We apply the phase field theory to regularise interfaces such as solid/crack in fracture mechanics, solid/liquid in corrosion, and solid/void in topology optimisation. For example, in fracture mechanics, it enables the smooth representation of discontinuities in the displacement field during crack propagation. In corrosion, it addresses discontinuities in metal ion concentration at the solid/liquid interface. Similarly, it regularises material-void boundaries in topology optimisation and captures solid-liquid transitions during phase transformations in solidification processes.

The strength of this method lies in its foundation on energy-based variational principles, which replace sharp interface conditions with continuous field descriptions. This eliminates the need for explicit interface tracking, providing a powerful and flexible framework for studying complex geometries and evolving topologies.

Our work seeks to enhance the phase field methodology by tackling challenges such as parameter calibration, computational efficiency, and coupling with multi-physics phenomena. These advancements aim to unlock deeper insights into interface-driven systems, offering transformative opportunities in both fundamental research and practical applications.