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05M-154

Integration of Reliability- And Possibility-Based Design Optimizations Using Performance Measure Approach

Byeng D. Youn Department of Mechanical Engineering, University of Detroit Mercy, Detroit, MI 48219-0900, USA

Kyung K. Choi, and Liu Du Center for Computer-Aided Design, Department of Mechanical & Industrial Engineering, College of Engineering, The

University of Iowa, Iowa City, IA 52242, USA

Copyright © 2004 SAE International

ABSTRACT

Since deterministic optimum designs obtained without considering uncertainty lead to unreliable designs, it is vital to develop design methods that take account of the input uncertainty. When the input data contain sufficient information to characterize statistical distribution, the design optimization that incorporates the probability method is called a reliability-based design optimization (RBDO). It involves evaluation of probabilistic output performance measures. The enriched performance measure approach (PMA+) has been developed for efficient and robust design optimization process. This is integrated with the enhanced hybrid mean value (HMV+) method for effective evaluation of non-monotone and/or highly nonlinear probabilistic constraints. When sufficient information of input data cannot be obtained due to restrictions of budgets, facilities, human, time, etc., the input statistical distribution is not believable. In this case, the probability method cannot be used for reliability analysis and design optimization. To deal with the situation that input uncertainties have insufficient information, a possibility (or fuzzy set) method should be used for structural analysis. A possibility-based design optimization (PBDO) method is proposed along with a new numerical method, called maximal possibility search (MPS), for fuzzy (or possibility) analysis and employing the performance measure approach (PMA) that improves numerical efficiency and stability in PBDO. The proposed RBDO and PBDO methods are applied to two examples to show their computational features. Also, RBDO and PBDO results are compared for implications of these methods in design optimization.

1. INTRODUCTION

Due to the extensive efforts of engineering disciplines over last three decades, design guidelines and/or standards have been modified to incorporate the concept of uncertainty into an early design stage. In

response to these new design requirements, various methods have been developed to treat uncertainties in engineering analysis and, more recently, to carry out design optimization with reliability.

There are two different types of uncertainties: aleatory and epistemic uncertainties [1]. Aleatory uncertainty is classified as objective and irreducible uncertainty, whereas epistemic uncertainty is a subjective and reducible uncertainty that stems from lack of knowledge on data. In general, a large amount of data is employed to construct aleatory uncertainty, such that an uncertainty is accurately quantified with a distribution type and its parameters. Often, it is very difficult to collect sufficient data for uncertainty quantification due to the restrictions of resources (budgets, facilities, human, time, etc.). Thus, it is desirable that different design methodology is utilized for different types of uncertainty.

For aleatory uncertainty, a reliability-based design optimization (RBDO) [2-5] has been used to consider engineering uncertainties in a design process. In RBDO, a primary concern has been how to make it computationally affordable, while maintaining numerical accuracy and stability. Computational efficiency and stability have been improved by using the enriched performance measure approach (PMA+), where four improvements are made over the original PMA: as a way to launch RBDO at a deterministic optimum design, as an efficient probabilistic feasibility check, as an enhanced hybrid-mean value (HMV+) method, and as a fast reliability analysis under the condition of design closeness [5].

In areas where it is not possible to produce accurate statistical information, the probabilistic methods are not appropriate for structural analysis and design optimization subject to physical uncertainties, since improper modeling of uncertainty could cause greater degree of statistical uncertainty than those of physical uncertainty [6]. To handle epistemic uncertainty when

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modeling physical uncertainty with insufficient information, possibility-based (or fuzzy set) methods have recently been introduced in structural analysis and design [7]. In aid of the operational framework of possibility theory [8-13], a fuzzy analysis entails the following four attractive features.

1) The fuzzy analysis preserves the intrinsic random nature of physical variables through their membership functions.

2) Extended fuzzy operations are simpler than those required to use probability, especially when a number of variables are involved.

3) It has been pointed out that, when little information is available for input data, the possibility-based method is better since it provides a more conservative design than the probabilistic design that is consistent with the limited available information [13,14]. This is a desirable merit, since a conservative optimum design is preferred when accurate statistical information is not available.

