Quasi-periodic responses composed of multiple base frequencies widely exist in science and engineering problems. The multiple harmonic balance (MHB) method is one of the most commonly used approaches for such problems. However, it is limited by low-order estimations due to complex symbolic operations in practical uses. Many variants have been developed to improve the MHB method, among which the time domain MHB-like methods are regarded as crucial improvements because of their high efficiency and simple derivation. But there is still one main drawback remaining to be addressed. The time domain MHB-like methods negatively suffer from non-physical solutions, which have been shown to be caused by aliasing (mixtures of the high-order into the low-order harmonics). Inspired by the collocation-based harmonic balancing framework recently established by our group, we herein propose a reconstruction multiple harmonic balance (RMHB) method to reconstruct the conventional MHB method using discrete time domain collocations. Our study shows that the relation between the MHB and time domain MHB-like methods is determined by an aliasing matrix, which is non-zero when aliasing occurs. On this basis, a conditional equivalence is established to form the RMHB method. Three numerical examples demonstrate that this new method is more robust and efficient than the state-of-the-art methods.

翻译：由多个基频组成的准周期响应广泛存在于科学与工程问题中。多重谐波平衡方法是处理此类问题最常用的方法之一，但由于实际应用中复杂的符号运算，其使用受限于低阶估计。研究人员已开发出多种改进型多重谐波平衡方法的变体，其中时域类多重谐波平衡方法因其高效性和简单推导而被视为关键改进，但仍存在一个主要缺陷有待解决：时域类多重谐波平衡方法易受非物理解影响，研究表明这是由混叠现象（高阶谐波混入低阶谐波）引起的。受本课题组近期建立的基于配置点的谐波平衡框架启发，我们提出一种重构多重谐波平衡方法（RMHB），通过离散时域配置重构传统多重谐波平衡方法。研究表明，多重谐波平衡方法与时域类多重谐波平衡方法之间的关系由混叠矩阵决定，当混叠发生时该矩阵非零。在此基础上建立了条件等价性以构建RMHB方法。三个数值算例表明，该新方法比现有最优方法具有更强的鲁棒性和更高的效率。