Moment-of-fluid analytic reconstruction on 3D rectangular hexahedrons

doi:10.1016/j.jcp.2020.109346

Journal of Computational Physics

Volume 409, 15 May 2020, 109346

https://doi.org/10.1016/j.jcp.2020.109346 Get rights and content

Highlights

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Parametrization of the centroid locus at fixed volume on rectangular hexahedrons.
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Analytic formulas for the objective function for the MOF optimization problem.
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Robust method up to 200 times faster than geometric approaches.
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Improvement of the flood algorithm for the MOF method in convex polyhedral cells.

Abstract

The moment-of-fluid method (MOF) is a second-order accurate interface reconstruction method which can be seen as an extension of the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). MOF involves a computationally intensive minimization problem that needs to be solved on every cell containing several materials. We propose a new fast and robust reconstruction algorithm to tackle this problem on rectangular hexahedral cells. Our approach uses explicit analytic formulas of the objective function that does not use any geometric computations such as half-space–polyhedron intersections. The numerical results show that the proposed method is more robust and more than 200 times faster than the original approach. Additionally, we propose a faster reconstruction algorithm on convex polyhedral cells. All the methods presented in this article have been implemented and verified on the open-source code Notus.

Introduction

Common engineering problems involve several materials interacting with each other. The numerical simulations of these phenomena require the tracking of the location of the materials over time. Across the interface between two materials, some physical phenomena must be described such as the heat or mass transfer. Any numerical errors on the location of the interface have an impact on the physics of the whole problem. As a result, the numerical simulations require accurate tracking methods. In this article we only consider numerical methods designed for the Eulerian framework where the velocity field is defined on the whole domain and where the motion of the materials is independent of the underlying mesh.

A lot numerical strategies have been developed to track the materials in this context such as the level-set method [1], the front-tracking method [2], and the volume-of-fluid method with piecewise linear interface construction (VOF-PLIC). For the latter, any cell containing two materials is partitioned by a linear interface such that the volume of each part contains exactly the same volume as the real location of the material. The most common application of the VOF-PLIC method is the advection of the materials which is composed of two steps. In the first step, the geometry of the partition is advected with a Lagrangian method, and then the volume of each material is computed from the intersection of the geometry with the underlying mesh. In the second step, named reconstruction, a new partition is computed using the volume in each cells and their neighborhood.

Recently the moment-of-fluid method (MOF) [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22] was introduced as a second-order accurate extension of the VOF-PLIC method for multi-material reconstruction. Besides the volume of the materials, MOF involves the centroids of each material which allows to reconstruct the partition with only the information contained in the cell. Furthermore, this method offers a straightforward way to represent $n \geq 2$ materials in the same cell [5], [6]. The improved accuracy of this method is at the expense of the time to partition the cell which involves a computationally intensive minimization problem. In this article, we present a fast and robust reconstruction strategy to solve this minimization problem on rectangular hexahedrons. Note that we only address the problem of the reconstruction, the advection can be done, for instance, using a Lagrangian remapping method [4] or a directional splitting method [12].

The currently proposed method falls under the continuity of the improvements made to MOF during the past decade. MOF was originally introduced in 2D on polygonal cells [3], [4]. It was quickly extended to multi-material reconstruction in 3D by Ahn & Shashkov [5] and in 2D by Dyadechko & Shashkov [6]. Note that the latter defines a convenient error criterion and provides a convergence study of the MOF reconstruction. Another effort to extend MOF to any coordinate system was made by Anbarlooei & Mazaheri [8] who have extended MOF to axisymmetric meshes. As MOF is a PLIC method, it is vulnerable to filaments that can not be advected. Jemison et al. [16] solved this problem by adapting MOF to filament capturing which can be done in a straightforward way by using a n-material reconstruction and allowing two materials to be identical in one cell. To improve the robustness of MOF, many solutions were proposed, such as the one advanced by Hill & Shashkov [13] which consists in changing slightly the minimization problem of MOF to minimize the centroid difference on both the material and its complementary. Since the objective function of MOF contains local minima where the minimization algorithms are prone to fall into, Qing et al. [22] have proposed a method in 2D that finds all the minima of the MOF problem and selects the best one. This gain of robustness is at the expense of the runtime of the algorithm.

MOF was designed to be used in conjunction with other methods. Ahn & Shashkov [7] proposed an interaction of MOF with an adaptive mesh refinement (AMR) strategy where the centroid difference is used as a criterion for mesh refinement. The coupling with an Arbitrary Lagrangian-Eulerian (ALE) strategy is also found among many authors [9], [11], [14]. Valuable implementation details can be found in many publications, for instance, in [10] MOF was coupled with a code based on the finite element method (FEM) and in [15] MOF is used in a compressible context. In [12] and [21], MOF is coupled with the levelset method (CLSMOF) and more recently, Kikinzon et al. [20] proposed a data structure to represent the partition of the multi-material reconstruction to simplify the interaction with other methods.

