Transcript Document

Reliability based design optimization
• Probabilistic vs. deterministic design
– Optimal risk allocation between two failure
modes.
• Laminate design example
– Stochastic, analysis, and design surrogates.
– Uncertainty reduction vs. extra weight.
Deterministic design for safety
• Like probabilistic design it needs to lead to low
probability of failure.
• Instead of calculating probabilities of failure
use array of conservative measures.
– Safety factors.
– Conservative material properties.
• Tests
• Accident investigations
– Risk allocation driven by history (accidents).
Pro and cons of probabilistic design
• Probabilistic design requires more data, that is
often not available or expensive to get.
• Probabilistic design may require to accept finite
probability of death or injury and my lead to legal
liabilities.
• Probabilistic design may allow more economical
risk allocation.
• Probabilistic design may allow trading measures
for compensating against uncertainty against
measures for reducing it.
Optimal risk allocation
• If there is a single failure mode, the chances are
that history has resulted in safety factors that
reflect the desired probability of failure.
• When there are multiple failure modes it makes
sense to have excessive protection against modes
that are cheap to protect against.
• Adding probabilities : If one mode has failure
probability p1 and a second p2, what is the
system failure probability if they are
independent?
Pfsystem  1  (1  p1 )(1  p2 )  p1  p2  p1 p2  p1  p2
Example
• An airplane wing weighs 10,000 lb and the tail weighs 1,000
lb. With a safety factor of 1.5, each has a failure probability of
1%, for a total failure probability of 2% (actually 1-0.99^2)
• For each component the relation between the probability of
failure and additional weight is
Pf  0.5100 W /W0 Pf 0
• Reduce the failure probability to 0.5% with minimum weight.
• Adding 200 lb to wing and 20 lb to tail reduces the
probabilities of each by a factor of 4 for 220 lbs.
• Adding 120 lb to the wing and 80 lb to the tail will lead to
0.435% wing failure probability plus 0.004% tail failure
probability. Safer and lighter.
• What is the optimum?
FORM vs. Monte Carlo
• FORM is much cheaper, but
– Does not give you good estimate of system
probability of failure when failure modes are
strongly coupled.
– Can have large errors when variables are far from
normal and limit state have multiple local MPPs.
– More difficult to allocate risk.
• MCS usually too expensive unless you fit a
surrogate to limit state function.
Deterministic Design of Composite
Laminates
•
Design of angle-ply laminate
– Maximum strain failure criterion
minimize h  4t1  t 2 
NAxial
2
such t hat
  1  
  2  
 12   12u
c
1
c
2
0.005  t 2
1


t
1
t
2
0.005 t1
y
NHoop
x
Load induced by internal pressure:
NHoop = 4,800 lb./in., NAxial = 2,400 lb./in.
.
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Summary of Deterministic Design
• Optimal ply-angles are 27 from hoop
direction
• Laminate thickness is 0.1 inch
• Probability of failure (510-4) is high with
safety factor 1.4.
.
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Reliability-based Laminate Design
minimize h  4t1  t2 
suchthat
P  Pt
•
4 Design Variables
– 1, 2, t1, t2
•
12 Normal Random Variables
–
–
–
–
–
0.005  t1
0.005  t 2
Pt = 10-4
•
.
Tzero (CV = 0.03)
1, 2 (CV = 0.035)
E1, E2, G12, 12 (CV = 0.035)
1c, 1t (CV = 0.06)
2c, 2t, 12u (CV = 0.09)
First ply failure principle
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Structural & Multidisciplinary Optimization Group
Response Surface Options
 Design response surface approximation (DRS)
– Response or Probability v.s. design variables: G=G(d)
– Used in optimization
 Stochastic response surface approximation (SRS)
– Response v.s. random variables: G=G(x)
– Used in probability calculation.
– Need to construct SRS at every point encountered in optimization
 Analysis response surfaces
– Response v.s. random variables + design variables: G=G(x, d)
– Advantage: improve efficiency of SRS
– Challenge:
• Construct RS in high dimensional space ( > 10 variables)
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Structural & Multidisciplinary Optimization Group
Analysis Response Surfaces (ARS)
Strain
• Fit strains in terms of 12 variables
• Design of experiments:
– Latin Hypercube Sampling (LHS)
R.V.
