This paper presents a convolution tensor decomposition based model reduction method for solving the Allen–Cahn equation. The Allen–Cahn equation is usually used to characterize phase separation or the motion of anti-phase boundaries in materials. Its solution is time-consuming when high-resolution meshes and large time scale integration are involved. To resolve these issues, the convolution tensor decomposition method is developed, in conjunction with a stabilized semi-implicit scheme for time integration. The development enables a powerful computational framework for high-resolution solutions of Allen–Cahn problems, and allows the use of relatively large time increments for time integration without violating the discrete energy law. To further improve the efficiency and robustness of the method, an adaptive algorithm is also proposed. Numerical examples have confirmed the efficiency of the method in both 2D and 3D problems. Orders-of-magnitude speedups were obtained with the method for high-resolution problems, compared to the finite element method. The proposed computational framework opens numerous opportunities for simulating complex microstructure formation in materials on large-volume high-resolution meshes at a deeply reduced computational cost.

中文翻译：

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卷积张量分解，用于 Allen-Cahn 方程的高效高分辨率解


本文提出了一种基于卷积张量分解的模型约简方法，用于求解 Allen-Cahn 方程。Allen-Cahn 方程通常用于表征材料中的相分离或反相边界的运动。当涉及高分辨率网格和大时间尺度集成时，其解决方案非常耗时。为了解决这些问题，开发了卷积张量分解方法，并结合稳定的半隐式时间积分方案。这一发展为高分辨率 Allen-Cahn 问题的解提供了一个强大的计算框架，并允许在不违反离散能量定律的情况下使用相对较大的时间增量进行时间积分。为了进一步提高该方法的效率和鲁棒性，该文还提出了一种自适应算法。数值示例证实了该方法在 2D 和 3D 问题中的效率。与有限元方法相比，使用该方法处理高分辨率问题时，获得了数量级的加速。所提出的计算框架为在大容量高分辨率网格上模拟材料中复杂微观结构的形成提供了许多机会，同时大大降低了计算成本。