考虑平稳随机响应的显式结构拓扑优化

随机载荷是工程结构在服役中经常承受的一种复杂的载荷形式,通常采用统计学特性对其进行描述。对随机载荷作用下的结构进行拓扑优化设计是一项极具挑战性的工作,其主要难点在于,(1) 传统隐式拓扑优化方法的设计变量数巨大,且用于结构动态性能拓扑优化问题时存在虚假模态等数值不稳定问题; (2) 对结构的随机动力响应统计量及其灵敏度进行计算需要极大的计算量; (3) 隐式拓扑优化框架下的分析模型与优化模型强耦合,导致结构有限元模型具有极高的自由度,进一步加剧了上述困难。本文基于移动可变形组件框架和虚拟激励法理论,提出了一种平稳随机载荷作用下结构的显式拓扑优化设计方法。通过将一系列可移动和可变形的结构组件作为优化的基础单元,实现了使用少量设计变量描述结构拓扑构型的目的。采用虚拟激励法、自由度删除技术和模态位移法有效降低了对结构进行随机振动分析和灵敏度分析的计算量。在此基础上,以结构柔顺度的标准差为目标函数、以设计域内实体材料的体积为约束条件,实现了限带白噪声作用下结构的拓扑优化设计,并通过数值算例验证了本文方法的有效性。

Explicit structural topology optimization considering structural stationary random responses

A random load is a complex load form borne by practical engineering structures in service,which is usually described in a statistical way.Contrasted to static topology optimization problems,structural random responses-oriented topology optimization problems are considered very challenging,and have pivotal issues to be addressed seriously.The difficulties can be summarized as follows.Firstly,under the traditional implicit topology optimization framework,a large number of design variables are involved in optimization,and there are numerical instability issues caused by the existence of grey elements,e. g.,highly spurious localized vibration eigenmodes.Secondly,a heavier computation effort for structural random response and corresponding sensitivity analysis needs to be paid.Last,due to the strong coupling between the analysis model and the optimization model under the implicit topology optimization framework,the structural finite element model has very high degrees of freedom (DOFs),which further aggravates the above difficulties.Under the Moving Morphable Component (MMC) based framework and pseudo excitation method (PEM),an explicit topology optimization method considering structural random responses is proposed in this paper.In this method,a series of moving morphable components are used as the basic building blocks of optimization,which renders the possibility of describing the structural topology by a small number of design variables.The PEM,DOF elimination technique (DET) and mode displacement method (MDM) are used to effectively reduce the computational expense associated with structural random response and corresponding sensitivity analysis.On this basis,the optimization problem with the standard deviation of flexibility as the objective function and the volume fraction of solid materials in the design domain as the constraint condition is determined.In numerical examples,for proving the effectiveness of the proposed method,the structural topology optimization problems under band limited white noise are solved.