Imprecise structural reliability analysis using PC-Kriging

The objective of the study described in this paper was to develop an efficient uncertainty quantification framework capable of analyzing uncertainty in integrated spacecraft system models. Specifically, this paper discusses the capabilities of the developed framework and the res ults when applied to the multidisciplinary analysis of a reusable launch veh icle (RLV). This particular framework is capable of efficiently propagating mixed (inherent and epistemic) uncertainties through complex simulation codes. The Second-Order Probability Theory utilizing a stochastic response surface obta ined with Point-Collocation NonIntrusive Polynomial Chaos was used for the propagation of the mixed uncertainties. This particular methodology was applied to the RLV analysis, and the uncertainty in the output parameters of interested was obtained in terms of intervals at various probability levels. This study has also demonstrated the feasibility of the developed uncertainty quantifica tion framework for efficient propagation of mixed uncertainties in the analysis of complex aerospace systems. I. Introduction ncertainties are generally ubiquitous in the analys is and design of highly complex engineering systems. Uncertainties can arise from the lack of k nowledge in physical modeling (epistemic uncertainty), inherent variations in the systems (a leatory uncertainty), and numerical errors in the computational procedures used for analysis. It is i mportant to account for all of these uncertainties in applications such as robust and reliable design of multi-disciplinary aerospace systems. A reusable la unch vehicle (RLV) is a highly complex aerospace system, which represents a cost viable option for access t o space missions due to its reusability aspect. Since an RLV system is composed of various subsystems which must work together to serve an over-arching p urpose or mission objective/constraint, a multidisciplinary approach for the analysis and des ign of RLV systems is required. Uncertainties are generally present in the models used in each discipline of an RLV multidisciplinary analysis framework. It is important to account for all of these uncertainties for accurate and reliabl e estimates of the RLV system performance. The primary purpose of this paper will be to demonstrate an efficient approach for uncertainty quantification in the multidiscipl inary analysis of a RLV, which has a mixture of aleatory (inherent) and epistemic input uncertainties. Uncertainty quantification will be performed on various output variables of interest f or an RLV system such as the weight and vehicle crossrange. For the propagation of mixed (aleatory-epist emic) uncertainty, Second-Order Probability Theory utilizing Point-Collocation Non-Intrusive Polynomia l Chaos (NIPC) 1 will be used. 2-3 In general, the NIPC methods, which are based on the spectral representa tion of uncertainty, are computationally more effic ient than traditional Monte Carlo methods for moderate number of uncertain variables and can give highly 1 Aerospace Engineer, 4020 Long Beach Blvd, Long Beach, CA, 90807, Member AIAA

PurposeBecause of the high computational efficiency, response surface method (RSM) has been widely used in structural reliability analysis. However, for a highly nonlinear limit state function (LSF), the approximate accuracy of the failure probability mainly depends on the design point, and the result is that the response surface function composed of initial experimental points rarely fits the LSF exactly. The inaccurate design points usually cause some errors in the traditional RSM. The purpose of this paper is to present a hybrid method combining adaptive moving experimental points strategy and RSM, describing a new response surface using downhill simplex algorithm (DSA-RSM).Design/methodology/approachIn DSA-RSM, the operation mechanism principle of the basic DSA, in which local descending vectors are automatically generated, was studied. Then, the search strategy of the basic DSA was changed and the RSM approximate model was reconstructed by combining the direct search advantage of DSA with the reliability mechanism of response surface analysis.FindingsThe computational power of the proposed method is demonstrated by solving four structural reliability problems, including the actual engineering problem of a car collision. Compared to specific structural reliability analysis methods, the approach of modified DSA interpolation response surface for structural reliability has a good convergent capability and computational accuracy.Originality/valueThis paper proposes a new RSM technology based on proxy model to complete the reliability analysis. The originality of this paper is to present an improved RSM that adjusts the position of the experimental points judiciously by using the DSA principle to make the fitted response surface closer to the actual limit state surface.

This paper discusses the important aspects of the reliability of systems with an imprecise general model of the structure function. It is assumed that the information about reliability behavior of components is restricted by the mean levels of component performance. In this case the classical reliability theory cannot provide a way for analyzing the reliability of systems. The theory of imprecise probabilities may be a basis in developing a general reliability theory which allows us to solve such problems. The basic tool for computing new reliability measures is the natural extension which can be regarded as a linear optimization problem. However, the linear programming computations will become impracticable when the number of components in the system is large. Therefore, the main aim of this paper is to obtain explicit expressions for computing the system reliability measures. We analyze the reliability of general structures and typical systems. The numerical examples illustrate the usefulness of the presented approach to reliability analyzing.