4) Possibility analysis provides a system-level possibility unlike reliability analysis [14].

Using PMA, a new formulation of PBDO has been formulated to improve numerical efficiency, stability, and accuracy [7]. To resolve disadvantages of the vertex method and the multilevel-cut method, a new maximal possibility search (MPS) method [7] has been proposed for a possibility (or fuzzy) analysis, such that it evaluates possibility constraints efficiently and accurately for nonlinear structural applications.

This paper presents an integrated design platform of both RBDO and PBDO that employ PMA for more effective design formulations. Two examples are used to show computational features of PMA+ for RBDO with aleatory input uncertainties and MPS for PBDO with epistemic input uncertainties. In addition, RBDO and PBDO results are compared for implications of these methods in design optimization.

2. RELIABILITY-BASED DESIGN OPTIMIZATION

2.1 RBDO MODEL FOR PERFORMANCE MEASURE APPROACH

For general engineering applications, the RBDO model [1-5] can be formulated as

β> − Φ − ≤ =

≤ ≤i t

L U

P G i npmin Cost( )s.t. ( ( ( )) 0) ( ) 0, 1, ,

dd X

d d d

(1)

where = = ∈T nid R[ ] ( )d µ X is the design vector,

= ∈T nriX R[ ]X is the random vector, and n, nr and np

are the number of design variables, random variables, and probabilistic constraints, respectively. The design constraints are described by the probability •P( ) of the failure event ≥iG ( ( )) 0d X .

The statistical description of the constraint violation is characterized by the cumulative distribution function

•iGF ( ) as

β≥ = − ≤ Φ −ii G tP G F( ( ) 0) 1 (0) ( )X (2)

where

≤

= =∫ ∫ …i i

G nGF f dx dx i np1( ) 0

(0) ( ) , 1, ,XXx (3)

In Eq. (3), f ( )X x is the joint probability density function of all random variables. Its evaluation requires a reliability analysis where multiple integrations are involved, as shown in Eq. (3). Some approximate probability integration methods have been developed to provide efficient solutions, such as the first-order reliability method (FORM) [15,16], or the asymptotic second-order reliability method (SORM) [17,18] with a rotationally invariant measure as the reliability. FORM often provides adequate accuracy and is widely used for design applications. Those reliability methods require a transformation T [19,20] from the original random parameter X to the standard normal random parameter U. The performance function G( )X in X-space can then be mapped onto G(X) = G(T(X)) ≡ G(U) in U-space.

The probabilistic constraint in Eq. (2) can be expressed as a performance measure through the inverse transformation of •GF ( ) as [3-5]

β

β

−

−

= − Φ −

= Φ ≤i i

i

p G t

G t

G F

F

1

1

( ( )) (1 ( ))

( ( )) 0

d X (4)

where ipG is the ith probabilistic constraint. In Eq. (4),

the probabilistic constraint in Eq. (1) can be replaced with the performance measure, which is referred to as the performance measure approach (PMA) [3-5]. Thus, the RBDO model using PMA can be redefined as

≤ =

≤ ≤ip

L U

G i npminimize Cost( )subject to ( ( )) 0, 1, 2, ,

dd X

d d d

(5)

2.2 RELIABILITY ANALYSIS MODEL OF PMA [3-5]

Reliability analysis in PMA can be formulated as the inverse of reliability analysis in the reliability index approach. The first-order probabilistic performance

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measure pG ,FORM is obtained from a nonlinear optimization problem in U-space, defined as

β= t

Gmaximize ( )subject to

UU

(6)

where the optimum point on the target reliability surface is identified as the most probable point (MPP) β β= t

*u with

the prescribed reliability β ββ ==tt

*u . The Karush-Kuhn-

Tucker (KKT) necessary condition of Eq. (6) is

β β β β β ββ= = == − ∇ ∇t t tt G G* * *( ) ( )u u u (7)

Any general optimization algorithm can be employed to solve the optimization problem in Eq. (6). However, an enhanced hybrid mean value (HMV+) first-order method is well suited for PMA due to its stability and efficiency [5,21].

