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Accelerating numerical methods for nonlinear acoustics using nested meshes
arXiv - CS - Computational Engineering, Finance, and Science Pub Date : 2020-11-05 , DOI: arxiv-2011.03009
Samuel P. Groth, Pierre G\'elat, Seyyed R. Haqshenas, Nader Saffari, Elwin van 't Wout, Timo Betcke, Garth N. Wells

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 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 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.

中文翻译:

使用嵌套网格加速非线性声学的数值方法

非线性超声的数值模拟在腹部高强度聚焦超声 (HIFU) 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 HIFU 换能器进行了数值实验,以确定准确表示谐波所需的单独网格的大小。特别是,随着计算域大小的修改,提出并采用快速体积势 (VP) 方法来执行收敛实验。VP 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。对均匀介质中的 HIFU 换能器进行了数值实验,以确定准确表示谐波所需的单独网格的大小。特别是,随着计算域大小的修改,提出并采用快速体积势 (VP) 方法来执行收敛实验。VP 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。对均匀介质中的 HIFU 换能器进行了数值实验,以确定准确表示谐波所需的单独网格的大小。特别是,随着计算域大小的修改,提出并采用快速体积势 (VP) 方法来执行收敛实验。VP 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。提出了一种快速体积势 (VP) 方法,并在修改计算域大小时采用该方法来执行收敛实验。VP 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。提出了一种快速体积势 (VP) 方法,并在修改计算域大小时采用该方法来执行收敛实验。VP 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。
更新日期:2020-11-09
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