Collision Finding for Particles on Non-linear Paths

Transcription

1 Collision Finding for Particles on Non-linear Paths Abstract For many N-body hard-sphere collisional simulations particle trajectories can not be assumed to be linear for the purposes of collision detection. This paper presents such a simulation and explores the relative efficiency of several methods of detecting collisions for non-linear paths. The methods can be categorized by how they approximate particle motion as well as by how they search for collisions. The motion can either use an analytic description of the exact motion or a cubic fit to that motion. The detection method can either use a standard Newton's method or a second order iterative method that makes new guessed based on the assumption of linear motion. Empirical testing found that for the system in question, using the method that makes guesses assuming linear motion and the full description of the motion can provide significant speed benefits across a wide range of simulation parameters. Keywords: N-body simulation, collision finding, optimization, physical simulation Mark C. Lewis Trinity University Background This work came about from a desire to find optimal methods for performing hard sphere collision simulations of planetary rings [!!! ref]. These simulations typically involve millions of particles and profiling has shown that over 90% of the runtime is often spent in the function that determines whether or not two particles actually collide. Given this, there are two obvious ways to speed up these simulations. The first is to do a better job of culling the pairs of particles that have to be checked so that this method is called fewer times during the course of a simulation. The second is to improve the speed of routines that determine whether or not a collision occurs. Some efforts have already been extended toward the first approach [!!! refs]. This paper explores some possibilities using the second approach. Collision detection has a long history that can only be briefly touched upon here. For the problem of colliding spheres, often referred to as the billiard ball problem, there are two fundamentally different approaches to handling the collisions: soft-sphere and hard-sphere. In a soft sphere simulation, a numerical integrator with a variable length time step is used. When particles begin to overlap a restoring force is applied [!!! ref]. Velocity dependence can be added to the force to model a dissipative coefficient of restitution. In the field of planetary ring dynamics, soft-sphere collisions are used in one of the codes developed by Salo and his collaborators [!!! refs]. Hard-sphere collisions are more similar to discrete event simulations than to continuous time simulations. The collisions between particles are handled as discrete events and the simulation code has to be able to detect the exact time at which two particles begin to overlap. Once this time has been found, the particles are advanced to that point in time and their velocities are altered in an appropriate manner to reflect a collision with the proper coefficient of restitution. Hard sphere collisions are used in the field of planetary rings in codes written by Richardson [!!! refs] and Lewis [!!! refs]. This paper focuses on the problem of processing hardsphere collisions and, in particular, the problem of determining whether two particles collide and when that collision happens. In many systems, the motions of the particles can be nicely approximated by straight lines. As long as the time steps are sufficiently short, a linear approximation can always be made to be sufficiently accurate. Of course, tiny time steps are not always numerically acceptable as the processing time required to run a simulation for a certain period of time typically scales as 1/ Δt. In collisional simulations this scaling law is complicated by the fact that the longer the time step, the larger a region each particle much check for its next potential collision. A full description of the scaling of processing time with time step for some different methods of detecting nearby collision pairs can be found elsewhere [!!! Lewis and Stewart, Lewis]. For our purposes here it is sufficient to say that the time scales roughly as A/Δt+B(Δt) d, where A and B are constants that depend on the methods used and d is the dimensionality of the particle distribution. This function will have a minimum that is the ideal time step to use. Unfortunately, the value of the optimal time step can depend somewhat on the nature of the simulation being considered and can even vary over the course of a simulation and the particle distribution evolves. Planetary rings happen to be a system where the optimal Δt value is larger than the safe limit for using a linear approximation of motion. This is primarily the case for perturbed systems where particles are more likely to undergo collisions at certain points in the simulation than in others. Figure 1 shows how this can be a problem. In this system, the particles travel on nearly sinusoidal paths. Linear approximations to this path are best near point B. If there are many collisions near point A, the particles wind up below where they should be. If they happen near point C, the particles would move upward. In perturbed planetary rings, the collisions happen in a very non-uniform, nonisotropic manner which can lead to systematic migration of particles if a linear trajectory is assumed. For this reason, some other, more sophisticated approach is required.!!! Further references in the collision detection literature. Methodology At its foundation, collision detection in hard-sphere models is nothing more than a root finding problem. We

