Two-scale and three-scale asymptotic computations of the Neumann-type eigenvalue problems for hierarchically perforated materials

A top-down strategy is proposed for analyzing the elliptic eigenvalue problems of the hierarchically perforated materials with three-scale periodic configurations. The heterogeneous structure considered is composed of perforated cells in the mesoscopic scale and composite cells in the microscopic scale, and Neumann boundary conditions are imposed on the boundaries of the cavities. By using the classical two-scale asymptotic expansion method, the homogenized eigenfunctions and eigenvalues are obtained and the first- and second-order auxiliary cell functions are defined firstly in the mesoscale. Then, the two-scale asymptotic analysis is furtherly applied to the mesoscopic cell problems and by expanding the meso cell functions to the second-order terms, the homogenized cell functions are derived and the relations between the homogenized coefficients and the coefficients of constituent materials in the three scale levels are established. Finally, the second-order three-scale asymptotic approximations of the eigenfunctions are presented and by the idea of “corrector equation”, the three-scale expressions of the eigenvalues are obtained. The corresponding finite element algorithm is established and the successively up-scaling procedures are established. Typical two-dimensional numerical examples are performed, and both the two-scale and three-scale computed approximations of the eigenvalues are compared with the ones obtained in the classical computation. By the least squares technique, it is demonstrated that the three-scale asymptotic solutions of the eigenfunctions are good approximations of the original eigensolutions corresponding to both the simple and multiple eigenvalues. This study offers an alternative approach to describe the physical and mechanical behaviors of the hierarchically structures with more than two scales and it is indicated that the second-order terms plays an important role not only in the derivation of the expansions but also in the practical computations to capture the local oscillations within the cells.