AFT Neural Function Approximators for 1D Nonlinear Force Laws

2026-09-24

AI Summary

This study addresses the high computational cost of the harmonic balance method caused by nonlinear contact and friction in assembled structures, as well as the difficulties encountered by alternating frequency-time schemes in handling non-smooth nonlinearities. To overcome these challenges, this work proposes a neural network-based surrogate model for nonlinear elements that directly maps displacements to the Fourier coefficients of nonlinear forces, thereby eliminating time-domain iterations. Parameter generalization is achieved through physics-informed normalization and phase alignment, while automatic differentiation is employed to generate Jacobian matrices seamlessly embedded into the original solver. Ultimately, this approach establishes a reusable surrogate library that significantly enhances computational efficiency and scalability for high-resolution analyses and systems with multiple nonlinear elements, all without modifying the underlying solution framework.

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.

Problem

• harmonic balance method
• alternating frequency-time scheme
• nonlinear vibration
• computational cost
• nonsmooth nonlinearities

Innovation

• Neural Function Approximator
• Harmonic Balance Method
• Alternating Frequency-Time
• Automatic Differentiation
• Nonlinear Dynamics