A Finite Operator Learning Technique for Mapping the Elastic Properties of Microstructures to Their Mechanical Deformations

International Journal for Numerical Methods in Engineering, vol. 126

Abstract

ABSTRACT To obtain fast solutions for governing physical equations in solid mechanics, we introduce a method that integrates the core ideas of the finite element method with physics‐informed neural networks and concept of neural operators. We propose directly utilizing the available discretized weak form in finite element packages to construct the loss functions algebraically, thereby demonstrating the ability to find solutions even in the presence of sharp discontinuities. Our focus is on micromechanics as an example, where knowledge of deformation and stress fields for a given heterogeneous microstructure is crucial for further design applications. The primary parameter under investigation is the Young's modulus distribution within the heterogeneous solid system. Our investigations reveal that physics‐based training yields higher accuracy compared with purely data‐driven approaches for unseen microstructures. Additionally, we offer two methods to directly improve the process of obtaining high‐resolution solutions, avoiding the need to use basic interpolation techniques. The first one is based on an autoencoder approach to enhance the efficiency for calculation on high resolution grid points. Next, Fourier‐based parametrization is utilized to address complex 2D and 3D problems in micromechanics. The latter idea aims to represent complex microstructures efficiently using Fourier coefficients. The proposed approach draws from finite element and deep energy methods but generalizes and enhances them by learning parametric solutions without relying on external data. Compared with other operator learning frameworks, it leverages finite element domain decomposition in several ways: (1) it uses shape functions to construct derivatives instead of automatic differentiation; (2) it automatically includes node and element connectivity, making the solver flexible for approximating sharp jumps in the solution fields; and (3) it can handle arbitrary complex shapes and directly enforce boundary conditions. We provided some initial comparisons with other well‐known operator learning algorithms, further emphasize the advantages of the newly proposed method.

Authors 9

1. Shahed Rezaei corresponding Aachen Access e.V Aachen Germany Access e.V

   Access

   Affiliation as printed

   ACCESS e.V. Aachen Germany

   ACCESS e.V., Aachen, Germany

2. Reza Najian Asl

   Technical University of Munich

   Affiliation as printed

   Chair of Structural Analysis Technical University of Munich München Germany

3. Shirko Faroughi corresponding

   Urmia University of Technology

   Affiliation as printed

   Department of Mechanical Engineering Urmia University of Technology Urmia Iran

   Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

4. Mahdi Asgharzadeh

   Urmia University of Technology

   Affiliation as printed

   Department of Mechanical Engineering Urmia University of Technology Urmia Iran

   Department of Mechanical Engineering, Urmia University of Technology, Urmia, Iran

5. Ali Harandi Aachen Institute of Applied Mechanics

   RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mechanics RWTH Aachen University Aachen Germany

   Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

6. Rasoul Najafi Koopas

   Helmut Schmidt University

   Affiliation as printed

   Institute of Solid Mechanics Helmut‐Schmidt University/University of the Federal Armed Forces Hamburg Germany

   Institute of Solid Mechanics, Helmut-Schmidt University/University of the Federal Armed Forces, Hamburg, Germany

7. Gottfried M. LASCHET Aachen Access e.V Aachen Germany Access e.V

   Access

   Affiliation as printed

   ACCESS e.V. Aachen Germany

   ACCESS e.V., Aachen, Germany

8. Stefanie Reese Aachen Institute of Applied Mechanics

   RWTH Aachen University

   Affiliation as printed

   Institute of Applied Mechanics RWTH Aachen University Aachen Germany

   Institute of Applied Mechanics, RWTH Aachen University, Aachen, Germany

9. Markus Apel Aachen Access e.V Aachen Germany Access e.V

   Access

   Affiliation as printed

   ACCESS e.V. Aachen Germany

   ACCESS e.V., Aachen, Germany