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Exploring the 3D architectures of deep material network in data-driven multiscale mechanics

Zeliang Liu, C. T. Wu

arXiv:1901.04832v3cs.CEcond-mat.mtrl-scics.LG

TL;DR

The paper addresses the challenge of modeling heterogeneous materials in general 3D settings with material and geometric nonlinearities. It develops a mechanistic deep material network trained from DNS-generated elastic data and validates it across nonlinear rubber, polycrystalline, and CFRP systems. The resulting framework includes three-scale CFRP homogenization and a reported computational advantage for complex composite modeling.

  • Problem

    Existing single-scale and macroscopic data-driven models have limited access to microscale physics, while heterogeneous 3D materials involve complex microstructural interactions, nonlinearities, and anisotropy.

  • Method

    DMN represents 3D heterogeneous materials with multilayer networks of mechanistic two-layer building blocks, trained offline on DNS-generated elastic datasets and extrapolated to nonlinear material behavior.

  • Results

    The framework is explored for hyperelastic rubber with Mullins effect, rate-dependent polycrystals, and CFRP composites, including three-scale CFRP homogenization; woven-composite average test errors are 6.12% for the linear FE model versus 3.48% for DMN.

  • Takeaways & Limitations

    DMN provides a data-driven framework for multiscale material modeling across complex 3D morphologies and nonlinear material laws, with networks that can be concatenated across CFRP scales.

  • Takeaways & Limitations

    For woven composites, the lowest training error was 2.19% with N = 8, and further error reduction may require more advanced learning techniques.

Abstract

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This paper extends the deep material network (DMN) proposed by Liu et al. (2019) to tackle general 3-dimensional (3D) problems with arbitrary material and geometric nonlinearities. It discovers a new way of describing multiscale heterogeneous materials by a multi-layer network structure and mechanistic building blocks. The data-driven framework of DMN is discussed in detail about the offline training and online extrapolation stages. Analytical solutions of the 3D building block with a two-layer structure in both small- and finite-strain formulations are derived based on interfacial equilibrium conditions and kinematic constraints. With linear elastic data generated by direct numerical simulations on a representative volume element (RVE), the network can be effectively trained in the offline stage using stochastic gradient descent and advanced model compression algorithms. Efficiency and accuracy of DMN on addressing the long-standing 3D RVE challenges with complex morphologies and material laws are validated through numerical experiments, including 1) hyperelastic particle-reinforced rubber composite with Mullins effect; 2) polycrystalline materials with rate-dependent crystal plasticity; 3) carbon fiber reinforced polymer (CFRP) composites with fiber anisotropic elasticity and matrix plasticity. In particular, we demonstrate a three-scale homogenization procedure of CFRP system by concatenating the microscale and mesoscale material networks. The complete learning and extrapolation procedures of DMN establish a reliable data-driven framework for multiscale material modeling and design.

1. Introduction

Heterogeneous materials challenge single-scale and analytical models because microstructural interactions, nonlinearities, and anisotropy are difficult to capture. This paper extends deep material networks to general 3D nonlinear problems and evaluates them across representative multiscale systems.

  • Single-scale empirical models can miss microstructural interactions and fail to capture nonlinear or anisotropic responses.
  • RVE homogenization provides an important approach for modeling multiscale materials, while analytical methods often rely on simplified geometries and material models.
  • Macroscopic data-driven models fit stress-strain or energy relations directly, but their extrapolation to unknown materials and loading spaces is limited by insufficient microscale physics.
  • Earlier DMN work represented challenging 2D RVEs with binary-tree networks of mechanistic two-layer building blocks while avoiding extensive offline sampling and extra calibration.
  • This paper extends DMN to 3D material and geometric nonlinearities and applies it to rubber composites, polycrystals, and CFRP systems.

