Approximation Of Limit State Function Research Articles

A Novel Sampling Method Based on Normal Search Particle Swarm Optimization for Active Learning Reliability Analysis

In active learning reliability methods, an approximation of limit state function (LSF) with high precision is the key to accurately calculating the failure probability (Pf). However, existing sampling methods cannot guarantee that candidate samples can approach the LSF actively, which lowers the accuracy and stability of the results and causes excess computational effort. In this paper, a novel candidate samples-generating algorithm was proposed, by which a group of evenly distributed candidate points on the predicted LSF of performance function (either the real one or the surrogate model) could be obtained. In the proposed method, determination of LSF is considered as an optimization problem in which the absolute value of performance function was considered as objective function. After this, a normal search particle swarm optimization (NSPSO) was designed to deal with such problems, which consists of a normal search pattern and a multi-strategy framework that ensures the uniform distribution and diversity of the solution that intends to cover the optimal region. Four explicit performance functions and two engineering cases were employed to verify the effectiveness and accuracy of NSPSO sampling method. Four state-of-the-art multi-modal optimization algorithms were used as competitive methods. Analysis results show that the proposed method outperformed all competitive methods and can provide candidate samples that evenly distributed on the LSF.

An efficient extreme value moment method combining adaptive Kriging model for time-variant imprecise reliability analysis

For time-variant imprecise reliability problem with parameterized probability-box (p-box), determining the upper and lower bounds of time-variant failure probability is usually more of concern for engineers. To compute these two bounds, this paper proposes a new extreme value moment method (EVMM) combining adaptive Kriging model. In the proposed method, the adaptive Kriging models are first employed to approximate limit state function (LSF) responses and the predicted values are mapped into the extreme value function (EVF) counterparts. Then, the weighted approach based on sparse grid numerical integration (WA-SGNI) is developed to estimate the statistical moments of the approximated EVF responses. The resulting moments are further used to decouple the calculation of bounds of time-variant failure probability into two deterministic optimization problems. The accuracy and efficiency of the proposed method for time-variant imprecise reliability analysis are demonstrated through three numerical examples with LSFs involving non-Gaussian processes and finite element analysis.

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.

Gaussian Process‐Based Response Surface Method for Slope Reliability Analysis

A new response surface method (RSM) for slope reliability analysis was proposed based on Gaussian process (GP) machine learning technology. The method involves the approximation of limit state function by the trained GP model and estimation of failure probability using the first‐order reliability method (FORM). A small amount of training samples were firstly built by the limited equilibrium method for training the GP model. Then, the implicit limit state function of slope was approximated by the trained GP model. Thus, the implicit limit state function and its derivatives for slope stability analysis were approximated by the GP model with the explicit formulation. Furthermore, an iterative algorithm was presented to improve the precision of approximation of the limit state function at the region near the design point which contributes significantly to the failure probability. Results of four case studies including one nonslope and three slope problems indicate that the proposed method is more efficient to achieve reasonable accuracy for slope reliability analysis than the traditional RSM.

An efficient evidence-based reliability analysis method via piecewise hyperplane approximation of limit state function

For evidence-based reliability analysis, whether a focal element belongs to the failure domain is commonly judged by the corresponding extreme values of a performance function in its response domain. In contrast, in this paper, an efficient method by which the ownership relationship between a focal element and the failure domain is directly determined in uncertain variable domain, is proposed via the piecewise hyperplane approximation of limit state function (LSF). The whole uncertainty domain is divided into several sub uncertainty domains on the defined reference direction. The approximate LSF is constructed by the piecewise hyperplane in each sub uncertainty domain, the belief measure and the plausibility measure of reliability analysis can be directly calculated in uncertainty domain through the approximate piecewise hyperplanes of LSF. The proposed evidence-based reliability analysis method is demonstrated by two numerical examples and two engineering applications.

