基于切比雪夫的稳定性分析和浦部的误差界对谐波平衡有用吗？
Are Chebyshev-based stability analysis and Urabe's error bound useful features for Harmonic Balance?
摘要：
谐波平衡法是计算非线性动力系统周期解的一种常用方法。在这项工作中，我们解决其两个主要缺点：首先，我们调查到什么程度的稳定性分析的计算负担可以减少一致使用切比雪夫多项式。其次，我们解决了一个严格的误差界的问题，据作者所知，已被忽略在所有的工程应用到目前为止。在这里，我们依靠浦部的误差界，并再次使用切比雪夫多项式的计算涉及的操作。我们使用的误差估计自动调整谐波截断阶数在数值连续，并面对一个国家的最先进的自适应谐波平衡实现的算法。在此基础上，我们首次严格证明了具有软化特征的强迫阻尼Duffing振子的孤立周期解的存在性。我们发现，努力获得一个严格的误差界，在其目前的形式，可能是太高，是有用的许多工程问题。基于一系列的数值例子得到的结果，我们得出结论，切比雪夫为基础的稳定性分析确实允许大幅加速。然而，与谐波平衡本身一样，当需要极高的截断阶数时，该方法变得低效，例如，存在（急剧规则化的）不连续性。
英文摘要：
Harmonic Balance is one of the most popular methods for computing periodic solutions of nonlinear dynamical systems. In this work, we address two of its major shortcomings: First, we investigate to what extent the computational burden of stability analysis can be reduced by consistent use of Chebyshev polynomials. Second, we address the problem of a rigorous error bound, which, to the authors' knowledge, has been ignored in all engineering applications so far. Here, we rely on Urabe's error bound and, again, use Chebyshev polynomials for the computationally involved operations. We use the error estimate to automatically adjust the harmonic truncation order during numerical continuation, and confront the algorithm with a state-of-the-art adaptive Harmonic Balance implementation. Further, we rigorously prove, for the first time, the existence of some isolated periodic solutions of the forced-damped Duffing oscillator with softening characteristic. We find that the effort for obtaining a rigorous error bound, in its present form, may be too high to be useful for many engineering problems. Based on the results obtained for a sequence of numerical examples, we conclude that Chebyshev-based stability analysis indeed permits a substantial speedup. Like Harmonic Balance itself, however, this method becomes inefficient when an extremely high truncation order is needed as, e.g., in the presence of (sharply regularized) discontinuities.
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