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

Abstract:

The multiscale asymptotic analysis and computational method for the Steklov elastic eigenvalue problems are developed in a periodically perforated domain. The first-order cell functions, the effective elastic coefficients, the homogenized elastic eigenvalue problems 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 expansions of the eigenvalues are 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, a finite element algorithm is 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.