Comments about the paper entitled “A generalized boundary integral equation for isotropic heat conduction with spatially varying thermal conductivity” by A. J. Kassab and E. Divo

doi:10.1016/S0955-7997(98)00046-0

Engineering Analysis with Boundary Elements

Volume 22, Issue 3, October 1998, Pages 235-240

https://doi.org/10.1016/S0955-7997(98)00046-0 Get rights and content

Abstract

An integral formulation for heat conduction problems in non-homogeneous media has recently been proposed by Kassab and Divo [Engineering Analysis with Boundary Elements 1996;18:273]. The goal of this communication is to revisit and clarify two key features of the formulation of Kassab and Divo. First, the contention that the integral equation formulation proposed by them does not possess the desired boundary-only character is made and substantiated; it is shown in particular that Eq. (10)therein does not hold owing to the fact that a crucial requirement for the fundamental solution, Eqs. (5c) and (5d) therein, is actually not met. Second, the limiting process associated with a vanishing neighbourhood in connection with the particular kernel function used therein is revisited.

Introduction

The purpose of the paper under discussion [1] was to establish a boundary-only integral formulation for heat conduction with isotropic but spatially varying conductivity $k(x)$ ( $x =(x, y)$ : field point), i.e. for temperature distributions T such that $div [k(x) ∇ T(x)]=0$ In order to do so, the variant kind of fundamental solution, denoted $E(x, ξ)$ , associated with a forcing term $D(x, ξ)$ containing a singular source at a given point $ξ =(x_{i}, y_{i})$ , is introduced: $div [k(x) ∇ E(x, ξ)]=−D(x, ξ)$

Section snippets

Examination of the sampling property

The following, crucial, requirement on the forcing term $D(x, ξ)$ (eqs. (5c) and (5d) of Ref. [1]) must be met in order to get rid of the domain integral that arises in the reciprocal theorem (Ω denotes an arbitrary domain required only to enclose the source point $ξ$ ): $∫ Ω T(x)D(x, ξ) d Ω(x)=T(ξ)A(ξ) with A(ξ)= ∫ Ω D(x, ξ) d Ω(x)$ The above relation is the analogue, for the particular fundamental solution at hand, of the sampling property of the Dirac distribution (the term “sifting property” was used in Ref. [1]

Limiting process of vanishing neighbourhood

Let $ξ$ be a fixed point on the boundary ∂Ω of a two-dimensional domain Ω. We consider an exclusion neighbourhood $v_{ε} (ξ)$ of $ξ$ , of radius ≤ε (Fig. 1). For any ε>0, $ξ$ is always an external point for the domain $Ω_{ε} (ξ)=Ω ν_{ε} (ξ)$ whose boundary ∂Ω_ε is given by $∂ Ω_{ε} =(∂ Ω−e_{ε})+s_{ε} =Γ_{ε} +s_{ε}$ where s_ε=Ω⋂∂v_ε, $e_{ε} = ∂ ω⋂ v ̄_{ε}$ , and Γ_ε=∂Ω−e_ε ( $v ̄_{ε}$ is the closure of v_ε).

The singular integral equation for heat conduction problems (T: unknown temperature field) in non-homogeneous media can then be sought as the limiting case, for ε

Discussion

From the analysis conducted in Section 3, the integral representation in Eq. (13)and the integral equation in Eq. (15)involve, in general, a domain term. It is thus not correct to state, as was done in Ref. [1], that recourse to the kernel E leads to boundary-only integral formulations for heat conduction problems in non-homogeneous media. However, since E is in some sense an “averaged” version associated with the “averaged” conductivity k̃, of the (unknown) fundamental solution associated with

Illustrative numerical examples

In order to illustrate the above considerations and have some quantitative idea of the approximate character of Eq. (3), two 2D examples are now considered.

The first example is analytical. We take k=(1+x)(1+y) and Ω={0≤x, y≤2}. A solution to Eq. (1)is T=(1+x)²−(1+y)². One then finds $k ̃ =(1+x_{i})(1+y_{i}) (i.e. constant) ∂ ∂ r (k ̃ −k)=(1+y_{i}) cos θ+(1+x_{i}) sin θ+2r cos θ sin θ$ To keep analytical calculations simple, we consider only source points on the horizontal median of Ω, i.e. such that y_i=1. After some

Conclusion

From the present analysis, our contention is that the approach proposed in Ref. [1], contrary to the claim made in that paper, does not lead to an integral equation without domain integrals. In particular, the integral equation, eq. (10) of Ref. [1], is found to be only approximately, not exactly, satisfied. The arguments detailed here for two-dimensional situations extend to three dimensions as well.

One may notice that the equations in Eq. (3)are not discussed or verified a posteriori in the

References (1)

A Kassab et al.
A generalized boundary integral solution for heat conduction problems in non-homogeneous media
Engineering Analysis with Boundary Elements
(1996)

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