Convolution tensor decomposition for efficient high-resolution solutions to the Allen-Cahn equation

TL;DR

This paper introduces a convolution tensor decomposition method combined with a stabilized semi-implicit scheme to efficiently solve high-resolution Allen-Cahn equations, significantly reducing computational costs for complex microstructure simulations.

Contribution

The paper develops a novel convolution tensor decomposition framework with an adaptive algorithm for efficient, high-resolution solutions to the Allen-Cahn equation, enabling large time steps and substantial speedups.

Findings

Achieved orders-of-magnitude speedups over finite element methods.

Successfully applied to 2D and 3D high-resolution problems.

Enabled large time increments without violating energy laws.

Abstract

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.