Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-23T20:13:44.526Z Has data issue: false hasContentIssue false
  • English
  • Français

Mathematical and numerical modelling of piezoelectric sensors

Published online by Cambridge University Press:  03 February 2012

Sebastien Imperiale  and
Patrick Joly
Sebastien Imperiale
Affiliation:
CEA, List, Saclay, 91191 Gif-sur-Yvette, France. [email protected] INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, 78153 Le Chesnay, France; [email protected]
Patrick Joly
Affiliation:
INRIA Rocquencourt, Domaine de Voluceau, Rocquencourt, 78153 Le Chesnay, France; [email protected]
Get access

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.

Keywords

Piezoelectricityquasi-static approximationultrasonic sensors
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 46 , Issue 4 , July 2012 , pp. 875 - 909
Copyright
© EDP Sciences, SMAI, 2012

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Références

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