Finite difference and finite volume methods for wave-based modelling of room acoustics
Finite difference and finite volume methods for wave-based modelling of room acoustics
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用于基于波的室内声学建模的有限差分法和有限体积法
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发表时间：
2016
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通讯作者：
B. Hamilton
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作者：
B. Hamilton
基于波的声音传播模型可用于预测和合成声音，就像在室内声学环境中自然听到的那样。由于聆听环境（例如礼堂和音乐厅）的巨大规模以及音频速率所需的时间分辨率（将频率解析到人类听力的极限），使用传统的基于时间步进网格的方法对此类模型进行数值模拟可能是一个昂贵的过程。有限差分方法为此类模拟提供了一个简单的起点，但众所周知，它们会遭受近似误差，可能需要昂贵的网格细化才能达到足够的精度水平。因此，大量研究致力于设计高精度同时保持计算效率的有限差分方法。由于室内声学模型需要复杂的边界条件来模拟非平凡几何形状上与频率相关的壁阻抗，因此设计和使用精确的有限差分方案的问题变得更加复杂。一段时间以来，以数值稳定的方式实现此类边界条件一直是一个挑战。过去已经制定了有限差分房间声学模拟的稳定边界条件，但通常它们仅适用于模拟琐碎的几何形状（例如，理想化的鞋盒大厅）。最近，有限体积方法已被证明是解决非平凡几何形状上的复杂边界条件问题的可行解决方案，并且它们还允许使用能量方法进行数值稳定性分析。有限体积方法自然适合完全非结构化网格，并且可以简化为有限差分方法中通常使用的网格类型。这允许室内声学仿真模型平衡空气中波传播的有限差分方法的简单性与复杂边界建模的有限体积方法的细节。本论文探索了这两种不同但相关的基于波的室内声学模拟方法。这项研究的首要主题是准确性、计算效率和数值稳定性之间的平衡。推导并比较了二维和三维空间维度的高阶优化方案，以期找到准确有效的有限差分方案。使用频域分析以及尽可能的能量技术来分析数值稳定性，从而允许适合室内声学建模的稳定且与频率相关的边界条件。在此过程中，研究了非笛卡尔网格的使用，探索了某些有限差分和有限体积方案之间的几何关系，并考虑了与边界阶梯效应相关的一些问题。此外，空气中的吸声模型也被纳入这些数值方案中，使用适合室内声学场景的物理参数。
Wave-based models of sound propagation can be used to predict and synthesize sounds as they would be heard naturally in room acoustic environments. The numerical simulation of such models with traditional time-stepping grid-based methods can be an expensive process, due to the sheer size of listening environments (e.g., auditoriums and concert halls) and due to the temporal resolution required by audio rates that resolve frequencies up to the limit of human hearing. Finite difference methods comprise a simple starting point for such simulations, but they are known to suffer from approximation errors that may necessitate expensive grid refinements in order to achieve sufficient levels of accuracy. As such, a significant amount of research has gone into designing finite difference methods that are highly accurate while remaining computationally efficient. The problem of designing and using accurate finite difference schemes is compounded by the fact that room acoustics models require complex boundary conditions to model frequency-dependent wall impedances over non-trivial geometries. The implementation of such boundary conditions in a numerically stable manner has been a challenge for some time. Stable boundary conditions for finite difference room acoustics simulations have been formulated in the past, but generally they have only been useful in modelling trivial geometries (e.g., idealised shoebox halls). Finite volume methods have recently been shown to be a viable solution to the problem of complex boundary conditions over non-trivial geometries, and they also allow for the use of energy methods for numerical stability analyses. Finite volume methods lend themselves naturally to fully unstructured grids and they can simplify to the types of grids typically used in finite difference methods. This allows for room acoustics simulation models that balance the simplicity of finite difference methods for wave propagation in air with the detail of finite volume methods for the modelling of complex boundaries. This thesis is an exploration of these two distinct, yet related, approaches to wave-based room acoustic simulations. The overarching theme in this investigation is the balance between accuracy, computational efficiency, and numerical stability. Higher-order and optimised schemes in two and three spatial dimensions are derived and compared, towards the goal of finding accurate and efficient finite difference schemes. Numerical stability is analysed using frequency-domain analyses, as well as energy techniques whenever possible, allowing for stable and frequency-dependent boundary conditions appropriate for room acoustics modelling. Along the way, the use of non-Cartesian grids is investigated, geometric relationships between certain finite difference and finite volume schemes are explored, and some problems associated to staircasing effects at boundaries are considered. Also, models of sound absorption in air are incorporated into these numerical schemes, using physical parameters that are appropriate for room acoustic scenarios.