Modelization and numerical approximation of piezoelectric thin shells: Part II: Approximation by finite element methods and numerical experiments

Abstract

A two-dimensional modelization of piezoelectric thin shells has been developed in the first part of this work. Three equivalent variational formulations have been considered: • an homogeneous one (with respect to the potential) the bilinear form of which is positive definite but not symmetric; • the associated nonhomogeneous one which is directly related to the natural boundary conditions on the potential; • a third one, whose bilinear form is now symmetric but no longer positive definite. In this second part of this work, the approximation of the second formulation by a conforming finite element method is analyzed. It takes into account the use of numerical integration techniques and gives criteria on the choice of suitable numerical schemes. Moreover, the drawbacks linked to the unsymmetrical character of the associated square matrix of the linear system are circumvent by using a condensation method which consists to eliminate the potential and then to only solve a symmetrical and elliptic linear system with respect to the displacement. This method is most effective than a full implementation of the third variational formulation. Finally some numerical experiments on plate and thin cylindrical shells prove the effectiveness of this method.