The multi-harmonic balance method combined with numerical continuation provides an efficient framework to compute a family of time-periodic solutions, or response curves, for large-scale, nonlinear mechanical systems. The predictor and corrector steps repeatedly solve a sequence of linear systems that scale by the model size and number of harmonics in the assumed Fourier series approximation. In this paper, a novel Newton–Krylov iterative method is embedded within the multi-harmonic balance and continuation algorithm to efficiently compute the approximate solutions from the sequence of linear systems that arise during the prediction and correction steps. Further, the method recycles, or reuses, both the preconditioner and the Krylov subspace generated by previous linear systems in the solution sequence. A delayed frequency preconditioner refactorizes the preconditioner only when the performance of the iterative solver deteriorates. The GCRO-DR iterative solver recycles a subset of harmonic Ritz vectors to initialize the solution subspace for the next linear system in the sequence. The performance of the iterative solver is demonstrated on two exemplars with contact-type nonlinearities and benchmarked against a direct solver with traditional Newton–Raphson iterations.
Kuether, Robert J. and Steyer, Andrew. "Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling." Nonlinear Dynamics, vol. 112, no. 5, Jan. 2024. https://doi.org/10.1007/s11071-023-09171-6
@article{osti_2311642,
author = {Kuether, Robert J. and Steyer, Andrew},
title = {Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling},
annote = {The multi-harmonic balance method combined with numerical continuation provides an efficient framework to compute a family of time-periodic solutions, or response curves, for large-scale, nonlinear mechanical systems. The predictor and corrector steps repeatedly solve a sequence of linear systems that scale by the model size and number of harmonics in the assumed Fourier series approximation. In this paper, a novel Newton–Krylov iterative method is embedded within the multi-harmonic balance and continuation algorithm to efficiently compute the approximate solutions from the sequence of linear systems that arise during the prediction and correction steps. Further, the method recycles, or reuses, both the preconditioner and the Krylov subspace generated by previous linear systems in the solution sequence. A delayed frequency preconditioner refactorizes the preconditioner only when the performance of the iterative solver deteriorates. The GCRO-DR iterative solver recycles a subset of harmonic Ritz vectors to initialize the solution subspace for the next linear system in the sequence. The performance of the iterative solver is demonstrated on two exemplars with contact-type nonlinearities and benchmarked against a direct solver with traditional Newton–Raphson iterations.},
doi = {10.1007/s11071-023-09171-6},
url = {https://www.osti.gov/biblio/2311642},
journal = {Nonlinear Dynamics},
issn = {ISSN 0924-090X},
number = {5},
volume = {112},
place = {United States},
publisher = {Springer},
year = {2024},
month = {01}}