Evaluation of failure probability under parameter epistemic uncertainty: application to aerospace system reliability assessment

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epistemic uncertainty: application to aerospace system

reliability assessment

Vincent Chabridon, Mathieu Balesdent, Jean-Marc Bourinet, Jérôme Morio,

Nicolas Gayton

To cite this version:

Vincent Chabridon, Mathieu Balesdent, Jean-Marc Bourinet, Jérôme Morio, Nicolas Gayton.

Eval-uation of failure probability under parameter epistemic uncertainty: application to aerospace

sys-tem reliability assessment.

Aerospace Science and Technology, Elsevier, 2017, 69, pp.526-537.

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Evaluation

of

failure

probability

under

parameter

epistemic

uncertainty:

Application

to

aerospace

system

reliability

assessment

Vincent Chabridon

a

,

c

,

∗

,

Mathieu Balesdent

a

,

Jean-Marc Bourinet

c

,

Jérôme Morio

b

,

Nicolas Gayton

c

a_{ONERA–TheFrenchAerospaceLab,BP80100,91123PalaiseauCedex,France} b_{ONERA–TheFrenchAerospaceLab,BP74025,31055ToulouseCedex,France}

c_{UniversitéClermontAuvergne,CNRS,SIGMAClermont,InstitutPascal,F-63000Clermont-Ferrand,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received2December2016

Receivedinrevisedform13June2017 Accepted13July2017

Availableonlinexxxx Keywords:

Distributionparameteruncertainty Rareeventsimulation

Predictivefailureprobability Launchvehiclestagefallbackzone Spacedebriscollision

Thispaper aimsatcomparingtwodifferentapproachestoperformareliabilityanalysisinacontextof

uncertainties affecting probability distributionparameters.Thefirst approachcalled “nestedreliability

approach”(NRA)isaclassicaldouble-loop-approachinvolvingasamplingphaseoftheparametersand

then a reliabilityanalysis for each sampled parameter value. A secondapproach, called “augmented

reliabilityapproach”(ARA),requirestosamplebothdistributionparametersandbasicrandomvariables

conditional tothem atthesame phaseand thenintegrate simultaneouslyoverbothdomains. Inthis

article,anumericalcomparisonisled.Possibilitiesofferedbybothapproachesareinvestigatedandthe

advantages ofthe ARAare illustratedthroughthe application ontwo academictest-casesillustrating

severalnumericaldifficulties(lowfailureprobability,nonlinearityofthelimit-statefunction,correlation

betweeninputbasicvariables)andtworealspacesystemcharacterization(alaunchvehiclestagefallback

zoneestimationandacollisionprobabilitybetweenaspacedebrisandasatelliteestimation)forwhich

onlytheARAistractable.

©2017ElsevierMassonSAS.Allrightsreserved.

1. Introduction

Reliability analysis appears to be one of the dedicated tools toquantify the risk of failure forcomplex aerospacesystems re-gardingtheuncertaintiesaffectingtheirbehaviorandtohelp engi-neerstomakemoreinformeddecisionsinthedesignphase.Under safety requirements, one needs to quantify a probability of fail-ure pf.However, failurescenariospossiblyimpactingthebehavior ofa systemoften lead to rare events, i.e.events associated to a verylow failure probability[1,2].Estimatingsuch aprobability is often burdensome since classical methods such as crude Monte Carlo(CMC) involvea large number ofmodel evaluations which make the calculations untractable [3]. Thus, the particular case of coupling between reliability analysis (underlying on multiple probabilisticanalyses)andexpensiveaerospacecomputercodesis definitelyawell-knownissue[4].

Simulationmodelsusedinaerospaceengineeringbecamemore andmorecomplexoverthelast decadesandreachedhighfidelity representation.However,theysufferfromvariouskeycomputer

ill-*

Correspondingauthor.

E-mailaddress:vincent.chabridon@onera.fr(V. Chabridon).

nesses:high-dimensionality, high-nonlinearitiesandmulti-physics combination. All in all, thecommon similaritybetween all these codesis theirexpensive-to-evaluateaspect.Ontopofthat, differ-entsources ofuncertaintyaffectthewayweevaluate thesystem performances(forinstance,anumberofinputvariablesare uncer-tain).Theseuncertainties canbe consideredasinherentfrom na-ture, duetovariousassumptions ornumericalapproximations, or finally stemingfrommeasurement errors.Fromapragmatic engi-neeringpointofview,uncertaintiescanbeseparatedintotwo cat-egories:aleatory and epistemic[5].Aleatoryuncertaintyrepresents naturalvariabilitywhichissupposedtobeirreducibleinaspecific context.Epistemicuncertaintyensuesfromthelackofknowledge or mathematical simplifications and can be reduced by adding more information orincreasing the modelfidelity. Both types of uncertainty require proper mathematical formalisms, numerical modeling and analyses. If aleatory uncertainty is often modeled using a probabilistic framework, several competing(and comple-mentary)formalismsareavailabletomodelepistemicuncertainty: probabilisticBayesiananalysis[6],intervalanalysis[7],Dempster– Shafer’sevidencetheory [8,9],possibilitytheory [10], probability-boxes [11,12].This non-exhaustive list mentions multiple frame-works which are commonly gathered under the global name of

impreciseprobabilities [13]. Mixing aleatory andepistemic

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

taintiescanleadtocombinesome oftheseframeworks.Choosing oneofthemdependsontheavailabletypeofdata(bounds, distri-bution,etc.)andhasanimpactonthepropagationofuncertainties throughthemodel.Inthispaper,weassume thatprobability dis-tributionsofinputdataareavailable buttheir parametersarenot knownpreciselyduetoalackofinformationorverylimiteddata. Aclassical approach,known asthe“nestedreliabilityapproach”

(NRA), is to consider sampling of uncertain distribution param-eters and to perform a nested reliability analysis for each real-ization of these parameters (see, for example [14] for reliability assessment under epistemic uncertainty on distribution parame-ters given by intervals, [15] in the context of probability-based tolerance analysisofproducts or[16] for a couplingwithSparse Polynomial Chaos Expansions). Nonetheless, it implies to repeat severaltimesa costly reliability analysis, whichcan be unafford-able for most cases in a complex industrial environment. Based on the context of complex aerospace systems design and taking intoaccountall theconstraintsmentionedearlier,assessing relia-bilitycoupledwithconsiderationontheuncertaintyaffectingthe distribution parameters seems to be quite challenging. However, some researchers proposed methods to incorporate this kind of uncertaintyaffecting distribution parameters intoa more general Bayesianframework[6,17–19].

Analternativeapproach,knownasthe“augmentedreliability

ap-proach” (ARA), aims at computing a different failure probability

(called“predictivefailureprobability”)whichtakesintoaccountthe uncertaintyinthedistributionparametersbyconsideringan

“aug-mentedinputspace”ofthebasicvariableswiththeiruncertain

dis-tributionparameters.Basedontheseconsiderations,theaimofthe paper is to describe the two approaches (nested vs. augmented) within the same framework and investigate different advantages offered by ARA. Moreover, explanations are given about the key pointofthetransformationbetweenthephysicalspaceofthebasic inputvariablesandthestandardnormalspace,whichiscommonly used in reliability analysis, under the consideration of this new “augmented space”. Thisarticle aims at givinga comparison be-tweenNRAandARAthroughnumericalapplicationtochallenging test-cases representing the main difficulties that aerospace engi-neering has to face to (nonlinear codes, correlated inputs and a low failure probability to estimate). Another goal is to highlight, throughnumericalresults,advantages anddrawbacksofboth ap-proaches coupled with advanced reliability methods tested on a benchmark representative of real world aerospace problems for whichonlytheARAisatractableapproach.

