Simulating free surface problems using Discrete Least Squares Meshless method

Abstract

A Discrete Least Squares Meshless (DLSM) method is presented here for the simulation of incompressible free surface flows. The governing equations of the mass and momentum conservations are solved in a Lagrangian form using a pressure projection method. Since there are no particles in the outer region of the free surface, the particle density will drop significantly. Free surfaces are, therefore, resolved by tracking particles with highly reduced density. A fully least squares approach is used in both function approximation and the discretization of the governing differential equations in space. The meshless shape functions are derived using the Moving Least Squares (MLS) method of function approximation. The discretized equations are obtained via a discrete least squares method in which the sum of the squared residuals are minimized with respect to unknown nodal parameters. The method enjoys the advantage of producing symmetric, positive definite matrixes for the cases considered. The method can be viewed as a truly meshless method since it does not need any mesh for both field variable approximation and the construction of system matrices. Two free surface problems namely dam break and evolution of a drop with an initial known velocity are solved to test the accuracy of the proposed method. The results show the ability of the proposed method to solve complex fluid dynamic problems with moving free surface boundaries.

Introduction

Free surface modeling is of great importance in the field of fluid mechanics. The difficulties of simulating these problems are mainly due to the complex governing equations, variable boundary conditions and also deformable computational domains. These difficulties make analytical approaches incompatible to deal with free surface problems. An encouraging alternative to investigate these complicated problems are numerical methods. Numerical methods can be classified in two main groups. The first is the mesh-based methods like Finite Difference Method (FDM), Finite Volume Method (FVM) and Finite Element Method (FEM). The second group is the meshless methods which do not use any grid structures. This property eliminates costly mesh generation procedure especially in modeling transient free surface problems.

Meshless methods are computationally more expensive than mesh-based methods. It is generally true, however, that the computational effort required by the mesh-based methods for both the simulation and mesh generation is more than that required by the meshless methods for the simulation alone. This is particularly true for the transient free surface problems where the mesh-based methods require frequent mesh updating.

Smoothed Particle Hydrodynamic (SPH) is one of the earliest meshless methods proposed by Gingold and Monaghan to simulate astrophysical hydrodynamics problems [1]. In this method, kernel approximation, based on the theory of integral interpolation, is used to interpolate unknown variables. The governing equations are transformed to discrete forms including moving particles as interpolation points. SPH has been successfully used to simulate different fluid mechanic problems such as study of gravity currents [2], free surface Newtonian and non-Newtonian flows [3], simulation of impulsive waves due to landslides [4] and wave propagation [5]. Ataie et al. modified the source term of the pressure Poisson equation to improve the stability and accuracy of the SPH method in solving free surface problems [6]. Koshizuka and Oka proposed a similar meshless method called Moving Particle Semi-implicit (MPS) method in which each particle is moved through interactions with neighboring particles using a kernel function [7]. Differential operators like Laplacian, Gradient and Divergence were transformed to account for the interaction among moving particles. This method has also been used to investigate dam break [8], vapor explosion [9], and wave breaking [10] problems. The MPS method, however, violates the momentum conservation due to the fact that the kernel functions used in MPS by Koshizuka et al. do not have continuous first derivatives [10], [11]. Ataie and Farhadi considered various kernel functions to improve the stability of MPS method [8].

Diffuse Element Method (DEM) is the first weak-form meshless method introduced by Nayroles et al. [12]. Shape functions of the DEM are created using MLS method [25] and the governing equations are discretized by a Galerkin weak-form over the problem domain. A set of background mesh is, however, needed to compute the integrals derived from the Galerkin approach. Element Free Galerkin (EFG) as the extended version of DEM method was proposed by Belytshko et al. [13]. The stiffness matrices are symmetric because of employment of the Galerkin procedure to discretize governing equations. Essential boundary conditions are implemented efficiently by penalty method. EFG is an efficient and accurate method in which the rate of convergence is higher than that of FEM. The EFG can be employed successfully on both regular and irregular nodal distribution. The EFG, however, is not a truly meshless method since the integrals are evaluated on a background cell structures. This method has been applied to investigate various engineering problems such as three dimensional structural analysis [26], static and dynamic analysis of shell structures [27], 2D elastic analysis [28] and unsteady nonlinear heat transfer [29].

The Meshless Local Petrov–Galerkin (MLPG) is a newly developed meshless method in which both interpolation and integration can be performed without requiring background mesh [14]. Background mesh is avoided because a local weak-form is used in this method. A local quadrature domain having a regular and simple shape is defined for each field node to implement the numerical integrations. As in the EFG, shape functions are created by the MLS approximation. The efficiency of MLPG is adversely affected by asymmetric coefficient matrix. Furthermore, MLPG encounters some difficulties when numerical integrations are carried out for nodes on and around the boundaries. The MLPG has been extensively employed to solve a wide range of problems such as fluid dynamic and heat transfer problems [30], solid mechanic problems [31] and analysis of rubber-like materials [32].

The Finite Point (FP) method has also been developed by Onate et al. to solve fluid mechanics problems [15]. In this method, interpolation is performed by a weighted least squares procedure while the integrals are evaluated using point collocation method. Stabilization of the convective terms and Neumann boundary condition are easily and efficiently accomplished in this method. However, Atluri and Zhu showed that the method is very sensitive to the choice of collocation points [14].

Recently a new class of meshless methods based on least squares approximations is proposed by different researchers. Afshar et al. extended Discrete Least Squares Meshless (DLSM) method for the solution of elliptic and convection-dominated problems [16], [17], [18], [19]. This method is based on minimizing a least squares functional with respect to the nodal parameters. The least squares functional is formed as the weighted squared residual of the governing differential equations and its boundary conditions at nodal points.

In this paper, DLSM method is extended to simulate free surface problems in a Lagrangian framework leading to complete elimination of numerical diffusion due to existence of first order spatial derivatives in an Eulerian formulation. The governing Navier–Stokes equations are solved in time by a two-step fractional method. The velocity field is integrated in time without enforcing incompressibility in the first step. In the second step, a Poisson equation of pressure is used to satisfy incompressibility condition. Boundary conditions are imposed efficiently by a penalty method. Two free surface problems of dam break and evolution of a drop are solved by the present method and the results are compared with the available analytical or experimental solutions. The results indicate the efficiency and accuracy of the proposed method to solve free surface problems considered.