# Mathematical and numerical modelling of piezoelectric sensors

Sebastien Imperiale; Patrick Joly

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 4, page 875-909
- ISSN: 0764-583X

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topImperiale, Sebastien, and Joly, Patrick. "Mathematical and numerical modelling of piezoelectric sensors." ESAIM: Mathematical Modelling and Numerical Analysis 46.4 (2012): 875-909. <http://eudml.org/doc/277852>.

@article{Imperiale2012,

abstract = {The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element/finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.},

author = {Imperiale, Sebastien, Joly, Patrick},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Piezoelectricity; quasi-static approximation; ultrasonic sensors; piezoelectricity; quasi-state approximation; ultrasonic sensor},

language = {eng},

month = {2},

number = {4},

pages = {875-909},

publisher = {EDP Sciences},

title = {Mathematical and numerical modelling of piezoelectric sensors},

url = {http://eudml.org/doc/277852},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Imperiale, Sebastien

AU - Joly, Patrick

TI - Mathematical and numerical modelling of piezoelectric sensors

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 4

SP - 875

EP - 909

AB - The present work aims at proposing a rigorous analysis of the mathematical and numerical modelling of ultrasonic piezoelectric sensors. This includes the well-posedness of the final model, the rigorous justification of the underlying approximation and the design and analysis of numerical methods. More precisely, we first justify mathematically the classical quasi-static approximation that reduces the electric unknowns to a scalar electric potential. We next justify the reduction of the computation of this electric potential to the piezoelectric domains only. Particular attention is devoted to the different boundary conditions used to model the emission and reception regimes of the sensor. Finally, an energy preserving finite element/finite difference numerical scheme is developed; its stability is analyzed and numerical results are presented.

LA - eng

KW - Piezoelectricity; quasi-static approximation; ultrasonic sensors; piezoelectricity; quasi-state approximation; ultrasonic sensor

UR - http://eudml.org/doc/277852

ER -

## References

top- N. Abboud, G. Wojcik and D.K. Vaughan, Finite element modeling for ultrasonic transducers. SPIE Int. Symp. Medical Imaging (1998).
- E. Canon and M. Lenczner, Models of elastic plates with piezoelectric inclusions part i : Models without homogenization. Math. Comput. Model.26 (1997) 79–106. Zbl0899.73224
- P. Challande, Optimizing ultrasonic transducers based on piezoelectric composites using a finite-element method. IEEE Trans. Ultrason. Ferroelectr. Freq. Control37 (2002) 135–140.
- G.C. Cohen, Higher-order numerical methods for transient wave equations. Springer (2002). Zbl0985.65096
- E. Dieulesaint and D. Royer, Elastic waves in solids, free and guided propagation. Springer (2000). Zbl0960.74002
- M. Durufle, P. Grob and P. Joly, Influence of gauss and gauss-lobatto quadrature rules on the accuracy of a quadrilateral finite element method in the time domain. Numer. Methods Partial Differ. Equ.25 (2009) 526–551. Zbl1167.65057
- Y. Gómez-Ullate Ricón and F.M. de Espinosa Freijo, Piezoelectric modelling using a time domain finite element program. J. Eur. Ceram. Soc.27 (2007) 4153–4157.
- T. Ikeda, Fundamentals of piezoelectricity. Oxford science publications (1990).
- N.A. Kampanis, V.A. Dougalis and J.A. Ekaterinaris, Effective computational methods for wave propagation. Chapman and Hall/CRC (2008).
- T. Lahrner, M. Kaltenbacher, B. Kaltenbacher, R. Lerch and E. Leder. Fem-based determination of real and complex elastic, dielectric, and piezoelectric moduli in piezoceramic materials. IEEE Trans. Ultrason. Ferroelectr. Freq. Control55 (2008) 465–475.
- R. Lerch, Simulation of piezoelectric devices by two-and three-dimensional finite elements. IEEE Trans. Ultrason. Ferroelectr. Freq. Control37 (2002) 233–247.
- S. Li, Transient wave propagation in a transversely isotropic piezoelectric half space. Z. Angew. Math. Phys.51 (2000) 236–266. Zbl1005.74032
- D. Mercier and S. Nicaise, Existence, uniqueness, and regularity results for piezoelectric systems. SIAM J. Math. Anal.37 (2005) 651–672. Zbl1127.35064
- J. San Miguel, J. Adamowski and F. Buiochi, Numerical modeling of a circular piezoelectric ultrasonic transducer radiating in water. ABCM Symposium Series in Mechatronics2 (2005) 458–464.
- P. Monk, Finite element methods for maxwell’s equations. Oxford science publications (2003). Zbl1024.78009
- J.C. Nédélec, Acoustic and electromagnetic equations : integral representations for harmonic problems. Springer (2001). Zbl0981.35002
- V. Priimenko and M. Vishnevskii, An initial boundary-value problem for model electromagnetoelasticity system. J. Differ. Equ.235 (2007) 31–55. Zbl1117.35078
- L. Schmerr Jr and S.J. Song, Ultrasonic nondestructive evaluation systems. Springer (2007).
- C. Weber and P. Werner, A local compactness theorem for maxwell’s equations. Math. Methods Appl. Sci.2 (1980) 12–25. Zbl0432.35032

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