Thispaperisorganized asfollows.Section2expounds a bibli-ographyreviewofreliabilityassessmentunderdistribution param-eteruncertaintyandaims atintroducing theformalconcepts and notations. Section 3 definesthe two approaches into a common frameworkandprovidesgenericalgorithmsforbothmethods. Sec-tion4will illustratethebenefitsofsuch an augmentedapproach throughanumericalcomparisonbetweenNRAandARAon differ-ent test-cases of increasing complexity (from academic toy-cases torealblack-boxcomputercodesissuedfromaerospaceresearch). Section 5 discusses limitations of those approaches and evokes possible enhancements. A conclusion gathering the most impor-tantresultsofthispaperisfinallygiveninSection6.

2. Formulationoffailureprobabilityestimationunder

distributionparameteruncertainty

2.1. Generictime-invariantreliabilityproblemstatement

Amodel

M(·)

isconsideredsuchthatitrepresentsastatic(i.e. time is not an explicit variable here) input–output system given by: y

=

M(

x

)

, where x

∈

D

X

⊆ R

d is a d-dimensional vector of

input variables and

M

:

D

X

→

D

Y

⊆ R

a given scalar mapping.

In general, thismapping can be eitherdefined using an analyti-cal expression oranumericalmodel.Inour case,thismodelisa computationallyexpensivesimulationcodewhichcanleadto con-sider itasablack-box function onlyknownpointwise. Intherest of the paper,

M(·)

is supposed to be a deterministic model,i.e. theunderlyingbehaviorofthemodelisnotstochastic.

Thescalarrandomvariables X1

,

X2

,

. . . ,

Xdrepresentthe

uncer-tain inputvariables(denotedasthebasicvariables inthefollowing,

see[20]and[21])ofthesystem.Thesebasicvariablesaregathered

in a d-dimensional random vector X of known continuous joint

probabilitydensityfunction (pdf) fX

: R

d

→ R

+.Inthespaceofthe

realizationsx

= (

x1

,

x2

,

. . . ,

xd

)

 oftherandomvectorX,failureis characterizedbytheuseofafunction g

: R

d

→ R

calledthe

limit-statefunction (lsf).Aclassicalformulationforthelsfinthecontext

ofstaticinput–outputmodelcanbe:

g

(

X

)

=

yth

−

M

(

X

)

(1)

whereyth

∈ R

isacharacteristicthresholdoutputvaluebeyondthe one thesystemfallsintoa failurestate.Thus,one candistinguish twodomainsassociatedtothebehaviorof g

(

·)

:thefailuredomain

givenby

F

x

= {

x

∈

D

X

:

g

(

x

)

≤

0

}

,which infactdoesincludethe

limit-state(hyper-)surface (LSS)

F

x0

= {

x

∈

D

X

:

g

(

x

)

=

0

}

splitting

the spaceintotwo,andthe safedomain

S

x

= {

x

∈

D

X

:

g

(

x

)

>

0

}

.

With no consideration ofany distribution parameter uncertainty,

thefailureprobability pfthereforereads:

pf

= P

[g

(

X

)

≤

0]

=



Fx fX

(

x

)

dx

=



DX

1

_Fx

(

x

)

fX

(

x

)

dx

= E

fX



1

_Fx

(

X

)



(2)

where

1

_Fx

(

·)

is the indicator function of thefailure domain

de-finedsuchthat

1

_Fx

(

x

)

=

1 ifx

∈

F

xand

1

Fx

(

x

)

=

0 otherwise.

Finally, estimating such a failure probability can be achieved using one ofthe classical methods available in thestructural re-liability literature [22].To doso, two distinct classesof methods have been developed: approximation methods such as the

First-Order Reliability Method (FORM) and the Second-OrderReliability

Method (SORM),which bothrely ontheconcept ofMost-Probable

Point (MPP); and simulationmethods based on Monte Carlo

sim-ulations [21]. Among this second class of methods, one can find more advancedsampling-based methodssuch asImportance

Sam-pling (IS) [3],DirectionalSampling (DS) [20], LineSampling (LS)or

SubsetSimulations (SS)[4].

Approximation methods (suchas FORM/SORM) havebeen de-veloped, following well-argued mathematical and historical rea-sons [23,21], in the so-called standardnormalspace (denoted as

U-space) inwhichall randomcomponentsof X become

indepen-dent standardGaussian variatesgatheredinthe vectorU.Among thesimulationmethods,the useofsuch astandardnormalspace isnotalwaysrequired(e.g.,CMCmethodisperformedinthe

orig-inal physicalspace, denoted asX-space).However, most advanced

sampling-based methodssuchasthosecitedabove arededicated, or have some adapted versions oftheir initial algorithms, to the standard normalspace. Thegeneralidea isto constructaregular transformation T

:

D

X

→ R

d allowing (intermsofprobability

dis-tributions)toget:

U

=

T

(

X

)

⇔

X

=

T−1

(

U

)

(3)

whereU

= (

U1

,

U2

,

. . . ,

Ud

)

isad-dimensionalstandardGaussian

vector of independent normal variates Ui with zero means and

unit standarddeviations.Then,onecandefineanewmappingfor thelsfinthestandardspaceconsidering G

: R

d

_{→ R}

_definedsuch

that: U

→

G

(

U

)

=



g

◦

T−1



(

U

)

(4)

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whichallowstorewritethefailureprobability:

pf

= P

[G

(

U

)

≤

0]

=



Fu

ϕ

d

(

u

)

du

=



Rd

1

_Fu

(

u

)

ϕ

d

(

u

)

du

= E

ϕd



1

_Fu

(

U

)



(5)

where

F

u

= {

u

∈ R

d

:

G

(

U

)

≤

0

}

stands for thefailure domain in

thestandard space, du

=

du1du2

. . .

dud and

ϕ

d

: R

d

→ R

+ isthe

d-dimensionalstandardGaussianpdfofU.

The choice of the transformation T

(

·)

generally depends on the available information. When only the marginal distributions

fXi

(

·)

andthelinearcorrelationsareknown,followingthe recom-mendations in [24,25], one should use the so-called Nataf

trans-formation[26].When thefull knowledgeofthe jointpdf fX

(

·)

is

available, itis advised [27] to better use theso-called Rosenblatt

transformation [28]. Thus, under the assumption of normal

cop-ula[29],withoutanyconsiderationofparameteruncertainty,both transformationscanbeusedsincetheyareidenticalinthisspecific case[30].

