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<p>This study presents an innovative numerical method utilizing a Non-Uniform Rational B-Splines (NURBS) technique hybridized with Galerkin for transient solutions of nonlinear phase-field problems. Particular focus is on phase transformations coupled with elasticity equations. The phase-field equations were formulated based on thermodynamic principles and further extended through energy expressions related to the martensitic phase transformation. This extension resulted in the coupling of the phase-field and elasticity equations. Subsequently, the NURBS method was implemented in combination with the Galerkin method to solve these nonlinear equations. The results address various scenarios, including the phase interface velocity, thickness, phase transformation in the presence of a nucleus, cases involving two and four nuclei with varying stress levels and temperatures affecting martensite development and suppression. The hybrid NURBS-Galerkin method determined that the phase interface thickness is 2.18&#xa0;nm, the uniaxial stress required for martensite growth at the equilibrium temperature is 2.71 <i>GPa</i>, and phase interface velocity under an equilibrium temperature and a uniaxial stress of 3 <i>GPa</i> is 3896&#xa0;m<i>/s</i>. The NURBS method was found to be more stable at larger time steps compared to the Bézier and Adams–Bashforth methods, offering a significant advantage in reducing computational costs. The presented method was validated through comparisons with COMSOL Multiphysics and established theoretical models, all of which highlight its excellent accuracy and convergence. The outcomes suggest that the NURBS-Galerkin technique has significant potential for solving complex physical problems, indicating its broader applicability in advanced scientific and engineering fields.</p>

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High-fidelity hybrid NURBS-Galerkin framework for nonlinear coupled phase-field elasticity: formulation and implementation

  • 期刊论文

  • 发表日期：

    2025年12月3日

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  • Ali Fattahi,
  • Mohammad Mohammadi Aghdam,
  • Younes Alizadeh Vaghasloo

摘要

This study presents an innovative numerical method utilizing a Non-Uniform Rational B-Splines (NURBS) technique hybridized with Galerkin for transient solutions of nonlinear phase-field problems. Particular focus is on phase transformations coupled with elasticity equations. The phase-field equations were formulated based on thermodynamic principles and further extended through energy expressions related to the martensitic phase transformation. This extension resulted in the coupling of the phase-field and elasticity equations. Subsequently, the NURBS method was implemented in combination with the Galerkin method to solve these nonlinear equations. The results address various scenarios, including the phase interface velocity, thickness, phase transformation in the presence of a nucleus, cases involving two and four nuclei with varying stress levels and temperatures affecting martensite development and suppression. The hybrid NURBS-Galerkin method determined that the phase interface thickness is 2.18 nm, the uniaxial stress required for martensite growth at the equilibrium temperature is 2.71 GPa, and phase interface velocity under an equilibrium temperature and a uniaxial stress of 3 GPa is 3896 m/s. The NURBS method was found to be more stable at larger time steps compared to the Bézier and Adams–Bashforth methods, offering a significant advantage in reducing computational costs. The presented method was validated through comparisons with COMSOL Multiphysics and established theoretical models, all of which highlight its excellent accuracy and convergence. The outcomes suggest that the NURBS-Galerkin technique has significant potential for solving complex physical problems, indicating its broader applicability in advanced scientific and engineering fields.