Phase-field modeling of fracture has gained popularity within the last decade due to the flexibility of the related computational framework in simulating three-dimensional arbitrarily complicated fracture processes. However, the numerical predictions are greatly affected by the presence of uncertainties in the mechanical properties of the material originating from unresolved heterogeneities and the use of noisy experimental data. The objective of this work is to apply the Bayesian approach to estimate bulk and shear moduli, tensile strength and fracture toughness of the phase-field model, thus improving accuracy of the simulations with the help of experimental data. Conventional approaches for estimating the Bayesian posterior probability density function adopt sampling schemes, which often require a large amount of model estimations to achieve the desired convergence, thus resulting in a high computational cost. In order to alleviate this problem, we employ a more efficient approach called sampling-free linear Bayesian update, which relies on the evaluation of the conditional expectation of parameters given experimental data. We identify the mechanical properties of cement mortar by conditioning on the experimental data of the three-point bending test (observations) in an online and offline manner. In the online approach the parameter values are sequentially updated on the fly as the new experimental information comes in. In contrast, the offline approach is used only when the whole history of experimental data is provided once the experiment is performed. Both versions of estimation are discussed and compared by validating the phase-field fracture model on an unused set of experimental data.

中文翻译：

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用于裂缝相场建模的参数识别：具有无采样更新的贝叶斯方法

由于相关计算框架在模拟三维任意复杂的断裂过程中的灵活性，断裂的相场建模在过去十年中得到了普及。然而，数值预测受到材料机械性能不确定性的很大影响，这些不确定性源于未解决的异质性和使用嘈杂的实验数据。这项工作的目的是应用贝叶斯方法来估计相场模型的体积和剪切模量、拉伸强度和断裂韧性，从而在实验数据的帮助下提高模拟的准确性。估计贝叶斯后验概率密度函数的传统方法采用采样方案，这通常需要大量的模型估计才能达到所需的收敛性，从而导致计算成本很高。为了缓解这个问题，我们采用了一种更有效的方法，称为无采样线性贝叶斯更新，它依赖于对给定实验数据的参数条件期望的评估。我们通过在线和离线方式调节三点弯曲试验（观察）的实验数据来确定水泥砂浆的机械性能。在在线方法中，随着新实验信息的出现，参数值按顺序动态更新。相比之下，离线方法仅在执行实验后提供整个实验数据历史时才使用。