A Deep Material Network for Multiscale Topology Learning and Accelerated Nonlinear Modeling of Heterogeneous Materials

Abstract In this paper, a new data-driven multiscale material modeling method, which we refer to as deep material network, is developed based on mechanistic homogenization theory of representative volume element (RVE) and advanced machine learning techniques. We propose to use a collection of connected mechanistic building blocks with analytical homogenization solutions to describe complex overall material responses which avoids the loss of essential physics in generic neural network. This concept is demonstrated for 2-dimensional RVE problems and network depth up to 7. Based on linear elastic RVE data from offline direct numerical simulations, the material network can be effectively trained using stochastic gradient descent with backpropagation algorithm, further enhanced by model compression methods. Importantly, the trained network is valid for any local material laws without the need for additional calibration or micromechanics assumption. Its extrapolations to unknown material and loading spaces for a wide range of problems are validated through numerical experiments, including linear elasticity with high contrast of phase properties, nonlinear history-dependent plasticity and finite-strain hyperelasticity under large deformations. By discovering a proper topological representation of RVE with fewer degrees of freedom, this intelligent material model is believed to open new possibilities of high-fidelity efficient concurrent simulations for a large-scale heterogeneous structure. It also provides a mechanistic understanding of structure–property relations across material length scales and enables the development of parameterized microstructural database for material design and manufacturing.

[1]  Dongdong Wang,et al.  An immersed particle modeling technique for the three-dimensional large strain simulation of particulate-reinforced metal-matrix composites , 2016 .

[2]  Julien Yvonnet,et al.  The reduced model multiscale method (R3M) for the non-linear homogenization of hyperelastic media at finite strains , 2007, J. Comput. Phys..

[3]  James H. Garrett,et al.  Knowledge-Based Modeling of Material Behavior with Neural Networks , 1992 .

[4]  P Kerfriden,et al.  A partitioned model order reduction approach to rationalise computational expenses in nonlinear fracture mechanics. , 2012, Computer methods in applied mechanics and engineering.

[5]  Ted Belytschko,et al.  Multiscale aggregating discontinuities: A method for circumventing loss of material stability , 2008 .

[6]  B. A. Le,et al.  Computational homogenization of nonlinear elastic materials using neural networks , 2015 .

[7]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[8]  M. Geers,et al.  Finite strain FFT-based non-linear solvers made simple , 2016, 1603.08893.

[9]  J. D. Eshelby.  The determination of the elastic field of an ellipsoidal inclusion, and related problems , 1957, Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences.

[10]  R. Hill.  A self-consistent mechanics of composite materials , 1965 .

[11]  Demis Hassabis,et al.  Mastering the game of Go without human knowledge , 2017, Nature.

[12]  V. G. Kouznetsova,et al.  Multi-scale computational homogenization: Trends and challenges , 2010, J. Comput. Appl. Math..

[13]  Tara N. Sainath,et al.  Deep Neural Networks for Acoustic Modeling in Speech Recognition: The Shared Views of Four Research Groups , 2012, IEEE Signal Processing Magazine.

[14]  Hervé Moulinec,et al.  A numerical method for computing the overall response of nonlinear composites with complex microstructure , 1998, ArXiv.

[15]  Karel Matous,et al.  A nonlinear manifold-based reduced order model for multiscale analysis of heterogeneous hyperelastic materials , 2016, J. Comput. Phys..

[16]  Guigang Zhang,et al.  Deep Learning , 2016, Int. J. Semantic Comput..

[17]  Toshio Mura,et al.  Micromechanics of defects in solids , 1982 .

[18]  Yoshua Bengio,et al.  Deep Sparse Rectifier Neural Networks , 2011, AISTATS.

[19]  Zeliang Liu,et al.  A statistical descriptor based volume-integral micromechanics model of heterogeneous material with arbitrary inclusion shape , 2015 .

[20]  Julien Yvonnet,et al.  COMPUTATIONAL HOMOGENIZATION METHOD AND REDUCED DATABASE MODEL FOR HYPERELASTIC HETEROGENEOUS STRUCTURES , 2013 .

[21]  J. Chaboche,et al.  FE2 multiscale approach for modelling the elastoviscoplastic behaviour of long fibre SiC/Ti composite materials , 2000 .

[22]  Emmanuel Roubin,et al.  Reduced order modeling strategies for computational multiscale fracture , 2017 .

[23]  G. Dvorak.  Transformation field analysis of inelastic composite materials , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[24]  Mark A Fleming,et al.  Microstructural material database for self-consistent clustering analysis of elastoplastic strain softening materials , 2018 .

[25]  Yann LeCun,et al.  The Loss Surfaces of Multilayer Networks , 2014, AISTATS.

[26]  Pierre Suquet,et al.  A model-reduction approach in micromechanics of materials preserving the variational structure of constitutive relations , 2016 .

[27]  J. Michel,et al.  Nonuniform transformation field analysis , 2003 .

[28]  Kun Wang,et al.  A multiscale multi-permeability poroplasticity model linked by recursive homogenizations and deep learning , 2018, Computer Methods in Applied Mechanics and Engineering.

[29]  Francisco Chinesta,et al.  Data-driven non-linear elasticity: constitutive manifold construction and problem discretization , 2017 .

[30]  Miguel A. Bessa,et al.  Self-consistent clustering analysis: An efficient multi-scale scheme for inelastic heterogeneous materials , 2016 .

[31]  Carsten Könke,et al.  Coupling of scales in a multiscale simulation using neural networks , 2008 .

[32]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy , 1958 .

[33]  Zeliang Liu,et al.  An extended micromechanics method for probing interphase properties in polymer nanocomposites , 2016 .

[34]  S. Shtrikman,et al.  A variational approach to the theory of the elastic behaviour of multiphase materials , 1963 .

[35]  J. E. Hilliard,et al.  Free Energy of a Nonuniform System. I. Interfacial Free Energy and Free Energy of a Nonuniform System. III. Nucleation in a Two‐Component Incompressible Fluid , 2013 .

[36]  M. Koishi,et al.  Three‐dimensional meshfree‐enriched finite element formulation for micromechanical hyperelastic modeling of particulate rubber composites , 2012 .

[37]  Wei Chen,et al.  A framework for data-driven analysis of materials under uncertainty: Countering the curse of dimensionality , 2017 .