A deep learning energy method for hyperelasticity and viscoelasticity

The potential energy formulation and deep learning are merged to solve partial differential equations governing the deformation in hyperelastic and viscoelastic materials. The presented deep energy method (DEM) is self-contained and meshfree. It can accurately capture the three-dimensional (3D) mechanical response without requiring any time-consuming training data generation by classical numerical methods such as the finite element method. Once the model is appropriately trained, the response can be attained almost instantly at any point in the physical domain, given its spatial coordinates. Therefore, the deep energy method is potentially a promising standalone method for solving partial differential equations describing the mechanical deformation of materials or structural systems and other physical phenomena.

[1]  Oscar Lopez-Pamies,et al.  A new I1-based hyperelastic model for rubber elastic materials , 2010 .

[2]  E Weinan,et al.  The Deep Ritz Method: A Deep Learning-Based Numerical Algorithm for Solving Variational Problems , 2017, Communications in Mathematics and Statistics.

[3]  S. Sadat,et al.  A machine learning based approach for phononic crystal property discovery , 2020 .

[4]  Xiaoqun Cao,et al.  Solving Partial Differential Equations Using Deep Learning and Physical Constraints , 2020, Applied Sciences.

[5]  Jan N. Fuhg,et al.  The mixed deep energy method for resolving concentration features in finite strain hyperelasticity , 2021, J. Comput. Phys..

[6]  Timon Rabczuk,et al.  An Energy Approach to the Solution of Partial Differential Equations in Computational Mechanics via Machine Learning: Concepts, Implementation and Applications , 2019, Computer Methods in Applied Mechanics and Engineering.

[7]  Wei Chen,et al.  Deep Generative Modeling for Mechanistic-based Learning and Design of Metamaterial Systems , 2020, Computer Methods in Applied Mechanics and Engineering.

[8]  Kurt Hornik,et al.  Approximation capabilities of multilayer feedforward networks , 1991, Neural Networks.

[9]  Hongwei Guo,et al.  A deep collocation method for heat transfer in porous media: Verification from the finite element method , 2020 .

[10]  Shuai Zheng,et al.  Generating three-dimensional structural topologies via a U-Net convolutional neural network , 2020 .

[12]  Paris Perdikaris,et al.  Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations , 2019, J. Comput. Phys..

[13]  Santanu Pattanayak,et al.  Pro Deep Learning with TensorFlow , 2017, Apress.

[14]  Xiaoying Zhuang,et al.  Analysis of three dimensional potential problems in non-homogeneous media with deep learning based collocation method , 2020, ArXiv.

[15]  Umberto Ravaioli,et al.  Prediction and optimization of mechanical properties of composites using convolutional neural networks , 2019, Composite Structures.

[16]  Natalia Gimelshein,et al.  PyTorch: An Imperative Style, High-Performance Deep Learning Library , 2019, NeurIPS.

[17]  Diab W. Abueidda,et al.  Meshless physics‐informed deep learning method for three‐dimensional solid mechanics , 2020, International Journal for Numerical Methods in Engineering.

[18]  Jia Zhao,et al.  Discovery of Governing Equations with Recursive Deep Neural Networks , 2020, Communications on Applied Mathematics and Computation.

[19]  A. Tartakovsky,et al.  Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport , 2019, Advances in Water Resources.

[20]  M Mozaffar,et al.  Deep learning predicts path-dependent plasticity , 2019, Proceedings of the National Academy of Sciences.

[21]  Yang Liu,et al.  Physics informed deep learning for computational elastodynamics without labeled data , 2020, Journal of Engineering Mechanics.

[22]  Diab W. Abueidda,et al.  Deep learning for plasticity and thermo-viscoplasticity , 2021 .

[23]  George Cybenko,et al.  Approximation by superpositions of a sigmoidal function , 1989, Math. Control. Signals Syst..

[24]  Seid Koric,et al.  Explicit coupled thermo‐mechanical finite element model of steel solidification , 2009 .

[25]  Ruben Juanes,et al.  A deep learning framework for solution and discovery in solid mechanics , 2020 .

[26]  Diab W. Abueidda,et al.  Deep Learning Sequence Methods in Multiphysics Modeling of Steel Solidification , 2021 .

[27]  Jorge Nocedal,et al.  On the limited memory BFGS method for large scale optimization , 1989, Math. Program..

[28]  Surya R. Kalidindi,et al.  Data-Driven Materials Investigations: The Next Frontier in Understanding and Predicting Fatigue Behavior , 2018 .

[29]  Ricard Borrell,et al.  Parallel mesh partitioning based on space filling curves , 2018, Computers & Fluids.

[30]  Andrew J. Meade,et al.  The numerical solution of linear ordinary differential equations by feedforward neural networks , 1994 .

[31]  J. D. Lawson.  An Order Five Runge-Kutta Process with Extended Region of Stability , 1966 .

[32]  K. Miller,et al.  Strong and weak form meshless methods in computational biomechanics , 2018 .

[33]  Xiaoying Zhuang,et al.  A deep energy method for finite deformation hyperelasticity , 2020 .

[34]  S. Jamali,et al.  Rheology-Informed Neural Networks (RhINNs) for forward and inverse metamodelling of complex fluids , 2021, Scientific Reports.

[35]  Hang Yang,et al.  Learning material law from displacement fields by artificial neural network , 2020 .

[36]  Seid Koric,et al.  Deep learning for topology optimization of 2D metamaterials , 2020 .

[37]  Marleen de Bruijne,et al.  Machine learning approaches in medical image analysis: From detection to diagnosis , 2016, Medical Image Anal..

[38]  Jun Hong,et al.  Investigation into the topology optimization for conductive heat transfer based on deep learning approach , 2018, International Communications in Heat and Mass Transfer.

[39]  Xiaoying Zhuang,et al.  An efficient optimization approach for designing machine learning models based on genetic algorithm , 2020, Neural Computing and Applications.

[40]  Dimitrios I. Fotiadis,et al.  Artificial neural networks for solving ordinary and partial differential equations , 1997, IEEE Trans. Neural Networks.

[41]  Wootaek Lim,et al.  Speech emotion recognition using convolutional and Recurrent Neural Networks , 2016, 2016 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA).