[American Institute of Aeronautics and Astronautics 8th Symposium on Multidisciplinary Analysis and...

(c)2000 American Institute of Aeronautics & Astronautics or Published with Permission of Author(s) and/or Author(s)' Sponsoring Organization.

AOO-40038 AIAA-2000-4759

System Reliability Prediction for a Design Environment:Use of Muti-Point Approximations

Ravi C. Penmetsa*, and Ramana V. Grandhif

pravi @ cs. wright. edu, rgrandhi @ cs. wright. eduWright State University, Dayton, OH 45435

AbstractThe objective of this paper is to develop a

method for the system reliability analysis that issuitable for design environment. System reliabilityinvolves more than one-failure criteria, which may befrom the same discipline, or from different disciplines.The reliability analysis using numerical integrationtechniques that produce exact results require largecomputational effort for a single limit-state (failurecriteria), therefore their use for multiple limit-states isnot viable. Due to this reason, various approximationshave been developed in the past to reduce the expensivecomputer simulations. Most of the system reliabilitymethods developed in the past are valid for linear orclose to linear limit-states. In this paper, a methodbased on nonlinear function approximations isdiscussed which is suitable for complex responses. Amulti-point approximation is used to capture the failuredomain accurately and then the Monte Carlo simulationis used to estimate the structural failure probability.Three examples are presented to show the applicabilityof the method.

Introduction

A real structure typically consists of manycomponents, of which, each has the potential to fail,and the individual component failure might lead tostructural failure. Even in simple structures composedof just one element, various failure modes such asbending action, buckling, axial stress, temperature,frequency, etc., may exist and be relevant in thesolution. The composition of many elements instructures is referred to as a "structural system" and asystem may be subject to many forms of loads, eithersingle or various combinations. Therefore, the

* Graduate Research Assistant' University Professor, Associate Fellow

This paper is declared a work of the U.S. Government and isnot subject to copyright protection in the United States.

reliability analysis of structural systems will involveconsideration of multiple, perhaps, correlated limit-states which can be defined in any discipline. Thesystem failure probability estimation involves largecomputational effort and methods are currently beingdeveloped to reduce this computational time.

One of the important applications of theprobabilistic methods is the evaluation of the systemreliability, which is made up of components withknown reliabilities. Most commonly, systems areclassified into two groups as (i) series system, and (ii)parallel system.

The series system is the one in which even ifone component fails to perform satisfactorily, the wholesystem will fail. This is also called a weakest linkmodel. As every component should functionsatisfactorily for the system to be reliable, the failureprobability of every component is estimated usingvarious approximation techniques. In this paper, FirstOrder Reliability Method (FORM) is used to estimatethe component failure probability inorder to comparethe results with the proposed method. It is evident that astatically determinate structure is a series system sincethe failure of any one of its members implies failure ofthe structure.

In the case of a parallel system, the systemsurvives even if one component has failed. The systemfails to function satisfactorily only when everycomponent of the system has failed to functionsatisfactorily. The parallel systems are sometimesreferred to as redundant systems. There are two types ofredundancy, active redundancy and passiveredundancy. Active redundancy occurs when redundant

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elements actively participate in structural behavior evenat low loading. In addition, passive redundancy occurswhen the redundant elements do not come into playuntil the structure has suffered a sufficient degree ofdegradation or failure of its elements. A system that is acombination of both series and parallel components iscalled a mixed system.

In structural system reliability analysis, thebound methods and numerical integration methods havepractical significance. In this, the failure probability ofthe individual components has to be estimated andFORM is considered in this paper. If the componentsof the system are assumed independent, then the systemfailure can be obtained easily. However, in practicalproblems, the failure conditions depend on the samerandom variables and therefore the components arecorrelated. Cornell1 has developed bounds on thesystem failure probability for systems subjected tomultiple failure modes. The upper bound on the systemfailure was obtained by assuming perfectly correlatedcomponents and this is obtained as

Upper Bound on Pf - Max [Component P, ]

The lower bound is obtained by assuming statisticallyindependent components.

nLower Bound on Pf - N [Component P, ], where n

iis the number of failure modes. If all the componentsare perfectly independent then the failure probability ofthe component can be determined by using theapproximation techniques developed by the authors inearlier work2.

