Abstract
Piezoelectric composites have found a wide range of applications in smart structures and devices and effective numerical methods should be developed to simulate not only the macroscopic coupled piezoelectric performances, but also the details of the local distributions of the stress and electric field. In this paper, we proposed a multi-scale asymptotic algorithm based on the Second-Order Two-Scale (SOTS) analysis method for the piezoelectric eigenvalue problem in perforated domain with periodic micro-configurations. The eigenfunctions and eigenvalues are expanded to the second-order terms and the homogenized eigensolutions; the expressions of the first- and second-order correctors are derived successively. The first- and second-order correctors of the eigenvalues are determined according to the integration forms of the correctors of the corresponding eigenfunctions. Explicit expressions of the homogenized material coefficients are derived for the laminated structures and the finite element procedures are proposed to compute the homogenized solutions and the correctors numerically. The error estimations for the approximations of eigenvalues are proved under some regularity assumptions and a typical numerical experiment is carried out for the two-dimensional perforated domain. The computed results show that the SOTS analysis method is efficient in identifying the piezoelectric eigenvalues accurately and reproducing the original eigenfunctions effectively. This approach also provides an efficient computational tool for piezoelectric eigenvalue analysis and can extend to other multi-physics problems with complex microstructures.