SPIE 2010 Spectral Element Presentation

8/3/2019 SPIE 2010 Spectral Element Presentation

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A time domain spectral elementmodel for piezoelectric

excitation of Lamb waves in isotropic plates

Ramy Mohamed & Patrice Masson

GAUS, Department of Mechanical EngineeringUniversit de Sherbrooke

Sherbrooke, QC, J1K 2R1, Canada.

March, 10, 2010

Patrice Masson (Universit de Sherbrooke) SEM of PZT excitation of Lamb Waves SPIE/Smart Materials & NDE 1 / 25
http://goforward/http://find/http://goback/


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Outline

Outline

1 IntroductionNumerical SimulationPrevious Work

2 Model DevelopmentFormulationSEM Discretization

3 ResultsTime Domain ResultsReSTFT Representation



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Introduction

Motivation

Optimization of the PZT actuators and sensors, placement andconfiguration, for SHM/NDE purposes:

Accurate simulation of wave propagation in complex geometrystructures, and boundary conditions.

Accurate representation of dynamic coupling between theactuators/sensors and inspected structure.

Optimization is -in general- an iterative process. Effectiveness in

terms of computational cost and time, is a major requirement.



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Introduction

Motivation


Accurate simulation of wave propagation in complex geometrystructures, and boundary conditions.





I d i


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Introduction

Motivation


Accurate simulation of wave propagation in complex geometry

structures, and boundary conditions.





I t d ti


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Introduction

Motivation


Accurate simulation of wave propagation in complex geometry

structures, and boundary conditions.





Introduction Numerical Simulation


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ChallengesNumerical Dispersion

2D Elastic Wave

P Relative error in P-wavephase velocity.

S Relative error in S-wavephase velocity.

n Number of grid points.

Wavelength. 20 10 5 4 3.3 2.85 2.5 2.2 210

5

0

5

10

15

20

25

n

P,

S

Non Conforming FE

Conforming FE

Legendre SE

SE- Seriani & Oliveira; Wave Motion, (45), 2008.

FE- Zyserman et. al. Int. J. Numer. Meth. Engng, (85), 2003.




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FEM vs SEMComputational Efficiency

Computational Cost

C = n

2

r1.0

tB

n

r

the required number of

grid points per wavelength for0.1 % disperison error.

B is the average number of

non-zero terms in a row in theproduct of matrices M1K.

The same procedure as in Dauksher & Emery; Int

J Numer Meth Engng (45), 1999.

2 x 2 3 x 3 4 x 4 5 x 5 6 x 6 7 x 7 8 x 8 9 x 9 10 x 10

0.1

0.5

1

5.0

10

Element dimensions

C/Cref

SEMt

tstable= 1

SEMt

tstable= 0.5

FEMt

tstable= 1

FEMt

tstable= 0.5Cref


Introduction Previous Work
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SEMHistory

Origin in CFD

1 Patera, A. T.; J Comput Phys 54, (1984).

2 Korczak, K. Z., and Patera, A. T.; J Comput Phys 62, (1986).

Computational Seismology

1 Seriani, G., and Priolo, E.; Finite Elements in Analysis and Design 16,(1994).

2 Komatitsch, D., and Villote, J. P.; Bullet Seis Soc Amer 54, (1998).

3 Komatitsch, D., and Tromp, J.; Geophys J Int 139, (1999).




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SEMLamb Wave

LW Propagation

1 Kudela, P., Krawczuka, M., and Ostachowicz, W.; J Sound Vibr 300,(2007).

2 Kudela, P., Zak, A., Krawczuka, M., and Ostachowicz, W.; J SoundVibr 302, (2007).

3 Peng, H., Meng, G., and Li, F.; J Sound Vibr 320, (2009).




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SEMLamb Wave

LW Propagation

Coupling Platewith

Actuator(s)

Coupling Platewith Sensor(s)

1 Kim, Y., Ha, S., and Chang, F. K.; AIAA Journal 46(3), (2008).

2 Ha, S., and Chang, F. K.; Smart Mater Struct 19, (2010).


Model Development Formulation


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p

Strong FormDomain Decomposition

s

p

g

e

x3

x

where:

p piezoceramic domain.

s plate domain.

g dynamic coupling interface.

e electroded boundary.




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Strong FormConstitutive Equations

PiezoceramicT = cES eTE

D = e S + SE

PlateT = cS

where:

T Mechanical stress tensor. S Infinitesimal strain tensor.cE Elasticity tensor of the piezoceramic. e Tensor of piezoelectric stress constants.c Elasticity tensor of the plate. D Electrical displacement vector.S Dielectric permittivity tensor. E Electric field vector.




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Strong FormGoverning Equations

Momentum conservation PZT

BT

cEBup + e

T

= pup x p

Charge conservation PZT

T

eBup S

= 0 x p

Momentum conservation Plate

BT

csBus = sus x s

Differential Operators

B =

x1 0

0 x3x3 x1

, =

x1x3




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Strong FormAssumptions & Boundary Conditions

p

g

e

s

The structure material s was modeled as purely elastic.

