Treeby Bradley E, Cox B T
Department of Medical Physics and Biomedical Engineering, University College London, Gower Street, London WC1E 6BT, United Kingdom.
J Acoust Soc Am. 2014 Oct;136(4):1499-510. doi: 10.1121/1.4894790.
The absorption of compressional and shear waves in many viscoelastic solids has been experimentally shown to follow a frequency power law. It is now well established that this type of loss behavior can be modeled using fractional derivatives. However, previous fractional constitutive equations for viscoelastic media are based on temporal fractional derivatives. These operators are non-local in time, which makes them difficult to compute in a memory efficient manner. Here, a fractional Kelvin-Voigt model is derived based on the fractional Laplacian. This is obtained by splitting the particle velocity into compressional and shear components using a dyadic wavenumber tensor. This allows the temporal fractional derivatives in the Kelvin-Voigt model to be replaced with spatial fractional derivatives using a lossless dispersion relation with the appropriate compressional or shear wave speed. The model is discretized using the Fourier collocation spectral method, which allows the fractional operators to be efficiently computed. The field splitting also allows the use of a k-space corrected finite difference scheme for time integration to minimize numerical dispersion. The absorption and dispersion behavior of the fractional Laplacian model is analyzed for both high and low loss materials. The accuracy and utility of the model is then demonstrated through several numerical experiments, including the transmission of focused ultrasound waves through the skull.
在许多粘弹性固体中，压缩波和剪切波的吸收已通过实验表明遵循频率幂律。现在已经确定，这种类型的损耗行为可以使用分数阶导数进行建模。然而，以前用于粘弹性介质的分数阶本构方程是基于时间分数阶导数的。这些算子在时间上是非局部的，这使得它们难以以内存高效的方式进行计算。在此，基于分数阶拉普拉斯算子推导了一个分数阶开尔文 - 沃伊特模型。这是通过使用并矢波数张量将粒子速度分解为压缩分量和剪切分量来实现的。这使得开尔文 - 沃伊特模型中的时间分数阶导数可以使用具有适当压缩波或剪切波速度的无损色散关系被空间分数阶导数所取代。该模型使用傅里叶配置谱方法进行离散化，这使得分数阶算子能够被高效计算。场分裂还允许使用k空间校正的有限差分格式进行时间积分，以最小化数值色散。针对高损耗和低损耗材料分析了分数阶拉普拉斯模型的吸收和色散行为。然后通过几个数值实验证明了该模型的准确性和实用性，包括聚焦超声波通过颅骨的传输。
