## M. Modal Analysis of a Periodic Piezoelectric Structure (MOD7)

In this analysis, only a two-dimensional or three-dimensionalcell of the active structure is meshed. The user specifies the wave vector,i.e., the wave number and the direction of propagation of the acoustic wave, aswell as the faces of the elementary cell on which the Bloch-Floquet conditionare applied (see Section III.D, PERIODIC, ANGLES, WAVENUMBER entries). For piezoelectric structures, the system of equations is reduced to:

where the stiffness and mass matrices have been modified after the assembly phase to take the specific implicit boundary conditions into account. The first equation system in this section can be rewritten, isolating the electrical potentials linked to the electrodes in the vector Fe and the internal electrical potentials in the vector Fi:

The short-circuit modal analysis, obtained by taking Fe = 0 for the potential of the electrodes, leads to computation of the resonance modes. The open-circuit modal analysis, obtainedby taking q = 0,leads to the computation of the antiresonance modes. The code computes thereal eigenvalues and complex eigenvectors of the linear hermitic system thatare the characteristic frequencies and modes of the structure for the givenwave number. The computation is done using real values with no internallosses. Note that this analysis is valid for magnetostrictive structures ifthe magnetic sources are also periodic (i.e., a constant magnetic field).

The matrices are assembled and stored to file by columns. Lanczos' algorithm for hermitic matrices is used to solve the problem. The problem can be solved only in double precision.

Calculating the resonance and antiresonance modes cannot becarried out in one step.

Note:

The electrical degrees-of-freedom are placed at the beginning of the assembled matrix. The numerical system can differ in size from a harmonic analysis.