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Fast Simulation of Laplacian Growth
March/April 2007 (vol. 27 no. 2)
pp. 68-76
Theodore Kim, University of North Carolina at Chapel Hill
Jason Sewall, University of North Carolina at Chapel Hill
Avneesh Sud, University of North Carolina at Chapel Hill
Ming C. Lin, University of North Carolina at Chapel Hill
Laplacian instability is the physical mechanism driving pattern formation in many disparate natural phenomena. Current algorithms for simulating this instability are slow and memory intensive. A new algorithm, based on the dielectric breakdown model from physics, is more than three orders of magnitude faster than previous methods and decreases memory use by two orders of magnitude.

1. B.K. Chakrabarti and L.G. Benguigui, Statistical Physics of Fracture and Breakdown in Disordered Systems, Oxford Univ. Press, 1997.
2. P. Prusinkiewicz and A. Lindenmayer, The Algorithmic Beauty of Plants, Springer Verlag, 1990.
3. C. Amitrano, "Fractal Dimensionality of the Model," Physical Rev. A, vol. 39, no. 12, 1989, pp. 6618–6620.
4. K. Moriarty, J. Machta, and R. Greenlaw, "Parallel Algorithm and Dynamic Exponent for Diffusion-Limited Aggregation," Physical Rev. E, vol. 55, no. 5, 1997, pp. 6211–6218.
5. T. Vicsek, Fractal Growth Phenomena, World Scientific, 1992.
6. J. Dongarra et al., "A Sparse Matrix Library in C++ for High Performance Architectures," Proc. 2nd Object Oriented Numerics Conf., 1992, pp. 214–218.
7. A. Fabri et al., "The CGAL Kernel: A Basis for Geometric Computation," Proc. 1st ACM Workshop Applications of Computer Geometry, vol. 1148, 1996, pp. 191–202.
8. R. Mech and P. Prusinkiewicz, "Visual Models of Plants Interacting with their Environment," Proc. Siggraph, ACM Press, 1996, pp. 397–410.
9. T. Reed and B. Wyvill, "Visual Simulation of Lightning," Proc. Siggraph, ACM Press, 1993, pp. 359–364.
10. T. Kim and M. Lin, "Physically Based Modeling and Rendering of Lightning," Proc. Pacific Graphics 2004, IEEE Press, 2004, pp. 267–275.
11. B. Mandelbrot and C. Evertsz, "The Potential Distribution around Growing Fractal Clusters," Nature, vol. 348, no. 6297, 1990, pp. L143–L145.
1. B. Mandelbrot, The Fractal Geometry of Nature, W H Freeman, 1982.
2. D. Ebert et al., Texturing and Modeling: A Procedural Approach, AP Professional, 1998.
3. T. Witten and L. Sander, "Diffusion-Limited Aggregation, A Kinetic Critical Phenomenon," Physical Rev. Letters, vol. 47, no. 19, 1981, pp. 1400–1403.
4. L. Niemeyer, L. Pietronero, and H.J. Wiesmann, "Fractal Dimension of Dielectric Breakdown," Physical Rev. Letters, vol. 52, no. 12, 1984, pp. 1033–1036.
5. M. Hastings and L. Levitov, "Laplacian Growth as One-Dimensional Turbulence," Physica D, vol. 116, no. 244, 1998, pp. 244–252.
6. T. Kim, M. Henson, and M. Lin, "A Hybrid Algorithm for Modeling Ice Formation," Proc. ACM Siggraph/ Eurographics Symp. Computer Animation, ACM Press, 2004, pp. 305–314.
7. B. Desbenoit, E. Galin, and S. Akkouche, "Simulating and Modeling Lichen Growth," Proc. Eurographics 2004, Blackwell Publishing, 2004, pp. 341–350.
8. T. Kim and M. Lin, "Physically Based Modeling and Rendering of Lightning," Proc. Pacific Graphics 2004, IEEE Press, 2004, pp. 267–275.
9. M. Eden, "A Two Dimensional Growth Process," Proc. 4th Berkeley Symp. Mathematical Statistics and Probability, Univ. of Calif. Press, 1961, pp. 223–239.
10. M. Hastings, "Fractal to Nonfractal Phase Transition in the Dielectric Breakdown Model," Physical Rev. Letters, vol. 87, no. 17, 2001, pp. 175502–175506.

Index Terms:
procedural texturing, natural phenomena, fractals, diffusion limited aggregation, dielectric breakdown model
Citation:
Theodore Kim, Jason Sewall, Avneesh Sud, Ming C. Lin, "Fast Simulation of Laplacian Growth," IEEE Computer Graphics and Applications, vol. 27, no. 2, pp. 68-76, March-April 2007, doi:10.1109/MCG.2007.33
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