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数学均匀化方法(MHM)一般需要通过有限元方法(FEM)来实现,摄动阶次和单元阶次直接影响计算结果。在解耦格式中,各阶摄动位移是相应阶次的影响函数和均匀化位移导数的乘积。单元阶次的选取取决于影响函数和均匀化位移的精度要求,而摄动阶次的选取则主要依赖于虚拟载荷的性质和均匀化位移各阶导数的计算精度;针对周期性复合材料杆的静力学问题,在施加不同阶次的载荷时,通过选择合适阶次的单元和摄动阶次得到了精确解。使用类似的方法研究了2D周期性复合材料静力学问题,指出了四边固支作为周期性单胞边界条件以及宏观位移求导精度对计算结果将有很大的影响。强调了二阶摄动对数学均匀化方法计算精度的作用;在数值结果中,应用最小势能原理评估了各阶摄动数学均匀化方法的计算精度,数值比较结果验证了结论的正确性。
The mathematical homogenization method (MHM) generally needs to be implemented by the finite element method (FEM), and the perturbation order and the unit order directly affect the calculation results. In the decoupled format, the perturbed displacements of the orders are the product of the influence function of the corresponding order and the uniform displacement derivative. The choice of unit order depends on the accuracy requirements of the influence function and the uniform displacement. The selection of perturbation order depends mainly on the nature of the virtual load and the calculation precision of the derivatives of the uniform displacement. For the periodic composite rod Static problems, when different loads are applied, are obtained by choosing the right order element and perturbation order. A similar method is used to study the static behavior of 2D periodic composites. It is pointed out that the quadripartite clamped as a periodic unit cell boundary condition and the precision of derivation of macroscopic displacements will have a great influence on the calculation results. The calculation accuracy of second order perturbation method is emphasized. In the numerical results, the calculation accuracy of the method for the uniform perturbation of each order is evaluated by the principle of minimum potential energy. The numerical results verify the correctness of the conclusion.