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This research applies neural networks to approximate nonlinear force laws in mechanical systems, replacing computationally expensive iterative solvers with direct mappings of Fourier coefficients. By training on individual elements like cubic springs rather than full system responses, the method enables scalable high-resolution analysis for complex assemblies with many nonlinear components. The approach effectively bridges computational engineering and machine learning to solve specific problems in dynamical systems without altering existing solver architectures.
AFT Neural Function Approximators for 1D Nonlinear Force Laws
arXiv:2609.29242v1 Announce Type: cross Abstract: Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.
Abstract:Nonlinear contacts and friction strongly influence the vibration response of assembled structures, but their accurate numerical treatment is computationally demanding. The harmonic balance method is widely used to compute periodic steady-state responses, yet the required alternating frequency-time scheme becomes costly for nonsmooth and hysteretic nonlinearities and must be repeated throughout the nonlinear solution process. Here we show that this procedure can be replaced by neural networks that directly map displacement Fourier coefficients to nonlinear force coefficients and provide the corresponding Jacobian through automatic differentiation. The surrounding solver and continuation algorithms remain unchanged for the computation of frequency response curves. The neural networks exclusively learn individual nonlinear elements rather than complete system responses. Physics-based nondimensionalization and phase normalization facilitate the learning process and enable a single trained network to cover a wide range of parameter combinations. Building on the cubic spring, unilateral spring, and Jenkins elements considered here, the approach points toward a reusable library of nonlinear-element surrogates that can be combined in arbitrary number and location within a mechanical system. By bypassing the iterative force evaluation in time domain, the method offers favorable computational scaling for high-resolution analyses and systems with many nonlinear elements.
Computational Engineering, Finance, and Science (cs.CE); Machine Learning (cs.LG); Dynamical Systems (math.DS)
arXiv:2609.29242 [cs.CE]
(or arXiv:2609.29242v1 [cs.CE] for this version)
https://doi.org/10.48550/arXiv.2609.29242 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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