On the phase-field algorithm for distinguishing connected regions in digital model

In this paper, we propose a novel model for the discrimination of complex three-dimensional connected regions. The modified model is grounded on the Allen–Cahn equation. The modified equation not only maintains the original interface dynamics, but also avoids the unbounded diffusion behavior of the original Allen–Cahn equation. This advantage enables us to accurately populate and extract the complex connectivity region of the target part. The model is discretized employing a semi-implicit Crank–Nicolson scheme, ensuring second-order accuracy in both time and space. This paper provides a rigorous proof of the unconditional energy stability of our method, thereby affirming the numerical stability and the physical rationality of the solution. We validate the discriminative ability of the proposed model for 3D complex connected regions.