Centre for Mathematics, University of Coimbra   Laboratory for Computational Mathematics  




Computing Resources






Mathematical Analysis of Piezoelectric Problems




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.


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 real-life applications.



at LCM
Research will be developed along the following lines:
  • Asymptotic and variational methods for the mathematical formulation, and related questions of existence and regularity of solutions.
  • Discretization schemes, using finite element and finite difference methods.
  • Analysis of error estimates.
  • Algorithms to solve the discrete problems, such as deterministic optimization algorithms and evolutionary algorithms.
  • Code development (envolving both finite elements and optimization).


[1] I. Figueiredo and G. Stadler, Frictional contact of an anisotropic piezoelectric plate, Preprint 07-16, Dep. Mathematics, University of Coimbra, 2007.

[2] Georg Stadler, Elliptic optimal control problems with L1-control cost and applications for the placement of control devices, Preprint 06-42 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, 7-8 (2007) 385–403.

[4] Isabel N. Figueiredo and Georg Stadler, Optimal control of piezoelectric anisotropic plates, in  CD-ROM Proceedings of the III
European Conference on Computational Mechanics: Solids, Structures and Coupled Problems in Engineering, C.A. Mota Soares et al. (eds.), Lisbon, Portugal, 5–8 June 2006.

[5] Isabel M. N. Figueiredo and Carlos M. F. Leal, A generalized piezoelectric Bernoulli-Navier anisotropic rod model, Journal of Elasticity Vol.85, 2 (2006) 85-106.

[6] L. Costa, P. Oliveira, I.N. Figueiredo and R. Leal, Actuator effect of a piezoelectric anisotropic plate model, Mechanics of Advanced Materials and Structures Vol. 13, 5 (2006) 403-417.

[7] Isabel N. Figueiredo, Approximation of bone remodeling models, Journal de Mathématiques Pures et Appliquées Vol. 84, 12 (2005) 1794-1812.

[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) 382-400.

[10] Isabel N. Figueiredo and Carlos F. Leal, A piezoelectric anisotropic plate model, Asymptotic Analysis Vol.44, 3-4 (2005) 327-346.





Contour plots of the electric potential for a square piezoelectric plate in frictional contact with a rigid obstacle (three different obstacles).




[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.




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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.





Isabel Maria Narra de Figueiredo, LCM-CMUC
Carlos M. Franco Leal, LCM-CMUC
Pedro N.F.P. Oliveira, Department of Production and Systems, University of Minho
Rogério A.C.P. Leal, Department of Mechanical Engineering, University of Coimbra
Georg Stadler, ICES, University of Texas at Austin, USA
José António Carvalho, Department of Mathematics, University of Coimbra
Lino Costa, Department of Production and Systems, University of Minho
Cecília S. Pinto, Department of Mathematics, IPV
Luis M.F. Roseiro, Department of Mechanical Engineering, ISEC
Urbano M.O. Ramos, Department of Mechanical Engineering, ISEC



    FCT Research Project - POCI/MAT/59502/2004