In engineering applications, such as turbine blade vibrations and rotor dynamics, steady-state responses of systems attract more attention than transient responses. Among various semi-analytical methods, the incremental harmonic balance (IHB) and alternating frequency/time domain (AFT) methods, which are based on the harmonic balance method, are most widely used. The IHB method is a combination of harmonic balance carried out by the Galerkin averaging procedure and Newton-Raphson method. It provides an attractive approach to obtain steady–state periodic responses of systems with multi-dimensions and strong nonlinearities. When the IHB method is used for a system with non-polynomial nonlinearities, numerical integration is needed in the Galerkin averaging procedure, which is usually very time-consuming.

In this work, an efficient Galerkin averaging-incremental harmonic balance (EGA-IHB) method is developed based on the fast Fourier transform (FFT) and tensor contraction. As a semi-analytical method, derivation is significantly simplified in the EGA-IHB method and it is very efficient and robust. The residual vector and Jacobian matrix corresponding to nonlinear terms are expressed using truncated Fourier series, and Fourier coefficient vectors are calculated using the FFT. Integrals in the Galerkin averaging procedure in the EGA-IHB method are calculated by multiplication of a constant tensor and Fourier coefficient vector with use of tensor contraction. Note that a constant tensor is only related to the number of truncated harmonic terms, which is determined for a particular nonlinear problem. Since constant tensors only need to be calculated once, which can be efficiently achieved by the FFT, efficiency of the EGA-IHB method is greatly increased. Since these tensors are highly sparse and can be unfolded into sparse matrices, the Jacobian matrix can be efficiently calculated using matrix-vector multiplication. Since the discrete Fourier transform (DFT) is used to generate every column of the Jacobian matrix in the AFT method, the number of usage of the DFT in each Newton iteration is twice of that of harmonic terms. The FFT is only used up to four times in the EGA-IHB method to generate the residual vector and Jacobian matrix, which makes the EGA-IHB method very efficient. Inaccurate results may be obtained from DFT-based methods when aliasing occurs, and the minimal non-aliasing sampling rate is determined for the EGA-IHB method. Several examples are analyzed using the EGA-IHB method, and its efficiency, convergence, and robustness are compared with those from the IHB, AFT, IIHB, and MIHB methods. The EGA-IHB method has considerable advantages in computation efficiency over the other methods, with very simple derivation.


An Efficient Galerkin Averaging-Incremental Harmonic Balance Method Based on the Fast Fourier Transform and Tensor Contraction

Division / Sponsors: 
Aerospace, Bioengineering, Applied Mechanics, Design Engineering Division, Advanced Energy Systems, Manufacturing Engineering Division, Noise Control and Acoustics, Nondestructive Evaluation, Diagnosis, and Prognosis (NDPD), Nuclear Engineering Division, Process Industries Division, Fluids Engineering, Heat Transfer, Materials, Microelectromechanical Systems, Safety Engineering and Risk Analysis, Computers & Information in Engineering


Paper Proceedings: https://asmedigitalcollection.asme.org/IMECE/IMECE2020/volume/84478

IMECE2020 - Virtual Conference

Supporting the need among researchers worldwide for additional online and multimedia content, ASME now offers the ASME Conference Video Collection. This platform will collect and provide long-term access to conference presentation videos, encompassing all technology areas within ASME's technical conference and events program. About 30,000-40,000 engineers across all mechanical engineering and related disciplines participate in ASME conferences each year. This includes authors of approximately 6,000 to 8,000 technical papers published by ASME, the majority of whom also submit video presentations for our virtual conferences. The video collection will be available as an add-on to existing product packages or as a separate subscription.