A simple absorbing layer implementation for transmission line matrix modeling

Abstract

An absorbing layer formulation for transmission line matrix modeling is proposed. The approach consists in attenuating the incident pulse propagating toward the absorbing layer only, using an attenuation factor which gradually decreases as the sound wave propagates along the absorbing medium. The formulation of the damping function followed by the attenuation factor along the absorbing layer is depicted and discussed. The efficiency of the present formulation is validated by comparison with another absorbing layer model and virtual boundary conditions proposed in the literature. Numerical simulations are also given in order to evaluate the effects of both the attenuation factor and the depth on the absorbing layer efficiency. As expected, results are consistent with absorbing layer implementation in other numerical methods; firstly, the attenuation at the entrance of the absorbing layer must be gentle, and secondly the efficiency increases with the layer depth. Lastly, it is shown that the unwanted reflection seems to vanish over the time when increasing the layer depth, meaning that reflections continuously occur within the absorbing layer and not on the geometrical limits of the absorbing layer. Although the approach is dedicated to outdoor sound propagation modeling (only an example on urban acoustics application is given), the proposed formulation of absorbing layers can be applied in other domains of acoustics. However, its application in shielded areas should be avoided because unwanted reflections due to an insufficient attenuation can be significant in comparison with the ambient noise in such quiet environments.

Introduction

The growing increase of noise annoyance in cities and their impact on resident health make major interest in the knowledge of sound propagation mechanisms in urban areas. Many numerical approaches can be employed for acoustical modeling. Most of them are limited considering moving and time-dependent sound sources and time-varying parameters like micrometeorological conditions. Thus, time-domain methods have attracted a lot of attention during the last decade [1], [2], due to the increase of computational resources and, overall, to their ability to model more realistic propagation conditions. Among these, the Transmission Line Matrix (TLM) method consists in an inherent discrete representation of wave propagation.

Initially developed by Johns and Beurle in electromagnetism [3], the TLM method was extended to acoustics by Saleh and Blanchfield in the beginning of the 1990s [4]. Recently, a few applications for outdoor sound propagation was proposed [5], [6], in particular for sound propagation over a porous ground [7], [8], [9]. Tsuchiya showed that the frequency-dependency of the atmospheric absorption can be taken into account including digital filters into the TLM numerical scheme [10]. Dutilleux introduced wind-induced sound speed gradients through the definition of the effective sound speed, which allows to account for micrometeorological effects [11].

Nevertheless, difficulties are encountered when modeling free-space propagation due to the lack of efficient absorbing boundary conditions. Thus, the computational domain is usually enlarged for outdoor applications in order to avoid unwanted reflections. A powerful direct TLM implementation of Bérenger's perfectly matched layer (PML) [12] was proposed by Dubard and Pompéi within the electromagnetism context [13], [14]. PMLs were applied to most of the numerical methods in acoustics [15], but no similar approach is available for TLM modeling of sound propagation yet. Instead, absorbing conditions for TLM models in acoustics are described by means of absorbing boundaries [16], [17], [8] and absorbing layers [18], [8], but their efficiency as well as their implementation is still not satisfactory.

As an alternative approach, a simple implementation of the absorbing layers concept, inspired by De Cogan et al. [18], is proposed in this paper, based on an intuitive formulation of absorbing layers. No rigorous PML implementation is formulated because the particle velocity would then be required, which can be avoided for TLM modeling in acoustics. Firstly, the implementation of absorbing conditions that are proposed in the literature is described and an alternative formulation is given (Section 2). The efficiency of the proposed formulation is then evaluated in Section 3 through a quantitative comparison with other artificial absorbing conditions, for several parameters of the absorbing layers and for several sound waves incident angles on the computational domain limit. The efficiency of the proposed formulation for long duration simulations and for realistic environment modeling is also discussed.