Exact finite volume particle method with spherical-support kernels

doi:10.1016/j.cma.2016.12.015

Computer Methods in Applied Mechanics and Engineering

Volume 317, 15 April 2017, Pages 102-127

https://doi.org/10.1016/j.cma.2016.12.015 Get rights and content

Highlights

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Presenting a new 3-D FVPM that features spherical-supported top-hat kernel.
•
Introducing an efficient algorithm for sphere surface partitioning using set operations.
•
Introducing an exact method for computing the area vector and surface area of the spherical sub-surfaces.
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Introducing an exact method for computing the volume of particle.

Abstract

The Finite Volume Particle Method (FVPM) is a meshless method for simulating fluid flows which includes many of the desirable features of mesh-based finite volume methods. In this paper, we develop a new 3-D FVPM formulation that features spherical kernel supports. The formulation is based on exact integration of interaction vectors constructed from top-hat kernels. The exact integration is obtained by an innovative surface partitioning algorithm as well as precise area computation of the sphere subsurfaces. Spherical-support FVPM improves the recently developed cubic-support version in two main aspects: spherical kernels have no directionality and result in smooth interactions between particles, leading to an improved method. We present three test cases that illustrate the improved accuracy and robustness brought by the spherical kernel. Although computations are 1.5 times slower on spherical support than cubic support, the cost is more than compensated by lower error with a higher convergence rate.

Introduction

The Finite Volume Particle Method (FVPM) is a meshless arbitrary Lagrangian–Eulerian (ALE) method introduced by Hietel et al. [1] in 2000 for compressible flows. This method has then been used in a wide range of applications such as incompressible flows [2], solid mechanics [3], fluid–structure interactions [4], [5], free-surface flows in Pelton turbines [6], [7] and silt erosion [8].

The FVPM includes many of the attractive features of both particle methods, such as Smoothed Particle Hydrodynamics (SPH) [9], and conventional mesh-based Finite Volume Methods (FVM) [10]. In FVPM, like in SPH, computational nodes usually move with the material velocity, which is compatible with the Lagrangian form of the equation of motion. This enables the method to handle moving interface problems like free-surface flows, without issues of mesh deformation or tangling. FVPM also does not require mesh generation which is a costly stage in simulation of flows with complex geometries.

Similarly to FVM, FVPM is locally conservative and consistent regardless of any variation in volume sizes. In fact, FVPM can be interpreted as a generalization of conventional mesh-based FVM [11]. In FVM, the computational domain is partitioned into finite control volumes with defined surfaces. The area vector of the surfaces is used as a weight for the flux exchanged between the control volumes. In FVPM, control volumes are replaced by overlapping particles and the exchange occurs through the interfaces defined by overlapping regions. For each pair of overlapping particles, two interaction vectors are defined and their difference is analogous to the area vector in FVM. The interaction vectors can be computed by either numerical or exact integration. Numerical integration is costly and approximate, and is mostly used for bell-shaped kernels [2], [12], [13]. Exact integration introduced by Quinlan et al. [5] is based on top-hat kernels, resulting in a reasonable compromise between efficiency and accuracy. This method was developed for 2-D computations using circular and rectangular top-hat kernels [14]. Recently, Jahanbakhsh et al. [15] developed a 3-D FVPM formulation which features exact integration based on cubic-supported top hat kernels.

Employing rectangular or cubic-supported top-hat kernels in FVPM, results in two important issues. First of all, these kernels have directionality, which destroys volume conservation in pure rigid body rotation and accordingly impairs the accuracy of the method. Secondly, the particles interaction can result in hard contact which causes high-frequency errors. Both issues disappear with circular or spherical-supported top-hat kernels. For the first time, we present in this paper a technique which uses spherical top-hat kernels for exact integration of the FVPM interaction vectors. This technique is based on an innovative surface partitioning algorithm accounting for logical set operations and precise area evaluation to handle complicated particle intersections robustly.

This paper is organized as follows. In the next section, we introduce the governing equations required for fluid flow computations. Then, a general introduction to FVPM is presented which highlights the technical challenges of spherical-supported kernel computations. Next, we present our solution to tackle the aforementioned challenges. In the fifth section, we present the time integration and flux discretization schemes and finally, in the last section, we present three test cases to validate the method and show its application for a real engineering problem.

