Reliability-Based Design Optimization Applied to

Reliability-Based Design Optimization Applied to Aircraft Initial Sizing

KSAS Conference, Spring 2017 Seul-Ki Kim

Shinseong Kang

Kyunghoon Lee

Department of Aerospace Engineering, Pusan National University

April 20, 2017

RBDO applied to the aircraft initial sizing

Outline

SK Kim Outline Introduction Motivation and Research Proposition Formulations Constraint Analysis Reliablity-Based Design Optimization Demonstration Summary and Conclusion

1 Introduction

Motivation and Research Proposition

2 Formulations

Constraint Analysis Reliablity-Based Design Optimization

3 Demonstration 4 Summary and Conclusion

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RBDO applied to the aircraft initial sizing SK Kim Outline Introduction Motivation and Research Proposition Formulations Constraint Analysis Reliablity-Based Design Optimization Demonstration Summary and Conclusion

Aircraft Initial Sizing

The overall shape, size, weight, and performance of the aircraft are determined.

Constraint Analysis

Designer determines a pair of thrust loading (TSL /WTO ) and wing loading (WTO /S) at takeo from constraint analysis with the following formula. f

TSL WTO , , T , P, CL , CD WTO S

=0

(source: Aircraft Engine Design)

(source: HowStuWorks)

Figure: Mission prole

Figure: Constraint analysis diagram

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Motivation and Research Proposition

Uncertainty in Flight

Uncertainty impacts the ight performance signicantly. A small variation of ight condition may cause a mission failure. 1

Aleatory uncertainty

Inherent randomness associated with a physical system or environment. Environmental factors: temperature, pressure, etc.

2

Epistemic uncertainty

Ignorance or limited data and knowledge. Aerodynamic data: CL , CD , K1 , K2 , etc.

Research Proposition Reliability-based design optimization (RBDO) in consideration of uncertainties. 4 / 22


Constraint Analysis

Master Equation

(source: Aircraft Engine Design)

Figure: Applied forces on aircraft

A master equation for ight performance can be derived from Newton's second law of forces applied to the aircraft as " 2 q S nβ WTO nβ WTO K1 + K2 β WTO q S q S Ps . +CD0 + CDR ] + V

TSL β = WTO α

(

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Constraint Analysis

Air-to-Air Fighter (AAF) Example Of the various aircraft, we adopt air-to-air ghter (AAF) and selected AAF specications are delineated as follows: Mission phases and segments

Performance requirements

1

Takeo, no obstacle

2 3

Supercruise Combat turn

4

Landing

2000 ft PA, 100 °F, 0.1M tR = 3 s, sTO = 1500 ft 1.5M/30kft, no afterburning 0.9M, two 360 deg 5g sustained turn, with afterburning 2000 ft PA, 100 °F tFR = 3 s, sL = 1500 ft Drag chute diameter 15.6 ft (source: Aircraft Engine Design)

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Constraint Analysis

Air-to-Air Fighter (AAF) Example

Mission Phase 1: Takeo

WTO S

(

=

−b +

√

b 2 + 4ac 2a

)2 ,

where ( #) " β α TSL CL,max log 1 − ξTO − µTO , 2 ρg0 ξTO β WTO kTO q b = tR kTO 2β/(ρCL,max ), c = sTO . a=−

Mission Phase 2: Supercruise TSL β = WTO α

β K1 q

WTO S

CD0 + β/q(WTO /S)

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Constraint Analysis

Air-to-Air Fighter (AAF) Example

Mission Phase 3: Combat Turn

β TSL = WTO α

K1 n2

β q

WTO S

+

CD0 β/q(WTO /S)

Mission Phase 4: Landing

WTO S

(

=

−b +

√

b 2 + 4ac 2a

)2 ,

where β a= ρg0 ξL

(

log 1 + ξL

b = tFR kTD

q

"

2β/(ρCL,max ),

(−α) TSL µB + β WTO

CL,max 2 kTD

#) ,

c = sL .

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Constraint Analysis

Air-to-Air Fighter (AAF) Example (source: Aircraft Engine Design)

Figure: Traditional constraint analysis; oset 9 / 22


Reliablity-Based Design Optimization (RBDO)

RBDO is a method that minimizes the objective function subject to predened probabilistic constraints.

Concept

Formulation minimize

d

f (d)

subject to P[gi (X) > 0] ≤ Pi , i = 1, · · · , Nc ,

d

lb

(source: Lecture note of Prof. Noh)

≤ d ≤ dub ,

where d = {d1 , . . . , dn } ∈ Rn is the design variable vector, X = {X1 , . . . , Xm } ∈ Rm is the random variable vector, and Pi is target probability of failure. 10 / 22



First-Order Reliability Method (FORM)

The probability of failure is dened by a multi-dimensional integral such that Pfailure = P[g (X) > 0] ≡

Z

Z ···

g (X)>0

fX (x)dx1 . . . dxm ,

which is the volume underneath the surface of the joint probability density function (PDF) fX (x) in the failure region g (X) > 0. In most cases, the numerical integration is so complicated that it requires considerable computation. To shorten computational burden, we approximate the performance function g (X) using rst-order reliability method (FORM).

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Rosenblatt Transformation T : X → U

The rst step of FORM requires a transformation T from the original random variables X to the standard normal random variables U such that g (T(X)) ≡ g (U).

