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一种有效的广义特征值分析方法
引用本文:刘寒冰,龚国庆,刘建设.一种有效的广义特征值分析方法[J].固体力学学报,2003,24(4):419-428.
作者姓名:刘寒冰  龚国庆  刘建设
作者单位:吉林大学南岭校区交通学院,长春,130025
基金项目:国家自然科学基金 (19872 0 2 9)资助
摘    要:提出了一种适合于自适应有限元分析中求解广义特征值问题的多重网格方法.这种方法充分利用了初始网格下的结果,通过插值或最小二乘拟合技术来得到网格变化后的新的近似特征向量,然后由多重网格迭代过程实现对结构广义特征值问题的求解.在多重网格迭代的光滑步中,选择了收敛梯度法以提高其收敛率;在粗网格校正步中,则导出了一种近似求解特征向量误差的方程.这种方法将网格离散过程和数值求解过程很好地相结合,建立了一个网格细分后广义特征值问题的快速重分析方法,与传统有限元方法相比较,具有计算简便、计算量少等特点,可以作为结构动力问题自适应有限元分析的一种十分有效的工具.

关 键 词:多重网格迭代  广义特征值  自适应有限元  结构动力学  插值法  最小二乘拟合法  收敛梯度法
修稿时间:2002年3月5日

AN EFFECTIVE ALGORITHM FOR GENERALIZED EIGENVALUE PROBLEMS
Liu Hanbing,Gong Guoqing,Liu Jianshe.AN EFFECTIVE ALGORITHM FOR GENERALIZED EIGENVALUE PROBLEMS[J].Acta Mechnica Solida Sinica,2003,24(4):419-428.
Authors:Liu Hanbing  Gong Guoqing  Liu Jianshe
Abstract:An effective multi-grid method fit for the adaptive finite element analysis for generalized eigenvalue problems is proposed. The method utilizes sufficiently the results in the original meshes and takes interpolation method or least-square approximation to obtain new eigenvectors in the changed meshes. And then it accomplishes the solution of the generalized eigenvalue problems by multi-grid iteration procedure. In the smooth process we select Conjugate Gradient method other than the traditional Gauss-Seidel or Jacobi iterations in order to improve the rate of convergence; in the coarse grid correction we deduct a formula to solve approximately the errors of the eigenvectors. It combines the grid discretization with the numerical solution and establishes a fast reanalysis procedure after remeshing. The numerical results show that the method is computationally simple and feasible as compared with the traditional finite element method, and can be used as a particularly effective tool for the generalized eigenvalue problems by adaptive finite element method.
Keywords:multi-grid iteration  generalized eigenvalue problem  adaptive FEM
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