基于特征值分析的多尺度结构优化设计方法

Multi-Scale Structure Optimization Design Based on Eigenvalue Analysis

基于特征值分析,提出了多尺度结构优化设计方法.该方法被用于分析宏观结构上作用有最不利荷载时,使宏观结构刚度最大的宏观拓扑结构和微观材料分布.引入约束条件为最不利荷载的Euclid范数等于1,根据Rayleigh-Ritz定理,可以将结构的柔顺度转换为一个与局部荷载向量维数相同的对称矩阵,这样就将作用有最不利荷载的柔顺度最小问题转换为求解对称矩阵的最大特征值最小问题,同时最不利荷载可以通过最大特征值矩阵的特征向量求得.最后通过算例验证所提多尺度结构优化设计方法的有效性,并说明宏观拓扑结构和微观材料分布的合理性.所提出的多尺度优化方法具有迭代稳定、收敛迅速等特点.该文拓扑优化中密度函数的更新是基于灵敏度分析和移动渐近线方法(method of moving asymptotes,MMA).

A multi-scale structure optimization method was proposed based on eigenvlue analysis,to find the macrostructure and microstructure of maximum macro stiffness under the worst load.The constraint that the Euclidian norm of the uncertain load is 1 was introduced,the structural compliance was calculated according to the Rayleigh-Ritz theorem,and the compliance was transformed to a symmetric matrix with the same dimensions as the local load vector.In this way,the compliance minimization problem under the worst load was transformed to the minimum problem of the maximum eigenvalue of the symmetric matrix.Moreover,the worst load case was determined with the eigenvector corresponding to the maximum eigenvalue of the matrix.Several numerical experiments demonstrated the validity of the proposed method,and illustrated the reasonability of the macro topological structure and the micro material distribution.The proposed multi-scale optimization method has virtues of iterative stability and rapid convergence.The update of the density function in the topological optimization was performed based on sensitivity analysis and the method of moving asymptotes(MMA).