若干周期性复合材料结构数学均匀化方法的计算精度

Accuracy analysis of mathematical homogenization method for several periodical composite structure

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） is generally implemented by finite element method （FEM） and its calculating accuracy depends completely on the order of perturbation and finite element, the perturbed displacements in uncoupled form are defined as the multiplications of influence functions and the derivatives of homogenized displacements. The order of elements depends on the accuracy requirements of influence function and homogenized displacements while the order of perturbations depends mainly on the accuracy of different-order derivatives of homogenized displacements and the properties of quasi loads. 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 loads. Then two-dimensional （2D） periodical composite structures are explored similarly, and the clamped boundary condition of periodical unite cell and the derivatives of homogenized displacements have great effects on the calculation accuracy of MHM. The effect of second-order perturbations is stressed for the accuracy of MHM. Finally, the potential energy functional is used to evaluate the accuracy of MHM and numerical comparisons validate the conclusions.