分段线性界面构造问题的解析解及其在流体体积模拟程序曲率计算中的应用
Analytic Solution to the Piecewise Linear Interface Construction Problem and its Application in Curvature Calculation for Volume-of-Fluid Simulation Codes
摘要：
自1984年以来，平面-立方体相交问题一直存在于文献中，并且从那时起，它的迭代解一直被用作计算流体动力学模拟代码中的分段线性界面构造（PLIC）的一部分。在许多情况下，PLIC是这些模拟计算时间的瓶颈，因此，对平面立方体相交的更快的解析解将大大减少此类模拟的计算时间。我们推导出所有交叉情况下的解析解，并将其与Scardovelli和Zaleski（Ruben Scardovelli和Stephane Zaleski）的前一个解进行比较。“解析关系连接线性界面和体积分数在矩形网格”。在：计算物理杂志164.1（2000），pp。228-237.），我们进一步改进以包括边缘情况，并进行微优化以减少算术运算和分支。然后，我们将计算时间和精度的比较扩展到包括两种不同的迭代解决方案。我们发现，最佳选择取决于所采用的硬件平台：在CPU上，牛顿-拉夫森是最快的矢量化，而分析解决方案执行更好。其原因是向量化指令集不包括解析解中使用的三角函数。在GPU上，最快的方法是我们优化版本的分析SZ解决方案。最后，我们提供详细的PLIC的应用之一：曲率计算的体积的流体模型用于自由表面流体模拟结合格子玻尔兹曼方法。
英文摘要：
The plane-cube intersection problem has been around in literature since 1984 and iterative solutions to it have been used as part of piecewise linear interface construction (PLIC) in computational fluid dynamics simulation codes ever since. In many cases, PLIC is the bottleneck of these simulations regarding compute time, so a faster, analytic solution to the plane-cube intersection would greatly reduce compute time for such simulations. We derive an analytic solution for all intersection cases and compare it to the one previous solution from Scardovelli and Zaleski (Ruben Scardovelli and Stephane Zaleski. "Analytical relations connecting linear interfaces and volume fractions in rectangular grids". In: Journal of Computational Physics 164.1 (2000), pp. 228-237.), which we further improve to include edge cases and micro-optimize to reduce arithmetic operations and branching. We then extend our comparison regarding compute time and accuracy to include two different iterative solutions as well. We find that the best choice depends on the employed hardware platform: on the CPU, Newton-Raphson is fastest with vectorization while analytic solutions perform better without. The reason for this is that vectorization instruction sets do not include trigonometric functions as used in the analytic solutions. On the GPU, the fastest method is our optimized version of the analytic SZ solution. We finally provide details on one of the applications of PLIC: curvature calculation for the Volume-of-Fluid model used for free surface fluid simulations in combination with the lattice Boltzmann method.
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