Nonlinear Limit State Function Research Articles

An improved analytical solution to outcrossing rate for scalar nonstationary and non-gaussian processes

For time-dependent reliability methods, there is a lack of analytical solutions to outcrossing rates of nonstationary and non-Gaussian processes. This paper aims to propose an analytical method to determine this type of outcrossing rate with improved accuracy and efficiency. The novelty of this proposed method is that it can consider the probabilistic properties of non-Gaussian processes by combining a two-component parallel system model with higher-order moments-based reliability indexes. The main contributions of this method are: (1) compared with PHI2+ method, it is more accurate for the calculation of the outcrossing rate of nonstationary and non-Gaussian processes with nonlinear limit state functions; (2) compared with MPHI2 method, it is analytical and not only applicable for the nonstationary stochastic processes but also insensitive to time increments; and (3) it is not only more accurate than the existing methods for nonstationary and non-Gaussian processes but also more computation efficient than the Monte Carlo simulation method. The proposed method has shown its advantages for practical structures with neither Gaussian processes nor linear limit state functions, which are beneficial for both researchers and asset managers to evaluate the time-dependent reliability of structures accurately and to develop risk-informed maintenance schemes with a view to prolonging their service life.

Modified control variates method based on second-order saddle-point approximation for practical reliability analysis

Abstract. A novel method is presented for efficiently analyzing the reliability of engineering components and systems with highly nonlinear complex limit state functions. The proposed method begins by transforming the integral of the limit state function into an integral of a highly correlated limit state function using the control variates method. The second-order reliability method is then employed within the control variates framework to approximate the highly correlated limit state function as a quadratic polynomial. Subsequently, the probability of failure is obtained through the estimation of the saddle-point approximation and a small number of samples generated by Latin hypercube sampling. To demonstrate the effectiveness of the proposed method, four examples involving mathematical functions and mechanical problems are solved. The results are compared with those obtained using the second-order reliability method (SORM), control variates based on Monte Carlo simulation (CVMCS) with second-order saddle-point approximation (SOSPA), importance sampling (IS) and Monte Carlo simulation (MCS). The findings demonstrate that, while maintaining high-precision reliability results, the proposed method significantly reduces the number of evaluations of the limit state function through a small number of initial samples. The method is capable of efficiently and accurately solving complex practical engineering reliability problems.

Time-variant reliability of interlayer damage of CRTS-II slab track by combining the second-order fourth-moment and outcrossing method

This study proposes the combining second-order fourth-moment and outcrossing method for evaluating the time-variant reliability of slab tracks’ interlayer damage. The merit of the method is to improve significantly the computing efficiency and accuracy for the implicit and strong nonlinear limit state function. Furthermore, the effects of narrow joints damage and temperature load on the time-variant reliability of slab tracks’ interlayer damage are investigated. Narrow joints damage is an inducement for the decrease of time-variant reliability index of slab tracks’ interlayer damage, and the index decreases more violently with the increase of temperature rise than that of positive temperature gradient.

Interval reliability sensitivity analysis using Monte Carlo simulation and mouth brooding fish algorithm (MBF)

In reliability analysis and reliability-based design approach, sensitivity analysis establishes a relation between the variation in random variable parameters and that in reliability. The analysis of sensitivity is also employed to determine key random variables with the greatest contribution to reliability. The sensitivity analysis of a non-linear limit state function is relatively complicated and time-consuming. Therefore, a novel approach is proposed in this study to calculate the reliability sensitivity parameter in terms of an interval. In the proposed algorithm, the non-linear limit state function is initially converted to a linear limit state function by the Monte Carlo simulation method. Then, the interval random variables are introduced to include both epistemic and aleatory uncertainties so as to obtain a realistic result. In the second step of the proposed methodology, the mouth-brooding fish algorithm (MBF) is applied to capture the upper and lower bounds of the reliability sensitivity parameters. Several illustrative examples were then presented, showing the capability of the proposed approach in obtaining results accurately and efficiently. According to the results, the percentage of improvement in the sensitivity analysis of random variables in the proposed method, compared to the classical method, by using the MBF optimization algorithm to the parameters of random variables due to the variation Coefficient is more than 40 percents. This means that the length of the response interval has increased.

