A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations
A Unified Variational Formulation for the Parabolic-Elliptic Eddy Current Equations
复制标题
抛物椭圆涡流方程的统一变分公式
DOI：
10.1137/110831477
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
B. Harrach
中科院分区：
文献类型：
--
作者：
L. Arnold;B. Harrach
瞬态激励电流产生电磁场，进而在近端导体中感应出电流。对于缓慢变化的场，这可以通过涡流方程来描述，该方程是通过忽略麦克斯韦方程中的电介质位移电流而获得的。涡流方程为抛物线椭圆型：在绝缘区域，场立即适应激励（准静止椭圆行为），而在导电区域，由于感应涡流（抛物线行为），这种适应需要一些时间。对于固定电导率，对方程进行了深入研究。然而，关于解对电导率的依赖性，特别是关于解对于方程从椭圆型变为抛物线型的敏感性的严格数学结果知之甚少。在这项工作中，我们推导了涡流方程的一个新的统一变分公式，该公式对于电导率具有均匀的矫顽力。然后，我们应用新的统一公式来研究电导率接近零时的情况，并相对于引入导电物体严格线性化非导电域周围的涡流方程。
Transient excitation currents generate electromagnetic fields which, in turn, induce electric currents in proximal conductors. For slowly varying fields, this can be described by 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 behavior), while in conducting regions, this adaptation takes some time due to the induced eddy currents (parabolic behavior). For fixed conductivity, the equations are well studied. However, little rigorous mathematical results are known for the solution's dependence on the conductivity, in particular for the solution's sensitivity with respect to the equation changing from elliptic to parabolic type. In this work, we derive a new unified variational formulation for the eddy current equations that is uniformly coercive with respect to the conductivity. We then apply our new unified formulation to study the case when the conductivity approaches zero and rigorously linearize the eddy current equations around a non-conducting domain with respect to the introduction of a conducting object.