2. The global framework of deep material network

DMN represents an RVE with a binary-tree network of mechanistic two-layer building blocks, trained offline on simulated elastic responses and extrapolated online to nonlinear materials and loading paths. Its weights encode geometric contributions, while each block combines homogenization with rotation; active networks can also be concatenated across material scales.

  • Offline training: The offline stage samples phase stiffness matrices, simulates RVE responses under six orthogonal loadings, and reserves test samples for generalization checks.Design of experiments explores the input space, while DNS methods such as finite elements or FFT generate effective stiffness matrices.
  • Offline training: DMN trains physically meaningful activations and rotation angles by minimizing a supervised fitting objective over the simulated RVE responses.Activations determine network weights, and rotation angles describe block orientations.
  • Online prediction: In online prediction, active bottom-layer nodes act as independent material points with their own loading paths and internal variables, supporting nonlinear extrapolation beyond the elastic training space.For finite-strain problems, the network propagates tangent stiffness and residual stress while retaining the mechanistic structure.
  • Multi-layer network structure: The network uses a binary-tree structure whose nodes can deactivate during training, while recursive weight summation preserves the physical interpretation of child-node contributions.ReLU activation enables automatic simplification and can increase training speed through vanishing gradients after deactivation.
  • Mechanistic building blocks: Each mechanistic building block applies homogenization to capture phase fractions and rotation to vary orientation, using analytical transformations for a two-layer 3D structure.The block receives stiffness inputs, homogenizes them, and rotates the resulting stiffness matrix.
  • Multi-scale extension: DMN networks can be concatenated without changing the online algorithms, enabling three-scale CFRP homogenization by attaching microscale networks to mesoscale woven-composite nodes.The demonstrated hierarchy connects microscale unidirectional-fiber and mesoscale woven-composite representations.

3. Mechanistic building block in 3D

The 3D DMN building block combines analytical homogenization with rotation for two-layer structures, extending from small-strain linear elasticity to finite-strain material and geometric nonlinearities.

  • Building-block operations: Each building block performs homogenization of two materials followed by rotation of the resulting structure.The two operations are represented by homogenization and rotation functions.
  • Small-strain formulation: The small-strain building block derives homogenization from interfacial equilibrium and kinematic constraints, then obtains the homogenized stiffness matrix.The derivation uses strain concentration tensors and averaged stress-strain relations.
  • Rotation operation: Three Tait-Bryan angles α, β, and γ parameterize the 3D rotation through successive X-, Y-, and Z-axis transformations.The rotated stiffness is expressed using the rotation matrix R(α, β, γ).
  • Finite-strain formulation: The finite-strain extension incorporates increments of deformation gradient, tangent elasticity, and residual first Piola-Kirchhoff stress.The nonlinear constitutive relation includes both stress and deformation increments, with residual stress required because of nonlinearity.
  • Finite-strain homogenization: Finite-strain homogenization derives functions HA and HP for tangent stiffness and residual stress from equilibrium, kinematic constraints, and averaging conditions.The homogenized quantities are then rotated using RA and RP.
  • Finite-strain rotation: The finite-strain rotation functions use a modified rotation matrix Rf to transform the tangent elasticity tensor and first Piola-Kirchhoff stress.The deformation gradient and stress are represented using all 9 tensor components because minor symmetries are absent.

4. Applications

The applications evaluate 3D DMN training and extrapolation across nonlinear multiscale material systems, with finite-strain online formulations used by default. A particle-reinforced composite example provides training-error, test-error, and volume-fraction comparisons with a linear finite-element model.

