Yu-ge DONG*,Hai-tao LUb,Ling-ling LI

aSchool of Mechanical and Automotive Engineering,Hefei University of Technology,Hefei 230009,Anhui,China

bSchool of Machinery Engineering,Huaiyin Institute of Technology,Huai'an 223003,Jiangsu,China

cProvince-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability,Hebei University of Technology,Tianjin 300130,China

Reliability sensitivity analysis based on multi-hyperplane combination method

Yu-ge DONGa,*,Hai-tao LUa,b,Ling-ling LIc

aSchool of Mechanical and Automotive Engineering,Hefei University of Technology,Hefei 230009,Anhui,China

bSchool of Machinery Engineering,Huaiyin Institute of Technology,Huai'an 223003,Jiangsu,China

cProvince-Ministry Joint Key Laboratory of Electromagnetic Field and Electrical Apparatus Reliability,Hebei University of Technology,Tianjin 300130,China

For a nonlinear limit state function,the frst-order reliability method(FORM)may cause large errors in the computation of not only the reliability index or failure probability but also the reliability sensitivity.In order to obtain more the accurate results of the reliability sensitivity analysis,a number of hyperplanes are built near the design point by frst-order Tayler series expansion,which replace the known nonlinear limit state hypersurface,and an equivalent computational method is utilized to construct an equivalent hyperplane of the obtained hyperplanes.And the reliability sensitivities can be estimated more accurately by the derived equations based on the equivalent hyperplane.An example shows that the method is effective and feasible.

Reliability sensitivity;Failure probability;Nonlinearity;Multi-hyperplane combination method;Equivalence

1.Introduction

Reliability sensitivity analysis plays an important role in reliability design,reliability-based optimization design and reliability-based robust design.Reliability sensitivities are useful to quantify the distribution parameters such as mean and standard deviation,and represent that the distribution parameters infuence the reliability to some degree.The reliability sensitivities are often expressed as the sensitivity of the computed failure probability to changes in the distribution parameters.

Among the methods available for the parametric sensitivity analysis,the method based on FORM is a fundamental and widely used approach due to the high computational effciency and acceptable accuracy of FORM[1-3].Obviously,FORM is to replace a certain limit state hypersurface with a hyperplane and makes it easy to calculate the failure or safety probability.Because of this,FORM may overestimate or underestimate the reliability when the hypersurface is nonlinear near the design point.The higher the curvature of the limit state hypersurface at the design point is,the bigger the computational error of the reliability is.Just as FORM usually overestimates or underestimates the reliability,it may underestimate or overestimate the degree to which the distribution parameters infuence the reliability.Therefore,the parametric sensitivity analysis based on FORM may be unsatisfactory or unacceptable.

Compared with FORM,SORM(second-order reliability method)can compute the reliability more accurately.But because of the diffculty in computing the reliability,the parametric sensitivity analysis based on SORM was seldom discussed[4].MCSM(Monte Carlo simulation method)can be used for the sensitivity analysis[5-8].The simulatedresults of the sensitivity analysis by MCSM are quite accurate, but one main shortcoming of MCSM is that the computational amount is too large.

MHCM(multi-hyperplane combination method)is an alternative to improve the calculation accuracy of the reliability for a nonlinear limit state function[9-11].In MHCM, some hyperplanes are utilized to replace the limit state hypersurface.Feng[9]frst studied MHCM for obtaining much more accurate computational result of reliability.Mahadevan et al.[10]used the idea of MHCM for a little more complex reliability calculation with multiple extreme points,but the handling method in Ref.[10]is rough and the calculation accuracy of the reliability is not high enough.Shin et al.[11] introduced a progressive importance sampling method based on multi-hyperplane combination and the importance sampling for the reliability analysis of the nonlinear performance function with multiple design points.Lu et al.[12]studied how to obtain some hyperplanes and how to compute reliability easily when MHCM was used.The computed results of MHCM are different when a different number of the hyperplanes and the different hyperplanes are selected.And the different way to calculate the reliability of the system consisting of the obtained hyperplanes may lead to the results with different calculation accuracies.Anyhow,compared to FORM, MHCM can improve the calculation accuracy of the reliability greatly.Compared to MCSM,MHCM requires less computational amount.

