Transcript Reliability-based optimization

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 may 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?
Top Hat question
• In a design problem g=r-c, and the costs of
changing the means of r or c by one unit are
the same. The standard deviation of r is twice
that of c. Which mean should we change to
reduce the Pf at minimum cost?
– Response
– Capacity
– Both
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 that
  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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Physical challenge in this problem
• In a cylinder under internal pressure the stresses in the
hoop directions are twice those in the axial direction,
and so you could put fibers in both directions, but
twice as many in the hoop direction.
• However, fibers shrink much less than matrix at low
temperatures, so the fibers in hoop direction will not
allow the matrix of the axial fibers to shrink, causing it
to crack.
• We have to compromise by having fibers in
intermediate directions with less than 90-degrees
between fibers in different layers of laminate.
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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Top Hat problem
• If the 27o design was built with 26o because of
manufacturing reliability that would
– Increase the chance of failure due to hoop stress
– Increase the chance of failure due to axial stress
– Increase the chance of failure due to matrix
cracking
– All of the above.
Reliability-based Laminate Design
minimize
h  4t1  t2 
such that
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
[email protected]
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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]
18
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
[email protected]
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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