Uncertainty quantification in reliability estimation with limit state surrogates

Model-based reliability analysis is affected by different types of epistemic uncertainty, due to inadequate data and modeling errors. When the physics-based simulation model is computationally expensive, a surrogate has often been used in reliability analysis, introducing additional uncertainty due to the surrogate. This paper proposes a framework to include statistical uncertainty and model uncertainty in surrogate-based reliability analysis. Two types of surrogates have been considered: (1) general-purpose surrogate models that compute the system model output over the desired ranges of the random variables; and (2) limit-state surrogates. A unified approach to connect the model calibration analysis using the Kennedy and O’Hagan (KOH) framework to the construction of limit state surrogate and to estimating the uncertainty in reliability analysis is developed. The Gaussian Process (GP) general-purpose surrogate of the physics-based simulation model obtained from the KOH calibration analysis is further refined at the limit state (local refinement) to construct the limit state surrogate, which is used for reliability analysis. An efficient single-loop sampling approach using the probability integral transform is used for sampling the input variables with statistical uncertainty. The variability in the GP prediction (surrogate uncertainty) is included in reliability analysis through correlated sampling of the model predictions at different inputs. The Monte Carlo sampling (MCS) error, which represents the error due to limited Monte Carlo samples, is quantified by constructing a probability density function. All the different sources of epistemic uncertainty are quantified and aggregated to estimate the uncertainty in the reliability analysis. Two examples are used to demonstrate the proposed techniques.     

A reliability-based multidisciplinary design optimization procedure based on combined probability and evidence theory

To address the reliability-based multidisciplinary design optimization (RBMDO) problem under mixed aleatory and epistemic uncertainties, an RBMDO procedure is proposed in this paper based on combined probability and evidence theory. The existing deterministic multistage-multilevel multidisciplinary design optimization (MDO) procedure MDF-CSSO, which combines the multiple discipline feasible (MDF) procedure and the concurrent subspace optimization (CSSO) procedure to mimic the general conceptual design process, is used as the basic framework. In the first stage, the surrogate based MDF is used to quickly identify the promising reliable regions. In the second stage, the surrogate based CSSO is used to organize the disciplinary optimization and system coordination, which allows the disciplinary specialists to investigate and optimize the design with the corresponding high-fidelity models independently and concurrently. In these two stages, the reliability-based optimization both in the system level and the disciplinary level are computationally expensive as it entails nested optimization and uncertainty analysis. To alleviate the computational burden, the sequential optimization and mixed uncertainty analysis (SOMUA) method is used to decompose the traditional double-level reliability-based optimization problem into separate deterministic optimization and mixed uncertainty analysis sub-problems, which are solved sequentially and iteratively until convergence is achieved. By integrating SOMUA into MDF-CSSO, the Mixed Uncertainty based RBMDO procedure MUMDF-CSSO is developed. The effectiveness of the proposed procedure is testified with one simple numerical example and one MDO benchmark test problem, followed by some conclusion remarks.     

Reliability based multidisciplinary design optimization of cooling turbine blade considering uncertainty data statistics

Considering the coupling among aerodynamic, heat transfer and strength, a reliability based multidisciplinary design optimization method for cooling turbine blade is introduced. Multidisciplinary analysis of cooling turbine blade is carried out by sequential conjugated heat transfer analysis and strength analysis with temperature and pressure interpolation. Uncertainty data including the blade wall, rib thickness, elasticity Modulus and rotation speed is collected. Data statistics display the probability models of uncertainty data follow three-parameter Weibull distribution. The thickness of blade wall, thickness and height of ribs are chosen as design variables. Kriging surrogate model is introduced to reduce time-consuming multidisciplinary reliability analysis in RBMDO loop. The reliability based multidisciplinary design optimization of a cooling turbine blade is carried out. Optimization results shows that the RBMDO method proposed in this work improves the performance of cooling turbine blade availably.

     

Confidence-based reliability assessment considering limited numbers of both input and output test data

Simulation-based methods can be used for accurate uncertainty quantification and prediction of the reliability of a physical system under the following assumptions: (1) accurate input distribution models and (2) accurate simulation models (including accurate surrogate models if utilized). However, in practical engineering applications, often only limited numbers of input test data are available for modeling input distribution models. Thus, estimated input distribution models are uncertain. In addition, the simulation model could be biased due to assumptions and idealizations used in the modeling process. Furthermore, only a limited number of physical output test data is available in the practical engineering applications. As a result, target output distributions, against which the simulation model can be validated, are uncertain and the corresponding reliabilities become uncertain as well. To assess the conservative reliability of the product properly under the uncertainties due to limited numbers of both input and output test data and a biased simulation model, a confidence-based reliability assessment method is developed in this paper. In the developed method, a hierarchical Bayesian model is formulated to obtain the uncertainty distribution of reliability. Then, we can specify a target confidence level. The reliability value at the target confidence level using the uncertainty distribution of reliability is the confidence-based reliability, which is the confidence-based estimation of the true reliability. It has been numerically demonstrated that the proposed method can predict the reliability of a physical system that satisfies the user-specified target confidence level, using limited numbers of input and output test data.     