As MHCM can get more accurate computational result of the reliability,it can obtain more accurate results of the reliability sensitivity analysis compared to FORM.In this paper, some equations of computing the sensitivities for the calculated failure probability were derived according to MHCM, and an example was given to show that the method of the sensitivity analysis based on MHCM is effective and feasible in improving the calculation accuracy of the sensitivity analysis.

2.Basic models of MHCM

Assuming thatx1,x2,x3,…,xnare basic random variables which are independently subjected to standard normal distribution and x=[x1,x2,x3,…,xn]is a 1×norder matrix,a nonlinear limit state function can be expressed byZ=g(x), and the limit state hypersurface can be expressed byZ=g(x)=0.To obtain a number of hyperplanes to replace the limit state hypersurface,the same number of points on the hypersurface and near the design point is obtained by extrapolation method and one-dimensional iterative search method[12].Then,a number of hyperplanes can be built at the obtained points by the frst-order Tayler series expansion.

In Fig.1,the hyperplane(for two dimensions,namely,two random variables,the hypersurface is reduced to a line)at the design point x0is acquired by the frst-order Tayler series expansion of the nonlinear limit state function,which is called as a main hyperplane,denoted byZl0=0.The other hyperplanes at the above obtained points except x0are called as subhyperplanes,denoted byZli=0(i=1,2,…,k).

Fig.1.Reliability computated by MHCM.

As shown in Fig.1,the dotted lines express the course of equivalent calculation,and each dotted line is expressed as a main equivalent approximate line by an equivalent calculation. The last main equivalent approximate line is expressed by a thick solid line.

The safety domainDSiof each hyperplane(main hyperplane or sub-hyperplanes)can be denoted by

Because the limit state functionZliis linear,the safety probabilityPSiin which the values of x are located inDSican be given accurately by

where Φ（·）is the accumulative integral function of standard normal distribution;and βiis the reliability index of the linear limit state functionZli.

In Fig.1(a),when the original nonlinear failure domain is a local convex set,the original nonlinear safety domainDScan be expressed approximately by the set union of allDSias follows

四是查處違法違規(guī)違紀(jì)行為。在項目采購人自查自糾中，對 “零報告”和“隱瞞、遲報、謊報”的，列為重點核查對象，一經(jīng)查實，將對采購人及當(dāng)事人進(jìn)行嚴(yán)肅追責(zé)。對于查處結(jié)果，發(fā)現(xiàn)存在一般性違規(guī)違紀(jì)問題的，督促項目采購主管部門和項目采購人整改到位；存在嚴(yán)重違法違規(guī)問題的，將按照法律法規(guī)和相關(guān)文件要求進(jìn)行處理，并對當(dāng)事人進(jìn)行嚴(yán)肅追責(zé)。同時，曝光一批群眾反映強(qiáng)烈、情節(jié)惡劣的違法違規(guī)違紀(jì)行為。對涉嫌違紀(jì)、職務(wù)違法、職務(wù)犯罪的，移送紀(jì)委監(jiān)委處理；涉嫌其他犯罪的，依法移交公安等司法機(jī)關(guān)查處。

According to the basic model of system reliability,the system consisting of the above hyperplanes is a parallel one. The safety probabilityPSin which the values of x are located inDScan be given by

在英語教學(xué)活動中，很多教師都切身感受到，學(xué)生們對英語精讀課缺乏積極學(xué)習(xí)的熱情，有的學(xué)生甚至對英語精讀課感到“厭煩”，有些學(xué)生感覺他們從精讀課上沒學(xué)到更多東西。這種現(xiàn)象，主要原因在于學(xué)生們沒有掌握學(xué)習(xí)要領(lǐng)，缺乏精讀學(xué)習(xí)的技巧，這就需要精讀課程的教師，要深入研究技巧的培育，不僅僅授學(xué)生以魚，更要授之以漁。

It is a little complicated to calculatePSdirectly using Eq. (4),because the operation of set union should be transformed into that of set intersection.Based on the duality of system reliability,when the original nonlinear failure domain is a convex set,the nonlinear failure domain can also be expressed by set intersection of failure domainDFiof each hyperplane,as shown in Fig.1(a).DFiis written as It is well known that not only the failure domain but also the safety domain of each hyperplane is a linear domain.The failure probabilityPFiin which the values of x are located in

