Stability conditions for the explicit integration of projection based nonlinear reduced-order and hyper reduced structural mechanics finite element models

TL;DR

This paper provides a theoretical analysis demonstrating that projection-based nonlinear reduced-order models in structural mechanics can be integrated with larger stable time steps than full models, enabling faster simulations.

Contribution

It proves that Galerkin projection preserves key matrix properties and that ROM eigenvalues separate from original eigenvalues, allowing larger stable time steps in explicit integration.

Findings

ROM eigenvalues separate from original eigenvalues

Stable time step size of ROM is always larger or equal to full model

Hyper-reduction methods can also extend stability

Abstract

Projection-based nonlinear model order reduction methods can be used to reduce simulation times for the solution of many PDE-constrained problems. It has been observed in literature that such nonlinear reduced-order models (ROMs) based on Galerkin projection sometimes exhibit much larger stable time step sizes than their unreduced counterparts. This work provides a detailed theoretical analysis of this phenomenon for structural mechanics. We first show that many desirable system matrix properties are preserved by the Galerkin projection. Next, we prove that the eigenvalues of the linearized Galerkin reduced-order system separate the eigenvalues of the linearized original system. Assuming non-negative Rayleigh damping and a time integration using the popular central difference method, we further prove that the theoretical linear stability time step of the ROM is in fact always larger than or equal to the critical time step of its corresponding full-order model. We also give mathematical expressions for computing the stable time step size. Finally, we show that under certain conditions this increase in the stability time step even extends to some hyper-reduction methods. The findings can be used to compute numerical stability time step sizes for the integration of nonlinear ROMs in structural mechanics, and to speed up simulations by permitting the use of larger time steps.