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Adaptive radial-based importance sampling method for structural reliability

Adaptive radial-based importance sampling method

for structural reliability

Frank Grooteman

National Aerospace Laboratory NLR,Anthony Fokkerweg 2,1059CM Amsterdam,The Netherlands

Received 2August 2007;received in revised form 5October 2007;accepted 10October 2007

Available online 3December 2007

Abstract

In this paper an adaptive radial-based importance sampling (ARBIS)method is presented.The radial-based importance sampling (RBIS)method,excluding a b -sphere from the sampling domain,is extended with an e?cient adaptive scheme to determine the optimal radius b of the sphere.The adaptive scheme is based on directional simulation.The underlying basic methods are presented brie?y.Several numerical examples demonstrate the e?ciency,accuracy and robustness of the scheme.As such,the ARBIS method can be applied as a black-box and is of particular interest in applications with a low probability of failure,for example in structural reliability,in combination with a small number of stochastic variables.ó2007Elsevier Ltd.All rights reserved.

Keywords:Importance sampling;Monte-Carlo simulation;Structural reliability;Failure probability;Adaptive

1.Introduction

The evaluation of the failure probability is a basic problem in structural reliability analyses.The failure probability can be formulated as:

p f ?P f G ex T60g ?Z

G ex T60

f ex Td x e1T

where x represents the vector of stochastic variables of the reliability problem and f (x )the joint probability density function in X -space.G (x )is the failure or limit-state function,de?ning a safe state when G >0and a failure state when G <0.The hyper-surface separating the safe from the failure domain G =0is called the limit-state.The integral represents the volume of the joint probability density function located in the failure domain.

In the past decades many methods have been presented to solve this integral equation,such as sampling methods based on Monte-Carlo simulation (MCS)and directional simulation (DS)[1,2]and methods based

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Structural Safety 30(2008)

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STRUCTURAL

SAFETY

534 F.Grooteman/Structural Safety30(2008)533–542

on an analytical solution of the integral equation:?rst-order reliability method(FORM)and second-order reliability method(SORM)[3].

FORM and SORM approximate the limit-state with,respectively,a?rst-order or an incomplete second-order function.Furthermore,the underlying solution method requires the solution of an optimization prob-lem to?nd the smallest distance to the limit-state.FORM,and to a lesser extent SORM,are often very e?-cient.However,neither method is robust in the case of a complex limit-state,such as a highly non-linear failure function,multiple failure points or a combination of failure functions(serial and parallel systems). An example of a series system having multiple design points is given in Fig.1.In general,the accuracy of the solution is unknown,because either narrow con?dence bounds cannot be obtained or they require an extra computational e?ort(e.g.importance sampling).

On the other hand,MCS and DS are very ine?cient compared with FORM and SORM,especially for small probability values.Nevertheless,convergence to the exact solution is guaranteed for an increasing num-ber of simulations,and con?dence bounds on the solution are available in the case of a?nite number of sim-ulations.Furthermore,these methods are very robust in the sense that they can handle complex limit-states.

Various methods have been presented to improve the e?ciency of the two basic sample methods(MCS and DS):for example[4–8],referred to as importance sampling techniques.The basic idea is to concentrate sam-pling near the most important part(s)of the limit-state(s),that is points on G(x)=0located closest to the origin in U-space.A widely applied approach is to shift the sampling centre from the origin to the design point.Often a FORM analysis,having the mentioned disadvantage,is applied?rst to obtain knowledge about the design point.An alternative strategy is to gather knowledge about the failure domain and thus limit-state(s)during sampling and use this knowledge to guide the sample domain towards the most important regions.This is called an adaptive method,e.g.[9].