2.2.Reliabilityanalysisunderdistributionparameteruncertainty

For complex systems such as aerospace ones, the joint pdf

fX

(

·)

is not accurately known [17,23]. For example, the choice

of a parametric model for the density fX

(

·)

can be based on

estimation of some distribution parameters (i.e. some moments of the pdf) which can introduce an important bias if the initial samples only provide some limitedinformation. Moreover, some expert-judgment-basedassumptionscanleadtoanapriorichoice ofsome values forthe parameters instead of others.The perfect knowledge of the joint pdf fX

(

·)

would require, from a general

point of view, the full knowledge of the marginal pdfs and the copula.However, theprobabilisticinformationavailableaboutthe inputrandomvectorX oftenreducestothemarginaldistributions and,inthecaseofdependentinputs,totheimperfect knowledge ofthelinearcorrelationmatrixR

= [

ρ

i j

]

i,j∈{1,...,d}[29].Thus,in

ad-ditiontothe first uncertaintylevelcharacterizing thebasic input variables, uncertainty may also affect both distribution parame-tersand the dependencestructure. Consequently, engineershave

toface what we call a bi-leveluncertainty.In this context,giving

back an hypothetic single measure of reliability taking only one levelintoaccountseemstobeinappropriate.

Suchatopichasbeenearlydiscussedamongthestructural re-liability community,mainly in the first investigations led by Der Kiureghian[6,17,31,32]andDitlevsen[18–20,33].Intheircommon paper[5], theseauthorsstress theneed ofa measureof reliabil-itythattakesintoaccountparameteruncertainty(the firstauthor proposed in[6] to call it“predictivereliabilitymeasure”,following Bayesiananalysisvocabulary,andprovidedaformaldefinitionthat willberecalledlater).

Assuming now that X is distributed according to the para-metricjoint pdf fX|

(

·|·)

,each random variable Xi isdistributed according to the marginal pdf fXi|i

(

·|·)

. In the case of depen-dent inputs, inthe normal copulacase, uncertaintyaffecting the correlation matrix could easily be considered in this framework. However,fromamoregeneralpointofview,uncertaintyaffecting the dependence structure (i.e. the copula) is not a widely stud-iedtopicinliterature.Moreover, froman engineeringperspective, this problem is really difficult to assess due to the crucial lack of information. In this paper, we will only consider distribution parameteruncertaintyandlet copulastructureuncertaintyto fu-turework.Indeedthevector



gathersalldistributionparameters ofthecorrespondingmarginalssuch that



= (

1

,



2

,

. . . ,



d

)

,

where each



i

,

i

∈ J

1

,

d

K

is a set of distribution parameters for

the i-th marginal (for instance, if Xi

∼

N (

μ

Xi

,

σ

Xi

)

, then



i

=

(

μ

Xi

,

σ

Xi

)

). Onecanimagine thatdepending onthedistribution type, all the marginal pdfs will not be defined with the same number ofparameters. Inthis paper, we assume that onlya set ofuncorrelateddistribution parametersareuncertainwhichleads to consider a generalcollectionof univariate random parameters givenby



= (

1

,



2

,

. . . ,



k

)



∈

D



⊆ R

k (which canbe either

momentsorbounds).Consequently,withoutanylossofgenerality, one can assumethe existence ofa jointpdf f

=



kj=1fj asa product ofthe marginal pdfs of the



j [34].Note herethat one

could alsoconsider adependencestructurebetweenthe distribu-tionparameters.However,theproblemwouldbefarmoredifficult andwouldimplytohave,atminimum,apriorinformationabout suchadependencestructure.Thistopicisbeyondthescopeofthis paper.Togetadeeperinsightaboutthepractical characterization of f

(θ )

basedonavailabledata(whichisnotthescopeofthis

pa-per),thereadermayreferto[35].Tosumup,inthispaper,onlya

priorprobabilitydistribution (forinstance,followinganexpert-based

judgment)willbeassumedfor



withoutanypurposeofBayesian reliabilityupdating[36].

Thus,anewformulationforthefailure probabilitycanbe pro-posed,following[17].Indeed,dueto thisbi-level uncertainty(on thevectorofbasicvariablesX andonthevectorofdistribution pa-rameters



),the failureprobability pf isnomoreadeterministic value.Itbecomesarandomvariable,denotedasPf,whichdepends ontherealization

θ

oftherandomvectorofuncertainparameters suchthat: Pf

(θ )

= P

[g

(

X

)

≤

0

|  = θ

] (6a)

=



DX

1

_Fx

(

x

)

fX|

(

x

|θ

dx (6b)

= E

fX|



1

_Fx

(

X

)

|  = θ



.

(6c)

Hence,byintegratingover

θ

,wegettheso-called“predictivefailure

probability”



Pfwhichisameasureofreliabilitytakingintoaccount

theeffectoftheuncertaincharacterizationofdistribution parame-ters:



Pf

= E

f[Pf

()

] (7a)

=



D Pf

(θ )

f

(θ )

d

θ

(7b)

=



D

⎛

⎜

⎝



DX

1

_Fx

(

x

)

fX_|

(

x

|θ)

dx

⎞

⎟

⎠

f

(θ )

d

θ .

(7c)

Eq.(7c) isthekeyequation whosesolving isunderconsideration inthispaper.Theideaisthatitcanbenumericallysolvedbytwo differentapproaches.

From anumericalpoint ofview,afirstwayofcomputingthis integral reliesonevaluating pointwisetheinner integralforeach realization

θ

of



[14–16]:thisleads tothenestedreliability ap-proach(presented in subsection 3.1). The second wayconsists in evaluatingitby treatingbothbasicvariablesanduncertain distri-bution parameters together andby integratingsimultaneouslyon both domains (butstill respecting the conditioning) assuggested in [17]: this is the augmented reliability approach (presented in subsection 3.2). The next section describes these approaches in details. As a remark, one can notice that this Bayesian frame-workprovideshereasinglereliabilitymeasure(thepredictive fail-ure probability).Nevertheless,thisquantity canhelp engineersto makemoreinformeddecisionsduringthedesignprocessandcan be coupledwiththeclassical reliabilitymeasure soasto analyze properlytheriskundertakenwithadesignchoice.Decisioncanbe thenenlightenedbysuchadditionalinformation[37–39].

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

Uptonow, toourknowledge,severalresearchersdeployed ef-forts to carry on the way of other approaches to compute this predictive failure probability (see anotherapproach by [40],used in[6]andin[41] onlywithFORMcalculations).Nevertheless,the trackofexploringtheaugmentedspace hasnotbeenover-exploited yet.In[42],theauthorrecommendedandimplementedthis strat-egy on a fracture mechanics test-case but limited his study to FORM algorithm. All these worksmainly focused on providing a globalreliabilityindex,robusttoparameteruncertainty,inthe spe-cific context of FORM. The use of an augmented space has also been exploitedby [43] fordesign sensitivity purpose while con-sideringuncertain design parameters. More recently, in [44],the authorsproposedinabroaderviewonthecomprehensionand in-terpretationofthedifferentlevelsofuncertaintyinvolvedinthese calculations and encouraged to use an augmented approach to solve asimilar integral problemgivenin Eq.(7c).However, their study did not aim at performing reliability assessment for rare eventfailureprobabilitiesofsomecomplexsimulationcodeswhich isthescopeofthepresentpaper.

