A novel numerical method named element differential method (EDM) is first presented to solve linear and nonlinear static electromagnetic problems. The main idea of this method is to use the direct differentiation formulation of the shape functions of Lagrange isoparametric elements to evaluate geometry and physical variables. A new collocation method is proposed to establish the system of equations, in which the governing equation is collocated at the internal nodes of the elements, the continuity conditions of tangential component of magnetic field intensity and normal component of electric displacement vector are collocated at interface nodes between elements, and Neumann boundary conditions are collocated at outer surface nodes. Compared with the finite element method, there is no need to use any variational or energy principles to establish the system of equations and there is no requirement for integral evaluation. The derived spatial derivatives can be directly substituted into the static electromagnetic governing equations and the boundary conditions to form the final system of algebraic equations. Three 2D numerical examples are used to verify the correctness and effectiveness of the presented method for solving linear and nonlinear static electromagnetic problems.

中文翻译：

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求解线性和非线性电磁问题的元微分法

首先提出了一种新的数值方法，称为元素微分法（EDM）来解决线性和非线性静态电磁问题。该方法的主要思想是利用拉格朗日等参单元形状函数的直接微分公式来评估几何和物理变量。提出了一种建立方程组的新配置方法，将控制方程配置在单元内部节点处，将磁场强度切向分量和电位移矢量法向分量的连续性条件配置在界面节点处。单元之间，Neumann 边界条件配置在外表面节点处。与有限元法相比，不需要使用任何变分或能量原理来建立方程组，也不需要积分评估。导出的空间导数可以直接代入静态电磁控制方程和边界条件，形成最终的代数方程组。三个二维数值例子用于验证所提出的解决线性和非线性静态电磁问题的方法的正确性和有效性。