bharani ravishankar, benjamin smarslok advisors dr. raphael t. haftka, dr. bhavani v. sankar...
TRANSCRIPT
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Bharani Ravishankar, Benjamin SmarslokAdvisors
Dr. Raphael T. Haftka,Dr. Bhavani V. Sankar
SEPARABLE SAMPLING OF THE LIMIT STATE FOR ACCURATE
MONTE CARLO SIMULATION
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Monte Carlo simulation-based techniques can require expensive calculations to obtain random samples
To improve the accuracy of pf estimate for complex limit states without performing additional expensive response computation?
Motivation - Probability of Failure Problems
2
Capacity
R C
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Outline & Objectives
Review Monte Carlo simulation techniques
- Crude Monte Carlo method
- Separable Monte Carlo method
Simple limit state example
- Explain the advantage of regrouping random variables
Complex (non-separable) limit state example - Tsai Wu Criterion
-Demonstrate regrouping & separable sampling of stress and strength
Compare the accuracy of the Monte Carlo methods
Conclusions
3
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Monte Carlo Simulations
Common way to propagate uncertainty from input to output & calculate probability of failure
Limit state function is defined as
Crude Monte Carlo (CMC)
- most commonly used
1 2( ) ( )R CX X
C Capacity (eg.Yield Strength)
R Response (eg. Stress)
, Failure
, Safe
R C
R C
1
1ˆ
N
cmc i ii
p I R CN
4
R C
Potential failure region
Response depends on a set of random variables X1
Capacity depends on a set of random variables X2
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5
Crude Monte Carlo Method
1
ˆ fcmc
f
pCV p
p N
x
y
z
100kPa
• isotropic material• diameter d, thickness t • Pressure P= 100 kPa
Limit state function
max 0Y
axial
hoop
max
2dP
t
Failure max Y
Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y
: 13, 1.5
10
f
C N
N
p
Y
0.062
Example:
I – Indicator function takes value 0 (not failed) or 1( failed)
Assuming Response ( ) involves Expensive computation (FEA)
R C Y
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Separable Monte Carlo Method If response and capacity are independent, we can use all of the possible combinations of random samples
1
1 ˆˆ ( )N
smc C ii
p F RN
Example:
Empirical CDF
10
10
f
N
M
p
0.062CMC
SMC
1 1
1ˆ
N M
smc i ji j
p I R CMN
6
R C Y
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7
Stress is a linear function of load P
u P u – Stress per unit load
P, d, t and Y are independent random variables
Regrouping the random variables
Regrouping the random variables
max
2dP
t
Random variables Response - Stress = f (P, d, t) Capacity - Yield Strength, Y
Regrouped variablesStresses per unit load u
Pressure load PYield Strength Y
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Monte Carlo Simulation Summary
Crude MC traditional method for estimating pf
– Looks at one-to-one evaluations of limit state
– Expensive for small pf
Separable MC uses the same amount of information as CMC, but is inherently more accurate
– Use when limit state components are independent
– Looks at all possible combinations of limit state R.V.s
– Permits different sample sizes for response and capacity
8
For a complex limit state, the accuracy of the pf estimate could be improved by regrouping and separable sampling of the RVs
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Complex limit state problem
Pressure vessel -1m dia. (deterministic) Thickness of each lamina
0.125 mm (deterministic) Lay up- [(+25/-25)]s
Internal Pressure Load, P= 100 kPa
9
x
y
z
xN
yN100kPa
Material Properties E1,E2,v12,G12
Loads PLaminate Stiffness
(FEA)
,x y
,x y
1 2 12, ,
yxy
x
Determination of Stresses
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Limit State - Tsai-Wu Failure Criterion
10
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F
11 1
22 2
11 2266 122
1 1 1
1 1 1
.1
2
L L L L
T T T T
LT
F FS S S S
F FS S S S
F FF F
S
F – Strength Coefficients
S – Strengths in Tension and Compression in the fiber and transverse direction
Limit state G = f (F, ); G < 0 safe G ≥ 0 failed
Non-separable limit state
obtained from Classical Laminate Theory (CLT)
F = f (Strengths S)
1 11 12
2 12 22
12 66
0 / 2
0 / 4
0 0 0ij
a a P
a a P a
a
P
=f (Laminate Stiffness aij, Pressure P)
No distinct response and capacity
Random Variables
( , )G S
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RVs - Uncertainty
11
Parameters Mean CV%
E1 (GPa) 159.1
5E2 (GPa) 8.3
G12 (GPa) 3.3
12 (no unit) 0.253
Pressure P (kPa) 100 15
S1T (MPa) 2312
10
S1C (MPa) 1809
S2T (MPa) 39.2
S2C (MPa) 97.2
S12 (MPa) 33.2
All the properties are assumed to have a normal distribution
CV(Pressure) > CV(Strengths) > CV(Stiffness Prop.)