The most expensive part of the MOF reconstruction is the evaluation of the objective function and its partial derivatives at each iteration of the minimization algorithm which involves computationally intensive geometrical manipulations. To tackle this problem, Chen & Zhang proposed analytic formulas for the partial derivatives of the objective function on convex polyhedral cells [17] and convex polygonal cells [19]. However, their methods requires a prior evaluation of the objective function with a geometric approach that still remains expensive. To completely avoid these computationally intensive geometric manipulations, another approach is to express the objective function with analytic formulas. This has been addressed by Lemoine et al. [18] in 2D for rectangular cells as we will discuss in section 2.3. In this paper, we propose a 3D extension of this method to rectangular hexahedral cells. This method can be applied to geometry tracking on any meshes composed of this kind of cells such as rectilinear grids with or without AMR.

Section snippets

The moment-of-fluid problem

The MOF problem can be summarized as follows. Consider $Ω \subset R^{3}$ a portion of space, for instance a polyhedral cell and $M \subset Ω$ the location of a given material in Ω, as depicted on the left of Fig. 1. Where the VOF-PLIC method requires the information of the volume of material from the cell and its neighborhood, MOF embeds all the information within the cell. Besides the volume of material, the MOF reconstruction involves the centroid of M denoted by $C (M)$ and defined by: $C (M) = \frac{1}{vol (M)} \int_{M} x d x$ As a PLIC

Description

In the remainder of this article, we consider a rectangular hexahedron $Ω = [0, c_{1}] \times [0, c_{2}] \times [0, c_{3}]$ . Recall that MOF consists in finding the best approximation of the material location $M \subset Ω$ . We define $V = vol (M)$ as the volume of M — or reference volume — and $C^{⋆}$ as its centroid — or reference centroid. From the definition of the set of the approximations $A$ in equation (2), any element of this set shares the same volume $V$ as the material location M. We introduce in the following equation the normal $n (θ, ϕ)$

Conclusion

In this article we have developed a new fast and robust approach to solve the minimization problem of the moment-of-fluid method in 3D on rectangular hexahedrons. The key idea is to find a global parametrization in spherical coordinates of the locus of the centroids at fixed volume. This allows to compute the objective function and its gradient with fully analytical formulas which avoid the use of a computationally intensive geometric flood algorithm. The numerical results show that our

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

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An efficient bilinear interface reconstruction algorithm and consistent multidimensional unsplit advection scheme for accurate capturing of highly-curved interfacial shapes on structured grids
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Citation Excerpt :
It has been shown [27] that the MOF method yields relatively low interface reconstruction errors and higher order of grid convergence of this error. There have been various implementations and extensions made in the recent past to the original MOF method [22], such as, the analytical interface reconstruction [28,29] in two dimensions [30,31] which was then extended to three dimensions [32], compressible flow simulation applications [33], accurate capture of the thin filaments [34], two-plane interface reconstruction [35], and to many other applications and grid types [36–44]. In a recent study, Asuri Mukundan et al. [45] compared the MOF and CLSVOF methods for variety of tests and found that the MOF method outperformed the CLSVOF method in terms of accuracy of interface reconstruction.
This paper presents a hybrid moment of fluid–level set (HyMOFLS) method of liquid/gas interface reconstruction for the application to simulate primary atomization of liquid fuel. This method combines the moment of fluid (MOF) method and the level set framework in the coupled level set volume of fluid (CLSVOF) method. In this hybrid method, the MOF and CLSVOF methods are used to reconstruct the interface in the under-resolved and the resolved regions of the flow, respectively. An interface surface resolution metric called interface resolution quality (IRQ) is introduced to identify and classify these two flow regions in the computational domain. Such a strategy classifies/tags each computational cell with MOF or CLSVOF method based on a threshold value for the IRQ. The MOF method uses liquid volume fraction as well as centroids of liquid and gas phases for the interface reconstruction. The CLSVOF method uses the level set for describing the interface and liquid volume fraction for conserving the mass. The phase centroids in the HyMOFLS method are computed and transported on-the-fly during the cell tagging process. The transport of the liquid volume fraction, level set, and the phase centroids are performed using a directionally split algorithm. This algorithm is coupled with the Navier-Stokes equations solver that uses ghost fluid method as well as consistent mass and momentum flux computation for the momentum equation. Various numerical tests that exhaustively assess the capabilities, accuracy, and computational time consumption of the HyMOFLS method under multiple flow conditions and configurations are presented. The results from these tests suggest that this hybrid framework is capable of well capturing the liquid/gas interface belonging to thin and under-resolved structures that are often encountered in simulations of turbulent atomization of liquids. Following these tests, a detailed parametric study on the threshold value of IRQ is presented to test its effect on the accuracy of interface reconstruction. Finally, this hybrid framework is employed to simulate turbulent jet injection and pre-filming planar Airblast atomization cases of engineering applications. The proposed HyMOFLS method is found to achieve a good balance between the accuracy and the computational cost of reconstructing the liquid/gas interface for complex and turbulent atomization configurations.

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Moment-of-fluid analytic reconstruction on 3D rectangular hexahedrons

Highlights

Abstract

Introduction

Section snippets

The moment-of-fluid problem

Description

Conclusion

Declaration of Competing Interest

J. Comput. Phys.

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Comput. Fluids

Comput. Fluids

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