3. Descriptionofthetwoapproaches

3.1. Thenestedreliabilityapproach(NRA)

Thisapproachisanested-loop-basedapproachsinceitinvolves the numerical estimation of two different quantities. The first (nested orinside)loop aims atcomputing a “conditional” failure probability whose numerical estimator is denoted as



Pf

(θ )

. This estimatorisameasureofreliabilityundertherealization

θ

ofthe randomvector



.Thesecond(outside)loopaimsatcomputingan estimatorofthepredictivefailureprobability,denotedas





Pf,by in-tegratingoverthesupportoftherandomvector



.Inpractice,it consistsincomputingseveral



Pf

(θ )

forarangeofrealizations

θ

of thevectorofuncertainparameters



.Ithasbeenwidelyused in literature, in various contexts, such asrare eventprobability es-timation with Kriging-based approach in [14], probability-based tolerance analysisof products in [15] oruncertainty propagation usingprobability-boxesandpolynomialchaosexpansionsin[16].

Algorithm1Nestedreliabilityapproach(NRA)withCMCfor

prob-abilityestimation.

A generic implementation of NRA framework coupled with a nested CMC method is given in the Algorithm 1. In the rectan-gular box at lines 5–9, one can choose any available reliability methodtoestimatetheconditionalfailure probability



Pf

(θ )

,from approximation methods (FORM, SORM) to most advanced simu-lation methods (IS, SS). Nevertheless, it seems more relevant to focusontheonesthat arestill,up-to-now,themostwidelyused either in aerospace industry or in research. Indeed CMC is still considered as the reference method for validation. FORM offers wide possibilities for practitioners who wantto perform reliabil-ityassessmentwithalowcomputationalcost,evenifthismethod only givesthe true failure probability for linear lsfs [21]. Finally,

Algorithm2NRAgenericbox(FORMorSS).

SS appeared to be a very powerful method to reach estimation ofrare eventfailureprobabilities, undertheconstraintof nonlin-earlsfs,witharathermoderatecomputationaleffort[45].Inbrief, therectangularboxcanbe seenasanon-intrusiveplug-in uncer-taintypropagationcodeforreliabilityassessment.Anexampleofa plug-in box(for FORM orSS)is giveninthe Algorithm 2.In nu-merous cases, an additional step is required: the transformation to the standard normal space (see subsection 2.1). Inthe nested case, the transformationis already included in theplug-in relia-bility rectangular box,i.e. classical transformations such asNataf orRosenblattonescanbebothused,andthedistribution parame-teruncertaintydoesnotchangeanythingtotheirimplementation. Nevertheless,one shouldnoticethat foreach sampledparameter, thealgorithmneeds torebuildandrecalculatethetransformation since it depends on the parameter value. Thus, for complicated transformations,withalargenumberofbasicvariables,the simu-lationcostinducedcanbeincreased.

Inthisnestedcase,wecandemonstratethattheestimator





Pfis unbiased.Moreover,themeanandvarianceof





Pfareestimatedby replicationofthealgorithm,usingthefollowingclassicalstatistics:

m_ Pf

=

1 Nrep Nrep



i=1





P(_fi) (8)

whichisthesamplemean withNrep thenumberofreplicationsof the predictivefailure probability estimation and S2



Pf

the unbiased

samplevariance definedby:

S_2 Pf

=

1 Nrep

−

1 Nrep



i=1







P(_fi)

−

m_P f



2

.

(9)

3.2. Theaugmentedreliabilityapproach(ARA)

Anotherapproachistoconsideranaugmentedinputrandom

vec-tor Zdef

= (,

X

)

 composedofthebasicvariables andtheir

distri-bution parametersasitappearsinEq.(7c) (see[16] fora similar definition). Thus, thisaugmentedinput spacehasa dimension of

k

+

d (k uncertaindistributionparameters



j andd randombasic

variables Xi).

A generic implementation framework is given in the Algo-rithm 3. Again, inthis algorithm, therectangular box can be re-placed byanynon-intrusiveplug-inuncertaintypropagationcode forreliabilityassessmentastheonescitedpreviously fortheNRA (see Algorithm 4as an example). This shows that the ARA does not sufferfromanymajordifferencewiththeclassicalnested ap-proachintermsofthevarietyofmethodsthatitcanhandle.Again, asfortheNRA,onecandemonstratethattheestimator





Pf is un-biased. Onemajordifferenceconcernsthetransformationto stan-dard normalspace:sincethereexists aconditioningbetweenthe

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

Algorithm3Augmentedreliabilityapproach (ARA)bluewith CMC

forprobabilityestimation.

Algorithm4ARAgenericbox(FORMorSS).

distributionparametersandthebasicinputvariables,Nataf trans-formationcannot beusedanymoreandRosenblatttransformation istheonlyonethatcanhandlethisconstraint.

Consideringuncertaintiesaffectingdistributionparametersleads toadapttheusualRosenblatttransformation.Weassumethejoint pdf f_X|

(

·|·)

isknown since we knowall the marginal pdfs and thecorrelation matrix(orthecovariancematrix)givingthelinear correlationstructurebetweenthebasicinputvariables(normalor Gaussian copula case [29]). In addition, we know the joint pdf

f

(

·)

asexplainedpreviously insubsection 2.2. Inthis case,

un-der theconsideration of theaugmentedspace ofdimension k

+

d

(k distribution parameters andd basic variables), one can apply Rosenblatttransformation[28]firsttothek componentsof



and

thentothed componentsofthevectorX

|

suchthat:

T_augRos

:



















R

k+d

_{−→ R}

k+d z

−→

u

=

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝ −1  F1 (θ1) . . . −1F_k(θk) . . . −1_{F X} 1|1,...,k(x1|θ1, . . . , θk)  . . . −1_{F X} d |1,...,k ,X1,...,Xd−1(xd|θ1, . . . , θk, x1, . . . , xd−1)  ⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠ (10)

where



−1

(

·)

is the normal inverse cumulative densityfunction

(CDF)and Fi

(

·)

, FXj|i

(

·|·)

respectivelythemarginalCDFsofthe parameters and the conditional marginal CDFs of the basic vari-ables.Inthecaseofcorrelatedinputs,onecanimplementthis reg-ulartransformationfollowingandadaptinggeneralformulasgiven in[28].Asaremark,oneshouldnoticethatfromanumericalpoint ofview,theinversetransformation



TRos

aug



−1

(

·)

(fromthestandard normalspacetothephysicalspace)canbethemostuseful (espe-ciallywhenFORM,SORMorSSmethodsareused).

3.3. Illustration

Fromanumericalpointofview,NRAandARAcanbeillustrated through atwo-dimensional examplewithone uncertain distribu-tion parameter. Letus call X1 and X2 the two basicinput vari-ablesmodeled astwoGaussianvariates suchthat X1

∼

N (

μ

X1

=

7

,

σ

X1

=

5

/

√

3

)

and X2

∼

N (

=

μ

X2

,

σ

X2

=

2

/

√

3

)

.Themeanof

X2 isconsideredasbeinguncertain(forexample,



∼

N (

2

,

1

.

5

)

). For thesake ofclarity, in Fig. 1(a), only three cloudsof samples are plottedforthree differentvaluesof

θ

(200 pointsper cloud). Indeed such a sequential samplingis the underlying principle of NRA.

As forARA, graphicalresultsplottedinFig. 1(b) bring out the underlying principle of this approach: covering in one algorithm steptheaugmented inputspace(forthe sakeofcomparison, 600 pointsareused,insteadof3

×

200 pointsforNRA).Onecanclearly notice thesame trendbetweenNRA andARA, the firstone by a sequential sampling strategy, the second one by a simultaneous samplingoverallthedimensionsoftheaugmentedinputspace.In brief,itappearsthatARAoffersbetterspace-fillingpropertiesthan NRA[46].

