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- Published: 09 January 2013

Wave-based modeling typically centers on predicting acoustic wave propagation in complex environments like concert halls, without some of the simplifying assumptions of geometrical acoustics. Wave-acoustic methods capture all relevant physical effects of sound fields---like diffraction, interference, and complex boundary interactions---automatically. These methods maximize physical fidelity at the expense of substantially more computation.

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 m^{3}, 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.

The above video is a demonstration of software developed at Aalto that simulates acoustic propagation in complex geometries using finite difference methods. The simulations account for absorption at boundaries, and all physical wave effects of linear acoustics. A unique aspect of finite difference simulation is that the three-dimensional, time-domain pressure field can be visualized as it propagates through space. The video shows projections of pressure onto several movable planes as it propagates through a complex living room geometry.

### Research on wave-based modeling at Aalto

We have worked with FDTD modeling for a long time, and it has resulted in contributions in several research problems.

#### 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.

#### References

[1] D Botteldooren: Finite-Difference Time-Domain Simulation of Low-Frequency Room Acoustic Problems. *Journal of the Acoustical Society of America* 98(6):3302–3308, 1995. BibTeX / Info

@article{Botteldooren_1995, author = "Botteldooren, D", journal = "Journal of the Acoustical Society of America", number = 6, pages = "3302--3308", title = "Finite-Difference Time-Domain Simulation of Low-Frequency Room Acoustic Problems", volume = 98, year = 1995 }

[2] L Savioja, T Rinne and T Takala. Simulation of room acoustics with a 3-D finite difference mesh. In *Proc. Int. Computer Music Conf.*. 1994, 463–466. BibTeX / Info

@inproceedings{Savioja_1994, address = "Aarhus, Denmark", author = "Savioja, L and Rinne, T and Takala, T", booktitle = "Proc. Int. Computer Music Conf.", keywords = "Wave-based models", pages = "463--466", title = "Simulation of room acoustics with a 3-{D} finite difference mesh", year = 1994 }

[3] S Sakamoto, H Nagatomo, A Ushiyama and H Tachibana: Calculation of impulse responses and acoustic parameters in a hall by the finite-difference time-domain method. *Acoustical Science and Technology* 29(4), 2008. BibTeX / Info

@article{Sakamoto_2008, author = "Sakamoto, S and Nagatomo, H and Ushiyama, A and Tachibana, H", journal = "Acoustical Science and Technology", number = 4, title = "Calculation of impulse responses and acoustic parameters in a hall by the finite-difference time-domain method", volume = 29, year = 2008 }

[4] K Kowalczyk and M Walstijn: Room acoustics simulation using 3-D compact explicit FDTD schemes. *IEEE Trans. Audio, Speech, Language Process.* 19(1):34–46, 2011. BibTeX / Info

@article{Kowalczyk_2011, author = "Kowalczyk, K and van Walstijn, M", journal = "IEEE Trans. Audio, Speech, Language Process.", number = 1, pages = "34--46", title = "Room acoustics simulation using {3-D} compact explicit {FDTD} schemes", volume = 19, year = 2011 }

[5] Kane Yee: Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media. *IEEE Transactions on Antennas and Propagation* 14(3):302–307, 1966. URL, DOI BibTeX / Info

@article{Yee1966, abstract = "Maxwell's equations are replaced by a set of finite difference equations. It is shown that if one chooses the field points appropriately, the set of finite difference equations is applicable for a boundary condition involving perfectly conducting surfaces. An example is given of the scattering of an electromagnetic pulse by a perfectly conducting cylinder.", author = "Yee, Kane", doi = "10.1109/TAP.1966.1138693", issn = "0018-926X", journal = "IEEE Transactions on Antennas and Propagation", month = "", number = 3, pages = "302--307", title = "Numerical solution of initial boundary value problems involving maxwell's equations in isotropic media", url = "http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=1138693", volume = 14, year = 1966 }

[6] J Strikwerda. *Finite Difference Schemes and Partial Differential Equations*. Chapman & Hall, 1989. BibTeX / Info

@book{Strikwerda_1989, address = "New York, NY", author = "Strikwerda, J", publisher = "Chapman \& Hall", title = "Finite Difference Schemes and Partial Differential Equations", year = 1989 }

[7] S Van Duyne and J O Smith. The 2-D digital waveguide mesh. In *Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics*. 1993. BibTeX / Info

@inproceedings{duy93a, address = "New Paltz, NY", author = "{Van Duyne}, S and Smith, J O", booktitle = "Proc. IEEE Workshop on Applications of Signal Processing to Audio and Acoustics", title = "The {2-D} digital waveguide mesh", year = 1993 }

