Non-probabilistic Reliability Analysis Research Articles

Non-probabilistic reliability analysis with both multi-super-ellipsoidal input and fuzzy state

In real-world engineering scenarios, incomplete uncertainty information and ambiguous failure states persist and pose significant challenges for structural reliability analysis. This paper introduces a non-probabilistic fuzzy reliability analysis (NPFRA) model featuring fuzzy output states, where the input uncertainties are quantified by a multi-super-ellipsoidal model. Initially, we define both reliability and failure indices of NPFRA, and provide the corresponding Monte Carlo simulation (MCS) solution. Additionally, an extended variable space (EVS) method is established to transform the NPFRA problem into a conventional non-probabilistic reliability analysis (NPRA) one, and MCS based on EVS is derived accordingly. To address the efficiency issue of MCS, a novel method called active learning kriging with norm-constrained expected risk function (ALK-NERF) is developed explicitly for NPFRA. Four examples are adopted to verify the rationality and effectiveness of the proposed ALK-NERF for NPFRA.

A non-probabilistic reliability analysis method with the fuzzy failure criterion

In engineering practice, the failure boundary of structures is not usually distinct due to some ambiguous information, which brings a tricky challenge to reliability assessment. The fuzzy theory is introduced to solve it with the traditional probability assessment method. For the non-probabilistic reliability system, however, this problem tends to be at a dead end because it lacks some necessary probability information. In this study, the non-probabilistic reliability model with fuzzy failure criterion is established, which involves two different evaluation indices in terms of the limit state surface. Based on this, a corresponding calculation method is proposed to deal with it, and the interval model and the ellipsoid model are shown for instance. In addition, the HRHL-RF method owning strong robustness is utilized to calculate the index of the ellipsoid model. Eventually, an established algorithm frame can achieve the calculation of the unified fuzzy reliability index, and three examples with the classical fuzzy membership functions are computed to show its feasibility and efficiency.

Non-Probabilistic Reliability Analysis of Slopes Based on a Multidimensional Parallelepiped Model

Aiming at the problem that the non-probabilistic reliability analysis method of slope engineering, which is based on an interval model, cannot consider the cross-correlation of geotechnical parameters, a non-probabilistic reliability analysis method of slopes based on a multidimensional parallelepiped model is proposed. This method can effectively alleviate the problem of difficult data survey in the field of geotechnical engineering. Using the limited sample data of soil parameters, the multidimensional parallelepiped model is constructed. The performance function of the slope is constructed based on Latin hypercube sampling and the quadratic response surface method. Then, the limit state equation of the slope can be standardized using the multidimensional parallelepiped model. The non-probabilistic reliability indexes of the slope are calculated based on the global optimal solution to judge the stability state of the slope. The example analysis verifies the feasibility of the proposed method. The results show that the correlation of shear strength parameters of soil has a great influence on the non-probabilistic reliability indexes of the slope. When the correlation coefficients of the shear strength parameters are between −1.0 and 0, the smaller the correlation coefficient is, the greater the non-probabilistic reliability index of the slope is; when the correlation coefficients of the shear strength parameters are between 0 and 1.0, the non-probabilistic reliability index of the slope does not change with the correlation coefficient. The non-probabilistic reliability indexes of the slope obtained using the multidimensional parallelepiped model are between the results obtained using an ellipsoid model and those obtained using an interval model, which are validated by Monte Carlo method and relatively more reasonable. In the absence of a large number of geotechnical sample data, this method provides a new way for slope stability analysis and expands the application field of calculation methods based on non-probabilistic theory.

