A least squares based meshfree technique for the numerical solution of the flow of viscoelastic fluids: A node enrichment strategy Mohsen Lashkarbolok a , Ebrahim Jabbari b,n , Jerry Westerweel c a Department of Engineering, Golestan University, Golestan, Iran b School of Civil Engineering, Iran University of Science and Technology, Iran c Laboratory for Aero and Hydrodynamics, Delft University of Technology, Netherlands article info Article history: Received 1 January 2013 Received in revised form 5 July 2014 Accepted 31 July 2014 Keywords: Meshfree method Least squares techniques Viscoelastic fluid Radial point interpolation methods Artificial compressibility abstract A fully implicit least-squares-based meshfree method is used to solve the governing equations of viscoelastic fluid flow. Here, pressure is connected to the continuity equation by an artificial compressibility technique. A radial point interpolation method is used to construct the meshfree shape functions. The method is used to solve two benchmark problems. Thanks to the flexibility of meshfree methods in domain discretization, a simple node enrichment strategy is used to discrete the problem domain more purposefully. It is shown that the introduced enrichment process have a positive effect on the accuracy of the results. & 2014 Elsevier Ltd. All rights reserved. 1. Introduction There are numerous investigations about the numerical simu- lation of viscoelastic fluid flow. Townsend used a finite difference technique to simulate the flow of viscoelastic fluid past stationary and rotating cylinders [1]. Viriyayuthakorn and Caswell simulated the flow of viscoelastic fluid using finite element method [2]. Darwish and Whiteman presented a staggered-grid, finite-volume method for the numerical simulation of isothermal viscoelastic liquids [3]. A comprehensive review in the field of numerical simulation of the viscoelastic fluid flow is performed by Owens and Timothy [4]. Most of the studies in this field are carried out by finite element and finite volume methods. These methods require mesh or grid to discretize domain of a problem. The subdivision of the domain into such components is laborious and difficult necessitating complex mesh or grid generation. Further, if adap- tivity processes are used, generally large areas of the problem have to be remeshed [5]. The main feature of a meshfree method is its ability to more easily discretize the domain of a problem using some scattered nodes instead of elements or grids. This ability is a promising approach to perform an effective refinement procedure. In the present study, a least-squares-based meshfree technique referred to as Collocated Discrete Least Squares (CDLS), that was presented in [6], is used to solve the governing equations. A Radial Point Interpolation Method (RPIM) using Multi-Quadratic Radial Basis Functions (MQ-RBF) is used to construct meshfree shape functions. By this kind of function approximation we suppose that the exponential behavior of the stresses can be captured better in comparison with polynomial basis functions usually used in conventional numerical methods [4]. Here, the equations are considered to be solved implicitly. It means that the evolution of the pressure, velocity and stresses are computed simultaneously at each time step. To connect the pressure to the continuity equation, conventional artificial compressibility technique is used. Although the problem is assumed to be steady state, the governing equa- tions are solved in time to the point that a steady-state solution is obtained. In this paper, a node enrichment strategy is used to discretize the domain of the problem according to some informa- tion from a prior solution. Some researchers have used adaptive refinement techniques to obtain more accurate solution in the simulation of viscoelastic fluid flow (for example [7,8]). However most of the efforts have been done using finite element and other mesh-based methods and authors believe that any adaptive process (including node enrichment) can be performed more simply in a meshfree technique. Adding nodes is simpler and more flexible than adding elements or grids to the computational domain of the problem. To assess the accuracy, the presented procedure is tested for two problems. In the first problem, an Oldroyd-B fluid creeping flow around a confined cylinder with a Contents lists available at ScienceDirect journal homepage: www.elsevier.com/locate/enganabound Engineering Analysis with Boundary Elements http://dx.doi.org/10.1016/j.enganabound.2014.07.011 0955-7997/& 2014 Elsevier Ltd. All rights reserved. n Corresponding author. E-mail addresses: mlbolok@iust.ac.ir (M. Lashkarbolok), jabbari@iust.ac.ir (E. Jabbari), J.westerweel@tudelft.nl (J. Westerweel). Engineering Analysis with Boundary Elements 50 (2015) 59–68