多工况线性结构稳健拓扑优化设计

ROBUST TOPOLOGY OPTIMIZATION DESIGN OF STRUCTURES WITH MULTIPLE-UNCERTAINTY LOAD CASES

针对实际工程中存在的多工况、载荷不确定的情况,研究了概率方法表示载荷不确定性的多工况线性结构稳健拓扑优化设计方法.基于线弹性位移叠加原理给出了多工况、不确定性条件下结构柔度均值与方差的计算方法,并在此基础上推导了结构灵敏度公式.对于承受M个工况的二维结构,根据每个工况下的柔度均值和方差以及灵敏度信息求出其结构整体的均值、方差及灵敏度信息;而结构在单工况n个不确定载荷下的均值方差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求得.提出了以结构整体柔度均值和标准差的加权和最小为目标、体积约束下的稳健拓扑优化设计方法.数值算例验证了所提方法的正确性和有效性以及载荷不确定、多工况条件下优化设计结果的稳健性.该设计方法可以很方便的推广到三维结构问题.

The uncertainties existed in practical applications have great effect on the performance of structures, so it is necessary to introduce uncertainty in structural conceptual design. Robust topology optimization under multiple load cases with uncertainty was studied, where the magnitude and direction of each load are treated as random variables and their probability density functions are given. The weighted sum of the mean and standard deviation of the structural compliance is minimized. 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 structure with M load cases, the expected compliance and variance of structures as well as sensitivity information can be obtained for each load case, and then the object function as well as sensitivity can be achieved readily. In each load case, 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 robust topology optimization to minimize the weighted sum of expectation and standard deviation of compliance under the constraint on the material volume was proposed and verified by numerical examples. The numerical examples also demonstrated the robustness of topology optimization results under multiple load cases with uncertainties. The proposed algorithm can be readily generalized into 3D cases.