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弹性结构相互作用(FPSI)模型的先进数值方法，这些模型在各种应用中起着至关重要的作用。例如，它们被广泛用于组织、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。研究小组将为一些FPSI模型提出、分析和实施有效和高效的数值算法。该项目将涉及博士后研究人员和HBCU的一些本科生，从而扩大代表不足的群体参与研究的范围。流体-孔弹性结构相互作用问题的数值模拟是具有挑战性的，因为它们是多区域、多尺度、多物理模型，子域模型类型不同，模拟物理规律的离散化格式很难设计，在将耦合模型的计算解耦的分区数值算法中很难保持稳定性和准确性。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，研究人员和团队成员将考虑基于整体公式的解耦预条件技术、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长格式、区域分解方法和可以加速数值算法收敛的多重网格方法。该项目有可能激发更多新的解耦算法，用于不同应用中的各种耦合的多域和多物理模型。预计该项目中的算法和相应的分析将对应用数学的几个分支，如数值分析和计算物理产生广泛的影响。开发的数值算法还将在生物力学计算和油藏工程模拟方面提供强大的工具。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。

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.