A novel shape transformation method via the nonlocal modified Allen-Cahn model: Analysis and numerical simulations

초록

In this study, we develop and analyze a new shape transformation method using a nonlocal modified Allen-Cahn (MAC) model, in which a fidelity term is incorporated to refine the phase-field value ensuring its convergence towards target position within the desired region, while simultaneously reducing its influence beyond the target area. The developed transformation method is fast, explicit, spectral accurate in space and second-order accurate in time. At each time step, it only requires solving several decoupled closed-form solutions. The unconditional maximum-bound-preservation and energy dissipation as well as the sharp convergence for the fully discrete scheme are established. Notably, the modified energy functional is close to the classical energy up to O(tau), where tau is the splitting step. This appears to be the first energy dissipation study for second-order three-operator splitting method. Extensive numerical experiments on complex shape transformations are performed to validate the theoretical analysis and the efficiency of the present method. It is observed that the nonlocal MAC model can provide much better results than the local model, especially in shape transformations involving corners or discontinuities.

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