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Reliability and Robustness in Engineering Design

Zissimos P. Mourelatos, Associate Prof.Jinghong Liang, Graduate Student

Mechanical Engineering Department Oakland University

Rochester, MI 48309, USAmourelat@oakland.edu

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Outline Definition of reliability-based design and robust design

Reliable / Robust design

Problem statement

Variability measure

Multi-objective optimization

Preference aggregation method

Indifferent designs

Examples

Summary and conclusions

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Reliable Design Problem Statement

Maximize Mean Performance

subject to :

Probabilistic satisfaction of performance targets

Reliability

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Robust Design Problem Statement

Minimize Performance Variation

subject to :

Deterministic satisfaction of performance targets

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Robust Design

A design is robust if performance is not sensitive to inherent variation/uncertainty.

Design Parameter

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Reliable & Robust Design under Uncertainty: Problem Statement

Maximize Mean Performance

Minimize Performance Variation

subject to :

Probabilistic satisfaction of performance targets Reliability

Robustness

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Reliable / Robust Design Problem Statement

Multi Objective

UL ddd

ULXXX μμμ

, ii RGP 0,, pXd ni ,...,1

mRX : vector of random design variables

qRp

kRd : vector of deterministic design variables

: vector of random design parameters

s.t.

where :

PXμd,

μμdX

,,min fR

PXμd,

μ,μd,X

fmin

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Reliable / Robust Design Problem:Issues

Variability Measure Calculation

Variance

Percentile Difference

Trade – offs in Multi – Objective Optimization

Preference Aggregation Method

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12%5%95RR

f ffR

PDFf

f

ΔRf

1%5Rf 2

%95Rf

Percentile Difference Approach

Advanced Mean Value (AMV) method is used

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Multi – Objective Optimization:Min – Min Problem

min f

min g

subject to constraints

min g

min f

g

f

utopia pt

Pareto set

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Multi – Objective Optimization:Issues

• Must calculate whole Pareto set

Series of RBDO problems

Visualize Pareto set

• Choose “best” point on Pareto set

Expensive

(How??)

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Preference Aggregation Method

• Capable of calculating whole Pareto set

• Use of Indifferent Designs to only get the “best” point on Pareto set

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Preference Functions

1

0weight

hw

1

0reliability

hr

Example: Trade – off between weight and reliability

Aggregate h(hw,hr) is maximized

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Preference Aggregation Axioms

Annihilation :

Idempotency :

Monotonicity : if

Commutativity :

Continuity :

0,0,,,,,0 211221 wwhhwhwh

12111 ,,, hwhwhh

2*2112211 ,,,,,, whwhhwhwhh *

22 hh

11222211 ,,,,,, whwhhwhwhh

221*12211 ,,,lim,,,

1*1

whwhhwhwhhhh

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sss

ww

hwhwwhwhh

1

21

22112211 ,,,

satisfies annihilation for 0s only.

2121

1

21wwww

prod hhhh 0sFor :Fully

compensating

21,min hhhsFor : Non - Compensating

Preference Aggregation Method

Aggregation is defined by

1

2, wwws

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Preference Aggregation Properties

• For any Pareto optimal point, there is always a set (s,w) to select it.

• For any fixed s, there are Pareto sets for which some Pareto points can never be selected for any choice of w.

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Indifferent Designs

h

h1=1

href

1

0

h2=a2h1=a1

h2=1

refhwahwah ;1,;,1 12

• Two designs are indifferent if they have the same overall preference

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Indifferent Designs

refhwahwah ;1,;,1 12

221 1 sref

sref

ssref

s hhaha ....s

sref

ssref

h

ahw

1

1

resulting in

and

The calculated (s,w) pair will select the “best” design on the Pareto set

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A Mathematical Example

10534min 22

41

31 xxxf x

x

045.621 xxG X

2,1,101 ixi

s.t.

fxμ

min

Xxμ

fRmin

RGP )0)(( X

RP 101 X

s.t.

Reliable/Robust Problem

12 RRf ffR

2,1,4.0,~ iNxixi

%952 R %51 R

R = 99.87%

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A Mathematical Example

RGP )0)(( X

fxμ

min

RP 101 X

s.t.