A dynamic Gaussian process surrogate model based on the grasshopper optimization algorithm for non-probabilistic reliability analysis of complex structures

In practical complex engineering structures, the performance function (PF) for reliability analysis is commonly implicit and highly nonlinear. Commonly used surrogate models are appropriate for structural reliability analysis with an implicit PF. However, these methods require hundreds of PF values, which are time-consuming to obtain by adopting numerical analysis, such as finite element analysis (FEA). Therefore, since non-probabilistic reliability analysis does not require a large number of samples, it has great development potential. In this paper, a dynamic Gaussian process (GP) surrogate model based on the grasshopper optimization algorithm (DGP-GOA) is proposed for the non-probabilistic reliability analysis. First, with the help of the scale factor of the convex set model, the non-probabilistic reliability analysis problem is transformed into an unconstrained optimization problem. Second, the DGP-GOA fits the PF by constructing a GP surrogate model with a small dataset. Third, the GOA is used to search for the global optimal solution to obtain a non-probabilistic reliability index. Then, a dynamic retraining strategy is proposed to improve the fitting accuracy and efficiency. The results demonstrate that the proposed method is highly applicable to the non-probabilistic structural reliability analysis of complex engineering structures.

Non-Probabilistic Reliability Analysis of Slopes Based on Fuzzy Set Theory

Aimed at the problem of fuzzy uncertainty of geotechnical parameters in slope stability analysis, a non-probabilistic reliability analysis method for slopes based on fuzzy set theory is proposed. Geotechnical parameters are described as fuzzy numbers, which are transformed into interval numbers at different cut set levels by taking fuzzy sets. The corresponding non-probabilistic reliability indexes and failure degrees of the slope are calculated by the non-probabilistic reliability analysis method based on the ellipsoidal model, and then the overall failure degree of the slope is obtained by weighted average to judge the stability state of the slope. The feasibility of the method was verified by a case analysis. The results show that the type and shape parameters of the fuzzy affiliation function of geotechnical parameters have a great influence on the non-probabilistic reliability of the slope. The slope failure degrees obtained from trapezoidal fuzzy numbers were larger, the slope failure degrees obtained from triangular fuzzy numbers and normal fuzzy numbers were medium, and the slope failure degrees obtained from lognormal fuzzy numbers were smaller. When considering soil parameters as triangular fuzzy numbers, normal fuzzy numbers, or lognormal fuzzy numbers, with the reduction of the shape parameters, the non-probabilistic reliability indexes of the slope increased while the failure degrees decreased. Additionally, adopting the overall failure degree to evaluate the stability of the slope can effectively solve the problem where the calculation results are too conservative (if the non-probabilistic reliability index is greater than 1) to judge the stability state of the slope in the traditional non-probabilistic reliability method.

Non-Probabilistic Reliability Analysis of Robot Accuracy under Uncertain Joint Clearance

The development of industrial robots in high-precision fields is currently constrained by the reliability of motion. Considering the influence of the joint clearance on the motion reliability of the industrial robot, the kinematic model of the industrial robot is established, the kinematic equation of the robot is deduced, and positive kinematic solutions are performed. The non-probability positioning accuracy reliability measure of a robot end-effector is proposed, based on the non-probability theory and method, combined with the prescribed permission interval and error interval, and different states of reliability can be judged according to the position relationship, the non-probability reliability properties are outlined, and the positioning accuracy reliability assessment model is established. Combined with the joint clearance modeling theory, the simulation of the robot end-effector under the influence of six joint clearances is carried out, and the displacement error interval of the end-effector under the preset motion path is analyzed for the industrial robot motion reliability problem. The motion path is split by time, and the end effector moves to different workspace areas in different time periods. The motion reliability of each segment is analyzed, and it is concluded that the reliability of the end-effector under the influence of uncertain joint clearance parameters changes in different working regions. Based on the above, the research direction of space division and partition parameter calibration is proposed, which lays a foundation for the study of partition non-probabilistic calibration of robot workspace.

