Automated Computation of Autonomous Spectral Submanifolds for Nonlinear Modal Analysis

TL;DR

This paper introduces an automated method for computing spectral submanifolds in nonlinear mechanical systems, enabling accurate, high-order analysis without breakdowns, and significantly speeding up backbone curve computations especially in high dimensions.

Contribution

The paper presents a novel automated algorithm for computing spectral submanifolds using the parameterization method, with error estimation and high accuracy, improving speed over traditional methods.

Findings

Algorithm provides high-accuracy SSM computation up to arbitrary orders.

Method does not break down when SSM folds over spectral subspace.

Significant speed-up in backbone curve computation, especially in high-dimensional systems.

Abstract

We discuss an automated computational methodology for computing two-dimensional spectral submanifolds (SSMs) in autonomous nonlinear mechanical systems of arbitrary degrees of freedom. In our algorithm, SSMs, the smoothest nonlinear continuations of modal subspaces of the linearized system, are constructed up to arbitrary orders of accuracy, using the parameterization method. An advantage of this approach is that the construction of the SSMs does not break down when the SSM folds over its underlying spectral subspace. A further advantage is an automated a posteriori error estimation feature that enables a systematic increase in the orders of the SSM computation until the required accuracy is reached. We find that the present algorithm provides a major speed-up, relative to numerical continuation methods, in the computation of backbone curves, especially in higher-dimensional problems. We illustrate the accuracy and speed of the automated SSM algorithm on lower- and higher-dimensional mechanical systems.