Section snippets

Governing equations

The equations of motion for isothermal and weakly compressible flows are derived from the mass and linear momentum conservation laws $\frac{d ρ}{d t} = - ρ \nabla \cdot C$ and $ρ \frac{d C}{d t} = \nabla \cdot (s - p I) + ρ g$ where $\frac{d}{d t}$ denotes the substantial time derivative, $ρ$ is the density, $C$ is the fluid velocity vector, $p$ is the static pressure, $s$ is the deviatoric stress tensor and $g$ represents the gravitational acceleration. For Newtonian fluids, $s$ reads $s = 2 μ (\dot{ε} - \frac{1}{3} tr (\dot{ε}) I)$ where $μ$ is the dynamic viscosity and $\dot{ε}$ is the deformation rate tensor

Formulation of the method

The FVPM formulation for conservation laws (6) reads $\frac{d}{d t} (U_{i} V_{i}) = \sum_{j} (U_{i j} \otimes {\dot{x}}_{i j} - F_{i j}) \cdot Δ_{i j} + (U_{b} \otimes {\dot{x}}_{b} - F_{b}) \cdot B_{i}$ and $\frac{d V_{i}}{d t} = \sum_{j} {\dot{x}}_{i j} \cdot Δ_{i j} + {\dot{x}}_{b} \cdot B_{i}$ with $Δ_{i j} = Γ_{i j} - Γ_{j i}$ ${\dot{x}}_{i j} = ({\dot{x}}_{j} \cdot Γ_{i j} - {\dot{x}}_{i} \cdot Γ_{j i}) \frac{Δ_{i j}}{Δ_{i j} \cdot Δ_{i j}}$ $B_{i} = - \sum_{j} Δ_{i j}$ where $U_{i}$ is the conserved variable of $i$ th particle, $V_{i}$ is its volume, $U_{i j}$ and $F_{i j}$ are the conserved variable and flux function at the interface of particles $i$ and $j$ , respectively, whereas ${\dot{x}}_{i j}$ is the velocity at which the interface moves. Similarly, $U_{b}$ , $F_{b}$ and ${\dot{x}}_{b}$ are the conserved variable, flux function and

Computation of exact geometric properties of spherical particles

The key innovations of the present work are presented in this section. The problem is the exact computation of particle interaction vectors $Γ_{i j}$ and volume $V_{i}$ for spherical particles using top-hat kernels. The new methods are presented in stages, beginning with the identification and representation of elementary surfaces formed by sphere intersections (Section 4.1). The second stage (Section 4.2) is the computation of surface area and projected area vector of these elementary surfaces, leading

Time integration and flux discretization

The surface partitioning algorithm for spherical kernel has been implemented in the computer program SPHEROS [8], [23]. For completeness, we introduce discretization and stabilizing parameters briefly hereafter. Readers are referred to [15] for more details, since these are the same as in the previously published work about FVPM with cubic particle.

In this study, we use the second-order explicit Runge–Kutta scheme for time integration. In this scheme, the field variables are updated for the

Validation

In order to validate the 3-D FVPM featuring spherical kernel, we present three test cases addressing key aspects of the method. We first verify the directionality of the method in 2-D and 3-D flow computations. For this purpose, rigid body rotational flow is studied for the convergence to the analytical solution. The analysis and comparison are performed for both spherical and cubic-supported kernels.

Secondly, to check the smoothness of the particle interaction, we study the behavior of the

Conclusion

The 3-D FVPM that features cubic-supported kernel is an attractive method for the computation of fluid flow problems including complex moving boundaries due to its consistency, conservation and ALE properties. However, directionality and hard interaction of the cubic-supported kernels can badly affect the accuracy and stability of the computations. Employing a spherical-supported top-hat kernel can alleviate these issues but it has not been used so far due to the lack of precise and fast

Acknowledgments

This work has been performed within the Swiss Competence Center on Energy Research–Supply of Electricity, GPU SPHEROS project Grant No. 17568.1 PFEN IW, with the support of the Swiss Commission for Technology and Innovation and GE Renewable Energy.

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Exact finite volume particle method with spherical-support kernels

Highlights

Abstract

Introduction

Section snippets

Governing equations

Formulation of the method

Computation of exact geometric properties of spherical particles

Time integration and flux discretization

Validation

Conclusion

Acknowledgments

J. Comput. Phys.

Comput. Methods Appl. Mech. Engrg.

Comput. Phys. Comm.

Comput. Methods Appl. Mech. Engrg.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

J. Comput. Phys.

A finite-volume particle method for compressible flows

Math. Models Methods Appl. Sci.

Fast exact evaluation of particle interaction vectors in the finite volume particle method

Flow simulation of a Pelton bucket using finite volume particle method

IOP Conf. Ser.: Earth Environ. Sci.

Flow simulation of jet deviation by rotating pelton buckets using finite volume particle method

ASME. J. Fluids Eng.

Simulation of Silt Erosion Using Particle-Based Methods

Smoothed particle hydrodynamics-theory and application to non-spherical stars

Mon. Not. R. Astron. Soc.