A collocation reliability analysis method for probabilistic and fuzzy mixed variables

Structural reliability analysis with fuzzy random variables using error principle

In structural reliability calculation, there are fuzzy uncertainties in the distribution parameters of random variables, which bring the problem of large computation and poor precision. In order to improve accuracy and efficiency of structural reliability, a novel structural reliability calculation method with fuzzy random variables is proposed from the perspective of perturbation principle. First, the fuzzy characteristic parameters of the structural fuzzy random variables are expressed as the sums of their true values and fuzzy perturbations. At the same time, the perturbations are equivalently selected. Second, the probability density function and the performance function are decomposed into a series of intervals under different level cut sets, and the Taylor series is expanded on the upper and lower limits of the intervals. Finally, according to the definition of structural reliability, based on the direct integration method, the sigmoid function is introduced into the reliability function to approximate the step function, and the fuzzy reliability is obtained. The simulation results show the effectiveness and accuracy of the proposed method, and it provides a feasible way for structural reliability analysis with fuzzy variables or fuzzy random variables.

The study on epistemic uncertainty due to the lack of knowledge has received increasing attention in risk assessment, reliability analysis, decision making, and design optimization. Different theories have been applied to model and quantify epistemic uncertainty. Research on sensitivity analysis for epistemic uncertainty has also been initialized. Sensitivity analysis can identify the contributions of individual input variables with epistemic uncertainty to the model output. It then helps guide the collection of more information to reduce the effect of epistemic uncertainty. In this paper, an effective sensitivity analysis method for epistemic uncertainty is proposed when both epistemic and aleatory uncertainties exist in model inputs. This method employs the unified uncertainty analysis framework to calculate the plausibility and belief measures. The gap between belief and plausibility measures is used as an indicator of the effect of epistemic uncertainty on the model output. The Kolmogorov-Smirnov distance between the two measures is used to quantify the main effect and the total effect of each independent variable with epistemic uncertainty. By the Kolmogorov-Smirnov distance, the importance of each variable is ranked. The feasibility and effectiveness of the proposed method is demonstrated with two engineering examples.

In modern engineering, physical processes are modelled using advanced computer simulation tools (e.g. finite element models) to assess and optimize their performance. Moreover, awareness is growing on concepts like structural reliability and robust design, hence making the efficient quantification and propagation of uncertainties a key challenge. A major part of those analyses is the characterization of uncertainty in the input, which is typically done with probabilistic variables. In the case of sparse data sets, however, probability theory is often not the optimal choice. Imprecise probabilities provide a more general framework accounting for both aleatory and epistemic uncertainties. The uncertainty propagation of imprecise probabilities leads to imprecise responses. In this context, an algorithm for solving imprecise structural reliability problems is presented. The algorithm transforms the imprecise problem into two precise structural reliability problems, which opens up possibilities for using traditional structural reliability analyses techniques. An adaptive experimental design algorithm based on Polynomial-Chaos-Kriging is used to efficiently estimate imprecise failure probabilities. The capabilities of the framework are illustrated through an application using an analytical function and a realistic engineering problem setting.

A sensitivity analysis estimates the effect of the change in the uncertain variable parameter on the probability of the structural failure. A novel fuzzy random reliability sensitivity measure of the failure probability is proposed to consider the effect of the epistemic and aleatory uncertainties. The uncertainties of the engineering variables are modeled as fuzzy random variables. Fuzzy quantities are treated using the y-cut approach. In fact, the fuzzy variables are transformed into the interval variables using the y-cut approach. Genetic approach considers different possible combinations within the search domain (y-cut) and calculates the parameter sensitivities for each of the combinations.

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from un- known statistical distributions, known as epistemic uncer- tainty. Existing methods in uncertainty-based design optimization have difficulty in handling both aleatory and epistemic uncertainties. To tackle design problems engaging both epistemic and aleatory uncertainties, reliability-based design optimization (RBDO) is integrated with Bayes theorem. It is referred to as Bayesian RBDO. However, Bayesian RBDO becomes extremely expensive when employing the first- or second-order reliability method (FORM/SORM) for reliability predictions. Thus, this paper proposes development of Bayesian RBDO methodology and its integration to a numerical solver, the eigenvector dimension reduction (EDR) method, for Bayesian reliabil- ity analysis. The EDR method takes a sensitivity-free approach for reliability analysis so that it is very efficient and accurate compared with other reliability methods such as FORM/SORM. Efficiency and accuracy of the Bayesian RBDO process are substantially improved after this integration.