Application of first-order reliability method with orthogonal plane sampling for high-dimensional series system reliability analysis

The increasing complexity of modern engineering systems has motivated a shift of research focus from component-level reliability to system reliability with interdependent components. There is a growing demand for efficient reliability methods to analyze high-dimensional systems that involve numerous dependent components and component variables. The first-order reliability method (FORM), which is widely used for component-level reliability analysis, becomes inaccurate for high-dimensional systems composed of numerous components, each with a nonlinear high-dimensional limit state function. By integrating the orthogonal plane sampling, this paper proposes an improved FORM-based method to tackle the curse of dimensionality for series systems. The idea is to construct secant hyperplanes using the orthogonal plane samples so as to reduce the FORM error for high-dimensional nonlinear limit state functions. The design points of secant hyperplanes are projected to high-dimensional system space using an efficient procedure based on the specified correlation matrix of variables. Finally, the series system reliability is computed as high-dimensional multi-normal integral, which is addressed by the equivalent component method. Four numerical examples are investigated to demonstrate the accuracy and efficiency of the proposed method. Results indicate that the proposed method is significantly more efficient than the Monte Carlo simulation and more accurate than the conventional FORM.

An innovative method for space-time-dependent reliability analysis

The spatiotemporal variability exists in almost all engineering problems, since the environmental actions and material properties may change both spatially and temporally in a stochastic manner. Therefore, reliability analysis is inherently space–time-dependent. In this study, an innovative space–time-dependent outcrossing rate (STOR) method is proposed for dealing with such reliability problems. The failure probability of a structure is determined based on outcrossing rate which is theoretically derived for the first time, considering the spatiotemporal variability of all basic variables. Then, an efficient numerical algorithm for computing the failure probability is proposed based on Gauss-Legendre quadrature. The application of STOR method is investigated through three examples, including non-Gaussian random processes, non-Gaussian random fields, strong nonlinear limit state functions and small failure probability problems. The proposed STOR method is found to be efficient for space–time-dependent reliability analysis with enough accuracy.

Uncertainty analysis and reliability improvement of planetary roller screw mechanism using active learning Kriging model

The uncertainties in the geometry, material and operation conditions may cause structural failure of the planetary roller screw mechanism (PRSM). The uncertainty analysis model is the key to the reliability assessment of the PRSM, however, the relevant studies have been rarely reported in the past. This paper focuses on establishing a preliminary mathematical model of the PRSM considering uncertain factors. The quasi-Monte Carlo (QMC) method is introduced to improve the solving efficiency of the multidimensional and nonlinear implicit limit state function (LSF). Then, the parameter sensitivities of the uncertain factors to the load distribution and contact characteristics are comprehensively ranked by the design of experiment (DoE). The computational cost for constructing the active learning Kriging (ALK) model of PRSM is decreased by only selecting the most sensitive variables. Moreover, the ALK model and QMC method (ALK-QMC) are combined to explore how the main factors affect the structural reliability of PRSM, which further guides the implementation of multi-objective optimization to improve the reliability by the developed NSGA-II-Downhill algorithm. Finally, the theoretical model and optimization results are verified by the finite element method.

Reliability assessment of passive residual heat removal system using FORM methodology with chaos control

Passive safety systems are adopted extensively to increase the inherent safety of the nuclear reactor. However, their stability of operation is more susceptive to thermal hydraulics uncertainty due to the relatively small driving force of passive systems. Coupled with a shortage of pertinent experimental and operational data, its reliability assessment is still a crucial and challenging task. So far, there are no accepted techniques for this problem and they are not yet frozen. Hasofer-Lind and Rackwitz-Flessler method (HL-RF) is an efficient iterative approach to identify the most probable point (MPP) and calculate the reliability. However, the approach may suffer from instability when dealing with highly nonlinear limit state functions (LSFs) or high-dimensional problems. In this paper, a straightforward and efficient method depending on chaos control was proposed to deal with numerical instability and chaos non-convergence problems. The proposed method was utilized to assess the reliability of the passive residual heat removal system (PRHRS) in Integral-type Pressurized Water Reactor (IPWR200). The results showed that it could efficiently and reliably locate MPP and accurately obtain the reliability index.

Reliability analysis of compressed timber studs on the buckling criterion

Introduction. Ensuring safety is a priority goal in the design, construction and operation of building and structures. A quantitative assessment of safety can be the failure probability of a structural element. The article presents algorithms for reliability analysis of timber studs by the buckling criterion under central compression force.
 
