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

doi:10.1016/j.jcp.2018.12.038

Journal of Computational Physics

Volume 384, 1 May 2019, Pages 60-76

https://doi.org/10.1016/j.jcp.2018.12.038 Get rights and content

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 $60 % \sim 66 %$ in solving the interface positioning equation and reduce the CPU time by $32 % \sim 39 %$ when implemented in the MoF method.

Section snippets

Introduction and motivation

The PLIC (Piecewise Linear Interface Calculation) is one of the most famous interface reconstruction methods in the multi-material flow simulation. The basic idea of the PLIC method is to use a planar interface in a mixed cell to approximate the material interface which can be expressed as $n \cdot x + d = 0$ where $n = (\sin θ \cos φ, \sin θ \sin φ, \cos θ)$ is the normal of the approximate interface, $(θ, φ)$ is the orientation of the interface and d is the planar constant. In the PLIC-VoF (Volume of Fluid) method, the

Centroid rotation rule

In this section, we will introduce the “centroid rotation rule” which is derived from the volume conservation requirement. This rule was mentioned in our previous paper [26], and it will be rigorously proved for an arbitrary polyhedron in this section. The “centroid rotation rule” is the foundation for deducing the analytical partial derivatives of the planar constant.

For an arbitrary polyhedron, some faces may be non-planar. And in order to deal with the non-planarity, we introduce an

The partial derivatives of the planar constant

With a given material volume $V^{ref}$ , the planar constant $d (θ, φ)$ is a function of the interface orientation $(θ, φ)$ . According to the definition of the partial derivatives, we have $\frac{\partial d}{\partial θ} = \lim_{Δ θ \to 0} \frac{d (θ + Δ θ, φ) - d (θ, φ)}{Δ θ}$ $\frac{\partial d}{\partial φ} = \lim_{Δ φ \to 0} \frac{d (θ, φ + Δ φ) - d (θ, φ)}{Δ φ}$ where Δθ and Δφ denote infinitesimal rotations of the interface.

Let us take the partial derivative with respect to θ as an example. Fig. 4 shows the two related interfaces $Γ_{1}$ ( $p_{1} p_{2} p_{3}$ ) and $Γ_{2}$ ( $q_{1} q_{2} q_{3}$ ), which are defined by $Γ_{1} : n (θ, φ) \cdot x + d (θ, φ) = 0$ and $Γ_{2} : n (θ + Δ θ, φ) \cdot x + d ($

The predicted Newton's iterative method

The Newton's method is a good choice for solving Eq. (2) but it also brings several problems in practice. Firstly, the $n^{th}$ iteration step of the Newton's method is $d^{n + 1} = d^{n} - \frac{V (d^{n})}{V^{'} (d^{n})}$ which involves the derivative $V^{'} (d^{n})$ and it must be obtained efficiently. Secondly, the potential divergence of the Newton's method must be avoided. Finally, an appropriate initial guess should be used because it will influence the efficiency of the Newton's method significantly.

In this section, we solved the three

Numerical result

A large number of numerical experiments are presented in this section. In this section, firstly, the analytical derivatives in Eq. (24) and Eq. (25) are verified on a polyhedron with non-planar faces. Then the efficiency and the robustness will be compared between the predicted-Newton's method and the secant/bisection method used by Ahn and Shashkov [8]. In their method, the secant method is stabilized with the bisection iteration to guarantee the convergence. Finally, the predicted-Newton's

Conclusion

In this paper, the interface positioning equation is studied from another perspective by focusing on the relation between the planar constant and the interface orientation for a given volume fraction. The analytical partial derivatives of the planar constant with respect to the interface orientation are deduced by the geometry analysis. These partial derivatives can be used to predict the planar constant whenever the interface orientation is changed which is useful for the iterative methods to