Physics-based sound radiation estimation from multiple speakers by combined lumped parameter and reduced-order finite element modeling

Abstract

In this paper a virtual sensor is derived that can perform full field estimation of the sound field radiating from a number of loudspeakers in the time domain. The virtual sensor scheme contains two parts: Firstly, a time-domain physics-based numerical system level model that includes both the electro-mechanical behavior of the loudspeaker and the resulting wave propagation. Secondly, a Kalman filter in which the numerical model and a limited set of microphone measurements are combined to perform full field estimation. The loudspeaker dynamics are captured in a lumped parameter model and the resulting wave propagation is captured in a finite element model. To reduce the calculational complexity of the system level model, Krylov subspace based reduced order modeling is applied on the finite element model, leading to a significant reduction of the number of states. The system level model is brought to a discrete time state–space format, which allows for efficient implementation in the Kalman filter. The method is validated using two setups that use both vented and closed box loudspeakers, leading to improved estimation results as compared to pure simulation results, thus showing the merits of the virtual sensor. Furthermore, the importance of including the lumped parameter speaker model in the system level model is shown by comparing the estimation results with and without the inclusion of the lumped parameter model.