Sequential optimization and reliability assessment for multidisciplinary systems design

With higher reliability and safety requirements, reliability-based design has been increasingly applied in multidisciplinary design optimization (MDO). A direct integration of reliability-based design and MDO may present tremendous implementation and numerical difficulties. In this work, a methodology of sequential optimization and reliability assessment for MDO is proposed to improve the efficiency of reliability-based MDO. The central idea is to decouple the reliability analysis from MDO with sequential cycles of reliability analysis and deterministic MDO. The reliability analysis is based on the first-order reliability method (FORM). In the proposed method, the reliability analysis and the deterministic MDO use two MDO strategies, the multidisciplinary feasible approach and the individual disciplinary feasible approach. The effectiveness of the proposed method is illustrated with two example problems.     

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.     

Probability box as a tool to model and control the effect of epistemic uncertainty in multiple dependent competing failure processes

Engineering components and systems are often subject to multiple dependent competing failure processes (MDCFPs). MDCFPs have been well studied in literature and various models have been developed to predict the reliability of MDCFPs. In practice, however, due to the limited resource, it is often hard to estimate the precise values of the parameters in the MDCFP model. Hence, the predicted reliability is affected by epistemic uncertainty. Probability box (P-box) is applied in this paper to describe the effect of epistemic uncertainty on MDCFP models. A dimension-reduced sequential quadratic programming (DRSQP) method is developed for the construction of P-box. A comparison to the conventional construction method shows that DRSQP method reduces the computational costs required for P-box constructions. Since epistemic uncertainty reflects the unsureness in the predicted reliability, a decision maker might want to reduce it by investing resource to more accurately estimate the value of each model parameter. A two-stage optimization framework is developed to allocate the resource among the parameters and ensure that epistemic uncertainty is reduced in a most efficient way. Finally, the developed methods are applied on a real case study, a spool valve, to demonstrate their validity.     

机电设备BIT 中传感层系统的不确定性动态分析􀀂

BIT(Built-in test)系统中传感层的不确定性反映了测试输出数据的可靠度，而数据的可信度影响着BIT系统进行决策的准确性和可信度，并且数据的不确定性是产生虚警的重要因素之一，对传感数据进行不确定性分析是获得数据可信度的有效的量化方法。传统的不确定性分析为静态分析过程，本文基于传感层系统的状态模式空间提出了传感层的不确定性动态分析方法，根据其可靠性模型，基于概率方法进行不确定性的动态计算，实现对传感层系统的执行和输出数据的可信度进行评估。同时对未覆盖故障集合对分析结果的影响进行了分析。最后对某电涡流位移传感器进行了不确定性分析的仿真。     

Nonparametric approach for uncertainty-based multidisciplinary design optimization considering limited data

Uncertainty-based multidisciplinary design optimization (UMDO) has been widely acknowledged as an advanced methodology to address competing objectives and reliable constraints of complex systems by coupling relationship of disciplines involved in the system. UMDO process consists of three parts. Two parts are to define the system with uncertainty and to formulate the design optimization problem. The third part is to quantitatively analyze the uncertainty of the system output considering the uncertainty propagation in the multidiscipline analysis. One of the major issues in the UMDO research is that the uncertainty propagation makes uncertainty analysis difficult in the complex system. The conventional methods are based on the parametric approach could possibly cause the error when the parametric approach has ill-estimated distribution because data is often insufficient or limited. Therefore, it is required to develop a nonparametric approach to directly use data. In this work, the nonparametric approach for uncertainty-based multidisciplinary design optimization considering limited data is proposed. To handle limited data, three processes are also adopted. To verify the performance of the proposed method, mathematical and engineering examples are illustrated.     

