We present a novel stochastic extended finite element (S-XFEM) method for solving fracture mechanics problems under uncertainty. The proposed S-XFEM couples XFEM and a novel multi-output Gaussian process based machine learning algorithm, referred to as the hybrid polynomial correlated function expansion (H-PCFE). With this approach, the underlying stochastic fracture mechanics problem is decoupled into multiple classical fracture mechanics problems. Since solution of a classical fracture mechanics problem is computationally expensive, we utilize the H-PCFE model as a surrogate to emulate the behavior of the system. Training samples required for training the H-PCFE model are generated by using the XFEM. Because of the intrinsic capability of XFEM in providing a mesh insensitive solution, the proposed approach can provide reasonable result from a coarse mesh. On the other hand, H-PCFE is capable of providing highly accurate solution from very few training samples. Overall, the proposed S-XFEM is highly efficient in solving stochastic fracture mechanics problems. The proposed approach is used for solving three stochastic fracture mechanics problems. Different case studies involving reliability analysis and stochastic fracture propagation have been reported. The procedure yields highly accurate results for all problems, indicating its possible applications to other large scale systems.

我们提出了一种新颖的随机扩展有限元（S-XFEM）方法来解决不确定性下的断裂力学问题。提出的S-XFEM将XFEM与一种新颖的基于多输出高斯过程的机器学习算法结合在一起，称为混合多项式相关函数展开（H-PCFE）。通过这种方法，潜在的随机断裂力学问题被分解为多个经典断裂力学问题。由于经典断裂力学问题的解决方案计算量很大，因此我们将H-PCFE模型用作模拟系统行为的替代方法。使用XFEM生成了训练H-PCFE模型所需的训练样本。由于XFEM具有提供网格不敏感解决方案的固有功能，所提出的方法可以从粗糙的网格中提供合理的结果。另一方面，H-PCFE能够从很少的训练样本中提供高度准确的解决方案。总体而言，建议的S-XFEM在解决随机断裂力学问题方面非常高效。所提出的方法用于解决三个随机断裂力学问题。已经报道了涉及可靠性分析和随机裂缝扩展的不同案例研究。该程序对所有问题均能得出高度准确的结果，表明它可能适用于其他大型系统。所提出的方法用于解决三个随机断裂力学问题。已经报道了涉及可靠性分析和随机裂缝扩展的不同案例研究。该程序对所有问题均产生高度准确的结果，表明其可能适用于其他大型系统。所提出的方法用于解决三个随机断裂力学问题。已经报道了涉及可靠性分析和随机裂缝扩展的不同案例研究。该程序对所有问题均能得出高度准确的结果，表明它可能适用于其他大型系统。