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Multiscale modeling of inelastic materials with Thermodynamics-based Artificial Neural Networks (TANN)

Filippo Masi, Ioannis Stefanou

arXiv:2108.13137v3cond-mat.mtrl-scics.CEcs.LGphysics.comp-ph

TL;DR

Complex inelastic microstructures are difficult to model, while detailed multiscale simulations are computationally expensive. The paper introduces thermodynamics-based artificial neural networks to discover internal state variables and model these materials, achieving accurate macroscopic and microstructural predictions and substantially reducing homogenized-model cost.

  • Problem

    Complex heterogeneous, multiscale materials are difficult to represent with heuristic constitutive models, while detailed multiscale calculations are computationally expensive.

  • Method

    TANN combines thermodynamics-based dimensionality reduction with encoder–decoder neural networks to autonomously identify internal state variables from microstructural coordinates and model dissipative behavior.

  • Results

    TANN accurately predicts macroscopic stress, free energy, dissipation rate, and microstructural fields, while FEM×TANN agrees excellently with micromechanical simulations and reduces computational cost by several orders of magnitude.

  • Takeaways & Limitations

    The identified internal state space enables efficient double-scale homogenization within classical finite-element workflows for complex inelastic microstructures.

  • Takeaways & Limitations

    The methodology cannot be used when scale separation fails, such as for strain localization at the macroscale, and its physical interpretation remains outside the current work.

Abstract

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The mechanical behavior of inelastic materials with microstructure is very complex and hard to grasp with heuristic, empirical constitutive models. For this purpose, multiscale, homogenization approaches are often used for performing reliable, accurate predictions of the macroscopic mechanical behavior of solids and structures. Nevertheless, the calculation cost of such approaches is extremely high and prohibitive for real-scale applications involving inelastic materials. Here, we propose the so-called Thermodynamics-based Artificial Neural Networks (TANN) for the constitutive modeling of materials with inelastic and complex microstructure. Our approach integrates thermodynamics-aware dimensionality reduction techniques and thermodynamics-based deep neural networks to identify, in an autonomous way, the constitutive laws and discover the internal state variables of complex inelastic materials. The efficiency and accuracy of TANN in predicting the average and local stress-strain response, the free-energy and the dissipation rate is demonstrated for both regular and perturbed two- and three-dimensional lattice microstructures in inelasticity. TANN manage to identify the internal state variables that characterize the inelastic deformation of the complex microstructural fields. These internal state variables are then used to reconstruct the microdeformation fields of the microstructure at a given state. Finally, a double-scale homogenization scheme (FEMxTANN) is used to solve a large scale boundary value problem. The high performance of the homogenized model using TANN is illustrated through detailed comparisons with microstructural calculations at large scale. An excellent agreement is shown for a variety of monotonous and cyclic stress-strain paths.

I. INTRODUCTION

Complex heterogeneous materials challenge heuristic constitutive modeling, while multiscale methods are accurate but computationally expensive. The paper proposes thermodynamics-aware neural networks with automatically discovered internal state variables to address this gap.

  • Multiscale modeling captures microstructural effects at the macroscopic scale but requires costly iterative representative-cell boundary-value calculations.
  • Artificial neural networks reduce calculation costs but may extrapolate poorly, require extensive low-noise data, and violate basic physical principles.
  • Earlier thermodynamics-based neural networks enforce relations among free energy, stress, dissipation, and internal variables but require those variables to be identified beforehand.
  • The proposed method discovers a reduced set of microstructural internal state variables using thermodynamics-based dimensionality reduction and combines them with TANN.
  • The paper develops the theoretical framework and benchmarks the proposed approach on complex microstructured materials.

B. Volume averaging under periodic conditions

The framework averages thermodynamic quantities over periodically constrained microstructures and represents their macroscopic behavior using state variables and internal state variables. This produces local-in-time constitutive restrictions suitable for homogenization.

  • Periodic displacement and anti-periodic traction conditions yield volume-averaged energy and dissipation relations for the microstructure.
  • The averaged quantities include stress, deformation, free energy, temperature, dissipation, and an entropy-related term used in the homogenized formulation.
  • Internal state variables encode process history through their evolution, allowing constitutive relations to depend on the current state rather than the full history.
  • The resulting averaged constitutive restrictions follow from requiring the dissipation inequality to hold for arbitrary rates and volumes.
  • Candidate internal variables must satisfy thermodynamic requirements, including evolution laws constrained by the second law.

III. LEARNING THE CONSTITUTIVE BEHAVIOR OF MATERIALS WITH THERMODYNAMICS-BASED ARTIFICIAL NEURAL NETWORKS

The neural-network formulation uses layered feed-forward mappings trained by gradient descent to minimize prediction error. After training, the network operates in inference mode to predict constitutive outputs.

