Incremental harmonic balance method based on asymptotic expansion and a trust-region strategy and application to strongly nonlinear systems

Wentao Zhou; Zeliang Liu; Huijian Li; Wentao Zhou; Zeliang Liu; Huijian Li

doi:10.3934/math.2026490

AIMS Mathematics

2026, Volume 11, Issue 4: 11948-11976. doi: 10.3934/math.2026490

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Incremental harmonic balance method based on asymptotic expansion and a trust-region strategy and application to strongly nonlinear systems

1.
School of Civil Engineering and Mechanics, Yanshan University, Qinhuangdao 066004, China
2.
Hebei Key Laboratory of Mechanical Reliability for Heavy Equipments and Large Structures, Yanshan University, Qinhuangdao 066004, China

Received: 18 February 2026 Revised: 08 April 2026 Accepted: 15 April 2026 Published: 28 April 2026
MSC : 37M05, 37M15

To address the narrow convergence basin and high sensitivity to initial guesses of the incremental harmonic balance method in the analysis of strongly nonlinear systems, this paper develops an enhanced approach that combines a trust region strategy with asymptotic expansion. In each incremental step, an artificial asymptotic parameter is introduced, and the nonlinear incremental equations are expanded into a series of linear recursive subproblems based on the Hessian matrix, enabling order-by-order decoupling of the response components. Furthermore, a trust region algorithm is employed as the inner solver to adaptively control the iteration step size, ensuring robustness and global convergence in solving each subproblem at each order. Numerical examples demonstrate that the proposed method preserves the accuracy of the traditional approach, while significantly widening the convergence basin, reducing dependence on initial guesses, and tracing the frequency response curve with fewer iterations and less computational time. The method, therefore, provides a semi-analytical framework with robust convergence and high computational efficiency for periodic response analysis of strongly nonlinear systems.
- incremental harmonic balance method,
- trust-region algorithm,
- asymptotic expansion,
- strongly nonlinear systems,
- Hessian matrix,
- periodic solutions
Citation: Wentao Zhou, Zeliang Liu, Huijian Li. Incremental harmonic balance method based on asymptotic expansion and a trust-region strategy and application to strongly nonlinear systems[J]. AIMS Mathematics, 2026, 11(4): 11948-11976. doi: 10.3934/math.2026490

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Abstract

To address the narrow convergence basin and high sensitivity to initial guesses of the incremental harmonic balance method in the analysis of strongly nonlinear systems, this paper develops an enhanced approach that combines a trust region strategy with asymptotic expansion. In each incremental step, an artificial asymptotic parameter is introduced, and the nonlinear incremental equations are expanded into a series of linear recursive subproblems based on the Hessian matrix, enabling order-by-order decoupling of the response components. Furthermore, a trust region algorithm is employed as the inner solver to adaptively control the iteration step size, ensuring robustness and global convergence in solving each subproblem at each order. Numerical examples demonstrate that the proposed method preserves the accuracy of the traditional approach, while significantly widening the convergence basin, reducing dependence on initial guesses, and tracing the frequency response curve with fewer iterations and less computational time. The method, therefore, provides a semi-analytical framework with robust convergence and high computational efficiency for periodic response analysis of strongly nonlinear systems.

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