The phase-field approach to brittle fracture provides a continuum framework for modeling crack initiation and propagation without explicit representation of discrete crack surfaces, provided the spatial discretization is fine enough to resolve the regularization length scale. However, uncertain local material parameters due to material defects can strongly influence simulation results, such as crack paths and remaining structural strength. At the same time, the ability to continuously monitor structures using sensors allows complementing modeling predictions with, e.g., displacement measurements. In this contribution, we connect these two complementary sources of information and present a Bayesian inference procedure that allows updating the current model state with incoming sensor data. We construct a Bayesian prior for the model state (both displacements and phase-field) and employ an ensemble Kalman filter (EnKF) to perform the update. In the EnKF, the update is computed by performing a Kalman shift on each ensemble member. Since the standard EnKF may produce assimilated states that violate common modeling assumptions, we present a phase field-based regularization technique as a proximal step correction toward model-consistent updates. 1D and 2D numerical examples demonstrate the performance and accuracy of the proposed method and show that the updated state matches the ground truth reasonably well. Unlike traditional Bayesian inversion techniques, which have already been applied to brittle fracture, we infer not the model parameters but the model state, i.e., the displacement field and the phase-field. Although only displacements are observed, the strong correlation between both fields also allows inference of the posterior phase-field.

翻译：脆性断裂的相场方法为裂纹萌生与扩展提供了连续介质建模框架，无需显式表示离散裂纹表面，前提是空间离散足够精细以解析正则化长度尺度。然而，材料缺陷导致的不确定局部参数会显著影响模拟结果，例如裂纹路径和剩余结构强度。与此同时，利用传感器持续监测结构的能力使得建模预测能够与位移测量等观测数据形成互补。本文中，我们将这两种互补信息源相结合，提出一种贝叶斯推断方法，能够利用传入的传感器数据更新当前模型状态。我们为模型状态（位移场与相场）构建贝叶斯先验分布，并采用集成卡尔曼滤波（EnKF）执行状态更新。在EnKF中，通过对每个集成成员执行卡尔曼偏移来计算更新量。由于标准EnKF可能产生违反常规建模假设的同化状态，我们提出一种基于相场的正则化技术，作为朝向模型一致性更新的邻近步校正。一维和二维数值算例验证了所提方法的性能与精度，表明更新后的状态与真实解吻合良好。与已应用于脆性断裂的传统贝叶斯反演技术不同，我们推断的不是模型参数而是模型状态（即位移场与相场）。尽管仅观测到位移，两个场之间的强相关性仍允许对后验相场进行推断。