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

这项研究致力于发展和严格分析非牛顿流体-结构相互作用(FSI)的稳定有效的数值格式。这项研究将集中于耦合模型系统的隐式类型的划分方案，该方案允许在固定/移动的区域设置中使用现有的局部求解器独立地解决每个子问题。数值方法的稳定性和精度将使用非牛顿流体和孔弹性/弹性结构模型进行研究。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，在这些应用中对数学支持有很大的需求。这项研究将为多物理环境下的非牛顿流动提供一个基本的数学基础。这个项目的技术目标是开发算法和分析两个耦合系统的数值格式：(A)准Newtoinan流体-孔弹性结构和(B)粘弹性流体-弹性结构。对于系统(A)，PI将集中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的界面条件的子系统解。与以前开发的最优化方法相比，这种方法的优点是可以灵活地选择非线性求解器，并且不需要额外的编码工作，因为伴随系统不再参与求解过程。由于期望线性化的算子不是自伴的，线性算子方程应该用迭代求解器来求解非自伴问题。当考虑分区格式来模拟粘弹性FSI时，由于缺乏关于沿移动边界的应力以及进出口边界沿两个子系统界面的移动的信息，因此会遇到额外的困难。PI将调查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(或在每个内部迭代中)要解决的流体问题的大小，这可能需要基于时间离散化方案(如分数时间步长方法)的算子分裂。为了解决这个问题，PI将调查各种时间推进计划。这一裁决反映了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。

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.