TWO GENERALIZED INCREMENTAL HARMONIC BALANCE METHODS WITH OPTIMIZATION FOR ITERATION STEP
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摘要: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.Abstract: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.
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图 1结合Backtracking line search (BLS)优化算法的IHB法(GIHB1)
Figure 1.Algorithm for a generalized IHB method with backtracking line search (BLS) method (GIHB1)
图 2结合BLS算法和狗腿算法的改进IHB法(GIHB2), 其中在狗腿算法的函数
$ {{F}}\left(\alpha\right) $ 中引入2个调节参数$ \alpha_1 $ 和$ \alpha_2 $ Figure 2.Algorithm for a generalized IHB method (GIHB2) combined with BLS method and dogleg method that contains two parameters
$ \alpha_1 $ and$ \alpha_2 $ in the function$ {{F}}\left(\alpha\right) $ 
图 3不同初值条件下用3种IHB法求解van der Pol振荡方程(32)迭代收敛情况, 其中,
$ \varepsilon=5.0 $ ,$ \lambda=0.87 $ . 初值分别为:(a)$ \omega_0=0.58 $ ,$ a_1=1.0 $ ,$ b_1=1.0 $ ; (b)$ \omega_0=0.58 $ ,$ a_1=1.0 $ ,$ b_1=1.4 $ ; (c)$ \omega_0=0.58 $ ,$ a_1=1.8 $ ,$ b_1=-0.7 $ Figure 3.Iteration convergence of solution of van der Pol Eq. (32) with different initial values using the three IHB methods, where
$ \varepsilon=5.0 $ ,$ \lambda=0.87 $ . Initial values are (a)$ \omega_0=0.58 $ ,$ a_1=1.0 $ ,$ b_1=1.0 $ ; (b)$ \omega_0=0.58 $ ,$ a_1=1.0 $ ,$ b_1=1.4 $ ; (c)$ \omega_0=0.58 $ ,$ a_1=1.8 $ ,$ b_1=-0.7 $ 
图 4用3种IHB法求得van der Pol振荡方程(32)解的初值
$ a_1 $ 和$ b_1 $ 的选定范围, 其他初值为$ \omega_0=0.58 $ 及其他谐波项系数设为零Figure 4.The regions for the initial value for
$ a_1 $ and$ b_1 $ of solutions of van der Pol Eq. (32) using the three IHB methods, the other initial value for frequncy is$ \omega_0=0.58 $ and the other initial harmonic terms are set to be zero
图 5用3种IHB法和四阶龙格-库塔数值法求得范德波尔振荡方程(32)的解, 其中
$ \lambda=0.87 $ 和$ \varepsilon=5.0 $ Figure 5.Solutions of van der Pol Eq. (32) by the three IHB methods and numerical integration using the fourth-order R-K method, where
$ \lambda=0.87 $ and$ \varepsilon=5.0 $ 
图 6不同初值条件下用3种IHB法求得耦合范德波尔振荡方程(38)的解, 其中,
$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ 和$ \gamma_2=0.08 $ Figure 6.The solutions of coupled van der Pol Eq. (38) with different initial values using the three IHB methods, where
$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ , and$ \gamma_2=0.08 $ 
图 7不同初值条件下用3种IHB法求解耦合van der Pol振荡方程(38)在
$ \omega\approx 1.0 $ 附近迭代收敛情况, 其中,$ \lambda=0.08 $ ,$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ 和$ \gamma_2=0.08 $ Figure 7.Iteration convergence of solution of coupled van der Pol Eq. (38) near
$ \omega\approx 1.0 $ with different initial values using the three IHB methods, where$ \lambda=0.08 $ ,$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ , and$ \gamma_2=0.08 $ 
8用3种IHB法和四阶龙格-库塔数值法求得耦合van der Pol振荡方程(38)的解, 其中,
$ \lambda=0.08 $ ,$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ 和$ \gamma_2 $ = 0.088.Solutions of coupled van der Pol Eq. (38) by the three IHB methods and numerical integration using the fourth-order R-K method, where
$ \lambda=0.08 $ ,$ \mu_1=-0.1 $ ,$ \mu_2=0.5 $ ,$ \gamma_1=0.1 $ , and$\gamma_2=$ 0.08 -
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