使用连续三次样条插值在一般多面体中进行迭代流体体积界面定位
Iterative Volume-of-Fluid interface positioning in general polyhedrons with Consecutive Cubic Spline interpolation
AI总结：
针对任意多面体单元中的几何VOF平面界面定位问题，提出复用函数值与导数的两点CCS迭代求根算法，平均两次迭代即达10^-12容差，且实现简单、几何无关、效率媲美最快方法。
摘要：
本文提出了一种直接且计算高效的连续三次样条（Consecutive Cubic Spline，CCS）迭代算法，用于在任意形状单元中定位非结构化几何流体体积（Volume-of-Fluid，VOF）方法的平面界面。CCS算法是一种专门为VOF界面定位问题设计的两点求根算法，在该问题中，体积分数函数在搜索区间端点处的导数趋于零。作为一种两点迭代算法，CCS复用先前迭代得到的函数值和导数，且不依赖区间括号界定。即使面对数值上极具挑战性的体积分数值，例如接近10^-9或1-10^-9，CCS算法平均也只需两次迭代即可将界面定位到10^-12的容差。所提出的CCS算法实现起来非常直接，因为其输入已经由每种几何VOF方法计算得到。它建立在预测性Newton方法的基础上并对其进行了显著改进，且独立于单元的几何模型及相关相交算法。该算法完全避免了其他当代方法所使用的截断体积几何参数化。就迭代次数而言，其计算效率可与迄今报道的最快方法相媲美。结果部分提供了CCS算法开源实现和性能测量数据的引用链接。
英文摘要：
A straightforward and computationally efficient Consecutive Cubic Spline (CCS) iterative algorithm is proposed for positioning the planar interface of the unstructured geometrical Volume-of-Fluid method in arbitrarily-shaped cells. The CCS algorithm is a two-point root-finding algorithm specifically designed for the VOF interface positioning problem, where the volume fraction function has diminishing derivatives at the ends of the search interval. As a two-point iterative algorithm, CCS re-uses function values and derivatives from previous iterations and does not rely on interval bracketing. The CCS algorithm only requires only two iterations on average to position the interface with a tolerance of $10^{-12}$, even with numerically very challenging volume fraction values, e.g. near $10^{-9}$ or $1-10^{-9}$. The proposed CCS algorithm is very straightforward to implement because its input is already calculated by every geometrical VOF method. It builds upon and significantly improves the predictive Newton method and is independent of the cell's geometrical model and related intersection algorithm. Geometrical parametrizations of truncated volumes used by other contemporary methods are completely avoided. The computational efficiency is comparable in terms of the number of iterations to the fastest methods reported so far. References are provided in the results section to the open-source implementation of the CCS algorithm and the performance measurement data.
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