RBDO Problem

4745.5* f

9471.5,2.2*xμ

Robust Problem

Xxμ

fRmin

RGP )0)(( X

RP 101 X

s.t.8982.2* fR

5332.5,4668.3*xμ

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A Mathematical Example

0

1

*f *3 f f

1h

0

1

*f *3 f f

1h

“cut-off”

*8 ff RR For h2 the “cut-off” value is

sss

w

whhh

1

21

1

hxμ

max

RGP )0)(( x

2,1,101 iRxP i

Final Optimization Problem

Single-Loop RBDO

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0.8

1.2

1.6

2

2.4

2.8

3.2

3.6

0.8 1 1.2 1.4 1.6 1.8 2

s=1, w=1~10, step=1

s=-1, w=1~10, step=1

s=-5, w=1~10, step=1

s=-5, w=0.1~1, step=0.1

s=-8, w=1~10, step=1

s=-8, w=0.1~1, step=0.1

ΔRf/ΔRf*

μf/μf*

Performance Optimum

Robust Optimum

Chosen Design

87.0refh

81.0,79.0 21 aa215.1,5 ws

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A Mathematical Example

.

*2*1minf

f

f

f

R

Rwwf

xμ

RGP )0)(( x

2,1,101 iRxP i

s.t.Weighted Sum

Approach

R=99.87%

121 ww

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A Mathematical Example

0.5

1

1.5

2

2.5

3

3.5

0.8 1 1.2 1.4 1.6 1.8 2

ΔRf/ΔRf*

μf/μf*

Reliable Optimum

Robust Optimum

Performance

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A Cantilever Beam Example

L=100 in w

Y

Z t

twμf

tw

,

min

ZYEtwRtw

,,,,min,

RGP )0)(( 1 X 5,0 tw

22

22

3

)()(4

),,,,(w

Z

t

Y

Ewt

LZYEtw

)*600

*600

(),,,,(221 Ztw

Ywt

ytwYZyG

,s.t.

where:

Reliable/Robust Formulation

• w,t : Normal R.V.’s

• y, E,Y,Z : Normal Random Parameters

• L : fixed

• R = 99.87%

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A Cantilever Beam Example

twμf

tw

,

min

RGP )0)(( 1 X 5,0 tw

)*600

*600

(),,,,(221 Ztw

Ywt

ytwYZyG

,s.t.

where:

RBDO Problem

2884.11* f

8369.3,9421.2, ** tw

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A Cantilever Beam Example

ZYEtwRtw

,,,,min,

RGP )0)(( 1 X 5,0 tw

22

22

3

)()(4

),,,,(w

Z

t

Y

Ewt

LZYEtw

)*600

*600

(),,,,(221 Ztw

Ywt

ytwYZyG

,s.t.

where:

1440.0* R

5,5, ** tw

Robust Problem

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0

1

2

3

4

5

6

7

0.75 0.95 1.15 1.35 1.55 1.75 1.95 2.15 2.35

s=1, w=0.1~1, step=0.1

s=-1, w=0.1~1, step=0.1

s=-5, w=0.1~1, step=0.1

ΔRδ/ΔRδ*

μf/μf*

Robust Optimum

Performance Optimum

Chosen Design

94.0refh

8.0,91.0 21 aa5895.0,5 ws

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Summary and Conclusions A methodology was presented for trading-off performance and robustness

A multi – objective optimization formulation was used

Preference aggregation method handles trade – offs

Variation is reduced by minimizing a percentile difference

AMV method is used to calculate percentiles

A single – loop probabilistic optimization algorithm identifies the reliable / robust design

Examples demonstrated the feasibility of the proposed method

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Q & A

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Design Under Uncertainty

Analysis /SimulationInput Output

Uncertainty (Quantified)

Uncertainty (Calculated)

1. Quantification

Propagation

2. Propagation

Design

3. Design

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Feasible Region

Increased Performance

x2

x1

f(x1,x2) contours

g1(x1,x2)=0

g2(x1,x2)=0

Deterministic Design Optimization and Reliability-Based Design Optimization

(RBDO)

Reliable Optimum

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pXμd,

μ,μd,X

fmin

UL ddd

ULXXX μμμ

, ii RGP 0,, pXd ni ,...,1

mRX : vector of random design variables

qRp

kRd : vector of deterministic design variables

: vector of random design parameters

s.t.

where :

RBDO Problem Statement

Single Objective

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Indifferent Designs

• Two designs are indifferent if they have the same overall preference

• Designer provides specific preferences a1=h1(xi) and a2=h2(xi) so that :

refhwahhhwhahh ;,1;1, 221211