Multi-fidelity information fusion based on prediction of kriging

In this paper, a novel kriging-based multi-fidelity method is proposed. Firstly, the model uncertainty of low-fidelity and high-fidelity models is quantified. On the other hand, the prediction uncertainty of kriging-based surrogate models(SM) is confirmed by its mean square error. After that, the integral uncertainty is acquired by math modeling. Meanwhile, the SMs are constructed through data from low-fidelity and high-fidelity models. Eventually, the low-fidelity (LF) and high-fidelity (HF) SMs with integral uncertainty are obtained and a proposed fusion algorithm is implemented. The fusion algorithm refers to the Kalman filter’s idea of optimal estimation to utilize the independent information from different models synthetically. Through several mathematical examples implemented, the fused SM is certified that its variance is decreased and the fused results tend to the true value. In addition, an engineering example about autonomous underwater vehicles’ hull design is provided to prove the feasibility of this proposed multi-fidelity method in practice. In the future, it will be a helpful tool to deal with reliability optimization of black-box problems and potentially applied in multidisciplinary design optimization.     

Bayesian reliability-based design optimization using eigenvector dimension reduction (EDR) method

In practical engineering design, most data sets for system uncertainties are insufficiently sampled from unknown statistical distributions, known as epistemic uncertainty. 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 reliability 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.     

Conservative reliability index for epistemic uncertainty in reliability-based design optimization

In this paper, a simple but efficient concept of epistemic reliability index (ERI) is introduced for sampling uncertainty in input random variables under conditions where the input variables are independent Gaussian, and samples are unbiased. The increased uncertainty due to the added epistemic uncertainty requires a higher level of target reliability, which is called the conservative reliability index (CRI). In this paper, it is assumed that CRI can additively be decomposed into the aleatory part (the target reliability index) and the epistemic part (the ERI). It is shown theoretically and numerically that ERI remains same for different designs, which is critically important for computational efficiency in reliability-based design optimization. Novel features of the proposed ERI include: (a) it is unnecessary to have a double-loop uncertainty quantification for handling both aleatory and epistemic uncertainty; (b) the effect of two different sources of uncertainty can be separated so that designers can better understand the optimization outcome; and (c) the ERI needs to be calculated once and remains the same throughout the design process. The proposed method is demonstrated with two analytical and one numerical examples.     

An extended probabilistic method for reliability analysis under mixed aleatory and epistemic uncertainties with flexible intervals

The reliability analysis approach based on combined probability and evidence theory is studied in this paper to address the reliability analysis problem involving both aleatory uncertainties and epistemic uncertainties with flexible intervals (the interval bounds are either fixed or variable as functions of other independent variables). In the standard mathematical formulation of reliability analysis under mixed uncertainties with combined probability and evidence theory, the key is to calculate the failure probability of the upper and lower limits of the system response function as the epistemic uncertainties vary in each focal element. Based on measure theory, in this paper it is proved that the aforementioned upper and lower limits of the system response function are measurable under certain circumstances (the system response function is continuous and the flexible interval bounds satisfy certain conditions), which accordingly can be treated as random variables. Thus the reliability analysis of the system response under mixed uncertainties can be directly treated as probability calculation problems and solved by existing well-developed and efficient probabilistic methods. In this paper the popular probabilistic reliability analysis method FORM (First Order Reliability Method) is taken as an example to illustrate how to extend it to solve the reliability analysis problem in the mixed uncertainty situation. The efficacy of the proposed method is demonstrated with two numerical examples and one practical satellite conceptual design problem.     

Multiobjective optimisation on a budget: Exploring surrogate modelling for robust multi-reservoir rules generation under hydrological uncertainty

Developing long term operation rules for multi-reservoir systems is complicated due to the number of decision variables, the non-linearity of system dynamics and the hydrological uncertainty. This uncertainty can be addressed by coupling simulation models with multi-objective optimisation algorithms driven by stochastically generated hydrological timeseries but the computational effort required imposes barriers to the exploration of the solution space. The paper addresses this by (a) employing a parsimonious multi-objective parameterization-simulation-optimization (PSO) framework, which incorporates hydrological uncertainty through stochastic simulation and allows the use of probabilistic objective functions and (b) by investigating the potential of multi-objective surrogate based optimisation (MOSBO) to significantly reduce the resulting computational effort. Three MOSBO algorithms are compared against two multi-objective evolutionary algorithms. Results suggest that MOSBOs are indeed able to provide robust, uncertainty-aware operation rules much faster, without significant loss of neither the generality of evolutionary algorithms nor of the knowledge embedded in domain-specific models.     

Model uncertainty approximation using a copula-based approach for reliability based design optimization

Reliability-based design optimization (RBDO) has been widely used to design engineering products with minimum cost function while meeting reliability constraints. Although uncertainties, such as aleatory uncertainty and epistemic uncertainty, have been well considered in RBDO, they are mainly considered for model input parameters. Model uncertainty, i.e., the uncertainty of model bias indicating the inherent model inadequacy for representing the real physical system, is typically overlooked in RBDO. This paper addresses model uncertainty approximation in a product design space and further integrates the model uncertainty into RBDO. In particular, a copula-based bias modeling approach is proposed and results are demonstrated by two vehicle design problems.     

