波导本征问题分析的比例边界有限元方法

引入了一种求解波导本征值问题的高效而精确算法-比例边界有限元方法SBFEM (Scaled Boundary Finite Element Method).该方法的一个特点是只需在边界上进行离散,问题降低一维,使计算工作量大大减少;另一特点是所建立的控制方程为二阶常微分方程,可以解析地求解,使计算精度得到了保证.论文利用变分原理并通过比例边界坐标变换,推导了TE波和TM波波导的比例边界有限元频域方程以及波导动剐度方程,同时给出了波导动刚度矩阵的连分式解形式,通过引入辅助变量进一步得出波导特征值方程并求出波导本征值.以矩形、L形波导和叶型加载矩形波导的本征问题分析为例,通过与解析解及其他数值方法比较,结果表明,此方法具有精度高、计算工作量小的优点,而且随着连分式阶数增加收敛速度快.进一步分析了一类角切四脊正方形波导的传输特性.

Scaled boundary finite element method for the eigen analysis of waveguide

Scaled boundary finite element method (SBFEM) is used for determining waveguide eigenva-lue.The key characteristic of SBFEM is that governing equations are discretized on boundary of computational domain by SBFEM,which reduces spatial dimension by one,and analytical procedure is applied in the reduced direction.A weighted residual approach and coordinate transformation between scaled and cartesian coordinate are used to derive the SBFEM equations for both TE and TM mode cases of waveguide based on Helmholtz Equation.The dynamic stiffness matrix of waveguide is also obtained as a continued fraction solution.By using this solution and introducing auxiliary variables,the standard procedures can be applied to perform eigen analysis of waveguide.The eigenvalues of rectangular,L-shaped and vaned rectangular waveguide are calculated and compared with analytic solution and other numerical methods.The results show that the present method yields excellent results,high precision and less amount of computation time. And rapid convergence is observed as the order of continued fraction increases.The transmission characteristics of a quadruple corner-cut ridged square waveguide have been analyzed.