SEPARABLE SAMPLING OF THE LIMIT STATE
FOR ACCURATE
MONTE CARLO SIMULATION
Bharani Ravishankar, Benjamin Smarslok
Advisors
Dr. Raphael T. Haftka,
Dr. Bhavani V. Sankar
Motivation - Probability of Failure Problems
Monte Carlo simulation-based techniques can require expensive calculations
to obtain random samples
Capacity
R
C
To improve the accuracy of pf estimate for complex limit states without performing
additional expensive response computation?
2
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
Monte Carlo Simulations
Common way to propagate uncertainty from input to output & calculate
probability of failure
 Limit state function is defined as
R  C, Failure
R( X1 )  C ( X2 )
R  C, Safe
Response depends on a set of random
variables X1
Capacity depends on a set of random
variables X2
R  Response (eg. Stress)
C  Capacity (eg.Yield Strength)
R
 Crude Monte Carlo (CMC)
- most commonly used
pˆ cmc
1

N
N
 I R  C 
i
C
Potential failure
region
i
i 1
4
Crude Monte Carlo Method
Assuming Response ( ) involves Expensive computation (FEA)
 Limit state function
• isotropic material
• diameter d, thickness t
• Pressure P= 100 kPa
 max  Y  0
 max 
2d
P
t
Failure   max  Y
z
y
Random variables
Response - Stress  = f (P, d, t)
Capacity - Yield Strength, Y
x
 hoop
100 kPa
R 
 axial
C Y
CV  pˆ cmc  
0.21
Example:
Pf estimate
0.19
Y  C : N 13, 1.5 
N  10
p f  0.062
1 pf
pf N

0.17
0.15
0.13
0.11
0.09
0.07
0
I – Indicator function
takes value 0 (not failed)
or 1( failed)
2000
4000
6000
Number of samples
5
8000
Separable Monte Carlo Method
If response and capacity are independent, we can use all of the possible combinations of
random samples
Empirical CDF
pˆ smc
1

MN
N
M
 I  R  C 
i
pˆ smc
j
i 1 j 1
1

N
N
 Fˆ (R )
C
i 1
Example:
C Y
N  10
M  10
p f  0.062
0.21
CMC
0.19
Pf estimate
R 
0.17
SMC
0.15
0.13
0.11
0.09
0.07
0
2000
4000
6000
Number of samples
6
8000
i
Regrouping the random variables
Random variables
Response - Stress  = f (P, d, t)
Capacity - Yield Strength, Y
 max 
2d
P
t
Regrouping the random variables
Stress  is a linear function of load P
P, d, t and Y are independent random variables
   uP
 u – Stress per unit load
Regrouped variables
Stresses per unit load  u
Pressure load P
Yield Strength Y
7
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
For a complex limit state, the accuracy of the pf estimate could be improved by
regrouping and separable sampling of the RVs
8
Complex limit state problem
Determination of Stresses
z
y
Material Properties
E1,E2,v12,G12
x
Ny
100 kPa
Laminate Stiffness
(FEA)
Nx
 Pressure vessel -1m dia. (deterministic)
 Thickness of each lamina
0.125 mm (deterministic)
 Lay up- [(+25/-25)]s
 Internal Pressure Load,
P= 100 kPa
Strains
 x , y
 xy
y
x
Stress
 x , y
Stress in each ply
1 , 2 ,12
9
Loads P
Limit State - Tsai-Wu Failure Criterion
Non-separable limit state
G( , S)  F1112  F22 22  F66122  F11  F2 2  2F121  2 1
No distinct response and capacity
Random Variables
F = f (Strengths S)
 =f (Laminate Stiffness aij, Pressure P)
F11 
1
1 1
F


