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Confidence Intervals for Reliability Estimated Using Response Surface Methods

S. Vittal* and P. Hajela**

Mechanical, Aerospace and Nuclear Engineering Rensselaer Polytechnic Institute, Troy, NY 12180

Abstract In this paper, two approaches for computing statistical confidence intervals for reliability estimated using response surface methods (RSM) are presented. In the first method, three limit states are used to obtain the upper, mean and lower probability of failure at the design point of interest. This approach is integrated with popular probabilistic design algorithms like the Mean-Value First-Order-Second-Moment (MV-FOSM), Monte Carlo (MC) and Hazofer-Lind-Rackwitz-Fiessler (HLRF) algorithms. In the second approach, a close form solution to predict confidence intervals for reliability obtained using the MV-FOSM approach is proposed. Analytical derivations for both methods are presented and the methods are validated via a representative case study. The proposed methods are applicable to any quadratic response surface approximation obtained using design-of-experiment methods.

Introduction Most methods for computing the reliability of a

structure require the explicit evaluation of a performance function (also called the limit state), that is an explicit function of the load and resistance related random variables. However, for most cases of practical importance, such performance functions are not available in explicit form, and usually have to be computed by numerical procedures like finite element analysis. Each evaluation of the limit state and its derivatives can be computationally expensive. To overcome this limitation of evaluating implicit response functions and limit states, three broad approaches are being developed – the use of “smart Monte Carlo” techniques like efficient sampling and variance reduction techniques, the use of sensitivity based analyses (the so-called

stochastic finite element methods) and finally, response surface techniques [1]. A detailed treatment of these methods is provided by Mahadevan and Haldar [2].

Reliability analysis using RSM

The key steps in implementing response surface methods in reliability engineering are shown below. Step 1: Select the range of values for the random variables required to evaluate the performance functions, ( )0g X , where ( )0 1,0 ,0, , kX x x= … , is

the vector of design variables evaluated at any design point. This is the “design of experiments” stage, and based on the number of design variables and their range, a suitable design is chosen. The popular ones currently used are the Full-Factorial, Central Composite, Box-Behnken and D-Optimal designs. Step 2: Evaluate the performance function using deterministic simulations based on the choice of design chosen from Step 1. It is important to note that the accuracy of the response surface is proportional to the number of deterministic function evaluations. Hence it is important to choose a design that provides the best trade-off between accuracy and the required number of function evaluations. Step 3: Construct first or second order polynomial approximation to the response of interest using standard regression analysis methods. This provides an approximate closed form solution to the response and forms the basis for the limit state Step 4: Evaluate the probability of violating the limit state using any standard explicit algorithm like MCS, MVFORM or HLRF.

In general, for a set of ‘n’ experiments and a model with ‘k’ design variables, the vector of response variables, Y can be written in matrix form as follows.

*Graduate Student, **Professor, Fellow AIAA Copyright 2002 by P. Hajela . Published by the American Institute of Aeronautics and Astronautics, Inc. with permission

9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization4-6 September 2002, Atlanta, Georgia

AIAA 2002-5475

Copyright © 2002 by the author(s). Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

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1 11 12 1 0 1

2 21 22 2 1 2

1 2

1 ...1 ...

,. . . . . . . .. . . . . . . .

1 ...

k

k

n n n nk k n

y x x x by x x x b

y x x x b

εε

ε

= • +

(1)

or, Y X b ε= + (2)

From linear regression theory, the least squares estimate for the regression coefficients is the vector b as shown below.

1( )T Tb X X X Y−= (3)

It is possible to obtain a confidence interval on the mean response obtained at a particular value of the design variable vector X. Let the values of the design variables at a particular point in the design space be defined as shown in (4).

( )0 1,0 2,0 ,01 ..T

nX x x x = (4) The estimated mean and standard deviation of

the response at that point are, ( ) 1 1,0 2 2,0 ,0.....o o n ny X b b x b x b x= + + + (5)

( ) ( ) 120 0T T

oStd y X X X X Xσ−

= (6)

where,

2T T TY Y b X Y

n pσ −

=−

(7)

The term 2σ refers to the variance of the errors.

Note that if the errors, ε are normally distributed (as should be the case for a good quality response surface) then b is normally distributed, i.e.,

T -1 2b ~ N[b, (X X) ]. σ If there are ‘k’ design variables, then the model will usually have ‘p’ parameters, bi, i = 1..p, where, p 1+2k+k(k-1)/2≥ .

