Theodore Kim Michael Henson Ming C.Lim. Jae ho Lim. Korea University Computer Graphics Lab.

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1 Theodore Kim Michael Henson Ming C.Lim Jae ho Lim

2 Abstract Movie Ice formation simulation Present a novel algorithm Motivated by the physical process of ice growth Hybrid algorithm by three techniques Diffusion limited aggregation(dla) Phase field method Stable fluid solvers Illustrate the results on arbitrary 3D surfaces Jae ho Lim 2010/6/02 # 2

3 Introduction Visual simulation and animation of ice formation is becoming increasingly popular in the visual effects industry The Day After Tomorrow, Harry Potter and the Prisoner of Azkaban, and Van Helsing. Jae ho Lim 2010/6/02 # 3

4 Introduction Very little investigation has been conducted on the modeling of ice formation in computer graphics Most research has focused on modeling and simulating dynamic fluid Water and smoke Relatively few have dealt with complex phase transition and solidication processes Exact mathematical model describing the physical process does not yet exist. Accurate simulation of solidification processes in computational physics and crystal growth communities Jae ho Lim 2010/6/02 # 4

5 Main contribution Present a novel, hybrid algorithm that synthesizes three simulation techniques Diffusion limited aggregation(dla) Phase field method Stable fluid solvers Each algorithms can accurately simulate only one stage of the crystallization process Combining all three techniques Accurately simulate the entire process Jae ho Lim 2010/6/02 # 5

6 Main contribution Simplification of one of the techniques Phase field method Problem as an advection-reaction-diffusion equation Present efficient solution method Accelerates the phase field method Show how the simulation can be parameterized User control Jae ho Lim 2010/6/02 # 6

7 Main contribution A physically-based modeling approach Motivated by the thermodynamics of ice formation A novel discrete-continuous method that combines three techniques Diffusion limited aggregation Phase field methods Stable fluid solvers; A faster, simplified formulation phase field method; Enables simple artistic control of visual results. Jae ho Lim 2010/6/02 # 7

8 Previous Work A physically based model for icicle growth. The Visual Computer 1993 Presented a simple approach for icicle formation Computer modeling of fallen snow SIGGRAPH 2000 Presented a method for simulating fallen snow Not the process of solidification Visual simulation of ice crystal growth Eurographics 2003 modeling solidification on 2D surfaces using the phase field method. Jae ho Lim 2010/6/02 # 8

9 Previous Work Directions in Condensed Matter Physics World Scientific 1986 context of solidification Modeling and numerical simulations of dendritic crystal growth Physica D Successfully used to simulate snowflake-like growth for the first time Jae ho Lim 2010/6/02 # 9

10 Previous Work A level set approach for the numerical simulation of dendritic growth. Journal of Scientific Computation 2003 Use level set method Achieve higher numerical accuracy Simplicity of implementation Nearly identical visual results Jae ho Lim 2010/6/02 # 10

11 Previous Work Diffusion-limited aggregation, a kinetic critical phenomenon Physical Review Letters 1981 The algorithm generalizes to the modeling of many other natural phenomena Deterministic growth model of pattern formation in dendritic solidification. Non-deterministic approach to anisotropic growth patterns with continuously tunable morphology: the fractal properties of some real snowakes. Journal of Physics 1987 Including snowflake growth Jae ho Lim 2010/6/02 # 11

12 3. The Process of Solidication 3.1. Three Stages of Freezing Chemical diffusion stage Surface kinetics stage Heat conduction stage Limited by slowest of the three stages Jae ho Lim 2010/6/02 # 12

13 3.1. Three Stages of Freezing First stage slowest Diffusion limited growth Crystal surrounded by water vapor Second stage slowest Kinetics limited growth Crystal is submerged in an undercooled liquid Third stage slowest Heat limited growth Crystal is surrounded by fluid flow Jae ho Lim 2010/6/02 # 13

14 3.2. Diffusion Limited Growth The diffusion limited growth case can be modeled by diffusion limited aggregation (DLA) first described by Witten and Sander Given a discrete 2D grid A single particle representing the crystal is placed in the center. A particle called the `walker Walker is placed at a random location along the grid perimeter and walks randomly along adjacent grid cells Either is adjacent to the crystal or falls off the grid If it is adjacent to the crystal, it sticks and becomes part of the crystal The process repeats until the aggregate achieves the size the user desires Jae ho Lim 2010/6/02 # 14

15 3.2. Diffusion Limited Growth On-Lattice algorithm Takes place on a 2D grid Susceptible to grid anisotropy artifacts Off-Lattice algorithm Do not suffer from this artifacts More expensive to compute Jae ho Lim 2010/6/02 # 15

16 3.3. Kinetics Limited Growth The phase field model simulates this process by tracking two quantities over a grid Temperature T Tracks the amount of heat within the grid cell Phase p Tracks the phase of the grid cell Defined over the continuous range [0,1] 0 represents water 1 represents ice Jae ho Lim 2010/6/02 # 16

