利用嵌套网格加速弱非线性聚焦超声频域数值方法
Accelerating frequency-domain numerical methods for weakly nonlinear focused ultrasound using nested meshes
摘要：
弱非线性超声的数值模拟在聚焦超声（FUS）治疗计划中具有重要意义。然而，大的域尺寸和在焦点处产生的高次谐波使得这些问题在计算上要求极高。数值方法通常采用均匀网格，网格足够精细，以解决问题中存在的最高谐波，从而导致非常大的自由度。本文提出了一种更有效的策略，其中每个谐波近似在一个单独的网格，其大小是成比例的谐波的波长。分辨较小波长所需的分辨率的增加通过域尺寸的减小来平衡。这种嵌套网格是可行的，由于越来越多的本地化性质的高次谐波附近的焦点。为了确定准确表示谐波所需的网格尺寸，对均匀介质中的FUS换能器进行了数值实验。特别是，提出了一种快速的\emph{volume potential}的方法，并进行收敛实验的计算域的大小被修改。这种方法允许每个谐波计算通过评估在域上的积分。使用中点规则离散此积分允许使用FFT快速执行计算。它表明，至少数量级减少内存消耗和计算时间可以实现嵌套网格。最后，它演示了如何将这种方法推广到非均匀传播域。
英文摘要：
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 in order to determine the size of the meshes required to accurately represent the harmonics. In particular, a fast \emph{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.
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