This paper introduces an efficient algorithm for the sequential positioning (or nested dissection) of two planar interfaces in an arbitrary polyhedron, such that, after each truncation, the respectively remaining polyhedron admits a prescribed volume. This task, among others, is frequently encountered in the numerical simulation of three-phase flows when resorting to the geometric Volume-of-Fluid method. For two-phase flows, the recent work of Kromer and Bothe (arXiv:2101.03861) addresses the positioning of a single plane by combining an implicit bracketing of the sought position with up to third-order derivatives of the volume fraction. An analogous application of their highly efficient root-finding scheme to three-material configurations requires computing the volume of a twice truncated arbitrary polyhedron. The present manuscript achieves this by recursive application of the Gaussian divergence theorem in appropriate form, which allows to compute the volume as a sum of quantities associated to the faces of the original polyhedron. With a suitable choice of the coordinate origin, accounting for the sequential character of the truncation, the volume parametrization becomes co-moving with respect to the planes. This eliminates the necessity to establish topological connectivity and tetrahedron decomposition after each truncation. After a detailed mathematical description of the concept, we conduct a series of carefully designed numerical experiments to assess the performance in terms of polyhedron truncations. The high efficiency of the two-phase positioning persists for sequential application, thereby being robust with respect to input data and possible intersection topologies. In comparison to an existing decomposition-based approach, the number of truncations was reduced by up to an order of magnitude, further highlighting the superiority of the divergence-based volume computation.

中文翻译：

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在非结构化多面网格上高效的三材料PLIC界面定位

本文介绍了一种有效的算法，用于在任意多面体中两个平面界面的顺序定位（或嵌套解剖），这样，在每次截断后，分别保留的多面体将容纳规定的体积。当诉诸几何流体体积法时，在三相流的数值模拟中经常遇到此任务。对于两相流，Kromer和Bothe（arXiv：2101.03861）的最新工作通过将所寻求位置的隐式括号与体积分数的最多三阶导数相结合，解决了单个平面的定位问题。将其高效的寻根方案类似地应用于三种材料的配置，需要计算两次截断的任意多面体的体积。本手稿通过以适当形式递归应用高斯散度定理来实现，从而可以将体积计算为与原始多面体的面相关的量之和。利用合适的坐标原点选择，占截断的顺序特征，体积参数化变得相对于平面共移动。这消除了在每次截断后建立拓扑连接性和四面体分解的必要性。在对该概念进行了详细的数学描述之后，我们进行了一系列精心设计的数值实验，以评估多面体截断的性能。两相定位的高效率​​持续适用于顺序应用，从而在输入数据和可能的相交拓扑方面具有鲁棒性。与现有的基于分解的方法相比，截断的数量最多减少了一个数量级，这进一步凸显了基于散度的体积计算的优越性。