4. NumericalcomparisonbetweenNRAandARA

To evaluate the efficiency of the ARA, a numerical validation benchmark has been performed with a systematic comparison to the classical NRA. Several test-cases of increasing complex-ityhave beenchosen (from two analytical casesto two different “real life” aerospace industrial applications) to check the validity ofthemethods.The choiceofthesetest-casesaimsatcoveringa range of classical problems encountered in reliability assessment of aerospacesystems: complex black-box models with numerous input variables, high nonlinearities, highcomputational cost, low probability of failure. Moreover, three reliability methods have been tested to calculate the failure probability: CMC, FORM and SS. The following numericalapplications havebeen implemented inMatlab®andperformedusingtheopen sourcetoolbox FERUM v4.1[47].

4.1. Methodologyandcomparison metrics

This paper aims at comparing results obtained for both NRA andARA.Foreachtypeofapproach,tworeliabilitymethods,FORM (when lsf is linear) andSS, will be used to estimate the predic-tivefailureprobability



Pf.Thesecombinedapproaches(NRA/FORM, NRA/SS, ARA/FORM, ARA/SS) will be respectively compared to a reference estimation performed using CMC (most of the time, a NRA/CMCwitha largenumberofsamples onbothdomains). Ta-ble 1givesabriefoverviewofthemethodology.

InTable 1,theblacksquares



standforsuccessfulcalculations ofthe test-casesandthe crosses

×

indicatethat FORM isclearly inappropriatesincethelsfisknownexplicitlytobenonlinear.Asa remark, one can notice that some specific cases are denoted as computationally “untractable”. Indeed, to overcome such a diffi-cultyandtogetareferenceresulttomakethecomparisonviable, specificcomputationalstrategieshavebeensetup.Forthesakeof clarity andtoavoidanyconfusion, thesestrategiesare presented anddiscussedinthededicatedsubsectionsofthetest-cases.

Oneneedstointroducethecomparisonmetricsusedinthe fol-lowing numerical benchmarks. Thus, following [4],we choose in this paperto characterize the quality of ourestimator





PfM of



Pf obtainedwiththemethodM bytheuseofthreeperformance met-ricscomputedwithrespecttothereferencecalculations,i.e.those obtainedbyCMC:

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

Fig. 1. Illustration of NRA and ARA simulation procedures on a two-dimensional problem.

Table 1

Overallmethodology.

Test-case Ref. NRA ARA

Ref. CMC CMC FORM SS CMC FORM SS

Correlated R−Sa _{(cf.4.2.1)       }

Nonlinear oscillatorb _{(cf.4.2.2)   ×   × }

Launch vehicle fallback zonec _{(cf.4.3) untractable      }

Collision between orbiting objectsd _{(cf.4.4) untractable untractable  untractable   }

a _{2correlatedbasicvariables,1uncertainparameter,g}

(·)linear,lowclassicalfailureprobability. b _{8independentbasicvariables,1uncertainparameter,g(·)nonlinear,lowclassicalfailureprobability.} c _{6independentbasicvariables,2uncertainparameters,g(·)nonlinear,lowclassicalfailureprobability.} d _{6correlatedbasicvariables,3uncertainparameters,g(·)nonlinear,lowclassicalfailureprobability.}

•

theRelativestandardError (RE):

RE







PM_f



=



V







PM_f



E







PM_f



(11) where

E







PM_f



and

V







PM_f



areestimatedonthesample ob-tainedbyreplicationoftheanalysis(seeEq.(8)andEq.(9));

•

theRelativeBias (RB):

RB







P_fM



=

E







PM_f



− 



PC MC_f





P_fC MC

.

(12)

Itgivesadescriptionofhowclosetheestimate





P_fM iscloseto thereferencevalue





PC MC_f .Inthefollowing(seeTables 3to 6), RBforNRAiscomputedwithreferenceto thequantity





Pf,ref (estimatedbyareferenceCMCoranothermethodwhenCMC isuntractable) while RBforARA iscomputed withreference to





PA R Af /C MC tomakethecomparisonrepresentative;

•

the efficiency

ν

M _{relatively to CMC estimate (respectively}

NRA/CMCandARA/CMC)definedsuchthat:

ν

M

=

N C MC sim NM sim (13) whereNC MC

sim istherequirednumberofCMCsimulationstoget

RE







PC MC_f



=

RE







PM_f



. Thus, the

ν

M _{ratio can be rewritten}

as:

ν

M

=



1

− 



P_fM



NM_sim

× 



P_fM

×

RE







P_fM



2

.

(14)

A value of

ν

M

_>

_{1 indicates that the method M ismore}

ef-ficient than CMCforthe giventest-case. Inother words,

ν

M

indicatesthequantitybywhichwecandividetheinitialCMC simulationbudgetforasamelevelofaccuracy.

4.2. Applicationontwoacademictest-cases

4.2.1. AfirstR

−

S examplewithcorrelatedbasicvariablesandlow

failureprobability

Description. Theaimofthisfirstacademictest-caseistocheckthe

validity of the two approachesregarding two difficulties: assum-ingastrongcorrelationintheinputprobabilisticmodelandtrying to estimatea low failureprobability withrespecttoa given sim-ulation budget.Table 2givestheinput data.The referencefailure probability withoutparameter uncertainty is pf,ref

=

8

.

84

×

10−8 (because of the linearlsf, the truefailure probability can be ob-tained usingFORM).The correlationcoefficient

ρ

=

0

.

9 expresses thelinearcorrelationbetweenthetwo basicvariables.Thefailure is considered whenthe sollicitation S overcomesthe resource R.

Thelsfthusreads:

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132 Table 2

Second-orderstatisticsanddistributionsofinputrandomvariablesforthe R−S test-case.

Variable Xia _{Distribution Meanμ}

Xi StdσXi X1=R Normal 12 5/ √ 3 X2=S Normal μX2 uncertain b ₂ /√3 =μX2 Normal 2 1.5

a _{LinearcorrelationbetweenX}

1andX2:ρ=0.9. b _{Forafixedvalueμ}

X2=2,pf,ref=8.84×10−8.

The conditional failure probability,i.e. Pf

(

= θ)

, can be written initsintegralformsincethejointconditionalpdf fX|

(

·|·)

canbe analyticallyderived.Onegets:

Pf

(θ

=

μ

X2

)

=



Fx 1 2

π σ

X1

σ

X2



1

−

ρ

2

×

exp



−

1 2

(

1

−

ρ

2

₎

×



(

x1

−

μ

X1

)

2

σ

2 X1

−

2

ρ

(

x1

−

μ

X1

)(

x2

− θ)

σ

X1

σ

X2

+

(

x2

− θ)

2

σ

2 X2



dx

.

(16)

Inthespecificcaseoftwocorrelatednormalvariablesandalinear lsf,theprobabilityoffailurebecomes:

Pf

(θ

=

μ

X2

)

=  (−β

C

)

= 

⎛

⎜

⎝−

_

μ

X1

− θ

σ

_X2₁

+

σ

_X2₂

−

2

ρσ

X1

σ

X2

⎞

⎟

⎠ ,

(17)

where

β

C is the Cornellreliabilityindex [48]. This simple

closed-formsolutioncanbeusedtocheckandvalidatenumericalresults obtainedforthiselementarytest-case.

