In this paper, a novel numerical algorithm for efficient modeling of three-dimensional shape transformation governed by the modified Allen-Cahn (A-C) equation is developed, which has important significance for computer science and graphics technology. The new idea of the proposed method is as follows. Firstly, the operator splitting method is used to decompose the three-dimensional problem into a series of one-dimensional subproblems that can be solved in parallel in the same direction. Secondly, a temporal p-adaptive strategy, which is based on the extrapolation technique, is proposed to improve the convergence order in time and preserve the computational efficiency simultaneously. Finally, a parallel least distance modification technique is developed to force the discrete maximum bound principle. The proposed method achieves high precision and high efficiency at the same time. Numerical examples include the effectiveness of the p-adaptive method and the bound preserving least distance modification, and a series of complex three-dimensional shape transformation modelings.

在本文中，开发了一种新的数值算法，用于高效建模由修正的 Allen-Cahn (AC) 方程控制的三维形状变换，这对计算机科学和图形技术具有重要意义。所提出方法的新思想如下。首先，采用算子分裂法，将三维问题分解为一系列可在同一方向上并行求解的一维子问题。其次，提出了一种基于外推技术的时间p自适应策略，以提高时间收敛顺序并同时保持计算效率。最后，开发了一种并行最小距离修改技术来强制离散最大界限原则。所提出的方法同时实现了高精度和高效率。数值示例包括 p 自适应方法的有效性和保界最小距离修改，以及一系列复杂的三维形状变换建模。