多相Allen-Cahn和Cahn-Hilliard模型及其离散化及 pairwise 表面张力的影响
Multiphase Allen-Cahn and Cahn-Hilliard Models and Their Discretizations with the Effect of Pairwise Surface Tensions
AI总结：
本文研究了多相模型的数学性质及数值离散化，证明了系数矩阵的unisolvent性质，并探讨了pairwise表面张力的物理条件，推导了多相Allen-Cahn和Cahn-Hilliard方程，展示了浓度动力学与相变量无关，并通过数值实验验证了表面张力的影响。
摘要：
本文研究了模拟N组分混合物相分离的多相模型的数学性质和数值离散化。对于一般的相变量选择，建立了基于pairwise表面张力的N相模型中系数矩阵的unisolvent性质。此外，系数矩阵在(N-1)维超平面上的对称正定性质被证明等价于某些pairwise表面张力的物理条件。由此可以推导出N相Allen-Cahn和Cahn-Hilliard方程。自然性质是所得到的浓度动力学与所选相变量无关。N相模型的有限元离散化可作为现有双相模型离散化的自然扩展。在某些时间步长限制下，可以证明并观察到数值方案的离散能量守恒定律。数值实验包括spinodal分解和三 junctions的演化，以研究pairwise表面张力的影响。
英文摘要：
In this paper, the mathematical properties and numerical discretizations of multiphase models that simulate the phase separation of an $N$-component mixture are studied. For the general choice of phase variables, the unisolvent property of the coefficient matrix involved in the $N$-phase models based on the pairwise surface tensions is established. Moreover, the symmetric positive-definite property of the coefficient matrix on an $(N-1)$-dimensional hyperplane --- which is of fundamental importance to the well-posedness of the models --- can be proved equivalent to some physical condition for pairwise surface tensions. The $N$-phase Allen-Cahn and $N$-phase Cahn-Hilliard equations can then be derived from the free-energy functional. A natural property is that the resulting dynamics of concentrations are independent of phase variables chosen. Finite element discretizations for $N$-phase models can be obtained as a natural extension of the existing discretizations for the two-phase model. The discrete energy law of the numerical schemes can be proved and numerically observed under some restrictions pertaining to time step size. Numerical experiments including the spinodal decomposition and the evolution of triple junctions are described in order to investigate the effect of pairwise surface tensions.
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