中文摘要：

针对三维共振腔的电磁场分析，利用Maxwell方程的对偶方程体系形式，从其相应的对偶变量变分原理出发，导出了三维电磁场辛有限单元的详细列式。为了有限元列式的保辛，变分原理被积函数可导向对于对偶变量为对称的形式。变分原理的边界积分项对于相邻单元相互抵消。由于采用了对偶变量的插值函数，使得电磁场单元构造可以在层面上进行，从而避免了所谓的连续性问题。无物理意义的零本征解可采用奇异值分解加以排除。文末分别对矩形及圆柱形的共振腔做了数值计算并与解析解和棱边元计算结果进行对比，算例表明了列式及算法的有效性。

英文摘要：

Based on the form of duality system of Maxwell＇s equations and the variational principle for dual variables, a symplectic finite element formulation for 3D electromagnetic fields is derived in details. In order to maintain the symplectic conservation conditions of FEM formulation, the integrand in the variational principle is rewritten in symmetrical form for duality variables. The boundary integral terms of the variational principle are cancelled mutually for the adjacent elements. Because the dual variables are interpolated independently at CO level, the difficulty of C1 continuity requirement is avoided. Zero eigenvalues can be eliminated using singular value decomposition （SVD） method. Numerical results for rectangular and cylinder cavity resonance demonstrate the efficiency of the symplectic finite element method.