A deep material network for multiscale topology learning and accelerated nonlinear modeling of heterogeneous materials

Abstract

In this paper, a new data-driven multiscale material modeling method, which we refer to as deep material network, is developed based on mechanistic homogenization theory of representative volume element (RVE) and advanced machine learning techniques. We propose to use a collection of connected mechanistic building blocks with analytical homogenization solutions which avoids the loss of essential physics in generic neural networks, and this concept is demonstrated for 2-dimensional RVE problems and network depth up to 7. Based on linear elastic RVE data from offline direct numerical simulations, the material network can be effectively trained using stochastic gradient descent with backpropagation algorithm, further enhanced by model compression methods. Importantly, the trained network is valid for any local material laws without the need for additional calibration or micromechanics assumption. Its extrapolations to unknown material and loading spaces for a wide range of problems are validated through numerical experiments, including linear elasticity with high contrast of phase properties, nonlinear history-dependent plasticity and finitestrain hyperelasticity under large deformations.By discovering a proper topological representation of RVE with fewer degrees of freedom, this intelligent material model is believed to open new possibilities of high-fidelity efficient concurrent simulations for a largescale heterogeneous structure. It also provides a mechanistic understanding of structure-property relations across material length scales and enables the development of parameterized microstructural database for material design and manufacturing. cost and accuracy. Analytical micromechanics methods [9,10,1,11,12,13] can be regarded as one type of reduced-order models with high efficiency. However, due to a loss of detailed physics in the microscale, they normally lose accuracy or require extensive model calibrations when irregular complex morphologies, nonlinear history-dependent properties or large deformations are presented. For heterogeneous hyperelastic materials, manifold-learning methods like isomap are used for nonlinear dimensionality reduction of microscopic strain fields [14]. The model reduction of history-dependent plastic materials can be more complex and challenging. Two examples are non-uniform transformation field analysis (NTFA) [15,16] and variants of the principle component analysis [17] or proper orthogonal decomposition (POD) [18,19,20]. However, they usually require extensive a priori simulations for interpolating nonlinear responses, and their extrapolation capability for new material inputs is usually limited, Recently, the self-consistent clustering analysis (SCA) [21,22] has demonstrated a powerful trade-off between accuracy and efficiency in predicting smallstrain elasto-plastic behavior though clustering techniques, and it only requires linear elastic simulations in the offline stage.Meanwhile, current advanced machine learning models (e.g. artificial neural networks and deep learning) hav...

Exploring the 3D architectures of deep material network in data-driven multiscale mechanics

“…Cost function. A cost function based on the mean square error (MSE) and a regularization term [1] is formulated to characterize how close is the model predictionC rve to the training referenceC dns : to remove the scaling effect. The operator ||...|| denotes the Frobenius matrix norm.…”

“…) (α) = r v (α); (A.1) Y (2,2) = 1, Y ([1,3,5],[1,3,5]) (β) = r p (−β), Y ([4,6],[4,6]) (β) = r v (−β); Z (3,3) = 1, Z ([1,2,6],[1,2,6]) (γ) = r p (γ), Z ([4,5],[4,5]) (γ) = r v (γ).The in-plane and output-plane rotation matrices r p and r v for an arbitrary angle θ are defined in Mandel notation asr p (θ) =   − sin 2 θ    , r v (θ) = cos θ − sin θ sin θ cos θ . (A.2)In finite-strain formulation, the elementary rotation matrices shown in Eq.…”

“…) (β) = r vf (−β); Z f (3,3) = 1, Z f ([1,2,8,9],[1,2,8,9]) (γ) = r pf (γ), Z f ([4,6],[4,6]) (γ) = Z f ([5,7],[5,7]) (γ) = r vf (γ),where the in-plane and output-plane rotation matrices are r pf (θ) = − sin θ cos θ − sin θ cos θ − sin θ cos θ sin θ cos θ cos 2 θ − sin 2 θ − sin θ cos θ sin θ cos θ −…”

A hybrid approach to simulate the homogenized irreversible elastic–plastic deformations and damage of foams by neural networks

“…The training itself may be computationally expensive, but it can be done offline, whereas the required online computation of the response function is not more expensive than conventional phenomenological models. Either experimental data can be used for training [1,13,19,45] or RVE simulations [11,15,23,27,28,34,43].…”

Stochastic Mechanical Characterization of Polysilicon MEMS: A Deep Learning Approach

“…As for the adoption of deep learning for data assimilation and, specifically, for assessing the effective properties of micro-structured materials, interesting results were recently discussed in [25,26]. Here, we propose a different approach in two distinctive directions: The NN is not trained to perfectly reproduce the results in terms of overall elastic properties of the film for any stochastic representation of it-termed the statistical volume element (SVE)-but instead to catch the statistical distributions of the mentioned properties; the NN is trained with a procedure similar to those adopted for image recognition, hence by handling a pool of pictures of the morphology of the film only.…”