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F ( , )G S
Separable Monte Carlo
1 1
1ˆ ( , ) 0j
N M
smci
ij
p I GMN
S
Crude Monte Carlo
1
1ˆ [ ( ), 0]i i
N
cmci
p I GN
S
{ } = {1, 2,12}T S = {S1T S1C S2T S2C S12 }
NN N M
Estimation of probability of failure
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12
CMC and SMC Comparison
N=500, repetitions = 10000
Expensive Responselimited to N=500 (CLT)Cheap Capacity varied M= 500, 5000 samples
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F ( , )G S
Actual Pf = 0.012
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Finite Element Analysis
13
Regrouping the expensive and inexpensive variables
( , )G SOriginal limit state
( , )uG P,SRegrouped limit state
Tsai – Wu Limit State Function
Stresses
Expensive
From Statistical distribution
Strengths S
Cheap
Expensive Cheap
Strengths SPressure Load P
Stresses per unit load u
Finite Element Analysis
Stresses per unit load
Load P
Cheap
From Statistical distribution
Expensive
u
u – Material Properties, P – Pressure Loads, S – Strengths
1 1
1ˆ ( , ) 0j
N M
smci
ij
p I GMN
S
1 1
,1
ˆ ( , ) 0.
N Musm j
u
i jjicp I G
M NP
S
N M
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14
Regrouping the random variables
Stresses
Material Properties
Load P
Strengths S
Cost Expensive Cheap Cheap
Uncertainty ~ 5% 15% 10%
( , )G S ( , )uG P,S
( , )u uG P,S
( , )u uG P,S
( , )G S
( , )G S
, ,u P,S -Mean values
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15
Comparison of the Methods
2 2 211 1 22 2 66 12 1 1 2 2 12 1 22 1F F F F F F
Expensive RVslimited to N=500 (CLT)Cheap RVs varied M= 500-50000 samples
N=500 repetitions = 10000
( , )G S
M
Crude Monte Carlo
Separable Monte Carlo
Separable Monte Carlo
regrouped RVs
500 40.0% 20.6% 36.3%1000 18.4% 26.0%
5000 16.2% 11.7%
10000 16.0% 8.2%
50000 15.6% 4.0%
ˆCV cmcp ˆCV smcp ˆCV usmcp
Actual Pf = 0.012
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ˆ ˆ&stdevboot bootmean p p
ˆbootp
Accuracy of probability of failure
For SMC, Bootstrapping – resampling with replacement
= error in pf estimate
Initial Sample size N
Re-sampling with replacement, N
Re-sampling with replacement, N
bootstrapped standard deviation/ CV
….…... ‘b’ bootstrap samples………..
pf estimate from bootstrap sample, pf estimate from bootstrap sample,
‘b’ estimates of ̂ bootp
k=1k=2
k= b
ˆbootp ˆ
bootp
ˆ ˆ/stdev boot bootp CV p
CMC
SMC
1ˆ f
cmcf
pCV p
p N
For CMC, accuracy of pf
ˆ ˆ/stdev boot bootp CV p
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Summary & Conclusions
17
Separable Monte Carlo was extended to non-separable limit state - Tsai-Wu failure criterion.
In Tsai-Wu Limit State, uncertainty in load affects the expensive stresses. By calculating response to unit loads, we can sample the effect of random loads more cheaply.
Statistical independence of the random variables enables appropriate sampling, thereby improving the accuracy of the estimate.
Shift uncertainty away from the expensive component furthers helps in accuracy gains.
Accuracy of the methods - for the same computational cost,
CMC SMC -original limit state SMC- Regrouped limit state
CV% 40% 16% 4%