Introduction to adaptive rectangular decomposition_adjustable rectangular decomposition for super lar-优快云博客

介绍了一种基于波数的数值声学方法——自适应矩形分解(ARD)，该方法通过将复杂空间分解为多个矩形区域并独立更新每个区域内的声场来提高内存和计算效率。与FDTD方法相比，ARD在保持精度的同时显著减少了采样数量。

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Introduction to adaptive rectangular decomposition

Knowledge dependencies: Wave equation, Finite-difference time-domain method, Discrete cosine transform

ARD (Adaptive Rectangular Decomposition) is a wave-based numerical acoustic method to solve the propagation of sound in a room proposed by Raghuvanshi et al. It decomposes a complex space into several rectangular partitions, and update sound field based on wave equation inside each partitions independently. This method gain a boost on both memory efficiency and computation efficiency comparing to FDTD by avoiding oversample in space.

FDTD always requires more than 10 samples per wavelength, but the pattern of wave propagation is simpler in rectangular space, the number of samples can be reduced to 2 per wavelength by manipulating a mathmetical trick, which makes ARD efficient and accurate.

The sound field can be represented as a superposition of a series of consine function in any rectangular room with rigid boundaries. At time $t$, the sound pressure values can be represented by modes as:

p (x, y, z, t) = \sum i = (i x, i y, i z) m i (t) Φ i (x, y, z)

$p(x,y,z,t)= \sum_{i=(i_x,i_y,i_z)}m_i(t)\Phi_i(x,y,z)$

Φ i (x, y, z) = cos (π i x l x x) cos (π i z l z z) cos (π i z l z z)

$\Phi_i(x,y,z)=\cos(\frac{\pi i_x}{l_x}x) \cos(\frac{\pi i_z}{l_z}z) \cos(\frac{\pi i_z}{l_z}z)$ $i$ is the index of samples and $l$ is the actual size of the space. This is essentially a discrete consine transform: $m(t)=DCT(p(t))$. This inspires us that we can update the sound field by updating modes $m(t)$ instead of updating pressure values directly which can avoid the error introducing by applying finite approximation to pressure values along space.

Now that the knowledge of sound behave in a rigid rectangular room, ARD assumes that each rectangular partition has rigid boundaries. At each time step, ARD updates the pressure values inside each partition independently by updating the modes, and compute the force terms on the boundary to compensate the rigid boundary assumption.

Reference:

Raghuvanshi N, Narain R, Lin M C. Efficient and accurate sound propagation using adaptive rectangular decomposition[J]. IEEE Transactions on Visualization and Computer Graphics, 2009, 15(5): 789-801.
Kuttruff H. Room acoustics[M]. Crc Press, 2016.