After the transformation, the probability of failure becomes Pfailure = P[g (U) > 0] ≡

Z g (U)>0

φU (u)d u.

MPP-based Method

In the second step of FORM, the performance function in U-space can be linearized at the most probable point (MPP) u ∗ that has the highest joint PDF such that g (U) ∼ = gL (U) = g (u∗ ) + ∇g T (u∗ )(U − u∗ ),

where ∇g (u∗ ) is the gradient of g (U) evaluated at u∗ .

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MPP-based FORM Method Maximizing the joint PDF φU (u) on the limit-state function gives the model for the MPP search such that minimize

u

m X

ui2 = kuk2

i=1

subject to g (u) = 0.

The minimum distance ku∗ k from the origin to the MPP is called reliability index β . (source: Lecture note of Prof. Noh)

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MPP-based FORM Method

The linearized limit-state function at the MPP (u∗ ) can be described such that g (U) ∼ = gL (U) =

n X

(Ui −

ui∗ )

i=1

∂g ∂Ui

, Ui =ui∗

since g (u∗ ) = 0. The mean and standard deviation of gL (U) is expressed as µgL =E [gL (U)] = −

n X i=1

σgL

ui∗

∂g ∂Ui

, Ui =ui∗

v  u uX p u n ∂g  = Var[gL (U)] = t ∂Ui i=1

2  . Ui =ui∗

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Consequently, the probability of failure using MPP-based FORM is given by µgL 0 − µgL = Φ . Pfailure = P[g (U) > 0] ∼ 1 − Φ = σgL

σgL

The probability of failure can be rewritten as Pfailure

where α =

∼ =Φ

µgL σgL

=Φ −

n X

! αi ui∗

= Φ −u∗T α ,

i=1

∇g (u∗ ) is the normalized gradient vector. k∇g (u∗ )k

Finally, the probability of failure can be simply expressed in terms of β such that Pfailure ∼ = Φ −u∗T α = Φ −βαT α = Φ(−β). 15 / 22




Probabilistic constraint P[g (X) > 0] ≤ P can be replaced by Φ(−β) ≤ Φ(−β target ),

where βtarget is called the target reliability index. In conclusion, the standard formulation of RBDO using MPP-based FORM is dened such that minimize

d

f (d)

subject to βitarget ≤ βi

d

lb

i = 1, · · · , Nc ,

≤ d ≤ dub .

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Demonstration

Random variables for AAF Example Mission phases

Random variables K1

Takeo

CL CD 0

Supercruise Combat turn

K1 CD 0 K1 CD 0 K1

Landing

CL CD 0

Mean value 0.180 1.389 0.014 0.270 0.028 0.180 0.016 0.180 1.210 0.014

Coecient of variation

Distribution type

0.1 " " " " " " " " "

Normal " " " " " " " " "

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Demonstration

Comparison of RBDO Results with Discrete Target Reliability 1.6 Solution space 1.4

Thrust Loading (TSL =WT O )


Takeoff Supercruise Combat turn Landing

Initial Point 99% RBDO 95% RBDO 90% RBDO

1.2

DO 1

0.8

0.6

0.4 20

30

40

50

60

70

80

90

100

110

120

Wing Loading (WT O =S)

Figure: Results of DO and RBDO1 1 DO : Deterministic optimum, RBDO : Reliability-based design optimum. 2 Non-linear optimizer: sequential quadratic programming (SQP)

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Demonstration

Comparison of RBDO Results with Discrete Target Reliability Methods

Costs

WTO S

∗

TSL WTO

∗

Constraints

g1 = −5.303 DO

0.365

55.911

1.055

g2 = 0 g3 = 0 g4 = −14.627 g1 = −10.985

RBDO (90%)

0.444

56.518

1.176

g2 = −0.131 g3 = −0.113 g4 = −14.020 g1 = −12.555

RBDO (95%)

0.467

56.665

1.209

g2 = −0.168 g3 = −0.146 g4 = −13.873 g1 = −17.214

RBDO (99.7%)

0.535

57.058

1.314

g2 = −0.279 g3 = −0.245 g4 = −13.480 19 / 22

RBDO applied to the aircraft initial sizing SK Kim

Demonstration

Comparison of RBDO Results with Discrete Target Reliability

Outline Introduction Motivation and Research Proposition Formulations Constraint Analysis Reliablity-Based Design Optimization Demonstration Summary and Conclusion

Methods

Sref (ft2 )

DO 429.254 RBDO (90%) 424.644 (1.074% ↓) RBDO (95%) 423.542 (1.331% ↓) RBDO (99.7%) 420.625 (2.010% ↓) *W TO = 24,000 lbf

TSL (lbf )

25,320 28,224 (11.470% ↑) 29,016 (14.597% ↑) 31,536 (24.550% ↑)

Findings 1

2

Compared to the deterministic design, the reliability-based design provides higher reliable results. Design with the higher target reliability tends to be oversized than that with the lower target reliability.

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Summary and Conclusion


Summary 1

2

Reliability-based design optimization is proposed based on MPP-based FORM. To consider uncertainty, a total of ten aerodynamic data were selected as random variables.

Conclusion 1

Reliability-based design optimization using MPP-based FORM provides a reliable optimum point for an aircraft initial sizing.

Future Work 1 2

Sensitivity analysis. MPP-based second-order reliability method (SORM).

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Thank you for your attention! Questions?

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