Mathematical and Computational Modeling of Interaction between Fluids and Poroelastic Structures

这项工作的目的是建模和模拟之间的相互作用的自由粘性流体和流动在相邻的可变形多孔介质通过一个界面分离的两个区域。这种物理现象发生在广泛的应用中，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，包括开发和分析新的数学模型及其解决方案的数值方法。高性能的计算软件设计运行在大规模并行计算机将开发和应用于模拟LDL运输和药物输送在心血管流量和跟踪有机和无机污染物在耦合的表面-地下水文系统。 自由流体流动由Stokes或Navier-Stokes方程模拟，而多孔弹性介质由多孔弹性的毕奥系统模拟。这两个区域通过动力学和运动学界面条件耦合，包括力的平衡，法向速度的连续性，以及无滑移或有摩擦的切向速度条件。该项目包括开发和分析：1）新的数学模型; 2）稳定，准确和鲁棒的结构保持数值方法; 3）有效的多尺度并行区域分解算法，用于解决由此产生的代数问题。 在第一个主要组成部分中，将开发和分析新FPSI模型的变分公式，包括Navier-Stokes -毕奥耦合、Brinkman和Forchheimer模型的使用、非牛顿模型和完全耦合的FPSI运输模型。这些新模型将扩展当前模型的能力，以更高的雷诺数的流动，非牛顿流变学的流体，并跟踪溶解在流体中的物种，包括浓度对流场的影响。模型解的存在性将采用半群理论和单调算子在希尔伯特或Banach空间设置的结果，加上不动点参数。第二个主要组成部分涉及新的离散化技术的FPSI模型的数值近似的调查，侧重于双混合离散与当地的质量守恒，当地的动量守恒，速度和应力的连续正常分量的精确近似，和相对于物理参数的鲁棒性。感兴趣的方法包括多点应力通量混合有限元法和局部应力模拟有限差分法，可以减少到正定的细胞为中心的计划，耦合通过砂浆有限元跨接口。在第三个主要组成部分，有效的多尺度区域分解算法FPSI将开发和分析。该方法将基于空间-时间变分公式，并允许每个区域内的多个子域，沿沿着子域界面具有不匹配的网格，以及不同子域的不同时间步长。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

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.