A phase-field approach for the fast simulation of shape transformations with obstacles

We develop a phase-field model and corresponding parallel numerical method to achieve fast simulation of shape or object transportation problems with obstacles in computer graphics and related areas. The phase-field model incorporates a convective Allen–Cahn equation and a penalty term that treats obstacles as an additional phase. A convection field attracts the evolving shape toward the target and guides it around obstacles, while a penalty term prevents unphysical penetration into obstacles. This reformulation can transform problems in complex regions with obstacles into problems in regular regions, and allows the use of a dimensional splitting strategy that decomposes the dynamics into one-dimensional subproblems suitable for parallel computation. A stabilizing term is further introduced to enhance robustness when the nonlinear term is treated explicitly. Numerical experiments in two and three dimensions demonstrate that the proposed approach can achieve smooth, stable evolution and accurate shape transformations with efficient computation.