Differential Equations and Particle Systems
Transcript of Differential Equations and Particle Systems
![Page 1: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/1.jpg)
Differential Equations
&
Particle Systems
Thanks to Trueille, Popovic, Baraff, Witkin
![Page 2: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/2.jpg)
Physics-based Animation
http://physbam.stanford.edu/~fedkiw/animations/large_pile.avi
![Page 3: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/3.jpg)
![Page 4: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/4.jpg)
![Page 5: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/5.jpg)
Describing Physics
source: http://people.rit.edu/andpph/exhibit-8.html
![Page 6: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/6.jpg)
What variables do we need?
• Position
• Velocity
• Radius
• Mass
• Racquet Info
Dynamic Static
http://p
eople
.rit.e
du/a
ndpph/e
xhib
it-8
.htm
l
• Rotation?
![Page 7: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/7.jpg)
What Happens Next?
• Position
• Velocity
http://p
eople
.rit.e
du/a
ndpph/e
xhib
it-8
.htm
l
?
Discrete Time:
Continuous Time:
![Page 8: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/8.jpg)
Differential Equations
![Page 9: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/9.jpg)
![Page 10: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/10.jpg)
![Page 11: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/11.jpg)
![Page 12: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/12.jpg)
![Page 13: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/13.jpg)
![Page 14: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/14.jpg)
![Page 15: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/15.jpg)
Two Problems
• Accuracy
• Instability
![Page 16: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/16.jpg)
Accuracy
Consider the equation:
What do the integral curves
look like?
![Page 17: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/17.jpg)
![Page 18: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/18.jpg)
Problem 2: Instability
• Consider the following system:
![Page 19: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/19.jpg)
Problem 2: Instability
to Neptune!
![Page 20: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/20.jpg)
Accuracy of Euler Method
Consider Taylor Expansion about x(t)...
constant linear everything
else
Euler step error
Euler’s method has error O(h2)... first order.
How can we get to O(h3) error?
![Page 21: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/21.jpg)
![Page 22: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/22.jpg)
The Midpoint Method
• Also known as second order Runge-Kutta:
![Page 23: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/23.jpg)
![Page 24: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/24.jpg)
4th-Order Runge-Kutta
Very popular
![Page 25: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/25.jpg)
q-Stage Runge-Kutta
General Form:
where:
Find the constant that ensure accuracty O(hn).
![Page 26: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/26.jpg)
![Page 27: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/27.jpg)
![Page 28: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/28.jpg)
![Page 29: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/29.jpg)
![Page 30: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/30.jpg)
![Page 31: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/31.jpg)
![Page 32: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/32.jpg)
![Page 33: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/33.jpg)
Particle Systems
![Page 34: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/34.jpg)
![Page 35: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/35.jpg)
![Page 36: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/36.jpg)
![Page 37: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/37.jpg)
![Page 38: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/38.jpg)
![Page 39: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/39.jpg)
![Page 40: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/40.jpg)
![Page 41: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/41.jpg)
![Page 42: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/42.jpg)
![Page 43: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/43.jpg)
![Page 44: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/44.jpg)
![Page 45: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/45.jpg)
![Page 46: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/46.jpg)
![Page 47: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/47.jpg)
![Page 48: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/48.jpg)
![Page 49: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/49.jpg)
![Page 50: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/50.jpg)
![Page 51: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/51.jpg)
![Page 52: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/52.jpg)
![Page 53: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/53.jpg)
![Page 54: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/54.jpg)
![Page 55: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/55.jpg)
![Page 56: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/56.jpg)
![Page 57: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/57.jpg)
![Page 58: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/58.jpg)
![Page 59: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/59.jpg)
![Page 60: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/60.jpg)
Example
http://www.youtube.com/watch?v=3_fLO4xjTqg
![Page 61: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/61.jpg)
![Page 62: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/62.jpg)
![Page 63: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/63.jpg)
![Page 64: Differential Equations and Particle Systems](https://reader030.fdocuments.us/reader030/viewer/2022012802/61bd16df61276e740b0f3d4d/html5/thumbnails/64.jpg)