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（如虚拟绘画和相变，如冰的形成和树突生长）都是由颜料、晶体或神经分支的扩散驱动的。用于捕获扩散的主要模型是傅立叶定律。然而，该公式阻止了异常扩散过程的模拟，其中扩散发生得比傅立叶定律预测的速率更快（超扩散）或更慢（亚扩散）。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者事件或由于全球变暖导致的永久冻土融化。该项目将推动计算机图形学中物理模拟的前沿，开发一个通用框架，用于有效模拟大规模应用中的各种扩散过程，从而能够对扩散参数进行表征，从而在真实的世界中进行具体的实验观察，或设计预防流动人群中疾病爆发的政策。项目成果将产生广泛的影响，支持可视化的这种复杂的物理过程在大大扩大规模。其他广泛的影响将来自于在商品工作站上运行高分辨率模拟的能力，这将使广大受众，特别是STEM学生，能够在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的企业级计算资源。Rutgers大学的多元化项目将招募和支持来自弱势群体的学生，该项目将通过开发一种新的分数导数扩散公式来推进计算机图形学的最新技术，该公式不仅可以模拟亚扩散和超扩散过程，还可以恢复传统傅里叶变换的最佳解算器的效率。基于扩散。将采用混合拉格朗日/欧拉表示法对微观和宏观相互作用进行建模，这两种相互作用强烈耦合在一起，同时考虑可能出现的裂缝等不连续性。为了扩展到大的问题大小，将开发一个自适应离散化方案，使用空间多项式区域，可以灵活地表示在任何不规则的域中的扩散通量的任意形状，使用多项式函数。对于快速数值解，该项目将使用多重网格方法开发一种高效的求解器，通过避免构建线性系统，同时在现代工作站上实现快速收敛，从而更好地利用硬件内存带宽。由此产生的框架将允许模拟扩散现象，如超扩散，没有被探索在计算机图形学或目前超出了现有的方法。所提出的方法的实施将作为开源软件包提供给社区，沿着一个轻量级客户端，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访问，特别是本科生和K-12学生，培养他们对STEM的早期兴趣。该奖项反映了NSF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.