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Real-time Simulations of Bubbles and Foam within a Shallow Water Framework

Real-time Simulations of Bubbles and Foam within a Shallow Water Framework. 논문 세미나 그래픽스 연구실 윤종철 2008.3.13. 목차. Abstract Introduction Related Work Overview The Shallow Water Equations Simulating Bubbles Spherical vortices Shallow Water Coupling Coalescence and Surface Animation

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Real-time Simulations of Bubbles and Foam within a Shallow Water Framework

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  1. Real-time Simulations of Bubbles and Foam within a Shallow Water Framework 논문 세미나 그래픽스 연구실 윤종철 2008.3.13

  2. 목차 Abstract • Introduction • Related Work • Overview • The Shallow Water Equations • Simulating Bubbles • Spherical vortices • Shallow Water Coupling • Coalescence and Surface Animation • Limitations • Simulating Foam • Smoothed Particle Hydrodynamics • Foam Model • Results • Conclusions

  3. Abstract • shallow water기반 particle model 제안 • 실시간 bubble과 foam 효과를 위해 SPH simulation 사용 • Bubble은 sphericalvortex 사용 • surface tension 포함 SPH와 Bubble과의 상호작용 • High frame rate

  4. 1. Introduction • Goal • 2d shallow water simulation에서 3d 효과 • Contribution • Shallow water 기반 Spherical vortices(3d flow field를 효율적으로 생성)bubble simulation model 제안 • SPH dynamics with surface tension 기반 Foam simulation 제안

  5. 2. Related work • Rapid, Stable Fluid Dynamics for Computer Graphics[KM90] • Dynamic Simulation of Splashing Fluids [OH95]

  6. 2. Related work • Interactive Animation of Ocean Waves [Lov03]

  7. 2. Related work • Stable fluids [Sta99] • Practical Animation of Liquids [FF01]

  8. 2. Related work • Efficient Simulation of Large Bodies of Water by Coupling Two and Three Dimensional Techniques [IGLF06]

  9. 2. Related work • Better with Bubbles: Enhancing the Visual Realism of Simulated Fluid [GH04] • Physics-Based Boiling Simulation [MUM06]

  10. 2. Related work • Simulation of bubbles [ZYP06] • Simulation and rendering of liquid foams [KVG02]

  11. 2. Related work • Particle-Based Fluid Simulation for Interactive Applications [MCG03] • Particle-Based Fluid-Fluid Interaction [MSRG05]

  12. Over view

  13. Over view • 현실 bubble 주변 flow는 very turbulent해서 계산 곤란. Real-timesimulation위해 간략화 • Section 3 • SW equation • Section 4 • Bubble은 particle로 simulation • Water surface, 다른 bubble과상호작용 • Section 5 • SPH로 Foam simulation

  14. 3. The Shallow Water Equations • SW equation은 full NS equation을 simplification하여 유도 • z축을 따라 velocity가 많이변하지 않는다고 가정 • 물 표면으로부터 바닥까지 일정한압력 변화 가정 • Very low viscosity 가정(viscosity term 무시) • 2D simulation은 x-y plane에 대응 • Gravity force는 z축에 작용 • Pressure와 gravity만이 flow를 제어하는 힘

  15. 3. The Shallow Water Equations • H : height of the water • v : horizontal velocity of fluid • g : gravitational force h y x

  16. 4. Simulating Bubbles • Bubble has the following properties: • Position • Velocity • Radius • Volume(or mass) • 상대적으로 낮은 bubble 속도의 flow와강한 surface tension을 가정 • bubble은 구 모양particle로 다룸 • Bubble position은 Euler step으로 update

  17. 4.1. Spherical vortices • Hill의 spherical vortex concept 사용 • [Bat67]에 의하면 bubble 주위에서 experimentally 발견 • bubble 주위 flow의 approximation이지만 사용

  18. 4.1. Spherical vortices • Hill의 spherical vortex는 보통 Polar coordinates 에서 stream function으로 주어짐 • a : radius of spherical vortex • But analytical solution 사용

  19. 4.1. Spherical vortices • Analytical solution [Saf94] • divergence-free velocity field를 생성 • 생성된 field의 Spherical region 밖은 zero vorticity

  20. 4.1. Spherical vortices • Center of spherical vortex : • Bubble center • Bubbleradius • Bubble velocity • Radius of spherical region a=3r a=r 1/2r r a