An importance sampling method originally proposed by Harbitz[6],referred to as the radial-based impor-tance sampling(RBIS)method,is to exclude an n-dimensional sphere called‘‘b-sphere’’from the safe part of the sampling domain.The remaining sampling domain is restricted to values outside the sphere located in the tail part of the joint probability density function.In principle no knowledge about the location of the design

point(s)is required.The method converges to the exact solution provided the sphere is located in the safe domain,which can be easily checked during sampling.However,the optimal choice is a sphere that touches the limit-state,maximising the excluded region.Hence the optimal sphere radius b is the smallest distance to the limit-state given by the most probable(design)point(MPP).

In this paper a very e?cient,accurate and robust adaptive scheme is presented to determine the optimal sphere radius.Because of these characteristics,the resulting method can be applied as a black-box and is for most structural reliability applications much more e?cient than crude Monte-Carlo method.

2.Adaptive Radial-Based Importance Sampling(ARBIS)

A set of dependent non-normal stochastic variables x can always be transformed to a set of independent standard normal variables u,called the U-space,by applying appropriate transformations[10–12].The remainder of the paper is therefore restricted to the U-space.Before presenting the adaptive scheme the ori-ginal idea of Harbitz[6]is brie?y presented.

2.1.Radial-based Importance Sampling(RBIS)

The method of Harbitz[6]is based on a simple but e?ective importance sampling method,excluding a b-sphere from the sample domain,Fig.1.The sphere has to be located inside the safe domain.The optimal radius b is equal to the distance to the Most Probable Point,which is the point on the failure surface (limit-state)that is closest to the origin.

The probability content of the excluded sphere is given by:

p?P fj U j6b g?P fj U j26b2g?v2

eb2Te2T

where v2

n is the chi-square distribution function with n degrees of freedom equal to the number of stochastic

variables.

The probability integral of Eq.(1)can be rewritten in terms of a conditional probability,yielding:

?P f G60jj U j>b g P fj U j>b ge3TThe?rst term can easily be obtained by Monte-Carlo sampling outside the sphere.The second term is given by Eq.(2),thus yielding:

p f ?

N fail

N sim

e1àv2

eb2TTe4T

In a crude MCS most sampling points would be located inside the sphere.Disregarding this part of the domain can save a huge amount of samples.Therefore,this method can be much more e?cient than MCS, especially for small probabilities of failure occurring in structural reliability analyses.The method has similar characteristics as MCS,such as convergence to the exact solution(accuracy)and the capability to handle com-plex limit-state(s)(robustness).

2.2.Adaptive scheme

The RBIS method proposed by Harbitz assumes that the MPP or design point is known.At the start of a probabilistic analysis,no information about the limit-state is available.The unknown MPP has to be deter-mined?rst.For this a FORM analysis can be used,which is e?cient but not robust.The latter makes it less suitable for practical applications.An alternative adaptive scheme is presented here that is robust,e?cient and guarantees an optimal radius b(accurate).The basic steps are depicted in the?ow chart.The dashed blocks in the?ow chart represent the new adaptive part.

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Basic steps of ARBIS method

An initial value of the radius b (b 0in Fig.2)is chosen such that the excluded sphere is located in the failure domain,contrary to the safe domain requirement of the previous section.This can be achieved by selecting an initial b that results in a low probability p 0of the sample domain outside the sphere.With Eq.(2)this yields:

b ????????????????????????

v à2n e1àp 0T

q e5T

The initial value of b opt is set to a very high value representing the unknown MPP.Next,the Monte-Carlo

method is initialised by selecting a start seed for the random number generator used to generate the sample points in U -space.For points outside the sphere (j U j >b )the limit-state function is evaluated (LSFE)and the result (failure or safe)is stored.If the sample point is located in the failure domain (dot in Fig.2),a line-search (see Section 2.3)is performed in this direction to determine the point on the limit-state (see

