  This paper presents a new boundary integral equation (BIE) method for
simulating particulate and multiphase flows through periodic channels of
arbitrary smooth shape in two dimensions. The authors consider a particular
system---multiple vesicles suspended in a periodic channel of arbitrary
shape---to describe the numerical method and test its performance. Rather than
relying on the periodic Green's function as classical BIE methods do, the
method combines the free-space Green's function with a small auxiliary basis,
and imposes periodicity as an extra linear condition. As a result, we can
exploit 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 and
product quadratures, while a first-order semi-implicit scheme evolves particles
by treating the vesicle-channel interactions explicitly. New
constraint-correction formulas are introduced that preserve reduced areas of
vesicles, independent of the number of time steps taken. By using two types of
fast algorithms, (i) the fast multipole method (FMM) for the computation of the
vesicle-vesicle and the vesicle-channel hydrodynamic interaction, and (ii) a
fast direct solver for the BIE on the fixed channel geometry, the computational
cost is reduced to $O(N)$ per time step where $N$ is the spatial discretization
size. Moreover, the direct solver inverts the wall BIE operator at $t = 0$,
stores its compressed representation and applies it at every time step to
evolve the vesicle positions, leading to dramatic cost savings compared to
classical 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.
