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.

瞬时(例如脉冲)励磁电流产生电磁场，而电磁场又在近端导体中感应电流。在数学上，这可以用偏微分方程组来描述，也就是忽略麦克斯韦方程中的介质位移电流而得到的涡流方程。涡流方程是抛物线-椭圆型的：在绝缘区，磁场瞬间适应激励(准稳定椭圆行为)，而在导电区，由于感应涡流(抛物线行为)，这种适应需要一段时间。涡流效应用于远程检测导电物体(例如，在地雷探测的情况下)和非侵入性地识别导体内部的缺陷(所谓的涡流检测)。在数学术语中，这导致了从(部分)解的知识重建涡流方程中的电导系数的反问题(S)。在所提出的项目中，我们的目标是利用抛物-椭圆型方程的统一变分理论来从理论上研究反问题中的可辨识性问题，并推导出严格合理的重构策略。

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