A meshless local boundary integral equation (LBIE) method for solving nonlinear problems

Abstract

A new meshless method for solving nonlinear boundary value problems, based on the local boundary integral equation (LBIE) method and the moving least squares approximation, is proposed in the present paper. The total formulation and a rate formulation are developed for the implementation of the present method. The present method does not need domain and boundary elements to deal with the volume and boundary integrals, which will cause some dif®culties for the conventional boundary element method (BEM) or the ®eld/boundary element method (FBEM), as the volume integrals are inevitable in dealing with nonlinear boundary value problems. This is the same for the element free Galerkin (EFG) method which also needs element-like cells in the entire domain to evaluate volume integrals. The``companion fundamental solution'' introduced in Zhu, Zhang and Atluri (1998) is used so that no derivatives of the shape functions are needed to construct the stiffness matrix for the interior nodes, as well as for those nodes with no parts of their local boundaries coinciding with the global boundary of the domain of the problem, where essential boundary conditions are speci®ed. It is shown that the satisfaction of the essential as well as natural boundary conditions is quite simple, and algorithmically very ef®-cient, in the present nonlinear LBIE approach. Numerical examples are presented for several problems, for which exact solutions are available. The present method converges fast to the ®nal solution with reasonably accurate results for both the unknown variable and its derivatives. No post processing procedure is required to compute the derivatives of the unknown variable (as in the conventional FBEM), since the solution from the present method, using the moving least squares approximation, is already smooth enough. The numerical results in these examples show that high rates of convergence for the Sobolev norms k Á k 0 and k Á k 1 are achievable, and that the values of the unknown variable and its derivatives are quite accurate.

An accurate, stable and efficient domain-type meshless method for the solution of MHD flow problems

“…Recent works have indicated that highly accurate results may be obtained with meshless methods, as compared to grid-based methods [1,2]. Over the last years, several meshless methods have been proposed, as the Smoothed Particle Hydrodynamics (SPH) [3], the Diffuse Element Method (DEM) [4], the Element Free Galerkin method (EFG) [5], the Reproducing Kernel Particle Method (RKPM) [6,7], the Partition of Unity Finite Element method (PUFEM) [8], the h-p Clouds [9], the Moving Least-Square Reproducing Kernel method (MLSRK) [10], the meshless Local Boundary Integral Equation method (LBIE) [11], the Meshless Local Petrov-Galerkin method (MLPG) [12], meshless point collocation methods using reproducing kernel approximations [13], the Merhod of Fundamental Solutions (MFS) [14], the Method of Particular Solutions (MPS) [15] and more. In the present work we imply the MPC method for the solution of equations that describe the MHD flow.…”

Meshless Local Petrov-Galerkin Method for Solving Contact, Impact and Penetration Problems

“…MLPG4 has been successfully applied to potential problems, elastostatics, elastodynamics, thermoelasticity, and plate bending problems [6,48,49,51,[53][54][55]68,69]. A summary of recent developments in the applications of MLPG4 can be found in [52].…”

Directional Method of Fundamental Solutions for Three-dimensional Laplace Equation

“…There are many papers concerned with the numerical solutions of elliptic type boundary value problems by using the meshless and mesh reduction methods, such as Zhu et al, 1998Zhu et al, , 1999Zhu, 1998a, 1998b;Atluri et al, 1999;Atluri and Shen, 2002;Li et al, 2007 andGhimire et al, 2016. The collocation techniques together with the expansion of trial solutions by utilizing different basis-functions were employed to solve the elliptic type boundary value problems; see, for example, Cheng et al, 2003;Hu et al, 2005;Tian et al, 2008, and Hu and Chen, 2008.…”