A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations
A Finite Element Method with Lagrange Multipliers for Low-Frequency Harmonic Maxwell Equations
复制标题
低频调和麦克斯韦方程组的拉格朗日乘子有限元法
DOI：
10.1137/s0036142901390780
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发表时间：
2002
期刊：
影响因子：
--
通讯作者：
P. Salgado
中科院分区：
文献类型：
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作者：
A. Bermúdez;R. Rodríguez;P. Salgado
本文的目的是分析一种有限元方法来解决低频谐波麦克斯韦方程组在一个有界区域包含导体和介质。这个偏微分方程系统是所谓的涡流问题的模型。在将这个问题写成磁场的形式后，在四面体网格上用Nederberg棱边有限元进行离散。如果在电介质域中的元素上施加无卷曲条件，则容易获得误差估计。
然后，无旋条件施加，在离散水平上，通过引入分段线性多值电位。由此产生的问题被证明是一个离散版本的其他连续配方中的磁场中的介电部分的域已被取代的磁势。此外，这种方法导致了一个重要的节省计算工作。与拓扑结构有关的问题也被认为是在考虑到有一个非简单连接的电介质域的可能性。
实施问题进行了讨论，包括一个负责任的程序施加的边界条件通过一个拉格朗日乘数。最后，该方法被应用到解决一个三维模型问题：一个圆柱形电极周围的电介质。
The aim of this paper is to analyze a finite element method to solve the low-frequency harmonic Maxwell equations in a bounded domain containing conductors and dielectrics. This system of partial differential equations is a model for the so-called eddy currents problem. After writing this problem in terms of the magnetic field, it is discretized by Nedelec edge finite elements on a tetrahedral mesh. Error estimates are easily obtained if the curl-free condition is imposed on the elements in the dielectric domain.
Then, the curl-free condition is imposed, at a discrete level, by introducing a piecewise linear multivalued potential. The resulting problem is shown to be a discrete version of other continuous formulation in which the magnetic field in the dielectric part of the domain has been replaced by a magnetic potential. Moreover, this approach leads to an important saving in computational effort. Problems related to the topology are also considered in that the possibility of having a nonsimply connected dielectric domain is taken into account.
Implementation issues are discussed, including an amenable procedure to impose the boundary conditions by means of a Lagrange multiplier. Finally, the method is applied to solve a three-dimensional model problem: a cylindrical electrode surrounded by dielectric.