An improved weighted response surface method based on vector projection sampling

Structural reliability theory stems from the nature of randomness, fuzziness, characteristic and some other uncertainties in the process of engineering structural design, construction and employment. With the rapid development of science technology and industry, many departments have realized the importance of structural reliability problem and its potential economic benefits. Solving the problem of structural reliability has quickly become an important issue in the field of academic research. Because of the complexity of the structure and the harsh working environment which lead to a complex structural reliability problem, the traditional quality analysis method can’t explain the failure problems in practical engineering. How to solve the problem of large-scale complex structural reliability, improve the accuracy and efficiency of reliability analysis method, and further obtain great economic benefits, have become one of the most important exploration areas for the enterprises and scholars at home and abroad. The response surface method, which can replace the implicit limit state function by small amount of computation arises at this time. Through a series of deterministic response surface method, it uses polynomial function to approximate implicit limit state function. By reasonably selecting sites and iteration strategy, it ensures that the polynomial function on the failure probability can converge to the failure probability of the implicit limit state function. Response surface method, with high efficiency of clarity and precision, strong operability, combined with finite element advantage, is a reliability method widely used at present. In the process of the implementation of the response surface method, it uses three key steps of selecting the response surface function forms, obtaining sample points by the experiment design and using the regression fitting model, which has direct impact on the degree of the response surface method approximating limit state function and determines the performance of the response surface method. This is a problem that the response surface method must solve. It is the paper’s original intention to conduct the research work based on the response surface method of structural reliability optimization, improving the above three steps and the efficiency of engineering structure .The paper aims at the implicit limit state function problem, studying the structural reliability analysis of the weighted response surface method, combining the advantage of obtaining better sites by vector projection method and increases weighted coefficient by weighted response surface method. The improved weighted response surface method based on vector projection sampling is proposed. It uses the vector of the gradient projection method to get new design point and sample points, giving the actual limit state function of sample points more weights to construct the quadratic response surface function, updating the iterative response surface function, and solves the problem of the structural reliability of the implicit limit state function. Example analysis shows the characteristics of the proposed method, a steady design point can be found, and the calculation of stability has been improved considerably. The classical response surface method is optimized and reduces the defects that the calculation results are seriously affected by the interpolation coefficient. To some extent, it improves the calculation accuracy and can get relatively better results. Improved weighted response surface method based on vector projection sampling combines the advantages of obtaining test sample points by vector projection and weighted regression method, which is effective, feasible and can operate directly. It extends the application field of the response surface method to some degree and gets better approximate fitting limit state function at design point. It has better stability and robustness and provides new approach and ideas to solve the implicit limit state function.

Efficient Assessment of Structural Reliability in Presence of Random and Fuzzy Uncertainties

This paper presents an efficient uncertainty analysis for estimating the possibility distribution of structural reliability in presence of mixed uncertain variables. The proposed method involves high dimensional model representation for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In this methodology, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods, thus the computational efficiency is accomplished. The proposed method is applicable for structural reliability estimation involving any number of fuzzy and random variables with any kind of distribution.

Comparison of the Function Regression Method and Data Classification Method for Limit State Function Approximation

To reduce the computational burden of the reliability analysis of complex engineering application, approximate method is always used to construct the surrogate model of the implicit limit state function. Since the limit state function is a classifier of the failure domain and safe domain, its approximation can be established by the function regression method and data classification method. In this paper, these two methods are tested to several limit state functions including linear function, highly nonlinear function, high dimensional function, series system and parallel system. Least squares support vector machines are used to construct the surrogate models. A detail comparison of function regression method and data classification method for limit state function approximation is given. The conclusions of this paper can give guidance for the engineers to choose an appropriate approximate method in the engineering applications.

A Moving Kriging Interpolation Response Surface Method for Structural Reliability Analysis

In order to obtain reliable structural design, it is of extreme importance to evaluate the failure probability, safety levels of structure (reliability analysis) and the effect of a change in a variable parameter on structural safety (sensitivity analysis) when uncertainties are considered. With a computationally cheaper approximation of the limit state function, various response surface methods (RSMs) have emerged as a convenient tool to solve this especially for complex problems. However, the traditional RSMs may produce large errors in some conditions especially for those highly non-linear limit state functions. Instead of the traditional least squares approximation, in the present paper, a new RSM is proposed which employs moving Kriging interpolation, based on the axial experimental points selected from the region where the most probable failure point (MPFP) is likely to exist to construct the substitute response surface. The proposed method is illustrated by comparing with the results obtained from first order reliability method (FORM) and other conventional RSMs with reference to specific structural reliability analysis problems. The results show that the proposed method improves the accuracy of the reliability analysis with a reasonable computational cost, and could more quickly approach the exact solution.