The paper describes an approach to obtaining prior imprecise reliability values of components and systems. The mathematical basis of the approach is the theory of coherent imprecise probabilities. The procedure of prior imprecise probability elicitation is based on analogical reasoning, and two cases of precise and imprecise reliabilities of prototypes are considered. Cases of combining different reliability judgements for the same component are analyzed. The remainder of the paper is devoted to imprecise reliability assessments of different system structures: components in parallel, in series and in series-parallel. The formulae obtained allow getting the lower and upper probabilities without the assumption of conditional independence. What we can expect from the use of this new tool is briefly discussed.

Fuzzy reliability analysis can be implemented using two discrete optimization maps in the processes of reliability and fuzzy analysis. Actually, the efficiency and robustness of the iterative reliability methods are two main factors in the fuzzy-based reliability analysis due to the huge computational burdens and unstable results. In the structural fuzzy reliability analysis, the first-order reliability method (FORM) using discrete nonlinear map can provide a C membership function. In this paper, a discrete nonlinear conjugate map is proposed using a relaxed-finite step size method for fuzzy structural reliability analysis, namely Fuzzy conjugate relaxed-finite step size method fuzzy CRS. A discrete conjugate map is stabilized using two adaptive factors to compute the relaxed factor and step size in FORM. The framework of the proposed fuzzy structural reliability method is established using two linked iterative discrete maps as an outer loop, which constructs the membership function of the response using alpha level set optimization based on genetic operator, and the inner loop, implemented for reliability analysis using proposed conjugate relaxed-finite step size method. The fuzzy CRS and fuzzy HL-RF methods are compared to evaluate the membership functions of five structural problems with highly nonlinear limit state functions. Results demonstrated that the fuzzy CRS method is computationally more efficient and is strongly more robust than the HL-RF for fuzzy-based reliability analysis of the nonlinear structural reliability problems.

In this paper the main results of DRUMS R.P3.3 Task ’Propagation of Uncertainty in Dependence Graphs’ are presented. It is divided in two parts. In the first one, it is considered a new calculus with imprecise probabilities, which is not as uninformative as classical one, [6, 7, 9]. In the second one, the axiomatic squeme given in [11] has been adapted to Pearl's causal networks, [10], by introducing three new axioms. Then, this axiomatic framework is particularized to the new calculus of upper and lower probabilities. It is shown that upper and lower probabilities may be propagated on causal networks on an analogous way to probabilities.

A stress-strength interference model in its interval form, with uncertain quantities of structures described by interval numbers, is used for reliability analysis. Such structural reliability analysis problem is called structural interval reliability analysis. A discrete possibility degree method for structural interval reliability is proposed. By comparing equally-spaced-points of structural interval stress value and structural interval strength value, a partition for comparison relations (big, equal or small) between interval stress and interval strength is built. Based on this partition, new concepts for interval reliability analysis are defined, such as structural safety possibility degree and structural failure possibility degree, etc. By one simple example, the feasibility and effectiveness of discrete possibility degree method for structural interval reliability are demonstrated.

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

The main bottleneck of the reliability analysis of structures with aleatory and epistemic uncertainties is the contradiction between the accuracy requirement and computational efforts. Existing methods are either computationally unaffordable or with limited application scope. Accordingly, a new technique for capturing the minimal and maximal point vectors instead of the extremum of the function is developed and thus a novel reliability analysis method for probabilistic and fuzzy mixed variables is proposed. The fuzziness propagation in the random reliability, which is the focus of the present study, is performed by the proposed method, in which the minimal and maximal point vectors of the structural random reliability with respect to fuzzy variables are calculated dimension by dimension based on the Chebyshev orthogonal polynomial approximation. First-Order, Second-Moment (FOSM) method which can be replaced by its counterparts is utilized to calculate the structural random reliability. Both the accuracy and efficiency of the proposed method are validated by numerical examples and engineering applications introduced in the paper.