A Grasshopper Optimization Algorithm-Based Response Surface Method for Non-Probabilistic Structural Reliability Analysis with an Implicit Performance Function

Non-probabilistic reliability analysis has great developmental potential in the field of structural reliability analysis, as it is often difficult to obtain enough samples to construct an accurate probability distribution function of random variables based on probabilistic theory. In practical engineering cases, the performance function (PF) is commonly implicit. Monte Carlo simulation (MCS) is commonly used for structural reliability analysis with implicit PFs. However, MCS requires the calculation of thousands of PF values. Such calculation could be time-consuming when the structural systems are complicated, and numerical analysis procedures such as the finite element method have to be adopted to obtain the PF values. To address this issue, this paper presents a grasshopper optimization algorithm-based response surface method (RSM). First, the method employs a quadratic polynomial to approximate the implicit PF with a small set of the actual values of the implicit PF. Second, the grasshopper optimization algorithm (GOA) is used to search for the global optimal solution of the scaling factor of the convex set since the problem of solving the reliability index is transformed into an unconstrained optimal problem. During the search process in the GOA, a dynamic response surface updating strategy is used to improve the approximate accuracy near the current optimal point to improve the computing efficiency. Two mathematical examples and two engineering structure examples that use the proposed method are given to verify its feasibility. The results compare favorably with those of MCS. The proposed method can be non-invasively combined with finite element analysis software to solve non-probabilistic reliability analysis problems of structures with implicit PF with high efficiency and high accuracy.

Structural reliability with credibility based on the non-probabilistic set-theoretic analysis

A structural reliability analysis method based on the non-probabilistic credible set model is proposed in this paper. Compared with the probabilistic method, the developed method overcomes the high requirement for the sample size of structural uncertain parameters. Moreover, it surpasses the traditional non-probabilistic reliability analysis method by relating the reliability index to its corresponding credibility. Via introducing the non-probabilistic credible set uncertainty quantification, the distribution range of the required credibility for structural uncertain parameters can be determined by scant samples. Accordingly, the improved uncertainty propagation analysis methods based on the non-probabilistic credible set uncertainty quantification are established to obtain the bounds of structural response by transforming into local coordinate system. The non-probabilistic credible reliability index is constructed and three advanced efficient approaches are developed to calculate that. Moreover, the procedure of structural non-probabilistic credible reliability analysis method is built by combining the above content. Besides, the influence of credibility on non-probabilistic credible reliability is discussed. A cantilever beam example, an aeroengine turbine disk case and a pressure vessel case are utilized to verify the feasibility of applying the proposed approach to various engineering problems.

Hybrid active learning method for non-probabilistic reliability analysis with multi-super-ellipsoidal model

Non-probabilistic reliability analysis is of great importance in both reliability measure and reliability based design, the efficiency and precision of non-probabilistic reliability analysis, as well as uncertainty quantification, have attracted great attention currently. In this study, considering the coexistence of correlated and independent uncertain-but-bounded variables in engineering applications, a multi-super-ellipsoidal model is used to quantify the uncertainties. Furthermore, inspired by both “norm” based reliability index and “volume-ratio” based reliability index, a hybrid non-probabilistic reliability index is derived to measure the reliability extent more accurately and intuitively. To improve the accuracy and efficiency of solving the hybrid non-probabilistic reliability index, an effective Kriging based hybrid active learning method (HALM) is further developed. Finally, four examples are used to verify the effectiveness and robustness of the proposed HALM. The results show that the selection of multi-super-ellipsoidal model has a certain effect on the estimation of reliability index. Compared with the sampling-based analysis method, HALM presents better performance in terms of the trade-of between in computational efficiency and accuracy.