The corresponding ( )100 1 α− % confidence

interval for the mean response is,

( ) ( ) ( )L U

o o oy X y X y X≤ ≤ (8) where,

( ) ( ) ( )( ) ( ) ( )

12/ 2,

12/ 2,

L T To o n p o o

U T To o n p o o

y X y X t X X X X

y X y X t X X X X

α

α

σ

σ

−

−

−

−

= −

= +(9)

Here, / 2,n ptα − is the standard ‘t’ statistic

computed at (n-p) degrees of freedom. Equations (8,9) will be subsequently used in computing confidence bounds for the reliability estimate obtained from the response surface model.

Confidence Intervals for Reliability using the

MCS algorithm

This approach can be explained with the help of Figure 1 shown below.

Figure 1

Using a well designed set of experimental runs, estimates of the response of interest are obtained and used to construct a quadratic response surface model. The model coefficients are estimated from (3). The average responses as well as their confidence intervals are obtained at the point of interest, also called the design point or checking point. Three limit states are constructed and the probabilities of violating those limit states are evaluated using any standard probabilistic design algorithm like the MCS, MV-FOSM and HLRF methods. These probabilities provide a bound for the probability of failure, and thereby provide a confidence interval for the reliability estimate as well. The various steps involved in the Monte Carlo version of this algorithm are as follows. Step 1: Establish an experimental design matrix using any standard design like Central Composite or Box-Behnken. After running ‘n’ simulations, set up Y and X matrices in the form Y = X b

Response Surface(Average)

Response Surface(Upper bound)

Response Surface(Lower Bound)

y

x1

x2

Response Surface(Average)

Response Surface(Upper bound)

Response Surface(Lower Bound)

y

x1

x2

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Step 2: Compute b = (XTX)-1XTY

Step 3: Compute pn

YXbYY TTT

−−

=2σ

Step 4: Compute the following

( ) 20 0 0 0 0 0

1 1

k k

i i ii i ij i ji i i j

Y X b b x b x b x x= = <

= + + +∑ ∑ ∑∑

( ) ( ) ( ) 12/ 2,

U T To o n p o oY X Y X t X X X Xα σ

−

−= +

( ) ( ) ( ) 12/ 2,

T To o n p o oL

Y X Y X t X X X Xα σ−

−= −

Step 5 : Compute primary and auxiliary limit states at the point of interest, where Y max is a vector of upper limits on the components of the response vector Y.

( ) ( )( ) ( )( ) ( )

0 0

0 0

0 0

max

max

max

UU

L L

g X Y Y X

g X Y Y X

g X Y Y X

= −

= −

= −

(10)

Step 6 : Compute the probability of violating these limit states. The total number of Monte Carlo trials is ‘N’. For each MCS trial, a random vector of design variables is generated, i.e.,

( )1, , kX rnd x x= … The limit states are evaluated for each vector of random design variables, and the total number of limit state violations are counted. The probability of failure is the total number of limit state violations divided by the total number of Monte Carlo trials. In pseudo code, this is written as follows.

[ ]

( )( )( )

1

1,, ,

0, 1

0, 1

0, 1

k

U U U

L L L

for i NX rnd x x

if g X fail fail

if g X fail fail

if g X fail failend

=

=

< ← +

< ← +

< ← +

…

Random numbers are generated from the distributions that characterize the various design variables. Step 7 : Evaluate the reliability estimate as well as the confidence bounds, where RU, R & RL denote the upper confidence interval, mean value and lower confidence interval of the reliability estimated from the response surface.

( )1

1

1

UU

LL

failR n

failR nfailR n

= −

= −

= −

Note that there is an additional source of error

in a Monte Carlo simulation, which is is a function of the number of trials ‘N’ used in the simulation. By choosing a large number of simulations, this can be kept quite small. However, if each function evaluation is expensive, then a limited number of Monte Carlo trials are performed and one must also account for the simulation error. The simulation error in MCS can be estimated as shown in (11). Note that 2Zα is the standard normal variate

obtained at 100(1-a)% level of confidence.

