METHODS FOR PROGNOSTICS AND RELIABILITY ...

Introduction

Fundamentals

Contribution 1

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Contribution 3

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METHODS FOR PROGNOSTICS AND RELIABILITY IN COMPOSITE MATERIALS USING STRUCTURAL HEALTH MONITORING PhD dissertation by:

Manuel Chiachío Ruano

University of Granada (Spain) 10/2014

Introduction

Fundamentals

Contribution 1

Contribution 2

Index Introduction Context Fundamentals Prognostics Bayesian filtering Particle-filters PF-based prognostics Contribution 1 Novel prognostics framework in composites Results for multi-step ahead prediction Results for EOL and RUL Contribution 2 Predicting RUL from time-dependent reliability Results for predicted long-term reliability Contribution 3 Novel algorithm for prognostics of rare-events (PFP-SUBSIM) Results for prognostics in composites using PFP-SUBSIM Conclusions Back slides

Contribution 3

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Context

◦ Composites are a class of engineering materials primarily used for light-weight structures ◦ High strength-to-weight ratio, corrosionless.

* Expected growth for the global carbon fiber marker of about 83% for the next five years (Lucintel (2014)).

Images for an offshore wind farm, composite bridge deck and the Solar Impulse ®, taken from Wikipedia.

Introduction

Fundamentals

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Conclusions

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Context ◦ Light-weight structures =⇒ fatigue degradation.

◦ In composites, fatigue represents one of the most important sources of uncertainty =⇒ safety and cost implications. 1500 Failed specimen Non-failed specimen

Maximum stress [M P a]

1400 1300 1200 1100 1000 900 800 700

1

10

102

103 104 Cycles

105

106

UGR-TUHH experimental program about fatigue in CFRP, 2012. Left: set of identical CRFP coupons. Right: deterministic SN-curve

107

Introduction

Fundamentals

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Conclusions

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Context ◦ Light-weight structures =⇒ fatigue degradation

◦ In composites, fatigue represents one of the most important sources of uncertainty =⇒ safety and cost implications

Coupon Coupon Coupon Coupon

3

#

cracks ] mm

4 3.5

1 2 3 4

Micro-cracks density

2.5 2 1.5 1 0.5 0 102

103

104 Cycles

105

UGR-TUHH experimental program about fatigue in CFRP, 2012. Left: set of identical CRFP coupons. Right: plot of microcracking vs. cycles.

106

Introduction

Fundamentals

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Conclusions

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Research objectives

* Conservative design approaches are adopted → Drawback. "The light-weight feature of CFRP cannot be utilized effectively if some damage features are not tolerated in the basic design concept" → propose design criteria to consider long-term durability given damage (Hosoi et al. (2010)). Central question Can we anticipate (1) the evolution of damage in composites and (2) the time to reach a certain damage threshold? Main contributions A novel prognostics framework to predict the damage progression of composite materials under fatigue conditions. 3 A method for reliability assessment along life-span. 3 An efficient algorithm for prognostics involving rare-events. 3

Introduction

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Contribution 2


Contribution 3

Conclusions

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Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

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Prognostics: the concept Prognostics is a core element of the Prognostics and Health Management (PHM) sciences (Farrar et al. 2007, Zio et al. 2013).

DETECT Sensing method (SHM for structural applications) capable to extract data of one or several fault modes of a component or system. DIAGNOSE Isolating any fault mode and assessment of the fault severity. PREDICT Remaining useful life (RUL), end of life (EOL).

• Timely decisions can be adopted to avoid unnecessary maintenance actions. • Improving life-cycle cost and increasing of availability. • Reducing risks by optimizing maintenance.

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Prognostics: the concept

n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

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Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Prognostics: The concept

n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


n

n+1

...

n+p−1 n+p

... ...

0

n n+1

n+p

n+2

n+p−1

...

n+3 ... n + 4

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Bayesian filtering Filtering algorithm Inputs/ Models

State transition equation p(zn |zn−1 )

Prediction p(zn |y0:n−1 )

SHM data (on-line) y0:n = [y0 , y1 , · · · , yn ]

Measurement equation p(yn |zn )

Updated damage state p(zn |y0:n )

p(zn+` |y0:n )

Total Probability Theorem Data Mean predicted states

Z Z

p(zn |zn−1 ) p(zn−1 |y0:n−1 ) dzn−1 {z } | Last update

Z-space

p(zn |y0:n−1 ) =

Bayes’ Theorem

n−1

p (zn−1 |y0:n−1 ) n

...

p(zn |y0:n ) = R

Z

n+`

p(yn |zn )p(zn |y0:n−1 )

p(yn |zn )p(zn |y0:n−1 )dzn

∝p(yn |zn )p(zn |y0:n−1 )

Introduction

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Conclusions

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p(zn+` |y0:n )


Z Z


Z-space

p(zn |y0:n−1 ) =

Bayes’ Theorem

n−1

n

p(zn |y0:n ) = R

p (zn |y0:n−1 ) ...

Z

n+`




Introduction

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Contribution 3

Conclusions

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p(zn+` |y0:n )


Z Z


Z-space

p(zn |y0:n−1 ) =

Bayes’ Theorem

n−1

n

p(zn |y0:n ) = R

p (zn |y0:n−1 ) ...

Z

n+`




Introduction

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Contribution 1

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Contribution 3

Conclusions

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p(zn+` |y0:n )


Z Z

p(zn |zn−1 ) p(zn−1 |y0:n−1 ) dzn−1 | {z } Last update

Z-space

p(zn |y0:n−1 ) =

p (yn |zn )

n−1

n

Bayes’ Theorem p(zn |y0:n ) = R

p (zn |y0:n−1 ) ...

Z

n+`




Introduction

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Contribution 3

Conclusions

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p(zn+` |y0:n )


Z Z


Z-space

p(zn |y0:n−1 ) =

Bayes’ Theorem p(zn |y0:n ) = R

p (zn |y0:n ) n−1

n

...

