非周期性多尺度问题的平均化方法

Homogenization Method of Non-Periodic Multi-Scale Problem

提出了解决复合材料非周期性多尺度问题的平均化理论框架及其有限元实现方法,指出材料的力学性能应该是大量原子、分子、晶粒等微粒行为的宏观体现,分析了算法平均的重要性,论证了平均化有限元方法是解决多尺度问题的有效方法.进而采用该理论框架对于含有相同数量级规模的典型小尺度结构的非均质材料进行了分析.研究结果表明,平均化有限元的计算规模远低于直接有限元法的计算规模,并且二者精度能保持一致.

In this paper, the basic idea and the finite element implementation of the homogenization theory applied in nonperiodic multi-scale problem were introduced. According to this, the mechanical properties are the macroscopic representation of particle behavior, such as atoms, molecules and grains. The importance of the arithmetic average is analyzed, and the homogenization method is proved to be the efficient method of solving multi-scale problems from algorithm analysis point of view. Furthermore, for heterogeneous materials with the same order of size of typical microscale structures, the calculation scale of finite element using the homogenization method is far less than that of the direct finite element method. The results show that the calculation accuracy of finite element using the homogenization method keeps consistent with that of the direct finite element method.