Common methods in acoustics include the finite element method (FEM), the boundary element method (BEM), and finite difference (FD) methods. Some methods seek solutions to the stationary problem in the frequency domain, which is useful for describing modal distributions in low frequencies. However, many interesting and useful properties of sound fields are captured or encoded in the transient, time-domain response. 

In room acoustics and more generally linear acoustics, the computational challenges are not the same as many other disciplines.  The physics of linear wave propagation is relatively straightforward, but the typical size of a problem and necessary boundary conditions present unique challenges.  For instance, the volume of a concert hall can be on the order of 10,000 m³, while the wavelengths of sound in the audible range are on the order of centimeters.  Capturing the wave physics on these scales requires massively parallel computing. Further inhibiting efficient computation, realistic geometries and boundary interactions also require careful design.  Ever-increasing computing power and demand for physics-based simulation make wave-based modeling a continually active area of research.

Finite-Difference Time-Domain Technique

The approach that is currently most promising for wide-band room acoustic simulation is the finite-difference time-domain (FDTD) technique. The first papers to discuss 3D, wave-based simulation in rooms, using FDTD date back to 1994 [1, 2].  Ever since, there has been a division between two separate approaches, and both of them are still valid and in use (see e.g., [3, 4]). The approach presented by Botteldooren [1] is based on the linearized Euler equations and it utilizes two variables to present the sound field, most typically sound pressure and volume velocity. The technique originates from the electro-magnetic literature where Yee presented this technique for simulation of electro-magnetic fields already in 1966 [5]. This approach has come to be known as the standard leapfrog scheme.

The other option is to start from the second-order, scalar wave equation, which requires only one field variable, such as sound pressure [2, 4] or velocity potential. This saves both in memory consumption and computation time, while still providing the same accuracy as the standard leapfrog scheme. The video below is computed with this technique.

A major problem in all the wave-based techniques is their heavy memory consumption at higher frequencies. Finite difference techniques require discretization in both time and space.  In particular, the room under study must be divided into small volumetric elements, called nodes, and the simulation progresses in discrete time steps in which all the nodes are updated according to given equations. The relation of the temporal and spatial discretization step sizes is dictated by the Courant condition [6].

Finite difference solvers can be further divided into explicit and implicit schemes. In the explicit case, the new state of a node can be computed based on its own history and on the past values of the neighboring nodes. This means that it is possible to perform all the updates in parallel thus suggesting it to be a good fit for parallel computation. In the implicit FDTD schemes, the new values of the nodes depend on the new values of their nearest neighbor nodes. Although there are ways to make this efficient, this approach is currently not favored as a general simulation tool. The main advantage of the implicit schemes is that they have less strict conditions on stability and they can be more accurate than explicit schemes, but due to the computational requirements the explicit ones are favored at the moment.

One of the first 3D FDTD implementations for room acoustics

The FDTD simulation for room acoustics has gained a great deal of popularity in recent years, but our earliest papers in the topic date back already to 1994 [2]. That paper was insipred by the 'Digital Waveguide Mesh' approach originally targeting physical modeling of musical instruments [7]. Although the technique is derived from signal processing viewpoint, it is equivalent to an FDTD in which the relation of the temporal sampling to the spatial sampling is set to the Courant limit [6].

Interpolated FDTD schemes

A fundamental problem in the FDTD simulations is dispersion that depends both on the frequency and on the propagation direction. In a rectangular mesh this is seen such that high frequencies get delayed in axial propagation directions in relation to the underlying grid whereas in diagonal directions all the frequencies propagate at the same constant speed. To overcome this limitation of direction-dependent dispersion, we introduced the interpolated digital waveguide mesh structure [8]. The proposed technique results in nearly isotropic propagation. This approach was later developed further by Kowalczyk et al. [4].

Frequency warping with FDTD

While the interpolated mesh improves over the original standard rectilinear scheme, it still has frequency-dependent dispersion error. Frequency warping [9] is a technique that can be used to post-process the obtained results such that this dispersion error is minimal [10, 11]. With this approach the usable frequency band in FDTD simulations can be remarkably increased [12].

Parallelization of a FDTD solver for real-time room acoustic simulation

FDTD is a technique that is easy to parallelize. Introduction of modern GPUs has had a remarkable impact on practical parallel computation in several areas including signal processing and acoustics [13]. Current GPUs are massively parallel processors that have made room acoustic simulation with FDTD a very attractive option. For small rooms it is possible to have even real-time performance for low- and mid-frequency simulation [14].

Frequency band extrapolation

As the computational load in FDTD simulation grows with respect to the fourth power of the frequency, simulation of high frequencies is still a major challenge for current computers. For this reason, we have developed an extrapolation technique that enables full-bandwidth auralization [15]. It is based on analysis of responses such that each reflection is detected from a spatial response, it's direction is estimated, and then extrapolated for full bandwidth. The process included simulation of air absorption that has a big role in sound attenuation at higher frequencies.