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High-order temporal parametric finite element methods for simulating solid-state dewetting

Xiaowen Gan, Yuqian Teng, Sisheng Wang

TL;DR

The paper addresses long-time simulation of 2D solid-state dewetting under a sharp-interface model by enhancing the ZJB energy-stable parametric FEM with temporally high-order schemes. It develops a predictor-corrector (PC-ZJB) and a family of Backward Differentiation Formula (BDFk-ZJB) time discretizations, both preserving mass/area and energy dissipation, and proves well-posedness with long-time mesh equidistribution. Numerical results demonstrate quadratic to quartic temporal convergence toward equilibrium shapes and confirm mesh quality maintenance, with dynamics converging to the Young angle and Wulff-like equilibria. This approach enables high-accuracy, long-time simulations of dewetting with isotropic surface energy, advancing efficient and reliable geometric evolution modeling in thin-film systems.

Abstract

We propose a class of temporally high-order parametric finite element methods for simulating solid-state dewetting of thin films in two dimensions using a sharp-interface model. The process is governed by surface diffusion and contact point migration, along with appropriate boundary conditions. By incorporating the predictor-corrector strategy and the backward differentiation formula for time discretization into the energy-stable parametric finite element method developed by Zhao et al. (2021), we successfully construct temporally high-order schemes. The resulting numerical scheme is semi-implicit, requiring the solution of a linear system at each time step. The well-posedness of the fully discretized system is established. Moreover, the method maintains the long-term mesh equidistribution property. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, measured by the manifold distance, while maintaining good mesh quality throughout the evolution.

High-order temporal parametric finite element methods for simulating solid-state dewetting

TL;DR

The paper addresses long-time simulation of 2D solid-state dewetting under a sharp-interface model by enhancing the ZJB energy-stable parametric FEM with temporally high-order schemes. It develops a predictor-corrector (PC-ZJB) and a family of Backward Differentiation Formula (BDFk-ZJB) time discretizations, both preserving mass/area and energy dissipation, and proves well-posedness with long-time mesh equidistribution. Numerical results demonstrate quadratic to quartic temporal convergence toward equilibrium shapes and confirm mesh quality maintenance, with dynamics converging to the Young angle and Wulff-like equilibria. This approach enables high-accuracy, long-time simulations of dewetting with isotropic surface energy, advancing efficient and reliable geometric evolution modeling in thin-film systems.

Abstract

We propose a class of temporally high-order parametric finite element methods for simulating solid-state dewetting of thin films in two dimensions using a sharp-interface model. The process is governed by surface diffusion and contact point migration, along with appropriate boundary conditions. By incorporating the predictor-corrector strategy and the backward differentiation formula for time discretization into the energy-stable parametric finite element method developed by Zhao et al. (2021), we successfully construct temporally high-order schemes. The resulting numerical scheme is semi-implicit, requiring the solution of a linear system at each time step. The well-posedness of the fully discretized system is established. Moreover, the method maintains the long-term mesh equidistribution property. Extensive numerical experiments demonstrate that our methods achieve the desired temporal accuracy, measured by the manifold distance, while maintaining good mesh quality throughout the evolution.
Paper Structure (10 sections, 4 theorems, 48 equations, 10 figures, 1 table, 1 algorithm)

This paper contains 10 sections, 4 theorems, 48 equations, 10 figures, 1 table, 1 algorithm.

Table of Contents

  1. Introduction
  2. Review of the ZJB scheme.
  3. High-order in time schemes
  4. Predictor-corrector time discretization
  5. Backward differentiation formulae time discretization
  6. Numerical results
  7. Cauchy test for the convergence order
  8. Time evolution of some geometric quantities
  9. Evolution of complex shapes
  10. Conclusions

Key Result

Theorem 3.1

(Well-posedness) Assume that the following conditions are satisfied: Then, the $N$-dimensional linear system PC is well-posed, i.e., there exists a unique solution $(\bm{X}^{m+1}, \kappa^{m+1}) \in \mathbb{X}^{h} \times \mathbb{K}^h$.

Figures (10)

  • Figure 1: A schematic illustration of an island film on a flat, rigid substrate in two dimensions with three interaces, i.e., island film/vapor (FV), island film/substrate (FS), and vapor/substrate (VS) interface, where $x_c^l$ and $x_c^r$ denote the $x$-cordinates of the left and right contact points, $\gamma_{FV}$ , $\gamma_{VS}$ and $\gamma_{FS}$ represent the film/vapor, vapor/substrate and film/substrate surface energy densities, respectively, $\bm{\tau}=(\partial_sx,\partial_sy)^{T}$ and $\bm{n}=(-\partial_sy,\partial_sx)^{T}$ are the unite tangential and normal vectors, and $\bm{\tau}_{sub}=(1,0)^{T}$ represents the unit vector along the x-coordinate.
  • Figure 2: Log-log plots of the numerical errors $E_{M}$ at four different times (i.e., $T = 0.25, 0.5, 1, 2$) are presented for: (a) the PC-ZJB scheme \ref{['PC']}, where the Cauchy path chosen as $\tau=0.05h$; (b) the BDF2-ZJB scheme\ref{['BDFk']}, with $\tau=0.05h$; (c) the BDF3-ZJB scheme\ref{['BDFk']} , with $\tau=0.01h^{\frac{2}{3}}$; and (d) the BDF4-ZJB scheme\ref{['BDFk']}, with $\tau=0.01h^{\frac{1}{2}}$. The Young's angle is chosen as $\theta_i=5\pi/6$.
  • Figure 3: Log-log plots of the numerical errors $E_{M}^{\mathrm{Wulff}}$ are presented for: (a) the ZJB scheme \ref{['eqn:dis2d']}, where the Cauchy path is chosen as $\tau=h^2$; (b) the PC-ZJB scheme \ref{['PC']} and the BDF2-ZJB scheme \ref{['BDFk']}, with $\tau=0.05h$. The Young's angle is chosen as $\theta_i=5\pi/6$.
  • Figure 4: Time evolution of the normalized energy $W(t)/W(0)$ and the relative area loss $\Delta A(t)/A(0)$ for the PC-ZJB scheme \ref{['PC']}: (a1, b1) under different mesh sizes $h=1/N$ with $\tau=0.01$ and $\theta_i=5\pi/6$; (a2, b2) under different time steps $\tau$ with $h=1/128$, $\tau_0=0.01$, and $\theta_i=5\pi/6$; (a3, b3) under different Young's angles $\theta_i$ with $h=1/128$ and $\tau=0.01$.
  • Figure 5: (a) Evolution of the mesh ratio indicator $\Psi(t)$ for the PC-ZJB scheme \ref{['PC']} under four different time steps, with $\tau_0 = 0.01$, $N = 128$, and Young's angle $\theta_i = 5\pi/6$; (b) evolution of the left contact angle $\theta_{d}^{l}(t)$ under four different mesh sizes, with $\tau = 0.01$ and Young's angle $\theta_i = 5\pi/6$.
  • ...and 5 more figures

Theorems & Definitions (8)

  • Theorem 3.1
  • proof
  • Proposition 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.2
  • proof