Triply periodic minimal surface using a modified Allen–Cahn equation

In this paper, we propose a fast and efficient model for triply periodic minimal surface. The proposed model is based on the Allen–Cahn equation with a Lagrange multiplier term. The Allen–Cahn equation has the motion of mean curvature. And the Lagrange multiplier term corresponding to the constant volume constraint also relates to the average of mean curvature. By combining two terms, the mean curvature will be constant everywhere on the surface at the equilibrium condition. The proposed numerical method with the second-order accuracy of time and space exhibits excellent stability. In addition, the resulting discrete system is solved by a fast numerical method such as a multigrid method. Various numerical experiments are performed to demonstrate the accuracy and robustness of the proposed method.