Accelerating numerical methods for nonlinear acoustics using nested meshes
Accelerating numerical methods for nonlinear acoustics using nested meshes
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发表时间：
2020-11
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通讯作者：
Samuel P. Groth;P. Gélat;S. R. Haqshenas;N. Saffari;E. V. Wout;T. Betcke;G. N. Wells
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作者：
Samuel P. Groth;P. Gélat;S. R. Haqshenas;N. Saffari;E. V. Wout;T. Betcke;G. N. Wells
非线性超声的数值模拟对于腹部高强度聚焦超声（HIFU）治疗的治疗计划非常重要。然而，大域尺寸和焦点处高次谐波的产生使得这些问题的计算要求极高。数值方法通常采用足够细的均匀网格来解决问题中存在的最高谐波，从而产生非常大的自由度。本文提出了一种更有效的策略，其中每个谐波在单独的网格上进行近似，其大小与谐波的波长成正比。解析较小波长所需的分辨率的增加通过域尺寸的减小来平衡。由于焦点附近高次谐波日益局部化的性质，这种嵌套网格划分是可行的。对均匀介质中的 HIFU 换能器进行数值实验，以确定准确表示谐波所需的单独网格的大小。特别是，提出了一种快速体积势（VP）方法，并用于在修改计算域大小时执行收敛实验。 VP 方法允许通过评估域上的积分来计算每个谐波。使用中点规则对该积分进行离散化可以使用 FFT 快速执行计算。结果表明，通过嵌套网格划分，可以实现内存消耗和计算时间至少减少一个数量级。
The numerical simulation of nonlinear ultrasound is important in the treatment planning for high-intensity focused ultrasound (HIFU) therapies in the abdomen. However, the large domain sizes and generation of higher harmonics at the focus make these problems extremely computationally demanding. Numerical methods typically employ a uniform mesh fine enough the resolve the highest harmonic present in the problem, leading to a very large number of degrees of freedom. This paper proposes a more efficient strategy in which each harmonic is approximated on a separate mesh, the size of which is proportional to wavelength of the harmonic. The increase in resolution required to resolve a smaller wavelength is balanced by a reduction in the domain size. This nested meshing is feasible owing to the increasingly localised nature of higher harmonics near the focus. Numerical experiments are performed for HIFU transducers in homogeneous media in order to determine the size of the separate meshes required to accurately represent the harmonics. In particular, a fast volume potential (VP) approach is proposed and employed to perform convergence experiments as the computation domain size is modified. The VP approach allows each harmonic to be computed via the evaluation of an integral over the domain. Discretising this integral using the midpoint rule allows the computations to be performed rapidly with the FFT. It is shown that at least an order of magnitude reduction in memory consumption and computation time can be achieved with nested meshing.