## Z. Transient analysis of an Electrically Driven Piezoelectric or Magnetostrictive Structure Radiating in an infinite Surrounding Fluid Space (TRA4)

In this analysis, known displacements U, electric potentials f and currents I are applied on the piezoelectric ormagnetostrictive structure. The system of equations becomes:

where × and ×× denote the first and second time derivative, respectively. w0is the pulsation at which the materials losses are defined. The Y vector defines the radiation condition,applied on the external surface S limiting the fluid domain, and which ismonopolar:

where R is the radius of the boundary surface S. [D] and[D'] are obtained by assembling the damping elements on that surface. n equals 1 for axisymmetric or 3D meshes, 2 forplain-strain meshes. This differential equation is solved by an iterativemethod, taking a constant time step D t. Three methods are implemented: the Central DifferenceMethod, the Newmark Method and the Wilson-q Method.The method, the time step and the method’s parameters are defined with the TRANSIENT command. The code computes the displacement field U, the pressure P,the electrical potential F, the reducedmagnetic potential f, and the currentsin magnetic sources I for the requested timesteps, together with reactions to prescribed displacements. The problem can besolved only in double precision.

Important notice: the Central Difference Method algorithm does not accept losses for electric or magnetic degrees of freedom, only for mechanical degrees of freedom. This means that the matrices [K"F], [K"uf], [K"uI], [K"FF], [K"fI], [K"ff] and [K"II] must be zero.