摘要本文把ICM方法中的过滤函数和变密度方法中的惩罚函数统称为映射函数，研究了该函数的选取问题，探讨了其选取对于结构拓扑优化优化迭代收敛效率的影响。为此，本文提出了高效率收敛的映射函数构造途径，写出了5类常见的具体映射函数形式，提出同高效率收敛映射函数MFHEC（Mapping function with highly efficient convergence）相配套的优化模型和寻优解法，先是自行比较了同类映射函数的过滤函数和准过滤函数寻优中收敛的快慢，然后相互比较了不同形式映射函数的快滤函数寻优收敛的快慢。以ICM方法求解位移约束下结构体积极小的拓扑优化问题为例，通过数值计算比较，印证了MFHEC函数的高效率收敛性。结果表明：同类函数比较中，快滤函数的收敛速度更快；5种不同类型映射函数比较中，幂函数形式的过滤函数收敛速度更快。最后需要强调的是：本文研究的映射函数的结论，包括ICM方法的过滤函数和变密度方法中的惩罚函数，二者都是同样适用的。
Abstract：In this paper, the filter function in ICM method and the penalty function in variable density method are both referred as the mapping function. The problem of how to select the mapping function is studied; and the influence of different mapping functions on the convergence efficiency of structural topology optimization is discussed. Therefore, an approach is proposed to construct a mapping function to achieve high-efficient convergence. Five common forms of mapping functions are given. An optimization model and optimization method matching MFHEC (Mapping function with highly efficient convergence) are proposed. Firstly, the convergence speed of the filter function and quasi-filter function of the same form of mapping functions is compared. Then the convergence speed of the fast filter function of different forms of mapping functions is compared. Taking the structural topology optimization problem minimizing structural volume under displacement constraints as an example, the ICM method is adopted to solve the problem. Through numerical comparison, the efficient convergence of MFHEC function is verified.The results show that the fast filter function has faster convergence rate than other functions in the same form functions. Compared with five different forms of mapping functions, the filter function of power function form converges fastest. Finally, it should be emphasized that the conclusions of the mapping function studied in this paper are equally applicable for the filter function of ICM method and the penalty function of variable density method.
2024, Vol. 45 
