基于能量有限元的薄板结构多目标拓扑优化

Multi-objective topology optimization of thin-platesbased on energy finite element analysis

拓扑优化设计是结构设计中降低成本和实现轻量化的关键环节。能量有限元法(energy finite element analysis,EFEA)是高频动响应分析的有力工具。在高频动力学拓扑优化中,用能量有限元法代替有限元法(finite element analysis,FEA)进行动响应计算可以获得较高的计算效率和计算精度。为了使能量有限元求解和拓扑优化过程能够有效衔接,促进基于能量有限元法的高频动力学拓扑优化的工程应用,建立了基于能量有限元法的多目标拓扑优化模型。基于能量有限元法,以最小能量柔度和感兴趣单元最小能量密度作为高频动力学拓扑优化的目标函数,采用“线性加权法”,建立多目标情况下的拓扑数学模型;再对模型进行离散化,进而对其进行灵敏度计算,通过灵敏度过滤和最优化准则得到多目标拓扑优化构型。最后通过算例,分析不同激励频率对拓扑构型的影响,揭示不同目标权重系数对拓扑构型的交叉影响规律。结果表明,当权重系数相同,激励频率不同时,中高频的减重百分比较低频大;当频率相同,权重系数比为1∶1时,减重百分比最大。基于能量有限元的高频动力学拓扑优化模型有利于发展飞行器等易诱发高频振动的结构的优化设计理论,优化设计结果对实际的结构设计具有一定的参考价值。

Topology optimization design can make for cost saving and achieving lightweight structures.The energy finite element analysis(EFEA)is a powerful tool to predict high-frequency dynamic responses.In the topology optimization for high-frequency dynamics problem,applying EFEA to predict dynamic responses,instead of finite element analysis(FEA),can achieve higher computational efficiency and computational accuracy.To promote the engineering application of high-frequency dynamic topology optimization based on EFEA,the following researches are performed.Based on EFEA,the minimum energy flexibility and the minimum energy density in the element of interest are taken as the objective function,and the“linear weighting method”is used to establish a topology mathematical model under the multi-objective situation.Then the model is discretized,and the sensitivity is calculated.Through the sensitivity filtering and optimized criterion,a multi-objective topology optimization configuration can be obtained.Finally,for a plate,the influences of different excitation frequencies and different weight coefficients on the topological configuration are analyzed.The results show that if the weight coefficient is the same but the excitation frequency is different,the weight loss percentage of the middle and high frequencies is higher than that of the low frequency.If the frequency is same and the weight coefficient ratio is 1∶1,the weight loss percentage is the largest.The high-frequency dynamic topology optimization design model based on EFEA is conducive to the development of optimal design theory for structures prone to high-frequency vibration,such as aircraft,and the optimal design results have a certain reference value for actual structural design.