Arquitectura de redes neuronales para el estudio de localización de deformaciones en una barra unidimensional mediante cinemática de discontinuidades débiles

Abstract

This work utilizes PINN models to solve the problem of strain localization in a one-dimensional bar with softening through its variational formulation. The solid's energy is incorporated into the loss function as an optimization problem. Two explored neural network architectures represent the problem's kinematics termed ReLU PINN and Kinematic Decomposition. Strain Localization is represented through weak discontinuity kinematics. The Kinematic Decomposition architecture approximates the displacement field as the sum of a regular part and a jump. On the other hand, the ReLU PINN architecture is used to approximate the regular part of the displacement field. The results demonstrate that the Kinematic Decomposition architecture allows for automatic learning of the localization band's location during training, which is the main distinctive contribution of this work. Thus, the jump's position does not need to be exactly prescribed but requires defining a perturbation delimiting the band formation region, achieved in this work by considering a bar with variable area. Another parameter identifying the jump's magnitude is also adequately learned during training. Furthermore, it was determined that the ReLU PINN architecture provides an approximation space that accurately represents continuous functions, such as the regular part of displacement. For future research, exploring these architectures in solving localization problems in two or more dimensions is suggested.