Collocation-based harmonic balance framework for highly accurate periodic solution of nonlinear dynamical system

TL;DR

This paper introduces a collocation-based harmonic balance framework that achieves highly accurate, high-order periodic solutions for nonlinear dynamical systems, overcoming limitations of existing methods in aliasing and approximation order.

Contribution

A novel framework unifying and extending harmonic balance methods with a new aliasing matrix and reconstruction approach for extremely high-order solutions.

Findings

Achieves >100 order solutions with high accuracy

Outperforms existing methods by 2-3 orders of magnitude in accuracy and speed

Successfully applied to complex nonlinear systems like cavitation bubbles and three-body problems

Abstract

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.