Probable Failure Point Research Articles

Improved Moving Least Square-Based Multiple Dimension Decomposition (MDD) Technique for Structural Reliability Analysis

This paper presents the state-of-the-art on different moving least square (MLS)-based dimension decomposition schemes for reliability analysis and demonstrates a modified version for high fidelity applications. The aim is to improve the performance of MLS-based dimension decomposition in terms of accuracy, number of function evaluations and computational time for large-dimensional problems. With this in view, multiple finite difference high dimension model representation (HDMR) scheme is developed. This anchored decomposition is implemented starting from an initial reference point and progressively evolving in successive iterations. Most probable point (MPP) of failure is identified in every iteration and is used as the reference point for the next decomposition until it converges. Hermite polynomials in MLS framework are used between the support points for efficient interpolation. The support points are generated sequentially using multiple sparse grids based on the Clenshaw–Curtis scheme. Once the global response surface is constructed using the support points generated in each iteration, importance sampling is employed for reliability analysis. Six different benchmark problems are solved to show its performance vis-à-vis other methods. Finally, reliability-based design of a composite plate is demonstrated, clearly showing the advantage and superiority of the proposed improvements in MLS-based multiple dimension decomposition (MDD).

Structure-MR damper reliability analysis using weighted uniform simulation method

In this study, the reliability assessment of the system of structure-MR damper in controlling seismic responses of the structure via semi-active control is studied. In this regard, numerical results are presented for two different examples including three and ten-storey shear structures. The limit state functions are defined based on the control responses obtained from the use of MR dampers in the structure stories. By defining several uncertainties in parameters of the structure-MR damper system, probability of failure of structure is obtained using weighted uniform simulation (WUS) method and compared with the results of traditional Monte Carlo Simulation (MCS). Moreover, the most probable point of failure, which cannot be calculated by MCS, is determined for each uncertain event. Graphs of system reliability are derived and interpreted. The results show satisfactory performance of the WUS method in the reliability analysis of smart structures. The results also indicate that reliability of the system is highly dependent on the mass uncertainty of the structure. Moreover, the highest probability of failure takes place when all parameters are considered to be simultaneously uncertain.

An Improved Structural Reliability Analysis Method Based on Local Approximation and Parallelization

The Kriging-based reliability method with a sequential design of experiments (DoE) has been developed in recent years for implicit limit state functions. Such methods include the efficient global reliability analysis, the active learning reliability method combining Kriging and MCS Simulations. In this research, a novel local approximation method based on the most probable failure point (MPFP) is proposed to improve such methods. In this method, the MPFP calculated in the last iteration is the center of the next sampling region. The size of the local region depends on the reliability index obtained by the First Order Reliability Method (FORM) and the deviation distance of the standard deviation. The proposed algorithm, which approximates the limit state function accurately near MPFP rather than in the whole design space, can avoid selecting samples in regions that have negligible effects on the reliability analysis results. In addition, a multi-point enrichment technique is also introduced to select multiple sample points in each iteration. After the high-quality approximation of limit state function is obtained, the failure probability is calculated by the Monte Carlo method. Four numerical examples are used to validate the accuracy and efficiency of the proposed method. Results show that the proposed method is very effective for an accurate evaluation of the failure probability.

Bayesian Optimized Mixture Importance Sampling for High-Sigma Failure Rate Estimation

In many application domains, in particular automotives, guaranteeing a very low failure rate is crucial to meet functional and safety standards. Especially, reliable operation of memory components such as SRAM cells is of essential importance. Due to aggressive technology downscaling, process and runtime variations significantly impact manufacturing yield as well as functionality. For this reason, a thorough memory failure rate assessment is imperative for correct circuit operation and yield improvement. In this regard, Monte Carlo (MC) simulations have been used as the conventional method to estimate the variability induced failure rate of memory components. However, MC methods become infeasible when estimating rare events such as high-sigma failure rates. To this end, importance sampling (IS) methods have been proposed which reduce the number of required simulations substantially. However, existing methods still suffer from inaccuracies and high computational efforts, in particular for high-sigma problems. In this article, we fill this gap by presenting an efficient mixture IS approach based on Bayesian optimization, which deploys a surface model of the objective function to find the most probable failure points. Its advantages include constant complexity independent of the dimensions of design space, the potential to find the global extrema, and the higher trustworthiness of the estimated failure rate by accurately exploring the design space. The approach is evaluated on a 6T-SRAM cell as well as a master–slave latch based on a 28-nm FDSOI process. The results show an improvement in accuracy, resulting in up to $63\times $ better accuracy in estimating failure rates compared to the best state-of-the-art solutions on a 28-nm technology node.

