A fast algorithm for simulating multiphase flows through periodic geometries of arbitrary shape

摘要

This paper presents a new boundary integral equation (BIE) method forsimulating particulate and multiphase flows through periodic channels ofarbitrary smooth shape in two dimensions. The authors consider a particularsystem---multiple vesicles suspended in a periodic channel of arbitraryshape---to describe the numerical method and test its performance. Rather thanrelying on the periodic Green's function as classical BIE methods do, themethod combines the free-space Green's function with a small auxiliary basis,and imposes periodicity as an extra linear condition. As a result, we canexploit existing free-space solver libraries, quadratures, and fast algorithms,and handle a large number of vesicles in a geometrically complex channel.Spectral accuracy in space is achieved using the periodic trapezoid rule andproduct quadratures, while a first-order semi-implicit scheme evolves particlesby treating the vesicle-channel interactions explicitly. Newconstraint-correction formulas are introduced that preserve reduced areas ofvesicles, independent of the number of time steps taken. By using two types offast algorithms, (i) the fast multipole method (FMM) for the computation of thevesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) afast direct solver for the BIE on the fixed channel geometry, the computationalcost is reduced to $O(N)$ per time step where $N$ is the spatial discretizationsize. Moreover, the direct solver inverts the wall BIE operator at $t = 0$,stores its compressed representation and applies it at every time step toevolve the vesicle positions, leading to dramatic cost savings compared toclassical approaches. Numerical experiments illustrate that a simulation with$N=128, 000$ can be evolved in less than a minute per time step on a laptop.