A novel discretization and numerical solver for non-fourier diffusion
A novel discretization and numerical solver for non-fourier diffusion
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DOI：
10.1145/3414685.3417863
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发表时间：
2020-11
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影响因子：
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通讯作者：
Tao Xue;Haozhe Su;Chengguizi Han;Chenfanfu Jiang;Mridul Aanjaneya
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作者：
Tao Xue;Haozhe Su;Chengguizi Han;Chenfanfu Jiang;Mridul Aanjaneya
我们将C - F扩散模型[安德森和塔玛2006；薛等人2018]引入计算机图形学，用于解决扩散驱动的问题，该模型具有几个吸引人的特性：(a)它从非平衡统计力学玻尔兹曼输运方程的角度从根本上解释了扩散；(b)与广泛使用的菲克/傅里叶定律不同，它允许扩散具有有限的传播速度；(c)它可以捕捉扩散驱动物理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区域、雪花和枝晶的形成，这些涵盖了从傅里叶型到非傅里叶型的扩散现象。我们使用该模型提出了一个统一的对流 - 扩散公式，将扩散量及其相关通量都视为主要未知量，并将传统的傅里叶型扩散作为一种极限情况恢复出来。我们在交错的MAC网格上为该公式设计了一种新颖的半隐式离散化方法，并设计了一种几何多重网格预条件共轭梯度求解器，以实现高效的数值求解。为了突出我们方法的有效性，我们展示了用物质点法(MPM)模拟的弹性多孔介质以及扩散驱动的欧拉不可压缩流体的端到端示例。
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.