A Neural Network-enhanced Reproducing Kernel Particle Method for Modeling Strain Localization

Modeling the localized intensive deformation in a damaged solid requires highly refined discretization for accurate prediction, which significantly increases the computational cost. Although adaptive model refinement can be employed for enhanced effectiveness, it is cumbersome for the traditional mesh-based methods to perform while modeling the evolving localizations. In this work, neural network-enhanced reproducing kernel particle method (NN-RKPM) is proposed, where the location, orientation, and shape of the solution transition near a localization is automatically captured by the NN approximation via a block-level neural network optimization. The weights and biases in the blocked parametrization network control the location and orientation of the localization. The designed basic four-kernel NN block is capable of capturing a triple junction or a quadruple junction topological pattern, while more complicated localization topological patters are captured by the superposition of multiple four-kernel NN blocks. The standard RK approximation is then utilized to approximate the smooth part of the solution, which permits a much coarser discretization than the high-resolution discretization needed to capture sharp solution transitions with the conventional methods. A regularization of the neural network approximation is additionally introduced for discretization-independent material responses. The effectiveness of the proposed NN-RKPM is verified by a series of numerical verifications.