Second-Order Three-Scale Asymptotic Analysis and Algorithms for Steklov Eigenvalue Problems in Composite Domain with Hierarchical Cavities

A top–down strategy based on the second-order asymptotic method is proposed for solving the Steklov eigenvalue problems on composite perforated materials with three-scale and two-periodic structures. Three different kinds of configurations are considered where the cavities are distributed only in the meso-scale, micro-scale, and both-scale representative cells respectively. Firstly, the second-order two-scale asymptotic expansion is performed between the macroscopic and the mesoscopic scale. Then, the second-order two-scale analysis is further developed on the mesoscopic cell functions at the microscopic level. While the asymptotic expansions of the first-order mesoscopic cell functions are similar for the three cases, the expansions of the second-order mesoscopic cell functions are distinguished from each other. It is interesting that when the holes with different scales are considered for the third case, the cell functions defined on the mesoscopic scale are dependent explicitly on the ratio between the mesoscopic and microscopic periods after homogenization. The three-scale asymptotic expansions of the eigenvalues are derived based on the "corrector equations" in a uniform manner and calculated in the integration form. The multi-scale finite element procedures are established based on these proposed asymptotic models and both the two- and three-dimensional asymptotic computations are carried out. By comparing the asymptotic computations with the classic finite element algorithm, it is demonstrated that this second-order three-scale asymptotic algorithm is effective in approximating the Steklov eigenvalues and reproducing the local oscillations of the eigenfunctions with less computational cost and the convergence of the second-order solutions are also confirmed. It is also instructive that when the parameters existing on the cavity boundaries are considered in the perforated materials, the second-order expansion terms should be included in the multi-scale analysis to reflect the asymptotic behavior of the structures correctly.