A Time-Shifting Algorithm for Alleviating Convergence Difficulties at Interior Acoustic Resonance Frequencies

Jui Hsiang Kao

doi:10.3390/app11062701

Abstract

This paper proposes a time-shifting boundary element method in the time domain to calculate the radiating pressures of an arbitrary object pulsating at eigenfrequencies of the interior (i.e., interior resonance frequencies). In this paper, the frequency shifting is time-step-dependent and could be viewed as an iterative, or relaxation, technique for the solution of the problem. The proposed method avoids numerical problems due to the internal resonance frequency by initializing the iteration with each scaled frequency. The scaled frequency is approximately equal to the true frequency at the last iterating time step. A sphere pulsating at the eigenfrequency in an infinite acoustic domain was calculated first; the result was compared with the analytical solution, and they were in good agreement. Moreover, two arbitrary-shaped radiators were taken as study cases to predict the radiating pressures at the interior resonance frequencies, and robustly convergent results were obtained. Finally, the accuracy of the proposed method was tested using a problem with a known solution. A point source was placed inside the object to compute the surface velocities; the computed surface pressures were identical to the pressures computed using the point source.

Highlights

Convergence Difficulties at InteriorThe radiating waves used for some specific purposes, such as remote sensing and military defense, are a topic worthy of research
The results show that the numerical instability at the internal resonance frequency is improved by shifting the inception of the first time step
It is noted that when KR is equal to π, the frequency 171.5 Hz is the first internal eigenfrequency

Summary

Introduction

Convergence Difficulties at InteriorThe radiating waves used for some specific purposes, such as remote sensing and military defense, are a topic worthy of research. The boundary element method (BEM) is widely used for solving problems involving radiating waves; incorrect results due to internal resonance frequencies constitute a problem to be solved. The well-known combined Helmholtz integral equation formulation (CHIEF) introduced by Schenck [1] has been widely applied to overcome non-unique solutions at eigenfrequencies in the frequency domain. Burton and Miller [2] solved the non-uniqueness problem at interior resonance frequencies by deriving a second integral equation (i.e., the hypersingular BIE) and combining it with the original boundary integral equation. According to the convolution quadrature rules, Lubich [3,4] proposed a numerical method to solve the integral equations.

Objectives

Results

Conclusion