论文标题
内核聚合快速多极方法：Laplace和Stokes内核函数的有效总和
Kernel Aggregated Fast Multipole Method: Efficient summation of Laplace and Stokes kernel functions
论文作者
论文摘要
Stokes流问题的许多不同的仿真方法涉及一项常见的计算强度任务 - $ O（n^2）$对点的内核函数的总和。一种流行的技术是内核独立的快速多极方法（KIFMM），该方法为所有点构建了空间自适应OCTREE，并在每个OCTREE盒周围构建了少数等效的多极和本地等效点，并使用$ O（n）$成本完成了核心总和。可以在这些等效点之间使用较简单的内核来提高KIFMM的效率。在这里，我们向该想法提供了进一步的扩展和应用，以实现各种内核的有效总和和柔性边界条件。我们称我们的方法为内核聚合快速多极方法（KAFMM），因为它在OCTREE遍历的不同阶段使用了不同的内核函数。我们已经基于高性能库PVFMM将方法作为开源软件库STKFMM，并支持Laplace内核，Stokeslet，Stokeslet，正规化Stokeslet，Rotne-Prager-Yamakawa（rpy）张量，以及Stokes Double layer and Traction Optorter。所有内核都支持开放和周期性的边界条件，并支持Stokeslet和Rpy张量的无滑膜边界条件。该软件包旨在可以使用，并且很容易扩展到其他内核。
Many different simulation methods for Stokes flow problems involve a common computationally intense task -- the summation of a kernel function over $O(N^2)$ pairs of points. One popular technique is the Kernel Independent Fast Multipole Method (KIFMM), which constructs a spatial adaptive octree for all points and places a small number of equivalent multipole and local equivalent points around each octree box, and completes the kernel sum with $O(N)$ cost, using these equivalent points. Simpler kernels can be used between these equivalent points to improve the efficiency of KIFMM. Here we present further extensions and applications to this idea, to enable efficient summations and flexible boundary conditions for various kernels. We call our method the Kernel Aggregated Fast Multipole Method (KAFMM), because it uses different kernel functions at different stages of octree traversal. We have implemented our method as an open-source software library STKFMM based on the high performance library PVFMM, with support for Laplace kernels, the Stokeslet, regularized Stokeslet, Rotne-Prager-Yamakawa (RPY) tensor, and the Stokes double-layer and traction operators. Open and periodic boundary conditions are supported for all kernels, and the no-slip wall boundary condition is supported for the Stokeslet and RPY tensor. The package is designed to be ready-to-use as well as being readily extensible to additional kernels.