  • Application scope: The applications cover particle-reinforced rubber with Mullins effect, polycrystalline materials, and CFRP composites.The CFRP example includes microscale and mesoscale RVEs for three-scale homogenization.
  • Training and extrapolation: DMN networks are trained with linear-elastic data and extrapolated online to nonlinear material models, using stochastic gradient descent and regularization.The online examples include Mooney-Rivlin hyperelasticity with Mullins effect, von Mises plasticity, and rate-dependent crystal plasticity.
  • Training and extrapolation: Finite-strain formulation with geometric nonlinearity is the default online-stage formulation for all RVEs.
  • Particle-reinforced composite: The particle-reinforced RVE contains 22.6% particles and is discretized with 84693 nodes and 59628 10-node tetrahedron elements.Its online phases use Mooney-Rivlin rubber with Mullins effect and Neo-Hookean particles 100 times harder than the matrix.
  • Particle-reinforced composite: The particle-composite comparison reports average training error, average test error, maximum test error, predicted particle volume fraction, and linear-FEM test errors.The table compares these quantities across DMN configurations and a linear finite-element reference.

4.1. Hyperelastic polymer composite with Mullins effect

The particle-reinforced rubber RVE tests DMN training, compression, and nonlinear extrapolation under Mullins-effect loading. Deeper networks improve accuracy, while the N = 8 network reproduces responses with a compressed representation.

  • RVE and material model: The RVE contains four spherical particles occupying 22.6% of the composite, discretized by 84,693 nodes and 59,628 ten-node tetrahedral elements.
  • Offline evaluation: For N = 4, training error stalls near 7.6%, whereas deeper N = 6 and N = 8 networks achieve good accuracy.
  • Offline evaluation: For N = 8, average training error reaches 0.53% after 40,000 epochs, with maximum test error of 2.41%.
  • Offline evaluation: Model compression reduces active bottom-layer nodes from 8, 32, and 128 to 4, 13, and 28 for N = 4, 6, and 8, respectively.
  • Offline evaluation: For N = 8, the predicted phase-1 volume fraction differs from full-field DNS by 0.88%, indicating that DMN extracts geometric information from stiffness data.

4.2. Polycrystalline materials with rate-dependent crystal plasticity

The 3D DMN represents polycrystalline RVEs with random or textured orientations and reproduces their rate-dependent crystal-plastic responses. Deeper networks reduce errors and recover mechanical behavior and orientation information more accurately.

  • RVE generation: Polycrystalline RVEs contain 415 equiaxed grains with either random or textured crystallographic orientation distributions.Both RVE types use the same nominal grain count but differ in their orientation distribution functions.
  • DMN formulation: The single-phase polycrystalline DMN assigns the same crystal stiffness input to all bottom-layer nodes while learning orientation-dependent network structure.The grains share one material model but have different orientations, so the offline network takes only one material input.
  • Offline evaluation: After 20000 epochs, depth N = 8 reduced training and test errors below 0.5% for both random and textured ODFs.The error slopes had not saturated, indicating that additional training could reduce errors further.
  • Offline evaluation: For the random ODF, N = 6 achieved approximately 1% mean errors and 3.64% maximum test error, while offering roughly fourfold lower computational cost than N = 8.The depth choice therefore trades representation accuracy against offline-training and online-extrapolation cost.
  • Orientation recovery: Sufficiently deep DMNs recovered the random ODF from mechanical data, while textured-ODF recovery remained more challenging but succeeded for N > 4.The comparison used pole figures generated from DNS and DMN predictions.
  • Online extrapolation: Networks with N = 6 and N = 8 predicted hardening behavior well at both strain rates, whereas N = 4 misestimated the yield stress for random and textured ODFs.The online tests used finite-strain rate-dependent crystal plasticity under uniaxial loading.

4.3. Carbon fiber-reinforced polymer with three scales

DMN models separate UD and woven CFRP RVEs and then concatenate their networks for three-scale homogenization. The models capture anisotropic, plastic, and geometrically nonlinear responses, although woven-RVE fitting is more difficult.

