Numerical Modeling of Multiphysics Systems

该研究的主要目标是设计、分析和实现精确、鲁棒和高效的偏微分方程数值方法，用于模拟科学和工程应用中出现的复杂多尺度多物理系统。特别是，研究将集中在几个关键组成部分的数学和数值模拟ofcoupled自由流体和多孔介质流：正确的数学模型的识别，特别是适当的界面条件，模型适定性的分析，稳定和精确的数值离散方法的设计和分析，包括非匹配网格上的有限元方法，内部收敛和后处理，和后验误差估计和自适应性，以及开发高效的求解算法，如并行区域分解和Newton-Krylov-多重网格求解器和预处理器。将强调地下水和地表水流动的耦合以及心血管系统的建模的应用。 地下和地表流动和输运的计算机建模可以为环境和能源问题提供可靠和经济有效的解决方案，如污染的地表水和地下水的修复以及提高石油采收率。目前的模拟器没有考虑地下和地表地质系统之间复杂的多尺度相互作用。这项研究将模拟这种相互作用，这对于准确预测流体行为至关重要。这项研究将导致开发强大和灵活的软件工具，用于对湿地、湖泊和河口的污染物治理进行精确的多处理器模拟。 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，在工业化国家，由心脏病发作和中风引起的死亡是主要原因。该领域研究的目的是通过计算模拟动脉粥样硬化，同时考虑到最近发现的炎症的关键作用。动脉及其壁内血流的宏观耦合模型将与急性炎症反应的细胞水平模型相结合。这项研究将导致开发优化的模拟工具，这将促进预防，检测和治疗的父亲。

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.