A Novel Second-Order Reliability Method (SORM) Using Non-Central or Generalized Chi-Squared Distributions

2012 ◽  
Author(s):  
Ikjin Lee ◽  
David Yoo ◽  
Yoojeong Noh
Keyword(s):  
Reliability Analysis ◽  
Linear Combination ◽  
Quadratic Function ◽  
Failure Surface ◽  
Probability Of Failure ◽  
Second Order ◽  
Cumulative Distribution ◽  
State Function ◽  
Reliability Method ◽  
Chi Squared

This paper proposes a novel second-order reliability method (SORM) using non-central or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point (MPP) in the standard normal U-space, (2) error due to approximating the quadratic function in U-space by a hyperbolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two errors by describing the quadratic failure surface with the linear combination of non-central chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function (PDF) over the linear failure surface and the second approach uses the cumulative distribution function (CDF) of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method (FORM), conventional SORM, and Monte Carlo simulation (MCS) results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

A Novel Second-Order Reliability Method (SORM) Using Noncentral or Generalized Chi-Squared Distributions

2012 ◽  
Vol 134 (10) ◽  
Author(s):  
Ikjin Lee ◽  
Yoojeong Noh ◽  
David Yoo
Keyword(s):  
Reliability Analysis ◽  
Linear Combination ◽  
Quadratic Function ◽  
Failure Surface ◽  
Probability Of Failure ◽  
Second Order ◽  
Cumulative Distribution ◽  
State Function ◽  
Reliability Method ◽  
Chi Squared

This paper proposes a novel second-order reliability method (SORM) using noncentral or general chi-squared distribution to improve the accuracy of reliability analysis in existing SORM. Conventional SORM contains three types of errors: (1) error due to approximating a general nonlinear limit state function by a quadratic function at most probable point in standard normal U-space, (2) error due to approximating the quadratic function in U-space by a parabolic surface, and (3) error due to calculation of the probability of failure after making the previous two approximations. The proposed method contains the first type of error only, which is essential to SORM and thus cannot be improved. However, the proposed method avoids the other two types of errors by describing the quadratic failure surface with the linear combination of noncentral chi-square variables and using the linear combination for the probability of failure estimation. Two approaches for the proposed SORM are suggested in the paper. The first approach directly calculates the probability of failure using numerical integration of the joint probability density function over the linear failure surface, and the second approach uses the cumulative distribution function of the linear failure surface for the calculation of the probability of failure. The proposed method is compared with first-order reliability method, conventional SORM, and Monte Carlo simulation results in terms of accuracy. Since it contains fewer approximations, the proposed method shows more accurate reliability analysis results than existing SORM without sacrificing efficiency.

Second-Order Reliability Method With First-Order Efficiency

2010 ◽  
Author(s):  
Xiaoping Du ◽  
Junfu Zhang
Keyword(s):  
Limit State ◽  
Saddlepoint Approximation ◽  
Quadratic Function ◽  
Second Order ◽  
State Function ◽  
Limit State Function ◽  
First Order ◽  
Reliability Method ◽  
Order Of Magnitude ◽  
Univariate Function

The widely used First Order Reliability Method (FORM) is efficient, but may not be accurate for nonlinear limit-state functions. The Second Order Reliability Method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second order reliability analysis method with first order efficiency. The method first performs the FORM and identifies the Most Probable Point (MPP). Then the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated as a quadratic function, which is created with the gradient information at the MPP and one more point near the MPP. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied for computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2018 ◽  
Vol 140 (3) ◽  
Author(s):  
Dimitrios I. Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer uses a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first- and second-order sensitivity derivatives of the limit state cumulant generating function (CGF), with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are calculated using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. Comparison with Monte Carlo simulation (MCS) shows that MVSOSA is more accurate than mean-value first-order saddlepoint approximation (MVFOSA) and more accurate than mean-value second-order second-moment (MVSOSM) method. Finally, the proposed RBTO-MVSOSA method for minimizing a compliance-based probability of failure is demonstrated using two two-dimensional beam structures under random loading. The density-based topology optimization based on the solid isotropic material with penalization (SIMP) method is utilized.

