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

Abstract

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 postprocessed finite element results.

Untitled

“…The solution Φ of the 3-D Laplace equation, is expressed in cylindrical coordinates and is defined in a domain with a boundary crack singularity along the z-axis or along a line parallel to this axis, as it is the case of the model problems which have been examined in the current study. It is written as follows (Costabel et al 2003, Christodoulou et al 2012b, Elliotis et al 2010, Elliotis 2016:…”

“…, is the so-called singular function, in which N is an additional parameter, selected so that N p <2N+1, the latter being the result of the substitution of Φ, expressed by ( 25), into the governing equation (Elliotis 2016) and asking the expression to satisfy the 3-D Laplace equation. Also, parameter c k,i is defined as…”

“…Following the standard procedure of the SFBIM for the 3-D case (Christodoulou et al 2012b, Elliotis et al 2010, Elliotis 2016, the governing equation is weighted by the singular functions V k (j) , in the Galerkin sense, to give (N p +1)N a +N λ discretized equations. Then, after applying Gauss divergence theorem and by considering that functions V k (j) satisfy exactly the boundary conditions along S A and S B and also by imposing the Dirichlet boundary condition on S C , by means of Lagrange multiplier μ(θ,z), we finally arrive at the following linear system of (N p +1)N a +N λ discretized equations:…”

“…It has been developed by G. Georgiou and co-workers and has been used in many studies to tackle planar harmonic and biharmonic equation problems and three-dimensional Laplace equation problems in the fields of theoretical and fracture Mechanics, fluid flow, etc. (Christodoulou et al 2012a, b, Elliotis et al 2010, Elliotis et al 2002, Elliotis et al 2005a, b, Elliotis et al 2006, Elliotis et al 2007, Elliotis et al 2014, Elliotis 2016, Elliotis 2019, Georgiou et al 1996. It exhibits exponential convergence, a feature which was observed in all previous applications of the method.…”

“…It exhibits exponential convergence, a feature which was observed in all previous applications of the method. Its application in three-dimensional problems (Christodoulou et al 2012b, Elliotis et al 2010, Elliotis 2016, has given much encouragement to extend the method to other threedimensional Laplacian problems of perfectly elastic solid bodies with a crack singularity. Also, in the last five or six years, researchers have worked on problems of Fracture Mechanics with crack singularities and have obtained interesting results (Yosibash and Mittelman 2016, Schapira and Yosibash 2020, Omer and Yosibash 2019, Yosibash and Schapira 2021, Chaumont-Frelet and Nicaise 2018, Dauge and Nicaise 2017, Woo and Kim 2018.…”