CAREER: Modeling and Simulating Generalized Diffusion for Computer Graphics and Computational Science

计算机图形学中出现的许多问题（如虚拟绘画和像冰形成和树突生长这样的相变）都是由颜料、晶体或神经分支扩散引起的。用来捕捉扩散的主要模型是傅立叶定律。然而，这个公式阻止了异常扩散过程的模拟，其中扩散发生的速度比傅里叶定律预测的速度更快（超扩散）或更慢（次扩散）。目前，需要有效地模拟和可视化超扩散现象，例如在COVID-19大流行期间目睹的疾病传播的超级传播事件或全球变暖导致的永久冻土融化。该项目将通过开发一个通用框架来有效地模拟大规模应用中的各种扩散过程，从而推动计算机图形学中物理模拟的前沿，从而使例如扩散参数的表征能够在现实世界中导致特定的实验观察，或设计防止疾病在移动人群中爆发的政策。项目成果将通过支持这种复杂物理过程的可视化而产生广泛的影响。额外的广泛影响将来自于在商品工作站上运行高分辨率模拟的能力，这将允许广大受众，特别是STEM专业的学生，在他们自己的工作站上模拟大规模问题，而这些问题以前可能需要较少访问的企业级计算资源。外展和教育活动，如研讨会，将利用罗格斯大学的多元化项目，招收和支持来自代表性不足群体的学生。该项目将通过使用分数阶导数开发一种新的扩散公式来推进计算机图形学的最新发展，该公式不仅可以模拟亚扩散和超扩散过程，还可以恢复传统基于傅里叶的扩散最著名的求解器的效率。拉格朗日/欧拉混合表示法将用于模拟微观和宏观相互作用，两者在考虑可能出现的裂缝等不连续时强耦合在一起。为了扩展到大的问题规模，将开发一种自适应离散化方案，该方案使用空间多项式区域，可以灵活地表示任意形状的任意不规则区域中的扩散通量。对于快速数值解，本项目将开发一种高效的求解器，使用Multigrid方法，通过避免线性系统的构建，更好地利用硬件内存带宽，同时导致现代工作站的快速收敛速率。由此产生的框架将允许模拟扩散现象，如超级扩散，这些现象要么没有在计算机图形学中探索过，要么目前超出了现有方法的范围。所提出的方法的实现将作为开源软件包提供给社区，以及一个轻量级客户端，该客户端支持来自浏览器的交互式用户反馈，而计算密集型模拟在远程服务器上运行，从而使这项研究广泛可访问，特别是本科生和K-12学生，以培养他们对STEM的早期兴趣。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Many problems that arise in computer graphics (such as virtual painting and phase changes like ice formation and dendrite growth) are driven by diffusion as pigment, crystals, or neural branches spread. The predominant model employed to capture diffusion is Fourier's law. However, this formulation prevents the simulation of anomalous diffusive processes, where diffusion occurs either faster (super-diffusion) or slower (sub-diffusion) than the rate predicted by Fourier's law. Currently, there is a need for efficiently simulating and visualizing super-diffusive phenomena, such as the super-spreader events for disease propagation witnessed during the COVID-19 pandemic or the melting of the permafrost due to global warming. This project will push the frontiers of physics simulation in computer graphics by developing a general framework for efficiently simulating all kinds of diffusive processes in large-scale applications, thereby enabling for example characterization of diffusion parameters that lead to specific experimental observations in the real world or the design of policies for preventing disease outbreaks in moving crowds. Project outcomes will have broad impact by supporting the visualization of such complex physical processes at greatly expanded scales. Additional broad impact will derive from the ability to run high resolution simulations on commodity workstations, which will allow a broad audience, particularly students in STEM, to simulate large-scale problems on their own workstations that previously may have required less-accessible enterprise-grade computational resources. Outreach and educational activities such as workshops will leverage diversity programs at Rutgers University to recruit and support students from under-represented groups.This project will advance the state-of-the-art in computer graphics by developing a novel formulation for diffusion using fractional derivatives that can not only simulate sub- and super-diffusive processes but also recover the efficiency of the best-known solvers for traditional Fourier-based diffusion. A hybrid Lagrangian/Eulerian representation will be adopted for modeling both micro- and macroscopic interactions, the two being strongly coupled together while accounting for discontinuities such as cracks that may emerge. To scale to large problem sizes, an adaptive discretization scheme will be developed using spatial polynomial regions that can flexibly represent the diffusion fluxes in any irregular domain of arbitrary shape using polynomial functions. For fast numerical solutions, this project will develop an efficient solver using Multigrid methods that better utilize the hardware memory bandwidth by avoiding construction of the linear system while leading to fast convergence rates on modern workstations. The resulting framework will allow the simulation of diffusive phenomena such as super-diffusion that have either not been explored in computer graphics or are currently beyond the reach of existing methods. Implementations of the proposed methodology will be made available to the community as open-source software packages, along with a lightweight client that supports interactive user feedback from the browser while the computationally intensive simulation runs on a remote server thereby making this research broadly accessible, in particular to undergraduate and K-12 students, to cultivate their early interest in STEM.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.