Monte Carlo Based Reliability Analysis - intales

Monte Carlo Based Reliability Analysis

Martin Schwarz

15 May 2014

Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 1 / 19

Plan of Presentation

Description of the problem

Monte Carlo Simulation

Sensitivity based Importance Sampling

Subset Simulation

Comparison

Prospects


Reliability

Probability of failure: Use a random Rn valued random variableX for describing the parameters of an input-output model of anengineering structure.

Pf := P(X ∈ F ),

F := {x ∈ Rn : x is a parameter combination leading to failure}

We define F by a function Φ(x):

x ∈ F ⇐⇒ Φ(x) > 1


Reliability


Pf := P(X ∈ F ),



x ∈ F ⇐⇒ Φ(x) > 1


Reliability


Pf := P(X ∈ F ),



x ∈ F ⇐⇒ Φ(x) > 1


The Small Launcher Model

FE-model of the Ariane 5 frontskirt.35 input parameters: Loads, E-moduli, yieldstresses etc.Considered as uniformly distributed with spread±15% around nominal value.


The Small Launcher Model

Φ(x) := max{

PEEQ(x)0.07 , SP(x)

180 , 0.001|EV (x)|

}PEEQ(x) > 0.07 plastification of metallic partSP(x) > 180 MPa rupture of composite part|EV (x)| < 0.001 buckling



Theorem

The probability of failure can be estimated by

Pf = P(X ∈ F ) ≈ PMCf :=

1

N

N∑i=1

1F (X ).

PMCf is an unbiased estimator and V(PMC

f ) = Pf (1−Pf )N







1 Monte Carlo Simulation with samplesize 5000

Nt = 5000 PSSf = 1.16% Bootstrap CoV κBS = 13.03%

Bayesian CoV κBA = 12.94%





























Importance Sampling

Use the following idea:∫1F · f dx =

∫1F

f

g· g dx

Theorem

Pf can be estimated by g -iid random variables (Y1, . . . ,YN).

Pf ≈ P ISf :=

1

N

N∑i=1

1F (Yi )f (Yi )

g(Yi ),

where P ISf is unbiased and V(P IS

f ) =

∫F

f 2

gdx−P2

f

N .


Importance Sampling

Use the following idea:∫1F · f dx =

∫1F

f

g· g dx

Theorem

Pf can be estimated by g -iid random variables (Y1, . . . ,YN).

Pf ≈ P ISf :=

1

N

N∑i=1

1F (Yi )f (Yi )

g(Yi ),

where P ISf is unbiased and V(P IS

f ) =

∫F

f 2

gdx−P2

f

N .


Importance Sampling

How to find a good g?

Idea: use correlation coefficient between parameters and output,use a function that pushes the realizations towards the critical area.


Importance Sampling




Importance Sampling




Importance Sampling

h(x , θ)


Importance Sampling

First approach: Use (rank) correlation coefficients from referencesolution (Monte Carlo with N = 5000).

Promising results:

N = 780 P ISf = 0.84% Bootstrap CoV κ = 19%


Importance Sampling

First approach: Use (rank) correlation coefficients from referencesolution (Monte Carlo with N = 5000).

Promising results:



Importance Sampling

Second approach: Estimate correlation with 99 realizationsand do Importance Sampling with these coefficients.



Importance Sampling

Second approach: Estimate correlation with 99 realizationsand do Importance Sampling with these coefficients.



Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1.

Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk

Estimate P0 by Monte Carlo Simulation.

Estimate Pk by Markov Chain Monte Carlo.

We choose αk such that

Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) =

P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)

︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)

︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation

Let α0 ≤ α1 ≤ . . . ≤ αm = 1. Then

Pf = P(X ∈ F ) = P(Φ(X ) > α0)︸︷︷︸:=P0

m∏k=1

P(Φ(X ) > αk |Φ(X ) > αk−1)︸︷︷︸:=Pk




Pk ≈ 0.2.


Subset Simulation


Subset Simulation


Subset Simulation


Subset Simulation


Subset Simulation

Theorem

The estimator P̃k is unbiased and the CoV κk is of order O(N−12 ).

Theorem

The estimator PSSf is consistent and the CoV κ is of order

O(N−12 ).


Subset Simulation

Theorem

The estimator P̃k is unbiased and the CoV κk is of order O(N−12 ).

Theorem

The estimator PSSf is consistent and the CoV κ is of order

O(N−12 ).


Subset Simulation

1 Subset Simulation with 900 realisations per level and 35parameters






3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters




Subset Simulation











Subset Simulation











Subset Simulation











Subset Simulation











Subset Simulation











Comparison

MC IS SS AS

Sample Size 1500 780 780 800

Estimated Pf 0.93% 1.24% 1.55% 0.83%

Bootstrap symmetric95%-confidence interval

[0.47%, 1.47%] [0.70%, 1.91%] [1.04%, 2.17%] [0.25%, 2.04%]

Bootstrap CoV 26.7% 25.0% 25.0% 53.19%

Bayesian symmetric95%-credibility interval

[0.56%, 1.56%] [1.12%, 2.25%] [0.35%, 2.77%]

Bayesian CoV 25.7% 17.7% 50.8%


Prospects

Winglet:

4.7 million DOFs

Composite failure by Yamada-Sun criterion (Main Joint)

Metallic failure by yielding and rupture criterion (Main Joint)

Currently running on HPC-system ”MACH”

3 parallel evaluations, 17 min

Pf ≈ 10−6


Thank you for your attention


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