In FORM, the limit-state is approximated witha linear function at the most probable failure point(MPP). The MPP is the point on the limit-state that isnearest to the origin in a standard normal spaceobtained by suitable transformations to the randomvariables. Due to rotational symmetry and exponentialdecay of the probability density in the standard normalspace, the MPP has the highest likelihood of failureamong all points in the failure domain. Therefore, theneighborhood of the MPP makes a major contributionto the failure probability integral. This property is thebasis for FORM, which in effect constructs anapproximation to the failure probability integral byusing the tangent plane at the design point as theintegration boundary. The Second-Order ReliabilityMethod (SORM) improves on this approximation by aquadratic surface using the second-order gradients. Insystem failure probability, the probability of failure of

each of the individual limit-states are estimated usingeither the FORM or SORM technique.

The method of narrow bounds presented byDitlevsen for the system failure probability had widerapplicability due to its high accuracy. These boundsconsidered the correlation between each of the twofailure modes, becoming more physically reasonable.Using this method, the system failure probability can beexpressed from the bounds of first-order or second-order joint probabilities. However, these bounds arequite accurate only when the limit-states are of linearform. In situations where this assumption is not valid,the alternate procedures have to be developed toestimate the failure probability.

Inorder to improve the accuracy of theDitlevsen's bounds both theoretically and practically,Feng has developed a method using third-order jointprobability for computing the system failureprobability. This method uses the first-, second-, andthird-order joint failure probabilities to estimate thefailure probability accurately. For problems, where thesecond and third-order joint probabilities can beestimated accurately using short computer run-times,the resulting accuracy is high.

The system failure probability obtained usingDitlevsen's method when correlation's among thefailure modes are less than 0.6, has narrow bounds,otherwise it has wide bounds. Similarly, the boundsobtained by the Feng's method are accurate when thejoint failure probabilities could be estimated accurately.However, in most circumstances the formulae forcomputing the second- and third-order jointprobabilities have large errors. Therefore, Song hasproposed a method using numerical integration in areduced domain of failure region. Song proposed toreduce the failure domain by a factor of safety index inevery direction and later used numerical integration inthe reduced domain. This method reduces the numberof actual simulations and gives accurate results for lessnumber of failure modes. The computer time of thismethod increases exponentially with the number offailure modes. Therefore, when the structure has manyfailure modes, this method cannot be directly used forcomputing the system failure probability. He hasproposed a method to deal with this drawback, howeverthat method required second and third order joint failureprobabilities. The alternate method uses the FORMfailure probability, which introduces errors.



When dealing with highly nonlinear problemswith a large number of non-normal random variablesand implicit limit-state functions, both the FORM andSORM approximations fail to give accurate results.Therefore, better approximations such as two-pointadaptive non-linear approximations (TANA2 orTANAS ) have to be used to approximate the limit-state functions. The approximations capture theinformation of the limit-state accurately in the vicinityof the MPP. When dealing with multiple limit-states,information about the MPP of each limit-state is vitalfor the accurate estimation of the system failureprobability. Therefore, the two-point approximation isused as a local approximation at each of the MPPs ofevery limit-state and then the multi-pointapproximations (MPA) are constructed . This multi-point approximation retains the information of each ofthe failure surfaces and constructs a joint failuredomain. Since this joint failure domain is constructedusing more accurate approximations of the individualfailure domains, it can be integrated using the MonteCarlo simulation technique to obtain the system failureprobability. The reduction in computational cost ofsystem reliability prediction greatly helps in thepreliminary design of multifunctional structures.

Basic Idea of the Proposed Method

The Monte Carlo simulation is the mostreliable method for component/system reliabilityprediction. Monte Carlo simulation can be performedon an approximate limit-state function if it matches thefunction value and the gradient at the MPP and it isaccurate enough over a large range of the failuredomain. Therefore, a Multi-Point approximation(MPA) described in appendix, is considered torepresent a larger failure region.

