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A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials

Zeliang Liu, C. T. Wu, M. Koishi

arXiv:1807.09829v4cs.CEphysics.comp-ph

TL;DR

Large-scale heterogeneous-material simulations need efficient RVE models without sacrificing microscale physics or requiring extensive calibration and sampling. The paper develops a deep material network of analytical mechanistic building blocks trained from linear elastic RVE data, then demonstrates extrapolation to nonlinear plasticity and finite-strain hyperelasticity. The resulting model supports efficient, accurate online prediction and parameterized multiscale material design.

  • Problem

    Direct RVE simulations are expensive, while reduced-order and generic learning methods can require extensive sampling, calibration, micromechanical assumptions, or risk physics loss and unsafe extrapolation.

  • Method

    The deep material network uses connected analytical homogenization building blocks in a hierarchical topology, trained with machine learning and compressed through nodal deletion and subtree merging.

  • Results

    The method provides accurate and efficient online predictions for linear elasticity, nonlinear history-dependent plasticity, and finite-strain hyperelasticity under large deformations.

  • Takeaways & Limitations

    The parameterized network can accelerate concurrent multiscale computation and support multiscale material design while avoiding numerical difficulty from large material distortion.

  • Takeaways & Limitations

    Training depth is constrained by optimization difficulty: with fixed training epochs, deeper networks do not always outperform shallower ones, and the optimal depth depends on training time and desired accuracy.

Abstract

from arXiv · show

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 which avoids the loss of essential physics in generic neural networks, and 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, 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. Introduction

The paper addresses the cost, accuracy, calibration, sampling, and extrapolation limitations of existing RVE model-reduction approaches by introducing a mechanistic deep material network. The method uses hierarchical building blocks and machine learning to represent RVE responses while targeting efficient prediction across nonlinear material behaviors.

  • Motivation: RVE homogenization provides macroscopic constitutive closure while retaining microscopic heterogeneity, but direct numerical simulations are computationally expensive.These costs limit direct simulation of large-scale heterogeneous structures in industrial applications.
  • Limitations of existing methods: Analytical micromechanics methods are efficient but can lose accuracy or require extensive calibration for complex morphologies, history dependence, and large deformations.
  • Limitations of existing methods: Generic machine-learning models can represent complex input-output relations, yet their mechanics applications include approaches that directly approximate RVE strain-energy surfaces.
  • Proposed approach: The deep material network represents an RVE DNS model with a hierarchical topology of mechanistic building blocks and requires only linear elastic RVE analysis for training.
  • Proposed approach: The proposed goals include avoiding extensive offline sampling, extra constitutive calibration, micromechanical assumptions, physics loss, and extrapolation danger.
  • Scope: The paper develops the theory, training procedure, and extrapolation studies for linear elasticity, nonlinear plasticity, and finite-strain hyperelasticity.

2. Theory of material network

The material network builds RVE homogenization from connected analytical two-layer building blocks arranged in a hierarchical binary tree. Compliance matrices and volume-fraction weights propagate through the network, whose bottom-layer parameters are trained and can be compressed by node deactivation or subtree simplification.

  • Core concept: Connected mechanistic building blocks represent complex RVE responses, while the network architecture records the homogenization path from constituent phases to macroscopic behavior.
  • Physically based building block: A two-layer building block performs analytical homogenization followed by rotation, transforming constituent compliance matrices and morphological descriptors into an updated compliance matrix.
  • Physically based building block: The building-block formulation uses compliance tensors because their analytical expressions are neater for the small-strain 2D plane-strain case, while stiffness-based formulations are equivalent.
  • Physically based building block: The two-phase, two-layer formulation stores compliance matrices in vectorized form with six independent variables and supplies analytical derivatives for gradient-based training.
  • Network architecture: A depth-N binary tree has 2^i nodes at layer i, with the output at layer 0 and constituent inputs at layer N+1.
  • Network architecture: Each building block combines two child nodes, while recursively summed weighting functions track volume fractions from bottom-layer nodes to parent nodes.
  • Network architecture: Only bottom-layer weights are independent fitting parameters, and ReLU activation can deactivate units permanently, simplifying the network and increasing training speed.
  • Network architecture: Directly fitting volume fractions can cause vanishing gradients in deep layers and unwanted early subtree deactivation, motivating bottom-layer latent parameters instead.

3. Machine learning of deep material network

The material network is trained against RVE data using a normalized mean-square-error objective, analytical gradients, stochastic mini-batches, and compression procedures that reduce redundancy and model complexity.

