## U. Harmonic Analysis of a Periodic Piezoelectric or MagnetostrictiveStructure (HAR8)

In this analysis, only a two-dimensional or three-dimensionalelementary cell of the solid structure and a part of the fluid domain thatsurrounds it are meshed. The matrix equation is reduced to:

where the various matrices have been modified after theassembly phase to take into account the specific boundary conditions and where Yi is the normal derivative of thepressure field incident on the mesh boundary. The single or double periodicityof the mesh is taken into account by a specific Bloch type phase relationbetween nodes separated by one mesh step. On the fluid mesh boundaries, thepressure field is matched to a homogeneous or evanescent plane-wave seriesexpansion. It is possible to model structures with fluid on both sides or onlyon the front side. Different fluids can be used on the front and the backsides.

Two types of analyses are available: the scattering and theradiation by a periodic piezoelectric structure. In both cases, the userspecifies the frequency, the angle of the incident wave (scattering) or ofobservation (radiation), the position of the hot electrode and the symmetryplanes of the elementary cell that correspond to the Bloch condition (seeParagraph I.3.E.2). The code computes the real and imaginary parts of thepressure and displacement fields, the electrical potential, the reducedmagnetic potential and the currents in the magnetic sources. It also providesthe transmission and reflection coefficients and, for piezoelectric structures,the free-field voltage sensitivity (scattering) or the transmitting voltageresponse (radiation).

The matrices are assembled and stored to file by columns. The phase relation between nodes separated by one mesh step as well as thecoupling with homogeneous and evanescent plane-waves on the mesh boundaries areincorporated in the matrices which become non symmetrical. Gaussian algorithmsare used to solve the problem, in single or double precision. The internallosses of the materials can be taken into account.