The MPA can be written using the followinggeneral formulation:

Naturally, the accuracy of a localapproximation is one of the primary factors on whichthe quality of the MPA is dependent. Therefore, Two-Point Adaptive Non-linear Approximations (TANA2)were used as local approximations to construct theMPA of each limit-state function. The TANA2 cancapture the information of the limit-state accurately inthe vicinity of the data points. MPA retains theinformation of each of the failure surface withoutincreasing the computational effort. Since this jointfailure domain is constructed using more accurateapproximations of the individual failure domains, it canbe integrated using the Monte Carlo simulationtechnique to obtain the system failure probability.

The MPA adaptively adjusts itself to behave asa local approximation when a design point is close toone of the data points. Function and gradient values ofthis MPA correspond directly with their exactcounterparts at the points where the localapproximations were generated.

Most reliability analysis methods begin withthe prediction of the most probable failure point. TheMPP of each limit-state function can be efficientlyestimated using the algorithm presented by Wang and

o

Grandhi . This algorithm uses a two-pointapproximation, TANA2, of the actual limit-state in thesearch procedure in-order to reduce the computationaltime. This method is very efficient when dealing withhighly nonlinear implicit problems with a large numberof random variables. In the process of searching for theMPP of each limit-state function, a series of data pointinformation including the function values and gradientshave been obtained. In this research, TANA2 is thelocal approximation constructed at the data points thatare obtained in the process of searching for the MPP.Once the local approximations are obtained, an MPA isconstructed that contains the information of all the localapproximations.

F ( X ) = W t ( X ) F , ( X ) (1)

where Fk(X) is a two-point local approximation andWk is a weighting function that adjusts the contributionof Fk(X) to F ( X ) i n Eq. 1. The evaluation of thisweighting function involves the selection of a blendingfunction and a power index "m". The procedural detailsfor evaluating the weighting function are discussed inappendix.

Main Steps of System ReliabilityCalculation

1. Estimate the MPP of each limit-state function. TheMPP is obtained by using the algorithm presented

Q

by Wang and Grandhi .

2. In the process of searching for the MPP of eachlimit-state, the information (function value andgradient) of a number of points of each limit-statefunction are obtained. Local TANA2



approximations at the points obtained in theprocess of searching for the MPP are used.

3. Once the local approximations are constructed,weighting functions that are required to constructthe MPA are evaluated. One weighting function isrequired for each of the local approximation. Theweighting function controls the influence of eachlocal approximation at a particular point in thedesign space. With the same process, a MPA canbe constructed for each limit-state function.

4. After the MPAs for each limit-state function areobtained, the Monte Carlo simulation is performedon the approximate limit-state functions, whichclosely represent the actual limit-states at the MPPand the data points.

Three examples are provided to show theapplicability of the proposed method. This method canbe applied for problems with multiple non-normalrandom variables and implicit/explicit limit-states. Theaccuracy of the method depends on the accuracy of theMPA of each limit-state function.

MPA and the Monte Carlo simulation is performed onthe approximate limit-state functions.

Example 1:In this example, a system reliability problem

with two limit-states is considered. The two limit-statesconsidered are functions of two variables, which arenormally distributed random variables where Xl has a

mean of 10.0 and a standard deviation of 4.0 and X2

has a mean of 10.0 and a standard deviation of 5.0. Thelimit-states considered are the following:

1.2.

= Xl3 +X2

3- 500. > 0.0 Safe= X1X2-40.>0.0 Safe

For each of the limit-state functions, two localapproximations (TANA2) are used to construct theMPA. This result is compared with the actualprobability of failure obtained using the Monte Carlosimulation.

3000

2500

2000

1500

1000 [

X2

10 12 14

-10

7

X,

10 12

Figure 3a: Limit State Gi(X) Figure 3b: Limit State G2(X)

Numerical ExamplesVarious examples have been studied to prove

the efficiency and accuracy of this method. Thismethod can produce an accurate value of the probabilityof failure unlike other methods, which only produce thebounds on its value. The final failure probability of thesystem estimated using MPA is compared with theresults obtained directly from the Monte Carlosimulation. Each limit-state is approximated using the

Method

Monte Carlosimulation

MPA (m=2.0)

