INVESTIGATION OF THE SOURCE AND IMPROVEMENT OF NON-PHYSICAL SOLUTIONS IN HIGH-ORDER HARMONIC BALANCE
摘要: 对于周期性非定常问题，高阶谐波平衡（High-order Harmonic Balance, HOHB）方法将非定常方程的解用Fourier 级数展开至一定阶次，从而消除其中的时间导数项，大大降低了计算消耗. 本文以达芬振子方程为例，探讨了HOHB 方法中非物理解的来源，分析结果表明：非物理解出现的原因是在推导过程中非线性项的简化处理导致方程左右两边并不严格相等. 根据非线性项的特点，在其处理过程中扩充子时间层上的时域解，并将非线性项中出现的更高阶谐波截断，使方程左右两边严格相等. 通过对达芬振子方程进行数值模拟发现：改进方法在消除非物理解的同时，也显著减少了计算所需谐波数. 对比参考文献发现，同阶改进方法的精度和原始谐波平衡方法基本相当，证明了本方法的可行性. 最后将本方法应用于具有立方刚度非线性的气动弹性系统中，验证本方法的工程适用性. 但是，当方程中非线性项较多时，本方法所需要的计算消耗会有所增加.Abstract: The time derivatives in unsteady equations are eliminated by high-order harmonic balance HOHB method by expanding solutions into Fourier series containing several harmonics, which can reduce computational consumes of periodic unsteady problems significantly. In this paper, the source of non-physical solutions in HOHB method is investigated by Duffing oscillator. It is illustrated that the left and right terms of equations are not strictly equal because of the processing of nonlinear terms in the derivation process, which induces non-physical solutions. According to the characteristics of nonlinear term, sub-time solutions are extended. Besides, higher order harmonics of nonlinear term are also truncated. Thus, the left and right sides of HOHB equations are enforced strictly to be equal. It is manifested that not only non-physical solutions are eliminated, but also the numbers of required harmonics are reduced through the numerical simulation of Duffing oscillator equation. Comparing with results in references, the accuracy and simulation ability of improved method and classical harmonic balance method with same number of harmonics are almost equivalent, which proves the feasibility of the improved method. Lastly the improved method is applied in nonlinear aeroelastic system with cubic nonlinearity, which validates its engineering applicability. However, when there are excessive number of nonlinear terms in dynamic system, the computational consume of improved method will increase.
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