It is referred to as Bayesian reliability. For each independent trial, reliability must be repeatedly computed. Then, reliability results can be aggregated to model ...
Bayesian Reliability Based Design Optimization under Both Aleatory and Epistemic Uncertainties Byeng D. Youn1 and Pingfeng Wang2 Department of Mechanical Engineering and Engineering Mechanics Michigan Technological University, Houghton, MI, 49931
[Abstract] In the last decade, considerable advances have been made in reliability-based design optimization. One assumption in reliability-based design optimization (RBD O) is that the complete information of input uncertainties is known. However, this assumption is not valid in practical engineering applications, since very few input uncertainties have been adequately modeled. This is mainly due to a lack of resources, such as facilities, labor, expenses, knowledge, etc. Possibility and evidence theories have been proposed to deal with situations having epistemic uncertainties with insufficient data. But, when involving uncertainties with both sufficient and insufficient data sizes (limited to sufficient) simultaneously, it is hard to deal with those data consistently in predictive modeling and design. To properly handle both sufficient and insufficient data for random inputs, this paper proposes an integration of the Bayesian approach to RBD O. When a design problem involvesinput uncertainties with both sufficient and insufficientinformation, reliability must be uncertain and subjective. Bayesian inference is used to model uncertain and subjective reliability. It can be modeled with a Beta distribution in a Bayesian sense, where the Beta distribution is bounded in 0 (0 % reliability) and 1 (100 % reliability). For design optimization, two requirements must be considered: 1) reliability must be uniquely defined and 2) reliability must be greater than target reliability when an exact reliability is realized. The exact reliability means the reliability evaluated with a complete data for input uncertainties. To satisfy both requirements, Bayesian reliability will be defined as the median value of the extreme distribution of the smallest reliability value. For the design optimization, continuum sensitivity for Bayesian reliability is also proposed in this paper. Bayesian RBD O willthus be conducted in aid of Bayesian reliability and its sensitivity. The methodology will be demonstrated with two examples (one mathematical example and one practical engineering problem).
Nomenclature PB
= Bayesian reliability
Xa
= Aleatory random variables vector
Xe
= Epistemic random variable vector
d Φ(
= Design variable vector = Standard Gaussian cumulative distribution function
)
Gi ( )
= Constraintfunction
P( A)
= Probability of event A
Pf
= Probability of failure
f X ( x)
= Probability density function of random vector X
1 Assistant Professor, Mechanical Engineering-Engineering Mechanics, Houghton, MI, 49931, AIAA Member. 2 Graduate Student, Mechanical Engineering-Engineering Mechanics, Houghton, MI, 49931. 1 A merican Institute of Aeronautics and Astronautics
R
I. Introduction
eliability is of critical importance in product and process design. Hence, various methods [1-3] have been developed to systematically treat uncertainties in engineering analysis and more recently, to carry out reliability-based design optimization (RBDO). In RBDO a design optimization strategy has been advanced to improve computational efficiency and stability [1-5]. As well, new methods for reliability assessment have been proposed to enhance numericalefficiency and stability [6-9]. Although making great progress in reliability analysis and RBDO, a major bottleneck of this study has been known to be data acquisition and process for uncertainty modeling.In practical engineering applications,itisrare to model uncertainties precisely. Generally an aleatory uncertainty is modeled with sufficientinformation, whereas an epistemic uncertainty results from insufficient data. The insufficiency of data is mainly due to a lack of resources, such as facilities, labor, expenses, knowledge, etc. Although uncertainties of material properties have been extensively studied, their statistical information availability is limited. Uncertainties in loads have not been investigated in a systematicalmanner,so RBDO has not been appliedtoindustrialapplications or,ifany, works may have been conducted by blindly modeling uncertainties. Moreover, while designing new product or process, uncertainty data are unavailable for new and revamped parts [10]. To deal with situations when having very few data for uncertainties, possibility [11-13] and evidence [14,15] theories have been recently proposed.In possibilitytheory, uncertainties are modeled with a fuzzy (or membership) function, while evidence theory modelsthem with a basic probability assignment (BPA). A new possibility analysis method was proposed for possibility-based design optimization (PBDO) [11]. As well,two different ways to model membership functionsin fuzzy operation are discussed in Refs. 11and12. Both probability and possibility are clearly explained from many different perspectives [13]. Uncertainties must be properly modeled with the consideration of a data size. For example, in the framework of possibility theory, there are many choices of membership functions [11]. However, no guideline is provided to properly model uncertainties for a given amount of data. Thus, it may lead to overly or less conservative optimum designs. This situation will be unchanged for modeling the BPA in the evidence theory, although evidence theory can well handle descriptive uncertainty. Therefore,a single solution from a specific uncertainty modeling could lead to a subjective decision. Although uncertainty modeling can be done reasonably, there is still a question of how to deal with aleatory and epistemic random variables [16] together without losing valuable information. In both theories, when having aleatory and epistemic uncertainties, aleatory uncertainties must be degenerated into epistemic uncertainties in order to perform possibility (or evidence) analysis [11]. Due to this fact, the challenge of possibility and evidence theories remaining is how to conduct reliabilitybased design with two distinctive types of uncertainties (e.g.,aleatory and epistemic uncertainties). To overcome the aforementioned challenge,this paper proposes an integration of a Bayesian inference to RBD O, thereby optimizing the design subject to Bayesian reliability. When analyzing reliability with both aleatory and epistemic input uncertainties, reliability becomes uncertain due to epistemic uncertainties. Uncertain reliability bounded in [0, 1] can be suitably modeled with Bayesian inference. Therefore,reliability can be modeled as a Beta distribution, r Beta ( x + 1, N − x + 1) where N is a total number of independent trials and x is the number of