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

涉及多相污染物在多孔介质中流动和输运的复杂物理现象通常由耦合的非线性偏微分方程组来模拟。最近在计算能力方面的进步(特别是随着新的并行体系结构的出现)极大地扩展了将更多物理知识结合到微分方程式中的可能性。虽然对模型的这种改变可以提高其准确表示潜在真实过程的能力，但这必须在建模过程的每个阶段使用物理、数学、数值和计算概念进行验证。研究人员和他的同事开发了描述多相污染物在多孔介质中流动和传输的微分方程组的分析和数值方法。第一个目标是对这些非线性偏微分方程组进行理论分析。要讨论的专题是规则性、稳定性和分叉现象。第二个目标是研究和发展能够捕捉尖锐解锋面、产生准确的流体速度、守恒质量、及时给出高阶近似和适合于局部网格加密的有限元方法。在这里，研究人员通过欧拉-拉格朗日局部伴随方法和耦合高阶时间推进的混合有限元方法的概念，发展了处理非线性输运-扩散方程组的有限元方法。第三个目标是开发高效的可并行迭代求解技术来求解离散问题。该项目继续开发精确的计数技术、高效的计算代码和支持数学分析，以模拟地下水水文学中的多相流和运移，并将其应用于修复和清理技术的设计。开发评估危险废物对地下水污染的改进方法变得越来越重要。事实上，地下水供应日益受到有机、无机和放射性污染物的威胁，这些污染物通过不当处置或意外释放进入环境。据估计，仅美国政府场所的补救费用就高达数千亿美元。保护地下水供应质量是一个具有广泛社会意义的重大挑战问题。这项拟议的研究也有许多工业应用。油藏的生产和管理受方程式和技术的支配，这些方程式和技术与开采的环水水文学几乎相同。用于汽车和航空航天的高质量纤维增强塑料复合材料的制造由与该项目研究的技术类似的技术控制。半导体的设计使用的方程式与地下水传输的方程式非常相似。在这些领域，技术转让可以很容易地进行。特别是，由于它们的知识含量和与行业的紧密联系，这类问题为学生提供了学习应用数学和计算数学技术的极好工具。

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.