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.

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.

A Closed-Form Second-Order Reliability Method Using Noncentral Chi-Squared Distributions

2014 ◽  
Vol 136 (10) ◽  
Author(s):  
Rami Mansour ◽  
Mårten Olsson
Keyword(s):  
Closed Form ◽  
Limit State ◽  
Random Variables ◽  
Probability Of Failure ◽  
Closed Form Expression ◽  
Form Expression ◽  
Reliability Method ◽  
Chi Squared ◽  
Rotational Transformation ◽  
State Functions

In the second-order reliability method (SORM), the probability of failure is computed for an arbitrary performance function in arbitrarily distributed random variables. This probability is approximated by the probability of failure computed using a general quadratic fit made at the most probable point (MPP). However, an easy-to-use, accurate, and efficient closed-form expression for the probability content of the general quadratic surface in normalized standard variables has not yet been presented. Instead, the most commonly used SORM approaches start with a relatively complicated rotational transformation. Thereafter, the last row and column of the rotationally transformed Hessian are neglected in the computation of the probability. This is equivalent to approximating the probability content of the general quadratic surface by the probability content of a hyperparabola in a rotationally transformed space. The error made by this approximation may introduce unknown inaccuracies. Furthermore, the most commonly used closed-form expressions have one or more of the following drawbacks: They neither do work well for small curvatures at the MPP and/or large number of random variables nor do they work well for negative or strongly uneven curvatures at the MPP. The expressions may even present singularities. The purpose of this work is to present a simple, efficient, and accurate closed-form expression for the probability of failure, which does not neglect any component of the Hessian and does not necessitate the rotational transformation performed in the most common SORM approaches. Furthermore, when applied to industrial examples where quadratic response surfaces of the real performance functions are used, the proposed formulas can be applied directly to compute the probability of failure without locating the MPP, as opposed to the other first-order reliability method (FORM) and the other SORM approaches. The method is based on an asymptotic expansion of the sum of noncentral chi-squared variables taken from the literature. The two most widely used SORM approaches, an empirical SORM formula as well as FORM, are compared to the proposed method with regards to accuracy and computational efficiency. All methods have also been compared when applied to a wide range of hyperparabolic limit-state functions as well as to general quadratic limit-state functions in the rotationally transformed space, in order to quantify the error made by the approximation of the Hessian indicated above. In general, the presented method was the most accurate for almost all studied curvatures and number of random variables.

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.

One-dimensional numerical study of compressible flow ejector

AIAA Journal ◽  
2002 ◽  
Vol 40 ◽  
pp. 1469-1472
Author(s):  
S. Han ◽  
J. Peddieson
Keyword(s):  
Compressible Flow ◽  
Numerical Study ◽  
One Dimensional

scholarly journals Topological correlators and surface defects from equivariant cohomology

2020 ◽  
Vol 2020 (9) ◽  
Author(s):  
Rodolfo Panerai ◽  
Antonio Pittelli ◽  
Konstantina Polydorou
Keyword(s):  
Quantum Mechanics ◽  
Equivariant Cohomology ◽  
Surface Defects ◽  
Star Product ◽  
Three Dimensional ◽  
Three Dimensions ◽  
Dimensional System ◽  
The Novel ◽  
One Dimensional ◽  
Dual Quantum

Abstract We find a one-dimensional protected subsector of $$ \mathcal{N} $$ N = 4 matter theories on a general class of three-dimensional manifolds. By means of equivariant localization we identify a dual quantum mechanics computing BPS correlators of the original model in three dimensions. Specifically, applying the Atiyah-Bott-Berline-Vergne formula to the original action demonstrates that this localizes on a one-dimensional action with support on the fixed-point submanifold of suitable isometries. We first show that our approach reproduces previous results obtained on S3. Then, we apply it to the novel case of S2× S1 and show that the theory localizes on two noninteracting quantum mechanics with disjoint support. We prove that the BPS operators of such models are naturally associated with a noncom- mutative star product, while their correlation functions are essentially topological. Finally, we couple the three-dimensional theory to general $$ \mathcal{N} $$ N = (2, 2) surface defects and extend the localization computation to capture the full partition function and BPS correlators of the mixed-dimensional system.

Synthesis of New Chalcogenide Materials. The Novel One-Dimensional Semiconductor K4 Ti3 S14

1987 ◽  
Vol 97 ◽  
Author(s):  
Steven A. Sunshine ◽  
Doris Kang ◽  
James A. Ibers
Keyword(s):  
Electrical Conductivity ◽  
Solid State ◽  
Alkali Metal ◽  
The Novel ◽  
Conductivity Measurements ◽  
One Dimensional ◽  
General Utility ◽  
Solid State Materials ◽  
Chalcogenide Materials

ABSTRACTThe use of A2 Q/Q melts (A - alkali metal, Q - S or Se) for the synthesis of new one-dimensional solid-state materials is found to be of general utility and is illustrated here for the synthesis of K4 Ti3 SI4. Reaction of Ti metal with a K2 S/S melt at 375°C for 50 h affords K4 Ti3 SI4. The structure possesses one-dimensional chains of seven and eightcoordinate Ti atoms with each chain isolated from all others by surrounding K atoms. There are six S-S pairs (dave - 2.069(3) Å) so that the compound is one of TiIV and may be described as K4 [Ti3 (S)2 (S2)6]. Electrical conductivity measurements indicate that this material is a semiconductor.

scholarly journals Self-Consistent Solution of the Multi Band Boltzmann, Poisson and Hole-Continuity Equations

VLSI Design ◽  
1998 ◽  
Vol 6 (1-4) ◽  
pp. 257-260
Author(s):  
Surinder P. Singh ◽  
Neil Goldsman ◽  
Isaak D. Mayergoyz
Keyword(s):  
Boltzmann Transport Equation ◽  
The Novel ◽  
Boltzmann Transport ◽  
One Dimensional ◽  
Bipolar Junction Transistor ◽  
Continuity Equations ◽  
The Poisson Equation ◽  
Curvilinear Grid ◽  
Consistent Solution ◽  
Junction Transistor

The Boltzmann transport equation (BTE) for multiple bands is solved by the spherical harmonic approach. The distribution function is obtained for energies greater than 3 eV. The BTE is solved self consistently with the Poisson equation for a one dimensional npn bipolar junction transistor (BJT). The novel features are: the use of boundary fitted curvilinear grid, and Scharfetter Gummel type discretization of the BTE.

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