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SC1 P HYSICALLY B ASED M ODELING Overview Overview One Lousy - PowerPoint PPT Presentation

Apr 18, 2023 •230 likes •500 views

S IGGRAPH 2001 C OURSE N OTES SC1 P HYSICALLY B ASED M ODELING Overview Overview One Lousy Particle Particle Systems Forces: gravity, springs Implementation and Interaction Simple collisions S IGGRAPH 2001 C OURSE N OTES

  1. S IGGRAPH 2001 C OURSE N OTES SC1 P HYSICALLY B ASED M ODELING

  2. Overview Overview • One Lousy Particle • Particle Systems • Forces: gravity, springs … • Implementation and Interaction • Simple collisions S IGGRAPH 2001 C OURSE N OTES SC2 P HYSICALLY B ASED M ODELING

  3. A Newtonian Particle A Newtonian Particle • Differential equation: f = ma • Forces can depend on: – Position, Velocity, Time = f x x t ( , , ) ɺ x ɺɺ m S IGGRAPH 2001 C OURSE N OTES SC3 P HYSICALLY B ASED M ODELING

  4. Second Order Equations Second Order Equations = f x x t Not in our standard form ( , , ) ɺ x ɺɺ because it has 2nd m derivatives x v =  Add a new variable, v, to get ɺ  = a pair of coupled 1st order v f m  ɺ equations. S IGGRAPH 2001 C OURSE N OTES SC4 P HYSICALLY B ASED M ODELING

  5. Phase Space Phase Space x Concatenate x and v to make   a 6-vector: Position in Phase   v   Space. x   ɺ Velocity in Phase Space :   v another 6-vector.   ɺ x v     ɺ =     A vanilla 1st-order differential v f m     ɺ equation. S IGGRAPH 2001 C OURSE N OTES SC5 P HYSICALLY B ASED M ODELING

  6. Particle Structure Particle Structure x Position Position in v Phase Space Velocity f Force Accumulator m mass S IGGRAPH 2001 C OURSE N OTES SC6 P HYSICALLY B ASED M ODELING

  7. Solver Interface Solver Interface x 6 Dim(state) v x v f Get/Set State v m f m Deriv Eval S IGGRAPH 2001 C OURSE N OTES SC7 P HYSICALLY B ASED M ODELING

  8. Particle Systems Particle Systems n time particles x x x x x x v v v v v v … f f f f f f m m m m m m S IGGRAPH 2001 C OURSE N OTES SC8 P HYSICALLY B ASED M ODELING

  9. Solver Interface Solver Interface Particle System n time particles Dim(State) Dim(State) Diffeq Solver Get/Set State Get/Set State 6n x 1 v 1 x 2 v 2 x n v n v 1 f 1 m 1 v 2 f 2 v n f n Deriv Eval Deriv Eval m 2 m n S IGGRAPH 2001 C OURSE N OTES SC9 P HYSICALLY B ASED M ODELING

  10. Deriv Eval Loop Deriv Eval Loop • Clear forces – Loop over particles, zero force accumulators. • Calculate forces – Sum all forces into accumulators. • Gather – Loop over particles, copying v and f /m into destination array. S IGGRAPH 2001 C OURSE N OTES SC10 P HYSICALLY B ASED M ODELING

  11. Forces Forces • Constant gravity • Position/time dependent force fields • Velocity-Dependent drag • n-ary springs S IGGRAPH 2001 C OURSE N OTES SC11 P HYSICALLY B ASED M ODELING

  12. Force Structures Force Structures • Unlike particles, forces are heterogeneous. • Force Objects: – black boxes – point to the particles they influence – add in their own forces (type dependent) • Global force calculation: – loop, invoking force objects S IGGRAPH 2001 C OURSE N OTES SC12 P HYSICALLY B ASED M ODELING

  13. Particle Systems, with forces Particle Systems, with forces particles n time forces nforces … F x x x F F F F v v v m … f f f A list of force objects to invoke m m S IGGRAPH 2001 C OURSE N OTES SC13 P HYSICALLY B ASED M ODELING

  14. Gravity Gravity Force Law: Particle system f G = m grav F G sys x apply_fun p v f m p->f += p->m * F->G p->f += p->m * F->G S IGGRAPH 2001 C OURSE N OTES SC14 P HYSICALLY B ASED M ODELING

  15. Viscous Drag Viscous Drag Force Law: Particle system f v = − k drag drag F k sys x apply_fun p v f m p->f -= F->k * p->v p->f -= F->k * p->v S IGGRAPH 2001 C OURSE N OTES SC15 P HYSICALLY B ASED M ODELING

  16. Damped Damped Spring Force Law: Spring     v x x ∆ ⋅∆ ∆ ( ) f x = − ∆ − + k r k     s d 1 x x ∆ ∆       Particle system f f = − 2 1 x F p1 sys v kd x f v ks apply_fun p2 f m m S IGGRAPH 2001 C OURSE N OTES SC16 P HYSICALLY B ASED M ODELING

  17. x x x Deriv Eval Loop Deriv Eval Loop 1 v v v m … f f f … F F F F F m m 2 Clear Force Invoke apply_force Accumulators functions x x x 3 v v v m … f f f Return [v, f/ m,… ] to solver. m m S IGGRAPH 2001 C OURSE N OTES SC17 P HYSICALLY B ASED M ODELING

  18. Solver Interface Solver Interface Dim(state) Solver System Get/Set State You are Here Deriv Eval S IGGRAPH 2001 C OURSE N OTES SC18 P HYSICALLY B ASED M ODELING

  19. Bouncing off the Walls Bouncing off the Walls • Later: rigid body collision and contact. • For now, just simple point-plane collisions. • Add-ons for a particle simulator. S IGGRAPH 2001 C OURSE N OTES SC19 P HYSICALLY B ASED M ODELING

  20. Normal and Tangential Components Normal and Tangential Components V N V T V N V N = ( N ⋅ V ) N V T = V - V N S IGGRAPH 2001 C OURSE N OTES SC20 P HYSICALLY B ASED M ODELING

  21. Collision Detection Collision Detection ( ) X - P ⋅ N < ε X N ⋅ V < 0 V N P • Within ε of the wall. • Within ε of the wall. • Heading in. • Heading in. S IGGRAPH 2001 C OURSE N OTES SC21 P HYSICALLY B ASED M ODELING

  22. Collision Response Collision Response V N -k r V N V T V T V ′ V Before After V ′ = V T - k r V N S IGGRAPH 2001 C OURSE N OTES SC22 P HYSICALLY B ASED M ODELING

  23. Conditions for Contact Conditions for Contact ( ) X - P ⋅ N < ε F X N ⋅ V < ε V P • On the wall N • On the wall • Moving along the wall • Moving along the wall • Pushing against the wall • Pushing against the wall S IGGRAPH 2001 C OURSE N OTES SC23 P HYSICALLY B ASED M ODELING

  24. Contact Force Contact Force F ′ = F T F -F N The wall pushes back, cancelling the normal component of F. F ′ (An example of a N constraint force. ) S IGGRAPH 2001 C OURSE N OTES SC24 P HYSICALLY B ASED M ODELING

  25. Basic 2-D Interaction Basic 2-D Interaction Cursor Cursor Operations: X X X Nail Mouse Nail Mouse – Create Spring Spring – Attach – Drag – Nail S IGGRAPH 2001 C OURSE N OTES SC25 P HYSICALLY B ASED M ODELING

  26. Try this at home! Try this at home! The notes give you everything you need to build a basic interactive mass/spring simulator—try it. S IGGRAPH 2001 C OURSE N OTES SC26 P HYSICALLY B ASED M ODELING

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