Enhancing Multiscale Simulations with Constitutive Relations-Aware Deep Operator Networks

TL;DR

This paper introduces a hybrid deep operator network approach that incorporates microscale constitutive relations to improve multiscale simulations in solid mechanics, achieving accurate results with limited data.

Contribution

It develops a constitutive relations-aware DeepONet for multiscale modeling, integrating microscale physics into surrogate models for enhanced accuracy.

Findings

Accurately predicts microscale strains and stresses from macroscale inputs.

Demonstrates robustness with limited training data.

Effective in quasi-static solid mechanics problems.

Abstract

Multiscale problems are widely observed across diverse domains in physics and engineering. Translating these problems into numerical simulations and solving them using numerical schemes, e.g. the finite element method, is costly due to the demand of solving initial boundary-value problems at multiple scales. On the other hand, multiscale finite element computations are commended for their ability to integrate micro-structural properties into macroscopic computational analyses using homogenization techniques. Recently, neural operator-based surrogate models have shown trustworthy performance for solving a wide range of partial differential equations. In this work, we propose a hybrid method in which we utilize deep operator networks for surrogate modeling of the microscale physics. This allows us to embed the constitutive relations of the microscale into the model architecture and to predict microscale strains and stresses based on the prescribed macroscale strain inputs. Furthermore, numerical homogenization is carried out to obtain the macroscale quantities of interest. We apply the proposed approach to quasi-static problems of solid mechanics. The results demonstrate that our constitutive relations-aware DeepONet can yield accurate solutions even when being confronted with a restricted dataset during model development.