Z

n+`




Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

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Bayesian Data filtering Mean predicted states

Filtering algorithm Inputs/ Models






Z-space

p(zn+` |y0:n )


Z Z


Z-space

p(zn |y0:n−1 ) =

Bayes’ Theorem p(yn |zn )p(zn |y0:n−1 )

p (zn |y0:n ) n−1

n

n−1

...

n

n+`

p(zn |y0:n ) = R p (zn |y0:n ) Z p(yn |zn )p(zn |y0:n−1 )dzn

...


n+`

Introduction

Fundamentals

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Contribution 3

Conclusions

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Future state prediction Future state prediction







p(zn+` |y0:n )

Data Mean predicted states

Future state prediction

Z-space

p(zn+` |y0:n ) = Z p(zn+` |zn:n+`−1 , y0:n )p(zn:n+`−1 |y0:n )dzn:n+`−1 Z



Z = Z

p (zn |y0:n ) n−1

n

...

p (zn+` |y0:n ) n+`

n+` Y

 t=n+1

 p(zt |zt−1 ) p(zn |y0:n )dzn:n+`−1

Introduction

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Conclusions

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Particle Filtering Data Mean predicted states

Z-space

Z-space

Particles in Z-space (Particle Filters) Data mean predicted states

p (zn |y0:n ) n−1

n

p (zn |y0:n )

...

n

n−1

n+`

PF-based approximation (Gordon et al. 1993) n o n o (i) N (i) N N samples or particles zn with associated weights ωn i=1

i=1

p(z0:n |y0:n ) ≈

N X i=1

(i)

(i)

ωn δ(z0:n − z0:n )

(i)

(i)

ω ˆn =

p(z0:n |y0:n ) (i)

q(z0:n |y0:n )

,

(i)

ωn = P N

1

i=1

(i)

ω ˆn

...

n+`

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

mean predicted Conclusions Back slides

Z-space

Particle filtering Particles in Z-space (Particle Filters) Data mean predicted states

Z-space

Z-space

Data Mean predicted states

p (zn |y0:n ) n−1

n

...

n−1

PF-based approximation (Gordon et al. 1993) n o n o (i) N (i) N N samples or particles zn with associated weights ωn i=1

i=1

p(z0:n |y0:n ) ≈

N X i=1

(i)

(i)

ωn δ(z0:n − z0:n )

(i)

(i)

ω ˆn =

p(z0:n |y0:n ) (i)

n

n−1

n+`

q(z0:n |y0:n )

,

(i)

ωn = P N

1

i=1

(i)

ω ˆn

...

n+`

n

...

Introduction

Fundamentals

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Conclusions

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Particle filtering Data Mean predicted states

Z-space

Z-space

Particles in Z-space (Particle Filters) Data mean predicted states

p (zn+` |y0:n )

p (zn |y0:n ) n−1

n

...

p (zn |y0:n ) n−1

n+`

n

...

PF-based approximation (Doucet, et al. 2001)

p zn+` |y0:n ≈ =

N X

Z

Z i=1 N X i=1

(i)

ω ˆn

(i)

(i)

(i)

p(zn+1 |zn )ω ˆ n δ(zn − zn ) Z

(i)

Z

p(zn+1 |zn )

n+` Y t=n+2

n+` Y t=n+2

p(zt |zt−1 )dzn+1:n+`−1

p(zt |zt−1 )dzn+1:n+`−1

n+`

Introduction

Fundamentals

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Contribution 2

Contribution 3

Conclusions

PF-based prognostics Filtering algorithm Inputs/ Models

State transition eq. p(zn |zn−1 )


Data (on-line) y0:n = [y0 , · · · , yn ]


Updated state p(zn |y0:n )

Z-space

U¯

zn+` ∈ U¯

`=`+1

EOLn = n + ` RU Ln = `

Prognostics p(EOLn |y0:n ) p(RU Ln |y0:n )

Key definitions

threshold

Performance function g : Z → R, g (z) > b

U

Failure region ¯ , {z ∈ Z : g (z) > b} U n−1

n

...

...

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Introduction

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Conclusions

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`=`+1

Future state prediction (Orchard et al. 2008)

U¯ Z-space

zn+` ∈ U¯

Extending the ith trajectory

threshold

(i)

(i)

z0:n → zn:n+` , i = 1, . . . , N

U

By simulation using the state transition eq.: (i) (i) (i) (i) (i) zt ∼ p(zt |zt−1 ) → z0:t = z0:t−1 , zt ; t = n + 1, . . . , n + ` ∈ N

n−1

n

...

...

Introduction

Fundamentals

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Conclusions

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Z-space

U¯


zn+` ∈ U¯


`=`+1

EOL and RUL calculation (ith particle)

threshold

U

n o (i) (i) EOLn = inf t ∈ N : t > n ∧ I(U¯ ) (zt ) = 1 (i)

n−1

z

n

RU Ln }|

...

(i)

{

(i)

RULn = EOLn − n (i)

EOLn

Introduction

Fundamentals

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Conclusions

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Z-space

U¯


zn+` ∈ U¯


`=`+1


threshold

U


(i)

RULn = EOLn − n n−1

n

...

...

Introduction

Fundamentals

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Conclusions

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Z-space

U¯


zn+` ∈ U¯


`=`+1


threshold

U


(i)


n

...

...

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

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Z-space

U¯


zn+` ∈ U¯


`=`+1


threshold

U


(i)


n

...

...

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

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zn+` ∈ U¯


`=`+1

PDF of end of life

Z-space

U¯

PDF of EOL and RUL

threshold

p(EOLn |y0:n ) ≈

U

p(RULn |y0:n ) ≈ p (zn |y0:n ) n−1

n

...

...

