Numerical modeling of localizations is a challenging task due to the evolving rough solution in which the localization paths are not predefined. Despite decades of efforts, there is a need for innovative discretization-independent computational methods to predict the evolution of localizations. In this work, an improved version of the neural network-enhanced Reproducing Kernel Particle Method (NN-RKPM) is proposed for modeling brittle fracture. In the proposed method, a background reproducing kernel (RK) approximation defined on a coarse and uniform discretization is enriched by a neural network (NN) approximation under a Partition of Unity framework. In the NN approximation, the deep neural network automatically locates and inserts regularized discontinuities in the function space. The NN-based enrichment functions are then patched together with RK approximation functions using RK as a Partition of Unity patching function. The optimum NN parameters defining the location, orientation, and displacement distribution across location together with RK approximation coefficients are obtained via the energy-based loss function minimization. To regularize the NN-RK approximation, a constraint on the spatial gradient of the parametric coordinates is imposed in the loss function. Analysis of the convergence properties shows that the solution convergence of the proposed method is guaranteed. The effectiveness of the proposed method is demonstrated by a series of numerical examples involving damage propagation and branching.

翻译：数值模拟局部化问题是一项具有挑战性的任务，因为解在演化过程中变得粗糙，且局部化路径并非预先定义。尽管经过数十年的努力，仍需要创新的、与离散化无关的计算方法来预测局部化的演化。本文提出了一种改进的神经网络增强再生核粒子法（NN-RKPM），用于模拟脆性断裂。在所提出的方法中，基于粗糙且均匀离散化定义的背景再生核（RK）近似，在划分单位框架下通过神经网络（NN）近似进行增强。在神经网络近似中，深度神经网络自动在函数空间中定位并插入正则化不连续性。随后，基于神经网络的增强函数与RK近似函数通过使用RK作为划分单位拼接函数进行组合。定义局部化位置、方向以及跨局部化位移分布的最优神经网络参数与RK近似系数，通过基于能量的损失函数最小化获得。为了正则化NN-RK近似，在损失函数中施加了对参数坐标空间梯度的约束。收敛性分析表明，所提出方法的解收敛性得以保证。通过一系列涉及损伤传播与分叉的数值算例，验证了该方法的有效性。