Strain = g(θ1, θ2, t1, t2, E1, E2, G12, 12, 1, 2, Tzero, Tservice)
D.V.
ARS
• Probabilities calculated by MCS based on fitted polynomials
– Reduce computational cost of MCS
[email protected]
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Structural & Multidisciplinary Optimization Group
Reliability-based Design Optimization
ARS
• Design Response Surface (DRS)
– Fit to Probability in terms of 4 D.V.
– Filter out noise generated by MCS
– Used in RBDO
DOE & MCS
DRS
Probability
Optimization
No
ti
i
POF = p(θ1, θ2, t1, t2)
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Converge?
Yes
Stop
No
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Structural & Multidisciplinary Optimization Group
Approximation
1
2
t1 (inch)
t2 (inch)
20 to 30
20 to 30
0.0125 to 0.03
0.0125 to 0.03
Design variables
Range
ARS Error Statistics
Quadratic ARS based on LHS 182 points
2 in 1
Rsquare Adj.
0.996
Standard error (millistrain)
0.060
Mean of Response (millistrain)
8.322
FCCCD 25 points
LHS 252 points
quadratic
5th order
Rsquare Adj.
0.686
0.998
Standard error (probability)
5.3e-4
0.12e-4
Mean of Response (probability)
3.2e-4
0.44e-4
DRS Error Statistics
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Structural & Multidisciplinary Optimization Group
Optimization
• Deterministic, Reliability-based, and Simplified designs
Ply Angles
Thickness (inch)
Deterministic
[(27.0)2/(27.0)3]S
0.10
Probability of
Failure
5e-4
Reliability
[(24.9)3/(25.2)3]S
0.12
0.55e-4
Simplified
[(25)6]S
0.12
0.57e-4
• The thickness is high for application
[email protected]
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Structural & Multidisciplinary Optimization Group
Improving Reliability-based Design
• Reliability-based design
– Thickness of 0.12 inch
– Probability of failure of 10-4 level
Must reduce uncertainties:
 Quality control (QC)
– Reject small numbers of poor specimen
– Truncate distribution of allowables at lower side (–2 )
 Reduce material scatter
– Reduce Coefficient of Variation (CV)
– Better manufacture process (Better curing process)
 Improve allowables
– Increase Mean Value of allowables
– New materials
[email protected]
15
Structural & Multidisciplinary Optimization Group
Change Distribution of 2 allowable
• Reduce scatter (CV) by 10%
(0.12 inch)
[(25)6]S
Probability of failure
CV = 0.09
CV = 0.081
0.57e-4
0.11e-4
• Increase allowable (Mean value) by 10%
(0.12 inch)
[(25)6]S
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Probability of failure
E(ε2u) = 0.0154
E(ε2u) = 0.01694
0.57e-4
0.03e-4
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Structural & Multidisciplinary Optimization Group
Quality Control (QC) on 2 allowable
• Reduce probability of failure
(0.12 inch)
[(25)6]S
Normal
0.57e-4
Probability of failure
Truncate at -3
(14 out of 10,000)
0.001e-4
Truncate at -2
(23 out of 1,000)
< 1e-7
0.10 inch
Truncate at –2.8
(26 out of 10,000)
0.9e-4
0.08 inch
Truncate at -1.35
(90 out of 1,000)
1e-4
• Reduce thickness
(POF=1e-4)
[(25)6]S
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0.12 inch
Normal
0.57e-4
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Structural & Multidisciplinary Optimization Group
Tradeoff Plot
1.0E+00
Nominal
Quality control to -2 Sigma
10% increase in allowable
10% reduction in variability
All
Series6
Failure Probability
1.0E-01
1.0E-02
1.0E-03
1.0E-04
1.0E-05
1.0E-06
1.0E-07
1.0E-08
0.06
0.08
0.1
0.12
0.14
0.16
Thickness (inch)
• To be chosen by the cost of implementing these methods
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