successful trials [17]. It is referred to as Bayesian reliability. For each independent trial, reliability must be repeatedly computed. Then, reliability results can be aggregated to model Bayesian reliability with the Beta distribution, where itis bounded in 0 (= 0% reliability) and 1 (= 100% reliability). However, Bayesian reliability must be uniquely defined for design optimization. At the same time, the Bayesian approach with incomplete information for input uncertainties should assure greater than target reliability when an exact reliability is realized. To satisfy both requirements,Bayesian reliability will be defined as the median value of the extreme distribution of the smallest reliability value. Since Bayesian reliability analysis requires evaluating reliabilities repeatedly, reliability analysis must be performed efficiently and accurately. A com mon challenge in reliability analysis (or uncertainty quantification) is a multi-dimensional integration to determine the probabilistic nature of a system. Itis almost impossible to conduct the multi-dimensional integration analytically. Direct numericalintegration is notfeasible due toits computational cost. Otherthan direct numerical or analyticalintegration methods,existing methods for uncertainty quantification can be categorized into three groups: 1) the sampling method; 2)the expansion method; and 3)the most probable point(MPP)-based method. The sampling method is the most comprehensive but expensive to use for estimating statistical moments of system responses. Monte Carlo Simulation (MCS) [18, 19] isthe most widely used sampling method but demands thousands of computational analyses (e.g., finite element analysis (FEA), crash analysis, etc.). To relieve the computational burden, other sampling methods have been developed, such as the quasi-MCS [20, 21], (adaptive) importance sampling [22-25], directional sampling [26], etc. Nevertheless, sampling methods are considerably expensive, so itis often used for verification of uncertainty analysis when alternative methods are employed. The 2 A merican Institute of Aeronautics and Astronautics
idea ofthe expansion method isto estimate the statistical moments of system responses with a small perturbation to simulateinput uncertainty. This expansion method includes Taylor expansion [27], perturbation method [28,29],the Neumann expansion method [30], etc. Taylor expansion and perturbation methods require high-order partial derivatives to maintain good accuracy. The Neumann expansion method employs Neumann series expansion of the inverse of random matrices, which requires an enormous amount of computational effort.In summary, allexpansion methods could become computationallyinefficient orinaccurate when the number orthe degree ofinput uncertainty is high. Moreover, since it requires high-order partial derivatives of system responses, it may not be practical for large-scale engineering applications. A rotationally invariant reliability index is introduced through a nonhomogeneous transformation [31]. It is referred to as the MPP-based method, which has been widely used to perform reliability analysis using derivatives of system responses.In this paper,the Hasofer-Lind-Rackwitz-Fiessler (HL-RF) method will be used for reliability analysis. Two examples (a mathematical example and a practical engineering problem) are used to carry out RBDO using the Bayesian approach.
II. Introduction of Bayesian Reliability-Based Design Optimization (Bayesian RBDO) Knowing that both aleatory and epistemic uncertainties exist in the system of interest, RBDO [1-3] can be formulated as
minimize C (X a , Xe ; d) subject to PB (Gi ( X a , X e ; d) ≤ 0) ≥ Φ( β ti ), i = 1, L
U
d ≤ d ≤ d , d∈R
nd
, np
and X a ∈ R , Xe ∈ R na
(1) ne
where PB (Gi ( X a , X e ; d) ≤ 0) = RiB ( X a , Xe ; d ) is Bayesian reliability where Gi ( X a , X e ; d) ≤ 0 is defined as a safety event, C (X a , X e ; d) isthe objective function, d = ( X) isthe design vector, X isthe random vector,its prescribed reliability targetβti , and np, nd, na, and ne are the numbers of probabilistic constraints, design variables, aleatory random variables, and epistemic random variables, respectively. Among all (both aleatory and epistemic) random variables,ifthe parameters describing a random variable are controllable,they are considered as design variables. For instance, a random variable with a normal distribution may have two design variables, mean and standard deviation. W hen modeling uncertaintieswithinsufficient data,the degree of statistical uncertainty could be greaterthan that of physical uncertainty. Then, although RBDO is successfully performed, the optimum design could be unreliable due to a lack of data. Hence, when involving epistemic uncertainties,the probabilistic constraints must be described in a Bayesian sense. Thisis shown as
RiB (X a , Xe ; d) =
Gi ( Xa Xe ) ≤ 0
f Xa Xe (x a xe )dx a dx e ≥ Φ ( βti ) ≥ Rit
(2)
or R ≥ R B i
t i
where RiB (X a , Xe ; d ) is the Bayesian reliability, Rit is a target reliability and f Xa Xe (x a xe ) is the joint Probability Density Function (PDF) of both aleatory and epistemic uncertainties. Due to lacking data for epistemic uncertainty, itis impossible to precisely model the joint PDF f Xe (xe ) for epistemic uncertainties. Modeling the joint PDF for epistemic uncertainties depends strongly on a set of data and is thus subjective. For example, let X1 and X2 be epistemic and aleatory uncertainties, respectively. X2 is assumed to be normally distributed with N ( µ = 5.0, σ = 1.0) whereas X1 is modeled with 20 samples generated from N (5.0,1.0) . The mean, standard deviation,skewness, and kurtosisfrom the data are 4.5789, 0.9294, 0.2908, and 2.4851,respectively.Itisfound that the Gam ma distribution, f ( x2 ) = b − a Γ −1 (a ) x a −1e− x b , provides the best fitto data with a = 25.4439 and b = 0.1800 . By assuming that both random variables are statisticallyindependent,the joint PDF can be plotted, as shown in Fig. 1. As shown in Fig. 1,the reliability estimate could be inaccurate due to insufficient data,leading to an unreliable design. This necessitates the Bayesian approach for reliability assessment. It is hypothesized that the reliability estimate becomes more accurate asthe data sizeis augmented.It will be shown thatthe resultfrom Bayesian RBD O asymptotically approaches to thatfrom the conventional(or Frequentist) RBDO. In other words, Frequentist RBDO is a special case of Bayesian RBD O, since Bayesian RBDO is able to handle aleatory and/or epistemic uncertainties. 3 A merican Institute of Aeronautics and Astronautics
This paper is focused on Bayesian RBD O, when employing both aleatory and epistemic uncertainties. Bayesian reliability will be defined in Section 3.1 and its analysis will be introduced in Section 3.2. One engineering problem (a torque arm) will be used for demonstration of Bayesian reliabilityin Section 3.3.
