Innovative Finite Element Methods for Modeling Multiphase Contaminant Flows in Porous Media

涉及多孔介质中多相污染物流动和输运的复杂物理现象通常用非线性偏微分方程的耦合系统来模拟。最近在计算能力方面的进步（特别是随着新的并行架构的出现）极大地扩展了将更多物理纳入微分方程的潜力。虽然这种对模型的更改可以增加其准确表示潜在真实过程的能力，但必须在建模过程的每个阶段利用物理，数学，数值和计算概念进行验证。研究者和他的同事开发了描述多孔介质中多相污染物流动和输送的微分方程的解析和数值方法。第一个目标是对这些非线性偏微分方程进行理论分析。要处理的特殊主题是不规则性、稳定性和分岔现象。第二个目标是研究和开发能够捕获尖锐解前沿的有限元方法，产生精确的流体速度，保存质量，及时给出高阶近似，并有效地适应局部网格细化。在这里，研究人员通过欧拉-拉格朗日局部伴随法和高阶时间步进程序耦合的混合有限元方法的概念，发展了处理非线性输运-扩散方程系统的有限元方法。第三个目标是为由此产生的离散问题开发有效的并行迭代解决技术。该项目继续发展精确的数值技术，高效的计算代码，以及支持地下水水文学中多相流和输沙模型的数学分析，这些模型在修复和清理技术的设计中有应用。发展评估危险废物对地下水污染的改进方法已变得越来越重要。事实上，地下水供应正日益受到有机、无机和放射性污染物的威胁，这些污染物由于处置不当或意外释放而进入环境。据估计，仅美国政府场地的修复费用就高达数千亿美元。保护地下水供应的质量是一个具有广泛社会重要性的重大挑战问题。该研究也有许多工业应用。油藏的生产和管理是由几乎与地下水水文学中开发的相同的方程和技术控制的。用于汽车和航空航天应用的高质量纤维增强塑料复合材料的制造是由类似于本项目研究的技术控制的。半导体的设计使用的方程与地下水输送的方程非常相似。在这些领域，技术转让可以很容易地进行。特别是，由于它们的智力内容和与工业的紧密联系，这些各种各样的问题为学生学习应用数学和计算数学技术提供了一个很好的工具。

Ewing9972147 Complex physical phenomena involving multiphase contaminantflows and transport in porous media are often modeled by coupledsystems of nonlinear partial differential equations. Recentadvances in computational capabilities (particularly with theadvent of new parallel architectures) have greatly expanded thepotential for incorporating more physics into the differentialequations. While such changes to the model can increase itsability to represent accurately the underlying real process, thismust be verified in each phase of the modeling process utilizingphysical, mathematical, numerical, and computational concepts.The investigator and his colleague develop analytical andnumerical methods for the differential equations describingmultiphase contaminant flows and transport in porous media. Thefirst objective is a theoretical analysis of these nonlinearpartial differential equations. Special topics to be treated areregularity, stability, and bifurcation phenomena. The secondobjective is to study and develop finite element methods capableof capturing sharp solution fronts, producing accurate fluidvelocities, conserving mass, giving high-order approximations intime, and being efficiently adapted in local grid refinement.Here the investigators develop finite element methods fortreating systems of nonlinear transport-diffusion equations viaconcepts of Eulerian-Lagrangian localized adjoint methods andmixed finite element methods coupled with high-ordertime-stepping procedures. The third objective is to developefficient parallelizable iterative solution techniques for theresulting discrete problems. This project continues ongoing development of accuratenumerical techniques, efficient computational codes, andsupporting mathematical analysis for modeling of multiphase flowsand transport in groundwater hydrology that has applications indesign of remediation and clean-up technologies. The developmentof improved methods for assessing groundwater contamination byhazardous wastes has become increasingly important. Indeed,groundwater supplies are increasingly threatened by organic,inorganic, and radioactive contaminants introduced into theenvironment by improper disposal or accidental release. Estimatesof remediation costs at U.S. government sites alone range intothe hundreds of billions of dollars. Protecting the quality ofgroundwater supplies is a grand challenge problem of broadsocietal importance. There are also a number of industrialapplications of the proposed research. The production andmanagment of oil reservoirs are governed by equations andtechnologies that are nearly identical to those exploited ingroundwater hydrology. The manufacture of high quality fiberreinforced plastic composites for automotive and aerospaceapplication is controlled by technology similar to that studiedin this project. Design of semiconductors uses equations veryanalogous to those for groundwater transport. In these areas,technology transfer can be easily carried out. In particular,because of their intellectual content and strong links toindustry, this variety of problems provides an excellent vehiclefor students to learn techniques in applied and computationalmathematics.