热力耦合问题数学均匀化方法的计算精度

Accuracy of the Mathematical Homogenization Method for Thermomechanical Problems

针对复合材料周期结构热力耦合问题,推导了数学均匀化方法(MHM)各阶摄动位移的全解耦格式和各阶影响函数控制方程,并使用加权残量方法将其转化为易于编程计算的有限元列式.在解耦格式中,各阶摄动位移是相应阶次的影响函数和宏观场导数的乘积,即影响函数和宏观场导数的计算精度共同决定摄动项的精度,其中影响函数的计算精度取决于单胞边界条件选取的适用性.针对2D复合材料周期结构静力学问题,使用超单胞边界条件和微分求积有限单元法,分别提高了影响函数和宏观场导数的求解精度.在此基础上,研究了高阶展开项对MHM真实位移精度的影响,确定了二阶摄动项的必要性.最后应用最小势能原理评估了各阶摄动MHM的计算精度,数值比较结果验证了结论的正确性.

For thermo-mechanical problems of periodical composite structures,the full decou pled scheme of each order perturbation and the governing equation of each order influence function for the mathematical homogenization method(MHM)were derived,then the weighted residual method was utilized to transform them into the conveniently programmable finite ele ment matrix form.The perturbation displacements in the uncoupled form were defined as the products of influence functions and the macro field derivatives,and the calculating accuracy of the perturbation displacements were determined by the accuracy of influence functions and the macro field derivatives,in turn the accuracy of influence functions depended mainly on the ap plicability of unit cell boundary conditions.For the static problems of 2D periodical composite structures,the super unit cell periodical boundary condition and the differential quadrature fi nite element method were applied to guarantee the calculating accuracy of the influence func tion and the macro field derivatives respectively.On this basis,the influence of the high-order perturbations on the true displacement of the MHM was studied,and the necessity of the 2nd order perturbation was emphasized.Finally,the potential energy functional was used to evalu ate the accuracy of the MHM.Numerical comparisons validate the conclusions.