2.3 ENHANCED HYBRID MEAN VALUE (HMV+) METHOD

Even though the hybrid mean value (HMV) method [3] performs well for mildly nonlinear monotone performance functions, it could fail to converge for highly nonlinear and non-monotone output performance functions. To improve numerical stability and efficiency, an enhanced HMV (HMV+) method has been proposed by revising the previous HMV algorithm [5,21]. In this HMV+ algorithm, if the value of the performance function is decreased at the next search point, the performance function is approximated along the arc (with constant probability level ��) between the current point and next search point to find a new search point where the approximated performance function attains its maximum value. Detail numerical procedure and numerical results can be found in refs. 5 and 21, where it has shown that the HMV+ method improves numerical efficiency and stability substantially in reliability analysis for highly nonlinear performance function.

The performance and its sensitivity values at two search points, −k k( ) ( 1)

HMV+ HMV+ and u u , are used to interpolate the performance function along the arc region between these two search points. For the interpolation, a parametric coordinate t is introduced as

β−= + = ≥k kts t t s t( 1) ( )( ) and , , 0U u u U (8)

where

ββ

ββ

− −+

− −−

− ⋅ + ⋅ + −=

− ⋅ − ⋅ + −=

k k k kt

t

k k k kt

t

t t ts t

t t ts t

( 1) ( ) ( 1) ( ) 2 2 4

2

( 1) ( ) ( 1) ( ) 2 2 4

2

( ) (1 )( )

( ) (1 )( )

u u u u

u u u u (9)

The sensitivity of the performance function with respect to the parametric coordinate t can be obtained using a chain rule as

−∂∂ ∂ ⎛ ⎞= = +⎜ ⎟∂ ∂ ∂ ⎝ ⎠k ki

i i

UdG G G dsdt U t U dt

( 1) ( )u u (10)

In Eq. (10), the sensitivities of performance function are evaluated at two search points −k k( 1) ( )and u u . Then,

−k kg g( 1) ( )( ), ( )u u , −k kdg dg

dt dt

( 1) ( )( ) ( ), and u u are used to interpolate the performance function using a cubic polynomial as = + + +G t a a t a t a t2 3

0 1 2 3( ) .

The next search point +k( 1)u is obtained where the approximated performance function G is maximum, as

β

β

+ −= + =

⎧ >⎪⎨

<⎪⎩

k k k

i

i

s t t t t

G t

G t

( 1) * ( 1) * ( ) *( ) at

( ) is maximum, for 0where

( ) is minimum, for 0

u u u (11)

Note that s(t) may not be unique for a given value of t, if

>t 1 (s, t>0; β

β −− ⋅≤ t

k kt

s t4

4 ( 1) ( ) 2( ),

u u), as shown in Eq. (9).

This could be true when the angle composed of three points a−k k( 1) (0) ( ), , nd u u u is more than 90°, which can

be expressed mathematically as − <ik k( 1) ( ) 0u u . One of these two s values, which yields greater performance function value will be selected.

2.4 ENRICHED PERFORMANCE MEASURE APPROACH [PMA+]

The enriched PMA (PMA+) has been proposed to enhance numerical efficiency while maintaining stability in the RBDO process [5]. PMA+ is an extension of PMA by integrating three key ideas: as a way to launch RBDO at a deterministic optimum design, as a probabilistic feasibility check, and as a fast reliability analysis under the condition of design closeness. The overall design procedure in PMA+ for RBDO is first to obtain the deterministic optimum design efficiently, and then carry out reliability-based design optimization. The feasibility of probabilistic constraints in RBDO can be identified by using the MV first-order method that provides an allowable degree of accuracy for the purpose of constraint violation. Once the feasibility status of probabilistic constraints is identified by the MV first-order method, a refined reliability analysis is performed using

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the enhanced hybrid mean value (HMV+) first-order method to evaluate ε-active and violate constraints. The MV first-order method based feasibility check for probabilistic constraints substantially improves the numerical efficiency of the RBDO process.