Results.Table 3illustratesthatNRAandARAgivesimilarresultsfor

estimatingthepredictive failure probability.Moreover, foralmost allthemethods (exceptNRA/CMCwhich suffersherefromalack ofpoints whilecomputingtheintegral over

D

) itdemonstrates

thatARAcanhandlebothrareeventprobabilitiesandstrong corre-lationbetweenbasicinputvariables.Ontheonehand,ARA/FORM seriouslychallengesothermethodssinceithasaverysmall num-berofsimulationcodeevaluationscomparedtoCMCandSSandit givesexactresultssincethelsfislinear.Ontheotherhand,ARA/SS definitelygivespromisingresultscomparedtoARA/CMCsincethe

ν

value (

ν

=

54

.

44) is high. In a classical context of rare event (oftenencounteredinaerospaceengineering),onecanseethe su-periority of ARA (coupled with FORM or SS) compared to other NRA-coupledmethods.Thistest-caseserves asapreliminary ver-ificationfor ARA beforetesting it ona real aerospacesimulation codesuchasthosepresentedinsubsections4.3and4.4.Italso re-vealshowhighcanbethevariationsbetweentheclassical failure probabilityestimateandthepredictiveoneconsideringparameters uncertainty(here,itdropsfrom108 _to105_).

4.2.2. Anonlinearoscillator

Description. Thisnonlinearoscillatorisawell-knownstructural

re-liabilitytest-casefirstlyproposedin[49]andthenusedfor bench-markingpurposesin[2,50,51].Theaimhere,istoassessreliability ofa two-degree-of-freedom primary-secondary system, asshown inFig. 2,underawhitenoisebaseacceleration.Thebasicvariables characterizing the physical behavior are the massesmp and ms,

springstiffnesseskp andks, naturalfrequencies

ω

p

= (

kp

/

mp

)

1/2 Tab

le 3 Re su lt s for the R − S te st -case. Appr oac h CMC a FO RM (e xa ct ) S S b m P f S 2  Pf RE m P f S 2  Pf RE RB ν m P f S 2  Pf RE RB ν NRA c 2 . 57 × 10 − 5 2 . 79 × 10 − 11 0 . 21 1 . 99 × 10 − 5 7 . 38 × 10 − 11 0 . 43 − 2 . 10 × 10 − 3 − 1 . 99 × 10 − 5 5 . 68 × 10 − 11 0 . 38 − 1 . 99 × 10 − 3 48 . 03 ARA 1 . 95 × 10 − 5 1 . 89 × 10 − 12 7 . 03 × 10 − 2 1 . 97 × 10 − 5 −− 7 . 84 × 10 − 3 − 1 . 99 × 10 − 5 7 . 26 × 10 − 11 0 . 43 1 . 91 × 10 − 2 54 . 44 a NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les | ARA: Nx ,θ = 10 6 sam ples. b NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les/s tep | ARA: Nx ,θ = 10 3 sam ples/s tep. c Re f. (NRA/CMC, Nθ = 10 6sam ples, Nx = 10 6 sam ples):  Pf, re f = 1 . 99 × 10 − 5. Ta b le 4 Re su lt s for the nonlinear oscillat o r te st -case. Appr oac h CMC a SS b m P f S 2  Pf RE m P f S 2  Pf RE RB ν NRA c 1 . 41 × 10 − 4 1 . 41 × 10 − 10 8 . 39 × 10 − 2 1 . 59 × 10 − 4 2 . 18 × 10 − 10 9 . 27 × 10 − 2 2 . 77 × 10 − 2 0 . 18 ARA 1 . 53 × 10 − 4 1 . 51 × 10 − 10 8 . 00 × 10 − 2 1 . 52 × 10 − 4 3 . 31 × 10 − 9 0 . 38 − 1 . 08 × 10 − 2 11 . 5 a NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les | ARA: Nx ,θ = 10 6 sam ples. b NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les/s tep | ARA: Nx ,θ = 10 3 sam ples/s tep. c Re f. (NRA/CMC, Nθ = 10 6sam ples, Nx = 10 6 sam ples):  Pf, re f = 1 . 55 × 10 − 4.

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

Fig. 2. Two-degree-of-freedomdampedoscillatorwithprimaryandsecondary sys-tems.

and

ω

s

= (

ks

/

ms

)

1/2anddampingratios

ζ

p and

ζ

s,wherethe

sub-scripts p and s respectively refer to the primary and secondary oscillators.IfFsdenotestheforcecapacityofthesecondaryspring, then thereliabilityof thesystemcan be evaluatedusing the fol-lowinglsf[49,52]: g

(

X

)

=

Fs

−

3ks

×









π

S0 4

ζ

s

ω

3s



ζ

a

ζ

s

ζ

p

ζ

s



4

ζ

a2

+

r2



+

γ

ζ

a2



ζ

p

ω

3p

+ ζ

s

ω

3s



ω

p 4

ζ

ω

4 a



(18)

whereS0 istheintensityofthewhitenoise,

γ

=

ms

/

mp themass ratio,

ω

a

= (

ω

p

+

ω

s

)/

2 the average frequency ratio,

ζ

a

= (ζ

p

+

ζ

s

)/

2 the average dampingratio and r

= (

ω

p

−

ω

s

)/

ω

a a tuning

parameter.TheprobabilisticmodelforX isdetailedinTable 7. Thetwo interesting characteristicsofthisapplicationtest-case areitssetofnon-normalbasicrandomvariablesandthefactthat it suffers froma highly nonlinear limit-statesurface [50] (which preventsfromusinganyFORM-basedapproach).Moreover, follow-ing[51],itseemsrelevanttoconsiderthemeanoftheforce capac-ity

μ

X7 asthemostinfluentdistributionparameteronthefailure probability.Thenominalvalue for

μ

X7 ischosento be21

.

5 N so asto reachareferenceprobabilitywithout parameteruncertainty

pf,ref equalto4

.

75

×

10−5 [51].

Results. NumericalresultssummarizedinTable 7showthat,forthe

samesimulationbudget,ARA/CMCismoreaccuratethanNRA/CMC toestimate the predictivefailure probability (the referenceresult





Pf,ref is provided below the table). As for ARA/SS, a significant gainisnoticeablereferringtothehigh

ν

valuescomparedtounity (

ν

>

10).Inbrief,that meanstheARA/SSisveryefficienttotreat thisproblemcomparedtoa classical MonteCarloapproach.A fi-nal remark concerns the comparisonbetween the two reference probabilitiespf,ref

=

4

.

75

×

10−5 and





Pf,ref

=

1

.

55

×

10−4:onecan seethat,inthiscase,consideringuncertaintyonadistribution pa-rametermakesthesystemlesssafe,whichcanbe,forexample,an importantindicatorfordesignorre-designpurposes.