The design of engineering systems is often based on analysis models with substantial errors in predicting failures, that is epistemic uncertainty. The epistemic uncertainty is reduced by post design tests, and the safety of unsafe designs restored by redesign. When this process of design, test and redesign is to be based on probabilistic analysis, there is some controversy on whether uncertainty associated with variability (aleatory uncertainty) should be treated differently than the epistemic uncertainty. In this paper we compare several approaches to design and redesign and treatments of the epistemic uncertainties. These include safety factors, probabilistic approach disregarding redesign and regarding redesign, treating epistemic uncertainty and aleatory uncertainty the same, and more conservative treatment of the epistemic uncertainty. We demonstrate that the proposed approach can allow tradeoff of system performance against development cost (probability of redesign), while a standard reliability based design optimization, which does not take into account future redesign, provides only a single point on the tradeoff curve. We also show that the tradeoff can be achieved even with the traditional safety factor approach, without any probabilistic optimization. Furthermore, we investigate different treatments of epistemic error for probability of failure calculation. We find that it is possible to design to the 95th percentile of the probability of failure with modest mass penalty compared to treating epistemic and aleatory uncertainty alike.

A novel method for importance measure analysis in the presence of epistemic and aleatory uncertainties

Aiming at providing more accurate information for the first restoration, overhaul and maintenance of airborne products, this paper gives a calendar life assessment method for airborne products considering epistemic and aleatory uncertainty on the basis of investigation and analysis for the factors which affect the uncertainty of calendar life assessment. The uncertainty of calendar life is modeled as two types: epistemic uncertainty and aleatory uncertainty. a aleatory uncertainty model of calendar life based on interval analysis and blind number theory is given with parameter sensitivity analysis. Considering the designer's problem of incomplete cognition of failure mechanism, a measurement method of epistemic uncertainty is given based on the concept of confidence reliability.

Statistical model calibration and design optimization under aleatory and epistemic uncertainty

An honest communication of uncertainty about quantities of interest enhances transparency in scientific assessments. To support this communication, risk assessors should choose appropriate ways to evaluate and characterize epistemic uncertainty. A full treatment of uncertainty requires methods that distinguish aleatory from epistemic uncertainty. Quantitative expressions for epistemic uncertainty are advantageous in scientific assessments because they are nonambiguous and enable individual uncertainties to be characterized and combined in a systematic way. Since 2019, the European Food Safety Authority (EFSA) recommends assessors to express epistemic uncertainty in conclusions of scientific assessments quantitatively by subjective probability. A subjective probability can be used to represent an expert judgment, which may or may not be updated using Bayes's rule to integrate evidence available for the assessment and could be either precise or approximate. Approximate (or bounded) probabilities may be enough for decision making and allow experts to reach agreement on certainty when they struggle to specify precise subjective probabilities. The difference between the lower and upper bound on a subjective probability can also be used to reflect someone's strength of knowledge. In this article, we demonstrate how to quantify uncertainty by bounded probability, and explicitly distinguish between epistemic and aleatory uncertainty, by means of robust Bayesian analysis, including standard Bayesian analysis through precise probability as a special case. For illustration, the two analyses are applied to an intake assessment.

Key Points Methods are provided to quantify information and uncertainty in data Best achievable model performance for a given data set is estimated Model adequacy is characterized in terms of Aleatory and Epistemic Uncertainties

An approach to combining unreliable pieces of evidence and their propagation in a system response analysis

Pressure systems contain hazardous fluids within industrial processes. Inspection plays a vital role in managing the reliability of these safety-critical systems. It allows engineers to identify, characterize, and manage degradation of pressure vessels, piping, and associated equipment to prevent failure and the associated consequences on people and the environment. Mixed uncertainty can affect decision-making in at least three distinct aspects of inspection practice: inspection planning, inspection data analysis, and integrity assessment. Despite this, the inspection engineering discipline lacks methodologies for handling both aleatory and epistemic uncertainties within analyses, which could be expensively misleading. This paper demonstrates the benefits of applying mixed uncertainty quantification and analysis techniques to pressure vessel inspection and integrity assessment through a worked example, which shows how the epistemic and aleatory uncertainty in inspection data can be represented using an imprecise probability approach. The limitations of empirical data are shown to pose challenges to implementing these techniques in practice, and so practical requirements for a framework for implementing uncertainty analysis methods in inspection are proposed. These include, for example, the ability to generate meaningful yet conservative results from even a limited amount of poor-quality data, while allowing results to be bounded more narrowly as more data is collected, findings from better data are pooled, or engineering judgment and assumptions are applied.

Geotechnical engineering involves significant aleatory and epistemic uncertainties, often inadequately addressed in traditional reliability analyses, which tend to neglect epistemic uncertainties, simplify dependencies among random variables, and rely on less efficient methods such as Monte Carlo simulations, FORM, or SORM. This paper presents a reliability analysis method that integrates both aleatory and epistemic uncertainties using random sets, copulas for variable dependence, and the subset simulation algorithm for enhanced efficiency. Rather than providing a single-point probability of failure, our methodology computes bounds on failure probability whose width reflects the epistemic uncertainties in the input variables. The evaluated methodology allows geotechnical engineers to assess whether the bounds on the probability of failure are adequate or if more and better data are needed to refine them, leading to more reliable models. The applicability of the approach is demonstrated through three practical geotechnical examples.