Unified Uncertainty Analysis Research Articles

Fast Convergence Reliability-Based Design Optimization Method Considering Random and Evidence Variables

An efficient approximate reliability-based design optimization method under both aleatory and epistemic uncertainty is presented. As stated by the unified uncertainty analysis based on the first-order reliability method (FORM-UUA), it is possible to merge the probability and evidence theory to quantify the belief and plausibility of a specific performance function under mixed uncertainty. When the number of evidence variables and the number of intervals increase, the number of evaluations grows dramatically. As a result, if the hybrid reliability analysis is included in an optimization problem, usually referred to as nested hybrid reliability-based design optimization (HRBDO), it becomes unmanageable due to its high computational cost. The strategy proposed allows to avoid the computation of the subplausibilities for each focal element as required in the FORM-UUA. This strategy decouples the nested HRBDO into an iterative process with a deterministic optimization and a reliability analysis phase consisting of two separate but connected reliability analyses that handle separately the random and evidence variables. Then, the optimum design obtained is checked and adjusted through the FORM-UUA method. One analytical and one numerical problem are presented to validate the proposed method.

Unified uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties

The uncertainty analysis of structures generally involves uncertain parameters of different types. In order to derive predictions regarding uncertain structural responses, it is crucial to represent the uncertainty appropriately according to the underlying information available. This paper presents a unified framework for uncertainty analysis under probabilistic, evidence, fuzzy and interval uncertainties, by which the quantities with sufficient data, sparse data, and subjective information can be simultaneously considered. A Taylor expansion-based unified uncertainty analysis (T-UUA) method is first proposed for small uncertainty problems. By temporarily neglecting the evidence, fuzzy and interval variables, the probability-evidence-interval-fuzzy model is degraded into a random problem, in which the expectations and variances of responses can be obtained as functions in terms of evidence, interval and fuzzy uncertainties. Then, through dealing with the evidence variables, the previous expectations and variances are further expressed as a summation of functions in terms of fuzzy and interval variables with basic probability assignments (BPAs). The fuzziness is then discretized by using α-cut technique and thus the expectations and variances are further expressed as functions of only intervals. Afterwards, by reconsidering the interval uncertainties, the bounds of the expectations and variances are computed via combining Taylor expansion with interval arithmetic. In addition, a dimensional reduction (DR)/efficient global optimization (EGO)-based unified uncertainty analysis (DR/EGO-UUA) method is also presented to solve the large uncertainty problems. The framework of DR/EGO-UUA is similar as T-UUA. However, in DR/EGO-UUA, the second moments of responses are computed by DR integrations, and their upper and lower bounds are calculated by the EGO. Finally, three numerical examples are investigated to demonstrate the effectiveness of the proposed methods.

An enhanced unified uncertainty analysis approach based on first order reliability method with single-level optimization

In engineering, there exist both aleatory uncertainties due to the inherent variation of the physical system and its operational environment, and epistemic uncertainties due to lack of knowledge and which can be reduced with the collection of more data. To analyze the uncertain distribution of the system performance under both aleatory and epistemic uncertainties, combined probability and evidence theory can be employed to quantify the compound effects of the mixed uncertainties. The existing First Order Reliability Method (FORM) based Unified Uncertainty Analysis (UUA) approach nests the optimization based interval analysis in the improved Hasofer–Lind–Rackwitz–Fiessler (iHLRF) algorithm based Most Probable Point (MPP) searching procedure, which is computationally inhibitive for complex systems and may encounter convergence problem as well. Therefore, in this paper it is proposed to use general optimization solvers to search MPP in the outer loop and then reformulate the double-loop optimization problem into an equivalent single-level optimization (SLO) problem, so as to simplify the uncertainty analysis process, improve the robustness of the algorithm, and alleviate the computational complexity. The effectiveness and efficiency of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.

On-orbit servicing system assessment and optimization methods based on lifecycle simulation under mixed aleatory and epistemic uncertainties

To assess the on-orbit servicing (OOS) paradigm and optimize its utilities by taking advantage of its inherent flexibility and responsiveness, the OOS system assessment and optimization methods based on lifecycle simulation under uncertainties are studied. The uncertainty sources considered in this paper include both the aleatory (random launch/OOS operation failure and on-orbit component failure) and the epistemic (the unknown trend of the end-used market price) types. Firstly, the lifecycle simulation under uncertainties is discussed. The chronological flowchart is presented. The cost and benefit models are established, and the uncertainties thereof are modeled. The dynamic programming method to make optimal decision in face of the uncertain events is introduced. Secondly, the method to analyze the propagation effects of the uncertainties on the OOS utilities is studied. With combined probability and evidence theory, a Monte Carlo lifecycle Simulation based Unified Uncertainty Analysis (MCS-UUA) approach is proposed, based on which the OOS utility assessment tool under mixed uncertainties is developed. Thirdly, to further optimize the OOS system under mixed uncertainties, the reliability-based optimization (RBO) method is studied. To alleviate the computational burden of the traditional RBO method which involves nested optimum search and uncertainty analysis, the framework of Sequential Optimization and Mixed Uncertainty Analysis (SOMUA) is employed to integrate MCS-UUA, and the RBO algorithm SOMUA-MCS is developed. Fourthly, a case study on the OOS system for a hypothetical GEO commercial communication satellite is investigated with the proposed assessment tool. Furthermore, the OOS system is optimized with SOMUA-MCS. Lastly, some conclusions are given and future research prospects are highlighted.

