A convective Allen-Cahn model for the two- and three-dimensional shape transformations of non-contact objects

Abstract

This paper proposes a shape transformation model based on the Allen-Cahn equation, and its numerical scheme. The model overcomes the limitations of previous shape transformation models by introducing a convective term, realizing a smooth and stable shape transformation when the initial shape is not in contact with the target shape. To solve the problem of high-dimensions and the complexity of nonlinear terms, the numerical scheme adopts the dimension-splitting method, which can accelerate the computation by parallel algorithm, and incorporate a first-order stabilization term to mitigate numerical instability from explicit nonlinear computations. The numerical experiments explore the effect of the attracting coefficient and illustrates the effectiveness of our method in dealing with the non-contact objects through model comparison. Finally, 2D and 3D transformations validate the robustness and effectiveness of the proposed model and algorithm.