The weak form of the phase-field model for grain growth and its application in numerical simulation

Grain growth is a fundamental process in materials science, influencing the mechanical and physical properties of polycrystalline materials, including electrical conductivity, thermal conductivity, and corrosion resistance. This study focuses on the classical phase-field model for grain growth, systematically deriving its weak form to facilitate numerical implementation in finite element software. The weak form transforms complex partial differential equations into integral forms suitable for numerical solutions, eliminating the need for custom code development and enabling seamless multi-physics coupling. Numerical simulations were conducted in two-dimensional and three-dimensional settings to validate the model’s feasibility and accuracy. The results demonstrate the consistency of grain growth behavior with classical theories, revealing dynamic grain boundary migration and topological changes. This work not only provides a flexible and efficient framework for phase-field simulations but also offers a practical reference for implementing other phase-field models in finite element methods, paving the way for future advancements in materials science.