III. Bayesian Reliability This paper proposes a Bayesian inference to deal with insufficient information. Reliability will be modeled in a Bayesian sense.It will be integrated to RBD O, thereby optimizing the design subject to Bayesian reliability. Section 3.1 defines Bayesian reliability that is modeled with Beta distribution. Section 3.2 provides detailed description on Figure 1. Joint PDF Contours: Black (Approximate Bayesian reliability analysis, which will be Joint PDF with Limited Data) and Blue (True Joint illustrated with atorque arm in Section 3.3. PDF) A. Definition of Reliability in a Bayesian Sense W hen modeling uncertainties with insufficient data, reliability must be uncertain and subjective. A question is how to model uncertain and subjective reliability in a probability sense. The following discussion will answer the question of modeling reliability using the Bayesian inference. In many engineering applications, outcomes of events from repeated trials can be a binary manner, such as occurrence or nonoccurrence, success or failure, good or bad, and so forth. In such cases, random behavior can be modeled with a discrete probability distribution model.In addition,ifthe events satisfy the additional requirements of a Bernoulli sequence, that is to say, if they are statistically independent and the probability of occurrence or nonoccurrence of eventsremains constant,they can be mathematicallyrepresented by the binomial distribution [37]. In other words, ifthe probability of an event occurrence in each trialis p and the probability of nonoccurrence is
(1 − p ) ,then the probability of x occurrences out of a total of N trials can be described by the probability mass function (PMF) of a Binomialdistribution as
Pr ( X = x, N | p ) =
N x
p x (1 − p )
N −x
x = 0,1, 2,
,N
W here p,the probability of occurrence in each trial,isthe parameter ofthe distribution, and
(3)
N N! is = x [ x !( N − x ) !]
the binomial coefficient,indicating the number of ways thatx occurrences out of atotal ofN trials are possible. In Eq.(3),the probability of x N (x occurrences out of N trials)can be calculated when a prior distribution on p is provided. This inference process seeks to update p based on the outcomes of the trials.Given x occurrences out of atotal of N trials,the probability distribution of p can be calculated using Bayes’ Rule as [38]
f ( p | x) =
f ( p) f ( x | p) 1 0
f ( p ) f ( x | p ) dp
(4)
where f ( p ) isthe prior distribution of p , f ( p | x ) isthe posterior distribution of p and f ( x | p ) isthe likelihood
of x given p .The integral in the denominator is a normalizing factor to make the probability distribution proper. The prior distribution is known about p priorto the currenttrials.In this paper, a uniform prior distribution is used to model p bounded in [0, 1]. When using the uniform prior and Binomial likelihood function in Eq. (4), the posterior distribution can be obtained as
4 A merican Institute of Aeronautics and Astronautics
f ( p | x) =
1 β −1 pα −1 (1 − p ) Beta (α , β )
(5)
where α = x + 1 and β = N − x + 1 . Indeed, it is found that the posterior distribution is a Beta distribution,
p ∼ Beta ( x + 1, N − x + 1) , wherethe two parameters of the Beta posterior distribution depend on the outcome of the trials. The posterior distribution, f ( p | x ) = Beta ( x + 1, N − x + 1) ,represents the probability distribution of reliability. In fact,the distribution is a function of x and N, the number of safety occurrences and the total number of trials, respectively. Figure 2 shows the dependence ofthe PDF of reliability on the number of safety occurrences, x, out of N trials. The largerthe number of safety occurrences for a given N trials,the greaterthe mean of reliability. Figure 3 exhibitsthe dependence of the PDF of reliability on the total number of trials (N) with the same ratio of x to N. As the total number oftrialsisincreased,the variation of reliabilityis decreased. In other words,the PDF of reliability asymptotically converges tothe exactreliability with the increase ofthe number oftrials.Itis also associated with a confidence of reliability model. The largerthe data size,the higherthe confidence level of the reliability model. The Bayesian updating process can be conveniently carried out,since the Beta distribution for uncertain reliability, p,is the conjugate distribution forthe Binominal distribution forthe number of safety occurrences, x.
