比例边界等几何分析方法I:波导本征问题

张勇; 林皋; 胡志强

doi:10.6052/0459-1879-2012-2-20120222

比例边界等几何分析方法I:波导本征问题

大连理工大学建设工程学部, 大连 116024

基金项目: 中德科学基金 (GZ566) 和国家自然科学基金 (90510018, 90915009, 51009019, 51109134) 资助项目.

详细信息

中图分类号: O242, O441, TN814
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出版历程
- 收稿日期: 2011-09-25
- 修回日期: 2011-11-01
- 刊出日期: 2012-03-17

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

Faculty of Infrastructure Engineering, Dalian University of Technology, Dalian 116024, China

Funds: The project was supported by the Chinese-Germany Science Foundation (GZ566) and the National Natural Science Foundation of China (90510018, 90915009, 51009019, 51109134).

摘要

摘要: 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.
- 比例边界等几何分析方法 /
- 比例边界有限元 /
- 波导 /
- 本征值 /
- 亥姆霍兹方程
Abstract: 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 exported to algebraic general eigenvalue equations, and thus the cutoff wavenumber of waveguide is obtained. 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.
- scaled boundary isogeometric analysis /
- scaled boundary finite element method /
- waveguide /
- eigenvalue /
- Helmholtz equation

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