A phase-field and Fourier neural operator-based method for predicting crack evolution in column-shell structures

With the increasing application of engineering structures under extreme conditions, accurately predicting their fracture behavior has become a critical challenge in materials science and fracture mechanics. Column-shell structures, as typical load-bearing components, are highly sensitive to crack initiation and propagation, which directly affect their safety and reliability. Although traditional finite element methods can provide accurate fracture evolution simulations, their high computational cost limits applicability in rapid prediction scenarios. To address this issue, a hybrid framework that integrates the phase-field method with the Fourier neural operator (FNO) is proposed for predicting the fracture evolution of column-shell structures. In the proposed framework, the phase-field method is first employed to describe crack initiation, propagation, and possible coalescence in a continuous manner, avoiding explicit crack tracking and enabling physically consistent simulations. Based on this formulation, a finite element model of the column-shell structure is established to generate high-fidelity fracture evolution data under various conditions, including different critical energy release rates, geometric configurations, and loading scenarios. Subsequently, a data-driven learning framework is developed using the FNO to approximate the nonlinear mapping between input parameters and fracture responses. The input of the model includes the spatial distribution of the critical energy release rate, geometric features, and applied loading conditions, while the output corresponds to the evolving phase-field variable that characterizes crack growth over time. A series of FNO models are constructed and trained in a sequential manner to separately capture crack initiation and propagation stages, forming a coupled prediction framework. The training process is carried out using the generated dataset, with appropriate normalization and validation strategies to ensure model robustness and generalization capability. The results demonstrate that the proposed method achieves high prediction accuracy under random critical energy release rates, varying geometries, and complex loading conditions, while significantly reducing computational cost compared to conventional finite element simulations. Once trained, the model enables near real-time prediction of fracture evolution.