Figure 2. Dependence of the PDF of Reliability on the number of safety occurrences, x
Figure 3. Dependence of the PDF of Reliability on the total number of trials,N
B. Bayesian Reliability Analysis W hen only epistemic uncertainties are engaged to assess reliability, its PDF can be modeled using the Beta distribution in Eq. (5) by counting the number of safety occurrences, x. However, both aleatory and epistemic uncertainties generally appearin most engineering design problems.In such situations,the PDF of reliability can be similarly obtained through Bayesian reliability analysis. It requires reliability analyses with the consideration of aleatory uncertainties at all sampled points for epistemic uncertainties. Different reliability levels are assessed at different sample pointsfor epistemic uncertainties. Section 3.2.1 explains how to modelthe PDF of reliability using the Beta distribution with both aleatory and epistemic uncertainties. For design optimization, Bayesian (random) reliability must be uniquely defined while ensuring atleast target reliability, even if having a lack of data for input uncertainties, compared to a Frequentist approach with completeinformation. With these criteria, Section 3.2.2 will define Bayesian reliability asthe median value ofthe extreme distribution ofthe smallestreliability value. 1) PDF of reliability with both epistemic and aleatory uncertainties As described in Section 2, when involving both aleatory and epistemic uncertainties,itis observed thatreliability is also uncertain. To assess Bayesian reliability,the PDF of reliability must be constructed with finite but limited number of samples. Two steps will be employed to generate the PDF for reliability. 5 A merican Institute of Aeronautics and Astronautics
First, reliability analyses are conducted at samples for epistemic uncertainties while considering aleatory uncertainties. When the data size is small, due to a lack of resources, reliability p = RiB becomes subjective, strongly dependent on the samples. This necessitates the Bayesian approach for reliability assessment. It is hypothesized that reliability measure become objective as a data size is augmented. Then the result of Bayesian RBD O asymptotically approaches to that of the conventional (or Frequentist) RBDO. In other words, Frequentist RBD O is a special case of Bayesian RBDO, since Bayesian RBDO is able to handle aleatory and/or epistemic uncertainties. Second,the results of reliability analyses will be used to model a PDF of reliability with the Beta distribution.It can be mathematically formulated as RiB (X a , X e ; d ) = Pr[ g i ( X a ) ≤ 0 | x e ( k )]
for k = 1,
,N
= f ( pi | x ) =
(6)
1 piα −1 1 − piα −1 Beta (α i , β i )
(
)
β −1
where N is the number of finite sampled data set for epistemic uncertainties. A detailed description is given in the following example. Example: Let g ( X 1 , X 2 ) = 1 − X 12 X 2 20 ≤ 0 be an inequality constraint with two random variables where X 1 is an epistemic random variable and X 2 is an aleatory random variable, X 2 purpose of comparison,X 1 is assumed to be distributed as X1
Normal ( µ X 2 = 2.8, σ X 2 0.2) . For the
N(µX1 = 2.9,σ X1 0.2) . Since X 1 is an epistemic
random variable, only 20 data are randomly sampled from its distribution model,as shown in Table 1. The table also shows the corresponding reliabilities Pr[ g ( X 2 ) ≤ 0 | X 1 ( k )] for k = 1, , 20 that are computed through reliability analyses [1-9]. As shown in Fig. 4, areliability distribution is constructed atthe design point, ( µ X1 , µ X 2 ) = (2.9, 2.8) . From Table 1,the expectation of feasible design points out of total number 20 can be obtained from the sum of all 20 probabilities which is 17.7066. The reliability then can be modeled by the Beta distribution as Beta(17.7066+1=18.7066, 20–17.7066+1=3.2934). As shown in Fig. 4, the actual reliability of the design point is 0.8488 thatis obtained from the Monte Carlo simulation with the assumed PDFs fortwo random variables. 2) Definition of Bayesian Reliability Bayesian reliability must be uniquely defined for design optimization. At the same time,the Bayesian approach with incomplete information for input uncertainties should assure greater than target reliability when an exact reliability is realized. To satisfy both requirements, Bayesian reliability is defined as the median value of the extreme distribution forthe smallest value derived from the Beta distribution in Eq. (6). Table 1. X 1 samples and probabilities X1
Probability
X1
Probability
3.1047
0.99986
2.7605
0.80983
2.7750
0.84471
3.1006
0.99984
2.8175
0.91969
2.5933
0.19228
2.9277
0.99018
2.9604
0.99520
3.1706
0.99997
2.9354
0.99168
3.4741
1.00000
2.9575
0.99488
2.9029
0.98353
2.9430
0.99295
2.8196
0.92247
2.9706
0.99618
2.7157
0.67025
2.6738
0.50480
2.8869
0.97730
2.8185
0.92101
Figure 4. Actual and Estimated Reliability for ( µ X1 , µ X 2 ) = (2.9, 2.8)
6 A merican Institute of Aeronautics and Astronautics
First, the extreme distribution for the smallest reliability value is constructed from the Beta distribution. For random reliability Ri with a mother distribution function FRi (ri ) ,let 1Ri be the smallest value among N samples for random reliability, Ri . Then the Cumulative Distribution Function (CDF) of the smallestreliability value, 1Ri , can be expressed as [37].
