Source-linked AI summary
Automated discovery of generalized standard material models with EUCLID
Moritz Flaschel, Siddhant Kumar, Laura De Lorenzis
TL;DR
Material-model discovery usually assumes a constitutive class in advance, while stress-strain labels are often unavailable from standard experiments. This paper extends EUCLID with generalized standard materials to discover free-energy and dissipation potentials from full-field displacements and reaction forces, recovering material classes and parameters on noisy synthetic benchmarks.
Problem
Existing approaches commonly require labeled stress-strain pairs or produce black-box predictions, while EUCLID previously assumed that the material class was known.
Method
EUCLID uses generalized standard materials, convex thermodynamic potentials, balance constraints, sparse regression, and full-field displacement plus net reaction-force data to discover constitutive behavior.
Results
EUCLID automatically distinguished elasticity, viscoelasticity, plasticity, viscoplasticity, and hardening behaviors from displacement and force data, while generally recovering material parameters with good agreement.
Takeaways & Limitations
Sparse regression selects a parsimonious set of model features and corresponding internal variables, yielding simple and interpretable generalized-standard-material descriptions.
Takeaways & Limitations
The study initially excludes potentials depending on spatial gradients or state variables in the dissipation potential, and automatic differentiation requires smooth approximations for non-smooth potentials.
Abstract
from arXiv · showhide
We extend the scope of our approach for unsupervised automated discovery of material laws (EUCLID) to the case of a material belonging to an unknown class of behavior. To this end, we leverage the theory of generalized standard materials, which encompasses a plethora of important constitutive classes. We show that, based only on full-field kinematic measurements and net reaction forces, EUCLID is able to automatically discover the two scalar thermodynamic potentials, namely, the Helmholtz free energy and the dissipation potential, which completely define the behavior of generalized standard materials. The a priori enforced constraint of convexity on these potentials guarantees by construction stability and thermodynamic consistency of the discovered model; balance of linear momentum acts as a fundamental constraint to replace the availability of stress-strain labeled pairs; sparsity promoting regularization enables the automatic selection of a small subset from a possibly large number of candidate model features and thus leads to a parsimonious, i.e., simple and interpretable, model. Importantly, since model features go hand in hand with the correspondingly active internal variables, sparse regression automatically induces a parsimonious selection of the few internal variables needed for an accurate but simple description of the material behavior. A fully automatic procedure leads to the selection of the hyperparameter controlling the weight of the sparsity promoting regularization term, in order to strike a user-defined balance between model accuracy and simplicity. By testing the method on synthetic data including artificial noise, we demonstrate that EUCLID is able to automatically discover the true hidden material model from a large catalog of constitutive classes, including elasticity, viscoelasticity, elastoplasticity, viscoplasticity, isotropic and kinematic hardening.
1. Introduction
EUCLID addresses the weakness of choosing a material class and model form in advance by discovering constitutive behavior from experimentally accessible full-field displacements and reaction forces. This work extends EUCLID toward unknown material classes using generalized standard materials as a broad modeling framework.
- A priori material-model selection is experience-dependent, and an inappropriate choice can prevent identifying the material response.
- Many data-driven methods require labeled stress-strain pairs that standard experiments do not provide and produce predictions that are difficult to interpret physically.
- EUCLID uses full-field displacement data and net reaction forces, avoiding the need for directly labeled stress-strain data.
- Earlier EUCLID versions selected models within known classes such as hyperelasticity, elastoplasticity, and viscoelasticity.
- This work targets materials whose constitutive class is unknown by leveraging generalized standard materials, a framework encompassing many important constitutive behaviors.
- Prior generalized-standard-material learning studies included neural-network or variational approaches, but one solid-material benchmark was restricted to one-dimensional viscoelasticity with one internal variable and quadratic dissipation.
2. Generalized standard materials
Generalized standard materials are characterized by free-energy and dissipation potentials whose convexity and regularity properties yield constitutive behavior with stability and thermodynamic consistency. Their state and complementary laws connect observable strains, internal variables, thermodynamic forces, and dissipation.
- Assumptions and potentials: The framework considers infinitesimal strains in three-dimensional, isothermal materials without phase transitions, with observable strain and non-observable internal variables describing irreversible phenomena.
- Assumptions and potentials: Generalized standard materials are defined by a strictly convex Helmholtz free-energy potential and a convex dissipation potential with non-negative values and zero value at the origin.
- Assumptions and potentials: The two potentials and their parameters completely characterize the material behavior, while their stated properties guarantee stability and thermodynamic consistency.
- Constitutive laws: The constitutive formulation separates reversible energetic contributions from irreversible dissipative contributions and supplements state laws with complementary laws for dissipative evolution.
- Dissipation and constitutive laws: The intrinsic dissipation is non-negative for any dissipation potential with the postulated properties, so the dissipation inequality is satisfied by construction.
