A Neural Network Constitutive Model and Automatic Stiffness Evaluation for Multiscale Finite Elements

A neural network model for a constitutive law in nonlinear structures is proposed in this paper. The artificial neural network (ANN) model is constructed based on a data set of responses from representative volume elements, which was calculated by finite elements and using an open scientific software machine learning platform. The tangential stiffness matrix, which can be used within a multiscale finite element analysis, is calculated via the method of automatic differentiation. Two types of constitutive neural networks are proposed. The first approach involves training a residual model with three respective surrogate models for the stress components, ensuring less computing cost. The second approach considers a separate ANN for each stress component, ensuring a high rate of convergence. The numerical results are compared with the given data set as well as with the results obtained after applying a polynomial regression. The loss function, without and including the Sobolev metrics, is considered. In addition, a physics-informed constitutive neural model, which enforces hyperelasticity principles, is also analyzed. The choice of the hyperparameters is discussed.