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An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems

Institute of Applied Mechanics, Chair of Materials Theory, University of Stuttgart, 70569 Stuttgart, Pfaffenwaldring 7, Germany
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Math. Comput. Appl. 2019, 24(2), 40; https://doi.org/10.3390/mca24020040
Received: 22 March 2019 / Revised: 9 April 2019 / Accepted: 9 April 2019 / Published: 17 April 2019
PDF [3268 KB, uploaded 17 April 2019]

Abstract

The present work addresses a solution algorithm for homogenization problems based on an artificial neural network (ANN) discretization. The core idea is the construction of trial functions through ANNs that fulfill a priori the periodic boundary conditions of the microscopic problem. A global potential serves as an objective function, which by construction of the trial function can be optimized without constraints. The aim of the new approach is to reduce the number of unknowns as ANNs are able to fit complicated functions with a relatively small number of internal parameters. We investigate the viability of the scheme on the basis of one-, two- and three-dimensional microstructure problems. Further, global and piecewise-defined approaches for constructing the trial function are discussed and compared to finite element (FE) and fast Fourier transform (FFT) based simulations.
Keywords: machine learning; artificial neural networks; computational homogenization machine learning; artificial neural networks; computational homogenization
This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited (CC BY 4.0).
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Göküzüm, F.S.; Nguyen, L.T.K.; Keip, M.-A. An Artificial Neural Network Based Solution Scheme for Periodic Computational Homogenization of Electrostatic Problems. Math. Comput. Appl. 2019, 24, 40.

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