Unified uncertainty analysis by the mean value first order saddlepoint approximation

Uncertainties exist in products or systems widely. In general, uncertainties are classified as epistemic uncertainty or aleatory uncertainty. This paper proposes a unified uncertainty analysis (UUA) method based on the mean value first order saddlepoint approximation (MVFOSPA), denoted as MVFOSPA-UUA, to estimate the systems probabilities of failure considering both epistemic and aleatory uncertainties simultaneously. In this method, the input parameters with epistemic uncertainty are modeled using interval variables while input parameters with aleatory uncertainty are modeled using probability distribution or random variables. In order to calculate the lower and upper bounds of system probabilities of failure, both the best case and the worst case scenarios of the system performance function need to be considered, and the proposed MVFOSPA-UUA method can handle these two cases easily. The proposed method is demonstrated to be more efficient, robust and in some situations more accurate than the existing methods such as uncertainty analysis based on the first order reliability method. The proposed method is demonstrated using several examples.

A Step towards a Complete Uncertainty Analysis of Residual Stress Determination Using Neutron Diffraction

Up to now the tendency in residual stress determination (using neutron diffraction) has been to assess the uncertainties in terms of the propagation of ‘fitting uncertainty’ of the Bragg peaks only. There are many other sources of uncertainty, some more obvious than others, that should be taken into account or at least be considered in terms of their impact when interpreting the data. These cover not just the instrument calibration and characteristics and the technique itself but also the properties of the sample. A discussion of how best to combine the uncertainty of all contributing factors will be made. These factors (on the sample side) will include variations in chemical composition, grain size related problems, surfaces/interfaces cutting through the sampling volume, texture variations within the sampling volume, presence of intergranular strains (plastic anisotropy) etc. The knowledge of appropriate elastic constants, for example, and their uncertainty is necessary for a more reliable stress determination. One should also be aware of the more subtle influences on the elastic constants such as texture or chemical variation. This should be a step in the right direction for a ‘unified uncertainty analysis’ covering all possible aspects of uncertainty in residual stress determination using neutron diffraction.

Reliability Analysis for Multidisciplinary Systems with Random and Interval Variables

Tremendous efforts have been devoted to developing efficient approaches to reliability analysis for multidisciplinary systems. Most of the approaches are only capable of dealing with random variables modeled by probability distributions. Both random and interval variables, however, may exist in multidisciplinary systems. Their propagation through coupled subsystems make s reliability analysis computationally expensive. In this work, a unified reliability analysis framework is proposed to deal with both random and interval variables in multidisciplinary systems. The framework is an extension of an existing unified uncertainty analysis framework for single-disciplinary problems. The new framework involves probabilistic analysis and interval analysis. Both probabilistic analysis and interval analysis are decoupled from each other and are performed sequentially. The first order reliability method is used for probabilistic analysis. Three supporting algorithms are developed. The effectiveness of the algorithms is demonstrated with a mathematical example and an engineering application.

Unified Uncertainty Analysis by the First Order Reliability Method

Two types of uncertainty exist in engineering. Aleatory uncertainty comes from inherent variations while epistemic uncertainty derives from ignorance or incomplete information. The former is usually modeled by the probability theory and has been widely researched. The latter can be modeled by the probability theory or nonprobability theories and is much more difficult to deal with. In this work, the effects of both types of uncertainty are quantified with belief and plausibility measures (lower and upper probabilities) in the context of the evidence theory. Input parameters with aleatory uncertainty are modeled with probability distributions by the probability theory. Input parameters with epistemic uncertainty are modeled with basic probability assignments by the evidence theory. A computational method is developed to compute belief and plausibility measures for black-box performance functions. The proposed method involves the nested probabilistic analysis and interval analysis. To handle black-box functions, we employ the first order reliability method for probabilistic analysis and nonlinear optimization for interval analysis. Two example problems are presented to demonstrate the proposed method.

Sensitivity Analysis with Mixture of Epistemic and Aleatory Uncertainties

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.