RG-Based Local Hopf Reduction and Slow-Manifold Reconstruction for Nonlinear Aeroelastic Systems

TL;DR

This paper introduces an RG-based reduction method for nonlinear aeroelastic systems that efficiently captures Hopf bifurcations and limit-cycle oscillations, improving local reduced-order modeling near flutter conditions.

Contribution

It develops a renormalization-group approach tailored for polynomial nonlinearities, enabling explicit Hopf bifurcation coefficients and slow-manifold approximations in large discretized models.

Findings

Provides explicit coefficients for Hopf threshold and LCO amplitude.

Offers a compatible slow-manifold approximation with stable modes.

Demonstrates the method on nonlinear aeroelastic examples.

Abstract

Self-excited limit-cycle oscillations (LCOs) from Hopf bifurcations are a key feature of nonlinear aeroelasticity and depend sensitively on structural and aerodynamic parameters. Classical center-manifold and normal-form theory describe this local behavior, but can be cumbersome to apply in large discretized models and standard reduced-order modeling (ROM) workflows. A renormalization-group (RG)-based reduction is developed that directly yields a Hopf-type amplitude equation on a local invariant manifold, specialized for polynomial nonlinearities in tensor-based discretizations and compatible with finite-element-type settings. The method provides explicit coefficients governing the Hopf threshold, criticality, and leading LCO amplitude/frequency trends, and admits a companion slow-manifold approximation with selected stable modes retained as static coordinates. Representative nonlinear-aeroelastic examples illustrate how the proposed framework supplies compact, parameter-aware Hopf/LCO descriptors suitable for local ROM construction near flutter.