Fig.2),usually requiring two-to-three G -function evaluations.The resulting distance ~b

opt is a ?rst approxima-tion of the distance to the MPP and is used to determine a new radius b (b 1in Fig.2)of the sphere.This new

radius is selected somewhat smaller than ~b

opt according to:536 F.Grooteman /Structural Safety 30(2008)533–542

p 0?1àv 2n e~b 2opt T

b ????????????????????????????????v à2n

1àp 0

p step

!v u u t e6T

in which 1àp step multiplied by 100denotes the percentage of samples located in the sample domain between

the spheres b and ~b

opt ,indicated by the dashed circles in Fig.2.Only a new failure point located in this part of the domain triggers a new line-search,ensuring that a limited number (two-to-seven)of line-searches are per-formed to converge to the MPP.This is important,because all the points evaluated in a line-search cannot be added to the Monte-Carlo set and are therefore extra points reducing e?ciency.After the line-search,the Monte-Carlo simulation is restarted using the same seed for the random number generator.In this way,the same set of sample points is regenerated and the information stored for points evaluated in a previous Monte-Carlo cycle is re-used.

The value of p step should be selected close to 1,minimising the sample domain between b and ~b

opt .This prevents unnecessary sampling after the MPP has been located,since all samples in this domain are redundant.However,a value of p step close to 1can result in locking of the adaptive part of the algorithm,thereby pro-ducing erroneous results.In that case,no sample point is obtained in the failure region between both spheres before convergence (see Section 4)has been reached.A value of p step =0.8has proven to be a good choice.The adaptive approach is robust and always converges to the MPP for any initial b -value,even if the ini-tially selected b value is too small,i.e.in the safe domain away from the MPP.If the sphere is far from the MPP,this can result in a large number of simulations before the ?rst failure point is found,because most sam-ple points are located close to the sphere.This reduces the e?ciency of the algorithm,because all sample points with a radius less than the ?nal b opt are not part of the ?nal sample set.The e?ciency is then vastly improved by performing a ?rst line-search in a direction with negative G -gradient,i.e.the sample point has a lower G -value than the origin,instead of postponing the line-search until a failure point has been found.A small initial p 0,which results in an initial sphere located far in the failure domain,is the best choice.In general,this quickly results in a ?rst failure point.In structural reliability the probability of failure is usually small (<10à5).Selection of an initial p 0of 10à6,denoting the probability content outside the sphere including the part in the safe domain,generally su?ces.

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538 F.Grooteman/Structural Safety30(2008)533–542

2.3.Line-search scheme

If a failure point(Fig.2)is found with a distance to the origin that is less than the current minimum dis-tance~b opt,a line-search is performed in that direction to locate the point on the limit-state that is an improved estimate of the MPP.The procedure is one-dimensional and schematically depicted in Fig.3.The LSF-value at the origin is determine once at the start of ARBIS(point0in Fig.3)and is used as a scaling value as well.A linear function is?tted through this point and the known failure point(point1),thereby determining a?rst estimate of the limit-state point(point2).Next,the G-function value is determined in this point and a qua-dratic?t is made resulting in an improved estimate.This procedure is repeated until the limit-state point is found having an absolute error tolerance of0.01.A higher accuracy is unnecessary,because Eq.(6)guarantees that the MPP is always located outside the https://www.wendangku.net/doc/0a6952602.html,ually the process converges in two-to-three iterations. The search is aborted after a maximum of?ve iterations,to prevent spending an excessive amount of analysis time in rare cases.

3.Numerical examples

The ARBIS method is here applied to a set of widely used test problems obtained from the literature,rep-resenting a broad range of possible limit-states that can occur in practice.The problems are summarized in Table1,in which the last column gives the corresponding reference.Since these problems are used by various authors,the reference is not necessarily the?rst one.Because of the simple nature of the limit-state functions they can be evaluated many times,making a near exact evaluation possible by crude Monte-Carlo.This near exact value is given in column5of Table1.Column6gives the?nal(optimal)b value.

The ARBIS method is examined on

?E?ciency.This is re?ected by the number of G-function evaluations necessary to obtain a converged solu-tion.This number is compared with crude Monte-Carlo and RBIS.