First-order SeriesBounds

Probability ofFailure0.2406

0.2353

0.1318 to0.2366

% Difference

-2.20

-45.22 to-1.66

Table 1: MPA and Monte Carlo Results



Table 1 shows the results obtained using theMPA and the Monte Carlo simulation. In this case,50,000 simulations were performed for the Monte Carlosimulation and the converged value of the probability offailure is taken as the exact system failure probability.The percent error is the percentage deviation of theMPA result from the Monte Carlo Simulation. For thisexample, the plots, shown in Figure 3a and Figure3b, ofthe exact limit-state functions and the MPA of the firstlimit-state are almost identical which shows that MPAapproximates the joint failure domain accurately. Thesecond limit-state function approximation is alsoaccurate around the MPP as shown in the figure. Theplots were generated at the MPP by varying X2.

Example 2; Cantilever BeamA cantilever beam shown in Fig. 4 is subjected to a tipload P = 80. Ib. Three failure modes, the displacement,stress and frequency are considered as

4. PU'Ebh3

12.PLbh2

and frequency

- 0.15 < 0.0, displacement

-104 <Q.Q, stress

ElpAL4

where L, b, and h are the length, width and height of thebeam with mean values of 30", 0.8359" and 2.5093",respectively and the Young's Modulus, E, is 107. Thelength, width and height of the beam are considered asthe random variables, and the standard deviations are,crL=0.3", <rb=0.08" and <rh=0.25" respectively.

In this example each of the limit-states areapproximated using MPA. To improve the accuracy ofthe approximations two additional points are added toeach of the MPA. These two points are the MPP of theother two limit-states when one MPA is beingconstructed. This procedure improves the accuracy ofthe MPA at each of the three MPP's. The localapproximations constructed at the intermediate pointsare TANA2 for displacement and stress constraints. Afirst-order approximation is constructed at each of theother two MPP's and then these local approximationsare added to the MPA.

LFigure 4: Cantilever Beam

The frequency constraint is not dependent onthe width of the beam therefore there is no change inthe value of the width from the mean and the gradient iszero. For that reason, a first-order approximation isconsidered instead of TANA2 approximation becauseTANA2 is not applicable when the gradients are zero.These first-order approximations are considered at eachof the intermediate design points and the MPP's of thedisplacement and stress constraints.

Method


MPA (m=2.0)



0.03097

0.02888 to0.03612

%Difference

3.89

-3.11 to20.51

Table 2: MPA and Monte Carlo Results for CantileverBeam

The above table clearly shows the accuracy ofthe proposed MPA method compared to the First-orderseries bounds. The MPA results are quite comparable tothe Monte Carlo simulation results and the error was3.89%. 100,000 Monte Carlo simulations are performedon the exact limit-state functions and the MPAapproximations. The first-order series bounds are notaccurate because neither of the limit-states consideredwere linear functions of the random variables.

Example 3: Ten-Bar Truss StructureThe system failure probability of a ten-bar

truss, shown in Fig. 4, was calculated in this example.The cross-sectional areas of all the ten truss membersare normally distributed random variables with 2.5 inmean value and 0.5 standard deviation. The Young's



modulus is 107 psi and the forces applied arePj — P2 = 10 lb., as shown in Fig. 4. Two limit-states have been considered to estimate the systemfailure probability. One is the displacement limit andthe other is the eigenvalue limit. The maximumdisplacement of the tip of the truss structure should beless than 1.8 in, and the eigenvalue must be greater than175 (rad/sec)2.

——-1-0>0.01.8

>0.0175.0

good local approximation of the individual limit-statesensures good system failure probability estimation.

360" 360"

360"

1.0E+041b 1.0E+041b

Method


MPA (m=2.0)



0.172

0.08622 to0.1524

%Difference

-6.52

-53. 19 to -17.32

Table 3: MPA and Monte Carlo Results for Ten-barTruss

The structural analysis is done usingASTROS, a finite element analysis program. Table 3compares the results obtained by different methods tothe Monte Carlo method. Since this is a problem withimplicit limit-state functions, TANA2 approximationsare constructed at the data points obtained in theprocess of searching for the MPP. These localapproximations are blended together using the MPAmethod. There are two MPAs, one of whichcorresponds to the displacement limit-state and theother corresponds to the fundamental frequency. In thiscase, 50,000 simulations were used to estimate thesystem failure probability using the Monte Carlomethod.

