Advances in Numerical Schemes for Phase Field Models
A special issue of Mathematics (ISSN 2227-7390). This special issue belongs to the section "C: Mathematical Analysis".
Deadline for manuscript submissions: 28 February 2026 | Viewed by 25
Special Issue Editor
Interests: applied mathematics; partial differential equations/analysis; modeling in materials science; (non-)Newtonian fluid dynamics and its applications
Special Issues, Collections and Topics in MDPI journals
Special Issue Information
Dear Colleagues,
Our understanding of the universe varies with the scale at which we observe it; accordingly, different models are formulated to address phenomena at these various scales. Among these, phase-field models have emerged as a powerful tool for both theoretical and numerical analyses at the mesoscale. They have been successfully applied to a wide range of complex phenomena, including as phase transformations, solidification, dendritic growth, crack propagation, battery simulations, and tumor growth.
A central idea in phase-field modeling is the introduction of one or more continuous order parameters that vary smoothly across interfaces. This framework has also been extended to the phase-field crystal method to study pattern formation and related phenomena. However, numerically solving phase-field equations remains challenging due to their strong non-linearity, stiffness, higher-order nature, and coupling with other physical equations, such as those governing mechanics and fluid flow.
Instead of pursuing analytical solutions, researchers commonly employ numerical schemes to solve these equations with sufficient efficiency and accuracy. This Special Issue (SI) aims to highlight significant recent developments in phase-field modeling and computational techniques. It serves as a forum for presenting research across various topics, including
- Modeling approaches: phase-field method, phase-field crystal method, etc.
- Numerical simulations: finite element method, finite difference method, spectral methods, adaptive techniques, machines learning, and more.
We are particularly interested in innovative numerical schemes based on artificial intelligence (AI) technologies, which hold great promise in enhancing the efficiency and scope of phase-field simulations, thereby advancing both scientific understanding and technological innovation.
We invite you to contribute your latest research to this Special Issue and help foster a vibrant exchange of ideas in this rapidly evolving field.
Prof. Dr. Peicheng Zhu
Guest Editor
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Keywords
- phase-field model
- phase-field crystal model
- numerical simulations
- finite element method
- finite difference method
- spectral method
- machine learning
- numerical schemes based up AI technology
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