( )

/ 2

1R RError R Z

Nα

− = ±

(11)

Confidence Intervals for Reliability using the

MV-FOSM algorithm The main disadvantage of MCS is the large computational cost involved in computing the probability of failure. In addition, this algorithm is difficult to incorporate within an optimization framework as the gradient of the limit state cannot be easily computed. The MVFOSM algorithm is widely used in reliability-based-optimization and problems where the random variables are normally or lognormally distributed with low coefficients of variation (COV < 0.15). In this approach, Steps 1 – 5 are identical to that described in the MCS approach. Once the limit states have been established, a reliability safety index is computed for each limit state. Step 6a : Compute partial derivatives of the primary and auxiliary limit state function with respect to each probabilistic design variable. Note the partial derivatives are evaluated at the mean values of the ‘k’ design variables. The maximum value of the limit state, Ymax could also be a random variable. Thus, we may also need to compute a partial derivative of the limit state with

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respect to Ymax as well, leading to a maximum of (k+1) partial derivative evaluations per limit state.

The design point is designated as 0X . Partial derivatives of each of the limit states with respect to the ‘k’ design variables and maximum response, Ymax are computed as shown below.

( )

ixXg

∂∂ 0 , ( )

i

U

xXg

∂∂ 0 , ( )

i

L

xXg

∂∂ 0

Step 7a : Compute the reliability estimate along with the upper and lower confidence bounds at the point of interest, X0. Reliability estimate, R, is as follows.

R = ( )( ) ( )

∂

∂∑=

k

ik

k

xVarxXg

Xg

1

2

0

0φ

Upper and lower confidence limits are as shown.

RU = ( )

( ) ( )

∂∂∑

=

k

ik

k

U

U

xVarxXg

Xg

1

2

0

0φ

RL = ( )( ) ( )

∂

∂∑=

k

ik

k

L

L

xVarxXg

Xg

1

20

0φ

This method is simple to implement, and can be

used within a multiobjective optimization formulation. Confidence Intervals for Reliability using the

HLRF algorithm

The HLRF algorithm is one of the most popular reliability estimation algorithms currently in use. The primary advantage of this algorithm is that it is not restricted to normally distributed random variables, but can handle most known probability distributions. The overall approach for computing reliability confidence intervals can also be applied to the HLRF technique as well. The general philosophy remains the same – the confidence limits of the mean response are computed from

the response surface equations, and three estimates of reliability are made using the HLRF algorithm. Here, Steps 1– 5 are the same as described in the MCS approach. Note that Steps 6b – 10b are used to obtain the probability of violating one limit state only, and need to be repeated for all applicable limit states.

Step 6b: Begin HLRF computations. Evaluate partial derivatives of the limit state function at the initial design points.

Compute ( )ix

Xg∂

∂ 0 for all probabilistic design

variables.

Compute equivalent normal approximations at the point of interest,

( )( )( )

( )( ) ( ) ( )( )

1 (0)

(0)

(0) 1 (0)

i

i

i

X i

iX i

i i i X i

F xx

f x

x x x F x

ϕ φσ

µ σ φ

−

−

=

= − •

The index i varies from1 to the maximum number of probabilistic design variables.

Note that FXi(xi(0)) is the cumulative density

function of the actual distribution of the design variable at the checking point, and fXi(xi

(0)) is the probability mass function (probability density function) of the design variable expressed in its original distribution.

Step 7b: Compute equivalent limit state functions

( ) ( )

( )( ) ( )( )

•

∂∂

=

•

∂∂

=

•

∂∂

=

∑

∑

∑

=

=

=

k

ii

Xi

k

ii

Xi

k

ii

Xi

xxgZ

xxgZ

xxgZ

i

i

i

1

2

2

2

1

1

)0(

)0(

)0(

)0(

σσ

µµ

Step 8b: Estimate the reliability safety index by computing the vector of direction cosines, ( )ixα , and using that to compute a better estimate of

HLRFβ

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( )( )

( )( )

(0 )1

(0) (0), ( )( )

i

k

ii i X

i

HLRF

g xx

xZ

Z g X b ZZ

σ

ασ

µβ

σ

=

∂ • ∂ = −

− −= −

∑

The mean values of the design variables are then updated as shown below.

( ) ( )(1) ( )i i i HLRF ix x x xµ α β σ= + • •

Step 9b: Assume a starting value of initialHLRFβ = 3.0.

