Learning constitutive relations using symmetric positive definite neural networks

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

doi:10.1016/j.jcp.2020.110072

Cited by 103 publications

(51 citation statements)

References 38 publications

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“…as well as open-cell topologies. Finally, instead of training the symmetric part of the stiffness tensor, one can try to predict the Cholesky factor of a tangent stiffness matrix to impose a weak convexity on the strain energy function [91]. The latter point is specifically interesting when it comes to non-linear material behavior.…”

Section: Discussionmentioning

confidence: 99%

Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence

Mianroodi¹,

Rezaei²,

Siboni³

et al. 2021

Preprint

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The elastic properties of materials derive from their electronic and atomic nature.However, simulating bulk materials fully at these scales is not feasible, so that typically homogenized continuum descriptions are used instead. A seamless and lossless transition of the constitutive description of the elastic response of materials between these two scales has been so far elusive. Here we show how this problem can be overcome by using Artificial Intelligence (AI). A Convolutional Neural Network (CNN) model is trained, by taking the structure image of a nanoporous material as input and the corresponding elasticity tensor, calculated from Molecular Statics (MS), as output. Trained with the atomistic data, the CNN model captures the size-and poredependency of the material's elastic properties which, on the physics side, can stem from surfaces and non-local effects. Such effects are often ignored in upscaling from atomistic to classical continuum theory. To demonstrate the accuracy and the efficiency of the trained CNN model, a Finite Element Method (FEM) based result of an elastically deformed nanoporous beam equipped with the CNN as constitutive law is compared with that by a full atomistic simulation. The good agreement between the atomistic simulations and the FEM-AI combination for a system with size and surface effects establishes a new lossless scale bridging approach to such problems. The trained CNN model deviates from the atomistic result by 9.6% for porosity scenarios of up to 90% but it is about 230 times faster than the MS calculation and does not require to change simulation methods between different scales. The efficiency of the CNN evaluation together with the preservation of important atomistic effects makes the trained model an effective atomistically-informed constitutive model for macroscopic simulations of nanoporous materials, optimization of nano-structures, and solving of inverse problems.

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Section: Discussionmentioning

confidence: 99%

Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence

Mianroodi¹,

Rezaei²,

Siboni³

et al. 2021

Preprint

View full text Add to dashboard Cite

show abstract

“…The development of machine learning techniques in recent years, especially deep learning, has provided new tools to extract complicated relationships from massive data in an efficient and flexible way, thus a new possibility to data-driven constitutive modeling. In light of such strengths and promises, many machine learning-based constitutive models have been developed in the past few years in a wide range of application areas such as turbulence modeling [1][2][3][4][5][6] and computational mechanics [7][8][9][10][11][12] in general.…”

Section: A Invariance Properties Of Constitutive Modelsmentioning

confidence: 99%

VCNN-e: A vector-cloud neural network with equivariance for emulating Reynolds stress transport equations

Han¹,

Zhou²,

Xiao³

2022

Preprint

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Developing robust constitutive models is fundamental and a longstanding problem for accelerating the simulation of complicated physics. Machine learning provides promising tools to construct constitutive models based on various calibration data. In this work, we propose a new approach to emulate constitutive tensor transport equations for tensorial quantities through a vector-cloud neural network with equivariance (VCNN-e). The VCNN-e respects all the invariance properties desired by constitutive models and faithfully reflects the region of influence in physics. By design, the model guarantees that the predicted tensor is invariant to the frame translation and ordering (permutation) of the neighboring points. Furthermore, it is equivariant to the frame rotation, i.e., the output tensor co-rotates with the coordinate frame. We demonstrate its performance on Reynolds stress transport equations, showing that the VCNN-e can effectively emulate the Reynolds stress transport model for Reynolds-averaged Navier-Stokes (RANS) equations. Such a priori evaluations of the proposed network pave the way for developing and calibrating robust and nonlocal, non-equilibrium closure models for the RANS equations.

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“…Starting from the influential works of Wu and Ghaboussi (1990); Ghaboussi et al (1990Ghaboussi et al ( , 1991 for concrete in a small-strain biaxial state of stress, these tools have been employed for different material models with increasing complexity over the years (Lefik and Schrefler, 2003;Jung and Ghaboussi, 2006;Huang et al, 2020;Fuhg et al, 2021a;Lu et al, 2021;Logarzo et al, 2021). Recently, in the hope of needing less data and in order to generate models with higher generalization capability, efforts have been made to train data-driven constitutive models that do not only train with raw stress-strain data but incorporate additional physics-based restrictions to the trained model (Liu and Wu, 2019;Heider et al, 2020;Xu et al, 2021;Linka et al, 2021). When dealing with hyperelastic materials, where no rate-dependence is considered, these models try to include some of the following physics-informed principles:…”

Section: Introductionmentioning

confidence: 99%

“…The idea behind physics-informed or physics-guided data-driven constitutive models is that the trained surrogate should abide to these conditions and not rely solely on raw data. Additionally, a large majority of the proposed works in the literature for physics-guided constitutive models are based on artificial neural networks (ANNs) (Liu and Wu, 2019;Heider et al, 2020;Xu et al, 2021;Masi et al, 2021). For general information about ANNs we refer to Goodfellow et al (2016).…”

Section: Introductionmentioning

confidence: 99%

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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Learning constitutive relations using symmetric positive definite neural networks

Cited by 103 publications

References 38 publications

Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence

Lossless Multi-Scale Constitutive Elastic Relations with Artificial Intelligence

VCNN-e: A vector-cloud neural network with equivariance for emulating Reynolds stress transport equations

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

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