Efficient calculation of finite element matrices for two-dimensional piezoelectric analyses

In this paper we demonstrate a efficient method of deriving the necessary matrices to implement a two-dimensional finite element analysis of multilayered piezoelectric microacoustic wave devices. Whereas other commercially available simulation packages are typically based on a modification of a structurally relevant problem, the notations and methods presented in this paper are characteristic of those used within the Ultrasonics community. The presented technique allows for a symbolic assembly of the structural stiffness-, mass-, force- and damping matrices avoiding time-consuming numerical integration. In the example problem presented the necessary integral calculations can be determined simply by inspection. The resulting matrices remain generally sparse and thus are well suited to generic matrix solvers. The proposed method has been formulated for both two- and three-dimensional problems and verified against existing commercial Finite Element Method package ANSYS, however only the two-dimensional implementation is discussed here. We also present preliminary results of our efforts to reduce computation time by the use of highly specialised Graphics Processing Units (GPUs).