If ( )initalHLRF HLRFβ β− is within an acceptable

error range, then the solutions have converged – else go to Step 8 , update the safety index and repeat till the values of HLRFβ have converged.

Step 10b: The reliability estimate can be obtained from the converged value of the reliability safety index, ( )1 HLRFR φ β= − −

Step 11b: Set g(X) = g(X0)U. Repeat steps 9 – 13 until the reliability safety index has converged. The upper bound is, ( )1U

HLRFR φ β= − −

Step 12b : Set g(X) = g(X0)L. Repeat steps 9 – 13 until the reliability safety index has converged. The lower bound is ( )1L HLRFR φ β= − −

This method is more tedious, but provides a good tradeoff between the highly accurate but computationally intensive Monte Carlo simulation approach and the relatively simple but inflexible MV-FOSM approach. In practice, it has been observed that design variables like manufacturing tolerances, part clearances and other dimensional variation are generally normally distributed. Material strength usually follows either a normal or lognormal distribution. Loads are generally non-normal, requiring the use of the HLRF method.

For extremely nonlinear limit states and complex probability distributions, the HLRF method may fail to converge. Another disadvantage in the HLRF approach is that it is difficult to implement this algorithm within a reliability based optimization formulation.

Confidence Intervals for Reliability using a closed-form approach

The following steps will illustrate the derivation of the safety index confidence limits. The limit states can also be described with the help of Figure 2 shown below.

Figure 2 Note that the “average” limit state is now bounded by an upper and lower limit state, which are computed by substituting the upper and lower confidence limit of the design point in the limit state equation. Analytic expressions for the reliability safety index confidence intervals are now derived. Consider the limit state gU(X). This can also be written as,

( )

( ) ( )

( )( ) ( )

( ) ( )

12/ 2,

12/ 2,

12/ 2,

( ) max

max

max

UUO O

T TO n p O O

T TO n p O O

T TO n p O O

g X y X Y

y X t X X X X Y

y X Y t X X X X

g X t X X X X

α

α

α

σ

σ

σ

−

−

−

−

−

−

= −

= + − = − +

= +

(12)

In a similar manner, the equation of the lower confidence interval limit state can be written as, ( ) ( ) 12

/ 2,( )L T TO O n p O Og X g X t X X X Xα σ

−

− = −

(13)

The reliability safety index, β , for the average limit state is computed as shown in (14)

g(x1,x2) = 0

gL(x1,x2) = 0

gU(x1,x2) = 0

x1

x2

bL bbU

g(x1,x2) = 0

gL(x1,x2) = 0

gU(x1,x2) = 0

x1

x2

bL bbU

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( )( ) ( )

2

1

O

nO

ii i

g X

g XVar x

x

β

=

=∂

∂ ∑

(14)

The upper confidence limit of the reliability

safety index is computed along similar lines,

( )( ) ( )

( ) ( )

( ) ( )

2

1

12/ 2,

2

1

UOU

nO

ii i

T TO n p O O

nO

ii i

g X

g XVar x

x

g X t X X X X

g XVar x

x

α

β

σ

=

−

−

=

=∂

∂

+ =

∂ ∂

∑

∑

(15)

This is simplified as shown below.

( )

( ) ( )

12/ 2,

2

1

T Tn p O O

U

nO

ii i

t X X X X

g XVar x

x

α σβ β

−

−

=

= +

∂ ∂

∑

(16)

Note that this requires the evaluation of only

one limit state, and reduces the computational cost involved. Proceeding along the same lines, the lower confidence limit for the reliability safety index is,

( )

( ) ( )

12/ 2,

2

1

T Tn p O O

L

nO

ii i

t X X X X

g XVar x

x

α σβ β

−

−

=

= −

∂ ∂

∑

(17)

i.e., the upper and lower limits can be obtained by computing a “confidence factor”, Db, such that, bU,L = b Ä Db. Note that the confidence factor can be written as follows.

( )

( ) ( )

12/ 2,

2

1

T Tn p O O

nO

ii i

t X X X X

g XVar x

x

α σβ

−

−

=

∆ =

∂ ∂

∑

(18)

It is often easier to estimate the standard error

from the numerical values of the response surface and actual model estimates. The standard error due to regression alone could be obtained as shown below.

( )2

,1

n

i RS ii

R

y y

n pσ =

−=

−

∑ (19)

Using the standard error directly in (19), we

obtain another form of the “confidence factor” as shown below.

