Application of Neural Operators to the Phase Field Modeling of Brittle Fractures

Brittle fracture is a common yet volatile phenomenon that occurs in hard substances. Its unpredictability poses a challenge in numerous domains, including medicine, engineering, and environmental science. Typically, brittle fracture is modeled using the finite element method (FEM), which is extremely slow. Recent breakthroughs in machine learning have led to the use of convolutional neural networks (CNNs) to model brittle fracture. Although a CNN is faster than FEM, its accuracy is limited by its focus on local relationships rather than the global relationships that determine fracture paths. In this work, we use a Fourier Neural Operator (FNO) to model brittle fracture. This method performs spectral convolutions in the frequency domain, thereby allowing the FNO to capture the global influence of cracks and cavities. We trained the FNO on 2080 input-output image pairs of size 512 <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">$\mathrm{x}512$</tex>, where the input image is a plate containing arbitrarily-shaped cavities, and the output image is the fracture path if the plate were to be loaded under tension. After training, the FNO predicted the fracture path of a previously unseen input in 2 seconds. Compared to FEM, which required 25 minutes, our FNO is 750 times faster. Also, the FNO is one-third more accurate than a CNN in predicting the correct fracture path. This work shows the potential of FNOs in brittle fracture analysis across diverse fields.