A Decoupling Strategy for Reliability Analysis of Multidisciplinary System with Aleatory and Epistemic Uncertainties

In reliability-based multidisciplinary design optimization, both aleatory and epistemic uncertainties may exist in multidisciplinary systems simultaneously. The uncertainty propagation through coupled subsystems makes multidisciplinary reliability analysis computationally expensive. In order to improve the efficiency of multidisciplinary reliability analysis under aleatory and epistemic uncertainties, a comprehensive reliability index that has clear geometric meaning under multisource uncertainties is proposed. Based on the comprehensive reliability index, a sequential multidisciplinary reliability analysis method is presented. The method provides a decoupling strategy based on performance measure approach (PMA), probability theory and convex model. In this strategy, the probabilistic analysis and convex analysis are decoupled from each other and performed sequentially. The probabilistic reliability analysis is implemented sequentially based on the concurrent subspace optimization (CSSO) and PMA, and the non-probabilistic reliability analysis is replaced by convex model extreme value analysis, which improves the efficiency of multidisciplinary reliability analysis with aleatory and epistemic uncertainties. A mathematical example and an engineering application are demonstrated to verify the effectiveness of the proposed method.

Ellipsoid non-probabilistic reliability analysis of the crack growth fatigue of a new titanium alloy used in deep-sea manned cabin

Reliability analysis of crack growth fatigue of titanium alloy is one of the important guarantees for the safety of deep-sea manned cabin. However, the traditional probabilistic fatigue reliability analysis methods often need accurate parameter statistics information. In this paper, under the condition of the limited test data, a first-order approximate ellipsoid non-probabilistic reliability analysis method for fatigue crack growth of a titanium alloy was proposed. A limit state equation of the crack growth correction model, considering the small crack effect, is formulated according to the fatigue life model. The uncertain parameters are constructed by the interval model, normal distribution, and Khachiyan algorithm. The different limit state equations are obtained by changing the coefficient κ. The first-order approximation and Monte Carlo methods are used to solve the reliability of the ellipsoid model. Calculations demonstrate that the results of the ellipsoid non-probabilistic reliability method are more conservative than that of the probabilistic reliability method. When κ is less than or equal to 2, the Khachiyan algorithm is preferred to build the ellipsoid model, and when κ is greater than 2, the interval model is preferred. Through parameter sensitivity analysis, the uncertain parameters KC, n, m, A, and ΔKthR are found to significantly affect the crack growth rate.

Non-Probabilistic Reliability Bounds for Series Structural Systems

Most of the current non-probabilistic reliability methods are applicable for an individual component of a structure. However, a system consisting of interconnected components is involved in many engineering problems and its non-probabilistic reliability analysis remains a major challenge. In this paper, a non-probabilistic reliability method using upper and lower bound techniques is proposed for a series structural system. Based on the failure areas and the joint failure areas of two failure modes, the bounds of the non-probabilistic failure degree for a series system are firstly formulated. A judgement criterion is then presented to determine the joint failure areas of two failure modes. An analytical solution for the joint failure area of two failure modes is further given to simplify the calculation of the bounds. Four numerical examples are used to demonstrate the validity and effectiveness of the presented method.

Long-term safety assessment of large-scale arch dam based on non-probabilistic reliability analysis

Long-term service status of a large-scale arch dam is affected by variations of loads and material properties with randomness and fuzziness. This study aims to resolve the problems of a variety of uncertain variables that coexist in the reliability assessment of arch dams. Firstly, a time-varying non-probabilistic reliability (N-PR) solution method of the arch dams is proposed based on the convex set model. Subsequently, non-probabilistic target reliability indexes are put forward according to the defined probabilistic reliability indexes during long-term operation of the arch dams. Afterwards, a fuzzy prediction index is formulated based on the mixed statistical model and the Info-gap model under the most unfavorable reservoir water depth, so as to timely find defects and avoid the potential failure of the arch dams. Numerical simulation on the Jinping I arch dam is conducted to validate the proposed methods. The possible failure modes and the long-term N-PR of the world’s highest arch dam is appraised, and the fuzzy prediction indexes are concluded. Research results can provide a feasible way for large-scale arch dam reliability and safety assessment.