Application of Partial Safety Factors for Fitness-for-Service Assessment of Pressure Equipment With Local Metal Loss

2017 ◽  
Author(s):  
Takuyo Kaida ◽  
Shinsuke Sakai
Keyword(s):  
Reliability Analysis ◽  
Limit State ◽  
Probability Of Failure ◽  
Pressure Vessels ◽  
Metal Loss ◽  
State Function ◽  
Safety Factors ◽  
Loss Assessment ◽  
Reliability Method ◽  
Partial Safety Factors

Reliability analysis considering data uncertainties can be used to make a rational decision as to whether to run or repair a pressure equipment that contains a flaw. Especially, partial safety factors (PSF) method is one of the most useful reliability analysis procedure and considered in a Level 3 assessment of a crack-like flaw in API 579-1/ASME FFS-1:2016. High Pressure Institute of Japan (HPI) formed a committee to develop a HPI FFS standard including PSF method. To apply the PSF method effectively, the safety factors for each dominant variable should be prepared before the assessment. In this paper, PSF for metal loss assessment of typical pressure vessels are derived based on first order reliability method (FORM). First, a limit state function and stochastic properties of random variables are defined. The properties of a typical pressure vessel are based on actual data of towers in petroleum and petrochemical plants. Second, probability of failure in several cases are studied by Hasofer-Lind method. Finally, PSF’s in each target probability of failure are proposed. HPI published a new technical report, HPIS Z 109 TR:2016, that provide metal loss assessment procedures based on FORM and the proposed PSF’s described in this paper.

scholarly journals Copula-Based Slope Reliability Analysis Using the Failure Domain Defined by theg-Line

2016 ◽  
Vol 2016 ◽  
pp. 1-11 ◽  
Author(s):  
Xiaoliang Xu ◽  
Jianlin Li ◽  
Jiawei Gong ◽  
Huafeng Deng ◽  
Liangpeng Wan
Keyword(s):  
Shear Strength ◽  
Reliability Analysis ◽  
Slope Failure ◽  
Limit State ◽  
Probability Of Failure ◽  
Function Approach ◽  
State Function ◽  
Shear Strength Parameters ◽  
Strength Parameters ◽  
Reliability Method

The estimation of the cross-correlation of shear strength parameters (i.e., cohesion and internal friction angle) and the subsequent determination of the probability of failure have long been challenges in slope reliability analysis. Here, a copula-based approach is proposed to calculate the probability of failure by integrating the copula-based joint probability density function (PDF) on the slope failure domain delimited with theg-line. Here, copulas are used to construct the joint PDF of shear strength parameters with specific marginal distributions and correlation structure. In the paper a failure (limit state) function approach is applied to investigate a system characterized by a homogeneous slope. The results show that the values obtained by using the failure function approach are similar to those calculated by means of conventional methods, such as the first-order reliability method (FORM) and Monte Carlo simulations (MC). In addition, an entropy weight (EW) copula is proposed to address the discrepancies of the results calculated by different copulas to avoid over- or underestimating the slope reliability.

A New Inverse Reliability Analysis Method Using MPP-Based Dimension Reduction Method (DRM)

2007 ◽  
Author(s):  
Ikjin Lee ◽  
Kyung K. Choi ◽  
Liu Du ◽  
David Gorsich
Keyword(s):  
Dimension Reduction ◽  
Reliability Analysis ◽  
Reduction Method ◽  
Probability Of Failure ◽  
Second Order ◽  
Dimension Reduction Method ◽  
Inverse Reliability Analysis ◽  
Highly Nonlinear ◽  
Reliability Method ◽  
Nonlinear Performance

There are two commonly used reliability analysis methods of analytical methods: linear approximation - First Order Reliability Method (FORM), and quadratic approximation - Second Order Reliability Method (SORM), of the performance functions. The reliability analysis using FORM could be acceptable for mildly nonlinear performance functions, whereas the reliability analysis using SORM is usually necessary for highly nonlinear performance functions of multi-variables. Even though the reliability analysis using SORM may be accurate, it is not desirable to use SORM for probability of failure calculation since SORM requires the second-order sensitivities. Moreover, the SORM-based inverse reliability analysis is very difficult to develop. This paper proposes a method that can be used for multi-dimensional highly nonlinear systems to yield very accurate probability of failure calculation without requiring the second order sensitivities. For this purpose, the univariate dimension reduction method (DRM) is used. A three-step computational process is proposed to carry out the inverse reliability analysis: constraint shift, reliability index (β) update, and the most probable point (MPP) approximation method. Using the three steps, a new DRM-based MPP is obtained, which computes the probability of failure of the performance function more accurately than FORM and more efficiently than SORM.

A Second-Order Reliability Method With First-Order Efficiency

2010 ◽  
Vol 132 (10) ◽  
Author(s):  
Junfu Zhang ◽  
Xiaoping Du
Keyword(s):  
Limit State ◽  
Saddlepoint Approximation ◽  
Quadratic Function ◽  
Second Order ◽  
State Function ◽  
Limit State Function ◽  
First Order ◽  
Reliability Method ◽  
Order Of Magnitude ◽  
Univariate Function

The first-order reliability method (FORM) is efficient but may not be accurate for nonlinear limit-state functions. The second-order reliability method (SORM) is more accurate but less efficient. To maintain both high accuracy and efficiency, we propose a new second-order reliability analysis method with first-order efficiency. The method first performs the FORM to identify the most probable point (MPP). Then, the associated limit-state function is decomposed into additive univariate functions at the MPP. Each univariate function is further approximated by a quadratic function. The cumulant generating function of the approximated limit-state function is then available so that saddlepoint approximation can be easily applied in computing the probability of failure. The accuracy of the new method is comparable to that of the SORM, and its efficiency is in the same order of magnitude as the FORM.

