[S-CG62] Rheology, fracture and friction in Earth and planetary sciences

[SCG62-P15] Computation of shrinkage-induced crack patterns based on variational-energy-type physics-informed deep learning

*Shin-ichi Ito¹ (1.The University of Tokyo)

Keywords:Fracture, Physics-informed deep learning, Pattern formation

Crack formation induced by contraction is a phenomenon commonly observed, such as mud drying cracks, thermal contraction-induced cracks in glass, paint cracking, and columnar joints seen along coastlines. Understanding the dynamics of these pattern formations is crucial for predicting and controlling fractures, and therefore, reproducing these patterns numerically is an essential task. However, the significant gap between the spatial scale of the crack width and that of the entire system poses computational challenges for quantitative numerical simulations. In this study, as one potential approach to alleviate these computational difficulties, we propose a numerical method for fracture pattern formation based on variational energy-type physics-informed deep learning. Specifically, we consider a thin elastic body adhered to a horizontal two-dimensional substrate and model the system using a phase-field crack method that accounts for the competition among the energies of elastic deformation, substrate adhesion, and surface tension. To obtain the steady-state solution of this model via deep learning, we represent the solution as a neural network and formulate the problem using the Deep Ritz method, which minimizes a loss function derived from the integration of the aforementioned energies. Because the Deep Ritz method allows us to obtain the solution as a continuous function without resorting to a mesh, it eliminates the computational difficulties found in conventional approaches and enables fast, memory-efficient solution acquisition. However, while the Deep Ritz method is well suited for steady-state problems, it becomes extremely computationally expensive for tracking time evolution since the network must be retrained at every small time step. To overcome this issue, we have devised a method that obtains the solution as a spatiotemporal function through a single training process by constructing a constrained loss function that treats the time variable as a parameter. This approach also facilitates the incorporation of data from directly observable quantities--such as crack patterns and displacement fields--into the optimization problem as a data assimilation method. Compared to existing methods, it reduces both implementation cost and computational effort, and thus holds promise as an inverse problem-solving technique for efficiently estimating quantities that are difficult to measure directly, such as the surface tension coefficient field. In this presentation, we will detail our method and numerical results, and discuss potential future applications to three-dimensional problems, such as columnar joints.