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
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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
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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I d i
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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 geometry
structures, 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.
Patrice Masson (Universit de Sherbrooke) SEM of PZT excitation of Lamb Waves SPIE/Smart Materials & NDE 3 / 25
I t d ti
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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 geometry
structures, 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.
Patrice Masson (Universit de Sherbrooke) SEM of PZT excitation of Lamb Waves SPIE/Smart Materials & NDE 3 / 25
Introduction Numerical Simulation
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Introduction Numerical Simulation
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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Introduction Numerical Simulation
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Introduction Numerical Simulation
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
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Introduction Previous Work
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Introduction Previous Work
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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Introduction Previous Work
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Introduction Previous Work
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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Introduction Previous Work
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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).
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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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Model Development Formulation
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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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Model Development Formulation
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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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Model Development Formulation
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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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Model Development Formulation
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Strong FormAssumptions & Boundary Conditions
s
p
e
g
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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Model Development Formulation
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Strong FormAssumptions & Boundary Conditions
s
p
g
e
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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Model Development Formulation
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Strong FormAssumptions & Boundary Conditions
s
p
g
e
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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Model Development Formulation
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Strong FormAssumptions & Boundary Conditions
s
p
g
e
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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Model Development Formulation
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Strong FormAssumptions & Boundary Conditions
s
p
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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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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Model Development SEM Discretization
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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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Model Development SEM Discretization
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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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Model Development SEM Discretization
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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
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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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Results Time Domain Results
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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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Results Time Domain Results
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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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Results Time Domain Results
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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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Results Time Domain Results
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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
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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
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Results ReSTFT Representation
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
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Results ReSTFT Representation
R lt Vi li ti
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Results Visualization
Excitation Frequency 250 kHz
3.5 cycles
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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.
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Summary
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Thank You
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