N X i=1 N X i=1

(i)

(i)

(i)

(i)

ωn δ(EOLn − EOLn )

ωn δ(RULn − RULn )

Introduction

Fundamentals

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Conclusions

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zn+` ∈ U¯


`=`+1

PDF of end of life

Z-space

U¯

PDF of EOL and RUL

threshold

p(EOLn |y0:n ) ≈

U

p(RULn |y0:n ) ≈

n−1

...

...

...

N X i=1 N X i=1

(i)

(i)

(i)

(i)



Introduction

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Contribution 1

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Contribution 3

Conclusions

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zn+` ∈ U¯


`=`+1

PDF of end of life

Z-space

U¯

PDF of EOL and RUL

threshold

p(EOLn |y0:n ) ≈

U

p(RULn |y0:n ) ≈

n−1

...

...

...

N X i=1 N X i=1

(i)

(i)

(i)

(i)



Introduction

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Contribution 3

Conclusions

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Introduction

Fundamentals

Contribution 1

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Conclusions

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Novel prognostics framework for composites Inputs: un = {σmax , h, . . .} Damage models Stochastic embedding

Modified Paris’ law ρn = f1 (ρn−1 , un , θ)

p(ρn |ρn−1 , θ) ∼ N f1 (ρn−1 , un , θ), σv1,n p(Dn |ρn , θ) ∼ N f2 (ρn , un , θ), σv2,n

Norm. effective stiffness Dn = f2 (ρn , un , θ)

Artificial dynamics p(θn |θn−1 ) Likelihood (Eq. 5.15a & 5.15b) ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n ˆ n |Dn ) ∼ N D ˆ n , σw2,n p(D

Measurement equation (Eq. 5.12) yn = xn + w n

ρb0 ρb1 ··· ρbn

b0 D b 1 ··· D bn D

Challenges

Transition equation p(zn |zn−1 ) = p(xn |xn−1 , θn ) p(θn |θn−1 ) = | {z } (Eq. 5.10)

xn = [ρn , Dn ] zn = (xn , θ)

Damage datah y0:n = [y0 , y1 , · · · , yn ] =


p(Dn |ρn , θn )p(ρn |ρn−1 , θn )p(θn |θn−1 ) Prognostics p(RU Ln |y0:n ) Updating p(zn |y0:n )

i

• • •

Pool of models and model parameterization (Chiachío et al. in IWSHM 2013, Chiachío et al., IJF 2014) Transition equation and PF-based formulation (Chiachío et al. in PHM 2013, Chiachío et al. in EPHM 2014, Chiachío et al. in IGI Global 2014, to appear ) Definition of damage thresholds for composites (Chiachío et al. in PHM 2013)

Introduction

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Conclusions

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Micro-to-macro scale modeling of fatigue damage Inputs: un = {σmax , h, . . .} Damage models Stochastic embedding




un

Transition equation p(zn |zn−1 ) = p(x |xn−1 , θn ) p(θn |θn−1 ) = | n {z } (Eq. 5.10)

Artificial dynamics p(θn |θn−1 )

xn = [ρn , Dn ] zn = (xn , θ)

Inputs: = {σmax , h, . . .}


p(Dn |ρn , θn )p(ρn |ρn−1 , θn )p(θn |θn−1 ) Prognostics p(RU Ln |y0:n )

Likelihood (Eq. 5.15a & 5.15b) ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n ˆ n |Dn ) ∼ N D ˆ n , σw2,n p(D




b1 D b 2 ··· D bn D

Updating p(zn |y0:n )

i

Damage models Stochastic embedding




1


p(Dn |ρn , θn )p(ρn |ρn−1 , θn )p(θn |θn−1 )


xn = [ρn , Dn ] zn = (xn , θ)

Likelihood (Eq. 5.15a & 5.15b)

Pool of models for damage evolution (Chiachío et al. IJF 2014) Measurement equation (Eq. 5.12) yn = xn + w n

f1 : ρn = ρn−1 + A ∆G (ρn−1 ) Damage datah 1 f2 : yD(ρ) =1 , y2 , · · · , yn ] = 0:n = [y

α


b 2 ··· D 1 + kρR(ρ, a,Db 1θ)

bn D

ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n 2 ˆ ˆ p(Dn |Dn ) ∼ N Dn , σσ w2,n xh

where,

i

G =

2ρt90 E0

1 D(ρ)

−

!

1 D(ρ/2)

Model parameters θ={ |{z} α , E1 , E2 , |{z} t , σv 1,n , σv 2,n } |{z} |{z} | {z } | {z } θ1

θ2

θ3

θ4

θ5

θ6

Introduction

Fundamentals

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Conclusions

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Inputs: un = {σmax , h, . . .}

Stochastic embedding2







xn = [ρn , Dn ] zn = (xn , θ)





un


Inputs: = {σmax , h, . . .}


b1 D b 2 ··· D bn D


i





1




xn = [ρn , Dn ] zn = (xn , θ)

Likelihood (Eq. 5.15a & 5.15b) ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ )θ)∼ N+ρˆn , σw1,n n xn = f(xn−1 ,ρnu|ρ , vn n |{z} {z | ˆ n |Dn ) }∼ N D ˆ n , σ|{z} p( D w 2,n system output modeling error model output


1 Damage data Principle of Maximum Information h ρb1 ρb2 ··· ρbn iEntropy → vn ∼ N (0, Σvn ) =⇒ y0:n = [y1 , y2 , · · · , yn ] = b b bn D1 D2 ··· D Σvn = diag σv2 , σv2 1,n

1 2

Jaynes (1957) Beck (2010)

2,n

xn ∼ N f(xn−1 , un , θ), Σvn ,

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Transition equation

Conclusions

Back slides

Inputs: un = {σmax , h, . . .} Damage models Stochastic embedding






xn = [ρn , Dn ] zn = (xn , θ)







b1 D b 2 ··· D bn D


i

Inputs: un = {σmax , h, . . .} 1








xn = [ρn , Dn ] zn = (xn , θ)

