Stability of PAIRS interactions


Use these tests to explore the stability and convergence behaviors of the PAIRS method. The first four tests use a PAIRS calculation of collision force for sphere-sphere and sphere-container collisions. The last test applies opposite forces to the ends of the segmented filaments that are the focus of SimFil.


Use the Control Panel to vary sphere number, dimension, and force magnitudes (open View -> Control Panel and examine the "stability demos" tab). You can change force magnitudes and the PAIRS collision coefficient "on the fly", whereas any change to sphere number or size requires a restart of the run (not of the program... use "Program -> Restart").


Test 1: A small sphere squeezed between two very large spheres (horizontal planes in the limit)


The two large spheres in this demonstration are completely immobile. Notice that if the small sphere is perfectly centered (as is the default), then the PAIRS method will converge to the appropriate equilibrium point for any collision coefficient less than 1.0. Try changing the initial position such that there is a non-zero y- or z-offset... in that case the small sphere moves out to a non-colliding position.



Test 2: Many spheres squeezed into a too-small container


Here many identical spheres (10 by default) are forced to overlap with one another through confinement in a spherical container. This test can demonstrate that the stability / convergence of the PAIRS method is highly dependent upon the degree of sphere overlap, and even the initial conditions. Try adjusting the PAIRS collision coefficient (to about 0.82) with the default values for sphere number and radius to see the spheres oscillate. With the more reasonable default collision coefficient of 0.4, you can make the spheres overlap to a ridiculous extent (by increasing the spherical radius) without loss of stability.



Test 3: Spheres packing together under a force field


Here many identical spheres (10 by default) experience a constant force in the x-direction. The spheres achieve an equilibrium position as a balance between the constant force, sphere-sphere forces, and sphere-container forces. Note that increasing the constant force and/or decreasing the PAIRS collision coefficient increases the degree of overlap of spheres and container, but does not result in any instability or failed convergence. Increasing the PAIRS collision coefficient can, however, result in oscillations about the equilibrium. This is in keeping with the general point of these demos: the PAIRS method is perfectly stable and convergent for actual pairwise interactions, but when each agent is subject to other forces, or multiple pairwise interactions, then the relevant PAIRS coefficient may need to be reduced to avoid non-converging oscillations.



Test 4: Spheres with a central attractor


This test is similar, in many ways, to Test 3. A central attractor at the origin exerts a force on each sphere that is linearly proportional, via a Hookean spring constant, to the distance of that sphere from the origin. This is the only demonstration with spheres in which an wild numerical instability can occur (as a result of the Hookean spring).



Test 5: A segmented filaments under traction forces


This single segmented filament, whose endpoints are linked by the special PAIRS translational and rotational springs described in Alberts (2008), experiences equal and opposite forces on each terminal segment. The default PAIRS coefficient for translation is 0.4 (here we reference the "Coeff. for PAIRS translation" found in the "main" tab of the Control Panel, NOT the PAIRS collision coefficient for spheres relevant in tests 1-4). This is the value recommended in Alberts (2008) when the time-constant for axial relaxations is shorter than the simulation time-step. With 0.4 as the PAIRS coefficient, the filament is unconditionally stable with respect to "reasonable" magnitudes of the applied force (up to many 10s of pN per filament). Try turning on the Brownian motion for the filament (uncheck "Tuning Tasks -> Filament Brownian Motion Off"). Of course, the equilibrium spacing between supposedly coincident segment endpoints becomes unreasonable large with large forces, but no oscillations are observed.


At large forces (try 2 nN or more) an instability occurs when Brownian motion is turned on, but can be rectified by reducing one or more PAIRS coefficient. This might be viewed as additional tuning constraint (i.e. if such large forces per filament are expected, a time-step reduction might be necessary), but I consider such large forces as too extreme to warrant discussion in Alberts (2008).