Tensor Basis Gaussian Process Models of Hyperelastic Materials

Frankel, Ari; Jones, Reese E.; Swiler, Laura Painton

doi:10.1615/jmachlearnmodelcomput.2020033325

Cited by 42 publications

(35 citation statements)

References 25 publications

(29 reference statements)

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“…For the isotropic case we follow we the approach proposed by Frankel et al (2020). The isotropic case is fully defined by the 3 invariants…”

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

“…On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling A PREPRINT which following Frankel et al (2020) allows to define an equation system for the unknown scalar values of the form…”

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

“…The matrix might be severely ill-conditioned depending on the number of unique principal strains. Frankel et al (2020) describe an algorithmic way to avoid this problem.…”

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

“…One solution for probably the most significant problem associated with GPR in the big data regime has recently been proposed by the authors in Fuhg et al (2021c) where local approximate GPR (laGPR) has been presented for constitutive modeling applications and corresponding multiscale calculations. Frankel et al (2020) presented the only work so far where GPR and physics-guided data-driven constitutive modeling is attempted. They propose an approach for isotropic hyperelastic materials which is generally based on building a surrogate model that maps from the space of invariants of the right Cauchy-Green deformation tensor to the space of coefficients linked with the stress generators.…”

Section: Introductionmentioning

confidence: 99%

“…They propose an approach for isotropic hyperelastic materials which is generally based on building a surrogate model that maps from the space of invariants of the right Cauchy-Green deformation tensor to the space of coefficients linked with the stress generators. In this work we utilize the approach discussed in Frankel et al (2020) as a starting point. We generalize it, extend it to anisotropic materials and by employing laGPR we allow the framework to be applicable in the big data context.…”

Section: Introductionmentioning

confidence: 99%

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On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Fuhg,

Bouklas

2021

Preprint

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Data-driven constitutive modeling is an emerging field in computational solid mechanics with the prospect of significantly relieving the computational costs of hierarchical computational methods. Additionally, this data-driven paradigm could enable a seamless connection of experimental data probing material responses with numerical simulations at the structural level. Traditionally, these surrogates have just been trained using datasets which map strain inputs to stress outputs directly. Data-driven constitutive models for elastic and inelastic materials have commonly been developed based on artificial neural networks (ANNs), which recently enabled the incorporation of physical laws in the construction of these models. However, ANNs do not offer convergence guarantees from an engineering point of view and are majorly reliant on user-specified parameters. In contrast to ANNs, Gaussian process regression (GPR) is based on nonparametric modeling principles as well as on fundamental statistical knowledge and hence allows for strict convergence guarantees. GPR however has the major disadvantage that it scales poorly as datasets get large. In this work we present a physics-informed data-driven constitutive modeling approach for isostropic and anisotropic materials at finite strain based on probabilistic machine learning that can be used in the big data context. This generalized approach is based on rewriting the stress output as a linear combination of an irreducible integrity basis. The trained GPR surrogates are able to respect physical principles such as material frame indifference, material symmetry, thermodynamic consistency, stress-free undeformed configuration, and the local balance of angular momentum. Furthermore, this paper presents the first sampling approach that directly generates space-filling points in the invariant space corresponding to a bounded domain of the gradient deformation tensor. The sampling technique is based on simulated annealing and provides more efficient and reliable physics-informed constitutive models. Overall, the presented approach is tested on synthetic data from isotropic and anisotropic constitutive laws and shows surprising accuracy even far beyond the limits of the training domain, indicating that the resulting surrogates can efficiently generalize as they incorporate knowledge about the underlying physics.

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“…For the isotropic case we follow we the approach proposed by Frankel et al (2020). The isotropic case is fully defined by the 3 invariants…”

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

“…The matrix might be severely ill-conditioned depending on the number of unique principal strains. Frankel et al (2020) describe an algorithmic way to avoid this problem.…”

Section: Physics-informed Mapping Approach For Isotropic Materialsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Fuhg,

Bouklas

2021

Preprint

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Modeling and Design of Zero‐Stiffness Elastomer Springs Using Machine Learning

Kim

Tawfick

King

2022

Advanced Intelligent Systems

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Architected metamaterials exhibit complex stress–strain responses, such as quasi‐zero stiffness (QZS) plateau response useful in vibration and energy absorption, by exploiting elastic instabilities such as buckling. However, the design of metamaterials with prescribed nonlinear mechanical properties is challenging due to the difficulty in accurately predicting nonlinear mechanical responses such as buckling and self‐contact, as well as geometric and material uncertainties. Herein, additive manufacturing (AM) with machine learning (ML) to demonstrate data‐driven modeling and the design of nonlinear elastomeric springs with a broad stress plateau is combined. 243 elastomer chi (χ) spring parts with different design parameters fabricated by AM are tested. A Gaussian process (GP) model is trained on key features of the measured mechanical response and predicts the mechanical response within a 3% error with as few as 97 parts. To account for the geometric and material uncertainties, the GP model to develop an uncertainty‐aware design optimization approach to tailor the mechanical response based on the covariance matrix adaptation evolution strategy (CMA‐ES) is utilized. Excellent agreement is demonstrated between the designed response and measured response over the range of plateau stress considered. The research illustrates the promise of experimental data‐driven approaches and AM in enabling complex material design.

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Meta‐modeling of fractional constitutive relationships for rocks based on physics‐induced machine learning

Zhang

Zhu

2023

Num Anal Meth Geomechanics

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A fractional constitutive meta-model for the mechanical behavior of rocks is proposed to bypass complex integration algorithms and consider the uncertainty of model/material parameters. First, a physics-induced database is constructed with the aid of a fractional constitutive model and two developed input-output strategies. Then three machine algorithms (i.e., random forests, extreme gradient boosting, and multilayer perceptron) together with two input-output strategies are employed to formulate six different meta-models. The grid-search and fivefold cross-validation methods are utilized to optimize key hyperparameters of meta-models. Through the comparisons of evaluation metrics and prediction results, the multilayer perceptron algorithm with strategy II becomes an optimal combination to construct the meta-model. The results of the feature's importance and sensitivity analyses indicate that the parameters' influences in this data-driven paradigm are in line with physical reality. The effectiveness of the proposed meta-model is validated by comparing the predicted results with experimental observations from the literature. Additionally, the uncertainty analyses from the meta-model and material/model parameters can be conducted based on a linear regression model and the Monte Carlo simulation. The study results show that the presented meta-model based on physics-guided synthetic data has the potential to replace the general fractional constitutive model.

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Tensor Basis Gaussian Process Models of Hyperelastic Materials

Cited by 42 publications

References 25 publications

On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

On physics-informed data-driven isotropic and anisotropic constitutive models through probabilistic machine learning and space-filling sampling

Modeling and Design of Zero‐Stiffness Elastomer Springs Using Machine Learning

Meta‐modeling of fractional constitutive relationships for rocks based on physics‐induced machine learning

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