Hybrid Reliability Research Articles

A combined projection-outline-based active learning Kriging and adaptive importance sampling method for hybrid reliability analysis with small failure probabilities

In this paper, the adaptive importance sampling (AIS) method is extended for hybrid reliability analysis under random and interval variables (HRA-RI) with small failure probabilities. In AIS, the design space is divided into random and interval variable subspaces. In random variable subspace, Markov Chain Monte Carlo (MCMC) is employed to generate samples which populate the failure regions. Then based on these samples, two kernel sampling density functions are established for estimations of the lower and upper bounds of failure probability. To improve the computational efficiency of AIS in cases with time-consuming performance functions, a combination method of projection-outline-based active learning Kriging and AIS, termed as POALK-AIS, is proposed in this paper. In this method, design of experiments is sequentially updated for the construction of Kriging metamodel with focus on the approximation accuracy of the projection outlines on the limit-state surface. During the procedure of POALK-AIS, multiple groups of sample points simulated by AIS are used to calculate the upper and lower bounds of failure probability. The accuracy, efficiency and robustness of POALK-AIS for HRA-RI with small failure probabilities are verified by five test examples.

Reliability analysis of structures based on a probability‐uncertainty hybrid model

AbstractThe traditional reliability analysis method based on probabilistic method requires probability distributions of all the uncertain parameters. However, in practical applications, the distributions of some parameters may not be precisely known due to the lack of sufficient sample data. The probabilistic theory cannot directly measure the reliability of structures with epistemic uncertainty, ie, subjective randomness and fuzziness. Hence, a hybrid reliability analysis (HRA) problem will be caused when the aleatory and epistemic uncertainties coexist in a structure. In this paper, by combining the probability theory and the uncertainty theory into a chance theory, a probability‐uncertainty hybrid model is established, and a new quantification method based on the uncertain random variables for the structural reliability is presented in order to simultaneously satisfy the duality of random variables and the subadditivity of uncertain variables; then, a reliability index is explored based on the chance expected value and variance. Besides, the formulas of the chance theory‐based reliability and reliability index are derived to uniformly assess the reliability of structures under the hybrid aleatory and epistemic uncertainties. The numerical experiments illustrate the validity of the proposed method, and the results of the proposed method can provide a more accurate assessment of the structural system under the mixed uncertainties than the ones obtained separately from the probability theory and the uncertainty theory.

Discretization Analysis Method of Hybrid Reliability Based on Evidence Theory

Aiming at the problem that various types of uncertainties, such as randomness, fuzziness, and interval, coexist in structure reliability analysis, a discretization analysis method of hybrid reliability for uncertain structures is proposed based on evidence theory (ET) in this article. Firstly, in order to establish a hybrid reliability model based on ET, a generalized density method (GDM) is developed to transform the fuzzy variables into equivalent random variables on the basis of the entropy equivalent method (EEM). Based on the discrete property of the basic probability assignment (BPA) in evidence theory, the random variables and fuzzy variables (equivalent random variables) are both discretized into subintervals according to six-sigma rule. Then, the BPA of each subinterval is solved and all focal elements are assigned BPA, so the evidence structure characterization of random and fuzzy variables is realized. Secondly, using Fmincon function based on the sequential quadratic programming (SQP) algorithm in MATLAB, the minimum and maximum values of performance function over each focal element can be acquired directly. Meanwhile, the production rules are used to judge the belonging of focal elements and classify them, so the numerical calculation of belief measure and plausibility measure is also realized. Finally, combined with the Monte Carlo Simulation (MCS) method, an engineering example is provided to demonstrate the feasibility and accuracy of the proposed method.

A novel projection outline based active learning method and its combination with Kriging metamodel for hybrid reliability analysis with random and interval variables

This paper focuses on the hybrid reliability analysis with both random and interval variables (HRA-RI). It is determined that a metamodel only accurately approximating the projection outlines on the limit-state surface can precisely estimate the lower and upper bounds of failure probability in HRA-RI. According to this idea, a novel projection outline based active learning (POAL) method is proposed to sequentially update design of experiments (DoE). Then, a HRA-RI method combining POAL and Kriging metamodel (POAL–Kriging) is developed. In this method, Kriging metamodel is refined based on the update samples, which are sequentially chosen using POAL from the vicinity of the projection outlines on the limit-state surface. In the end, the lower and upper bounds of failure probability in HRA-RI are precisely estimated. Compared to the approximation of the whole limit-state surface, the proposed method only approximates the projection outlines on the limit-state surface, and therefore few DoE are needed to build a high quality metamodel. The accuracy, efficiency and robustness of the proposed method for HRA-RI are illustrated by four examples.