2 have the function f(t)=d(t)-(r 1+r 2), where d(t) is the distance between two particles as a function of time and the r-values are particle radii. We want to find the value of t such that f(t)=0. For this reason, Newton's method automatically jumps out as an approach in any situation where a direct solution can't be found. To remind the reader, the way that Newton's method works is to assume a linear approximation to the function at the point of the current guess and set the next guess to the point where the linear approximation would cross zero. That is to say that g n1 =g n f g n, (1) f ' g n where g n is the n th guess. In practice we use, f(t)=d 2 (t)- (r 1+r 2) 2, because it has the same roots and avoids the requirement of having to calculate or deal with a square root. This simplified the analytic treatments and improves performance for the numerics. Of course, Newton's method has several standard challenges that one must deal with. Namely, it only works well if one is sufficiently close to the root and it can often find any of several roots if there is more than one near the initial guess. Newton's method can also behave very poorly in a situation where the function has a local minimum that doesn't pass below zero. All of these cause problems for a collision detection routine. One way to address these problems is to use a method that assumes linear motion. To see why this helps, let us look at the behavior of two particles as they pass near one another. For this example we will assume linear motion as it not only illustrates the problems one encounters with Newton's method, it also shows how those problems are resolved when linear motion is assumed. In figure 2 we plot f(t)=d(t)-(r 1+r 2) for two particles showing the three basic cases that we have to consider: a solid hit, a grazing hit, and a miss. The difference between the first two is one of magnitude. Both have roots, but in the grazing hit there is only a small period of time where the value is negative and as such there are two roots that are very close to one another. The truth is that this distinction in the type of hit isn't significant as it is the scale of the particle sizes to the relative distance traveled by the particles during a time step that truly matters. If the particles travel a small distance relative to the sizes of the particles during a time step, Newton's method is likely to work well. The initial guess will inevitably be fairly close to the root and the separation between the two roots is more likely to be large relative the the time step. However, for grazing collisions or systems where the time step is large and particles travel long distances relative to the particle sizes, the two roots will be close together. In such a system, small changes in curvature of the path can lead to Newton's method finding the exit time instead of the collision time. In the third case, Newton's method can run through many iterations jumping from one side of the minimum to the other. To prevent these situations from occurring, the Newton's method must be altered from the normal form. To insure that the first root is found, the step can be damped. If the step to the next guess ever leads to a time at which the derivative of the distance is positive, the normal Newton step is halved repeatedly until a point where the derivative of distance with time is still negative is found. One must also put carefully selected constraints on the main loop of the Newton's method to prevent it from running more iterations than are truly required. The stopping conditions include things such as moving too far from the initial guess or taking too many iterations to converge. Figure 1: This figure shows a sinusoidal path for a particle with linear approximations to the motion. A linear approximation is good at point B, but it is offset from the true path at points A and C. If the system has more collisions in one part of the motion than others the result is an artificial, systematic drift of particles. Figure 2: This shows the three possible cases for particles passing close to one another on nearly linear trajectories. They can have a solid hit, a grazing hit, or a miss.