Traction and displacement continuity corresponding (i. e. idealbonding) at the interface g.

Traction free external boundaries.Isolated non-electroded electrical boundaries.

Uniform excitation voltage distribution on the electroded boundary.

Zero initial conditions.




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s

p

e

g









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s

p

g

e









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s

p

g

e









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s

p

g

e









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s

p







Model Development SEM Discretization


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SEM2D Shape Functions

The 2D shape functions is the tensor product of the 1D Lagrange polynomials. Thedisplacement ueN|e and electric potential

e

N|e :

ueN

(, ) =N

m=0

Nn=0

uemn(lNm () l

Nn ()) = Lu

e

eN

(, ) =N

m=0

Nn=0

emn

(lNm

() lNn

()) = Le

Corner function Boundary function Interior function




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Spatial Integration SchemeLGL Numerical Integration

Finite Element (p=6)Gauss Quadrature

x quadrature node, collocation node.

Legendre Spectral ElementLGL Quadrature

quadrature nodes is the collocation nodes.




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Matrix EquationsStrong Coupling

Semidiscrete EquationsM 0

0 0

u

+

Kuu Ku

KTu K

u

=

0

fe

Condensed Form

Mu +

KuuKuK1K

T

u

u = K1 fe MU + KU = F

Time Integration

MUn+1 + (1 + )KUn+1 KUn = F(tn+)

Time step limited by CFL condition




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Matrix Equations

Iterative weak coupling per time step implemented by Kim, Y.,

Ha, S., and Chang, F. K.; AIAA Journal 46(3), (2008)

Ku K = Pin p

Mu = FextFint

in s+p

@ tn1u @ tn1

(u,) @ tn

One step simultaneous solution based on strong coupling in

condensed form used in the present study

MU + KU = Fin s+p


Results Time Domain Results


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Experimental Setup

Notched rectangular aluminum plate 1.54 mm thick, 700 mm long and 70 mm wide.

Excitation signal 3.5 tone burst modulated by a Hanning window.

3 patches of BM 500 PZT (Sensor Technology Ltd.), 0.5 mm thick, 7 mm wide.

Excitation voltage amplitude 10 V.

70

Sensor 2Sensor 1Actuator

330210

1.54

700

0.8

0.8

all dimensions in mm


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Simulation ResultsNodal Forces Excitation

S0 simulated interaction with the notch

0 50 100 150 200 250 300 350 400

0

100

200

300

400

500

600

700

Time (s)

Position

in

x1

direction

(mm

)

Transmitted S0

Converted A0Incident S0

Reflected S0




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Simulation ResultsNodal Forces Excitation

A0 simulated interaction with the notch

0 50 100 150 200 250 300 350 400

0

100

200

300

400

500

600

700

Time (s)

Position

in

x1

direction

(mm)

Converted S0

Converted S0

Reflected A0

Incident A0

Transmitted A0




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Time TraceSensor 1 at 250 kHz

Sensor response was approximated as proportional to the longitudinal strain

55 60 65 70 75

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

NormalizedVoltage

SEM

Experimental

80 90 100 110 120 130 140 150 160

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

NormalizedVoltage

SEM

Experimental

Incident A0

Reflected S0from incident S0

Converted A0from incident S0


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Time TraceSensor 1 at 450 kHz

Sensor response was approximated as proportional to the longitudinal strain

55 60 65 70 75

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

NormalizedVoltage

90 100 110 120 130 140 150 160

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time (s)

NormalizedVoltage

SEM

Experimental

Incident A0

Converted A0

Reflected S0


Results ReSTFT Representation

S l d S l
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Simulated SignalSensor 1 at 300 kHz

Time (s)

Frequency(MHz)

0 50 100 150 200 250 300 350 400 450 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 105

0 50 100 150 200 250 300 350 400 450 500

1

0

1

A0

S0



E i l Si l


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Experimental SignalSensor 1 at 300 kHz

Time (s)

Fre

quency(MHz)

0 50 100 150 200 250 300 350 400 450 500

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

x 105

0 50 100 150 200 250 300 350 400 450 500

1

0

1

S0A0



R lt Vi li ti


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Results Visualization

Excitation Frequency 250 kHz

3.5 cycles


Summary

S


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Summary

The strong coupling condensed form formulation offers acomputational cost advantage. Enabling a fast implementation whenplugged into an iterative optimization process.

Less numerical dispersion is achievable with slight increase incomputational requirements. Valuable when modeling complex

geometry, material anisotropy, and heterogeneity.The simulation results agrees well with the experimentalmeasurements. Although no attempt to model the attenuation, or thebonding layer was made.

Outlook

Inclusion of material attenuation, and bonding layer.Modeling composite structures.Optimizing the actuator for selectivity, for different geometries.


Summary


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Thank You


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SPIE 2010 Spectral Element Presentation

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