DFican be obtained The original nonlinear failure domainDFcan be expressed approximately by the set intersection of allDFias follows The failure probabilityPFin which the values of x are located inDFcan be given by

Assuming thatDis the full probability space,we haveDF∪DS=D,DF∩DS=φ,andPF+PS=1.So,PForPScan be computed to obtain the reliability.From the above discussion,it is known that,when the original nonlinear failure domain is the convex set,it is much more easy to compute the failure probabilityPFof the original nonlinear limit state function,that is to say,to compute the joint probability of the failure domains of the obtained hyperplanes.

Accordingly,as shown in Fig.1(b),when the original nonlinear safety domain is a local convex set,it is easy to compute the safety probability of the original nonlinear safety domain by using the safety domain of each hyperplane.The safety domain is replaced approximately by the set intersection of the safety domains of the above obtained hyperplanes, namelyPScan be obtained as

再比如《黃鶴樓送別》描寫李白站在江邊目送孤帆遠(yuǎn)去，可不可以添寫李白的內(nèi)心獨白？答案是畫蛇添足。老師可以告訴學(xué)生這叫“無聲勝有聲”。有時候一個眼神、一個姿勢可以表達(dá)無盡情誼，語言反倒顯得蒼白。

In fact,the computation of the safety probability is transformed into that of series system consisting of the above hyperplanes.

Therefore,computingPForPSdepends on whether the failure domain or the safety domain is a convex domain.It is much more easy to calculate the probability of a convex domain.

無論是PBL教學(xué)法、TBL教學(xué)法、虛擬現(xiàn)實教學(xué)法還是標(biāo)準(zhǔn)化病人教學(xué)法，都有著優(yōu)勢與不足。但是即使教學(xué)模式產(chǎn)生了巨大變化，它的根本目的也不會改變，就是讓學(xué)生在最少的時間里最大程度的掌握醫(yī)學(xué)知識和臨床實踐技能。為了達(dá)到這種目的，教師在課堂上也可以嘗試將幾種教學(xué)方法相結(jié)合，實現(xiàn)優(yōu)勢互補(bǔ)。

If neither the failure domain nor the safety domain of the original nonlinear limit state function is a convex domain,the system consisting of the above hyperplanes is neither a parallel one nor a series one,and it is much more complicated to compute the system reliability.In the paper,the case is not discussed for the time being.

3.An algorithm of MHCM

It is not diffcult to calculatePFiorPSiof each hyperplane using Eq.(6)or Eq.(2),but it is comparatively diffcult to compute the joint probabilityPForPSof all the obtained hyperplanes by using Eq.(8)or Eq.(10).In the paper,the failure or safety probability of the system consisting of the obtained hyperplanes is gotten by a method,called as a sequential equivalent computing algorithm[12].

Afterk+1 hyperplanes are acquired,they are arranged in the increasing order of the distances between the origin and the obtained points.The arranged hyperplanes areZl0=0,Zl1=0,…,Zlk=0,whereZl0=0 is the obtained main hyperplane,andZl1=0,…,Zlk=0 are the obtained subhyperplanes.

整整三個月，阿東沒有出門，他在家里讀書。老巴很高興，他喜歡兒子這樣。就算沒有賺錢，但讀書畢竟是正經(jīng)事。而且，心里有大志向的人，才會這樣去讀書。

Theith(i=2,3,…,k)equivalence hyperplane=0 is gotten using the(i-1)th equivalent hyperplane=0 and theith sub-hyperplaneZli=0.Therefore,thekth equivalence hyperplane or the last equivalence hyperplane=0 is gotten by using=0 and the last sub-hyperplaneZlk=0.

