最优步长因子法在桁架结构尺寸优化方面的应用

Application of optimal step-size factor methods in truss sizing optimization

针对桁架结构尺寸优化的特性,依据原约束优化问题的对偶函数关于KKT乘子的一阶偏导数确定乘子的寻优方向;依据对偶函数的极值必要条件和约束优化问题的KKT条件,推导乘子迭代的最优步长因子;依据广义Lagrange函数关于各杆横截面积一阶偏导数应为零的极值必要条件,推导出求解该非线性方程组的优化迭代求解式及其步长因子;通过2种不同约束条件的10杆桁架结构尺寸优化算例验证了本文方法可自动确定各迭代求解式中的步长因子;与已有文献采用序列二次规划法的算例相比,本文方法无需采用一维搜索法寻找步长因子,可大幅度减少计算时间。

Based on the characteristics of the sizing optimization for trusses, the searching direction of KKT multipliers is determined by the first order partial differentiation of the dual function of the original constraint optimization question with respect to the multipliers. The optimal step-size factor of multipliers is derived by the essential extremum conditions of the dual function and the KKT conditions of the constraint optimization question. The solution of iterative optimization and its step-size factor for solving non-linear equations are derived by the essential extremum conditions, that the first order partial differentiation of the augmented Lagrangian function with respect to the bar cross-sectional areas is zero. The numerical examples for the ten-bar trusses with two different constraint conditions show that the methods can automatically determine the step-size factors of the iterative solutions, and greatly reduce the computing time without one dimensional search method to search the step-size factors, which was used in existing literatures with the sequential quadratic programming.