?Robustness.This re?ects how the method performs in the case of a complex limit-state function:noisy fail-ure function,highly non-linear failure function,multiple failure points and/or multiple failure functions.?Accuracy.The method always converges to the‘exact’solution provided enough samples are taken into account.

Sampling is ended when the maximum relative error in the probability value is below a threshold value. Hence an equal accuracy level is obtained with the Monte-Carlo,ARBIS and RBIS method and therefore their e?ciencies can be compared.The maximum relative error is given by:

E rel max

?z a =2COV p f ?U à1c t1

COV p f

e7T

where c is the con?dence level.For each failure point the current value of the COV p f is checked against a threshold value,where COV p f is given by:

COV P f ???????????????1àP f

N sim P f

s e8T

Table 1

Limit-state function descriptions Case Limit-state function Stochastic variables Description

P f e$P ARBIS f

Tb opt Ref.

g ?x 1t2x 2t2x 3tx 4à5x 5à5x 6t0:001P

6i ?1sin e100x i Tx 1...4:LN e120;12T

Linear LS with noise term 1.22e à02

2.361[4]

x 5:LN e50;15T(1.32e à02)

x 6:LN e40;12T2g =x 1x 2à146.14

x 1:N e78064:4;11709:7TMultiple failure points 1.46e à07

5.443[4]x 2:N e0:0104;0:00156T(1.11e à07)3g ?2t0:015P 9i ?1x 2

i àx 10

x 1...10:N e0;1TQuadratic LS 10

terms 5.34e à03 2.103[3](5.6e à03)4

g ?0:1ex 1àx 2T2àex 1tx 2T

??2

p t2:5x 1:N e0;1TQuadratic LS with mixed term,convex LS

4.16e à03 2.481[14]

x 2:N e0;1T(3.71e à03)5g ?à0:5ex 1àx 2T2àex 1tx 2T

??2p t3x 1:N e0;1TConcave LS 1.05e à01 1.625[14]x 2:N e0;1T(1.12e à01)6g ?2àx 2à0:1x 21t0:06x 3

x 1:N e0;1TNon-linear LS with saddle point

3.47e à02 1.996[3]x 2:N e0;1T(3.58e à02)

7g ?2:5à0:2357ex 1àx 2Tt0:00463ex 1tx 2à20T4x 1:N e10;3THighly non-linear LS 2.86e à03

2.431[15]x 2:N e10;3T(2.60e à03)8g =3àx 2+(4x 1)4x 1:N e0;1THighly non-linear LS 1.80e à04

2.925[16]x 2:N e0;1T(2.03e à04)9

g 1?2:677àx 1àx 2x 1...5:N e0;1T

Parallel system

2.11e à04 2.738[16]

g 2?2:500àx 2àx 3(1.96e à04)

g 3?2:323àx 3àx 4g 4?2:250àx 4àx 5g ?max eg 1;g 2;g 3;g 4T

g 1?àx 1àx 2àx 3t3???

3p x 1:N e0;1TSeries system

2.57e à03 2.953[2]

g 2?àx 3t3x 2:N e0;1T(2.81e à03)min eg 1;g 2T

x 3:N e0;1T11

g 1?àx 1àx 2àx 3t3???

3p x 1:N e0;1TParallel system 1.23e à04 3.434[2]

g 2?àx 3t3x 2:N e0;1T(1.11e à04)max eg 1;g 2T

x 3:N e0;1T12

g 1?2àx 2texp eà0:1x 21Tte0:2x 1T4

x 1:N e0;1T

Series system 3.54e à03 2.925[16]

g 2?4:5àx 1x 2Multiple failure points (4.51e à03)min eg 1;g 2T

g 1?2àx 2texp eà0:1x 21Tte0:2x 1T4

x 1:N e0;1TParallel system

2.50e à04

3.219[16]