Reliability bounds through random sets: Non-parametric methods and geotechnical applications

Random sets form a well-established, general tool for modelling epistemic uncertainty in engineering. They can be seen as encompassing probability theory, fuzzy sets and interval analysis. Random set models for data uncertainty are typically used to obtain robust upper and lower bounds for the reliability of structures in engineering models. The goal of this paper is to show how random set models can be constructed from measurement data by non-parametric methods using inequalities of Tchebycheff type. Relations with sensitivity analysis will also be high-lighted. We demonstrate the application of the methods in an FE-model for the excavation of a cantilever sheet pile wall.     

Dynamics of entanglement and uncertainty relation in coupled harmonic oscillator system: exact results

The dynamics of entanglement and uncertainty relation is explored by solving the time-dependent Schrödinger equation for coupled harmonic oscillator system analytically when the angular frequencies and coupling constant are arbitrarily time dependent. We derive the spectral and Schmidt decompositions for vacuum solution. Using the decompositions, we derive the analytical expressions for von Neumann and Rényi entropies. Making use of Wigner distribution function defined in phase space, we derive the time dependence of position–momentum uncertainty relations. To show the dynamics of entanglement and uncertainty relation graphically, we introduce two toy models and one realistic quenched model. While the dynamics can be conjectured by simple consideration in the toy models, the dynamics in the realistic quenched model is somewhat different from that in the toy models. In particular, the dynamics of entanglement exhibits similar pattern to dynamics of uncertainty parameter in the realistic quenched model.     

Spatio-temporal uncertainty in Spatial Decision Support Systems: A case study of changing land availability for bioenergy crops in Mozambique

Spatial Decision Support Systems (SDSSs) often include models that can be used to assess the impact of possible decisions. These models usually simulate complex spatio-temporal phenomena, with input variables and parameters that are often hard to measure. The resulting model uncertainty is, however, rarely communicated to the user, so that current SDSSs yield clear, but therefore sometimes deceptively precise outputs. Inclusion of uncertainty in SDSSs requires modeling methods to calculate uncertainty and tools to visualize indicators of uncertainty that can be understood by its users, having mostly limited knowledge of spatial statistics. This research makes an important step towards a solution of this issue. It illustrates the construction of the PCRaster Land Use Change model (PLUC) that integrates simulation, uncertainty analysis and visualization. It uses the PCRaster Python framework, which comprises both a spatio-temporal modeling framework and a Monte Carlo analysis framework that together produce stochastic maps, which can be visualized with the Aguila software, included in the PCRaster Python distribution package. This is illustrated by a case study for Mozambique in which it is evaluated where bioenergy crops can be cultivated without endangering nature areas and food production now and in the near future, when population and food intake per capita will increase and thus arable land and pasture areas are likely to expand. It is shown how the uncertainty of the input variables and model parameters effects the model outcomes. Evaluation of spatio-temporal uncertainty patterns has provided new insights in the modeled land use system about, e.g., the shape of concentric rings around cities. In addition, the visualization modes give uncertainty information in an comprehensible way for users without specialist knowledge of statistics, for example by means of confidence intervals for potential bioenergy crop yields. The coupling of spatio-temporal uncertainty analysis to the simulation model is considered a major step forward in the exposure of uncertainty in SDSSs.     

An efficient strategy for reliability-based multidisciplinary design optimization using BLISS

This paper presents an efficient reliability-based multidisciplinary design optimization (RBMDO) strategy. The conventional RBMDO has tri-level loops: the first level is an optimization in the deterministic space, the second one is a reliability analysis in the probabilistic space, and the third one is the multidisciplinary analysis. Since it is computationally inefficient when high-fidelity simulation methods are involved, an efficient strategy is proposed. The strategy [named probabilistic bi-level integrated system synthesis (ProBLISS)] utilizes a single-level reliability-based design optimization (RBDO) approach, in which the reliability analysis and optimization are conducted in a sequential manner by approximating limit state functions. The single-level RBDO is associated with the BLISS formulation to solve RBMDO problems. Since both the single-level RBDO and BLISS are mainly driven by approximate models, the accuracy of models can be a critical issue for convergence. The convergence of the strategy is guaranteed by employing the trust region–sequential quadratic programming framework, which validates approximation models in the trust region radius. Two multidisciplinary problems are tested to verify the strategy. ProBLISS significantly reduces the computational cost and shows stable convergence while maintaining accuracy.