载荷不确定条件下的结构拓扑优化算法

研究了采用概率方法表示载荷不确定性时的连续体结构拓扑优化方法.基于线弹性体的位移叠加原理给出了结构柔度均值与方差的计算方法，并在此基础上给出了结构灵敏度分析方法.对于承受n个不确定载荷的平面结构，其柔度均值与方差以及灵敏度信息可以通过求解其在2n个确定性载荷工况下的位移而获得.给出了载荷不确定条件下以结构柔度均值与标准差的加权和最小为目标的拓扑优化算法，并通过数值算例验证了该方法的有效性及载荷不确定条件下结构拓扑优化结果的稳健性.该算法可以推广到三维结构问题.

Algorithm of structural topology optimization under loading uncertainty

Structural topology optimization under loading uncertainty was studied, where the uncertainty was described by the probabilistic approach. According to the superposition principle of linear theory, computational method for expected and variance of structural compliance was proposed. Sensitivity analysis method was developed based on the expressions of the expected and variance of compliance. For 2D cases, the expected compliance and variance of structures as well as sensitivity information can be obtained by solving the equilibrium equation under 2n deterministic load cases, where n is the number of uncertain loads. Algorithm of structural topology optimization to minimize the weighted sum of expectation and standard deviation of compliance was proposed and verified by numerical examples. The numerical examples also demonstrate the robustness of topology optimization results under loading uncertainty. The proposed algorithm can be readily generalized into 3D cases.