3 Using an approximation of linear motion has the capability to improve upon the standard Newton's method in a few ways. First,because it is basically using a quadratic approximation to the distance, it is more capable of working further from the point of the collision. Second, because the solution is the solution to a quadratic, it is easy to pick the first root where the particles are heading toward one another. Lastly, this method is far less likely to oscillate about a local minimum. Unfortunately, the trade off is that is can simply get stuck in a local minimum, but that is far easier to test for as the value of the guess stops changing without finding a collision. The simulation code that is used for this work approximates the motion of particles between collisions using an analytic solution to the restricted Hill equations [!!! ref] called guiding center motion that was developed by Stewart (1991)[!!! ref]. This solution is accurate to first order in eccentricity and works well for planetary ring systems where eccentricities rarely get above The motion of a particle in local Cartesian coordinates, ignoring the oblateness of the planet is specified by the following equations: x=x e cos y=y 2e sin z=i sin where Ẋ=ė= i=0 = =1 (2) Ẏ= 3 2 X. (3) Given two particles, it is straightforward to calculate the distance between them as well as the derivative of the distance between them. Using subscripts to denote the two particles, the value of d 2 (t) and its derivative are given by the following: Figure 3: This is a plot showing particles in one of the initial conditions that was used where particles are clumped together. d 2 t= x 1 x 2 2 y 1 y 2 2 z 1 z 2 2 = x 2 y 2 z 2 x=x 1 X 2 e 1 cos 1 e 2 cos 2 y=y 1 Y 2 2e 1 sin 1 2e 2 sin 2 z=i 1 sin 1 i 2 sin 2 d 2 ' t =2 x x2 y y2 z z x=e 1 sin 1 e 2 sin 2 y= 3 2 X X 2 2 e 1 cos 1 2e 2 cos 2. (4) z=i 1 cos 1 i 2 cos 2 So in this system, a full analytic solution can be used for the root finding algorithm. Unfortunately, this formula includes numerous sine and cosine functions which can hurt performance on many systems. Proper use of temporary variables limits it to eight trigonometric function calls each time a new guess is evaluated, but that is still a fairly high cost.. An alternative to using the full analytic description of the motion is to fit cubic curves to the motion and run Newton's method or another iterative method on those curves. The cubic fits allow us to fit both position and velocity at the beginning and the end of the time step. This generally provides a fit that is very close to the full analytic solution, but without the trigonometric functions. One might be tempted to think that having a cubic description would allow us to find a closed form solution. After all, there is a closed form solution to the roots of a cubic polynomial. However, the fit has a separate cubic for x, y, and z and the distance formula uses the square of this so it actually involved a sixth order polynomial, for which there is not a closed form solution. The benefit of this method comes from that fact that the trigonometric functions only have to be called when determining the cubic fit. After that, all the steps that are required can be done without further calls to trigonometric functions. Of course, the standard first-order Newton's method is not the only method of finding roots. This work also explores one other approach. It is similar to a second order variation on the Newton's method, but instead of working with the distance function, it works directly with the particles and uses our knowledge about the motion of the particles to make more intelligent guesses as to when a collision might occur. Instead of using the linear approximation to d(t) to find the next guess at the root, this alternate method assumes linear motion for the particles to make the next guess. It can be more efficient than general applications of Newton's method because velocities are a natural part of the particle description and the method includes inherent knowledge of the system that a general Newton's method does not. There are challenges with this approach of course. The main one is that it isn't as well behaved when two particles don't collide. The iteration method always moves either to the point where the linear motion would cause a collision, or to the closest point of approach assuming that motion if it they don't hit on those trajectories. In the situation where the particles truly don't collide, this second case can easily lead to an infinite loop. Care has to be taken to insure that this is exited nicely. Another factor that we are interested in exploring is how

4 well the different methods work when the particles are taking different length time steps. A formula for the scaling of runtime with time step was given in the first section. This scaling of runtime is based wholly on the search algorithm!!! Discuss impact of time step on run time. Results A number of different tests were run to test the effectiveness of the different methods on different types of systems. Each method was used on four different setups using five different time steps. The time steps were chosen to span a range that would likely include or come close to including the optimum value. The different system configurations included two with a single particle size and two with a particle size distribution. In each case one run was done when the particles were fairly uniformly distributed and the other was done after the system had evolved for a while with particle self-gravity so that the particles were clustered together. Figure 3 shows the initial configuration of one of the simulations where the particles are clumped together. All the simulations were run for a tenth of an orbit and timed. In one run statistics were kept for how many iterations the collision finding algorithm went through before either a collision was found or the possibility of a collision was rejected. In total 100 distinct simulations were performed six times in order to get averages for the times. That total number comes from the fact that there were five methods, five time steps, and four system configurations. Figure 4 shows the timing results for the 100 different simulations. Each of the four plots shows the results for a single system configuration. The color and size of dot indicate the method used. The smallest and darkest dots represent the original method. Next up in size and lighter in color is the Newton's method using the full equations of motion. This is followed by the Newton's method using a cubic approximation to the motion and the linear method using the full equations of motion. The largest dots and lightest color represent the linear method with a cubic approximation to the motion. Some of the trends in this plot are roughly what were expected. The times for any given method and simulation tend to have a minimum somewhere in the time range. Often it is 1/1000 th of an orbit period, but in some cases it moves up to 3/1000 th or even 1/100 th of an orbital period. All of the methods have curves with very similar shapes though they Figure 4: These four scatter plots show the average run times of the 100 different systems that were looked at. Each plot is for a different system configuration and the separate data points are for the different methods and time steps that were used.

