An Efficient Implicit Mesh-Free Method to Solve Two-Dimensional Compressible Euler Equations

Abstract

Local radial basis function-based differential quadrature (RBF-DQ) method is a natural mesh-free approach, in which any derivative of a function at a point is approximated by a weighted linear sum of functional values at its surrounding scattered points. In this paper, the weighting coefficients in the spatial derivative approximation of the Euler equation are determined by using a weighted least-square procedure in the frame of RBFs, which enhances the flexibility of distributing points in the computational domain. An upwind method is further introduced to cope with discontinuities by using Roe's approximate Riemann solver for estimation of the inviscid flux on the virtual mid-point between the reference knot and its surrounding knot. The lower-upper symmetric GaussSeidel (LU-SGS) algorithm, which is implemented in a matrix-free form like the one used in the finite-volume method, is introduced in the work to speed up the convergence. The proposed approach is validated by its application to simulate transonic flows over a NACA 0012 airfoil. It was found that the present mesh-free results agree very well with available data in the literature, and the implicit LU-SGS algorithm can greatly save the computational time as compared with explicit time marching methods.

A graphics processing unit-accelerated meshless method for two-dimensional compressible flows

“…In meshless methods (Batina, 1993;Chen & Shu, 2005;Katz & Jameson, 2010), scattered points are first distributed in the physical domain to be solved, as shown in Figure 2(a). For each node in the domain, several nearest neighbors are selected to form a local cloud of points, establishing connectivity throughout the domain.…”

“…Recently, in order to improve accuracy and efficiency, adaptive meshless methods based on point adding/removing (Oñate, Idelsohn, Zienkiewicz, & Taylor, 1996) and point moving (Ma, Chen, & Zhou, 2008;Zhou & Xu, 2010), implicit meshless methods (Chen & Shu, 2005;Singh, Ramesh, & Balakrishnan, 2015) and multi-cloud meshless methods (Katz & Jameson, 2009) have been reported on, with successful applications. However, their computational efficiencies remain largely uncompetitive with mesh-based methods.…”

A study of gridless method with dynamic clouds of points for solving unsteady CFD problems in aerodynamics

“…Let ∇W = W k −W i . The flux limiter can be expressed as [6] A small value (ε = 10 −12 ) is used to prevent division by zero in smooth regions where the differences are very small. Elements of the Jacobian matrix A are calculated by using Roe's averages.…”

A meshless finite point method for three‐dimensional analysis of compressible flow problems involving moving boundaries and adaptivity

“…In the last two decades, numerous meshless techniques aimed at dealing with steady and unsteady compressible flow problems [3][4][5][6][7][8][9][10] have been developed (cf. [11] for a comparative analysis of popular discretization approaches).…”