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，如结冰和树枝生长)是由颜料、晶体或神经分支扩散时的扩散驱动的。用来捕捉扩散的主要模型是傅里叶定律。然而，这一公式阻止了对异常扩散过程的模拟，在异常扩散过程中，扩散发生得比傅里叶定律预测的速度快(超扩散)或慢(次扩散)。目前，需要有效地模拟和可视化超扩散现象，例如在新冠肺炎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。该项目将推动计算机图形学中物理模拟的前沿，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，从而例如能够表征导致在现实世界中进行具体实验观察的扩散参数，或者设计防止在流动人群中爆发疾病的政策。项目成果将产生广泛的影响，因为它支持在极大范围内可视化这种复杂的物理过程。其他广泛的影响将来自在商用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以前可能需要较少访问的企业级计算资源的大规模问题。推广和教育活动，如研讨会，将利用罗格斯大学的多样性计划来招收和支持代表不足的群体的学生。该项目将通过开发一种使用分数导数的新扩散公式来推动计算机图形学的发展，该公式不仅可以模拟亚扩散和超扩散过程，而且还可以恢复传统傅立叶扩散最著名的求解器的效率。将采用拉格朗日/欧拉混合表示来模拟微观和宏观相互作用，两者强耦合在一起，同时考虑可能出现的裂缝等不连续性。为了适应大规模的问题规模，提出了一种空间多项式区域的自适应离散化方案，该方案可以灵活地用多项式函数来表示任意形状的任意不规则区域内的扩散通量。对于快速数值解，该项目将开发一种使用多重网格方法的高效求解器，通过避免构建线性系统而更好地利用硬件内存带宽，同时在现代工作站上实现快速收敛。由此产生的框架将允许模拟扩散现象，如超扩散，这些现象要么没有在计算机图形学中探索，要么目前超出了现有方法的范围。建议方法的实施将以开源软件包的形式提供给社区，并提供一个轻量级客户端，该客户端支持来自浏览器的交互式用户反馈，同时计算密集型模拟在远程服务器上运行，从而使这项研究能够广泛访问，特别是对本科生和K-12年级的学生，以培养他们对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智力优势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.