空间有界几何不确定性结构稳健性拓扑优化

doi:10.3901/JME.2023.11.159

摘要/Abstract

摘要： 在结构拓扑优化设计中，考虑到实际工程结构中存在的与材料性能、工作载荷等相关的不确定性，近年来发展了一系列稳健性拓扑优化方法。目前，稳健性拓扑优化的研究大多是基于结构确定的几何边界，事实上，加工制造误差或测量误差往往会带来结构边界的不确定性，忽略这类几何不确定性可能对导致结构设计对边界的微小波动非常敏感。针对设计域空间结构边界扰动的有界特质，采用区间场度量结构的几何不确定性，并基于切比雪夫多项式展开提出了一种高效的稳健性拓扑优化方法。首先，通过Heaviside密度过滤中投影阈值的区间场描述表征结构的边界扰动，并构造了基于最坏情况的稳健性拓扑优化模型；其次，基于区间KL(Karhunen-Loève)展开将区间场近似离散为有限个区间变量，并结合切比雪夫多项式展开方法求解稳健性目标函数及约束；随后，推导了稳健性目标函数及约束对设计变量的灵敏度信息，并采用基于梯度的优化算法更新拓扑设计变量；最后，通过多个数值算例验证所提出方法的有效性。数值算例分析结果表明，结构边界的几何不确定性波动对结构性能具有重要影响。相比于确定边界下的拓扑优化设计，考虑不确定性的拓扑优化设计在结构边界波动下具有更好的稳健性。

关键词: 稳健性拓扑优化, 几何不确定性, 区间场, KL展开, 切比雪夫多项式

Abstract: In structural topology optimization design, considering the uncertainties related to material properties and working loads in practical engineering structures, a series of robust topology optimization methods have been developed in recent years. At present, most robust topology optimization methods are based on deterministic geometry boundary of structures. In fact, manufacturing errors or measurement errors often lead to the uncertainty of structure boundaries. The structural design may be very sensitive to the small fluctuation of the boundary if ignoring the geometric uncertainty. Considering the spatially bounded characteristic of the boundary uncertainty, the interval field is used to measure the geometric uncertainty, and an efficient robust topology optimization method based on Chebyshev polynomial expansion is developed. Firstly, the boundary disturbance of the structure is described by modelling the projection threshold variable in the Heaviside filter as an interval field, and a robust topology optimization model under the worst case is then constructed. Secondly, based on the interval KL (Karhunen-Loève) expansion, the interval field is approximately discretized into finite interval variables, and the robust objective function and constraint are evaluated using the Chebyshev polynomial expansion method. Thirdly, the sensitivities of the robust objective function and constraint with respect to the design variables are derived, and the gradient-based optimization algorithm is used to update the topology design variables. Finally, several numerical examples are provided to verify the effectiveness of the proposed method. The analysis results of numerical examples show that the geometric uncertainty fluctuation of the structural boundary has an important impact on the structural performance. Compared with the topology optimization design under the deterministic boundary, the robust topology optimization design considering uncertainty has better robustness when considering the fluctuation of the structural boundary.

Key words: robust topology optimization, geometric uncertainty, interval field, KL expansion, Chebyshev polynomial

中图分类号:

O342

郑静, 丁少楠, 姜潮. 空间有界几何不确定性结构稳健性拓扑优化[J]. 机械工程学报, 2023, 59(11): 159-170.

ZHENG Jing, DING Shaonan, JIANG Chao. Robust Topology Optimization for Structures Considering Spatially Bounded Geometric Uncertainties[J]. Journal of Mechanical Engineering, 2023, 59(11): 159-170.

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