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

Ye Lu, Chaoqian Yuan, Han Guo

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.

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

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.

Paper Structure

This paper contains 21 sections, 1 theorem, 64 equations, 18 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Problem formulation
  3. Strong form
  4. Weak form
  5. Semi-discretized form
  6. Time integration
  7. Stabilized semi-implicit formulation
  8. The proposed CTD formulation for the AC equation
  9. Separation of variables
  10. Convolution approximation
  11. Solution algorithm
  12. Numerical results
  13. 2D AC problem
  14. 3D AC problem
  15. Influence of different controlling parameters
  16. ...and 6 more sections

Key Result

Theorem 1

The solution to the formulation eq:AC-discrete-semi-stable is energetically stable for arbitrary $\Delta t$, if $\alpha$ is chosen according to

Figures (18)

  • Figure 1: Supporting nodes for FE and CFE approximations, left: $A^e$, right: $A^e_s$, the square in the left figure represents the domain of an element
  • Figure 2: Examples of supporting nodes for $A^e$, $A_s^i$, and $A^e_s$, $s$ is the patch size that defines the number of layers outside an element
  • Figure 3: 1D CFE and FE shape functions in the natural coordinate system $\xi\in [-1,1]$, left: CFE shape functions $\tilde{{N}}_k({\xi})$ with $s=2, p=1, a=1$, right: linear FE shape functions ${{N}}_i({\xi})$
  • Figure 4: CFE shape function in global coordinate system, $\tilde{{N}}_k({x})$ with $x\in [-5,5]$, patch size $s=2$
  • Figure 5: Computational domain and initial condition
  • ...and 13 more figures

Theorems & Definitions (2)

  • Theorem 1
  • proof