基于能量统一格式的多尺度有限元法

Multi-scale finite element method based on unified energy frame

为分析简单晶体多尺度有限元计算的能量构成,利用能量最小原理得到在统一理论框架下多尺度有限元计算的统一格式,表明有限元计算可以在微观原子尺度下和在宏观连续介质尺度下进行多尺度有限元计算.基于简单晶体变形的特点说明过渡单元设计应遵守的原则,并指出理想过渡单元应该是类似于晶体结构的单元,对于较复杂的晶体,则应该利用空间群方法充分研究具有230种空间群的过渡单元的性质.引用EIDEL等的纳米压痕计算结果作为算例,表明在计算中无虚拟插值点,多级晶胞单元具有与原单胞相同的点群操作且位移场插值受晶体中原子间键长的约束.

To analyze the energy of single-crystal in multi-scale finite element computation,the minimized energy principle is used to obtain the unified format under unified theory frame.It is shown that finite element computation can be used for multi-scale analysis under both micro-atomic scale or macro-continuum scale.Based on the deformation characteristics of single-crystal,a design principle for transitional elements is introduced and the optimal transitional elements should be analogous to crystal structure.For complicated crystal,space group method can be used for a further study on the character of transitional elements which contain 230 kinds of space groups.The paradigm of nano-indentation computation results obtained by EIDEL et al is cited to verify that no virtual interpolation point is required in the simulation and multi-level cell elements could have the same point-group symmetry operations as those of the primitive cell.Furthermore the displacement interpolation field must be constrained by the length of bonds between atoms in the crystal.