17 3.3. Kinetics Limited Growth Jae ho Lim 2010/6/02 # 17

18 3.3. Kinetics Limited Growth and computed by replacing the derivatives with finite differences Constants value table Time step Simplification that allows a larger time step in Section. 5 Jae ho Lim 2010/6/02 # 18

19 3.4. Heat Limited Growth Flow of heat around a crystal can significantly influence its final shape Using the fluid solver Stable fluid Provides a simple, practical alternative for modeling heat limited growth Jae ho Lim 2010/6/02 # 19

20 4. A Hybrid Algorithm for Ice Growth Assumption Diffusion limited growth Presence of water vapor Absence of liquid water and fluid flow Kinetics limited growth Presence of liquid water Absence of water vapor and fluid flow Heat limited growth Presence of fluid flow and liquid water Absence of water vapor Jae ho Lim 2010/6/02 # 20

21 4. A Hybrid Algorithm for Ice Growth To properly simulate ice growth account for all of these factors All three factors by coupling the simulation techniques for each of the three growth types Phase field methods and DLA Phase fields and fluid flow DLA and fluid flow. Jae ho Lim 2010/6/02 # 21

22 4.1. Phase Fields and DLA Necessary to three new step Placement of the walker onto the p (phase) field Release of heat when a walker sticks Introduction of a humidity term. Jae ho Lim 2010/6/02 # 22

23 4.1. Phase Fields and DLA In the original DLA algorithm Crystal can only grow when a walker sticks to the crystal In the hybrid simulation Phase field simulation may have also altered the position of the crystal Jae ho Lim 2010/6/02 # 23

24 4.1. Phase Fields and DLA Placement of the walker onto the p (phase) field Walks on the grid For the p variable in the phase field simulation If the walker is adjacent to a cell with p > 0.5 Particle sticks Set the value of that cell to p = 1 Release of heat When a walker sticks Forms hydrogen bonds with the crystal Release a small amount of heat The freezing walker will release less heat than if the liquid has frozen Jae ho Lim 2010/6/02 # 24

25 4.1. Phase Fields and DLA Modify Equation 2 to account for this heat release K is a latent heat constant. L = K 6 Bonds have only formed along one face of the hexagonal grid cell Humidity term, H Discussed further in 4.4 Jae ho Lim 2010/6/02 # 25

26 4.2 Phase field method and Fluid flow Derived a model that couples the phase field equations and the Navier-Stokes equations Anderson, et al. (Physica 2000) Does not present any simulation results visually Major features of solidification in a flow simply advecting the heat field with the Stable Fluid solver Jae ho Lim 2010/6/02 # 26

27 4.2 Phase field method and Fluid flow Following features of growth in a flow Fast growth in regions facing upstream (into flow) Stunted growth in regions facing downstream (away from flow) Asymmetric growth in regions perpendicular to the flow. Set any grid cell with p > 0.5 to an obstacle in the fluid domain. Set the velocities in the obstacle interior to zero Along the obstacle boundary to the no-slip condition Jae ho Lim 2010/6/02 # 27

28 4.2 Phase field method and Fluid flow The velocity field u is then advanced Described by Stable Fluids Joe Stam 1999 Implementing internal obstacles and various boundary conditions Numerical Simulation in Fluid Dynamics Griebel. M et al 1997 Jae ho Lim 2010/6/02 # 28

29 4.2 Phase field method and Fluid flow The resultant velocity field u can be used to advance a density field Density field Temperature field T from the phase field simulation Diffusion constant for the density field It must be set to zero contain the diffusion operator 2 PDE for a temperature field T PDE for a moving density field ρ Jae ho Lim 2010/6/02 # 29

30 4.2 Phase field method and Fluid flow If diffusion constant k is nonzero, temperature field T will incorrectly be diffused twice. once by Eqn. 2 once by Eqn. 7 If k is set to zero in Eqn. 7, the correct result is obtained Jae ho Lim 2010/6/02 # 30

31 4.3 DLA and Fluid flow The integration of DLA and simplified fluid flow has been studied by the physics community in the past. Morphological changes in convection-diffusionlimited deposition. NAGATANI T et al. Physical Review 1991 Deposition of particles in a two-dimensional lattice gas flow TOUISSAINT J et al Physical Review 1992 Jae ho Lim 2010/6/02 # 31

32 4.3 DLA and Fluid flow Simplification DLA and phase field simulations share the p field Phase field methods and with the fluid solver DLA with the full set of Navier-Stokes equations Fluid velocities should influence the walker Fluid velocity of the current grid cell is also added to that direction Jae ho Lim 2010/6/02 # 32

33 4.4. User Control Visual Simulation of Ice growth Kim T et al. Eurographics 2003 Suggested a seed crystal map and melting temperature map Can be effectively controlled using these same parameters Tunable morphology control humidity term H Jae ho Lim 2010/6/02 # 33