Using the MPA method, the system failureprobability obtained was 0.172, which is a 6.52%difference from the actual value. As seen in Table 3, thefirst-order series bounds are not accurate becauseneither of the limit-states are linear functions. However,the MPA approximations were accurate enough and theMPA approximation could integrate the individualfailure domains to model the joint failure domain. A

Figure 5: Ten-bar Truss

SummaryThe use of MPA has enabled modeling of the

n-dimensional joint failure domain for using the MonteCarlo simulation. This approximation reduces aconsiderable amount of computational effort (MPA ofeach limit-state function is explicit) without sacrificingmuch accuracy. Because information at more points isused to construct MPA of each limit-state function,MPA is accurate over a larger region. It is possible togive a good prediction of the intersection point ofdifferent limit-state functions.

Using the available methods in literature, thebounds on the system failure can be obtained. Thesebounds are estimated by using approximationtechniques unlike the n-dimensional integration, whichis more accurate. This may lead into an additionaluncertainty in the bounds. In MPA, the failureprobability of the system is available as a single valueand it takes into account the correlation between thelimit-states. After solving a class of problems withMPA, it is possible to understand where the resultstands in comparison to the Monte Carlo simulation,and provide the expected level of accuracy using MPA.

Based on experience a bound can be specified on theaccuracy of this approximate result and this willproduce a considerably small range to make a decisionon the safety of the system.

MPA has a tremendous potential for problemswhere the limit-states are not unimodal and exhibit highnonlinearity. In those cases, the MPP search starts froma mean point and approaches from one side of the



nonlinear surface. All the points generated in the searchmay represent very small region of the nonlineardomain. If an approximation is built using the searchedfailure points, then the system reliability may notinclude the complete failure region. In cases whereTANA2 converges rapidly to the MPP, the entiredomain may not have been investigated. In thosesituations, a design of experiments approach forchoosing the points for building several localapproximations is appropriate. The idea is to capturethe failure region accurately using multipleapproximations.

AcknowledgementsThis research work has been sponsored by

U.S. Air Force under contract F33615-98-C-2895. Thesupport for the Graduate Research Assistant wasprovided by the Dayton Area Graduate Studies Institute(DAGSI).

References1. Cornell, C. A., "Bounds on the Reliability of

Structural Systems," Journal of Struct. ASCE, Vol.93 (1), February 1967, pp. 171-200

2. Penmetsa, R. C., Zhou, L., and Grandhi, R. V.,"Adaptation of Fast Fourier Transformations toEstimate the Structural Failure Probability," 41stAIAA/ASME/ASCE/AHS/ASC Structures,Structural Dynamics, and Materials Conference,Atlanta, Georgia, April 3-6, 2000

3. Ditilevsen, O., "Narrow Reliability Bounds forStructural System," Journal of StructuralMechanics, Vol.7, 1979, pp. 453-472

4. Feng, Y., "A Method for Computing StructuralSystem Reliability With High Accuracy,"Computers and Structures, Vol. 33 (1), 1989, pp.l-5

5. Song, B. F., "A Numerical Integration Method inAffine Space and a Method with High Accuracyfor Computing Structural System Reliability,"Computers and Structures, Vol. 42 (2), 1996,pp255-262

6. Wang, L. P., and Grandhi, R. V., "Improved Two-Point Function Approximations for DesignOptimization", AIAA Journal, Vol. 33(9), 1995, pp.1720-1727

7. Xu Suqiang, and Grandhi, R. V., "Multi-PointApproximation for Reducing the Response SurfaceModel Development Cost in Optimization",

Proceedings of the 1st ASMO UK/ISSMOConference on Engineering Design Optimization,Ilkley, West Yorkshire, UK, July 8-9, 1999, pp.381-388

8. Wang, L. P., and Grandhi, R. V., "Safety IndexCalculation Using Intervening Variables forStructural Reliability Analysis", Computers andStructures, Vol. 59(6), 1996, pp. 1139-1148

9. Rasmussen, J., "Nonlinear Programming byCumulative Approximation Refinement",Structural Optimization, Vol.l5(l), 1998, pp. 1-7

Appendix: Multi-point ApproximationBased on Local Approximations

The multi-point approximation can beregarded as the connection of many localapproximations. With function and sensitivityinformation already available at a series of points, onelocal approximation is built at each point. All localapproximations are then integrated into a multi-pointapproximation by the use of a weighting function. Theweighting functions are selected such that theapproximation reproduces function and gradientinformation at the known data points.

