论文标题
基于搭配的谐波平衡框架，用于高度准确的非线性动力学系统的周期性解决方案
Collocation-based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system
论文作者
论文摘要
周期性的动态系统普遍存在，科学和工程中存在。谐波平衡（HB）方法及其变体是此类系统最广泛使用的方法，但要么局限于低阶近似值，要么因混信和不当采样问题而受到损害。在这里，我们提出了一个基于搭配的谐波平衡框架，以成功统一和重建类似于HB的方法。在此框架下，通过引入一种新型的混叠矩阵，发现了一种新的条件身份，该框架恰好弥合了频域和时域谐波分析之间的差距。在实施将异叠矩阵消失的混杂矩阵后，我们提出了一种强大的重建谐波平衡（RHB）方法，该方法获得了以前被认为是范围内的极高的（> 100）的非陈述溶液，用于一系列复杂的非线性系统，包括空穴泡泡方程和三体问题。我们表明，目前的方法是2-3个数量级比最新方法更快，同时更快。因此，它在寻求高度准确的周期性解决方案的多学科问题中立即应用。
Periodic dynamical systems ubiquitously exist in science and engineering. The harmonic balance (HB) method and its variants have been the most widely-used approaches for such systems, but are either confined to low-order approximations or impaired by aliasing and improper-sampling problems. Here we propose a collocation-based harmonic balance framework to successfully unify and reconstruct the HB-like methods. Under this framework a new conditional identity, which exactly bridges the gap between frequency-domain and time-domain harmonic analyses, is discovered by introducing a novel aliasing matrix. Upon enforcing the aliasing matrix to vanish, we propose a powerful reconstruction harmonic balance (RHB) method that obtains extremely high-order (>100) non-aliasing solutions, previously deemed out-of-reach, for a range of complex nonlinear systems including the cavitation bubble equation and the three-body problem. We show that the present method is 2-3 orders of magnitude more accurate and simultaneously much faster than the state-of-the-art method. Hence, it has immediate applications in multi-disciplinary problems where highly accurate periodic solutions are sought.
