Numerical Analysis: A fast summation method for fiber simulations
A numerical method for large scale simulations of fibers immersed in a Stokesian fluid has been developed. The mathematical description is based on a boundary integral formulation with a slender body approximation. Periodic boundary conditions are introduced through the so-called Ewald summation formulas for the fundamental solutions. This decomposes the sum, i.e. the matrix-vector product, into one part in real-space and one in Fourier space. The computation of the Fourier space sum is accelerated using a FFT based method, the Spectral Ewald (SE) method, which allows for the use of significantly smaller FFT grids relative to established methods for fast Ewald summation. The real space treatment is based on the Linked Cell List method, and is modified to enable analytical integration for fibers in close vicinity to each other, which ensures better accuracy. By appropriately adjusting the computational load of the sums in the decomposition as the problem scales up, the resulting method is rendered to have a complexity of O(N log N ), where N is the number of fibers. Provided error bounds offer a straight-forward parameter selection. Results from simulations of a few thousand fibers in different configurations have been obtained and analyzed.