Kernel Aggregated Fast Multipole Method: Efficient summation of Laplace and Stokes kernel functions
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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 and places a small number of equivalent multipole and local points around each octree box, and completes the kernel sum with O(N) performance. However, the KIFMM cannot be used directly with nonlinear kernels, can be inefficient for complicated linear kernels, and in general is difficult to implement compared to less-efficient alternatives such as Ewald-type methods. Here we present the Kernel Aggregated Fast Multipole Method (KAFMM), which overcomes these drawbacks by allowing different kernel functions to be used for specific stages of octree traversal. In many cases a simpler linear kernel suffices during the most extensive stage of octree traversal, even for nonlinear kernel summation problems. The KAFMM thereby improves computational efficiency in general and also allows efficient evaluation of some nonlinear kernel functions such as the regularized Stokeslet. We have implemented our method as an open-source software library STKFMM 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. Massive parallelism is supported with mixed OpenMP and MPI.
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