During RBDO iterations, sufficient information is generated while evaluating the cost and probabilistic constraints, and updating the design. Some of this information could be reused to evaluate probabilistic constraints efficiently at the next design iteration using the condition of design closeness. In other words, under the condition that two consecutive RBDO design iterations are close enough, the reliability analysis can be efficiently carried out by starting from the MPP obtained at the previous iteration, instead of at the mean value point of the current design iteration. This fast reliability analysis method is integrated with the HMV+ method to evaluate probabilistic constraints efficiently.

3. POSSIBILITY-BASED DESIGN OPTIMIZATION

3.1 PBDO MODEL FOR PERFORMANCE MEASURE APPROACH

For general engineering applications, the PBDO model can be formulated as [7]

αΠ > ≤ =

≤ ≤i t

L U

G i npminimize Cost( )subject to ( ( ( )) 0) , 1, ,

dd X

d d d

(12)

where = ∈T nid R[ ]d is the design vector,

= ∈T nriX R[ ]X is the fuzzy random vector where the

fuzzy random variable iX has the membership function Π

iX ix( ) and the maximal grade [13] Π =iX i ix dmax{ ( )} ,

αt is a target possibility of failure, and n, nr, and np are the number of design variables, fuzzy random variables, and possibility constraints, respectively. Fuzzy random parameters considered in this paper are assumed to be non-interactive. The possibility constraints are described by a possibility αΠ • ≤ t( ) for the failure event

>iG ( ( )) 0d X .

The PMA approach, which is developed for RBDO as shown in Section 2, is applicable to the PBDO model to formulate as [7]

Π ≤ =

≤ ≤i

L U

G i npminimize Cost( )subject to ( ( )) 0, 1, 2, ,

dd X

d d d (13)

3.2 GENERATION OF MEMBERSHIP FUNCTION

The generation of the input membership functions of the fuzzy variables using the available limited set of data is a

very important step of the possibility analysis and PBDO. Several methods have been proposed depending on the number and the kind of the data available. This paper introduces two procedures: 1) generating the pseudo-probability density function of the fuzzy variable from the available data; 2) generating the membership function of the fuzzy variable from the pseudo-probability density function.

3.2.1 Generation of Pseudo-Probability Density Function

If the only available information for the input fuzzy variable is the judgment of experts (subjective) with the most likely value and the interval corresponding to the certain confidence levels, the pseudo-probability density function can be generated using the framework of Program-Evaluation and Review Technique (PERT) analysis [22]. If the fuzzy variable has a random nature but the available data are not sufficient to assign the probability of elementary events, the pseudo-probability density function can be estimated using kernel-smoothing method [23]. Notice that, since the input data is not sufficient, the pseudo-probability density function cannot be used directly in the probability-based method.

3.2.2 Generation of Membership Function

To generate the membership function of the fuzzy variable from the pseudo-probability density function, two methods are used in this paper.

The probability-possibility consistent principle and the least conservative principle are used to generate the membership function from the pseudo-probability density function. The probability-possibility consistent principle says that, the probability of one event cannot be less than the possibility of this event [14]. The membership function satisfying the probability-possibility consistent principle is not unique. The least conservative one is chosen such that the membership function is not too much conservative.

If the pseudo-cumulative distribution function of a fuzzy variable is XF x( ) , then the membership function of the fuzzy variable satisfying the probability-possibility consistent principle and the least conservative principle is unique:

∈ ≤⎧

Π = ⎨ − ∈ >⎩X X

XX X

F x x x F xx

F x x x F x2 ( ) { : ( ) 0.5}

( )2 2 ( ) { : ( ) 0.5}

(14)

If the membership function generating by the above way is not conservative enough, Savoia’s way is a good alternative one [8]:

Π = + − +X X L X R L X Rx F x f x x x F x( ) ( ) ( )( ) ( ) (15)

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where Xf x( ) is the pseudo-probability density function and Lx and Rx is choosing such that

= =X L X R Xf x f x f x( ) ( ) ( ) .