- Dual dissipation potential: Dual complementary laws use the Legendre-Fenchel transform of the dissipation potential to express irreversible evolution in terms of thermodynamic forces.
- Modeling assumptions: This study restricts the dissipation potential to be independent of strain rate, while rate dependence can remain through the evolution of internal variables.
- Numerical solution: The numerical treatment is formulated for strain-controlled histories and may use analytical derivatives or automatic differentiation, with automatic differentiation requiring smooth approximations for non-smooth potentials.
3. Automated discovery of generalized standard material models
EUCLID constructs a broad library of generalized standard material models and jointly selects a concise constitutive model and its parameters from data. The library combines thermodynamically consistent viscoelastic and viscoplastic mechanisms while allowing sparsity to deactivate unnecessary features and internal variables.
- Approach: EUCLID searches a broad generalized standard material library, making the material-behavior category itself an outcome rather than a predefined input.The approach combines model construction, weak momentum-balance constraints, and sparse regression for simultaneous model selection and parameter identification.
- Material model library: The library combines generalized Maxwell viscoelasticity with viscoplasticity, including isotropic and kinematic hardening mechanisms.It is three-dimensional and isotropic and covers a large portion of the models in the cited catalogue.
- Internal variables: The model uses nMW+3 internal variables: Maxwell viscous strains, plastic strain, and isotropic and kinematic hardening variables.The library may introduce more internal variables than ultimately needed, allowing sparsity regularization to deactivate unnecessary ones.
- Scope of the library: Compared with prior single-class studies, the present library is reduced within each class to emphasize discrimination among material classes.The authors note that this sacrifices within-class versatility, which was established in their earlier work, and leaves further extensions for future developments.
- Thermodynamic constraints: All material parameters are constrained to be non-negative to ensure convexity of the thermodynamic potentials.Reciprocal parameterizations also simplify limiting cases, such as purely viscoelastic behavior when yield stress tends to infinity.
3.2. Available data
EUCLID uses full-field displacement measurements and global reaction forces as input, reconstructs strains through finite-element interpolation, and fits material parameters by enforcing weak linear momentum balance. The resulting cost function combines interior and boundary force residuals, with a fixed reaction-force weighting.
- Available measurements: The input consists of two-dimensional full-field displacement data and global reaction-force measurements collected during a mechanical test.Displacements are recorded at spatial points over time, while reaction data contain the measured force components at constrained boundaries.
- Field reconstruction: Finite-element shape functions interpolate the measured displacements and provide the displacement field used to compute the infinitesimal strain field.The mesh nodes correspond to the points where displacement measurements are known.
- Identification constraint: EUCLID uses weak linear momentum balance as a physical constraint to identify model parameters without labeled stress-strain pairs.The objective is built from squared unbalanced-force residuals in the specimen interior and at displacement-constrained boundaries.
- Equilibrium residuals: At free degrees of freedom, internal nodal forces satisfy equilibrium, while constrained boundaries require their summed internal forces to match measured global reactions.The free and displacement-constrained degrees of freedom are explicitly partitioned to impose these two conditions.
- Cost-function weighting: The reaction-force contribution is weighted with λr = 100 because fewer reaction measurements are available than free degrees of freedom.The authors state that λr should be sufficiently larger than one and that its precise choice is not crucial for success.
3.4. Sparsity promoting regularization
EUCLID uses sparsity-promoting regularization to select parsimonious material models while balancing accuracy against simplicity. An automated hyperparameter procedure identifies a low-cost, low-complexity solution and removes negligible model terms during postprocessing.
- Regularization and sparsity: Sparsity-promoting regularization produces parameter vectors with many zero entries and therefore simpler thermodynamic potentials.The regularization term is large for dense vectors and small for sparse vectors.
- Regularization and sparsity: The weighting parameter λp controls the compromise between model accuracy and sparsity, while p determines the regularization curvature.The method chooses p = 1 to limit computational complexity and uses convex L1 regularization.
- Regularization and sparsity: The regularization sum is treated as dimensionless using fixed units because combining parameters with different units is physically unmotivated.The weighting factor λp is therefore assigned units of kN2 under the adopted convention.
- Hyperparameter selection: The automated procedure solves the optimization for successively larger λp values and selects a sparse solution among those below a prescribed cost threshold.The study considers 24 values, λp ∈ {10^-4 · 2^j kN2 : j = 0, . . . , 23}.
- Hyperparameter selection: Across cases, automated selection yields a solution with both low cost and low regularization, balancing model accuracy and sparsity.After selection, parameters below θth = 10^-4 are reset to their minimal values, removing unnecessary potential terms and supporting material classification.
4. Numerical benchmarks
Synthetic finite-element benchmarks tested EUCLID across five viscoelastic, elastoplastic, and viscoplastic material combinations under heterogeneous loading and artificial noise. EUCLID generally identified the hidden material class, calibrated parameters closely, and reproduced responses under unseen loading paths.
