A predicted-Newton's method for solving the interface positioning equation in the MoF method on general polyhedrons

Abstract

Solving the interface positioning equation is the key procedure of the PLIC-VoF methods. Most of previous research only focused on the planar constant calculation and paid less attention to the relationship between the planar constant and the approximate interface orientation. The latter issue is important especially for the second order iteration based PLIC-VoF method, such as the MoF and LVIRA method. In these methods, the most accurate interface orientation is calculated through an iterative procedure, so the interface positioning equation has to be solved multiple times for the given volume fraction with different interface orientations. In this situation, if the incremental relation between the planar constant and the interface orientation is known, a predicted planar constant can be estimated. In this paper, we deduce the analytical partial derivatives of the planar constant with respect to the interface orientation and use them to predict the planar constant. A predicted-Newton's method is proposed to solve the interface positioning equation which takes the predicted planar constant as the initial guess. A great deal of numerical tests are also presented in this paper to verify the robustness of the new scheme. The efficiency of the proposed predicted-Newton's method is compared with the commonly used secant/bisection method by Ahn and Shashkov, and the numerical results indicate that the new method can reduce the iteration steps by 60%∼66% in solving the interface positioning equation and reduce the CPU time by 32%∼39% when implemented in the MoF method.