Structural shape parametric optimization for an internal structural-acoustic problem

A structural shape optimization problem, with respect to the structural aspect ratio is developed in the context of an axisymmetric structural-acoustic system, consisting of an elastic dome coupled with an internal acoustic cavity, is analyzed in the low- and medium-frequency ranges. The dome is a thin shell considered as a three-dimensional continuum with a dissipative constitutive equation. The internal fluid is a dissipative acoustic fluid. The dome is submitted to an external wall pressure field modeled by a stochastic field. The cost function is related to the pressure field over an internal axisymmetric observation surface inside the acoustic cavity. We are interested in minimizing the internal noise over the observation surface with respect to the structural aspect ratio defining the geometric shape of the dome. This paper develops an analysis of the structural-acoustic shape optimization problem to determine wheter or not there exist values of the dome aspect ratio for which the internal noise is a minimum. The frequency response functions of the structural-acoustic system are calculated to construct the cost function. In this context, the Fourier series expansions of the structural displacement field and the internal fluid velocity potential are carried out with respect to the polar angle variable. For each fixed circumferential wave number, a reduced matrix model is constructed using the structural modes of the structure in vacuo and the acoustic modes of the internal acoustic cavity with rigid wall. The structural modes and the acoustic modes are computed by the finite element method. The optimization parameter is the aspect ratio of the structure. The analysis presented shows that the structural shape optimization problem of the dome with respect to its aspect ratio parameter has a clear solution which minimizes internal noise in the low- and medium-frequency ranges.