A Novel Discretization and Numerical Solver for Non-Fourier Diffusion
Tao Xue*, Haozhe Su*, Chengguizi Han, Chenfanfu Jiang and Mridul Aanjaneya
ACM Transactions on Graphics, (SIGGRAPH Asia proceedings), 39, 6, 178:1-14, (2020)
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Abstract
We introduce the C-F diffusion model [Anderson and Tamma 2006; Xue et al. 2018] to computer graphics for diffusion-driven problems that has several attractive properties: (a) it fundamentally explains diffusion from the perspective of the non-equilibrium statistical mechanical Boltzmann Transport Equation, (b) it allows for a finite propagation speed for diffusion, in contrast to the widely employed Fick's/Fourier's law, and (c) it can capture some of the most characteristic visual aspects of diffusion-driven physics, such as hydrogel swelling, limited diffusive domain for smoke flow, snowflake and dendrite formation, that span from Fourier-type to non-Fourier-type diffusive phenomena. We propose a unified convection-diffusion formulation using this model that treats both the diffusive quantity and its associated flux as the primary unknowns, and that recovers the traditional Fourier-type diffusion as a limiting case. We design a novel semi-implicit discretization for this formulation on staggered MAC grids and a geometric Multigrid-preconditioned Conjugate Gradients solver for efficient numerical solution. To highlight the efficacy of our method, we demonstrate end-to-end examples of elastic porous media simulated with the Material Point Method (MPM), and diffusion-driven Eulerian incompressible fluids.
BibTeX
@article{Xue:2020:NFD,
author = {Xue, Tao and Su, Haozhe and Han, Chengguizi and Jiang, Chenfanfu and Aanjaneya, Mridul},
title = {A Novel Discretization and Numerical Solver for Non-Fourier Diffusion},
year = {2020},
issue_date = {December 2020},
publisher = {Association for Computing Machinery},
address = {New York, NY, USA},
volume = {39},
number = {6},
issn = {0730-0301},
url = {https://doi.org/10.1145/3414685.3417863},
doi = {10.1145/3414685.3417863},
journal = {ACM Trans. Graph.},
month = nov,
articleno = {178},
numpages = {14},
keywords = {fick's law, C-F diffusion, MPM, convection-diffusion, multigrid solver, non-fourier, incompressble flow}
}
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