Advances in piezoelectric finite element modeling of adaptive structural elements: a survey

Abstract

This paper makes a first attempt to survey and discuss the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements. For most contributions, the specific assumptions, in particular those of electrical type, and the characteristics of the elements are precised. The informations are illustrated in tables and figures for helpful use by the researchers as well as the designers interested in this growing field of smart materials and structures. Focus is put on the development of adaptive piezoelectric finite elements only. However, papers on other applications and active systems are also listed for completeness purpose. In total, more than 100 papers were found in the open literature. Taking this number as a measure of research activity, trends and ideas for future research are identified and outlined.

Introduction

Since the early 70s, many finite element models have been proposed for the analysis of piezoelectric structural elements. They were mainly devoted to the design of ultrasonic transducers till the early 90s [113], [28], [37], [3], [4], [25], [49], [50], [63], [70], [107]. By the late 80s, interests have been directed towards applications in smart materials and structures [98]. During the last two decades, several review papers and bibliographies have appeared in the open literature on the finite element technology [71] and modeling of structural elements [69]. These include sandwich plates [36], thin [111] and moderately thick [34] shells, and layered anisotropic composite plates and shells [79]. However, careful analysis of these survey papers and those on the relatively new field of ‘intelligent’ or smart materials and structures [26], [76], [77] indicates that the finite element modeling of adaptive structural elements does not retain the expected attention. In fact, this highly active application area of finite element methods is in continuous growth, particularly during the last five years (Fig. 1). Hence, it gains a certain maturity so that some piezoelectric elements have become available in commercial finite element codes [67], [72].

It is the objective of this paper to make a first attempt to survey and discuss the advances and trends in the formulations and applications of the finite element modeling of adaptive structural elements, namely, solids, shells, plates and beams. The underlying assumptions, in particular those of electrical type, and the characteristics of the elements such as their shapes, independent variables, interpolation functions degrees, and nodal degrees of freedom (dofs) are precised for the main contributions. The primary interest is the analysis of piezoelectric-based rather than other active materials-based adaptive structural finite element formulations. The informations are presented in tables and figures for helpful use by the researchers as well as the designers interested in this continuously growing field of smart materials and structures.

In the following, common theoretical formulations used for finite element development are first discussed according to the variational equations used, and the specific assumptions made to take into account the electro-mechanical coupling. Then, the piezoelectric finite element characteristics are detailed for solids, shells, plates and beams, separately. Particular attention is paid to the use or not of electric dof. Next, applications and current trends in smart finite element modeling are briefly discussed. As a closure of this survey, some ideas are outlined for future research directions. More than 100 papers are listed alphabetically at the references section. Therefore, this survey is surely incomplete and the author wishes to apologize, in advance, for any inadvertent omission of relevant publications.