Fast Adaptive Finite Element Methods for Electromagnetic Applications

有限元（finite Element， FE）分析在各种工程应用中广泛应用于电磁场的计算机模拟。虽然在过去二十年中取得了重大进展，但仍然存在相当大的计算困难。通常需要非常精细的有限元网格，包含数十万或更多的元素，以达到理想的精度，并且计算时间可能高得令人望而却步。在本项目中，将采用多级预调节器自适应有限元方法来解决微磁学和地球物理中的各种电磁问题，从而实现电磁有限元分析的定性进展。快速自适应方法将大大提高电磁场计算的速度和精度。该项目本质上是跨学科的，旨在弥合最先进的计算方法和电磁应用之间的差距。预条件分层基多网格方法将应用于磁记录问题，特别是多粒子系统（磁带、薄膜介质）和磁记录磁头。将模拟由Landau-Lifshitz-Gilbert方程描述的静磁场（“退磁”）以及动态过程。多电平预调节器也将应用于地球物理学中的静态和涡流问题。自适应网格细化将系统地应用于多个应用领域。在微磁学中，它将提供一种追踪畴壁运动的方法。在地球物理应用中，对于非常大的计算域是典型的，自适应网格细化应该导致在可管理的尺寸网格上的合理精度。四面体元素和其他类型元素的网格细化策略将根据申请人提出的元素形状的精确先验特征进行评估：一般最大特征值准则，以及一阶四面体元素的最小奇异值条件“边缘形状矩阵”。两名研究生将参与拟议的项目。对这些学生来说，一个明显的好处是他们将接触到现代计算方法、电磁场分析和各种实际应用。几家公司已经对申请人的研究表现出了浓厚的兴趣。在拟议的项目过程中，极有可能发展与磁记录行业、石油勘探服务和有限元软件公司的工业合作。认为本项目的研究成果将具有理论和实践意义：它将促进磁记录介质和磁头的快速三维场分析、地球物理学中的电磁感应测井，并将对无损检测、优化等领域产生重要影响

9812895TsukermanFinite Element (FE) analysis is very widely used for computer simulation of electromagnetic fields in various engineering applications. Although significant progress has been achieved over the last two decades, considerable computational difficulties remain. Very fine FE meshes, with hundreds of thousands of elements or more, are often needed to achieve a desirable accuracy, and the computational time may be prohibitively high.In the proposed project, a qualitative advancement in electromagnetic FE analysis will be achieved by applying adaptive finite element methods with multilevel preconditioners to a variety of electromagnetic problems in micromagnetics and geophysics. Fast adaptive methods will lead to a drastic increase in the speed and accuracy of electromagnetic field computation. The project is interdisciplinary in its nature and is intended to bridge the gap between state-of-the-art computational methods and electromagnetic applications.Preconditioned Hierarchical Basis Multigrid methods will be applied to magnetic recording problems, in particular to multiparticle systems (magnetic tapes, thin film media) and to magnetic recording heads. Magnetostatic ("demagnetizing") fields as well as dynamic processes described by the Landau-Lifshitz-Gilbert equation will be modeled. Multilevel Preconditioners will also be applied to static and eddy current problems in geophysics.Adaptive mesh refinement will be systematically implemented in several application areas. In micromagnetics, it will provide a way to trace the domain wall motion. In geophysical applications, for which very large computational domains are typical, adaptive mesh refinement should lead to a reasonable accuracy on meshes of manageable size.Mesh refinement strategies for tetrahedral elements and other types of elements will be evaluated on the basis of precise a priori characterization of element shapes proposed by the applicant: the general maximum eigenvalue criterion and, for first order tetrahedral elements, the minimum singular value condition for the element 'edge shape matrix'.Two graduate students are to be involved in the proposed project. An obvious benefit to these students will be their exposure to modern computational methods, electromagnetic field analysis and various practical applications.Several companies have already expressed significant interest in the applicant's research. Industrial collaboration with companies in magnetic recording industry, oil exploration services and finite element software is very likely to develop in the course of the proposed project.It is felt that the results of this project would be of both theoretical and practical significance: they would facilitate fast three-dimensional field analysis of magnetic recording media and heads, galvanic and induction logging in geophysics, and will have important implications for nondestructive testing, optimization, and in other areas.***