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

点击翻译按钮获取中文摘要

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.