On the numerical approximation of a phase-field volume reconstruction model: Linear and energy-stable leap-frog finite difference scheme

IF 3.8 2区数学 Q1 MATHEMATICS, APPLIED

Communications in Nonlinear Science and Numerical Simulation Pub Date : 2025-07-11 DOI:10.1016/j.cnsns.2025.109104

Boyi Fu, Dongting Cai, Xiangjie Kong, Renjun Gao, Junxiang Yang

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引用次数: 0

Abstract

Three-dimensional (3D) volume reconstruction plays a vital role in industries like 3D printing, prosthetic devices, medical imaging, and computer vision. Our paper is aiming to engineer a stable and precise leap-frog scheme that guarantees desired accuracy for a phase-field model aimed at 3D volume reconstruction. Utilizing scattered data points from the target object, we reconstruct a smooth, narrow volume by solving an Allen–Cahn-type equation enhanced with a control function. The non-negative attribute of this control function links the evolution of the main equation to an energy dissipation law, thereby ensuring energy stability. By developing and validating a leap-frog numerical scheme, we design a linear, second-order accurate, and unconditionally energy-stable method for advancing the solution in time. The spatial domain is discretized employing the finite difference method, with the fully discrete energy stability being analytically assessed. Through comprehensive numerical experiments, we confirm the algorithm’s accuracy, stability, and proficiency in reconstructing diverse 3D volumes. Furthermore, by evaluating the surfaces of reconstructed volumes, we identify how specific parameter selections influence the performance of the leap-frog numerical scheme.

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一种相场体积重构模型的数值近似：线性和能量稳定的跨越式有限差分格式

三维（3D）体重建在3D打印、假肢设备、医学成像和计算机视觉等行业中起着至关重要的作用。我们的论文旨在设计一个稳定和精确的跨越式方案，以保证针对三维体重建的相场模型所需的精度。利用来自目标物体的分散数据点，我们通过求解带有控制函数增强的allen - cahn型方程来重建光滑的窄体。该控制函数的非负属性将主方程的演化与能量耗散规律联系起来，从而保证了能量稳定。通过发展和验证一种跨越式数值格式，我们设计了一种线性、二阶精确、无条件能量稳定的方法来及时推进解。采用有限差分法对空间域进行离散化，并对完全离散的能量稳定性进行分析评估。通过全面的数值实验，我们证实了该算法的准确性、稳定性和对各种三维体重建的熟练程度。此外，通过评估重建体的表面，我们确定了特定参数选择如何影响跨越式数值方案的性能。

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来源期刊

Communications in Nonlinear Science and Numerical Simulation MATHEMATICS, APPLIED-MATHEMATICS, INTERDISCIPLINARY APPLICATIONS

CiteScore

6.80

自引率

7.70%

发文量

378

审稿时长

78 days

期刊介绍： The journal publishes original research findings on experimental observation, mathematical modeling, theoretical analysis and numerical simulation, for more accurate description, better prediction or novel application, of nonlinear phenomena in science and engineering. It offers a venue for researchers to make rapid exchange of ideas and techniques in nonlinear science and complexity. The submission of manuscripts with cross-disciplinary approaches in nonlinear science and complexity is particularly encouraged. Topics of interest: Nonlinear differential or delay equations, Lie group analysis and asymptotic methods, Discontinuous systems, Fractals, Fractional calculus and dynamics, Nonlinear effects in quantum mechanics, Nonlinear stochastic processes, Experimental nonlinear science, Time-series and signal analysis, Computational methods and simulations in nonlinear science and engineering, Control of dynamical systems, Synchronization, Lyapunov analysis, High-dimensional chaos and turbulence, Chaos in Hamiltonian systems, Integrable systems and solitons, Collective behavior in many-body systems, Biological physics and networks, Nonlinear mechanical systems, Complex systems and complexity. No length limitation for contributions is set, but only concisely written manuscripts are published. Brief papers are published on the basis of Rapid Communications. Discussions of previously published papers are welcome.