Serviceability reliability analysis of prestressed concrete bridges

An effective method for serviceability reliability analysis of prestressed concrete bridges is presented in this paper. The method integrates the advantages of the Artificial Neural Network (ANN) method, First Order Reliability Method (FORM) and the importance sampling updating method. A distinctive feature of the present method is the introduction of an explicit approximate limit state function. The explicit formulation of the approximate limit state function is derived by using the parameters of the developed ANN model. Once the explicit limit state function is obtained, the failure probability can be easily estimated by using a hybrid reliability method consisting of the FORM and the importance sampling updating method. The accuracy and efficiency of the robust method is demonstrated through two numerical examples. As a practical engineering example, the serviceability reliability of a prestressed concrete bridge is illustrated. The obtained results clearly show the applicability and merits of the present method.

High dimensional model representation for structural reliability bounds estimation under mixed uncertainties

This paper presents an efficient uncertain analysis method for estimating the bounds on the reliability of structural systems in the presence of mixed uncertain variables. The proposed method involves high dimensional model representation (HDMR) for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through numerical examples.

MEMBERSHIP FUNCTION OF FAILURE PROBABILITY USING MULTICUT-HIGH DIMENSIONAL MODEL REPRESENTATION

The structural reliability analysis in the presence of mixed uncertain variables demands more computation as the entire configuration fuzzy variables needs to be explored. Moreover the existence of multiple design points deviate the accuracy of results as the optimization algorithms may converge to a local design point by neglecting the main contribution from the global design point. Therefore, in this paper a novel uncertainty analysis method for estimating the membership function of failure probability of structural systems involving multiple design points in the presence of mixed uncertain variables is presented. The proposed method involves Multicut-High Dimensional Model Representation technique for the limit state function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by sample points, as compared to full scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability analysis involving any number of fuzzy and random variables. The accuracy and efficiency of the proposed method is demonstrated through three examples.

Structural Reliability Assessment by Local Approximation of Limit State Functions Using Adaptive Markov Chain Simulation and Support Vector Regression

Abstract: The surrogate model method is widely used in structural reliability analysis to approximate complex limit state functions. Accurate results can only be obtained when the surrogate model for the limit state function is approximated sufficiently close to the failure region. This study develops a novel local approximation method for efficient structural reliability assessment. The adaptive Markov chain simulation is utilized to generate samples in the failure region (the “region of most interest”). The support vector regression technique is then used to obtain an explicit approximation of the original complex limit state function around the region of most interest. Four examples are given to demonstrate the application and efficiencies of the proposed method.

An Efficient Response Surface Method Using Givens Transformation for Structural Reliability Analysis

The response surface method was proposed as a collection of statistical and mathematical techniques that are useful for modeling and analyzing a system which is influenced by several input variables. This method gives an explicit approximation of the implicit limit state function of the structure through a number of deterministic structural analyses. However, the position of the experimental points is very important to improve the accuracy of the evaluation of failure probability. In the paper, the experimental points are obtained by using Givens transformation in such way these experimental points nearly close to limit state function. A Numerical example is presented to demonstrate the improved accuracy and computational efficiency of the proposed method compared to the classical response surface method. As seen from the result of the example, the proposed method leads to a better approximation of the limit state function over a large region of the design space, and the number of experimental points using the proposed method is less than that of classical response surface method.

Multicut‐High Dimensional Model Representation for Structural Reliability Bounds Estimation Under Mixed Uncertainties

Abstract: In reliability analysis of structural systems involving both aleatory and epistemic uncertainties, in conjunction with multiple design points, every configuration of the interval variables is to be explored to determine the bounds on reliability. To reduce the computational cost involved, this article presents a novel uncertain analysis method for estimating the bounds on reliability of structural systems involving multiple design points in the presence of mixed uncertain (both random and fuzzy) variables. The proposed method involves Multicut‐High Dimensional Model Representation (MHDMR) technique for the limit state/performance function approximation, the transformation technique to obtain the contribution of the fuzzy variables to the convolution integral and fast Fourier transform for solving the convolution integral. The limit state function approximation is obtained by linear and quadratic approximations of the first‐order HDMR component functions at the most probable point. In the proposed method, efforts are required in evaluating conditional responses at a selected input determined by the sample points, as compared to full‐scale simulation methods. Therefore, the proposed technique estimates the failure probability accurately with significantly less computational effort compared to the direct Monte Carlo simulation. The methodology developed is applicable for structural reliability estimation involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through four examples involving explicit/implicit performance functions.

