Solving the incompressible fluid flows by a high‐order mesh‐free approach

Abstract

Summary In this paper, we propose for the first time to extend the application field of the high‐order mesh‐free approach to the stationary incompressible Navier‐Stokes equations. This approach is based on a high‐order algorithm, which combines a Taylor series expansion, a continuation technique, and a moving least squares (MLS) method. The Taylor series expansion permits to transform the nonlinear problem into a succession of continuous linear ones with the same tangent operator. The MLS method is used to transform the succession of continuous linear problems into discrete ones. The continuation technique allows to compute step‐by‐step the whole solution of the discrete problems. This mesh‐free approach is tested on three examples: a flow around a cylindrical obstacle, a flow in a sudden expansion, and the standard benchmark lid‐driven cavity flow. A comparison of the obtained results with those computed by the Newton‐Raphson method with MLS, the high‐order continuation with finite element method, and those of literature is presented.

Bifurcation points and bifurcated branches in fluids mechanics by high‐order mesh‐free geometric progression algorithms

“…Applying the penalty method, 7,39 Equation (1b) is reformulated by eliminating the pressure term P introducing a penalty parameter as follows: …”

“…In this section, the numerical modeling procedure of the HO‐MLS‐GPM is made in three steps. The first one describes the resolution steps of the steady Navier–Stokes equation (5) by the HO‐MLS approach where the detailed resolution procedure is presented in Reference 39. In the second step, we present the coupling of the HO‐MLS approach with a geometric progression (GP) to detect the bifurcation points.…”

“…The resolution of this nonlinear equation in a stationary case is performed by adopting the mesh‐free approach presented in Reference 39. Remember that the strategy of resolution adopted is described as follows.…”

“…It is well known that the difficulties encountered in this formulation are mainly related to the pressure boundary conditions. In order to overcome the aforementioned difficulties, we have chosen the implementation of the penalty method 7,8,39 to ensure numerically the incompressibility condition by eliminating the pressure term. The penalized stationary Navier–Stokes equations are solved by the HO‐MLS approach or HO‐EFG approach and the GP is added to indicate the critical Reynolds values.…”

Radial point interpolation method and high-order continuation for solving nonlinear transient heat conduction problems

“…Based on the FEM, many applications of high-order continuation (HOC) coupled with homotopy transformation show the performance of this technique with respect to computation time, automatic adaptability of the step length, and the exactness of solutions in structural and fluid mechanics [16][17][18][19]. In addition, the meshless method such as MLS approximation is coupled with HOC for the resolution of dynamic or static nonlinear problems [20][21][22][23][24][25][26][27][28]. Meshless methods such as MLS approximation have some difficulties in the treatment of essential boundary conditions.…”