Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity
Geometric deep learning for computational mechanics Part I: Anisotropic Hyperelasticity
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DOI：
10.1016/j.cma.2020.113299
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发表时间：
2020-01
期刊：
影响因子：
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通讯作者：
Nikolaos N. Vlassis;R. Ma;WaiChing Sun
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作者：
Nikolaos N. Vlassis;R. Ma;WaiChing Sun
我们提出了一种机器学习方法，该方法集成了几何深度学习和Sobolev训练，以生成一系列有限应变各向异性超弹性模型，这些模型可以预测以前在训练过程中看不到的多晶体的均匀响应。虽然手工制作的超弹性模型通常包含微观结构属性的均匀度量，例如孔隙度或成分的平均取向，但这些度量可能不能充分代表属性的拓扑结构。我们通过引入加权图的概念来填补这一知识空白，加权图作为一种新的高维描述符来表示拓扑信息，例如组合体中各向异性颗粒的连通性。通过利用Frankel等人（2019）先前使用的混合机器学习架构中的图卷积深度神经网络，人工智能从加权图中提取低维特征，随后学习这些低维特征对所得存储的弹性能量泛函的影响。为了确保平滑性并防止无意中产生非凸存储能量泛函，我们对神经网络采用Sobolev训练方法，通过对训练的能量泛函进行方向导数隐式地获得应力测量。数值实验结果表明，Sobolev训练能够生成超弹性能量泛函数，该泛函数比最小化l2范数的经典训练更准确地预测弹性能量和应力测量。利用几何学习生成的弹性能量函数，对未见基准FFT模拟和相场裂缝模拟进行验证，以证明预测的质量。
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 L 2 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.