3.3 POSSIBILITY ANALYSIS MODEL OF PMA

In this paper, it is assumed that the non-interactive input fuzzy variables iX have its membership function Π

iX ix( ) satisfying three properties [8,14,24]: (1) unity, (2) strong convexity, and (3) boundedness. These three properties enable non-interactive input fuzzy variables

iX to be uniquely transformed to an isosceles triangular membership function as

( ) + − ≤ ≤⎧Π = = − ≤⎨ − ≤ ≤⎩i

i iU i i i

i i

u uu u u

u u1, 1 0

1 , 11 , 0 1

(16)

The transformation can be written as

Π − ≤⎧⎪= ⎨ − Π >⎪⎩

i

i

X L i i ii

X R i i i

X X dU

X X d,

,

( ) 1

1 ( ) (17)

where ΠiX L ix, ( ) and Π

iX R ix, ( ) are the left side and right side of the membership function of the input fuzzy variable iX , respectively, and id is the maximal grade of the membership function. Thus, evaluation of the possibility constraint requires an inverse fuzzy analysis using PMA, which is formulated as

( )U

U α∞≤ − t

Gmaximize

subject to 1 (18)

where the optimum point on the target possibility domain U α

∞≤ − t1 is identified as the most possible point

(MPP) αt

*u with the prescribed possibility of failure αt .

The difference between reliability analysis [3,5] and fuzzy analysis [8-14,24-26] is that, the most probable point (MPP) in reliability analysis is based on FORM, which means the related probability is first order approximation, whereas the most possible point (MPP) in possibility analysis is exact, along with the related possibility. Another difference is the search domain enclosed by the target confidence level. The reliability analysis has nr-dimensional hyper-sphere as its search domain, U β≤ t2 , whereas the possibility analysis has nr-dimensional hyper-cube as its search domain, U α

∞≤ − t1 , that makes the numerical computation

simpler, compared to the reliability analysis. Since the MPP search in possibility analysis is different from the one in reliability analysis, a new numerical method has

been proposed in the next section to solve the problem (16).

3.4 GENERATION OF PSEUDO-PROBABILITY DENSITY FUNCTION

For a non-monotonic response within the range of input fuzzy parameters, the fuzzy analysis could be inaccurate using the vertex method [8] or computationally expensive using the multilevel-cut method or other computational schemes such as a global optimization method [26]. Thus, in this paper, the maximal possibility search (MPS) method has been proposed for fuzzy analysis to ensure numerical efficiency and accuracy in PBDO. This method first attempts to find an MPP using the maximal possibility search, since in majority of cases the MPP is likely to be on the vertex of the target possibility domain U α

∞≤ − t1 . If the maximal

possibility search does not yield a solution at a vertex, then the maximal possibility search is integrated with an interpolation in search of the MPP on the edge or in the interior domain of the hyper-cube.

3.4.1 Maximal Possibility Search

The maximal possibility search algorithm is as following:

Step 1. Set the iteration counter =k 0 with the convergence parameter ε −= 310 . Set the pointer =j 1.

Let =(0)u 0 . Calculate the performance G (0)( )u and the

sensitivity ∇G (0)( )u . Let = ∇k kG( ) ( )( )d u .

Step 2. Compute the next point as ( ) ( )π+ = ⋅k kt

1 sgn( )u d where π α= −t t1 and

= nrX X X1 2sgn( ) (sgn( ),sgn( ), ,sgn( ))X if

= ∈T nriX R[ ]X . Let = +k k 1.

Step 3. Calculate the performance kG ( )( )u and the

sensitivity ∇ kG ( )( )u . Let β −= ∇ +k k kG( ) ( ) ( 1)( )d u d where

β −= ∇ ∇k kG G( ) ( 1) 2( ( ) / ( ) )u u . If

∇ =k kG ( ) ( )sgn( ( )) sgn( )u u , it is the maximum point and

stop. If ≥k jG G( ) ( )( ) ( )u u , let =j k and go to Step 2. Otherwise, go to Step 4.

Step 4. Go to Step 5, with j( )u , jG ( )( )u and ∇ jG ( )( )u .

3.4.2 Maximal Possibility Search with an Interpolation

The proposed maximal possibility search with an interpolation algorithm is as following:

Step 5. Let =l 0 and = ∇l jG( ) ( )( )d u . Go to Step 6.

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Step 6. Calculate the new point +k( 1)u on the boundary of the domain using the start point j( )u and the search direction l( )d . Let = +k k 1.

Step 7. Calculate the performance kG ( )( )u and the

sensitivity ∇ kG ( )( )u . If

u

u

π π

ε π π

∂⎧ = = = −⎪ ∂⎪⎨

∂⎪ < − < <⎪ ∂⎩

ki i t i t

i

kt i t

i

G u u ux

G ux

( )

( )

sgn( ( )) sgn( ), for or

( ) , for

then it is the maximum point and stop. Otherwise, go to Step 8.

Step 8. Using jG ( )( )u , kG ( )( )u , ∇ jG ( )( )u and ∇ kG ( )( )u to construct the third order polynomial GP t( ) on the

straight line between j( )u , and k( )u where t is the parameter for the line. Calculate the maximum point t * for this polynomial. Let ( )+k 1u be the point on the line

corresponding to t * . Let = +k k 1.

Step 9. Calculate the performance kG ( )( )u and the

sensitivity ∇ kG ( )( )u . Check the convergent criteria using the equation in Step 7. If convergent, stop. Otherwise, let the new conjugate direction be

β+ = ∇ +l k lG( 1) ( ) ( )( )d u d where

β −= ∇ ∇k kG G( ) ( 2) 2( ( ) / ( ) )u u . Let =j k , = +l l 1, and

go to Step 6.

The proposed MPS method is sufficient for the monotonic responses. The proposed MPS method with an interpolation will be used only when the maximal possibility search fails. Since in most cases the maximal possibility search finds MPP efficiently, the MPS method will be efficient while robust for nonlinear and non-monotonic responses.

4. INTEGRATION OF RBDO AND PBDO

4.1 WHY PBDO FOR EPISTEMIC UNCERTAINTY?

It is apparent that RBDO must be performed when the input statistical information is sufficient. However, without modeling physical input uncertainties properly due to restriction of resources, a probabilistic approach may lead to an unsafe design, as shown in Fig. 1. Histograms of two uncertainties are built with insufficient raw data (50 samples from normal distribution: X N X N2 2

1 2~ (2.0,1.0 ), ~ (4.0,1.0 ) ), and they are

presumably modeled as a normal distribution through a fitting technique. An occupant safety constraint with a linear model is defined as ≤G 500 . Ninety-two percent reliability with imprecise statistical data turns out to be seventy-seven reliability with precise statistical information. Less conservative (or aggressive) estimation of failure rate due to lack of data could lead to a dangerous design. System nonlinearity and smaller amount of data will increase the error associated with a reliability level.

CDF PDF

Occupant Safety PerformanceG(X) = 10X1+120X2+390

22 ~ (3.64,0.93 )X N2

1 ~ (1.75,0.99 )X N

Figure 1. Erroneous Design Employing Lack of Information

4.2 STRUCTURE OF INTEGRATED RBDO AND PBDO

RBDO and PBDO can be integrated in one design platform, as shown in Fig. 2. Aleatory uncertainty must be dealt with a probability approach, whereas epistemic uncertainty can entail a possibility (or fuzzy) approach. Both RBDO and PBDO employ PMA to improve numerical efficiency, stability, and accuracy.

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AleatoryUncertainty

ReliabilityAnalysis:

HMV+RBDO

OptimumDesign w/

Confidence

PossibilityAnalysis:

MPSPBDOEpistemic

Uncertainty

PMA

Data Level Analysis Design Optimum

Integrated Frameworko f RBDO & PBDO

Figure 2. Structure of Integrated RBDO and PBDO

5. NUMERICAL EXAMPLES OF RBDO AND PBDO

5.1 NONLINEAR MATHEMATICAL EXAMPLE [5]

Consider the following mathematical model for both RBDO and PBDO with design variables

