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Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling

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Abstract

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. 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.

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Data availability

The datasets generated during this study are not currently publicly available but will be made available from the corresponding author upon reasonable request.

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Acknowledgements

This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. Department of Energy or the United States Government. Supported by the Laboratory Directed Research and Development program at Sandia National Laboratories, a multimission laboratory managed and operated by National Technology and Engineering Solutions of Sandia LLC, a wholly owned subsidiary of Honeywell International Inc. for the U.S. Department of Energy’s National Nuclear Security Administration under contract DE-NA0003525. The authors would like to thank Christopher Van Damme for the many technical discussions and suggestions regarding multi-harmonic balance. The authors appreciate discussions with Daniel Rixen who suggested the idea to explore subspace recycling concepts within iterative solvers. Finally, the authors would like to thank Mike Parks for his help understanding and deploying the GCRO-DR solver.

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Kuether, R.J., Steyer, A. Large-scale harmonic balance simulations with Krylov subspace and preconditioner recycling. Nonlinear Dyn 112, 3377–3398 (2024). https://doi.org/10.1007/s11071-023-09171-6

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