On hypersingular surface integrals in the symmetric Galerkin boundary element method: application to heat conduction in exponentially graded materials

Abstract

SUMMARYA symmetric Galerkin formulation and implementation for heat conduction in a three-dimensional functionally graded material is presented. The Green's function of the graded problem, in which the thermal conductivity varies exponentially in one co-ordinate, is used to develop a boundary-only formulation without any domain discretization. The main task is the evaluation of hypersingular and singular integrals, which is carried out using a direct 'limit to the boundary' approach. However, due to complexity of the Green's function for graded materials, the usual direct limit procedures have to be modified, incorporating Taylor expansions to obtain expressions that can be integrated analytically. Several test examples are provided to verify the numerical implementation. The results of test calculations are in good agreement with exact solutions and corresponding finite element method simulations.

The simple boundary element method for multiple cracks in functionally graded media governed by potential theory: a three‐dimensional Galerkin approach

“…A number of transformations and analytical integration(s) are carried out to treat this integral. Details of this technique can be found in the papers by Sutradhar et al [38] and Gray et al [49].…”

“…The range of co-ordinates has been chosen for the sake of ease of computational implementation when dealing with singular integration [38].…”

“…As a result, each time a new computer code has to be implemented. Moreover, if the treatment of singularity involves analytical integration, then the process becomes much involved [38]. Several approaches have been developed to evaluate domain integrals associated with boundary element formulations including approximate particular solution methods [17,18], dual reciprocity methods [19], and multiple reciprocity methods [20,21].…”

Green's function evaluation for three‐dimensional exponentially graded elasticity

“…For the analytic integration, the exponential prefactor in (8) is a minor complication. As discussed in [35], the exponential is easily handled using a Taylor series in . Thus, for the coincident integral over an element…”

“…These modifications are primarily for handling the exponential factor in (8), and as these are essentially the same as the procedures employed for FGM thermal analysis [35], the discussion is kept to a minimum.…”