System reliability-based design optimization with interval parameters by sequential moving asymptote method

Reliability-based design optimization (RBDO) offers a powerful tool to deal with the structural design with heterogeneous interval parameters concurrently. However, it is time-consuming in the practical engineering design. Therefore, a novel sequential moving asymptote method (SMAM) is proposed to improve the computational efficiency for convex model in this study, in which the nested double-loop optimization problem is decoupled to a sequence of deterministic suboptimization problems based on the method of moving asymptotes. In addition, the sensitivity of reliability index is derived, so the finite difference for the nested optimization loop can be avoided to tremendously improve the computational efficiency. Then, the accuracy of the SMAM is proved based on the error analysis. Furthermore, the Kreisselmeier-Steinhauser (KS) function is used to assemble the multiple constraints to deal with the parallel and series RBDO problems. One benchmark mathematical example, three numerical examples, and one complex civil engineering example, i.e., tower crane, are tested to demonstrate the efficiency of the proposed method by comparison with other existing methods, and the results indicate that SMAM offers a general and effective tool for non-probabilistic reliability analysis and optimization.

A framework for the use of reliability methods in deep urban excavations analysis

Deep excavations in urban areas impose deformation to adjacent structures; hence the reliability of deformation analysis for the real deep excavation projects is very important to be assessed. In this study a framework is presented for the use of reliability methods in deformation analysis of deep urban excavations. The suggested framework is applied for 5 real deep excavation projects implemented during last 10 years. All studied cases were recognized as projects of high importance in urban areas, and were monitored during the excavation process. A non-probabilistic reliability analysis procedure, Random set method, in combination with finite element numerical modeling is applied to obtain the probability of unsatisfactory performance for each case. The reliability analysis results are confirmed by field observations and measurements. Typical results for the probability of analytical deformations exceeding the acceptable values along with the site observations and measured displacements for 5 real deep excavation projects show that the reliability analysis could be a beneficial tool for designer. It is concluded that applying the suggested framework in the design stage of deep excavation projects may lead to design more appropriate systems compared to common deterministic design methods.

Non-probabilistic Integrated Reliability Analysis of Structures with Fuzzy Interval Uncertainties using the Adaptive GPR-RS Method

Due to its weak dependence on the quantity of variable samples, the non-probabilistic reliability analysis method based on the convex set model is applicable to practical problems in structural engineering with inherent uncertainties. However, when dealing with the black-box limit-state function issues in practical complex structural engineering, the traditional quadratic polynomial response surface (QP-RS) method has the problem of insufficient precision in approximating a highly nonlinear function. Meanwhile, fixing the limits of interval variables is trickier in case of scant samples and meager statistical information. To remedy the above deficiencies, this paper introduces a reasonable integrated reliability analysis approach. First, an adaptive Gaussian process regression response surface (GPR-RS) method that dynamically improves the fitted accuracy near the design point of the black-box limit-state function is formulated. Furthermore, the integrated reliability index with consideration of fuzzy interval uncertainties is presented. Three validation and two application examples are employed, which have justified the approach as a more reasonable assessor of practical complex structural reliability with safer results.

Non-Probabilistic Time-Varying Reliability-Based Analysis of Corroded Pipelines Considering the Interaction of Multiple Uncertainty Variables

Reliability analysis of corroded pipelines is critical to the integrity and safe working of pipeline infrastructure. Aiming at less probability information is obtained for corrosion pipeline engineering, and the mechanical properties of pipeline with corrosion defects deteriorate caused by the accumulative effect of corrosion growth. Based on the quasi-static analysis method and non-probability theory, this paper presents a reliability model for assessing corroded pipelines with corrosion growth. In fact, reliability analysis of corroded pipelines needs to consider the interaction of multiple uncertainty variables. By introducing interaction theory, a mathematical model of corrosion defects considering the interaction of variables is put forward. Moreover, this paper develops a non-probabilistic time-varying reliability method for pipeline systems with multiple defects. Thus, several numerical examples are investigated to discuss the effectiveness of the proposed methodology. The results show that a two-dimensional or even three-dimensional ellipsoid model with correlation has more accurate results to evaluate corroded pipelines under the interaction of multiple corroded defects with poor information. Furthermore, a non-probabilistic time-varying reliability model is established according to the time-varying characteristics of the corroded pipeline under the influence of multiple factors. An effective complement to the theory of non-probabilistic reliability analysis of system is investigated. The analysis of the results suggests that interaction of corroded pipeline has a negligible impact on reliability. It also provides a theoretical basis for maintenance and is of great significance for risk- and reliability-informed decisions regarding buried oil and gas pipelines.