1 − F1 R ( ri ) = P ( 1Ri > ri ) = P ( 1Ri > ri , 2 Ri > ri ,
, N Ri > ri )
i
(7)
For identically distributed and statistically independentRi ,the CDF becomes
F1 R ( ri ) = 1 − 1 − FRi (ri )
N
(8)
i
Bayesian reliability, RiB is defined as the median value of the reliability distribution. That is to say, Bayesian
( )
reliabilityisthe solution ofthe nonlinear equation in Eq. (8) by setting F1 R RiB = 0.5 . i
( )
RiB = FR−i 1 1 − N 1 − F1 R ri m i
= FR−i 1 1 − N 0.5
(9)
The PDF of the extreme distribution with N samples can be obtained as
dF1 R
f 1 R ( ri ) =
i
( r) 1
i
1
d ri
i
=
{
d 1 − 1 − FRi (ri ) dri
N
}
(10)
where
FRi (ri ) =
ri 0
1 β −1 θ αi −1 (1 − θ ) i dθ B (α i , β i )
(11)
It can be rewritten as f 1 R ( ri ) = i
dF1 R ( ri ) i
dri
= N 1−
= N 1 − FRi ( ri )
N −1
ri 0
1
B (α i , β i )
N −1
θ iα
i
−1
(1 − θ i )
β i −1
dθ
1 β −1 riα i −1 (1 − ri ) i B (α i , β i )
(12)
f Ri ( ri )
3) Procedure of Bayesian Reliability Analysis Based on the discussion in Section 3.2.2,the procedure to conduct Bayesian reliability analysisfollows as: (1) Collectlimited randomly sampled data for a set of epistemic uncertainties. (2) Calculate probabilities with the consideration of aleatory uncertainties corresponding to randomly sampled data. (3) Constructthe Beta distribution to model reliability using probabilities measured atrandomly sampled data. (4) Constructthe extreme distribution based on the Beta distribution obtained in Step 3. (5) Get the Bayesian reliability using Eq. (9). Let us considerthe example in Section 3.2.1. Using Eq. (10),the extreme distribution for the smallestreliability value is obtained as
F1 R ( r ) = 1 − 1 −
1
0
R
1 2.2934 θ 17.7066 (1 − θ ) B (18.7066, 3.2934 )
20
From Eq. (9), Bayesian reliability is calculated as PBi = 0.6884 . The Beta distribution for reliability and its extreme distribution forthe smallestreliability value are graphically shown in Fig. 5.
7 A merican Institute of Aeronautics and Astronautics
Figure 5. Bayesian Reliability
C. Results of Bayesian reliability analysis A torque arm is often used in mechanical, aerospace, and civil structures. This example is considered for Bayesian reliability analysis and design optimization. There are three geometric parameters:two shape designs and one sizing design (thickness). Two shape design parameters determine the shape of the torque arm while maintaining the symmetric geometry. The first parameter is defined to control the outer shape of the torque arm, whereas the second is defined to change the slotlength. Using a morphing technique,the design domain is specified inthe dashed box, as shown in Fig. 6. The torque arm is modeled as a plane stress problem using 1693 nodes, 2800 elements, and 10158 DOF with a thickness of 5mm. The fixed boundary condition is imposed atthe left hole. The torque arm is made of Aluminum 6061 with E=67.6 GPa, ν=0.3, and the ultimate stress sU =300 MPa. For FE and design modeling, GENESIS 8.0 is used in the paper. Continuum sensitivity is provided forthe design optimization.
G5-G9
G1-G4
Figure 6. Torque FE model and design domain Two shape variables (X1 and X2) are considered as aleatory uncertainty with the complete statisticalinformation, as shown in Table 2. One sizing variable (X3)isregarded as an epistemic uncertain variable with a finite number of data. From the given random properties forthe epistemic uncertain variable X3 in Table 2, 50 samples are randomly chosen for X3. Table 2. Random Propertiesin Torque Arm Model Random Variable X1 (shape) X2 (shape) X3 (sizing)
Mean 0.00 0.00 5.00
Standard Deviation 1.0 1.0 0.1
Distribution Type Normal Normal Normal
8 A merican Institute of Aeronautics and Astronautics
Nine constraints are defined on two critical regions using the von Mises stress in Fig. 6. For those constraints, Bayesian reliabilities are defined as
RiB (X a , Xe ; d) = PB (Gi ( X) =
si ( X) − 1 ≤ 0) sU
(13)
The PDFs for reliabilities at the critical spots are estimated using Bayesian inference. The PDFs are plotted in the dotted curve in Figs. 7-10. The extreme distributions (solid curves) of the reliability PDFs are presented in the figures. The median values of the extreme distribution are then defined as the Bayesian reliabilities for different constraints. As illustratedin Figs. 7-9, G1 to G4 (the most criticalspots) arelessreliablethan G5 to G9 (the secondary critical spots). This observation is consistent with a stress contour in Fig. 6,since the stresses in
Figure 7. Bayesian Reliability for G1
Figure 9. Bayesian Reliability for G4
Figure 8. Bayesian Reliability for G2 and G3
Figure 10. Bayesian Reliability for G5 to G9
G1 to G4 are relatively high. W hen a target Bayesian reliabilityis setto 90%, G1 to G4 are violated, whilethe others are inactive.
