基于Logistic插值模型的约束阻尼结构拓扑动力学优化

Topology dynamic optimization of damping plates based on logistic material interpolation model

建立阻尼板有限元动力学方程,求解模态损耗因子算式。构建以模态损耗因子倒数为目标函数,以阻尼单元相对密度为拓扑变量,以阻尼材料用量及频率为约束的优化模型,采用优化准则法解算该模型。运用序列凸规划法构造目标函数的凸性逼近函数而对准则法进行改进,基于SIMP(SolidIstropicMaterialwithPenalization)和RAMP(RationalApproximationofMaterial Properties)材料插值模型推导出逼近函数对于拓扑变量的灵敏度,从而实现了两种插值模型下的改进准则法(OC),并验证了改进法具有全域性优化特性。针对这两种插值模型在对阻尼单元的密度进行惩罚时存在偏惠性问题,结合Logistic(LOGS)函数的几何特征,提出可以对单元密度范围进行选择性惩罚的LOGS模型,编程实现了基于LOGS插值的改进准则法。研究表明:LOGS模型比SIMP和RAMP具有更佳的单元惩罚特性,能更均衡地将单元中间密度惩罚至0或1,从而能得到更具全域性、减振效果更佳、更便于实施的阻尼与约束层优化构型。

To lay some foundations for vibration reduction of plates, dynamic topology optimization of damping plates ware investigated. By building dynamic finite element equation for damping plate, a modal loss factor formula is derived. To facilitate its solution by variable density method, a topology optimal model for damping plate is proposed, which takes damping elements’ density as design variable, the reciprocal value of modal loss factor as objective function, damping material volume and the modal frequencies as constraints. Considering the objective function’s non-convexity, both the objective function and traditional optimal criterion are improved by sequence convex technique, so that objective function is of strict mathematical convexity. Based on SIMP and RAMP density interpolation models, the objective function sensitivity with respect to topological variables are derived and the improved OC methods based on SIMP and RAMP are used to optimize a damping plate. It shows that the improved OC as the advantages of global convergence. On account that both the SIMP and RAMP generate too many intermediate density values in iteration, a LOGS-typed interpolation model using logistic function is proposed, which punishes the elements’ density selectively. The improved OC with LOGS model is implemented for a damping plate to validate LOGS model work well. It is found that LOGS has better polarization characteristics. It can punish intermediate density values to 0 or 1 well, global optimization results are obtained and a better damping layout scheme is achieved.