The local approximations discussed in thissection are TANA26 approximations. The

^)f?{ V\

function F(X) and gradient ———— information isdx

available at-X*= (x^h x2,b —, xn,k)T, k- 1,2,..., K.The multi-point approximation can be written in termsof the local approximations as,

where Wj. is a weighting function

(A2)

and Fk (X) is the TANA2 approximation.



Wk( X ) adjusts the contribution of Fk (X) to

F (X)in Eq. Al. 0k(X) is called a blendingfunction and has its maximum of 1 at X^ and vanisheswhen Xk is very far from X.

The important details of the TANA26

approximation are presented below. Further details canbe found in Ref. 6. The physical variables aretransformed to the intervening variables using therelation

= x i = 1,2,..., n

where the exponents pt represent the nonlinear indicesand are different for each variable. Information at twopoints, namely, the comparison point (X^and theexpansion point (X2), is used in building theapproximation. The approximation is obtained byexpanding the function at the expansion point X2 as

1=1

(A3)This equation is a second-order Taylor series

expansion in terms of the intervening variablesJi (Vi ~ xi' ) ' m which the Hessian matrix has onlydiagonal elements of the same value £ . Therefore, thisapproximation does not need the calculation of thesecond-order derivatives. Unlike the original second-order approximation, this approximation is expanded interms of the intervening variables y{, to improve theaccuracy. The error from the approximate Hessianmatrix is partially corrected by adjusting thenonlinearity index pi. In contrast to the true quadraticapproximation, this approximation is closer to theactual function for highly nonlinear problems due to itsadaptability.

Eq. A3 has n +1 unknown constants, so n +1equations are required. By differentiating Eq. A3,n equations are obtained by matching the derivativesavailable at the previous point, Xl:

dx

i,2'i,l

i = l,2,...,n (A4)Another equation is obtained by matching the exact andapproximate function values with the previous pointX,:

(A5)There are many ways to solve these n + l equations assimultaneous equations. Here a simple adaptive searchtechnique was used to solve these equations .

Both the function and the derivative values at twopoints were used to construct the approximation. Theexact function and derivative values are equal to theapproximate function and derivative values,respectively, at the previous and current points. Thistwo-point approximation demonstrated a higheraccuracy compared to other forms of two-pointapproximations. This was accomplished using severalexplicit functions and structural shape optimizationproblems6.

New blending functions in Eq. A2 were used tomake the MPA reproduce the exact function andgradient values at the data points where the localapproximation was built. There are at least threeblending functions that could meet this requirement.They are

1 (A6)Exp(hk)-l

1

and <pk

where

(A7)

(A8)

(Xi-xijcr)m (A9)i=i

where m is a positive integer. Additionally, it isrecommended, from computational consideration, thatthe design space be normalized as xte [0,1] to measure



the weighting function. Equation A8 was used in thiswork.

With each of Eqs. A6-A9, the weightingfunction of Eq. A2 has the properties

(A10)= 8kj

0<Wk(Xj)<l

K (All)

(A12)

The weighting function varies between 0 and1, and the summation of all weighting functions is 1.The following properties can be shown for eachblending function given in Eqs. A6-A8.

(A13)

dx,

i = 1, 2, ..., n (A14)

From Eqs. Al, A10, A13 and A14, the following areobtained as

(A15)

(A16)

j = 1, 2,..., K

dF(Xj} =

dx; dx,ti = 1, 2,.... n; j=l, 2, ..., K

Equations A15 and A16 show that the multi-pointapproximation has the same zero-order and first-orderinformation as the original function at the data points.

(All)k=i

Differentiating Eq. 1. The MPA is an average value of all TANA-2estimations when a design point is far from every datapoint.


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