Enriching harmonic balance with non-smooth Bernoulli bases for accelerated convergence of non-smooth periodic systems

The harmonic balance method(HBM) has been widely applied to get the periodic solution of nonlinear systems,however, its convergence rate as well as computation efficiency is dramatically degraded when the system is highly non-smooth, e.g., discontinuous. In order to accelerate the convergence, an enriched HBM is developed in this paper where the non-smooth Bernoulli bases are additionally introduced to enrich the conventional Fourier bases. The basic idea behind is that the convergence rate of the HB solution, as a truncated Fourier series, can be improved if the smoothness of the solution becomes finer. Along this line, using non-smooth Bernoulli bases can compensate the highly non-smooth part of the solution and then, the smoothness of the residual part for Fourier approximation is improved so as to achieve accelerated convergence. Numerical examples are conducted on systems with non-smooth restoring and/or external forces. The results confirm that the proposed enriched HBM indeed increases the convergence rate and the increase becomes more significant if more non-smooth bases are used.