Two lower boundedness-preservity auxiliary variable methods for a phase-field model of 3D narrow volume reconstruction

Abstract

Three-dimensional (3D) volume reconstruction remains a fundamental technique with wide applications in fields such as 3D printing, medical diagnostics, and industrial design. This paper presents two novel lower boundedness-preserving auxiliary variable methods designed for the phase-field model of 3D narrow volume reconstruction. By employing scattered point data, our approach reconstructs smooth narrow volumes using a phase-field Allen–Cahn model with a control function. The proposed method ensures energy dissipation and smooth surface throughout the reconstruction process. By introducing an auxiliary variable, the nonlinear term in the governing equation is reformulated, allowing for efficient time-marching schemes. The fully discrete scheme is linear, and its unconditional stability is rigorously estimated. Numerical experiments are conducted to demonstrate the energy stability, accuracy, and effectiveness of our proposed methods in various 3D reconstruction tasks, establishing its broad applicability.