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.

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.

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.

Most Probable Point-Based Approximated Dimension Reduction Method in Reliability-Based Design Optimization

2018 ◽  
Author(s):  
Yongsu Jung ◽  
Hyunkyoo Cho ◽  
Ikjin Lee
Keyword(s):  
Dimension Reduction ◽  
Design Optimization ◽  
Reliability Analysis ◽  
Reduction Method ◽  
Dimension Reduction Method ◽  
Reliability Based Design Optimization ◽  
Method Performance ◽  
Most Probable Point ◽  
Reliability Based Design ◽  
Function Information

The conventional most probable point (MPP)-based dimension reduction method (DRM) and following researches show high accuracy in reliability analysis and thus have been successfully applied to reliability-based design optimization (RBDO). However, improvement in accuracy usually leads to reduction in efficiency. The MPP-based DRM is certainly better from the perspective of accuracy than first-order reliability methods (FORM). However, it requires additional function evaluations which could require heavy computational cost such as finite element analysis (FEA) to improve accuracy of probability of failure estimation. Therefore, in this paper, we propose MPP-based approximated DRM (ADRM) that performs one more approximation at MPP to maintain accuracy of DRM with efficiency of FORM. In the proposed method, performance functions will be approximated in original X-space with simplified bivariate DRM and linear regression using available function information such as gradients obtained during the previous MPP searches. Therefore, evaluation of quadrature points can be replaced by the proposed approximation. In this manner, we eliminate function evaluations at quadrature points for reliability analysis, so that the proposed method requires function evaluations for MPP search only, which is identical with FORM. In RBDO where sequential reliability analyses in different design points are necessary, ADRM becomes more powerful due to accumulated function information, which will lead to more accurate approximation. To further improve efficiency of the proposed method, several techniques, such as local window and adaptive initial point, are proposed as well. Numerical study verifies that the proposed method is as accurate as DRM and as efficient as FORM by utilizing available function information obtained during MPP searches.

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