Excellence in Research: Numerical Algorithms for Fluid Poroelastic Structure Interaction Models

该研究项目旨在开发复杂流体-孔隙-弹性结构相互作用（FPSI）模型的先进数值方法，在各种应用中发挥关键作用。例如，它们被广泛应用于组织、牙齿和骨骼的生物力学建模。其他应用领域包括环境科学和油藏工程。研究小组将提出，分析和实现一些FPSI模型的有效和高效的数值算法。该项目将涉及HBCU的博士后研究人员和一些本科生，从而扩大代表性不足的群体在研究中的参与。研究和教育的努力有望帮助参与者从这样一个具有挑战性的计算科学主题中获得经验。流体-孔隙弹性结构相互作用问题的数值模拟具有多域、多尺度、多物理场模型、子域模型类型不同、模拟物理规律的离散化方案设计困难、分割数值算法解耦耦合模型计算难以保持稳定性和精度等特点。该项目的目标是开发能够解决这些困难的高效和有效的数值算法。特别是，研究者和团队成员将考虑基于单片公式的解耦预处理技术，结合Robin-Neumann迭代的双网格方法，在不同子域模型中使用不同时间步长的多速率时间步进方案，域分解方法，以及可以加速数值算法收敛的多网格方法。该项目有潜力为不同应用中的各种耦合多域和多物理模型激发更多新颖的解耦算法。预计本课题的算法和相应的分析将对数值分析和计算物理等应用数学的几个分支产生广泛的影响。所开发的数值算法也将为生物力学计算和油藏工程模拟提供有力的工具。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 research project aims at developing advanced numerical methods for the complex fluid poroelastic structure interaction (FPSI) models, which play a critical role in various applications. For example, they are widely used in biomechanic modeling of tissues, teeth, and bones. Other application areas include environmental sciences and reservoir engineering. The research team will propose, analyze, and implement effective and efficient numerical algorithms for some FPSI models. The project will involve postdoc researchers and some undergraduate students at a HBCU and therefore broaden the participation of underrepresented groups in research. The research and educational efforts are expected to help the participants to gain experience from such a challenging computational science topic.Numerical simulations of fluid poroelastic structure interaction problems are challenging because they are multi-domain, multi-scale, and multi-physics models, the subdomain models are of different types, discretization schemes that mimic physical laws are difficult to design, and the stability and accuracy are hard to preserve in partitioned numerical algorithms which decouple the computations of the coupled models. The objective of this project aims at developing efficient and effective numerical algorithms that can address the these difficulties. In particular, the investigator and the team members will consider decoupled preconditioning techniques based on a monolithic formulation, two-grid methods combined with the Robin-Neumann iteration, multi-rate time-stepping schemes which use different time step sizes in different subdomain models, domain decomposition methods, and multigrid methods that can accelerate the convergence of the numerical algorithms. The project has the potential of stimulating more novel decoupled algorithms for various coupled multi-domain and multi-physics models in different applications. It is expected that the algorithms and the corresponding analysis in this project will have a broad impact on several branches of applied mathematics such as numerical analysis and computational physics. The developed numerical algorithms will also provide powerful tools in biomechanic computation and reservoir engineering simulation.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.