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使用波動方程式的擬真和互動式水面模擬

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. Introduction. What can fluid simulation do for us? What is the difficulty of the topic?. Objective. Real-time and visual quality Interact with users. Height Field. h.

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使用波動方程式的擬真和互動式水面模擬

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  1. 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

  2. Introduction • What can fluid simulation do for us? • What is the difficulty of the topic?

  3. Objective • Real-time and visual quality • Interact with users

  4. Height Field h present water surface as a 2D function h(x,y) 2D array h[i,j] y x h j i

  5. Height Field • Pros: Process from 3D to 2D reduce computational time • Cons: One location (x,y) maps to exactly one height no breaking waves

  6. Hello World!

  7. Algorithm • Use 2 arrays A[N][M] , B[N][M] of the same size • Initialize A[i][j] with arbitrary values • Initialize 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

  8. Algorithm • It simulate the wave propagation phenomenon travel direction

  9. Pros and Cons • Pros: 1). Simple 2). Computational time is independentof wave source • Cons: 1). Unnatural shape of wave propagation 2). Slow wave propagation demo

  10. Addition of Sine Waves

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

  12. Algorithm • summate all sine waves • v = λ x f is a constant , and λ increases with t • Phase increases constantly to animate wave propagation

  13. Pros and Cons • Pros: Better look of shape of wave propagation • Cons: Computational time is proportional to wave source number demo

  14. Wave Equation

  15. Governing Equation h: height of water surface t: time c: velocity x: position of x direction y: position of y direction

  16. Damping • With damping • Where k is damping factor

  17. Finite Difference Where h is a insignificant and non-zero value

  18. Discrete Wave Equation

  19. Discrete Wave Equation

  20. stability • How 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

  21. Result • CPU : AMD Athlon™3500+2.2GHZ • Graphics Card: Radeon HD 3650 • RAM : 1G • About 60 fps in grid size of 360 x 360 • demo 1demo 2

  22. END • Thanks for listening.

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