In distinct element model (DEM), the grain's position is updated by Newton's second law of motion. The grain only knows which way the force is pushing it, which is perfect for modeling fast, high-inertial events like avalanches, but not so much for slow, low-inertial systems like tectonic deformation. To make DEM work for these systems, heavy artificial damping is required to drain the kinetic energy.
Kishino's quasi-static DEM (qsDEM) solves low-inertial systems by removing mass and time entirely. qsDEM replaces the time-integration of Newton's law with a geometric matrix inversion. When a grain experiences an unbalanced generalized force (f), qsDEM calculates the forward step (Δu) required to deform the surrounding constraints to cancel out the unbalanced force. It solves this by using the contact stiffness matrix (S), and the optimal step forward is calculated by multiplying the force by the inverse of this matrix, scaled by a relaxation factor α that prevents overshooting the target.
What exactly is the contact stiffness matrix (S)? This is the summation of stiffness contributions from every constraint it is currently touching. Intuitively, before the grain moves, the stiffness matrix (S) acts as a map of its local confines, and by multiplying the force by the inverse of the stiffness, the forward step Δu is rotated away from the stiffest boundaries, automatically steering the grain down the optimal path of least resistance.
To see what this looks like, below is a simple sandbox comparing DEM with qsDEM. Slide the right wall to load the pile.