The equivalent failure probability of a parallel system consisting of two hyperplanes is given by

The equivalent reliability of a series system consisting of two hyperplanes is given by

在傳播傳統(tǒng)體育文化的過程中，大眾媒體的作用仍然極為關(guān)鍵。在我國電視媒體普遍具有一定的公益性，其中中央電視臺的報道視角，更加傾向于傳統(tǒng)的民族文化。因此在傳播民族體育文化的過程中，可通過中央電視臺第五頻道，打造民族體育文化的傳播平臺，例如，該頻道可設(shè)置賽龍舟、民間武術(shù)以及舞獅等體育運動的專題節(jié)目。通過央視媒體的傳播，傳統(tǒng)運動將得到更好的推廣。再有我國的地方媒體，也應(yīng)當(dāng)對本地區(qū)的特色體育項目進(jìn)行集中報道。例如，在對民間武術(shù)的報道過程中，地方媒體可突出武術(shù)派別的地域差異，通過這樣的宣傳，民間愛好者的鄉(xiāng)土情結(jié)將得到激發(fā)，對于該項賽事的關(guān)注度也將得到提升。

where Φ-1（·）is the inverse function of accumulative integral function of standard normal distribution.

Theith equivalent reliability index is

（三）教學(xué)形式單一枯燥。教學(xué)形式的好壞直接關(guān)系到學(xué)生的學(xué)習(xí)興趣，以及高中教學(xué)的創(chuàng)新性，進(jìn)而對學(xué)生的學(xué)習(xí)情況有所影響。我國當(dāng)前的高中音樂教學(xué)過程中，存在著明顯的教學(xué)形式單一枯燥等現(xiàn)象，大多數(shù)音樂教師在高中音樂的教學(xué)過程中，多以音樂的教學(xué)為主，而忽視了培養(yǎng)學(xué)生的音樂鑒賞能力以及鑒賞水平。單一枯燥的教學(xué)形式大大削弱了高中學(xué)生們的學(xué)習(xí)積極性，為新課改形勢下高中音樂的教學(xué)帶來了阻礙。

According to the sequential equivalent computing algorithm,the frst equivalence computation is obtained by the main hyperplaneZl0=0 and the sub-hyperplaneZl1=0. Therefore,the frst equivalent hyperplane=0 can be obtained.The equivalent conditions are as follow:(1)the obtained equivalenthyperplane is parallelto the main hyperplane;and(2)the reliability of the equivalent hyperplane is equal to that of system consisting of the two hyperplanes.

where αeis a 1×norder matrix,which is the same as α0of the main hyperplane,because=0 is parallel to the main hyperplaneZl0=0.αeis denoted by

Obviously,it is easy to know that

The process of equivalence is shown in Fig.1(a).

Especially,the comprehensive failure probabilityof the system consisting of the obtained hyperplanesZl0=0,Zl1=0,…,Zlk=0 can be estimated by

The failure probabilityPFof the original limit state function can be estimated byPF≈And the reliability index β can be given by β=Φ-1（1-PF）.

Accordingly,as shown in Fig.1(b),for the convex safety domain,the method of obtaining any equivalent hyperplane is as follows.

Theith equivalent reliability indexis

Theith equivalent hyperplane=0 is

Theith equivalent hyperplane=0 can also be expressed by Eq.(13).

研究指出，抑郁癥具有一定的遺傳因素，可能與基因有關(guān)，也可能與血液有關(guān)，抑郁癥家族人員患抑郁癥的比例要比其他家庭高10倍以上，血緣關(guān)系越近，抑郁癥的發(fā)病率越高。筆者在近幾年的工作中發(fā)現(xiàn)，重度抑郁的學(xué)生多來自抑郁癥家族遺傳。

PSandPFof the original limit state function can be estimated by

4.Sensitivity analysis using MHCM

From the section above,it is known that the fnal equivalent hyperplane can be expressed by

When the random variables are not standard normal variables,they should be transformed into standard normal random variables.Assuming thata random variableyi（i=1，2，3，…，n）is a normal variable,and its mean value and standard deviation are μiand σi,respectively.Let two 1×norder matrixes be

And,σ is ann×norder diagonal matrix,of which the diagonal elements are σ1,σ2,σ3,…,σnrespectively.

where Φ（·）is the probability density function of the standard normal distribution.

