Finite element calculation of wave propagation and excitation in periodic piezoelectric systems



Many sensors and actuators in technical systems consist of quasiperiodic structures which are constructed by successive repetition of a base cell. Typical examples are piezoelectric composites used as ultrasonic transducers or surface acoustic wave (SAW) devices utilized in telecommunication systems. The precise numerical simulation of such devices including all physical effects is currently beyond the capacity of high end computation. Therewith, we have to restrict the numerical analysis to the periodic substructure and have to introduce special boundaries to account for the periodicity. Due to the fact, that wave propagation phenomena have to be considered for SAW applications the peri-odic boundary condition (PBC) has to be able to model each possible mode within the periodic structure. That means this condition must hold for each phase difference existing at the periodic boundaries. To fulfill this difficult criteria we have introduced the Floquet theorem to the PBCs offering two different solution strategies. The first method leads to a quadratic eigenvalue problem. Therein, a huge matrix in-cluding all nodes not belonging to the periodic boundaries has to be inverted due to a Schur-Complement Hofer et al. formulation. The application of the second method results in a general non symmetric eigenvalue prob-lem including the inner and the nodes at one periodic boundary. These techniques consider every kind of propagating mode automatically. The advantage of the latter scheme is that the matrices keep sparse. With the use of an efficient eigensolver utilizing the Arnoldi method, which calculates only the needed eigenvalues and takes advantage of the sparsity of the matrices, the second approach is much faster as the scheme with the Schur-Complement which has to deal with dense matrices . In the first part of this paper we describe the basic theory of our new PBC methods. In the second part we show simulation results for the eigensolution of a periodic SAW structure which can be expressed efficiently in a dispersion diagram. Such diagrams give valuable information to SAW designers such as wave velocity and reflectivity of electrodes and help them to analyze the interaction of surface acoustic waves (SAW) or leaky surface acoustic waves (LSAW) with radiating bulk waves. Finally, we calculate the charge distribution of SAW structures by solving the inhomogeneous piezoelectric partial differential equations with incorporated PBCs. Therewith, we determine the electrical admittance which character-izes the electrical behavior of a SAW device. Further on, the computation of a voltage excitation on arbitrary electrodes will be demonstrated using our newly developed PBCs.

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