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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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 2 / 19
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 3 / 19
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 3 / 19
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 3 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 4 / 19
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 4 / 19
Monte Carlo Simulation
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 5 / 19
Monte Carlo Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 6 / 19
Monte Carlo Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 6 / 19
Monte Carlo Simulation
1 Monte Carlo Simulation with samplesize 5000
Nt = 5000 PSSf = 1.16% Bootstrap CoV κBS = 13.03%
Bayesian CoV κBA = 12.94%
2 Monte Carlo Simulation with samplesize 1500
Nt = 1500 PSSf = 0.93% Bootstrap CoV κBS = 26.7%
Bayesian CoV κBA = 25.7%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 7 / 19
Monte Carlo Simulation
1 Monte Carlo Simulation with samplesize 5000
Nt = 5000 PSSf = 1.16% Bootstrap CoV κBS = 13.03%
Bayesian CoV κBA = 12.94%
2 Monte Carlo Simulation with samplesize 1500
Nt = 1500 PSSf = 0.93% Bootstrap CoV κBS = 26.7%
Bayesian CoV κBA = 25.7%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 7 / 19
Monte Carlo Simulation
1 Monte Carlo Simulation with samplesize 5000
Nt = 5000 PSSf = 1.16% Bootstrap CoV κBS = 13.03%
Bayesian CoV κBA = 12.94%
2 Monte Carlo Simulation with samplesize 1500
Nt = 1500 PSSf = 0.93% Bootstrap CoV κBS = 26.7%
Bayesian CoV κBA = 25.7%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 7 / 19
Monte Carlo Simulation
1 Monte Carlo Simulation with samplesize 5000
Nt = 5000 PSSf = 1.16% Bootstrap CoV κBS = 13.03%
Bayesian CoV κBA = 12.94%
2 Monte Carlo Simulation with samplesize 1500
Nt = 1500 PSSf = 0.93% Bootstrap CoV κBS = 26.7%
Bayesian CoV κBA = 25.7%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 7 / 19
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 .
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 8 / 19
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 .
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 8 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 9 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 9 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 9 / 19
Importance Sampling
h(x , θ)
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 10 / 19
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%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 11 / 19
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%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 11 / 19
Importance Sampling
Second approach: Estimate correlation with 99 realizationsand do Importance Sampling with these coefficients.
N = 780 P ISf = 1.24% Bootstrap CoV κ = 25%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 12 / 19
Importance Sampling
Second approach: Estimate correlation with 99 realizationsand do Importance Sampling with these coefficients.
N = 780 P ISf = 1.24% Bootstrap CoV κ = 25%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 12 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
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.
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 13 / 19
Subset Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 14 / 19
Subset Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 14 / 19
Subset Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 14 / 19
Subset Simulation
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 14 / 19
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 ).
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 15 / 19
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 ).
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 15 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
Subset Simulation
1 Subset Simulation with 900 realisations per level and 35parameters
Nt = 2340 PSSf = 1.30% Bootstrap CoV κBS = 13.1%
Bayesian CoV κBA = 10.6%
2 Subset Simulation with 300 realisations per level and 35parameters
Nt = 780 PSSf = 1.55% Bootstrap CoV κBS = 25.0%
Bayesian CoV κBA = 17.7%
3 Subset Simulation with 300 realisations per level and 10 mostsensitive parameters
Nt = 780 PSSf = 1.20% Bootstrap CoV κBS = 21.3%
Bayesian CoV κBA = 18.4%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 16 / 19
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%
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 17 / 19
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
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 18 / 19
Thank you for your attention
Martin Schwarz Monte Carlo Based Reliability Analysis 15 May 2014 19 / 19
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