A computationally efficient methodology for propagating the epistemic uncertainty in the median ground motion in probabilistic seismic hazard analysis is developed using the polynomial chaos (PC) approach. For this application, the epistemic uncertainty in the median ground motion for a specific scenario is assumed to be lognormally distributed and fully correlated across earthquake scenarios. In the hazard calculation, a single central ground‐motion model (GMM) is used for the median along with the epistemic standard error of the median for each scenario. A set of PC coefficients is computed for each scenario and each test ground‐motion level. The additional computation burden in computing these PC coefficients depends on the order of the approximation but is less than computing the median ground motion from one additional GMM. With the PC method, the mean and fractiles of the hazard due to the epistemic uncertainty distribution of the median ground motion are computed as a postprocess that is very fast computationally. For typical values of the standard deviation of epistemic uncertainty in the median ground motion (<0.2 natural log units), the methodology accurately estimates the epistemic uncertainty distribution of the hazard over the 1%–99% range. This full epistemic range is not well modeled with just a small number of GMM branches uses in the traditional logic‐tree approach. The PC method provides more accuracy, faster computation, and reduced memory requirements than the traditional approach. For large values of the epistemic uncertainty in the median ground motion, a higher order of the PC expansion may be needed to be included to capture the full range of the epistemic uncertainty.

Some transmission structures exhibit important geometric and material nonlinearities that should be considered in their analysis and design. Unless simplifying assumptions are made, reliability analysis of these structures is complex and can be very computer-time intensive. This paper describes an approximate method to conduct reliability analysis of nonlinear transmission structures. It is shown that for most transmission structures, the design point is close to the 50-year return period loads. Using this result, approximate g-functions for nonlinear problems are obtained with minimum computational effort. It is also shown that when the material strengths are highly correlated, g-functions that include interaction of axial load and bending moments may be reduced to g-functions with no interaction. These simplifying results are implemented in a microcomputer program, RELIAB, which can handle complex transmission structures such as large latticed structures, tension-only structures, or structures that exhibit nonlinearities. RELIAB's accuracy is ascertained by comparing results with another computer program, TRANSREL, which accounts for the residual strength of latticed towers following progressive failure of one or more components.

Structural reliability analysis with fuzzy random variables using error principle

: The Aircraft Structural Reliability and Risk Handbook provides: a) A reference on basic statistics, probability, and reliability techniques required for conducting structural risk and reliability analyses, and b) Illustrative examples of how to apply these techniques to problems involving the nucleation and growth of fatigue cracks in metallic airframe structure. This particular volume describes the fundamentals of probabilistic analysis and structural reliability analysis for the next flight.

The increasing frequency and severity of Natech accidents warn us to investigate the occurrence mechanism of these events. Cascading disasters chain magnifies the impact of natural hazards due to its propagation through critical infrastructures and socio-economic networks. In order to manipulate imprecise probabilities of cascading events in Natech scenarios, this work proposes an improved Bayesian network (BN) combining with evidence theory to better deal with epistemic uncertainty in Natech accidents than traditional BNs. Effective inference algorithms have been developed to propagate system faulty in a socio-economic system. The conditional probability table (CPT) of BN in the traditional probability approach is modified by utilizing an OR/AND gate to obtain the belief mass propagation in the framework of evidence theory. Our improved Bayesian network methodology makes it possible to assess the impact and damage of Natech accidents under the environment of complex interdependence among accidents with insufficient data. Finally, a case study of Guangdong province, an area prone to natural disasters, is given. The modified Bayesian network is carried out to analyze this area’s Natech scenario. After diagnostic analysis and sensitivity analysis of human factors and the natural factor, we are able to locate the key nodes in the cascading disaster chain. Findings can provide useful theoretical support for urban managers of industrial cities to enhance disaster prevention and mitigation ability.

Structural health monitoring data has been widely acknowledged as a significant source for evaluating the performance and health conditions of structures. However, a holistic framework that efficiently incorporates monitored data into structural identification and, in turn, provides a realistic life-cycle performance assessment of structures is yet to be established. There are different sources of uncertainty, such as structural parameters, computer model bias and measurement errors. Neglecting to account for these factors results in unreliable structural identifications, consequent financial losses, and a threat to the safety of structures and human lives. This paper proposes a new framework for structural performance assessment that integrates a comprehensive probabilistic finite element model updating approach, which deals with various structural identification uncertainties and structural reliability analysis. In this framework, Gaussian process surrogate models are replaced with a finite element model and its associate discrepancy function to provide a computationally efficient and all-round uncertainty quantification. Herein, the structural parameters that are most sensitive to measured structural dynamic characteristics are investigated and used to update the numerical model. Sequentially, the updated model is applied to compute the structural capacity with respect to loading demand to evaluate its as-is performance. The proposed framework’s feasibility is investigated and validated on a large lab-scale box girder bridge in two different health states, undamaged and damaged, with the latter state representing changes in structural parameters resulted from overloading actions. The results from the box girder bridge indicate a reduced structural performance evidenced by a significant drop in the structural reliability index and an increased probability of failure in the damaged state. The results also demonstrate that the proposed methodology contributes to more reliable judgment about structural safety, which in turn enables more informed maintenance decisions to be made.