4.3. Applicationonalaunchvehiclestagefallbackzoneestimation

Description. Spacelaunchercomplexityarisesfromthecoupling

be-tween several subsystems such as stages or boosters and other embedded systems. Optimal trajectory assessmentis a key disci-pline since it is one of the cornerstones of the mission success (for ascent as well as for re-entry trajectories). However, during therealflight,aleatoryuncertaintiescanaffectthedifferentflight phasesatdifferent levels (e.g.,on the dynamicsperturbations or stage combustion) andbe combined to lead to a failure state of the space vehicle trajectory. After their propelled phase, the dif-ferentstagesreachsuccessivelytheir separationaltitudesandmay

fall back intothe ocean (see Fig. 3). Such a dynamic phase is of Tab

le 5 Re su lt s for the la unc h ve h ic le sta g e fallbac k te st -case. Appr oac h CMC a FO RM S S b m P f S 2  Pf RE m P f S 2  Pf RE RB ν m P f S 2  Pf RE RB ν NRA c 1 . 18 × 10 − 4 8 . 82 × 10 − 11 7 . 97 × 10 − 2 9 . 16 × 10 − 3 1 . 11 × 10 − 4 1 . 15 75 . 32 − 1 . 13 × 10 − 4 4 . 98 × 10 − 9 0 . 63 − 6 . 09 × 10 − 2 5 . 65 × 10 − 2 ARA 1 . 27 × 10 − 4 1 . 05 × 10 − 10 8 . 07 × 10 − 2 8 . 28 × 10 − 5 −− − 0 . 35 − 1 . 25 × 10 − 4 1 . 52 × 10 − 9 0 . 31 − 1 . 57 × 10 − 2 20 . 6 a NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les | ARA: Nx ,θ = 10 6 sam ples. b NRA: Nθ = 10 3sam ples, Nx = 10 3 sam p les/s tep | ARA: Nx ,θ = 10 3 sam ples/s tep. c Re f. (Ga uss–Hermit e with Nθ = 1009 sam p les + SS with Nx = 10 4 sam ples/s tep):  Pf, re f = 1 . 20 × 10 − 4. Ta b le 6 Re su lt s for the collision te st -case. Appr oac h CMC FORM S S m P f S 2  Pf RE m  Pf S 2  Pf RE RB ν m P f S 2  Pf RE RB ν NRA untr a ctable −− untr actable −− − − untr actable −− −− ARA 3 . 55 × 10 − 4 a 4 . 33 × 10 − 9 0 . 19 0 . 10 −− 293 − 3 . 68 × 10 − 4 b 6 . 96 × 10 − 8 0 . 72 3 . 74 × 10 − 2 1 . 32 a ARA: Nx ,θ = 10 5sam ples. b ARA: Nx ,θ = 2 × 10 − 3 sam ples/s tep.

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

Fig. 3. Illustration scheme of the first stage fallback phase into the Atlantic Ocean.

Table 7

Second-orderstatisticsanddistributionsofinputrandomvariablesforthenonlinear oscillatortest-case.

Variable Xia _{Distribution Meanμ}

Xi CvδXi b X1=mp(kg) Lognormal 1.5 10% X2=ms(kg) Lognormal 0.01 10% X3=kp(N m−1) Lognormal 1 20% X4=ks(N m−1) Lognormal 0.01 20% X5= ζp(1) Lognormal 0.05 40% X6= ζs(1) Lognormal 0.02 50% X7=Fs(N) Lognormal μX7uncertain c _10% X8=S0(m s−2) Lognormal 100 10% =μX7 (N) Normal 21.5 10%

a _{Thebasicvariablesareindependent.}

b _{Notethatthecoefficientofvariation(Cv)ofX}

i isdefinedbyδXi=σXi/|μXi| forμXi =0.

c _{Forafixedvalueμ}

X7=21.5,pf,ref=4.75×10−5.

utmost importance in terms of launcher safety since the conse-quenceofamistake inthepredictionofthefallback zonecan be dramatic in terms ofhuman security and environmental impact. Forthatreason, the handlingofuncertainties plays acrucial role inthecomprehensionandpredictionoftheglobalsystem behav-ior. Thatis thereason whyit is ofutmost importance to take it intoaccountduringthereliabilityanalysis.

Theblack-boxmodel

M(·)

consideredhereisatrajectory sim-ulationcode ofthedynamic fallback phase ofa generic launcher firststage. Fortheinterested reader,two differentbutclose test-cases(withdifferentnumericalvalues)areusedin[4]andin[53]. Asaninputvectorofthesimulationcode,thefollowingbasic vari-ables representing the initial conditions and the launch vehicle characteristicswillbepassedthroughthecode:

X1:stagealtitudeperturbationatseparation(



h (m)); X2:velocityperturbationatseparation(



v (m s−1)); X3:flightpathangleperturbationatseparation(



γ

(rad)); X4:azimuthangleperturbationatseparation(

ψ

(rad)); X5:propellantmassperturbationatseparation(



m (kg)); X6:dragforceerrorperturbation(



Cd dimensionless).

Moreover,inthiscase, themean valuesofthebasicvariables X2 andX3areconsideredasuncertain(seeTable 8)astheyare phys-icalquantitiesdifficulttomeasureandto controlinreality.As an output,thecodewillgivebackthedistanceDcode

=

M(

X

)

,which is alsoa random variable, betweenthe theoretical fallback

posi-Table 8

Second-orderstatisticsanddistributionsofinputrandomvariablesforthelaunch vehiclestagefallbacktest-case.

Variable Xia _{Distribution Meanμ}

Xi StdσXi

X1= h (m) Normal 0 1650

X2= v (m s−1) Normal μX2uncertain

b _3.7

X3= γ(rad) Normal μX3uncertain 0.001

X4= ψ(rad) Normal 0 0.0018

X5= m (kg) Normal 0 70

X6= Cd(1) Normal 0 0.1

2=μX2(m s−1) Normal 0 3.7

3=μX3(rad) Normal 0 0.001

a _{Thebasicvariablesareindependent.} b _{Forfixedvaluesμ}

X2=0 andμX3=0,pf,ref=2.31×10−7.

tionintotheoceanandtheestimatedoneduetotheuncertainty propagation.ThesystemfailureisconsideredifthedistanceDcode exceedsathresholdsafetydistancedsafe:

g

(

X

)

=

dsafe

−

M

(

X

)

=

dsafe

−

Dcode

.

(19)

Inthenumericalexperiment,thethresholdsafetydistancedsafe is chosen tobe equal to 20 km so astoreach a reference prob-ability withoutparameter uncertainty pf,ref equal to 2

.

31

×

10−7 (estimated by CMC with108 samples and confirmedby SS with 103 samples/step).

Results. Numerical results gathered in Table 5 show that both

NRA/CMC and ARA/CMC give similar results andmanage to cor-rectly estimatethe predictivefailure probability(whose reference value is given in Table 5). NRA/SS andARA/SS even if a signifi-cant value of the efficiency (

ν

>

20) for ARA/SS which indicates howpromising istheuseofARA/SSwithsuch anindustrial test-case.AsforNRA/FORMandARA/FORM,theybothgivepoorresults. ARA/FORMgivesatleastan order ofmagnitudeofthe predictive failureprobabilityquiteclosetothereferenceone.

