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Enhancing Multiscale Simulations with Constitutive Relations-Aware Deep Operator Networks

Hamidreza Eivazi, Mahyar Alikhani, Jendrik-Alexander Tröger, Stefan Wittek, Stefan Hartmann, Andreas Rausch

TL;DR

This work proposes a hybrid method in which deep operator networks are utilized for surrogate modeling of the microscale physics and allows 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.

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.

Enhancing Multiscale Simulations with Constitutive Relations-Aware Deep Operator Networks

TL;DR

This work proposes a hybrid method in which deep operator networks are utilized for surrogate modeling of the microscale physics and allows 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.

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.
Paper Structure (13 sections, 11 equations, 3 figures, 1 table)

This paper contains 13 sections, 11 equations, 3 figures, 1 table.

Table of Contents

  1. Introduction
  2. Multiscale computations
  3. Operator learning
  4. Our contribution
  5. Methodology
  6. Deep Operator Networks (DeepONets)
  7. POD basis functions and Galerkin projection
  8. Numerical experiments
  9. Problem setup
  10. Training
  11. RVE simulations
  12. Multiscale simulations
  13. Conclusions

Figures (3)

  • Figure 1: A schematic view of the proposed method. (a) representation of the hybrid multiscale method. (b) the representative volume element (RVE) employed in this study; light gray indicates the matrix and dark gray shows the fiber. (c) and (d) depict DeepONet and POD-DeepONet architectures, respectively.
  • Figure 2: The first 10 POD basis functions for displacement in $y$ direction.
  • Figure 3: POD-DeepONet simulation results of the microscale RVE for a test sample of the prescribed macroscale strains ${\hat{\bm{\varepsilon}}} = \lbrace -0.011, \, -0.036, \, 0.017 \rbrace^\top$. The first row shows reference data obtained from the FE simulation. The second row depicts the predicted solution. Errors are reported in the third row, where contours show the absolute error and titles show the relative $\ell_2$-norm of errors.