Likelihood (Eq. 5.15a & 5.15b) ˆ n |Dn3): p(ˆ ρn |ρn )p(D n |xn , θ) = transition Joint state-modelp(y parameter PDF Measurement equation (Eq. 5.12) yn = xn + w n p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n ˆnn), σwp(θ p(zn |zn−1 ) = p(Dn |ρn , θp( )ˆp(ρ |θ n−1 ) ∼n−1 N;θ D nD n |Dnn)|ρ 2,n n

|

Damage datah

i

{z

stiffness loss

}|

{z

} |

{z

}

micro-cracks evol. model adaptation

3y = [y , y , · · · , y ] = ρb1 ρb2 ··· ρbn 0:n 1 et. 2 n Proceedings Chiachio al, In b1 D b 2 ··· D b n of the Structural Health Monitoring, 2013. D

Introduction

Fundamentals

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Conclusions

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Sequential damage estimation Inputs: un = {σmax , h, . . .} Damage models Stochastic embedding






xn = [ρn , Dn ] zn = (xn , θ)






Dn−1


b1 D b 2 ··· D bn D

i

Dn+`

Dn

1

ρ0

ρn−1

ρn

θ0

θ n−1

θn

···

···

ρn+`

θ n+`

p(zn |zn−1 ) = p(Dn |ρn , θ n ) p(ρn |ρn−1 ; θ n ) p(θ n |θ n−1 ) | {z }| {z } | {z } stiffness loss



Introduction

Fundamentals

Contribution 1

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Contribution 3

Conclusions

Back slides

Measurement equation Inputs: un = {σmax , h, . . .}

Inputs: un = {σmax , h, . . .}






Stochastic embedding




ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n ˆ n |Dn ) ∼ N D ˆ n , σw2,n p(D i

Transition equation p(zn |zn−1 ) = p(xn |xn−1 , θn ) p(θn |θn−1 ) = | {z } 1

Artificial dynamics p(θn |θn−1 ) Likelihood (Eq. 5.15a & 5.15b) ˆ n |Dn ) p(yn |xn , θ) = p(ˆ ρn |ρn )p(D p(ˆ ρn |ρn ) ∼ N ρˆn , σw1,n ˆ n |Dn ) ∼ N D ˆ n , σw2,n p(D




b1 D b 2 ··· D bn D

(Eq. 5.10)

xn = [ρn , Dn ] zn = (xn , θ)

b0 D b 1 ··· D bn D

Transition equation p(zn |zn−1 ) = p(x |xn−1 , θn ) p(θn |θn−1 ) = | n {z }



Likelihood (Eq. 5.15a & 5.15b)

Damage models


(Eq. 5.10)


xn = [ρn , Dn ] zn = (xn , θ)


i



Introduction

Fundamentals

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Conclusions

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Measurement equation

Inputs: un = {σmax , h, . . .} Damage models





Prognostics p(RU Ln |y0:n )







b0 D b 1 ··· D bn D


xn = [ρn , Dn ] zn = (xn , θ)



i

RECALL: Total Probability Theorem

p(zn |y0:n−1 ) =

Z

Bayes’ Theorem

p(zn |zn−1 ) p(zn−1 |y0:n−1 ) dzn−1 | {z } Z Last update

p(zn |y0:n ) = R

Z




Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Experimental set-up: on-line measurements4

• • • •

Torayca T700G 15.24 [cm] x 25.4 [cm] dogbone [02 /904 ]s coupons Frequency f = 5 Hz; σmax = 0.8σu ; R = 0.14 Sensing: PZT sensors → micro-cracks; Rosette strain gauges → stiffness decrease. U = (ρ, D) ∈ [0, 0.418] × [1, 0.875] ⊂ R2

4 A. Saxena, K. Goebel, C.C. Larrosa, and F.K. Chang, "CFRP Composites dataset", NASA Ames Prognostics Data Repository, NASA Ames, Moffett Field, CA.

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Multi-step ahead prediction of damage states Z N X (i) p zn+` |y0:n ≈ ω ˆn i=1

(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)

ˆ n |D ) ωn ∝ ωn−1 p(ρˆn |ρn )p(D n

0.6

1 collected data upcoming data updated state filtered prediction (median) 50% probability bands 95% probability bands

0.5 0.95 D

ρ

0.4 0.3 collected data upcoming data updated state filtered prediction (median) 50% probability bands 95% probability bands

0.2 0.1 0

0

2

4

6

8

0.9

10

n

4

0.85

0

2

4

6

8

Figure :

10

n

x 10

Sequential prediction for microcrack density (left), and normalized effective stiffness (right) at ˆ n = 0.920 n = 1 · 104 . Measured damage: ρˆn = 208, D

4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


(i)

Z

p(zn+1 |zn )

(i)

(i)

n+` Y t=n+2

(i)

p(zt |zt−1 )dzn+1:n+`−1 ; (i)


0.6

1

0.5 0.95 D

ρ

0.4 0.3 0.2

0.9

0.1 0

0.85 0

2

4

6

8

10

n

4

0

2

4

6

8

Figure :

10

n

x 10


4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Model adaptation 11

9

x 10 2.5

x 10

1.6

11

1.5 10

α 1.5

E2

1.4 E1

2

1.3

9

1.2 8 1.1

1

1 0

2

4

6

8

10

n

7 0

2

4

6

8

10

n

4

0

2

4

6

8

10

n

4

x 10

4

x 10

x 10

−4

−3

x 10

x 10

1.8

1.5

3

1.7 σ v 1, n

t

σ v 2, n

2.5

1.6

1

2

1.5 1.5

1.4 1.3

0.5 0

2

4

6

8

10

n x 10

From left to right, top to bottom:

1 0

2

4

6

8

10

n

4

4

x 10

θ1 , θ2 , θ3 , θ4 , θ5 , θ6

=

0

2

4

6

8

10

n

α, E1 , E2 , t, σv1,n , σv2,n . The dashed lines

represent the 25% − 50% probability bands.

