Advanced Discretizations and Domain Decomposition Algorithms for Multiphysics Couplings of Fluid Flows and Solid Mechanics

将开发一个计算框架来模拟耦合的物理过程。它将被应用于具有社会重要性的地球科学和生物医学问题。这项研究将研究地下和地表的耦合流动，以模拟受污染的含水层、河流、湖泊和湿地之间的相互作用。该项目将对裂隙和易变形油藏中的流动进行建模，以提供水力压裂和碳汇的预测性模拟，包括地面下沉、井筒坍塌和断层激活。该项目将进一步模拟动脉内的流动，考虑动脉壁内的流动。这会影响管腔内的血流速度和压力波的速度，以及低密度脂蛋白(LDL)的运输和药物进入组织的过滤。这项研究将导致开发模拟工具，以促进药物输送以及预防、检测和治疗动脉粥样硬化等心血管疾病。本项目的目标是对具有多尺度输入参数的耦合流动和力学问题的多物理系统进行数学和计算建模。模拟域被分解成子域的联合，每个子域与一个物理、数学和数值模型相关联。有物理意义的界面条件通过迫击炮有限元或惩罚方法施加在离散水平上。该公式为多物理和多数值耦合提供了极大的灵活性。此外，当与粗尺度的砂浆单元相结合时，它提供了一种多尺度的近似，为并行地解决粗网格问题提供了有效的途径。该项目将开发1)严格的数学和物理意义的多物理模型；2)稳健、准确和高效的多尺度离散化技术；3)高效的并行区域分解求解器和预条件；4)高效的非迭代时间分割算法。这项研究的两个主要组成部分是：A)混合弹性公式和离散化，以及它们与多物理框架中的混合流离散的耦合；B)时空多区域变分公式和离散，允许在不同的子域中进行不同的时间步长。将开发一个计算框架，并将其应用于地球科学和生物医学问题。该研究将发展耦合自由和多孔介质流动与多孔固体变形的偏微分方程组的变分公式。自由流体模型，如Stokes、Brinkman或Navier-Stokes方程，将通过具有物理意义的界面条件与Darcy流动相耦合。达西渗流穿过可变形的多孔介质的区域将由孔弹性的Biot系统来模拟。非牛顿流体的非线性模型以及降维裂缝模型也将被研究。重点将放在混合弹性配方以及混合斯托克斯和达西配方上。PI将研究变分公式的适定性，并将开发稳定和准确的离散化。新的单元中心混合有限元方法的弹性和孔洞弹性将被研究。基本类型的界面条件将通过砂浆有限元在粗略尺度上施加。PI将进行稳定性和先验多尺度误差分析。PI将通过将全局问题简化为一个粗略的界面问题来开发高效的并行非重叠区域分解算法来求解所产生的代数系统。PI将分析接口算子的条件数，并开发有效的预条件子来加速接口迭代。PI还将研究惩罚方法，如Nitsche的耦合方法，以施加界面条件，导致可修改为有效的非迭代时间分割算法的松散耦合公式。PI将研究这些方法的稳定性和准确性，以及它们作为整体方案的预条件的特性。PI将进一步开发多物理模型的多区域时空变分公式和离散化，在时间上将空间非重叠区域分解方法与Galerkin类型近似相结合，允许与不同区域和不同类型的物理相关联的不同时间步长。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。

A computational framework will be developed for modeling coupled physical processes. It will be applied to geoscience and biomedical problems of societal importance. The research will investigate coupled subsurface and surface flows to model interactions between contaminated aquifers, rivers, lakes, and wetlands. The project will model flows in fractured and deformable reservoirs to provide predictive simulations of hydraulic fracturing and carbon sequestration, including surface subsidence, wellbore collapse, and fault activation. The project will further model flow in arteries, accounting for flow within the arterial wall. This has an effect on the blood velocity in the lumen and the speed of the pressure wave, as well as low density lipoproteins (LDL) transport and drugs filtered into the tissue. The research will lead to the development of simulation tools that advance drug delivery as well prevention, detection, and therapy of cardiovascular diseases such as atherosclerosis. The objective of this project is mathematical and computational modeling of multiphysics systems of coupled flow and mechanics problems with multiscale input parameters. The simulation domain is decomposed into a union of subdomains, each one associated with a physical, mathematical, and numerical model. Physically meaningful interface conditions are imposed on the discrete level via mortar finite elements or penalty methods. The formulation provides great flexibility for multiphysics and multinumerics couplings. Furthermore, when combined with coarse scale mortar elements, it provides a multiscale approximation and an efficient way to solve the coarse grid problem in parallel. The project will develop 1) Mathematically rigorous and physically meaningful multiphysics models; 2) Robust, accurate and efficient multiscale discretization techniques; 3) Efficient parallel domain decomposition solvers and preconditioners; 4) Efficient non-iterative time-partitioned algorithms. Two main components of the proposed research are A) mixed elasticity formulations and discretizations, and their coupling with mixed flow discretizations in the multiphysics framework; B) space-time multidomain variational formulations and discretizations allowing for different time stepping in different subdomains. A computational framework will be developed and applied to geoscience and biomedical problems. The research will develop variational formulations of Partial differential Equations systems coupling free and porous media fluid flows with deformations of the porous solids. Free fluid models such as Stokes, Brinkman, or Navier-Stokes equations will be coupled through physically meaningful interface conditions with Darcy flow. Regions with Darcy flow through deformable porous media will be modeled by the Biot system of poroelasticity. Nonlinear models for non Newtonian fluids, as well as reduced-dimension fracture models will also be investigated. An emphasis will be placed on mixed elasticity formulations coupled with mixed Stokes and Darcy formulations. The PI will study well-posedness of the variational formulations and will develop stable and accurate discretizations. Novel cell-centered mixed finite element methods for elasticity and poroelasticity will be investigated. The essential-type interface conditions will be imposed on a coarse scale via mortar finite elements. The PI will carry out stability and a priori multiscale error analysis. the PI will develop efficient parallel non-overlapping domain decomposition algorithms for the solution of the resulting algebraic systems by reducing the global problem to a coarse scale interface problem. The PI will analyze the condition number of the interface operator and will develop efficient preconditioners for speeding up the interface iteration. The PI will also study penalty methods, such as the Nitsche's coupling method, to impose interface conditions, resulting in loosely coupled formulations amendable to efficient non-iterative time-partitioned algorithms. The PI will study the stability and accuracy of the methods, as well as their properties as preconditioners for monolithic schemes. The PI will further develop multidomain space-time variational formulations and discretizations for the multiphysics models, coupling spatial non-overlapping domain decomposition methods with Galerkin-type approximations in time, allowing for different time steps associated with different regions and different types of physics.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.