A generalized incremental harmonic balance method by combining a data-driven framework for initial value selection of strongly nonlinear dynamic systems

Abstract

The incremental harmonic balance (IHB) method is a semi-analytical and semi-numerical method that consists of the harmonic balance process and the Newton–Raphson iteration process, which can accurately obtain solutions of strongly nonlinear dynamic systems, and has been successfully applied to many practical problems. However, it is difficult to choose proper initial values for the Newton–Raphson iteration process, especially for strongly nonlinear dynamic systems, which can cause divergence of the IHB method. A novel generalized IHB method by combining a data-driven framework to obtain feasible initial values for strongly nonlinear dynamic systems is proposed in this work. In the proposed data-driven framework, an artificial neural network (ANN) is trained to learn the mapping between small system parameters of a weakly nonlinear dynamic system and corresponding solutions of the IHB method. The amount of data required for training is determined by the number of system parameters, which is usually only a few hundreds to a few thousands, making the training process very expeditious. Once the trained ANN receives other system parameters of a nonlinear dynamic system, it can immediately output a set of feasible initial values for the IHB method, which can make the IHB method quickly converge. Results from extensive testing indicate that the mapping learned by the ANN is effective not only for weakly nonlinear dynamic systems that are characterized by smaller system parameters, but also for strongly nonlinear dynamic systems in most cases that involve larger system parameters. During the testing for strongly nonlinear dynamic systems, successful convergences generate new training data. These data can be leveraged to further fine-tune the ANN to yield an improved ANN, which allows continuous refinement of the ANN and enhances prediction of initial values for strongly nonlinear dynamic systems. With the introduction of this simple yet highly efficient data-driven initial value selection framework, the applicability of the IHB method can be significantly enhanced. Three numerical examples are presented to show the efficiency and advantages of the present methodology.