4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Results for EOL and RUL prediction 4

x 10

x 10

−5

n =2 0 0 0 0

1.5 [(1−0.2)RUL*,(1+0.2)RUL*] [(1−0.1)RUL*,(1+0.1)RUL*] RUL* Estimated RUL (median) Estimated RUL (mean)

9 8

1

7 0.5

RU L n

6

Figure :

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10

α-λ performance metric a results for RUL for predicting matrix-micro-crack density

10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Results for EOL and RUL prediction −5

4

2 [(1−0.2)RUL*,(1+0.2)RUL*] [(1−0.1)RUL*,(1+0.1)RUL*] RUL* Estimated RUL (median) Estimated RUL (mean)

9 8

1.5

1

RU L n

7

Figure :

n =5 0 0 0 0

x 10

x 10

6

0.5

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2


Contribution 3

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Reliability in composites

Composites: Part B 43 (2012) 902–913

Contents lists available at SciVerse ScienceDirect

Composites: Part B journal homepage: www.elsevier.com/locate/compositesb

Reliability in composites – A selective review and survey of current development Manuel Chiachio ⇑, Juan Chiachio, Guillermo Rus Dept. Structural Mechanics and Hydraulic Engineering, University of Granada, ETS de Ingenieros de Caminos, Canales y Puertos, Campus de Fuentenueva s/n, 18071 Granada, Spain

a r t i c l e

i n f o

Article history:

Excerpts from the25 reliability review Received July 2011 Received in revised form 26 August 2011 Accepted 18 October 2011 Available online 29 October 2011

a b s t r a c t As a response to the rampant increase in research activity within reliability in the past few decades, and to the lack of a conclusive framework for composite applications, this article attempts to identify the most relevant reliability topics to composite materials and provide a selective review. Available reliability assessment methods are briefly explained, referenced and compared within an unified formulation. Recent developments to confer efficiency in computing reliability in large composite structures are also highlighted. Finally, some general conclusions are derived along with an overview of future directions of research within reliability of composite materials and their influence on design and optimization. ! 2011 Elsevier Ltd. All rights reserved.

1. Numerous studies have been published. Keywords:

2. LackA. Lamina/ply of a consensual framework → failure criteria. A. Laminates

C. Statistical properties/methods 3. Need to consider progressive damage in the reliability formulation of composites. Reliability

1. Introduction The need to incorporate uncertainties in engineering design has long been recognized. In contrast to the traditional approach of using safety coefficients, the probabilistic design allows the esti-

adopted. The reasons for that are multiple: (1) there are a wide range of possibles failure functions to adopt, (2) numerous influencing random variables need being incorporated, (3) several reliability methods arise and (4) there are different ways to consider reliability for a laminate, as shown in Fig. 1.

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Particles in Long-term Z-space (Particle Filters) reliability

Conclusions

5

Particles in Z-space (Particle Filters)

Z-space

U¯

U¯ threshold

Z-space

threshold U

U n−1

p (zn |y0:n ) ...

n

n+`

Mathematical definition: Rn+`|n ≡ P(zn+` ∈ U |y0:n ) =

Z U

p(zn+` |y0:n )dzn+`

Based on particle-filter approximation of p(zn+` |y0:n ), an estimation of Rn+`|n is readily obtained as:

p (zn |y0:n N ) X n−1 5

n

Rn+`|n ≈

...

(i)

i=1

(i)

ωn IU (zn+` )

Chiachío et al. 2014. Best Paper Award EPHM, Nantes, France

n+`

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Reliability-based prognostics6 Predicted particle trajectories Particles in Z-space measured data

U¯ threshold

Z-space

Rational connection between long-term reliability and RUL provided that: → Key concept:

U p (zn+` |y0:n )

(i) ¯ and RUL(i) Events zn+` ∈ U n 6 ` occur with the same probability. p (zn |y0:n )

z }| { ¯ |y0:n ) =⇒ P(RULn 6 `|y0:n ) = P(zn+` ∈ U = 1 − P(zn+` ∈ U |y0:n ) | {z }

1.0

0 P (RU Ln 6 `|y0:n )

0.9 Reliability (Rn )

Rn+`|n

FRULn = 1 − Rn+`|n

n

n−1

FRUL (`−n) n

0.8 0.7 0.6 0.5 0.4 0.3

n+` z

` }|

{

-

Long-term reliability prediction

0.5

0.2 0.1 1

0 n−1

6

Rn+`|n

? n

n+` Cycles

Chiachío et al. 2014. Best Paper Award EPHM, Nantes, France. Also submitted to Rel. Eng. & Syst. Saf.

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Reliability-based prognostics6 Predicted particle trajectories Particles in Z-space measured data

U¯ threshold

Z-space

Rational connection between long-term reliability and RUL provided that: → Key concept:

U p (zn+` |y0:n )

(i) ¯ and RUL(i) Events zn+` ∈ U n 6 ` occur with the same probability. p (zn |y0:n )

z }| { ¯ |y0:n ) =⇒ P(RULn 6 `|y0:n ) = P(zn+` ∈ U = 1 − P(zn+` ∈ U |y0:n ) | {z }

1.0

0 P (RU Ln 6 `|y0:n )

0.9 Reliability (Rn )

Rn+`|n

FRULn = 1 − Rn+`|n

n

n−1

FRUL (`−n) n

0.8 0.7 0.6 0.5 0.4 0.3

n+` z

` }|

{

-

Long-term reliability prediction

0.5

0.2 0.1 1

0 n−1

6

Rn+`|n

? n

n+` Cycles

Chiachío et al. 2014. Best Paper Award EPHM, Nantes, France. Also submitted to Rel. Eng. & Syst. Saf.