Substituting Eq.(24)into Eq.(21),we have

Eq.(21)can be rewritten as

Based on Eq.(27),μiand σican be used to obtain the equivalent mean valuethe standard deviationthe reliability indexand the failure probabilityof the equivalent hyperplane

“為了滿足工業(yè)建設(shè)的迫切需要，20世紀(jì)50年代要將各種電壓等級從最高22千伏統(tǒng)一升至35千伏，每天忙得頭疼;六七十年代隨著時代發(fā)展需要，又逐步升到110千伏和220千伏，只用了十來年甚至就是幾年時間，且昆明電網(wǎng)一直保持著西南甚至全國高海拔地區(qū)，最早設(shè)計投運當(dāng)時國內(nèi)最高電壓等級的紀(jì)錄?！被貞浧鹦轮袊闪⒁詠矸旄驳氐淖兓?，老專家覃國成對許多細(xì)節(jié)記憶猶新，很是自豪。

本文詳細(xì)闡述并實現(xiàn)了視差貼圖技術(shù)，該技術(shù)在運動視差、自遮擋和表面自陰影方面有著更好的表達(dá)，使得繪制的物體看起來更加逼真。其中，著重描述了計算高度場與觀察射線交點的算法以及計算表面自陰影的算法。

From Eqs.(15)and(26),we know that the sensitivities of μiand σi（i=1，2，3，…，n）for the computed failure probabilitycan be gotten by

Eq.(24)can be introduced.

Referring to Fig.1,using Eqs.(32)and(33)can come to the conclusion that the absolute values of the sensitivities computed based on MHCM are usually smaller than those computed based on FORM when the failure domain is a convex one.The reason is that the reliability index computed by using MHCM is bigger than that computed by using FORM.When the safety domain is a convex one,the opposite conclusion is obtained.

蜂窩汽封與其他各種形式的“硬齒”汽封相比，它的汽封齒最薄，硬度也最低，因此碰磨時不會象其他“硬齒”汽封一樣在轉(zhuǎn)子表面留下顯著的磨痕。同時，蜂窩汽封與其他“硬齒”汽封不同，在與轉(zhuǎn)子碰磨時象不會產(chǎn)生“刀片”切割現(xiàn)象，這是它不會磨傷轉(zhuǎn)子的主要原因。梳齒類汽封的每一根汽封齒，在與轉(zhuǎn)子碰磨時猶如車刀，很容易在轉(zhuǎn)軸或覆環(huán)表面割出溝槽。此外，蜂窩汽封由于它曲折的網(wǎng)狀結(jié)構(gòu)使其表面面積大大增加，又極大地提高了它的耐磨性，因此蜂窩汽封既耐磨又不易磨傷轉(zhuǎn)子是它突出的一個特點。

5.An example

A pull rod with equal diameter is subjected to a constant loadF=1.5e+5N.The mean values and standard deviations of the diameterdand yield limitrare μd=32 mm, σd=3 mm,and μr=295 MPa,σr=25 MPa,respectively. Let's compute the failure probability and parametric sensitivities of the pull rod.

For the convex failure domain,the method of obtaining any equivalent hyperplane is as follows.

Letx1,x2～N(0,1),we have

Table 1Parameters of lines.

Substituting Eq.(35)into Eq.(34)yields

FORM is used to obtain the reliability index β=2.044 659 4 and the failure probabilityPF=0.020 444 2. The limit state function expressed by Eq.(36)is nonlinear so that the failure and safety domains are nonlinear,too.Because of this,the failure probability computed using FORM may be not accurate enough.It cannot ensure that the sensitivities of FORM failure probability are accurate enough.

In this reliability problem,the most probable failure domain can be thought to be only located near the design point,and the safety domain near the design point is a convex one.Clearly,forn=2,the hyperplanes are reduced to the lines.In order to illustrate the effectiveness of the method proposed in the paper,one main lineZl0=0 and four sub-linesZl1=0,Zl2=0,Zl3=0,Zl4=0 are gotten to replace the limit state curve.It is not diffcult to know that the fve lines consist of a series system.

According to the fve lines,four corresponding reliability indexescanbecomputedbyusingofEqs.(17)and(18).Also,four equivalent lines can be obtained by using Eq.(13),and they are parallel to the main line.The parameters σ and β of the fve obtained lines and four equivalent lines are listed in Table 1.

It is known that the original limit state curve is replaced equivalently by the last equivalent lineZe4=0.The computational results of β,PFand the errors ofPFare given in Table 2.The simulation number of MCSM is 107

It can be seen from Table 2 that β andPFcomputed by MHCM is very close to those obtained by MCSM.Compared with MCSM,the error ofPFcomputed by MHCM or the last equivalent line is only-0.083%.The error is less than that computed by SORM,and far less than that computed by FORM.

