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Finite Element Modelling of piezo shells

Contact : André Preumont


The use of piezoelectric materials as actuators and sensors for noise and vibration control has been demonstrated extensively over the past few years. The design of control systems involving piezoelectric actuators and sensors requires an accurate knowledge of the transfer functions between the inputs and the outputs of the system. These are not easy to determine numerically, particularly for shell structures with embedded distributed actuators and sensors. The situation where they are nearly collocated is particularly critical, because the zeros of the transfer functions are dominated by local effects which can only be accounted for by finite elements.

Finite Element Modelling

A full implementation of a library of piezoelectric elements (triangular and rectangular shells, brick, prism and tetrahedron volume elements) into a commercial finite element package is readily available (Samcef V8.1-4 and above (SAMTECH)) and is suitable for industrial problems resolution.

For shell elements (Mindlin formulation), it is assumed that the electric field and displacement are uniform across the thickness and aligned on the normal to the mid-plane. The electrical degrees of freedom are the voltages across the piezoelectric layers; it is assumed that the voltage is constant over each element (this implies that the finite element mesh follows the shape of the electrodes). One electrical degree of freedom of type "voltage" per piezoelectric layer is defined. The assembly takes into account the equipotentiality condition of the electrodes; this reduces the number of electric variables to the number of electrodes.

For volume elements, one additional degree of freedom of type "electric potential" is defined in each node of the piezoelectric volume element.

The idea behind modelling structures embedding piezoelectric actuators and sensors using finite elements is to gather the necessary informations to design a good control strategy. A state space representation can be deduced from a linear finite element analysis and is easily implemented in a control oriented software allowing the designer to extract the various frequency response functions and use the control design tools.

Some benchmarks

(in pdf format)

Related contract

1997 : Convention RW 9713549: Modélisation du couplage piézoélectrique dans les structures actives (Région Wallonne - Direction Générale des Techniques, de la Recherche et de l'Environnement)