Numerical Modeling of Multiphysics Systems

提出的研究的主要目标是设计，分析和实现精确，稳健和高效的数值方法来模拟科学和工程应用中出现的复杂多尺度多物理系统的偏微分方程。特别是，研究将集中在耦合自由流体和多孔介质流动的数学和数值模拟的几个关键组成部分：识别正确的数学模型，特别是合适的界面条件，分析模型的适定性，设计和分析稳定和准确的数值离散化方法，包括非匹配网格上的有限元方法，内部收敛和后处理，后验误差估计和自适应，以及开发有效的求解算法，如并行域分解和牛顿-克雷洛夫多网格求解器和预调节器。在本研究中发展的数值技术是通用的，并解决了偏微分方程数值分析的基本问题。应用于地下水和地表水流动的耦合，以及心血管系统的建模将被强调。地下和地表流动和运输的计算机建模可以为污染地表水和地下水修复以及提高石油采收率等环境和能源问题提供可靠和经济的解决方案。电流模拟器没有考虑到地表和地下地质系统之间复杂的多尺度相互作用。本研究将模拟这种相互作用，这对准确预测流体行为至关重要。该研究将导致开发强大灵活的软件工具，用于执行湿地，湖泊和河口污染物修复的精确逼真的多处理器模拟。心血管系统的数学和计算建模是一个相对较新的但发展迅速的研究领域。这一应用对社会具有巨大的重要性。例如，由动脉硬化引起的心脏病发作和中风是工业化国家死亡的主要原因。在这一领域提出的研究的目的是计算动脉粥样硬化模型，考虑到最近发现的炎症的关键影响。动脉及其壁内血流的耦合宏观模型将与急性炎症反应的细胞水平模型相结合。这项研究将导致优化模拟工具的发展，这将促进动脉粥样硬化的预防、检测和治疗。

The main objectives of the proposed research are to design, analyze,and implement accurate, robust, and efficient numerical methods forpartial differential equations which model complex multiscalemultiphysics systems arising in science and engineeringapplications. In particular, the research will focus on severalcritical components of the mathematical and numerical modeling ofcoupled free fluid and porous media flows: identification of a correctmathematical model, especially appropriate interface conditions,analysis of the well-posedness of the model, design and analysis ofstable and accurate numerical discretization methods, including finiteelement methods on non-matching grids, interior convergence andpostprocessing, and a posteriori error estimation and adaptivity, anddevelopment of efficient solution algorithms such as parallel domaindecomposition and Newton-Krylov-multigrid solvers and preconditioners.The numerical techniques to be developed in this research are generaland address fundamental questions of numerical analysis of partialdifferential equations. Applications to coupling of groundwater andsurface water flows, as well as modeling of the cardiovascular systemwill be emphasized. Computer modeling of subsurface and surface flowand transport can provide reliable and cost-effective solutions toenvironmental and energy problems such as contaminant surface waterand groundwater remediation and enhanced oil recovery. Currentsimulators do not take into account the complex multiscale interactionbetween subsurface and surface geosystems. This research will modelthis interaction, which is crucial for the accurate prediction offluid behavior. The research will result in the development of robustand flexible software tools for performing accurate realisticmultiprocessor simulations of contaminant remediation of wetlands,lakes and estuaries. Mathematical and computational modeling of thecardiovascular system is a relatively new but rapidly growing area ofresearch. This application has a vast importance to society. Forexample, the heart attacks and strokes that result fromatherosclerosis are the major cause of death in industrialnations. The aim of the proposed research in this area is to modelcomputationally atherosclerosis taking into account the recentlydiscovered, but critical effect of inflammation. Coupled macroscopicmodels of blood flow within the artery and its wall will be combinedwith cellular level models of acute inflammatory response. Theresearch will lead to the development of optimized simulation toolswhich will advance prevention, detection, and therapy ofatherosclerosis.