## V. Harmonic Analysis of a radiating Piezoelectric or MagnetostrictiveStructure Using a Coupled FEM-BEM Method (HAR9)

In this analysis, finite-element modeling is used inassociation with an external boundary-element code. This code must have adescription of the solid-fluid interfaces similar to the description used inthe CHIEF program (constant pressure and velocity oneach surface). First, the boundary-element code is used to solve numericallythe Helmholtz equation and to compute the mutual impedance matrix [Z] thatrelates pressures and velocities on the radiating surfaces:

Then, finite element modeling is used to solve the coupledproblem. The matrix [Z] is an additional entry described in the JOB.ZRAD file(see section III.D, CHIEF entry).

Then, finite-element modeling is used to solve the coupledproblem. The set of equations is:

where [X] is a coupling matrix which prescribes thecontinuity of the pressure and of the velocity at the interface in an averagesense. The user usually prescribes the applied electrical potential. The codecomputes the displacement field, the electrical potential, the reduced magneticpotential, the currents in the magnetic sources, the electrical impedance ofthe structure and the average velocity of the radiating surfaces. The usercontrols the frequency scan.

Finally, the average velocities can be introduced into theboundary-element program to compute the acoustic field in the fluid medium andto provide the transmitting voltage response and directivity patterns.

The matrices are assembled and stored in a file by columns. Gaussian algorithms are used to solve the problem, in single or doubleprecision. The internal losses in the materials can be taken into account.More general problems including prescribed excitations (displacements,electrical potentials, currents in the magnetic sources) and/or loads (forces,electrical charges) can also be solved.