We present a machine learning approach that integrates geometric deep learning and Sobolev training to generate a family of finite strain anisotropic hyperelastic models that predict the homogenized ...responses of polycrystals previously unseen during the training. While hand-crafted hyperelasticity models often incorporate homogenized measures of microstructural attributes, such as the porosity or the averaged orientation of constituents, these measures may not adequately represent the topological structures of the attributes. We fill this knowledge gap by introducing the concept of the weighted graph as a new high-dimensional descriptor that represents topological information, such as the connectivity of anisotropic grains in an assemble. By leveraging a graph convolutional deep neural network in a hybrid machine learning architecture previously used in Frankel et al. (2019), the artificial intelligence extracts low-dimensional features from the weighted graphs and subsequently learns the influence of these low-dimensional features on the resultant stored elastic energy functionals. To ensure smoothness and prevent unintentionally generating a non-convex stored energy functional, we adopt the Sobolev training method for neural networks such that a stress measure is obtained implicitly by taking directional derivatives of the trained energy functional. Results from numerical experiments suggest that Sobolev training is capable of generating a hyperelastic energy functional that predicts both the elastic energy and stress measures more accurately than the classical training that minimizes L2 norms. Verification exercises against unseen benchmark FFT simulations and phase-field fracture simulations that employ the geometric learning generated elastic energy functional are conducted to demonstrate the quality of the predictions.
We introduce a deep learning framework designed to train smoothed elastoplasticity models with interpretable components, such as the stored elastic energy function, yield surface, and plastic flow ...that evolve based on a set of deep neural network predictions. By recasting the yield function as an evolving level set, we introduce a deep learning approach to deduce the solutions of the Hamilton–Jacobi equation that governs the hardening/softening mechanism. This machine learning hardening law may recover any classical hand-crafted hardening rules and discover new mechanisms that are either unbeknownst or difficult to express with mathematical expressions. Leveraging Sobolev training to gain control over the derivatives of the learned functions, the resultant machine learning elastoplasticity models are thermodynamically consistent, interpretable, while exhibiting 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 those obtained from black-box deep neural network models such as the recurrent neural network, the 1D convolutional neural network, and the multi-step feed-forward model.
For material modeling and discovery, synthetic microstructures play a critical role as digital twins. They provide stochastic samples upon which direct numerical simulations can be conducted to ...populate material databases. A large ensemble of simulation data on synthetic microstructures may provide supplemental data to inform and refine macroscopic material models, which might not be feasible from physical experiments alone. However, synthesizing realistic microstructures with realistic microstructural attributes is highly challenging. Thus, it is often oversimplified via rough approximations that may yield an inaccurate representation of the physical world. Here, we propose a novel deep learning method that can synthesize realistic three-dimensional microstructures with controlled structural properties using the combination of generative adversarial networks (GAN) and actor-critic (AC) reinforcement learning. The GAN-AC combination enables the generation of microstructures that not only resemble the appearances of real specimens but also yield user-defined physical quantities of interest (QoI). Our validation experiments confirm that the properties of synthetic microstructures generated by the GAN-AC framework are within a 5% error margin with respect to the target values. The scientific contribution of this paper resides in the novel design of the GAN-AC microstructure generator and the mathematical and algorithmic foundations therein. The proposed method will have a broad and substantive impact on the materials community by providing lenses for analyzing structure-property-performance linkages and for implementing the notion of 'materials-by-design'.
We present a machine learning framework to train and validate neural networks to predict the anisotropic elastic response of a monoclinic organic molecular crystal known as β$$ \beta $$‐HMX in the ...geometrical nonlinear regime. A filtered molecular dynamic (MD) simulations database is used to train neural networks with a Sobolev norm that uses the stress measure and a reference configuration to deduce the elastic stored free energy functional. To improve the accuracy of the elasticity tangent predictions originating from the learned stored free energy, a transfer learning technique is used to introduce additional tangential constraints from the data while necessary conditions (e.g., strong ellipticity, crystallographic symmetry) for the correctness of the model are either introduced as additional physical constraints or incorporated in the validation tests. Assessment of the neural networks is based on (1) the accuracy with which they reproduce the bottom‐line constitutive responses predicted by MD, (2) the robustness of the models measured by detailed examination of their stability and uniqueness, and (3) the admissibility of the predicted responses with respect to mechanics principles in the finite‐deformation regime. We compare the training efficiency of the neural networks under different Sobolev constraints and assess the accuracy and robustness of the models against MD benchmarks for β$$ \beta $$‐HMX.
Featured Cover Vlassis, Nikolaos N.; Zhao, Puhan; Ma, Ran ...
International journal for numerical methods in engineering,
09/2022, Letnik:
123, Številka:
17
Journal Article
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The cover image is based on the Original Article Molecular dynamics inferred transfer learning models for finite‐strain hyperelasticity of monoclinic crystals: Sobolev training and validations ...against physical constraints by Nikolaos Vlassis et al., https://doi.org/10.1002/nme.6992.
This paper presents a computational framework that generates ensemble predictive mechanics models with uncertainty quantification (UQ). We first develop a causal discovery algorithm to infer causal ...relations among time-history data measured during each representative volume element (RVE) simulation through a directed acyclic graph. With multiple plausible sets of causal relationships estimated from multiple RVE simulations, the predictions are propagated in the derived causal graph while using a deep neural network equipped with dropout layers as a Bayesian approximation for UQ. We select two representative numerical examples (traction-separation laws for frictional interfaces, elastoplasticity models for granular assembles) to examine the accuracy and robustness of the proposed causal discovery method for the common material law predictions in civil engineering applications.
