This PDE is Beautiful - Phase Field Modeling

Phase-field simulations are not only visually appealing but also extremely useful. They are widely used to model physical processes such as solidification and melting, phase separation, and fracture. This video provides an introduction to phase-field modeling. We begin with the Allen–Cahn equation and then briefly explore extensions such as the Cahn–Hilliard equation and coupled systems.

You can find code for solving the Allen–Cahn and Cahn–Hilliard equations on my GitHub page: https://github.com/DrSimulate/phase-field

I would like to acknowledge the use of a phase-field fracture implementation phasefield-jr-py by Nathan Shauer: https://github.com/nathanshauer/phasefield-jr-py

Please also see the accompanying paper: https://www.sciencedirect.com/science/article/abs/pii/S0965997825001516

I also would like to acknowledge the use of FiPy, developed by J. E. Guyer, D. Wheeler, and J. A. Warren, as well as their example gallery to generate the simulations of dendritic growth: https://pages.nist.gov/fipy/en/latest/generated/examples.phase.anisotropy.html

Keywords: engineering, physics, continuum physics, continuum mechanics, solid mechanics, fluid mechanics, phase-field simulation, phase-field modeling, Allen-Cahn, Cahn-Hilliard, Ginzburg-Landau, double-well potential, solidification, crystal growth, dendritic growth, phase separation, fracture, coupled problems

Music:
The Palace Gardens - Asher Fulero
Variable Circumstance - Dan