Algorithm Development and Analysis for Non-Newtonian Fluids Interacting with Elastic and Poroelastic Structures

本研究关注非牛顿流体-结构相互作用（FSI）的稳定和有效的数值格式的发展和严格的分析。研究将集中在耦合模型系统的隐式划分方案上，该方案允许在固定/移动域设置中使用现有的局部求解器独立解决每个子问题。将使用非牛顿流体和孔弹性/弹性结构模型研究数值方法的稳定性和精度特性。理论研究将为数值算法的进一步发展提供坚实的基础和指导。由于许多重要的生物和工程过程涉及非牛顿流体流动，因此在这些应用中需要大量的数学支持。这项研究将为多物理环境下的非牛顿流体提供潜在的数学基础。本课题的技术目标是研究准牛顿流体-孔弹性结构和粘弹性流体-弹性结构两种耦合系统的算法和数值格式分析。对于系统(a)， PI将侧重于非线性算子的开发和分析，其中算子方程的解产生满足整个耦合系统界面条件的子系统解。与先前开发的优化方法相比，该方法的优点是可以灵活地选择非线性求解器，并且由于伴随系统不再涉及求解过程，因此不需要额外的编码工作。由于期望线性化算子不自伴随，因此对于非自伴随问题，线性算子方程应采用迭代求解器求解。当考虑分区方案来模拟粘弹性流固耦合时，由于缺乏两个子系统沿运动边界的应力和沿界面的进出口边界的运动信息，会遇到额外的困难。PI将研究应力边界值的各种选择及其对FSI解的影响。粘弹性FSI的另一个问题是在每个时间步长（或每个内部迭代）中要解决的流体问题的大小，这可能需要基于时间离散方案（如分数时间步长方法）的算子分裂。为了解决这个问题，PI将研究各种时间步进方案。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。

This research concerns the development and rigorous analysis of stable and efficient numerical schemes for non-Newtonian fluid - structure interactions (FSI). The study will be focused on partitioning schemes of an implicit type for the coupled model systems that allow each subproblem to be solved independently using existing local solvers in a fixed/moving domain setting. Stability and accuracy properties of numerical methods will be investigated using non-Newtonian fluid and poroelastic/elastic structure models. The theoretical investigation will provide a solid foundation and guidance to the further development of numerical algorithms. Because of the many important biological and engineering processes involving non-Newtonian fluid flows, there is a great demand for mathematical support in these applications. This research will provide an underlying mathematical foundation for non-Newtonian flows in a multiphysical setting. The technical goal of this project is to develop algorithms and analyze numerical schemes for two coupled systems: (a) quasi-Newtoinan fluid - poroelastic structure and (b) viscoelastic fluid - elastic structure. For system (a) the PI will focus on the development and analysis of a nonlinear operator, where a solution to the operator equation yields subsystem solutions satisfying interface conditions of the whole coupled system. Advantages of this approach over the previously developed optimization approach are the flexibility to choose a nonlinear solver and that no extra coding effort is needed as the adjoint system is no longer involved in a solution process. Since it is expected that the linearized operator is not self-adjoint, the linear operator equation should be solved by an iterative solver for a non-self-adjoint problem. When a partitioned scheme is considered for simulating viscoelastic FSI, extra difficulty is encountered due to the lack of information on the stress along the moving boundary and movement of inlet and outlet boundaries along the interface of two subsystems. The PI will investigate various choices for a stress boundary value and their effect on a solution of FSI. Another issue with the viscoelastic FSI is the size of the fluid problem to be solved in each time step (or in each internal iteration), which may require an operator splitting based on a temporal discretization scheme such as a fractional time step method. To tackle this, the PI will investigate various time stepping schemes.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.