[8] L Savioja and V Välimäki. Improved Discrete-Time Modeling of Multi-Dimensional Wave Propagation Using the Interpolated Digital Waveguide Mesh. In *Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing* 1. 1997, 459–462. BibTeX / Info

@inproceedings{sav97, address = "Munich, Germany", author = {Savioja, L and V\"{a}lim\"{a}ki, V}, booktitle = "Proc. IEEE Int. Conf. Acoust., Speech, Signal Processing", keywords = "Wave-based models", pages = "459--462", title = "Improved Discrete-Time Modeling of Multi-Dimensional Wave Propagation Using the Interpolated Digital Waveguide Mesh", volume = 1, year = 1997 }

[9] A Härmä, M Karjalainen, L Savioja, V Välimäki, U K Laine and J Huopaniemi: Frequency-Warped Signal Processing for Audio Applications. *Journal of the Audio Engineering Society* 48(11):1011–1031, 2000. BibTeX / Info

@article{har00b, author = {H\"{a}rm\"{a}, A and Karjalainen, M and Savioja, L and V\"{a}lim\"{a}ki, V and Laine, U K and Huopaniemi, J}, journal = "Journal of the Audio Engineering Society", keywords = "Wave-based models", number = 11, pages = "1011--1031", title = "Frequency-Warped Signal Processing for Audio Applications", volume = 48, year = 2000 }

[10] L Savioja and V Välimäki: Reduction of the Dispersion Error in the Triangular Digital Waveguide Mesh Using Frequency Warping. *IEEE Signal Processing Letters* 6(3):58–60, 1999. BibTeX / Info

@article{sav99c, author = {Savioja, L and V\"{a}lim\"{a}ki, V}, journal = "IEEE Signal Processing Letters", keywords = "Wave-based models", number = 3, pages = "58--60", title = "Reduction of the Dispersion Error in the Triangular Digital Waveguide Mesh Using Frequency Warping", volume = 6, year = 1999 }

[11] L Savioja and V Välimäki: Reducing the dispersion error in the digital waveguide mesh using interpolation and frequency-warping techniques. *IEEE Transactions on Speech and Audio Processing* 8(2):184–194, 2000. BibTeX / Info

@article{Savioja_2000, author = {Savioja, L and V\"{a}lim\"{a}ki, V}, journal = "IEEE Transactions on Speech and Audio Processing", keywords = "Wave-based models", number = 2, pages = "184--194", title = "Reducing the dispersion error in the digital waveguide mesh using interpolation and frequency-warping techniques", volume = 8, year = 2000 }

[12] L Savioja and V Välimäki: Interpolated rectangular 3-D digital waveguide mesh algorithms with frequency warping. *IEEE Trans. on Speech and Audio Processing* 11(6):783–790, 2003. DOI BibTeX / Info

@article{Savioja_2003, author = {Savioja, L and V\"{a}lim\"{a}ki, V}, doi = "10.1109/TSA.2003.818028", journal = "IEEE Trans. on Speech and Audio Processing", keywords = "Wave-based models", number = 6, pages = "783--790", title = "Interpolated rectangular 3-{D} digital waveguide mesh algorithms with frequency warping", volume = 11, year = 2003 }

[13] L Savioja, V Välimäki and J O Smith: Audio Signal Processing Using Graphics Processing Units. *Journal of the Audio Engineering Society* 59(1/2):3–19, 2011. BibTeX / Info

@article{Savioja_2011, author = {Savioja, L and V\"{a}lim\"{a}ki, V and Smith, J O}, journal = "Journal of the Audio Engineering Society", keywords = "GPU compute", number = "1/2", pages = "3--19", title = "Audio Signal Processing Using Graphics Processing Units", volume = 59, year = 2011 }

[14] L Savioja. Real-time 3D finite-difference time-domain simulation of low- and mid-frequency room acoustics. In *Proc. Int. Conf. Digital Audio Effects*. 2010. BibTeX / Info

@inproceedings{Savioja_2010b, address = "Graz, Austria", author = "Savioja, L", booktitle = "Proc. Int. Conf. Digital Audio Effects", keywords = "Wave-based models", title = "Real-time {3D} finite-difference time-domain simulation of low- and mid-frequency room acoustics", year = 2010 }

[15] A Southern and L Savioja. Spatial High Frequency Extrapolation Method for Room Acoustic Auralization. In *Proc. Int. Conf. on Digital Audio Effects (DAFx-12)*. 2012, 145–149. BibTeX / Info

@inproceedings{Southern_2012, address = "York, UK", author = "Southern, A and Savioja, L", booktitle = "Proc. Int. Conf. on Digital Audio Effects (DAFx-12)", keywords = "Auralization,Wave-based models", pages = "145--149", title = "Spatial High Frequency Extrapolation Method for Room Acoustic Auralization", year = 2012 }