 Materials and methods. The FOSM (First Order Second Moment) method is a classic approach for solving many reliability analyses tasks in the engineering sector. This method uses the second statistical moments of random variables (mathematical expectation and variance) and the method of statistical linearization with the decomposition of the function into a Taylor series of the first order. However, it is often necessary to deal with nonlinear limit state functions, where the statistical linearization method can lead to incorrect results. In this case, it is necessary to use other algorithms for reliability analysis: the Hasofer – Lind method, SORM (Second Order Reliability Method), etc.
 
 Results. The numerical example of reliability analysis for compressed timber studs on the buckling criterion is considered. To illustrate the problem of the research, the dimensions of the cross-section of the studs and the strength of the wood are taken as random variables, and the load is considered a deterministic value. In the following, an algorithm for reliability analysis is described for a timber studs under random load. The classical FOSM approach showed an overestimated reliability index by 35 % compared to the results of the Monte Carlo numerical experiment and the analytical solution using the Hasofer – Lind algorithm.
 
 Conclusions. The use of the traditional FOSM method for reliability analysis of wooden studs according to the buckling criterion can lead to overestimated reliability index estimates. It is unacceptable from the point of view of the structural safety. The calculation and analysis of the reliability of timber studs should be carried out on the basis of the Hasofer – Lind algorithm, the SORM method or other optimization methods.

Reliability Analysis of Concrete Gravity Dams Based on Least Squares Support Vector Machines with an Improved Particle Swarm Optimization Algorithm

A reliability analysis method based on least squares support vector machines with an improved particle swarm optimization algorithm (IPSO-LSSVM) is proposed to calculate the reliability of concrete gravity dams when explicit nonlinear limit-state functions are difficult to obtain accurately. First, the main failure modes of concrete gravity dams and their influencing factors are determined. Second, Latin hypercube sampling is used to create samples. A finite element calculation batch program of concrete gravity dams is written to calculate the safety indexes of each sample. Third, based on the samples, the IPSO-LSSVM model is established to replace the finite element calculation. Finally, the failure probability of concrete gravity dams is obtained by using the Monte Carlo (MC) method. The case study for a typical concrete gravity dam in the Yunnan Province of China shows that the dam is reliable because the failure probability is 8.87 × 10−5. The proposed reliability analysis method is efficient and feasible for calculating the reliability of concrete gravity dams.

A first order reliability method based on hybrid conjugate approach with adaptive Barzilai–Borwein steps

In the first order reliability method (FORM), the Hasofer–Lind and Rackwitz–Flessler (HL–RF) algorithm sometimes encounters numerical instability problems due to the highly nonlinear limit state function (LSF). In this paper, an improved HL–RF algorithm introducing the hybrid conjugate gradient method with adaptive Barzilai–Borwein step sizes is developed to enhance the robustness and efficiency of the original HL–RF method. The proposed algorithm is composed of two stages, the steepest descent method is performed in the first stage to move to the vicinity of the most probable failure point (MPFP) and provide a good initial location for the second stage. Along with the hybrid conjugate search direction defined by the descent direction of the non-differential merit function, the second stage quantizes the adaptive Barzilai–Borwein step sizes to accelerate the process of locating the final MPFP under the nonmonotone line search rule. Eight illustrative examples with nonlinear LSFs are analyzed in detail to validate the performance of the proposed algorithm compared with other first order reliability methods. The results indicate that the proposed algorithm is not only computationally efficient but also robust in terms of convergence, especially for those problems with super nonlinear LSFs and with nonlinear LSFs involving high-frequency noise terms.

Data-driven reduced order model and simplicial homology global optimization for reliability analysis and application

A novel framework of reliability analysis was developed in this study to consider the uncertainty of geomaterials and geological conditions by combining the reduced-order model (ROM), reliability analysis, and numerical model. The reliability method was used to determine the reliability index using the simplicial homology global optimization (SHGO) based on the ROM. The developed method was verified and illustrated using three numerical examples and a simple slope. The limit state curve in all three numerical examples was in excellent agreement with the actual curve. The reliability index and failure probability were also in excellent agreement with those of the actual limit state function using the first-order reliability method (FORM) and Monte Carlo simulation, respectively, indicating that the ROM method can present the limit state function well. The results showed that the developed method is feasible and effective for reliability analysis of geotechnical and geological engineering problems with a complex, nonlinear, and implicit limit state function. Furthermore, the developed method is effective, efficient, and accurate for reliability analysis. It provides an excellent way to approximate the limit state function to avoid the time-consuming numerical model in a practical engineering system.

