Using Spectral Submanifolds for Optimal Mode Selection in Model Reduction

TL;DR

This paper introduces a rigorous, curvature-based criterion for selecting modes in nonlinear model reduction using Spectral Submanifolds, enabling automatic and reliable projection of complex systems.

Contribution

It develops explicit formulas for SSM curvature and provides an open-source tool for automatic mode selection in nonlinear model reduction.

Findings

Accurately reproduces forced-response curves in complex models

Identifies critical modes with highest nonlinear sensitivity

Provides an automated mode selection procedure

Abstract

Model reduction of large nonlinear systems often involves the projection of the governing equations onto linear subspaces spanned by carefully-selected modes. The criteria to select the modes relevant for reduction are usually problem-specific and heuristic. In this work, we propose a rigorous mode-selection criterion based on the recent theory of Spectral Submanifolds (SSM), which facilitates a reliable projection of the governing nonlinear equations onto modal subspaces. SSMs are exact invariant manifolds in the phase space that act as nonlinear continuations of linear normal modes. Our criterion identifies critical linear normal modes whose associated SSMs have locally the largest curvature. These modes should then be included in any projection-based model reduction as they are the most sensitive to nonlinearities. To make this mode selection automatic, we develop explicit formulas for the scalar curvature of an SSM and provide an open-source numerical implementation of our mode-selection procedure. We illustrate the power of this procedure by accurately reproducing the forced-response curves on three examples of varying complexity, including high-dimensional finite element models.