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Numerical Integration

Explore how games use differential equations for physics simulations and solve complex equations through numerical integration. Learn about projectile and mass-spring motions models, implicit vs explicit Euler methods, Verlet integrator, and more. Enhance your understanding of stability, performance, and accuracy in integration techniques.

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Numerical Integration

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  1. Numerical Integration Erin CattoBlizzard Entertainment

  2. Basic Idea • Games use differential equations for physics. • These equations are hard to solve exactly. • We can use numerical integration to solve them approximately.

  3. Overview • Differential Equations • Numerical Integrators • Demos

  4. Typical Game Loop Start t = 0 Choose Dt Player Input t = t + Dt Simulate Dt Render

  5. Simulation • Animation • AI • Physics • Differential Equations

  6. What is a differential equation? • An equation involving derivatives. rate of change of a variable = a function

  7. Anatomy of Differential Equations • State • Dependent variables • Independent variables • Initial Conditions • Model • The differential equation itself

  8. Projectile Motion • State • Independent variable: time (t) • Dependent variables: position (y) and velocity (v) • Initial Conditions • t0, y0, v0

  9. Projectile Motion • Model: vertical motion y x

  10. First Order Form • Numerical integrators need differential equations to be put into a special format.

  11. First Order Form • Arrays of equations work too.

  12. Projectile MotionFirst Order Form y x

  13. Projectile MotionFirst Order Form y x

  14. Mass-Spring Motion • Consider the vertical motion of a character.

  15. Mass-Spring Motion • State • time: t • position: x • velocity: v • Initial Conditions • t0, x0, v0 x

  16. Mass-Spring Motion • Idealized model m x k ground

  17. Mass-Spring Motion • First Order Form

  18. Solving Differential Equations • Sometimes we can solve our DE exactly. • Many times our DE is too complicated to be solved exactly.

  19. Hard Problems • Nonlinear equations • Multiple variables

  20. Hard Problems • Projectile with air resistance

  21. Hard Problems • Mass-spring in 3D

  22. Hard Problems • Numerical integration can help! • Handles nonlinearities • Handles multiple variables

  23. Numerical Integration • Start with our first order form

  24. t t+h A Simple Idea • Approximate the slope. x t

  25. A Simple Idea • Forward difference:

  26. A Simple Idea • Shuffle terms:

  27. A Simple Idea • Using this formula, we can make a time step h to find the new state. • We can continue making time steps as long as we want. • The time step is usually small

  28. Explicit Euler • This is called the Explicit Euler method. • All terms on the right-hand side are known. • Substitute in the known values and compute the new state.

  29. What If … • This is called the Implicit Euler method. • The function depends on the new state. • But we don’t know the new state!

  30. Implicit Euler • We have to solve for the new state. • We may have to solve a nonlinear equation. • Can be solved using Newton-Raphson. • Usually impractical for games.

  31. Implicit vs Explicit • Explicit is fast. • Implicit is slow. • Implicit is more stable than explicit. • More on this later.

  32. Opening the Black Box • Explicit and Implicit Euler don’t know about position or velocity. • Some numerical integrators work with position and velocity to gain some advantages.

  33. The Position ODE • This equation is trivially linear in velocity. • We can exploit this to our advantage.

  34. Symplectic Euler • First compute the new velocity. • Then compute the new position using the new velocity.

  35. Symplectic Euler • We get improved stability over Explicit Euler, without added cost. • But not as stable as Implicit Euler

  36. Verlet • Assume forces only depend on position. • We can eliminate velocity from Symplectic Euler.

  37. Verlet • Write two position formulas and one velocity formula.

  38. Verlet • Eliminate velocity to get:

  39. Newton • Assume constant force: • Exact for projectiles (parabolic motion).

  40. Demos • Projectile Motion • Mass-Spring Motion

  41. Integrator Quality • Stability • Performance • Accuracy

  42. Stability • Extrapolation • Interpolation • Mixed • Energy

  43. Performance • Derivative evaluations • Matrix Inversion • Nonlinear equations • Step-size limitation

  44. Accuracy • Accuracy is measured using the Taylor Series.

  45. Accuracy • First-order accuracy is usually sufficient for games. • You can safely ignore RK4, BDF, Midpoint, Predictor-Corrector, etc. • Accuracy != Stability

  46. Further Reading &Sample Code • http://www.gphysics.com/downloads/ • Hairer, Geometric Numerical Integration

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