周期多孔结构的Steklov弹性特征值问题的多尺度渐近分析

本文针对周期多孔结构的Steklov弹性特征值问题发展了一种多尺度渐近分析与计算方法,通过对特征函数进行二阶双尺度渐近展开,依次推导得到了一阶单胞函数、材料等效弹性系数、均匀化弹性特征值问题及二阶单胞函数.该多尺度渐近模型的特点是均匀化特征值出现在控制微分方程中而不在孔洞边界上.通过对特征值进行二阶渐近展开并利用校正方程思想,本文得到了特征值的一阶与二阶校正表达式,给出了多尺度特征值的误差估计.最后,基于多尺度渐近展开模型本文进行了有限元计算.数值算例结果显示了多尺度分析在预测Steklov弹性特征值与特征函数的有效性及二阶校正的必要性.

Multiscale asymptotic analysis for Steklov elastic eigenvalue problem in periodically perforated domain

The multiscale asymptotic analysis and computational method for the Steklov elastic eigenvalue problem is developed in the periodically perforated domain. The first-order cell functions, the effective elastic coefficient, the homogenized elastic eigenvalue problem and the second-order cell functions are derived successively by performing the second-order two-scale(SOTS) asymptotic expansions for the eigenfunctions. The feature of the multiscale asymptotic model is that the derived homogenized eigenvalues appear in the homogenized equation instead of on the boundaries of cavities. The SOTS asymptotic expansion of the eigenvalues is also carried out, and the first- and second-order correctors for the eigenvalues are obtained by the idea of corrector equation and the error estimations of the multiscale eigenvalues are proved. Finally, the finite element algorithms are proposed based on the multiscale asymptotic expansion model. The numerical results show the effectiveness of the SOTS analysis method for predicting the Steklov elastic eigenvalues and eigenfunctions and the necessity of the second-order correctors is also demonstrated.