基于速度场水平集的结构应力最小化鲁棒性拓扑优化
VELOCITY FIELD LEVEL SET-BASED ROBUST TOPOLOGY OPTIMIZATION FOR STRUCTURAL STRESS MINIMIZATION
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摘要: 针对传统确定性拓扑优化方法在处理载荷不确定性时可能导致结构应力敏感的问题, 本文提出一种基于速度场水平集框架的结构应力最小化鲁棒性拓扑优化方法. 首先, 采用P范数函数对结构最大应力进行全局度量. 进而通过多项式混沌展开法(Polynomial Chaos Expansion, PCE)建立载荷不确定性下结构最大应力随机响应的代理模型, 并直接从PCE展开系数解析提取最大应力的均值与标准差, 建立以二者线性组合最小化为目标的鲁棒性拓扑优化模型, 从而兼顾应力性能的平均表现与波动程度. 在优化过程中, 采用速度场水平集方法描述结构边界的演化, 结合直接微分法与伴随变量法, 推导了全局最大应力统计矩对速度设计变量的解析灵敏度, 为拓扑演化提供精确梯度信息, 并引入全局收敛移动渐近线法(Globally Convergent Method of Moving Asymptotes, GCMMA)实现设计变量的高效更新. 为验证所提方法的有效性与稳定性, 本文通过两个典型数值算例开展系统仿真, 并结合蒙特卡罗模拟(Monte Carlo Simulation, MCS)对优化结果进行对比分析. 此外, 本文还讨论了权重系数、载荷变异系数及体积约束限值对最终构型与应力性能的影响规律.Abstract: To address the issue that traditional deterministic topology optimization methods may lead to stress concentration-sensitive structures under load uncertainties, this paper proposes a robust topology optimization method (RTO) for structural stress minimization based on a velocity field level set framework. First, the P-norm function is adopted to globally measure the maximum stress in the structure. Then, a surrogate model for the stochastic response of the maximum stress under uncertain loads is constructed using the Polynomial Chaos Expansion (PCE) method. The mean and standard deviation of the maximum stress are analytically derived directly from the PCE coefficients, and a robust topology optimization model is established with the objective of minimizing the linear combination of these two statistical moments, thereby balancing both the average performance and the fluctuation of stress behavior. During the optimization process, the velocity field level set method is employed to describe the evolution of structural boundaries. By combining the direct differentiation method and the adjoint variable method, analytical sensitivities of the statistical moments of the global maximum stress with respect to the velocity design variables are derived, providing accurate gradient information for topological evolution. The Globally Convergent Method of Moving Asymptotes (GCMMA) is introduced to efficiently update the design variables. To verify the effectiveness and stability of the proposed method, two typical numerical examples are systematically simulated and analyzed, with comparison results validated by Monte Carlo Simulation (MCS). Furthermore, the influences of weighting coefficients, load variation coefficients, and volume constraint limits on the final configurations and stress performance are investigated.