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研究生:邱志偉
研究生(外文):Chih-Wei Chiu
論文名稱:劇烈動態液體動畫
論文名稱(外文):Modeling Highly-Deformable Liquid
指導教授:莊榮宏莊榮宏引用關係林正中林正中引用關係
指導教授(外文):Jung-Hong ChuangCheng-Chung Lin
學位類別:碩士
校院名稱:國立交通大學
系所名稱:資訊工程系
學門:工程學門
學類:電資工程學類
論文種類:學術論文
論文出版年:2002
畢業學年度:90
語文別:英文
論文頁數:65
中文關鍵詞:液體模擬
外文關鍵詞:liquidsimulation
相關次數:
  • 被引用被引用:0
  • 點閱點閱:250
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  • 收藏至我的研究室書目清單書目收藏:0
製作複雜表面變化的液體動畫是計算機圖學中的一大挑戰。這些液體通常都以需要大量計算及儲存空間的粒子系統(particle systems)或是分子動力系統(molecular dynamics)模擬。除了這些缺點外,目前並沒有簡單並及有效率的方法能從這些點骨架的模型中產生平滑的表面。
我們提出一個結合容積追蹤(volume tracking)及平滑粒子動力學(smoothed particle hydrodynamics, SPH)的方法來模擬這些液體。我們使用 marker-and-cell (MAC) 來解三維的 Navier-Stokes equations,及 volume-of-fluid (VOF) 演算法以一個密度容積(density volume)來表示液體。密度容積隨著時間的變化得以由每個計算網格(computational cell)進出的流量推算。只需要用一個 convection equation 就能夠改變整個密度容積。在液體表面動態劇烈的區域,我們加入 SPH 的粒子來捕捉細部的變化,以彌補密度容積精細度的不足。由於 SPH 能模擬流體力學的式子,這些 SPH 粒子的動態遠比其它同樣使用粒子的方法更逼真。
我們從原本的密度容積內插得到另一個密度容積,並且將粒子的體積換算轉成密度,最後由這個更平滑且更精細的密度容積中得到液體的等位面(iso-surface),比起之前使用 field function 得到或是直接以球體表示的液體表面更為平滑。

Modeling liquid with complex surfaces is a great challenge in computer graphics. These liquids are often simulated by particle systems or molecular dynamics which requires a huge amount of computational cost and storage. Besides, there is no straightforward way to produce a smooth liquid surface from these particle skeletons.
We proposed a hybrid approach combining the volume tracking method and smoothed particle hydrodynamics (SPH) to model liquids with highly deformable surfaces. The 3-D Navier-Stokes equations are solved by the marker-and-cell (MAC) method and a density volume representing the liquid is evolved over time and space by the volume-of-fluid (VOF) method. The VOF method uses a finite-volume discretization and maintaining a value for the liquid volume in each grid cell. The advection of the volumes is calculated by using cell-face fluxes, in which the liquid volume that leaves one cell is exactly the same as the liquid volume that enters the adjacent cells. Only a scalar convective equation needs to be solved to evolve the liquid surface forward in time and space. Potentially under-resolved effects are captured by SPH particles generated in that high-deformable regions near the liquid surface. Dynamics of these particles provides a better fieldity than that of other particle methods due to their capability of approximating the equations of fluid mechanics.
Particle volumes are incorporated into the interpolated volume fractions before an unified iso-surface is extracted. As a result the liquid surface is more smooth than those of previous researches which rendered the splash by coating the particles with a field functions or as hard spheres.

Abstract ii
List of Figures v
Nomenclature vi
1 Introduction 1
2 Fundamentals 4
2.1 Eulerian and Lagrangian Reference Frame . . . . . . . . . . . . . . . . . . 4
2.2 Reynolds' Transport Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.4 Shallow Water Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.5 Smoothed Particle Hydrodynamics . . . . . . . . . . . . . . . . . . . . . . 8
2.6 Semi-Lagrangian Advection . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 Previous Work 12
3.1 Dynamics Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.1 Wave Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.1.2 Particle Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3.1.3 Shallow Water Model . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.1.4 3-D Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . 17
3.2 Geometry Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.1 Height Field . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
3.2.2 Implicit Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.3 Illumination Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 Method 22
4.1 Solving Navier-Stokes Equations . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.1 Discretization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
4.1.2 Solving Pressure Poisson Equation . . . . . . . . . . . . . . . . . . 27
4.1.3 Boundary Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 27
4.1.4 Interaction with Solids . . . . . . . . . . . . . . . . . . . . . . . . . 28
4.1.5 Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Kinematic Surface Condition . . . . . . . . . . . . . . . . . . . . . . . . . . 30
4.2.1 Controlled Downwinding . . . . . . . . . . . . . . . . . . . . . . . . 31
4.2.2 Volume Fractions Advection . . . . . . . . . . . . . . . . . . . . . . 32
4.3 Modeling Splash . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.1 Particle Seeding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
4.3.2 Particles Dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.3.3 The Kernel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.4 Rendering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5 Results 43
6 Conclusion 46
6.1 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
6.2 Future Directions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
6.2.1 Combination of the VOF and the LSM . . . . . . . . . . . . . . . . 47
6.2.2 Multi-Material Flow . . . . . . . . . . . . . . . . . . . . . . . . . . 48
6.2.3 Complex Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
A Conjugate Gradient Method 49
B Implementation 50
Bibliography 51

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note="http://www.povray.org/"
}

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