[Paper Review] Iterative Volume-of-Fluid interface positioning in general polyhedrons with Consecutive Cubic Spline interpolation

Tomislav Marić|arXiv (Cornell University)|Dec 3, 2020

Fluid Dynamics and Heat Transfer29 references9 citations

TL;DR

This paper introduces the Consecutive Cubic Spline (CCS) algorithm, a highly efficient iterative method for positioning the planar interface in unstructured polyhedral cells within the Volume-of-Fluid (VOF) method. By leveraging existing geometric data and employing a two-point root-finding approach with cubic spline interpolation, CCS achieves an average of just two iterations to converge within a tolerance of 10−12, even for extreme volume fractions near 0 or 1, significantly outperforming existing methods in computational efficiency and ease of implementation.

ABSTRACT

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.

Motivation & Objective

• To develop a computationally efficient and robust interface positioning algorithm for the Volume-of-Fluid (VOF) method in general unstructured polyhedral cells.
• To address the challenge of high computational cost and numerical instability in interface positioning, especially for extreme volume fractions (e.g., near 0 or 1).
• To eliminate the need for complex geometric parameterizations of truncated volumes used in prior methods.
• To provide a method that is both highly efficient and straightforward to integrate into existing geometrical VOF solvers without modifying cell data structures.

Proposed method

• The CCS algorithm uses a scalar parameterization of the interface position along the normal vector, mapping the interface position to a scalar s along the cell’s normal direction.
• It formulates the interface positioning problem as a root-finding task for the function ˜α(s) = αc(s) − αc, where αc(s) is the normalized truncated volume as a function of s.
• The method applies a two-point iterative root-finding scheme based on consecutive cubic spline interpolation, reusing function values and derivatives from previous iterations to accelerate convergence.
• It avoids interval bracketing and does not require geometric parameterization of the truncated volume, relying only on the volume fraction and cell geometry already computed in standard VOF methods.
• The algorithm is designed to handle diminishing derivatives at the interval boundaries (smin and smax), which commonly occur in VOF due to the nature of volume truncation.
• It is independent of the underlying cell geometry and intersection algorithm, making it compatible with any unstructured mesh and cell model.

Experimental results

Research questions

• RQ1 Can a simple, iterative interface positioning algorithm be developed that maintains high accuracy and efficiency even for volume fractions near 0 or 1?
• RQ2 How does the proposed CCS algorithm compare in convergence speed and robustness to existing methods like Newton’s method, Brent’s method, and CIBRAVE?
• RQ3 To what extent can the number of volume truncation operations be reduced without introducing complex geometric parameterizations?
• RQ4 Can the algorithm be implemented with minimal changes to existing geometrical VOF codes, using only data already available in the solver?
• RQ5 Does the absence of geometric parameterization lead to a significant reduction in computational cost and improved cache efficiency?

Key findings

• The CCS algorithm achieves an average of only 2 iterations to converge to a tolerance of 10−12, even for volume fractions as extreme as 10−9 or 1−10−9.
• The method outperforms Newton’s method, Brent’s method, and the stabilized secant-bisection method in terms of convergence speed and robustness.
• CCS matches the best-performing method, CIBRAVE, in terms of average number of volume truncations, but without requiring complex geometric parameterizations.
• The algorithm is computationally efficient and highly portable, requiring no modification to existing cell data structures or mesh connectivity.
• The CPU time distribution shows that the majority of the cost comes from volume truncation, so reducing the number of truncations via CCS leads to significant speedups.
• The method is robust even when the cell normal is collinear with a face normal, a case that breaks many conventional methods, and handles it via a special-case treatment in the algorithm.