure mode is considered.
Two types of methods are available for reliability analysis. Monte Carlo sampling methods are easy to use, but such methods are often computationally prohibitive because of the number of performance model evaluations that they require. This has given rise to a number of alternative, “analytical” methods that are based on the concept of a most probable point (MPP) of failure. Such methods are drastically more efficient, but they also introduce additional approximations that are sometimes unacceptable.
For large structures requiring time-consuming finite element analysis, component-level reliability analysis can be a computationally expensive task and system-level reliability analysis can be even more so. Monte Carlo sampling methods can be used for system analysis, but the expense is increased n-fold when applied to a system made up of n components if each response function is defined in a separate model. The MPP-based efficient analytical reliability methods have also been adapted for system analysis by developing an approximation of the combined system failure region. However, these analytical methods already involve approximations at the component level, and calculating the reliability given this combined failure region involves yet another approximation that has the potential to introduce substantial error.
The efficiency and accuracy tradeoffs between sampling and analytical methods have been well studied for component reliability analysis, but an additional feature arises with system reliability analysis: some component failure modes may contribute more than others to the probability of system failure. For example, in a series system, the probability of failure is driven primarily by the weakest component. However, the previously proposed sampling and analytical methods for system reliability analysis treat all of the component performance functions equally, ignoring the fact that in many cases one or more failure modes are significantly more important than the others. In such a situation, the computational effort expended in evaluating the performance functions associated with unimportant failure modes or in locating the corresponding most probable point of failure is wasted.
This paper presents an approach based on the use of surrogate models that is designed to take advantage of this special feature of system analysis. The performance models associated with the various failure modes are only evaluated as necessary to construct an accurate surrogate model of the system failure surface. The method is an extension of previous work on component reliability analysis, and it is based on the creation of Gaussian process surrogate models that are required to be locally accurate only in the regions of the component limit states that contribute to system failure.
Section 2 provides an introduction to reliability analysis methods for component- and system-level problems. Section 3 describes the use of Gaussian process models for component reliability analysis. Section 4 reformulates this approach for reliability analysis at the system level. Three potential methods are explored, but one is determined to be clearly superior to the others. Section 5 applies this newly derived method to three system problems including both series and parallel systems, comparing its accuracy to sampling methods and its efficiency, where available, to results from the literature. Section 6 provides some conclusions on the proposed method.
Section snippets
Reliability analysis
The goal of reliability analysis is to determine the probability that an engineered component or system will fail in service given that its behavior is affected by random inputs. This behavior is defined by a response function , where is the vector of random inputs with the joint probability density function . Failure is defined to occur when the response function exceeds (or fails to exceed) some threshold value . The probability of failure, pf, is then computed by
Efficient global reliability analysis
Efficient global reliability analysis was recently introduced by the authors in Ref. [13]. EGRA locates multiple points in the vicinity of the limit state and uses these points to construct a Gaussian process model that provides a global approximation for the entire limit state. By adaptively focusing the samples in the only region of the space where accuracy is important (near the limit state) an effective model is created with a minimal number of samples. The resulting model is then used as
System-level formulations for EGRA
There are multiple ways in which EGRA might be extended to the system-level reliability analysis problem. This section will explore three of them, detailing the advantages and disadvantages of each.
Computational experiments
The third method detailed in the previous section for using EGRA to solve system-level reliability analysis problems is clearly more promising than the others, so only this method is applied to the collection of test problems explored in this section. EGRA is compared to first-order analytical methods and varying levels of Latin hypercube sampling to demonstrate its efficiency and accuracy.
The example problems explored here were chosen to emulate the type of complex response functions for which
Conclusions
This paper presented the application of the efficient global reliability analysis method to system-level reliability analysis. Three formulations for applying EGRA to this class of problems were explored, but one was identified as the best option. This formulation uses independent Gaussian process models for each of the component response functions, and selects the training data for these models based on a search for the composite limit state. At each new training point selected by EGRA, only
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