





Mathematical Analysis of Piezoelectric Problems


Problem 
Piezoelectricity can be defined as an interaction between two phenomena: the direct piezoelectric effect (a mechanical deformation generates an electric field in the material) and the inverse piezoelectric effect (the application to the material of an electric field or of a potential difference generates a deformation), cf. T. Ikeda, Fundamentals of Piezoelectricity, Oxford University Press, Oxford, 1990. Therefore, a single piezoelectric device is both an actuator and a sensor, and consequently, piezoelectric materials belong to a class of smart or intelligent materials, that are very important in many applications as, for example, biomechanics, biomedicine, structural mechanics, etc.
 
Modelling & Computational Challenges 
The scope of this project is essentially to acquire a better mathematical knowledge of some particular piezoelectric models, as adaptive rod models and composite laminated plate models. This research project will lead to a better understanding of the mechanical and electric behavior of these problems and, consequently, to an improvement of reallife applications.
 
Research at LCM 
Research will be developed along the following lines:


Papers & Reports 
[1] I. Figueiredo and G. Stadler, Frictional contact of an anisotropic piezoelectric plate, Preprint 0716, Dep. Mathematics, University of Coimbra, 2007.
[2] Georg Stadler, Elliptic optimal control problems with L1control cost and applications for the placement of control devices, Preprint 0642 of the Department of Mathematics, University of Coimbra, 2006. [3] L. Costa, I. Figueiredo, R. Leal, P. Oliveira, G. Stadler, Modeling and numerical study of actuator and sensor effects for a laminated piezoelectric plate, Computers and Structures, Vol. 85, 78 (2007) 385–403. [7] Isabel N. Figueiredo, Approximation of bone remodeling models, Journal de Mathématiques Pures et Appliquées Vol. 84, 12 (2005) 17941812. [8] Isabel N. Figueiredo; Carlos F. Leal and Cecília S. Pinto, Shape analysis of an adaptive elastic rod model, SIAM Journal on Applied Mathematics Vol.66, 1 (2005) 153 173. [9] Isabel N. Figueiredo; Carlos F. Leal and Cecília S. Pinto, Conical differentiability for bone remodeling contact rod models, ESAIM: Control, Optimisation and Calculus of Variations Vol.11, 3 (2005) 382400. [10] Isabel N. Figueiredo and Carlos F. Leal, A piezoelectric anisotropic plate model, Asymptotic Analysis Vol.44, 34 (2005) 327346.
 
...... Contour plots of the electric potential for a square piezoelectric plate in frictional contact with a rigid obstacle (three different obstacles).


Software  [1] Patches  Finite Element Code for Elastic Plates with Piezoelectric Patches (MATLAB code for the software COMSOL MULTIPHYSICS 3.3)  available under request . [2] Lampiezo.m  Finite Element Code for a Laminated Piezoelectric Plate (MATLAB code for the MATLAB Toolbox CALFEM)  available under request. [3] Piezo.m  Finite Element Code for a Piezoelectric Plate (MATLAB code for the MATLAB Toolbox CALFEM)  available under request.


... ... ... Optimal control actuator problem: location and intensity of the applied electric potential (colored squares) for a rectangular piezoelectric plate obliged to have a desired displacement . The four plots illustrate the optimal solutions for four different values of the problem's parameter; this parameter controls the number of finite elements where the applied electric potential is nonzero.


Project

Isabel Maria Narra de Figueiredo, LCMCMUC


FCT Research Project  POCI/MAT/59502/2004  

