优化迭代步长的两种改进增量谐波平衡法

增量谐波平衡法(IHB法)是一个半解析半数值的方法, 其最大优点是适合于强非线性系统振动的高精度求解. 然而, IHB法与其他数值方法一样, 也存在如何选择初值的问题, 如初值选择不当, 会存在不收敛的情况. 针对这一问题, 本文提出了两种基于优化算法的IHB法: 一是结合回溯线搜索优化算法(BLS)的改进IHB法(GIHB1), 用来调节IHB法的迭代步长, 使得步长逐渐减小满足收敛条件; 二是引入狗腿算法的思想并结合BLS算法的改进IHB法(GIHB2), 在牛顿-拉弗森(Newton-Raphson)迭代中引入负梯度方向, 并在狗腿算法中引入2个参数来调节BSL搜索方式用于调节迭代的方式, 使迭代方向沿着较快的下降方向, 从而减少迭代的步数, 提升收敛的速度. 最后, 给出的两个算例表明两种改进IHB法在解决初值问题上的有效性. 

TWO GENERALIZED INCREMENTAL HARMONIC BALANCE METHODS WITH OPTIMIZATION FOR ITERATION STEP

As a semi-analytial and semi-numerical method, the incremental harmonic balance (IHB) method is capable of dealing with strongly nonlinear systems to any desired accuracy. However, as it is often in case numerical method, there exists initial value problem that can cause divergence with using the IHB method. To solve the initial value problem, two generalized IHB method are presented in this work. The first one (GIHB1) is combined with backtracking line search (BLS) optimization algorithm, which adjust the iteration step to decrease for the convergence of the solutions. The second one (GIHB2) is combined with BLS optimization algorithm and the dogleg method, which is an iterative optimization algorithm for the solution of non-linear least squares problems. The GIHB2 method is adopted for the Newton-Raphson iteration with gradient descent such that the convergence of the solutions increases monotonically along the path with gradient descent way with two parameters. At the end, two examples are presented to show the efficiency and the advantages of the two GIHB methods for initial value problem.