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.

中文翻译：

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一种新颖的非傅立叶扩散离散化和数值求解器

我们介绍CF扩散模型[安德森和塔玛 2006；薛等人。2018] 用于解决扩散驱动问题的计算机图形学，该问题具有几个吸引人的特性：（a）它从非平衡统计机械玻尔兹曼输运方程的角度从根本上解释扩散，（b）它允许有限的扩散传播速度，与广泛使用的菲克/傅里叶定律相比，(c) 它可以捕捉到扩散驱动物理学的一些最具特征的视觉方面，例如水凝胶膨胀、烟雾流动的有限扩散域、雪花和枝晶形成，跨越从傅立叶型到非傅立叶型扩散现象。我们使用该模型提出了一个统一的对流-扩散公式，该模型同时处理了扩散量和其相关的通量作为主要未知数，并且恢复了传统的傅里叶型扩散作为极限情况。我们为交错 MAC 网格上的该公式设计了一种新颖的半隐式离散化，并为有效的数值求解设计了几何多重网格预处理的共轭梯度求解器。为了突出我们方法的有效性，我们展示了使用材料点方法 (MPM) 和扩散驱动的欧拉不可压缩流体模拟的弹性多孔介质的端到端示例。