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Reliability-based prognostics Rn+`|n ≡ P(zn+` ∈ U |y0:n ) =

Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)

n z }| { }| { z P(RULn 6 `|y0:n ) = 1 − P(zn+` ∈ U |y0:n ) 4

x 10

[(1−0.2)RUL*,(1+0.2)RUL*] [(1−0.1)RUL*,(1+0.1)RUL*] RUL* Estimated RUL (median)

9

1 R n | n +ℓ

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10

Long-term reliability (left) and RUL (right) predicted at cycle (n = 1 · 103 )

Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)

n z }| { }| { z P(RULn 6 `|y0:n ) = 1 − P(zn+` ∈ U |y0:n )

4

x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions


Z U

p(zn+` |y0:n )dzn+` ≈

N X i=1

(i)

(i)

ωn 1U (zn+` )

Rn+`|n

FRUL (`−n)


x 10


9

1

8 0.8

7 6

RU L n

0.6

R

Introduction

5 4

0.4 3 2

0.2

1 0 0

2

4

6

8

cycles

Figure :

10

12

0

14 4

x 10

1

2

3

4

5

6

7

8

9

n

4

x 10


Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2


Contribution 3

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Prognostics of an asymptotic process Predicted particle trajectories Particles in Z-space Data

U¯

U¯

Z-space

Z-space

Predicted particle trajectories Particles in Z-space measured data

threshold

U

threshold

U

p (zn+` |y0:n )

p (zn+` |y0:n )

p (zn |y0:n ) n−1

n

···

p (zn |y0:n ) n+`−1

n+`

n−1

n

...

The vast majority of samples are employed to simulate the useful domain High threshold (unprovable damage event)

n+`

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


U¯

U¯

Z-space

Z-space

Predicted particle trajectories Particles in Z-space Data

threshold

U

threshold

U

p (zn+` |y0:n )

n

···

}|

{

p (zn+` |y0:n )

p (zn |y0:n ) n−1

z

p (zn |y0:n ) n+`−1

n+`

n−1

n

...


n+`

Introduction

trajectories Z-space

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


U¯

U¯

Z-space

Z-space

Predicted particle trajectories Particles in Z-space Data

threshold

U

p (zn+` |y0:np (z) |y n

n−1

n+`−1

n

threshold

U

p (zn+` |y0:n )

}|

{

p (zn+` |y0:n )

0:n )

···

z

p (zn |y0:n ) n+`−1

n+`

n−1

n

...

n+`


n+`

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

PF-based prognostics using Subset Simulation

Z-space

REGIONS

U¯1 . . . ⊃ U¯j . . . ⊃ U¯m U¯m ≡ U¯

final threshold

¯m ≡ U ¯ U

¯ U j

n−1

7

P(U¯) = P

n

S.K. Au and J.L. Beck,2001

···

7

NESTED SEQUENCE OF INTERMEDIATE FAILURE


¯ U j+1

Back slides

m \

j=1

U¯j ;

P(U¯j ) P(U¯) n+`−1

n+`

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

PFP-SubSim algorithm


Z-space

U¯

final threshold

U

M particles

n−1

8

n

···

n+`−1

n+`

Nomination to Best Paper Awards, EPHM 2014, Nantes, France

8

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3



Z-space

U¯

n−1

9

n

···

n+`−1

n+`


9

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


Conclusions

Back slides

10


Z-space

U¯

U¯j

INTERMEDIATE FAILURE REGION

U¯j , {z ∈ Z : g (z) > bj },

MP0 Particles

bj < bj+1 ∈ R, P0 ∈ (0, 1)

Uj

n−1

10

n

···

n+`−1

n+`


Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


Conclusions

Back slides

11


By MCMC: Z-space

U¯

p(zt |U¯j ) ∝ p(zt )IU¯j (zt ), j = 1, . . . , m

U¯j

6 seeds

Uj

n−1

11

n

···

I(U¯ ) (z) = n+`−1

n+`


(

1, 0,

if z ∈ U¯ if z ∈ U

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3



Z-space

U¯

M particles

n−1

12

final threshold

n

···

n+`−1

n+`


12

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3


Conclusions

Back slides

13 −5

x 10 4.5 4 3.5 3 2.5


2 1.5 1 0.5

Z-space

U¯

0

final threshold

0

M particles

x 10

2

4

6 8 ( cycles)

10

12 4

x 10

−5

7 6 5 4 3

n−1

n

···

n+`−1

n+`

2 1 0 0

13


2

4

6 8 ( cy cl es )

10

12 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Multi-step forward modeling using FPF-SubSim

1

14

800

0.95

600

0.9

ρn

Dn

b = 418 0. 88

0.85

400

200

0.8 0.75

0 0

2

4

6

8

n ( cy cl es )

10

0

2

4

6

n ( cy cl es )

4

x 10

4

Prediction cycle n = 1 · 10 . Observe the simulation using three subsets

14

Back slides

Chiachío et al. 2014 in Proc. ASC Conf., San Diego, CA, (USA)

8

10 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Efficiency gained for rare-events

Unit covariance: (4) Any stochastic algorithm for estimating PU¯ has a c.o.v. of the form c.o.v. = √ 4

NTotal

Algorithm 6 PFP-SUBSIM

350 300 Unit cov (4)

0.012

250 200

0.022 0.047

150

0.098 0.16 0.24

100

0.33

0.4

50

600

575

550

525

500

475

Threshold value ρ [ #

• •

450

425

cracks ] m

Different values for micro-crack density are employed as prognostic threshold for each calculation. The corresponding failure probability is labelled by the text box

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2


Contribution 3

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Conclusions and future work Conclusions