An Efficient Regional Sensitivity Analysis Method Based on Failure Probability with Hybrid Uncertainty

The application of reliability sensitivity analysis (RSA) to the high voltage direct current (HVDC) transmission systems is one of the hot topics in the future. A regional RSA method, the contribution to failure probability (CFP) plot, is investigated in this paper. This CFP plot contains both aleatory and epistemic uncertain variables modeled as random variables by probability theory and interval variables by evidence theory, respectively. A surrogate model of second-level limit state function needs to be established for each joint focal element (JFE), which is a time-consuming process. Additionally, an excessive number of Monte Carlo simulations (MCS) and optimizations may exceed the computing power of modern computers. In order to deal with the above problems and further decrease the computational cost, a more effective CFP calculation method under the framework of random-evidence hybrid reliability analysis is proposed. Three important improvements in the proposed method make the calculation of CFP more efficient and easy to implement. Firstly, an active learning kriging (ALK) based on the symbol prediction idea is employed to directly establish a surrogate model rather than a second-level limit state function with fewer function calls, which greatly simplifies construction of the model. Secondly, a random set-based Monte Carlo simulation (RS-MCS) is used to handle the issue of oversized optimization caused by too many JFEs. Thirdly, for further reducing the size of optimizations and improving the efficiency of the CFP calculation, a Karush-Kuhn-Tucker-based optimization (KKTO) method is recommended in the proposed method to solve the extreme value of performance function. A numerical example and an engineering example were studied to verify the accuracy, effectiveness and practicality of the proposed method. It can be seen from the results that regardless of whether it is modeling or computational efficiency, the proposed method is better than the original method.

Hybrid Reliability Parameter Selection Method Based on Text Mining, Frequent Pattern Growth Algorithm and Fuzzy Bayesian Network

Reliability parameter selection is very important in the period of equipment project design and demonstration. In this paper, the problem in selecting the reliability parameters and their number is proposed. In order to solve this problem, the thought of text mining is used to extract the feature and curtail feature sets from text data firstly, and frequent pattern tree (FPT) of the text data is constructed to reason frequent item-set between the key factors by frequent patter growth (FPG) algorithm. Then on the basis of fuzzy Bayesian network (FBN) and sample distribution, this paper fuzzifies the key attributes, which forms associated relationship in frequent item-sets and their main parameters, eliminates the subjective influence factors and obtains condition mutual information and maximum weight directed tree among all the attribute variables. Furthermore, the hybrid model is established by reason fuzzy prior probability and contingent probability and concluding parameter learning method. Finally, the example indicates the model is believable and effective.

Hybrid Reliability Analysis Framework for Reliability Analysis of Tunnels

Numerical modelling of tunnels, generally carried out by using some numerical analysis tools [e.g., finite element (FE) method, discrete element method], is often computationally expensive. Hence, it becomes extremely difficult to perform reliability analysis of such models using conventional reliability analysis tools. This necessitates development of efficient techniques for reliability analysis. In this work, a novel framework, referred to as a hybrid reliability analysis framework (HRAF), is developed for reliability analysis of tunnels. The proposed approach utilizes a hybrid polynomial correlated function expansion (H-PCFE), a distribution adaptive sequential experimental design (DA-SED), and an adaptive algorithm for further refining the estimates of DA-SED based H-PCFE. The primary idea of HRAF is to use DA-SED based H-PCFE in zones where the probability of misclassification is less and to use actual simulation for zones with a higher probability of misclassification. As a consequence, results obtained are highly accurate and, at that too, from a reasonably lower number of training points and/or actual simulations. Two tunnel problems have been presented to illustrate the performance of the proposed approach. Results obtained have been benchmarked against results obtained using a full-scale Monte Carlo simulation (MCS) with 105 sample points. Results obtained indicate the excellent performance (both in accuracy and efficiency) of the proposed framework.

