  We consider suspensions of rigid bodies in a two-dimensional viscous fluid.
Even with high-fidelity numerical methods, unphysical contact between particles
occurs because of spatial and temporal discretization errors. We apply the
method of Lu et al. [Journal of Computational Physics, 347:160-182, 2017] where
overlap is avoided by imposing a minimum separation distance. In its original
form, the method discretizes interactions between different particles
explicitly. Therefore, to avoid stiffness, a large minimum separation distance
is used. In this paper, we extend the method of Lu et al. by treating all
interactions implicitly. This new time stepping method is able to simulate
dense suspensions with large time step sizes and a small minimum separation
distance. The method is tested on various unbounded and bounded flows, and
rheological properties of the resulting suspensions are computed.
