Improvement of the Stokesian Dynamics method for systems with a finite number of particles

Abstract

An improvement of the Stokesian Dynamics method for many-particle systems is presented. A direct calculation of the hydrodynamic interaction is used rather than imposing periodic boundary conditions. The two major difficulties concern the accuracy and the speed of calculations. The accuracy discussed in this work is not concerned with the lubrication correction but, rather, focuses on the multipole expansion which until now has only been formulated up to the so-called FTS version or the first order of force moments. This is improved systematically by a real-space multipole expansion with force moments and velocity moments evaluated at the centre of the particles, where the velocity moments are calculated through the velocity derivatives; the introduction of the velocity derivatives makes the formulation and its extensions straightforward. The reduction of the moments into irreducible form is achieved by the Cartesian irreducible tensor. The reduction is essential to form a well-defined linear set of equations as a generalized mobility problem. The order of truncation is not limited in principle, and explicit calculations of two-body problems are shown with order up to 7. The calculating speed is improved by a conjugate-gradient-type iterative method which consists of a dot-product between the generalized mobility matrix and the force moments as a trial value in each iteration. This provides an O(N²) scheme where N is the number of particles in the system. Further improvement is achieved by the fast multipole method for the calculation of the generalized mobility problem in each iteration, and an O(N) scheme for the non-adaptive version is obtained. Real problems are studied on systems with N = 400 000 particles. For mobility problems the number of iterations is constant and an O(N) performance is achieved; however for resistance problems the number of iterations increases as almost N^1/2 with a high accuracy of 10⁻⁶ and the total cost seems to be O(N^3/2).