Hybrid reliability analysis and optimization for spacecraft structural system with random and fuzzy parameters

In aerospace engineering, the incomplete environment conditions have been realized to be a significant factor affecting the system safety assessment. Considering the hybrid random and fuzzy uncertainties in system inputs, the paper proposes a novel numerical procedure for reliability analysis and reliability-based optimization. Random parameters are adopted to denote the aleatory uncertainties with sufficient sample information; whereas fuzzy parameters are used to quantify the epistemic uncertainties associated with expert opinions. Using the level-cut operation, fuzzy parameters are converted into interval variables, and a satisfaction degree-based interval ranking strategy is utilized to precisely quantify the interval safety possibility. Then the system safety possibility is calculated by the multiple integral, where cut levels of different fuzzy parameters are treated as independent variables. Subsequently based on the given safety index, a hybrid reliability optimization model is established. To avoid the huge computational burden caused by nested-loop optimization, a modified interval Monte Carlo method (MIMC) is proposed for limit state function evaluation. Eventually, a numerical example about the refractory ceramics tile of spacecraft verifies the feasibility of proposed method for hybrid reliability analysis and optimization design in practical aerospace engineering.

A hybrid reliability algorithm using PSO-optimized Kriging model and adaptive importance sampling

This paper aims to reduce the computational cost of reliability analysis. A new hybrid algorithm is proposed based on PSO-optimized Kriging model and adaptive importance sampling method. Firstly, the particle swarm optimization algorithm (PSO) is used to optimize the parameters of Kriging model. A typical function is fitted to validate improvement by comparing results of PSO-optimized Kriging model with those of the original Kriging model. Secondly, a hybrid algorithm for reliability analysis combined optimized Kriging model and adaptive importance sampling is proposed. Two cases from literatures are given to validate the efficiency and correctness. The proposed method is proved to be more efficient due to its application of small number of sample points according to comparison results.

MPP-Based Dimension Reduction Method for Accurate Prediction of the Probability of Failure of a Performance Function

A hybrid reliability analysis method is proposed to yield very accurate failure rate calculation of a performance function when dealing with highly nonlinear electromagnetic systems in the presence of uncertainties. To achieve this goal, the first-order reliability method called reliability index approach is first conducted for searching a most probable failure point (MPP) at a given design. However, its result may have significant errors especially for nonlinear or multi-dimensional performance functions. To overcome the drawback, the univariate dimension reduction method is additionally executed at the obtained MPP, and then the probability of failure of a performance function is recalculated through additively decomposing an n-dimensional function into n 1-D functions. A mathematical example and TEAM workshop problem 22 are provided to demonstrate numerical efficiency and accuracy of the proposed method by comparison with the existing reliability methods.

Structural Reliability Analysis of Corroded Pipeline made in X60 Steel Based on M5 Model Tree Algorithm and Monte Carlo Simulation

Accurate determination of the failure probability of oil and gas pipeline is very important in integrity assessment and work conditions of such structure. In this paper, the failure probability of the corroded pipelines which is made by X60 steel grade is evaluated. The burst corroded performance function is developed using the M5 model tree for this complex real engineering failure problem. The structural reliability of a the pressurized gas pipeline containing external corrosion defects has been evaluated using hybrid reliability method combined by the M5 model tree and Monte Carlo simulation. The results indicated that increasing the defects depth are strongly reduced the safety levels of this problem.

Hybrid Structural Reliability Analysis under Multisource Uncertainties Based on Universal Grey Numbers

Nondeterministic parameters of certain distribution are employed to model structural uncertainties, which are usually assumed as stochastic factors. However, model parameters may not be precisely represented due to some factors in engineering practices, such as lack of sufficient data, data with fuzziness, and unknown‐but‐bounded conditions. To this end, interval and fuzzy parameters are implemented and an efficient approach to structural reliability analysis with random‐interval‐fuzzy hybrid parameters is proposed in this study. Fuzzy parameters are first converted to equivalent random ones based on the equal entropy principle. 3σ criterion is then employed to transform the equivalent random and the original random parameters to interval variables. In doing this, the hybrid reliability problem is transformed into the one only with interval variables, in other words, nonprobabilistic reliability analysis problem. Nevertheless, the problem of interval extension existed in interval arithmetic, especially for the nonlinear systems. Therefore, universal grey mathematics, which can tackle the issue of interval extension, is employed to solve the nonprobabilistic reliability analysis problem. The results show that the proposed method can obtain more conservative results of the hybrid structural reliability.

