Spectral Element Fast Solver for Piezoelectric Structures

8/3/2019 Spectral Element Fast Solver for Piezoelectric Structures

1/31

Time Domain SpectralElement Fast Solver for

Piezoelectric Structures

Ramy Mohamed & Patrice Masson

GAUS, Department of Mechanical Engineering

Universit de Sherbrooke

Nov, 3, 2011
http://find/http://goback/


2/31

Outline

Outline

1 IntroductionHistoryContextMotivation

2 Model DevelopmentPrevious WorkNumerical SimulationFormulation

3 ResultsTime Domain Results

Ramy Mohamed (UdS-GAUS) SEM Fast Solver for PZT Structures 3 Nov 2011 2 / 23


3/31

Introduction History

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).



4/31

Introduction Context

Active SHM System

An ultrasound pulse isinjected into the structurethrough an integratedpiezoelectric element.

Then a sensor collects thesignal carryinginformation about theregion being queried.

C. E. S. Cesnik and A. Raghavan, Ch 3, Encyclopedia of SHM.


I d i C


5/31


Guided Waves for SHM Application

Piezoceramic (PZT) for SHM

Low cost, small size, and easily integrated into the structure.

strain coupled with the structure, and operates at different frequencies.

Guided Waves for SHM

Two dimensional, enabling scanning the whole cross section from few

locations.

Propagation for long distance with low losses of energy.

multi-modal and dispersive, making interpretation of the signals very

difficult.


I t d ti C t t


6/31


Active-SHM via Guided Waves

The frequency (wavelength) is dictated by the smallest detectabledamage.

Relatively narrow band frequency content.


Introduction Motivation


7/31


High vs Low Frequency

General Elastodynamics Equations

Axial and flexural waves are lowfrequency approximations of

fundamental Lamb modes S0,and A0.

For accurate simulations at highfrequency we need to solve the

general elastodynamicsequations.0 2 4 6 8 10

0

5

10

15

PhaseVelocity(Km

/s)

Frequency x thickness (MHz mm)

Flexural Wave (Approx. A0)

Axial Wave (Approx. S0)

S0 Lamb Mode

A0 Lamb Mode




8/31


Challenges at High Frequency: Numerical Dispersion

2D Elastic Wave

P Relative error in P-wavephase velocity, and S Relativeerror in S-wave phase velocity.

n Number of nodes perWavelength.

Minimum number of nodes perminimum wavelength is 5 for

SEM as opposed to 20 for theFEM for the same numericaldispersion error (0.1%).

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

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

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


Model Development Previous Work


9/31


SEM: Lamb Wave

LW Propagation

1 Kudela, P.,et al.; J Sound Vibr 300, (2007).

2 Kudela, P., et al.; J Sound Vibr 302, (2007).

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




10/31


SEM: Lamb 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 Numerical Simulation


11/31

p

FEM vs SEM: Computational Efficiency

2D Case

Circular frequency

Period time = 2

Wavelength = cp 1

Average element size h 1

Time Step t h 1

Number of elements Nel A

h2

2

Number of time steps Nt =Tsimt

Total ops O(3)




12/31

p


2D Case

Circular frequency

Period time = 2

Wavelength = cp 1

Average element size h 1

Time Step t h 1

Number of elements Nel Ah2 2

Number of time steps Nt = Tsimt

Total ops O(3)




13/31

SEM: 2D Shape Functions

The 2D shape functions is the tensor product of the 1D Lagrange polynomials. The

displacementueN|e and electric potential

eN|e

Mode shape of the A0 @ 450kHz

Mode shape of the S0 @ 450kHz




14/31

SEM: Mapping

x

y

Ji,j = J(i, j)

F (x, y)




15/31

SEM: Mapping

x

y

Ji,j = J(i, j)

F (x, y)




16/31

Spatial Integration Scheme: LGL Numerical Integration

Finite Element (p=6)Gauss Quadrature

x quadrature node, collocation node.

Legendre Spectral ElementLGL Quadrature

quadrature nodes is the collocation nodes.




17/31


Computational Cost

C =

n

2

r1.0

tB

n

r

the required number of grid

points per wavelength for 0.1 %disperison error.

B is the average number of non-zeroterms in a row in the product 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

FEM

ttstable = 0.5Cref


Model Development Formulation
http://goback/http://find/http://find/http://goback/


18/31

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.




19/31


s

p

e

g









20/31


s

p

g

e









21/31


s

p

g

e









22/31


s

p

g

e









23/31


s

p







Results


24/31

Benchmark setup


Results Time Domain Results


25/31

Results @ 200 kHz

The results of the FE simulation (ANSYS), and the SEM (5.5 toneburst).



26/31

Results Time Domain Results


27/31

CPU Time

The solver was written in Fortran 2003. The solver were built using Intel Fortran compiler, under Windows 7 with a 2.26 double

core processor, and 4 GB RAM. Both cores were used in both ANSYS, and SEM solver (through the auto-parallelization of the

compiler).

SEM FEM

Frequency (kHz) 200 450 200 450

Number of Elements 845 1520 9123 45132

CPU time (min) 14 32 32 54


Conclusions


28/31

Conclusions

The strong coupling of piezoelectric equations in SEM settings offersa higher accuracy in the simulation.

SEM is more accurate and computationally efficient than the FEMwith respect to modeling GW propagation in relatively complex thin

walled structures, especially at high frequencies. Less numerical dispersion is achievable with linear increase in

computational requirements. Valuable when modeling complexgeometry, material anisotropy, and heterogeneity.

Versatile optimizations for different kinds of structures, for examplethe proposed element is optimized for thin walled sections (3 nodes inthe thickness direction and 6 in the longitudinal direction).


Conclusions


29/31

This work has been supported by the Natural Sciences and Engineering ResearchCouncil of Canada (NSERC).

http://groups.google.com/group/semsolve

Thank You


Conclusions

M E


30/31

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


Conclusions

M i E i


31/31

Matrix Equations

Iterative 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


Date post:	06-Apr-2018
Category:	Documents
Upload:	ramy
View:	222 times
Download:	0 times

Spectral Element Fast Solver for Piezoelectric Structures

Documents