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.

中文翻译：

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数字模型中区分连通区域的相场算法


在本文中，我们提出了一种用于区分复杂三维连通区域的新模型。修改后的模型基于 Allen-Cahn 方程。修改后的方程不仅保持了原始界面动力学，而且避免了原始Allen-Cahn方程的无界扩散行为。这一优势使我们能够准确地填充和提取目标部分的复杂连接区域。该模型采用半隐式 Crank-Nicolson 格式进行离散化，确保时间和空间上的二阶精度。本文严格证明了我们方法的无条件能量稳定性，从而肯定了该解的数值稳定性和物理合理性。我们验证了所提出的模型对 3D 复杂连通区域的判别能力。