  • Multiscale architecture: The CFRP hierarchy homogenizes a microscale UD fiber composite into the yarn phase of a mesoscale woven composite before obtaining overall properties.The UD and woven RVEs have distinct finite-element discretizations and phase volume fractions.
  • Offline evaluation: A sufficiently deep DMN extracted phase volume fractions from mechanical data, and the linear FEM reference for the woven composite had 6.12% average and 28.3% maximum test error.The paper compares these results with DMNs of varying depths and active-node counts.
  • UD online extrapolation: DMNs with N ≥ 7 captured UD loading-unloading responses across two tension and two shear directions.The fiber direction is nearly elastic because its Young’s modulus is 245.0 GPa versus 3.8 GPa for epoxy.
  • Woven online extrapolation: Finite-strain DMN reproduced woven-composite stiffening after ε11 reached 1.5%, whereas the small-strain formulation did not.The stiffening is associated with yarn straightening and local rotations despite overall deformation below 2.0%.
  • Three-scale results: The woven response preserves microstructural effects, including matrix-dominated out-of-plane shear and additional hardening in the three-scale model.The three-scale in-plane tension response also shows fiber-straightening stiffening, while the two-scale model misses some shear plasticity.
  • Three-scale homogenization: Three-scale homogenization concatenates each active yarn node in the woven DMN with a copy of the UD DMN, leaving only fiber and matrix constitutive models as online inputs.The demonstrated example uses N_ud = 7 and N_woven = 8.

5. A comment on computational cost

DMN substantially reduces online computational cost relative to DNS and comparable-accuracy linear FE models, while training cost increases with network depth.

  • 400 training samples required 39.5 h to generate, averaging 356 s per sample.
  • DMN training times for N = 4, 6, and 8 were 5.4, 16.7, and 43.0 h, respectively.
  • 6.0 s online DMN time at N = 8 was 810 times faster than the 4860 s DNS computation for the hyperelastic particle-reinforced RVE.
  • At similar accuracy, DMN took 2.6 s versus 2420 s for linear FE, making it around 930 times faster.DMN average test error was 1.38%, compared with 2.30% for the linear FE model.
  • For woven-composite homogenization, DMN required 0.65 s versus 212 s for linear FE, with average test errors of 3.48% and 6.12%, respectively.

6. Conclusions

The paper develops a 3D DMN framework for nonlinear heterogeneous materials and validates it across complex material systems. Its demonstrations include efficient learning, accurate nonlinear prediction, and three-scale CFRP homogenization.

  • The framework models general 3D heterogeneous materials with material and geometric nonlinearities using a derived two-layer building block.
  • DMN training combines DNS-generated offline elastic datasets with stochastic gradient descent and model compression algorithms.
  • Three representative systems—Mullins-effect rubber, rate-dependent polycrystals, and elasto-plastic CFRP—demonstrate representation of complex RVEs and prediction of highly nonlinear behavior.
  • DMN can discover hidden geometric information, including phase volume fraction and orientation distributions, from purely mechanical data.

Appendix A. Elementary rotation matrices

Appendix A specifies elementary rotation matrices for small- and finite-strain formulations. The matrices are assembled from in-plane and output-plane rotations parameterized by arbitrary angles.

  • Small-strain formulation: Small-strain elementary rotations X, Y, and Z place identity entries on the corresponding coordinate and use rp and rv blocks for rotated components.
  • Small-strain formulation: The in-plane and output-plane rotation matrices rp and rv are defined in Mandel notation for an arbitrary angle θ.
  • Finite-strain formulation: Finite-strain formulation likewise defines elementary rotation matrices and their in-plane and output-plane rotation blocks for arbitrary angles.

Appendix B. Gradients for training

Appendix B provides the derivative components needed to compute training gradients. It differentiates homogenization, volume-fraction, and rotation operations using chain-rule-based decompositions.

  • The training gradient uses derivatives of building-block quantities obtained through the chain rule.
  • Derivatives of (Ĉ345)^−1 with respect to variables such as volume fraction or stiffness components support computation of the strain concentration tensor s1 in hC.
  • The appendix gives derivatives of volume fraction f1 with respect to weights w1 and w2.
  • Rotation derivatives are simplified by decomposing the rotation function rC into three steps and differentiating stiffness matrices with respect to rotation angles.
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