g 2?4:5àx 1x 2x 2:N e0;1T(2.03e à04)max eg 1;g 2T

14g 1?0:1ex 1àx 2T2àex 1tx 2T

??2p t3x 1:N e0;1TSeries system 2.18e à03 2.925[14]

g 2?0:1ex 1àx 2T2

tex 1tx 2T??2

p t3x 2:N e0;1T

Multiple failure points

(2.72e à03)

g 3?x 1àx 2t3:5???2p g 4?àx 1tx 2t3:5???2p g ?min eg 1;g 2;g 3;g 4T

F.Grooteman /Structural Safety 30(2008)533–542

539

For all problems the threshold coe?cient of variation (COV)for the probability of failure p f was set to 0.1.This means that with 95%con?dence the relative error in the estimate of the probability of failure p f is less than:

E p f max ?1:96COV p f %20%

e9T

This accuracy is acceptable for most engineering applications.In general,the real error will be less than 10%,which is often better than the errors produced in other parts of the analysis (e.g.accuracy of the under-lying deterministic model and numerical model).Reducing the COV value reduces the error at the expense of more simulation.

The value obtained for the probability of failure with the ARBIS method,given the above accuracy level,is presented in column 5of Table 1between parentheses.The required number of simulations is presented in Table 2,columns 2–4,for respectively the Monte-Carlo method,ARBIS and RBIS with optimal radius.The ?fth column of Table 2presents the gain in e?ciency of ARBIS over the Monte-Carlo method.RBIS was applied using the ?nal b -value obtained in ARBIS,which is the optimal radius being close to the MPP.The di?erence in e?ciency with ARBIS,presented in column 6,is therefore a measure of the e?ciency of the adaptive scheme and re?ects the number of G -function analyses spent in the subsequent line-searches to determine the MPP.These values show that the adaptive scheme is very e?cient,because only a small number of additional G -function analyses are required compared with the optimal RBIS method.3.1.Discussion

The various problems serve to demonstrate the e?ciency and robustness of the ARBIS method.In all prob-lems the same value for p 0of 10à6and p step of 0.8was used,see Section 2.2.As explained in the previous sec-tion,all results have a similar accuracy level,by selecting a ?xed value for the coe?cient of variation of p f .The relative error in p f was well below the maximum expected error of 20%for all problems.

Robustness is demonstrated by the noisy limit-state of problem 1,multiple failure points of problem 2,highly non-linear limit-states of problems 6–8;multiple failure functions of problems 9–14,where multiple failure points are present in problems 12and 14as well.For all problems the ARBIS method proved to be very robust.

Fig.4shows sample plots obtained with ARBIS for the two-dimensional problems,clearly demonstrating the approach.The sample points inside the excluded sphere,see for example problems 5,6and 12,including the point in the origin,are related to the line-search method and are left out of the sample set used to calculate the probability value.

As expected,ARBIS is much more e?cient for most problems than MCS.The gain in e?ciency is less for problems 1and 3,which have an increased number of variables combined with a high probability of failure value.The e?ciency strongly depends on the probability value.The probability value is determined by the

Table 2

Number of deterministic analyses required by the di?erent stochastic methods Case MCS ARBIS RBIS opt D eMCS àARBIS TD eARBIS àRBIS T17655352034984157222>1096760>1097317830166741655512751194270961215114225954735942155141801146273430728124532673683519141900349351483541304867478934934178936170167427673452943568210376591096108636573101156372344844472559251121224902216190247122613351660193019073497532314

41220

465

413

40807

540 F.Grooteman /Structural Safety 30(2008)533–542

location of the MPP (b opt )and shape of the limit-state.A rough estimate of the minimum number of required sample points is given by:

N >

z a =2c 21àv 2n eb 2opt T

p f e10TThe farther away the location of the MPP,the larger is b opt ,reducing the number of simulations.This is

re?ected by the numerator of the above equation.Problem 2shown in Fig.4is an example,demonstrating an extreme reduction in samples for small probabilities.On the other hand,for a very narrow shaped limit-state

F.Grooteman /Structural Safety 30(2008)533–542

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the contribution of the failure domain to the whole domain outside the sphere is small,increasing the required number of simulations.This is re?ected by the denominator of the above equation.Problem8shown in Fig.4 gives an example of a very narrow limit-state.The probability value for more realistic problems is in general less than10à5.The gain in e?ciency is therefore expected to be(much)higher for more realistic problems than shown for some of the test problems.

