Numerical Modeling of Stokes Flow in a Circular Cavity by Variational Multiscale Element Free Galerkin Method

Ping Zhang,Xiaohua Zhang

doi:10.1155/2014/451546

Abstract

The variational multiscale element free Galerkin method is extended to simulate the Stokes flow problems in a circular cavity as an irregular geometry. The method is combined with Hughes’s variational multiscale formulation and element free Galerkin method; thus it inherits the advantages of variational multiscale and meshless methods. Meanwhile, a simple technique is adopted to impose the essential boundary conditions which makes it easy to solve problems with complex area. Finally, two examples are solved and good results are obtained as compared with solutions of analytical and numerical methods, which demonstrates that the proposed method is an attractive approach for solving incompressible fluid flow problems in terms of accuracy and stability, even for complex irregular boundaries.

Highlights

The Stokes flow can be considered for a range of engineering processed and natural phenomena, and numerical solution of the Stokes equations for incompressible viscous fluids has been usually dominated by meshbased methods such as finite difference method (FDM), finite volume method (FVM), and finite element method (FEM)
The ways to deal with the Babuska-Brezzi condition for most of meshless methods based on Galerkin weak from are used with the ideas of finite element method (FEM) to solve such problems, such as streamline-upwind Petrov-Galerkin [7, 8], pressure-stabilizing Petrov-Galerkin (PSPG) [9, 10], Galerkin least-squares (GLS) [11, 12], projection or fractional step method [13, 14], and finite calculus approach (FIC) [15, 16]
In the EFG method, the field variable u(x) is approximated by moving least squares (MLS) approximation, which consists of three parts: a basis function, a group of nonconstant coefficients, and a weight function associated with each node

Summary

Introduction

The Stokes flow can be considered for a range of engineering processed and natural phenomena, and numerical solution of the Stokes equations for incompressible viscous fluids has been usually dominated by meshbased methods such as finite difference method (FDM), finite volume method (FVM), and finite element method (FEM). The ways to deal with the Babuska-Brezzi condition for most of meshless methods based on Galerkin weak from are used with the ideas of finite element method (FEM) to solve such problems, such as streamline-upwind Petrov-Galerkin [7, 8], pressure-stabilizing Petrov-Galerkin (PSPG) [9, 10], Galerkin least-squares (GLS) [11, 12], projection or fractional step method [13, 14] (e.g., characteristic-based split algorithm), and finite calculus approach (FIC) [15, 16] These methods are related to a stabilization parameter and it depends on the problem under consideration and the chosen numerical method.

Review on the Element Free Galerkin Method

The VMEFG Method for Stokes Problem

Numerical Examples

Conclusion