Most Probable Failure Point Update Method for Accurate First-Order Reliability-Based Electromagnetic Designs

A most probable failure point update method is proposed to obtain an accurate reliability-based design of electromagnetic devices or systems in the presence of uncertainties. The first-order reliability method has been recently adopted to solve electromagnetic design problems. However, its result could be very erroneous especially for nonlinear or multi-dimensional performance functions. To overcome the drawback, a three-step computational procedure is additionally executed to ensure prescribed design feasibility at an optimum obtained from the conventional first-order reliability method: failure rate calculation, reliability index update, and most probable point update. A mathematical example and a blushless DC motor design problem are provided to demonstrate numerical accuracy of the proposed method by comparison with the conventional method.

Integration of a Hybrid Method into an Efficient Reliability-Based Design Optimization Technique for Large Structures

The computational burden imposed by the repetitive and lengthy nature of the structural analyses of large structures has lead researches to continuously devise efficient methods for conducting their reliability analysis in addition to powerful methods for solving the design optimization problem. The presence of a considerably large number of random variables in the reliability analyses of large structure is another issue that tends to slow down the design optimization problem. In this paper, a hybrid method that was recently presented by the first author will be integrated into a structural reliability-based design optimization technique for large structures. In this technique, modified concepts of the weighted average simulation method are used to determine the most probable point of failure in a computationally efficient manner. The most probable point of failure is then transferred into the standard normal space, where the reliability index is calculated in closed-form. The firefly algorithm will be used for solving the optimization problem. It will be shown that the integrated approach significantly reduces the computational resources required for solving the design problem. The technique will be tested on a truss bridge. The results will shed light on the efficiency of the proposed technique in solving reliability-based design optimization problems.

An effective first order reliability method based on Barzilai–Borwein step

In nonlinear problems, the Hasofer–Lind–Rackwitz–Fiessler algorithm of the first order reliability method sometimes is puzzled by its non-convergence. A new Hasofer–Lind–Rackwitz–Fiessler algorithm incorporating Barzilai–Borwein step is investigated in this paper to speed up the rate of convergence and performs in a stable manner. The algorithm is essentially established on the basis of the global Barzilai–Borwein gradient method, which is dealt with two stages. The first stage, implemented by the traditional steepest descent method with specific decayed step sizes, prepares a good initial point for the global Barzilai–Borwein gradient algorithm in the second stage, which takes the merit function as the objective to locate the most probable failure point. The efficiency and convergence of the proposed method and some other reliability analysis methods are presented and discussed in details by several numerical examples. It is found that the proposed method is stable and very efficient in the nonlinear problems except those super nonlinear ones, even more accurate than the descent direction method with step sizes following the fixed exponential decay strategy.

AKOIS: An adaptive Kriging oriented importance sampling method for structural system reliability analysis

A major issue in the structural reliability analysis is to determine an accurate estimation result of the failure probability ideally based on a small number of model evaluations. In this regard, the active-learning Kriging based importance sampling method has been received considerable attentions. However, the utility of the most probable failure point (MPP) as the unique sampling center has limited its potential applications for multi-MPP problems. To this end, the paper presents an adaptive Kriging oriented importance sampling (AKOIS) approach. The outer active-learning loop of the AKOIS procedure is used to identify importance sampling centers, whereas its inner-loop is realized based on a rather small subregion centering at the sampling center to gain new training samples. Besides, numerical convergence of local active-learning iterations will immediately trigger another round of outer global search for a new importance sampling center. In this regard, the determined importance sampling centers are able to adaptively cover all branches of the investigated limit-state surface for structural system reliability analysis. Engineering applications of the data-driven importance sampling approach are demonstrated by several system reliability examples in the literature.