The reliability-based design and optimization procedures for a heavy-duty pallet system

Shipping heavy-duty goods or equipment is being a new sector recently, and the pallet is rather important. In this article, a comparison of various materials revealed that laminated veneer lumber was ideal for heavy-duty packaging industry. For various pallets with random parameters, a hybrid reliability optimization design method was proposed, which aimed to provide an effective computational tool for reliability design of pallet system. With first-order second-moment method, the separation and combination of component and system failure probabilities were obtained. Several case studies were included to illustrate the computational procedures, as well as to discuss the effects of dimension and quantity of members. Using design of experiment method, the sensitivities of each parameter were significant, and the optimization of whole system was realized. To reduce the cost, recycling and standardization were worth considering in the future.

Reliability analysis of motion mechanism under three types of hybrid uncertainties

Hybrid reliability analysis of motion mechanism, in which the inputs contain both random variables and interval variables, is investigated in this paper. Firstly, the hybrid uncertainty is divided into three categories. In the first category, the inputs that include both random variables and interval variables are considered. In the second category, the inputs that include random variables with interval distribution parameters are considered. In the third category, the inputs that include both random variables with interval distribution parameters, and interval variables are considered. The above three types of uncertainties outline the general situation of hybrid uncertainties. In order to achieve the reliability analysis of motion mechanism under the above three hybrid uncertainties, this paper presents a method of combining the Poisson's assumption-based first passage method and a double loop iterative algorithm. The specific calculation process is described in detail by a numerical example. Simultaneously, the accuracy and efficiency of the proposed is also shown and discussed by comparing the Monte Carlo simulation (MCS) results in this numerical example. Finally, the proposed method is applied to two practical engineering examples, a crank slider mechanism and a rack-and-pinion steering linkage.

Hybrid reliability analysis with uncertain statistical variables, sparse variables and interval variables

ABSTRACTThere are differences among sampling data and representation types of uncertain statistical variables, sparse variables and interval variables, which increase the complexity of structure reliability analysis. Therefore, a hybrid first order reliability analysis method considering the three types of uncertain variables is demonstrated in this article. First, distribution types and distribution parameters of sparse variables are identified and probabilistically estimated. Secondly, interval variables are transformed into probabilistic types using a uniformity approach. Thirdly, a unified hybrid reliability calculation method considering these uncertain variables simultaneously is demonstrated. The most probable point (MPP) is searched for using the first order reliability method, and then a linear approximation function of performance function is constructed in the neighbourhood of the MPP. Finally, the belief and plausibility measures of the reliability index are efficiently calculated using the theoretical analytical method. Three examples are investigated to demonstrate the effectiveness of the proposed method.

Structural optimization oriented time-dependent reliability methodology under static and dynamic uncertainties

Uncertainty with characteristics of time-dependency, multi-sources and small-samples extensively exists in the whole process of structural design. Associated with frequent occurrences of material aging, load varying, damage accumulating, traditional reliability-based design optimization (RBDO) approaches by combination of the static assumption and the probability theory will be no longer applicable when dealing with the design problems for lifecycle structural models. In view of this, a new non-probabilistic time-dependent RBDO method under the mixture of time-invariant and time-variant uncertainties is investigated in this paper. Enlightened by the first-passage concept, the hybrid reliability index is firstly defined, and its solution implementation relies on the technologies of regulation and the interval mathematics. In order to guarantee the stability and efficiency of the optimization procedure, the improved ant colony algorithm (ACA) is then introduced. Moreover, by comparisons of the models of the safety factor-based design as well as the instantaneous RBDO design, the physical means of the proposed optimization policy are further discussed. Two numerical examples are eventually presented to demonstrate the validity and reasonability of the developed methodology.