Graphic abstract
Experimental data are often costly to obtain, which makes it difficult to calibrate complex models. For many models an experimental design that produces the best calibration given a limited ...experimental budget is not obvious. This paper introduces a deep reinforcement learning (RL) algorithm for design of experiments that maximizes the information gain measured by Kullback–Leibler divergence obtained via the Kalman filter (KF). This combination enables experimental design for rapid online experiments where manual trial-and-error is not feasible in the high-dimensional parametric design space. We formulate possible configurations of experiments as a decision tree and a Markov decision process, where a finite choice of actions is available at each incremental step. Once an action is taken, a variety of measurements are used to update the state of the experiment. This new data leads to a Bayesian update of the parameters by the KF, which is used to enhance the state representation. In contrast to the Nash–Sutcliffe efficiency index, which requires additional sampling to test hypotheses for forward predictions, the KF can lower the cost of experiments by directly estimating the values of new data acquired through additional actions. In this work our applications focus on mechanical testing of materials. Numerical experiments with complex, history-dependent models are used to verify the implementation and benchmark the performance of the RL-designed experiments.
Abstract Background Lipoprotein-associated Phospholipase A2 (Lp-PLA2), has a powerful inflammatory and atherogenic action in the vascular wall and is an independent marker of poor prognosis in ...coronary artery disease (CAD). We investigate the association of Lp-PLA2 with markers of vascular dysfunction and atherosclerosis with proven prognostic value in CAD. Methods In 111 patients with angiographically documented chronic CAD, we measured 1) carotid intima-media thickness (CIMT), 2) reactive hyperemia using fingertip peripheral arterial tonometry (RH-PAT), 3) coronary flow reserve (CFR), by Doppler echocardiography 4) pulse wave velocity (PWV) and 5) blood levels of Lp-PLA2. Results Patients with Lp-PLA2 concentration >234.5 ng/ml (50th percentile) had higher CIMT (1.44 ± 0.07 vs. 1.06 ± 0.06 mm), PWV (11.0 ± 2.36 vs. 9.7 ± 2.38 m/s) and lower RH-PAT(1.24 ± 0.25 vs. 1.51 ± 0.53) and CFR (2.39 ± 0.75 vs. 2.9 ± 0.86) compared to those with lower Lp-PLA ( p < 0.05 for all comparisons). Lp-PLA2 was positively associated with CIMT (regression coefficient b: 0.30 per unit of Lp-PLA2, p = 0.02), PWV (b:0.201, p = 0.04) and inversely with RHI-PAT (b: −0.371, p < 0.001) and CFR (b:−0.32, p = 0.002). In multivariate analysis, Lp-PLA2 was an independent determinant of RHI-PAT, CFR, CIMT and PWV in a model including age, sex, smoking, diabetes, dyslipidemia and hypertension ( p < 0.05 for all vascular markers). Lp-PLA2, RHI-PAT and CFR were independent predictors of cardiac events during a 3-year follow-up. Conclusions Elevated Lp-PLA2 concentration is related with endothelial dysfunction, carotid atherosclerosis, impaired coronary flow reserve and increased arterial stiffness and adverse outcome in CAD patients. These findings suggest that the prognostic role of Lp-PLA2 in chronic CAD may be explained by a generalized detrimental effect of this lipase on endothelial function and arterial wall properties.
We introduce a denoising diffusion algorithm to discover microstructures with nonlinear fine-tuned properties. Denoising diffusion probabilistic models are generative models that use diffusion-based ...dynamics to gradually denoise images and generate realistic synthetic samples. By learning the reverse of a Markov diffusion process, we design an artificial intelligence to efficiently manipulate the topology of microstructures to generate a massive number of prototypes that exhibit constitutive responses sufficiently close to designated nonlinear constitutive behaviors. To identify the subset of microcstructures with sufficiently precise fine-tuned properties, a convolutional neural network surrogate is trained to replace high-fidelity finite element simulations to filter out prototypes outside the admissible range. Results of this study indicate that the denoising diffusion process is capable of creating microstructures of fine-tuned nonlinear material properties within the latent space of the training data. More importantly, this denoising diffusion algorithm can be easily extended to incorporate additional topological and geometric modifications by introducing high-dimensional structures embedded in the latent space. Numerical experiments are conducted on the open-source mechanical MNIST data set (Lejeune, 2020). Consequently, this algorithm is not only capable of performing inverse design of nonlinear effective media, but also learns the nonlinear structure–property map to quantitatively understand the multiscale interplay among the geometry, topology, and their effective macroscopic properties.
The history-dependent behaviors of classical plasticity models are often driven by internal variables evolved according to phenomenological laws. The difficulty to interpret how these internal ...variables represent a history of deformation, the lack of direct measurement of these internal variables for calibration and validation, and the weak physical underpinning of those phenomenological laws have long been criticized as barriers to creating realistic models. In this work, geometric machine learning on graph data (e.g. finite element solutions) is used as a means to establish a connection between nonlinear dimensional reduction techniques and plasticity models. Geometric learning-based encoding on graphs allows the embedding of rich time-history data onto a low-dimensional Euclidean space such that the evolution of plastic deformation can be predicted in the embedded feature space. A corresponding decoder can then convert these low-dimensional internal variables back into a weighted graph such that the dominating topological features of plastic deformation can be observed and analyzed.
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