Sequential Stochastic Response Surface Method Using Moving Least Squares-Based Sparse Grid Scheme for Efficient Reliability Analysis

The present work demonstrates an efficient method for reliability analysis using sequential development of the stochastic response surface. Here, orthogonal Hermite polynomials are used whose unknown coefficients are evaluated using moving least square technique. To do so, collocation points in the conventional stochastic response surface method (SRSM) are replaced by the sparse grid scheme so as to reduce the number of function evaluations. Moreover, the domain is populated sequentially by the sparse grid based on the outcome of the optimization to find out the most probable failure point. Hence, the support points are generated based on a coupled effect of the optimization for failure region and the sub-grids hierarchy. Continuous and differentiable penalty function is imposed to determine multiple failure points, if any, by repeating the optimization. Once the response surface is developed, reliability analysis is carried out using importance sampling. Five different benchmark examples are presented in this study to validate the performance of the proposed modeling. As the accuracy of the method is established, two reliability-based design examples involving nonlinear finite element (FE) analysis of plates are demonstrated. Numerical study shows the efficiency of the proposed sequential SRSM in terms of accuracy and number of time-exhaustive evaluation of the original performance function, as compared to other methods available in the literature. Based on these results, it may be concluded that the proposed method works satisfactorily for a large class of reliability-based design problems.

Augmented Single Loop Single Vector Algorithm Using Nonlinear Approximations of Constraints in Reliability-Based Design Optimization

Abstract Reliability-based design optimization (RBDO) aims at minimizing a function of probabilistic design variables, given a maximum allowed probability of failure. The most efficient methods available for solving moderately nonlinear problems are single loop single vector (SLSV) algorithms that use a first-order approximation of the probability of failure in order to rewrite the inherently nested structure of the loop into a more efficient single loop algorithm. The research presented in this paper takes off from the fundamental idea of this algorithm. An augmented SLSV algorithm is proposed that increases the rate of convergence by making nonlinear approximations of the constraints. The nonlinear approximations are constructed in the following way: first, the SLSV experiments are performed. The gradient of the performance function is known, as well as an estimate of the most probable failure point (MPP). Then, one extra experiment, a probe point, per performance function is conducted at the first estimate of the MPP. The gradient of each performance function is not updated but the probe point facilitates the use of a natural cubic spline as an approximation of an augmented MPP estimate. The SLSV algorithm using probing (SLSVP) also incorporates a simple and effective move limit (ML) strategy that also minimizes the heuristics needed for initiating the optimization algorithm. The size of the forward finite difference design of experiment (DOE) is scaled proportionally with the change of the ML and so is the relative position of the MPP estimate at the current iteration. Benchmark comparisons against results taken from the literature show that the SLSVP algorithm is more efficient than other established RBDO algorithms and converge in situations where the SLSV algorithm fails.

Improved Nonprobabilistic Global Optimal Solution Method and Its Application in Bridge Reliability Assessment

Utilizing the improved one‐dimensional optimization algorithm conveniently solved the nonprobabilistic reliability index, however, only searching the part of probable failure points. Utilizing the global optimal solution method produced heavy computation, although it was capable of searching all probable failure points. This paper, based on these two methods, proposed an improved nonprobabilistic global optimal solution method. The presented method possessed the advantages of searching all probable failure points and generating less computation, by means of which the values of interval variables were determined based on the monotonicity of performance function to the corresponding variables. Without losing any probable failure points, this method was contributive to reducing root value point equations, lowering the computational complexity, and improving the computational efficiency. The effectiveness and feasibility of the presented method were verified by two examples. The proposed method was also introduced to build the nonprobabilistic reliability evaluation process of existing bridges. Taking the Longganhu bridge for example, the nonprobabilistic reliability index of it was calculated using the presented nonprobabilistic reliability evaluation process. The computed nonprobabilistic reliability index η = 0.737 < 1 indicated that the Longganhu bridge was unreliable and needed to be reinforced. Reinforcement measures were carried out for a hollow slab and bridge deck of the Longganhu bridge, respectively. The reinforced bridge was reevaluated as η = 2.159 > 1. The results showed that the bridge was reliable after reinforcement. The study illustrated the application and feasibility of the improved nonprobabilistic global optimal solution method and nonprobabilistic reliability evaluation process in reliability assessment and reinforcement of existing bridges.

