Mathematical and Computational Modeling of Interaction between Fluids and Poroelastic Structures

这项工作的目的是对自由粘性流体与相邻可变形多孔介质中通过分隔这两个区域的界面的流动之间的相互作用进行建模和模拟。这种物理现象存在于广泛的应用中，包括地球科学、生物医学科学和工业设计。该项目在这一问题的数学和计算建模方面取得了进展，包括为解决这些问题开发和分析新的数学模型和数值方法。基于大规模并行计算机的高性能计算软件将被开发并应用于模拟低密度脂蛋白在心血管流动中的传输和药物传输，以及追踪地表-地下水文系统中的有机和无机污染物。本项目的目标是流体-孔弹性结构相互作用的数学和计算建模。自由流体的渗流用Stokes方程或Navier-Stokes方程模拟，而多孔弹性介质用Biot系统模拟。这两个区域通过动力和运动界面条件耦合，包括力的平衡、法向速度的连续性和无滑动或有摩擦的切向速度条件。该项目包括1)新的数学模型的开发和分析；2)稳定、准确和健壮的结构保持的数值方法；以及3)用于解决所产生的代数问题的高效的多尺度并行区域分解算法。在第一个主要部分中，将发展和分析新的FPSI模型的变分公式，包括Navier-Stokes-Biot耦合，Brinkman和Forchheimer模型的使用，非牛顿模型，以及完全耦合的FPSI-输运模型。这些新模型将把当前的模型能力扩展到具有较高雷诺数的流动、具有非牛顿流变性的流体以及跟踪溶解在流体中的物种，包括浓度对流场的影响。模型解的存在性将利用半群理论和Hilbert或Banach空间中的单调算子的结果，结合不动点变元来建立。第二个主要部分是研究新的离散化技术，用于FPSI模型的数值近似，重点是局部质量守恒的双重混合离散，局部动量守恒，速度和应力的连续法向分量的精确近似，以及对物理参数的稳健性。方法包括多点应力通量混合有限元方法和局部应力模拟有限差分法，这些方法可以归结为正定的单元中心格式，通过界面上的砂浆有限元进行耦合。在第三个主要部分中，我们将开发和分析高效的多尺度区域分解算法。该方法将基于时空变分公式，并将允许每个区域内的多个子域在子域界面上具有不匹配的网格，以及不同子域中的不同时间步长。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 objective of this work is modeling and simulation of the interaction between a free viscous fluid and flow in an adjacent deformable porous medium through an interface separating the two regions. Such physical phenomenon occurs in a broad range of applications, including geosciences, biomedical sciences, and industrial design. The project makes advances in the mathematical and computational modeling of this problem, including development and analysis of new mathematical models and numerical methods for their solution. High performance computational software designed to run on massively parallel computers will be developed and applied for modeling LDL transport and drug delivery in cardiovascular flows and tracing organic and inorganic contaminants in coupled surface-subsurface hydrological systems.The goal of this project is mathematical and computational modeling of fluid-poroelastic structure interaction (FPSI). The free fluid flow is modeled by the Stokes or the Navier-Stokes equations, while the poroelastic medium is modeled by the Biot system of poroelasticity. The two regions are coupled via dynamic and kinematic interface conditions, including balance of forces, continuity of normal velocity, and no-slip or slip with friction tangential velocity condition. The project includes development and analysis of 1) new mathematical models; 2) stable, accurate, and robust structure-preserving numerical methods; and 3) efficient multiscale parallel domain decomposition algorithms for the solution of the resulting algebraic problems. In the first main component, variational formulations of new FPSI models, including Navier-Stokes - Biot couplings, use of Brinkman and Forchheimer models, non-Newtonian models, and fully coupled FPSI-transport models will be developed and analyzed. These new models will extend current model capabilities to flows with higher Reynolds numbers, fluids with non-Newtonian rheology, and tracking species dissolved in the fluid, including the effect of the concentration on the flow field. Existence of model solutions will be established employing results from semigroup theory and monotone operators in Hilbert or Banach space setting, coupled with fixed point arguments. The second main component involves investigation of novel discretization techniques for the numerical approximation of the FPSI models, focusing on dual mixed discretizations with local conservation of mass, local momentum conservation, accurate approximations with continuous normal components for velocities and stresses, and robustness with respect to physical parameters. Methods of interest include multipoint stress-flux mixed finite element methods and local-stress mimetic finite difference methods that can be reduced to positive definite cell-centered schemes, coupled through mortar finite elements across the interface. In the third main component, efficient multiscale domain decomposition algorithms for FPSI will be developed and analyzed. The methodology will be based on space-time variational formulations and will allow for multiple subdomains within each region with non-matching grids along subdomain interfaces, as well different time steps in different subdomains.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.