Research in Robust and Efficient Computational Methods for Partial Differential Equations Arising in Fluid Flows and Electromagnetics

该项目旨在开发、分析和数值验证用于流体流动和电磁学的新有限元方法。 流体流动方法的研究将集中于非标准最小二乘原理在构造数值算法中的应用。 虽然对于不可压缩粘性流，传统的最小二乘原理已成为混合Galerkin方法的一种可行的替代方法，但在高Re或跨音速流的情况下，这种原理并不完全令人满意。 为了克服L2-范数方法存在的计算缺陷，如对奇异性敏感，缺乏范数等价性等，新算法将基于网格相关和负范数最小二乘原理。 第二个研究方向的重点将是在三维涡流计算。 为了解决3D模拟算法的计算复杂性，将使用控制方程的等效势公式。 适当的规范选择的潜力的问题将得到彻底和系统的检查，以确保良定的微分方程组。 数学的发展将特别注意相关边界和界面条件的有效实现，其中最小二乘项将被用来执行这些条件weaken.The共同的线程连接的两个主要研究方向ofthis项目的重点是新的计算算法偏微分方程。 这种方程出现在几乎每一个科学和工程领域，其有效的数值解是至关重要的，我们能够进行计算机模拟的物理过程，从大气运动到超导体中的电流流动。 因此，开发高性能的计算工具，使逼真的模拟将发挥越来越重要的作用，在未来的科学和技术的进步。 这种计算算法的影响将不仅体现在通过用虚拟计算机实验取代现场实验而可能节省的巨大成本和/或时间方面，而且体现在以下事实中：在某些情况下，计算机模型可能是唯一可行的设计方法。 该项目的重点是开发用于流体流动和电磁场建模中产生的微分方程的工具。 我们想到的两个具体应用是三维涡流的计算和跨音速和高雷诺数流动的模拟。 我们研究的一个动机是这些问题的实际相关性。 例如，涡电流方程的解决方案出现在不同的领域，如聚变电源的环向场磁体的开发，等离子体物理现象的建模，以及磁带磁头的设计，而跨音速和高Re流与飞机的设计和污染物扩散的建模有关。 与此同时，在现实的三维环境中数值求解这些问题仍然是一个突出的和具有挑战性的计算任务.因此，我们的研究也是出于真实的和现有的需要，以开发高效和强大的计算工具，电磁学和流体流动的应用，可以在这样的现实环境中使用。

This project is directed towards the development, analysis and numerical validation of new finite element methods for fluid flows and electromagnetics. The research in methods for fluid flows will focus on applications of nonstandard least-squares principles in the constructionof the numerical algorithms. While for incompressible viscous flows conventional least-squares principles have established themselves as a viable alternative to mixed Galerkin methods, such principles are not completely satisfactory in the context of high Re or transonic flows. To circumvent existing computational defects of L2-norm methods such as sensitivity to singularities, lack of norm equivalence and etc., the new algorithms will be based on mesh-dependent and negative-norm least-squares principles. The emphasis of the second research direction will be on eddy current computations in three dimensions. To address computational complexity of 3D simulations algorithms will use equivalent potential formulations of the governing equations. The issue of proper gauge selection for the potentials will receive a thorough and systematic examination so as to ensure well-posed sets of differential equations. Algorithmic development will pay special attention to the efficient implementation of the relevant boundary and interface conditions where least-squares terms will be used to enforce these conditions weakly.The common thread which links the two principal research directions ofthis project is the focus on new computational algorithms for partial differential equations. Such equations arise in virtually every field of science and engineering and their efficient numerical solution is critical for our ability to conduct computer simulations of physical processes ranging from atmospheric motions to flows of current in superconductors. As a result, development of high performance computational tools which enable realistic simulations will play an increasingly important role for the future advances in science and technology. The impact of such computational algorithms will be felt not only in terms of tremendous cost and/or time savings made possible by replacing field experiments by virtual, computer experiments, but also by the fact that in some instances computer models may be the only feasible design approach. In the focus of this project is the development of such tools for differential equations arising in modeling of fluid flows and electromagnetic fields. Two specific applications that we have in mind are computation of three-dimensional eddy currents and simulations of transonic and high Reynolds number flows. One motivation for our research is the practical relevance of these problems. For instance, solution of the eddy current equations arises in such varied areas as development of toroidal field magnets for fusion power, modeling of plasma physics phenomena, and design of tape heads, while transonic and high Re flows are relevant to design of aircraft and modeling of dispersion of pollutants. At the same time numerical solution of these problems in realistic three-dimensional settings continues to be an outstanding and challenging computationaltask. Thus, our research is also motivated by the real and existing need to develop efficient and robust computational tools for electromagnetics and fluid flow applications which can be used in such realistic settings.