Adaptive fourth-order phase field analysis using deep energy minimization

Phase field modeling of fracture is computationally expensive as it demands a very fine mesh to resolve the damage region. Hence, the practical application of such models are severely limited. Local refinement techniques are often necessary. In our recent work (Goswami et al., 2019), for solving brittle fracture problems using physics informed neural network (PINN), the crack path is resolved by minimizing the variational energy of the system. However, in Goswami et al. (2019) we used a pre-refined domain based on prior information of the failure path, which is not always available. In this work, we propose an adaptive h-refinement scheme to locally refine the domain along the path of the growth of the crack. The phase field parameter, phi and a residual-based posteriori error estimator are the proposed convenient measures to determine the need for refinement. For phi, a critical threshold is chosen such that it is lower than the value at which crack nucleation occurs and the fracture region is easily identified. On the other hand, for the residual-based error estimator, elements contributing to the highest error are marked for refinement. The proposed algorithm takes as input the geometry described using NURBS patches. For the evaluation of the basis functions, we develop a procedure based on the Bezier representation and integrate it with the adaptive refinement formulation. The results obtained using the adaptive refinement integrated variational energy based PINN approach is validated with the available analytical solution for several examples from the literature. The proposed approach is implemented on several two and three-dimensional examples to illustrate the effectiveness of the formulation. Code and data necessary for replicating the results of the examples in the article will be made available through a GitHub repository.