Multiscale domain decomposition methods for flow and mechanics problems

将开发一个计算框架来模拟不同物理现象的相互作用。它将应用于具有社会重要性的地球科学和生物医学问题。耦合地下和地表流动和运输将被调查，以模拟污染含水层，河流，湖泊和湿地之间的相互作用。裂缝性和可变形油藏中的流动将进行建模，以更好地理解和预测水力压裂和碳封存过程中的重要过程，包括地表沉降、孔隙坍塌、空腔生成和井筒坍塌。另一个令人感兴趣的应用是动脉内的流动，即动脉壁内的流动。这对管腔内的血流速度和压力波的速度，以及低密度脂蛋白（LDL）的运输和冠状动脉血流过程中过滤到组织中的药物都有影响。我们期望对动脉血流建模的研究能够导致优化模拟工具的发展，这将促进药物输送以及心血管疾病的预防、检测和治疗。教育活动将与研究活动相结合并得到加强。研究生和博士后将通过研究工作组或论文工作积极参与研究项目。最先进的研究成果将纳入课程。这项工作的主要目标是开发一个计算框架，用于模拟具有多尺度输入参数的耦合流动和力学问题的多物理场系统。研究方法基于多块域分解方法。仿真域被分解为子域的联合，每个子域都与一个物理、数学和数值模型相关联。通过砂浆有限元对离散层施加有物理意义的界面条件。该公式为多物理场和多数值耦合提供了很大的灵活性。此外，该方法结合粗尺度砂浆单元，提供了一种多尺度逼近和并行求解粗网格问题的有效方法。该项目将开发1)数学上严谨、物理上有意义的多物理场模型；2)稳健、准确、高效的多尺度离散化技术；3)高效的多尺度并行域分解求解器和预处理器。计算框架将应用于地球科学和生物医学问题。我们将发展将自由和多孔介质流体流动与多孔固体变形耦合的偏微分方程组的变分公式。这些公式将通过物理上有意义的界面条件耦合自由流体模型，如Stokes， Brinkman或具有单相或多相达西流的Navier-Stokes方程。在含有可变形多孔介质的区域，达西流动将与弹性耦合，并采用Biot孔隙弹性系统进行建模。我们将研究变分公式的适定性。我们将为这些多物理场变分公式开发稳定和精确的多尺度砂浆离散方法。我们将采用适当的混合有限元、有限体积和不连续伽辽金方法在精细尺度上对子域方程进行离散化。将利用砂浆有限元空间在粗尺度上施加界面条件。我们将对这些方法进行先验的多尺度误差分析。我们还将开发有效的并行无重叠区域分解算法，通过将耦合全局多尺度问题简化为粗尺度界面问题来求解所得到的代数系统。我们将分析接口运算符的条件数，并开发有效的预处理以加速接口迭代。

A computational framework will be developed for modeling interactions of different physical phenomena. It will be applied to geoscience and biomedical problems of societal importance. Coupling subsurface and surface flow and transport will be investigated to model interactions between contaminated aquifers, rivers, lakes, and wetlands. Flows in fractured and deformable reservoirs will be modeled to provide improved understanding and predictive simulations of important processes occurring in hydraulic fracturing and carbon sequestration, including surface subsidence, pore collapse, cavity generation, and wellbore collapse. Another application of interest is 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 during coronary artery flow. We expect the research on modeling arterial flows to lead to the development of optimized simulation tools which will advance drug delivery as well prevention, detection, and therapy of cardiovascular diseases. Educational activities will be integrated with and enhanced by research activities. Graduate students and postdocs will participate actively in research projects through research working groups or dissertation work. State-of-the-art research results will be incorporated into the curriculum. The primary objective of this work is to develop a computational framework for modeling multiphysics systems of coupled flow and mechanics problems with multiscale input parameters. The research approach is based on a multiblock domain decomposition methodology. 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. The formulation provides great flexibility for multiphysics and multinumerics couplings. Furthermore, this domain decomposition approach, combined with coarse scale mortar elements, 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 multiscale parallel domain decomposition solvers and preconditioners. The computational framework will be applied to geoscience and biomedical problems. We will develop variational formulations of systems of partial differential equations coupling free and porous media fluid flows with deformations of the porous solids. These formulations will couple through physically meaningful interface conditions free fluid models such as Stokes, Brinkman, or Navier-Stokes equations with single phase or multiphase Darcy flow. In regions involving deformable porous media the Darcy flow will be coupled with elasticity and modeled by the Biot system of poroelasticity. We will study well posedness of the variational formulations. We will develop stable and accurate multiscale mortar discretization methods for these multiphysics variational formulations. We will employ suitable mixed finite element, finite volume, and discontinuous Galerkin methods for the discretization of the subdomain equations on a fine scale. A mortar finite element space will be utilized to impose interface conditions on a coarse scale. We will carry out a priori multiscale error analysis for these methods. We will also develop efficient parallel non-overlapping domain decomposition algorithms for the solution of the resulting algebraic systems by reducing the coupled global multiscale problem to a coarse scale interface problem. We will analyze the condition number of the interface operator and will develop efficient preconditioners for speeding up the interface iteration.