( )( )

( ) ( )

2

, 11/ 2,

2

1

n

i RS iT Ti

n p O O

nO

ii i

y yt X X X X

n p

g XVar x

x

α

β

−=

−

=

−

− ∆ =

∂ ∂

∑

∑

(20)

The methods proposed in this chapter are now validated with the help of a case study

Case Study: Reliability of a Two-Bar Truss

The two-bar truss optimization problem layout is shown in Figure 3.

Figure 3 The objective of the design exercise is to

minimize the weight of the design, f1, as well as its deflection at node 3, f2 . The reliability of the truss structure, f3. is computed from the reliabilities of the two truss members. A truss member is considered to have failed if the tensile stress exceeds the yield stress of the member. The design is considered to be a “series-reliability” system in which failure of any of its members caused the whole design to fail.

h

w

x

y

P45 o

1

3

2

h

w

x

y

P45 o

1

3

2

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The weight of the design to be minimized is obtained as,

21 2 12 1f hx xρ= + (21)

and the displacement at node 3 to be minimized is as follows.

( ) ( )1.5 0.52 41 1

2 21 2

1 1

2 2

Ph x xf

Ex x

+ += (22)

The reliability of the structure, f3 is derived from

the limit state equations for the problem. The deterministic stress constraints are as shown in (23) and (24).

( )( )0.521 1

1 01 2

1 10

2 2

P x xg

x xσ

+ +≡ − ≤ (23)

( )( )0.521 1

2 01 2

1 10

2 2

P x xg

x xσ

− +≡ − ≤ (24)

The side bounds on the design variables are

applied in context to the type of problem being solved. Typical values are shown below. 1

2

0.1 2.2290.5 2.498

xx

≤ ≤

(25)

The design variables are defined as,

1wxh

= (26)

2min

AxA

= (27)

The nominal values for the design and problem

variables are, Load P = 10000 lbs, Yield Stress ‘s0’ = 20000 lbs/in2, Density ‘r’=0.283 lbs/in3, Young’s Modulus ‘E’ = 30 x 106 lbs/in2, Cross section Amin = 1.0 in2 For this case study, a nominal design point, 0X was chosen as shown below.

10

2

0.76801.5580

xX

x

= =

(28)

The next step was to reformulate the same problem using a surrogate model, which is a quadratic response surface, developed over the region of interest. The well-known central composite design (CCD) was used to generate the response surface equations.

The allowable range for the two design variables used in developing the response surface are shown in Table 1.

x1 x2 Low 0.7300 0.9000 High 0.8000 1.7500

Coded Low 1 1

Coded High -1 -1 Table 1: Table of design values

The response surface coefficients are estimated

using the steps 1 - 7, and the final form of the quadratic approximations for the stress in truss 1 & 2 are shown in (29-30)

1, 1 2

2 21 2 1 2

19819.47 21670.686 22551.686

17020.408 5400.692 1690.753RSS x x

x x x x

= + −

− + +(29)

2, 1 2

2 21 2 1 2

13409.947 14434.009 6214.439

1975.510 719.834 4418.1551RSS x x

x x x x

= − + +

− − −(30)

Equations (29) and (30) are subsequently used

in formulating the limit states and in obtaining confidence bounds on the reliability estimated. The probability of failure of any truss member is the probability that applied stress exceeds strength, i.e.

0 1, 0 2,0 0RS RSP S or P Sσ σ − < − <

For truss member 1, the limit state equations at the design point are,

1, 0 1

22 1

22 1 2

(19819.47 21670.686

22551.686 17020.408

5400.692 1690.753 )

RSS x

x x

x x x

σ= − +

−

+ +

(31)

1, 0 1

22 1

22 1 2 1,72

(19819.47 21670.686

22551.686 17020.408

5400.692 1690.753 ) )

URS

G

S x

x x

x x x t Eα

σ= − +

− −

+ + −

(32)

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1, 0 1

22 1

22 1 2 1,72

(19819.47 21670.686

22551.686 17020.408

5400.692 1690.753 ) )

LRS

G

S x

x x

x x x t Eα

σ= − +

− −

+ + +

(33)

Proceeding along similar lines, the mean, upper

and lower bound limit states for the second truss member are shown in (34-36)

2, 0 1

22 1

22 1 2

( 13409.947 14434.009

6214.439 1975.510

719.834 4418.1551 )