  • Dataset and cost function: Training data are generated from high-fidelity DNS or experiments, with Latin hypercube design used to generate the input space.The authors note that experimental data generation may be challenging because experimental tests are limited.
  • Dataset and cost function: The training objective is a normalized mean-square-error cost over samples, with an additional activation constraint term to make optimization well-posed.The normalization uses the squared norm of the homogenized compliance matrix, while the constraint term controls activation magnitudes.
  • Backpropagation and SGD: Analytical backpropagation propagates output errors through network layers to compute gradients with respect to rotation angles and bottom-layer activations.The procedure uses the chain rule and the analytical derivatives of the network's homogenization functions.
  • Backpropagation and SGD: Stochastic gradient descent estimates gradients from randomly selected mini-batches rather than the full dataset at every update.The learning rate is updated with a Bold driver algorithm, which the authors report effectively improved training speed in their numerical study.
  • Model compression and parameter reduction: Model compression removes redundant structure through ReLU-based node deactivation, parent-node deletion, and subtree merging after network reordering.A parent node with one child can be deleted, while similar subtrees can be merged; deleting a node may require updating a rotation angle.

4. Prediction and extrapolation

The trained network extrapolates beyond linear elasticity by propagating constitutive information through mechanistic homogenization and de-homogenization operations. This framework supports nonlinear history-dependent materials and finite-strain problems while keeping operation counts proportional to network degrees of freedom.

  • Nonlinear small-strain plasticity: Each bottom-layer node stores an independent material response, allowing the network to perform online prediction of nonlinear plasticity rather than only linear elasticity.For history-dependent materials, internal variables are stored at the active bottom-layer nodes.
  • Nonlinear small-strain plasticity: For nonlinear materials, residual strain arises from material nonlinearity and need not vanish during iterative loading-step updates.If all phases are linear elastic, residual strain is zero and the training-process homogenization procedure applies.
  • Nonlinear small-strain plasticity: Compliance or stiffness information and residual strain or stress are propagated forward through building blocks, while incremental stresses and strains are de-homogenized backward.A Newton iteration combines one forward homogenization process with one backward de-homogenization process.
  • Computational cost: The material network's operation count per Newton iteration is proportional to its number of degrees of freedom.This scaling is contrasted with FEM, whose operation count is proportional to (Ndof)^2.x, and integral equation methods, proportional to (Ndof)^3.
  • Finite-strain problem: Finite-strain extrapolation uses the deformation gradient F and first Piola-Kirchhoff stress P as strain and stress measures.The stiffness information and residual first Piola-Kirchhoff stress are propagated through homogenization and rotation operations to the output layer.

A11 A12 A13 A14 A22 A23 A24 A33 A34 sym A44

The finite-strain material-network formulation propagates constitutive information through analytical homogenization and rotation operations while preserving rotational information and enforcing interface constraints. Incremental deformation and stress are solved at each loading step, with Newton iteration used for convergence.

  • Incremental constitutive response: At each loading step, incremental stress ∆P and strain ∆F are related through stiffness and residual first Piola-Kirchhoff stress.The residual quantity is denoted δP in the constitutive relation.
  • Network formulation: The forward homogenization process propagates stiffness tensors and residual stresses from constitutive laws at the bottom layer to the output layer.Homogenization and rotation operations are applied at each building block.
  • Building-block homogenization: The two-layer homogenized stiffness depends on constituent stiffness tensors A1 and A2 and volume fractions f1 and f2 = 1 − f1.Analytical expressions are derived from equilibrium and kinematic constraints.
  • Finite-strain measures: Finite-strain modeling uses first Piola-Kirchhoff stress P and deformation gradient F to retain rotational information.The formulation avoids symmetric measures such as second Piola-Kirchhoff stress and Green strain.
  • Interface constraints: At least 4 degrees of freedom must be constrained at the 2D building-block interface to satisfy equilibrium and kinematic conditions.This constraint is specified for finite-strain problems.
  • Rotation operation: The finite-strain rotation matrix depends on the prescribed angle θ, and the rotated stiffness matrix and residual stress are computed through the rotation operation.The rotation operation is part of the building-block transfer between network layers.
  • Solution procedure: After the macroscopic boundary-value problem is solved, ∆F and ∆P are propagated backward through the network, with Newton’s method finding convergence at each loading step.This backward procedure is described for finite-strain response and parallels the nonlinear-plasticity procedure.