IV. Implementation of Bayesian RBDO Although Bayesian reliability analysis is successfully performed, many technical difficulties are found in Bayesian RBDO. To resolve those technical challenges, three topics are discussed in this section: (1) sensitivity analysis for Bayesian reliability, (2) format of Bayesian reliability, and (3)target Bayesian reliability. 9 A merican Institute of Aeronautics and Astronautics
A. Sensitivity Analysis for Bayesian Reliability Analysis Let the ithreliability follow the Beta distribution, Beta (α i , β i ) , where
α i = 1 + x = 1 + R1 + R2 + βi = 2 + N − α i
+ RN
(14)
Recallthe definition of Bayesian Reliabilityin Eq. (9).
FRi ( RiB ) = 1 − N 0.5 =
( g (R
RiB 0
1 β −1 θ αi −1 (1 − θ ) i dθ B (α i , β i )
) can be defined as 1 ,α ) = θ (1 − θ ) B (α , 2 + N − α )
(15)
From Eq. (18), a function g RiB , α i B i
RiB
i
1+ N −α i
α i −1
0
i
dθ + N 0.5 − 1
(16)
i
Then the sensitivity for Bayesian reliability with respectto d j can be expressed as
∂RiB dRiB ∂α i = ⋅ ∂d j d α i ∂d j
(17)
dRiB ∂g ∂α i =− i dα i ∂gi ∂RiB
(18)
where
∂α i ∂P1 ∂P2 = + + ∂d j ∂d j ∂d j
+
∂PN ∂d j
(19)
)
(20)
Numerator and denominatorin Eq. (18) can be rewritten as α i −1 ∂gi 1 1 − RiBi = RiBi B Bei ∂Ri
( ) (
∂gi = ∂α i where Bei = Beta (α i , N + 2 − α i ) =
0
dBei−1 αi −1 1 θ 1+ N −α i ln + θ (1 − θ ) dθ Bei 1 + θ dα i
RiB 0
1 α −1 i
t
1+ N −α i
(t )
N +1−α i
(21)
dt and
dBe −1 1 = 2 dα i Bei
1 0
θ α −1 (1 − θ )
1+ N −α i
i
ln
θ 1+θ
dθ
(22)
Equation (18) can be expressed as RiB
dRiB =− dα i
0
θ α −1 (1 − θ )
1+ N −α i
i
ln
θ 1+θ
+ Bei
( R ) (1 − R ) α i −1 B
B i
i
dBei−1 dθ dα i
1+ N −α i
(23)
The sensitivity for Bayesian reliability with respecttothe jthdesign variable, d j ,can be express as RiB
∂RiB =− ∂d j
0
θ α −1 (1 − θ )
1+ N −α i
i
ln
θ 1+θ
( R ) (1 − R ) α i −1 B
i
B i
+ Bi
1+ N −α i
dBei−1 dθ dα i
N
∂Pj
j =1
∂d j
(24)
Noted that the integration part in Eq. (24) encounters numerical singularity when computing sensitivity of Bayesian reliability. As well,itfollows a complicated mathematical derivation and implementation. A more simple way is sought to calculate Bayesian reliability sensitivity. The idea comes from an one-to-one mapping between Bayesian reliability (the median value of the extreme distribution) and the mean value of the Beta distribution for reliability, RiB = RiB ( M i ) or M i = M i ( RiB ) . Instead of Bayesian reliability, the corresponding mean value of the Beta distribution for reliability will be used for design optimization. Suppose thatthe Beta distribution Beta (α , β ) be used to model reliability. Its mean value,M i = M i ( RiB ) ,can be expressed as 10 A merican Institute of Aeronautics and Astronautics
Mi =
αi
α i + βi
αi
=
(25)
N +2
The sensitivity ofits mean value to design variable,d j ,can be expressed as
∂M i 1 ∂α i = ∂d j N + 2 ∂d j
(26)
From Eq. (19), Eq.(26) can be expressed as
∂M i ∂P1 ∂P2 1 = + + ∂d j N + 2 ∂d j ∂d j
+
∂PN ∂d j
(27)
Results of reliability mean, M i = M i ( RiB ) , can be converted to a reliability index. Then, the sensitivity can be developed for the format of the reliability index βiB , where βiB = Φ −1 ( M i ) . Correspondingly, all reliabilities,
Pi , i = 1 ~ N ,can be transformed into the reliabilityindices, βi . The sensitivity of Bayesian reliabilityindex can be expressed as
∂βiB ∂M i = ∂d j ∂d j
∂M i ∂βiB
(28)
where
∂M i = ∂β iB
1 2π
e
−
(β )
B 2 i
2
(29)
Similarly,
∂M i ∂P1 ∂P2 1 = + + ∂d j N + 2 ∂d j ∂d j
+
∂PN ∂d j
∂P1 ∂β1 1 = + N + 2 ∂β1 ∂d j
∂P ∂β N + N ∂β N ∂d j
∂Pi ∂P ∂β i = i = ∂d j ∂β i ∂d j
e
(30)
where
1 2π
−
β i2 2
∂βi ∂d j
(31)
By substituting the sensitivity of reliability index, ∂β i ∂d j into Eq. (31),the sensitivity of Bayesian reliability,
∂β iB ∂d j ,can be obtained as
(β )
B 2 i
2
∂βiB e 2 N − β2k ∂β k = e ∂d j N + 2 k =1 ∂d j where i = 1,
, nc , j = 1,
, nd , k = 1,
,N .