µ µ= =T Td d1 2 1 2[ , ] [ , ]d . The RBDO problem is defined as

β

α

+ − − += − −

> ≤ Φ − =

Π > ≤ =

≤ ≤ ≤ ≤

i

i

i t

i t

d d d d

P G i

G i

d d1 2

2 21 2 1 2( 10) ( 10)

30 120minimize Cost( )

subject to ( ( ) 0) ( ), 1,2,3

or ( ( ) 0) , 1,2,3

0 10 & 0 10

d

XY

(19)

where β =t 2.0 (2σ design), α =t 0.023 , and nonlinear performances are defined as

+ +

= −

= − + − + − − − +

= −

= +

= −

X X

X X

G

G Y Y Y Z

G

Y X XZ X X

1

2 3 42

3

1 2

1 2

21 2

21 2

20

80( 8 5)

( ) 1

( ) 1 ( 6) ( 6) 0.6( 6)

( ) 1

0.9063 * 0.4226 *0.4226 * 0.9063 *

X

X

X (20)

The initial design is = T(0) [5.0,5.0]d . Random properties are µ µX N X N1 1 2 2~ ( ,0.3) and ~ ( ,0.3) for RBDO, and fuzzy variables are modeled using Eq. (14) for PBDO, as shown in Fig. 3.

Figure 3. Membership Function of Random Input

Tables 1 and 2 display the numerical results of RBDO and PBDO, respectively. In these tables, NFE represent the number of function evaluations, with the same number of sensitivity analyses. The RBDO provided 2� design, which is component level reliability. Unlike RBDO, it has been found that the deterministic optimization does not improve the PBDO efficiency, due to the difference in HMV+ and MPS. Even though the second constraint is non-monotonic and highly nonlinear, both HMV+ method for RBDO and MPS method for PBDO behave very well in terms of numerical stability. Note that the same input random properties are used for optimum designs of RBDO and PBDO. However, when a reliability analysis is carried out for the PBDO result using the Monte Carlo simulation method the PBDO result, the system level reliability is 99.4 % (2.52-sigma). In other words, PBDO provides a much conservative design in terms of cost and confidence level, compared to RBDO. However, they share the similar feasibility behavior of the probabilistic constraints. Optimization histories of RBDO and PBDO are shown in Figs. 4 and 5, respectively.

Table 1. RBDO Results of Mathematical Problem

Iter. Cost d1 d2 G1 G2 G3 NFE0 -0.833 5.000 5.000 -3.259 -1.220 0.014 NADet. -2.290 5.195 0.742 -0.002 -0.004 -1.109 121 -2.290 5.195 0.742 0.811 1.069 -0.738 102 -2.107 4.696 1.141 0.422 0.220 -0.827 83 -1.905 4.644 1.562 0.019 -0.031 -0.715 84 -1.900 4.672 1.571 -0.003 -0.001 -0.702 65 -1.900 4.671 1.569 0.000 -0.001 -0.703 6Opt. -1.900 4.671 1.569 Act. Act. Inact. 50

Table 2. PBDO Results of Mathematical Problem

Iter. Cost d1 d2 G1 G2 G3 NFE0 -0.833 5.000 5.000 -3.259 -1.220 0.014 71 -2.221 5.515 0.873 0.670 1.406 -0.476 42 -2.055 4.577 1.261 0.477 0.003 -0.714 183 -1.768 4.575 1.866 -0.000 -0.042 -0.553 104 -1.774 4.597 1.852 0.000 -0.000 -0.550 5Opt. -1.774 4.597 1.852 Act. Act. Inact. 44

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Figure 4. RBDO History for Mathematical Example

Figure 5. PBDO History for Mathematical Example

5.2 RBDO EXAMPLE FOR VEHICLE SIDE IMPACT [27]

The large-scale design application shown in Fig. 6 is used to investigate computational features of the RBDO with aleatory input uncertainties and the PBDO with epistemic input uncertainties. The design objective is to enhance overall performance of side-crash with a target confidence level while minimizing the vehicle weight. Detail discussions of side-crash modes are summarized in Ref. 27. The optimal Latin hypercube method used a total of 33 samples to generate a sample of design points for constructing the stepwise regression response surface [28]. In this study, approximations of the responses obtained from response surface method are regarded as exact responses of vehicle side impact to investigate computational features of the RBDO with aleatory input uncertainties and the PBDO with epistemic input uncertainties. Normal distribution and 5 % coefficient of variation is used for random variables, and fuzzy variables are modeled using Eq. (14) for PBDO, as shown in Fig. 3.