5. Evaluations

The material network achieves accurate RVE predictions across elastic, plastic, and finite-strain cases while using reduced degrees of freedom and substantially less computation than DNS. Accuracy depends on sufficient depth, initialization, training convergence, and the sampling range for high-contrast materials.

  • Network training: After 10,000 epochs, average training errors were below 0.7% for all RVEs at N = 7, with negligible training–validation differences indicating no over-fitting.A 2% accuracy level was reached after 1,000 epochs for N = 5 and 7; N = 3 was generally insufficient except for the uniform RVE.
  • Network representation: N ≥ 5 represented DNS accurately with fewer degrees of freedom, while maximum training errors remained below 2.5% for all RVEs.The trained networks also extracted geometric information such as phase volume fraction from homogenized mechanical-property data.
  • Linear elasticity: For N = 7, average testing errors were below 1% for matrix-inclusion and anisotropic RVEs and 1.7% for the amorphous RVE.Maximum testing error stayed below 2% for the anisotropic RVE but reached 9.5% for matrix-inclusion and amorphous RVEs; broader sampling was suggested for high phase-property contrasts.
  • Small-strain nonlinear plasticity: The networks captured nonlinear plastic behavior for hard, soft, and anisotropic RVEs under uniaxial tension, pure shear, and loading–unloading paths when N ≥ 5.These predictions matched DNS while using significantly fewer degrees of freedom.
  • Limitations and practical choices: Deeper networks can improve representational accuracy but may train more slowly because their cost functions contain more local minima.The optimum depth depends on available training time and desired accuracy, and different initializations can produce different trained topologies.
  • Computational efficiency: For 2D amorphous RVEs, N = 5 reduced computation time to 0.49 s versus about 79 s for typical finite-element simulations, a 150-fold speed-up.Evaluation time increased with active-node count, reaching 1.58 s for N = 7 with Na = 86.
  • Finite-strain hyperelasticity: The method predicted global and local RVE responses accurately under large deformations, with good agreement between DNS and network P11–F11 curves.Amorphous-RVE DNS runs terminated early because of severe element distortion, whereas the networks were evaluated for the prescribed deformation comparisons.

6. Conclusion

The paper concludes that deep material networks combine mechanistic building blocks, hierarchical topology, and machine learning to reduce RVE models and accelerate challenging multiscale predictions. The method addresses several limitations of prior reduced-order approaches while retaining applications to nonlinear and finite-strain problems.

  • 6. Conclusion: Deep material networks use mechanistic building blocks and hierarchical topology to learn RVE representations for accelerated online prediction.The building blocks have analytical homogenization solutions, while training uses SGD with backpropagation, nodal deletion, and subtree merging.
  • 6. Conclusion: The training procedure can use offline DNS data or experimental testing data to construct reduced-order RVE models.
  • 6. Conclusion: The method targets efficient and accurate homogenization predictions for nonlinear history-dependent plasticity and finite-strain hyperelasticity under large deformations.
  • 6. Conclusion: Future work includes improving data sampling and training theory, extending the present 2D building block to 3D and multiphysics, and applying the method to more complex industrial RVEs.
  • 6. Conclusion: The parameterized material network is positioned to accelerate concurrent computation and reduce numerical difficulty from large material distortion in conventional RVE approaches.

Appendix A. Analytical solutions of 2D building block in finite strain

The appendix derives analytical finite-strain homogenization solutions for the two-layer building block by enforcing interface equilibrium and kinematic constraints. It formulates the homogenized response and concentration tensors from an imposed overall deformation gradient and phase constitutive laws.

  • Appendix A. Analytical solutions of 2D building block in finite strain: The finite-strain two-layer building block is derived from equilibrium and kinematic constraints at the interface between layers.
  • Appendix A. Analytical solutions of 2D building block in finite strain: The derivation begins by prescribing an arbitrary overall deformation gradient after homogenization.
  • Appendix A. Analytical solutions of 2D building block in finite strain: Homogenization definitions and phase constitutive laws are combined with equilibrium to derive the layer concentration tensor.
  • Appendix A. Analytical solutions of 2D building block in finite strain: The appendix derives the homogenized stress and stiffness response, including a formulation that accounts for residual stress.
  • Appendix A. Analytical solutions of 2D building block in finite strain: The same derivation procedure can extend to analytical 3D two-layer building-block solutions in small-strain and finite-strain formulations.
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