A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach

Abstract

The Galerkin ®nite element method (GFEM) owes its popularity to the local nature of nodal basis functions, i.e., the nodal basis function, when viewed globally, is non-zero only over a patch of elements connecting the node in question to its immediately neighboring nodes. The boundary element method (BEM), on the other hand, reduces the dimensionality of the problem by one, through involving the trial functions and their derivatives, only in the integrals over the global boundary of the domain; whereas, the GFEM involves the integration of the``energy'' corresponding to the trial function over a patch of elements immediately surrounding the node. The GFEM leads to banded, sparse and symmetric matrices; the BEM based on the global boundary integral equation (GBIE) leads to full and unsymmetrical matrices. Because of the seemingly insurmountable dif®culties associated with the automatic generation of element-meshes in GFEM, especially for 3-D problems, there has been a considerable interest in element free Galerkin methods (EFGM) in recent literature. However, the EFGMs still involve domain integrals over shadow elements and lead to dif®culties in enforcing essential boundary conditions and in treating nonlinear problems.The object of the present paper is to present a new method that combines the advantageous features of all the three methods: GFEM, BEM and EFGM. It is a meshless method. It involves only boundary integration, however, over a local boundary centered at the node in question; it poses no dif®culties in satisfying essential boundary conditions; it leads to banded and sparse system matrices; it uses the moving least squares (MLS) approximations. The method is based on a Local Boundary Integral Equation (LBIE) approach, which is quite general and easily applicable to nonlinear problems, and non-homogeneous domains.The concept of a``companion solution'' is introduced so that the LBIE for the value of trial solution at the source point, inside the domain X of the given problem, involves only the trial function in the integral over the local boundary oX s of a sub-domain X s centered at the node in question. This is in contrast to the traditional GBIE which involves the trial function as well as its gradient over the global boundary C of X. For source points that lie on C, the integrals over oX s involve, on the other hand, both the trial function and its gradient. 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 LBIE approach.In the example problems dealing with Laplace and Poisson's equations, high rates of convergence for the Sobolev norms jj Á jj 0 and jj Á jj 1 have been found.In essence, the present EF-LBIE (Element Free-Local Boundary Integral Equation) approach is found to be a simple, ef®cient, and attractive alternative to the EFG methods that have been extensively popularized in recent literature.

Numerical simulation of the nonlinear ultrasonic pressure wave propagation in a cavitating bubbly liquid inside a sonochemical reactor

“…Next, one integral equation for the potential, arising from the application of Green's identities, is written at each source node. The 'companion solution' approach [33] is applied in order to eliminate the integral containing the gradient of the potential. The method encounters boundary integrals and domain integrals for the source nodes distributed over the global solution domain.…”

“…for the surface integral and the domain integral, is the elimination related to the normal derivative by using the companion solution approach [32,33].…”

“…If Neuman BCs are given on the part of the global boundary where ‫ݎ‬ is located, the integral containing the gradient of the potential has to be evaluated, as it does not vanish on the global boundary [33,35]. The following equation is applied:…”

Localized boundary-domain integral formulations for problems with variable coefficients

“…This results, after discretization, in sparsely populated systems of linear algebraic equations, which can be solved by well known efficient methods. The local boundary integral equation method of [9,10] can be considered as a particular realization of the localized BDIE/BDIDE method described here.…”

Mesh-based numerical implementation of the localized boundary-domain integral-equation method to a variable-coefficient Neumann problem

“…The mesh-less method based on the MLS approximation, c.f. [7,8], and the element-wise polynomial approximation were used for numerical implementations in [6]. A localized boundary-domain integro-differential equation, which is equivalent to use of the piece-wise constant cut-off function, c.f.…”