Abstract 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 equations”, 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.

中文翻译：

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分层穿孔材料诺依曼型特征值问题的两尺度和三尺度渐近计算

摘要 提出了一种自顶向下的策略来分析具有三尺度周期构型的分层穿孔材料的椭圆特征值问题。所考虑的异质结构由细观尺度的穿孔单元和微观尺度的复合单元组成，并在空腔边界施加诺依曼边界条件。利用经典的双尺度渐近展开法，得到均质化的特征函数和特征值，并在中尺度上首先定义一阶和二阶辅助单元函数。然后，将双尺度渐近分析进一步应用于细观细胞问题，并通过将细观细胞功能扩展到二阶项，推导了均质化单元函数，建立了均质化系数与三个尺度层次的构成材料系数之间的关系。最后，给出了特征函数的二阶三尺度渐近逼近，并通过“校正方程”的思想，得到了特征值的三尺度表达式。建立了相应的有限元算法，并建立了逐级放大的程序。执行典型的二维数值示例，并将特征值的两尺度和三尺度计算近似值与经典计算中获得的近似值进行比较。通过最小二乘法，证明了特征函数的三尺度渐近解是对应于简单和多特征值的原始特征解的良好近似。这项研究提供了一种替代方法来描述具有两个以上尺度的分层结构的物理和力学行为，并表明二阶项不仅在扩展的推导中而且在实际计算中都起着重要作用捕捉细胞内的局部振荡。