A deep energy method for finite deformation hyperelasticity

authored by: Vien Minh Nguyen-Thanh, Xiaoying Zhuang, Timon Rabczuk
Abstract: We present a deep energy method for finite deformation hyperelasticitiy using deep neural networks (DNNs). The method avoids entirely a discretization such as FEM. Instead, the potential energy as a loss function of the system is directly minimized. To train the DNNs, a backpropagation dealing with the gradient loss is computed and then the minimization is performed by a standard optimizer. The learning process will yield the neural network's parameters (weights and biases). Once the network is trained, a numerical solution can be obtained much faster compared to a classical approach based on finite elements for instance. The presented approach is very simple to implement and requires only a few lines of code within the open-source machine learning framework such as Tensorflow or Pytorch. Finally, we demonstrate the performance of our DNNs based solution for several benchmark problems, which shows comparable computational efficiency such as FEM solutions.
Organisation(s): Institute of Continuum Mechanics
External Organisation(s): Ton Duc Thang University
Type: Article
Journal: European Journal of Mechanics, A/Solids
Volume: 80
ISSN: 0997-7538
Publication date: 03.2020
Publication status: Published
Peer reviewed: Yes
ASJC Scopus subject areas: General Materials Science, Mechanics of Materials, Mechanical Engineering, General Physics and Astronomy
Electronic version(s): https://doi.org/10.1016/j.euromechsol.2019.103874 (Access: Closed)
: Details in the research portal "Research@Leibniz University"

BibTeX

@article{738164dd6fa74b1a9f77656ff46ba804,
title = "A deep energy method for finite deformation hyperelasticity",
abstract = "We present a deep energy method for finite deformation hyperelasticitiy using deep neural networks (DNNs). The method avoids entirely a discretization such as FEM. Instead, the potential energy as a loss function of the system is directly minimized. To train the DNNs, a backpropagation dealing with the gradient loss is computed and then the minimization is performed by a standard optimizer. The learning process will yield the neural network's parameters (weights and biases). Once the network is trained, a numerical solution can be obtained much faster compared to a classical approach based on finite elements for instance. The presented approach is very simple to implement and requires only a few lines of code within the open-source machine learning framework such as Tensorflow or Pytorch. Finally, we demonstrate the performance of our DNNs based solution for several benchmark problems, which shows comparable computational efficiency such as FEM solutions.",
keywords = "Artificial neural networks (ANNs), Deep energy method, Hyperelasticity, Machine learning, Partial differential equations (PDEs)",
author = "Nguyen-Thanh, {Vien Minh} and Xiaoying Zhuang and Timon Rabczuk",
note = "Funding Information: The first and second authors owe the gratitude to the sponsorship from Sofja Kovalevskaja Programme of Alexander von Humboldt Foundation. The first author also would like to thank MSc. Somdatta Goswami and especially Dr. Cosmin Anitescu, Dr. Simon Hoell for the first code and the fruitful discussions during his research stay at Bauhaus Universit{\"a}t Weimar.",
year = "2020",
month = mar,
doi = "10.1016/j.euromechsol.2019.103874",
language = "English",
volume = "80",
journal = "European Journal of Mechanics, A/Solids",
issn = "0997-7538",
publisher = "Elsevier BV",
}