Integral Equations & Model Reduction For Fast Computation of Nonlinear Periodic Response

TL;DR

This paper introduces a reformulated integral equations method combined with model reduction techniques to efficiently compute the steady-state response of nonlinear mechanical systems under periodic forcing, achieving faster and more reliable solutions.

Contribution

It presents a new reformulation of existing integral equations for nonlinear response computation, along with convergence analysis and the integration of model reduction for enhanced performance.

Findings

Reformulated equations improve speed and convergence.

Model reduction significantly enhances computational efficiency.

Demonstrated effectiveness on nonlinear finite-element models.

Abstract

We propose a reformulation for the integral equations approach of Jain, Breunung \& Haller [Nonlinear Dyn. 97, 313--341 (2019)] to steady-state response computation for periodically forced nonlinear mechanical systems. This reformulation results in additional speed-up and better convergence. We show that the solutions of the reformulated equations are in one-to-one correspondence with those of the original integral equations and derive conditions under which a collocation type approximation converges to the exact solution in the reformulated setting. Furthermore, we observe that model reduction using a selected set of vibration modes of the linearized system substantially enhances the computational performance. Finally, we discuss an open-source implementation of this approach and demonstrate the gains in computational performance using three examples that also include nonlinear finite-element models.