An improved node moving technique for adaptive analysis using collocated discrete least squares meshless method

Highlights

• Collocated Discrete Least Squares Meshless (CDLSM) method is applied for the simulation that is a pure meshless method with symmetric and positive-definite properties.

• A residual-based error estimator coinciding well with the CDLSM method is used and therefore no furthe cost is required for error estimation process.

• An improved node moving algorithm is proposed to capture the nonlinear nature of the node positioning algorithm.

Abstract

In this paper, an improved node moving technique was developed for adaptive solution of some flow problems. Collocated Discrete Least Squares Meshless (CDLSM) method was applied for simulation. This method is a truly meshless method and also enjoys the symmetric and positive-definite property. Then, an improved node moving technique was developed based on the spring analogy. In this mechanism, each node was assumed to be connected to its neighboring nodes via some virtual springs. Then, higher values of stiffness were allocated to the springs located at higher error areas. In the previously published works, the corresponding system of equations was assumed to be linear. In this work, first, it was shown that this assumption may lead to inaccurate results in some cases and then, an improved method was proposed considering the nonlinear nature of the system of equations. Some numerical examples were used to illustrate the ability of the proposed node moving technique for some simple examples and benchmark problems in the Computational Fluid Dynamics (CFD) context. The Results of the study showed a considerable improvement in comparison with those of previously published works.

Introduction

Adaptive refinement techniques are essential tools for improving the efficiency of the numerical methods. Meshless methods introduced recently have great advantages in adaptive analysis in comparison to the conventional mesh-based numerical ones. Some detailed reviews on meshless methods can be found in [1], [2], [3].

In recent years, along with the introduction of new meshless methods, several adaptive analysis techniques have also been proposed for them. An error estimate method has been proposed by Gavete et al. [4]. There are several different error estimator methods, including the residual-based error estimators and the recovery-based error estimators [5].

Liu and Tu [6] proposed an error estimator based on the energy of the background cells used in some meshless methods. Several error estimation techniques based on background cells for the Element Free Galerkin (EFG) were presented by Lee and Zhou [7,8]. A node generation procedure was proposed for the adaptive analysis of the hyperbolic problems by Park et al. [9]. Liu et al. [10] developed a node enrichment strategy for adaptive analysis using the stabilized radial point collocation method (LS-RPCM) in which, the interpolation error proposed in [11] was used as a measure of error, and a node enrichment strategy was suggested according to the estimated error. Another node enrichment technique was also proposed in [12] using a residual-based error estimate for the Regularized Least-Squares Radial Point Collocation Method (RLS-RPCM) for elliptic partial differential equations. After that, Silva calculated the error estimate based on the L2 norm and solved Poisson's problems [13]. An adaptive analysis of the solid mechanics problems based on node enrichment has been proposed for the Gradient Smoothing Method (GSM) using a Delaunay diagram [14]. Further, an adaptive implementation of Finite Point Method (FPM) for solving boundary value problems was presented in [15]. The strategy of adaptation was based on an error estimation of the potential and its gradient, together with a refinement procedure using the concept of Voronoi neighbors to insert new points in the domain. Fallah and Ebrahimnejad [16] provided an adaptive refinement procedure based on the meshless finite volume (MFV) method for the effective analysis of elasticity problems. They used a posteriori error estimator developed by Zienkiewicz and Zhu [17]. Another class of refinement techniques, named node-moving, has been recently applied for adaptive analysis in meshless methods. Such refinement techniques are more efficient than others since the number of nodes and, therefore, the number of degrees of freedom (DOFs(remains fixed in them. Afshar and Lashckarbolok [18] proposed an adaptive refinement strategy based on the node-moving methodology in conjunction with the Collocated Discrete Least Squares Meshless (CDLSM) method for one-dimensional hyperbolic problems. Afterwards, the method was extended for adaptive analysis of two-dimensional hyperbolic problems by Afshar and Firoozjaee [19]. In these techniques, an analogy with the spring systems was used to implement the movement of nodes according to the distribution of errors represented by the error estimator. Each nodal point was assumed to be connected to the neighboring nodes via virtual springs, while the stiffness of each spring was assumed to be proportional to the errors estimated at its two endpoints. The system of springs was let to reach an equilibrium state from which the new positions of the nodal points were found. The assembled system of algebraic equations was assumed to be linear in all these works.

In this work, first, it is shown that the assumption of the linear system of algebraic equations for the system of springs is not acceptable in all cases and in some cases; the above-mentioned assumption may lead to inaccurate and unsuitable solutions. Then, an improved adaptive refinement strategy for capturing the nonlinear nature of the problem is also introduced. In addition, several benchmark examples are presented to assess the ability and efficiency of the proposed algorithm.