数学均匀化方法（MHM)一般需要通过有限元方法来实现，摄动阶次和单元阶次直接影响计算结果，其中各阶摄动项的解耦形式是相应阶次的影响函数和均匀化位移导数的乘积。影响函数精确解的阶次由其控制方程右端项的虚拟载荷阶次决定，外载荷阶次决定了均匀化位移及其各阶导数精确解的阶次；单元阶次的选取同时依赖于影响函数和均匀化位移精确解的阶次，而摄动阶次的选取则主要依赖于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，在周期性复合材料杆上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对计算结果有很大的影响。最后应用最小势能原理评估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) is generally implemented by finite element method, and its calculating accuracy depends completely on the order of perturbation and finite element，the perturbations in uncoupled form are defined as the multiplications of influence functions and the derivatives of homogenized displacements. The order of exact solutions for influence function depend on the order of pseudo load which is the right term of the governing equation whereas the exact solution of homogenized displacements and its different orders derivatives depend on the external loads; The order of elements depends on the exact solution of influence function and homogenized displacements simultaneously while the order of perturbations depend mainly on the accuracy of different order derivatives of homogenized displacements; For the static problems of periodical composite rod, the exact solutions can be obtained using correct order of MHM and finite element for the static problem of periodic composite rod subjected to different order of load. Then two dimensional(2D) periodical composite are explored similarly，and the clamped boundary con-dition of periodical unite cell have great influence for calculating accuracy of MHM. Finally, the potential en-ergy functional is used to evaluate the accuracy of MHM, and numerical comparisons validate the conclu-sions.
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