An importance learning method for non-probabilistic reliability analysis and optimization

With the time-consuming computations incurred by nested double-loop strategy and multiple performance functions, the enhancement of computational efficiency for the non-probabilistic reliability estimation and optimization is a challenging problem in the assessment of structural safety. In this study, a novel importance learning method (ILM) is proposed on the basis of active learning technique using Kriging metamodel, which builds the Kriging model accurately and efficiently by considering the influence of the most concerned point. To further accelerate the convergence rate of non-probabilistic reliability analysis, a new stopping criterion is constructed to ensure accuracy of the Kriging model. For solving the non-probabilistic reliability-based design optimization (NRBDO) problems with multiple non-probabilistic constraints, a new active learning function is further developed based upon the ILM for dealing with this problem efficiently. The proposed ILM is verified by two non-probabilistic reliability estimation examples and three NRBDO examples. Comparing with the existing active learning methods, the optimal results calculated by the proposed ILM show high performance in terms of efficiency and accuracy.

Non-probabilistic reliability analysis and design optimization for valve-port plate pair of seawater hydraulic pump for underwater apparatus

Seawater hydraulic axial piston pumps are widely used in underwater apparatus systems in recent years. Reliability of the valve-port plate pair is vital to the performance of seawater hydraulic axial piston pumps. As the distribution information of the material combinations and working conditions of seawater hydraulic axial piston pumps are insufficient, the probability reliability analysis methods are not available for the valve-port plate pair. This paper develops a non-probabilistic reliability convex model for the valve-port plate pair. Firstly, the rotation of the pump cylinder is partitioned into six stages. Based on this partition, the dynamic models of contact pressure and pv value (the product of contact specific pressure and linear velocity) of the valve-port plate pair are derived by accounting for the pre-compression angle. Secondly, the marginal interval for the limiting [pv] value (It is the usual criterion to evaluate the critical operating conditions under which the material fails) of the material combination was obtained through adhesive wear tests. Thirdly, a limit-state function is then established based on the stress-strength interference model. Finally, the analysis and sea test results indicate that the reliability of the valve-port plate pair has significantly improved by adding auxiliary support on the port plate.

New target performance approach for a super parametric convex model of non-probabilistic reliability-based design optimization

Non-probabilistic reliability-based design optimization (NRBDO) is an essential tool to deal with an optimization model with uncertain-but-bounded variables. A major disadvantage of NRBDO is that it conventionally suffers from excessively high computational costs. In the present study, a new target performance approach (TPA) is proposed to eliminate high computational cost incurred by complicated equality constraints in non-probabilistic reliability analysis in which the target performance constraint is applied to substitute for the non-probabilistic reliability index constraint. To further enhance the optimization efficiency, a new NRBDO algorithm is developed based on the proposed TPA in which the target performance iterative method (TPIM) is established to search for the target concern point while a convex approximate method is used to further improve the convergence. A mathematical example, three numerical examples, and two complex engineering examples, i.e., an axially compressed stiffened shell in a launch vehicle and a tower crane in civil engineering, are utilized to demonstrate the effectiveness of the proposed method in comparison with that of other existing methods. The results indicate that the proposed method is promising in terms of complicated engineering problems in the absence of prior knowledge of uncertain parameters.