论文标题
使用对称积极确定的神经网络学习本构关系
Learning Constitutive Relations using Symmetric Positive Definite Neural Networks
论文作者
论文摘要
我们介绍了用于对动态方程中组成关系建模的Cholesky因素对称阳性确定的神经网络（SPD-NN）。 SPD-NN没有直接预测应力，而是训练神经网络，以预测切线刚度矩阵的潮汐因子，以增量形式计算应力。由于特殊的结构，SPD-NN弱对应变能函数施加了凸度，满足路径依赖性材料的时间一致性，因此提高了数值稳定性，尤其是当在有限元模拟中使用SPD-NN时。根据可用数据的类型，我们提出了两种训练方法，即对应变和压力对的直接训练以及用于负载和位移对的间接训练。我们证明了SPD-NN对来自固体力学的超弹性，弹性和多尺度纤维增强板问题的有效性。 SPD-NN的一般性和鲁棒性使其成为广泛构造建模应用程序的有前途的工具。
We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations. Instead of directly predicting the stress, the SPD-NN trains a neural network to predict the Cholesky factor of a tangent stiffness matrix, based on which the stress is calculated in the incremental form. As a result of the special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of the SPD-NN make it a promising tool for a wide range of constitutive modeling applications.