Throughthistest-case,onecan illustratethebudgetallocation problem which appears in NRA. Classical reference failure prob-ability withoutparameter uncertainty pf,ref is very low (order of magnitude of 10−7, see Table 8) andcan be time-consuming to get,especially withan expensive computer model.Adding a sec-ondintegrationloopover

D

θ togetthereferencepredictivefailure

probability





Pf,refcanbeuntractableinacontextofrareevent.The problemconcerningthesimulationbudgetallocation(betweenthe twodomains,

D

X and

D

)canbeanobstacletoanaccurate

1 67 2 68 3 69 4 70 5 71 6 72 7 73 8 74 9 75 10 76 11 77 12 78 13 79 14 80 15 81 16 82 17 83 18 84 19 85 20 86 21 87 22 88 23 89 24 90 25 91 26 92 27 93 28 94 29 95 30 96 31 97 32 98 33 99 34 100 35 101 36 102 37 103 38 104 39 105 40 106 41 107 42 108 43 109 44 110 45 111 46 112 47 113 48 114 49 115 50 116 51 117 52 118 53 119 54 120 55 121 56 122 57 123 58 124 59 125 60 126 61 127 62 128 63 129 64 130 65 131 66 132

methods.Forinstance,forafixedsimulationbudget,oneneedsto decidewhether toallow a substantialbudget (i.e. moresamples) togetabetterprecision overtheintegralon

D

X oroverthe

inte-gralon

D

.Ontheonehand,becausethesimulationbudgetover

D

X isnot easilyreducibleasitdirectlyaffectstheestimation

ac-curacyofthe failure probability,itcomes thataddingthe second integrationbudgetover

D

canbecomputationallycritical.Onthe

otherhand,samplingwithonly afewnumberofpoints over

D



mayintroduceabiasinthefinal measureofreliabilityby advan-tagingsomeparametervalueswhichinfluencethefinalprobability measurewithouttakingtheirrelative weightintoaccount.Forall thesereasons,adesignofexperiments (DOE)hasbeenconductedso astooptimizethesamplingon

D

.Todoso,wedecidedtousea

quadraturescheme-basedDOEover

D

.Theideaistoapproximate

ak-variateintegralover

D



⊆ R

koftheform:

I

[

Pf

(θ )

] =



D

Pf

(θ )

f

(θ )

d

θ

(20)

where f

(θ )

≡

w

(θ )

is a density (or weight) function which

is evaluated on gridpoints. The quadrature rule provides an ap-proximation usinga combinationof theseweight functions such that[54]:

I

[

Pf

(θ )

]

≈

M1



j1=1 M2



j2=1

· · ·

Mk



jk=1

(

wj1

⊗

wj2

⊗ · · · ⊗

wjk

)

×

Pf

(θ

₁(j1)

, θ

₂(j2)

, . . . , θ

_k(jk)

)

(21) with wj the weights and

⊗

thetensor productoperator. The

in-dicesM1

,

. . . ,

Mk representsthenumberofpointsineach

dimen-sion.Inthespecific caseoflaunchvehiclestage fallback,wehave two uncertaindistribution parameters, which means that the in-tegration domain is

R

2. The quadrature type is chosen to be a

Gauss–Hermitequadraturescheme,whichmeansthat weuse

Gaus-sian weights [55,56]. Depending on the problem dimensionality, one canchoose an accuracylevel Macc which allowsto integrate complete polynomials oftotal order 2Macc

−

1 exactly. Here, we

choseanaccuracylevelMacc

=

14 soastoprovideenoughsamples

(here,itcorrespondsexactlyto1009 samples[55])tocoverthe do-main

D

.Suchachoiceisconstrainedbytheexpensiveaspectof

thecomputer code.However, fordifferentapplications,onecould choose another accuracy level. Finally, coupling this DOE with a SSmethodwith104 samples/stepallowstoestimatethereference predictivefailureprobability





Pf,ref.Alastremarkconcernsthefact thattakingonlytwoparametersoutofsixbasicvariablesasbeing uncertainimpliestoincrease thefailureprobability ofthree loga-rithmicdecadesin termsofmagnitudecompared tothe classical referenceestimate.Again,thatemphasizeshowcrucialtaking dis-tribution parameters uncertainty is during the reliability analysis phase.

4.4. Applicationonestimationofcollisionprobabilitybetweenorbiting

objects

Description. Because of the drastic growth of the number of

or-biting objects (cataloged anduncataloged space debris) over the pastfewdecades,thenumberofpotentialcollisionbetween satel-litesandotherorbitingobjectsincreased(seeFig. 4).Spacedebris surveillance and management is one of the key issue and is di-rectlylinkedtotherareeventprobabilityestimationtopic.

In this test-case, the failure scenario concerns a collision be-tween a space debris and a satellite, both orbiting around an Earth-centeredinertialreferenceframe.Thedynamicalmodelused

Fig. 4. Pictureextractedfromavideo(Thestoryofspacedebris)byESA© (http://

www.esa.int).

Table 9

Second-orderstatisticsanddistributionsofinputrandomvariablesforthecollision test-case.

Variable Xia _{Distribution Meanμ}

Xi StdσXi

X1=rdeb,1(km) Normal μX1 uncertain

b _10%

X2=rdeb,2(km) Normal μX2 uncertain 10%

X3=rdeb,3(km) Normal μX3 uncertain 20%

X4=vdeb,1(km s−1) Normal 0 20% X5=vdeb,2(km s−1) Normal 0 40% X6=vdeb,3(km s−1) Normal 0 50% 1=μX1(km) Normal 0 0.0625 2=μX2(km) Normal 0 0.04 3=μX3(km) Normal 0 0.000625

a _{LinearcorrelationstructuregivenbythematrixR inEq.(22).} b _{Forfixedvaluesμ}

X1=μX2=μX3=0,pf,ref=3.4×10−4.

hereisadeterministicmodel

M(·)

computingtheminimum dis-tance Dmin between the debris and the satellite during a given time span

τ

. For the interested reader, two other applications (withdifferentnumericalvalues)ofasimilartest-casearetreated in[4]andin[34]andanotherapproachtoestimatethe probabil-ity ofcollision can be found in [57]. Assuming that the position andthespeedofthesatellite areperfectlyknown,theinput vec-tor of basicvariables gathers thethree components ofthe space debrispositionvectorandthethreecomponentsofitsspeed vec-tor[58].Here,themeanvaluesofthespacedebrispositionvector are uncertain (see Table 9) because of the difficulty to measure accurately thesequantities.Asforthecorrelationstructure ofthe basicvariables, itisgivenby thefollowinglinearcorrelation ma-trixR

= [

ρ

i j

]

i,j∈{1,...,d}: R

=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

1 0

.

97 0

.

12 0

.

84

−

0

.

85

−

0

.

98 1 0

.

10 0

.

73 0 0 1

−

0

.

14

−

0

.

47

−

0

.

11 1

−

0

.

79

−

0

.

81

(

sym

.)

1 0

.

83 1

⎤

⎥

⎥

⎥

⎥

⎥

⎦

.

(22)

Thusthelsfcanbeexpressedasfollows:

g

(

X

)

=

dcollision

−

M

(

X

)

=

dcollision

−

Dmin

.

(23) In the numerical experiment, the threshold collision distance dcollision is chosen to be equal to 20 m so as to reach a ref-erence probability without parameter uncertainty pf,ref equal to 3

.

4

×

10−4 (estimated by CMC with 106 samples andconfirmed withaSSwith104 samples/step).

Results. This industrial test-case can be considered as the worst

case here since it involves sixcorrelated basicrandom variables withthreeuncertaindistributionparameters.Numericalresultsare given in Table 6. Again, since we know the conditional marginal