Domain Decomposition Methods for Coupled Models of Non-Newtonian Fluids and Solid Structures

人们对流体与孔隙弹性结构相互作用的数值研究越来越感兴趣；然而，该领域的大多数数值和分析研究都集中在牛顿流体与固体结构的耦合上，而对非牛顿流体与弹性/孔弹性固体相互作用的分析和数值研究很少。非牛顿流体-孔隙弹性结构相互作用（FPSI）问题本身就引起了相关科学和工程界的兴趣，因为对这些强耦合非线性问题的研究可以更好地理解和统一描述复杂的现实生活过程。这项研究将为多物理环境下的非牛顿流提供潜在的数学基础，同时促进教学，培养研究生，并使本科生参与研究经验。提出的研究还将通过为重要过程的数值模拟提供改进的算法，从而使生物医学、聚合物、材料和石油工业受益。研究活动在于为FPSI问题开发严谨、高效和稳定的数值格式。目前大多数FPSI问题的研究成果讨论了在整体或分区设置下子问题之间具有匹配网格和时间步长的数值方法。本研究的目标是为非牛顿FPSI系统设计允许局部空间和时间离散化的域分解方法。同时，对数值格式的理论精度和稳定性进行了分析。具体来说，这个项目有五个主要目标：(i)为非牛顿FPSI系统开发和分析基于全局实时、无重叠域分解的并行、准确、高效的解耦算法；（ii）及时开发和分析精确的离散化和投影方法，以有效地处理本地时间尺度差异较大的非牛顿FPSI系统；（iii）研究粘弹性FPSI体系的适定性；（iv）基于加权最小二乘有限元法，开发和分析粘弹性FPSI系统精确、高效的离散化方法；（五）教育和培养研究生和本科生。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。

There has been increasing interest in numerical studies on fluids interacting with poroelastic structures; however, most numerical and analytical studies in this area have been focused on Newtonian fluids coupled with solid structures, and few analytical and numerical studies have been undertaken on the interaction of non-Newtonian fluids and elastic/poroelastic solids. Non-Newtonian fluid-poroelastic structure interaction (FPSI) problems are themselves of interest to related scientific and engineering communities, as the study of these strongly coupled nonlinear problems leads to a better understanding and a unifying description of complex real-life processes. This research will provide an underlying mathematical foundation for non-Newtonian flows in a multi-physical setting while promoting teaching, training graduate students, and involving undergraduates in research experiences. The proposed research will also benefit the biomedical, polymer, material, and oil industries by providing improved algorithms for the numerical simulation of important processes. The research activity lies in the development of rigorous, efficient, and stable numerical schemes for FPSI problems. Most current research results of FPSI problems discuss numerical methods with matching grids and time steps between subproblems in either a monolithic or partitioned setting. The goal of the research is to design domain decomposition methods for non-Newtonian FPSI systems that allow local space and time discretization. At the same time, the theoretical accuracy and stability properties of the numerical schemes will be analyzed. Specifically, we have five main objectives for this project: (i) develop and analyze parallel, accurate, efficient decoupling algorithms based on global-in-time, non-overlapping domain decomposition for non-Newtonian FPSI systems; (ii) develop and analyze accurate discretization and projection methods in time to effectively handle non-Newtonian FPSI systems with a large difference in local time scales; (iii) study the well-posedness of the viscoelastic FPSI system; (iv) develop and analyze accurate and efficient discretization methods for viscoelastic FPSI systems based on weighted least-squares finite element methods; (v) educate and train graduate and undergraduate students.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.