Mathematische Theorie direkter und inverser transienter Wirbelstromprobleme

正向和逆向瞬态涡流问题的数学理论

• 批准号: 183973053
• 负责人: Professor Dr. Bastian von Harrach
• 金额: --
• 依托单位: Schwerpunkt Analysis und Numerik
• 依托单位国家: 德国
• 项目类别: Research Grants
• 财政年份: 2010
• 资助国家: 德国
• 起止时间: 2009-12-31 至 2013-12-31
• 项目状态: 已结题
• 来源: https://gepris.dfg.de/gepris/projekt/183973053?language=en
• 关键词: Mathematische; Theorie; direkter; und; inverser

Transient (e.g. pulsed) excitation currents generate electromagnetic fields which in turn induce electric currents in proximal conductors. Mathematically, this can be described by partial differential equations, the eddy-current equations, which are obtained by neglecting the dielectric displacement currents in Maxwell’s equations. The eddy-current equations are of parabolic-elliptic type: In insulating regions, the field instantaneously adapts to the excitation (quasistationary elliptic behaviour), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behaviour). Eddy current effects are used for remotely detecting conducting objects (e.g. in the context of land mine detection) and to non-invasively identify flaws inside a conductor (so-called eddy-current testing). In mathematical terms this leads to the inverse problem of reconstructing the conductivity coefficient in the eddy current equations from (partial) knowledge of the solution(s). In the proposed project we aim to utilize a unified variational theory for the parabolic-elliptic equations to theoretically study identifiability questions in the inverse problem and derive rigorously justified reconstruction strategies.

瞬态（例如脉冲）激励电流产生电磁场，电磁场又在近端导体中感应电流。在数学上，这可以通过偏微分方程、涡流方程来描述，该方程通过忽略麦克斯韦方程中的介电位移电流而获得。涡电流方程是抛物线椭圆型的：在绝缘区域，场瞬时适应激励（准稳态椭圆行为），而在导电区域，由于感应涡电流（抛物线行为），这种适应需要一些时间。涡流效应用于远程检测导电物体（例如在地雷检测的背景下）和非侵入性地识别导体内部的缺陷（所谓的涡流测试）。在数学术语中，这导致从解的（部分）知识重构涡流方程中的传导系数的逆问题。在建议的项目中，我们的目标是利用统一的变分理论的抛物椭圆方程的反问题，从理论上研究可识别性问题，并得到严格合理的重建策略。

项目成果

Unique shape detection in transient eddy current problems

• DOI: 10.1088/0266-5611/29/9/095004
• 发表时间: 2013-09
• 期刊: Inverse Problems
• 影响因子: 2.1
• 作者: L. Arnold;B. Harrach

A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations

抛物椭圆涡流方程的统一变分公式

• DOI: 10.1137/110831477
• 发表时间: 2012
• 期刊: SIAM J. Appl. Math.
• 作者: L. Arnold;B. Harrach
• 通讯作者: B. Harrach