• A novel model-based prognostics framework which combines • • •

Scientific work

• • • • • •

micro and macro scale information from physics-based models. Predictions of damage events along with long-term reliability are possible in composites. PFP-SubSim algorithm can be considered a general purpose prognostics algorithm It has the added feature of being efficient for the prognostics of processes involving rare-event simulations → asymptotic processes 4 2 9 2 8 2

journal papers (+2 submitted for publication) book chapters (+1 submitted for publication) conference papers (full) patents talks at international conferences technical seminars at international institutions

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Conclusions and future work Conclusions

• A novel model-based prognostics framework which combines • • •

Scientific work

• • • • • •

micro and macro scale information from physics-based models. Predictions of damage events along with long-term reliability are possible in composites. PFP-SubSim algorithm can be considered a general purpose prognostics algorithm It has the added feature of being efficient for the prognostics of processes involving rare-event simulations → asymptotic processes 4 2 9 2 8 2

journal papers (+2 submitted for publication) book chapters (+1 submitted for publication) conference papers (full) patents talks at international conferences technical seminars at international institutions

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Conclusions and future work

Future work

• Consideration of other damage modes (delamination) and different lay-ups for validation

• Optimal scaling strategies for PFP-SubSim (proposal PDF for model adaptation and prediction)

• Extend the prognostics methodology to full-scale structures → integration SHM and PHM

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Conclusions and future work

Future work

• Consideration of other damage modes (delamination) and different lay-ups for validation

• Optimal scaling strategies for PFP-SubSim (proposal PDF for model adaptation and prediction)

• Extend the prognostics methodology to full-scale structures → integration SHM and PHM

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

THANK YOU

Conclusions

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Fatigue damage in composites

• • •

Several fracture modes (matrix-cracks, local delamination, global delamination, fiber breakage, etc) Multi-stage process Noticeable from beginning of lifespan

Figure :

Example of damage pattern

Cross-ply laminates

Quasi-isotropic laminates

[0n /90m ]S

[0n /45r / − 45t /90m ]S

STAGE I 1% to 10% of stiffness decrease

Transverse cracks saturation (90o plies)

Transverse cracks growth (90o ,+45o ,−45o plies)

STAGE I 1% to 5% of stiffness decrease

STAGE II 0.5% to 1% of stiffness decrease

Longitudinal splits in 0o plies and delamination onset in 0/90 int. (small influence in stiffness)

Edge delamination (all possible interfaces) and cracks saturation

STAGE II Linear-smooth stiffness lost by delam. 5% to 15%

STAGE III sudden lost of stiffness.

Splits saturation and nucleation of delamination areas (random fibre-breakage)

Rapid growth of delaminations, family of micro-cracks (random fibre-breakage)

STAGE III Increase in stiffness lost.

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Transition equation Inputs: un = {σmax , h, . . .} Damage models Stochastic embedding






xn = [ρn , Dn ] zn = (xn , θ)





y0:n

Damage datah = [y1 , y2 , · · · , yn ] =


b1 D b 2 ··· D bn D

i

State and model parameter transition PDF15 :

p(zn |zn−1 ) = p(Dn |ρn , θ n ) p(ρn |ρn−1 ; θ n ) p(θ n |θ n−1 ) | {z }| {z } | {z } stiffness loss


 ρn − f1 (ρn−1 , θ) 2  2πσv1,n 2σv2 1,n   1 (Dn − f2 (ρn , θ))2  p(Dn |ρn , θ) = √ exp − 2πσv2,n 2σv2

p(ρn |ρn−1 , θ) = √

1



exp −

2,n

p(θ|θ n−1 ) = N (θ n−1 , Σξn ) Parameter regeneration by “artificial dynamics” 16 : θ n ∼ p(θ|θ n−1 ) = N (θ n−1 , Σξn ), Σξn ∈ Rnθ ×nθ 15 16

Stochastic independence inherited from independence of model errors v1n , v2n Shrinkage over Σξn following Daigle and Goebel (2011), Daigle and Goebel (2013)

1


Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

EXPERIMENTAL SET-UP: ON-LINE MEASUREMENTS17

OFF-LINE PZT data X-Ray data Feature extraction (Signal processing)

Damage mapping

Model inputs FEA Surrogate model Models Stiffness

Lamb waves

US model

Microcracks → ρˆ Delaminations → a ˆ

Micro-scale damage models ρn = f1 (ρn−1 ) an = f2 (ρn , an−1 ) Macro-scale damage model En = f2 (ρn , an )

ˆ Stiffness → E

model Measurement ˆ = h(ρn , an , En ) ρˆ, a ˆ, E

Probabilistic model prediction p(xn |xn−1 ) x = {E, a, ρ}

Strain gauges data

• •

15.24 [cm] x 25.4 [cm] dogbone [02 /904 ]s coupons frequency f = 5 Hz; σmax = 0.8σu ; R = 0.14

17 A. Saxena, K. Goebel, C.C. Larrosa, and F.K. Chang, "CFRP Composites dataset", NASA Ames Prognostics Data Repository, NASA Ames, Moffett Field, CA.

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

First approach

Damage models


Conclusions

18

Probabilistic damage models

Matrix-crack evolution model ρn = g1 (ρn−1 ; θn )

ρn = g1 (ρn−1 , θn ) + e1

Matrix-crack evolution model p(ρn |ρn−1 , θn )

Stiffness reduction model Dn = g2 (ρn , θn )

Dn = g2 (ρn , θn ) + e2

Stiffness reduction model p(Dn |ρn , θn )

Parameter regeneration θn = θn−1 + ξn


State transition equation

State transition equation p(zn |zn−1 ) zn = {ρn , Dn , θn }







p(zn+` |y0:n )

18

Back slides

In proceedings of the Conference of the Prognostics and Health Management Society, New Orleans, 2013

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Defining prognostics thresholds in composites19 [ 0 2/ 9 0 4] s TC LD GD

