Open Access

ABSTRACT

A Nonlinear Meshless Local Petrov-Galerkin (MLPG) Approach Based on the Nonlinear Regular Local Boundary Integral Equation

1 Department of Mechanical Engineering, Widener University, One University Place, Chester, PA, 9013, USA

The International Conference on Computational & Experimental Engineering and Sciences 2008, 7(2), 101-106. https://doi.org/10.3970/icces.2008.007.101

Abstract

A nonliear meshless local Petrov-Galerkin (NMLPG) method for solving nonlinear boundary value problems, based on the nonlinear regular local boundary integral equation (NRLBIE) and the moving least squares approximation, is proposed in the present paper. No special integration scheme is needed to evaluate the volume and boundary integrals. The integrals in the present method are evaluated only over regularly-shaped sub-domains and their boundaries. This flexibility in choosing the size and the shape of the local sub-domain will lead to a more convenient formulation in dealing with the nonlinear problems. Compared to the original meshless local Petrov-Galerkin (MLPG) method that has been extensively popularized in recent literature, the present method does not need the derivatives of the shape functions in constructing the system stiffness matrix, for those nodes with no displacement specified on their local boundaries. This is very attractive in engineering applications as the calculation of derivatives of the shape functions from the MLS approximation is quite costly. Also compared to the meshless local boundary integral equation (MLBIE) method, the present approach does not involve singular or hyper-singular integrals which have to be tackled in the MLBIE method. Thus, the present method possesses high accuracy, and is cost effective. Numerical examples show that the present method converges fast to the final solution with reasonably accurate results for both the unknown variable and its derivatives.

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