1 reliability and robustness in engineering design zissimos p. mourelatos, associate prof. jinghong...
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1
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]
2
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
7
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
8
Reliable / Robust Design Problem:Issues
Variability Measure Calculation
Variance
Percentile Difference
Trade – offs in Multi – Objective Optimization
Preference Aggregation Method
9
12%5%95RR
f ffR
PDFf
f
ΔRf
1%5Rf 2
%95Rf
Percentile Difference Approach
Advanced Mean Value (AMV) method is used
10
Multi – Objective Optimization:Min – Min Problem
min f
min g
subject to constraints
min g
min f
g
f
utopia pt
Pareto set
11
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
13
Preference Functions
1
0weight
hw
1
0reliability
hr
Example: Trade – off between weight and reliability
Aggregate h(hw,hr) is maximized
14
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
15
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
16
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.
17
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
18
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
19
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%
20
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
21
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
22
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
23
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
24
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
25
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%
26
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
27
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
28
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
29
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
31
Design Under Uncertainty
Analysis /SimulationInput Output
Uncertainty (Quantified)
Uncertainty (Calculated)
1. Quantification
Propagation
2. Propagation
Design
3. Design
32
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
33
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