Inverse structural reliability analysis under mixed uncertainties using high dimensional model representation and fast Fourier transform

This paper presents a novel solution procedure for inverse reliability problems with implicit response functions without requiring the derivatives of the response functions with respect to the uncertain variables, that can be used to determine the unknown design parameters such that prescribed reliability indices are attained in the presence of mixed uncertain (both random and fuzzy) variables. The proposed computational procedure involves three steps: (i) probability of failure calculation using High Dimensional Model Representation (HDMR) for the limit state/performance function approximation, transformation technique to obtain the contribution of the fuzzy variables to the convolution integral, and fast Fourier transform for solving the convolution integral, (ii) reliability index update, and (iii) most probable point MPP update. The limit state function approximation is obtained by linear and quadratic approximations of the first-order HDMR component functions at most probable point. This is a versatile method that can solve even highly nonlinear problems or the problems with multiple parameters. The methodology developed is applicable for inverse reliability analysis involving any number of fuzzy variables and random variables with any kind of distribution. The accuracy and efficiency of the proposed method is demonstrated through four examples involving explicit/implicit performance functions.

Reliability-based topology optimization using a standard response surface method for three-dimensional structures

In this paper, a reliability-based topology optimization (RBTO) for 3-D structures was performed using bi-directional evolutionary structural optimization (BESO) and the standard response surface method (SRSM). In order to get a stable optimal topology, the most recently-developed filter scheme was implemented with BESO, and SRSM was used to generate an approximate limit state function. These results were compared with the recently announced results of RBTO for 2-D structures, and the differences between the results for the 3-D and 2-D structures were examined. A cantilever beam and an MBB beam were selected as the numerical examples. The comparison showed that the optimal topologies of deterministic topology optimization (DTO) and RBTO for the 2-D and 3-D MBB beams, respectively, are very different. Specifically, the two-support member on the left hand side comes into being along the width for the 3-D case, but not for the 2-D case. This shows that RBTO for 3-D structures should be performed as part of the design process.

An artificial neural network based genetic algorithm for estimating the reliability of long span suspension bridges

An accurate and efficient artificial neural network (ANN) based genetic algorithm (GA) is presented for estimating the reliability of long span suspension bridges. In this method, the training datasets for establishing an ANN model are generated by uniform design method and are distributed uniformly over the entire design space. The explicit formulation of the approximate limit state function is then derived by using the parameters of the developed ANN model. Once the explicit limit state function is obtained, the failure probability can be easily estimated by using an improved GA that introduces new approach for the penalty function and coded method and GA operators. A numerical example involving a detailed computational model of a long span suspension bridge with a main span of 1108 m is presented to demonstrate the applicability and merits of the present method. Finally, several important parameters in the present method are discussed.

Hybrid genetic algorithms for structural reliability analysis

The traditional genetic algorithms (GA) involve step-by-step numerical iterations for searching the minimum reliability index of a structural system, and therefore require a relatively long computation time. In practice the size of a design problem can be very large, the limit state functions are usually implicit in terms of the random variables. When using the traditional genetic algorithms, one can encounter problems with the immense effort required in coding ones own finite element code (or for integration with other commercial finite element software) when using the traditional genetic algorithms. For convenient practical applications of the GA in engineering, two new GA methods, namely, a hybrid GA method consisting of artificial neural network (ANN) and a hybrid GA method consisting of ANN and Monte Carlo simulation with importance sampling are proposed in the present study. A distinctive feature of these proposed methods is the introduction of an explicit approximate limit state function. The explicit formulation of the approximate limit state function is derived by using the parameters of the ANN model. By introducing the derived approximate limit state function, the failure probability can be easily calculated, practically when the limit state functions are not explicitly known. These proposed methods are investigated and their accuracy and efficiency are demonstrated using numerical examples. Finally, some important parameters in these proposed methods are also discussed.