ROBUST DYNAMIC TOPOLOGY OPTIMIZATION OF CONTINUUM STRUCTURE SUBJECTED TO HARMONIC EXCITATION OF LOADING POSITION UNCERTAINTY
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摘要: 研究当外载荷作用位置不确定时, 连续体结构动态稳健性拓扑优化设计. 在减小结构对简谐激励动响应的同时, 有效降低其对外载荷作用点随机扰动的敏感性. 首先基于非概率凸模型的方法, 将外激励作用位置的不确定性用有界区间变量表示. 其次通过对加载位置的导数分析, 获得了在激励位置扰动情况下结构动柔顺度的二阶泰勒展开式. 基于变密度方法, 推导出了动柔顺度对拓扑设计变量的一阶灵敏度显性表达式. 最后在材料体积约束下, 采用移动渐近优化算法并结合载荷扰动区间内灵敏度的最大绝对值, 对连续体结构进行动态稳健性拓扑优化设计, 并与传统载荷位置固定条件下的确定性优化结果进行对比, 充分展示考虑外激励作用位置扰动对结构拓扑构型设计及其动柔顺度变化的影响. 数值优化结果表明, 采用文中提出的方法所获得的结构动响应的稳健性更高, 能有效抵抗外激励作用位置的随机扰动. 只要少许增大材料的体积, 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: The robust topology optimization of an elastic continuum structure is performed under the loading position uncertainty of a dynamic excitation subject to the material volume constraint. The design purpose in this work is to minimize the structural dynamic compliance while reducing its sensitivity to the external load position perturbations in a certain region. First, on the basis of the non-probabilistic convex representation of an uncertainty, the stochastic variation of the loading position is indicated simply with an uncertain-but-bounded interval variable. Then, with the density variable method of the RAMP (rational approximation of material properties) model, the design sensitivity analyses of the structural dynamic compliance with respect to the topological variables are conducted according to the quadratic Taylor series expansion once the loading position moves locally. Finally, by using the gradient-based density approach of the standard MMA (method of moving asymptotes) upon the choice of the maximal absolute value of the design sensitivities over the uncertain position interval, the robust dynamic topology optimization designs can be implemented within a single-level optimization procedure for computational efficiency. The optimal configurations of two benchmark examples loaded with the harmonic excitation are compared comprehensively with those obtained under the fixed loading position of the excitation. Numerical results show that the present dynamic topology optimizations can essentially provide higher robustness to the loading point disturbances than the equivalent deterministic topology optimization solutions. 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 interval.
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