Computational acoustics studies the propagation of sound through a medium and may be roughly classified into Geometric Acoustics and Numerical Acoustics depending on how wave propagation is modeled. There has also been effort to combine the two techniques.
2.2.1 Numerical Acoustic Techniques
Accurate, numerical acoustic simulations typically solve the acoustic wave equation using numerical methods. The Finite Difference Time Domain (FDTD) method was originally proposed to model electromagnetic waves (Yee, 1966; Taflove and Hagness, 2005). It discretizes space as a uniform grid and solves for the field values at each cell for discrete time steps. It has been an adopted to room acoustics problems (Botteldooren, 1994, 1995) and has recently been applied to medium sized 3D scenes (Sakamoto et al., 2002, 2004, 2006). The Finite Element Method (FEM) (Zienkiewicz et al., 2006; Thompson, 2006) and the Boundary Element Method (BEM) (Cheng and Cheng, 2005; Gumerov and Duraiswami, 2009) discretize the scene’s volume and surface into elements respectively. They are usually employed to solve the steady-state frequency domain response, with FEM applied mainly to interior and BEM to exterior acoustic problems (Kleiner et al., 1993). Digital Waveguide Mesh approaches (Van Duyne and Smith, 1993) roots in musical synthesis and use discrete waveguide elements to propagate acoustic waves along a single dimension (Savioja, 1999; Karjalainen and Erkut, 2004; Murphy et al., 2007). Recently Raghuvanshi et al. proposed a method based on adaptive rectangular decomposition (2009a). It achieves high accuracy with a coarse spatial discretization.
These techniques, however, require the volume or boundary of the scene to be discretized at least twice the Nyquist frequency, and their time and space complexity increases as a third or fourth power of frequencies. Hence, these techniques often require many hours of simulation time and gigabytes of storage to model low frequencies in large scenes with static sources, and they scale as the third or fourth power of frequency. Despite recent advances, they remain impractical for many real-time applications.
Equivalent source method, also called the Method of Fundamental solutions (Ochmann, 1995, 1999), expresses the solution fields of the wave equation in terms of a linear combination of points sources of various order (monopoles, dipoles, etc). The main idea behind this technique is to choose the positions and amplitudes of these elementary sources such that the boundary condition is satisfied. Thus, the resulting solution satisfies the wave equation. Recently, Mehra et al. (2013) proposed a novel sound propagation technique for large outdoor scenes based on equivalent sources. James et al. (2006b) solved a relatedsound radiationproblem, using equivalent sources to represent the radiation field generated by a vibrating object.
2.2.2 Geometric Acoustic Techniques
Most acoustics simulation software and commercial systems are based on geometric tech- niques (Funkhouser et al., 1998; Vorlander, 1989) that assume sound travels along linear rays (Funkhouser et al., 2004). These methods are often based on stochastic ray tracing (Vorlander, 1989) or image sources (Borish, 1984). They frequently take advantage of recent advances in CPU- and/or GPU- based ray tracing techniques (Taylor et al., 2009, 2012) or frustum tracing (Chandak et al., 2008; Lauterbach et al., 2007) to efficiently approximate sound propagation in complex, dynamic scenes. The simplified assumption of rays limits these methods to accurately capture specular and diffuse reflections only at high frequencies. Diffraction is typically modeled by identifying individual diffract- ing edges (Svensson et al., 1999; Tsingos et al., 2001). These ray-based techniques can interactively model early reflections and first order edge-diffraction (Taylor et al., 2012); however, they cannot interactively model the reverberation of the impulse response explicitly, since that would require high-order reflections and wave effects such as scattering, interference, and diffraction. Hence, many commercial systems approximate reverberation using the parameters of simple statistical models (Eyring, 1930).
While ray-tracing has been successfully used in many interactive acoustics systems (Lentz et al., 2007), the number of rays traced has to be limited for scenes with moving listeners in order to maintain real-time performance. As the worst-case complexity of image source methods scales exponentially with the number of polygons in the scene, some interactive systems often group the polygons to simplify the scene representation (Alarcao et al., 2010; Joslin and Magnenat-Thalmann, 2003).
2.2.3 Hybrid Techniques
Several methods for combining geometric and numerical acoustic techniques have been proposed. One line of work is based onfrequency decomposition: dividing the frequencies to be modeled into low and high frequencies. Low frequencies are modeled by numerical acoustic techniques, and high frequencies are treated by geometric methods, including the finite difference time domain method (FDTD) (Southern et al., 2011; Lokki et al., 2011), the digital waveguide mesh method (DWM) (Murphy et al., 2008), and the finite element method (FEM) (Granier et al., 1996; Aretz,
2012). However, these methods use numerical methods at lower frequencies over the entire domain. As a result, they are limited to offline applications and may not scale to very large scenes.
Another method of hybridization is based onspatial decomposition. The entire simulation domain is decomposed to different regions: near-object regions are handled by numerical acoustic techniques to simulate wave effects, while far-field regions are handled by geometric acoustic techniques. Hampel et al. (2008) combine the boundary element method (BEM) and geometric acoustics using a spatial decomposition. Their method provides a one-way coupling from BEM to ray tracing, converting pressures in the near-object region (computed by BEM) to rays that enter the far-field region containing the listener. In electromagnetic wave propagation, Wang et al. (2000) propose a hybrid technique combining ray tracing and FDTD. Their technique is also based on a one-way coupling, where rays are traced in the far-field region and collected at the boundaries of the near-object regions. The pressures are then evaluated and serve as the boundary condition for the FDTD method. These one-way coupling methods do not allow rays to enter and exit the near-object regions of an object, and therefore acoustic effects of that object will not be propagated to the far-field regions. Barbone et al. (1998) propose a two-way coupling that combines the acoustic field generated using ray-tracing and FEM. Jean et al. (2008) present a hybrid BEM/beam tracing approach to compute the radiation of tyre noise. However, these methods do not describe how multiple entrance of rays into near-object regions of different objects is handled, which is crucial when simulating interaction between multiple objects.
2.2.4 Acoustic Kernel-Based Interactive Techniques
There has been work in enabling interactive auralization for acoustic simulations through precomputation. At a high level, these techniques tend to precompute an acoustic kernel, which is used at runtime for interactive propagation in static environments. Raghuvanshi et al.(2010) precompute acoustic responses on a sampled spatial grid using a numerical solver. They then encode perceptually salient information to perform interactive sound rendering. Mehra et al. (2013) proposed an interactive sound propagation technique for large outdoor scenes based on equivalent sources. Other techniques use geometric methods to precompute high-order reflections or reverberation (Tsingos, 2009; Antani et al., 2012) and compactly store the results for interactive sound propagation at runtime. Our method
can be integrated into any of these systems as an acoustic kernel that can efficiently capture wave effects in a large scene.
CHAPTER 3: SOUND SYNTHESIS FROM FLUID SIMULATION
In this chapter, I discuss my work on performing sound synthesis from fluid simulation. The rest of this chapter is organized as follows – in the next section I describe the physical principles of liquid sound. After that, I describe how liquid sound can be simulated by integrating various kinds of fluid simulators. Following this, I discuss the implementation details and the results obtained with my approach. Finally I conclude with a summary of my contributions and a discussion of limitations of my approach and possible directions of future work.