Influence of bone anisotropy on reliability assessment of mini-plate fixation system stabilization in symphysis mandibular fractures: Two studied cases under convalescence period

The reliability analysis is used to in order to measure the stability of the mini-plate fixation system used in the human fractured mandibles after the chirurgical operation. The failure is assumed to take place when the Most Probable failure Point (MPP) is found. In this work, two studied cases of 3-dimensional finite element models are considered for the same fracture situation. A successful fracture healing requires that a number of constraints which are influenced by the loading conditions are fulfilled, and since muscle activity is difficult to evaluate, there is a strong need to introduce loading uncertainties in order to obtain a reliable design. Several categories of critical failure scenarios are considered in this study: The first category of failure scenarios is that of the relative displacement between two fracture surfaces should not exceed a critical threshold to ensure rapid healing. The second category is failure of the mini-plates which in this work is interpreted as when the yield stress within the mini-plates is reached, and the third category of failure scenarios is that of the yield stress in the mandible bone tissues should not be exceeded. Two fractured mandibles are studied under the convalescence period: Case I (a single isotropic bone tissue) and Case II (composite anisotropic bone tissues). During the fixation of the mini-plates, the drilling positions in the two cases can vary when considering a composite bone tissue mandible relative to a homogeneous bone tissue one. To show the effect of the bone anisotropy, an analytical formulation is developed as a helpful technique to analyze the effect of the mini-plate position changes. The bone properties used in the anisotropic case (Case II) are orthotropic. The results show that the reliability indices are very affected when considering the bone anisotropy.

Refined first-order reliability method using cross-entropy optimization method

Generally, the first-order reliability method (FORM) is an efficient and accurate reliability method for problems with linear limit state functions (LSFs). It is showed that the FORM formula may produce inaccurate results when the LSF is defined by mathematical forms introduced as gray function. Thus, the original FORM formula may provide the results with huge errors. In this paper, a probabilistic optimization model as refined FORM (R-FORM) is presented to search most probable failure point (MPP) with the accurate results for gray LSFs. The cross-entropy optimization (CEO) method is utilized to search MPP in proposed R-FORM model. Several reliability problems are applied to illustrate the accuracy of the R-FORM compared to the conventional FORM formula. Results illustrate that the R-FORM provides more accurate results than the FORM for gray performance functions.

Bayesian inference in validation of global MPP for the reliability analysis of composite structures

Realistic analysis of structure failures under uncertainties are due to be associated with the use of probabilistic methods. One of the main problems in structural reliability analysis of composite laminate structures is the possible existence of multiple MPP (Most Probable Failure Point). In this work, we propose a numerical inverse technique for the global MPP search as a function of the anisotropy of laminated composites. Therefore, the analysis considers the maximum loading capability of the composite structure for a prescribed reliability level. This is equivalent to solving a target reliability-based design optimization problem. A Bayesian method to estimate the probability of failure based on Monte Carlo simulation provides the validation of the results. The validation process demonstrates that the proposed methodology is adequate to estimate the probability of failure of the laminated composite structures. Furthermore, the paper outlines and discusses the sensitivity of reliability index under maximum loading variability for angle ply composites.

A surrogate‐based iterative importance sampling method for structural reliability analysis

AbstractIn this paper, a surrogate‐based iterative importance sampling (IS) method is proposed to efficiently evaluate failure probability (FP) of engineering structures. The surrogate is constructed by the training points centered at the most probable failure point (MPP), which is estimated by an approximate algorithm without any additional evaluations of the real limit state function (LSF), and the IS method using the estimated MPP is further obtained; and an iterative procedure combined with the surrogate and the IS method is finally proposed to improve the accuracy of the estimated result for the FP. Three numerical examples are employed to demonstrate the accuracy of the proposed method, and further, an inside flap of an aircraft is employed to demonstrate the practical application of the proposed method in engineering problems. The results show that the reasonable estimated result of the FP within the range of 10−3‐10−5 can be evaluated by the proposed method with only hundreds of the evaluations of the real LSF.

