Bharani Ravishankar, Benjamin SmarslokAdvisors

Dr. Raphael T. Haftka,Dr. Bhavani V. Sankar

SEPARABLE SAMPLING OF THE LIMIT STATE FOR ACCURATE

MONTE CARLO SIMULATION

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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%