  • A depth-K feed-forward network propagates signals through an input layer, K−1 hidden layers, and an output layer.
  • Weights and biases are optimized by gradient descent using mean absolute error between benchmark outputs and predictions.
  • Once trained, the network enters recall mode and produces output predictions.

A. Thermodynamics-based Artificial Neural Networks with a priori determined internal state variables

The original TANN formulation predicts thermodynamic constitutive behavior when internal state variables are known, whereas microstructured materials require discovering an adequate state representation. The extended approach uses internal coordinates and dimensionality reduction to address this problem.

  • A. Thermodynamics-based Artificial Neural Networks with a priori determined internal state variables: Existing TANN takes the previous material state, strain increment, and time increment as inputs and predicts updated internal variables, energy, stress increment, and dissipation.
  • A. Thermodynamics-based Artificial Neural Networks with a priori determined internal state variables: In finite-element inference, TANN outputs updated internal state variables and stresses, with prior work demonstrating efficient and robust inelastic-material modeling.
  • B. Internal coordinates and dimensionality reduction: The original formulation can fail when users choose inadequate internal state variables for microstructured materials.
  • B. Internal coordinates and dimensionality reduction: Internal coordinates are microscopic quantities such as displacement, velocity, or momentum fields that describe the material state and differ from macroscopic internal state variables.
  • B. Internal coordinates and dimensionality reduction: Autoencoders provide lower-dimensional representations of high-dimensional internal coordinates, but conventional latent representations need not respect thermodynamics.

C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables

The extended TANN discovers thermodynamically consistent internal state variables from latent representations of microscopic internal coordinates, without specifying them a priori. An encoder–decoder architecture and two-step training procedure enable prediction of material behavior and optional reconstruction of microstructural fields without requiring full process histories.

  • C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables: Latent representations of internal coordinates automatically identify the internal state variables, so the extended TANN requires no a priori choice of their nature or values.The identification is unsupervised and uses thermodynamics-based neural networks for microstructured materials.
  • C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables: The encoder reduces high-dimensional internal coordinates to latent variables, while the decoder reconstructs updated coordinates from the updated latent state.The identified variables enter the free-energy and dissipation-rate descriptions, thereby satisfying the model’s thermodynamic restrictions.
  • C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables: Training proceeds in two steps: the first discovers internal state variables, and fine-tuning then trains the thermodynamic networks using those discovered variables.During fine-tuning, decoder weights are frozen and the trained encoder generates the internal-state inputs before optimization.
  • C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables: The decoder can be omitted during recall when microstructural-field reconstruction is unnecessary, reducing the computation required in large multiscale analyses.When reconstruction is needed, recall mode uses the decoder to predict internal coordinates from the material-state internal variables.
  • C. Thermodynamics-based Artificial Neural Networks without a priori determined internal state variables: The formulation does not require process histories, which the authors identify as an advantage for data generation and efficient large multiscale simulations.This history-free formulation follows from representing evolution through the discovered internal state variables.

IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS

The applications use inelastic 2D and 3D lattice cells modeled at small strain with periodic boundary conditions and homogenized microscopic responses. The benchmark structures comprise elasto-plastic bars, while microscopic displacements and bar forces serve as internal coordinates.

  • IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS: The lattice calculations assume small strains, with |∇u| ≪1 and the stress and strain measures reduced to their infinitesimal forms.The benchmark uses p ≈σ and Sym(f) ≈ε −I under the stated small-strain assumptions.
  • IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS: Microscopic displacement fields are solved on the lattice cell, and microscopic quantities are volume-averaged to obtain macroscopic responses.The average stress is computed from microscopic stress and boundary tractions, while the average strain is obtained from microscopic strain and boundary displacement data.
  • IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS: Periodic boundary conditions are applied so the lattice-cell model can be used incrementally in the subsequent asymptotic-homogenization boundary-value problem.This connects the microscopic cell calculation to the macroscopic multiscale analysis.
  • IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS: The benchmarks use 2D and 3D lattice structures made from 6 and 40 elasto-plastic bars, respectively.Each bar has Young modulus 100 GPa, yield strength 300 MPa, hardening parameter 10 GPa, and circular cross-section A = 1 cm2.
  • IV. APPLICATIONS TO MICROSTRUCTURED, LATTICE MATERIALS: The internal coordinates are microscopic nodal displacements and uniform internal forces along each bar, although other coordinate choices are possible.The approach is therefore not restricted to this particular representation of the microstructural state.

A. Data generation

TANN are trained on generated material-state transitions and then evaluated on unseen loading paths, predicting macroscopic and microscopic responses for regular lattices.