B. Format of Bayesian Reliability 11 A merican Institute of Aeronautics and Astronautics
(32)
Let us recallthe formulation of Bayesian RBDO here.
minimize C (X a , Xe ; d) subject to RiB = FGBi (0) ≥ Rit = Φ ( β ti ), i = 1, L
U
d ≤ d ≤ d , d∈R
nd
, np
and X a ∈ R , Xe ∈ R na
(33) ne
To be consistent with the sensitivity analysis for Bayesian reliability, Bayesian reliability constraints can be formulated in terms ofthe Bayesian reliabilityindex as
RiB ≥ Rit
⇔
Φ −1
M i ≥ M it
⇔
βiB ≥ Φ −1 ( M it )
(34)
Finally,the Bayesian RBDO can be reformulated as
minimize C (X a , Xe ; d)
subject to βiB ≥ Φ −1 ( M it ) , i = 1, L
U
d ≤ d ≤ d , d∈R
nd
, np
(35)
and X a ∈ R , Xe ∈ R na
ne
C. A Setup of Target Reliability for Bayesian RBD O Target reliability must depend on the data size of epistemic uncertainties. With few data for uncertainties, setting to 99.9% target reliability does not make sense. Although high reliability is achieved through RBD O, the confidence of reliability will be extremely low. This section provides a guideline to settargetreliabilityin Bayesian RBD O, which depends on a data size of epistemic uncertainties. To assist the setup of target reliability, the maximum Bayesian reliability, PBmax will be determined for a given sample size, N.Itcan be obtained when the Beta distribution for reliability is distributed to be Beta (1 + N ,1) with all safe samples. Then, the maximum Bayesian reliability can be defined as
PBmax = median F1 R ( ri ) i
where Ri ~ Beta (1 + N ,1)
(36)
which is graphically shown in Fig. 11. As the data size rises, PBmax rapidly increases to 90% and then slowly increases. Target reliability must be set lower than the maximum Bayesian reliability for a given data size. For example,the targetreliability with 50 data setfor epistemic uncertainties must be lower than 92%.
Figure 11. Target Bayesian Reliability with Sample Size, N
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V. Result of Bayesian RBDO A. M athematical Example Consider the following mathematical RBDO problem with three random variables. Two of them are aleatory with X i Normal ( µi , 0.6) and one of them is epistemic with N samples. In this paper, aleatory random variables are considered as design variables, d = [d1 , d 2 ]T = [ µ1 = µ ( X 1 ), µ 2 = µ ( X 2 )]T . On the contrary, epistemic random variable, X 3 is not considered as a design variable, since none of its statistical property is known. Then, RBDO problem is defined as
Minimize d1 + d 2 Subject to PB (Gi ( X ) ≤ 0) = FGBi (0) ≥ Φ ( β ti ), i = 1, 2,3
(37)
0 ≤ d1 & d 2 ≤ 10 where
G1 ( X) = X 12 X 2 X 3 / 20 − 1 G2 ( X) = ( X 1 + X 2 + X 3 − 5) 2 / 30 + ( X 1 − X 2 − X 3 − 11) 2 /120 − 1
(38)
G3 ( X) = 80 /( X + 8 X 2 X 3 + 5) − 1 2 1
In this study, target reliability is set toΦ ( βti ) = 90% .Itis hypothesized that the design based on the Bayesian principle asymptotically approaches to that based on the Frequentist as the sample size is increased. To verify the hypothesis, Bayesian RBDO is carried outfor different sample sizes with N = 50, 100, 500, 1000, and 10000. Those samples are randomly generated during the design optimization by assuming X 3 Normal (1.0, 0.1) . For each sample size, Bayesian RBDO was performed 20 times to get more objective result.Itis apparentin Fig. 12 that the hypothesis is valid, since the Bayesian reliability-based optimum design with 10000 samples is very close to the Frequentistreliability-based optimum design. Smaller data size yields more reliable optimum designs. As increasing the data size from 50 to 10000, the optimum designs are narrowly spread out and move toward the Frequentist reliability-based optimum design. Figure 12 can provide an insightful idea to classify aleatory and epistemic uncertainties even though the classification of aleatory and epistemic types is not discussed in the paper. Bayesian RBD O with more than 500 samples producesthe optimum design close tothe Frequentistreliability-based optimum design. The Pareto frontier of the optimum design can be constructed along the optimum design trajectory as the data size isincreased, as shown in Fig. 12.