1
S L S L
S L S L
F22 
1
1 1
F

 
2
 

ST ST
ST ST
F66 
F11.F22
1
F

12
2
S LT
2
obtained from Classical Laminate Theory (CLT)
 1   a11
  
 2    a12
   0
 12  
a12
a22
0
0   P / 2


0   P / 4   aij P
a66   0 
F – Strength Coefficients
Limit state G = f (F, ); G < 0 safe
G ≥ 0 failed
S – Strengths in Tension and
Compression in the fiber and
transverse direction
10
Estimation of probability of failure
G( , S)
 F1112  F22 22  F66122  F11  F2 2  2F121 2 1
RVs - Uncertainty
{ } = {1, 2,12}T
Parameters
Mean
E1 (GPa)
159.1
E2 (GPa)
8.3
G12 (GPa)
3.3
12 (no unit)
CV%
Crude Monte Carlo
5
1 N
pˆ cmc   I [G( i , Si )  0]
N i 1
0.253
Pressure P (kPa)
100
S1T (MPa)
2312
S1C (MPa)
1809
S2T (MPa)
39.2
S2C (MPa)
97.2
S12 (MPa)
33.2
S = {S1T S1C S2T S2C S12 }
N
15
10
CV(Pressure) > CV(Strengths) > CV(Stiffness Prop.)
All the properties are assumed to have a normal distribution
11
N
Separable Monte Carlo
pˆ smc
1

MN
N
M
 I G( , S )  0
i
i 1 j 1
N
M
j
CMC and SMC Comparison
G( , S)  F1112  F22 22  F66122  F11  F2 2  2F121 2 1
Coefficient of variation
45
40
SMC
35
CMC
N=500,
repetitions = 10000
30
25
20
Actual Pf = 0.012
15
10
5
0
500
5000
50000
M Samples
Expensive Response
limited to N=500 (CLT)
Cheap Capacity
varied M= 500, 5000 samples
12
Tsai – Wu Limit State Function
Original limit state
G ( , S)
pˆ smc
1

MN
N
M
 I G( , S )  0
Stresses 
i
i 1 j 1
Strengths S
Stresses per
Load P
u
Element Analysis
unit load Finite

From Statistical
Finite Element AnalysisExpensive
Expensive
N
j
From Statistical
distribution
distribution
Cheap
Cheap
Regrouping the expensive and inexpensive variables
Regrouped limit state
G( u , P,S)
u
pˆ sm
c
1

M .N
Expensive
Stresses per unit load
N
M
 I G(
u
i , Pj , S j )
i 1 j 1
Cheap
u
Pressure
Load P
 u – Material Properties, P – Pressure Loads, S – Strengths
13
Strengths S
 0

M
Regrouping the random variables
Stresses 
Cost
Uncertainty
Material
Properties
Load
P
Strengths
S
Expensive
Cheap
Cheap
~ 5%
15%
10%
G ( , S)
G( u , P,S)
G ( , S)
Gu ( u , P,S)
G( , S)
Gu ( u , P,S)
 , u , P,S -Mean values
14
Comparison of the Methods
2
G( , S)  F1112  F22 22  F6612
 F11  F2 2  2F121  2 1
Expensive RVs
limited to N=500 (CLT)
Cheap RVs
varied M= 500-50000 samples
SMC
M
SMC-unit load
CMC
45
Coefficient of Variation
40
500
1000
5000
10000
50000
Crude Monte
Carlo
Separable
Monte Carlo
CV  pˆ cmc 
CV  pˆ smc 
40.0%
20.6%
18.4%
16.2%
16.0%
15.6%
Separable
Monte Carlo
regrouped RVs
35
30
25
N=500
repetitions = 10000
20
15
10
Actual Pf = 0.012
5
0
500
5000
50000
M Samples
15
u
CV  pˆ smc

36.3%
26.0%
11.7%
8.2%
4.0%
Accuracy of probability of failure
For CMC, accuracy of pf
0.21
CMC
0.19
pf N
For SMC, Bootstrapping – resampling with replacement
Pf estimate
CV  pˆ cmc  
1 pf
0.17
SMC
0.15
0.13
0.11
0.09
Initial Sample size N
0.07
0
k=1
Re-sampling with
replacement, N
….…... ‘b’ bootstrap samples………..
ˆ boot
pf estimate from bootstrap sample, p
4000
6000
8000
Number of samples
k= b
k=2
2000
Re-sampling with
replacement, N
ˆ boot
pf estimate from bootstrap sample, p
‘b’ estimates of pˆ boot
pˆ boot
bootstrapped standard
deviation/ CV
stdev  pˆ boot  / CV  pˆ boot 
stdev  pˆ boot  / CV  pˆ boot  = error in pf estimate
mean  pˆ boot  &stdev  pˆ boot 
Summary & Conclusions
 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
CV%
40%
SMC -original limit state
SMC- Regrouped limit state
16%
4%
17