Figure 6. Side Impact Crashworthiness

The vehicle side impact is employed to show computational features of PMA+ for RBDO with aleatory input uncertainties and MPS for PBDO with epistemic input uncertainties. Ten performance measures are used to determine the occupant safety. The design objective is to enhance side impact crash performance with a 2σ reliability, while minimizing the vehicle weight. Other than the 2� reliability requirement, other features of the RBDO problem of crashworthiness for side impact is defined the same way as in Refs. 27, 29, and 0. Table 3 lists the initial, deterministic optimum, reliability-based optimum, and possibility-based optimum designs. The RBDO provided 2� design (97.7% reliability), which is component level reliability. When a reliability analysis is carried out for the PBDO result using the Monte Carlo simulation method, the system level reliability is 98 %. Again, PBDO provides a much conservative design in terms of cost and confidence level, compared to RBDO. The objective histories for RBDO and PBDO are shown in Fig. 7. Again, PBDO yields a conservative design in terms of the cost and confidence level, compared to RBDO.

Table 3. Designs for Side Impact Crashworthiness

bi Initial Det. Opt.

Rel. Opt.

Pos. Opt.

1 1.000 0.500 0.574 0.829 2 1.000 1.226 1.350 1.349 3 1.000 0.500 0.500 0.648 4 1.000 1.207 1.462 1.486 5 1.000 1.238 1.480 1.632 6 1.000 1.200 1.200 1.200 7 1.000 0.400 0.400 0.421 8 0.300 0.345 0.345 0.345 9 0.300 0.300 0.192 0.201

Figure 7. RBDO and PBDO Cost History for Side Impact

6. DISCUSSIONS AND CONCLUSIONS

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This paper presented an integrated design formulation for RBDO and PBDO using the performance measure approach (PMA). When the input data contain sufficient information to characterize statistical distribution, RBDO with the enriched performance measure approach (PMA+) was employed for a design optimization. On the other hand, when sufficient information of input data is not available due to restrictions of resources, the input statistical distribution is not accurate. In this situation, the probability method could provide an unreliable design due to imprecise statistical information. For inaccurate statistical information, the possibility-based design optimization (PBDO) has been introduced by integrating a possibility (or fuzzy set) method to the design platform. For PBDO, the maximal possibility search (MPS) method was developed, and showed its efficiency and stability for highly nonlinear and non-monotonic performance responses. PBDO provides more conservative design than RBDO, in terms of confidence level and cost. Using PMA, aleatory and epistemic uncertainties can be dealt with by RBDO and PBDO in the integrated design platform. Future research will address a challenge of how to handle situations that an engineering system contains both aleatory and epistemic uncertainties simultaneously.

7. ACKNOWLEDGEMENTS

Research is partially supported by the Automotive Research Center sponsored by the U.S. Army TARDEC.

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9. DEFINITIONS, ACRONYMS, ABBREVIATIONS

X Random or fuzzy variable; X = [X1, X2,…, Xn]T x Realization of X; x = [x1, x2,…, xn]T U Independent and standard normal random

variable u Realization of U; u = [u1, u2,…, un]T V Non-interactive fuzzy random variable with an

isosceles triangular membership function v Realization of V; v = [v1, v2,…, vn]T β Reliability index α Possibility index G Performance function; considered “fail” if G > 0 P A( ) Probability of event A Φ •( ) Standard normal distribution Π A( ) Possibility of event A

*u Most probable point (MPP) for reliability analysis *v Most possible point (MPP) for possibility analysis

t, s(t) Parametric coordinates for MPP search T Transformation matrix from X space to U space θ Conjugate direction factor for maximal possibility

search