Table 2Comparisons of failure probabilities.

Table 3Comparisons of sensitivities.

In the example,because FORM is to replace the convex nonlinear safety domain with a linear safety domain which is determined by a tangent line of the limit state curve through the design point,the actually calculated safety domain of FORM is far greater than safety domain of the original limit state function,and FORM overestimates reliability.So,the safety probability computed by FORM is far bigger than the truesafety probability.That is,thefailureprobability computed by FORM is far smaller than the true failure probability(see the last row of Table 2).It can be seen from Fig.1(b)that the safety domain calculated by MHCM is a little greater than the original safety domain,so the failure probability computed by MHCM is a little smaller than the true failure probability.

MHCM can be used to obtain more accurate computational results of not only the failure probability but also sensitivity analysis.The computed sensitivities and the absolute value errors of the computed sensitivities are listed in Table 3.The simulation number of MCSM is also 107.

From Table 3,some conclusions can be drawn as follows.

(1)The absolute values of the sensitivities by MHCM are bigger than those by FORM.

對套管瓷套表面受力情況進(jìn)行分析，在變壓器抽真空期間(在真空度為4mbar以下的狀態(tài)下)，計算是按照變壓器內(nèi)部為絕對真空進(jìn)行計算，油箱承受101Kpa(約為1個大氣壓)的內(nèi)外壓差，變壓器油箱的材料為Q235B的鋼板，厚度約為20mm厚，利用力學(xué)分析中常用的方法，采用PRO/engineer Wildfire建立的模型，結(jié)構(gòu)強(qiáng)度分析采用ANSYS軟件，在進(jìn)行數(shù)值分析時，部分單元格采用ANSYS的Mesh模型結(jié)構(gòu)和計算機(jī)硬件系統(tǒng)設(shè)定合適的單元劃分。

(2)Compared with the sensitivities computed by FORM,the sensitivities computed by MHCM are closer to the sensitivities simulated by MCSM

(3)Although FORM overestimates the reliability(see Table 2, the reliability index computed by using FORM is bigger than the true value,or the failure probability computed by using FORM is smaller the true value),it underestimates the degree to which the distribution parameters infuence the reliability.

(4)The absolute values of the sensitivities computed by MHCM may be bigger or smaller than the ones simulated by MCSM.The reason is that the algorithm of MHCM in the paper may cause the errors that can not be easy to know and control.

(5)The sensitivities of the parameters μd,σdand μrcomputed by MHCM are much more easier to be close to the sensitivities estimated by MCSM.In the example,the fve lines can be used to make the failure probability relatively accurate and the sensitivities are close to those by MCSM, but it can not make all the sensitivities accurate enough.

6.Conclusions

If the failure probability of a nonlinear limit state function cannot be computed accurately by a method,the sensitivitiesfor the failure probability cannot be computed accurately by the method either.In the case of the nonlinear limit state function,using FORM to analyze the sensitivities almost certainly underestimates or overestimates the degree to which the parameters of random variables infuence the reliability. By replacing the original limit state hypersurface with some hyperplanes,MHCM can improve the calculation accuracy of the reliability and the sensitivities,and makes all the computational results closer to their corresponding true values.

To obtain much more accurate computational results of the reliability and the sensitivities,more hyperplanes may be selected if necessary.In practical applications,not so many hyperplanes are required to get enough accurate results of computing the reliability and the sensitivities.For a certain nonlinear limit state function with single design point,2n+1 hyperplanes are a good choice.It is undoubted that how to select the number of hyperplanes and how to satisfy the requirement of the given accuracy in calculating the reliability and the sensitivities should be made in the further research.

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Received 15 October 2013;revised 7 July 2014;accepted 30 July 2014 Available online 4 September 2014

*Corresponding author.

E-mail address:dyghfut@163.com(Y.G.DONG).

Peer review under responsibility of China Ordnance Society.

http://dx.doi.org/10.1016/j.dt.2014.07.010

2214-9147/Copyright?2014,China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.

Copyright?2014,China Ordnance Society.Production and hosting by Elsevier B.V.All rights reserved.