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Numerical investigation into the dependence of the Allen–Cahn equation on the free energy
Advances in Computational Mathematics ( IF 1.7 ) Pub Date : 2022-05-27 , DOI: 10.1007/s10444-022-09955-1
Yunho Kim, Dongsun Lee

Phase-field modeling is strongly influenced by the shape of a free energy functional. In the theory of thermodynamics, it is a logarithmic type potential that is legitimate for modeling and simulating binary systems. Nevertheless, a tremendous amount of works have been dedicated to phase-field equations driven by 4-th double-well potentials as a polynomial approximation to the logarithmic type, which is valid only in the critical temperature regime. For comprehensive understanding of polynomial and logarithmic potentials and their relationship in the context of phase-field modeling, we provide and analyze a numerical framework for the Allen–Cahn equation derived from the Ginzburg–Landau functional with a logarithmic potential and its polynomial approximations. We prove that our numerical schemes guarantee the boundedness and energy dissipation properties of the solutions. As for the logarithmic free energy, we characterize different morphological changes of numerical solutions under various atomic binding energy configurations. Comparison of the logarithmic potential with its 2n-th order polynomial approximations reveals difference in the dynamics of spinodal decomposition. In particular, unlike the 6-th order or higher polynomials, the most studied solution by the 4-th order polynomial approximation using the 4-th double-well potential turns out to violate the logarithmic energy dissipation law. The geometric aspect of the Allen–Cahn equation with the logarithmic potential is also confirmed numerically. In summary, this study demonstrates the validity and applicability of the numerical framework for logarithmic and polynomial potentials and supports the need for further mathematical analysis on the logarithmic model.



中文翻译:

艾伦-卡恩方程对自由能依赖性的数值研究

相场建模受自由能泛函形状的强烈影响。在热力学理论中,它是一种对数型势,适用于建模和模拟二元系统。然而,大量的工作已经致力于由 4-th 双阱势驱动的相场方程作为对数类型的多项式近似,这仅在临界温度范围内有效。为了全面了解多项式和对数势及其在相场建模背景下的关系,我们提供并分析了从具有对数势及其多项式近似的 Ginzburg-Landau 泛函导出的 Allen-Cahn 方程的数值框架。我们证明了我们的数值方案保证了解的有界性和能量耗散特性。至于对数自由能,我们描述了在各种原子结合能配置下数值解的不同形态变化。对数电位与其 2 的比较n阶多项式近似揭示了旋节线分解动力学的差异。特别是,与 6 阶或更高阶多项式不同,研究最多的使用 4 阶双阱势的 4 阶多项式逼近解决方案违反了对数能量耗散定律。具有对数势的 Allen-Cahn 方程的几何方面也得到了数值证实。总之,这项研究证明了对数和多项式势的数值框架的有效性和适用性,并支持对对数模型进行进一步数学分析的需要。

更新日期:2022-05-27
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