Shape transformation on curved surfaces using a phase-field model

Shape transformation on evolving curved surfaces is essential for its diverse applications across various scientific disciplines and facilitates the deeper understanding of natural phenomena, the development of new materials, and engineering design optimization. In this study, we develop a phase-field model and its numerical methods for shape transformation on curved surfaces. A modified surface Allen–Cahn (AC) equation with a fidelity term is proposed to simulate shape transformation on curved surfaces. To numerically solve the modified surface AC equation on curved surfaces, we propose a fully explicit scheme and an unconditionally stable method. The proposed stable approach is not only simple and efficient to implement numerically but is also unconditionally stable and eliminates the restrictive temporal time step size constraints. Through numerical experiments using the proposed approach, we demonstrate that shape transformation on evolving curved surfaces can be implemented on both simple and complex curved surfaces.