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Svensson, 2015). Acoustic radiance transfer is a further development of basic radiosity incorporating arbitrary reflection characteristics into the model, lifting the original limitation of only ideally diffuse reflections (Siltanen et al., 2007). The result of these simulations is energy RIRs at surfaces that can all be precomputed. In the interactive simulation, the responses to a given receiver are gathered and processed for auralization similarly as with BEM.
Modeling Based on Transport Theory
Sound particles propagating in enclosed spaces can be described by the transport equation, with field quantities being sound energy density and energy flux. The particles propagate along straight lines and strike partially absorptive walls or objects that also partially scatter the incoming particles. Therefore, the transport-equation modeling is still classified under geometrical acoustics. In room acoustics, Jing and Xiang (2010) seem to be the first to have solved this transport equation for simulating a long space and also to experimentally validate their solutions. They demonstrate both mathematically and experimentally that the transport- equation model is capable of incorporating arbitrary wall properties, including absorption, specular, and diffuse reflection (see Table 1).
Navarro et al. (2010) independently explained exactly the same transport theory in a comprehensible manner, calling it the radiactive transfer model. The transport equation can be simplified asymptotically to the diffusion equation. Room-acoustic modeling using the diffusion equation was first reported by Valeau et al. (2006). Jing and Xiang (2008) proposed a rigorous boundary condition, making the diffusion equation applicable in broader room-acoustic conditions. The recent decade has witnessed a stream of room-acoustic applications using the diffusion equation, partially reflected also in the recent special issue (Savioja and Xiang, 2019).
One of attractive features of the diffusion equation lies in either finite-element or finite-difference implementation, yet the mean-free path length of the space under consideration, rather than wavelengths, primarily dictates the mashing condition in enclosures of proportionated dimension. The simulation can, therefore, be implemented extremely efficiently. Another important feature is that room simulations based on the diffusion equation allow for outputting sound energy flux without
extra computational expense due to its deep root in Fick’s law (Jing and Xiang, 2008).
Hybrid Models
From the perceptual viewpoint of a listener, the direct sound along with the early reflections, as illustrated in Figure 3, are the most important ones, and so the modeling of the early reflections deserve more attention than detailed modeling of the late reverberation. Some modeling techniques are at their best in accurately modeling accurately this early part, whereas some other techniques can efficiently model the later part. This suggests that a hybrid model combining different techniques can provide an optimal solution. In practice, it is advantageous to use a hybrid technique in the time domain. The technique separately calculates the direct sound and the early reflections by the image-source method, even in real time, and the late part is gathered from precomputed responses or exploiting its random nature by artificial approximation (Xiang et al. 2019), enabling real-time auralization.
Similar division also takes place in the frequency domain. The wave-based models excel at the low frequencies, whereas the geometrical-acoustic models are better
suited for higher frequencies. Basically, the wave- based models provide an accurate solution, but their computational load gets excessive at higher frequencies, and for this reason, a somewhat less accurate but more efficient model is better suited for that range.
One additional phenomenon worth consideration is air absorption. In practice, air acts as a low-pass filter, causing higher frequencies to be attenuated much more than the lower range as a function of propagation distance. Modeling this by wave-based solvers increases their computational load, and it is often advantageous to switch to geometrical acoustic models on frequencies in which the air absorption is notable. In those energy- based models, air absorption is straightforward to implement as postprocessing.
Concluding Remarks
Initially conducted in the 1960s, room-acoustic simulations have been progressed remarkably. New modeling paradigms have emerged, and the old ones have been developed to be more efficient. They are capable of highly complex geometries and boundaries.
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