IE-AK: A novel adaptive sampling strategy based on information entropy for Kriging in metamodel-based reliability analysis

This article focuses on the adaptive Kriging metamodel-based reliability analysis for reducing a sequential number of calls of the complex original functions. To avoid the repetitive and tedious deterministic response analysis with stochastic simulation method (including Monte Carlo Simulation and its various improvement, such as importance sampling, subset simulation) in reliability analysis, herein a novel sequential sampling strategy related to Kriging metamodel is proposed, which is implemented based on information entropy theory. In addition, the generalized F-discrepancy method is simultaneously quoted to further optimize the candidate pool to improve the effectiveness of the training metamodel. Finally, a new structural reliability analysis method is proposed, which continuously reduces the number of deterministic analysis of structures without sacrificing accuracy. To highlight the applicability of the method and verify its accuracy and effectiveness, a series of typical examples are tested and compared, including highly nonlinear limit state functions, high-dimension performance function with analytic expressions and dynamic reliability analysis of nonlinear engineering structures subject to seismic excitation with implicit performance function. Numerical results show that significant computational savings and desired accuracy can be achieved when dealing with different reliability analysis cases.

Sequential Active Learning of Low-Dimensional Model Representations for Reliability Analysis

To date, the analysis of high-dimensional, computationally expensive engineering models remains a difficult challenge in risk and reliability engineering. We use a combination of dimensionality reduction and surrogate modeling termed partial least squares--driven polynomial chaos expansion (PLS-PCE) to render such problems feasible. Standalone surrogate models typically perform poorly for reliability analysis. Therefore, in a previous work, we have used PLS-PCEs to reconstruct the intermediate densities of a sequential importance sampling approach to reliability analysis. Here, we extend this approach with an active learning procedure that allows for improved error control at each importance sampling level. To this end, we formulate an estimate of the combined estimation error for both the subspace identified in the dimension reduction step and the surrogate model constructed therein. With this, it is possible to adapt the training set so as to optimally learn the subspace representation and the surrogate model constructed therein. The approach is gradient-free and thus can be directly applied to black box--type models. We demonstrate the performance of this approach with a series of low- (2 dimensions) to high- (869 dimensions) dimensional example problems featuring a number of well-known caveats for reliability methods besides high dimensions and expensive computational models: strongly nonlinear limit-state functions, multiple relevant failure regions, and small probabilities of failure.

Dimensional Reduction-Based Moment Model for Probabilistic Slope Stability Analysis

Uncertainty is an inevitable factor that influences the function analysis, design, and safe operation in engineering systems. Due to the complexity property and unclear failure mechanism, uncertainty is an intrinsic property of slope engineering. Hence, stability analysis and design cannot meet the demands of slope engineering based on the traditional deterministic method, which cannot deal with uncertainty. In this study, a practical reliability approach was developed to consider the uncertainty factor in slope stability analysis by combining the multiplicative dimensional reduction method (MDRM) and first-order second moment (FOSM). MDRM was used to approximate the complex, nonlinear, high-dimensional, and implicit limit state function. The statistical moment of safety factor was estimated based on the moment method using MDRM. FOSM is adopted to compute the reliability index based on the statistical moment of the safety factor. The proposed method was illustrated and verified by an infinite slope with an analytical solution. The reliability index and failure probability were compared with Monte Carlo simulations (MCS) in various cases. Then, it was applied to a slope based on numerical solutions. The results show that the proposed method is feasible and effective for probabilistic slope stability analysis. The reliability index obtained from the proposed method shows high consensus with the traditional response surface method (RSM). It shows that the proposed method is effective, efficient, and accurate. MDRM provides a practical, simple, and efficient probabilistic slope stability analysis approach.

The GLO method: An efficient algorithm for time-dependent reliability analysis based on outcrossing rate

Outcrossing method has been widely applied in time-dependent reliability analysis (TRA), where the cumulative failure probability is calculated based on the numerical integration of outcrossing rate. The efficiency of outcrossing method is influenced by two factors, i.e., the number of discretized instants for numerical integration and the computational time for solving outcrossing rate at each instant. Focusing on these factors, an efficient Gauss-Legendre quadrature based outcrossing (GLO) method is proposed in this study. The main innovation of the proposed method consists of two aspects. First, the cumulative failure probability is evaluated by weighted sum of outcrossing rate at only three to five instants with the aid of Gauss-Legendre quadrature, which avoids the time-consuming numerical integration with a large number of instants discretized. Second, an efficient algorithm for computing outcrossing rate is developed from a recently proposed system reliability method based on the well-known FORM/SORM. Three examples are presented to demonstrate the application of the proposed GLO method, including explicit, implicit, and strong nonlinear limit state functions. It is shown that, reliability results by the proposed GLO method are in close agreement with those by Monte Carlo simulation, with the computational time significantly reduced.

