Multiscale numerical modeling of multiphysics systems

本项目主要研究复杂多物理场问题的多尺度随机数值方法。将开发一个用于模拟各种多尺度应用的并行计算框架。感兴趣的模型将基于Stokes或naver -Stokes方程与单相或多相达西流的耦合。流动系统将与反应输运相结合。研究方法基于多块域分解方法。仿真域是子域（块）的联合。每个块都与物理、数学和数值模型相关联。物理上有意义的界面条件通过砂浆有限元施加在离散水平上。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，该区域分解方法与粗尺度砂浆单元相结合，提供了一种多尺度近似和并行求解粗网格问题的有效方法。物理特性的不确定性通过随机偏微分方程建模。利用随机有限元和配点法实现了概率空间的近似。本研究的主要内容为：(1)建立多尺度数学模型，确定合适的界面条件，并分析模型的适定性；2)新的随机偏微分方程离散化技术；3)物理和随机空间的后验误差估计和自适应网格细化算法；(4)高效的并行域分解求解器和预处理器。该研究方法可用于设计、分析和实现科学和工程应用中大量问题的计算方法。该项目将强调精确、稳健和有效的数值方法来解决在空间和时间的广泛尺度上表现出行为的问题。该研究还旨在通过随机建模来量化由于输入参数的不确定性而导致的预测中的不确定性。研究成果将应用于两个主要应用领域：地表水与地下水流动的耦合，以及模拟人体炎症反应中发生的复杂多尺度过程。地下和地表流动和运输的计算机模拟对环境和能源工业具有重大的经济影响。由于地下水和地表水系统的邻近，河流、湖泊和湿地的污染会导致含水层的污染，反之亦然。模拟这些复杂的相互作用是一个主要的计算挑战。急性炎症是对创伤或感染的反应，可能导致全身性炎症或败血症。败血症经常导致器官衰竭和死亡（在美国每年有超过50万例死亡病例）。由于这一过程的复杂性，目前缺乏有效的治疗方法。数学和计算机建模正在成为了解炎症过程的动力学和空间特征的一种更常用的工具。

This project focuses on multiscale and stochastic numerical methodsfor complex multiphysics problems. A parallel computational frameworkfor modeling various multiscale applications will be developed. Themodels of interest will be based on coupling the Stokes or theNavier-Stokes equations with single phase or multiphase Darcy flow.The flow system will be coupled with reactive transport. The researchapproach is based on a multiblock domain decomposition methodology.The simulation domain is a union of subdomains (blocks). Each blockis associated with a physical, mathematical, and numericalmodel. Physically meaningful interface conditions are imposed on thediscrete level via mortar finite elements. The formulation providesgreat flexibility for multiphysics and multinumericscouplings. Furthermore, this domain decomposition approach, combinedwith coarse scale mortar elements, provides a multiscale approximationand an efficient way to solve the coarse grid problem in parallel.Uncertainty in the physical characteristics is modeled via stochasticpartial differential equations. Approximation in probability space isachieved using stochastic finite element and collocation methods. Themain components of the research will be 1) development of multiscalemathematical models, identifying appropriate interface conditions, andanalysis of the well-posedness of the models; 2) novel discretizationtechniques for stochastic partial differential equations; 3) aposteriori error estimates and adaptive mesh refinement algorithms inphysical and stochastic space; and (4) efficient parallel domaindecomposition solvers and preconditioners.The research methodology can be applied for the design, analysis, andimplementation of computational methods for a large class of problemsarising in science and engineering applications. The project willemphasize accurate, robust, and efficient numerical methods forproblems exhibiting behavior on a wide range of scales in space andtime. The research also aims to quantify uncertainty in the predictiondue to to uncertainty in the input parameters through stochasticmodeling. The research results will be applied to two mainapplication areas: couplings of surface water with groundwater flowsand modeling complex multiscale processes occurring in theinflammatory response in the human body. Computer modeling ofsubsurface and surface flow and transport has a major economic impacton environmental and energy industries. Due to the proximity ofgroundwater and surface water systems, contamination of rivers, lakes,and wetlands can lead to contamination of aquifers andvice-versa. Simulating these complex interactions is a majorcomputational challenge. Acute inflammation, which occurs in responseto trauma or infection, may lead to a systemic inflammation orsepsis. Sepsis frequently leads to organ failure and death (with morethat 500,000 fatal cases per year in the U.S.). Effective treatmentsare lacking, due to the complexity of the process. Mathematical andcomputer modeling is becoming a more common tool in the effort to gaininsight of the dynamics and spatial characteristics of theinflammatory process.