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Energy-stable parametric finite element approximations for regularized solid-state dewetting in strongly anisotropic materials

Meng Li, Chunjie Zhou

TL;DR

This paper addresses stable numerical simulation of solid-state dewetting under strongly anisotropic surface energy by formulating a regularized sharp-interface model with Willmore energy and proving energy-dissipation. It introduces a novel geometric system based on surface-energy matrices $\boldsymbol{B}_q(\theta)$, establishing an equivalent conservative formulation that underpins two energy-stable parametric finite element methods (PFEMs): ES-PFEM and AC-PFEM. The authors prove area conservation (AC-PFEM) and energy stability (both ES-PFEM and AC-PFEM) and develop a full Newton-Raphson solver for the implicit schemes. Numerical experiments show convergence $O(\tau + h^2)$, improved mesh quality due to Willmore regularization, and robust long-time behavior, highlighting the practical impact for simulating anisotropic dewetting phenomena. Overall, the work advances reliable, structure-preserving simulations of thin-film evolution with strong anisotropy, with potential extensions to axisymmetric and 3D settings.

Abstract

In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of film/vapor interfacial energy, substrate energy, and regularized Willmore energy. By introducing two geometric relations, we innovatively establish an equivalent regularized sharp-interface model and further construct an energy-stable parametric finite element algorithm for this equivalent model. We provide a detailed proof of the energy stability of the numerical scheme, addressing a gap in the relevant theory. Additionally, we develop another structure-preserving parametric finite element scheme that can preserve both area conservation and energy stability. Finally, we present several numerical simulations to show accuracy and efficiency as well as some structure-preserving properties of the proposed numerical methods. More importantly, extensive numerical simulations reveal that our schemes provide better mesh quality and are more suitable for long-term computations.

Energy-stable parametric finite element approximations for regularized solid-state dewetting in strongly anisotropic materials

TL;DR

This paper addresses stable numerical simulation of solid-state dewetting under strongly anisotropic surface energy by formulating a regularized sharp-interface model with Willmore energy and proving energy-dissipation. It introduces a novel geometric system based on surface-energy matrices , establishing an equivalent conservative formulation that underpins two energy-stable parametric finite element methods (PFEMs): ES-PFEM and AC-PFEM. The authors prove area conservation (AC-PFEM) and energy stability (both ES-PFEM and AC-PFEM) and develop a full Newton-Raphson solver for the implicit schemes. Numerical experiments show convergence , improved mesh quality due to Willmore regularization, and robust long-time behavior, highlighting the practical impact for simulating anisotropic dewetting phenomena. Overall, the work advances reliable, structure-preserving simulations of thin-film evolution with strong anisotropy, with potential extensions to axisymmetric and 3D settings.

Abstract

In this work, we aim to develop energy-stable parametric finite element approximations for a sharp-interface model with strong surface energy anisotropy, which is derived from the first variation of an energy functional composed of film/vapor interfacial energy, substrate energy, and regularized Willmore energy. By introducing two geometric relations, we innovatively establish an equivalent regularized sharp-interface model and further construct an energy-stable parametric finite element algorithm for this equivalent model. We provide a detailed proof of the energy stability of the numerical scheme, addressing a gap in the relevant theory. Additionally, we develop another structure-preserving parametric finite element scheme that can preserve both area conservation and energy stability. Finally, we present several numerical simulations to show accuracy and efficiency as well as some structure-preserving properties of the proposed numerical methods. More importantly, extensive numerical simulations reveal that our schemes provide better mesh quality and are more suitable for long-term computations.
Paper Structure (7 sections, 3 theorems, 57 equations, 14 figures)

This paper contains 7 sections, 3 theorems, 57 equations, 14 figures.

Table of Contents

  1. Introduction
  2. A new geometric system
  3. Variational formulation
  4. Parametric finite element approximations
  5. An iterative solver
  6. Numerical results
  7. Conclusions

Key Result

Lemma 3.1

(Area conservation & Energy dissipation). Let $\left(\vec{X}(\cdot, t), \mu(\cdot, t),\kappa(\cdot, t)\right)\in\mathbb X \times H^1(\mathbb{I}) \times H_0^1(\mathbb{I})$ be a solution of the variational formulation eqn:vf, then the area $A(t)$ is conservative during the evolution, i.e., and the energy $W_c(t)$ is dissipative during the evolution, i.e.,

Figures (14)

  • Figure 1: Plot of numberical errors employing ES-PFEM at $t_m = 1$(left panel) and $t_m = 2$(right panel) for $2$-fold anisotropy. The parameters are selected as $\eta = 100$, $\sigma = -0.6$.
  • Figure 2: Plot of numberical errors employing ES-PFEM at $t_m = 1$ (left panel) and $t_m = 2$ (right panel) for $4$-fold anisotropy. The parameters are selected as $\eta = 100$, $\sigma = -0.6$.
  • Figure 3: The time history of the energy $E(t)/E(0)$ using ES-PFEM with $2$-fold (left panel) and $4$-fold (right panel) anisotropy.
  • Figure 4: The time history of the area loss $\triangle A(t)$ and the energy $E(t)/E(0)$. The blue line represents the results obtained using the AC-PFEM, while the red line represents the results obtained using the ES-PFEM for $2$-fold anisotropy. The degree of anisotropy is chosen as $\beta = 9/24$.
  • Figure 5: The time history of the area loss $\triangle A(t)$ and the energy $E(t)/E(0)$. The blue line represents the results obtained using the AC-PFEM, while the red line represents the results obtained using the ES-PFEM for $4$-fold anisotropy. The degree of anisotropy is chosen as $\beta = 1/10$.
  • ...and 9 more figures

Theorems & Definitions (9)

  • Lemma 3.1
  • proof
  • Remark 1
  • Theorem 4.1
  • proof
  • Remark 2
  • Theorem 4.2
  • proof
  • Remark 3