Collaborative Research: Multidimensional couplings for flow and transport in porous media

项目重点研究不同空间尺度耦合的物理现象。这些耦合问题的数学和数值分析是具有挑战性的，因为在解决缺乏平滑性。这些耦合问题的应用很多；例如，器官血流的数学模型对于理解器官灌注、栓塞和药物输送的机制非常重要。该项目将培养研究生和本科生在计算和应用数学方面的能力。研究成果将发表在研究期刊上、在线上并在科学会议上展示。该项目的总体目标是制定、分析和应用不连续伽辽金方法来解决多孔介质中一维和三维耦合流动和输运过程。由于弱解在线源上表现出奇异性，因此数值分析具有挑战性。研究小组将开发和分析具有奇异数据的模型问题的数值方法。该方法将新颖高效的时间步进算法与不连续有限元方法相结合。研究人员将把算法应用于器官和脉管系统的多维耦合。基于物理的计算模型的开发和验证将以流动和输送成像为指导。基于神经网络的图像图像分割提取器官和血管的几何形状。学生将参与研究，他们将接受数值分析和科学计算方面的培训。拓展活动将让高中生了解计算数学的最新发展。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。

The project focuses on the study of physical phenomena coupling different spatial scales. The mathematical and numerical analysis of these coupled problems is challenging because of the lack of smoothness in the solution. Applications of these coupled problems are many; for instance the mathematical modeling of blood flow in organs is important in understanding the mechanisms of organ perfusion, embolization and drug delivery. The project will train graduate students and undergraduate students in computational and applied mathematics. Research outcomes will be published in research journals, online and presented at scientific meetings.The overall goal of the project is the formulation, analysis and application of discontinuous Galerkin methods for the solution of coupled one-dimensional and three-dimensional flow and transport processes in porous media. The numerical analysis is challenging because weak solutions exhibit a singularity on the line source. The research team will develop and analyze numerical methods for model problems with singular data. The methods will combine novel efficient time-stepping algorithms with discontinuous finite element methods. The investigators will apply the algorithms to multidimensional couplings in organ and vasculature. The development and validation of physics-based computational models will be guided by imaging of flow and transport. Neural network based image image segmentation extracts the geometry of the organ and blood vessels. Students will be involved in the research and they will be trained in numerical analysis and scientific computing. Outreach activities will engage high school students with recent developments in computational mathematics.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.