Second-order accurate and unconditionally stable algorithm with unique solvability for a phase-field model of 3D volume reconstruction

Abstract

Three-dimensional (3D) volume reconstruction is an important technique in the fields of 3D printing, artificial limb, and medical diagnostic. This work aims to develop an unconditionally stable algorithm with desired accuracy and unique solvability for a phase-field model of 3D volume reconstruction. Based on scattered points of a target object, a smooth narrow volume is reconstructed by solving a Allen–Cahn-type equation with control function. Since the non-negative property of control function, the evolution of governing equation can associate with an energy dissipation law (i.e., energy stability). By modifying the nonlinear term in governing equation into an appropriate truncated form and utilizing a stabilized Crank–Nicolson-type discretization in time, we develop a linear, second-order accurate, and unconditionally energy-stable time-marching scheme to update the solution. The space is discretized with the finite difference method and the fully discrete energy stability is analytically estimated. We also prove the unique solvability of our proposed time-marching scheme. The accuracy, stability, and capability of reconstructing various 3D volumes are verified via extensive numerical simulations. To facilitate the interested readers, we provide the open source codes of 3D volume reconstruction of a teapot on corresponding author's personal website: https://cfdyang521.github.io.