  • A. Data generation: The encoding of internal coordinates identifies 12 internal state variables, substantially fewer than the lattice structures’ internal coordinates.This reduced representation supports prediction of the material state and reconstruction of updated internal coordinates.
  • A. Data generation: The two-step training procedure reaches convergence after approximately 2500 epochs in its first step and 1000 epochs in its second step.Figure 6 reports dimensionless MAE on training and validation sets, with shaded validation regions.
  • B. 2D regular lattices: TANN predicts stresses, free energy, dissipation rate, internal state variables, and microscopic internal coordinates for a 2 × 2 lattice with extremely good performance.The comparison uses a uni-axial strain path with increasing amplitude and respects the first and second laws of thermodynamics.
  • B. 2D regular lattices: TANN accurately predicts stress behavior and internal coordinates under combined cyclic uni-axial loading and monotonous shearing.With shearing, most internal state variables oscillate between evolving amplitudes, while three variables resemble those from uni-axial loading.
  • A. Data generation: The physical nature of the computed internal state variables is not examined because feature-extraction analysis is outside the work’s scope.The paper identifies the variables but does not provide their physical interpretation.
  • B. 2D regular lattices: For the 6 × 6 lattice, TANN predicts stress-strain behavior and internal-coordinate evolution, although the discovered internal state variables differ from those of the 2 × 2 lattice.The difference is attributed to the non-uniqueness of thermodynamically admissible internal state variables.

C. 2D non-regular lattices

TANN is tested on geometrically perturbed 2D lattices, where it reproduces micromechanical behavior despite non-regularity and permanent inelastic deformation.

  • C. 2D non-regular lattices: The non-regular lattices are generated by perturbing regular lattice coordinates with a zero-mean normal distribution having standard deviation 0.2.The same TANN architecture is retained across the non-regular cases.
  • C. 2D non-regular lattices: TANN excellently predicts the material behavior of 2 × 2 and 6 × 6 non-regular lattices under cyclic uni-axial loading.The comparison is made against micromechanical simulations despite the lack of geometric periodicity and the resulting higher number of internal state variables.
  • C. 2D non-regular lattices: Internal state variables remain nonzero during unloading because permanent inelastic deformations are present and captured by the proposed approach.This behavior is reported for the cyclic loading response of the non-regular lattices.

V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH

FEM×TANN replaces classical Gauss-point constitutive models with trained thermodynamics-based networks for large-scale double-scale homogenization. Across elastic and inelastic loading cases, the homogenized model converges toward micromechanical simulations while reconstructing microscopic displacements and stresses.

  • V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH: FEM×TANN uses trained TANN at finite-element Gauss points to solve large-scale homogenized analyses while separating macroscopic and auxiliary microscale problems.The network receives stress, deformation, and internal state variables and outputs their updated values for each increment.
  • V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH: The benchmarks compare homogenized finite-element predictions with full micromechanical simulations across unit-cell ratios ϵ ∈ [1, 1/20] and uniform, compression, torsion, and history loading scenarios.The 3D benchmarks use linear tetrahedral elements with uniform 0.5 cm size.
  • V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH: For uniform compression, the comparisons evaluate a control-point displacement, total free energy, and dissipation rate against micromechanical results.The displacement uses the first-order homogenization approximation uϵ = u(0) + ϵu(1).
  • V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH: 0.1% displacement error at ϵ = 1/20 accompanies energy errors of 0.04% and 0.07% in elastic and inelastic uniform compression, respectively.The dissipation-rate error is 0.12%, and displacement error remains below 0.8% for ϵ ≤ 1/8.
  • V. APPLICATION TO DOUBLE-SCALE HOMOGENIZATION: THE FEM×TANN APPROACH: The method reconstructs first-order microscopic displacements and zeroth-order microscopic stresses by decoding internal coordinates.Microscopic stresses are obtained by dividing decoded internal forces by the bars’ cross-sectional areas.

B. Uni-axial compression (lcC)

Under uni-axial compression, FEM×TANN agrees closely with micromechanical simulations for both elastic and inelastic cases. The comparison remains accurate in a boundary-layer region with high strain and stress gradients, although first-order homogenization cannot fully resolve such effects.

  • B. Uni-axial compression (lcC): At ϵ = 1/20, relative errors are 1.6% for control-point displacement, 0.05% for total energy, and 0.35% for microstresses.These results cover both the 2 cm elastic and 5 cm inelastic compression cases.
  • B. Uni-axial compression (lcC): The compression benchmark applies 2 cm elastic and 5 cm inelastic uni-axial displacements to the same cubic lattice structure used previously.The reported comparisons concern the homogenized model and micromechanical simulations.
  • B. Uni-axial compression (lcC): The displacement comparison occurs inside a boundary layer with high strain and stress gradients, where first-order asymptotic homogenization cannot capture details exactly.Despite this limitation, the reported errors remain limited for the application.
  • C. Torsional deformation (lcT): The broader torsional comparisons show increasing errors under stronger warping and non-negligible boundary-layer effects, reaching 4.2% in dissipation rate at ω = 60° and ϵ = 1/20.At ω = 40°, displacement and total-energy errors are 0.03% and 1.3%; at ω = 60°, displacement and energy errors are 0.68% and 1.9%.