A Study on Computational Efficiency Improvement of Novel SORM Using the Convolution Integration

2017 ◽  
Vol 140 (2) ◽  
Author(s):  
Jeong Woo Park ◽  
Ikjin Lee
Keyword(s):  
Linear Combination ◽  
Computational Efficiency ◽  
Integral Method ◽  
Numerical Study ◽  
Probability Of Failure ◽  
Computational Time ◽  
The Novel ◽  
One Dimensional ◽  
Reliability Method ◽  
Chi Squared

This paper proposes to apply the convolution integral method to the novel second-order reliability method (SORM) to further improve its computational efficiency. The novel SORM showed better accuracy in estimating the probability of failure than conventional SORMs by utilizing a linear combination of noncentral or general chi-squared random variables. However, the novel SORM requires significant computational time when integrating the linear combination to calculate the probability of failure. In particular, when the dimension of performance functions is higher than three, the computational time for full integration increases exponentially. To reduce this computational burden for the novel SORM, we propose to obtain the distribution of the linear combination using the convolution and to use the distribution for the probability of failure estimation. Since it converts an N-dimensional full integration into one-dimensional integration, the proposed method is computationally very efficient. Numerical study illustrates that the accuracy of the proposed method is almost the same as the full integral method and Monte Carlo simulation (MCS) with much improved efficiency.

Reliability-Based Topology Optimization Using Mean-Value Second-Order Saddlepoint Approximation

2017 ◽  
Author(s):  
Dimitrios Papadimitriou ◽  
Zissimos P. Mourelatos
Keyword(s):  
Topology Optimization ◽  
Limit State ◽  
Random Variables ◽  
Saddlepoint Approximation ◽  
Probability Of Failure ◽  
Mean Value ◽  
Second Order ◽  
State Function ◽  
Sensitivity Derivatives ◽  
Reliability Method

A reliability-based topology optimization (RBTO) approach is presented using a new mean-value second-order saddlepoint approximation (MVSOSA) method to calculate the probability of failure. The topology optimizer is based on a discrete adjoint formulation. MVSOSA is based on a second-order Taylor expansion of the limit state function at the mean values of the random variables. The first and second-order sensitivity derivatives of the limit state cumulant generating function with respect to the random variables in MVSOSA, are computed using direct-differentiation of the structural equations. Third-order sensitivity derivatives, including the sensitivities of the saddlepoint, are computed using the adjoint approach. The accuracy of the proposed MVSOSA reliability method is demonstrated using a nonlinear mathematical example. The results are compared with the available mean value first-order saddlepoint approximation (MVFOSA) method and Monte Carlo simulation. Finally, the proposed RBTO-MVSOSA method for minimizing compliance-based probability of failure, is demonstrated using two 2D beam structures under random loading.

Inverse Reliability Analysis for Approximated Second-Order Reliability Method Using Hessian Update

2014 ◽  
Author(s):  
Jongmin Lim ◽  
Byungchai Lee ◽  
Ikjin Lee
Keyword(s):  
Reliability Analysis ◽  
Reliability Index ◽  
Numerical Study ◽  
Computational Cost ◽  
Probability Of Failure ◽  
Second Order ◽  
Good Efficiency ◽  
Study Results ◽  
Highly Nonlinear ◽  
Reliability Method

According to order of approximation, there are two types of analytical reliability analysis methods; first-order reliability method and second-order reliability method. Even though FORM gives acceptable accuracy and good efficiency for mildly nonlinear performance functions, SORM is required in order to accurately estimate the probability of failure of highly nonlinear functions due to its large curvature. Despite its necessity, SORM is not commonly used because the calculation of the Hessian is required. To resolve the heavy computational cost in SORM due to the Hessian calculation, a quasi-Newton approach to approximate the Hessian is introduced in this study instead of calculating the Hessian directly. The proposed SORM with the approximated Hessian requires computations only used in FORM leading to very efficient and accurate reliability analysis. The proposed SORM also utilizes the generalized chi-squared distribution in order to achieve better accuracy. Furthermore, an SORM-based inverse reliability method is proposed in this study as well. A reliability index corresponding to the target probability of failure is updated using the proposed SORM. Two approaches in terms of finding more accurate most probable point using the updated reliability index are proposed and compared with existing methods through numerical study. The numerical study results show that the proposed SORM achieves efficiency of FORM and accuracy of SORM.

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