一种有效的广义特征值分析方法

AN EFFECTIVE ALGORITHM FOR GENERALIZED EIGENVALUE PROBLEMS

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 .

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.