HW2: Cloth & Fluid Simulation Rigid Body Dynamics, Collision Response, & Deformation Last Time? Today • Navier-Stokes • Rigid Body Dynamics Equations • Collision Response • Conservation of • Non-Rigid Objects Momentum & Mass • Finite Element Method • Incompressible Flow • Deformation • Level-of-Detail Rigid Body Dynamics Degree of Freedom (DOF) • How do we simulate this object’s motion over time? • Rotations: • We could discretize the object into many particles… – But a rigid body does not deform – Only a few degrees of freedom f1(t) • Instead, we use f (t) v(t) only one particle 2 at the center of mass x(t) • Compute net force Net Torque f (t) & net torque 3 Net Force • Translations count too… 6 DOF Nice Reference Material: http://www.pixar.com/companyinfo/research/pbm2001/ 1 Collisions Collision Response • Detection • tangential velocity vt unchanged • Response • normal velocity vn reflects: • Overshooting problem v N (when we enter the solid) • coefficient of restitution – 1 for elastic – 0 for plastic • change of velocity = -(1+є)v • change of momentum Impulse = -m(1+є)v Collisions - Overshooting Energy & Rigid Body Collisions • Usually, we detect collision when it’s too late: • Total Energy = we’re already inside Kinetic Energy + Potential Energy + • Solutions: back up Rotational Energy – Compute intersection point backtracking • Total Energy stays – Compute response there constant if there is no – Advance for remaining fixing damping and no friction fractional time step • Rotational Energy • Other solution: is constant between Quick and dirty fixup collisions – Just project back to object closest point http://www.myphysicslab.com/ collision.html Collision Between Two Objects Center of Mass & Moment of Inertia • Suppose a vertex on body A • Center of Mass: mean location of all mass is colliding into an edge of in the system body B at point P. Define • Moment of Inertia: the following variables: a measure of an object's resistance http://en.wikipedia.org/wiki/Fosbury_Flop to changes to its rotation • If a solid cylinder & a hollow tube have the same radius & the same Liu & Popović mass, which will reach the bottom of the ramp first? http://solomon.physics.sc.edu/~tedeschi/demo/demo12.html http://www.myphysicslab.com/collision.html http://hyperphysics.phy-astr.gsu.edu/hbase/hoocyl2.html 2 Advanced Collisions Rigid Body Dynamics • What about Friction? • Physics • What if the contact between two objects is not a – Velocity single point? – Acceleration • What if more than two objects collide – Angular simultaneously? Momentum • Collisions • Friction from: Darren Lewis http://www-cs-students.stanford.edu/~dalewis/cs448a/rigidbody.html Victor J. Milenkovic & Harald Schmidl Collisions Optimization-Based Animation Today SIGGRAPH 2001 • We know how to simulate • Rigid Body Dynamics bouncing really well • Collision Response • But resting • Non-Rigid Objects collisions • Finite Element Method are harder • Deformation to manage • Level-of-Detail Guendelman, Bridson & Fedkiw Nonconvex Rigid Bodies with Stacking SIGGRAPH 2003 Simulation of Non-Rigid Objects Strain & Stress • We modeled string & cloth using mass-spring • Stress systems. Can we do the same? – the internal distribution of forces within a body that • Yes… balance and react to the • But a more physically accurate model uses loads applied to it volumetric elements: – normal stress & shear stress • Strain http://en.wikipedia.org/wiki/Image:Stress_tensor.png – material deformation caused by stress. – measured by the change in length of a line or by the change in angle between two lines Image from O’Brien et al. 1999 3 Finite Element Method Today • To solve the continuous problem • Rigid Body Dynamics (deformation of all points of the object) • Collision Response – Discretize the problem – Express the interrelationship • Non-Rigid Objects – Solve a big linear system • Finite Element Method • More principled than Mass-Spring • Deformation • Level-of-Detail object finite elements large matricial system Diagram from Debunne et al. 2001 Gilles Debunne , Mathieu Desbrun, Level of Detail Marie-Paule Cani, & Alan H. Barr Multi-Resolution Deformation Dynamic Real-Time Deformations using Space & Time Adaptive Sampling • Use Voronoi diagrams to match parent & child vertices. • Interactive shape SIGGRAPH 2001 deformation • Interpolate values for inactive interface vertices from • Use high- active parent/child vertices resolution model only in areas of extreme deformation • Need to avoid interference of vibrations between simulations at different resolutions Debunne et al. “Dynamic Real-Time Deformations using Space & Time Adaptive Sampling”, 2001 Mueller, Dorsey, McMillan, Pre-computation & Simulation Multiple Materials Jagnow, & Cutler Stable Real-Time Deformations • FEM matrix pre-computed Symposium on Computer Animation 2002 • Level of detail coupling pre-computed for rest topology • What to do if connectivity of elements changes? – Cloth is cut or torn – Surgery simulation 4 Tree Stump Images from Cutler et al. 2002 Image from Cutler et al. 2002 Readings for Today: “Deformable Objects Alive!” Coros, Martin, Thomaszewski, Schumacher, & Sumner, SIGGRAPH 2012 “Coupling Water and Smoke to Thin Deformable and Rigid Shells”, Guendelman, Selle, Losasso, & Fedkiw, SIGGRAPH 2005. Image from Cutler et al. 2002 How to read a research paper? Components of a well-written research paper? (especially an advanced paper in a new area) • Motivation/context/related work • Multiple readings are often necessary • Contributions of this work • Don't necessarily read from front to back • Clear description of algorithm • Lookup important terms – Sufficiently-detailed to allow work to be reproduced • Target application & claimed contributions – Work is theoretically sound (hacks/arbitrary constants discouraged) • Experimental procedure • Results • How well results & examples support the claims – well chosen examples • Scalability of the technique (order notation) – clear tables/illustrations/visualizations • Limitations of technique, places for future research • Conclusions • Possibilities for hybrid systems with other work – limitations of the method are clearly stated 5 … or read the other paper for today Reading for Friday: (the one that you didn’t read) • James O’Brien & Jessica Hodgins “Graphical Modeling and Animation of Brittle Fracture” SIGGRAPH 1999. • Fracture threshhold • Material properties • Remeshing • Parameter tuning – need connectivity info! 6 .