5 are shifted a bit in absolute time. These shifts give us the relative overall performance of the methods. Clearly not all of the methods performed equivalently. The linear method with the full description of the motion ran the fastest across the board. The original method competed with the Newton's method using the cubic approximation for last place. It is a bit surprising at first that the cubic methods perform as poorly as they do. This is likely because of the overhead in making the cubic fits. The code, as it stands, does not store the cubic fits in the particles. This is a trade off of speed vs. memory. The cubic methods might fair better if the cubic fits were cached for the particles. Of course, they would have to be calculated intelligently so that they caches would remain valid without the cost of updating those values becoming too significant. For other codes that are based on linear approximations to motion during a time step, these methods might fair better than they do here. In order to understand why the timing results are the way they are, we can look at the statistics on the number of cycles taken for a few of the different simulations. Figure 5 shows histograms of the number of times a call to the collision finding algorithm exited for different reasons. For efficiency reasons, we would like the number of iterations to be as low as possible. Because each one of the 100 simulations can produce its own histogram plot, only certain simulations have been selected to illustrate certain features of the different methods. Figure 5 shows the single size system at the early time with a time step of 1/1000 th of an orbital period. Looking at figure 5 it is clear why the original method ran slowly. The overly conservative guessing method takes iterations before it finds any collisions. All the other methods would typically find a collision in five or fewer iterations. As was discussed above, the linear approximation methods have a problem determining when they can rule out a collision. This is seen here as a low level tail that extends to fairly high number of iterations. Just looking at figure 5 one might expect that the Newton's method with the full equations of motion would be the fastest followed by either the Newton's method with cubic motion or the linear approximation method with full equations of motion. We know from figure 4, however, that in this particular system setup it is the full linear method that wins out. The reason for this becomes more clear with figure 6. This figure shows a slightly different histogram for the three methods that use the full equations of motion of the same simulations shown in figure 5. The difference is that this histogram shows how many times the equations of motion were evaluated instead of just the number of iterations of the approximation method. These values can be very different because if a normal step goes too far, either into a region where the distances are increasing or where the distance is larger than it had been the previous iteration, the method will test a smaller step. This adaptation of the step can occur many times for each iteration of the outer loop. When this happens using the full equations of motion it requires trigonometric function evaluations. For that reason, the code was instrumented to keep track of this. That count is shown in figure 6. With this plot, the reason that the linear method beats out the Newton's method becomes fairly clear. The Newton's method spends a lot of time scaling back the step while the linear method doesn't do so nearly as much. This is likely a result of some of the difficulties in root finding for collisions that were discussed earlier. The fact that most cases where there is a root there are actually two, very close together and we care which one we find adversely impacts Newton's method. The linear method does a better job of guessing the location of the near root on the first try. At this point we seem to have an adequate explanation for the relative performance of the methods that use the full equations of motion. What about the relative performance of the methods that use a cubic approximation? In general those methods were slower than their counterparts using the full description of motion. It was speculated above that this was Figure 5: This plot shows a set of histograms that plot the number of times the collision finding algorithm exited after a particular number of iterations. The bars are colored by the reason for exiting the loop. The darkest bars indicate instances where the routine actually found a collision. The lighter gray is for a call that exited with a non-collision. This is for the single sized, early simulation using a time step of 1/1000 th of an orbit.

6 Figure 6: The three plots in this figure show the number of times that the particle locations/distance/derivative of distance was evaluated before exiting the collision finding loop for the three methods that use the full description of the motion. The dark colored parts of the bars indicate that the loop terminated when a collision was found. The lighter color indicates that a collision was not found. This is for the same simulation shown in figure 5. the result of the setup costs in making the cubic fits. The truth of this speculation is not well illustrated by figure 5, but that is mainly because of the choice of the range for the vertical axis. There is actually a third exit criteria plotted in figure 5. It is the calls that exited before any iterations of the root finding loop. Most of the time this was because the particles were heading apart from one another. These early terminations are by far the most common reason for a call to the collision detection to terminate. If the plots in figure 5 were scaled to show these the vertical axis would have to go above 10 8 instead of the current 3*10 6. This wasn't done because having the plot go so high makes everything except the first bar impossible to see. While at first it might seen that something like a check for the relative velocities of the particles should be a simple dot product on their initial velocities and should be the same for all methods, this isn't actually the case. This is because when the collision detection routines are called it is possible that the particle positions are for different points in time because one of both of the particles have been involved in an earlier collision during that time step. For this reason, the routines must first adjust the particle positions and velocities so that they have a consistent time frame. This is done in different ways for the different methods so that the work that goes into it is useful for the rest of the calculation. In the case of the cubic motion, the cubic curves are fit to particles that have been rolled back to the beginning of the time step and particles advanced to the end of the time step. This constitutes a fair bit of work that is required setup for the iterative method, but it is not the fastest method of determining if two particles are heading away from one another at a shared time. Once again, other simulation codes that are based on the assumption of cubic motion will have the cubic coefficients stored for each particle and would not incur this extra cost. Conclusions Blah!!! Acknowledgments This work was supported by a grant from the NASA PG&G program. All figures in this paper were made with SwiftVis, which was developed under a NASA AISRP grant. Bibliography Lewis papers Richardson papers Salo papers Stewart 1991