ROBUST DYNAMIC TOPOLOGY OPTIMIZATION OF CONTINUUM STRUCTURE SUBJECTED TO HARMONIC EXCITATION WITH LOADING DIRECTION UNCERTAINTY
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摘要: 采用一种高效的方法开展了在外载荷作用方向不确定条件下, 连续体结构动态稳健性拓扑优化设计研究, 有效降低了结构的稳态动响应对简谐激励作用方向随机扰动的敏感性. 首先基于概率模型, 将动载荷作用方向的不确定性用正态分布函数表示. 其次通过二阶泰勒展开式, 高效地计算出在激励方向扰动情形下结构动柔顺度的均值和方差, 进而推导出了其对拓扑设计变量的一阶导数灵敏度显性表达式. 最后在材料体积约束下, 以动柔顺度概率特征指标的加权和为设计目标, 基于变密度方法, 对连续体结构进行动态稳健性拓扑优化设计, 并与传统载荷方向固定条件下的确定性优化结果进行对比, 充分展示了考虑外激励作用方向随机扰动对结构拓扑构型设计及其动柔顺度变化的影响. 对优化数值结果进一步分析表明, 采用文章提出的方法所得结构的动响应稳健性更高, 能有效抵抗外激励作用方向的随机扰动. 只需少许增加材料, 稳健性优化设计的动响应将在整个载荷扰动区域内优于确定性优化结果.Abstract: An efficient method is proposed in this paper to study the dynamic robust topology optimization of a continuum structure under the loading direction uncertainty of a harmonic excitation. The design purpose is to reduce the sensitivity of the structural steady-state response of the topological layout to the external load direction perturbations. Firstly, on the basis of the probability representation of an uncertainty, the stochastic variation of the loading direction is described reasonably by a normal distribution, and the external excitation is decomposed along the two essential axes. Then, both the mean and variance of the structural dynamic compliance under the load direction variation are represented efficiently through the quadratic Taylor series expansions with respect to the nominal loading state. Furthermore, the design sensitivity analyses of those stochastic characteristics to a topological variable are performed readily upon the quadratic Taylor expressions such that the explicit formulae are achieved without any extra implementations. Finally, the weighted sum of the mean and standard deviation of the dynamic structural compliance is accordingly defined as the objective function and the coefficient of variation of the dynamic compliance is introduced as the measurement of the structural robustness. Then the robust dynamic topology optimization can be performed with a gradient-based density approach on the RAMP (rational approximation of material properties) model. Both the robust and deterministic topology optimizations of two benchmark examples loaded with the harmonic excitation are conducted respectively, and the structural layouts are compared comprehensively. Numerical results show that the robust dynamic topology optimal configurations can essentially provide a higher robustness against the load direction disturbances than the corresponding deterministic ones and exhibit much stronger resilience. As the material volume constraint is relaxed a little, the dynamic compliance of the robust topology optimization will be smaller than that of the deterministic topology optimization over the whole load uncertain variation range.
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图 1 结构受具有方向不确定外载荷作用的示意图
Figure 1. Schematic of a load with the direction uncertainty imposed to a structure
图 3 确定性和稳健性设计策略结构拓扑构型对比
Figure 3. Comparison of the topology optimizations on two different design strategies of DTO and RTO
图 4 结构动柔顺度随外激励作用方向变化情况
Figure 4. Variation of the structural dynamic compliance caused by the external loading direction disturbances
图 5 两种设计策略结构拓扑优化在高频外激励条件下结果对比
Figure 5. Comparison of the topology optimizations on two different design strategies under an external load of a higher frequency
图 7 两种设计策略结构拓扑优化结果对比
Figure 7. Comparison of the topology optimizations on two different design strategies
图 8 结构动柔顺度随外激励作用方向变化情况
Figure 8. Variation of the structural dynamic compliance caused by the external loading direction disturbances
图 9 两种设计策略拓扑优化收敛过程
Figure 9. Convergence curves of the dynamic compliance on the two design strategies
表 1 两种方法所得结构动柔顺度概率特征指标对比
Table 1. Comparison of the obtained stochastic measurements of the structural dynamic compliance
Method $ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J $ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J CPU time/s MCS 8.233 0 0.168 3 27.890 6 proposed method 8.224 5 0.170 8 0.015 6 different ratio/% 0.103 2 1.485 4 −99.944 1 
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表 2 两种设计策略结构拓扑优化结果对比
Table 2. Comparison of the numerical results with the two optimization strategies
Loading direction Cd0/J $ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J $ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J J/J cv fixed 0.961 7 2.461 6 2.181 4 6.824 5 0.842 1 uncertain 1.108 9 1.170 3 0.016 0 1.202 3 0.013 7 different ratio/% 15.302 1 −52.458 9 −99.266 4 −82.382 9 −98.376 1
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表 3 简支平板结构前3阶固有频率比较
Table 3. Comparison of the first three natural frequencies of the rectangular panel structure
Design status Natural frequency /Hz first second third initial 265.64 285.96 468.53 deterministic optimization 483.12 496.45 613.87 robust optimization 482.73 489.39 595.12
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表 4 两种设计策略结构拓扑在高频外激励条件下优化结果对比
Table 4. Comparison of the optimized results under an external load of a higher frequency
Loadingdirection Cd0/J $ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J $ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J J/J cv fixed 0.816 4 1.090 4 0.079 1 1.248 6 0.072 5 uncertain 0.908 0 0.896 8 0.015 8 0.928 4 0.017 6 different ratio/% 11.220 0 −17.755 0 −80.025 3 −25.644 7 −75.724 1
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表 5 结构拓扑优化结果对比
Table 5. Comparison of the optimized results
Loading direction Cd0/J $ {\text{μ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J $ {\text{σ}}_{{\text{C}}_{\text{d}}\text{(}\text{θ}\text{)}} $/J cv (10−2) fixed 2.967 7 3.001 2 0.047 5 1.581 7 uncertain β = 1 3.085 4 2.989 7 0.006 6 0.220 8 different ratio/% 3.631 2 −0.383 9 −86.097 1 −86.043 5 uncertain β = 2 3.008 9 2.995 8 0.004 5 0.149 3 different ratio/% 1.389 2 −0.180 7 −90.577 0 −90.559 9
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