identifiability analysis of finite element models for ... · 2015 informs philadelphia 4 dynamic...

Identifiability Analysis of Finite Element Models for Vibration-Response Based

Structural Damage Detection

Yuhang Liu1

Professor Shiyu Zhou1

Professor Jiong Tang2

University of Wisconsin1

University of Connecticut2

2015 INFORMS Philadelphia

*The financial support of this work is provided by Air Force Office of Scientific Research

under the program Dynamic Data Driven Application Systems (DDDAS)

Agenda

• Research Motivation Structural Health Monitoring (SHM)

Finite Element Model (FEM) in SHM

Identifiability Issue in FEM

• Identifiability in FEM-based structural damage detection Identifiability of the damage severity at a given location

Identifiability of damage location

• Numerical Study

• Conclusion & Future Work


Structural Health Monitoring

3

SHM

Reduced Maintena

nce

Improved Safety

Increased Longevity

Level 1: Detection

Level 2: Localization & Evaluation

Level 3: Prognosis of Remaining Lifetime

Level 2: Localization & Evaluation


Dynamic Data Driven Application Systems (DDDAS)

2015 INFORMS Philadelphia 4

Dynamic Feedback & Control Loop

Applications

Pie

zo

ele

ctr

ic t

ran

sd

ucer

Vi

i

Applications measurements systems and methods

Tunable piezoelectric impedance sensor systems

Applications modeling

Data driven sensor tuning and structural weakness estimation

Data collection

Data-driven Sensor tuning

Mathematical and statistical algorithms

Structural weakness estimation

Aircraft structure

Failure threshold

Stru

ctu

ral w

eak

ne

sse

s

Failure time prognosis

Time

Structural weakness growth modeling and prognosis

Mathematical and Statistical Algorithms

FEM-Based Damage Detection


𝑴 𝒙 𝒕 + 𝑲(𝜽)𝒙(𝒕) = 𝑭(𝒕)Finite Element Model with unknown parameters

Sensor based Measurements

min𝜃

yobs − ymod(𝜃)2

Minimize differences between measurements and FEM predictions

max𝜃

𝑝(𝜃|𝑦𝑜𝑏𝑠)

Maximize the posterior distribution

𝜃1𝜃2𝜃3

Identifiability of the damage severity

𝛾2

𝛾1

Frequency

Log

Am

plit

ud

e

Frequency Response


?System Response

Identifiability of the damage location

𝑝2, 𝛾2

𝑝1, 𝛾1

Frequency

Log

Am

plit

ud

e

Frequency Response


?System Response

Parameter Identifiability in Linear Time Invariant System

8

𝑼(𝑠) 𝒀(𝑠)𝑯(𝑠)

𝑯 𝑠 = 𝑪(𝜽) 𝑠𝑰 − 𝑨(𝜽) −1𝑩(𝜽) + 𝑫(𝜽)

𝒀 𝑠 = 𝑯 𝑠 𝑼 𝑠 ,

Definition

Let 𝐴, 𝐵, 𝐶, 𝐷 𝜃 be a parametrization of the system matrices 𝐴, 𝐵, 𝐶, 𝐷 . This

parametrization is said to be parameter-identifiable if:

𝑪 𝜽𝟏 𝑠𝑰 − 𝑨 𝜽𝟏−𝟏

𝑩 𝜽𝟏 + 𝑫 𝜽𝟏

= 𝑪 𝜽𝟐 𝑠𝑰 − 𝑨 𝜽𝟐−𝟏

𝑩 𝜽𝟐 + 𝑫 𝜽𝟐

for all 𝑠 ∈ ℂ, implying 𝜽𝟏 = 𝜽𝟐.


Formulate FEM of a beam structure into LTI System

9

𝑯 𝑠 = 𝑬𝒄 −𝑠𝟐𝑴+𝑲(𝜽)−𝟏

𝑬𝑩,

𝑲𝑝 = 𝛾 × 𝐸𝐼

12

𝑙3

6

𝑙2−

12

𝑙3

6

𝑙2

4

𝑙−

6

𝑙2

2

𝑙12

𝑙2−

6

𝑙2

𝑠𝑦𝑚4

𝑙

Element

Node 1 Node 2

𝑑2𝑣1

𝑴𝒆 = 𝜌

13𝑙

35

11𝑙2

210

9𝑙

70−

13𝑙2

420𝑙3

105

13𝑙2

420−

𝑙2

14013𝑙

35−

11𝑙2

210

𝑠𝑦𝑚𝑙3

105

The transfer function of the LTI system formulated from FEM of a beam structure is:

𝜽 = (𝒑, 𝜸)

1

⋱

2

4 × 4

𝑴𝑜𝑟 𝑲 =

𝟎

𝟎

Joint nodes

Damage location Damage severity




Lemma 𝑯𝑗|(𝑝, 𝛾1) ≠ 𝑯𝑗|(𝑝, 𝛾2) if 𝛾1 ≠ 𝛾2 for 𝑝 = 1,2,…𝑛 and

𝑗 = 1,2, , … 𝑛 + 1, where 𝑯𝑗|(𝑝, 𝛾1) represents the matrix 𝑯𝑗 given parameters

(𝑝, 𝛾1).

Input

𝑗𝑡ℎ 𝑝Output

11

−𝑠𝟐𝑴+𝑲 𝜽 = 𝑽 =

𝑹1

−𝑸2

−𝑸2𝑻

𝑹2

⋱

−𝑸3𝑻

⋱−𝑸𝒏

⋱𝑹𝒏

−𝑸𝒏+1

−𝑸𝒏+1𝑻

𝑹𝒏+1

𝜴𝒊 = 𝜟𝒊, 𝑖 < 𝑗𝜴𝒊 = 𝜮𝒊, 𝑖 > 𝑗𝜴𝒋 = 𝜟𝒋 + 𝜮𝒋 − 𝑹𝒋

𝑽𝑗−1 = 𝜴𝒋

−1

𝜟1 = 𝑹1,

𝜟𝒊 = 𝑹𝒊 −𝑸𝒊𝜟𝒊−1−1 𝑸𝒊

𝑻,

𝜮𝒏+1 = 𝑹𝒏+1,

𝜮𝒊 = 𝑹𝒊 − 𝑸𝒊+1𝑻 𝜮𝒊+1

−1 𝑸𝒊+1.Input

Output damage

Input

Output

Input

Output

𝑗 ≤ 𝑝 − 1

𝑗 = 𝑝 𝑜𝑟 𝑝 + 1

𝑗 > 𝑝 + 1

𝜴𝒋 = 𝜟𝒋 + 𝜮𝒋 − 𝑹𝒋



Lemma

Analytical analysis can be messy and untraceable for identifiability in damage location

Proof Sketch of the Lemma



Sufficient condition based on natural frequencies 𝑓


Frequency

Log

Am

plit

ud

e

Frequency Response

𝐻 𝑠 =𝑐1𝑐2

𝜆 − 𝑧1)(𝜆 − 𝑧2)(𝜆 − 𝑧3)⋯ (𝜆 − 𝑧𝑚−1)(𝜆 − 𝑧𝑚

𝜆 − 𝑓12)(𝜆 − 𝑓2

2)(𝜆 − 𝑓32)⋯ (𝜆 − 𝑓𝑛−1

2 )(𝜆 − 𝑓𝑛2 𝑓𝑖

2 = 𝜆𝑖 , eigenvalue of 𝑴−𝟏𝑲(𝒑, 𝜸)



𝛾0.2 0.4 0.6 0.8 1

0.94

0.95

0.96

0.97

0.98

0.99

1

𝒇𝒊(𝜽)/𝐦𝐚𝐱(𝒇

𝒊)

𝑝∗ = 23

0.2 0.4 0.6 0.8 10.93

0.94

0.95

0.96

0.97

0.98

0.99

1

𝛾

𝒇𝒊(𝜽)/𝐦𝐚𝐱(𝒇

𝒊)

