使用波動方程式的擬真和互動式水面模擬

496410032 吳金龍Advisor: Damon Shing-Min Liu

IntroductionWhat can fluid simulation do for us?

What is the difficulty of the topic?

ObjectiveReal-time and visual quality

Interact with users

Height Field present water surface as a

2D function h(x,y)

2D array h[i,j]

x

y

h

h

i

j

Height FieldPros: Process from 3D to 2D reduce computational time

Cons: One location (x,y) maps to exactly one height no breaking waves

Hello World!

AlgorithmUse 2 arrays A[N][M] , B[N][M] of the same sizeInitialize A[i][j] with arbitrary valuesInitialize B[i][j] with 0

loop

forall i,j do B[i,j] += (A[i-1,j] + A[i+1,j] + A[i,j-1]

+ A[i,j+1])/4- A[i,j];

forall i,j do B[i,j] *= 0.99;

forall i,j do A[i,j] += B[i,j];

endloop

AlgorithmIt simulate the wave propagation

phenomenontravel direction

Pros and Cons Pros: 1). Simple

2). Computational time is independent of wave source

Cons: 1). Unnatural shape of wave propagation 2). Slow wave propagation

demo

Addition of Sine Waves

Governing Equation

h : height of water surface A : amplitude ω : angular velocity t : time φ : phase λ : wave length f : frequency

)cos(0

iii

sourceNum

i

twAh

Algorithmsummate all sine waves

v = λ x f is a constant , and λ increases with t

Phase increases constantly to animate wave propagation

Pros and Cons Pros: Better look of shape of wave

propagation

Cons: Computational time is proportional to wave source number

demo

Wave Equation

Governing Equation

)yh

xh

(cth

2

2

2

22

2

2

h: height of water surfacet: timec: velocityx: position of x directiony: position of y direction

DampingWith damping

Where k is damping factor

)yh

xh

(cth

kth

2

2

2

22

2

2

Finite Difference

hh

)f(xh)f(xf ii

2iii

h)h)f(x)2f(xh)f(x

(2

2

h

f

Where h is a insignificant and non-zero value

Discrete Wave Equation

yx

y

hhh

y

h

x

hhh

x

h

t

hhh

t

h

nyx

nyx

nyx

nyx

nyx

nyx

nyx

nyx

nyx

)(

2

)(

2

)(

2

2

1,,

1,

2

2

2

1,,

1,

2

2

2

1,,

1,

2

2

Discrete Wave Equation

))4(2

1

4(

))(1(

)4(

))(1(

,1,11,11,11,1

,1,1,,1,12

22

,,,,

,1,1,,1,12

22

,,,,

tyx

tyx

tyx

tyx

tyx

tyx

tyx

tyx

tyx

tyx

ttyx

tyx

tyx

ttyx

tyx

tyx

tyx

tyx

tyx

ttyx

tyx

tyx

ttyx

hhhhh

hhhhhx

tc

hhtkhh

or

hhhhhx

tc

hhtkhh

stabilityHow to choose ?

Courant-Friedrichs-Lewy stability condition holds which says that information must not travel more than one grid cell per time step, i.e. c t < x

2

22

x

tc

ResultCPU : AMD Athlon™ 3500+ 2.2 GHZGraphics Card: Radeon HD 3650RAM : 1 GAbout 60 fps in grid size of 360 x 360

demo 1 demo 2

END

Thanks for listening.