Particle Systems. g(x,t) x. Reading. Particle in a flow field. What are particle systems? CSE 457 Winter 2014

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1 Reading article Systems CSE 457 Winter 2014 Required: Witkin, article System Dynamics, SIGGRAH 01 course notes on hysically Based Modeling. Witkin and Baraff, Differential Equation Basics, SIGGRAH 01 course notes on hysically Based Modeling. Optional Hockney and Eastwood. Computer simulation using particles. Adam Hilger, ew York, Gain Miller. The motion dynamics of snakes and worms. Computer Graphics 22: , What are particle systems? A particle system is a collection of point masses that obeys some physical laws (e.g, graity, heat conection, spring behaiors, ). article systems can be used to simulate all sorts of physical phenomena: article in a flow field We begin with a single particle with: osition, Velocity, Suppose the elocity is actually dictated by a driing function, a ector flow field, g: y g(,t) If a particle starts at some point in that flow field, how should it moe? 3 4 1

2 Diff eqs and integral cures The equation Euler s method One simple approach is to choose a time step, Δt, and take linear steps along the flow: is actually a first order differential equation. We can sole for through time by starting at an initial point and stepping along the ector field: Start Here Writing as a time iteration: with This approach is called Euler s method and looks like: This is called an initial alue problem and the solution is called an integral cure. 5 roperties: Simplest numerical method Bigger steps, bigger errors. Error ~ O(Δt 2 ). eed to take pretty small steps, so not ery efficient. Better (more complicated) methods eist, e.g., adaptie timesteps, Runge-Kutta, and implicit integration. 6 article in a force field Second order equations ow consider a particle in a force field f. In this case, the particle has: Mass, m Acceleration, The particle obeys ewton s law: So, gien a force, we can sole for the acceleration: This equation: is a second order differential equation. Our solution method, though, worked on first order differential equations. We can rewrite the second order equation as: The force field f can in general depend on the position and elocity of the particle as well as time. Thus, with some rearrangement, we end up with: where we substitute in and its deriatie to get a pair of coupled first order equations

3 hase space Differential equation soler Starting with: Concatenate and to make a 6-ector: position in phase space. Applying Euler s method: Taking the time deriatie: another 6-ector. And making substitutions: A anilla 1 st -order differential equation. Writing this as an iteration, we hae: with Again, performs poorly for large Δt article structure Single particle soler interface How do we represent a particle? getdim osition in phase space position elocity force accumulator mass derieal getstate setstate

4 article systems article system soler interface In general, we hae a particle system consisting of n particles to be managed oer time: For n particles, the soler interface now looks like: particles n time particles n time get/setstate getdim derieal article system diff. eq. soler Forces We can sole the eolution of a particle system again using the Euler method: Each particle can eperience a force which sends it on its merry way. Where do these forces come from? Some eamples: Constant (graity) osition/time dependent (force fields) Velocity-dependent (drag) -ary (springs) How do we compute the net force on a particle?

5 article systems with forces Graity and iscous drag Force objects are black boes that point to the particles they influence and add in their contributions. We can now isualize the particle system with force objects: particles n time forces nf The force due to graity is simply: p->f += p->m * F->G Often, we want to slow things down with iscous drag: F 1 F 2 F nf p->f -= F->k * p-> Damped spring A spring is a simple eamples of an -ary force. Recall the equation for the force due to a 1D spring: With damping: In 2D or 3D, we get: r r = rest length derieal 1. Clear forces Loop oer particles, zero force accumulators 2. Calculate forces Sum all forces into accumulators 3. Return deriaties Loop oer particles, return and f/m 1 Clear force accumulators F 1 F 2 F 3 F nf Apply forces to particles 2 ote: stiff spring systems can be ery unstable under Euler integration. Simple solutions include heay damping (may not look good), tiny time steps (slow), or better integration (Runge-Kutta is straightforward) Return deriaties to soler 20 5

6 Bouncing off the walls Handling collisions is a useful add-on for a particle simulator. Collision Detection How do you decide when you e made eact contact with the plane? For now, we ll just consider simple point-plane collisions. A plane is fully specified by any point on the plane and its normal ormal and tangential elocity Collision Response To compute the collision response, we need to consider the normal and tangential components of a particle s elocity. before after The response to collision is then to immediately replace the current elocity with a new elocity: The particle will then moe according to this elocity in the net timestep

7 Collision without contact In general, we don t sample moments in time when particles are in eact contact with the surface. There are a ariety of ways to deal with this problem. The most epensie is backtracking: determine if a collision must hae occurred, and then roll back the simulation to the moment of contact. A simple alternatie is to determine if a collision must hae occurred in the past, and then pretend that you re currently in eact contact. Very simple collision response How do you decide when you e had a collision during a timestep? A problem with this approach is that particles will disappear under the surface. We can reduce this problem by essentially offsetting the surface: 3 3 Also, the response may not be enough to bring a particle to the other side of a wall In that case, detection should include a elocity check: More complicated collision response article frame of reference Another solution is to modify the update scheme to: Let s say we had our robot arm eample and we wanted to launch particles from its tip. detect the future time and point of collision reflect the particle within the time-step How would we go about starting the particles from the right place? First, we hae to look at the coordinate systems in the OpenGL pipeline

8 The OpenGL geometry pipeline rojection and modeliew matrices Any piece of geometry will get transformed by a sequence of matrices before drawing: p = M project M iew M model p The first matri is OpenGL s GL_ROJECTIO matri. The second two matrices, taken as a product, are maintained on OpenGL s GL_MODELVIEW stack: M modeliew = M iew M model Robot arm code, reisited Recall that the code for the robot arm looked something like: glrotatef( theta, 0.0, 1.0, 0.0 ); base(h1); gltranslatef( 0.0, h1, 0.0 ); glrotatef( phi, 0.0, 0.0, 1.0 ); upper_arm(h2); gltranslatef( 0.0, h2, 0.0 ); glrotatef( psi, 0.0, 0.0, 1.0 ); lower_arm(h3); All of the GL calls here modify the modeliew matri. ote that een before these calls are made, the modeliew matri has been modified by the iewing transformation, M iew. Computing the particle launch point To find the world coordinate position of the end of the robot arm, you need to follow a series of steps: 1. Figure out what M iew is before drawing your model. 2. Draw your model and add one more transformation to the tip of the robot arm. 3. Compute Mat4f Miew = glgetmodelviewmatri(); gltranslatef( 0.0, h3, 0.0 ); Mat4f particlexform = getworldxform(miew); 4. Transform a point at the origin by the resulting matri. Vec3f particleorigin = particlexform * Vec3f(0,0,0); ow you re ready to launch a particle from that last computed point!

9 Summary What you should take away from this lecture: The meanings of all the boldfaced terms Euler method for soling differential equations Combining particles into a particle system hysics of a particle system Various forces acting on a particle Simple collision detection How to hook your particle system into the coordinate frame of your model 33 9