In the present study, the multiphase volume distribution problem, where there can be an arbitrary number of phases, is addressed using a consistent and conservative volume distribution algorithm. The proposed algorithm satisfies the summation constraint, the conservation constraint, and the consistency of reduction. The first application of the volume distribution algorithm is to determine the Lagrange multipliers in multiphase Phase-Field models that enforce the mass conservation, and a multiphase conservative Allen-Cahn model that satisfies the consistency of reduction is developed. A corresponding consistent and conservative numerical scheme is developed for the model. The multiphase conservative Allen-Cahn model has a better ability than the multiphase Cahn-Hilliard model to preserve under-resolved structures. The second application is to develop a numerical procedure, called the boundedness mapping, to map the order parameters, obtained numerically from a multiphase model, into their physical interval, and at the same time to preserve the physical properties of the order parameters. Along with the consistent and conservative schemes for the multiphase Phase-Field models, the numerical solutions of the order parameters are reduction consistent, conservative, and bounded, which are theoretically analyzed and numerically validated. Then, the multiphase Phase-Field models are coupled with the momentum equation by satisfying the consistency of mass conservation and the consistency of mass and momentum transport, thanks to the consistent formulation. It is demonstrated that the proposed model and scheme converge to the sharp-interface solution and are capable of capturing the complicated multiphase dynamics even when there is a large density and/or viscosity ratio.

在本研究中，多相体积分布问题，其中可以有任意数量的相，使用一致和保守的体积分布算法来解决。该算法满足求和约束、守恒约束和约简一致性。体积分布算法的第一个应用是确定强制质量守恒的多相相场模型中的拉格朗日乘子，以及满足减少一致性的多相保守 Allen-Cahn 模型被开发。为该模型开发了相应的一致和保守的数值方案。多相保守 Allen-Cahn 模型比多相 Cahn-Hilliard 模型具有更好的保留欠分辨结构的能力。第二个应用是开发一种称为有界映射的数值程序，将从多相模型中以数值方式获得的阶次参数映射到它们的物理区间，同时保留阶次参数的物理特性。连同多相相场模型的一致和保守方案，阶参数的数值解是归约一致、保守和有界的，对其进行了理论分析和数值验证。然后，由于一致的公式，质量守恒的一致性以及质量和动量传输的一致性。结果表明，所提出的模型和方案收敛于锐界面解，即使在密度和/或粘度比很大的情况下，也能够捕获复杂的多相动力学。