Water droplet simulation
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Water Droplet Simulation. Avneesh Sud. Problem. Realistic Animation of individual water droplets on surfaces Lot of work on ‘large’ water bodies (Napier-Stokes, Stam) Water droplets tougher Gravity Interfacial , surface tension Air resistance Accurate rendering

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Problem
Problem

  • Realistic Animation of individual water droplets on surfaces

    • Lot of work on ‘large’ water bodies (Napier-Stokes, Stam)

  • Water droplets tougher

    • Gravity

    • Interfacial , surface tension

    • Air resistance

    • Accurate rendering

  • Two Problems : Flow and Rendering


Work presented
Work Presented

  • Kaneda93 : Water Droplets on Glass Plate

  • Kaneda96 : Droplets on a Curved Surface


Flow of water droplets
Flow of Water Droplets

  • Physics studied by Janosi89

  • Droplet runs down glass plate if mass exceeds critical mass

  • Route of stream determined by impurities and presence of other droplets

  • Traces of water left behind on surface : “Wetting”


Simulation of flow
Simulation of Flow

  • Surface divided into a nxn lattice

  • An affinity for water fi,jassigned to each lattice point

    • Depends on surface properties, impurities etc.


Simulation of flow1
Simulation of Flow

  • When does droplet move?

  • Which direction does it move in?

  • What speed does it move with?

  • How much moves/left behind?

  • When does it stop?


Critical mass
Critical Mass

  • Each lattice point associated with a mass mi,j

  • Droplet moves if

msc() is the static critical weight when inclination angle is 


Direction of motion
Direction of Motion

  • From (i,j) to (i-1,j+1), (i,j+1) or (i+1,j+1)

  • If water exists in one of the 3 directions, then moves to that direction

    • If exists in more than 1 direction, then to (i,j+1)

    • If not exist in (i,j+1), then to direction with larger mass

  • If not exist in any direction, then tendency to move in direction (i+k,j+1) is

    w1(,k) is the inclination weighting function


Speed of water
Speed of water

  • Depends in wetness of new direction and inclination

  • Acceleration ai,j() can be dry acceleration ad() or wet acceleration aw(),

    aw() > ad()

    t is time from generation/collision


Mass of moving water
Mass of Moving Water

  • Mass of remaining water m’i,jdepends on fi,j, mmin, mmax

  • Mass of water at new position


Merging
Merging

  • Conservation of mass and momentum


Stopping
Stopping

  • With no water ahead, decelerates and stops when mass is less than dynamic critical mass mdc()

  • Speed during deceleration given by


Shape of droplet
Shape of Droplet

  • Shape of static droplet on glass plate determined by interfacial, surface tensions (de Gennes 85)

  • Dynamic droplets complicated – different contact angles at head & tail, several contact angles to minimize energy


Shape of droplet1
Shape of Droplet

  • Approximated by a sphere

  • Radius of sphere dependent on mass


Rendering
Rendering

  • Background textures generated for environment mapping

    • Front face at higher resolution


Rendering1
Rendering

  • Tracing reflected/refracted rays

    • Most rays either reflected or refracted : follow one principle direction


Rendering2
Rendering

  • Determining pixel color using the background texture at the intersection point




Curved surfaces
Curved Surfaces

  • Modeled as Bezier Patches

  • Discretized to a quad mesh – approximated by a plane

  • A more generic method that reduces discretization artifacts


Critical mass1
Critical Mass

  • Movement Test : Sum of external (gravity, wind) forces greater than affinity force

Fi,j is the static critical force

s is degree of affinity coefficient


Direction of motion1
Direction of Motion

  • “Roulette” of 8 adjacent meshes used

  • Probability assigned to each mesh

  • Using probabilities allows to simulate random ‘meandering’ of water



Direction of motion3
Direction of Motion

  • Weight of each direction determined by

    • Newton’s Laws of Motion

    • Degree of affinity for water

    • Wetness of neighboring meshes



Newton s laws
Newton’s Laws

  • Compute principle velocity vector VP from Newton's mechanics.

    Solve to find position Pwhere droplet crosses boundary and the velocity VPat crossing point


Newton s laws1
Newton’s Laws

  • Probabilities distributed between 2 nearest neighboring directions of VP (l, l+1)


Affinity
Affinity

  • Probability distributed to directions whose angle from VPis less than 90 deg

ckis degree of affinity in neighboring mesh

u(x) is unit step function


Wetness
Wetness

  • Probability distributed to directions whose angle from principle vector is less than 90 deg

    • Prevents a droplet from suddenly changing direction back to a wet spot

g(d*k) is a function to check wetness


Determining direction
Determining direction

  • All 3 probabilities weighted and summed up

    1 controls regularity, 2 controls meandering

  • Direction depends on generated random number r

    p is the desired direction of movement


Speed of travel
Speed of Travel

  • Project acceleration in direction of principle vector

  • Internal forces depend on affinity


Mass of moving drop
Mass of moving drop

  • “Wetting” : Mass of remaining water

    h(c) is a function for remaining water based on affinity

  • “Merging” : Follow conservation of mass, momentum


Algorithm
Algorithm

  • Make a discretized surface model, specify degree of affinity for each mesh element

  • Put new droplets at mesh points, specify weight and initial speed

  • Initialize timekeepers for all droplets to accumulate time taken for a droplet to travel from one mesh to another

  • If droplet moves, goto step (5) else repeat (4) for all droplets

  • Determine mesh droplet will visit next

  • Calculate time required for droplet to move to next mesh

  • Move the droplet to next mesh based on timekeeper value

  • Repeat steps (4)-(7) for all droplets till end of frame

  • Repeat (2)-(8) for duration of animation


Rendering3
Rendering

  • Similar to glass plate –efficient bezier intersection tests used

  • Higher quality method based on metaballs

    • Efficient intersection between ray and metaball using bezier clipping




References
References

  • Janosi IM, Horvath VK Dynamics of Water Droplets on a Window Pane. Physical Review 40(9): 5232-5237 (1989)

  • De gennes PG. Wetting: Statucs and Dynamics. Rev. Mod. Phys. 57(3): 827-863 (1985)

  • Kaneda K, Kagawa T, Yamashita H. Animation of Wwater Droplets on a Glass Plate. Proc. Computer Animation’93: 177-189 (1993)

  • Kaneda K, Zuyama Y, Yamashita H, Nishita T. Animation of Water Droplet Flow on Curved Surfaces. Proc. PACIFIC GRAPHICS '96: 50-65 (1996).

  • Yu YJ, Jung HY, Cho HG. A New Rendering Techniques for Water Droplet using Metaball in the Gravitation Force. WCGS'98 (1998)


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