Imposing The Boundary Conditions In The
LBIE/RBF Method: A Simplified Approach
Author(s)
E. H. Ooi & V. Popov
Abstract
A simple approach for imposing the boundary conditions in the local boundary
integral equation (LBIE) method is proposed. The proposed approach maintains
the weak formulation on the boundary by enforcing the integral equation derived
from the Green’s second identity and the fundamental solution of the Laplace
equation. Unlike in the LBIE, the subdomains at the boundary in the proposed
method preserve their circular shapes, such that difficulties associated with the
evaluation of near-singular and singular integrals and the determination of
intersection between the global and local boundaries can be avoided. The
proposed approach is compared with the conventional LBIE by solving the
convection-diffusion equation. The unknown field variables were approximated
with the RBF approximations. Numerical results showed that the proposed
method, despite its simplicity, yielded results of comparable accuracy with the
LBIE when third order RBF was used.
Keywords: local boundary integral equation, weak formulation, radial basis
functions, meshless methods, companion solution.
1 Introduction
The local boundary integral equation (LBIE) method is a meshless method
developed by Zhu et al. [1] that combine s th e advantag es of se veral
numerical methods including the Galerkin finite element method, the boundary
element method and the element-free Galerkin method. Implementation of the
LBIE involves the distribution of collocation nodes over the boundary and
interior of the solution domain, whereby a circular (spherical in 3D) subdomain
centred on each node is generated. In each subdomain, the integral equation
derived from the governing equation holds. The LBIE introduces the concept of
Keywords
local boundary integral equation, weak formulation, radial basis
functions, meshless methods, companion solution.
Related Book
Other papers in this volume
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