- Benchmark design: Five benchmark models combined elasticity, viscosity, plasticity, and isotropic or kinematic hardening in different configurations.The listed VEEP, EVP, and VEVP models included mixed rate-dependent and hardening mechanisms.
- Benchmark design: A square specimen with two elliptic holes and varied loading rates generated heterogeneous strain fields for two-dimensional plane-strain simulations.Displacement-controlled tension and compression were applied, with the loading parameter and rates varied during data generation.
- Discovered models: EUCLID automatically discriminated elastic, viscoelastic, plastic, and viscoplastic behavior, including isotropic versus kinematic hardening, from a single test.It failed to recover the true class only for the VEEP model at σ = 0.3 µm, while still producing a parsimonious model.
- Validation: Parameter agreement was very good, although discrepancies increased with noise, and true and discovered responses agreed well under uniaxial tension and simple shear.The response discrepancies likewise increased as the noise level increased.
5. Conclusions
The paper extends EUCLID to materials from unknown constitutive classes by discovering generalized-standard-material potentials from displacement and reaction-force data. Convexity, momentum balance, and sparse regression support physically consistent, parsimonious models, while broader libraries and experimental validation remain future goals.
- Contributions: EUCLID discovers the Helmholtz free energy and dissipation potential for materials belonging to unknown constitutive classes.These two scalar thermodynamic potentials completely define generalized standard materials.
- Contributions: Convexity guarantees stability and thermodynamic consistency by construction, while balance of linear momentum replaces unavailable stress-strain labels in the inverse problem.Momentum balance is enforced locally in the interior and globally at constrained domain sides.
- Contributions: Sparse regression selects a parsimonious set of model features and corresponding internal variables, with an automatic procedure choosing the sparsity-weight hyperparameter.The procedure targets a user-defined balance between model accuracy and simplicity.
- Future scope: The current model library does not yet cover phenomena such as anisotropy, heterogeneity, softening, or coupled thermo- and electromechanical effects.The authors identify these extensions as directions for future generalization.
Appendix A. Solving the constitutive equations
The constitutive equations are solved in strain control using an implicit-Euler time discretization and a viscoelastic-predictor/viscoplastic-corrector algorithm.
- Solution procedure: Given previous state variables and the current strain, the solver first computes internal variables and then the current stress.The update uses Equations (19), (22), and (16).
- Solution procedure: Implicit Euler discretization converts the constitutive evolution equations into time-discrete equations for the update.The time derivative is approximated using the current and previous time steps.
- Solution procedure: A viscoelastic predictor is followed by a viscoplastic corrector when the trial state violates admissibility.The predictor/corrector scheme is used to solve the discretized constitutive equations.
Appendix A.1. Viscoelastic predictor step
The viscoelastic predictor assumes no viscoplastic evolution, updates viscoelastic internal variables, computes trial stress, and tests admissibility before invoking correction.
- Predictor step: During the trial step, the viscoelastic assumption keeps viscoplastic internal variables unchanged when f_t,trial ≤ 0.Only the viscoelastic internal variables are updated under this assumption.
- Predictor step: The viscoelastic update is split into deviatoric and volumetric contributions to obtain explicit internal-variable expressions.This splitting supplies the update formulas used in the predictor.
- Predictor step: The current trial stress is computed from the updated internal variables before admissibility is checked.The resulting stress enters the evaluation of f_t,trial.
- Predictor step: If f_t,trial > 0, the viscoelastic predictor is inadmissible and the viscoplastic corrector is applied.The yield or admissibility test determines whether correction is necessary.
Appendix A.2. Viscoplastic corrector step
The viscoplastic corrector step introduces a scalar internal variable to simplify computation of the internal-variable updates. Once this scalar increment is computed, the remaining internal variables and stress follow from the derived update relations.
- A scalar internal variable γ is introduced to take advantage of characteristics of the viscoplastic corrector problem.
- The increment ∆γ is computed explicitly from variables at the previous time step, after which the other viscoplastic updates are obtained from ∆γ.The resulting relations provide a computationally efficient implementation of the viscoplastic corrector.
- After computing ∆γ, the remaining internal variables and the stress can be computed.
Appendix B. Consistent tangent
The consistent tangent is implemented for forward finite element simulations, with separate treatment depending on whether the viscoelastic predictor is admissible. It is required for the forward problem but not for EUCLID’s inverse problem.
- The material model library requires computation of the consistent tangent for forward finite element simulations.
- The consistent-tangent derivation considers two cases, beginning with an admissible viscoelastic predictor.
- The consistent tangent is needed for solving the forward finite element problem, but not for the EUCLID inverse problem.
Appendix C. Numerical settings
Table C.3 lists the parameters and hyperparameters used in both the finite element simulations and the EUCLID inverse discovery process.
- The numerical settings include parameters used during finite element data generation.
- The listed settings also include hyperparameters for the EUCLID inverse discovery process.
- Table C.3 consolidates the settings for simulation and inverse discovery.