Simple and efficient volume merging method for triply periodic minimal structures

Abstract

Triply periodic minimal surfaces (TPMSs), which are periodic in all three directions and are surfaces of zero mean curvature, have been proven experimentally to be highly suitable for tissue scaffolds. However, simply gluing different TPMS units with different porosities and pore sizes could induce discontinuous structures and destroy the physical properties. In this study, we propose a simple and efficient volume merging method for triply periodic minimal structures. The proposed method can be divided into two steps. The first step is a novel merging algorithm for unit triply periodic minimal structures in the implicit function framework. The composite scaffold can be designed by merging different unit structures to satisfy the properties of internal connectivity. The second step is to optimize the designed composite scaffolds to satisfy the properties of TPMSs. A modified Allen–Cahn-type equation with a correction term is proposed. The mean curvature on the surface is constant at all points in the equilibrium state. Typically, the obtained structure is smooth owing to the motion by mean curvature flow. Therefore, the quality of the structure is significantly improved. Based on the operator splitting method, the proposed algorithm consists of two analytical evaluations for the ordinary differential equations and one numerical solution for the implicit Poisson-type equation. The proposed numerical scheme can be applied in a straightforward manner to a GPU-accelerated discrete cosine transform (DCT) implementation, which can be executed multiple times faster than CPU-only alternatives. Computational experiments are presented to demonstrate the efficiency and robustness of the proposed method.