  21. 4.1. Spherical vortices • Bubble j와 bubble i의 spherical vortex center와의 거리 : • Bubble j에 영향을 주는 velocity : • : linear fall off function • z 방향 속도 :

  22. 4.2. Shallow Water Coupling • Water surface에서 bubble이 터지기 전까지 SW simulation과 bubble과 coupling은 noeffect • Normalized spline kernel W(h,d) 사용 • Bubble 전체가 물에 잠겨있는 동안은 fluid surface를 다른 것으로 대체할 만큼 smooth하다고 가정

  23. 4.2. Shallow Water Coupling • Bubble이 잠겨있는 동안, 로 대체 • 는 water surface를 rendering하기 위한 offset으로만 사용, SW simulation에는 영향 안줌 • However, 일 때, integral of 는 bubble의 volume과 같음

  24. 4.2. Shallow Water Coupling • Surface에서 bubble이 터지면 bubble 주위에 circular wave가 생기도록 변경 • SW simulation의 fluid height에 더함 • (10)은 sw simulation의 질량 보존

  25. 4.2. Shallow Water Coupling • Bubble에 작용하는 힘 • Only buoyancy and velocity forces : • Buoyancy force • : gas와 fluid 사이의 density 차이 ( 사용)

  26. 4.3. Coalescence and Surface Animation • Coalescence of bubbles • 두 bubble i, j의 거리 <ri + rj이면,simulation에서 삭제후, volume m(mi + mj)의 newbubble 생성 • bubble의 다른 속성은 volume에 따라 interpolation • Ex) velocity • Radius • Bubble surface animation • Bubble 이동 방향에 Sinusoidal offset 더함 • Bubblevelocity와 size로 frequency를 정함 • Bubble이 merge될 때, frequency,amplitude 일시적 증가 uj

  27. 4.4. Limitations • fluid의 inertia effect 다룰 수 없음 • Bubble이 fluid에 존재할 동안만, bubble 주위에서만 flow 생성 • bubble없이, SW simulation은 오직 fluid volume • 3차원 flow effect 불가 • 결과 velocity field는 divergence-free하지 않음

  28. 5. Simulating Foam • fluid surface상의 Foam structure는 bubble에 의해 생성 • surface tension effect가 깨지는 것이 delay되어서 생김 • surface tension은 foam bubble이 서로 typical clustering하도록 이끔 그래서 Foamsimulation을 위해 surface tension algorithm과 SPHsimulation 사용

  29. 5.1. SPH • Use scalar kernel function W(r) • Symmetric: W(|x-xi|) • Normalized:  W(x) dx = 1 xi r x

  30. 5.2. Foam Model • Foam의 overall volume을 SPH로 나타냄 • Surface tension은 bubble의 덩어리를 모델링 • SPH particle size는 section4의 bubble의 평균 size로 초기화 • user가 정한 확률로 foam particle 생성 • 삭제된 bubble의 위치에서 foam particle 생성 • SPH particle size는 고정, but virtual size 저장 • Rendering, surface force에 사용 • surface상의 foam bubble은 반 구형 모양 • randomizedlife time • 터지면 equation(10)으로 wave 생성

  31. 5.2. Foam Model • Surface tension • Normal 생성 위한 Colorfiled function 사용 • Normal의 divergence는 curvature, surface tension force의세기를 구하기 위해 사용 • 는 surface tension의 세기를 제어

  32. Surface tension Consider the following iso-curves (level sets). What sign does the laplacian have? 0 <0 >0

  33. Surface tension [MCG03] • Color field : smoothed color field • 1 at particle locations • 0 everywhere else • Gradient field of smoothed color field : • Curvature of the surface : • Surface tension :

  34. 5.2. Foam Model • Foam이 surface에서 떠다니게 하기 위해, SPH update step동안 각 particle에 gravity 방향에 힘을 더함 • 위치 xi에서 SPH particle을 height field의 z쪽 위치로 particle을 이동시키기위한 힘

  35. 6. Results • Core2Duo CPU(2.4), geforce 7900 gpu(single thread).

  36. 7. Conclusions • Bubble과 foam을 real-time simulation 알고리즘 제안 • SW와 particle기반 bubble,foam과의 coupling으로 높은 performance • Spherical vortex를 사용, 효율적으로 vortex를 생성 • SPH simulation으로 foam의 clustering을 제안 • Future work • Foam을 랜더링하기 위한 shader를 만들고 싶다. • equation(15)로 foam의 thick layer 다루기.

  37. END

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