Dimension adaptive finite difference decomposition using multiple sparse grids for stochastic computation

The present study aims to investigate uncertainty quantification followed by reliability analysis of structure with homogeneous non-normal random fields. In stochastic finite element formulation, these continuous fields are discretized by different methods (e.g. Karhunen-Loève Expansion) which transformed it into a set of random variables. However, this discretization often leads to large number of random variables, especially for multiple random fields. With this in view, two different meta-model based approaches are presented in this study using high dimensional model representation (HDMR) for efficient stochastic computation. First, an adaptive multiple finite difference HDMR (AMFD-HDMR) is proposed that decomposes the original performance function into summands of smaller dimensions. These subfunctions are modeled by polynomial chaos expansion (PCE) using moving least square technique which utilizes the benefits of orthogonality of the basis functions and provides adaptive interpolation between the support points. These support points are generated in a sparse grid framework based on the hierarchial tensor products of the sub-grids and the appropriate statistical properties. An iterative scheme is developed with the aim to create new support points in the desired locations such as most probable failure point and/or maxima/minima. Later, a dimension adaptive multiple finite difference HDMR (dAMFD-HDMR) is proposed utilizing sensitivity analysis to further improve the efficiency and accuracy. In the second proposal, an intermittent HDMR formulation is suggested based on the individual and mutual contributions of the significant dimensions. Once the meta-model is built, Monte Carlo simulation is performed over it, thus bypassing the time exhaustive computation of the original performance function. Numerical studies are carried out using composite plate to prove the merits of the proposed algorithms compared to other methods available in the literature.

Global kriging surrogate modeling for general time-variant reliability-based design optimization problems

While design optimization under uncertainty has been widely studied in the last decades, time-variant reliability-based design optimization (t-RBDO) is still an ongoing research field. The sequential and mono-level approaches show a high numerical efficiency. However, this might be to the detriment of accuracy especially in case of nonlinear performance functions and non-unique time-variant most probable failure point (MPP). A better accuracy can be obtained with the coupled approach, but this is in general computationally prohibitive. This work proposes a new t-RBDO method that overcomes the aforementioned limitations. The main idea consists in performing the time-variant reliability analysis on global kriging models that approximate the time-dependent limit state functions. These surrogates are built in an artificial augmented reliability space and an efficient adaptive enrichment strategy is developed that allows calibrating the models simultaneously. The kriging models are consequently only refined in regions that may potentially be visited by the optimizer. It is also proposed to use the same surrogates to find the deterministic design point with no extra computational cost. Using this point to launch the t-RBDO guarantees a fast convergence of the optimization algorithm. The proposed method is demonstrated on problems involving nonlinear limit state functions and non-stationary stochastic processes.

Failure probability of a FinFET-based SRAM cell utilizing the most probable failure point

Application requirements along with the unceasing demand for ever-higher scale of device integration, has driven technology towards an aggressive downscaling of transistor dimensions. This development is confronted with variability challenges, mainly the growing susceptibility to time-zero and time-dependent variations. To model such threats and estimate their impact on a system's operation, the reliability community has focused largely on Monte Carlo-based simulations and methodologies. When assessing yield and failure probability metrics, an essential part of the process is to accurately capture the lower tail of a distribution. Nevertheless, the incapability of widely-used Monte Carlo techniques to achieve such a task has been identified and recently, state-of-the-art methodologies focusing on a Most Probable Failure Point (MPFP) approach have been presented. However, to strictly prove the correctness of such approaches and utilize them on large scale, an examination of the concavity of the space under study is essential. To this end, we develop an MPFP methodology to estimate the failure probability of a FinFET-based SRAM cell, studying the concavity of the Static Noise Margin (SNM) while comparing the results against a Monte Carlo methodology.

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.

Theoretical analysis of non-probabilistic reliability based on interval model

The aim of this paper is to propose a theoretical approach for performing the non-probabilistic reliability analysis of structure. Due to a great deal of uncertainties and limited measured data in engineering practice, the structural uncertain parameters were described as interval variables. The theoretical analysis model was developed by starting from the 2-D plane and 3-D space. In order to avoid the loss of probable failure points, the 2-D plane and 3-D space were respectively divided into two parts and three parts for further analysis. The study pointed out that the probable failure points only existed among extreme points and root points of the limit state function. Furthermore, the low-dimensional analytical scheme was extended to the high-dimensional case. Using the proposed approach, it is easy to find the most probable failure point and to acquire the reliability index through simple comparison directly. A number of equations used for calculating the extreme points and root points were also evaluated. This result was useful to avoid the loss of probable failure points and meaningful for optimizing searches in the research field. Finally, two kinds of examples were presented and compared with the existing computation. The good agreements show that the proposed theoretical analysis approach in the paper is correct. The efforts were conducted to improve the optimization method, to indicate the search direction and path, and to avoid only searching the local optimal solution which would result in missed probable failure points.