The Schwarz alternating method for the seamless coupling of nonlinear reduced order models and full order models

TL;DR

This paper introduces a Schwarz alternating method-based approach for seamlessly coupling nonlinear reduced order models with full order models, enhancing robustness, stability, and integration into multi-physics simulations.

Contribution

It extends the Schwarz method to enable efficient coupling of ROMs and FOMs in nonlinear solid mechanics with hyper-reduction, addressing stability and integration challenges.

Findings

Effective ROM-FOM coupling in nonlinear hyper-elasticity demonstrated

Improved stability and accuracy in predictive regimes shown

Method achieves computational efficiency with hyper-reduction

Abstract

Projection-based model order reduction allows for the parsimonious representation of full order models (FOMs), typically obtained through the discretization of certain partial differential equations (PDEs) using conventional techniques where the discretization may contain a very large number of degrees of freedom. As a result of this more compact representation, the resulting projection-based reduced order models (ROMs) can achieve considerable computational speedups, which are especially useful in real-time or multi-query analyses. One known deficiency of projection-based ROMs is that they can suffer from a lack of robustness, stability and accuracy, especially in the predictive regime, which ultimately limits their useful application. Another research gap that has prevented the widespread adoption of ROMs within the modeling and simulation community is the lack of theoretical and algorithmic foundations necessary for the "plug-and-play" integration of these models into existing multi-scale and multi-physics frameworks. This paper describes a new methodology that has the potential to address both of the aforementioned deficiencies by coupling projection-based ROMs with each other as well as with conventional FOMs by means of the Schwarz alternating method. Leveraging recent work that adapted the Schwarz alternating method to enable consistent and concurrent multi-scale coupling of finite element FOMs in solid mechanics, we present a new extension of the Schwarz formulation that enables ROM-FOM and ROM-ROM coupling in nonlinear solid mechanics. In order to maintain efficiency, we employ hyper-reduction via the Energy-Conserving Sampling and Weighting approach. We evaluate the proposed coupling approach in the reproductive as well as in the predictive regime on a canonical test case that involves the dynamic propagation of a traveling wave in a nonlinear hyper-elastic material.