Because of its robustness,ARBIS can be applied as a black-box and is of particular interest in situations where a low probability of failure is expected,such as in structural reliability analyses,consisting of a small number of stochastic variables.Like most sampling methods,ARBIS is suitable for application on a parallel computer,which is an ongoing trend[13],compensating for its lower e?ciency compared with FORM and SORM by simultaneous analyses.Because of this,the robustness of a stochastic method becomes of increasing importance.

4.Conclusion

Importance sampling methods are more e?cient than Monte-Carlo Simulation and Directional Simulation, but require information about the location of the limit-state(s),especially the part closest to the origin in U-space.Gathering this information can be expensive and can fail to locate all the important parts.In this paper, the Radial-Based Importance Sampling method has been extended with a very e?cient and robust adaptive scheme that automatically determines the optimal radius of the excluded sphere.For this reason the method can be applied as a black-box and is of particular interest in applications with a low probability of failure,such as structural reliability,in combination with a small number of stochastic variables.Furthermore,the method is suitable to be applied on a parallel computer.

References

[1]Bjerager P.Probability integration by directional simulation.J Eng Mech1988;114(8):1285–302.

[2]Nie J,Ellingwood BR.Directional methods for structural reliability analysis.Struct Safety2000;22:233–49.

[3]Der Kiureghian A,Lin HZ,Hwang SJ.Second-order reliability approximations.J Eng Mech Div ASCE1987;113(8):1208–25.

[4]Engelund S,Rackwitz R.A benchmark study on importance sampling techniques in structural reliability.Struct Safety

1993;12:255–76.

[5]Melchers RE.Importance sampling in structural systems.Struct Safety1989;6:3–10.

[6]Harbitz A.An e?cient sampling method for probability of failure calculation.Struct Safety1986;3:109–15.

[7]Melchers RE.Radial importance sampling for structural reliability.J Eng Mech Div ASCE1990;116(1):189–203.

[8]Au SK,Beck JL.Importance sampling in high dimensions.Struct Safety2003;25:139–63.

[9]Bucher CG.Adaptive sampling–an iterative fast Monte-Carlo procedure.Struct Safety1988;5:119–26.

[10]Rosenblatt M.Remarks on a multivariate transformation.Ann Math Statist1952;23(3):470–2.

[11]Der Kiureghian A,Liu PL.Structural reliability under incomplete probability information.J Eng Mech Div ASCE1986;112:85–104.

[12]Wu YT,Burnside OH,Cruse TA.Probabilistic methods for structural response https://www.wendangku.net/doc/0a6952602.html,putational mechanics of probabilistic

and reliability analysis.Elmepress International;1989[Chapter7].

[13]Schue¨ller https://www.wendangku.net/doc/0a6952602.html,putational stochastic mechanics–recent https://www.wendangku.net/doc/0a6952602.html,put Struct2001;79:2225–34.

[14]Borri A,Speranzini E.Structural reliability analysis using a standard deterministic?nite element code.Struct Safety

1997;19(4):361–82.

[15]Grandhi RV,Wang L.Higher-order failure probability calculation using nonlinear https://www.wendangku.net/doc/0a6952602.html,put Methods Appl Mech

Eng1999;168:185–206.

[16]Au SK,Beck JL.A new adaptive importance sampling scheme for reliability calculations.Struct Safety1999;21:135–58.