Energy Gain in Wall Collisions in Perfectly Elastic Setting

[BKaperick]

Here the ELASTICITY parameter is set to 1, to indicate that collisions with walls are perfectly elastic. We appear to gain a small (relative to system) amount of total energy with each hit. This effect likely exists with other choices of ELASTICITY, but this should be the simplest case to analyze.

[BKaperick]

Currently, we use a fixed time step dt. If - in between two states - a collision occurred, the step is reversed, and we advance at increment dt/2. This is repeated, incrementing at dt/2^k until no collisions exist above COLLISION_TOL or the modified time increment dt/2^k is below EPSILON. We only seem to be gaining energy when the second condition fails:

Fizz-2D/src/physics.py

Lines 91 to 114 in 5c8ac7a

	while max_collision_overlap > COLLISION_TOL and dt > EPSILON:
	if first_iter:
	first_iter = False
	else:
	dt /= 2
	for obj in self.objs:
	obj.reverse_update()
	# Do first pass of position, velocity and acceleration updates for
	# each object in the world
	for obj in self.objs:
	obj.pre_update([], gdf, dt)
	# check_collisions() compiles a list of collisions with information:
	# (1,2) the two objects contained in the collision
	# (3) the unit vector normal to the surface of collision
	# (4) magnitude of normal vector
	# (5) a flag indicating a sign change for the normal vector
	collisions = self.check_collisions()
	magnitudes = [abs(magnitude) for _,__,___,magnitude,____ in collisions]
	if collisions:
	max_collision_overlap = max(magnitudes)
	else:
	max_collision_overlap = 0

This makes perfect sense, as this is exactly the scenario in which the collision resolution "fails" and has to bail out for fear of instability with a small time step.

[BKaperick]

Interestingly, changing the EPSILON parameter results in only negligible improvement in the magnitude of energy error incurred.
With GRAVITY=[0, 100] and input file simulations/BOX_1SQUARE.in:

EPSILON	Relative Energy Gain in Collision
1e-4	1.83360663%
1e-8	1.82640794%
1e-9	1.82640756%
1e-10	1.82640751%

[BKaperick]

Reducing time discretization can make this energy gain arbitrarily small, at the cost of more computational time.