The meshless Galerkin boundary node method for Stokes problems in three dimensions

Abstract

SUMMARYThe Galerkin boundary node method (GBNM) is a boundary only meshless method that combines an equivalent variational formulation of boundary integral equations for governing equations and the moving least-squares (MLS) approximations for generating the trial and test functions. In this approach, boundary conditions can be implemented directly and easily despite of the fact that the MLS shape functions lack the delta function property. Besides, the resulting formulation inherits the symmetry and positive definiteness of the variational problems. The GBNM is developed in this paper for solving three-dimensional stationary incompressible Stokes flows in primitive variables. The numerical scheme is based on variational formulations for the first-kind integral equations, which are valid for both interior and exterior problems simultaneously. A rigorous error analysis and convergence study of the method for both the velocity and the pressure is presented in Sobolev spaces. The capability of the method is also illustrated and assessed through some selected numerical examples.

A Meshfree Method for Signorini Problems Using Boundary Integral Equations

“…Since the nodes lie on Γ in the present method, the point interpolation is only needed on Γ. As in [10,[18][19][20], it is assumed that the boundary Γ is the union of piecewise smooth segments Γ ( = 1, 2, . .…”

“…To carry out integrations numerically, the boundary Γ is discretized into cells. It has been shown that one node per cell and the position of the node at the center of the cell can yield excellent results [15,18,19]. This nodal placement is used in this research.…”

“…Boundary integral equations (BIEs) are attractive computational techniques as they can reduce the dimensionality of the original problem by one. The idea of meshfree has also been applied in boundary integral equations (BIEs) [10,11,[15][16][17][18][19][20][21][22][23]. The first BIEs-based meshfree method is the boundary node method (BNM) [10,15], which takes the advantages of both BIEs in dimension reduction and moving least-squares (MLS) approximations in elements elimination.…”

The Error Estimates of the Interpolating Element-Free Galerkin Method for Two-Point Boundary Value Problems

“…Error estimation for meshless method is certainly important to increase the reliability and reduce the cost of numerical computations in many engineering problems. Some error analyses have been done for the MLS approximation and the meshless method based on it [69][70][71][72][73][74]. Krysl and Belytschko studied the convergence of the continuous and discontinuous shape functions of the second-order elliptic partial differential equations [75].…”

On error estimator and adaptivity in the meshless Galerkin boundary node method

“…Another salient feature of the GBNM is the conservation of the symmetry and positive definiteness of variational problems in the process of numerical implementation. The GBNM has been successfully applied to problems in potential theory [11,31,32], linear elasticity [33] and fluid mechanics [34][35][36]. The corresponding convergence analysis and a priori error estimate have also been established.…”