High dimensional model representation for hybrid reliability analysis with dependent interval variables constrained within ellipsoids

Random variables could be dependent, and so could interval variables. To accommodate dependent interval variables, this work develops an efficient hybrid reliability analysis method to handle both random and dependent interval input variables. Due to the dependent interval variables, the reliability analysis needs to perform the probability analysis for every combination of dependent interval variables. This involves a nested double-loop procedure and dramatically decreases the efficiency. The proposed method decomposes the nested double loops into sequential probability analysis (PA) loop and interval analysis (IA) loop. An efficient IA algorithm based on the cut-HDMR (High Dimensional Model Representation) expansion is developed and a sampling strategy with the leave-one-out technique without extra calls of the limit-state function is proposed. The proposed method has good accuracy and efficiency as demonstrated by two engineering examples.

A Hybrid Reliability Evaluation Method for Meshed VSC-HVDC Grids

High-voltage direct current (HVDC) grids are emerging, and their reliability has been an increasing concern for the utilities. HVDC grids are different from typical two-terminal HVDC transmission systems due to the loops in their topology, which makes it difficult to evaluate the reliability by conventional analytical methods. This paper proposes an innovative hybrid method to evaluate the reliability of meshed HVDC grids. First, steady-state models and reliability models are established for the components in HVDC grids, especially for converters and power flow controllers. In the models, virtual buses are introduced to represent the external AC connections to the HVDC grid. Then a hybrid reliability evaluation method is proposed based on an analytical approach and Monte Carlo simulation. One innovation of the paper is the application of an analytical analysis method to accelerate state evaluation in Monte Carlo simulation by skipping unnecessary optimization. The proposed models and methods are verified on two HVDC grids. Test results show that HVDC grids under most failure states (approximately 70%) tend to shed no load except on buses connected to faulted converters, and the application of the analytical method could promote evaluation efficiency significantly.

Structural reliability analysis based on probability and probability box hybrid model

In many structural reliability analysis problems, probability approach is often used to quantify the uncertainty, while it needs a great amount of information to construct precise distributions of the uncertain parameters. However, in many practical engineering applications, distributions of some uncertain variables may not be precisely known due to lack of sufficient sample data. Hence, a complex hybrid reliability problem will be caused when the random and non-precise probability variables both exist in a same structure. In this paper, a new hybrid reliability analysis method is developed based on probability and probability box (p-box) models. Random distributions are used to deal with the uncertain parameters with sufficient information, while the probability box models are employed to deal with the non-precise probability variables. Due to the existence of the p-box parameters, a limit-state band will be resulted and the corresponding reliability index will belong to an interval instead of a fixed value. According to the interval analysis, the hybrid reliability model based on random and probability box variables is constructed and the complex nesting optimization problem will be involved in this hybrid reliability analysis. In order to obtain the minimal and maximal reliability index, the corresponding solution strategy is developed, in which the intergeneration projection genetic algorithm (IP-GA) with fine global convergence performance is employed as inner and outer optimization solver. Four numerical examples are investigated to demonstrate the effectiveness of the present method.

Novel reliability-based optimization method for thermal structure with hybrid random, interval and fuzzy parameters

• New reliability model with hybrid random, interval and fuzzy parameters. • Reliability is defined by novel interval ranking strategy and integral operation. • Extra safety information in the transition state is fully utilized. • Cut levels of different fuzzy variables are assumed to be independent. • Subinterval vertex method is presented for extrema of interval limit state function. In this paper, novel reliability-based optimization model and method are proposed for thermal structure design with random, interval and fuzzy uncertainties in material properties, external loads and boundary conditions. Random variables are used to quantify the probabilistic uncertainty with sufficient sample data; whereas, interval variables and fuzzy variables are adopted to model the non-probabilistic uncertainty associated with objective limited information and subjective expert opinions, respectively. Using the interval ranking strategy, the level-cut limit state function is precisely quantified to represent the safety state. The eventual safety possibility is derived based on multiple integral, where the cut levels of different fuzzy variables are considered to be independent. Then a hybrid reliability-based optimization model is established with considerable computational cost caused by three-layer nested loop. To improve the computational efficiency, a subinterval vertex method is presented to replace the inner-loop and middle-loop. Comparing numerical results with traditional reliability model, a mono-objective example and a multi-objective example are provided to demonstrate the feasibility of proposed method for hybrid reliability analysis and optimization in practical engineering.