比例边界等几何分析方法Ⅰ:波导本征问题

SCALED BOUNDARY ISOGEOMETRIC ANALYSIS AND ITS APPLICATION I:EIGENVALUE PROBLEM OF WAVEGUIDE

提出比例边界等几何方法(scaled boundary isogeometric analysis,SBIGA),并用以求解波导本征值问题.在比例边界等几何坐标变换的基础上,利用加权余量法将控制偏微分方程进行离散处理,半弱化为关于边界控制点变量的二阶常微分方程,即TE波或TM波波导的比例边界等几何分析的频域方程以及波导动刚度方程,同时利用连分式求解波导动刚度矩阵.通过引入辅助变量进一步得出波导本征方程.该方法只需在求解域的边界上进行等几何离散,使问题降低一维,计算工作量大为节约,并且由于边界的等几何离散,使得解的精度更高,进一步节省求解自由度.以矩形和L形波导的本征问题分析为例,通过与解析解和其他数值方法比较,结果表明该方法具有精度高、计算工作量小的优点.

Scaled boundary isogeometric analysis （SBIGA） is approved and applied for waveguide eigenvalue problem. Based on scaled boundary isogeometric transformation, the governing partial differential equations （PDEs） for waveguide eigenvalue problem are semi-weakened to a set of 2nd order ordinary differential equations （ODEs） by weighted residual methods, and transformed to a set of 1st order ODEs about the dynamic stiffness matrix in wavenumber domain. Approximating the dynamic stiffness matrix in the continued fraction expression and introducing auxiliary variables, the ODEs are finally and thus the cutoff wavenumber of waveguide is obtained. exported to algebraic general eigenvalue equations, The main property of SBIGA is that the governing PDEs are isogeometricly discretized on domain boundary, which reduces the spatial dimension by one and analytical feature in the radial direction like traditional SBFEM, additionally, boundary is exactly discretized as its geometry design. The numerical examples, including rectangular and L-shaped waveguides, are presented and compared with analytic solution and other numerical methods. The results show that SBIGA yields high precision results with fewer amounts of DOFs than other methods do.