用于脆性断裂相场建模的无网格多分辨率深度能量方法
A mesh-free multiresolution deep energy method with phase-field modeling of brittle fracture
- University of New South Wales（新南威尔士大学）
- Central Queensland University（中央昆士兰大学）
- Bauhaus-Universität Weimar（魏玛包豪斯大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出一种结合多分辨率特征编码与分层蒙特卡洛积分的无网格多分辨率深度能量方法，用于脆性断裂相场建模，在多个测试问题上的结果优于基准方法，零样本分类随机多裂纹状态的准确率达90%。
AI中文摘要：
脆性断裂的相场建模通过将裂纹演化重构为能量泛函的最小化过程，无需显式追踪裂纹。但该方法要求离散化足够密集，以解析由正则化长度设定宽度的局部化带，且该带的路径预先未知。本文提出一种无网格离散化方法，其中单个神经网络表示位移场和相场，并通过直接最小化增量能量进行训练。坐标通过基于$C^1$二次B样条网格构建的多分辨率特征编码输入网络，因此表示的最精细尺度由选择决定，而非通过缓慢训练获得；能量则通过在每个优化器迭代时重新采样的点上进行分层蒙特卡洛积分来估计。这种配对被证明至关重要：当积分点固定或编码过于粗糙无法表示局部化带时，裂纹无法扩展；而当另一要素到位后，每个要素都能容忍广泛的设置范围。由于该表示是全局$C^1$的，二阶和四阶断裂能密度可在相同离散化上计算。针对六个问题（从单边缺口拉伸和剪切到基于单个样条片的厚壁环），计算得到的载荷-位移曲线与匹配正则化长度的交错有限元参考结果吻合，单边缺口测试的峰值载荷误差约1%，裂纹图案拓扑变化处的峰值载荷误差约8%。在随机多裂纹构型的公开基准数据集上，该方法在20次零样本运行中对90%的植入裂纹的活跃或休眠状态进行了分类，而数据集作者提出的深度Ritz基线方法未能做到这一点。
英文摘要：
Phase-field modeling of brittle fracture removes the need to track cracks explicitly by recasting their evolution as the minimization of an energy functional. In return it requires a discretization dense enough to resolve a localization band whose width is set by a regularization length and whose path is not known in advance. We propose a mesh-free discretization in which a single neural network represents the displacement and phase fields and is trained by minimizing the incremental energy directly. The coordinates enter the network through a multiresolution feature encoding built from $C^1$ quadratic B-spline grids, so the finest scale the representation can express is set by choice rather than reached through slow training, and the energy is estimated by stratified Monte Carlo integration on points redrawn at every optimizer iteration. This pairing proves critical, since the crack fails to advance both when the integration points are held fixed and when the encoding is too coarse to represent the band, while each ingredient tolerates a wide range of settings once the other is in place. Because the representation is globally $C^1$, the second- and the fourth-order fracture energy densities run on the identical discretization. Across six problems, from single-edge-notched tension and shear to a thick-walled ring on a single spline patch, the computed load-displacement curves follow staggered finite element references at matched regularization length, with peak loads within about 1% on the single-edge-notched tests and within 8% where the crack pattern changes topology. On a public benchmark dataset of random multi-crack configurations the method classifies the active or dormant state of 90% of the seeded cracks in twenty zero-shot runs, where the deep Ritz baseline of the dataset authors fails.
