The numerical simulation of weakly nonlinear ultrasound is important in treatment planning for focused ultrasound (FUS) therapies. 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 to 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 the 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 FUS transducers in homogeneous media to determine the size of the meshes required to accurately represent the harmonics. In particular, a fast volume potential approach is proposed and employed to perform convergence experiments as the computation domain size is modified. This 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. Finally, it is demonstrated how to generalise this approach to inhomogeneous propagation domains.

中文翻译：

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使用嵌套网格加速弱非线性聚焦超声的频域数值方法

弱非线性超声的数值模拟在聚焦超声 (FUS) 治疗的治疗计划中很重要。然而，大域尺寸和焦点处高次谐波的产生使得这些问题对计算的要求极高。数值方法通常采用足够精细的均匀网格来解决问题中存在的最高谐波，从而产生大量的自由度。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近似，网格的大小与谐波的波长成正比。解析较小波长所需的分辨率增加与域尺寸的减小相平衡。由于焦点附近的高次谐波越来越局部化，这种嵌套网格划分是可行的。对均匀介质中的 FUS 换能器进行了数值实验，以确定准确表示谐波所需的网格大小。特别是一个快速随着计算域大小的修改，提出并采用体积势方法来执行收敛实验。这种方法允许通过对域上的积分进行评估来计算每个谐波。使用中点规则离散该积分允许使用 FFT 快速执行计算。结果表明，嵌套网格划分可以使内存消耗和计算时间至少减少一个数量级。最后，演示了如何将这种方法推广到非均匀传播域。