RSS x

x x

x x x

σ= − − +

+ −

− −

(34)

2, 0 1

22 1

22 1 2 2,72

( 13409.947 14434.009

6214.439 1975.510

719.834 4418.1551 )

URS

G

S x

x x

x x x t Eα

σ= − − +

+ −

− − −

(35)

2, 0 1

22 1

22 1 2 2,72

( 13409.947 14434.009

6214.439 1975.510

719.834 4418.1551 )

LRS

G

S x

x x

x x x t Eα

σ= − − +

+ −

− − +

(36)

The terms, EG1 and EG2 are evaluated at the

corresponding level of confidence. The next step is to use (31-36) to compute the design reliability and its confidence limits.

MCS algorithm results Monte Carlo simulations were run for the

corresponding limit states. To minimize error, 50,000 trials were run for each case. A Monte Carlo simulation was also performed on the original problem to compute the “exact” or “true” reliability estimate for error analysis purposes.

The results can also be explained with the help

of the two graphs, Figures 4 and 5 shown below. Figure 4 shows the “exact” reliability of the truss bounded by upper and lower confidence bounds obtained by Monte Carlo simulations from the surrogate response surface approximations. Figure 5 shows the response surface estimate of reliability bounded by confidence intervals, which were obtained from response surfaces.

Figure 4 : “Exact Reliability” with confidence bounds, Monte Carlo method

Figure 5: Response surface reliability with confidence bounds, Monte Carlo

From the figures, the following key observations

can be made. The jitter in the mean estimate of reliability can be explained by the simulation error inherent in any Monte Carlo procedure. In actual practice, the number of simulations needs to be extremely large to allow the reliability estimates to stabilize. For the 50,000 trials conducted for each of the six limit states, the computational cost was 6,500,000 flops

The proposed algorithm works well in providing upper and lower bounds at higher levels of confidence. Confidence intervals are usually estimated at 90 - 95 % levels of confidence, and results indicate that the true probability of survival is bounded by these intervals. Although computationally intensive, the proposed approach can be used in providing a useful lower bound on the reliability of a system

0.976000

0.978000

0.980000

0.982000

0.984000

0.986000

0.988000

0.990000

0.992000

50 60 70 80 90 95 99

Confidence Level (%)

Reli

abilit

y

Upper LimitLower LimitTrue Estimate

0.976000

0.978000

0.980000

0.982000

0.984000

0.986000

0.988000

0.990000

0.992000

50 60 70 80 90 95 99Confidence Level (%)

Reli

abilit

y

Upper LimitLower LimitTrue Estimate

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MV-FOSM algorithm results In this section, the MV-FOSM approach to

estimating confidence bounds is discussed. There were two types of MV-FOSM confidence bound models proposed – In this section the approach based on evaluating primary and auxiliary limit states is discussed. As the surrogate models are quadratic polynomial functions, the partial derivatives can be computed easily in close form. The reliability safety indices for each limit state are initially computed and combined to obtain system level estimates of the probability of failure.

If required, an equivalent “safety index” at the system level can also be obtained by the inverse normal transformation of the probability of failure. Results from the MV-FOSM algorithm are summarized in Figures 6-9.

Figure 6: “Exact safety index” with confidence intervals, MV-FOSM

As shown in Figures 6 and 7, the “true

reliability safety index” and the “estimated” safety index remain stationary. Safety index bounds are obtained at various levels of confidence, and they “bound” the true and estimated safety indices in an approximately symmetric manner.

The reliability safety indices are converted into

probabilities of survival (reliability) and can be visualized as shown in Figures 8 and 9. The term “exact” implies the reliability was obtained using the analytic limit state as opposed to the response surface approximation of the same quantity.

On average, this algorithm predicts a slightly higher reliability than the Monte Carlo approach. This is due to nature of the generating response surface as well as limitations in the MV-FOSM algorithm

Figure 7: Estimated safety index with confidence bounds, MV-FOSM

Figure 8: Exact reliability estimates with confidence bounds, MV-FOSM

Figure 9: Response surface reliability estimate with confidence bounds, MV-FOSM

This algorithm is also limited to normally

distributed design variables only. The “exact” and “expected” values of reliability remain constant regardless of the level of confidence. The upper confidence limit tends asymptotically to unity as the confidence level increases. The lower survival probability confidence bounds drops sharply as the level of confidence increases.

