This paper establishes the conserved N-component Allen–Cahn model on evolving surfaces and conducts numerical simulations of the model. In mathematical modeling, since the surface motion velocity causes local contraction or expansion of the surface, it is hard to simultaneously fulfill the componential mass conservation and the point-wise mass conservation as the usual case on the static domain. Therefore, according to these two types of conservation, three models are established: the componential mass conservation model, the point-wise mass conservation model, and the componential and point-wise mass conservation model. For the numerical simulation, the evolving surface finite element method is used to discretize the model in time and space. To achieve a stable, linear, high-accuracy and decoupled numerical scheme, the evolving surface finite element method has been enhanced by incorporating the stabilized semi-implicit approach. Furthermore, the stability analysis has been undertaken to validate the robustness of the devised numerical scheme. Through the validation of numerous numerical simulations, the reasonableness of the proposed model and the efficacy of the numerical approach are evaluated.

中文翻译：

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在演化曲面上对守恒 N 分量 Allen-Cahn 模型进行建模和仿真


本文在演化表面上建立了守恒 N 分量 Allen-Cahn 模型，并对该模型进行了数值模拟。在数学建模中，由于表面运动速度会导致表面的局部收缩或膨胀，因此很难像静态域上通常的情况那样同时实现分量质量守恒和逐点质量守恒。因此，根据这两种类型的守恒，建立了三种模型：分量质量守恒模型、逐点质量守恒模型以及分量和逐点质量守恒模型。对于数值模拟，使用演化表面有限元方法在时间和空间上离散模型。为了实现稳定、线性、高精度和解耦的数值方案，通过引入稳定半隐式方法，增强了演化表面有限元方法。此外，还进行了稳定性分析以验证所设计的数值方案的鲁棒性。通过大量数值模拟的验证，评估了所提模型的合理性和数值方法的有效性。