Digital Twin Digital Twin WorkshopWorkshop

Harry MillwaterHarry Millwater

University of Texas at San AntonioUniversity of Texas at San Antonio

Sept. 10-11, 2012Sept. 10-11, 2012

1

TopicsTopics

FAA Probabilistic Damage FAA Probabilistic Damage Tolerance AnalysisTolerance Analysis

System Reliability AnalysisSystem Reliability Analysis Sensitivity MethodsSensitivity Methods

2

Damage Tolerance Damage Tolerance MethodologyMethodology

Geometry

…54

321

n

Sensitivities

Insp 1

Insp 2

Time

RS

EIFS

Insp 1

Insp 2

Time

Crack Size

Loading EVD POD

SFPOF & CTPOF

Material Prop.

POF CalculationsPOF Calculations

4

€

SFPOF(T) = FEVD

KC

β(ao, t) πa(ao, t)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

t =1

T −1

∏ ⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥1 − FEVD

KC

β(ao,T) πa(ao,T)

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

⎡

⎣ ⎢ ⎢

⎤

⎦ ⎥ ⎥fa0

(a0) fKc(Kc )da0dKc

0

∞

∫−∞

∞

∫

Survival t-1 POF Flight t

€

CTPOF(t) =1

N1 − (1 − SFPOF(t))

t =1

T

∏ ⎡

⎣ ⎢

⎤

⎦ ⎥

i=1

N

∑

R(t) =1 − CTPOF(t)

hz(t) =1

R(t)SFPOF(t)

EVD ResultsEVD Results

5

Flight 1 Flight 2

...Flight n

FrechetGumbelWeibull

EVD ResultsEVD Results

6

Usage Location Scale Shape DistributionSingle Engine Unpress. Basic

Inst.14.49 1.75 0.01 Frechet

Single Engine Unpress. Personal

12.03 1.15 0.00 Gumbel

Single Engine Unpress. Executive

11.68 0.90 -0.04 Weibull

Twin Engine Unpress. Basic Inst.

13.93 1.42 0.00 Gumbel

Twin Engine Unpress. General

12.05 1.16 0.02 Frechet

Pressurized 13.41 1.26 0.02 FrechetAgric/Special 12.07 0.94 -0.12 Weibull

Special Usage Survey 14.56 1.32 -0.05 Weibull

7

POF contours

Probabilistic Probabilistic Modeling of SystemsModeling of Systems

L. Domyancic, D. Sparkman, and H.R. Millwater, L. Smith and D. Wieland, “A Fast First-Order Method for Filtering Limit States,” 50 th AIAA AIAA Nondeterministic Approaches Conference, May 4-7, 2009, Palm Springs, CA, AIAA 2009-2260

Pair-wise Filtering Pair-wise Filtering CriteriaCriteria

Pair-wise filtering – neglect limit states Pair-wise filtering – neglect limit states that don’t contribute significantly to the that don’t contribute significantly to the system POF system POF relative to relative to the dominant limit the dominant limit statesstates

8

€

γi =P g1 giU[ ] − P g1[ ]

P g1[ ]i = 2,n

g1 – dominant limit state, serves as a benchmark

Filtering ProcedureFiltering Procedure

Pair-wise comparison of each limit state with dominantPair-wise comparison of each limit state with dominant Each limit state is assigned a filtering error, i.e. the Each limit state is assigned a filtering error, i.e. the

relative error incurred if the limit state were filteredrelative error incurred if the limit state were filtered

γi P[E1 E i ] P[E1]

P[E1] i 2,n

9

is calculated using the bivariate normalis calculated using the bivariate normal

P[E1 E i ]1 2(;)

P[E1 E i ]

User decides how much error to allow and set tolerance

(typical values: 1E-4 – 1E-6)

€

γi < γ TOL filter limit state

γ i ≥ γ TOL keep limit state

Second Order BoundsSecond Order Bounds

Examine the cumulative effect of the filtered Examine the cumulative effect of the filtered limit states using 2limit states using 2ndnd order bounds order bounds

Compare bounds from a) all limit states vs. b) only Compare bounds from a) all limit states vs. b) only unfiltered limit statesunfiltered limit states

10

P1 max Pi Pij ;0j1

i 1

i2

m

Pfs Pi

i1

m

maxji

Piji2

m

Aircraft ModelAircraft Model

Lower wing skin analyzed, ~900 limit statesLower wing skin analyzed, ~900 limit states

Root radius Element 292 and 277 left unfiltered (Root radius Element 292 and 277 left unfiltered (γγTOLTOL =1E- =1E-6)6)

Bounds on entire system vs. only critical locations widen Bounds on entire system vs. only critical locations widen by 1E-5by 1E-5

11

g1

LS # Beta Corr. Filt. Error292 0.297 1 0277 0.65 0.99 2.50E-04256 0.949 0.98 2.20E-09

Sensitivity MethodsSensitivity Methods

Numerical Differentiation (*)Numerical Differentiation (*) VisualVisual Linear RegressionLinear Regression Score Function partial derivativesScore Function partial derivatives Local probabilistic partial derivatives Local probabilistic partial derivatives

(*)(*) Global Sensitivity AnalysisGlobal Sensitivity Analysis

Local Prob. Sensitivity

Original CDFModified CDF

ZFEM Complex ZFEM Complex Variable FE MethodVariable FE Method

€

P

0

⎧ ⎨ ⎩

⎫ ⎬ ⎭=

EI

L3

12 6L

6L 4L2

⎡

⎣ ⎢

⎤

⎦ ⎥δ

φ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

P

0

⎧ ⎨ ⎩

⎫ ⎬ ⎭=

EI

(L + ih)3

12 6(L + ih)

6(L + ih) 4(L + ih)2

⎡

⎣ ⎢

⎤

⎦ ⎥δ

φ

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

δ =PL3

3EI−

PLh2

EI

⎧ ⎨ ⎩

⎫ ⎬ ⎭+ i

PL2h

EI−

Ph3

3EI

⎧ ⎨ ⎩

⎫ ⎬ ⎭

€

∂δ∂L

= Im[δ ]/h

Abaqus 2D Verification

Analytical Abaqus

€

∂σθ

∂ri

Strain Energy Release Rate

Double Cantilever Beam (DBC)

Exact = 0.800

Exact = 0.8

€

G = −∂U

∂a

Higher Order Sensitivities

€

∂2σ θ

∂ro2

€

∂2σ r

∂p∂ro

Thick walled cylinder