Many different simulation methods for Stokes flow problems involve a common computationally intense task—the summation of a kernel function over O(N²) 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.

斯托克斯流问题的许多不同模拟方法都涉及一项常见的计算密集型任务——核函数在O ( N ² ) 对点上的求和。一种流行的技术是核独立快速多极方法（KIFMM），它为所有点构造一个空间自适应八叉树，并在每个八叉树框周围放置少量等效多极和局部等效点，并用O ( N) 成本，使用这些等效点。可以在这些等效点之间使用更简单的内核来提高 KIFMM 的效率。在这里，我们提出了对这个想法的进一步扩展和应用，以实现对各种内核的有效求和和灵活的边界条件。我们称我们的方法为核聚合快速多极方法（KAFMM），因为它在八叉树遍历的不同阶段使用不同的核函数。我们已经实现了我们的方法作为一个开源软件库STKFMM基于高性能库PVFMM，支持拉普拉斯核、Stokeslet、正则化 Stokeslet、Rotne-Prager-Yamakawa (RPY) 张量以及 Stokes 双层和牵引算子。所有内核都支持开放和周期性边界条件，Stokeslet 和 RPY 张量支持无滑移壁边界条件。该软件包旨在即用型，并且易于扩展到其他内核。