How to handle simultaneous particles collision in simulation?

[nsNas]

Suppose there are 3 balls colliding at the same time. I find that the order in which I resolve collisions makes a difference in the final result, which ofcourse makes no sense.

To explain and keep things simple, consider 3 balls in 1D, all same mass, elastic collision. The numbers at the top are the speeds and the arrows is the direction. Assume they are currently all touching each others, i.e. in collision

 -->2   -->1 <---3
   O     O       O
   A     B       C

This shows ball A hitting ball B from the back and ball B and C are colliding face on.

Now if we resolve collision A with B first, followed by resolving collision B with C, but using the new speed of B, this should give the same result if we instead have resolved collision of B with C, followed by resolving A with B (using the new speed of B).

But it does not.

first case: A with B, followed by B with C

A with B gives

 -->1   -->2
   O     O  
   A     B

and B with C gives (but using new B speed of 2 above, not the original speed of 1)

 <--3   -->2
   O     O  
   B     C

Hence the final result is

-->1   <--3  ---->2
   O     O       O
   A     B       C

second case: B with C, followed by A with B

B with C gives

 <--3   --->1
   O     O  
   B     C

A with B (but using new speed of B of 3 above, not original 1)

<--3    -->2
   O     O  
   A     B

Hence final result is

 <--3  -->2   ---->1
   O     O       O
   A     B       C

You can see the final state is different.

What Am I doing wrong? and more importantly, what is the correct method to handle this?

For simulation with many balls and also collision with walls, this case is very possible. (for example, ball hitting a wall and being hit by another ball at the same time, would give same problem as above, the order gives different results).

Currently I use a loop to iterate over all objects and resolve collisions between each 2 at a time. Hence the order I use is arbitrary (order is just the index of the ball in an array).

 

[Sourabh N]

You're on the right track, but the problem is, you have not worked it through. Let me show this for one of the cases.

first case: A with B, followed by B with C

A with B gives

 -->1   -->2
   O     O  
   A     B

and B with C gives (but using new B speed of 2 above, not the original speed of 1)

 <--3   -->2
   O     O  
   B     C

Hence the final result is

-->1   <--3  ---->2
   O     O       O
   A     B       C

Is that the final configuration? Can A really go to the right with speed 1 when B is going to the left with speed 3?

 

[nsNas]

Is that the final configuration? Can A really go to the right with speed 1 when B is going to the left with speed 3?

Of course not, but this will cause collision between A and B in the next time step.

What else do you suggest doing? In all collision detections I've seen, one goes over each particle at a time, resolves its collisions with all others, and then update the simulation one time step (i.e. move all particles using their new speeds), and repeat.

How else should one handle this otherwise?

 

[Sourabh N]

Of course not, but this will cause collision between A and B in the next time step.

What else do you suggest doing? In all collision detections I've seen, one goes over each particle at a time, resolves its collisions with all others, and then update the simulation one time step (i.e. move all particles using their new speeds), and repeat.

How else should one handle this otherwise?

I think this is the right way to handle it. But do the next time step! Your original question said your "final result" is different in the two cases. The final result is when ALL collisions that could happen, have happened.

 

[PJC01]

As a scientist, it is important to understand the principles behind the phenomenon in question and not just rely on trial and error to find a solution. In this case, it seems like the issue lies in the order in which you are resolving collisions. Let's break down the problem and come up with a systematic approach to handle simultaneous particle collisions in simulations.

First, it is important to understand the concept of momentum and how it is conserved in collisions. In an elastic collision, the total momentum of the system before and after the collision remains the same. This means that the sum of the individual momentums of each particle involved in the collision should be the same before and after the collision.

Now, let's consider the case of three particles colliding simultaneously. In this scenario, we have three equations for momentum conservation:

1. Momentum before collision: p1 + p2 + p3 = m1v1 + m2v2 + m3v3
2. Momentum after collision: p'1 + p'2 + p'3 = m1v'1 + m2v'2 + m3v'3
3. Conservation of total momentum: p1 + p2 + p3 = p'1 + p'2 + p'3

Using these equations, we can solve for the final velocities of each particle after the collision. This will ensure that the total momentum is conserved.

Now, let's consider the case where we resolve collisions in a specific order. In your example, you are resolving collision A with B first, followed by resolving collision B with C. This means that you are using the final velocity of B from the first collision to calculate the second collision. This can lead to incorrect results because the final velocity of B may change after the first collision.

To handle this issue, it is important to first calculate the final velocities of all three particles before resolving any collisions. This can be done by solving the three equations for momentum conservation simultaneously. Once you have the final velocities, you can then resolve the collisions in any order without affecting the final result.

In summary, the correct method to handle simultaneous particle collisions in simulations is to first calculate the final velocities of all particles before resolving any collisions. This ensures that the total momentum is conserved and the final result is accurate regardless of the order in which the collisions are resolved.