Figure 12. Bayesian Reliability-Based Optimum Designs with Sample Size N In orderto observe the feasibility of Bayesian reliability constraints, Figs. 13-15 show constraintfeasibility atthe base design and optimum designs with N = 50 and 10,000 samples. As shown in Fig. 13,two constraints (G1 and G2) 13 A merican Institute of Aeronautics and Astronautics
(a) G1
( b ) G2
( c) G3
Figure 13. Bayesian Reliability Constraints at the Base Design with N=50
(a) G1
( b ) G2
( c) G3
Figure 14. Bayesian Reliability Constraints at the Optimum Design with N = 50
( a) G1
( b ) G2
( c) G3
Figure 15. Bayesian Reliability Constraints at the Optimum Design with N = 10,000 14 A merican Institute of Aeronautics and Astronautics
are feasible atthe base design whereas the third constraint(G3) isinfeasible. At both optimu m designs with N = 50 and 10,000 samples,the firsttwo constraints(G1 and G2) become active whereas the third constraintisfeasible.Itis evident thatthe PDF and its extreme distribution of reliability with large samples have smaller variation than those with small samples. In other words, reliability estimates become more confident with the increase of samples for input uncertainties. Therefore, a wider distribution of reliability constraints forces the design to be away from the failure surface, as shown in Fig. 12. Itis observed that the optimization results with smaller samples require more design iterations and itis mainly due to alack of data. B. Torque Arm The torque arm used in Section 3.3 will be used for Bayesian RBDO. Three random variables are used: two shape parameters and one sizing parameter (thickness). Two shape variables (X1 and X2) are considered as aleatory uncertainty withthe complete statisticalinformation, as shown in Table 3. One sizing variable (X3)isregarded as an epistemic uncertain variable with a finite number of data. The epistemic uncertain variable, X3,is modeled with 50 samples at every updated design during Bayesian RBDO. These samples are randomly generated assuming X 3 Normal (5.0, 0.1) .In this example, aleatory random variables are considered as design variables, whereas epistemic random variable,
X 3 is not considered as a design variable. For FE and design modeling,
GENESIS 8.0 is used in the paper. Continuum sensitivity isprovided forthe design optimization. Table 3 Random Propertiesin Torque Arm Model Random Variable X1 (shape) X2 (shape) X3 (sizing)
dL -20.0 -20.0 NA
•X =d (Mean) 0.00 0.00 5.00
dU 20.0 20.0 NA
Standard Deviation 1.0 1.0 0.1
Distribution Type Normal Normal Normal
Bayesian reliabilities are defined for nine constraints. The Bayesian RBDO can be formulated as
Minimize Mass Subject to
PB (Gi ( X) =
si ( X) − 1 ≤ 0) = FGBi (0) ≥ Φ ( β ti ), i = 1, sU
,9
(39)
In this study,targetreliability issettoΦ ( βti ) = 90% . Figures 16 and 17 display the history ofthe objective function and the Bayesian reliability constraints. Fourteen design iterations are taken to get the Bayesian reliability-based optimum design. As shown in Fig. 17, the third constraint becomes active and others are feasible. The constraints G1, G4-G9 have 91.87% of Bayesian reliability, which is the maximum Bayesian reliability with N=50. In other words, all fifty samples turn out to be 100% safe. Numerically, β = 6 is set to 100% safety. Figures 18 and 19 illustrate the reliability PDFs and Bayesian reliabilities at the optimum design. Indeed, Bayesian reliabilities of G2 and G3 maintain more than 90% reliability. Finally, the initial design and three optimum designs are compared in Table 4 and Fig. 20. The deterministic optimum design hasthe smallest mass. The reliability-based optimum designs attain reliability greaterthan the targetreliability while sacrificing their mass. Due to alack of data, Bayesian RBD O produces the optimum design that satisfies more than the target reliabilityin any circumstance.
15 A merican Institute of Aeronautics and Astronautics
Figure 16. History of Objective Function
Figure 17. History of Bayesian Reliabilities
Figure 18. G2 Bayesian Reliability at the Optimu m
Figure 19. G3 Bayesian Reliability at the Optimu m
Table 4 Design Comparison Designs
Initial
X1 (shape) X2 (shape) X3 (sizing) Mass
0.0000 0.0000 5.0000 0.0570
( a) Frequentist Approach
Deterministic Optimum 5.4314 5.7122 5.0000 0.0502
Frequentist Optimum 3.5255 6.6462 5.0000 0.0532
Bayesian Optimum 1.9622 0.7294 5.0000 0.0542
( b ) Bayesian Approach
Figure 20. Reliability-Based Optimu m Designs of Torque Arm
16 A merican Institute of Aeronautics and Astronautics
VI. Conclusion Practical engineering systems have both sufficient(aleatory) and insufficient(epistemic) data for random inputs, such as geometric tolerances, material properties,loads, etc. A fundamental difficulty has been found in employing Frequentist statistics to deal with both aleatory and epistemic uncertainties simultaneously. To resolve such a difficulty, this paper proposes an integration of a Bayesian approach to Reliability-Based Design Optimization (RBDO). When a design problem involves both aleatory and epistemic types of input uncertainties,reliability must be uncertain and subjective. Bayesian inference is used to model uncertain and subjective reliability. Its PDF is successfully modeled with the Beta distribution. For design optimization,two requirements must be considered: 1) reliability must be uniquely defined and 2) reliability must be greaterthan targetreliability when an exactreliability is realized. To satisfy both requirements, Bayesian reliability is defined as the median value of the extreme distribution ofthe smallestreliability value. Numerical procedure is presented along with thetorque arm problem. It is shown that a confidence level of reliability model is strongly associated with the data size for epistemic uncertainties. The largerthe data size,the higher the confidence level of a reliability model. Detailed description of Bayesian reliability analysis is presented with a numerical procedure and the torque arm example. For design optimization,three topics are discussed in this section: (1) sensitivity analysisfor Bayesian reliability,(2) format of Bayesian reliability, and (3) target Bayesian reliability. Two examples (one mathematical example and the torque arm) are used to demonstrate the feasibility of Bayesian RBD O. Itis hypothesized in the paper thatthe result from Bayesian RBDO asymptotically approaches to that from the conventional (or Frequentist) RBDO. Through the mathematical example, it is examined that the hypothesis is valid, since the Bayesian optimum design becomes closer to the Frequentist one as the data size is increased. Finally, in the torque arm example, the Bayesian reliability-based optimum design issuccessfullyfound and turns outto be more reliable,compared tothe Frequentist RBD O.
Acknowledgments Research is partially supported by the STAS contract(TCN-05122) sponsored by the U.S. Army TARDEC.
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