采用非协调元的连续体拓扑优化设计　

TOPOLOGY OPTIMIZATION OF CONTINUUM STRUCTURES USING INCOMPATIBLE FINITE ELEMENTS

介绍了满足分片检验条件的一种非协调元,推导了结构拓扑优化设计中数值计算和敏度分析的基本方程,给出了数值算例,并对协调等参元和非协调元的拓扑优化结果进行了对照.最后的优化结果表明:采用非协调元所得的优化解已经能够使用;如果再实施过滤技术,设计区域中的中间密度单元明显减少,优化结果会更加精致;使用两类单元的求解效率和优化迭代次数相近;非协调元比等参元具有更高精度的拓扑优化结果,能进一步克服棋盘格式.

Topology design has been one of the main research topics in the structural optimization area since Bendsoe and Kikuchi revived its interest using homogenization method and material distribution technique in their landmark paper in 1988. Till now, there are still some numerical difficulties in solving a topology optimization problem. A well-known error is the checkerboard pattern that occurs because of the artificial high stiffness and certain elements in the optimal process whose stability are not guaranteed. In recent years, different ways are presented to prevent the checkerboard pattern. One is using the higher-order elements, but this increases the cpu time in the iteration process due to the large freedoms; the others are using the filtering techniques or density redistribution algorithm based on the low-order isoparametric compatible elements, however there are still some gray area in the final layout structure which makes the engineering applications very difficult. To overcome the numerical problems, the topology design using incompatible finite elements generated according to PTC is carried out. And expressions have been derived for analytical response sensitivity computation and numerical computation in topology optimization process. The final optimal results indicate that the solutions for the layout structure in the domain using 4-node incompatible elements can be used directly in engineering fields. Further, the optimal layout structure using the filtering technique and incompatibles elements shows less intermediate density or gray area compared with that using 4-node isoparametric compatible elements. Moreover, the iteration times and cpu time for topology optimization using the two kinds of elements are close to each other. Thus the improvement in the accuracy of the final optimal results using incompatible elements is validated and the checkerboard pattern problem is also overcome completely.