Eigenvalue perturbation


𝑲− 𝜆𝑖𝑴 𝝓𝑖 = 0

𝑲+ 𝚫𝑲 − 𝜆𝑖 + Δ𝜆𝑖 𝑴 𝝓𝒊 + 𝚫𝝓𝒊 = 0

Δ𝜆𝑖 =𝝓𝒊

𝑻𝚫𝑲𝝓𝒊

𝝓𝒊𝑻𝐌𝝓𝒊

= 𝛾 − 1 𝐸𝐼 ×𝝓𝒊

𝑻𝑮𝝓𝒊

𝝓𝒊𝑻𝑴𝝓𝒊

≤ 0

𝑮𝑝 =

12

𝑙36

𝑙2−12

𝑙36

𝑙2

4

𝑙−

6

𝑙22

𝑙12

𝑙2−

6

𝑙2

𝑠𝑦𝑚4

𝑙

≽ 0 𝛾 ∈ [0,1

Relationship between 𝑓 and 𝑝, 𝛾


𝛾𝑝

𝑓1

1

60

0.1

1

110

135

𝛾

𝑓1

0.1 1110

135

for each location


0.2 0.4 0.6 0.8 1110

115

120

125

130

135

0.2 0.4 0.6 0.8 11.48

1.49

1.5

1.51

1.52

1.53

1.54x 10

5

0.2 0.4 0.6 0.8 116

17

18

19

20

21

0.2 0.4 0.6 0.8 11.42

1.43

1.44

1.45

1.46

1.47

1.48x 10

5

𝑓1

𝑓50 𝑓50

𝑓1

𝛾 𝑝𝛾

Relationship between 𝑓 and 𝑝, 𝛾



Define 𝑓𝑖(𝒑,𝜸)

the 𝑖𝑡ℎ natural frequency (sorted from smallest to largest) for

parameters (𝒑, 𝜸) and 𝑆𝑘 = min𝜸

𝑖∈ℒ 𝑓𝑖𝒑∗,𝜸∗ − 𝑓𝑖

𝒑𝒌,𝜸2

Numerical Algorithm of Location Identifiability of Scalar Valued 𝒑 and 𝜸

1 Input (𝑝∗, 𝛾∗), and update 𝑲(𝑝∗,𝛾∗),

2 Compute 𝑓𝑖(𝑝∗,𝛾∗),

by taking the square root the eigenvalues of 𝑴−𝟏𝑲(𝑝∗,𝛾∗),

3 Define the set ℒ and a threshold 𝑡

4 For 𝑘 = 1:𝑛

𝑝𝑘 = 𝑘 and compute 𝑆𝑘

End

5 Sort 𝑆𝑘 in the ascending order as 𝑆𝑘(𝑗 )

, i.e., 𝑆(1) ≤ 𝑆(2) ≤ ⋯ ≤ 𝑆(𝑛)

6 If 𝑆(2) − 𝑆(1) > 𝑡, claim the parameter (𝑝∗, 𝛾∗) is identifiable,

otherwise (𝑝∗, 𝛾∗) is not identifiable in the sense of natural frequency.


𝑘

𝑆𝑘

1 600

700

𝑆𝑘 = min𝜸

𝑖∈ℒ

𝑓𝑖𝒑∗,𝜸∗ − 𝑓𝑖

𝒑𝒌,𝜸2

e.g. 𝑝∗, 𝛾∗ = 23,0.4 𝓛 = {𝟏: 𝟓}

𝑆𝑘2− 𝑆𝑘

1> threshold


𝑆 1 ≤ 𝑆 2 ≤ 𝑆 3 … .


0 20 40 600

0.5

1

1.5

2

X: 1

Y: 0.0661

0 20 40 600

100

200

300

400

500

600

700

0 20 40 600

0.5

1

1.5

2

2.5

3

3.5

4

4.5x 10

8

0 20 40 600

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0 20 40 600

100

200

300

400

500

600

700

0 20 40 600

1

2

3

4

5

6

7

8

9x 10

8

𝓛 = {𝟏} 𝓛 = {𝟏: 𝟓} 𝓛 = {𝐚𝐥𝐥}

𝑆𝑘

𝑝∗, 𝛾∗ = 23,0.4

𝑘

Identifiability of the damage locationIdentifiability Power vs 𝓛

𝑆𝑘 = min𝜸

𝑖∈ℒ

𝑓𝑖𝒑∗,𝜸∗ − 𝑓𝑖

𝒑𝒌,𝜸2


0.2 0.4 0.6 0.8 10

100

200

300

400

500

600

700

800

0.2 0.4 0.6 0.8 10

500

1000

1500

2000

2500

𝛾

𝑆𝑘2 − 𝑆𝑘

1

𝓛 = {𝟏: 𝟓}

𝛾

for each location

Identifiability Power vs 𝛾


𝑆𝑘2 − 𝑆𝑘

1

Conclusion and Future Work

• Conclusion Theoretically proof the identifiability of damage at a fixed location

in the FEM-based structure Investigate the change of natural frequencies as a function of

damage severity A numerical algorithm is proposed to numerically check and

validate the location identifiability. Easy to extend to non-uniform or multiple damages beam structure

• Future Work Theoretically investigate of the identifiability issue in more

complicated structures Explore the identifiability when measurements are contaminated by

noise in practice


identifiability analysis of finite element models for ... · 2015 informs philadelphia 4 dynamic...

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