Engineering Analysis with Boundary Elements 88 (2018) 64–79 Contents lists available at ScienceDirect Engineering Analysis with Boundary Elements journal homepage: www.elsevier.com/locate/enganabound Mixed Discrete Least Squares Meshfree method for solving the incompressible Navier–Stokes equations S. Faraji Gargari a , M. Kolahdoozan a,∗ , M.H. Afshar b a Department of Civil and Environmental Engineering, Amirkabir University of Technology (Tehran, Polytechnic), 424, Hafez Ave, Tehran, Iran b Department of Civil Engineering, Iran University of Science and Technology, Narmak, Tehran, Iran a r t i c l e i n f o Keywords: Mixed Discrete Least Squares Meshfree Discrete Least Squares Meshfree MLS Navier–Stokes a b s t r a c t A Mixed Discrete Least Squares Meshfree (MDLSM) method is proposed in this paper for the solution of in- compressible Navier–Stokes equations. A semi-incremental two-step fractional projection method is first used to discretize the incompressible Navier–Stokes equations, followed by a mixed formulation used to solve the pressure equations. Using the mixed formulation, it is expected that the accuracy of the pressure approximation and in particular the pressure gradients are improved compared with that of conventional solution methods and in particular Discrete Least Squares Meshfree (DLSM) method. DLSM method is based on minimizing the least squares functional defined as the weighted summation of the squared residuals of the differential equation and its boundary conditions. The method is not subject to the Ladyzenskaja–Babuska–Brezzi (LBB) condition since it formulates the problem in the form of a minimization problem rather than a saddle-point problem. A number of numerical experiments are used to evaluate the efficiency of the proposed MDLSM method and to compare its accuracy against the DLSM method. From the results, it is found that the proposed MDLSM method can efficiently simulate the incompressible fluid flow problems. Furthermore, it can be concluded that the MDLSM method has higher accuracy compared with the DLSM method. © 2017 Elsevier Ltd. All rights reserved. 1. Introduction Meshfree methods have attracted many researchers’ attentions for solving the partial differential equations (PDEs) over the last few decades because of their merits compared to mesh-based methods. Al- though the mesh-based methods such as finite volume method (FVM) and finite element method (FEM) are widely used to simulate the engi- neering problems, these methods suffer from the inherent limitation of having to rely on meshes with a properly defined connectivity. Unlike the mesh-based methods, only a set of nodes is used in meshfree meth- ods to discretize the problem domain. The meshfree methods, therefore, can be an efficient alternative to overcome the meshing difficulties in mesh-based methods [1]. The meshfree methods are normally categorized into two major groups regarding the approximation approach; namely kernel function method, and polynomial series method. Smoothed particle hydrody- namic (SPH) and moving particle semi-implicit (MPS) are the most known meshfree methods using the kernel approximation. The SPH method has been efficiently used to solve the flow problems such as free surface flows [2], viscous and heat conducting flows [3], two-fluid modeling [4], multiphase flow [5], and sediment scouring and flushing ∗ Corresponding author. E-mail addresses: saebfaraji@aut.ac.ir (S. Faraji Gargari), mklhdzan@aut.ac.ir (M. Kolahdoozan), mhafshar@iust.ac.ir (M.H. Afshar). [6]. The MPS method is mainly similar to the SPH method. In this method the spatial derivatives are calculated without recording to the gradient of Kernel function. The method successfully has been applied for simulating the free surface [7], wave breaking [8] and multi-phase flow [9, 10] problems. The computational effort of the kernel function method is less than the polynomial method; however the consistency and accuracy of results are higher when the polynomial series method is used [1]. Furthermore, when polynomial series are used, higher-order accuracy and consistency can be achieved by increasing the order of basic functions. This property is more useful when dealing with the complex problems. The methods such as element free Galerkin (EFG), meshless local Petrov–Galerkin (MLPG), and DLSM are the methods using the series polynomial method. Meshfree methods, which are based on the polynomial series approx- imation, fall into two major categories regarding the discretization form; namely weak-form and strong-form. In the weak-form formulations, re- quired consistency of the trial function can be reduced via integration by parts. Such a property, which is also useful for improving the accu- racy of the results, is absent in the strong-form formulations. Therefore, the required order of the basic function is lower in the weak-form com- pared to the strong-form. However, weak-form methods require back- https://doi.org/10.1016/j.enganabound.2017.12.018 Received 2 April 2017; Received in revised form 4 December 2017; Accepted 27 December 2017 0955-7997/© 2017 Elsevier Ltd. All rights reserved.