Machine learning-accelerated time integration of plasticity models

Finite element simulations of structures with nonlinear material behavior require advanced material models to provide accurate predictions. However, the computational costs of these models can be high, as they solve coupled differential algebraic equations at each integration point, in each equilibrium iteration, in every time step. In this study, we propose a machine learning-based framework to accelerate these computations by explicitly calculating the state variable updates with neural networks, enabling large time steps with low computational costs. The neural networks operate on invariants, and the necessary and sufficient evolution directions are determined analytically based on training data. Furthermore, the proposed framework enforces exact fulfillment of the plastic consistency condition. To evaluate the proposed framework, two prototype material models based on the von Mises yield criterion and different nonlinear kinematic hardening laws are chosen. Only 10 cycles of two-dimensional multiaxial proportional loading are used to generate the training and validation data, while 100 cycles of three-dimensional loading are used as test data. After evaluating the proposed framework in material point simulations, we incorporate it into finite element simulations to assess its accuracy and computational efficiency in a boundary value problem. The results from both material point and finite element simulations show very promising numerical performance of the neural network-based time integrator. It achieves high accuracy and numerical stability, generalizes from two-dimensional training data to three-dimensional cyclic loading over longer loading histories, and provides a noticeable reduction in computational time for a single strain increment per load segment.