Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

Xu, Kailai; Huang, Daniel Z.; Darve, Eric

doi:10.48550/arxiv.2004.00265

Cited by 10 publications

(18 citation statements)

References 33 publications

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“…∂ n+1 = H 0, and The symmetric positive definiteness of the tangent stiffness matrix H ensures that the strain energy is weak convex. The weak convexity is beneficial for stabilizing both training and predictive modeling [41]. Equivalently, Equation ( 20) can be written in the following form…”

Section: Neural-network-based Viscous Constitutive Relationsmentioning

confidence: 99%

Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning

Xu,

Tartakovsky,

Burghardt

et al. 2020

Preprint

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We propose a novel approach to model viscoelasticity materials using neural networks, which capture rate-dependent and nonlinear constitutive relations. However, inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input-output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multi-physics interactions. We show that limited displacement data hold sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computationmodeling viscoelasticity properties from observed displacement data-as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and physics constrained learning. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

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Section: Neural-network-based Viscous Constitutive Relationsmentioning

confidence: 99%

Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning

Xu,

Tartakovsky,

Burghardt

et al. 2020

Preprint

Self Cite

View full text Add to dashboard Cite

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“…ANN models are widely used in approximating complex mappings between input and output as an universal approximation theorem [17], which allows to approximate any functions in a form-free manner. In addition to the universal function approximation, the ANN model also features several other advantages in constructing a constitutive model [18] such as better performance for unevenly distributed data, approximating a non-smooth function, and mapping a high dimensional input-output. The early work of ANN-based constitutive models were carried out by Ghaboussi and his co-workers [19][20][21][22][23][24].…”

Section: B Applicationsmentioning

confidence: 99%

“…In other words, the input of the ANN model and the ANN model itself need to be determined simultaneously during the training [28,42]. However, this approach brings another benefit that the physical constraints are implicitly imposed into the training process as the ANN model must go through the FE model to generate correct input data [18,30]. Some examples of using indirectly measurable data for training ANN models are given in Table 1.…”

Section: Coupling Ann With Mechanical Modelsmentioning

confidence: 99%

How machine learning can help the design and analysis of composite materials and structures?

Liu¹,

Tian²,

Tao³

et al. 2020

Preprint

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Machine learning models are increasingly used in many engineering fields thanks to the widespread digital data, growing computing power, and advanced algorithms. Artificial neural networks (ANN) is the most popular machine learning model in recent years. Although many ANN models have been used in the design and analysis of composite materials and structures, there are still some unsolved issues that hinder the acceptance of ANN models in the practical design and analysis of composite materials and structures. Moreover, the emerging machine learning techniques are posting new opportunities and challenges in the data-based design paradigm. This paper aims to give a state-of-the-art literature review of ANN models in the nonlinear constitutive modeling, multiscale surrogate modeling, and design optimization of composite materials and structures. This review has been designed to focus on the discussion of the general frameworks and benefits of ANN models to the above problems. Moreover, challenges and opportunities in each key problem are identified and discussed. This paper is expected to open the discussion of future research scope and new directions to enable efficient, robust, and accurate data-driven design and analysis of composite materials and structures.

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“…Recently, there have been attempts to rectify the limitations of machine learning models that do not distinguish or partition the elastic and plastic strain. Xu et al (2020), for instance, introduce a differentiable transition function to create a smooth transition between the elastic and plastic range for an incremental constitutive law generated from supervised learning. Mozaffar et al (2019) and later Zhang and Mohr (2020) inroduce the machine learning to deduce the yield function and enable linear and distortion hardening using loss functions that minimizing the L 2 norm of the yield function discrepancy.…”

Section: Why Sobolev Training For Plasticitymentioning

confidence: 99%

Sobolev training of thermodynamic-informed neural networks for smoothed elasto-plasticity models with level set hardening

Vlassis¹,

Sun²

2020

Preprint

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We introduce a deep learning framework designed to train smoothed elastoplasticity models with interpretable components, such as a smoothed stored elastic energy function, a yield surface, and a plastic flow that are evolved based on a set of deep neural network predictions. By recasting the yield function as an evolving level set, we introduce a machine learning approach to predict the solutions of the Hamilton-Jacobi equation that governs the hardening mechanism. This machine learning hardening law may recover classical hardening models and discover new mechanisms that are otherwise very difficult to anticipate and hand-craft. This treatment enables us to use supervised machine learning to generate models that are thermodynamically consistent, interpretable, but also exhibit excellent learning capacity. Using a 3D FFT solver to create a polycrystal database, numerical experiments are conducted and the implementations of each component of the models are individually verified. Our numerical experiments reveal that this new approach provides more robust and accurate forward predictions of cyclic stress paths than these obtained from black-box deep neural network models such as a recurrent GRU neural network, a 1D convolutional neural network, and a multi-step feedforward model.

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Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

Cited by 10 publications

References 33 publications

Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning

Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning

How machine learning can help the design and analysis of composite materials and structures?

Sobolev training of thermodynamic-informed neural networks for smoothed elasto-plasticity models with level set hardening

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