The Singular Function Boundary Integral Method for the 2-D and 3-D Laplace Equation Problems in Mechanics, with a Boundary Singularity

Miltiades C Elliotis

doi:10.11648/j.pamj.20160506.14

Abstract

In this study the Singular Function Boundary Integral Method (SFBIM) is implemented in the case of a planar elliptic boundary value problem in Mechanics, with a point boundary singularity. The method is also extended in the case of a typical problem of Solid Mechanics, concerning the Laplace equation problem in three dimensions, defined in a domain with a straight edge singularity on the surface boundary. In both the 2-D and 3-D cases, the general solution of the Laplace equation is approximated by the leading terms (which contain the singular functions) of the local asymptotic solution expansion. The singular functions are used to weight the governing equation in the Galerkin sense. For the 2-D Laplacian model problem of this study, which is defined over a domain with a re-entrant corner, the resulting discretized equations are reduced to boundary integrals by means of Green’s second identity. For the 3-D model problem of this work, the volume integrals of the discretized equations are reduced to surface integrals by implementing Gauss’ divergence theorem. The Dirichlet boundary conditions are then weakly enforced by means of Lagrange multipliers. The values of the latter are calculated together with the singular coefficients, in the 2-D case or the Edge Flux Intensity Functions (EFIFs), in the 3-D model problem, which appear in the local solution expansion. For the planar problem, the numerical results are favorably compared with the analytic solution. Especially for the extension of the method in three dimensions, the preliminary numerical results compare favorably with available post-processed finite element results.

Highlights

In the past few decades, many numerical methods have been proposed for the solution of 2-D elliptic boundary value problems in Mechanics, with a boundary singularity
The local solution is characterized by the presence of certain eigenpairs and the socalled Edge Flux Intensity Functions (EFIFs), which are the primary unknowns in the 3-D case [30]
The outline of the rest of this paper is as follows: in Section 2 we present both a 2-D Laplace equation problem, with a point boundary singularity and a 3-D Laplace equation problem with a straight-edge singularity

Summary

Introduction

In the past few decades, many numerical methods have been proposed for the solution of 2-D elliptic boundary value problems in Mechanics, with a boundary singularity. In [18] a singular finite element method is developed for Stoke’ s flow problems, in which special elements are employed, in the neighborhood of the singularity, with which the radial form of the local expansion is utilized, in order to resolve the convergence difficulties and improve the accuracy of the global solution The interest in such two-dimensional Laplace equation problems with a boundary singularity [20], is motivated by Miltiades C. The local solution is characterized by the presence of certain eigenpairs (arising from the 2-D problem) and the socalled Edge Flux Intensity Functions (EFIFs), which are the primary unknowns in the 3-D case [30] The interest in such a problem is motivated by the need to compute generalized stress intensity functions for the V-notched solids loaded by static loads.

Governing Equation and Local Solution in 2-D and 3-D

The 3-D Model Problem of the Laplace Equation

The SFBIM in the 2-D Case

The SFBIM in the 3-D Case

Numerical Results for the 2-D Problem

Conclusions