Reduction of Low Frequency Acoustical Resonances Inside Bounded Space Using Eigenvalue Problem Solutions and Topology Optimization

Abstract

The chapter deals with the problem of a space with an acoustical source, which forms a field of some values. All apply to an acoustic field characterized by an acoustic pressure p. In the low-frequency range and high values of boundary impedance, the modal approach is successfully applied. In this case, the field variation in all points of space is described by a specific time-dependent variable w(t). The field shape is related to eigenfunctions $\varPsi (r)$, which are the solution of the eigenvalue problem. Eventually, the acoustic pressure distribution p(r, t) is defined by a sum over a set of a space’s eigenfunctions $\varPsi (r)$ and time components w(t). Each w(t) contains the source factor Q, which is an integral of the strength source multiplied by the related eigenfunction values in points where the source is located. Thereafter, if the integration is calculated over a region, where the value of the eigenfunction $\varPsi _m$ is zero, the source factor Q is zero as well. Considering the above, the aim of this research is to obtain the space where as many points as possible exist, where eigenfunction $\varPsi _m$ values are equal to zero, for as many eigenfrequency $\omega _m$ as possible. In order to find the specific configuration of the topology, an optimization problem is formulated. The eigenfunctions are considered as design variables. A minimum of multiobjective functions, based on eigenvalue problem solutions is searched. As the result of the optimization, the shape of space and point locations is obtained. The specified point is a possible source location, which guarantees reduction of resonances in a particular frequency range.