reliability and robustness in engineering design zissimos p. mourelatos, associate prof
DESCRIPTION
Reliability and Robustness in Engineering Design Zissimos P. Mourelatos, Associate Prof. Jinghong Liang, Graduate Student Mechanical Engineering Department Oakland University Rochester, MI 48309, USA [email protected]. Outline. Definition of reliability-based design and robust design - PowerPoint PPT PresentationTRANSCRIPT
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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, [email protected]
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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
3
Reliable Design Problem Statement
Maximize Mean Performance
subject to :
Probabilistic satisfaction of performance targets
Reliability
4
Robust Design Problem Statement
Minimize Performance Variation
subject to :
Deterministic satisfaction of performance targets
5
Robust Design
A design is robust if performance is not sensitive to inherent variation/uncertainty.
Design Parameter
6
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??)
12
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