Analysis of the incompressibility constraint in the Smoothed Particle Hydrodynamics method

Smoothed particle hydrodynamics is a particle-based, fully Lagrangian, method for fluid-flow simulations. In this work, fundamental concepts of the method are first briefly recalled. Then, we present a thorough comparison of three different incompressibility treatments in SPH: the weakly compressible approach, where a suitably-chosen equation of state is used; and two truly incompressible methods, where the velocity field projection onto a divergence-free space is performed. A noteworthy aspect of the study is that, in each incompressibility treatment, the same boundary conditions are used (and further developed) which allows a direct comparison to be made. Problems associated with implementation are also discussed and an optimal choice of the computational parameters has been proposed and verified. Numerical results show that the present state-of-the-art truly incompressible method (based on a velocity correction) suffer from density accumulation errors. To address this issue, an algorithm, based on a correction for both particle velocities and positions, is presented. The usefulness of this density correction is examined and demonstrated in the last part of the paper

On Meshfree GFDM Solvers for the Incompressible Navier-Stokes Equations

Meshfree solution schemes for the incompressible Navier--Stokes equations are usually based on algorithms commonly used in finite volume methods, such as projection methods, SIMPLE and PISO algorithms. However, drawbacks of these algorithms that are specific to meshfree methods have often been overlooked. In this paper, we study the drawbacks of conventionally used meshfree Generalized Finite Difference Method~(GFDM) schemes for Lagrangian incompressible Navier-Stokes equations, both operator splitting schemes and monolithic schemes. The major drawback of most of these schemes is inaccurate local approximations to the mass conservation condition. Further, we propose a new modification of a commonly used monolithic scheme that overcomes these problems and shows a better approximation for the velocity divergence condition. We then perform a numerical comparison which shows the new monolithic scheme to be more accurate than existing schemes

Simulation of flows with violent free surface motion and moving objects using unstructured grids

This is the peer reviewed version of the following article: [Löhner, R. , Yang, C. and Oñate, E. (2007), Simulation of flows with violent free surface motion and moving objects using unstructured grids. Int. J. Numer. Meth. Fluids, 53: 1315-1338. doi:10.1002/fld.1244], which has been published in final form at https://doi.org/10.1002/fld.1244. This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Self-Archiving.A volume of fluid (VOF) technique has been developed and coupled with an incompressible Euler/Navier–Stokes solver operating on adaptive, unstructured grids to simulate the interactions of extreme waves and three-dimensional structures. The present implementation follows the classic VOF implementation for the liquid–gas system, considering only the liquid phase. Extrapolation algorithms are used to obtain velocities and pressure in the gas region near the free surface. The VOF technique is validated against the classic dam-break problem, as well as series of 2D sloshing experiments and results from SPH calculations. These and a series of other examples demonstrate that the ability of the present approach to simulate violent free surface flows with strong nonlinear behaviour.Peer ReviewedPostprint (author's final draft

Modelling Shear Flows with SPH and Grid Based Methods

Given the importance of shear flows for astrophysical gas dynamics, we study the evolution of the Kelvin-Helmholtz instability (KHI) analytically and numerically. We derive the dispersion relation for the two-dimensional KHI including viscous dissipation. The resulting expression for the growth rate is then used to estimate the intrinsic viscosity of four numerical schemes depending on code-specific as well as on physical parameters. Our set of numerical schemes includes the Tree-SPH code VINE, an alternative SPH formulation developed by Price (2008), and the finite-volume grid codes FLASH and PLUTO. In the first part, we explicitly demonstrate the effect of dissipation-inhibiting mechanisms such as the Balsara viscosity on the evolution of the KHI. With VINE, increasing density contrasts lead to a continuously increasing suppression of the KHI (with complete suppression from a contrast of 6:1 or higher). The alternative SPH formulation including an artificial thermal conductivity reproduces the analytically expected growth rates up to a density contrast of 10:1. The second part addresses the shear flow evolution with FLASH and PLUTO. Both codes result in a consistent non-viscous evolution (in the equal as well as in the different density case) in agreement with the analytical prediction. The viscous evolution studied with FLASH shows minor deviations from the analytical prediction.Comment: 16 pages, 17 figure

Simulating water-entry/exit problems using Eulerian-Lagrangian and fully-Eulerian fictitious domain methods within the open-source IBAMR library

In this paper we employ two implementations of the fictitious domain (FD) method to simulate water-entry and water-exit problems and demonstrate their ability to simulate practical marine engineering problems. In FD methods, the fluid momentum equation is extended within the solid domain using an additional body force that constrains the structure velocity to be that of a rigid body. Using this formulation, a single set of equations is solved over the entire computational domain. The constraint force is calculated in two distinct ways: one using an Eulerian-Lagrangian framework of the immersed boundary (IB) method and another using a fully-Eulerian approach of the Brinkman penalization (BP) method. Both FSI strategies use the same multiphase flow algorithm that solves the discrete incompressible Navier-Stokes system in conservative form. A consistent transport scheme is employed to advect mass and momentum in the domain, which ensures numerical stability of high density ratio multiphase flows involved in practical marine engineering applications. Example cases of a free falling wedge (straight and inclined) and cylinder are simulated, and the numerical results are compared against benchmark cases in literature.Comment: The current paper builds on arXiv:1901.07892 and re-explains some parts of it for the reader's convenienc

Performance of algebraic multigrid methods for non-symmetric matrices arising in particle methods

Large linear systems with sparse, non-symmetric matrices arise in the modeling of Markov chains or in the discretization of convection-diffusion problems. Due to their potential to solve sparse linear systems with an effort that is linear in the number of unknowns, algebraic multigrid (AMG) methods are of fundamental interest for such systems. For symmetric positive definite matrices, fundamental theoretical convergence results are established, and efficient AMG solvers have been developed. In contrast, for non-symmetric matrices, theoretical convergence results have been provided only recently. A property that is sufficient for convergence is that the matrix be an M-matrix. In this paper, we present how the simulation of incompressible fluid flows with particle methods leads to large linear systems with sparse, non-symmetric matrices. In each time step, the Poisson equation is approximated by meshfree finite differences. While traditional least squares approaches do not guarantee an M-matrix structure, an approach based on linear optimization yields optimally sparse M-matrices. For both types of discretization approaches, we investigate the performance of a classical AMG method, as well as an AMLI type method. While in the considered test problems, the M-matrix structure turns out not to be necessary for the convergence of AMG, problems can occur when it is violated. In addition, the matrices obtained by the linear optimization approach result in fast solution times due to their optimal sparsity.Comment: 16 pages, 7 figure