LI Hongpeng, LING Song, QI Zhenbiao, JIANG Keru, CHEN Lei. Accuracy of the Mathematical Homogenization Method for Thermomechanical Problems[J]. Applied Mathematics and Mechanics, 2020, 41(1): 54-69. doi: 10.21656/1000-0887.400119
 Citation: LI Hongpeng, LING Song, QI Zhenbiao, JIANG Keru, CHEN Lei. Accuracy of the Mathematical Homogenization Method for Thermomechanical Problems[J]. Applied Mathematics and Mechanics, 2020, 41(1): 54-69.

# Accuracy of the Mathematical Homogenization Method for Thermomechanical Problems

##### doi: 10.21656/1000-0887.400119
• Rev Recd Date: 2019-11-18
• Publish Date: 2020-01-01
• For thermo-mechanical problems of periodical composite structures, the full decoupled 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 element 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 applicability 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 finite element method were applied to guarantee the calculating accuracy of the influence function 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 evaluate the accuracy of the MHM. Numerical comparisons validate the conclusions.
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