34 4.4. User Control The melting temperature map User-specified field whose values range over [0,1] 0 : Fully suppressed growth 1 : Fully promoted growth Intermediate values : varying degrees of desired growth The melting temperature map Sticking probability map for DLA When the walker is adjacent to the crystal Random number over [0,1] is chosen Number is less than a sticking probability Freezing Otherwise it continues walking In basic DLA, the `sticking probability is essentially set to 1 everywhere Jae ho Lim 2010/6/02 # 34

35 4.4. User Control Alternately desire different growth types from the crystal morphology Random Lichen-like growth Snowflake-like growth Controlled using the multiple-hit averaging technique Non-deterministic approach to anisotropic growth patterns with continuously tunable morphology NITTMANN J et al Journal of Physics 1987 In order for a grid cell to freeze, n walkers must stick at that cell In basic DLA, n=1 Increasing n, increasingly regular growth patterns are obtained. Jae ho Lim 2010/6/02 # 35

36 4.4. User Control The humidity control A way of controlling how `branchy' or `frosty' the results appear Very high humidity extreme branchiness of the DLA algorithm Very low humidity smooth features of the phase field algorithm Jae ho Lim 2010/6/02 # 36

37 5. Faster Phase Field Methods The performance of our hybrid algorithm is limited by the timestep restriction of the phase field methods Visual Simulation of Ice growth Midpoint and RK4 are unable to increase the timestep enough to justify their expense Other than linear multistep methods are required Jae ho Lim 2010/6/02 # 37

38 5. Faster Phase Field Methods Only the diagonal entries of the Hessian are use Omitting Noticeably smoother Branching features remained the same Jae ho Lim 2010/6/02 # 38

39 5. Faster Phase Field Methods A simplified phase PDE can now be written If we apply the identity The schemes will only be shown in the x direction, with the y direction following by symmetry Jae ho Lim 2010/6/02 # 39

40 5.1. Second Order Accuracy In Time The Lax-Wendroff scheme is applied to the advection term Old method With Lax-Wendroff scheme Jae ho Lim 2010/6/02 # 40

41 5.1. Second Order Accuracy In Time Crank-Nicolson discretization is applied to the diffusion term Old method With Crank-Nicolson Jae ho Lim 2010/6/02 # 41

42 5.1. Second Order Accuracy In Time Red-Black Gauss-Seidel iteration is the best solution method. working precision in less than 10 iterations Multigrid solver not likely give better performance Conjugate gradient Cannot be applied. system is not symmetric SOR is difficult Matrix eigenvalues change every iteration Jae ho Lim 2010/6/02 # 42

43 5.2. Performance Analysis Euler timestep is , second order timestep is WP : Solved to working precision RP : Solved to reduced precision Speedup : RP over Euler Jae ho Lim 2010/6/02 # 43

44 6. Implementation and Results We implement one step of the hybrid algorithm as: The DLA simulation Walk on the p field Insert heat into the T field Account for fluid velocities Rendered in 3DS Max 5 Jae ho Lim 2010/6/02 # 44

45 6. Implementation and Results Figure 6, the humidity was started at 300 and increased by 50 after the 75th timestep Jae ho Lim 2010/6/02 # 45

46 7. Results and Discussion 7.1. Comparisons DLA, Phase field and our method 7.2. Limitations 7.3. Generalization Jae ho Lim 2010/6/02 # 46

47 7.2. Limitations Limited by the 2D treatment of fluid flow Velocities are roughly parallel to the simulation domain To handle the perpendicular case A full 3D fluid solver is necessary Add this feature to our existing framework Cannot handle thick features such as icicles physical process of icicle formation is still unknown presents an interesting research challenge Jae ho Lim 2010/6/02 # 47

48 7.3. Generalization Developed a novel method of texture synthesis phase field equations non-linear advection-reaction-diffusion system represent a more general class of phenomena In the absence of a fluid flow and with isotropic growth settings Laplacian growth algorithm Anisotropy and fluid flow non-laplacian growth algorithm It has potential to increase the realism of other Laplacian phenomena Jae ho Lim 2010/6/02 # 48

49 8. Conclusions and Future Work Presented a novel, hybrid algorithm modeling ice formation a set of parameters for the algorithm method of accelerating one of its main components Our method has higher degree of realism than any previous technique. Jae ho Lim 2010/6/02 # 49

50 8. Conclusions and Future Work Phase field methods Unconditionally stable algorithm non-linear nature of the equations is difficult DLA Arbitrary anisotropy function could be imposed on a square grid Recent work has produced true isotropy on a square grid Jae ho Lim 2010/6/02 # 50

51 8. Conclusions and Future Work For a large humidity DLA can be the slowest component of the simulation faster alternative solution methods dielectric breakdown Hastings-Levitov conformal mapping Rendering issues Ice is composed of highly anisotropic mesofacets that exhibit strong spectral dispersion. making realistic rendering difficult Jae ho Lim 2010/6/02 # 51