120

J m

2

¤

100

E RR, G

£

80 60 40 20 0 0

0.5 1 £ ¤ Cr ac k d e n s i t y m m − 1

1.5

Model-based process reconstruction

• • •

∆GTC > {∆GLD , ∆GGD } =⇒ Transverse cracks

∆GLD > {∆GTC , ∆GGD } =⇒ Local delamination

∆GGD > {∆GLD , ∆GTC } =⇒ Global delamination

19

(a) At first ramp → Initial damage as matrix cracks

(b) Critical crack density → Microcracking stops, local delamination starts (c) critical phase → Local delaminations stops, global (or edge) delamination starts

Chiachío et al., in PHM 2013, Chiachío et al., in ASME Conf. San Diego (USA), 2013

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Results for EOL and RUL prediction 4

x 10

x 10

−5

n =2 0 0 0 0

1.5 [(1−0.2)RUL*,(1+0.2)RUL*] [(1−0.1)RUL*,(1+0.1)RUL*] RUL* Estimated RUL (median) Estimated RUL (mean)

9 8

1

7 0.5

RU L n

6

Figure :

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

1.5

1

RU L n

7

Figure :

n =3 0 0 0 0

x 10

x 10

6

0.5

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

1.5

1

RU L n

7

Figure :

n =4 0 0 0 0

x 10

x 10

6

0.5

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

1.5

1

RU L n

7

Figure :

n =5 0 0 0 0

x 10

x 10

6

0.5

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

1.5

1

RU L n

7

Figure :

n =6 0 0 0 0

x 10

x 10

6

0.5

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

4 3

RU L n

7

Figure :

n =7 0 0 0 0

x 10

x 10

2

6

1

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

6

4

RU L n

7

Figure :

n =8 0 0 0 0

x 10

x 10

6

2

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides


4


9 8

0.8 0.6

RU L n

7

Figure :

n =9 0 0 0 0

x 10

x 10

0.4

6

0.2

5

0 0

4

2

4

6

8

E OL n

3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

4

x 10


10

12

14

16 4

x 10

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Prognostics of an asymptotic process

Predicted particle trajectories Particles in Z-space measured data U¯

Z-space

Introduction

U¯ ≡ {z ∈ Z : g (z) > b}, b ∈ R P(U¯) → 0

final threshold

1. Increase samples.

U

2. Choosing a conservative threshold.

M particles

n−1

n

···

n+`−1

3. New efficient simulation methods (few works).

n+`

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

Prognostics of an asymptotic process

Predicted particle trajectories Particles in Z-space measured data U¯

Z-space

Introduction

U¯ ≡ {z ∈ Z : g (z) > b}, b ∈ R P(U¯) → 0

final threshold

1. Increase samples.

U

2. Choosing a conservative threshold.

M particles

n−1

n

···

n+`−1

3. New efficient simulation methods (few works).

n+`

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

PF-based prognostics using Subset Simulation

Back slides

20

NESTED SEQUENCE OF INTERMEDIATE FAILURE REGIONS

Z-space


U¯1 . . . ⊃ U¯j . . . ⊃ U¯m U¯m ≡ U¯

final threshold

¯m ≡ U ¯ U ¯ U j+1

P(U¯) = P

¯ U j

P(U¯1 ) | {z }

m Y

M.C. j=2

n−1

n

···

n+`−1

n+`

m \

j=1

U¯j

S.K. Au and J.L. Beck,2001

=

P(U¯j |U¯j−1 ) | {z } MCMC

P(U¯j |U¯j−1 ) = P0 ; P0 ∈ (0, 1)

20

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

PREDICTED RUL USING PFP-SUBSIM 4

x 10 9

[(1−0.2)RUL*,(1+0.2)RUL*] [(1−0.1)RUL*,(1+0.1)RUL*] RUL* estimated RUL

8 7

RU L n

6

Recall: U = (ρ, D) ∈ [0, 0.418] × [1, 0.875] ⊂ R2

5 4 3 2 1 0 0

1

2

3

4

5

6

7

8

9

n

RUL∗ means true RUL from data

4

x 10

Back slides

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

PREDICTED RUL COMPARISON ∗ 2

Entropy = 1/2 ln |(2πe) [var (RULn )] |

Accuracy : |RULn − RULn k2

5

x 10

8 − R U L ∗k k 22

PFP−SUBSIM Algorithm 6

2.5 1.75

k

6

kR U L

Entropy

7

5

1 0.25

4

0

2

4

6

cycles

8

10

0

2

4

x 10

b = 418 [#cracks · m−1 ]; D = 0.875

4

6

cycles

8

10 4

x 10

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

PREDICTED RUL COMPARISON ∗ 2

Entropy = 1/2 ln |(2πe) [var (RULn )] |

Accuracy : |RULn − RULn k2

7

0.05 0.04 ¯ P ( U)

Entropy

6

5 4

3

0.03 0.02 0.01

0

2

4

6

cycles

8

10

0

0

2

4

x 10

b = 600 [#cracks · m−1 ]; D = 0.86

4

6

cycles

8

10 x 10

4

Introduction

Fundamentals

Contribution 1

Contribution 2

Contribution 3

Conclusions

Back slides

PREDICTED SYSTEM FAILURE PROBABILITY ¯) = P(U

Z ¯ U

h i n+` p(zn+1:n+` )dzn+1:n+` = Efn I(U¯ ) {zt }t=n+1

7

10

1 Algorithm 6 PFP−SUBSIM

6

10

0.6

NT

¯ P ( U)

0.8

5

10

0.4

4

10 0.2

3

0

2

4

6

cycles

Figure :

8

10 4

x 10

10

0

2

4

6

cycles

8

10 4

x 10

¯ ) estimation using PFP-SubSim algorithm in comparison to the standard Upper panel: Plot for P(U PF-based prognostics algorithm used as benchmark. Bottom panel: Total number of simulations required by each algorithm.

METHODS FOR PROGNOSTICS AND RELIABILITY ...

METHODS FOR PROGNOSTICS AND RELIABILITY ...

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