Ring simulation: A novel simple and efficient simulation method for structural reliability analysis

In this paper, a novel simulation method, referred to as “ring simulation”, is proposed. The method estimates the failure probability only using the information from the importance interval of the radius, i.e., the hyper-spherical ring, where the contributions to the failure probability are dominant. A stepwise computing procedure is proposed to determine the importance interval and simultaneously obtain the failure probability, in which samples are uniformly generated in the importance interval. The variability of the proposed method for the evaluation of the failure probability is also investigated. The accuracy, efficiency, and robustness of the proposed method are demonstrated through four examples involving the multiple design points, existence of local minima, small failure probability, and highly nonlinear, implicit, and high-dimensional limit state functions. The results reveal that the proposed method is almost as simple and applicable as crude Monte Carlo simulation method, but more efficient. Compared with directional simulation, subset simulation, and weighted average simulation, the proposed method has shown its advantage.

A probabilistic approach to estimate the remaining life and reliability of corroded pipelines

Considering corrosion rate during the remaining life assessment of aging pipelines is fundamental to calculate the interval between two consecutive inspections. A total of 798 internal and external corrosion defects have been detected, using the Magnetic Flux Leakage intelligent inspection tool, over 48 km of a pipeline length located in the west region of Algeria. The statistical analysis has shown that there is a strong correlation between the corrosion defect length and the corrosion defect circumferential width, with a significant correlation coefficient equal to 82.87%. A probabilistic methodology is presented for the assessment of the remaining life of a corroded pipeline transporting gas, and a finite element method (FEM) was used to assess the pipeline failure pressure. The numerical FEM modeling results were compared with the commonly used codes-models for calculating limit pressure to establish a more realistic and accurate engineering model. The reliability analysis of an API 5L X60 steel made Algerian natural gas pipeline, in service for thirty years, and exposed to active corrosion attack, is presented. The sensitivity analysis of the basic random variables within the nonlinear limit state function was carried out to bring out the relative contribution of each variable affecting the remaining life of corroded pipelines. The reliability analysis is carried out by using Breitung's formula, based on the second-order reliability method (SORM). The reliability assessment of the corroded pipeline is based on the usage of the notched failure assessment diagram (NFAD), different codes for the calculation of the failure pressure, and the numerical results using the finite element analyses (FEA) software ANSYS.

Positioning accuracy reliability analysis of industrial robots based on differential kinematics and saddlepoint approximation

This paper proposes a new positioning accuracy reliability analysis method based on the differential kinematics and saddlepoint approximation to evaluate the influence of kinematic parameter uncertainties on the robotic motion. Firstly, considering the reliability analysis instability caused by complex nonlinear limit state function, the reliability model of positioning accuracy is established based on the differential kinematics and error propagation. Subsequently, the parameters of the position error distribution are deduced analytically to combine with the eigen-decomposition to complete the modeling of the cumulant generating function, thereby the kinematic reliability is yielded according to the saddlepoint approximation method. Finally, the comparative analysis of a six degrees of freedom serial industrial robot is conducted and the results of which demonstrate that the proposed method provides a better performance than the existing methods in terms of accuracy and efficiency for kinematic reliability analysis.

A New SORM Method for Structural Reliability with Hybrid Uncertain Variables

(1) Background: in practical applications, probabilistic and non-probabilistic information often simultaneously exit. For a complex system with a nonlinear limit-state function, the analysis and evaluation of the reliability are imperative yet challenging tasks. (2) Methods: an improved second-order method is proposed for reliability analysis in the presence of both random and interval variables, where a novel polar transformation is employed. This method enables a unified reliability analysis taking both random variables and bounded intervals into account, simplifying the calculation by transforming a high-dimension limit-state function into a bivariate state function. The obtained nonlinear probability density functions of two variables in the function inherit the statistic characteristics of interval and random variables. The proposed method does not require any strong assumptions and so it can be used in various practical engineering applications. (3) Results: the proposed method is validated via two numerical examples. A comparative study towards a contemporary algorithm in state-of-the-art literature is carried out to demonstrate the benefits of our method. (4) Conclusions: the proposed method outperforms existing methods both in efficiency and accuracy, especially for cases with strong nonlinearity.