D. Cyclic loading (lcH)

The homogenized TANN model agrees closely with micromechanical simulations under demanding cyclic torsional loading, including residual displacement and dissipation after unloading, while substantially reducing computational cost.

  • D. Cyclic loading (lcH): The homogenized TANN model agrees very well with micromechanical results for vertical displacement and total dissipation under cyclic torsional loading.The comparison covers the first-order control-point displacement and zeroth-order total dissipation rate.
  • D. Cyclic loading (lcH): The model accurately predicts the residual non-zero displacement and dissipation rate after sudden unloading from ω = 60°.The unloading increment reveals permanent displacements associated with plasticity.
  • D. Cyclic loading (lcH): The FE model using TANN compares displacement, total energy, and dissipation between elastic and plastic torsional cases.The cases use ω = 40° for uniform elastic deformation and ω = 60° for uniform plastic deformation.
  • D. Cyclic loading (lcH): The identified internal state variables remain nonzero after unloading, reflecting their dissipative character and microscopic stress-strain configurations.Each internal variable represents a particular stress-strain configuration at the microscopic level.
  • E. Computational cost: The computational cost ratio ranges from approximately 1 to 300 while preserving small relative errors in total free-energy.The ratio compares constitutive-function calls between the homogenized and micromechanical models across representative loading scenarios and mesh sizes.
  • E. Computational cost: For sufficiently small ϵ, the computational cost ratio increases considerably without significantly affecting finite-element accuracy.Adaptive mesh refinement and higher-order homogenization are identified as possible further improvements.

VI. CONCLUSIONS

The paper develops a thermodynamics-aware TANN framework that discovers microstructural internal state variables and supports efficient multiscale finite-element simulations. Across regular and non-regular lattice examples, the homogenized model closely matches detailed micromechanical results while greatly reducing computational cost, although applications beyond elastoplastic lattices remain future work.

  • VI. CONCLUSIONS: The method discovers the necessary internal state variables from microstructural internal coordinates using a thermodynamics-trained encoder-decoder architecture.Internal coordinates may include microscopic displacements, velocities, stresses, and other microstructural fields.
  • VI. CONCLUSIONS: The framework is demonstrated on two- and three-dimensional regular and non-regular elastoplastic lattice structures with hardening.The authors state that the broader framework can apply to inelastic microstructures for which computer models are available.
  • VI. CONCLUSIONS: The homogenized TANN model shows excellent agreement with exact micromechanical simulations at both macro- and microscales across varied stress-strain paths.It predicts average stresses, strains, elastic energy, dissipation, and microstructural displacement and force fields.
  • VI. CONCLUSIONS: The computational cost of the homogenized problem is several orders of magnitude lower than that of the detailed micromechanical model.The approach can be used in classical finite-element codes and combined with established acceleration techniques.
  • VI. CONCLUSIONS: The study focuses on elastoplastic lattice structures, while practical applications to other materials and microstructures are left for future work.The authors describe broader material coverage as a capability of the methodology rather than a demonstrated application here.
  • VI. CONCLUSIONS: TANN may also support constitutive modeling, physical interpretation of the learned state space, and macro-element development, but these applications exceed the current scope.Feature extraction for interpreting internal variables is identified as future work.

APPENDIX A

The appendix formulates the microstructural auxiliary problem variationally, imposes periodic displacement constraints, and reduces equilibrium to nonlinear nodal equations solved by Newton-Raphson iteration. Mesh sensitivity is assessed using displacement, energy, and dissipation errors across loading scenarios.

  • APPENDIX A: The auxiliary microstructure problem is formulated from the variational balance equations in the absence of bulk and inertia forces.The formulation uses microscopic stress and strain, macroscopic average strain, tractions, and time-incremented quantities.
  • APPENDIX A: Periodic displacement fields are imposed on the lattice boundary through Lagrange multipliers, with opposite-boundary values defining the periodic jump.Antiperiodic tractions and periodic displacements make the corresponding boundary term vanish.
  • APPENDIX A: Assuming homogeneous strains within lattice elements yields a weak equilibrium form in nodal displacements and forces.The formulation identifies nodal force and displacement variables and distinguishes boundary nodes from the total node set.
  • APPENDIX A: The resulting nonlinear nodal equations are solved using Newton-Raphson’s method.Equilibrium is satisfied when the residual associated with every nodal variation vanishes.
  • APPENDIX A: Mesh sensitivity is evaluated through relative errors in control-point displacement, total energy, and dissipation rate for three loading scenarios.The analysis informs the choice of a tetrahedral finite-element mesh for the numerical simulations.
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