  We describe and summarize a class of minimal numerical models emerged from
recent development of simulation methods for dense particle suspensions in
overdamped linear flows. The main ingredients include (i) a frame-invariant,
short-range lubrication model for spherical particles, and (ii) a soft-core,
stick/slide frictional contact model activated when particles overlap. We
implement a version of the model using a modified velocity-Verlet algorithm
that explicitly solves the $N$-body dynamical system in $\mathcal{O}(cN)$
operations, where $c$ is a kernel constant depending on the cutoff of particle
interactions. The implementation is validated against literature results on
jamming transition and shear thickening suspensions from 40% to 64% volume
fractions. Potential strategies to extend the present methodology to
non-spherical particles are also suggested for very concentrated suspensions.