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American Institute of Aeronautics and Astronautics

10

The MV-FOSM approach can also be directly used in a “design-for-reliability” algorithm, in which the lower bound on relibility provides a conservative estimate of risk. It could also be used within a reliability-based design optimization formulation in which the reliability bounds are imposed as constaints.

Results from the MV-FOSM “Confidence

Factor” approach

The results from the simulations are identical to that obtained by the previous MV-FOSM algorithm, except that only two limit states were evaluated in the confidence factor approach as compared to six in the previous section. It however requires the calculation of an additional “confidence factor” per limit state, which brings the total computational cost for the case study in the range of 5000-6000 flops. Owing to the extremely simple form of the limit state, there appears to no particular computational advantage. However, the savings in the current method become significant as the limit states become more complex, and when they have to be evaluated using implicit techniques like finite element simulations. Results for the probabilities obtained from the “exact” and response surface approaches are shown in Figures 10 and 11 respectively.

Figure 10: “Exact” Reliability & confidence intervals, confidence factor approach

The proposed “confidence factor” approach has all the numerical advantages and disadvantages of the full MV-FOSM based approach for confidence bounds. It however directly estimates the upper and lower bounds on the safety index, thereby reducing the required number of limit state function evaluations by 66%. This can result in significant computational advantages for complicated and implicit limit states.

Figure 11: Estimated reliability and confidence intervals, confidence factor

Figure 12: “Exact” safety index with confidence bounds, confidence factor

Figures 12 and 13 show the “true” reliability safety index and the estimated safety index respectively, flanked by the confidence bounds estimated from the proposed algorithm.

Figure 13: Estimated safety index with confidence bounds, confidence factor

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American Institute of Aeronautics and Astronautics

11

In the next section, results from the HLRF approach are discussed. It is important to note that the Monte Carlo and HLRF approaches remain the most flexible in terms of modeling a wide variety of design variable distributions.

Results from the HLRF approach

All of the preceding algorithms, with the exception of the Monte Carlo method, are based on the underlying assumption that the design variables are normally distributed. The Hazofer-Lind-Rackwitz-Fiessler (HLRF) algorithm is based on the concept of an “equivalent normal” transformation at the design point (checking point). This is an iterative approach which is computationally more demanding than the MV-FOSM method. In general, there is no guarantee that the HLRF method will converge. However, this approach is far more accurate and flexible than the MV-FOSM method, and should be used if the design variables are non-normal. Results from the truss case study are detailed below. Plots of reliability safety index and the reliability vs. confidence level are shown in figures 14 and 15 respectively.

Figure 14: HLRF results for reliability safety

index and confidence intervals

They follow a trend similar to what was observed in the full MV-FOSM and MV-FOSM confidence factor approaches. The only advantage of this algorithm over the others is it’s ability to model various kinds of distributions like the Weibull, Lognormal, Normal, etc. For complex probabilistic design problems, this may be the only computationally feasible and accurate algorithm. It is possible to include this within an overall DFR algorithm, or within a probabilistic optimization framework. The Monte Carlo technique would not work in such an environment

owing to the large sample sizes required and the inability to obtain numerical gradients during the solution process.

Figure 15: HLRF results for reliability & confidence intervals

Summary & Discussion

In this paper, algorithms that provide confidence intervals on reliability estimated from response surface models are presented. These intervals can be easily incorporated within a design-for-reliability process and can be used as constraints in a reliability-based optimization formulation. The case studies indcate that the proposed methods provide a useful bound on the risk inherent due to model approximation as well as model variability. The approaches described in this paper could also be extended to include other global approximation techniques like kriging.

References

[1] Myers, R.H., and Montgomery, D.C., Response Surface Methodology, John Wiley & Sons, 1995 [2] Haldar, A., and Mahadevan, S., Probability, Reliability and Statistical Methods in Engineering Design, John Wiley & Sons, 1995 [3] Fox, E.P., “The Pratt & Whitney probabilistic design system”, AIAA-94-1442-CP, 1994 [4] Madsen, H.O., Krenk, S., and Lind, N.C., Methods of structural safety, Prentice Hall Inc, 1986. [5] Haftka, R.T., and Gurdal, Z., Elements of structural optimization, Kluwer Academic Publishers, 1993

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